LMFDB - Embedded newform 8512.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{3} +2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{9} +O(q^{10})$	$q-0.381966 q^{3} +2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{9} -3.38197 q^{11} -1.00000 q^{13} -0.854102 q^{15} +1.85410 q^{17} -1.00000 q^{19} -0.381966 q^{21} +3.00000 q^{23} +2.23607 q^{27} +3.38197 q^{29} -5.85410 q^{31} +1.29180 q^{33} +2.23607 q^{35} -2.70820 q^{37} +0.381966 q^{39} +2.61803 q^{41} -2.00000 q^{43} -6.38197 q^{45} +11.1803 q^{47} +1.00000 q^{49} -0.708204 q^{51} +10.0902 q^{53} -7.56231 q^{55} +0.381966 q^{57} -12.7082 q^{59} -6.70820 q^{61} -2.85410 q^{63} -2.23607 q^{65} -13.5623 q^{67} -1.14590 q^{69} +1.47214 q^{71} +10.8541 q^{73} -3.38197 q^{77} +10.0000 q^{79} +7.70820 q^{81} -1.14590 q^{83} +4.14590 q^{85} -1.29180 q^{87} +15.7082 q^{89} -1.00000 q^{91} +2.23607 q^{93} -2.23607 q^{95} -12.4164 q^{97} +9.65248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + 2 * q^7 + q^9	$ 2 q - 3 q^{3} + 2 q^{7} + q^{9} - 9 q^{11} - 2 q^{13} + 5 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 9 q^{29} - 5 q^{31} + 16 q^{33} + 8 q^{37} + 3 q^{39} + 3 q^{41} - 4 q^{43} - 15 q^{45} + 2 q^{49} + 12 q^{51} + 9 q^{53} + 5 q^{55} + 3 q^{57} - 12 q^{59} + q^{63} - 7 q^{67} - 9 q^{69} - 6 q^{71} + 15 q^{73} - 9 q^{77} + 20 q^{79} + 2 q^{81} - 9 q^{83} + 15 q^{85} - 16 q^{87} + 18 q^{89} - 2 q^{91} + 2 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^7 + q^9 - 9 * q^11 - 2 * q^13 + 5 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 9 * q^29 - 5 * q^31 + 16 * q^33 + 8 * q^37 + 3 * q^39 + 3 * q^41 - 4 * q^43 - 15 * q^45 + 2 * q^49 + 12 * q^51 + 9 * q^53 + 5 * q^55 + 3 * q^57 - 12 * q^59 + q^63 - 7 * q^67 - 9 * q^69 - 6 * q^71 + 15 * q^73 - 9 * q^77 + 20 * q^79 + 2 * q^81 - 9 * q^83 + 15 * q^85 - 16 * q^87 + 18 * q^89 - 2 * q^91 + 2 * q^97 - 12 * q^99

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
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	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
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−3.38197	−1.01970	−0.509851	−	0.860263i	$-0.670299\pi$
$11$	−3.38197	−1.01970	−0.509851	+	0.860263i	$0.670299\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−0.854102	−0.220528
$16$	0	0
$17$	1.85410	0.449686	0.224843	−	0.974395i	$-0.427813\pi$
$17$	1.85410	0.449686	0.224843	+	0.974395i	$0.427813\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.381966	−0.0833518
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	3.38197	0.628015	0.314008	−	0.949420i	$-0.398328\pi$
$29$	3.38197	0.628015	0.314008	+	0.949420i	$0.398328\pi$
$30$	0	0
$31$	−5.85410	−1.05143	−0.525714	−	0.850661i	$-0.676202\pi$
$31$	−5.85410	−1.05143	−0.525714	+	0.850661i	$0.676202\pi$
$32$	0	0
$33$	1.29180	0.224873
$34$	0	0
$35$	2.23607	0.377964
$36$	0	0
$37$	−2.70820	−0.445226	−0.222613	−	0.974907i	$-0.571459\pi$
$37$	−2.70820	−0.445226	−0.222613	+	0.974907i	$0.571459\pi$
$38$	0	0
$39$	0.381966	0.0611635
$40$	0	0
$41$	2.61803	0.408868	0.204434	−	0.978880i	$-0.434465\pi$
$41$	2.61803	0.408868	0.204434	+	0.978880i	$0.434465\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−6.38197	−0.951367
$46$	0	0
$47$	11.1803	1.63082	0.815410	−	0.578884i	$-0.196511\pi$
$47$	11.1803	1.63082	0.815410	+	0.578884i	$0.196511\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.708204	−0.0991684
$52$	0	0
$53$	10.0902	1.38599	0.692996	−	0.720942i	$-0.256290\pi$
$53$	10.0902	1.38599	0.692996	+	0.720942i	$0.256290\pi$
$54$	0	0
$55$	−7.56231	−1.01970
$56$	0	0
$57$	0.381966	0.0505926
$58$	0	0
$59$	−12.7082	−1.65447	−0.827234	−	0.561858i	$-0.810087\pi$
$59$	−12.7082	−1.65447	−0.827234	+	0.561858i	$0.810087\pi$
$60$	0	0
$61$	−6.70820	−0.858898	−0.429449	−	0.903091i	$-0.641292\pi$
$61$	−6.70820	−0.858898	−0.429449	+	0.903091i	$0.641292\pi$
$62$	0	0
$63$	−2.85410	−0.359583
$64$	0	0
$65$	−2.23607	−0.277350
$66$	0	0
$67$	−13.5623	−1.65690	−0.828450	−	0.560063i	$-0.810777\pi$
$67$	−13.5623	−1.65690	−0.828450	+	0.560063i	$0.810777\pi$
$68$	0	0
$69$	−1.14590	−0.137950
$70$	0	0
$71$	1.47214	0.174710	0.0873552	−	0.996177i	$-0.472158\pi$
$71$	1.47214	0.174710	0.0873552	+	0.996177i	$0.472158\pi$
$72$	0	0
$73$	10.8541	1.27038	0.635188	−	0.772357i	$-0.280923\pi$
$73$	10.8541	1.27038	0.635188	+	0.772357i	$0.280923\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.38197	−0.385411
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−1.14590	−0.125779	−0.0628893	−	0.998021i	$-0.520032\pi$
$83$	−1.14590	−0.125779	−0.0628893	+	0.998021i	$0.520032\pi$
$84$	0	0
$85$	4.14590	0.449686
$86$	0	0
$87$	−1.29180	−0.138495
$88$	0	0
$89$	15.7082	1.66507	0.832533	−	0.553975i	$-0.186890\pi$
$89$	15.7082	1.66507	0.832533	+	0.553975i	$0.186890\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.23607	0.231869
$94$	0	0
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−12.4164	−1.26070	−0.630348	−	0.776313i	$-0.717088\pi$
$97$	−12.4164	−1.26070	−0.630348	+	0.776313i	$0.717088\pi$
$98$	0	0
$99$	9.65248	0.970110
$100$	0	0
$101$	−4.52786	−0.450539	−0.225270	−	0.974296i	$-0.572326\pi$
$101$	−4.52786	−0.450539	−0.225270	+	0.974296i	$0.572326\pi$
$102$	0	0
$103$	−8.70820	−0.858045	−0.429022	−	0.903294i	$-0.641142\pi$
$103$	−8.70820	−0.858045	−0.429022	+	0.903294i	$0.641142\pi$
$104$	0	0
$105$	−0.854102	−0.0833518
$106$	0	0
$107$	−7.47214	−0.722359	−0.361179	−	0.932496i	$-0.617626\pi$
$107$	−7.47214	−0.722359	−0.361179	+	0.932496i	$0.617626\pi$
$108$	0	0
$109$	−16.4164	−1.57241	−0.786203	−	0.617968i	$-0.787956\pi$
$109$	−16.4164	−1.57241	−0.786203	+	0.617968i	$0.787956\pi$
$110$	0	0
$111$	1.03444	0.0981849
$112$	0	0
$113$	−10.8541	−1.02107	−0.510534	−	0.859858i	$-0.670552\pi$
$113$	−10.8541	−1.02107	−0.510534	+	0.859858i	$0.670552\pi$
$114$	0	0
$115$	6.70820	0.625543
$116$	0	0
$117$	2.85410	0.263862
$118$	0	0
$119$	1.85410	0.169965
$120$	0	0
$121$	0.437694	0.0397904
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−19.7082	−1.74882	−0.874410	−	0.485187i	$-0.838751\pi$
$127$	−19.7082	−1.74882	−0.874410	+	0.485187i	$0.838751\pi$
$128$	0	0
$129$	0.763932	0.0672605
$130$	0	0
$131$	−1.09017	−0.0952486	−0.0476243	−	0.998865i	$-0.515165\pi$
$131$	−1.09017	−0.0952486	−0.0476243	+	0.998865i	$0.515165\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	7.52786	0.643149	0.321574	−	0.946884i	$-0.395788\pi$
$137$	7.52786	0.643149	0.321574	+	0.946884i	$0.395788\pi$
$138$	0	0
$139$	−20.7082	−1.75645	−0.878223	−	0.478251i	$-0.841271\pi$
$139$	−20.7082	−1.75645	−0.878223	+	0.478251i	$0.841271\pi$
$140$	0	0
$141$	−4.27051	−0.359642
$142$	0	0
$143$	3.38197	0.282814
$144$	0	0
$145$	7.56231	0.628015
$146$	0	0
$147$	−0.381966	−0.0315040
$148$	0	0
$149$	−20.8885	−1.71126	−0.855628	−	0.517591i	$-0.826829\pi$
$149$	−20.8885	−1.71126	−0.855628	+	0.517591i	$0.826829\pi$
$150$	0	0
$151$	−21.5623	−1.75472	−0.877358	−	0.479837i	$-0.840696\pi$
$151$	−21.5623	−1.75472	−0.877358	+	0.479837i	$0.840696\pi$
$152$	0	0
$153$	−5.29180	−0.427816
$154$	0	0
$155$	−13.0902	−1.05143
$156$	0	0
$157$	7.85410	0.626826	0.313413	−	0.949617i	$-0.398528\pi$
$157$	7.85410	0.626826	0.313413	+	0.949617i	$0.398528\pi$
$158$	0	0
$159$	−3.85410	−0.305650
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−9.14590	−0.716362	−0.358181	−	0.933652i	$-0.616603\pi$
$163$	−9.14590	−0.716362	−0.358181	+	0.933652i	$0.616603\pi$
$164$	0	0
$165$	2.88854	0.224873
$166$	0	0
$167$	−1.47214	−0.113917	−0.0569587	−	0.998377i	$-0.518140\pi$
$167$	−1.47214	−0.113917	−0.0569587	+	0.998377i	$0.518140\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.85410	0.218259
$172$	0	0
$173$	−2.18034	−0.165768	−0.0828841	−	0.996559i	$-0.526413\pi$
$173$	−2.18034	−0.165768	−0.0828841	+	0.996559i	$0.526413\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.85410	0.364857
$178$	0	0
$179$	1.09017	0.0814831	0.0407416	−	0.999170i	$-0.487028\pi$
$179$	1.09017	0.0814831	0.0407416	+	0.999170i	$0.487028\pi$
$180$	0	0
$181$	21.5623	1.60271	0.801357	−	0.598187i	$-0.204112\pi$
$181$	21.5623	1.60271	0.801357	+	0.598187i	$0.204112\pi$
$182$	0	0
$183$	2.56231	0.189411
$184$	0	0
$185$	−6.05573	−0.445226
$186$	0	0
$187$	−6.27051	−0.458545
$188$	0	0
$189$	2.23607	0.162650
$190$	0	0
$191$	−9.32624	−0.674823	−0.337411	−	0.941357i	$-0.609551\pi$
$191$	−9.32624	−0.674823	−0.337411	+	0.941357i	$0.609551\pi$
$192$	0	0
$193$	−1.56231	−0.112457	−0.0562286	−	0.998418i	$-0.517908\pi$
$193$	−1.56231	−0.112457	−0.0562286	+	0.998418i	$0.517908\pi$
$194$	0	0
$195$	0.854102	0.0611635
$196$	0	0
$197$	18.2705	1.30172	0.650860	−	0.759198i	$-0.274408\pi$
$197$	18.2705	1.30172	0.650860	+	0.759198i	$0.274408\pi$
$198$	0	0
$199$	−3.29180	−0.233349	−0.116675	−	0.993170i	$-0.537223\pi$
$199$	−3.29180	−0.233349	−0.116675	+	0.993170i	$0.537223\pi$
$200$	0	0
$201$	5.18034	0.365393
$202$	0	0
$203$	3.38197	0.237367
$204$	0	0
$205$	5.85410	0.408868
$206$	0	0
$207$	−8.56231	−0.595121
$208$	0	0
$209$	3.38197	0.233935
$210$	0	0
$211$	−6.56231	−0.451768	−0.225884	−	0.974154i	$-0.572527\pi$
$211$	−6.56231	−0.451768	−0.225884	+	0.974154i	$0.572527\pi$
$212$	0	0
$213$	−0.562306	−0.0385286
$214$	0	0
$215$	−4.47214	−0.304997
$216$	0	0
$217$	−5.85410	−0.397402
$218$	0	0
$219$	−4.14590	−0.280154
$220$	0	0
$221$	−1.85410	−0.124720
$222$	0	0
$223$	26.4164	1.76897	0.884487	−	0.466565i	$-0.154509\pi$
$223$	26.4164	1.76897	0.884487	+	0.466565i	$0.154509\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.6180	−1.36847	−0.684233	−	0.729263i	$-0.739863\pi$
$227$	−20.6180	−1.36847	−0.684233	+	0.729263i	$0.739863\pi$
$228$	0	0
$229$	9.41641	0.622254	0.311127	−	0.950368i	$-0.399294\pi$
$229$	9.41641	0.622254	0.311127	+	0.950368i	$0.399294\pi$
$230$	0	0
$231$	1.29180	0.0849939
$232$	0	0
$233$	−16.0344	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$233$	−16.0344	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$234$	0	0
$235$	25.0000	1.63082
$236$	0	0
$237$	−3.81966	−0.248114
$238$	0	0
$239$	−13.4721	−0.871440	−0.435720	−	0.900082i	$-0.643506\pi$
$239$	−13.4721	−0.871440	−0.435720	+	0.900082i	$0.643506\pi$
$240$	0	0
$241$	−17.7082	−1.14069	−0.570343	−	0.821407i	$-0.693190\pi$
$241$	−17.7082	−1.14069	−0.570343	+	0.821407i	$0.693190\pi$
$242$	0	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	2.23607	0.142857
$246$	0	0
$247$	1.00000	0.0636285
$248$	0	0
$249$	0.437694	0.0277377
$250$	0	0
$251$	1.20163	0.0758460	0.0379230	−	0.999281i	$-0.487926\pi$
$251$	1.20163	0.0758460	0.0379230	+	0.999281i	$0.487926\pi$
$252$	0	0
$253$	−10.1459	−0.637867
$254$	0	0
$255$	−1.58359	−0.0991684
$256$	0	0
$257$	27.9787	1.74526	0.872632	−	0.488378i	$-0.162411\pi$
$257$	27.9787	1.74526	0.872632	+	0.488378i	$0.162411\pi$
$258$	0	0
$259$	−2.70820	−0.168280
$260$	0	0
$261$	−9.65248	−0.597473
$262$	0	0
$263$	3.43769	0.211977	0.105989	−	0.994367i	$-0.466199\pi$
$263$	3.43769	0.211977	0.105989	+	0.994367i	$0.466199\pi$
$264$	0	0
$265$	22.5623	1.38599
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−14.5623	−0.887879	−0.443940	−	0.896057i	$-0.646420\pi$
$269$	−14.5623	−0.887879	−0.443940	+	0.896057i	$0.646420\pi$
$270$	0	0
$271$	−19.2705	−1.17060	−0.585300	−	0.810817i	$-0.699023\pi$
$271$	−19.2705	−1.17060	−0.585300	+	0.810817i	$0.699023\pi$
$272$	0	0
$273$	0.381966	0.0231176
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.7082	1.06398	0.531991	−	0.846750i	$-0.321444\pi$
$277$	17.7082	1.06398	0.531991	+	0.846750i	$0.321444\pi$
$278$	0	0
$279$	16.7082	1.00029
$280$	0	0
$281$	−8.88854	−0.530246	−0.265123	−	0.964215i	$-0.585413\pi$
$281$	−8.88854	−0.530246	−0.265123	+	0.964215i	$0.585413\pi$
$282$	0	0
$283$	8.56231	0.508976	0.254488	−	0.967076i	$-0.418093\pi$
$283$	8.56231	0.508976	0.254488	+	0.967076i	$0.418093\pi$
$284$	0	0
$285$	0.854102	0.0505926
$286$	0	0
$287$	2.61803	0.154538
$288$	0	0
$289$	−13.5623	−0.797783
$290$	0	0
$291$	4.74265	0.278019
$292$	0	0
$293$	12.7082	0.742421	0.371211	−	0.928549i	$-0.378943\pi$
$293$	12.7082	0.742421	0.371211	+	0.928549i	$0.378943\pi$
$294$	0	0
$295$	−28.4164	−1.65447
$296$	0	0
$297$	−7.56231	−0.438809
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	1.72949	0.0993566
$304$	0	0
$305$	−15.0000	−0.858898
$306$	0	0
$307$	−2.43769	−0.139127	−0.0695633	−	0.997578i	$-0.522161\pi$
$307$	−2.43769	−0.139127	−0.0695633	+	0.997578i	$0.522161\pi$
$308$	0	0
$309$	3.32624	0.189223
$310$	0	0
$311$	−4.90983	−0.278411	−0.139205	−	0.990264i	$-0.544455\pi$
$311$	−4.90983	−0.278411	−0.139205	+	0.990264i	$0.544455\pi$
$312$	0	0
$313$	−2.29180	−0.129540	−0.0647700	−	0.997900i	$-0.520631\pi$
$313$	−2.29180	−0.129540	−0.0647700	+	0.997900i	$0.520631\pi$
$314$	0	0
$315$	−6.38197	−0.359583
$316$	0	0
$317$	−25.4164	−1.42753	−0.713764	−	0.700386i	$-0.753011\pi$
$317$	−25.4164	−1.42753	−0.713764	+	0.700386i	$0.753011\pi$
$318$	0	0
$319$	−11.4377	−0.640388
$320$	0	0
$321$	2.85410	0.159300
$322$	0	0
$323$	−1.85410	−0.103165
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.27051	0.346760
$328$	0	0
$329$	11.1803	0.616392
$330$	0	0
$331$	−1.43769	−0.0790228	−0.0395114	−	0.999219i	$-0.512580\pi$
$331$	−1.43769	−0.0790228	−0.0395114	+	0.999219i	$0.512580\pi$
$332$	0	0
$333$	7.72949	0.423573
$334$	0	0
$335$	−30.3262	−1.65690
$336$	0	0
$337$	11.2705	0.613944	0.306972	−	0.951719i	$-0.400684\pi$
$337$	11.2705	0.613944	0.306972	+	0.951719i	$0.400684\pi$
$338$	0	0
$339$	4.14590	0.225174
$340$	0	0
$341$	19.7984	1.07214
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.56231	−0.137950
$346$	0	0
$347$	−9.43769	−0.506642	−0.253321	−	0.967382i	$-0.581523\pi$
$347$	−9.43769	−0.506642	−0.253321	+	0.967382i	$0.581523\pi$
$348$	0	0
$349$	−24.2705	−1.29917	−0.649585	−	0.760289i	$-0.725057\pi$
$349$	−24.2705	−1.29917	−0.649585	+	0.760289i	$0.725057\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	9.38197	0.499352	0.249676	−	0.968329i	$-0.419676\pi$
$353$	9.38197	0.499352	0.249676	+	0.968329i	$0.419676\pi$
$354$	0	0
$355$	3.29180	0.174710
$356$	0	0
$357$	−0.708204	−0.0374821
$358$	0	0
$359$	31.7426	1.67531	0.837656	−	0.546198i	$-0.183925\pi$
$359$	31.7426	1.67531	0.837656	+	0.546198i	$0.183925\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.167184	−0.00877490
$364$	0	0
$365$	24.2705	1.27038
$366$	0	0
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	0	0
$369$	−7.47214	−0.388984
$370$	0	0
$371$	10.0902	0.523856
$372$	0	0
$373$	34.2705	1.77446	0.887230	−	0.461328i	$-0.152627\pi$
$373$	34.2705	1.77446	0.887230	+	0.461328i	$0.152627\pi$
$374$	0	0
$375$	4.27051	0.220528
$376$	0	0
$377$	−3.38197	−0.174180
$378$	0	0
$379$	17.2918	0.888220	0.444110	−	0.895972i	$-0.353520\pi$
$379$	17.2918	0.888220	0.444110	+	0.895972i	$0.353520\pi$
$380$	0	0
$381$	7.52786	0.385664
$382$	0	0
$383$	15.7639	0.805499	0.402750	−	0.915310i	$-0.368055\pi$
$383$	15.7639	0.805499	0.402750	+	0.915310i	$0.368055\pi$
$384$	0	0
$385$	−7.56231	−0.385411
$386$	0	0
$387$	5.70820	0.290164
$388$	0	0
$389$	−10.7984	−0.547499	−0.273750	−	0.961801i	$-0.588264\pi$
$389$	−10.7984	−0.547499	−0.273750	+	0.961801i	$0.588264\pi$
$390$	0	0
$391$	5.56231	0.281298
$392$	0	0
$393$	0.416408	0.0210050
$394$	0	0
$395$	22.3607	1.12509
$396$	0	0
$397$	34.5410	1.73356	0.866782	−	0.498687i	$-0.166184\pi$
$397$	34.5410	1.73356	0.866782	+	0.498687i	$0.166184\pi$
$398$	0	0
$399$	0.381966	0.0191222
$400$	0	0
$401$	−8.61803	−0.430364	−0.215182	−	0.976574i	$-0.569034\pi$
$401$	−8.61803	−0.430364	−0.215182	+	0.976574i	$0.569034\pi$
$402$	0	0
$403$	5.85410	0.291614
$404$	0	0
$405$	17.2361	0.856467
$406$	0	0
$407$	9.15905	0.453997
$408$	0	0
$409$	−5.14590	−0.254448	−0.127224	−	0.991874i	$-0.540607\pi$
$409$	−5.14590	−0.254448	−0.127224	+	0.991874i	$0.540607\pi$
$410$	0	0
$411$	−2.87539	−0.141832
$412$	0	0
$413$	−12.7082	−0.625330
$414$	0	0
$415$	−2.56231	−0.125779
$416$	0	0
$417$	7.90983	0.387346
$418$	0	0
$419$	−23.2918	−1.13788	−0.568939	−	0.822379i	$-0.692646\pi$
$419$	−23.2918	−1.13788	−0.568939	+	0.822379i	$0.692646\pi$
$420$	0	0
$421$	21.2918	1.03770	0.518849	−	0.854866i	$-0.326361\pi$
$421$	21.2918	1.03770	0.518849	+	0.854866i	$0.326361\pi$
$422$	0	0
$423$	−31.9098	−1.55151
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.70820	−0.324633
$428$	0	0
$429$	−1.29180	−0.0623685
$430$	0	0
$431$	22.3607	1.07708	0.538538	−	0.842601i	$-0.318977\pi$
$431$	22.3607	1.07708	0.538538	+	0.842601i	$0.318977\pi$
$432$	0	0
$433$	31.5410	1.51576	0.757882	−	0.652391i	$-0.226234\pi$
$433$	31.5410	1.51576	0.757882	+	0.652391i	$0.226234\pi$
$434$	0	0
$435$	−2.88854	−0.138495
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−2.85410	−0.135910
$442$	0	0
$443$	−12.3820	−0.588285	−0.294142	−	0.955762i	$-0.595034\pi$
$443$	−12.3820	−0.588285	−0.294142	+	0.955762i	$0.595034\pi$
$444$	0	0
$445$	35.1246	1.66507
$446$	0	0
$447$	7.97871	0.377380
$448$	0	0
$449$	−14.6738	−0.692498	−0.346249	−	0.938143i	$-0.612545\pi$
$449$	−14.6738	−0.692498	−0.346249	+	0.938143i	$0.612545\pi$
$450$	0	0
$451$	−8.85410	−0.416923
$452$	0	0
$453$	8.23607	0.386964
$454$	0	0
$455$	−2.23607	−0.104828
$456$	0	0
$457$	24.5623	1.14898	0.574488	−	0.818513i	$-0.305201\pi$
$457$	24.5623	1.14898	0.574488	+	0.818513i	$0.305201\pi$
$458$	0	0
$459$	4.14590	0.193514
$460$	0	0
$461$	−34.0902	−1.58774	−0.793869	−	0.608089i	$-0.791936\pi$
$461$	−34.0902	−1.58774	−0.793869	+	0.608089i	$0.791936\pi$
$462$	0	0
$463$	−30.7082	−1.42713	−0.713566	−	0.700588i	$-0.752921\pi$
$463$	−30.7082	−1.42713	−0.713566	+	0.700588i	$0.752921\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	24.9230	1.15330	0.576649	−	0.816992i	$-0.304360\pi$
$467$	24.9230	1.15330	0.576649	+	0.816992i	$0.304360\pi$
$468$	0	0
$469$	−13.5623	−0.626249
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	6.76393	0.311006
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−28.7984	−1.31859
$478$	0	0
$479$	−41.4508	−1.89394	−0.946969	−	0.321325i	$-0.895872\pi$
$479$	−41.4508	−1.89394	−0.946969	+	0.321325i	$0.895872\pi$
$480$	0	0
$481$	2.70820	0.123483
$482$	0	0
$483$	−1.14590	−0.0521402
$484$	0	0
$485$	−27.7639	−1.26070
$486$	0	0
$487$	−9.58359	−0.434274	−0.217137	−	0.976141i	$-0.569672\pi$
$487$	−9.58359	−0.434274	−0.217137	+	0.976141i	$0.569672\pi$
$488$	0	0
$489$	3.49342	0.157978
$490$	0	0
$491$	−10.4721	−0.472601	−0.236300	−	0.971680i	$-0.575935\pi$
$491$	−10.4721	−0.472601	−0.236300	+	0.971680i	$0.575935\pi$
$492$	0	0
$493$	6.27051	0.282410
$494$	0	0
$495$	21.5836	0.970110
$496$	0	0
$497$	1.47214	0.0660343
$498$	0	0
$499$	3.14590	0.140830	0.0704149	−	0.997518i	$-0.477568\pi$
$499$	3.14590	0.140830	0.0704149	+	0.997518i	$0.477568\pi$
$500$	0	0
$501$	0.562306	0.0251220
$502$	0	0
$503$	−17.8885	−0.797611	−0.398805	−	0.917036i	$-0.630575\pi$
$503$	−17.8885	−0.797611	−0.398805	+	0.917036i	$0.630575\pi$
$504$	0	0
$505$	−10.1246	−0.450539
$506$	0	0
$507$	4.58359	0.203564
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	10.8541	0.480157
$512$	0	0
$513$	−2.23607	−0.0987248
$514$	0	0
$515$	−19.4721	−0.858045
$516$	0	0
$517$	−37.8115	−1.66295
$518$	0	0
$519$	0.832816	0.0365566
$520$	0	0
$521$	−32.2361	−1.41229	−0.706144	−	0.708068i	$-0.749567\pi$
$521$	−32.2361	−1.41229	−0.706144	+	0.708068i	$0.749567\pi$
$522$	0	0
$523$	13.5836	0.593969	0.296985	−	0.954882i	$-0.404019\pi$
$523$	13.5836	0.593969	0.296985	+	0.954882i	$0.404019\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.8541	−0.472812
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	36.2705	1.57401
$532$	0	0
$533$	−2.61803	−0.113400
$534$	0	0
$535$	−16.7082	−0.722359
$536$	0	0
$537$	−0.416408	−0.0179693
$538$	0	0
$539$	−3.38197	−0.145672
$540$	0	0
$541$	−13.4164	−0.576816	−0.288408	−	0.957508i	$-0.593126\pi$
$541$	−13.4164	−0.576816	−0.288408	+	0.957508i	$0.593126\pi$
$542$	0	0
$543$	−8.23607	−0.353444
$544$	0	0
$545$	−36.7082	−1.57241
$546$	0	0
$547$	16.2705	0.695677	0.347838	−	0.937555i	$-0.386916\pi$
$547$	16.2705	0.695677	0.347838	+	0.937555i	$0.386916\pi$
$548$	0	0
$549$	19.1459	0.817127
$550$	0	0
$551$	−3.38197	−0.144077
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	2.31308	0.0981849
$556$	0	0
$557$	18.3262	0.776508	0.388254	−	0.921552i	$-0.373078\pi$
$557$	18.3262	0.776508	0.388254	+	0.921552i	$0.373078\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	2.39512	0.101122
$562$	0	0
$563$	−17.1803	−0.724065	−0.362032	−	0.932165i	$-0.617917\pi$
$563$	−17.1803	−0.724065	−0.362032	+	0.932165i	$0.617917\pi$
$564$	0	0
$565$	−24.2705	−1.02107
$566$	0	0
$567$	7.70820	0.323714
$568$	0	0
$569$	−24.5967	−1.03115	−0.515575	−	0.856845i	$-0.672422\pi$
$569$	−24.5967	−1.03115	−0.515575	+	0.856845i	$0.672422\pi$
$570$	0	0
$571$	−16.1246	−0.674794	−0.337397	−	0.941362i	$-0.609546\pi$
$571$	−16.1246	−0.674794	−0.337397	+	0.941362i	$0.609546\pi$
$572$	0	0
$573$	3.56231	0.148817
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.8541	0.868168	0.434084	−	0.900872i	$-0.357072\pi$
$577$	20.8541	0.868168	0.434084	+	0.900872i	$0.357072\pi$
$578$	0	0
$579$	0.596748	0.0248000
$580$	0	0
$581$	−1.14590	−0.0475399
$582$	0	0
$583$	−34.1246	−1.41330
$584$	0	0
$585$	6.38197	0.263862
$586$	0	0
$587$	38.9443	1.60740	0.803701	−	0.595033i	$-0.202861\pi$
$587$	38.9443	1.60740	0.803701	+	0.595033i	$0.202861\pi$
$588$	0	0
$589$	5.85410	0.241214
$590$	0	0
$591$	−6.97871	−0.287066
$592$	0	0
$593$	−15.0000	−0.615976	−0.307988	−	0.951390i	$-0.599656\pi$
$593$	−15.0000	−0.615976	−0.307988	+	0.951390i	$0.599656\pi$
$594$	0	0
$595$	4.14590	0.169965
$596$	0	0
$597$	1.25735	0.0514601
$598$	0	0
$599$	4.09017	0.167120	0.0835599	−	0.996503i	$-0.473371\pi$
$599$	4.09017	0.167120	0.0835599	+	0.996503i	$0.473371\pi$
$600$	0	0
$601$	−14.5623	−0.594009	−0.297004	−	0.954876i	$-0.595988\pi$
$601$	−14.5623	−0.594009	−0.297004	+	0.954876i	$0.595988\pi$
$602$	0	0
$603$	38.7082	1.57632
$604$	0	0
$605$	0.978714	0.0397904
$606$	0	0
$607$	40.1246	1.62861	0.814304	−	0.580439i	$-0.197119\pi$
$607$	40.1246	1.62861	0.814304	+	0.580439i	$0.197119\pi$
$608$	0	0
$609$	−1.29180	−0.0523462
$610$	0	0
$611$	−11.1803	−0.452308
$612$	0	0
$613$	−13.6869	−0.552809	−0.276405	−	0.961041i	$-0.589143\pi$
$613$	−13.6869	−0.552809	−0.276405	+	0.961041i	$0.589143\pi$
$614$	0	0
$615$	−2.23607	−0.0901670
$616$	0	0
$617$	24.2148	0.974850	0.487425	−	0.873165i	$-0.337936\pi$
$617$	24.2148	0.974850	0.487425	+	0.873165i	$0.337936\pi$
$618$	0	0
$619$	13.4377	0.540107	0.270053	−	0.962845i	$-0.412959\pi$
$619$	13.4377	0.540107	0.270053	+	0.962845i	$0.412959\pi$
$620$	0	0
$621$	6.70820	0.269191
$622$	0	0
$623$	15.7082	0.629336
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	−1.29180	−0.0515894
$628$	0	0
$629$	−5.02129	−0.200212
$630$	0	0
$631$	1.29180	0.0514256	0.0257128	−	0.999669i	$-0.491814\pi$
$631$	1.29180	0.0514256	0.0257128	+	0.999669i	$0.491814\pi$
$632$	0	0
$633$	2.50658	0.0996275
$634$	0	0
$635$	−44.0689	−1.74882
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−4.20163	−0.166214
$640$	0	0
$641$	−40.6869	−1.60704	−0.803518	−	0.595280i	$-0.797041\pi$
$641$	−40.6869	−1.60704	−0.803518	+	0.595280i	$0.797041\pi$
$642$	0	0
$643$	−48.4164	−1.90936	−0.954678	−	0.297639i	$-0.903801\pi$
$643$	−48.4164	−1.90936	−0.954678	+	0.297639i	$0.903801\pi$
$644$	0	0
$645$	1.70820	0.0672605
$646$	0	0
$647$	17.0689	0.671047	0.335524	−	0.942032i	$-0.391087\pi$
$647$	17.0689	0.671047	0.335524	+	0.942032i	$0.391087\pi$
$648$	0	0
$649$	42.9787	1.68706
$650$	0	0
$651$	2.23607	0.0876384
$652$	0	0
$653$	18.5967	0.727747	0.363873	−	0.931448i	$-0.381454\pi$
$653$	18.5967	0.727747	0.363873	+	0.931448i	$0.381454\pi$
$654$	0	0
$655$	−2.43769	−0.0952486
$656$	0	0
$657$	−30.9787	−1.20859
$658$	0	0
$659$	26.5066	1.03255	0.516275	−	0.856423i	$-0.327318\pi$
$659$	26.5066	1.03255	0.516275	+	0.856423i	$0.327318\pi$
$660$	0	0
$661$	16.8328	0.654721	0.327360	−	0.944900i	$-0.393841\pi$
$661$	16.8328	0.654721	0.327360	+	0.944900i	$0.393841\pi$
$662$	0	0
$663$	0.708204	0.0275044
$664$	0	0
$665$	−2.23607	−0.0867110
$666$	0	0
$667$	10.1459	0.392851
$668$	0	0
$669$	−10.0902	−0.390109
$670$	0	0
$671$	22.6869	0.875819
$672$	0	0
$673$	−18.2705	−0.704276	−0.352138	−	0.935948i	$-0.614545\pi$
$673$	−18.2705	−0.704276	−0.352138	+	0.935948i	$0.614545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.7426	1.33527	0.667634	−	0.744489i	$-0.267307\pi$
$677$	34.7426	1.33527	0.667634	+	0.744489i	$0.267307\pi$
$678$	0	0
$679$	−12.4164	−0.476498
$680$	0	0
$681$	7.87539	0.301786
$682$	0	0
$683$	−28.5279	−1.09159	−0.545794	−	0.837919i	$-0.683772\pi$
$683$	−28.5279	−1.09159	−0.545794	+	0.837919i	$0.683772\pi$
$684$	0	0
$685$	16.8328	0.643149
$686$	0	0
$687$	−3.59675	−0.137224
$688$	0	0
$689$	−10.0902	−0.384405
$690$	0	0
$691$	−18.2918	−0.695853	−0.347926	−	0.937522i	$-0.613114\pi$
$691$	−18.2918	−0.695853	−0.347926	+	0.937522i	$0.613114\pi$
$692$	0	0
$693$	9.65248	0.366667
$694$	0	0
$695$	−46.3050	−1.75645
$696$	0	0
$697$	4.85410	0.183862
$698$	0	0
$699$	6.12461	0.231654
$700$	0	0
$701$	37.5279	1.41741	0.708704	−	0.705506i	$-0.249280\pi$
$701$	37.5279	1.41741	0.708704	+	0.705506i	$0.249280\pi$
$702$	0	0
$703$	2.70820	0.102142
$704$	0	0
$705$	−9.54915	−0.359642
$706$	0	0
$707$	−4.52786	−0.170288
$708$	0	0
$709$	−11.2918	−0.424072	−0.212036	−	0.977262i	$-0.568009\pi$
$709$	−11.2918	−0.424072	−0.212036	+	0.977262i	$0.568009\pi$
$710$	0	0
$711$	−28.5410	−1.07037
$712$	0	0
$713$	−17.5623	−0.657714
$714$	0	0
$715$	7.56231	0.282814
$716$	0	0
$717$	5.14590	0.192177
$718$	0	0
$719$	−5.88854	−0.219606	−0.109803	−	0.993953i	$-0.535022\pi$
$719$	−5.88854	−0.219606	−0.109803	+	0.993953i	$0.535022\pi$
$720$	0	0
$721$	−8.70820	−0.324310
$722$	0	0
$723$	6.76393	0.251553
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−43.0000	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$727$	−43.0000	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−3.70820	−0.137153
$732$	0	0
$733$	22.8328	0.843349	0.421675	−	0.906747i	$-0.361442\pi$
$733$	22.8328	0.843349	0.421675	+	0.906747i	$0.361442\pi$
$734$	0	0
$735$	−0.854102	−0.0315040
$736$	0	0
$737$	45.8673	1.68954
$738$	0	0
$739$	22.4164	0.824601	0.412300	−	0.911048i	$-0.364725\pi$
$739$	22.4164	0.824601	0.412300	+	0.911048i	$0.364725\pi$
$740$	0	0
$741$	−0.381966	−0.0140319
$742$	0	0
$743$	−14.8885	−0.546208	−0.273104	−	0.961985i	$-0.588050\pi$
$743$	−14.8885	−0.546208	−0.273104	+	0.961985i	$0.588050\pi$
$744$	0	0
$745$	−46.7082	−1.71126
$746$	0	0
$747$	3.27051	0.119662
$748$	0	0
$749$	−7.47214	−0.273026
$750$	0	0
$751$	−28.8541	−1.05290	−0.526451	−	0.850206i	$-0.676477\pi$
$751$	−28.8541	−1.05290	−0.526451	+	0.850206i	$0.676477\pi$
$752$	0	0
$753$	−0.458980	−0.0167262
$754$	0	0
$755$	−48.2148	−1.75472
$756$	0	0
$757$	−5.41641	−0.196863	−0.0984313	−	0.995144i	$-0.531382\pi$
$757$	−5.41641	−0.196863	−0.0984313	+	0.995144i	$0.531382\pi$
$758$	0	0
$759$	3.87539	0.140668
$760$	0	0
$761$	41.1803	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$761$	41.1803	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$762$	0	0
$763$	−16.4164	−0.594314
$764$	0	0
$765$	−11.8328	−0.427816
$766$	0	0
$767$	12.7082	0.458867
$768$	0	0
$769$	10.8328	0.390641	0.195321	−	0.980739i	$-0.437425\pi$
$769$	10.8328	0.390641	0.195321	+	0.980739i	$0.437425\pi$
$770$	0	0
$771$	−10.6869	−0.384880
$772$	0	0
$773$	5.34752	0.192337	0.0961685	−	0.995365i	$-0.469341\pi$
$773$	5.34752	0.192337	0.0961685	+	0.995365i	$0.469341\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.03444	0.0371104
$778$	0	0
$779$	−2.61803	−0.0938008
$780$	0	0
$781$	−4.97871	−0.178152
$782$	0	0
$783$	7.56231	0.270255
$784$	0	0
$785$	17.5623	0.626826
$786$	0	0
$787$	15.4164	0.549536	0.274768	−	0.961511i	$-0.411399\pi$
$787$	15.4164	0.549536	0.274768	+	0.961511i	$0.411399\pi$
$788$	0	0
$789$	−1.31308	−0.0467470
$790$	0	0
$791$	−10.8541	−0.385927
$792$	0	0
$793$	6.70820	0.238215
$794$	0	0
$795$	−8.61803	−0.305650
$796$	0	0
$797$	−46.6869	−1.65374	−0.826868	−	0.562396i	$-0.809880\pi$
$797$	−46.6869	−1.65374	−0.826868	+	0.562396i	$0.809880\pi$
$798$	0	0
$799$	20.7295	0.733357
$800$	0	0
$801$	−44.8328	−1.58409
$802$	0	0
$803$	−36.7082	−1.29540
$804$	0	0
$805$	6.70820	0.236433
$806$	0	0
$807$	5.56231	0.195802
$808$	0	0
$809$	54.4853	1.91560	0.957800	−	0.287434i	$-0.0928022\pi$
$809$	54.4853	1.91560	0.957800	+	0.287434i	$0.0928022\pi$
$810$	0	0
$811$	−44.2492	−1.55380	−0.776900	−	0.629624i	$-0.783209\pi$
$811$	−44.2492	−1.55380	−0.776900	+	0.629624i	$0.783209\pi$
$812$	0	0
$813$	7.36068	0.258150
$814$	0	0
$815$	−20.4508	−0.716362
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	2.85410	0.0997304
$820$	0	0
$821$	31.4164	1.09644	0.548220	−	0.836334i	$-0.315306\pi$
$821$	31.4164	1.09644	0.548220	+	0.836334i	$0.315306\pi$
$822$	0	0
$823$	13.1246	0.457495	0.228748	−	0.973486i	$-0.426537\pi$
$823$	13.1246	0.457495	0.228748	+	0.973486i	$0.426537\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0557	0.419219	0.209609	−	0.977785i	$-0.432781\pi$
$827$	12.0557	0.419219	0.209609	+	0.977785i	$0.432781\pi$
$828$	0	0
$829$	−46.8328	−1.62657	−0.813285	−	0.581865i	$-0.802323\pi$
$829$	−46.8328	−1.62657	−0.813285	+	0.581865i	$0.802323\pi$
$830$	0	0
$831$	−6.76393	−0.234638
$832$	0	0
$833$	1.85410	0.0642408
$834$	0	0
$835$	−3.29180	−0.113917
$836$	0	0
$837$	−13.0902	−0.452462
$838$	0	0
$839$	−16.3607	−0.564833	−0.282417	−	0.959292i	$-0.591136\pi$
$839$	−16.3607	−0.564833	−0.282417	+	0.959292i	$0.591136\pi$
$840$	0	0
$841$	−17.5623	−0.605597
$842$	0	0
$843$	3.39512	0.116934
$844$	0	0
$845$	−26.8328	−0.923077
$846$	0	0
$847$	0.437694	0.0150393
$848$	0	0
$849$	−3.27051	−0.112244
$850$	0	0
$851$	−8.12461	−0.278508
$852$	0	0
$853$	−16.5623	−0.567083	−0.283541	−	0.958960i	$-0.591509\pi$
$853$	−16.5623	−0.567083	−0.283541	+	0.958960i	$0.591509\pi$
$854$	0	0
$855$	6.38197	0.218259
$856$	0	0
$857$	−54.8673	−1.87423	−0.937115	−	0.349021i	$-0.886514\pi$
$857$	−54.8673	−1.87423	−0.937115	+	0.349021i	$0.886514\pi$
$858$	0	0
$859$	24.4377	0.833803	0.416902	−	0.908952i	$-0.363116\pi$
$859$	24.4377	0.833803	0.416902	+	0.908952i	$0.363116\pi$
$860$	0	0
$861$	−1.00000	−0.0340799
$862$	0	0
$863$	−18.3820	−0.625729	−0.312865	−	0.949798i	$-0.601289\pi$
$863$	−18.3820	−0.625729	−0.312865	+	0.949798i	$0.601289\pi$
$864$	0	0
$865$	−4.87539	−0.165768
$866$	0	0
$867$	5.18034	0.175934
$868$	0	0
$869$	−33.8197	−1.14725
$870$	0	0
$871$	13.5623	0.459541
$872$	0	0
$873$	35.4377	1.19938
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	−16.8754	−0.569841	−0.284921	−	0.958551i	$-0.591967\pi$
$877$	−16.8754	−0.569841	−0.284921	+	0.958551i	$0.591967\pi$
$878$	0	0
$879$	−4.85410	−0.163725
$880$	0	0
$881$	51.1033	1.72171	0.860857	−	0.508846i	$-0.169928\pi$
$881$	51.1033	1.72171	0.860857	+	0.508846i	$0.169928\pi$
$882$	0	0
$883$	−51.8328	−1.74431	−0.872157	−	0.489227i	$-0.837279\pi$
$883$	−51.8328	−1.74431	−0.872157	+	0.489227i	$0.837279\pi$
$884$	0	0
$885$	10.8541	0.364857
$886$	0	0
$887$	1.58359	0.0531718	0.0265859	−	0.999647i	$-0.491536\pi$
$887$	1.58359	0.0531718	0.0265859	+	0.999647i	$0.491536\pi$
$888$	0	0
$889$	−19.7082	−0.660992
$890$	0	0
$891$	−26.0689	−0.873340
$892$	0	0
$893$	−11.1803	−0.374136
$894$	0	0
$895$	2.43769	0.0814831
$896$	0	0
$897$	1.14590	0.0382604
$898$	0	0
$899$	−19.7984	−0.660313
$900$	0	0
$901$	18.7082	0.623261
$902$	0	0
$903$	0.763932	0.0254221
$904$	0	0
$905$	48.2148	1.60271
$906$	0	0
$907$	−26.1246	−0.867453	−0.433727	−	0.901044i	$-0.642802\pi$
$907$	−26.1246	−0.867453	−0.433727	+	0.901044i	$0.642802\pi$
$908$	0	0
$909$	12.9230	0.428628
$910$	0	0
$911$	9.05573	0.300030	0.150015	−	0.988684i	$-0.452068\pi$
$911$	9.05573	0.300030	0.150015	+	0.988684i	$0.452068\pi$
$912$	0	0
$913$	3.87539	0.128257
$914$	0	0
$915$	5.72949	0.189411
$916$	0	0
$917$	−1.09017	−0.0360006
$918$	0	0
$919$	44.7082	1.47479	0.737394	−	0.675463i	$-0.236056\pi$
$919$	44.7082	1.47479	0.737394	+	0.675463i	$0.236056\pi$
$920$	0	0
$921$	0.931116	0.0306813
$922$	0	0
$923$	−1.47214	−0.0484559
$924$	0	0
$925$	0	0
$926$	0	0
$927$	24.8541	0.816316
$928$	0	0
$929$	39.3262	1.29025	0.645126	−	0.764076i	$-0.276805\pi$
$929$	39.3262	1.29025	0.645126	+	0.764076i	$0.276805\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	1.87539	0.0613975
$934$	0	0
$935$	−14.0213	−0.458545
$936$	0	0
$937$	−32.9787	−1.07737	−0.538684	−	0.842508i	$-0.681078\pi$
$937$	−32.9787	−1.07737	−0.538684	+	0.842508i	$0.681078\pi$
$938$	0	0
$939$	0.875388	0.0285672
$940$	0	0
$941$	−13.6393	−0.444629	−0.222315	−	0.974975i	$-0.571361\pi$
$941$	−13.6393	−0.444629	−0.222315	+	0.974975i	$0.571361\pi$
$942$	0	0
$943$	7.85410	0.255765
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	25.7984	0.838335	0.419167	−	0.907909i	$-0.362322\pi$
$947$	25.7984	0.838335	0.419167	+	0.907909i	$0.362322\pi$
$948$	0	0
$949$	−10.8541	−0.352339
$950$	0	0
$951$	9.70820	0.314810
$952$	0	0
$953$	−31.7426	−1.02825	−0.514123	−	0.857717i	$-0.671882\pi$
$953$	−31.7426	−1.02825	−0.514123	+	0.857717i	$0.671882\pi$
$954$	0	0
$955$	−20.8541	−0.674823
$956$	0	0
$957$	4.36881	0.141224
$958$	0	0
$959$	7.52786	0.243087
$960$	0	0
$961$	3.27051	0.105500
$962$	0	0
$963$	21.3262	0.687228
$964$	0	0
$965$	−3.49342	−0.112457
$966$	0	0
$967$	20.1459	0.647848	0.323924	−	0.946083i	$-0.394998\pi$
$967$	20.1459	0.647848	0.323924	+	0.946083i	$0.394998\pi$
$968$	0	0
$969$	0.708204	0.0227508
$970$	0	0
$971$	−29.1803	−0.936442	−0.468221	−	0.883611i	$-0.655105\pi$
$971$	−29.1803	−0.936442	−0.468221	+	0.883611i	$0.655105\pi$
$972$	0	0
$973$	−20.7082	−0.663875
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.81966	−0.122202	−0.0611009	−	0.998132i	$-0.519461\pi$
$977$	−3.81966	−0.122202	−0.0611009	+	0.998132i	$0.519461\pi$
$978$	0	0
$979$	−53.1246	−1.69787
$980$	0	0
$981$	46.8541	1.49594
$982$	0	0
$983$	47.0689	1.50126	0.750632	−	0.660720i	$-0.229749\pi$
$983$	47.0689	1.50126	0.750632	+	0.660720i	$0.229749\pi$
$984$	0	0
$985$	40.8541	1.30172
$986$	0	0
$987$	−4.27051	−0.135932
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	18.8754	0.599596	0.299798	−	0.954003i	$-0.403081\pi$
$991$	18.8754	0.599596	0.299798	+	0.954003i	$0.403081\pi$
$992$	0	0
$993$	0.549150	0.0174268
$994$	0	0
$995$	−7.36068	−0.233349
$996$	0	0
$997$	30.8541	0.977159	0.488580	−	0.872519i	$-0.337515\pi$
$997$	30.8541	0.977159	0.488580	+	0.872519i	$0.337515\pi$
$998$	0	0
$999$	−6.05573	−0.191595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.j.1.2		2
4.3	odd	2		8512.2.a.be.1.1		2
8.3	odd	2		133.2.a.a.1.2	✓	2
8.5	even	2		2128.2.a.o.1.1		2
24.11	even	2		1197.2.a.j.1.1		2
40.19	odd	2		3325.2.a.q.1.1		2
56.3	even	6		931.2.f.j.324.1		4
56.11	odd	6		931.2.f.k.324.1		4
56.19	even	6		931.2.f.j.704.1		4
56.27	even	2		931.2.a.d.1.2		2
56.51	odd	6		931.2.f.k.704.1		4
152.75	even	2		2527.2.a.e.1.1		2
168.83	odd	2		8379.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.a.1.2	✓	2	8.3	odd	2
931.2.a.d.1.2		2	56.27	even	2
931.2.f.j.324.1		4	56.3	even	6
931.2.f.j.704.1		4	56.19	even	6
931.2.f.k.324.1		4	56.11	odd	6
931.2.f.k.704.1		4	56.51	odd	6
1197.2.a.j.1.1		2	24.11	even	2
2128.2.a.o.1.1		2	8.5	even	2
2527.2.a.e.1.1		2	152.75	even	2
3325.2.a.q.1.1		2	40.19	odd	2
8379.2.a.bn.1.1		2	168.83	odd	2
8512.2.a.j.1.2		2	1.1	even	1	trivial
8512.2.a.be.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} +2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{9} +O(q^{10})\)	\(q-0.381966 q^{3} +2.23607 q^{5} +1.00000 q^{7} -2.85410 q^{9} -3.38197 q^{11} -1.00000 q^{13} -0.854102 q^{15} +1.85410 q^{17} -1.00000 q^{19} -0.381966 q^{21} +3.00000 q^{23} +2.23607 q^{27} +3.38197 q^{29} -5.85410 q^{31} +1.29180 q^{33} +2.23607 q^{35} -2.70820 q^{37} +0.381966 q^{39} +2.61803 q^{41} -2.00000 q^{43} -6.38197 q^{45} +11.1803 q^{47} +1.00000 q^{49} -0.708204 q^{51} +10.0902 q^{53} -7.56231 q^{55} +0.381966 q^{57} -12.7082 q^{59} -6.70820 q^{61} -2.85410 q^{63} -2.23607 q^{65} -13.5623 q^{67} -1.14590 q^{69} +1.47214 q^{71} +10.8541 q^{73} -3.38197 q^{77} +10.0000 q^{79} +7.70820 q^{81} -1.14590 q^{83} +4.14590 q^{85} -1.29180 q^{87} +15.7082 q^{89} -1.00000 q^{91} +2.23607 q^{93} -2.23607 q^{95} -12.4164 q^{97} +9.65248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + 2 * q^7 + q^9	\( 2 q - 3 q^{3} + 2 q^{7} + q^{9} - 9 q^{11} - 2 q^{13} + 5 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 9 q^{29} - 5 q^{31} + 16 q^{33} + 8 q^{37} + 3 q^{39} + 3 q^{41} - 4 q^{43} - 15 q^{45} + 2 q^{49} + 12 q^{51} + 9 q^{53} + 5 q^{55} + 3 q^{57} - 12 q^{59} + q^{63} - 7 q^{67} - 9 q^{69} - 6 q^{71} + 15 q^{73} - 9 q^{77} + 20 q^{79} + 2 q^{81} - 9 q^{83} + 15 q^{85} - 16 q^{87} + 18 q^{89} - 2 q^{91} + 2 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^7 + q^9 - 9 * q^11 - 2 * q^13 + 5 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 9 * q^29 - 5 * q^31 + 16 * q^33 + 8 * q^37 + 3 * q^39 + 3 * q^41 - 4 * q^43 - 15 * q^45 + 2 * q^49 + 12 * q^51 + 9 * q^53 + 5 * q^55 + 3 * q^57 - 12 * q^59 + q^63 - 7 * q^67 - 9 * q^69 - 6 * q^71 + 15 * q^73 - 9 * q^77 + 20 * q^79 + 2 * q^81 - 9 * q^83 + 15 * q^85 - 16 * q^87 + 18 * q^89 - 2 * q^91 + 2 * q^97 - 12 * q^99

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