LMFDB - Embedded newform 9072.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4536)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.00000 q^{5} +1.00000 q^{7} -0.267949 q^{11} -4.00000 q^{13} -3.46410 q^{17} +0.535898 q^{19} -3.46410 q^{23} -1.00000 q^{25} +6.92820 q^{29} +4.92820 q^{31} +2.00000 q^{35} +1.92820 q^{37} -10.0000 q^{41} -9.39230 q^{43} +5.46410 q^{47} +1.00000 q^{49} -5.19615 q^{53} -0.535898 q^{55} +3.46410 q^{59} -9.46410 q^{61} -8.00000 q^{65} +11.3923 q^{67} +0.267949 q^{71} -2.92820 q^{73} -0.267949 q^{77} -3.53590 q^{79} +3.07180 q^{83} -6.92820 q^{85} -12.0000 q^{89} -4.00000 q^{91} +1.07180 q^{95} +3.46410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 4 * q^5 + 2 * q^7	$ 2 q + 4 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 8 q^{19} - 2 q^{25} - 4 q^{31} + 4 q^{35} - 10 q^{37} - 20 q^{41} + 2 q^{43} + 4 q^{47} + 2 q^{49} - 8 q^{55} - 12 q^{61} - 16 q^{65} + 2 q^{67} + 4 q^{71} + 8 q^{73} - 4 q^{77} - 14 q^{79} + 20 q^{83} - 24 q^{89} - 8 q^{91} + 16 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 8 * q^19 - 2 * q^25 - 4 * q^31 + 4 * q^35 - 10 * q^37 - 20 * q^41 + 2 * q^43 + 4 * q^47 + 2 * q^49 - 8 * q^55 - 12 * q^61 - 16 * q^65 + 2 * q^67 + 4 * q^71 + 8 * q^73 - 4 * q^77 - 14 * q^79 + 20 * q^83 - 24 * q^89 - 8 * q^91 + 16 * q^95

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.267949	−0.0807897	−0.0403949	−	0.999184i	$-0.512862\pi$
$11$	−0.267949	−0.0807897	−0.0403949	+	0.999184i	$0.512862\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0.535898	0.122944	0.0614718	−	0.998109i	$-0.480421\pi$
$19$	0.535898	0.122944	0.0614718	+	0.998109i	$0.480421\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	1.92820	0.316995	0.158497	−	0.987359i	$-0.449335\pi$
$37$	1.92820	0.316995	0.158497	+	0.987359i	$0.449335\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−9.39230	−1.43231	−0.716157	−	0.697940i	$-0.754100\pi$
$43$	−9.39230	−1.43231	−0.716157	+	0.697940i	$0.754100\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.46410	0.797021	0.398511	−	0.917164i	$-0.369527\pi$
$47$	5.46410	0.797021	0.398511	+	0.917164i	$0.369527\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.19615	−0.713746	−0.356873	−	0.934153i	$-0.616157\pi$
$53$	−5.19615	−0.713746	−0.356873	+	0.934153i	$0.616157\pi$
$54$	0	0
$55$	−0.535898	−0.0722605
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	−9.46410	−1.21175	−0.605877	−	0.795558i	$-0.707178\pi$
$61$	−9.46410	−1.21175	−0.605877	+	0.795558i	$0.707178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	11.3923	1.39179	0.695896	−	0.718143i	$-0.255008\pi$
$67$	11.3923	1.39179	0.695896	+	0.718143i	$0.255008\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.267949	0.0317997	0.0158999	−	0.999874i	$-0.494939\pi$
$71$	0.267949	0.0317997	0.0158999	+	0.999874i	$0.494939\pi$
$72$	0	0
$73$	−2.92820	−0.342720	−0.171360	−	0.985208i	$-0.554816\pi$
$73$	−2.92820	−0.342720	−0.171360	+	0.985208i	$0.554816\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.267949	−0.0305356
$78$	0	0
$79$	−3.53590	−0.397820	−0.198910	−	0.980018i	$-0.563740\pi$
$79$	−3.53590	−0.397820	−0.198910	+	0.980018i	$0.563740\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.07180	0.337173	0.168587	−	0.985687i	$-0.446080\pi$
$83$	3.07180	0.337173	0.168587	+	0.985687i	$0.446080\pi$
$84$	0	0
$85$	−6.92820	−0.751469
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.07180	0.109964
$96$	0	0
$97$	3.46410	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.92820	0.291367	0.145684	−	0.989331i	$-0.453462\pi$
$101$	2.92820	0.291367	0.145684	+	0.989331i	$0.453462\pi$
$102$	0	0
$103$	7.46410	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$103$	7.46410	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.26795	−0.412598	−0.206299	−	0.978489i	$-0.566142\pi$
$107$	−4.26795	−0.412598	−0.206299	+	0.978489i	$0.566142\pi$
$108$	0	0
$109$	−3.07180	−0.294225	−0.147112	−	0.989120i	$-0.546998\pi$
$109$	−3.07180	−0.294225	−0.147112	+	0.989120i	$0.546998\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.80385	0.263764	0.131882	−	0.991265i	$-0.457898\pi$
$113$	2.80385	0.263764	0.131882	+	0.991265i	$0.457898\pi$
$114$	0	0
$115$	−6.92820	−0.646058
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	−10.9282	−0.993473
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−11.3923	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$127$	−11.3923	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.39230	0.733239	0.366620	−	0.930371i	$-0.380515\pi$
$131$	8.39230	0.733239	0.366620	+	0.930371i	$0.380515\pi$
$132$	0	0
$133$	0.535898	0.0464683
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.0526	−1.62777	−0.813885	−	0.581026i	$-0.802651\pi$
$137$	−19.0526	−1.62777	−0.813885	+	0.581026i	$0.802651\pi$
$138$	0	0
$139$	−19.4641	−1.65092	−0.825462	−	0.564458i	$-0.809085\pi$
$139$	−19.4641	−1.65092	−0.825462	+	0.564458i	$0.809085\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.07180	0.0896281
$144$	0	0
$145$	13.8564	1.15071
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.7321	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$149$	−13.7321	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$150$	0	0
$151$	−6.46410	−0.526041	−0.263021	−	0.964790i	$-0.584719\pi$
$151$	−6.46410	−0.526041	−0.263021	+	0.964790i	$0.584719\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.85641	0.791686
$156$	0	0
$157$	−6.92820	−0.552931	−0.276465	−	0.961024i	$-0.589163\pi$
$157$	−6.92820	−0.552931	−0.276465	+	0.961024i	$0.589163\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	−25.3923	−1.98888	−0.994439	−	0.105310i	$-0.966416\pi$
$163$	−25.3923	−1.98888	−0.994439	+	0.105310i	$0.966416\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.92820	0.690885	0.345443	−	0.938440i	$-0.387729\pi$
$167$	8.92820	0.690885	0.345443	+	0.938440i	$0.387729\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.53590	−0.192801	−0.0964004	−	0.995343i	$-0.530733\pi$
$173$	−2.53590	−0.192801	−0.0964004	+	0.995343i	$0.530733\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.3205	−0.995622	−0.497811	−	0.867286i	$-0.665863\pi$
$179$	−13.3205	−0.995622	−0.497811	+	0.867286i	$0.665863\pi$
$180$	0	0
$181$	4.92820	0.366310	0.183155	−	0.983084i	$-0.441369\pi$
$181$	4.92820	0.366310	0.183155	+	0.983084i	$0.441369\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.85641	0.283529
$186$	0	0
$187$	0.928203	0.0678769
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.1244	0.732573	0.366286	−	0.930502i	$-0.380629\pi$
$191$	10.1244	0.732573	0.366286	+	0.930502i	$0.380629\pi$
$192$	0	0
$193$	−8.92820	−0.642666	−0.321333	−	0.946966i	$-0.604131\pi$
$193$	−8.92820	−0.642666	−0.321333	+	0.946966i	$0.604131\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.80385	0.484754	0.242377	−	0.970182i	$-0.422073\pi$
$197$	6.80385	0.484754	0.242377	+	0.970182i	$0.422073\pi$
$198$	0	0
$199$	−16.3923	−1.16202	−0.581010	−	0.813897i	$-0.697342\pi$
$199$	−16.3923	−1.16202	−0.581010	+	0.813897i	$0.697342\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.92820	0.486265
$204$	0	0
$205$	−20.0000	−1.39686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.143594	−0.00993257
$210$	0	0
$211$	27.2487	1.87588	0.937939	−	0.346799i	$-0.112732\pi$
$211$	27.2487	1.87588	0.937939	+	0.346799i	$0.112732\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.7846	−1.28110
$216$	0	0
$217$	4.92820	0.334548
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.8564	0.932083
$222$	0	0
$223$	−19.8564	−1.32968	−0.664842	−	0.746984i	$-0.731501\pi$
$223$	−19.8564	−1.32968	−0.664842	+	0.746984i	$0.731501\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−18.3923	−1.21540	−0.607699	−	0.794168i	$-0.707907\pi$
$229$	−18.3923	−1.21540	−0.607699	+	0.794168i	$0.707907\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	10.9282	0.712877
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.7321	0.758883	0.379442	−	0.925216i	$-0.376116\pi$
$239$	11.7321	0.758883	0.379442	+	0.925216i	$0.376116\pi$
$240$	0	0
$241$	20.3923	1.31358	0.656792	−	0.754072i	$-0.271913\pi$
$241$	20.3923	1.31358	0.656792	+	0.754072i	$0.271913\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−2.14359	−0.136394
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.5359	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$251$	14.5359	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$252$	0	0
$253$	0.928203	0.0583556
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.14359	−0.133714	−0.0668568	−	0.997763i	$-0.521297\pi$
$257$	−2.14359	−0.133714	−0.0668568	+	0.997763i	$0.521297\pi$
$258$	0	0
$259$	1.92820	0.119813
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.6603	1.64394	0.821971	−	0.569530i	$-0.192875\pi$
$263$	26.6603	1.64394	0.821971	+	0.569530i	$0.192875\pi$
$264$	0	0
$265$	−10.3923	−0.638394
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.9282	−0.910189	−0.455094	−	0.890443i	$-0.650394\pi$
$269$	−14.9282	−0.910189	−0.455094	+	0.890443i	$0.650394\pi$
$270$	0	0
$271$	2.53590	0.154045	0.0770224	−	0.997029i	$-0.475459\pi$
$271$	2.53590	0.154045	0.0770224	+	0.997029i	$0.475459\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.267949	0.0161579
$276$	0	0
$277$	−10.0718	−0.605156	−0.302578	−	0.953125i	$-0.597847\pi$
$277$	−10.0718	−0.605156	−0.302578	+	0.953125i	$0.597847\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.5885	0.929929	0.464965	−	0.885329i	$-0.346067\pi$
$281$	15.5885	0.929929	0.464965	+	0.885329i	$0.346067\pi$
$282$	0	0
$283$	−6.53590	−0.388519	−0.194259	−	0.980950i	$-0.562230\pi$
$283$	−6.53590	−0.388519	−0.194259	+	0.980950i	$0.562230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.3205	−0.661351	−0.330676	−	0.943744i	$-0.607277\pi$
$293$	−11.3205	−0.661351	−0.330676	+	0.943744i	$0.607277\pi$
$294$	0	0
$295$	6.92820	0.403376
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.8564	0.801337
$300$	0	0
$301$	−9.39230	−0.541363
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.9282	−1.08383
$306$	0	0
$307$	3.85641	0.220097	0.110048	−	0.993926i	$-0.464899\pi$
$307$	3.85641	0.220097	0.110048	+	0.993926i	$0.464899\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	34.2487	1.94207	0.971033	−	0.238945i	$-0.0768015\pi$
$311$	34.2487	1.94207	0.971033	+	0.238945i	$0.0768015\pi$
$312$	0	0
$313$	18.7846	1.06177	0.530884	−	0.847444i	$-0.321860\pi$
$313$	18.7846	1.06177	0.530884	+	0.847444i	$0.321860\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	−1.85641	−0.103939
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.85641	−0.103293
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.46410	0.301246
$330$	0	0
$331$	−1.07180	−0.0589113	−0.0294556	−	0.999566i	$-0.509377\pi$
$331$	−1.07180	−0.0589113	−0.0294556	+	0.999566i	$0.509377\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.7846	1.24486
$336$	0	0
$337$	17.9282	0.976611	0.488306	−	0.872673i	$-0.337615\pi$
$337$	17.9282	0.976611	0.488306	+	0.872673i	$0.337615\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.32051	−0.0715095
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.26795	0.229116	0.114558	−	0.993417i	$-0.463455\pi$
$347$	4.26795	0.229116	0.114558	+	0.993417i	$0.463455\pi$
$348$	0	0
$349$	23.3205	1.24832	0.624159	−	0.781297i	$-0.285442\pi$
$349$	23.3205	1.24832	0.624159	+	0.781297i	$0.285442\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.8564	0.631053	0.315526	−	0.948917i	$-0.397819\pi$
$353$	11.8564	0.631053	0.315526	+	0.948917i	$0.397819\pi$
$354$	0	0
$355$	0.535898	0.0284425
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.26795	0.225254	0.112627	−	0.993637i	$-0.464074\pi$
$359$	4.26795	0.225254	0.112627	+	0.993637i	$0.464074\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.85641	−0.306538
$366$	0	0
$367$	−9.07180	−0.473544	−0.236772	−	0.971565i	$-0.576089\pi$
$367$	−9.07180	−0.473544	−0.236772	+	0.971565i	$0.576089\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.19615	−0.269771
$372$	0	0
$373$	−30.8564	−1.59768	−0.798842	−	0.601541i	$-0.794554\pi$
$373$	−30.8564	−1.59768	−0.798842	+	0.601541i	$0.794554\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−27.7128	−1.42728
$378$	0	0
$379$	−34.3205	−1.76293	−0.881463	−	0.472253i	$-0.843441\pi$
$379$	−34.3205	−1.76293	−0.881463	+	0.472253i	$0.843441\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.7846	−1.67522	−0.837608	−	0.546272i	$-0.816046\pi$
$383$	−32.7846	−1.67522	−0.837608	+	0.546272i	$0.816046\pi$
$384$	0	0
$385$	−0.535898	−0.0273119
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.0718	−0.662766	−0.331383	−	0.943496i	$-0.607515\pi$
$389$	−13.0718	−0.662766	−0.331383	+	0.943496i	$0.607515\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.07180	−0.355821
$396$	0	0
$397$	8.39230	0.421198	0.210599	−	0.977573i	$-0.432459\pi$
$397$	8.39230	0.421198	0.210599	+	0.977573i	$0.432459\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.1244	1.80396	0.901982	−	0.431773i	$-0.142112\pi$
$401$	36.1244	1.80396	0.901982	+	0.431773i	$0.142112\pi$
$402$	0	0
$403$	−19.7128	−0.981965
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.516660	−0.0256099
$408$	0	0
$409$	9.32051	0.460869	0.230435	−	0.973088i	$-0.425985\pi$
$409$	9.32051	0.460869	0.230435	+	0.973088i	$0.425985\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.46410	0.170457
$414$	0	0
$415$	6.14359	0.301577
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.0718	0.736305	0.368153	−	0.929765i	$-0.379990\pi$
$419$	15.0718	0.736305	0.368153	+	0.929765i	$0.379990\pi$
$420$	0	0
$421$	−20.7128	−1.00948	−0.504740	−	0.863271i	$-0.668412\pi$
$421$	−20.7128	−1.00948	−0.504740	+	0.863271i	$0.668412\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−9.46410	−0.458000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.60770	0.462786	0.231393	−	0.972860i	$-0.425672\pi$
$431$	9.60770	0.462786	0.231393	+	0.972860i	$0.425672\pi$
$432$	0	0
$433$	−39.1769	−1.88272	−0.941361	−	0.337401i	$-0.890452\pi$
$433$	−39.1769	−1.88272	−0.941361	+	0.337401i	$0.890452\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.85641	−0.0888040
$438$	0	0
$439$	33.4641	1.59715	0.798577	−	0.601892i	$-0.205586\pi$
$439$	33.4641	1.59715	0.798577	+	0.601892i	$0.205586\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.3923	−0.493753	−0.246877	−	0.969047i	$-0.579404\pi$
$443$	−10.3923	−0.493753	−0.246877	+	0.969047i	$0.579404\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.4449	−1.20082	−0.600409	−	0.799693i	$-0.704995\pi$
$449$	−25.4449	−1.20082	−0.600409	+	0.799693i	$0.704995\pi$
$450$	0	0
$451$	2.67949	0.126172
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−5.92820	−0.277310	−0.138655	−	0.990341i	$-0.544278\pi$
$457$	−5.92820	−0.277310	−0.138655	+	0.990341i	$0.544278\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.3205	0.527249	0.263624	−	0.964625i	$-0.415082\pi$
$461$	11.3205	0.527249	0.263624	+	0.964625i	$0.415082\pi$
$462$	0	0
$463$	−12.6077	−0.585929	−0.292965	−	0.956123i	$-0.594642\pi$
$463$	−12.6077	−0.585929	−0.292965	+	0.956123i	$0.594642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16.9282	−0.783344	−0.391672	−	0.920105i	$-0.628103\pi$
$467$	−16.9282	−0.783344	−0.391672	+	0.920105i	$0.628103\pi$
$468$	0	0
$469$	11.3923	0.526048
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.51666	0.115716
$474$	0	0
$475$	−0.535898	−0.0245887
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.46410	−0.158279	−0.0791394	−	0.996864i	$-0.525217\pi$
$479$	−3.46410	−0.158279	−0.0791394	+	0.996864i	$0.525217\pi$
$480$	0	0
$481$	−7.71281	−0.351674
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.92820	0.314594
$486$	0	0
$487$	−20.4641	−0.927317	−0.463658	−	0.886014i	$-0.653464\pi$
$487$	−20.4641	−0.927317	−0.463658	+	0.886014i	$0.653464\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.3923	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.267949	0.0120192
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.9282	−0.487264	−0.243632	−	0.969868i	$-0.578339\pi$
$503$	−10.9282	−0.487264	−0.243632	+	0.969868i	$0.578339\pi$
$504$	0	0
$505$	5.85641	0.260607
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.3205	−0.590421	−0.295211	−	0.955432i	$-0.595390\pi$
$509$	−13.3205	−0.590421	−0.295211	+	0.955432i	$0.595390\pi$
$510$	0	0
$511$	−2.92820	−0.129536
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.9282	0.657815
$516$	0	0
$517$	−1.46410	−0.0643911
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.3205	−0.758825	−0.379413	−	0.925228i	$-0.623874\pi$
$521$	−17.3205	−0.758825	−0.379413	+	0.925228i	$0.623874\pi$
$522$	0	0
$523$	−6.53590	−0.285795	−0.142897	−	0.989738i	$-0.545642\pi$
$523$	−6.53590	−0.285795	−0.142897	+	0.989738i	$0.545642\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.0718	−0.743659
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	−8.53590	−0.369039
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.267949	−0.0115414
$540$	0	0
$541$	14.7128	0.632553	0.316277	−	0.948667i	$-0.397567\pi$
$541$	14.7128	0.632553	0.316277	+	0.948667i	$0.397567\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.14359	−0.263163
$546$	0	0
$547$	22.1769	0.948216	0.474108	−	0.880467i	$-0.342771\pi$
$547$	22.1769	0.948216	0.474108	+	0.880467i	$0.342771\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.71281	0.158171
$552$	0	0
$553$	−3.53590	−0.150362
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.6603	1.21437	0.607187	−	0.794559i	$-0.292298\pi$
$557$	28.6603	1.21437	0.607187	+	0.794559i	$0.292298\pi$
$558$	0	0
$559$	37.5692	1.58901
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.4641	0.820314	0.410157	−	0.912015i	$-0.365474\pi$
$563$	19.4641	0.820314	0.410157	+	0.912015i	$0.365474\pi$
$564$	0	0
$565$	5.60770	0.235918
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.73205	−0.240300	−0.120150	−	0.992756i	$-0.538338\pi$
$569$	−5.73205	−0.240300	−0.120150	+	0.992756i	$0.538338\pi$
$570$	0	0
$571$	−22.9282	−0.959515	−0.479758	−	0.877401i	$-0.659275\pi$
$571$	−22.9282	−0.959515	−0.479758	+	0.877401i	$0.659275\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	−17.4641	−0.727040	−0.363520	−	0.931586i	$-0.618425\pi$
$577$	−17.4641	−0.727040	−0.363520	+	0.931586i	$0.618425\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.07180	0.127440
$582$	0	0
$583$	1.39230	0.0576634
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.3205	−1.21019	−0.605093	−	0.796154i	$-0.706864\pi$
$587$	−29.3205	−1.21019	−0.605093	+	0.796154i	$0.706864\pi$
$588$	0	0
$589$	2.64102	0.108821
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.2487	−0.831515	−0.415757	−	0.909476i	$-0.636483\pi$
$593$	−20.2487	−0.831515	−0.415757	+	0.909476i	$0.636483\pi$
$594$	0	0
$595$	−6.92820	−0.284029
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.9090	1.58978	0.794889	−	0.606755i	$-0.207529\pi$
$599$	38.9090	1.58978	0.794889	+	0.606755i	$0.207529\pi$
$600$	0	0
$601$	20.2487	0.825962	0.412981	−	0.910740i	$-0.364488\pi$
$601$	20.2487	0.825962	0.412981	+	0.910740i	$0.364488\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.8564	−0.888589
$606$	0	0
$607$	−0.392305	−0.0159232	−0.00796158	−	0.999968i	$-0.502534\pi$
$607$	−0.392305	−0.0159232	−0.00796158	+	0.999968i	$0.502534\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.8564	−0.884216
$612$	0	0
$613$	−45.9282	−1.85502	−0.927511	−	0.373795i	$-0.878056\pi$
$613$	−45.9282	−1.85502	−0.927511	+	0.373795i	$0.878056\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.7128	1.27671	0.638355	−	0.769742i	$-0.279615\pi$
$617$	31.7128	1.27671	0.638355	+	0.769742i	$0.279615\pi$
$618$	0	0
$619$	0.392305	0.0157681	0.00788403	−	0.999969i	$-0.497490\pi$
$619$	0.392305	0.0157681	0.00788403	+	0.999969i	$0.497490\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.67949	−0.266329
$630$	0	0
$631$	−20.7846	−0.827422	−0.413711	−	0.910408i	$-0.635768\pi$
$631$	−20.7846	−0.827422	−0.413711	+	0.910408i	$0.635768\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.7846	−0.904180
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.8756	0.469060	0.234530	−	0.972109i	$-0.424645\pi$
$641$	11.8756	0.469060	0.234530	+	0.972109i	$0.424645\pi$
$642$	0	0
$643$	−31.7128	−1.25063	−0.625316	−	0.780372i	$-0.715030\pi$
$643$	−31.7128	−1.25063	−0.625316	+	0.780372i	$0.715030\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.07180	0.278021	0.139011	−	0.990291i	$-0.455608\pi$
$647$	7.07180	0.278021	0.139011	+	0.990291i	$0.455608\pi$
$648$	0	0
$649$	−0.928203	−0.0364352
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.73205	0.224312	0.112156	−	0.993691i	$-0.464224\pi$
$653$	5.73205	0.224312	0.112156	+	0.993691i	$0.464224\pi$
$654$	0	0
$655$	16.7846	0.655829
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.3731	1.65062	0.825310	−	0.564680i	$-0.191000\pi$
$659$	42.3731	1.65062	0.825310	+	0.564680i	$0.191000\pi$
$660$	0	0
$661$	21.8564	0.850116	0.425058	−	0.905166i	$-0.360254\pi$
$661$	21.8564	0.850116	0.425058	+	0.905166i	$0.360254\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.07180	0.0415625
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.53590	0.0978973
$672$	0	0
$673$	−30.8564	−1.18943	−0.594714	−	0.803938i	$-0.702735\pi$
$673$	−30.8564	−1.18943	−0.594714	+	0.803938i	$0.702735\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.1769	1.65942	0.829712	−	0.558192i	$-0.188505\pi$
$677$	43.1769	1.65942	0.829712	+	0.558192i	$0.188505\pi$
$678$	0	0
$679$	3.46410	0.132940
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.0526	0.958610	0.479305	−	0.877649i	$-0.340889\pi$
$683$	25.0526	0.958610	0.479305	+	0.877649i	$0.340889\pi$
$684$	0	0
$685$	−38.1051	−1.45592
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.7846	0.791831
$690$	0	0
$691$	−16.9282	−0.643979	−0.321990	−	0.946743i	$-0.604352\pi$
$691$	−16.9282	−0.643979	−0.321990	+	0.946743i	$0.604352\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.9282	−1.47663
$696$	0	0
$697$	34.6410	1.31212
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.5885	0.588768	0.294384	−	0.955687i	$-0.404886\pi$
$701$	15.5885	0.588768	0.294384	+	0.955687i	$0.404886\pi$
$702$	0	0
$703$	1.03332	0.0389724
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.92820	0.110126
$708$	0	0
$709$	−9.00000	−0.338002	−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000	−0.338002	−0.169001	+	0.985616i	$0.554054\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17.0718	−0.639344
$714$	0	0
$715$	2.14359	0.0801659
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−35.4641	−1.32259	−0.661294	−	0.750127i	$-0.729992\pi$
$719$	−35.4641	−1.32259	−0.661294	+	0.750127i	$0.729992\pi$
$720$	0	0
$721$	7.46410	0.277978
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.92820	−0.257307
$726$	0	0
$727$	−48.1051	−1.78412	−0.892060	−	0.451917i	$-0.850740\pi$
$727$	−48.1051	−1.78412	−0.892060	+	0.451917i	$0.850740\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	32.5359	1.20338
$732$	0	0
$733$	6.39230	0.236105	0.118053	−	0.993007i	$-0.462335\pi$
$733$	6.39230	0.236105	0.118053	+	0.993007i	$0.462335\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.05256	−0.112442
$738$	0	0
$739$	35.2487	1.29664	0.648322	−	0.761366i	$-0.275471\pi$
$739$	35.2487	1.29664	0.648322	+	0.761366i	$0.275471\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.1244	0.371427	0.185713	−	0.982604i	$-0.440540\pi$
$743$	10.1244	0.371427	0.185713	+	0.982604i	$0.440540\pi$
$744$	0	0
$745$	−27.4641	−1.00621
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.26795	−0.155947
$750$	0	0
$751$	−38.1769	−1.39310	−0.696548	−	0.717510i	$-0.745282\pi$
$751$	−38.1769	−1.39310	−0.696548	+	0.717510i	$0.745282\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.9282	−0.470505
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.6077	0.710778	0.355389	−	0.934718i	$-0.384348\pi$
$761$	19.6077	0.710778	0.355389	+	0.934718i	$0.384348\pi$
$762$	0	0
$763$	−3.07180	−0.111207
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	49.3205	1.77854	0.889272	−	0.457380i	$-0.151212\pi$
$769$	49.3205	1.77854	0.889272	+	0.457380i	$0.151212\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	46.1051	1.65829	0.829143	−	0.559037i	$-0.188829\pi$
$773$	46.1051	1.65829	0.829143	+	0.559037i	$0.188829\pi$
$774$	0	0
$775$	−4.92820	−0.177026
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.35898	−0.192006
$780$	0	0
$781$	−0.0717968	−0.00256909
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.8564	−0.494556
$786$	0	0
$787$	5.60770	0.199893	0.0999464	−	0.994993i	$-0.468133\pi$
$787$	5.60770	0.199893	0.0999464	+	0.994993i	$0.468133\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.80385	0.0996933
$792$	0	0
$793$	37.8564	1.34432
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.9282	−1.52059	−0.760297	−	0.649575i	$-0.774947\pi$
$797$	−42.9282	−1.52059	−0.760297	+	0.649575i	$0.774947\pi$
$798$	0	0
$799$	−18.9282	−0.669632
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.784610	0.0276883
$804$	0	0
$805$	−6.92820	−0.244187
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.05256	0.107322	0.0536611	−	0.998559i	$-0.482911\pi$
$809$	3.05256	0.107322	0.0536611	+	0.998559i	$0.482911\pi$
$810$	0	0
$811$	49.5692	1.74061	0.870305	−	0.492513i	$-0.163921\pi$
$811$	49.5692	1.74061	0.870305	+	0.492513i	$0.163921\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−50.7846	−1.77891
$816$	0	0
$817$	−5.03332	−0.176094
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.1962	−1.15855	−0.579277	−	0.815131i	$-0.696665\pi$
$821$	−33.1962	−1.15855	−0.579277	+	0.815131i	$0.696665\pi$
$822$	0	0
$823$	−29.0718	−1.01338	−0.506690	−	0.862129i	$-0.669131\pi$
$823$	−29.0718	−1.01338	−0.506690	+	0.862129i	$0.669131\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.9808	−1.80755	−0.903774	−	0.428010i	$-0.859215\pi$
$827$	−51.9808	−1.80755	−0.903774	+	0.428010i	$0.859215\pi$
$828$	0	0
$829$	−45.5692	−1.58268	−0.791342	−	0.611373i	$-0.790617\pi$
$829$	−45.5692	−1.58268	−0.791342	+	0.611373i	$0.790617\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.46410	−0.120024
$834$	0	0
$835$	17.8564	0.617946
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.6410	1.81737	0.908685	−	0.417483i	$-0.137088\pi$
$839$	52.6410	1.81737	0.908685	+	0.417483i	$0.137088\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.00000	0.206406
$846$	0	0
$847$	−10.9282	−0.375498
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.67949	−0.228970
$852$	0	0
$853$	0.392305	0.0134323	0.00671613	−	0.999977i	$-0.497862\pi$
$853$	0.392305	0.0134323	0.00671613	+	0.999977i	$0.497862\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.32051	−0.181745	−0.0908725	−	0.995863i	$-0.528966\pi$
$857$	−5.32051	−0.181745	−0.0908725	+	0.995863i	$0.528966\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.2679	−1.50690	−0.753449	−	0.657506i	$-0.771611\pi$
$863$	−44.2679	−1.50690	−0.753449	+	0.657506i	$0.771611\pi$
$864$	0	0
$865$	−5.07180	−0.172446
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.947441	0.0321397
$870$	0	0
$871$	−45.5692	−1.54405
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	7.14359	0.241222	0.120611	−	0.992700i	$-0.461515\pi$
$877$	7.14359	0.241222	0.120611	+	0.992700i	$0.461515\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−38.7846	−1.30669	−0.653343	−	0.757062i	$-0.726634\pi$
$881$	−38.7846	−1.30669	−0.653343	+	0.757062i	$0.726634\pi$
$882$	0	0
$883$	18.6077	0.626199	0.313099	−	0.949720i	$-0.398633\pi$
$883$	18.6077	0.626199	0.313099	+	0.949720i	$0.398633\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.7846	−0.563572	−0.281786	−	0.959477i	$-0.590927\pi$
$887$	−16.7846	−0.563572	−0.281786	+	0.959477i	$0.590927\pi$
$888$	0	0
$889$	−11.3923	−0.382086
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.92820	0.0979886
$894$	0	0
$895$	−26.6410	−0.890511
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.1436	1.13875
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.85641	0.327638
$906$	0	0
$907$	−45.3923	−1.50723	−0.753613	−	0.657318i	$-0.771691\pi$
$907$	−45.3923	−1.50723	−0.753613	+	0.657318i	$0.771691\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−51.4641	−1.70508	−0.852541	−	0.522661i	$-0.824940\pi$
$911$	−51.4641	−1.70508	−0.852541	+	0.522661i	$0.824940\pi$
$912$	0	0
$913$	−0.823085	−0.0272402
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.39230	0.277138
$918$	0	0
$919$	2.60770	0.0860199	0.0430100	−	0.999075i	$-0.486305\pi$
$919$	2.60770	0.0860199	0.0430100	+	0.999075i	$0.486305\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.07180	−0.0352786
$924$	0	0
$925$	−1.92820	−0.0633989
$926$	0	0
$927$	0	0
$928$	0	0
$929$	55.0333	1.80558	0.902792	−	0.430077i	$-0.141513\pi$
$929$	55.0333	1.80558	0.902792	+	0.430077i	$0.141513\pi$
$930$	0	0
$931$	0.535898	0.0175634
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.85641	0.0607110
$936$	0	0
$937$	21.0718	0.688386	0.344193	−	0.938899i	$-0.388153\pi$
$937$	21.0718	0.688386	0.344193	+	0.938899i	$0.388153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.8564	−0.842895	−0.421447	−	0.906853i	$-0.638478\pi$
$941$	−25.8564	−0.842895	−0.421447	+	0.906853i	$0.638478\pi$
$942$	0	0
$943$	34.6410	1.12807
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43.4641	−1.41239	−0.706197	−	0.708016i	$-0.749591\pi$
$947$	−43.4641	−1.41239	−0.706197	+	0.708016i	$0.749591\pi$
$948$	0	0
$949$	11.7128	0.380214
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.1436	0.717301	0.358651	−	0.933472i	$-0.383237\pi$
$953$	22.1436	0.717301	0.358651	+	0.933472i	$0.383237\pi$
$954$	0	0
$955$	20.2487	0.655233
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.0526	−0.615239
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.8564	−0.574818
$966$	0	0
$967$	38.6410	1.24261	0.621306	−	0.783568i	$-0.286603\pi$
$967$	38.6410	1.24261	0.621306	+	0.783568i	$0.286603\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.5359	−0.659028	−0.329514	−	0.944151i	$-0.606885\pi$
$971$	−20.5359	−0.659028	−0.329514	+	0.944151i	$0.606885\pi$
$972$	0	0
$973$	−19.4641	−0.623990
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	0	0
$979$	3.21539	0.102764
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−44.1051	−1.40673	−0.703367	−	0.710826i	$-0.748321\pi$
$983$	−44.1051	−1.40673	−0.703367	+	0.710826i	$0.748321\pi$
$984$	0	0
$985$	13.6077	0.433577
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.5359	1.03458
$990$	0	0
$991$	−17.6795	−0.561608	−0.280804	−	0.959765i	$-0.590601\pi$
$991$	−17.6795	−0.561608	−0.280804	+	0.959765i	$0.590601\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−32.7846	−1.03934
$996$	0	0
$997$	−22.7846	−0.721596	−0.360798	−	0.932644i	$-0.617496\pi$
$997$	−22.7846	−0.721596	−0.360798	+	0.932644i	$0.617496\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bo.1.2		2
3.2	odd	2		9072.2.a.z.1.1		2
4.3	odd	2		4536.2.a.q.1.1	yes	2
12.11	even	2		4536.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4536.2.a.l.1.2	✓	2	12.11	even	2
4536.2.a.q.1.1	yes	2	4.3	odd	2
9072.2.a.z.1.1		2	3.2	odd	2
9072.2.a.bo.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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.00000 q^{5} +1.00000 q^{7} -0.267949 q^{11} -4.00000 q^{13} -3.46410 q^{17} +0.535898 q^{19} -3.46410 q^{23} -1.00000 q^{25} +6.92820 q^{29} +4.92820 q^{31} +2.00000 q^{35} +1.92820 q^{37} -10.0000 q^{41} -9.39230 q^{43} +5.46410 q^{47} +1.00000 q^{49} -5.19615 q^{53} -0.535898 q^{55} +3.46410 q^{59} -9.46410 q^{61} -8.00000 q^{65} +11.3923 q^{67} +0.267949 q^{71} -2.92820 q^{73} -0.267949 q^{77} -3.53590 q^{79} +3.07180 q^{83} -6.92820 q^{85} -12.0000 q^{89} -4.00000 q^{91} +1.07180 q^{95} +3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 4 * q^5 + 2 * q^7	\( 2 q + 4 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 8 q^{19} - 2 q^{25} - 4 q^{31} + 4 q^{35} - 10 q^{37} - 20 q^{41} + 2 q^{43} + 4 q^{47} + 2 q^{49} - 8 q^{55} - 12 q^{61} - 16 q^{65} + 2 q^{67} + 4 q^{71} + 8 q^{73} - 4 q^{77} - 14 q^{79} + 20 q^{83} - 24 q^{89} - 8 q^{91} + 16 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 8 * q^19 - 2 * q^25 - 4 * q^31 + 4 * q^35 - 10 * q^37 - 20 * q^41 + 2 * q^43 + 4 * q^47 + 2 * q^49 - 8 * q^55 - 12 * q^61 - 16 * q^65 + 2 * q^67 + 4 * q^71 + 8 * q^73 - 4 * q^77 - 14 * q^79 + 20 * q^83 - 24 * q^89 - 8 * q^91 + 16 * q^95

Introduction

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