LMFDB - Embedded newform 3072.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.765367$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.19891 q^{5} +0.613126 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.19891 q^{5} +0.613126 q^{7} +1.00000 q^{9} -1.53073 q^{11} +1.24943 q^{13} -1.19891 q^{15} -4.35916 q^{17} +1.29769 q^{19} -0.613126 q^{21} -4.00000 q^{23} -3.56261 q^{25} -1.00000 q^{27} +0.965872 q^{29} -6.77791 q^{31} +1.53073 q^{33} +0.735084 q^{35} +10.8052 q^{37} -1.24943 q^{39} +8.68873 q^{41} -8.68873 q^{43} +1.19891 q^{45} -9.65685 q^{47} -6.62408 q^{49} +4.35916 q^{51} -11.4184 q^{53} -1.83522 q^{55} -1.29769 q^{57} -9.04789 q^{59} +1.52432 q^{61} +0.613126 q^{63} +1.49796 q^{65} +12.1094 q^{67} +4.00000 q^{69} -3.56940 q^{71} +5.15800 q^{73} +3.56261 q^{75} -0.938533 q^{77} +7.49623 q^{79} +1.00000 q^{81} +7.18759 q^{83} -5.22625 q^{85} -0.965872 q^{87} -0.672715 q^{89} +0.766057 q^{91} +6.77791 q^{93} +1.55582 q^{95} +4.71832 q^{97} -1.53073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} + 8 q^{13} + 8 q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} - 8 q^{31} + 16 q^{35} + 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{59} + 24 q^{61} - 8 q^{63} - 8 q^{65} - 16 q^{67} + 16 q^{69} - 16 q^{71} - 8 q^{73} - 4 q^{75} - 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} - 16 q^{91} + 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 + 8 * q^13 + 8 * q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 - 8 * q^31 + 16 * q^35 + 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 + 16 * q^59 + 24 * q^61 - 8 * q^63 - 8 * q^65 - 16 * q^67 + 16 * q^69 - 16 * q^71 - 8 * q^73 - 4 * q^75 - 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 - 16 * q^91 + 8 * q^93 - 32 * q^95 - 16 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.19891	0.536170	0.268085	−	0.963395i	$-0.413609\pi$
$5$	1.19891	0.536170	0.268085	+	0.963395i	$0.413609\pi$
$6$	0	0
$7$	0.613126	0.231740	0.115870	−	0.993264i	$-0.463034\pi$
$7$	0.613126	0.231740	0.115870	+	0.993264i	$0.463034\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.53073	−0.461534	−0.230767	−	0.973009i	$-0.574123\pi$
$11$	−1.53073	−0.461534	−0.230767	+	0.973009i	$0.574123\pi$
$12$	0	0
$13$	1.24943	0.346529	0.173265	−	0.984875i	$-0.444568\pi$
$13$	1.24943	0.346529	0.173265	+	0.984875i	$0.444568\pi$
$14$	0	0
$15$	−1.19891	−0.309558
$16$	0	0
$17$	−4.35916	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$17$	−4.35916	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$18$	0	0
$19$	1.29769	0.297711	0.148856	−	0.988859i	$-0.452441\pi$
$19$	1.29769	0.297711	0.148856	+	0.988859i	$0.452441\pi$
$20$	0	0
$21$	−0.613126	−0.133795
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.56261	−0.712522
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.965872	0.179358	0.0896790	−	0.995971i	$-0.471416\pi$
$29$	0.965872	0.179358	0.0896790	+	0.995971i	$0.471416\pi$
$30$	0	0
$31$	−6.77791	−1.21735	−0.608674	−	0.793420i	$-0.708298\pi$
$31$	−6.77791	−1.21735	−0.608674	+	0.793420i	$0.708298\pi$
$32$	0	0
$33$	1.53073	0.266467
$34$	0	0
$35$	0.735084	0.124252
$36$	0	0
$37$	10.8052	1.77637	0.888186	−	0.459484i	$-0.151966\pi$
$37$	10.8052	1.77637	0.888186	+	0.459484i	$0.151966\pi$
$38$	0	0
$39$	−1.24943	−0.200069
$40$	0	0
$41$	8.68873	1.35695	0.678476	−	0.734623i	$-0.262641\pi$
$41$	8.68873	1.35695	0.678476	+	0.734623i	$0.262641\pi$
$42$	0	0
$43$	−8.68873	−1.32502	−0.662509	−	0.749054i	$-0.730509\pi$
$43$	−8.68873	−1.32502	−0.662509	+	0.749054i	$0.730509\pi$
$44$	0	0
$45$	1.19891	0.178723
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	−6.62408	−0.946297
$50$	0	0
$51$	4.35916	0.610405
$52$	0	0
$53$	−11.4184	−1.56843	−0.784217	−	0.620486i	$-0.786935\pi$
$53$	−11.4184	−1.56843	−0.784217	+	0.620486i	$0.786935\pi$
$54$	0	0
$55$	−1.83522	−0.247460
$56$	0	0
$57$	−1.29769	−0.171884
$58$	0	0
$59$	−9.04789	−1.17794	−0.588968	−	0.808157i	$-0.700465\pi$
$59$	−9.04789	−1.17794	−0.588968	+	0.808157i	$0.700465\pi$
$60$	0	0
$61$	1.52432	0.195169	0.0975845	−	0.995227i	$-0.468888\pi$
$61$	1.52432	0.195169	0.0975845	+	0.995227i	$0.468888\pi$
$62$	0	0
$63$	0.613126	0.0772466
$64$	0	0
$65$	1.49796	0.185799
$66$	0	0
$67$	12.1094	1.47939	0.739697	−	0.672940i	$-0.234969\pi$
$67$	12.1094	1.47939	0.739697	+	0.672940i	$0.234969\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−3.56940	−0.423610	−0.211805	−	0.977312i	$-0.567934\pi$
$71$	−3.56940	−0.423610	−0.211805	+	0.977312i	$0.567934\pi$
$72$	0	0
$73$	5.15800	0.603698	0.301849	−	0.953356i	$-0.402396\pi$
$73$	5.15800	0.603698	0.301849	+	0.953356i	$0.402396\pi$
$74$	0	0
$75$	3.56261	0.411375
$76$	0	0
$77$	−0.938533	−0.106956
$78$	0	0
$79$	7.49623	0.843392	0.421696	−	0.906737i	$-0.361435\pi$
$79$	7.49623	0.843392	0.421696	+	0.906737i	$0.361435\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.18759	0.788940	0.394470	−	0.918909i	$-0.370928\pi$
$83$	7.18759	0.788940	0.394470	+	0.918909i	$0.370928\pi$
$84$	0	0
$85$	−5.22625	−0.566867
$86$	0	0
$87$	−0.965872	−0.103552
$88$	0	0
$89$	−0.672715	−0.0713076	−0.0356538	−	0.999364i	$-0.511351\pi$
$89$	−0.672715	−0.0713076	−0.0356538	+	0.999364i	$0.511351\pi$
$90$	0	0
$91$	0.766057	0.0803046
$92$	0	0
$93$	6.77791	0.702837
$94$	0	0
$95$	1.55582	0.159624
$96$	0	0
$97$	4.71832	0.479073	0.239536	−	0.970887i	$-0.423005\pi$
$97$	4.71832	0.479073	0.239536	+	0.970887i	$0.423005\pi$
$98$	0	0
$99$	−1.53073	−0.153845
$100$	0	0
$101$	5.68419	0.565598	0.282799	−	0.959179i	$-0.408737\pi$
$101$	5.68419	0.565598	0.282799	+	0.959179i	$0.408737\pi$
$102$	0	0
$103$	−15.5381	−1.53101	−0.765506	−	0.643428i	$-0.777511\pi$
$103$	−15.5381	−1.53101	−0.765506	+	0.643428i	$0.777511\pi$
$104$	0	0
$105$	−0.735084	−0.0717369
$106$	0	0
$107$	0.0773278	0.00747556	0.00373778	−	0.999993i	$-0.498810\pi$
$107$	0.0773278	0.00747556	0.00373778	+	0.999993i	$0.498810\pi$
$108$	0	0
$109$	−3.23585	−0.309938	−0.154969	−	0.987919i	$-0.549528\pi$
$109$	−3.23585	−0.309938	−0.154969	+	0.987919i	$0.549528\pi$
$110$	0	0
$111$	−10.8052	−1.02559
$112$	0	0
$113$	−7.65685	−0.720296	−0.360148	−	0.932895i	$-0.617274\pi$
$113$	−7.65685	−0.720296	−0.360148	+	0.932895i	$0.617274\pi$
$114$	0	0
$115$	−4.79565	−0.447197
$116$	0	0
$117$	1.24943	0.115510
$118$	0	0
$119$	−2.67271	−0.245007
$120$	0	0
$121$	−8.65685	−0.786987
$122$	0	0
$123$	−8.68873	−0.783436
$124$	0	0
$125$	−10.2658	−0.918203
$126$	0	0
$127$	−20.9008	−1.85465	−0.927325	−	0.374257i	$-0.877898\pi$
$127$	−20.9008	−1.85465	−0.927325	+	0.374257i	$0.877898\pi$
$128$	0	0
$129$	8.68873	0.765000
$130$	0	0
$131$	7.72061	0.674552	0.337276	−	0.941406i	$-0.390494\pi$
$131$	7.72061	0.674552	0.337276	+	0.941406i	$0.390494\pi$
$132$	0	0
$133$	0.795649	0.0689916
$134$	0	0
$135$	−1.19891	−0.103186
$136$	0	0
$137$	7.42063	0.633987	0.316994	−	0.948428i	$-0.397327\pi$
$137$	7.42063	0.633987	0.316994	+	0.948428i	$0.397327\pi$
$138$	0	0
$139$	−12.2522	−1.03922	−0.519611	−	0.854403i	$-0.673923\pi$
$139$	−12.2522	−1.03922	−0.519611	+	0.854403i	$0.673923\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	0	0
$143$	−1.91254	−0.159935
$144$	0	0
$145$	1.15800	0.0961663
$146$	0	0
$147$	6.62408	0.546345
$148$	0	0
$149$	−3.54206	−0.290177	−0.145088	−	0.989419i	$-0.546347\pi$
$149$	−3.54206	−0.290177	−0.145088	+	0.989419i	$0.546347\pi$
$150$	0	0
$151$	−13.0019	−1.05808	−0.529039	−	0.848598i	$-0.677447\pi$
$151$	−13.0019	−1.05808	−0.529039	+	0.848598i	$0.677447\pi$
$152$	0	0
$153$	−4.35916	−0.352417
$154$	0	0
$155$	−8.12612	−0.652706
$156$	0	0
$157$	−1.14840	−0.0916519	−0.0458260	−	0.998949i	$-0.514592\pi$
$157$	−1.14840	−0.0916519	−0.0458260	+	0.998949i	$0.514592\pi$
$158$	0	0
$159$	11.4184	0.905536
$160$	0	0
$161$	−2.45250	−0.193284
$162$	0	0
$163$	18.4821	1.44763	0.723815	−	0.689994i	$-0.242387\pi$
$163$	18.4821	1.44763	0.723815	+	0.689994i	$0.242387\pi$
$164$	0	0
$165$	1.83522	0.142871
$166$	0	0
$167$	−20.2104	−1.56393	−0.781964	−	0.623324i	$-0.785782\pi$
$167$	−20.2104	−1.56393	−0.781964	+	0.623324i	$0.785782\pi$
$168$	0	0
$169$	−11.4389	−0.879917
$170$	0	0
$171$	1.29769	0.0992371
$172$	0	0
$173$	−10.1374	−0.770736	−0.385368	−	0.922763i	$-0.625925\pi$
$173$	−10.1374	−0.770736	−0.385368	+	0.922763i	$0.625925\pi$
$174$	0	0
$175$	−2.18433	−0.165120
$176$	0	0
$177$	9.04789	0.680081
$178$	0	0
$179$	20.1094	1.50304	0.751522	−	0.659708i	$-0.229320\pi$
$179$	20.1094	1.50304	0.751522	+	0.659708i	$0.229320\pi$
$180$	0	0
$181$	9.57900	0.712001	0.356001	−	0.934486i	$-0.384140\pi$
$181$	9.57900	0.712001	0.356001	+	0.934486i	$0.384140\pi$
$182$	0	0
$183$	−1.52432	−0.112681
$184$	0	0
$185$	12.9545	0.952437
$186$	0	0
$187$	6.67271	0.487957
$188$	0	0
$189$	−0.613126	−0.0445983
$190$	0	0
$191$	−2.30767	−0.166977	−0.0834885	−	0.996509i	$-0.526606\pi$
$191$	−2.30767	−0.166977	−0.0834885	+	0.996509i	$0.526606\pi$
$192$	0	0
$193$	2.03278	0.146323	0.0731613	−	0.997320i	$-0.476691\pi$
$193$	2.03278	0.146323	0.0731613	+	0.997320i	$0.476691\pi$
$194$	0	0
$195$	−1.49796	−0.107271
$196$	0	0
$197$	−26.4025	−1.88110	−0.940551	−	0.339653i	$-0.889690\pi$
$197$	−26.4025	−1.88110	−0.940551	+	0.339653i	$0.889690\pi$
$198$	0	0
$199$	10.1927	0.722538	0.361269	−	0.932462i	$-0.382344\pi$
$199$	10.1927	0.722538	0.361269	+	0.932462i	$0.382344\pi$
$200$	0	0
$201$	−12.1094	−0.854128
$202$	0	0
$203$	0.592201	0.0415644
$204$	0	0
$205$	10.4170	0.727557
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−1.98642	−0.137404
$210$	0	0
$211$	−7.06147	−0.486131	−0.243066	−	0.970010i	$-0.578153\pi$
$211$	−7.06147	−0.486131	−0.243066	+	0.970010i	$0.578153\pi$
$212$	0	0
$213$	3.56940	0.244571
$214$	0	0
$215$	−10.4170	−0.710435
$216$	0	0
$217$	−4.15571	−0.282108
$218$	0	0
$219$	−5.15800	−0.348545
$220$	0	0
$221$	−5.44646	−0.366369
$222$	0	0
$223$	−18.9646	−1.26996	−0.634982	−	0.772527i	$-0.718992\pi$
$223$	−18.9646	−1.26996	−0.634982	+	0.772527i	$0.718992\pi$
$224$	0	0
$225$	−3.56261	−0.237507
$226$	0	0
$227$	−15.5945	−1.03504	−0.517521	−	0.855670i	$-0.673145\pi$
$227$	−15.5945	−1.03504	−0.517521	+	0.855670i	$0.673145\pi$
$228$	0	0
$229$	−6.37465	−0.421249	−0.210624	−	0.977567i	$-0.567550\pi$
$229$	−6.37465	−0.421249	−0.210624	+	0.977567i	$0.567550\pi$
$230$	0	0
$231$	0.938533	0.0617509
$232$	0	0
$233$	−1.13880	−0.0746050	−0.0373025	−	0.999304i	$-0.511877\pi$
$233$	−1.13880	−0.0746050	−0.0373025	+	0.999304i	$0.511877\pi$
$234$	0	0
$235$	−11.5777	−0.755247
$236$	0	0
$237$	−7.49623	−0.486933
$238$	0	0
$239$	−25.6652	−1.66014	−0.830071	−	0.557657i	$-0.811700\pi$
$239$	−25.6652	−1.66014	−0.830071	+	0.557657i	$0.811700\pi$
$240$	0	0
$241$	19.2172	1.23789	0.618944	−	0.785435i	$-0.287561\pi$
$241$	19.2172	1.23789	0.618944	+	0.785435i	$0.287561\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.94169	−0.507376
$246$	0	0
$247$	1.62138	0.103166
$248$	0	0
$249$	−7.18759	−0.455495
$250$	0	0
$251$	−6.46927	−0.408336	−0.204168	−	0.978936i	$-0.565449\pi$
$251$	−6.46927	−0.408336	−0.204168	+	0.978936i	$0.565449\pi$
$252$	0	0
$253$	6.12293	0.384946
$254$	0	0
$255$	5.22625	0.327281
$256$	0	0
$257$	−22.1094	−1.37914	−0.689572	−	0.724217i	$-0.742201\pi$
$257$	−22.1094	−1.37914	−0.689572	+	0.724217i	$0.742201\pi$
$258$	0	0
$259$	6.62498	0.411656
$260$	0	0
$261$	0.965872	0.0597860
$262$	0	0
$263$	22.8914	1.41155	0.705773	−	0.708438i	$-0.250600\pi$
$263$	22.8914	1.41155	0.705773	+	0.708438i	$0.250600\pi$
$264$	0	0
$265$	−13.6896	−0.840947
$266$	0	0
$267$	0.672715	0.0411695
$268$	0	0
$269$	−23.4776	−1.43145	−0.715726	−	0.698381i	$-0.753904\pi$
$269$	−23.4776	−1.43145	−0.715726	+	0.698381i	$0.753904\pi$
$270$	0	0
$271$	25.5837	1.55410	0.777049	−	0.629440i	$-0.216716\pi$
$271$	25.5837	1.55410	0.777049	+	0.629440i	$0.216716\pi$
$272$	0	0
$273$	−0.766057	−0.0463639
$274$	0	0
$275$	5.45341	0.328853
$276$	0	0
$277$	13.6692	0.821300	0.410650	−	0.911793i	$-0.365302\pi$
$277$	13.6692	0.821300	0.410650	+	0.911793i	$0.365302\pi$
$278$	0	0
$279$	−6.77791	−0.405783
$280$	0	0
$281$	21.1116	1.25941	0.629707	−	0.776832i	$-0.283175\pi$
$281$	21.1116	1.25941	0.629707	+	0.776832i	$0.283175\pi$
$282$	0	0
$283$	17.0479	1.01339	0.506696	−	0.862125i	$-0.330867\pi$
$283$	17.0479	1.01339	0.506696	+	0.862125i	$0.330867\pi$
$284$	0	0
$285$	−1.55582	−0.0921589
$286$	0	0
$287$	5.32729	0.314460
$288$	0	0
$289$	2.00228	0.117781
$290$	0	0
$291$	−4.71832	−0.276593
$292$	0	0
$293$	−16.1047	−0.940845	−0.470422	−	0.882441i	$-0.655898\pi$
$293$	−16.1047	−0.940845	−0.470422	+	0.882441i	$0.655898\pi$
$294$	0	0
$295$	−10.8476	−0.631573
$296$	0	0
$297$	1.53073	0.0888222
$298$	0	0
$299$	−4.99772	−0.289025
$300$	0	0
$301$	−5.32729	−0.307060
$302$	0	0
$303$	−5.68419	−0.326548
$304$	0	0
$305$	1.82752	0.104644
$306$	0	0
$307$	−6.91858	−0.394864	−0.197432	−	0.980317i	$-0.563260\pi$
$307$	−6.91858	−0.394864	−0.197432	+	0.980317i	$0.563260\pi$
$308$	0	0
$309$	15.5381	0.883931
$310$	0	0
$311$	20.4389	1.15899	0.579493	−	0.814977i	$-0.303251\pi$
$311$	20.4389	1.15899	0.579493	+	0.814977i	$0.303251\pi$
$312$	0	0
$313$	11.6241	0.657032	0.328516	−	0.944498i	$-0.393452\pi$
$313$	11.6241	0.657032	0.328516	+	0.944498i	$0.393452\pi$
$314$	0	0
$315$	0.735084	0.0414173
$316$	0	0
$317$	−21.5368	−1.20963	−0.604814	−	0.796367i	$-0.706752\pi$
$317$	−21.5368	−1.20963	−0.604814	+	0.796367i	$0.706752\pi$
$318$	0	0
$319$	−1.47849	−0.0827797
$320$	0	0
$321$	−0.0773278	−0.00431601
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−4.45123	−0.246910
$326$	0	0
$327$	3.23585	0.178943
$328$	0	0
$329$	−5.92087	−0.326428
$330$	0	0
$331$	−13.8435	−0.760910	−0.380455	−	0.924799i	$-0.624233\pi$
$331$	−13.8435	−0.760910	−0.380455	+	0.924799i	$0.624233\pi$
$332$	0	0
$333$	10.8052	0.592124
$334$	0	0
$335$	14.5181	0.793206
$336$	0	0
$337$	33.8428	1.84353	0.921767	−	0.387744i	$-0.126745\pi$
$337$	33.8428	1.84353	0.921767	+	0.387744i	$0.126745\pi$
$338$	0	0
$339$	7.65685	0.415863
$340$	0	0
$341$	10.3752	0.561847
$342$	0	0
$343$	−8.35327	−0.451034
$344$	0	0
$345$	4.79565	0.258189
$346$	0	0
$347$	−10.6560	−0.572041	−0.286021	−	0.958223i	$-0.592333\pi$
$347$	−10.6560	−0.572041	−0.286021	+	0.958223i	$0.592333\pi$
$348$	0	0
$349$	15.6009	0.835097	0.417548	−	0.908655i	$-0.362889\pi$
$349$	15.6009	0.835097	0.417548	+	0.908655i	$0.362889\pi$
$350$	0	0
$351$	−1.24943	−0.0666896
$352$	0	0
$353$	−28.2978	−1.50614	−0.753071	−	0.657939i	$-0.771428\pi$
$353$	−28.2978	−1.50614	−0.753071	+	0.657939i	$0.771428\pi$
$354$	0	0
$355$	−4.27939	−0.227127
$356$	0	0
$357$	2.67271	0.141455
$358$	0	0
$359$	−3.41474	−0.180223	−0.0901116	−	0.995932i	$-0.528722\pi$
$359$	−3.41474	−0.180223	−0.0901116	+	0.995932i	$0.528722\pi$
$360$	0	0
$361$	−17.3160	−0.911368
$362$	0	0
$363$	8.65685	0.454367
$364$	0	0
$365$	6.18399	0.323685
$366$	0	0
$367$	−10.0916	−0.526778	−0.263389	−	0.964690i	$-0.584840\pi$
$367$	−10.0916	−0.526778	−0.263389	+	0.964690i	$0.584840\pi$
$368$	0	0
$369$	8.68873	0.452317
$370$	0	0
$371$	−7.00090	−0.363469
$372$	0	0
$373$	14.9610	0.774649	0.387325	−	0.921943i	$-0.373399\pi$
$373$	14.9610	0.774649	0.387325	+	0.921943i	$0.373399\pi$
$374$	0	0
$375$	10.2658	0.530125
$376$	0	0
$377$	1.20679	0.0621528
$378$	0	0
$379$	−12.9681	−0.666128	−0.333064	−	0.942904i	$-0.608082\pi$
$379$	−12.9681	−0.666128	−0.333064	+	0.942904i	$0.608082\pi$
$380$	0	0
$381$	20.9008	1.07078
$382$	0	0
$383$	−3.81527	−0.194951	−0.0974755	−	0.995238i	$-0.531077\pi$
$383$	−3.81527	−0.194951	−0.0974755	+	0.995238i	$0.531077\pi$
$384$	0	0
$385$	−1.12522	−0.0573464
$386$	0	0
$387$	−8.68873	−0.441673
$388$	0	0
$389$	−28.1239	−1.42594	−0.712968	−	0.701196i	$-0.752650\pi$
$389$	−28.1239	−1.42594	−0.712968	+	0.701196i	$0.752650\pi$
$390$	0	0
$391$	17.4366	0.881809
$392$	0	0
$393$	−7.72061	−0.389453
$394$	0	0
$395$	8.98733	0.452201
$396$	0	0
$397$	32.0862	1.61036	0.805180	−	0.593031i	$-0.202069\pi$
$397$	32.0862	1.61036	0.805180	+	0.593031i	$0.202069\pi$
$398$	0	0
$399$	−0.795649	−0.0398323
$400$	0	0
$401$	−14.9545	−0.746794	−0.373397	−	0.927672i	$-0.621807\pi$
$401$	−14.9545	−0.746794	−0.373397	+	0.927672i	$0.621807\pi$
$402$	0	0
$403$	−8.46852	−0.421847
$404$	0	0
$405$	1.19891	0.0595744
$406$	0	0
$407$	−16.5400	−0.819855
$408$	0	0
$409$	10.5298	0.520667	0.260333	−	0.965519i	$-0.416168\pi$
$409$	10.5298	0.520667	0.260333	+	0.965519i	$0.416168\pi$
$410$	0	0
$411$	−7.42063	−0.366033
$412$	0	0
$413$	−5.54750	−0.272974
$414$	0	0
$415$	8.61729	0.423006
$416$	0	0
$417$	12.2522	0.599995
$418$	0	0
$419$	20.7853	1.01543	0.507713	−	0.861526i	$-0.330491\pi$
$419$	20.7853	1.01543	0.507713	+	0.861526i	$0.330491\pi$
$420$	0	0
$421$	−14.0179	−0.683192	−0.341596	−	0.939847i	$-0.610967\pi$
$421$	−14.0179	−0.683192	−0.341596	+	0.939847i	$0.610967\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	15.5300	0.753315
$426$	0	0
$427$	0.934599	0.0452284
$428$	0	0
$429$	1.91254	0.0923385
$430$	0	0
$431$	−13.8318	−0.666253	−0.333126	−	0.942882i	$-0.608104\pi$
$431$	−13.8318	−0.666253	−0.333126	+	0.942882i	$0.608104\pi$
$432$	0	0
$433$	−14.8476	−0.713531	−0.356766	−	0.934194i	$-0.616121\pi$
$433$	−14.8476	−0.713531	−0.356766	+	0.934194i	$0.616121\pi$
$434$	0	0
$435$	−1.15800	−0.0555217
$436$	0	0
$437$	−5.19077	−0.248308
$438$	0	0
$439$	35.7840	1.70787	0.853937	−	0.520376i	$-0.174208\pi$
$439$	35.7840	1.70787	0.853937	+	0.520376i	$0.174208\pi$
$440$	0	0
$441$	−6.62408	−0.315432
$442$	0	0
$443$	33.0311	1.56936	0.784678	−	0.619903i	$-0.212828\pi$
$443$	33.0311	1.56936	0.784678	+	0.619903i	$0.212828\pi$
$444$	0	0
$445$	−0.806526	−0.0382330
$446$	0	0
$447$	3.54206	0.167534
$448$	0	0
$449$	−7.27775	−0.343458	−0.171729	−	0.985144i	$-0.554935\pi$
$449$	−7.27775	−0.343458	−0.171729	+	0.985144i	$0.554935\pi$
$450$	0	0
$451$	−13.3001	−0.626279
$452$	0	0
$453$	13.0019	0.610882
$454$	0	0
$455$	0.918436	0.0430569
$456$	0	0
$457$	14.3096	0.669376	0.334688	−	0.942329i	$-0.391369\pi$
$457$	14.3096	0.669376	0.334688	+	0.942329i	$0.391369\pi$
$458$	0	0
$459$	4.35916	0.203468
$460$	0	0
$461$	−23.4923	−1.09414	−0.547072	−	0.837086i	$-0.684258\pi$
$461$	−23.4923	−1.09414	−0.547072	+	0.837086i	$0.684258\pi$
$462$	0	0
$463$	29.7704	1.38355	0.691773	−	0.722115i	$-0.256830\pi$
$463$	29.7704	1.38355	0.691773	+	0.722115i	$0.256830\pi$
$464$	0	0
$465$	8.12612	0.376840
$466$	0	0
$467$	−23.4398	−1.08467	−0.542333	−	0.840164i	$-0.682459\pi$
$467$	−23.4398	−1.08467	−0.542333	+	0.840164i	$0.682459\pi$
$468$	0	0
$469$	7.42456	0.342834
$470$	0	0
$471$	1.14840	0.0529153
$472$	0	0
$473$	13.3001	0.611541
$474$	0	0
$475$	−4.62317	−0.212126
$476$	0	0
$477$	−11.4184	−0.522812
$478$	0	0
$479$	24.2104	1.10620	0.553101	−	0.833115i	$-0.313445\pi$
$479$	24.2104	1.10620	0.553101	+	0.833115i	$0.313445\pi$
$480$	0	0
$481$	13.5004	0.615565
$482$	0	0
$483$	2.45250	0.111593
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	−10.0498	−0.455399	−0.227699	−	0.973732i	$-0.573120\pi$
$487$	−10.0498	−0.455399	−0.227699	+	0.973732i	$0.573120\pi$
$488$	0	0
$489$	−18.4821	−0.835789
$490$	0	0
$491$	36.6848	1.65556	0.827781	−	0.561052i	$-0.189603\pi$
$491$	36.6848	1.65556	0.827781	+	0.561052i	$0.189603\pi$
$492$	0	0
$493$	−4.21039	−0.189626
$494$	0	0
$495$	−1.83522	−0.0824868
$496$	0	0
$497$	−2.18849	−0.0981672
$498$	0	0
$499$	1.98005	0.0886393	0.0443196	−	0.999017i	$-0.485888\pi$
$499$	1.98005	0.0886393	0.0443196	+	0.999017i	$0.485888\pi$
$500$	0	0
$501$	20.2104	0.902934
$502$	0	0
$503$	12.0739	0.538348	0.269174	−	0.963092i	$-0.413249\pi$
$503$	12.0739	0.538348	0.269174	+	0.963092i	$0.413249\pi$
$504$	0	0
$505$	6.81485	0.303257
$506$	0	0
$507$	11.4389	0.508021
$508$	0	0
$509$	4.48892	0.198968	0.0994838	−	0.995039i	$-0.468281\pi$
$509$	4.48892	0.198968	0.0994838	+	0.995039i	$0.468281\pi$
$510$	0	0
$511$	3.16250	0.139901
$512$	0	0
$513$	−1.29769	−0.0572946
$514$	0	0
$515$	−18.6288	−0.820883
$516$	0	0
$517$	14.7821	0.650115
$518$	0	0
$519$	10.1374	0.444984
$520$	0	0
$521$	4.75428	0.208289	0.104145	−	0.994562i	$-0.466790\pi$
$521$	4.75428	0.208289	0.104145	+	0.994562i	$0.466790\pi$
$522$	0	0
$523$	34.7343	1.51883	0.759413	−	0.650609i	$-0.225486\pi$
$523$	34.7343	1.51883	0.759413	+	0.650609i	$0.225486\pi$
$524$	0	0
$525$	2.18433	0.0953319
$526$	0	0
$527$	29.5460	1.28704
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−9.04789	−0.392645
$532$	0	0
$533$	10.8560	0.470223
$534$	0	0
$535$	0.0927092	0.00400817
$536$	0	0
$537$	−20.1094	−0.867783
$538$	0	0
$539$	10.1397	0.436748
$540$	0	0
$541$	16.2981	0.700709	0.350354	−	0.936617i	$-0.386061\pi$
$541$	16.2981	0.700709	0.350354	+	0.936617i	$0.386061\pi$
$542$	0	0
$543$	−9.57900	−0.411074
$544$	0	0
$545$	−3.87950	−0.166180
$546$	0	0
$547$	8.84339	0.378116	0.189058	−	0.981966i	$-0.439457\pi$
$547$	8.84339	0.378116	0.189058	+	0.981966i	$0.439457\pi$
$548$	0	0
$549$	1.52432	0.0650563
$550$	0	0
$551$	1.25341	0.0533969
$552$	0	0
$553$	4.59613	0.195448
$554$	0	0
$555$	−12.9545	−0.549890
$556$	0	0
$557$	34.3335	1.45476	0.727379	−	0.686236i	$-0.240738\pi$
$557$	34.3335	1.45476	0.727379	+	0.686236i	$0.240738\pi$
$558$	0	0
$559$	−10.8560	−0.459158
$560$	0	0
$561$	−6.67271	−0.281722
$562$	0	0
$563$	−27.8788	−1.17495	−0.587475	−	0.809243i	$-0.699878\pi$
$563$	−27.8788	−1.17495	−0.587475	+	0.809243i	$0.699878\pi$
$564$	0	0
$565$	−9.17990	−0.386201
$566$	0	0
$567$	0.613126	0.0257489
$568$	0	0
$569$	22.1525	0.928682	0.464341	−	0.885656i	$-0.346291\pi$
$569$	22.1525	0.928682	0.464341	+	0.885656i	$0.346291\pi$
$570$	0	0
$571$	34.4845	1.44313	0.721566	−	0.692345i	$-0.243422\pi$
$571$	34.4845	1.44313	0.721566	+	0.692345i	$0.243422\pi$
$572$	0	0
$573$	2.30767	0.0964042
$574$	0	0
$575$	14.2504	0.594284
$576$	0	0
$577$	−22.3488	−0.930391	−0.465196	−	0.885208i	$-0.654016\pi$
$577$	−22.3488	−0.930391	−0.465196	+	0.885208i	$0.654016\pi$
$578$	0	0
$579$	−2.03278	−0.0844794
$580$	0	0
$581$	4.40690	0.182829
$582$	0	0
$583$	17.4785	0.723885
$584$	0	0
$585$	1.49796	0.0619329
$586$	0	0
$587$	−14.3096	−0.590621	−0.295311	−	0.955401i	$-0.595423\pi$
$587$	−14.3096	−0.590621	−0.295311	+	0.955401i	$0.595423\pi$
$588$	0	0
$589$	−8.79565	−0.362418
$590$	0	0
$591$	26.4025	1.08605
$592$	0	0
$593$	−27.3439	−1.12288	−0.561440	−	0.827517i	$-0.689753\pi$
$593$	−27.3439	−1.12288	−0.561440	+	0.827517i	$0.689753\pi$
$594$	0	0
$595$	−3.20435	−0.131366
$596$	0	0
$597$	−10.1927	−0.417157
$598$	0	0
$599$	23.4684	0.958891	0.479446	−	0.877572i	$-0.340838\pi$
$599$	23.4684	0.958891	0.479446	+	0.877572i	$0.340838\pi$
$600$	0	0
$601$	−20.4717	−0.835058	−0.417529	−	0.908664i	$-0.637104\pi$
$601$	−20.4717	−0.835058	−0.417529	+	0.908664i	$0.637104\pi$
$602$	0	0
$603$	12.1094	0.493131
$604$	0	0
$605$	−10.3788	−0.421959
$606$	0	0
$607$	32.6633	1.32576	0.662881	−	0.748725i	$-0.269333\pi$
$607$	32.6633	1.32576	0.662881	+	0.748725i	$0.269333\pi$
$608$	0	0
$609$	−0.592201	−0.0239972
$610$	0	0
$611$	−12.0656	−0.488120
$612$	0	0
$613$	7.11562	0.287397	0.143699	−	0.989622i	$-0.454100\pi$
$613$	7.11562	0.287397	0.143699	+	0.989622i	$0.454100\pi$
$614$	0	0
$615$	−10.4170	−0.420055
$616$	0	0
$617$	20.5619	0.827789	0.413895	−	0.910325i	$-0.364168\pi$
$617$	20.5619	0.827789	0.413895	+	0.910325i	$0.364168\pi$
$618$	0	0
$619$	35.4412	1.42450	0.712251	−	0.701925i	$-0.247676\pi$
$619$	35.4412	1.42450	0.712251	+	0.701925i	$0.247676\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−0.412459	−0.0165248
$624$	0	0
$625$	5.50523	0.220209
$626$	0	0
$627$	1.98642	0.0793301
$628$	0	0
$629$	−47.1018	−1.87807
$630$	0	0
$631$	3.05731	0.121709	0.0608547	−	0.998147i	$-0.480617\pi$
$631$	3.05731	0.121709	0.0608547	+	0.998147i	$0.480617\pi$
$632$	0	0
$633$	7.06147	0.280668
$634$	0	0
$635$	−25.0583	−0.994408
$636$	0	0
$637$	−8.27631	−0.327920
$638$	0	0
$639$	−3.56940	−0.141203
$640$	0	0
$641$	10.8890	0.430089	0.215045	−	0.976604i	$-0.431010\pi$
$641$	10.8890	0.430089	0.215045	+	0.976604i	$0.431010\pi$
$642$	0	0
$643$	−23.6729	−0.933567	−0.466783	−	0.884372i	$-0.654587\pi$
$643$	−23.6729	−0.933567	−0.466783	+	0.884372i	$0.654587\pi$
$644$	0	0
$645$	10.4170	0.410170
$646$	0	0
$647$	42.3417	1.66462	0.832311	−	0.554309i	$-0.187017\pi$
$647$	42.3417	1.66462	0.832311	+	0.554309i	$0.187017\pi$
$648$	0	0
$649$	13.8499	0.543657
$650$	0	0
$651$	4.15571	0.162875
$652$	0	0
$653$	−48.3535	−1.89222	−0.946109	−	0.323850i	$-0.895023\pi$
$653$	−48.3535	−1.89222	−0.946109	+	0.323850i	$0.895023\pi$
$654$	0	0
$655$	9.25633	0.361675
$656$	0	0
$657$	5.15800	0.201233
$658$	0	0
$659$	23.0023	0.896042	0.448021	−	0.894023i	$-0.352129\pi$
$659$	23.0023	0.896042	0.448021	+	0.894023i	$0.352129\pi$
$660$	0	0
$661$	−21.4790	−0.835437	−0.417719	−	0.908576i	$-0.637170\pi$
$661$	−21.4790	−0.835437	−0.417719	+	0.908576i	$0.637170\pi$
$662$	0	0
$663$	5.44646	0.211523
$664$	0	0
$665$	0.953914	0.0369912
$666$	0	0
$667$	−3.86349	−0.149595
$668$	0	0
$669$	18.9646	0.733214
$670$	0	0
$671$	−2.33333	−0.0900771
$672$	0	0
$673$	17.6569	0.680622	0.340311	−	0.940313i	$-0.389468\pi$
$673$	17.6569	0.680622	0.340311	+	0.940313i	$0.389468\pi$
$674$	0	0
$675$	3.56261	0.137125
$676$	0	0
$677$	32.2287	1.23865	0.619325	−	0.785135i	$-0.287406\pi$
$677$	32.2287	1.23865	0.619325	+	0.785135i	$0.287406\pi$
$678$	0	0
$679$	2.89293	0.111020
$680$	0	0
$681$	15.5945	0.597582
$682$	0	0
$683$	−18.4037	−0.704198	−0.352099	−	0.935963i	$-0.614532\pi$
$683$	−18.4037	−0.704198	−0.352099	+	0.935963i	$0.614532\pi$
$684$	0	0
$685$	8.89668	0.339925
$686$	0	0
$687$	6.37465	0.243208
$688$	0	0
$689$	−14.2665	−0.543509
$690$	0	0
$691$	12.5138	0.476048	0.238024	−	0.971259i	$-0.423500\pi$
$691$	12.5138	0.476048	0.238024	+	0.971259i	$0.423500\pi$
$692$	0	0
$693$	−0.938533	−0.0356519
$694$	0	0
$695$	−14.6894	−0.557199
$696$	0	0
$697$	−37.8756	−1.43464
$698$	0	0
$699$	1.13880	0.0430732
$700$	0	0
$701$	4.41159	0.166623	0.0833117	−	0.996524i	$-0.473450\pi$
$701$	4.41159	0.166623	0.0833117	+	0.996524i	$0.473450\pi$
$702$	0	0
$703$	14.0219	0.528846
$704$	0	0
$705$	11.5777	0.436042
$706$	0	0
$707$	3.48513	0.131072
$708$	0	0
$709$	40.7819	1.53159	0.765797	−	0.643082i	$-0.222345\pi$
$709$	40.7819	1.53159	0.765797	+	0.643082i	$0.222345\pi$
$710$	0	0
$711$	7.49623	0.281131
$712$	0	0
$713$	27.1116	1.01534
$714$	0	0
$715$	−2.29297	−0.0857523
$716$	0	0
$717$	25.6652	0.958484
$718$	0	0
$719$	38.4170	1.43271	0.716357	−	0.697734i	$-0.245808\pi$
$719$	38.4170	1.43271	0.716357	+	0.697734i	$0.245808\pi$
$720$	0	0
$721$	−9.52680	−0.354797
$722$	0	0
$723$	−19.2172	−0.714695
$724$	0	0
$725$	−3.44102	−0.127796
$726$	0	0
$727$	1.48610	0.0551165	0.0275583	−	0.999620i	$-0.491227\pi$
$727$	1.48610	0.0551165	0.0275583	+	0.999620i	$0.491227\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.8756	1.40088
$732$	0	0
$733$	−7.74829	−0.286190	−0.143095	−	0.989709i	$-0.545705\pi$
$733$	−7.74829	−0.286190	−0.143095	+	0.989709i	$0.545705\pi$
$734$	0	0
$735$	7.94169	0.292934
$736$	0	0
$737$	−18.5362	−0.682790
$738$	0	0
$739$	−42.6392	−1.56851	−0.784254	−	0.620440i	$-0.786954\pi$
$739$	−42.6392	−1.56851	−0.784254	+	0.620440i	$0.786954\pi$
$740$	0	0
$741$	−1.62138	−0.0595627
$742$	0	0
$743$	−43.6516	−1.60142	−0.800711	−	0.599051i	$-0.795545\pi$
$743$	−43.6516	−1.60142	−0.800711	+	0.599051i	$0.795545\pi$
$744$	0	0
$745$	−4.24662	−0.155584
$746$	0	0
$747$	7.18759	0.262980
$748$	0	0
$749$	0.0474117	0.00173238
$750$	0	0
$751$	−43.3854	−1.58315	−0.791577	−	0.611069i	$-0.790740\pi$
$751$	−43.3854	−1.58315	−0.791577	+	0.611069i	$0.790740\pi$
$752$	0	0
$753$	6.46927	0.235753
$754$	0	0
$755$	−15.5881	−0.567310
$756$	0	0
$757$	22.6861	0.824539	0.412270	−	0.911062i	$-0.364736\pi$
$757$	22.6861	0.824539	0.412270	+	0.911062i	$0.364736\pi$
$758$	0	0
$759$	−6.12293	−0.222248
$760$	0	0
$761$	−31.0047	−1.12392	−0.561960	−	0.827164i	$-0.689953\pi$
$761$	−31.0047	−1.12392	−0.561960	+	0.827164i	$0.689953\pi$
$762$	0	0
$763$	−1.98398	−0.0718251
$764$	0	0
$765$	−5.22625	−0.188956
$766$	0	0
$767$	−11.3047	−0.408189
$768$	0	0
$769$	43.3767	1.56420	0.782102	−	0.623150i	$-0.214148\pi$
$769$	43.3767	1.56420	0.782102	+	0.623150i	$0.214148\pi$
$770$	0	0
$771$	22.1094	0.796249
$772$	0	0
$773$	−5.42745	−0.195212	−0.0976059	−	0.995225i	$-0.531118\pi$
$773$	−5.42745	−0.195212	−0.0976059	+	0.995225i	$0.531118\pi$
$774$	0	0
$775$	24.1470	0.867387
$776$	0	0
$777$	−6.62498	−0.237670
$778$	0	0
$779$	11.2753	0.403980
$780$	0	0
$781$	5.46380	0.195510
$782$	0	0
$783$	−0.965872	−0.0345175
$784$	0	0
$785$	−1.37683	−0.0491410
$786$	0	0
$787$	12.6824	0.452077	0.226039	−	0.974118i	$-0.427422\pi$
$787$	12.6824	0.452077	0.226039	+	0.974118i	$0.427422\pi$
$788$	0	0
$789$	−22.8914	−0.814957
$790$	0	0
$791$	−4.69462	−0.166921
$792$	0	0
$793$	1.90453	0.0676318
$794$	0	0
$795$	13.6896	0.485521
$796$	0	0
$797$	−24.7038	−0.875054	−0.437527	−	0.899205i	$-0.644146\pi$
$797$	−24.7038	−0.875054	−0.437527	+	0.899205i	$0.644146\pi$
$798$	0	0
$799$	42.0958	1.48924
$800$	0	0
$801$	−0.672715	−0.0237692
$802$	0	0
$803$	−7.89552	−0.278627
$804$	0	0
$805$	−2.94034	−0.103633
$806$	0	0
$807$	23.4776	0.826449
$808$	0	0
$809$	−6.81166	−0.239485	−0.119743	−	0.992805i	$-0.538207\pi$
$809$	−6.81166	−0.239485	−0.119743	+	0.992805i	$0.538207\pi$
$810$	0	0
$811$	2.81804	0.0989546	0.0494773	−	0.998775i	$-0.484244\pi$
$811$	2.81804	0.0989546	0.0494773	+	0.998775i	$0.484244\pi$
$812$	0	0
$813$	−25.5837	−0.897259
$814$	0	0
$815$	22.1584	0.776175
$816$	0	0
$817$	−11.2753	−0.394473
$818$	0	0
$819$	0.766057	0.0267682
$820$	0	0
$821$	−36.5756	−1.27650	−0.638249	−	0.769830i	$-0.720341\pi$
$821$	−36.5756	−1.27650	−0.638249	+	0.769830i	$0.720341\pi$
$822$	0	0
$823$	5.15309	0.179625	0.0898126	−	0.995959i	$-0.471373\pi$
$823$	5.15309	0.179625	0.0898126	+	0.995959i	$0.471373\pi$
$824$	0	0
$825$	−5.45341	−0.189863
$826$	0	0
$827$	35.6433	1.23944	0.619719	−	0.784824i	$-0.287247\pi$
$827$	35.6433	1.23944	0.619719	+	0.784824i	$0.287247\pi$
$828$	0	0
$829$	42.3283	1.47012	0.735061	−	0.678001i	$-0.237153\pi$
$829$	42.3283	1.47012	0.735061	+	0.678001i	$0.237153\pi$
$830$	0	0
$831$	−13.6692	−0.474178
$832$	0	0
$833$	28.8754	1.00047
$834$	0	0
$835$	−24.2305	−0.838531
$836$	0	0
$837$	6.77791	0.234279
$838$	0	0
$839$	48.7768	1.68396	0.841981	−	0.539507i	$-0.181389\pi$
$839$	48.7768	1.68396	0.841981	+	0.539507i	$0.181389\pi$
$840$	0	0
$841$	−28.0671	−0.967831
$842$	0	0
$843$	−21.1116	−0.727124
$844$	0	0
$845$	−13.7143	−0.471785
$846$	0	0
$847$	−5.30774	−0.182376
$848$	0	0
$849$	−17.0479	−0.585082
$850$	0	0
$851$	−43.2210	−1.48160
$852$	0	0
$853$	−40.3965	−1.38315	−0.691576	−	0.722304i	$-0.743083\pi$
$853$	−40.3965	−1.38315	−0.691576	+	0.722304i	$0.743083\pi$
$854$	0	0
$855$	1.55582	0.0532079
$856$	0	0
$857$	−25.2977	−0.864153	−0.432076	−	0.901837i	$-0.642219\pi$
$857$	−25.2977	−0.864153	−0.432076	+	0.901837i	$0.642219\pi$
$858$	0	0
$859$	−35.1412	−1.19900	−0.599502	−	0.800373i	$-0.704635\pi$
$859$	−35.1412	−1.19900	−0.599502	+	0.800373i	$0.704635\pi$
$860$	0	0
$861$	−5.32729	−0.181553
$862$	0	0
$863$	−29.4193	−1.00144	−0.500722	−	0.865608i	$-0.666932\pi$
$863$	−29.4193	−1.00144	−0.500722	+	0.865608i	$0.666932\pi$
$864$	0	0
$865$	−12.1539	−0.413245
$866$	0	0
$867$	−2.00228	−0.0680011
$868$	0	0
$869$	−11.4747	−0.389254
$870$	0	0
$871$	15.1298	0.512653
$872$	0	0
$873$	4.71832	0.159691
$874$	0	0
$875$	−6.29424	−0.212784
$876$	0	0
$877$	3.89312	0.131461	0.0657307	−	0.997837i	$-0.479062\pi$
$877$	3.89312	0.131461	0.0657307	+	0.997837i	$0.479062\pi$
$878$	0	0
$879$	16.1047	0.543197
$880$	0	0
$881$	−21.6049	−0.727887	−0.363943	−	0.931421i	$-0.618570\pi$
$881$	−21.6049	−0.727887	−0.363943	+	0.931421i	$0.618570\pi$
$882$	0	0
$883$	−23.0974	−0.777290	−0.388645	−	0.921387i	$-0.627057\pi$
$883$	−23.0974	−0.777290	−0.388645	+	0.921387i	$0.627057\pi$
$884$	0	0
$885$	10.8476	0.364639
$886$	0	0
$887$	56.1260	1.88453	0.942263	−	0.334873i	$-0.108693\pi$
$887$	56.1260	1.88453	0.942263	+	0.334873i	$0.108693\pi$
$888$	0	0
$889$	−12.8149	−0.429796
$890$	0	0
$891$	−1.53073	−0.0512815
$892$	0	0
$893$	−12.5316	−0.419355
$894$	0	0
$895$	24.1094	0.805887
$896$	0	0
$897$	4.99772	0.166869
$898$	0	0
$899$	−6.54659	−0.218341
$900$	0	0
$901$	49.7745	1.65823
$902$	0	0
$903$	5.32729	0.177281
$904$	0	0
$905$	11.4844	0.381754
$906$	0	0
$907$	26.2683	0.872223	0.436112	−	0.899893i	$-0.356355\pi$
$907$	26.2683	0.872223	0.436112	+	0.899893i	$0.356355\pi$
$908$	0	0
$909$	5.68419	0.188533
$910$	0	0
$911$	22.8560	0.757251	0.378626	−	0.925550i	$-0.376397\pi$
$911$	22.8560	0.757251	0.378626	+	0.925550i	$0.376397\pi$
$912$	0	0
$913$	−11.0023	−0.364122
$914$	0	0
$915$	−1.82752	−0.0604161
$916$	0	0
$917$	4.73370	0.156321
$918$	0	0
$919$	0.319035	0.0105240	0.00526200	−	0.999986i	$-0.498325\pi$
$919$	0.319035	0.0105240	0.00526200	+	0.999986i	$0.498325\pi$
$920$	0	0
$921$	6.91858	0.227975
$922$	0	0
$923$	−4.45971	−0.146793
$924$	0	0
$925$	−38.4949	−1.26570
$926$	0	0
$927$	−15.5381	−0.510338
$928$	0	0
$929$	57.3735	1.88236	0.941182	−	0.337900i	$-0.109716\pi$
$929$	57.3735	1.88236	0.941182	+	0.337900i	$0.109716\pi$
$930$	0	0
$931$	−8.59602	−0.281723
$932$	0	0
$933$	−20.4389	−0.669140
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	39.8100	1.30054	0.650268	−	0.759705i	$-0.274657\pi$
$937$	39.8100	1.30054	0.650268	+	0.759705i	$0.274657\pi$
$938$	0	0
$939$	−11.6241	−0.379337
$940$	0	0
$941$	45.2038	1.47360	0.736801	−	0.676110i	$-0.236336\pi$
$941$	45.2038	1.47360	0.736801	+	0.676110i	$0.236336\pi$
$942$	0	0
$943$	−34.7549	−1.13178
$944$	0	0
$945$	−0.735084	−0.0239123
$946$	0	0
$947$	43.1772	1.40307	0.701535	−	0.712635i	$-0.252498\pi$
$947$	43.1772	1.40307	0.701535	+	0.712635i	$0.252498\pi$
$948$	0	0
$949$	6.44455	0.209199
$950$	0	0
$951$	21.5368	0.698379
$952$	0	0
$953$	15.1594	0.491060	0.245530	−	0.969389i	$-0.421038\pi$
$953$	15.1594	0.491060	0.245530	+	0.969389i	$0.421038\pi$
$954$	0	0
$955$	−2.76669	−0.0895280
$956$	0	0
$957$	1.47849	0.0477929
$958$	0	0
$959$	4.54978	0.146920
$960$	0	0
$961$	14.9401	0.481938
$962$	0	0
$963$	0.0773278	0.00249185
$964$	0	0
$965$	2.43712	0.0784537
$966$	0	0
$967$	−16.1170	−0.518287	−0.259143	−	0.965839i	$-0.583440\pi$
$967$	−16.1170	−0.518287	−0.259143	+	0.965839i	$0.583440\pi$
$968$	0	0
$969$	5.65685	0.181724
$970$	0	0
$971$	−4.15965	−0.133489	−0.0667447	−	0.997770i	$-0.521261\pi$
$971$	−4.15965	−0.133489	−0.0667447	+	0.997770i	$0.521261\pi$
$972$	0	0
$973$	−7.51217	−0.240829
$974$	0	0
$975$	4.45123	0.142553
$976$	0	0
$977$	41.2552	1.31987	0.659935	−	0.751323i	$-0.270584\pi$
$977$	41.2552	1.31987	0.659935	+	0.751323i	$0.270584\pi$
$978$	0	0
$979$	1.02975	0.0329109
$980$	0	0
$981$	−3.23585	−0.103313
$982$	0	0
$983$	−32.0257	−1.02146	−0.510730	−	0.859741i	$-0.670625\pi$
$983$	−32.0257	−1.02146	−0.510730	+	0.859741i	$0.670625\pi$
$984$	0	0
$985$	−31.6543	−1.00859
$986$	0	0
$987$	5.92087	0.188463
$988$	0	0
$989$	34.7549	1.10514
$990$	0	0
$991$	9.94041	0.315768	0.157884	−	0.987458i	$-0.449533\pi$
$991$	9.94041	0.315768	0.157884	+	0.987458i	$0.449533\pi$
$992$	0	0
$993$	13.8435	0.439311
$994$	0	0
$995$	12.2201	0.387403
$996$	0	0
$997$	−18.0161	−0.570574	−0.285287	−	0.958442i	$-0.592089\pi$
$997$	−18.0161	−0.570574	−0.285287	+	0.958442i	$0.592089\pi$
$998$	0	0
$999$	−10.8052	−0.341863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.j.1.3		4
3.2	odd	2		9216.2.a.ba.1.2		4
4.3	odd	2		3072.2.a.s.1.3		4
8.3	odd	2		3072.2.a.m.1.2		4
8.5	even	2		3072.2.a.p.1.2		4
12.11	even	2		9216.2.a.bm.1.2		4
16.3	odd	4		3072.2.d.e.1537.6		8
16.5	even	4		3072.2.d.j.1537.7		8
16.11	odd	4		3072.2.d.e.1537.3		8
16.13	even	4		3072.2.d.j.1537.2		8
24.5	odd	2		9216.2.a.z.1.3		4
24.11	even	2		9216.2.a.bl.1.3		4
32.3	odd	8		1536.2.j.j.1153.3	yes	8
32.5	even	8		1536.2.j.f.385.4	yes	8
32.11	odd	8		1536.2.j.j.385.3	yes	8
32.13	even	8		1536.2.j.f.1153.4	yes	8
32.19	odd	8		1536.2.j.e.1153.2	yes	8
32.21	even	8		1536.2.j.i.385.1	yes	8
32.27	odd	8		1536.2.j.e.385.2	✓	8
32.29	even	8		1536.2.j.i.1153.1	yes	8
96.5	odd	8		4608.2.k.bj.3457.1		8
96.11	even	8		4608.2.k.be.3457.4		8
96.29	odd	8		4608.2.k.bc.1153.4		8
96.35	even	8		4608.2.k.be.1153.4		8
96.53	odd	8		4608.2.k.bc.3457.4		8
96.59	even	8		4608.2.k.bh.3457.1		8
96.77	odd	8		4608.2.k.bj.1153.1		8
96.83	even	8		4608.2.k.bh.1153.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.2	✓	8	32.27	odd	8
1536.2.j.e.1153.2	yes	8	32.19	odd	8
1536.2.j.f.385.4	yes	8	32.5	even	8
1536.2.j.f.1153.4	yes	8	32.13	even	8
1536.2.j.i.385.1	yes	8	32.21	even	8
1536.2.j.i.1153.1	yes	8	32.29	even	8
1536.2.j.j.385.3	yes	8	32.11	odd	8
1536.2.j.j.1153.3	yes	8	32.3	odd	8
3072.2.a.j.1.3		4	1.1	even	1	trivial
3072.2.a.m.1.2		4	8.3	odd	2
3072.2.a.p.1.2		4	8.5	even	2
3072.2.a.s.1.3		4	4.3	odd	2
3072.2.d.e.1537.3		8	16.11	odd	4
3072.2.d.e.1537.6		8	16.3	odd	4
3072.2.d.j.1537.2		8	16.13	even	4
3072.2.d.j.1537.7		8	16.5	even	4
4608.2.k.bc.1153.4		8	96.29	odd	8
4608.2.k.bc.3457.4		8	96.53	odd	8
4608.2.k.be.1153.4		8	96.35	even	8
4608.2.k.be.3457.4		8	96.11	even	8
4608.2.k.bh.1153.1		8	96.83	even	8
4608.2.k.bh.3457.1		8	96.59	even	8
4608.2.k.bj.1153.1		8	96.77	odd	8
4608.2.k.bj.3457.1		8	96.5	odd	8
9216.2.a.z.1.3		4	24.5	odd	2
9216.2.a.ba.1.2		4	3.2	odd	2
9216.2.a.bl.1.3		4	24.11	even	2
9216.2.a.bm.1.2		4	12.11	even	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.19891 q^{5} +0.613126 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.19891 q^{5} +0.613126 q^{7} +1.00000 q^{9} -1.53073 q^{11} +1.24943 q^{13} -1.19891 q^{15} -4.35916 q^{17} +1.29769 q^{19} -0.613126 q^{21} -4.00000 q^{23} -3.56261 q^{25} -1.00000 q^{27} +0.965872 q^{29} -6.77791 q^{31} +1.53073 q^{33} +0.735084 q^{35} +10.8052 q^{37} -1.24943 q^{39} +8.68873 q^{41} -8.68873 q^{43} +1.19891 q^{45} -9.65685 q^{47} -6.62408 q^{49} +4.35916 q^{51} -11.4184 q^{53} -1.83522 q^{55} -1.29769 q^{57} -9.04789 q^{59} +1.52432 q^{61} +0.613126 q^{63} +1.49796 q^{65} +12.1094 q^{67} +4.00000 q^{69} -3.56940 q^{71} +5.15800 q^{73} +3.56261 q^{75} -0.938533 q^{77} +7.49623 q^{79} +1.00000 q^{81} +7.18759 q^{83} -5.22625 q^{85} -0.965872 q^{87} -0.672715 q^{89} +0.766057 q^{91} +6.77791 q^{93} +1.55582 q^{95} +4.71832 q^{97} -1.53073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{7} + 4 q^{9} + 8 q^{13} + 8 q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} - 8 q^{31} + 16 q^{35} + 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{59} + 24 q^{61} - 8 q^{63} - 8 q^{65} - 16 q^{67} + 16 q^{69} - 16 q^{71} - 8 q^{73} - 4 q^{75} - 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} - 16 q^{91} + 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 4 * q^9 + 8 * q^13 + 8 * q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 - 8 * q^31 + 16 * q^35 + 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 + 16 * q^59 + 24 * q^61 - 8 * q^63 - 8 * q^65 - 16 * q^67 + 16 * q^69 - 16 * q^71 - 8 * q^73 - 4 * q^75 - 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 - 16 * q^91 + 8 * q^93 - 32 * q^95 - 16 * q^97

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