LMFDB - Embedded newform 8112.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.561553 q^{5} -3.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.561553 q^{5} -3.56155 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.561553 q^{15} +2.56155 q^{17} -1.12311 q^{19} +3.56155 q^{21} -2.00000 q^{23} -4.68466 q^{25} -1.00000 q^{27} -5.68466 q^{29} -1.56155 q^{31} +2.00000 q^{33} +2.00000 q^{35} -3.43845 q^{37} -2.56155 q^{41} -0.438447 q^{43} -0.561553 q^{45} -8.24621 q^{47} +5.68466 q^{49} -2.56155 q^{51} +11.6847 q^{53} +1.12311 q^{55} +1.12311 q^{57} -11.1231 q^{59} +12.1231 q^{61} -3.56155 q^{63} +0.438447 q^{67} +2.00000 q^{69} +14.0000 q^{71} +1.87689 q^{73} +4.68466 q^{75} +7.12311 q^{77} -9.56155 q^{79} +1.00000 q^{81} -9.12311 q^{83} -1.43845 q^{85} +5.68466 q^{87} -13.1231 q^{89} +1.56155 q^{93} +0.630683 q^{95} +4.43845 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 4 q^{11} - 3 q^{15} + q^{17} + 6 q^{19} + 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + q^{31} + 4 q^{33} + 4 q^{35} - 11 q^{37} - q^{41} - 5 q^{43} + 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} - 6 q^{57} - 14 q^{59} + 16 q^{61} - 3 q^{63} + 5 q^{67} + 4 q^{69} + 28 q^{71} + 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} - 10 q^{83} - 7 q^{85} - q^{87} - 18 q^{89} - q^{93} + 26 q^{95} + 13 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 4 * q^11 - 3 * q^15 + q^17 + 6 * q^19 + 3 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + q^31 + 4 * q^33 + 4 * q^35 - 11 * q^37 - q^41 - 5 * q^43 + 3 * q^45 - q^49 - q^51 + 11 * q^53 - 6 * q^55 - 6 * q^57 - 14 * q^59 + 16 * q^61 - 3 * q^63 + 5 * q^67 + 4 * q^69 + 28 * q^71 + 12 * q^73 - 3 * q^75 + 6 * q^77 - 15 * q^79 + 2 * q^81 - 10 * q^83 - 7 * q^85 - q^87 - 18 * q^89 - q^93 + 26 * q^95 + 13 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	0	0
$7$	−3.56155	−1.34614	−0.673070	−	0.739579i	$-0.735025\pi$
$7$	−3.56155	−1.34614	−0.673070	+	0.739579i	$0.735025\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.561553	0.144992
$16$	0	0
$17$	2.56155	0.621268	0.310634	−	0.950530i	$-0.399459\pi$
$17$	2.56155	0.621268	0.310634	+	0.950530i	$0.399459\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	3.56155	0.777195
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	−1.56155	−0.280463	−0.140232	−	0.990119i	$-0.544785\pi$
$31$	−1.56155	−0.280463	−0.140232	+	0.990119i	$0.544785\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−3.43845	−0.565277	−0.282639	−	0.959226i	$-0.591210\pi$
$37$	−3.43845	−0.565277	−0.282639	+	0.959226i	$0.591210\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.56155	−0.400047	−0.200024	−	0.979791i	$-0.564102\pi$
$41$	−2.56155	−0.400047	−0.200024	+	0.979791i	$0.564102\pi$
$42$	0	0
$43$	−0.438447	−0.0668626	−0.0334313	−	0.999441i	$-0.510643\pi$
$43$	−0.438447	−0.0668626	−0.0334313	+	0.999441i	$0.510643\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	−8.24621	−1.20283	−0.601417	−	0.798935i	$-0.705397\pi$
$47$	−8.24621	−1.20283	−0.601417	+	0.798935i	$0.705397\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	−2.56155	−0.358689
$52$	0	0
$53$	11.6847	1.60501	0.802506	−	0.596645i	$-0.203500\pi$
$53$	11.6847	1.60501	0.802506	+	0.596645i	$0.203500\pi$
$54$	0	0
$55$	1.12311	0.151440
$56$	0	0
$57$	1.12311	0.148759
$58$	0	0
$59$	−11.1231	−1.44811	−0.724053	−	0.689745i	$-0.757723\pi$
$59$	−11.1231	−1.44811	−0.724053	+	0.689745i	$0.757723\pi$
$60$	0	0
$61$	12.1231	1.55220	0.776102	−	0.630607i	$-0.217194\pi$
$61$	12.1231	1.55220	0.776102	+	0.630607i	$0.217194\pi$
$62$	0	0
$63$	−3.56155	−0.448713
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.438447	0.0535648	0.0267824	−	0.999641i	$-0.491474\pi$
$67$	0.438447	0.0535648	0.0267824	+	0.999641i	$0.491474\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	1.87689	0.219674	0.109837	−	0.993950i	$-0.464967\pi$
$73$	1.87689	0.219674	0.109837	+	0.993950i	$0.464967\pi$
$74$	0	0
$75$	4.68466	0.540938
$76$	0	0
$77$	7.12311	0.811753
$78$	0	0
$79$	−9.56155	−1.07576	−0.537879	−	0.843022i	$-0.680774\pi$
$79$	−9.56155	−1.07576	−0.537879	+	0.843022i	$0.680774\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.12311	−1.00139	−0.500695	−	0.865624i	$-0.666922\pi$
$83$	−9.12311	−1.00139	−0.500695	+	0.865624i	$0.666922\pi$
$84$	0	0
$85$	−1.43845	−0.156022
$86$	0	0
$87$	5.68466	0.609459
$88$	0	0
$89$	−13.1231	−1.39105	−0.695523	−	0.718504i	$-0.744827\pi$
$89$	−13.1231	−1.39105	−0.695523	+	0.718504i	$0.744827\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.56155	0.161925
$94$	0	0
$95$	0.630683	0.0647067
$96$	0	0
$97$	4.43845	0.450656	0.225328	−	0.974283i	$-0.427655\pi$
$97$	4.43845	0.450656	0.225328	+	0.974283i	$0.427655\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−3.43845	−0.342138	−0.171069	−	0.985259i	$-0.554722\pi$
$101$	−3.43845	−0.342138	−0.171069	+	0.985259i	$0.554722\pi$
$102$	0	0
$103$	7.56155	0.745062	0.372531	−	0.928020i	$-0.378490\pi$
$103$	7.56155	0.745062	0.372531	+	0.928020i	$0.378490\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−8.24621	−0.797191	−0.398596	−	0.917127i	$-0.630502\pi$
$107$	−8.24621	−0.797191	−0.398596	+	0.917127i	$0.630502\pi$
$108$	0	0
$109$	−17.8078	−1.70567	−0.852837	−	0.522177i	$-0.825120\pi$
$109$	−17.8078	−1.70567	−0.852837	+	0.522177i	$0.825120\pi$
$110$	0	0
$111$	3.43845	0.326363
$112$	0	0
$113$	−14.8078	−1.39300	−0.696499	−	0.717558i	$-0.745260\pi$
$113$	−14.8078	−1.39300	−0.696499	+	0.717558i	$0.745260\pi$
$114$	0	0
$115$	1.12311	0.104730
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.12311	−0.836314
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.56155	0.230967
$124$	0	0
$125$	5.43845	0.486430
$126$	0	0
$127$	−9.56155	−0.848451	−0.424225	−	0.905557i	$-0.639454\pi$
$127$	−9.56155	−0.848451	−0.424225	+	0.905557i	$0.639454\pi$
$128$	0	0
$129$	0.438447	0.0386031
$130$	0	0
$131$	17.3693	1.51756	0.758782	−	0.651345i	$-0.225795\pi$
$131$	17.3693	1.51756	0.758782	+	0.651345i	$0.225795\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0.561553	0.0483308
$136$	0	0
$137$	1.43845	0.122895	0.0614474	−	0.998110i	$-0.480428\pi$
$137$	1.43845	0.122895	0.0614474	+	0.998110i	$0.480428\pi$
$138$	0	0
$139$	−10.9309	−0.927144	−0.463572	−	0.886059i	$-0.653433\pi$
$139$	−10.9309	−0.927144	−0.463572	+	0.886059i	$0.653433\pi$
$140$	0	0
$141$	8.24621	0.694456
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.19224	0.265101
$146$	0	0
$147$	−5.68466	−0.468863
$148$	0	0
$149$	6.56155	0.537543	0.268772	−	0.963204i	$-0.413382\pi$
$149$	6.56155	0.537543	0.268772	+	0.963204i	$0.413382\pi$
$150$	0	0
$151$	15.3693	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$151$	15.3693	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$152$	0	0
$153$	2.56155	0.207089
$154$	0	0
$155$	0.876894	0.0704339
$156$	0	0
$157$	−4.36932	−0.348709	−0.174355	−	0.984683i	$-0.555784\pi$
$157$	−4.36932	−0.348709	−0.174355	+	0.984683i	$0.555784\pi$
$158$	0	0
$159$	−11.6847	−0.926654
$160$	0	0
$161$	7.12311	0.561379
$162$	0	0
$163$	15.8078	1.23816	0.619080	−	0.785328i	$-0.287506\pi$
$163$	15.8078	1.23816	0.619080	+	0.785328i	$0.287506\pi$
$164$	0	0
$165$	−1.12311	−0.0874337
$166$	0	0
$167$	−6.24621	−0.483346	−0.241673	−	0.970358i	$-0.577696\pi$
$167$	−6.24621	−0.483346	−0.241673	+	0.970358i	$0.577696\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1.12311	−0.0858860
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	16.6847	1.26124
$176$	0	0
$177$	11.1231	0.836064
$178$	0	0
$179$	13.1231	0.980867	0.490433	−	0.871479i	$-0.336838\pi$
$179$	13.1231	0.980867	0.490433	+	0.871479i	$0.336838\pi$
$180$	0	0
$181$	9.68466	0.719855	0.359927	−	0.932980i	$-0.382801\pi$
$181$	9.68466	0.719855	0.359927	+	0.932980i	$0.382801\pi$
$182$	0	0
$183$	−12.1231	−0.896166
$184$	0	0
$185$	1.93087	0.141960
$186$	0	0
$187$	−5.12311	−0.374639
$188$	0	0
$189$	3.56155	0.259065
$190$	0	0
$191$	0.876894	0.0634499	0.0317249	−	0.999497i	$-0.489900\pi$
$191$	0.876894	0.0634499	0.0317249	+	0.999497i	$0.489900\pi$
$192$	0	0
$193$	−19.4924	−1.40310	−0.701548	−	0.712623i	$-0.747507\pi$
$193$	−19.4924	−1.40310	−0.701548	+	0.712623i	$0.747507\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.3693	0.810030	0.405015	−	0.914310i	$-0.367266\pi$
$197$	11.3693	0.810030	0.405015	+	0.914310i	$0.367266\pi$
$198$	0	0
$199$	23.1771	1.64298	0.821490	−	0.570223i	$-0.193143\pi$
$199$	23.1771	1.64298	0.821490	+	0.570223i	$0.193143\pi$
$200$	0	0
$201$	−0.438447	−0.0309257
$202$	0	0
$203$	20.2462	1.42101
$204$	0	0
$205$	1.43845	0.100466
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	2.24621	0.155374
$210$	0	0
$211$	−7.31534	−0.503609	−0.251804	−	0.967778i	$-0.581024\pi$
$211$	−7.31534	−0.503609	−0.251804	+	0.967778i	$0.581024\pi$
$212$	0	0
$213$	−14.0000	−0.959264
$214$	0	0
$215$	0.246211	0.0167915
$216$	0	0
$217$	5.56155	0.377543
$218$	0	0
$219$	−1.87689	−0.126829
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−4.68466	−0.312311
$226$	0	0
$227$	−1.12311	−0.0745431	−0.0372716	−	0.999305i	$-0.511867\pi$
$227$	−1.12311	−0.0745431	−0.0372716	+	0.999305i	$0.511867\pi$
$228$	0	0
$229$	−0.246211	−0.0162701	−0.00813505	−	0.999967i	$-0.502589\pi$
$229$	−0.246211	−0.0162701	−0.00813505	+	0.999967i	$0.502589\pi$
$230$	0	0
$231$	−7.12311	−0.468666
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	4.63068	0.302072
$236$	0	0
$237$	9.56155	0.621090
$238$	0	0
$239$	−0.630683	−0.0407955	−0.0203977	−	0.999792i	$-0.506493\pi$
$239$	−0.630683	−0.0407955	−0.0203977	+	0.999792i	$0.506493\pi$
$240$	0	0
$241$	−2.80776	−0.180864	−0.0904320	−	0.995903i	$-0.528825\pi$
$241$	−2.80776	−0.180864	−0.0904320	+	0.995903i	$0.528825\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.19224	−0.203944
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.12311	0.578153
$250$	0	0
$251$	30.7386	1.94021	0.970103	−	0.242695i	$-0.0780314\pi$
$251$	30.7386	1.94021	0.970103	+	0.242695i	$0.0780314\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	1.43845	0.0900791
$256$	0	0
$257$	−16.1771	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$257$	−16.1771	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$258$	0	0
$259$	12.2462	0.760943
$260$	0	0
$261$	−5.68466	−0.351872
$262$	0	0
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	−6.56155	−0.403073
$266$	0	0
$267$	13.1231	0.803121
$268$	0	0
$269$	3.36932	0.205431	0.102715	−	0.994711i	$-0.467247\pi$
$269$	3.36932	0.205431	0.102715	+	0.994711i	$0.467247\pi$
$270$	0	0
$271$	−1.06913	−0.0649450	−0.0324725	−	0.999473i	$-0.510338\pi$
$271$	−1.06913	−0.0649450	−0.0324725	+	0.999473i	$0.510338\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.36932	0.564991
$276$	0	0
$277$	17.6847	1.06257	0.531284	−	0.847194i	$-0.321710\pi$
$277$	17.6847	1.06257	0.531284	+	0.847194i	$0.321710\pi$
$278$	0	0
$279$	−1.56155	−0.0934877
$280$	0	0
$281$	2.80776	0.167497	0.0837486	−	0.996487i	$-0.473311\pi$
$281$	2.80776	0.167497	0.0837486	+	0.996487i	$0.473311\pi$
$282$	0	0
$283$	1.31534	0.0781889	0.0390945	−	0.999236i	$-0.487553\pi$
$283$	1.31534	0.0781889	0.0390945	+	0.999236i	$0.487553\pi$
$284$	0	0
$285$	−0.630683	−0.0373584
$286$	0	0
$287$	9.12311	0.538520
$288$	0	0
$289$	−10.4384	−0.614026
$290$	0	0
$291$	−4.43845	−0.260186
$292$	0	0
$293$	−24.5616	−1.43490	−0.717451	−	0.696609i	$-0.754691\pi$
$293$	−24.5616	−1.43490	−0.717451	+	0.696609i	$0.754691\pi$
$294$	0	0
$295$	6.24621	0.363668
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.56155	0.0900064
$302$	0	0
$303$	3.43845	0.197534
$304$	0	0
$305$	−6.80776	−0.389811
$306$	0	0
$307$	10.1922	0.581702	0.290851	−	0.956768i	$-0.406062\pi$
$307$	10.1922	0.581702	0.290851	+	0.956768i	$0.406062\pi$
$308$	0	0
$309$	−7.56155	−0.430162
$310$	0	0
$311$	10.8769	0.616772	0.308386	−	0.951261i	$-0.400211\pi$
$311$	10.8769	0.616772	0.308386	+	0.951261i	$0.400211\pi$
$312$	0	0
$313$	−1.31534	−0.0743475	−0.0371738	−	0.999309i	$-0.511835\pi$
$313$	−1.31534	−0.0743475	−0.0371738	+	0.999309i	$0.511835\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	23.0540	1.29484	0.647420	−	0.762133i	$-0.275848\pi$
$317$	23.0540	1.29484	0.647420	+	0.762133i	$0.275848\pi$
$318$	0	0
$319$	11.3693	0.636560
$320$	0	0
$321$	8.24621	0.460259
$322$	0	0
$323$	−2.87689	−0.160075
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.8078	0.984772
$328$	0	0
$329$	29.3693	1.61918
$330$	0	0
$331$	−23.8078	−1.30859	−0.654297	−	0.756238i	$-0.727035\pi$
$331$	−23.8078	−1.30859	−0.654297	+	0.756238i	$0.727035\pi$
$332$	0	0
$333$	−3.43845	−0.188426
$334$	0	0
$335$	−0.246211	−0.0134520
$336$	0	0
$337$	2.12311	0.115653	0.0578265	−	0.998327i	$-0.481583\pi$
$337$	2.12311	0.115653	0.0578265	+	0.998327i	$0.481583\pi$
$338$	0	0
$339$	14.8078	0.804247
$340$	0	0
$341$	3.12311	0.169126
$342$	0	0
$343$	4.68466	0.252948
$344$	0	0
$345$	−1.12311	−0.0604660
$346$	0	0
$347$	13.6155	0.730920	0.365460	−	0.930827i	$-0.380912\pi$
$347$	13.6155	0.730920	0.365460	+	0.930827i	$0.380912\pi$
$348$	0	0
$349$	13.8078	0.739113	0.369556	−	0.929208i	$-0.379510\pi$
$349$	13.8078	0.739113	0.369556	+	0.929208i	$0.379510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.6847	−0.941259	−0.470630	−	0.882331i	$-0.655973\pi$
$353$	−17.6847	−0.941259	−0.470630	+	0.882331i	$0.655973\pi$
$354$	0	0
$355$	−7.86174	−0.417258
$356$	0	0
$357$	9.12311	0.482846
$358$	0	0
$359$	15.3693	0.811162	0.405581	−	0.914059i	$-0.367069\pi$
$359$	15.3693	0.811162	0.405581	+	0.914059i	$0.367069\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−1.05398	−0.0551676
$366$	0	0
$367$	−20.0540	−1.04681	−0.523404	−	0.852084i	$-0.675338\pi$
$367$	−20.0540	−1.04681	−0.523404	+	0.852084i	$0.675338\pi$
$368$	0	0
$369$	−2.56155	−0.133349
$370$	0	0
$371$	−41.6155	−2.16057
$372$	0	0
$373$	3.63068	0.187990	0.0939948	−	0.995573i	$-0.470036\pi$
$373$	3.63068	0.187990	0.0939948	+	0.995573i	$0.470036\pi$
$374$	0	0
$375$	−5.43845	−0.280840
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.3153	0.581230	0.290615	−	0.956840i	$-0.406140\pi$
$379$	11.3153	0.581230	0.290615	+	0.956840i	$0.406140\pi$
$380$	0	0
$381$	9.56155	0.489853
$382$	0	0
$383$	26.7386	1.36628	0.683140	−	0.730287i	$-0.260614\pi$
$383$	26.7386	1.36628	0.683140	+	0.730287i	$0.260614\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−0.438447	−0.0222875
$388$	0	0
$389$	−3.05398	−0.154843	−0.0774213	−	0.996998i	$-0.524669\pi$
$389$	−3.05398	−0.154843	−0.0774213	+	0.996998i	$0.524669\pi$
$390$	0	0
$391$	−5.12311	−0.259087
$392$	0	0
$393$	−17.3693	−0.876166
$394$	0	0
$395$	5.36932	0.270160
$396$	0	0
$397$	12.0540	0.604972	0.302486	−	0.953154i	$-0.402184\pi$
$397$	12.0540	0.604972	0.302486	+	0.953154i	$0.402184\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−18.5616	−0.926920	−0.463460	−	0.886118i	$-0.653392\pi$
$401$	−18.5616	−0.926920	−0.463460	+	0.886118i	$0.653392\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.561553	−0.0279038
$406$	0	0
$407$	6.87689	0.340875
$408$	0	0
$409$	18.3693	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$409$	18.3693	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$410$	0	0
$411$	−1.43845	−0.0709534
$412$	0	0
$413$	39.6155	1.94935
$414$	0	0
$415$	5.12311	0.251483
$416$	0	0
$417$	10.9309	0.535287
$418$	0	0
$419$	17.7538	0.867329	0.433665	−	0.901074i	$-0.357220\pi$
$419$	17.7538	0.867329	0.433665	+	0.901074i	$0.357220\pi$
$420$	0	0
$421$	−14.7538	−0.719056	−0.359528	−	0.933134i	$-0.617062\pi$
$421$	−14.7538	−0.719056	−0.359528	+	0.933134i	$0.617062\pi$
$422$	0	0
$423$	−8.24621	−0.400945
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−43.1771	−2.08949
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.87689	−0.138575	−0.0692876	−	0.997597i	$-0.522073\pi$
$431$	−2.87689	−0.138575	−0.0692876	+	0.997597i	$0.522073\pi$
$432$	0	0
$433$	25.2462	1.21326	0.606628	−	0.794986i	$-0.292522\pi$
$433$	25.2462	1.21326	0.606628	+	0.794986i	$0.292522\pi$
$434$	0	0
$435$	−3.19224	−0.153056
$436$	0	0
$437$	2.24621	0.107451
$438$	0	0
$439$	1.31534	0.0627778	0.0313889	−	0.999507i	$-0.490007\pi$
$439$	1.31534	0.0627778	0.0313889	+	0.999507i	$0.490007\pi$
$440$	0	0
$441$	5.68466	0.270698
$442$	0	0
$443$	−14.7386	−0.700254	−0.350127	−	0.936702i	$-0.613862\pi$
$443$	−14.7386	−0.700254	−0.350127	+	0.936702i	$0.613862\pi$
$444$	0	0
$445$	7.36932	0.349339
$446$	0	0
$447$	−6.56155	−0.310351
$448$	0	0
$449$	−8.24621	−0.389163	−0.194581	−	0.980886i	$-0.562335\pi$
$449$	−8.24621	−0.389163	−0.194581	+	0.980886i	$0.562335\pi$
$450$	0	0
$451$	5.12311	0.241238
$452$	0	0
$453$	−15.3693	−0.722113
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.6155	1.33858	0.669289	−	0.743002i	$-0.266599\pi$
$457$	28.6155	1.33858	0.669289	+	0.743002i	$0.266599\pi$
$458$	0	0
$459$	−2.56155	−0.119563
$460$	0	0
$461$	36.8078	1.71431	0.857154	−	0.515060i	$-0.172230\pi$
$461$	36.8078	1.71431	0.857154	+	0.515060i	$0.172230\pi$
$462$	0	0
$463$	−26.6847	−1.24014	−0.620071	−	0.784546i	$-0.712896\pi$
$463$	−26.6847	−1.24014	−0.620071	+	0.784546i	$0.712896\pi$
$464$	0	0
$465$	−0.876894	−0.0406650
$466$	0	0
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	−1.56155	−0.0721058
$470$	0	0
$471$	4.36932	0.201327
$472$	0	0
$473$	0.876894	0.0403196
$474$	0	0
$475$	5.26137	0.241408
$476$	0	0
$477$	11.6847	0.535004
$478$	0	0
$479$	−6.24621	−0.285397	−0.142698	−	0.989766i	$-0.545578\pi$
$479$	−6.24621	−0.285397	−0.142698	+	0.989766i	$0.545578\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.12311	−0.324113
$484$	0	0
$485$	−2.49242	−0.113175
$486$	0	0
$487$	−1.12311	−0.0508928	−0.0254464	−	0.999676i	$-0.508101\pi$
$487$	−1.12311	−0.0508928	−0.0254464	+	0.999676i	$0.508101\pi$
$488$	0	0
$489$	−15.8078	−0.714852
$490$	0	0
$491$	19.7538	0.891476	0.445738	−	0.895163i	$-0.352941\pi$
$491$	19.7538	0.891476	0.445738	+	0.895163i	$0.352941\pi$
$492$	0	0
$493$	−14.5616	−0.655819
$494$	0	0
$495$	1.12311	0.0504798
$496$	0	0
$497$	−49.8617	−2.23660
$498$	0	0
$499$	−28.4924	−1.27550	−0.637748	−	0.770245i	$-0.720134\pi$
$499$	−28.4924	−1.27550	−0.637748	+	0.770245i	$0.720134\pi$
$500$	0	0
$501$	6.24621	0.279060
$502$	0	0
$503$	11.7538	0.524076	0.262038	−	0.965058i	$-0.415605\pi$
$503$	11.7538	0.524076	0.262038	+	0.965058i	$0.415605\pi$
$504$	0	0
$505$	1.93087	0.0859226
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.80776	−0.301749	−0.150874	−	0.988553i	$-0.548209\pi$
$509$	−6.80776	−0.301749	−0.150874	+	0.988553i	$0.548209\pi$
$510$	0	0
$511$	−6.68466	−0.295712
$512$	0	0
$513$	1.12311	0.0495863
$514$	0	0
$515$	−4.24621	−0.187110
$516$	0	0
$517$	16.4924	0.725336
$518$	0	0
$519$	−3.75379	−0.164773
$520$	0	0
$521$	−37.9309	−1.66178	−0.830891	−	0.556436i	$-0.812169\pi$
$521$	−37.9309	−1.66178	−0.830891	+	0.556436i	$0.812169\pi$
$522$	0	0
$523$	23.8617	1.04340	0.521701	−	0.853129i	$-0.325298\pi$
$523$	23.8617	1.04340	0.521701	+	0.853129i	$0.325298\pi$
$524$	0	0
$525$	−16.6847	−0.728178
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−11.1231	−0.482702
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.63068	0.200202
$536$	0	0
$537$	−13.1231	−0.566304
$538$	0	0
$539$	−11.3693	−0.489711
$540$	0	0
$541$	29.7386	1.27856	0.639282	−	0.768972i	$-0.279232\pi$
$541$	29.7386	1.27856	0.639282	+	0.768972i	$0.279232\pi$
$542$	0	0
$543$	−9.68466	−0.415608
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	24.9309	1.06597	0.532984	−	0.846126i	$-0.321071\pi$
$547$	24.9309	1.06597	0.532984	+	0.846126i	$0.321071\pi$
$548$	0	0
$549$	12.1231	0.517402
$550$	0	0
$551$	6.38447	0.271988
$552$	0	0
$553$	34.0540	1.44812
$554$	0	0
$555$	−1.93087	−0.0819609
$556$	0	0
$557$	14.0691	0.596128	0.298064	−	0.954546i	$-0.403659\pi$
$557$	14.0691	0.596128	0.298064	+	0.954546i	$0.403659\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	5.12311	0.216298
$562$	0	0
$563$	−1.36932	−0.0577098	−0.0288549	−	0.999584i	$-0.509186\pi$
$563$	−1.36932	−0.0577098	−0.0288549	+	0.999584i	$0.509186\pi$
$564$	0	0
$565$	8.31534	0.349829
$566$	0	0
$567$	−3.56155	−0.149571
$568$	0	0
$569$	40.7386	1.70785	0.853926	−	0.520394i	$-0.174215\pi$
$569$	40.7386	1.70785	0.853926	+	0.520394i	$0.174215\pi$
$570$	0	0
$571$	−19.3693	−0.810581	−0.405290	−	0.914188i	$-0.632830\pi$
$571$	−19.3693	−0.810581	−0.405290	+	0.914188i	$0.632830\pi$
$572$	0	0
$573$	−0.876894	−0.0366328
$574$	0	0
$575$	9.36932	0.390728
$576$	0	0
$577$	29.6847	1.23579	0.617894	−	0.786261i	$-0.287986\pi$
$577$	29.6847	1.23579	0.617894	+	0.786261i	$0.287986\pi$
$578$	0	0
$579$	19.4924	0.810077
$580$	0	0
$581$	32.4924	1.34801
$582$	0	0
$583$	−23.3693	−0.967858
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6307	0.603873	0.301936	−	0.953328i	$-0.402367\pi$
$587$	14.6307	0.603873	0.301936	+	0.953328i	$0.402367\pi$
$588$	0	0
$589$	1.75379	0.0722636
$590$	0	0
$591$	−11.3693	−0.467671
$592$	0	0
$593$	−44.4233	−1.82425	−0.912123	−	0.409917i	$-0.865558\pi$
$593$	−44.4233	−1.82425	−0.912123	+	0.409917i	$0.865558\pi$
$594$	0	0
$595$	5.12311	0.210027
$596$	0	0
$597$	−23.1771	−0.948575
$598$	0	0
$599$	0.384472	0.0157091	0.00785455	−	0.999969i	$-0.497500\pi$
$599$	0.384472	0.0157091	0.00785455	+	0.999969i	$0.497500\pi$
$600$	0	0
$601$	−35.9309	−1.46565	−0.732825	−	0.680417i	$-0.761799\pi$
$601$	−35.9309	−1.46565	−0.732825	+	0.680417i	$0.761799\pi$
$602$	0	0
$603$	0.438447	0.0178549
$604$	0	0
$605$	3.93087	0.159813
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−20.2462	−0.820418
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−22.8617	−0.923377	−0.461688	−	0.887042i	$-0.652756\pi$
$613$	−22.8617	−0.923377	−0.461688	+	0.887042i	$0.652756\pi$
$614$	0	0
$615$	−1.43845	−0.0580038
$616$	0	0
$617$	10.8078	0.435104	0.217552	−	0.976049i	$-0.430193\pi$
$617$	10.8078	0.435104	0.217552	+	0.976049i	$0.430193\pi$
$618$	0	0
$619$	−24.3002	−0.976707	−0.488353	−	0.872646i	$-0.662402\pi$
$619$	−24.3002	−0.976707	−0.488353	+	0.872646i	$0.662402\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	46.7386	1.87254
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	−2.24621	−0.0897050
$628$	0	0
$629$	−8.80776	−0.351189
$630$	0	0
$631$	−14.4384	−0.574786	−0.287393	−	0.957813i	$-0.592788\pi$
$631$	−14.4384	−0.574786	−0.287393	+	0.957813i	$0.592788\pi$
$632$	0	0
$633$	7.31534	0.290759
$634$	0	0
$635$	5.36932	0.213075
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	26.1771	1.03393	0.516966	−	0.856006i	$-0.327061\pi$
$641$	26.1771	1.03393	0.516966	+	0.856006i	$0.327061\pi$
$642$	0	0
$643$	38.5464	1.52012	0.760061	−	0.649852i	$-0.225169\pi$
$643$	38.5464	1.52012	0.760061	+	0.649852i	$0.225169\pi$
$644$	0	0
$645$	−0.246211	−0.00969456
$646$	0	0
$647$	47.6155	1.87196	0.935980	−	0.352054i	$-0.114517\pi$
$647$	47.6155	1.87196	0.935980	+	0.352054i	$0.114517\pi$
$648$	0	0
$649$	22.2462	0.873240
$650$	0	0
$651$	−5.56155	−0.217974
$652$	0	0
$653$	14.8769	0.582178	0.291089	−	0.956696i	$-0.405982\pi$
$653$	14.8769	0.582178	0.291089	+	0.956696i	$0.405982\pi$
$654$	0	0
$655$	−9.75379	−0.381112
$656$	0	0
$657$	1.87689	0.0732246
$658$	0	0
$659$	−14.2462	−0.554954	−0.277477	−	0.960732i	$-0.589498\pi$
$659$	−14.2462	−0.554954	−0.277477	+	0.960732i	$0.589498\pi$
$660$	0	0
$661$	−30.3693	−1.18123	−0.590615	−	0.806954i	$-0.701115\pi$
$661$	−30.3693	−1.18123	−0.590615	+	0.806954i	$0.701115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.24621	−0.0871043
$666$	0	0
$667$	11.3693	0.440222
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−24.2462	−0.936015
$672$	0	0
$673$	−6.75379	−0.260339	−0.130170	−	0.991492i	$-0.541552\pi$
$673$	−6.75379	−0.260339	−0.130170	+	0.991492i	$0.541552\pi$
$674$	0	0
$675$	4.68466	0.180313
$676$	0	0
$677$	25.6155	0.984485	0.492242	−	0.870458i	$-0.336177\pi$
$677$	25.6155	0.984485	0.492242	+	0.870458i	$0.336177\pi$
$678$	0	0
$679$	−15.8078	−0.606646
$680$	0	0
$681$	1.12311	0.0430375
$682$	0	0
$683$	36.1080	1.38163	0.690816	−	0.723030i	$-0.257251\pi$
$683$	36.1080	1.38163	0.690816	+	0.723030i	$0.257251\pi$
$684$	0	0
$685$	−0.807764	−0.0308631
$686$	0	0
$687$	0.246211	0.00939355
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.30019	0.0875032	0.0437516	−	0.999042i	$-0.486069\pi$
$691$	2.30019	0.0875032	0.0437516	+	0.999042i	$0.486069\pi$
$692$	0	0
$693$	7.12311	0.270584
$694$	0	0
$695$	6.13826	0.232837
$696$	0	0
$697$	−6.56155	−0.248537
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	19.3693	0.731569	0.365785	−	0.930700i	$-0.380801\pi$
$701$	19.3693	0.731569	0.365785	+	0.930700i	$0.380801\pi$
$702$	0	0
$703$	3.86174	0.145648
$704$	0	0
$705$	−4.63068	−0.174402
$706$	0	0
$707$	12.2462	0.460566
$708$	0	0
$709$	−25.4924	−0.957388	−0.478694	−	0.877982i	$-0.658890\pi$
$709$	−25.4924	−0.957388	−0.478694	+	0.877982i	$0.658890\pi$
$710$	0	0
$711$	−9.56155	−0.358586
$712$	0	0
$713$	3.12311	0.116961
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.630683	0.0235533
$718$	0	0
$719$	−1.36932	−0.0510669	−0.0255335	−	0.999674i	$-0.508128\pi$
$719$	−1.36932	−0.0510669	−0.0255335	+	0.999674i	$0.508128\pi$
$720$	0	0
$721$	−26.9309	−1.00296
$722$	0	0
$723$	2.80776	0.104422
$724$	0	0
$725$	26.6307	0.989039
$726$	0	0
$727$	−39.6695	−1.47126	−0.735630	−	0.677383i	$-0.763114\pi$
$727$	−39.6695	−1.47126	−0.735630	+	0.677383i	$0.763114\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.12311	−0.0415396
$732$	0	0
$733$	−53.4924	−1.97579	−0.987894	−	0.155131i	$-0.950420\pi$
$733$	−53.4924	−1.97579	−0.987894	+	0.155131i	$0.950420\pi$
$734$	0	0
$735$	3.19224	0.117747
$736$	0	0
$737$	−0.876894	−0.0323008
$738$	0	0
$739$	−6.24621	−0.229771	−0.114885	−	0.993379i	$-0.536650\pi$
$739$	−6.24621	−0.229771	−0.114885	+	0.993379i	$0.536650\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.3693	−1.37095	−0.685474	−	0.728097i	$-0.740405\pi$
$743$	−37.3693	−1.37095	−0.685474	+	0.728097i	$0.740405\pi$
$744$	0	0
$745$	−3.68466	−0.134995
$746$	0	0
$747$	−9.12311	−0.333797
$748$	0	0
$749$	29.3693	1.07313
$750$	0	0
$751$	30.1080	1.09865	0.549327	−	0.835607i	$-0.314884\pi$
$751$	30.1080	1.09865	0.549327	+	0.835607i	$0.314884\pi$
$752$	0	0
$753$	−30.7386	−1.12018
$754$	0	0
$755$	−8.63068	−0.314103
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	15.3693	0.557137	0.278569	−	0.960416i	$-0.410140\pi$
$761$	15.3693	0.557137	0.278569	+	0.960416i	$0.410140\pi$
$762$	0	0
$763$	63.4233	2.29608
$764$	0	0
$765$	−1.43845	−0.0520072
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	16.1771	0.582603
$772$	0	0
$773$	7.75379	0.278884	0.139442	−	0.990230i	$-0.455469\pi$
$773$	7.75379	0.278884	0.139442	+	0.990230i	$0.455469\pi$
$774$	0	0
$775$	7.31534	0.262775
$776$	0	0
$777$	−12.2462	−0.439330
$778$	0	0
$779$	2.87689	0.103075
$780$	0	0
$781$	−28.0000	−1.00192
$782$	0	0
$783$	5.68466	0.203153
$784$	0	0
$785$	2.45360	0.0875728
$786$	0	0
$787$	−1.17708	−0.0419584	−0.0209792	−	0.999780i	$-0.506678\pi$
$787$	−1.17708	−0.0419584	−0.0209792	+	0.999780i	$0.506678\pi$
$788$	0	0
$789$	−15.3693	−0.547162
$790$	0	0
$791$	52.7386	1.87517
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.56155	0.232714
$796$	0	0
$797$	−41.6155	−1.47410	−0.737049	−	0.675840i	$-0.763781\pi$
$797$	−41.6155	−1.47410	−0.737049	+	0.675840i	$0.763781\pi$
$798$	0	0
$799$	−21.1231	−0.747282
$800$	0	0
$801$	−13.1231	−0.463682
$802$	0	0
$803$	−3.75379	−0.132468
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−3.36932	−0.118606
$808$	0	0
$809$	37.3002	1.31140	0.655702	−	0.755019i	$-0.272373\pi$
$809$	37.3002	1.31140	0.655702	+	0.755019i	$0.272373\pi$
$810$	0	0
$811$	−1.56155	−0.0548335	−0.0274168	−	0.999624i	$-0.508728\pi$
$811$	−1.56155	−0.0548335	−0.0274168	+	0.999624i	$0.508728\pi$
$812$	0	0
$813$	1.06913	0.0374960
$814$	0	0
$815$	−8.87689	−0.310944
$816$	0	0
$817$	0.492423	0.0172277
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.4924	0.924592	0.462296	−	0.886726i	$-0.347026\pi$
$821$	26.4924	0.924592	0.462296	+	0.886726i	$0.347026\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	−9.36932	−0.326198
$826$	0	0
$827$	34.7386	1.20798	0.603990	−	0.796992i	$-0.293577\pi$
$827$	34.7386	1.20798	0.603990	+	0.796992i	$0.293577\pi$
$828$	0	0
$829$	−19.4924	−0.677000	−0.338500	−	0.940966i	$-0.609919\pi$
$829$	−19.4924	−0.677000	−0.338500	+	0.940966i	$0.609919\pi$
$830$	0	0
$831$	−17.6847	−0.613474
$832$	0	0
$833$	14.5616	0.504528
$834$	0	0
$835$	3.50758	0.121385
$836$	0	0
$837$	1.56155	0.0539752
$838$	0	0
$839$	−19.6155	−0.677203	−0.338602	−	0.940930i	$-0.609954\pi$
$839$	−19.6155	−0.677203	−0.338602	+	0.940930i	$0.609954\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	−2.80776	−0.0967045
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24.9309	0.856635
$848$	0	0
$849$	−1.31534	−0.0451424
$850$	0	0
$851$	6.87689	0.235737
$852$	0	0
$853$	6.12311	0.209651	0.104826	−	0.994491i	$-0.466572\pi$
$853$	6.12311	0.209651	0.104826	+	0.994491i	$0.466572\pi$
$854$	0	0
$855$	0.630683	0.0215689
$856$	0	0
$857$	31.4384	1.07392	0.536958	−	0.843609i	$-0.319573\pi$
$857$	31.4384	1.07392	0.536958	+	0.843609i	$0.319573\pi$
$858$	0	0
$859$	−20.4384	−0.697351	−0.348675	−	0.937244i	$-0.613368\pi$
$859$	−20.4384	−0.697351	−0.348675	+	0.937244i	$0.613368\pi$
$860$	0	0
$861$	−9.12311	−0.310915
$862$	0	0
$863$	−2.49242	−0.0848430	−0.0424215	−	0.999100i	$-0.513507\pi$
$863$	−2.49242	−0.0848430	−0.0424215	+	0.999100i	$0.513507\pi$
$864$	0	0
$865$	−2.10795	−0.0716725
$866$	0	0
$867$	10.4384	0.354508
$868$	0	0
$869$	19.1231	0.648707
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4.43845	0.150219
$874$	0	0
$875$	−19.3693	−0.654802
$876$	0	0
$877$	−19.4384	−0.656390	−0.328195	−	0.944610i	$-0.606440\pi$
$877$	−19.4384	−0.656390	−0.328195	+	0.944610i	$0.606440\pi$
$878$	0	0
$879$	24.5616	0.828441
$880$	0	0
$881$	−37.9309	−1.27792	−0.638962	−	0.769239i	$-0.720636\pi$
$881$	−37.9309	−1.27792	−0.638962	+	0.769239i	$0.720636\pi$
$882$	0	0
$883$	11.8078	0.397363	0.198681	−	0.980064i	$-0.436334\pi$
$883$	11.8078	0.397363	0.198681	+	0.980064i	$0.436334\pi$
$884$	0	0
$885$	−6.24621	−0.209964
$886$	0	0
$887$	−49.3693	−1.65766	−0.828830	−	0.559501i	$-0.810993\pi$
$887$	−49.3693	−1.65766	−0.828830	+	0.559501i	$0.810993\pi$
$888$	0	0
$889$	34.0540	1.14213
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	9.26137	0.309920
$894$	0	0
$895$	−7.36932	−0.246329
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.87689	0.296061
$900$	0	0
$901$	29.9309	0.997142
$902$	0	0
$903$	−1.56155	−0.0519652
$904$	0	0
$905$	−5.43845	−0.180780
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−3.43845	−0.114046
$910$	0	0
$911$	10.7386	0.355787	0.177893	−	0.984050i	$-0.443072\pi$
$911$	10.7386	0.355787	0.177893	+	0.984050i	$0.443072\pi$
$912$	0	0
$913$	18.2462	0.603861
$914$	0	0
$915$	6.80776	0.225058
$916$	0	0
$917$	−61.8617	−2.04285
$918$	0	0
$919$	−44.4924	−1.46767	−0.733835	−	0.679328i	$-0.762271\pi$
$919$	−44.4924	−1.46767	−0.733835	+	0.679328i	$0.762271\pi$
$920$	0	0
$921$	−10.1922	−0.335846
$922$	0	0
$923$	0	0
$924$	0	0
$925$	16.1080	0.529626
$926$	0	0
$927$	7.56155	0.248354
$928$	0	0
$929$	−12.8078	−0.420209	−0.210105	−	0.977679i	$-0.567380\pi$
$929$	−12.8078	−0.420209	−0.210105	+	0.977679i	$0.567380\pi$
$930$	0	0
$931$	−6.38447	−0.209243
$932$	0	0
$933$	−10.8769	−0.356094
$934$	0	0
$935$	2.87689	0.0940845
$936$	0	0
$937$	−3.43845	−0.112329	−0.0561646	−	0.998422i	$-0.517887\pi$
$937$	−3.43845	−0.112329	−0.0561646	+	0.998422i	$0.517887\pi$
$938$	0	0
$939$	1.31534	0.0429245
$940$	0	0
$941$	2.49242	0.0812507	0.0406253	−	0.999174i	$-0.487065\pi$
$941$	2.49242	0.0812507	0.0406253	+	0.999174i	$0.487065\pi$
$942$	0	0
$943$	5.12311	0.166831
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	10.7386	0.348959	0.174479	−	0.984661i	$-0.444176\pi$
$947$	10.7386	0.348959	0.174479	+	0.984661i	$0.444176\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−23.0540	−0.747576
$952$	0	0
$953$	−34.9848	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$953$	−34.9848	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$954$	0	0
$955$	−0.492423	−0.0159344
$956$	0	0
$957$	−11.3693	−0.367518
$958$	0	0
$959$	−5.12311	−0.165434
$960$	0	0
$961$	−28.5616	−0.921340
$962$	0	0
$963$	−8.24621	−0.265730
$964$	0	0
$965$	10.9460	0.352365
$966$	0	0
$967$	−9.12311	−0.293379	−0.146690	−	0.989183i	$-0.546862\pi$
$967$	−9.12311	−0.293379	−0.146690	+	0.989183i	$0.546862\pi$
$968$	0	0
$969$	2.87689	0.0924192
$970$	0	0
$971$	−52.9848	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$971$	−52.9848	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$972$	0	0
$973$	38.9309	1.24807
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.8229	−0.506220	−0.253110	−	0.967438i	$-0.581453\pi$
$977$	−15.8229	−0.506220	−0.253110	+	0.967438i	$0.581453\pi$
$978$	0	0
$979$	26.2462	0.838833
$980$	0	0
$981$	−17.8078	−0.568558
$982$	0	0
$983$	−27.6155	−0.880799	−0.440399	−	0.897802i	$-0.645163\pi$
$983$	−27.6155	−0.880799	−0.440399	+	0.897802i	$0.645163\pi$
$984$	0	0
$985$	−6.38447	−0.203426
$986$	0	0
$987$	−29.3693	−0.934836
$988$	0	0
$989$	0.876894	0.0278836
$990$	0	0
$991$	−40.3542	−1.28189	−0.640946	−	0.767586i	$-0.721458\pi$
$991$	−40.3542	−1.28189	−0.640946	+	0.767586i	$0.721458\pi$
$992$	0	0
$993$	23.8078	0.755517
$994$	0	0
$995$	−13.0152	−0.412608
$996$	0	0
$997$	20.6155	0.652900	0.326450	−	0.945214i	$-0.394147\pi$
$997$	20.6155	0.652900	0.326450	+	0.945214i	$0.394147\pi$
$998$	0	0
$999$	3.43845	0.108788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bo.1.1		2
4.3	odd	2		507.2.a.d.1.1		2
12.11	even	2		1521.2.a.m.1.2		2
13.4	even	6		624.2.q.h.289.2		4
13.10	even	6		624.2.q.h.529.2		4
13.12	even	2		8112.2.a.bk.1.2		2
39.17	odd	6		1872.2.t.r.289.1		4
39.23	odd	6		1872.2.t.r.1153.1		4
52.3	odd	6		507.2.e.g.22.2		4
52.7	even	12		507.2.j.g.361.1		8
52.11	even	12		507.2.j.g.316.4		8
52.15	even	12		507.2.j.g.316.1		8
52.19	even	12		507.2.j.g.361.4		8
52.23	odd	6		39.2.e.b.22.1	yes	4
52.31	even	4		507.2.b.d.337.4		4
52.35	odd	6		507.2.e.g.484.2		4
52.43	odd	6		39.2.e.b.16.1	✓	4
52.47	even	4		507.2.b.d.337.1		4
52.51	odd	2		507.2.a.g.1.2		2
156.23	even	6		117.2.g.c.100.2		4
156.47	odd	4		1521.2.b.h.1351.4		4
156.83	odd	4		1521.2.b.h.1351.1		4
156.95	even	6		117.2.g.c.55.2		4
156.155	even	2		1521.2.a.g.1.1		2
260.23	even	12		975.2.bb.i.724.4		8
260.43	even	12		975.2.bb.i.874.1		8
260.127	even	12		975.2.bb.i.724.1		8
260.147	even	12		975.2.bb.i.874.4		8
260.179	odd	6		975.2.i.k.451.2		4
260.199	odd	6		975.2.i.k.601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.1	✓	4	52.43	odd	6
39.2.e.b.22.1	yes	4	52.23	odd	6
117.2.g.c.55.2		4	156.95	even	6
117.2.g.c.100.2		4	156.23	even	6
507.2.a.d.1.1		2	4.3	odd	2
507.2.a.g.1.2		2	52.51	odd	2
507.2.b.d.337.1		4	52.47	even	4
507.2.b.d.337.4		4	52.31	even	4
507.2.e.g.22.2		4	52.3	odd	6
507.2.e.g.484.2		4	52.35	odd	6
507.2.j.g.316.1		8	52.15	even	12
507.2.j.g.316.4		8	52.11	even	12
507.2.j.g.361.1		8	52.7	even	12
507.2.j.g.361.4		8	52.19	even	12
624.2.q.h.289.2		4	13.4	even	6
624.2.q.h.529.2		4	13.10	even	6
975.2.i.k.451.2		4	260.179	odd	6
975.2.i.k.601.2		4	260.199	odd	6
975.2.bb.i.724.1		8	260.127	even	12
975.2.bb.i.724.4		8	260.23	even	12
975.2.bb.i.874.1		8	260.43	even	12
975.2.bb.i.874.4		8	260.147	even	12
1521.2.a.g.1.1		2	156.155	even	2
1521.2.a.m.1.2		2	12.11	even	2
1521.2.b.h.1351.1		4	156.83	odd	4
1521.2.b.h.1351.4		4	156.47	odd	4
1872.2.t.r.289.1		4	39.17	odd	6
1872.2.t.r.1153.1		4	39.23	odd	6
8112.2.a.bk.1.2		2	13.12	even	2
8112.2.a.bo.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.561553 q^{5} -3.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.561553 q^{5} -3.56155 q^{7} +1.00000 q^{9} -2.00000 q^{11} +0.561553 q^{15} +2.56155 q^{17} -1.12311 q^{19} +3.56155 q^{21} -2.00000 q^{23} -4.68466 q^{25} -1.00000 q^{27} -5.68466 q^{29} -1.56155 q^{31} +2.00000 q^{33} +2.00000 q^{35} -3.43845 q^{37} -2.56155 q^{41} -0.438447 q^{43} -0.561553 q^{45} -8.24621 q^{47} +5.68466 q^{49} -2.56155 q^{51} +11.6847 q^{53} +1.12311 q^{55} +1.12311 q^{57} -11.1231 q^{59} +12.1231 q^{61} -3.56155 q^{63} +0.438447 q^{67} +2.00000 q^{69} +14.0000 q^{71} +1.87689 q^{73} +4.68466 q^{75} +7.12311 q^{77} -9.56155 q^{79} +1.00000 q^{81} -9.12311 q^{83} -1.43845 q^{85} +5.68466 q^{87} -13.1231 q^{89} +1.56155 q^{93} +0.630683 q^{95} +4.43845 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 4 q^{11} - 3 q^{15} + q^{17} + 6 q^{19} + 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + q^{31} + 4 q^{33} + 4 q^{35} - 11 q^{37} - q^{41} - 5 q^{43} + 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} - 6 q^{57} - 14 q^{59} + 16 q^{61} - 3 q^{63} + 5 q^{67} + 4 q^{69} + 28 q^{71} + 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} - 10 q^{83} - 7 q^{85} - q^{87} - 18 q^{89} - q^{93} + 26 q^{95} + 13 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 4 * q^11 - 3 * q^15 + q^17 + 6 * q^19 + 3 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + q^31 + 4 * q^33 + 4 * q^35 - 11 * q^37 - q^41 - 5 * q^43 + 3 * q^45 - q^49 - q^51 + 11 * q^53 - 6 * q^55 - 6 * q^57 - 14 * q^59 + 16 * q^61 - 3 * q^63 + 5 * q^67 + 4 * q^69 + 28 * q^71 + 12 * q^73 - 3 * q^75 + 6 * q^77 - 15 * q^79 + 2 * q^81 - 10 * q^83 - 7 * q^85 - q^87 - 18 * q^89 - q^93 + 26 * q^95 + 13 * q^97 - 4 * q^99

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