LMFDB - Embedded newform 6720.2.a.cz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6720,2,Mod(1,6720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

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

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

chi := DirichletCharacter("6720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6720.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:	$53.6594701583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.46410 q^{11} +3.46410 q^{13} +1.00000 q^{15} +2.00000 q^{17} -1.46410 q^{19} +1.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -1.46410 q^{31} -1.46410 q^{33} +1.00000 q^{35} -2.00000 q^{37} +3.46410 q^{39} +8.92820 q^{41} +4.00000 q^{43} +1.00000 q^{45} +2.92820 q^{47} +1.00000 q^{49} +2.00000 q^{51} +7.46410 q^{53} -1.46410 q^{55} -1.46410 q^{57} +13.8564 q^{59} -8.92820 q^{61} +1.00000 q^{63} +3.46410 q^{65} +6.92820 q^{67} -6.92820 q^{69} +4.39230 q^{71} +7.46410 q^{73} +1.00000 q^{75} -1.46410 q^{77} -4.00000 q^{79} +1.00000 q^{81} +1.07180 q^{83} +2.00000 q^{85} +2.00000 q^{87} +0.928203 q^{89} +3.46410 q^{91} -1.46410 q^{93} -1.46410 q^{95} +12.5359 q^{97} -1.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} + 8 q^{53} + 4 q^{55} + 4 q^{57} - 4 q^{61} + 2 q^{63} - 12 q^{71} + 8 q^{73} + 2 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} + 4 q^{87} - 12 q^{89} + 4 q^{93} + 4 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 + 8 * q^53 + 4 * q^55 + 4 * q^57 - 4 * q^61 + 2 * q^63 - 12 * q^71 + 8 * q^73 + 2 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 + 4 * q^87 - 12 * q^89 + 4 * q^93 + 4 * q^95 + 32 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	8.92820	1.39435	0.697176	−	0.716900i	$-0.254440\pi$
$41$	8.92820	1.39435	0.697176	+	0.716900i	$0.254440\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	7.46410	1.02527	0.512637	−	0.858606i	$-0.328669\pi$
$53$	7.46410	1.02527	0.512637	+	0.858606i	$0.328669\pi$
$54$	0	0
$55$	−1.46410	−0.197419
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	3.46410	0.429669
$66$	0	0
$67$	6.92820	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$67$	6.92820	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	4.39230	0.521271	0.260635	−	0.965437i	$-0.416068\pi$
$71$	4.39230	0.521271	0.260635	+	0.965437i	$0.416068\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.46410	−0.166850
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.07180	0.117645	0.0588225	−	0.998268i	$-0.481265\pi$
$83$	1.07180	0.117645	0.0588225	+	0.998268i	$0.481265\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	0	0
$93$	−1.46410	−0.151820
$94$	0	0
$95$	−1.46410	−0.150214
$96$	0	0
$97$	12.5359	1.27283	0.636414	−	0.771348i	$-0.280417\pi$
$97$	12.5359	1.27283	0.636414	+	0.771348i	$0.280417\pi$
$98$	0	0
$99$	−1.46410	−0.147148
$100$	0	0
$101$	11.8564	1.17976	0.589878	−	0.807492i	$-0.299176\pi$
$101$	11.8564	1.17976	0.589878	+	0.807492i	$0.299176\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	5.07180	0.490309	0.245155	−	0.969484i	$-0.421161\pi$
$107$	5.07180	0.490309	0.245155	+	0.969484i	$0.421161\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−10.3923	−0.977626	−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923	−0.977626	−0.488813	+	0.872389i	$0.662570\pi$
$114$	0	0
$115$	−6.92820	−0.646058
$116$	0	0
$117$	3.46410	0.320256
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	8.92820	0.805029
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.9282	−1.67961	−0.839803	−	0.542891i	$-0.817330\pi$
$127$	−18.9282	−1.67961	−0.839803	+	0.542891i	$0.817330\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	10.9282	0.954802	0.477401	−	0.878686i	$-0.341579\pi$
$131$	10.9282	0.954802	0.477401	+	0.878686i	$0.341579\pi$
$132$	0	0
$133$	−1.46410	−0.126954
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.46410	−0.637701	−0.318851	−	0.947805i	$-0.603297\pi$
$137$	−7.46410	−0.637701	−0.318851	+	0.947805i	$0.603297\pi$
$138$	0	0
$139$	−1.46410	−0.124183	−0.0620917	−	0.998070i	$-0.519777\pi$
$139$	−1.46410	−0.124183	−0.0620917	+	0.998070i	$0.519777\pi$
$140$	0	0
$141$	2.92820	0.246599
$142$	0	0
$143$	−5.07180	−0.424125
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	7.07180	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$149$	7.07180	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$150$	0	0
$151$	−1.07180	−0.0872216	−0.0436108	−	0.999049i	$-0.513886\pi$
$151$	−1.07180	−0.0872216	−0.0436108	+	0.999049i	$0.513886\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−1.46410	−0.117599
$156$	0	0
$157$	−7.46410	−0.595700	−0.297850	−	0.954613i	$-0.596270\pi$
$157$	−7.46410	−0.595700	−0.297850	+	0.954613i	$0.596270\pi$
$158$	0	0
$159$	7.46410	0.591942
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−1.46410	−0.113980
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−1.46410	−0.111963
$172$	0	0
$173$	11.0718	0.841773	0.420887	−	0.907113i	$-0.361719\pi$
$173$	11.0718	0.841773	0.420887	+	0.907113i	$0.361719\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	13.8564	1.04151
$178$	0	0
$179$	−9.46410	−0.707380	−0.353690	−	0.935363i	$-0.615073\pi$
$179$	−9.46410	−0.707380	−0.353690	+	0.935363i	$0.615073\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	−8.92820	−0.659992
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−2.92820	−0.214131
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.3923	−0.896676	−0.448338	−	0.893864i	$-0.647984\pi$
$191$	−12.3923	−0.896676	−0.448338	+	0.893864i	$0.647984\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	0	0
$195$	3.46410	0.248069
$196$	0	0
$197$	12.5359	0.893146	0.446573	−	0.894747i	$-0.352644\pi$
$197$	12.5359	0.893146	0.446573	+	0.894747i	$0.352644\pi$
$198$	0	0
$199$	20.3923	1.44557	0.722786	−	0.691072i	$-0.242861\pi$
$199$	20.3923	1.44557	0.722786	+	0.691072i	$0.242861\pi$
$200$	0	0
$201$	6.92820	0.488678
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	8.92820	0.623573
$206$	0	0
$207$	−6.92820	−0.481543
$208$	0	0
$209$	2.14359	0.148275
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	4.39230	0.300956
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−1.46410	−0.0993897
$218$	0	0
$219$	7.46410	0.504377
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	−13.0718	−0.875352	−0.437676	−	0.899133i	$-0.644198\pi$
$223$	−13.0718	−0.875352	−0.437676	+	0.899133i	$0.644198\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	0	0
$229$	18.7846	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$229$	18.7846	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$230$	0	0
$231$	−1.46410	−0.0963308
$232$	0	0
$233$	−12.5359	−0.821254	−0.410627	−	0.911803i	$-0.634690\pi$
$233$	−12.5359	−0.821254	−0.410627	+	0.911803i	$0.634690\pi$
$234$	0	0
$235$	2.92820	0.191015
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−15.3205	−0.991001	−0.495501	−	0.868608i	$-0.665015\pi$
$239$	−15.3205	−0.991001	−0.495501	+	0.868608i	$0.665015\pi$
$240$	0	0
$241$	26.7846	1.72535	0.862674	−	0.505760i	$-0.168788\pi$
$241$	26.7846	1.72535	0.862674	+	0.505760i	$0.168788\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.07180	−0.322711
$248$	0	0
$249$	1.07180	0.0679224
$250$	0	0
$251$	29.8564	1.88452	0.942260	−	0.334883i	$-0.108697\pi$
$251$	29.8564	1.88452	0.942260	+	0.334883i	$0.108697\pi$
$252$	0	0
$253$	10.1436	0.637722
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−16.9282	−1.05595	−0.527976	−	0.849259i	$-0.677049\pi$
$257$	−16.9282	−1.05595	−0.527976	+	0.849259i	$0.677049\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−9.85641	−0.607772	−0.303886	−	0.952708i	$-0.598284\pi$
$263$	−9.85641	−0.607772	−0.303886	+	0.952708i	$0.598284\pi$
$264$	0	0
$265$	7.46410	0.458516
$266$	0	0
$267$	0.928203	0.0568051
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−6.53590	−0.397028	−0.198514	−	0.980098i	$-0.563611\pi$
$271$	−6.53590	−0.397028	−0.198514	+	0.980098i	$0.563611\pi$
$272$	0	0
$273$	3.46410	0.209657
$274$	0	0
$275$	−1.46410	−0.0882886
$276$	0	0
$277$	−15.8564	−0.952719	−0.476360	−	0.879251i	$-0.658044\pi$
$277$	−15.8564	−0.952719	−0.476360	+	0.879251i	$0.658044\pi$
$278$	0	0
$279$	−1.46410	−0.0876535
$280$	0	0
$281$	10.7846	0.643356	0.321678	−	0.946849i	$-0.395753\pi$
$281$	10.7846	0.643356	0.321678	+	0.946849i	$0.395753\pi$
$282$	0	0
$283$	−9.85641	−0.585903	−0.292951	−	0.956127i	$-0.594637\pi$
$283$	−9.85641	−0.585903	−0.292951	+	0.956127i	$0.594637\pi$
$284$	0	0
$285$	−1.46410	−0.0867259
$286$	0	0
$287$	8.92820	0.527015
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.5359	0.734867
$292$	0	0
$293$	11.0718	0.646821	0.323411	−	0.946259i	$-0.395170\pi$
$293$	11.0718	0.646821	0.323411	+	0.946259i	$0.395170\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	0	0
$297$	−1.46410	−0.0849558
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	11.8564	0.681133
$304$	0	0
$305$	−8.92820	−0.511227
$306$	0	0
$307$	−9.07180	−0.517755	−0.258877	−	0.965910i	$-0.583353\pi$
$307$	−9.07180	−0.517755	−0.258877	+	0.965910i	$0.583353\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−17.3205	−0.979013	−0.489506	−	0.872000i	$-0.662823\pi$
$313$	−17.3205	−0.979013	−0.489506	+	0.872000i	$0.662823\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	21.3205	1.19748	0.598740	−	0.800944i	$-0.295668\pi$
$317$	21.3205	1.19748	0.598740	+	0.800944i	$0.295668\pi$
$318$	0	0
$319$	−2.92820	−0.163948
$320$	0	0
$321$	5.07180	0.283080
$322$	0	0
$323$	−2.92820	−0.162930
$324$	0	0
$325$	3.46410	0.192154
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	2.92820	0.161437
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	6.92820	0.378528
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	−10.3923	−0.564433
$340$	0	0
$341$	2.14359	0.116082
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.92820	−0.373002
$346$	0	0
$347$	−13.8564	−0.743851	−0.371925	−	0.928263i	$-0.621302\pi$
$347$	−13.8564	−0.743851	−0.371925	+	0.928263i	$0.621302\pi$
$348$	0	0
$349$	−3.07180	−0.164430	−0.0822148	−	0.996615i	$-0.526199\pi$
$349$	−3.07180	−0.164430	−0.0822148	+	0.996615i	$0.526199\pi$
$350$	0	0
$351$	3.46410	0.184900
$352$	0	0
$353$	31.8564	1.69555	0.847773	−	0.530360i	$-0.177943\pi$
$353$	31.8564	1.69555	0.847773	+	0.530360i	$0.177943\pi$
$354$	0	0
$355$	4.39230	0.233119
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	17.4641	0.921720	0.460860	−	0.887473i	$-0.347541\pi$
$359$	17.4641	0.921720	0.460860	+	0.887473i	$0.347541\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	−8.85641	−0.464841
$364$	0	0
$365$	7.46410	0.390689
$366$	0	0
$367$	−32.7846	−1.71134	−0.855671	−	0.517520i	$-0.826855\pi$
$367$	−32.7846	−1.71134	−0.855671	+	0.517520i	$0.826855\pi$
$368$	0	0
$369$	8.92820	0.464784
$370$	0	0
$371$	7.46410	0.387517
$372$	0	0
$373$	32.9282	1.70496	0.852479	−	0.522762i	$-0.175098\pi$
$373$	32.9282	1.70496	0.852479	+	0.522762i	$0.175098\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	6.92820	0.356821
$378$	0	0
$379$	−17.0718	−0.876919	−0.438460	−	0.898751i	$-0.644476\pi$
$379$	−17.0718	−0.876919	−0.438460	+	0.898751i	$0.644476\pi$
$380$	0	0
$381$	−18.9282	−0.969721
$382$	0	0
$383$	−5.85641	−0.299248	−0.149624	−	0.988743i	$-0.547806\pi$
$383$	−5.85641	−0.299248	−0.149624	+	0.988743i	$0.547806\pi$
$384$	0	0
$385$	−1.46410	−0.0746175
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−22.7846	−1.15523	−0.577613	−	0.816311i	$-0.696016\pi$
$389$	−22.7846	−1.15523	−0.577613	+	0.816311i	$0.696016\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	0	0
$393$	10.9282	0.551255
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−4.53590	−0.227650	−0.113825	−	0.993501i	$-0.536310\pi$
$397$	−4.53590	−0.227650	−0.113825	+	0.993501i	$0.536310\pi$
$398$	0	0
$399$	−1.46410	−0.0732968
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−5.07180	−0.252644
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	2.92820	0.145146
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−7.46410	−0.368177
$412$	0	0
$413$	13.8564	0.681829
$414$	0	0
$415$	1.07180	0.0526124
$416$	0	0
$417$	−1.46410	−0.0716974
$418$	0	0
$419$	−32.7846	−1.60163	−0.800816	−	0.598910i	$-0.795601\pi$
$419$	−32.7846	−1.60163	−0.800816	+	0.598910i	$0.795601\pi$
$420$	0	0
$421$	−11.8564	−0.577846	−0.288923	−	0.957352i	$-0.593297\pi$
$421$	−11.8564	−0.577846	−0.288923	+	0.957352i	$0.593297\pi$
$422$	0	0
$423$	2.92820	0.142374
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−8.92820	−0.432066
$428$	0	0
$429$	−5.07180	−0.244869
$430$	0	0
$431$	26.2487	1.26436	0.632178	−	0.774823i	$-0.282161\pi$
$431$	26.2487	1.26436	0.632178	+	0.774823i	$0.282161\pi$
$432$	0	0
$433$	28.5359	1.37135	0.685674	−	0.727909i	$-0.259508\pi$
$433$	28.5359	1.37135	0.685674	+	0.727909i	$0.259508\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	10.1436	0.485234
$438$	0	0
$439$	−7.32051	−0.349389	−0.174694	−	0.984623i	$-0.555894\pi$
$439$	−7.32051	−0.349389	−0.174694	+	0.984623i	$0.555894\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0.928203	0.0440011
$446$	0	0
$447$	7.07180	0.334485
$448$	0	0
$449$	−3.85641	−0.181995	−0.0909975	−	0.995851i	$-0.529006\pi$
$449$	−3.85641	−0.181995	−0.0909975	+	0.995851i	$0.529006\pi$
$450$	0	0
$451$	−13.0718	−0.615527
$452$	0	0
$453$	−1.07180	−0.0503574
$454$	0	0
$455$	3.46410	0.162400
$456$	0	0
$457$	26.7846	1.25293	0.626466	−	0.779449i	$-0.284501\pi$
$457$	26.7846	1.25293	0.626466	+	0.779449i	$0.284501\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−31.8564	−1.48370	−0.741850	−	0.670565i	$-0.766052\pi$
$461$	−31.8564	−1.48370	−0.741850	+	0.670565i	$0.766052\pi$
$462$	0	0
$463$	−21.0718	−0.979289	−0.489645	−	0.871922i	$-0.662874\pi$
$463$	−21.0718	−0.979289	−0.489645	+	0.871922i	$0.662874\pi$
$464$	0	0
$465$	−1.46410	−0.0678961
$466$	0	0
$467$	−17.0718	−0.789989	−0.394994	−	0.918684i	$-0.629253\pi$
$467$	−17.0718	−0.789989	−0.394994	+	0.918684i	$0.629253\pi$
$468$	0	0
$469$	6.92820	0.319915
$470$	0	0
$471$	−7.46410	−0.343928
$472$	0	0
$473$	−5.85641	−0.269278
$474$	0	0
$475$	−1.46410	−0.0671776
$476$	0	0
$477$	7.46410	0.341758
$478$	0	0
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	0	0
$483$	−6.92820	−0.315244
$484$	0	0
$485$	12.5359	0.569226
$486$	0	0
$487$	24.7846	1.12310	0.561549	−	0.827444i	$-0.310206\pi$
$487$	24.7846	1.12310	0.561549	+	0.827444i	$0.310206\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	6.53590	0.294961	0.147480	−	0.989065i	$-0.452884\pi$
$491$	6.53590	0.294961	0.147480	+	0.989065i	$0.452884\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−1.46410	−0.0658065
$496$	0	0
$497$	4.39230	0.197022
$498$	0	0
$499$	−9.07180	−0.406109	−0.203055	−	0.979167i	$-0.565087\pi$
$499$	−9.07180	−0.406109	−0.203055	+	0.979167i	$0.565087\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	11.8564	0.527603
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	33.7128	1.49429	0.747147	−	0.664659i	$-0.231423\pi$
$509$	33.7128	1.49429	0.747147	+	0.664659i	$0.231423\pi$
$510$	0	0
$511$	7.46410	0.330192
$512$	0	0
$513$	−1.46410	−0.0646417
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−4.28719	−0.188550
$518$	0	0
$519$	11.0718	0.485998
$520$	0	0
$521$	−40.6410	−1.78052	−0.890258	−	0.455457i	$-0.849476\pi$
$521$	−40.6410	−1.78052	−0.890258	+	0.455457i	$0.849476\pi$
$522$	0	0
$523$	12.7846	0.559032	0.279516	−	0.960141i	$-0.409826\pi$
$523$	12.7846	0.559032	0.279516	+	0.960141i	$0.409826\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−2.92820	−0.127555
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	13.8564	0.601317
$532$	0	0
$533$	30.9282	1.33965
$534$	0	0
$535$	5.07180	0.219273
$536$	0	0
$537$	−9.46410	−0.408406
$538$	0	0
$539$	−1.46410	−0.0630633
$540$	0	0
$541$	−27.8564	−1.19764	−0.598820	−	0.800883i	$-0.704364\pi$
$541$	−27.8564	−1.19764	−0.598820	+	0.800883i	$0.704364\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−25.0718	−1.07199	−0.535996	−	0.844220i	$-0.680064\pi$
$547$	−25.0718	−1.07199	−0.535996	+	0.844220i	$0.680064\pi$
$548$	0	0
$549$	−8.92820	−0.381046
$550$	0	0
$551$	−2.92820	−0.124746
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	−30.3923	−1.28776	−0.643882	−	0.765125i	$-0.722677\pi$
$557$	−30.3923	−1.28776	−0.643882	+	0.765125i	$0.722677\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	−2.92820	−0.123629
$562$	0	0
$563$	25.8564	1.08972	0.544859	−	0.838528i	$-0.316583\pi$
$563$	25.8564	1.08972	0.544859	+	0.838528i	$0.316583\pi$
$564$	0	0
$565$	−10.3923	−0.437208
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−19.8564	−0.832424	−0.416212	−	0.909268i	$-0.636643\pi$
$569$	−19.8564	−0.832424	−0.416212	+	0.909268i	$0.636643\pi$
$570$	0	0
$571$	−4.78461	−0.200230	−0.100115	−	0.994976i	$-0.531921\pi$
$571$	−4.78461	−0.200230	−0.100115	+	0.994976i	$0.531921\pi$
$572$	0	0
$573$	−12.3923	−0.517696
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	−1.32051	−0.0549735	−0.0274867	−	0.999622i	$-0.508750\pi$
$577$	−1.32051	−0.0549735	−0.0274867	+	0.999622i	$0.508750\pi$
$578$	0	0
$579$	15.8564	0.658970
$580$	0	0
$581$	1.07180	0.0444656
$582$	0	0
$583$	−10.9282	−0.452600
$584$	0	0
$585$	3.46410	0.143223
$586$	0	0
$587$	17.8564	0.737013	0.368506	−	0.929625i	$-0.379869\pi$
$587$	17.8564	0.737013	0.368506	+	0.929625i	$0.379869\pi$
$588$	0	0
$589$	2.14359	0.0883252
$590$	0	0
$591$	12.5359	0.515658
$592$	0	0
$593$	45.7128	1.87720	0.938600	−	0.345007i	$-0.112123\pi$
$593$	45.7128	1.87720	0.938600	+	0.345007i	$0.112123\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	20.3923	0.834601
$598$	0	0
$599$	33.4641	1.36731	0.683653	−	0.729807i	$-0.260390\pi$
$599$	33.4641	1.36731	0.683653	+	0.729807i	$0.260390\pi$
$600$	0	0
$601$	15.0718	0.614791	0.307396	−	0.951582i	$-0.400542\pi$
$601$	15.0718	0.614791	0.307396	+	0.951582i	$0.400542\pi$
$602$	0	0
$603$	6.92820	0.282138
$604$	0	0
$605$	−8.85641	−0.360064
$606$	0	0
$607$	−21.0718	−0.855278	−0.427639	−	0.903950i	$-0.640655\pi$
$607$	−21.0718	−0.855278	−0.427639	+	0.903950i	$0.640655\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	10.1436	0.410366
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	8.92820	0.360020
$616$	0	0
$617$	−34.3923	−1.38458	−0.692291	−	0.721618i	$-0.743399\pi$
$617$	−34.3923	−1.38458	−0.692291	+	0.721618i	$0.743399\pi$
$618$	0	0
$619$	−11.6077	−0.466553	−0.233276	−	0.972410i	$-0.574945\pi$
$619$	−11.6077	−0.466553	−0.233276	+	0.972410i	$0.574945\pi$
$620$	0	0
$621$	−6.92820	−0.278019
$622$	0	0
$623$	0.928203	0.0371877
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.14359	0.0856069
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−39.7128	−1.58094	−0.790471	−	0.612499i	$-0.790164\pi$
$631$	−39.7128	−1.58094	−0.790471	+	0.612499i	$0.790164\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−18.9282	−0.751143
$636$	0	0
$637$	3.46410	0.137253
$638$	0	0
$639$	4.39230	0.173757
$640$	0	0
$641$	2.78461	0.109985	0.0549927	−	0.998487i	$-0.482486\pi$
$641$	2.78461	0.109985	0.0549927	+	0.998487i	$0.482486\pi$
$642$	0	0
$643$	−3.21539	−0.126803	−0.0634013	−	0.997988i	$-0.520195\pi$
$643$	−3.21539	−0.126803	−0.0634013	+	0.997988i	$0.520195\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	−13.8564	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$647$	−13.8564	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$648$	0	0
$649$	−20.2872	−0.796342
$650$	0	0
$651$	−1.46410	−0.0573827
$652$	0	0
$653$	20.5359	0.803632	0.401816	−	0.915720i	$-0.368379\pi$
$653$	20.5359	0.803632	0.401816	+	0.915720i	$0.368379\pi$
$654$	0	0
$655$	10.9282	0.427000
$656$	0	0
$657$	7.46410	0.291202
$658$	0	0
$659$	−30.5359	−1.18951	−0.594755	−	0.803907i	$-0.702751\pi$
$659$	−30.5359	−1.18951	−0.594755	+	0.803907i	$0.702751\pi$
$660$	0	0
$661$	20.9282	0.814013	0.407006	−	0.913425i	$-0.366573\pi$
$661$	20.9282	0.814013	0.407006	+	0.913425i	$0.366573\pi$
$662$	0	0
$663$	6.92820	0.269069
$664$	0	0
$665$	−1.46410	−0.0567754
$666$	0	0
$667$	−13.8564	−0.536522
$668$	0	0
$669$	−13.0718	−0.505385
$670$	0	0
$671$	13.0718	0.504631
$672$	0	0
$673$	15.0718	0.580975	0.290488	−	0.956879i	$-0.406183\pi$
$673$	15.0718	0.580975	0.290488	+	0.956879i	$0.406183\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	30.7846	1.18315	0.591574	−	0.806250i	$-0.298507\pi$
$677$	30.7846	1.18315	0.591574	+	0.806250i	$0.298507\pi$
$678$	0	0
$679$	12.5359	0.481084
$680$	0	0
$681$	6.92820	0.265489
$682$	0	0
$683$	21.8564	0.836312	0.418156	−	0.908375i	$-0.362677\pi$
$683$	21.8564	0.836312	0.418156	+	0.908375i	$0.362677\pi$
$684$	0	0
$685$	−7.46410	−0.285189
$686$	0	0
$687$	18.7846	0.716678
$688$	0	0
$689$	25.8564	0.985051
$690$	0	0
$691$	−42.2487	−1.60722	−0.803608	−	0.595158i	$-0.797089\pi$
$691$	−42.2487	−1.60722	−0.803608	+	0.595158i	$0.797089\pi$
$692$	0	0
$693$	−1.46410	−0.0556166
$694$	0	0
$695$	−1.46410	−0.0555365
$696$	0	0
$697$	17.8564	0.676360
$698$	0	0
$699$	−12.5359	−0.474151
$700$	0	0
$701$	7.85641	0.296732	0.148366	−	0.988932i	$-0.452599\pi$
$701$	7.85641	0.296732	0.148366	+	0.988932i	$0.452599\pi$
$702$	0	0
$703$	2.92820	0.110439
$704$	0	0
$705$	2.92820	0.110283
$706$	0	0
$707$	11.8564	0.445906
$708$	0	0
$709$	−17.7128	−0.665219	−0.332609	−	0.943065i	$-0.607929\pi$
$709$	−17.7128	−0.665219	−0.332609	+	0.943065i	$0.607929\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	10.1436	0.379881
$714$	0	0
$715$	−5.07180	−0.189674
$716$	0	0
$717$	−15.3205	−0.572155
$718$	0	0
$719$	24.7846	0.924310	0.462155	−	0.886799i	$-0.347076\pi$
$719$	24.7846	0.924310	0.462155	+	0.886799i	$0.347076\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	26.7846	0.996130
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	2.92820	0.108601	0.0543005	−	0.998525i	$-0.482707\pi$
$727$	2.92820	0.108601	0.0543005	+	0.998525i	$0.482707\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−10.3923	−0.383849	−0.191924	−	0.981410i	$-0.561473\pi$
$733$	−10.3923	−0.383849	−0.191924	+	0.981410i	$0.561473\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−10.1436	−0.373644
$738$	0	0
$739$	−28.7846	−1.05886	−0.529429	−	0.848354i	$-0.677594\pi$
$739$	−28.7846	−1.05886	−0.529429	+	0.848354i	$0.677594\pi$
$740$	0	0
$741$	−5.07180	−0.186317
$742$	0	0
$743$	1.85641	0.0681049	0.0340525	−	0.999420i	$-0.489159\pi$
$743$	1.85641	0.0681049	0.0340525	+	0.999420i	$0.489159\pi$
$744$	0	0
$745$	7.07180	0.259091
$746$	0	0
$747$	1.07180	0.0392150
$748$	0	0
$749$	5.07180	0.185319
$750$	0	0
$751$	37.5692	1.37092	0.685460	−	0.728110i	$-0.259601\pi$
$751$	37.5692	1.37092	0.685460	+	0.728110i	$0.259601\pi$
$752$	0	0
$753$	29.8564	1.08803
$754$	0	0
$755$	−1.07180	−0.0390067
$756$	0	0
$757$	32.9282	1.19680	0.598398	−	0.801199i	$-0.295804\pi$
$757$	32.9282	1.19680	0.598398	+	0.801199i	$0.295804\pi$
$758$	0	0
$759$	10.1436	0.368189
$760$	0	0
$761$	−44.9282	−1.62865	−0.814323	−	0.580412i	$-0.802892\pi$
$761$	−44.9282	−1.62865	−0.814323	+	0.580412i	$0.802892\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	40.6410	1.46555	0.732776	−	0.680470i	$-0.238224\pi$
$769$	40.6410	1.46555	0.732776	+	0.680470i	$0.238224\pi$
$770$	0	0
$771$	−16.9282	−0.609654
$772$	0	0
$773$	−4.92820	−0.177255	−0.0886276	−	0.996065i	$-0.528248\pi$
$773$	−4.92820	−0.177255	−0.0886276	+	0.996065i	$0.528248\pi$
$774$	0	0
$775$	−1.46410	−0.0525921
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	−13.0718	−0.468346
$780$	0	0
$781$	−6.43078	−0.230111
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−7.46410	−0.266405
$786$	0	0
$787$	−28.7846	−1.02606	−0.513030	−	0.858371i	$-0.671477\pi$
$787$	−28.7846	−1.02606	−0.513030	+	0.858371i	$0.671477\pi$
$788$	0	0
$789$	−9.85641	−0.350897
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−30.9282	−1.09829
$794$	0	0
$795$	7.46410	0.264724
$796$	0	0
$797$	−36.9282	−1.30806	−0.654032	−	0.756467i	$-0.726924\pi$
$797$	−36.9282	−1.30806	−0.654032	+	0.756467i	$0.726924\pi$
$798$	0	0
$799$	5.85641	0.207185
$800$	0	0
$801$	0.928203	0.0327964
$802$	0	0
$803$	−10.9282	−0.385648
$804$	0	0
$805$	−6.92820	−0.244187
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	2.78461	0.0979017	0.0489508	−	0.998801i	$-0.484412\pi$
$809$	2.78461	0.0979017	0.0489508	+	0.998801i	$0.484412\pi$
$810$	0	0
$811$	−21.1769	−0.743622	−0.371811	−	0.928308i	$-0.621263\pi$
$811$	−21.1769	−0.743622	−0.371811	+	0.928308i	$0.621263\pi$
$812$	0	0
$813$	−6.53590	−0.229224
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	−5.85641	−0.204890
$818$	0	0
$819$	3.46410	0.121046
$820$	0	0
$821$	29.7128	1.03698	0.518492	−	0.855082i	$-0.326494\pi$
$821$	29.7128	1.03698	0.518492	+	0.855082i	$0.326494\pi$
$822$	0	0
$823$	−32.7846	−1.14280	−0.571400	−	0.820672i	$-0.693599\pi$
$823$	−32.7846	−1.14280	−0.571400	+	0.820672i	$0.693599\pi$
$824$	0	0
$825$	−1.46410	−0.0509735
$826$	0	0
$827$	2.92820	0.101824	0.0509118	−	0.998703i	$-0.483787\pi$
$827$	2.92820	0.101824	0.0509118	+	0.998703i	$0.483787\pi$
$828$	0	0
$829$	23.8564	0.828567	0.414284	−	0.910148i	$-0.364032\pi$
$829$	23.8564	0.828567	0.414284	+	0.910148i	$0.364032\pi$
$830$	0	0
$831$	−15.8564	−0.550053
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−1.46410	−0.0506068
$838$	0	0
$839$	−8.78461	−0.303278	−0.151639	−	0.988436i	$-0.548455\pi$
$839$	−8.78461	−0.303278	−0.151639	+	0.988436i	$0.548455\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.7846	0.371442
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−8.85641	−0.304310
$848$	0	0
$849$	−9.85641	−0.338271
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	−2.39230	−0.0819110	−0.0409555	−	0.999161i	$-0.513040\pi$
$853$	−2.39230	−0.0819110	−0.0409555	+	0.999161i	$0.513040\pi$
$854$	0	0
$855$	−1.46410	−0.0500712
$856$	0	0
$857$	−24.9282	−0.851531	−0.425766	−	0.904833i	$-0.639995\pi$
$857$	−24.9282	−0.851531	−0.425766	+	0.904833i	$0.639995\pi$
$858$	0	0
$859$	−44.3923	−1.51465	−0.757323	−	0.653041i	$-0.773493\pi$
$859$	−44.3923	−1.51465	−0.757323	+	0.653041i	$0.773493\pi$
$860$	0	0
$861$	8.92820	0.304272
$862$	0	0
$863$	46.9282	1.59745	0.798727	−	0.601693i	$-0.205507\pi$
$863$	46.9282	1.59745	0.798727	+	0.601693i	$0.205507\pi$
$864$	0	0
$865$	11.0718	0.376452
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	5.85641	0.198665
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	12.5359	0.424276
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	11.0718	0.373442
$880$	0	0
$881$	5.21539	0.175711	0.0878555	−	0.996133i	$-0.471999\pi$
$881$	5.21539	0.175711	0.0878555	+	0.996133i	$0.471999\pi$
$882$	0	0
$883$	−42.6410	−1.43498	−0.717492	−	0.696567i	$-0.754710\pi$
$883$	−42.6410	−1.43498	−0.717492	+	0.696567i	$0.754710\pi$
$884$	0	0
$885$	13.8564	0.465778
$886$	0	0
$887$	−10.9282	−0.366933	−0.183467	−	0.983026i	$-0.558732\pi$
$887$	−10.9282	−0.366933	−0.183467	+	0.983026i	$0.558732\pi$
$888$	0	0
$889$	−18.9282	−0.634832
$890$	0	0
$891$	−1.46410	−0.0490492
$892$	0	0
$893$	−4.28719	−0.143465
$894$	0	0
$895$	−9.46410	−0.316350
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	−2.92820	−0.0976610
$900$	0	0
$901$	14.9282	0.497331
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	28.7846	0.955777	0.477889	−	0.878420i	$-0.341402\pi$
$907$	28.7846	0.955777	0.477889	+	0.878420i	$0.341402\pi$
$908$	0	0
$909$	11.8564	0.393252
$910$	0	0
$911$	−26.2487	−0.869659	−0.434829	−	0.900513i	$-0.643191\pi$
$911$	−26.2487	−0.869659	−0.434829	+	0.900513i	$0.643191\pi$
$912$	0	0
$913$	−1.56922	−0.0519336
$914$	0	0
$915$	−8.92820	−0.295157
$916$	0	0
$917$	10.9282	0.360881
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	−9.07180	−0.298926
$922$	0	0
$923$	15.2154	0.500821
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−16.6410	−0.545974	−0.272987	−	0.962018i	$-0.588012\pi$
$929$	−16.6410	−0.545974	−0.272987	+	0.962018i	$0.588012\pi$
$930$	0	0
$931$	−1.46410	−0.0479840
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−2.92820	−0.0957625
$936$	0	0
$937$	35.1769	1.14918	0.574590	−	0.818442i	$-0.305162\pi$
$937$	35.1769	1.14918	0.574590	+	0.818442i	$0.305162\pi$
$938$	0	0
$939$	−17.3205	−0.565233
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−61.8564	−2.01432
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	25.8564	0.839334
$950$	0	0
$951$	21.3205	0.691365
$952$	0	0
$953$	−14.6795	−0.475515	−0.237758	−	0.971324i	$-0.576412\pi$
$953$	−14.6795	−0.475515	−0.237758	+	0.971324i	$0.576412\pi$
$954$	0	0
$955$	−12.3923	−0.401006
$956$	0	0
$957$	−2.92820	−0.0946554
$958$	0	0
$959$	−7.46410	−0.241028
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	5.07180	0.163436
$964$	0	0
$965$	15.8564	0.510436
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	−2.92820	−0.0940674
$970$	0	0
$971$	−37.8564	−1.21487	−0.607435	−	0.794369i	$-0.707801\pi$
$971$	−37.8564	−1.21487	−0.607435	+	0.794369i	$0.707801\pi$
$972$	0	0
$973$	−1.46410	−0.0469369
$974$	0	0
$975$	3.46410	0.110940
$976$	0	0
$977$	−34.3923	−1.10031	−0.550154	−	0.835063i	$-0.685431\pi$
$977$	−34.3923	−1.10031	−0.550154	+	0.835063i	$0.685431\pi$
$978$	0	0
$979$	−1.35898	−0.0434333
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	0.784610	0.0250252	0.0125126	−	0.999922i	$-0.496017\pi$
$983$	0.784610	0.0250252	0.0125126	+	0.999922i	$0.496017\pi$
$984$	0	0
$985$	12.5359	0.399427
$986$	0	0
$987$	2.92820	0.0932057
$988$	0	0
$989$	−27.7128	−0.881216
$990$	0	0
$991$	−59.4256	−1.88772	−0.943859	−	0.330350i	$-0.892833\pi$
$991$	−59.4256	−1.88772	−0.943859	+	0.330350i	$0.892833\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	20.3923	0.646480
$996$	0	0
$997$	−48.2487	−1.52805	−0.764026	−	0.645185i	$-0.776780\pi$
$997$	−48.2487	−1.52805	−0.764026	+	0.645185i	$0.776780\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.cz.1.1		2
4.3	odd	2		6720.2.a.cq.1.2		2
8.3	odd	2		3360.2.a.bf.1.1	yes	2
8.5	even	2		3360.2.a.bb.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bb.1.2	✓	2	8.5	even	2
3360.2.a.bf.1.1	yes	2	8.3	odd	2
6720.2.a.cq.1.2		2	4.3	odd	2
6720.2.a.cz.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:	\( N \)	\(=\)	\( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.46410 q^{11} +3.46410 q^{13} +1.00000 q^{15} +2.00000 q^{17} -1.46410 q^{19} +1.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -1.46410 q^{31} -1.46410 q^{33} +1.00000 q^{35} -2.00000 q^{37} +3.46410 q^{39} +8.92820 q^{41} +4.00000 q^{43} +1.00000 q^{45} +2.92820 q^{47} +1.00000 q^{49} +2.00000 q^{51} +7.46410 q^{53} -1.46410 q^{55} -1.46410 q^{57} +13.8564 q^{59} -8.92820 q^{61} +1.00000 q^{63} +3.46410 q^{65} +6.92820 q^{67} -6.92820 q^{69} +4.39230 q^{71} +7.46410 q^{73} +1.00000 q^{75} -1.46410 q^{77} -4.00000 q^{79} +1.00000 q^{81} +1.07180 q^{83} +2.00000 q^{85} +2.00000 q^{87} +0.928203 q^{89} +3.46410 q^{91} -1.46410 q^{93} -1.46410 q^{95} +12.5359 q^{97} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{31} + 4 q^{33} + 2 q^{35} - 4 q^{37} + 4 q^{41} + 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} + 4 q^{51} + 8 q^{53} + 4 q^{55} + 4 q^{57} - 4 q^{61} + 2 q^{63} - 12 q^{71} + 8 q^{73} + 2 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} + 4 q^{87} - 12 q^{89} + 4 q^{93} + 4 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 2 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^31 + 4 * q^33 + 2 * q^35 - 4 * q^37 + 4 * q^41 + 8 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 + 4 * q^51 + 8 * q^53 + 4 * q^55 + 4 * q^57 - 4 * q^61 + 2 * q^63 - 12 * q^71 + 8 * q^73 + 2 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 + 4 * q^87 - 12 * q^89 + 4 * q^93 + 4 * q^95 + 32 * q^97 + 4 * q^99

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