LMFDB - Embedded newform 9576.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{11} -3.00000 q^{13} +1.61803 q^{17} -1.00000 q^{19} +1.76393 q^{23} +0.854102 q^{29} -4.38197 q^{31} +2.23607 q^{35} -4.70820 q^{37} +1.14590 q^{41} -3.52786 q^{43} +7.47214 q^{47} +1.00000 q^{49} -10.3262 q^{53} -6.38197 q^{55} +3.00000 q^{59} -15.1803 q^{61} -6.70820 q^{65} +2.85410 q^{67} +15.1803 q^{71} +2.32624 q^{73} -2.85410 q^{77} +0.472136 q^{79} -9.56231 q^{83} +3.61803 q^{85} -5.70820 q^{89} -3.00000 q^{91} -2.23607 q^{95} -9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + q^{11} - 6 q^{13} + q^{17} - 2 q^{19} + 8 q^{23} - 5 q^{29} - 11 q^{31} + 4 q^{37} + 9 q^{41} - 16 q^{43} + 6 q^{47} + 2 q^{49} - 5 q^{53} - 15 q^{55} + 6 q^{59} - 8 q^{61} - q^{67} + 8 q^{71} - 11 q^{73} + q^{77} - 8 q^{79} + q^{83} + 5 q^{85} + 2 q^{89} - 6 q^{91} - 18 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + q^11 - 6 * q^13 + q^17 - 2 * q^19 + 8 * q^23 - 5 * q^29 - 11 * q^31 + 4 * q^37 + 9 * q^41 - 16 * q^43 + 6 * q^47 + 2 * q^49 - 5 * q^53 - 15 * q^55 + 6 * q^59 - 8 * q^61 - q^67 + 8 * q^71 - 11 * q^73 + q^77 - 8 * q^79 + q^83 + 5 * q^85 + 2 * q^89 - 6 * q^91 - 18 * 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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.85410	−0.860544	−0.430272	−	0.902699i	$-0.641582\pi$
$11$	−2.85410	−0.860544	−0.430272	+	0.902699i	$0.641582\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.76393	0.367805	0.183903	−	0.982944i	$-0.441127\pi$
$23$	1.76393	0.367805	0.183903	+	0.982944i	$0.441127\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.854102	0.158603	0.0793014	−	0.996851i	$-0.474731\pi$
$29$	0.854102	0.158603	0.0793014	+	0.996851i	$0.474731\pi$
$30$	0	0
$31$	−4.38197	−0.787024	−0.393512	−	0.919319i	$-0.628740\pi$
$31$	−4.38197	−0.787024	−0.393512	+	0.919319i	$0.628740\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.23607	0.377964
$36$	0	0
$37$	−4.70820	−0.774024	−0.387012	−	0.922075i	$-0.626493\pi$
$37$	−4.70820	−0.774024	−0.387012	+	0.922075i	$0.626493\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.14590	0.178959	0.0894796	−	0.995989i	$-0.471480\pi$
$41$	1.14590	0.178959	0.0894796	+	0.995989i	$0.471480\pi$
$42$	0	0
$43$	−3.52786	−0.537994	−0.268997	−	0.963141i	$-0.586692\pi$
$43$	−3.52786	−0.537994	−0.268997	+	0.963141i	$0.586692\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.47214	1.08992	0.544962	−	0.838461i	$-0.316544\pi$
$47$	7.47214	1.08992	0.544962	+	0.838461i	$0.316544\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3262	−1.41842	−0.709209	−	0.704998i	$-0.750948\pi$
$53$	−10.3262	−1.41842	−0.709209	+	0.704998i	$0.750948\pi$
$54$	0	0
$55$	−6.38197	−0.860544
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−15.1803	−1.94364	−0.971822	−	0.235717i	$-0.924256\pi$
$61$	−15.1803	−1.94364	−0.971822	+	0.235717i	$0.924256\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.70820	−0.832050
$66$	0	0
$67$	2.85410	0.348684	0.174342	−	0.984685i	$-0.444220\pi$
$67$	2.85410	0.348684	0.174342	+	0.984685i	$0.444220\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.1803	1.80157	0.900787	−	0.434260i	$-0.142990\pi$
$71$	15.1803	1.80157	0.900787	+	0.434260i	$0.142990\pi$
$72$	0	0
$73$	2.32624	0.272266	0.136133	−	0.990691i	$-0.456533\pi$
$73$	2.32624	0.272266	0.136133	+	0.990691i	$0.456533\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.85410	−0.325255
$78$	0	0
$79$	0.472136	0.0531194	0.0265597	−	0.999647i	$-0.491545\pi$
$79$	0.472136	0.0531194	0.0265597	+	0.999647i	$0.491545\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.56231	−1.04960	−0.524800	−	0.851226i	$-0.675860\pi$
$83$	−9.56231	−1.04960	−0.524800	+	0.851226i	$0.675860\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.70820	−0.605068	−0.302534	−	0.953139i	$-0.597833\pi$
$89$	−5.70820	−0.605068	−0.302534	+	0.953139i	$0.597833\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	9.47214	0.933317	0.466659	−	0.884438i	$-0.345458\pi$
$103$	9.47214	0.933317	0.466659	+	0.884438i	$0.345458\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.76393	−0.363873	−0.181937	−	0.983310i	$-0.558237\pi$
$107$	−3.76393	−0.363873	−0.181937	+	0.983310i	$0.558237\pi$
$108$	0	0
$109$	13.4721	1.29040	0.645198	−	0.764015i	$-0.276775\pi$
$109$	13.4721	1.29040	0.645198	+	0.764015i	$0.276775\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.61803	−0.622572	−0.311286	−	0.950316i	$-0.600760\pi$
$113$	−6.61803	−0.622572	−0.311286	+	0.950316i	$0.600760\pi$
$114$	0	0
$115$	3.94427	0.367805
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.61803	0.148325
$120$	0	0
$121$	−2.85410	−0.259464
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−18.6525	−1.65514	−0.827570	−	0.561363i	$-0.810277\pi$
$127$	−18.6525	−1.65514	−0.827570	+	0.561363i	$0.810277\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.90983	−0.691085	−0.345543	−	0.938403i	$-0.612305\pi$
$131$	−7.90983	−0.691085	−0.345543	+	0.938403i	$0.612305\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.47214	−0.382080	−0.191040	−	0.981582i	$-0.561186\pi$
$137$	−4.47214	−0.382080	−0.191040	+	0.981582i	$0.561186\pi$
$138$	0	0
$139$	−6.52786	−0.553686	−0.276843	−	0.960915i	$-0.589288\pi$
$139$	−6.52786	−0.553686	−0.276843	+	0.960915i	$0.589288\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.56231	0.716016
$144$	0	0
$145$	1.90983	0.158603
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.4164	−1.50873	−0.754365	−	0.656455i	$-0.772055\pi$
$149$	−18.4164	−1.50873	−0.754365	+	0.656455i	$0.772055\pi$
$150$	0	0
$151$	−11.7984	−0.960138	−0.480069	−	0.877231i	$-0.659388\pi$
$151$	−11.7984	−0.960138	−0.480069	+	0.877231i	$0.659388\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.79837	−0.787024
$156$	0	0
$157$	−13.3262	−1.06355	−0.531775	−	0.846886i	$-0.678475\pi$
$157$	−13.3262	−1.06355	−0.531775	+	0.846886i	$0.678475\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.76393	0.139017
$162$	0	0
$163$	8.90983	0.697872	0.348936	−	0.937147i	$-0.386543\pi$
$163$	8.90983	0.697872	0.348936	+	0.937147i	$0.386543\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.2361	0.792091	0.396045	−	0.918231i	$-0.370382\pi$
$167$	10.2361	0.792091	0.396045	+	0.918231i	$0.370382\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.6525	0.809893	0.404946	−	0.914340i	$-0.367290\pi$
$173$	10.6525	0.809893	0.404946	+	0.914340i	$0.367290\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.6180	−1.61581	−0.807904	−	0.589314i	$-0.799398\pi$
$179$	−21.6180	−1.61581	−0.807904	+	0.589314i	$0.799398\pi$
$180$	0	0
$181$	5.03444	0.374207	0.187104	−	0.982340i	$-0.440090\pi$
$181$	5.03444	0.374207	0.187104	+	0.982340i	$0.440090\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.5279	−0.774024
$186$	0	0
$187$	−4.61803	−0.337704
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.326238	−0.0236057	−0.0118029	−	0.999930i	$-0.503757\pi$
$191$	−0.326238	−0.0236057	−0.0118029	+	0.999930i	$0.503757\pi$
$192$	0	0
$193$	14.3820	1.03524	0.517618	−	0.855612i	$-0.326819\pi$
$193$	14.3820	1.03524	0.517618	+	0.855612i	$0.326819\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.3820	−1.52340	−0.761701	−	0.647929i	$-0.775635\pi$
$197$	−21.3820	−1.52340	−0.761701	+	0.647929i	$0.775635\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.854102	0.0599462
$204$	0	0
$205$	2.56231	0.178959
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.85410	0.197422
$210$	0	0
$211$	−6.32624	−0.435516	−0.217758	−	0.976003i	$-0.569874\pi$
$211$	−6.32624	−0.435516	−0.217758	+	0.976003i	$0.569874\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.88854	−0.537994
$216$	0	0
$217$	−4.38197	−0.297467
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.85410	−0.326522
$222$	0	0
$223$	−3.65248	−0.244588	−0.122294	−	0.992494i	$-0.539025\pi$
$223$	−3.65248	−0.244588	−0.122294	+	0.992494i	$0.539025\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.7984	1.31406	0.657032	−	0.753863i	$-0.271812\pi$
$227$	19.7984	1.31406	0.657032	+	0.753863i	$0.271812\pi$
$228$	0	0
$229$	−3.52786	−0.233128	−0.116564	−	0.993183i	$-0.537188\pi$
$229$	−3.52786	−0.233128	−0.116564	+	0.993183i	$0.537188\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.79837	0.117815	0.0589077	−	0.998263i	$-0.481238\pi$
$233$	1.79837	0.117815	0.0589077	+	0.998263i	$0.481238\pi$
$234$	0	0
$235$	16.7082	1.08992
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.18034	−0.335088	−0.167544	−	0.985865i	$-0.553584\pi$
$239$	−5.18034	−0.335088	−0.167544	+	0.985865i	$0.553584\pi$
$240$	0	0
$241$	1.23607	0.0796221	0.0398111	−	0.999207i	$-0.487324\pi$
$241$	1.23607	0.0796221	0.0398111	+	0.999207i	$0.487324\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	3.00000	0.190885
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.14590	−0.577284	−0.288642	−	0.957437i	$-0.593204\pi$
$251$	−9.14590	−0.577284	−0.288642	+	0.957437i	$0.593204\pi$
$252$	0	0
$253$	−5.03444	−0.316513
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.8541	−0.739439	−0.369719	−	0.929144i	$-0.620546\pi$
$257$	−11.8541	−0.739439	−0.369719	+	0.929144i	$0.620546\pi$
$258$	0	0
$259$	−4.70820	−0.292554
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.85410	0.607630	0.303815	−	0.952731i	$-0.401740\pi$
$263$	9.85410	0.607630	0.303815	+	0.952731i	$0.401740\pi$
$264$	0	0
$265$	−23.0902	−1.41842
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.09017	0.188411	0.0942055	−	0.995553i	$-0.469969\pi$
$269$	3.09017	0.188411	0.0942055	+	0.995553i	$0.469969\pi$
$270$	0	0
$271$	18.7426	1.13853	0.569267	−	0.822152i	$-0.307227\pi$
$271$	18.7426	1.13853	0.569267	+	0.822152i	$0.307227\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.7082	−0.703478	−0.351739	−	0.936098i	$-0.614410\pi$
$277$	−11.7082	−0.703478	−0.351739	+	0.936098i	$0.614410\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.58359	0.333089	0.166545	−	0.986034i	$-0.446739\pi$
$281$	5.58359	0.333089	0.166545	+	0.986034i	$0.446739\pi$
$282$	0	0
$283$	19.7426	1.17358	0.586789	−	0.809740i	$-0.300392\pi$
$283$	19.7426	1.17358	0.586789	+	0.809740i	$0.300392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.14590	0.0676402
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.2361	−0.831680	−0.415840	−	0.909438i	$-0.636512\pi$
$293$	−14.2361	−0.831680	−0.415840	+	0.909438i	$0.636512\pi$
$294$	0	0
$295$	6.70820	0.390567
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.29180	−0.306032
$300$	0	0
$301$	−3.52786	−0.203343
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−33.9443	−1.94364
$306$	0	0
$307$	−0.201626	−0.0115074	−0.00575371	−	0.999983i	$-0.501831\pi$
$307$	−0.201626	−0.0115074	−0.00575371	+	0.999983i	$0.501831\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.32624	−0.0752041	−0.0376020	−	0.999293i	$-0.511972\pi$
$311$	−1.32624	−0.0752041	−0.0376020	+	0.999293i	$0.511972\pi$
$312$	0	0
$313$	15.1246	0.854894	0.427447	−	0.904041i	$-0.359413\pi$
$313$	15.1246	0.854894	0.427447	+	0.904041i	$0.359413\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.47214	0.251180	0.125590	−	0.992082i	$-0.459918\pi$
$317$	4.47214	0.251180	0.125590	+	0.992082i	$0.459918\pi$
$318$	0	0
$319$	−2.43769	−0.136485
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.61803	−0.0900298
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.47214	0.411952
$330$	0	0
$331$	14.9098	0.819518	0.409759	−	0.912194i	$-0.365613\pi$
$331$	14.9098	0.819518	0.409759	+	0.912194i	$0.365613\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.38197	0.348684
$336$	0	0
$337$	33.3262	1.81540	0.907698	−	0.419624i	$-0.137838\pi$
$337$	33.3262	1.81540	0.907698	+	0.419624i	$0.137838\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.5066	0.677269
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.79837	0.257590	0.128795	−	0.991671i	$-0.458889\pi$
$347$	4.79837	0.257590	0.128795	+	0.991671i	$0.458889\pi$
$348$	0	0
$349$	−32.9787	−1.76531	−0.882655	−	0.470021i	$-0.844246\pi$
$349$	−32.9787	−1.76531	−0.882655	+	0.470021i	$0.844246\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.5623	0.668624	0.334312	−	0.942462i	$-0.391496\pi$
$353$	12.5623	0.668624	0.334312	+	0.942462i	$0.391496\pi$
$354$	0	0
$355$	33.9443	1.80157
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.56231	0.240789	0.120395	−	0.992726i	$-0.461584\pi$
$359$	4.56231	0.240789	0.120395	+	0.992726i	$0.461584\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.20163	0.272266
$366$	0	0
$367$	−36.5967	−1.91033	−0.955167	−	0.296066i	$-0.904325\pi$
$367$	−36.5967	−1.91033	−0.955167	+	0.296066i	$0.904325\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.3262	−0.536112
$372$	0	0
$373$	8.50658	0.440454	0.220227	−	0.975449i	$-0.429320\pi$
$373$	8.50658	0.440454	0.220227	+	0.975449i	$0.429320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.56231	−0.131965
$378$	0	0
$379$	8.41641	0.432322	0.216161	−	0.976358i	$-0.430646\pi$
$379$	8.41641	0.432322	0.216161	+	0.976358i	$0.430646\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.4721	−0.994980	−0.497490	−	0.867470i	$-0.665745\pi$
$383$	−19.4721	−0.994980	−0.497490	+	0.867470i	$0.665745\pi$
$384$	0	0
$385$	−6.38197	−0.325255
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−37.0344	−1.87772	−0.938860	−	0.344298i	$-0.888117\pi$
$389$	−37.0344	−1.87772	−0.938860	+	0.344298i	$0.888117\pi$
$390$	0	0
$391$	2.85410	0.144338
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.05573	0.0531194
$396$	0	0
$397$	−11.2361	−0.563922	−0.281961	−	0.959426i	$-0.590985\pi$
$397$	−11.2361	−0.563922	−0.281961	+	0.959426i	$0.590985\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.6180	0.680052	0.340026	−	0.940416i	$-0.389564\pi$
$401$	13.6180	0.680052	0.340026	+	0.940416i	$0.389564\pi$
$402$	0	0
$403$	13.1459	0.654844
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.4377	0.666082
$408$	0	0
$409$	6.03444	0.298384	0.149192	−	0.988808i	$-0.452333\pi$
$409$	6.03444	0.298384	0.149192	+	0.988808i	$0.452333\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	−21.3820	−1.04960
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.3607	1.14124	0.570622	−	0.821213i	$-0.306702\pi$
$419$	23.3607	1.14124	0.570622	+	0.821213i	$0.306702\pi$
$420$	0	0
$421$	16.2361	0.791298	0.395649	−	0.918402i	$-0.370520\pi$
$421$	16.2361	0.791298	0.395649	+	0.918402i	$0.370520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−15.1803	−0.734628
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8885	0.765324	0.382662	−	0.923888i	$-0.375007\pi$
$431$	15.8885	0.765324	0.382662	+	0.923888i	$0.375007\pi$
$432$	0	0
$433$	−37.6525	−1.80946	−0.904731	−	0.425983i	$-0.859928\pi$
$433$	−37.6525	−1.80946	−0.904731	+	0.425983i	$0.859928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.76393	−0.0843803
$438$	0	0
$439$	−25.8885	−1.23559	−0.617796	−	0.786338i	$-0.711974\pi$
$439$	−25.8885	−1.23559	−0.617796	+	0.786338i	$0.711974\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.2148	1.05546	0.527728	−	0.849413i	$-0.323044\pi$
$443$	22.2148	1.05546	0.527728	+	0.849413i	$0.323044\pi$
$444$	0	0
$445$	−12.7639	−0.605068
$446$	0	0
$447$	0	0
$448$	0	0
$449$	37.4508	1.76741	0.883707	−	0.468040i	$-0.155040\pi$
$449$	37.4508	1.76741	0.883707	+	0.468040i	$0.155040\pi$
$450$	0	0
$451$	−3.27051	−0.154002
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.70820	−0.314485
$456$	0	0
$457$	−18.0344	−0.843616	−0.421808	−	0.906685i	$-0.638604\pi$
$457$	−18.0344	−0.843616	−0.421808	+	0.906685i	$0.638604\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.7984	1.15498	0.577488	−	0.816399i	$-0.304033\pi$
$461$	24.7984	1.15498	0.577488	+	0.816399i	$0.304033\pi$
$462$	0	0
$463$	−17.4721	−0.811999	−0.406000	−	0.913873i	$-0.633077\pi$
$463$	−17.4721	−0.811999	−0.406000	+	0.913873i	$0.633077\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.4377	−0.482999	−0.241499	−	0.970401i	$-0.577639\pi$
$467$	−10.4377	−0.482999	−0.241499	+	0.970401i	$0.577639\pi$
$468$	0	0
$469$	2.85410	0.131790
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.0689	0.462968
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.7426	1.77020	0.885098	−	0.465404i	$-0.154091\pi$
$479$	38.7426	1.77020	0.885098	+	0.465404i	$0.154091\pi$
$480$	0	0
$481$	14.1246	0.644027
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.1246	−0.913812
$486$	0	0
$487$	33.5410	1.51989	0.759944	−	0.649988i	$-0.225226\pi$
$487$	33.5410	1.51989	0.759944	+	0.649988i	$0.225226\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.4721	1.01415	0.507077	−	0.861901i	$-0.330726\pi$
$491$	22.4721	1.01415	0.507077	+	0.861901i	$0.330726\pi$
$492$	0	0
$493$	1.38197	0.0622406
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.1803	0.680931
$498$	0	0
$499$	−10.3262	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$499$	−10.3262	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	6.70820	0.298511
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.8885	1.41343	0.706717	−	0.707496i	$-0.250175\pi$
$509$	31.8885	1.41343	0.706717	+	0.707496i	$0.250175\pi$
$510$	0	0
$511$	2.32624	0.102907
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.1803	0.933317
$516$	0	0
$517$	−21.3262	−0.937927
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.70820	0.381513	0.190757	−	0.981637i	$-0.438906\pi$
$521$	8.70820	0.381513	0.190757	+	0.981637i	$0.438906\pi$
$522$	0	0
$523$	−13.1803	−0.576336	−0.288168	−	0.957580i	$-0.593046\pi$
$523$	−13.1803	−0.576336	−0.288168	+	0.957580i	$0.593046\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.09017	−0.308853
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.43769	−0.148903
$534$	0	0
$535$	−8.41641	−0.363873
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.85410	−0.122935
$540$	0	0
$541$	−10.5836	−0.455024	−0.227512	−	0.973775i	$-0.573059\pi$
$541$	−10.5836	−0.455024	−0.227512	+	0.973775i	$0.573059\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.1246	1.29040
$546$	0	0
$547$	−32.2148	−1.37740	−0.688702	−	0.725044i	$-0.741819\pi$
$547$	−32.2148	−1.37740	−0.688702	+	0.725044i	$0.741819\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.854102	−0.0363860
$552$	0	0
$553$	0.472136	0.0200773
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.742646	−0.0314669	−0.0157335	−	0.999876i	$-0.505008\pi$
$557$	−0.742646	−0.0314669	−0.0157335	+	0.999876i	$0.505008\pi$
$558$	0	0
$559$	10.5836	0.447638
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.8328	1.00443	0.502217	−	0.864742i	$-0.332518\pi$
$563$	23.8328	1.00443	0.502217	+	0.864742i	$0.332518\pi$
$564$	0	0
$565$	−14.7984	−0.622572
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.2361	−0.429118	−0.214559	−	0.976711i	$-0.568831\pi$
$569$	−10.2361	−0.429118	−0.214559	+	0.976711i	$0.568831\pi$
$570$	0	0
$571$	32.7771	1.37168	0.685839	−	0.727753i	$-0.259435\pi$
$571$	32.7771	1.37168	0.685839	+	0.727753i	$0.259435\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.7426	0.488853	0.244426	−	0.969668i	$-0.421400\pi$
$577$	11.7426	0.488853	0.244426	+	0.969668i	$0.421400\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.56231	−0.396711
$582$	0	0
$583$	29.4721	1.22061
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.4164	−1.70944	−0.854719	−	0.519091i	$-0.826271\pi$
$587$	−41.4164	−1.70944	−0.854719	+	0.519091i	$0.826271\pi$
$588$	0	0
$589$	4.38197	0.180556
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.94427	−0.326232	−0.163116	−	0.986607i	$-0.552155\pi$
$593$	−7.94427	−0.326232	−0.163116	+	0.986607i	$0.552155\pi$
$594$	0	0
$595$	3.61803	0.148325
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.09017	−0.371414	−0.185707	−	0.982605i	$-0.559458\pi$
$599$	−9.09017	−0.371414	−0.185707	+	0.982605i	$0.559458\pi$
$600$	0	0
$601$	−45.8541	−1.87043	−0.935214	−	0.354083i	$-0.884793\pi$
$601$	−45.8541	−1.87043	−0.935214	+	0.354083i	$0.884793\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.38197	−0.259464
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.4164	−0.906871
$612$	0	0
$613$	−15.2705	−0.616770	−0.308385	−	0.951262i	$-0.599789\pi$
$613$	−15.2705	−0.616770	−0.308385	+	0.951262i	$0.599789\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.6180	−0.628758	−0.314379	−	0.949297i	$-0.601796\pi$
$617$	−15.6180	−0.628758	−0.314379	+	0.949297i	$0.601796\pi$
$618$	0	0
$619$	−4.43769	−0.178366	−0.0891830	−	0.996015i	$-0.528426\pi$
$619$	−4.43769	−0.178366	−0.0891830	+	0.996015i	$0.528426\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.70820	−0.228694
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.61803	−0.303751
$630$	0	0
$631$	−37.4721	−1.49174	−0.745871	−	0.666090i	$-0.767967\pi$
$631$	−37.4721	−1.49174	−0.745871	+	0.666090i	$0.767967\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−41.7082	−1.65514
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.5623	−1.36513	−0.682565	−	0.730825i	$-0.739135\pi$
$641$	−34.5623	−1.36513	−0.682565	+	0.730825i	$0.739135\pi$
$642$	0	0
$643$	−21.7639	−0.858286	−0.429143	−	0.903237i	$-0.641184\pi$
$643$	−21.7639	−0.858286	−0.429143	+	0.903237i	$0.641184\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.4164	1.43168	0.715838	−	0.698266i	$-0.246045\pi$
$647$	36.4164	1.43168	0.715838	+	0.698266i	$0.246045\pi$
$648$	0	0
$649$	−8.56231	−0.336100
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.7639	−1.55608	−0.778041	−	0.628213i	$-0.783787\pi$
$653$	−39.7639	−1.55608	−0.778041	+	0.628213i	$0.783787\pi$
$654$	0	0
$655$	−17.6869	−0.691085
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.03444	−0.274023	−0.137011	−	0.990569i	$-0.543750\pi$
$659$	−7.03444	−0.274023	−0.137011	+	0.990569i	$0.543750\pi$
$660$	0	0
$661$	24.8328	0.965885	0.482942	−	0.875652i	$-0.339568\pi$
$661$	24.8328	0.965885	0.482942	+	0.875652i	$0.339568\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.23607	−0.0867110
$666$	0	0
$667$	1.50658	0.0583349
$668$	0	0
$669$	0	0
$670$	0	0
$671$	43.3262	1.67259
$672$	0	0
$673$	−15.2705	−0.588635	−0.294317	−	0.955708i	$-0.595092\pi$
$673$	−15.2705	−0.588635	−0.294317	+	0.955708i	$0.595092\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.14590	0.0824736	0.0412368	−	0.999149i	$-0.486870\pi$
$677$	2.14590	0.0824736	0.0412368	+	0.999149i	$0.486870\pi$
$678$	0	0
$679$	−9.00000	−0.345388
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.0689	−0.729651	−0.364825	−	0.931076i	$-0.618871\pi$
$683$	−19.0689	−0.729651	−0.364825	+	0.931076i	$0.618871\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	30.9787	1.18020
$690$	0	0
$691$	−11.1246	−0.423200	−0.211600	−	0.977356i	$-0.567867\pi$
$691$	−11.1246	−0.423200	−0.211600	+	0.977356i	$0.567867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.5967	−0.553686
$696$	0	0
$697$	1.85410	0.0702291
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21.3050	−0.804677	−0.402338	−	0.915491i	$-0.631802\pi$
$701$	−21.3050	−0.804677	−0.402338	+	0.915491i	$0.631802\pi$
$702$	0	0
$703$	4.70820	0.177573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	−16.2361	−0.609758	−0.304879	−	0.952391i	$-0.598616\pi$
$709$	−16.2361	−0.609758	−0.304879	+	0.952391i	$0.598616\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.72949	−0.289472
$714$	0	0
$715$	19.1459	0.716016
$716$	0	0
$717$	0	0
$718$	0	0
$719$	47.7771	1.78178	0.890892	−	0.454215i	$-0.150080\pi$
$719$	47.7771	1.78178	0.890892	+	0.454215i	$0.150080\pi$
$720$	0	0
$721$	9.47214	0.352761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.70820	−0.100442	−0.0502209	−	0.998738i	$-0.515993\pi$
$727$	−2.70820	−0.100442	−0.0502209	+	0.998738i	$0.515993\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.70820	−0.211126
$732$	0	0
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.14590	−0.300058
$738$	0	0
$739$	−36.1246	−1.32887	−0.664433	−	0.747348i	$-0.731327\pi$
$739$	−36.1246	−1.32887	−0.664433	+	0.747348i	$0.731327\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.8197	0.396935	0.198467	−	0.980107i	$-0.436404\pi$
$743$	10.8197	0.396935	0.198467	+	0.980107i	$0.436404\pi$
$744$	0	0
$745$	−41.1803	−1.50873
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.76393	−0.137531
$750$	0	0
$751$	−12.2148	−0.445724	−0.222862	−	0.974850i	$-0.571540\pi$
$751$	−12.2148	−0.445724	−0.222862	+	0.974850i	$0.571540\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.3820	−0.960138
$756$	0	0
$757$	18.3607	0.667330	0.333665	−	0.942692i	$-0.391715\pi$
$757$	18.3607	0.667330	0.333665	+	0.942692i	$0.391715\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.1246	1.01952	0.509758	−	0.860318i	$-0.329735\pi$
$761$	28.1246	1.01952	0.509758	+	0.860318i	$0.329735\pi$
$762$	0	0
$763$	13.4721	0.487724
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	24.9443	0.899513	0.449757	−	0.893151i	$-0.351511\pi$
$769$	24.9443	0.899513	0.449757	+	0.893151i	$0.351511\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.6525	−1.03056	−0.515279	−	0.857023i	$-0.672312\pi$
$773$	−28.6525	−1.03056	−0.515279	+	0.857023i	$0.672312\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.14590	−0.0410561
$780$	0	0
$781$	−43.3262	−1.55033
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29.7984	−1.06355
$786$	0	0
$787$	−8.36068	−0.298026	−0.149013	−	0.988835i	$-0.547610\pi$
$787$	−8.36068	−0.298026	−0.149013	+	0.988835i	$0.547610\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.61803	−0.235310
$792$	0	0
$793$	45.5410	1.61721
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.9230	−0.599443	−0.299721	−	0.954027i	$-0.596894\pi$
$797$	−16.9230	−0.599443	−0.299721	+	0.954027i	$0.596894\pi$
$798$	0	0
$799$	12.0902	0.427719
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.63932	−0.234297
$804$	0	0
$805$	3.94427	0.139017
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.12461	0.0746974	0.0373487	−	0.999302i	$-0.488109\pi$
$809$	2.12461	0.0746974	0.0373487	+	0.999302i	$0.488109\pi$
$810$	0	0
$811$	48.8328	1.71475	0.857376	−	0.514691i	$-0.172093\pi$
$811$	48.8328	1.71475	0.857376	+	0.514691i	$0.172093\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19.9230	0.697872
$816$	0	0
$817$	3.52786	0.123424
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.3607	1.68780	0.843900	−	0.536501i	$-0.180254\pi$
$821$	48.3607	1.68780	0.843900	+	0.536501i	$0.180254\pi$
$822$	0	0
$823$	28.0689	0.978420	0.489210	−	0.872166i	$-0.337285\pi$
$823$	28.0689	0.978420	0.489210	+	0.872166i	$0.337285\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.1803	0.945153	0.472577	−	0.881290i	$-0.343324\pi$
$827$	27.1803	0.945153	0.472577	+	0.881290i	$0.343324\pi$
$828$	0	0
$829$	33.8885	1.17700	0.588499	−	0.808498i	$-0.299719\pi$
$829$	33.8885	1.17700	0.588499	+	0.808498i	$0.299719\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.61803	0.0560616
$834$	0	0
$835$	22.8885	0.792091
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.5279	0.605129	0.302565	−	0.953129i	$-0.402157\pi$
$839$	17.5279	0.605129	0.302565	+	0.953129i	$0.402157\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.94427	−0.307692
$846$	0	0
$847$	−2.85410	−0.0980681
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.30495	−0.284690
$852$	0	0
$853$	41.0902	1.40690	0.703450	−	0.710744i	$-0.251642\pi$
$853$	41.0902	1.40690	0.703450	+	0.710744i	$0.251642\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.4508	−0.971863	−0.485931	−	0.873997i	$-0.661519\pi$
$857$	−28.4508	−0.971863	−0.485931	+	0.873997i	$0.661519\pi$
$858$	0	0
$859$	−45.6180	−1.55647	−0.778234	−	0.627975i	$-0.783884\pi$
$859$	−45.6180	−1.55647	−0.778234	+	0.627975i	$0.783884\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.0344	−1.02238	−0.511192	−	0.859466i	$-0.670796\pi$
$863$	−30.0344	−1.02238	−0.511192	+	0.859466i	$0.670796\pi$
$864$	0	0
$865$	23.8197	0.809893
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.34752	−0.0457116
$870$	0	0
$871$	−8.56231	−0.290123
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	35.0132	1.18231	0.591155	−	0.806558i	$-0.298672\pi$
$877$	35.0132	1.18231	0.591155	+	0.806558i	$0.298672\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.2705	−0.986149	−0.493074	−	0.869987i	$-0.664127\pi$
$881$	−29.2705	−0.986149	−0.493074	+	0.869987i	$0.664127\pi$
$882$	0	0
$883$	−17.6525	−0.594053	−0.297027	−	0.954869i	$-0.595995\pi$
$883$	−17.6525	−0.594053	−0.297027	+	0.954869i	$0.595995\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.819660	−0.0275215	−0.0137607	−	0.999905i	$-0.504380\pi$
$887$	−0.819660	−0.0275215	−0.0137607	+	0.999905i	$0.504380\pi$
$888$	0	0
$889$	−18.6525	−0.625584
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.47214	−0.250045
$894$	0	0
$895$	−48.3394	−1.61581
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.74265	−0.124824
$900$	0	0
$901$	−16.7082	−0.556631
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.2574	0.374207
$906$	0	0
$907$	30.8885	1.02564	0.512819	−	0.858497i	$-0.328601\pi$
$907$	30.8885	1.02564	0.512819	+	0.858497i	$0.328601\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.3050	−1.16971	−0.584853	−	0.811140i	$-0.698848\pi$
$911$	−35.3050	−1.16971	−0.584853	+	0.811140i	$0.698848\pi$
$912$	0	0
$913$	27.2918	0.903227
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.90983	−0.261206
$918$	0	0
$919$	−23.9443	−0.789849	−0.394924	−	0.918714i	$-0.629229\pi$
$919$	−23.9443	−0.789849	−0.394924	+	0.918714i	$0.629229\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−45.5410	−1.49900
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.7295	0.352023	0.176012	−	0.984388i	$-0.443680\pi$
$929$	10.7295	0.352023	0.176012	+	0.984388i	$0.443680\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.3262	−0.337704
$936$	0	0
$937$	−10.9230	−0.356838	−0.178419	−	0.983955i	$-0.557098\pi$
$937$	−10.9230	−0.356838	−0.178419	+	0.983955i	$0.557098\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.3607	−0.337749	−0.168874	−	0.985638i	$-0.554013\pi$
$941$	−10.3607	−0.337749	−0.168874	+	0.985638i	$0.554013\pi$
$942$	0	0
$943$	2.02129	0.0658221
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6180	1.12493	0.562467	−	0.826819i	$-0.309852\pi$
$947$	34.6180	1.12493	0.562467	+	0.826819i	$0.309852\pi$
$948$	0	0
$949$	−6.97871	−0.226539
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−57.6180	−1.86643	−0.933216	−	0.359317i	$-0.883010\pi$
$953$	−57.6180	−1.86643	−0.933216	+	0.359317i	$0.883010\pi$
$954$	0	0
$955$	−0.729490	−0.0236057
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.47214	−0.144413
$960$	0	0
$961$	−11.7984	−0.380593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.1591	1.03524
$966$	0	0
$967$	−15.6869	−0.504457	−0.252229	−	0.967668i	$-0.581164\pi$
$967$	−15.6869	−0.504457	−0.252229	+	0.967668i	$0.581164\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	0	0
$973$	−6.52786	−0.209274
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.4853	−1.64716	−0.823580	−	0.567200i	$-0.808027\pi$
$977$	−51.4853	−1.64716	−0.823580	+	0.567200i	$0.808027\pi$
$978$	0	0
$979$	16.2918	0.520688
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.58359	0.0505087	0.0252544	−	0.999681i	$-0.491960\pi$
$983$	1.58359	0.0505087	0.0252544	+	0.999681i	$0.491960\pi$
$984$	0	0
$985$	−47.8115	−1.52340
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.22291	−0.197877
$990$	0	0
$991$	−33.5967	−1.06724	−0.533618	−	0.845726i	$-0.679168\pi$
$991$	−33.5967	−1.06724	−0.533618	+	0.845726i	$0.679168\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.1803	−0.354441
$996$	0	0
$997$	45.6738	1.44650	0.723251	−	0.690585i	$-0.242647\pi$
$997$	45.6738	1.44650	0.723251	+	0.690585i	$0.242647\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bn.1.2		2
3.2	odd	2		1064.2.a.c.1.2	✓	2
12.11	even	2		2128.2.a.j.1.1		2
21.20	even	2		7448.2.a.ba.1.1		2
24.5	odd	2		8512.2.a.v.1.1		2
24.11	even	2		8512.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.c.1.2	✓	2	3.2	odd	2
2128.2.a.j.1.1		2	12.11	even	2
7448.2.a.ba.1.1		2	21.20	even	2
8512.2.a.o.1.2		2	24.11	even	2
8512.2.a.v.1.1		2	24.5	odd	2
9576.2.a.bn.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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{11} -3.00000 q^{13} +1.61803 q^{17} -1.00000 q^{19} +1.76393 q^{23} +0.854102 q^{29} -4.38197 q^{31} +2.23607 q^{35} -4.70820 q^{37} +1.14590 q^{41} -3.52786 q^{43} +7.47214 q^{47} +1.00000 q^{49} -10.3262 q^{53} -6.38197 q^{55} +3.00000 q^{59} -15.1803 q^{61} -6.70820 q^{65} +2.85410 q^{67} +15.1803 q^{71} +2.32624 q^{73} -2.85410 q^{77} +0.472136 q^{79} -9.56231 q^{83} +3.61803 q^{85} -5.70820 q^{89} -3.00000 q^{91} -2.23607 q^{95} -9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + q^{11} - 6 q^{13} + q^{17} - 2 q^{19} + 8 q^{23} - 5 q^{29} - 11 q^{31} + 4 q^{37} + 9 q^{41} - 16 q^{43} + 6 q^{47} + 2 q^{49} - 5 q^{53} - 15 q^{55} + 6 q^{59} - 8 q^{61} - q^{67} + 8 q^{71} - 11 q^{73} + q^{77} - 8 q^{79} + q^{83} + 5 q^{85} + 2 q^{89} - 6 q^{91} - 18 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + q^11 - 6 * q^13 + q^17 - 2 * q^19 + 8 * q^23 - 5 * q^29 - 11 * q^31 + 4 * q^37 + 9 * q^41 - 16 * q^43 + 6 * q^47 + 2 * q^49 - 5 * q^53 - 15 * q^55 + 6 * q^59 - 8 * q^61 - q^67 + 8 * q^71 - 11 * q^73 + q^77 - 8 * q^79 + q^83 + 5 * q^85 + 2 * q^89 - 6 * q^91 - 18 * q^97

Introduction

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