LMFDB - Embedded newform 9576.2.a.bn.1.1

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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-2.23607 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.23607 q^{5} +1.00000 q^{7} +3.85410 q^{11} -3.00000 q^{13} -0.618034 q^{17} -1.00000 q^{19} +6.23607 q^{23} -5.85410 q^{29} -6.61803 q^{31} -2.23607 q^{35} +8.70820 q^{37} +7.85410 q^{41} -12.4721 q^{43} -1.47214 q^{47} +1.00000 q^{49} +5.32624 q^{53} -8.61803 q^{55} +3.00000 q^{59} +7.18034 q^{61} +6.70820 q^{65} -3.85410 q^{67} -7.18034 q^{71} -13.3262 q^{73} +3.85410 q^{77} -8.47214 q^{79} +10.5623 q^{83} +1.38197 q^{85} +7.70820 q^{89} -3.00000 q^{91} +2.23607 q^{95} -9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.85410	1.16206	0.581028	−	0.813884i	$-0.302651\pi$
$11$	3.85410	1.16206	0.581028	+	0.813884i	$0.302651\pi$
$12$	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
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.23607	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$23$	6.23607	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.85410	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$29$	−5.85410	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$30$	0	0
$31$	−6.61803	−1.18863	−0.594317	−	0.804231i	$-0.702578\pi$
$31$	−6.61803	−1.18863	−0.594317	+	0.804231i	$0.702578\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.23607	−0.377964
$36$	0	0
$37$	8.70820	1.43162	0.715810	−	0.698295i	$-0.246058\pi$
$37$	8.70820	1.43162	0.715810	+	0.698295i	$0.246058\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.85410	1.22660	0.613302	−	0.789848i	$-0.289841\pi$
$41$	7.85410	1.22660	0.613302	+	0.789848i	$0.289841\pi$
$42$	0	0
$43$	−12.4721	−1.90198	−0.950991	−	0.309217i	$-0.899933\pi$
$43$	−12.4721	−1.90198	−0.950991	+	0.309217i	$0.899933\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.47214	−0.214733	−0.107367	−	0.994220i	$-0.534242\pi$
$47$	−1.47214	−0.214733	−0.107367	+	0.994220i	$0.534242\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.32624	0.731615	0.365808	−	0.930691i	$-0.380793\pi$
$53$	5.32624	0.731615	0.365808	+	0.930691i	$0.380793\pi$
$54$	0	0
$55$	−8.61803	−1.16206
$56$	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
$61$	7.18034	0.919348	0.459674	−	0.888088i	$-0.347966\pi$
$61$	7.18034	0.919348	0.459674	+	0.888088i	$0.347966\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.70820	0.832050
$66$	0	0
$67$	−3.85410	−0.470853	−0.235427	−	0.971892i	$-0.575649\pi$
$67$	−3.85410	−0.470853	−0.235427	+	0.971892i	$0.575649\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.18034	−0.852150	−0.426075	−	0.904688i	$-0.640104\pi$
$71$	−7.18034	−0.852150	−0.426075	+	0.904688i	$0.640104\pi$
$72$	0	0
$73$	−13.3262	−1.55972	−0.779859	−	0.625955i	$-0.784709\pi$
$73$	−13.3262	−1.55972	−0.779859	+	0.625955i	$0.784709\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.85410	0.439216
$78$	0	0
$79$	−8.47214	−0.953190	−0.476595	−	0.879123i	$-0.658129\pi$
$79$	−8.47214	−0.953190	−0.476595	+	0.879123i	$0.658129\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.5623	1.15936	0.579682	−	0.814843i	$-0.303177\pi$
$83$	10.5623	1.15936	0.579682	+	0.814843i	$0.303177\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.70820	0.817068	0.408534	−	0.912743i	$-0.366040\pi$
$89$	7.70820	0.817068	0.408534	+	0.912743i	$0.366040\pi$
$90$	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
$103$	0.527864	0.0520120	0.0260060	−	0.999662i	$-0.491721\pi$
$103$	0.527864	0.0520120	0.0260060	+	0.999662i	$0.491721\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.23607	−0.796211	−0.398105	−	0.917340i	$-0.630332\pi$
$107$	−8.23607	−0.796211	−0.398105	+	0.917340i	$0.630332\pi$
$108$	0	0
$109$	4.52786	0.433691	0.216845	−	0.976206i	$-0.430423\pi$
$109$	4.52786	0.433691	0.216845	+	0.976206i	$0.430423\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.38197	−0.412221	−0.206110	−	0.978529i	$-0.566081\pi$
$113$	−4.38197	−0.412221	−0.206110	+	0.978529i	$0.566081\pi$
$114$	0	0
$115$	−13.9443	−1.30031
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.618034	−0.0566551
$120$	0	0
$121$	3.85410	0.350373
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	12.6525	1.12273	0.561363	−	0.827570i	$-0.310277\pi$
$127$	12.6525	1.12273	0.561363	+	0.827570i	$0.310277\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.0902	−1.66792	−0.833958	−	0.551828i	$-0.813930\pi$
$131$	−19.0902	−1.66792	−0.833958	+	0.551828i	$0.813930\pi$
$132$	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	$-0.438814\pi$
$137$	4.47214	0.382080	0.191040	+	0.981582i	$0.438814\pi$
$138$	0	0
$139$	−15.4721	−1.31233	−0.656165	−	0.754618i	$-0.727822\pi$
$139$	−15.4721	−1.31233	−0.656165	+	0.754618i	$0.727822\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.5623	−0.966889
$144$	0	0
$145$	13.0902	1.08708
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.41641	0.689499	0.344749	−	0.938695i	$-0.387964\pi$
$149$	8.41641	0.689499	0.344749	+	0.938695i	$0.387964\pi$
$150$	0	0
$151$	12.7984	1.04152	0.520758	−	0.853704i	$-0.325649\pi$
$151$	12.7984	1.04152	0.520758	+	0.853704i	$0.325649\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	14.7984	1.18863
$156$	0	0
$157$	2.32624	0.185654	0.0928270	−	0.995682i	$-0.470410\pi$
$157$	2.32624	0.185654	0.0928270	+	0.995682i	$0.470410\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.23607	0.491471
$162$	0	0
$163$	20.0902	1.57358	0.786792	−	0.617219i	$-0.211741\pi$
$163$	20.0902	1.57358	0.786792	+	0.617219i	$0.211741\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.76393	0.446026	0.223013	−	0.974815i	$-0.428411\pi$
$167$	5.76393	0.446026	0.223013	+	0.974815i	$0.428411\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.6525	−1.57018	−0.785089	−	0.619383i	$-0.787383\pi$
$173$	−20.6525	−1.57018	−0.785089	+	0.619383i	$0.787383\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.3820	−1.44868	−0.724338	−	0.689445i	$-0.757855\pi$
$179$	−19.3820	−1.44868	−0.724338	+	0.689445i	$0.757855\pi$
$180$	0	0
$181$	−24.0344	−1.78647	−0.893233	−	0.449594i	$-0.851569\pi$
$181$	−24.0344	−1.78647	−0.893233	+	0.449594i	$0.851569\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19.4721	−1.43162
$186$	0	0
$187$	−2.38197	−0.174187
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.3262	1.10897	0.554484	−	0.832194i	$-0.312916\pi$
$191$	15.3262	1.10897	0.554484	+	0.832194i	$0.312916\pi$
$192$	0	0
$193$	16.6180	1.19619	0.598096	−	0.801424i	$-0.295924\pi$
$193$	16.6180	1.19619	0.598096	+	0.801424i	$0.295924\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.6180	−1.68272	−0.841358	−	0.540479i	$-0.818243\pi$
$197$	−23.6180	−1.68272	−0.841358	+	0.540479i	$0.818243\pi$
$198$	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
$203$	−5.85410	−0.410877
$204$	0	0
$205$	−17.5623	−1.22660
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.85410	−0.266594
$210$	0	0
$211$	9.32624	0.642045	0.321022	−	0.947072i	$-0.395974\pi$
$211$	9.32624	0.642045	0.321022	+	0.947072i	$0.395974\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.8885	1.90198
$216$	0	0
$217$	−6.61803	−0.449261
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.85410	0.124720
$222$	0	0
$223$	27.6525	1.85175	0.925873	−	0.377834i	$-0.123331\pi$
$223$	27.6525	1.85175	0.925873	+	0.377834i	$0.123331\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.79837	−0.318479	−0.159240	−	0.987240i	$-0.550904\pi$
$227$	−4.79837	−0.318479	−0.159240	+	0.987240i	$0.550904\pi$
$228$	0	0
$229$	−12.4721	−0.824182	−0.412091	−	0.911143i	$-0.635201\pi$
$229$	−12.4721	−0.824182	−0.412091	+	0.911143i	$0.635201\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.7984	−1.49357	−0.746786	−	0.665065i	$-0.768404\pi$
$233$	−22.7984	−1.49357	−0.746786	+	0.665065i	$0.768404\pi$
$234$	0	0
$235$	3.29180	0.214733
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.1803	1.11130	0.555652	−	0.831415i	$-0.312469\pi$
$239$	17.1803	1.11130	0.555652	+	0.831415i	$0.312469\pi$
$240$	0	0
$241$	−3.23607	−0.208453	−0.104227	−	0.994554i	$-0.533237\pi$
$241$	−3.23607	−0.208453	−0.104227	+	0.994554i	$0.533237\pi$
$242$	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
$251$	−15.8541	−1.00070	−0.500351	−	0.865823i	$-0.666796\pi$
$251$	−15.8541	−1.00070	−0.500351	+	0.865823i	$0.666796\pi$
$252$	0	0
$253$	24.0344	1.51103
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.14590	−0.320992	−0.160496	−	0.987036i	$-0.551309\pi$
$257$	−5.14590	−0.320992	−0.160496	+	0.987036i	$0.551309\pi$
$258$	0	0
$259$	8.70820	0.541101
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.14590	0.193984	0.0969922	−	0.995285i	$-0.469078\pi$
$263$	3.14590	0.193984	0.0969922	+	0.995285i	$0.469078\pi$
$264$	0	0
$265$	−11.9098	−0.731615
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8.09017	−0.493266	−0.246633	−	0.969109i	$-0.579324\pi$
$269$	−8.09017	−0.493266	−0.246633	+	0.969109i	$0.579324\pi$
$270$	0	0
$271$	−23.7426	−1.44226	−0.721132	−	0.692798i	$-0.756378\pi$
$271$	−23.7426	−1.44226	−0.721132	+	0.692798i	$0.756378\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.70820	0.102636	0.0513180	−	0.998682i	$-0.483658\pi$
$277$	1.70820	0.102636	0.0513180	+	0.998682i	$0.483658\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.4164	1.93380	0.966900	−	0.255154i	$-0.0821262\pi$
$281$	32.4164	1.93380	0.966900	+	0.255154i	$0.0821262\pi$
$282$	0	0
$283$	−22.7426	−1.35191	−0.675955	−	0.736943i	$-0.736269\pi$
$283$	−22.7426	−1.35191	−0.675955	+	0.736943i	$0.736269\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.85410	0.463613
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.76393	−0.570415	−0.285207	−	0.958466i	$-0.592063\pi$
$293$	−9.76393	−0.570415	−0.285207	+	0.958466i	$0.592063\pi$
$294$	0	0
$295$	−6.70820	−0.390567
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.7082	−1.08192
$300$	0	0
$301$	−12.4721	−0.718882
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.0557	−0.919348
$306$	0	0
$307$	−24.7984	−1.41532	−0.707659	−	0.706554i	$-0.750249\pi$
$307$	−24.7984	−1.41532	−0.707659	+	0.706554i	$0.750249\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.3262	0.812366	0.406183	−	0.913792i	$-0.366859\pi$
$311$	14.3262	0.812366	0.406183	+	0.913792i	$0.366859\pi$
$312$	0	0
$313$	−25.1246	−1.42013	−0.710064	−	0.704138i	$-0.751334\pi$
$313$	−25.1246	−1.42013	−0.710064	+	0.704138i	$0.751334\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.47214	−0.251180	−0.125590	−	0.992082i	$-0.540082\pi$
$317$	−4.47214	−0.251180	−0.125590	+	0.992082i	$0.540082\pi$
$318$	0	0
$319$	−22.5623	−1.26325
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.618034	0.0343883
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.47214	−0.0811615
$330$	0	0
$331$	26.0902	1.43405	0.717023	−	0.697050i	$-0.245504\pi$
$331$	26.0902	1.43405	0.717023	+	0.697050i	$0.245504\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.61803	0.470853
$336$	0	0
$337$	17.6738	0.962751	0.481376	−	0.876514i	$-0.340137\pi$
$337$	17.6738	0.962751	0.481376	+	0.876514i	$0.340137\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.5066	−1.38126
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.7984	−1.06283	−0.531416	−	0.847111i	$-0.678340\pi$
$347$	−19.7984	−1.06283	−0.531416	+	0.847111i	$0.678340\pi$
$348$	0	0
$349$	13.9787	0.748263	0.374132	−	0.927376i	$-0.377941\pi$
$349$	13.9787	0.748263	0.374132	+	0.927376i	$0.377941\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.56231	−0.402501	−0.201250	−	0.979540i	$-0.564501\pi$
$353$	−7.56231	−0.402501	−0.201250	+	0.979540i	$0.564501\pi$
$354$	0	0
$355$	16.0557	0.852150
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.5623	−0.821347	−0.410674	−	0.911782i	$-0.634706\pi$
$359$	−15.5623	−0.821347	−0.410674	+	0.911782i	$0.634706\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.7984	1.55972
$366$	0	0
$367$	12.5967	0.657545	0.328772	−	0.944409i	$-0.393365\pi$
$367$	12.5967	0.657545	0.328772	+	0.944409i	$0.393365\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.32624	0.276524
$372$	0	0
$373$	−29.5066	−1.52779	−0.763896	−	0.645339i	$-0.776716\pi$
$373$	−29.5066	−1.52779	−0.763896	+	0.645339i	$0.776716\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.5623	0.904505
$378$	0	0
$379$	−18.4164	−0.945987	−0.472994	−	0.881066i	$-0.656827\pi$
$379$	−18.4164	−0.945987	−0.472994	+	0.881066i	$0.656827\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.5279	−0.537949	−0.268974	−	0.963147i	$-0.586685\pi$
$383$	−10.5279	−0.537949	−0.268974	+	0.963147i	$0.586685\pi$
$384$	0	0
$385$	−8.61803	−0.439216
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.96556	−0.403870	−0.201935	−	0.979399i	$-0.564723\pi$
$389$	−7.96556	−0.403870	−0.201935	+	0.979399i	$0.564723\pi$
$390$	0	0
$391$	−3.85410	−0.194910
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.9443	0.953190
$396$	0	0
$397$	−6.76393	−0.339472	−0.169736	−	0.985490i	$-0.554292\pi$
$397$	−6.76393	−0.339472	−0.169736	+	0.985490i	$0.554292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.3820	0.568388	0.284194	−	0.958767i	$-0.408274\pi$
$401$	11.3820	0.568388	0.284194	+	0.958767i	$0.408274\pi$
$402$	0	0
$403$	19.8541	0.989003
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.5623	1.66362
$408$	0	0
$409$	−23.0344	−1.13898	−0.569490	−	0.821998i	$-0.692859\pi$
$409$	−23.0344	−1.13898	−0.569490	+	0.821998i	$0.692859\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	−23.6180	−1.15936
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.3607	−1.04354	−0.521769	−	0.853087i	$-0.674728\pi$
$419$	−21.3607	−1.04354	−0.521769	+	0.853087i	$0.674728\pi$
$420$	0	0
$421$	11.7639	0.573339	0.286669	−	0.958030i	$-0.407452\pi$
$421$	11.7639	0.573339	0.286669	+	0.958030i	$0.407452\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.18034	0.347481
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8885	−0.957997	−0.478999	−	0.877816i	$-0.659000\pi$
$431$	−19.8885	−0.957997	−0.478999	+	0.877816i	$0.659000\pi$
$432$	0	0
$433$	−6.34752	−0.305043	−0.152521	−	0.988300i	$-0.548739\pi$
$433$	−6.34752	−0.305043	−0.152521	+	0.988300i	$0.548739\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.23607	−0.298312
$438$	0	0
$439$	9.88854	0.471954	0.235977	−	0.971759i	$-0.424171\pi$
$439$	9.88854	0.471954	0.235977	+	0.971759i	$0.424171\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.2148	−1.38804	−0.694018	−	0.719958i	$-0.744161\pi$
$443$	−29.2148	−1.38804	−0.694018	+	0.719958i	$0.744161\pi$
$444$	0	0
$445$	−17.2361	−0.817068
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.4508	−0.870749	−0.435375	−	0.900249i	$-0.643384\pi$
$449$	−18.4508	−0.870749	−0.435375	+	0.900249i	$0.643384\pi$
$450$	0	0
$451$	30.2705	1.42538
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.70820	0.314485
$456$	0	0
$457$	11.0344	0.516169	0.258085	−	0.966122i	$-0.416909\pi$
$457$	11.0344	0.516169	0.258085	+	0.966122i	$0.416909\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.201626	0.00939066	0.00469533	−	0.999989i	$-0.498505\pi$
$461$	0.201626	0.00939066	0.00469533	+	0.999989i	$0.498505\pi$
$462$	0	0
$463$	−8.52786	−0.396323	−0.198162	−	0.980169i	$-0.563497\pi$
$463$	−8.52786	−0.396323	−0.198162	+	0.980169i	$0.563497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.5623	−1.41425	−0.707127	−	0.707086i	$-0.750009\pi$
$467$	−30.5623	−1.41425	−0.707127	+	0.707086i	$0.750009\pi$
$468$	0	0
$469$	−3.85410	−0.177966
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−48.0689	−2.21021
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.74265	−0.171006	−0.0855029	−	0.996338i	$-0.527250\pi$
$479$	−3.74265	−0.171006	−0.0855029	+	0.996338i	$0.527250\pi$
$480$	0	0
$481$	−26.1246	−1.19118
$482$	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	$-0.774774\pi$
$487$	−33.5410	−1.51989	−0.759944	+	0.649988i	$0.774774\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.5279	0.610504	0.305252	−	0.952272i	$-0.401259\pi$
$491$	13.5279	0.610504	0.305252	+	0.952272i	$0.401259\pi$
$492$	0	0
$493$	3.61803	0.162948
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.18034	−0.322082
$498$	0	0
$499$	5.32624	0.238435	0.119218	−	0.992868i	$-0.461961\pi$
$499$	5.32624	0.238435	0.119218	+	0.992868i	$0.461961\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	−6.70820	−0.298511
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.88854	−0.172357	−0.0861783	−	0.996280i	$-0.527465\pi$
$509$	−3.88854	−0.172357	−0.0861783	+	0.996280i	$0.527465\pi$
$510$	0	0
$511$	−13.3262	−0.589518
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.18034	−0.0520120
$516$	0	0
$517$	−5.67376	−0.249532
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.70820	−0.206270	−0.103135	−	0.994667i	$-0.532887\pi$
$521$	−4.70820	−0.206270	−0.103135	+	0.994667i	$0.532887\pi$
$522$	0	0
$523$	9.18034	0.401428	0.200714	−	0.979650i	$-0.435674\pi$
$523$	9.18034	0.401428	0.200714	+	0.979650i	$0.435674\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.09017	0.178171
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−23.5623	−1.02060
$534$	0	0
$535$	18.4164	0.796211
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.85410	0.166008
$540$	0	0
$541$	−37.4164	−1.60866	−0.804329	−	0.594185i	$-0.797475\pi$
$541$	−37.4164	−1.60866	−0.804329	+	0.594185i	$0.797475\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.1246	−0.433691
$546$	0	0
$547$	19.2148	0.821565	0.410782	−	0.911733i	$-0.365256\pi$
$547$	19.2148	0.821565	0.410782	+	0.911733i	$0.365256\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.85410	0.249393
$552$	0	0
$553$	−8.47214	−0.360272
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.7426	1.76869	0.884346	−	0.466831i	$-0.154605\pi$
$557$	41.7426	1.76869	0.884346	+	0.466831i	$0.154605\pi$
$558$	0	0
$559$	37.4164	1.58255
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.8328	−1.25730	−0.628652	−	0.777687i	$-0.716393\pi$
$563$	−29.8328	−1.25730	−0.628652	+	0.777687i	$0.716393\pi$
$564$	0	0
$565$	9.79837	0.412221
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.76393	−0.241637	−0.120818	−	0.992675i	$-0.538552\pi$
$569$	−5.76393	−0.241637	−0.120818	+	0.992675i	$0.538552\pi$
$570$	0	0
$571$	−38.7771	−1.62277	−0.811385	−	0.584512i	$-0.801286\pi$
$571$	−38.7771	−1.62277	−0.811385	+	0.584512i	$0.801286\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−30.7426	−1.27983	−0.639917	−	0.768444i	$-0.721031\pi$
$577$	−30.7426	−1.27983	−0.639917	+	0.768444i	$0.721031\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.5623	0.438198
$582$	0	0
$583$	20.5279	0.850177
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.5836	−0.601929	−0.300965	−	0.953635i	$-0.597309\pi$
$587$	−14.5836	−0.601929	−0.300965	+	0.953635i	$0.597309\pi$
$588$	0	0
$589$	6.61803	0.272691
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.94427	0.408362	0.204181	−	0.978933i	$-0.434547\pi$
$593$	9.94427	0.408362	0.204181	+	0.978933i	$0.434547\pi$
$594$	0	0
$595$	1.38197	0.0566551
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.09017	0.0854020	0.0427010	−	0.999088i	$-0.486404\pi$
$599$	2.09017	0.0854020	0.0427010	+	0.999088i	$0.486404\pi$
$600$	0	0
$601$	−39.1459	−1.59679	−0.798397	−	0.602131i	$-0.794318\pi$
$601$	−39.1459	−1.59679	−0.798397	+	0.602131i	$0.794318\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.61803	−0.350373
$606$	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
$611$	4.41641	0.178669
$612$	0	0
$613$	18.2705	0.737939	0.368969	−	0.929442i	$-0.379711\pi$
$613$	18.2705	0.737939	0.368969	+	0.929442i	$0.379711\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.3820	−0.538738	−0.269369	−	0.963037i	$-0.586815\pi$
$617$	−13.3820	−0.538738	−0.269369	+	0.963037i	$0.586815\pi$
$618$	0	0
$619$	−24.5623	−0.987242	−0.493621	−	0.869677i	$-0.664327\pi$
$619$	−24.5623	−0.987242	−0.493621	+	0.869677i	$0.664327\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.70820	0.308823
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.38197	−0.214593
$630$	0	0
$631$	−28.5279	−1.13568	−0.567838	−	0.823140i	$-0.692220\pi$
$631$	−28.5279	−1.13568	−0.567838	+	0.823140i	$0.692220\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−28.2918	−1.12273
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14.4377	−0.570255	−0.285127	−	0.958490i	$-0.592036\pi$
$641$	−14.4377	−0.570255	−0.285127	+	0.958490i	$0.592036\pi$
$642$	0	0
$643$	−26.2361	−1.03465	−0.517325	−	0.855789i	$-0.673072\pi$
$643$	−26.2361	−1.03465	−0.517325	+	0.855789i	$0.673072\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.58359	0.376770	0.188385	−	0.982095i	$-0.439675\pi$
$647$	9.58359	0.376770	0.188385	+	0.982095i	$0.439675\pi$
$648$	0	0
$649$	11.5623	0.453860
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−44.2361	−1.73109	−0.865546	−	0.500830i	$-0.833028\pi$
$653$	−44.2361	−1.73109	−0.865546	+	0.500830i	$0.833028\pi$
$654$	0	0
$655$	42.6869	1.66792
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.0344	0.858340	0.429170	−	0.903224i	$-0.358806\pi$
$659$	22.0344	0.858340	0.429170	+	0.903224i	$0.358806\pi$
$660$	0	0
$661$	−28.8328	−1.12147	−0.560733	−	0.827996i	$-0.689481\pi$
$661$	−28.8328	−1.12147	−0.560733	+	0.827996i	$0.689481\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.23607	0.0867110
$666$	0	0
$667$	−36.5066	−1.41354
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.6738	1.06833
$672$	0	0
$673$	18.2705	0.704276	0.352138	−	0.935948i	$-0.385455\pi$
$673$	18.2705	0.704276	0.352138	+	0.935948i	$0.385455\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.85410	0.340291	0.170145	−	0.985419i	$-0.445576\pi$
$677$	8.85410	0.340291	0.170145	+	0.985419i	$0.445576\pi$
$678$	0	0
$679$	−9.00000	−0.345388
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.0689	1.49493	0.747465	−	0.664302i	$-0.231271\pi$
$683$	39.0689	1.49493	0.747465	+	0.664302i	$0.231271\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.9787	−0.608741
$690$	0	0
$691$	29.1246	1.10795	0.553976	−	0.832532i	$-0.313110\pi$
$691$	29.1246	1.10795	0.553976	+	0.832532i	$0.313110\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34.5967	1.31233
$696$	0	0
$697$	−4.85410	−0.183862
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.3050	1.56007	0.780033	−	0.625738i	$-0.215202\pi$
$701$	41.3050	1.56007	0.780033	+	0.625738i	$0.215202\pi$
$702$	0	0
$703$	−8.70820	−0.328436
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.00000	0.112827
$708$	0	0
$709$	−11.7639	−0.441804	−0.220902	−	0.975296i	$-0.570900\pi$
$709$	−11.7639	−0.441804	−0.220902	+	0.975296i	$0.570900\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−41.2705	−1.54559
$714$	0	0
$715$	25.8541	0.966889
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.7771	−0.886736	−0.443368	−	0.896340i	$-0.646216\pi$
$719$	−23.7771	−0.886736	−0.443368	+	0.896340i	$0.646216\pi$
$720$	0	0
$721$	0.527864	0.0196587
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.7082	0.397145	0.198573	−	0.980086i	$-0.436369\pi$
$727$	10.7082	0.397145	0.198573	+	0.980086i	$0.436369\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.70820	0.285098
$732$	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
$737$	−14.8541	−0.547158
$738$	0	0
$739$	4.12461	0.151726	0.0758631	−	0.997118i	$-0.475829\pi$
$739$	4.12461	0.151726	0.0758631	+	0.997118i	$0.475829\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.1803	1.21727	0.608634	−	0.793451i	$-0.291718\pi$
$743$	33.1803	1.21727	0.608634	+	0.793451i	$0.291718\pi$
$744$	0	0
$745$	−18.8197	−0.689499
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.23607	−0.300939
$750$	0	0
$751$	39.2148	1.43097	0.715484	−	0.698629i	$-0.246206\pi$
$751$	39.2148	1.43097	0.715484	+	0.698629i	$0.246206\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28.6180	−1.04152
$756$	0	0
$757$	−26.3607	−0.958095	−0.479048	−	0.877789i	$-0.659018\pi$
$757$	−26.3607	−0.958095	−0.479048	+	0.877789i	$0.659018\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.1246	−0.439517	−0.219758	−	0.975554i	$-0.570527\pi$
$761$	−12.1246	−0.439517	−0.219758	+	0.975554i	$0.570527\pi$
$762$	0	0
$763$	4.52786	0.163920
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	7.05573	0.254436	0.127218	−	0.991875i	$-0.459395\pi$
$769$	7.05573	0.254436	0.127218	+	0.991875i	$0.459395\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.65248	0.0954029	0.0477015	−	0.998862i	$-0.484810\pi$
$773$	2.65248	0.0954029	0.0477015	+	0.998862i	$0.484810\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.85410	−0.281402
$780$	0	0
$781$	−27.6738	−0.990245
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.20163	−0.185654
$786$	0	0
$787$	36.3607	1.29612	0.648059	−	0.761590i	$-0.275581\pi$
$787$	36.3607	1.29612	0.648059	+	0.761590i	$0.275581\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.38197	−0.155805
$792$	0	0
$793$	−21.5410	−0.764944
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.9230	1.69752	0.848760	−	0.528779i	$-0.177350\pi$
$797$	47.9230	1.69752	0.848760	+	0.528779i	$0.177350\pi$
$798$	0	0
$799$	0.909830	0.0321875
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−51.3607	−1.81248
$804$	0	0
$805$	−13.9443	−0.491471
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.1246	−1.34039	−0.670195	−	0.742185i	$-0.733790\pi$
$809$	−38.1246	−1.34039	−0.670195	+	0.742185i	$0.733790\pi$
$810$	0	0
$811$	−4.83282	−0.169703	−0.0848516	−	0.996394i	$-0.527042\pi$
$811$	−4.83282	−0.169703	−0.0848516	+	0.996394i	$0.527042\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−44.9230	−1.57358
$816$	0	0
$817$	12.4721	0.436345
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.63932	0.127013	0.0635066	−	0.997981i	$-0.479772\pi$
$821$	3.63932	0.127013	0.0635066	+	0.997981i	$0.479772\pi$
$822$	0	0
$823$	−30.0689	−1.04814	−0.524068	−	0.851677i	$-0.675586\pi$
$823$	−30.0689	−1.04814	−0.524068	+	0.851677i	$0.675586\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.81966	0.167596	0.0837980	−	0.996483i	$-0.473295\pi$
$827$	4.81966	0.167596	0.0837980	+	0.996483i	$0.473295\pi$
$828$	0	0
$829$	−1.88854	−0.0655918	−0.0327959	−	0.999462i	$-0.510441\pi$
$829$	−1.88854	−0.0655918	−0.0327959	+	0.999462i	$0.510441\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.618034	−0.0214136
$834$	0	0
$835$	−12.8885	−0.446026
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.4721	0.913920	0.456960	−	0.889487i	$-0.348938\pi$
$839$	26.4721	0.913920	0.456960	+	0.889487i	$0.348938\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.94427	0.307692
$846$	0	0
$847$	3.85410	0.132429
$848$	0	0
$849$	0	0
$850$	0	0
$851$	54.3050	1.86155
$852$	0	0
$853$	29.9098	1.02409	0.512047	−	0.858958i	$-0.328888\pi$
$853$	29.9098	1.02409	0.512047	+	0.858958i	$0.328888\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.4508	0.937703	0.468852	−	0.883277i	$-0.344668\pi$
$857$	27.4508	0.937703	0.468852	+	0.883277i	$0.344668\pi$
$858$	0	0
$859$	−43.3820	−1.48017	−0.740087	−	0.672511i	$-0.765216\pi$
$859$	−43.3820	−1.48017	−0.740087	+	0.672511i	$0.765216\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.965558	−0.0328680	−0.0164340	−	0.999865i	$-0.505231\pi$
$863$	−0.965558	−0.0328680	−0.0164340	+	0.999865i	$0.505231\pi$
$864$	0	0
$865$	46.1803	1.57018
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−32.6525	−1.10766
$870$	0	0
$871$	11.5623	0.391774
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	−41.0132	−1.38492	−0.692458	−	0.721458i	$-0.743472\pi$
$877$	−41.0132	−1.38492	−0.692458	+	0.721458i	$0.743472\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.27051	0.143877	0.0719386	−	0.997409i	$-0.477081\pi$
$881$	4.27051	0.143877	0.0719386	+	0.997409i	$0.477081\pi$
$882$	0	0
$883$	13.6525	0.459442	0.229721	−	0.973256i	$-0.426219\pi$
$883$	13.6525	0.459442	0.229721	+	0.973256i	$0.426219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.1803	−0.778320	−0.389160	−	0.921170i	$-0.627235\pi$
$887$	−23.1803	−0.778320	−0.389160	+	0.921170i	$0.627235\pi$
$888$	0	0
$889$	12.6525	0.424350
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.47214	0.0492632
$894$	0	0
$895$	43.3394	1.44868
$896$	0	0
$897$	0	0
$898$	0	0
$899$	38.7426	1.29214
$900$	0	0
$901$	−3.29180	−0.109666
$902$	0	0
$903$	0	0
$904$	0	0
$905$	53.7426	1.78647
$906$	0	0
$907$	−4.88854	−0.162321	−0.0811607	−	0.996701i	$-0.525863\pi$
$907$	−4.88854	−0.162321	−0.0811607	+	0.996701i	$0.525863\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.3050	0.904653	0.452327	−	0.891852i	$-0.350594\pi$
$911$	27.3050	0.904653	0.452327	+	0.891852i	$0.350594\pi$
$912$	0	0
$913$	40.7082	1.34724
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.0902	−0.630413
$918$	0	0
$919$	−6.05573	−0.199760	−0.0998800	−	0.994999i	$-0.531846\pi$
$919$	−6.05573	−0.199760	−0.0998800	+	0.994999i	$0.531846\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21.5410	0.709031
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	44.2705	1.45247	0.726234	−	0.687447i	$-0.241269\pi$
$929$	44.2705	1.45247	0.726234	+	0.687447i	$0.241269\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.32624	0.174187
$936$	0	0
$937$	53.9230	1.76159	0.880794	−	0.473500i	$-0.157010\pi$
$937$	53.9230	1.76159	0.880794	+	0.473500i	$0.157010\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34.3607	1.12013	0.560063	−	0.828450i	$-0.310777\pi$
$941$	34.3607	1.12013	0.560063	+	0.828450i	$0.310777\pi$
$942$	0	0
$943$	48.9787	1.59497
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.3820	1.05227	0.526136	−	0.850400i	$-0.323640\pi$
$947$	32.3820	1.05227	0.526136	+	0.850400i	$0.323640\pi$
$948$	0	0
$949$	39.9787	1.29776
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55.3820	−1.79400	−0.896999	−	0.442033i	$-0.854258\pi$
$953$	−55.3820	−1.79400	−0.896999	+	0.442033i	$0.854258\pi$
$954$	0	0
$955$	−34.2705	−1.10897
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.47214	0.144413
$960$	0	0
$961$	12.7984	0.412851
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−37.1591	−1.19619
$966$	0	0
$967$	44.6869	1.43703	0.718517	−	0.695509i	$-0.244821\pi$
$967$	44.6869	1.43703	0.718517	+	0.695509i	$0.244821\pi$
$968$	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
$973$	−15.4721	−0.496014
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.4853	1.07129	0.535645	−	0.844443i	$-0.320069\pi$
$977$	33.4853	1.07129	0.535645	+	0.844443i	$0.320069\pi$
$978$	0	0
$979$	29.7082	0.949478
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.4164	0.906343	0.453171	−	0.891423i	$-0.350293\pi$
$983$	28.4164	0.906343	0.453171	+	0.891423i	$0.350293\pi$
$984$	0	0
$985$	52.8115	1.68272
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−77.7771	−2.47317
$990$	0	0
$991$	15.5967	0.495447	0.247724	−	0.968831i	$-0.420318\pi$
$991$	15.5967	0.495447	0.247724	+	0.968831i	$0.420318\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.1803	0.354441
$996$	0	0
$997$	61.3262	1.94222	0.971111	−	0.238629i	$-0.0766981\pi$
$997$	61.3262	1.94222	0.971111	+	0.238629i	$0.0766981\pi$
$998$	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.1.1		2
3.2	odd	2		1064.2.a.c.1.1	✓	2
12.11	even	2		2128.2.a.j.1.2		2
21.20	even	2		7448.2.a.ba.1.2		2
24.5	odd	2		8512.2.a.v.1.2		2
24.11	even	2		8512.2.a.o.1.1		2

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

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.23607 q^{5} +1.00000 q^{7} +3.85410 q^{11} -3.00000 q^{13} -0.618034 q^{17} -1.00000 q^{19} +6.23607 q^{23} -5.85410 q^{29} -6.61803 q^{31} -2.23607 q^{35} +8.70820 q^{37} +7.85410 q^{41} -12.4721 q^{43} -1.47214 q^{47} +1.00000 q^{49} +5.32624 q^{53} -8.61803 q^{55} +3.00000 q^{59} +7.18034 q^{61} +6.70820 q^{65} -3.85410 q^{67} -7.18034 q^{71} -13.3262 q^{73} +3.85410 q^{77} -8.47214 q^{79} +10.5623 q^{83} +1.38197 q^{85} +7.70820 q^{89} -3.00000 q^{91} +2.23607 q^{95} -9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 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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