LMFDB - Embedded newform 3072.2.d.j.1537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1536)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.2
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.j.1537.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.19891i q^{5} -0.613126 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.19891i q^{5} -0.613126 q^{7} -1.00000 q^{9} +1.53073i q^{11} +1.24943i q^{13} -1.19891 q^{15} -4.35916 q^{17} +1.29769i q^{19} +0.613126i q^{21} +4.00000 q^{23} +3.56261 q^{25} +1.00000i q^{27} +0.965872i q^{29} -6.77791 q^{31} +1.53073 q^{33} +0.735084i q^{35} -10.8052i q^{37} +1.24943 q^{39} -8.68873 q^{41} +8.68873i q^{43} +1.19891i q^{45} -9.65685 q^{47} -6.62408 q^{49} +4.35916i q^{51} +11.4184i q^{53} +1.83522 q^{55} +1.29769 q^{57} +9.04789i q^{59} +1.52432i q^{61} +0.613126 q^{63} +1.49796 q^{65} +12.1094i q^{67} -4.00000i q^{69} +3.56940 q^{71} -5.15800 q^{73} -3.56261i q^{75} -0.938533i q^{77} +7.49623 q^{79} +1.00000 q^{81} +7.18759i q^{83} +5.22625i q^{85} +0.965872 q^{87} +0.672715 q^{89} -0.766057i q^{91} +6.77791i q^{93} +1.55582 q^{95} +4.71832 q^{97} -1.53073i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q + 16 * q^7 - 8 * q^9	$ 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 8 * q^9 + 32 * q^23 - 8 * q^25 - 16 * q^31 + 16 * q^39 - 32 * q^47 + 8 * q^49 + 32 * q^55 - 16 * q^63 - 16 * q^65 + 32 * q^71 + 16 * q^73 - 48 * q^79 + 8 * q^81 + 16 * q^89 - 64 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 1.19891i	− 0.536170i	−0.963395	−	0.268085i	$-0.913609\pi$
$5$	− 1.19891i	− 0.536170i	0.963395	−	0.268085i	$-0.0863908\pi$
$6$	0	0
$7$	−0.613126	−0.231740	−0.115870	−	0.993264i	$-0.536966\pi$
$7$	−0.613126	−0.231740	−0.115870	+	0.993264i	$0.536966\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.53073i	0.461534i	0.973009	+	0.230767i	$0.0741234\pi$
$11$	1.53073i	0.461534i	−0.973009	+	0.230767i	$0.925877\pi$
$12$	0	0
$13$	1.24943i	0.346529i	0.984875	+	0.173265i	$0.0554316\pi$
$13$	1.24943i	0.346529i	−0.984875	+	0.173265i	$0.944568\pi$
$14$	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
$19$	1.29769i	0.297711i	0.988859	+	0.148856i	$0.0475590\pi$
$19$	1.29769i	0.297711i	−0.988859	+	0.148856i	$0.952441\pi$
$20$	0	0
$21$	0.613126i	0.133795i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.56261	0.712522
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0.965872i	0.179358i	0.995971	+	0.0896790i	$0.0285841\pi$
$29$	0.965872i	0.179358i	−0.995971	+	0.0896790i	$0.971416\pi$
$30$	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
$35$	0.735084i	0.124252i
$36$	0	0
$37$	− 10.8052i	− 1.77637i	−0.459484	−	0.888186i	$-0.651966\pi$
$37$	− 10.8052i	− 1.77637i	0.459484	−	0.888186i	$-0.348034\pi$
$38$	0	0
$39$	1.24943	0.200069
$40$	0	0
$41$	−8.68873	−1.35695	−0.678476	−	0.734623i	$-0.737359\pi$
$41$	−8.68873	−1.35695	−0.678476	+	0.734623i	$0.737359\pi$
$42$	0	0
$43$	8.68873i	1.32502i	0.749054	+	0.662509i	$0.230509\pi$
$43$	8.68873i	1.32502i	−0.749054	+	0.662509i	$0.769491\pi$
$44$	0	0
$45$	1.19891i	0.178723i
$46$	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
$51$	4.35916i	0.610405i
$52$	0	0
$53$	11.4184i	1.56843i	0.620486	+	0.784217i	$0.286935\pi$
$53$	11.4184i	1.56843i	−0.620486	+	0.784217i	$0.713065\pi$
$54$	0	0
$55$	1.83522	0.247460
$56$	0	0
$57$	1.29769	0.171884
$58$	0	0
$59$	9.04789i	1.17794i	0.808157	+	0.588968i	$0.200465\pi$
$59$	9.04789i	1.17794i	−0.808157	+	0.588968i	$0.799535\pi$
$60$	0	0
$61$	1.52432i	0.195169i	0.995227	+	0.0975845i	$0.0311116\pi$
$61$	1.52432i	0.195169i	−0.995227	+	0.0975845i	$0.968888\pi$
$62$	0	0
$63$	0.613126	0.0772466
$64$	0	0
$65$	1.49796	0.185799
$66$	0	0
$67$	12.1094i	1.47939i	0.672940	+	0.739697i	$0.265031\pi$
$67$	12.1094i	1.47939i	−0.672940	+	0.739697i	$0.734969\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	3.56940	0.423610	0.211805	−	0.977312i	$-0.432066\pi$
$71$	3.56940	0.423610	0.211805	+	0.977312i	$0.432066\pi$
$72$	0	0
$73$	−5.15800	−0.603698	−0.301849	−	0.953356i	$-0.597604\pi$
$73$	−5.15800	−0.603698	−0.301849	+	0.953356i	$0.597604\pi$
$74$	0	0
$75$	− 3.56261i	− 0.411375i
$76$	0	0
$77$	− 0.938533i	− 0.106956i
$78$	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
$83$	7.18759i	0.788940i	0.918909	+	0.394470i	$0.129072\pi$
$83$	7.18759i	0.788940i	−0.918909	+	0.394470i	$0.870928\pi$
$84$	0	0
$85$	5.22625i	0.566867i
$86$	0	0
$87$	0.965872	0.103552
$88$	0	0
$89$	0.672715	0.0713076	0.0356538	−	0.999364i	$-0.488649\pi$
$89$	0.672715	0.0713076	0.0356538	+	0.999364i	$0.488649\pi$
$90$	0	0
$91$	− 0.766057i	− 0.0803046i
$92$	0	0
$93$	6.77791i	0.702837i
$94$	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
$99$	− 1.53073i	− 0.153845i
$100$	0	0
$101$	− 5.68419i	− 0.565598i	−0.959179	−	0.282799i	$-0.908737\pi$
$101$	− 5.68419i	− 0.565598i	0.959179	−	0.282799i	$-0.0912630\pi$
$102$	0	0
$103$	15.5381	1.53101	0.765506	−	0.643428i	$-0.222489\pi$
$103$	15.5381	1.53101	0.765506	+	0.643428i	$0.222489\pi$
$104$	0	0
$105$	0.735084	0.0717369
$106$	0	0
$107$	− 0.0773278i	− 0.00747556i	−0.999993	−	0.00373778i	$-0.998810\pi$
$107$	− 0.0773278i	− 0.00747556i	0.999993	−	0.00373778i	$-0.00118977\pi$
$108$	0	0
$109$	− 3.23585i	− 0.309938i	−0.987919	−	0.154969i	$-0.950472\pi$
$109$	− 3.23585i	− 0.309938i	0.987919	−	0.154969i	$-0.0495278\pi$
$110$	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
$115$	− 4.79565i	− 0.447197i
$116$	0	0
$117$	− 1.24943i	− 0.115510i
$118$	0	0
$119$	2.67271	0.245007
$120$	0	0
$121$	8.65685	0.786987
$122$	0	0
$123$	8.68873i	0.783436i
$124$	0	0
$125$	− 10.2658i	− 0.918203i
$126$	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
$131$	7.72061i	0.674552i	0.941406	+	0.337276i	$0.109506\pi$
$131$	7.72061i	0.674552i	−0.941406	+	0.337276i	$0.890494\pi$
$132$	0	0
$133$	− 0.795649i	− 0.0689916i
$134$	0	0
$135$	1.19891	0.103186
$136$	0	0
$137$	−7.42063	−0.633987	−0.316994	−	0.948428i	$-0.602673\pi$
$137$	−7.42063	−0.633987	−0.316994	+	0.948428i	$0.602673\pi$
$138$	0	0
$139$	12.2522i	1.03922i	0.854403	+	0.519611i	$0.173923\pi$
$139$	12.2522i	1.03922i	−0.854403	+	0.519611i	$0.826077\pi$
$140$	0	0
$141$	9.65685i	0.813254i
$142$	0	0
$143$	−1.91254	−0.159935
$144$	0	0
$145$	1.15800	0.0961663
$146$	0	0
$147$	6.62408i	0.546345i
$148$	0	0
$149$	3.54206i	0.290177i	0.989419	+	0.145088i	$0.0463466\pi$
$149$	3.54206i	0.290177i	−0.989419	+	0.145088i	$0.953653\pi$
$150$	0	0
$151$	13.0019	1.05808	0.529039	−	0.848598i	$-0.322553\pi$
$151$	13.0019	1.05808	0.529039	+	0.848598i	$0.322553\pi$
$152$	0	0
$153$	4.35916	0.352417
$154$	0	0
$155$	8.12612i	0.652706i
$156$	0	0
$157$	− 1.14840i	− 0.0916519i	−0.998949	−	0.0458260i	$-0.985408\pi$
$157$	− 1.14840i	− 0.0916519i	0.998949	−	0.0458260i	$-0.0145920\pi$
$158$	0	0
$159$	11.4184	0.905536
$160$	0	0
$161$	−2.45250	−0.193284
$162$	0	0
$163$	18.4821i	1.44763i	0.689994	+	0.723815i	$0.257613\pi$
$163$	18.4821i	1.44763i	−0.689994	+	0.723815i	$0.742387\pi$
$164$	0	0
$165$	− 1.83522i	− 0.142871i
$166$	0	0
$167$	20.2104	1.56393	0.781964	−	0.623324i	$-0.214218\pi$
$167$	20.2104	1.56393	0.781964	+	0.623324i	$0.214218\pi$
$168$	0	0
$169$	11.4389	0.879917
$170$	0	0
$171$	− 1.29769i	− 0.0992371i
$172$	0	0
$173$	− 10.1374i	− 0.770736i	−0.922763	−	0.385368i	$-0.874075\pi$
$173$	− 10.1374i	− 0.770736i	0.922763	−	0.385368i	$-0.125925\pi$
$174$	0	0
$175$	−2.18433	−0.165120
$176$	0	0
$177$	9.04789	0.680081
$178$	0	0
$179$	20.1094i	1.50304i	0.659708	+	0.751522i	$0.270680\pi$
$179$	20.1094i	1.50304i	−0.659708	+	0.751522i	$0.729320\pi$
$180$	0	0
$181$	− 9.57900i	− 0.712001i	−0.934486	−	0.356001i	$-0.884140\pi$
$181$	− 9.57900i	− 0.712001i	0.934486	−	0.356001i	$-0.115860\pi$
$182$	0	0
$183$	1.52432	0.112681
$184$	0	0
$185$	−12.9545	−0.952437
$186$	0	0
$187$	− 6.67271i	− 0.487957i
$188$	0	0
$189$	− 0.613126i	− 0.0445983i
$190$	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
$195$	− 1.49796i	− 0.107271i
$196$	0	0
$197$	26.4025i	1.88110i	0.339653	+	0.940551i	$0.389690\pi$
$197$	26.4025i	1.88110i	−0.339653	+	0.940551i	$0.610310\pi$
$198$	0	0
$199$	−10.1927	−0.722538	−0.361269	−	0.932462i	$-0.617656\pi$
$199$	−10.1927	−0.722538	−0.361269	+	0.932462i	$0.617656\pi$
$200$	0	0
$201$	12.1094	0.854128
$202$	0	0
$203$	− 0.592201i	− 0.0415644i
$204$	0	0
$205$	10.4170i	0.727557i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−1.98642	−0.137404
$210$	0	0
$211$	− 7.06147i	− 0.486131i	−0.970010	−	0.243066i	$-0.921847\pi$
$211$	− 7.06147i	− 0.486131i	0.970010	−	0.243066i	$-0.0781531\pi$
$212$	0	0
$213$	− 3.56940i	− 0.244571i
$214$	0	0
$215$	10.4170	0.710435
$216$	0	0
$217$	4.15571	0.282108
$218$	0	0
$219$	5.15800i	0.348545i
$220$	0	0
$221$	− 5.44646i	− 0.366369i
$222$	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
$227$	− 15.5945i	− 1.03504i	−0.855670	−	0.517521i	$-0.826855\pi$
$227$	− 15.5945i	− 1.03504i	0.855670	−	0.517521i	$-0.173145\pi$
$228$	0	0
$229$	6.37465i	0.421249i	0.977567	+	0.210624i	$0.0675497\pi$
$229$	6.37465i	0.421249i	−0.977567	+	0.210624i	$0.932450\pi$
$230$	0	0
$231$	−0.938533	−0.0617509
$232$	0	0
$233$	1.13880	0.0746050	0.0373025	−	0.999304i	$-0.488123\pi$
$233$	1.13880	0.0746050	0.0373025	+	0.999304i	$0.488123\pi$
$234$	0	0
$235$	11.5777i	0.755247i
$236$	0	0
$237$	− 7.49623i	− 0.486933i
$238$	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
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	7.94169i	0.507376i
$246$	0	0
$247$	−1.62138	−0.103166
$248$	0	0
$249$	7.18759	0.455495
$250$	0	0
$251$	6.46927i	0.408336i	0.978936	+	0.204168i	$0.0654490\pi$
$251$	6.46927i	0.408336i	−0.978936	+	0.204168i	$0.934551\pi$
$252$	0	0
$253$	6.12293i	0.384946i
$254$	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
$259$	6.62498i	0.411656i
$260$	0	0
$261$	− 0.965872i	− 0.0597860i
$262$	0	0
$263$	−22.8914	−1.41155	−0.705773	−	0.708438i	$-0.749400\pi$
$263$	−22.8914	−1.41155	−0.705773	+	0.708438i	$0.749400\pi$
$264$	0	0
$265$	13.6896	0.840947
$266$	0	0
$267$	− 0.672715i	− 0.0411695i
$268$	0	0
$269$	− 23.4776i	− 1.43145i	−0.698381	−	0.715726i	$-0.746096\pi$
$269$	− 23.4776i	− 1.43145i	0.698381	−	0.715726i	$-0.253904\pi$
$270$	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
$275$	5.45341i	0.328853i
$276$	0	0
$277$	− 13.6692i	− 0.821300i	−0.911793	−	0.410650i	$-0.865302\pi$
$277$	− 13.6692i	− 0.821300i	0.911793	−	0.410650i	$-0.134698\pi$
$278$	0	0
$279$	6.77791	0.405783
$280$	0	0
$281$	−21.1116	−1.25941	−0.629707	−	0.776832i	$-0.716825\pi$
$281$	−21.1116	−1.25941	−0.629707	+	0.776832i	$0.716825\pi$
$282$	0	0
$283$	− 17.0479i	− 1.01339i	−0.862125	−	0.506696i	$-0.830867\pi$
$283$	− 17.0479i	− 1.01339i	0.862125	−	0.506696i	$-0.169133\pi$
$284$	0	0
$285$	− 1.55582i	− 0.0921589i
$286$	0	0
$287$	5.32729	0.314460
$288$	0	0
$289$	2.00228	0.117781
$290$	0	0
$291$	− 4.71832i	− 0.276593i
$292$	0	0
$293$	16.1047i	0.940845i	0.882441	+	0.470422i	$0.155898\pi$
$293$	16.1047i	0.940845i	−0.882441	+	0.470422i	$0.844102\pi$
$294$	0	0
$295$	10.8476	0.631573
$296$	0	0
$297$	−1.53073	−0.0888222
$298$	0	0
$299$	4.99772i	0.289025i
$300$	0	0
$301$	− 5.32729i	− 0.307060i
$302$	0	0
$303$	−5.68419	−0.326548
$304$	0	0
$305$	1.82752	0.104644
$306$	0	0
$307$	− 6.91858i	− 0.394864i	−0.980317	−	0.197432i	$-0.936740\pi$
$307$	− 6.91858i	− 0.394864i	0.980317	−	0.197432i	$-0.0632602\pi$
$308$	0	0
$309$	− 15.5381i	− 0.883931i
$310$	0	0
$311$	−20.4389	−1.15899	−0.579493	−	0.814977i	$-0.696749\pi$
$311$	−20.4389	−1.15899	−0.579493	+	0.814977i	$0.696749\pi$
$312$	0	0
$313$	−11.6241	−0.657032	−0.328516	−	0.944498i	$-0.606548\pi$
$313$	−11.6241	−0.657032	−0.328516	+	0.944498i	$0.606548\pi$
$314$	0	0
$315$	− 0.735084i	− 0.0414173i
$316$	0	0
$317$	− 21.5368i	− 1.20963i	−0.796367	−	0.604814i	$-0.793248\pi$
$317$	− 21.5368i	− 1.20963i	0.796367	−	0.604814i	$-0.206752\pi$
$318$	0	0
$319$	−1.47849	−0.0827797
$320$	0	0
$321$	−0.0773278	−0.00431601
$322$	0	0
$323$	− 5.65685i	− 0.314756i
$324$	0	0
$325$	4.45123i	0.246910i
$326$	0	0
$327$	−3.23585	−0.178943
$328$	0	0
$329$	5.92087	0.326428
$330$	0	0
$331$	13.8435i	0.760910i	0.924799	+	0.380455i	$0.124233\pi$
$331$	13.8435i	0.760910i	−0.924799	+	0.380455i	$0.875767\pi$
$332$	0	0
$333$	10.8052i	0.592124i
$334$	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
$339$	7.65685i	0.415863i
$340$	0	0
$341$	− 10.3752i	− 0.561847i
$342$	0	0
$343$	8.35327	0.451034
$344$	0	0
$345$	−4.79565	−0.258189
$346$	0	0
$347$	10.6560i	0.572041i	0.958223	+	0.286021i	$0.0923326\pi$
$347$	10.6560i	0.572041i	−0.958223	+	0.286021i	$0.907667\pi$
$348$	0	0
$349$	15.6009i	0.835097i	0.908655	+	0.417548i	$0.137111\pi$
$349$	15.6009i	0.835097i	−0.908655	+	0.417548i	$0.862889\pi$
$350$	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
$355$	− 4.27939i	− 0.227127i
$356$	0	0
$357$	− 2.67271i	− 0.141455i
$358$	0	0
$359$	3.41474	0.180223	0.0901116	−	0.995932i	$-0.471278\pi$
$359$	3.41474	0.180223	0.0901116	+	0.995932i	$0.471278\pi$
$360$	0	0
$361$	17.3160	0.911368
$362$	0	0
$363$	− 8.65685i	− 0.454367i
$364$	0	0
$365$	6.18399i	0.323685i
$366$	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
$371$	− 7.00090i	− 0.363469i
$372$	0	0
$373$	− 14.9610i	− 0.774649i	−0.921943	−	0.387325i	$-0.873399\pi$
$373$	− 14.9610i	− 0.774649i	0.921943	−	0.387325i	$-0.126601\pi$
$374$	0	0
$375$	−10.2658	−0.530125
$376$	0	0
$377$	−1.20679	−0.0621528
$378$	0	0
$379$	12.9681i	0.666128i	0.942904	+	0.333064i	$0.108082\pi$
$379$	12.9681i	0.666128i	−0.942904	+	0.333064i	$0.891918\pi$
$380$	0	0
$381$	20.9008i	1.07078i
$382$	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
$387$	− 8.68873i	− 0.441673i
$388$	0	0
$389$	28.1239i	1.42594i	0.701196	+	0.712968i	$0.252650\pi$
$389$	28.1239i	1.42594i	−0.701196	+	0.712968i	$0.747350\pi$
$390$	0	0
$391$	−17.4366	−0.881809
$392$	0	0
$393$	7.72061	0.389453
$394$	0	0
$395$	− 8.98733i	− 0.452201i
$396$	0	0
$397$	32.0862i	1.61036i	0.593031	+	0.805180i	$0.297931\pi$
$397$	32.0862i	1.61036i	−0.593031	+	0.805180i	$0.702069\pi$
$398$	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
$403$	− 8.46852i	− 0.421847i
$404$	0	0
$405$	− 1.19891i	− 0.0595744i
$406$	0	0
$407$	16.5400	0.819855
$408$	0	0
$409$	−10.5298	−0.520667	−0.260333	−	0.965519i	$-0.583832\pi$
$409$	−10.5298	−0.520667	−0.260333	+	0.965519i	$0.583832\pi$
$410$	0	0
$411$	7.42063i	0.366033i
$412$	0	0
$413$	− 5.54750i	− 0.272974i
$414$	0	0
$415$	8.61729	0.423006
$416$	0	0
$417$	12.2522	0.599995
$418$	0	0
$419$	20.7853i	1.01543i	0.861526	+	0.507713i	$0.169509\pi$
$419$	20.7853i	1.01543i	−0.861526	+	0.507713i	$0.830491\pi$
$420$	0	0
$421$	14.0179i	0.683192i	0.939847	+	0.341596i	$0.110967\pi$
$421$	14.0179i	0.683192i	−0.939847	+	0.341596i	$0.889033\pi$
$422$	0	0
$423$	9.65685	0.469532
$424$	0	0
$425$	−15.5300	−0.753315
$426$	0	0
$427$	− 0.934599i	− 0.0452284i
$428$	0	0
$429$	1.91254i	0.0923385i
$430$	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
$435$	− 1.15800i	− 0.0555217i
$436$	0	0
$437$	5.19077i	0.248308i
$438$	0	0
$439$	−35.7840	−1.70787	−0.853937	−	0.520376i	$-0.825792\pi$
$439$	−35.7840	−1.70787	−0.853937	+	0.520376i	$0.825792\pi$
$440$	0	0
$441$	6.62408	0.315432
$442$	0	0
$443$	− 33.0311i	− 1.56936i	−0.619903	−	0.784678i	$-0.712828\pi$
$443$	− 33.0311i	− 1.56936i	0.619903	−	0.784678i	$-0.287172\pi$
$444$	0	0
$445$	− 0.806526i	− 0.0382330i
$446$	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
$451$	− 13.3001i	− 0.626279i
$452$	0	0
$453$	− 13.0019i	− 0.610882i
$454$	0	0
$455$	−0.918436	−0.0430569
$456$	0	0
$457$	−14.3096	−0.669376	−0.334688	−	0.942329i	$-0.608631\pi$
$457$	−14.3096	−0.669376	−0.334688	+	0.942329i	$0.608631\pi$
$458$	0	0
$459$	− 4.35916i	− 0.203468i
$460$	0	0
$461$	− 23.4923i	− 1.09414i	−0.837086	−	0.547072i	$-0.815742\pi$
$461$	− 23.4923i	− 1.09414i	0.837086	−	0.547072i	$-0.184258\pi$
$462$	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
$467$	− 23.4398i	− 1.08467i	−0.840164	−	0.542333i	$-0.817541\pi$
$467$	− 23.4398i	− 1.08467i	0.840164	−	0.542333i	$-0.182459\pi$
$468$	0	0
$469$	− 7.42456i	− 0.342834i
$470$	0	0
$471$	−1.14840	−0.0529153
$472$	0	0
$473$	−13.3001	−0.611541
$474$	0	0
$475$	4.62317i	0.212126i
$476$	0	0
$477$	− 11.4184i	− 0.522812i
$478$	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
$483$	2.45250i	0.111593i
$484$	0	0
$485$	− 5.65685i	− 0.256865i
$486$	0	0
$487$	10.0498	0.455399	0.227699	−	0.973732i	$-0.426880\pi$
$487$	10.0498	0.455399	0.227699	+	0.973732i	$0.426880\pi$
$488$	0	0
$489$	18.4821	0.835789
$490$	0	0
$491$	− 36.6848i	− 1.65556i	−0.561052	−	0.827781i	$-0.689603\pi$
$491$	− 36.6848i	− 1.65556i	0.561052	−	0.827781i	$-0.310397\pi$
$492$	0	0
$493$	− 4.21039i	− 0.189626i
$494$	0	0
$495$	−1.83522	−0.0824868
$496$	0	0
$497$	−2.18849	−0.0981672
$498$	0	0
$499$	1.98005i	0.0886393i	0.999017	+	0.0443196i	$0.0141120\pi$
$499$	1.98005i	0.0886393i	−0.999017	+	0.0443196i	$0.985888\pi$
$500$	0	0
$501$	− 20.2104i	− 0.902934i
$502$	0	0
$503$	−12.0739	−0.538348	−0.269174	−	0.963092i	$-0.586751\pi$
$503$	−12.0739	−0.538348	−0.269174	+	0.963092i	$0.586751\pi$
$504$	0	0
$505$	−6.81485	−0.303257
$506$	0	0
$507$	− 11.4389i	− 0.508021i
$508$	0	0
$509$	4.48892i	0.198968i	0.995039	+	0.0994838i	$0.0317192\pi$
$509$	4.48892i	0.198968i	−0.995039	+	0.0994838i	$0.968281\pi$
$510$	0	0
$511$	3.16250	0.139901
$512$	0	0
$513$	−1.29769	−0.0572946
$514$	0	0
$515$	− 18.6288i	− 0.820883i
$516$	0	0
$517$	− 14.7821i	− 0.650115i
$518$	0	0
$519$	−10.1374	−0.444984
$520$	0	0
$521$	−4.75428	−0.208289	−0.104145	−	0.994562i	$-0.533210\pi$
$521$	−4.75428	−0.208289	−0.104145	+	0.994562i	$0.533210\pi$
$522$	0	0
$523$	− 34.7343i	− 1.51883i	−0.650609	−	0.759413i	$-0.725486\pi$
$523$	− 34.7343i	− 1.51883i	0.650609	−	0.759413i	$-0.274514\pi$
$524$	0	0
$525$	2.18433i	0.0953319i
$526$	0	0
$527$	29.5460	1.28704
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 9.04789i	− 0.392645i
$532$	0	0
$533$	− 10.8560i	− 0.470223i
$534$	0	0
$535$	−0.0927092	−0.00400817
$536$	0	0
$537$	20.1094	0.867783
$538$	0	0
$539$	− 10.1397i	− 0.436748i
$540$	0	0
$541$	16.2981i	0.700709i	0.936617	+	0.350354i	$0.113939\pi$
$541$	16.2981i	0.700709i	−0.936617	+	0.350354i	$0.886061\pi$
$542$	0	0
$543$	−9.57900	−0.411074
$544$	0	0
$545$	−3.87950	−0.166180
$546$	0	0
$547$	8.84339i	0.378116i	0.981966	+	0.189058i	$0.0605434\pi$
$547$	8.84339i	0.378116i	−0.981966	+	0.189058i	$0.939457\pi$
$548$	0	0
$549$	− 1.52432i	− 0.0650563i
$550$	0	0
$551$	−1.25341	−0.0533969
$552$	0	0
$553$	−4.59613	−0.195448
$554$	0	0
$555$	12.9545i	0.549890i
$556$	0	0
$557$	34.3335i	1.45476i	0.686236	+	0.727379i	$0.259262\pi$
$557$	34.3335i	1.45476i	−0.686236	+	0.727379i	$0.740738\pi$
$558$	0	0
$559$	−10.8560	−0.459158
$560$	0	0
$561$	−6.67271	−0.281722
$562$	0	0
$563$	− 27.8788i	− 1.17495i	−0.809243	−	0.587475i	$-0.800122\pi$
$563$	− 27.8788i	− 1.17495i	0.809243	−	0.587475i	$-0.199878\pi$
$564$	0	0
$565$	9.17990i	0.386201i
$566$	0	0
$567$	−0.613126	−0.0257489
$568$	0	0
$569$	−22.1525	−0.928682	−0.464341	−	0.885656i	$-0.653709\pi$
$569$	−22.1525	−0.928682	−0.464341	+	0.885656i	$0.653709\pi$
$570$	0	0
$571$	− 34.4845i	− 1.44313i	−0.692345	−	0.721566i	$-0.743422\pi$
$571$	− 34.4845i	− 1.44313i	0.692345	−	0.721566i	$-0.256578\pi$
$572$	0	0
$573$	2.30767i	0.0964042i
$574$	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
$579$	− 2.03278i	− 0.0844794i
$580$	0	0
$581$	− 4.40690i	− 0.182829i
$582$	0	0
$583$	−17.4785	−0.723885
$584$	0	0
$585$	−1.49796	−0.0619329
$586$	0	0
$587$	14.3096i	0.590621i	0.955401	+	0.295311i	$0.0954231\pi$
$587$	14.3096i	0.590621i	−0.955401	+	0.295311i	$0.904577\pi$
$588$	0	0
$589$	− 8.79565i	− 0.362418i
$590$	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
$595$	− 3.20435i	− 0.131366i
$596$	0	0
$597$	10.1927i	0.417157i
$598$	0	0
$599$	−23.4684	−0.958891	−0.479446	−	0.877572i	$-0.659162\pi$
$599$	−23.4684	−0.958891	−0.479446	+	0.877572i	$0.659162\pi$
$600$	0	0
$601$	20.4717	0.835058	0.417529	−	0.908664i	$-0.362896\pi$
$601$	20.4717	0.835058	0.417529	+	0.908664i	$0.362896\pi$
$602$	0	0
$603$	− 12.1094i	− 0.493131i
$604$	0	0
$605$	− 10.3788i	− 0.421959i
$606$	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
$611$	− 12.0656i	− 0.488120i
$612$	0	0
$613$	− 7.11562i	− 0.287397i	−0.989622	−	0.143699i	$-0.954100\pi$
$613$	− 7.11562i	− 0.287397i	0.989622	−	0.143699i	$-0.0458996\pi$
$614$	0	0
$615$	10.4170	0.420055
$616$	0	0
$617$	−20.5619	−0.827789	−0.413895	−	0.910325i	$-0.635832\pi$
$617$	−20.5619	−0.827789	−0.413895	+	0.910325i	$0.635832\pi$
$618$	0	0
$619$	− 35.4412i	− 1.42450i	−0.701925	−	0.712251i	$-0.747676\pi$
$619$	− 35.4412i	− 1.42450i	0.701925	−	0.712251i	$-0.252324\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	−0.412459	−0.0165248
$624$	0	0
$625$	5.50523	0.220209
$626$	0	0
$627$	1.98642i	0.0793301i
$628$	0	0
$629$	47.1018i	1.87807i
$630$	0	0
$631$	−3.05731	−0.121709	−0.0608547	−	0.998147i	$-0.519383\pi$
$631$	−3.05731	−0.121709	−0.0608547	+	0.998147i	$0.519383\pi$
$632$	0	0
$633$	−7.06147	−0.280668
$634$	0	0
$635$	25.0583i	0.994408i
$636$	0	0
$637$	− 8.27631i	− 0.327920i
$638$	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
$643$	− 23.6729i	− 0.933567i	−0.884372	−	0.466783i	$-0.845413\pi$
$643$	− 23.6729i	− 0.933567i	0.884372	−	0.466783i	$-0.154587\pi$
$644$	0	0
$645$	− 10.4170i	− 0.410170i
$646$	0	0
$647$	−42.3417	−1.66462	−0.832311	−	0.554309i	$-0.812983\pi$
$647$	−42.3417	−1.66462	−0.832311	+	0.554309i	$0.812983\pi$
$648$	0	0
$649$	−13.8499	−0.543657
$650$	0	0
$651$	− 4.15571i	− 0.162875i
$652$	0	0
$653$	− 48.3535i	− 1.89222i	−0.323850	−	0.946109i	$-0.604977\pi$
$653$	− 48.3535i	− 1.89222i	0.323850	−	0.946109i	$-0.395023\pi$
$654$	0	0
$655$	9.25633	0.361675
$656$	0	0
$657$	5.15800	0.201233
$658$	0	0
$659$	23.0023i	0.896042i	0.894023	+	0.448021i	$0.147871\pi$
$659$	23.0023i	0.896042i	−0.894023	+	0.448021i	$0.852129\pi$
$660$	0	0
$661$	21.4790i	0.835437i	0.908576	+	0.417719i	$0.137170\pi$
$661$	21.4790i	0.835437i	−0.908576	+	0.417719i	$0.862830\pi$
$662$	0	0
$663$	−5.44646	−0.211523
$664$	0	0
$665$	−0.953914	−0.0369912
$666$	0	0
$667$	3.86349i	0.149595i
$668$	0	0
$669$	18.9646i	0.733214i
$670$	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
$675$	3.56261i	0.137125i
$676$	0	0
$677$	− 32.2287i	− 1.23865i	−0.785135	−	0.619325i	$-0.787406\pi$
$677$	− 32.2287i	− 1.23865i	0.785135	−	0.619325i	$-0.212594\pi$
$678$	0	0
$679$	−2.89293	−0.111020
$680$	0	0
$681$	−15.5945	−0.597582
$682$	0	0
$683$	18.4037i	0.704198i	0.935963	+	0.352099i	$0.114532\pi$
$683$	18.4037i	0.704198i	−0.935963	+	0.352099i	$0.885468\pi$
$684$	0	0
$685$	8.89668i	0.339925i
$686$	0	0
$687$	6.37465	0.243208
$688$	0	0
$689$	−14.2665	−0.543509
$690$	0	0
$691$	12.5138i	0.476048i	0.971259	+	0.238024i	$0.0764997\pi$
$691$	12.5138i	0.476048i	−0.971259	+	0.238024i	$0.923500\pi$
$692$	0	0
$693$	0.938533i	0.0356519i
$694$	0	0
$695$	14.6894	0.557199
$696$	0	0
$697$	37.8756	1.43464
$698$	0	0
$699$	− 1.13880i	− 0.0430732i
$700$	0	0
$701$	4.41159i	0.166623i	0.996524	+	0.0833117i	$0.0265497\pi$
$701$	4.41159i	0.166623i	−0.996524	+	0.0833117i	$0.973450\pi$
$702$	0	0
$703$	14.0219	0.528846
$704$	0	0
$705$	11.5777	0.436042
$706$	0	0
$707$	3.48513i	0.131072i
$708$	0	0
$709$	− 40.7819i	− 1.53159i	−0.643082	−	0.765797i	$-0.722345\pi$
$709$	− 40.7819i	− 1.53159i	0.643082	−	0.765797i	$-0.277655\pi$
$710$	0	0
$711$	−7.49623	−0.281131
$712$	0	0
$713$	−27.1116	−1.01534
$714$	0	0
$715$	2.29297i	0.0857523i
$716$	0	0
$717$	25.6652i	0.958484i
$718$	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
$723$	− 19.2172i	− 0.714695i
$724$	0	0
$725$	3.44102i	0.127796i
$726$	0	0
$727$	−1.48610	−0.0551165	−0.0275583	−	0.999620i	$-0.508773\pi$
$727$	−1.48610	−0.0551165	−0.0275583	+	0.999620i	$0.508773\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 37.8756i	− 1.40088i
$732$	0	0
$733$	− 7.74829i	− 0.286190i	−0.989709	−	0.143095i	$-0.954295\pi$
$733$	− 7.74829i	− 0.286190i	0.989709	−	0.143095i	$-0.0457054\pi$
$734$	0	0
$735$	7.94169	0.292934
$736$	0	0
$737$	−18.5362	−0.682790
$738$	0	0
$739$	− 42.6392i	− 1.56851i	−0.620440	−	0.784254i	$-0.713046\pi$
$739$	− 42.6392i	− 1.56851i	0.620440	−	0.784254i	$-0.286954\pi$
$740$	0	0
$741$	1.62138i	0.0595627i
$742$	0	0
$743$	43.6516	1.60142	0.800711	−	0.599051i	$-0.204455\pi$
$743$	43.6516	1.60142	0.800711	+	0.599051i	$0.204455\pi$
$744$	0	0
$745$	4.24662	0.155584
$746$	0	0
$747$	− 7.18759i	− 0.262980i
$748$	0	0
$749$	0.0474117i	0.00173238i
$750$	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
$755$	− 15.5881i	− 0.567310i
$756$	0	0
$757$	− 22.6861i	− 0.824539i	−0.911062	−	0.412270i	$-0.864736\pi$
$757$	− 22.6861i	− 0.824539i	0.911062	−	0.412270i	$-0.135264\pi$
$758$	0	0
$759$	6.12293	0.222248
$760$	0	0
$761$	31.0047	1.12392	0.561960	−	0.827164i	$-0.310047\pi$
$761$	31.0047	1.12392	0.561960	+	0.827164i	$0.310047\pi$
$762$	0	0
$763$	1.98398i	0.0718251i
$764$	0	0
$765$	− 5.22625i	− 0.188956i
$766$	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
$771$	22.1094i	0.796249i
$772$	0	0
$773$	5.42745i	0.195212i	0.995225	+	0.0976059i	$0.0311185\pi$
$773$	5.42745i	0.195212i	−0.995225	+	0.0976059i	$0.968882\pi$
$774$	0	0
$775$	−24.1470	−0.867387
$776$	0	0
$777$	6.62498	0.237670
$778$	0	0
$779$	− 11.2753i	− 0.403980i
$780$	0	0
$781$	5.46380i	0.195510i
$782$	0	0
$783$	−0.965872	−0.0345175
$784$	0	0
$785$	−1.37683	−0.0491410
$786$	0	0
$787$	12.6824i	0.452077i	0.974118	+	0.226039i	$0.0725776\pi$
$787$	12.6824i	0.452077i	−0.974118	+	0.226039i	$0.927422\pi$
$788$	0	0
$789$	22.8914i	0.814957i
$790$	0	0
$791$	4.69462	0.166921
$792$	0	0
$793$	−1.90453	−0.0676318
$794$	0	0
$795$	− 13.6896i	− 0.485521i
$796$	0	0
$797$	− 24.7038i	− 0.875054i	−0.899205	−	0.437527i	$-0.855854\pi$
$797$	− 24.7038i	− 0.875054i	0.899205	−	0.437527i	$-0.144146\pi$
$798$	0	0
$799$	42.0958	1.48924
$800$	0	0
$801$	−0.672715	−0.0237692
$802$	0	0
$803$	− 7.89552i	− 0.278627i
$804$	0	0
$805$	2.94034i	0.103633i
$806$	0	0
$807$	−23.4776	−0.826449
$808$	0	0
$809$	6.81166	0.239485	0.119743	−	0.992805i	$-0.461793\pi$
$809$	6.81166	0.239485	0.119743	+	0.992805i	$0.461793\pi$
$810$	0	0
$811$	− 2.81804i	− 0.0989546i	−0.998775	−	0.0494773i	$-0.984244\pi$
$811$	− 2.81804i	− 0.0989546i	0.998775	−	0.0494773i	$-0.0157555\pi$
$812$	0	0
$813$	− 25.5837i	− 0.897259i
$814$	0	0
$815$	22.1584	0.776175
$816$	0	0
$817$	−11.2753	−0.394473
$818$	0	0
$819$	0.766057i	0.0267682i
$820$	0	0
$821$	36.5756i	1.27650i	0.769830	+	0.638249i	$0.220341\pi$
$821$	36.5756i	1.27650i	−0.769830	+	0.638249i	$0.779659\pi$
$822$	0	0
$823$	−5.15309	−0.179625	−0.0898126	−	0.995959i	$-0.528627\pi$
$823$	−5.15309	−0.179625	−0.0898126	+	0.995959i	$0.528627\pi$
$824$	0	0
$825$	5.45341	0.189863
$826$	0	0
$827$	− 35.6433i	− 1.23944i	−0.784824	−	0.619719i	$-0.787247\pi$
$827$	− 35.6433i	− 1.23944i	0.784824	−	0.619719i	$-0.212753\pi$
$828$	0	0
$829$	42.3283i	1.47012i	0.678001	+	0.735061i	$0.262847\pi$
$829$	42.3283i	1.47012i	−0.678001	+	0.735061i	$0.737153\pi$
$830$	0	0
$831$	−13.6692	−0.474178
$832$	0	0
$833$	28.8754	1.00047
$834$	0	0
$835$	− 24.2305i	− 0.838531i
$836$	0	0
$837$	− 6.77791i	− 0.234279i
$838$	0	0
$839$	−48.7768	−1.68396	−0.841981	−	0.539507i	$-0.818611\pi$
$839$	−48.7768	−1.68396	−0.841981	+	0.539507i	$0.818611\pi$
$840$	0	0
$841$	28.0671	0.967831
$842$	0	0
$843$	21.1116i	0.727124i
$844$	0	0
$845$	− 13.7143i	− 0.471785i
$846$	0	0
$847$	−5.30774	−0.182376
$848$	0	0
$849$	−17.0479	−0.585082
$850$	0	0
$851$	− 43.2210i	− 1.48160i
$852$	0	0
$853$	40.3965i	1.38315i	0.722304	+	0.691576i	$0.243083\pi$
$853$	40.3965i	1.38315i	−0.722304	+	0.691576i	$0.756917\pi$
$854$	0	0
$855$	−1.55582	−0.0532079
$856$	0	0
$857$	25.2977	0.864153	0.432076	−	0.901837i	$-0.357781\pi$
$857$	25.2977	0.864153	0.432076	+	0.901837i	$0.357781\pi$
$858$	0	0
$859$	35.1412i	1.19900i	0.800373	+	0.599502i	$0.204635\pi$
$859$	35.1412i	1.19900i	−0.800373	+	0.599502i	$0.795365\pi$
$860$	0	0
$861$	− 5.32729i	− 0.181553i
$862$	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
$867$	− 2.00228i	− 0.0680011i
$868$	0	0
$869$	11.4747i	0.389254i
$870$	0	0
$871$	−15.1298	−0.512653
$872$	0	0
$873$	−4.71832	−0.159691
$874$	0	0
$875$	6.29424i	0.212784i
$876$	0	0
$877$	3.89312i	0.131461i	0.997837	+	0.0657307i	$0.0209378\pi$
$877$	3.89312i	0.131461i	−0.997837	+	0.0657307i	$0.979062\pi$
$878$	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
$883$	− 23.0974i	− 0.777290i	−0.921387	−	0.388645i	$-0.872943\pi$
$883$	− 23.0974i	− 0.777290i	0.921387	−	0.388645i	$-0.127057\pi$
$884$	0	0
$885$	− 10.8476i	− 0.364639i
$886$	0	0
$887$	−56.1260	−1.88453	−0.942263	−	0.334873i	$-0.891307\pi$
$887$	−56.1260	−1.88453	−0.942263	+	0.334873i	$0.891307\pi$
$888$	0	0
$889$	12.8149	0.429796
$890$	0	0
$891$	1.53073i	0.0512815i
$892$	0	0
$893$	− 12.5316i	− 0.419355i
$894$	0	0
$895$	24.1094	0.805887
$896$	0	0
$897$	4.99772	0.166869
$898$	0	0
$899$	− 6.54659i	− 0.218341i
$900$	0	0
$901$	− 49.7745i	− 1.65823i
$902$	0	0
$903$	−5.32729	−0.177281
$904$	0	0
$905$	−11.4844	−0.381754
$906$	0	0
$907$	− 26.2683i	− 0.872223i	−0.899893	−	0.436112i	$-0.856355\pi$
$907$	− 26.2683i	− 0.872223i	0.899893	−	0.436112i	$-0.143645\pi$
$908$	0	0
$909$	5.68419i	0.188533i
$910$	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
$915$	− 1.82752i	− 0.0604161i
$916$	0	0
$917$	− 4.73370i	− 0.156321i
$918$	0	0
$919$	−0.319035	−0.0105240	−0.00526200	−	0.999986i	$-0.501675\pi$
$919$	−0.319035	−0.0105240	−0.00526200	+	0.999986i	$0.501675\pi$
$920$	0	0
$921$	−6.91858	−0.227975
$922$	0	0
$923$	4.45971i	0.146793i
$924$	0	0
$925$	− 38.4949i	− 1.26570i
$926$	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
$931$	− 8.59602i	− 0.281723i
$932$	0	0
$933$	20.4389i	0.669140i
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−39.8100	−1.30054	−0.650268	−	0.759705i	$-0.725343\pi$
$937$	−39.8100	−1.30054	−0.650268	+	0.759705i	$0.725343\pi$
$938$	0	0
$939$	11.6241i	0.379337i
$940$	0	0
$941$	45.2038i	1.47360i	0.676110	+	0.736801i	$0.263664\pi$
$941$	45.2038i	1.47360i	−0.676110	+	0.736801i	$0.736336\pi$
$942$	0	0
$943$	−34.7549	−1.13178
$944$	0	0
$945$	−0.735084	−0.0239123
$946$	0	0
$947$	43.1772i	1.40307i	0.712635	+	0.701535i	$0.247502\pi$
$947$	43.1772i	1.40307i	−0.712635	+	0.701535i	$0.752498\pi$
$948$	0	0
$949$	− 6.44455i	− 0.209199i
$950$	0	0
$951$	−21.5368	−0.698379
$952$	0	0
$953$	−15.1594	−0.491060	−0.245530	−	0.969389i	$-0.578962\pi$
$953$	−15.1594	−0.491060	−0.245530	+	0.969389i	$0.578962\pi$
$954$	0	0
$955$	2.76669i	0.0895280i
$956$	0	0
$957$	1.47849i	0.0477929i
$958$	0	0
$959$	4.54978	0.146920
$960$	0	0
$961$	14.9401	0.481938
$962$	0	0
$963$	0.0773278i	0.00249185i
$964$	0	0
$965$	− 2.43712i	− 0.0784537i
$966$	0	0
$967$	16.1170	0.518287	0.259143	−	0.965839i	$-0.416560\pi$
$967$	16.1170	0.518287	0.259143	+	0.965839i	$0.416560\pi$
$968$	0	0
$969$	−5.65685	−0.181724
$970$	0	0
$971$	4.15965i	0.133489i	0.997770	+	0.0667447i	$0.0212613\pi$
$971$	4.15965i	0.133489i	−0.997770	+	0.0667447i	$0.978739\pi$
$972$	0	0
$973$	− 7.51217i	− 0.240829i
$974$	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
$979$	1.02975i	0.0329109i
$980$	0	0
$981$	3.23585i	0.103313i
$982$	0	0
$983$	32.0257	1.02146	0.510730	−	0.859741i	$-0.329375\pi$
$983$	32.0257	1.02146	0.510730	+	0.859741i	$0.329375\pi$
$984$	0	0
$985$	31.6543	1.00859
$986$	0	0
$987$	− 5.92087i	− 0.188463i
$988$	0	0
$989$	34.7549i	1.10514i
$990$	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
$995$	12.2201i	0.387403i
$996$	0	0
$997$	18.0161i	0.570574i	0.958442	+	0.285287i	$0.0920890\pi$
$997$	18.0161i	0.570574i	−0.958442	+	0.285287i	$0.907911\pi$
$998$	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.d.j.1537.2		8
4.3	odd	2		3072.2.d.e.1537.6		8
8.3	odd	2		3072.2.d.e.1537.3		8
8.5	even	2	inner	3072.2.d.j.1537.7		8
16.3	odd	4		3072.2.a.m.1.2		4
16.5	even	4		3072.2.a.j.1.3		4
16.11	odd	4		3072.2.a.s.1.3		4
16.13	even	4		3072.2.a.p.1.2		4
32.3	odd	8		1536.2.j.e.385.2	✓	8
32.5	even	8		1536.2.j.f.1153.4	yes	8
32.11	odd	8		1536.2.j.j.1153.3	yes	8
32.13	even	8		1536.2.j.i.385.1	yes	8
32.19	odd	8		1536.2.j.j.385.3	yes	8
32.21	even	8		1536.2.j.i.1153.1	yes	8
32.27	odd	8		1536.2.j.e.1153.2	yes	8
32.29	even	8		1536.2.j.f.385.4	yes	8
48.5	odd	4		9216.2.a.ba.1.2		4
48.11	even	4		9216.2.a.bm.1.2		4
48.29	odd	4		9216.2.a.z.1.3		4
48.35	even	4		9216.2.a.bl.1.3		4
96.5	odd	8		4608.2.k.bj.1153.1		8
96.11	even	8		4608.2.k.be.1153.4		8
96.29	odd	8		4608.2.k.bj.3457.1		8
96.35	even	8		4608.2.k.bh.3457.1		8
96.53	odd	8		4608.2.k.bc.1153.4		8
96.59	even	8		4608.2.k.bh.1153.1		8
96.77	odd	8		4608.2.k.bc.3457.4		8
96.83	even	8		4608.2.k.be.3457.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.2	✓	8	32.3	odd	8
1536.2.j.e.1153.2	yes	8	32.27	odd	8
1536.2.j.f.385.4	yes	8	32.29	even	8
1536.2.j.f.1153.4	yes	8	32.5	even	8
1536.2.j.i.385.1	yes	8	32.13	even	8
1536.2.j.i.1153.1	yes	8	32.21	even	8
1536.2.j.j.385.3	yes	8	32.19	odd	8
1536.2.j.j.1153.3	yes	8	32.11	odd	8
3072.2.a.j.1.3		4	16.5	even	4
3072.2.a.m.1.2		4	16.3	odd	4
3072.2.a.p.1.2		4	16.13	even	4
3072.2.a.s.1.3		4	16.11	odd	4
3072.2.d.e.1537.3		8	8.3	odd	2
3072.2.d.e.1537.6		8	4.3	odd	2
3072.2.d.j.1537.2		8	1.1	even	1	trivial
3072.2.d.j.1537.7		8	8.5	even	2	inner
4608.2.k.bc.1153.4		8	96.53	odd	8
4608.2.k.bc.3457.4		8	96.77	odd	8
4608.2.k.be.1153.4		8	96.11	even	8
4608.2.k.be.3457.4		8	96.83	even	8
4608.2.k.bh.1153.1		8	96.59	even	8
4608.2.k.bh.3457.1		8	96.35	even	8
4608.2.k.bj.1153.1		8	96.5	odd	8
4608.2.k.bj.3457.1		8	96.29	odd	8
9216.2.a.z.1.3		4	48.29	odd	4
9216.2.a.ba.1.2		4	48.5	odd	4
9216.2.a.bl.1.3		4	48.35	even	4
9216.2.a.bm.1.2		4	48.11	even	4

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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.19891i q^{5} -0.613126 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.19891i q^{5} -0.613126 q^{7} -1.00000 q^{9} +1.53073i q^{11} +1.24943i q^{13} -1.19891 q^{15} -4.35916 q^{17} +1.29769i q^{19} +0.613126i q^{21} +4.00000 q^{23} +3.56261 q^{25} +1.00000i q^{27} +0.965872i q^{29} -6.77791 q^{31} +1.53073 q^{33} +0.735084i q^{35} -10.8052i q^{37} +1.24943 q^{39} -8.68873 q^{41} +8.68873i q^{43} +1.19891i q^{45} -9.65685 q^{47} -6.62408 q^{49} +4.35916i q^{51} +11.4184i q^{53} +1.83522 q^{55} +1.29769 q^{57} +9.04789i q^{59} +1.52432i q^{61} +0.613126 q^{63} +1.49796 q^{65} +12.1094i q^{67} -4.00000i q^{69} +3.56940 q^{71} -5.15800 q^{73} -3.56261i q^{75} -0.938533i q^{77} +7.49623 q^{79} +1.00000 q^{81} +7.18759i q^{83} +5.22625i q^{85} +0.965872 q^{87} +0.672715 q^{89} -0.766057i q^{91} +6.77791i q^{93} +1.55582 q^{95} +4.71832 q^{97} -1.53073i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q + 16 * q^7 - 8 * q^9	\( 8 q + 16 q^{7} - 8 q^{9} + 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{39} - 32 q^{47} + 8 q^{49} + 32 q^{55} - 16 q^{63} - 16 q^{65} + 32 q^{71} + 16 q^{73} - 48 q^{79} + 8 q^{81} + 16 q^{89} - 64 q^{95} - 32 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 8 * q^9 + 32 * q^23 - 8 * q^25 - 16 * q^31 + 16 * q^39 - 32 * q^47 + 8 * q^49 + 32 * q^55 - 16 * q^63 - 16 * q^65 + 32 * q^71 + 16 * q^73 - 48 * q^79 + 8 * q^81 + 16 * q^89 - 64 * q^95 - 32 * q^97

Introduction

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