LMFDB - Embedded newform 6160.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} -1.46410 q^{13} +2.73205 q^{15} -3.46410 q^{17} -6.73205 q^{19} +2.73205 q^{21} +8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -4.73205 q^{29} -2.00000 q^{31} +2.73205 q^{33} +1.00000 q^{35} +0.732051 q^{37} +4.00000 q^{39} -2.19615 q^{41} -2.00000 q^{43} -4.46410 q^{45} -6.92820 q^{47} +1.00000 q^{49} +9.46410 q^{51} -7.26795 q^{53} +1.00000 q^{55} +18.3923 q^{57} -6.92820 q^{59} -4.92820 q^{61} -4.46410 q^{63} +1.46410 q^{65} +4.00000 q^{67} -22.3923 q^{69} -9.46410 q^{71} -14.3923 q^{73} -2.73205 q^{75} +1.00000 q^{77} +12.1962 q^{79} -2.46410 q^{81} +16.3923 q^{83} +3.46410 q^{85} +12.9282 q^{87} +3.46410 q^{89} +1.46410 q^{91} +5.46410 q^{93} +6.73205 q^{95} +14.5885 q^{97} -4.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 - 10 * q^19 + 2 * q^21 + 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 2 * q^35 - 2 * q^37 + 8 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^49 + 12 * q^51 - 18 * q^53 + 2 * q^55 + 16 * q^57 + 4 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 - 24 * q^69 - 12 * q^71 - 8 * q^73 - 2 * q^75 + 2 * q^77 + 14 * q^79 + 2 * q^81 + 12 * q^83 + 12 * q^87 - 4 * q^91 + 4 * q^93 + 10 * q^95 - 2 * q^97 - 2 * 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
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	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
$9$	4.46410	1.48803
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−6.73205	−1.54444	−0.772219	−	0.635356i	$-0.780853\pi$
$19$	−6.73205	−1.54444	−0.772219	+	0.635356i	$0.780853\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	8.19615	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−4.73205	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$29$	−4.73205	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	2.73205	0.475589
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	0.732051	0.120348	0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051	0.120348	0.0601742	+	0.998188i	$0.480834\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.19615	−0.342981	−0.171491	−	0.985186i	$-0.554858\pi$
$41$	−2.19615	−0.342981	−0.171491	+	0.985186i	$0.554858\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.46410	1.32524
$52$	0	0
$53$	−7.26795	−0.998330	−0.499165	−	0.866507i	$-0.666360\pi$
$53$	−7.26795	−0.998330	−0.499165	+	0.866507i	$0.666360\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	18.3923	2.43612
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−22.3923	−2.69572
$70$	0	0
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	12.1962	1.37217	0.686087	−	0.727519i	$-0.259327\pi$
$79$	12.1962	1.37217	0.686087	+	0.727519i	$0.259327\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	16.3923	1.79929	0.899645	−	0.436623i	$-0.143826\pi$
$83$	16.3923	1.79929	0.899645	+	0.436623i	$0.143826\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	12.9282	1.38605
$88$	0	0
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	5.46410	0.566601
$94$	0	0
$95$	6.73205	0.690694
$96$	0	0
$97$	14.5885	1.48123	0.740617	−	0.671928i	$-0.234533\pi$
$97$	14.5885	1.48123	0.740617	+	0.671928i	$0.234533\pi$
$98$	0	0
$99$	−4.46410	−0.448659
$100$	0	0
$101$	−7.85641	−0.781742	−0.390871	−	0.920446i	$-0.627826\pi$
$101$	−7.85641	−0.781742	−0.390871	+	0.920446i	$0.627826\pi$
$102$	0	0
$103$	−12.3923	−1.22105	−0.610525	−	0.791997i	$-0.709042\pi$
$103$	−12.3923	−1.22105	−0.610525	+	0.791997i	$0.709042\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	−7.85641	−0.759507	−0.379754	−	0.925088i	$-0.623991\pi$
$107$	−7.85641	−0.759507	−0.379754	+	0.925088i	$0.623991\pi$
$108$	0	0
$109$	−15.6603	−1.49998	−0.749990	−	0.661449i	$-0.769942\pi$
$109$	−15.6603	−1.49998	−0.749990	+	0.661449i	$0.769942\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	7.85641	0.739069	0.369534	−	0.929217i	$-0.379517\pi$
$113$	7.85641	0.739069	0.369534	+	0.929217i	$0.379517\pi$
$114$	0	0
$115$	−8.19615	−0.764295
$116$	0	0
$117$	−6.53590	−0.604244
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.07180	−0.0951066	−0.0475533	−	0.998869i	$-0.515142\pi$
$127$	−1.07180	−0.0951066	−0.0475533	+	0.998869i	$0.515142\pi$
$128$	0	0
$129$	5.46410	0.481087
$130$	0	0
$131$	−5.66025	−0.494539	−0.247269	−	0.968947i	$-0.579533\pi$
$131$	−5.66025	−0.494539	−0.247269	+	0.968947i	$0.579533\pi$
$132$	0	0
$133$	6.73205	0.583743
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−0.928203	−0.0793018	−0.0396509	−	0.999214i	$-0.512625\pi$
$137$	−0.928203	−0.0793018	−0.0396509	+	0.999214i	$0.512625\pi$
$138$	0	0
$139$	−13.6603	−1.15865	−0.579324	−	0.815097i	$-0.696683\pi$
$139$	−13.6603	−1.15865	−0.579324	+	0.815097i	$0.696683\pi$
$140$	0	0
$141$	18.9282	1.59404
$142$	0	0
$143$	1.46410	0.122434
$144$	0	0
$145$	4.73205	0.392975
$146$	0	0
$147$	−2.73205	−0.225336
$148$	0	0
$149$	−7.26795	−0.595414	−0.297707	−	0.954657i	$-0.596222\pi$
$149$	−7.26795	−0.595414	−0.297707	+	0.954657i	$0.596222\pi$
$150$	0	0
$151$	−11.1244	−0.905287	−0.452644	−	0.891692i	$-0.649519\pi$
$151$	−11.1244	−0.905287	−0.452644	+	0.891692i	$0.649519\pi$
$152$	0	0
$153$	−15.4641	−1.25020
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	4.53590	0.362004	0.181002	−	0.983483i	$-0.442066\pi$
$157$	4.53590	0.362004	0.181002	+	0.983483i	$0.442066\pi$
$158$	0	0
$159$	19.8564	1.57472
$160$	0	0
$161$	−8.19615	−0.645947
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−2.73205	−0.212690
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−30.0526	−2.29818
$172$	0	0
$173$	0.928203	0.0705700	0.0352850	−	0.999377i	$-0.488766\pi$
$173$	0.928203	0.0705700	0.0352850	+	0.999377i	$0.488766\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	18.9282	1.42273
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	13.4641	0.995295
$184$	0	0
$185$	−0.732051	−0.0538214
$186$	0	0
$187$	3.46410	0.253320
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	12.3923	0.892018	0.446009	−	0.895029i	$-0.352845\pi$
$193$	12.3923	0.892018	0.446009	+	0.895029i	$0.352845\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−24.2487	−1.72765	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	−24.2487	−1.72765	−0.863825	+	0.503793i	$0.831938\pi$
$198$	0	0
$199$	−2.92820	−0.207575	−0.103787	−	0.994600i	$-0.533096\pi$
$199$	−2.92820	−0.207575	−0.103787	+	0.994600i	$0.533096\pi$
$200$	0	0
$201$	−10.9282	−0.770816
$202$	0	0
$203$	4.73205	0.332125
$204$	0	0
$205$	2.19615	0.153386
$206$	0	0
$207$	36.5885	2.54307
$208$	0	0
$209$	6.73205	0.465666
$210$	0	0
$211$	−26.9282	−1.85381	−0.926907	−	0.375291i	$-0.877543\pi$
$211$	−26.9282	−1.85381	−0.926907	+	0.375291i	$0.877543\pi$
$212$	0	0
$213$	25.8564	1.77165
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	39.3205	2.65703
$220$	0	0
$221$	5.07180	0.341166
$222$	0	0
$223$	25.4641	1.70520	0.852601	−	0.522562i	$-0.175024\pi$
$223$	25.4641	1.70520	0.852601	+	0.522562i	$0.175024\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	24.3923	1.61189	0.805944	−	0.591991i	$-0.201658\pi$
$229$	24.3923	1.61189	0.805944	+	0.591991i	$0.201658\pi$
$230$	0	0
$231$	−2.73205	−0.179756
$232$	0	0
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	6.92820	0.451946
$236$	0	0
$237$	−33.3205	−2.16440
$238$	0	0
$239$	−1.26795	−0.0820168	−0.0410084	−	0.999159i	$-0.513057\pi$
$239$	−1.26795	−0.0820168	−0.0410084	+	0.999159i	$0.513057\pi$
$240$	0	0
$241$	3.26795	0.210507	0.105254	−	0.994445i	$-0.466435\pi$
$241$	3.26795	0.210507	0.105254	+	0.994445i	$0.466435\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	9.85641	0.627148
$248$	0	0
$249$	−44.7846	−2.83811
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−8.19615	−0.515288
$254$	0	0
$255$	−9.46410	−0.592665
$256$	0	0
$257$	−23.6603	−1.47589	−0.737943	−	0.674863i	$-0.764203\pi$
$257$	−23.6603	−1.47589	−0.737943	+	0.674863i	$0.764203\pi$
$258$	0	0
$259$	−0.732051	−0.0454874
$260$	0	0
$261$	−21.1244	−1.30756
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	7.26795	0.446467
$266$	0	0
$267$	−9.46410	−0.579194
$268$	0	0
$269$	28.3923	1.73111	0.865555	−	0.500814i	$-0.166966\pi$
$269$	28.3923	1.73111	0.865555	+	0.500814i	$0.166966\pi$
$270$	0	0
$271$	−0.392305	−0.0238308	−0.0119154	−	0.999929i	$-0.503793\pi$
$271$	−0.392305	−0.0238308	−0.0119154	+	0.999929i	$0.503793\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	7.07180	0.424903	0.212452	−	0.977172i	$-0.431855\pi$
$277$	7.07180	0.424903	0.212452	+	0.977172i	$0.431855\pi$
$278$	0	0
$279$	−8.92820	−0.534518
$280$	0	0
$281$	22.3923	1.33581	0.667906	−	0.744245i	$-0.267191\pi$
$281$	22.3923	1.33581	0.667906	+	0.744245i	$0.267191\pi$
$282$	0	0
$283$	31.7128	1.88513	0.942566	−	0.334021i	$-0.108406\pi$
$283$	31.7128	1.88513	0.942566	+	0.334021i	$0.108406\pi$
$284$	0	0
$285$	−18.3923	−1.08947
$286$	0	0
$287$	2.19615	0.129635
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−39.8564	−2.33642
$292$	0	0
$293$	21.4641	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$293$	21.4641	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$294$	0	0
$295$	6.92820	0.403376
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	21.4641	1.23308
$304$	0	0
$305$	4.92820	0.282188
$306$	0	0
$307$	−24.3923	−1.39214	−0.696071	−	0.717973i	$-0.745070\pi$
$307$	−24.3923	−1.39214	−0.696071	+	0.717973i	$0.745070\pi$
$308$	0	0
$309$	33.8564	1.92602
$310$	0	0
$311$	−24.9282	−1.41355	−0.706774	−	0.707439i	$-0.749850\pi$
$311$	−24.9282	−1.41355	−0.706774	+	0.707439i	$0.749850\pi$
$312$	0	0
$313$	22.1962	1.25460	0.627300	−	0.778777i	$-0.284160\pi$
$313$	22.1962	1.25460	0.627300	+	0.778777i	$0.284160\pi$
$314$	0	0
$315$	4.46410	0.251524
$316$	0	0
$317$	30.5885	1.71802	0.859009	−	0.511960i	$-0.171080\pi$
$317$	30.5885	1.71802	0.859009	+	0.511960i	$0.171080\pi$
$318$	0	0
$319$	4.73205	0.264944
$320$	0	0
$321$	21.4641	1.19801
$322$	0	0
$323$	23.3205	1.29759
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	0	0
$327$	42.7846	2.36599
$328$	0	0
$329$	6.92820	0.381964
$330$	0	0
$331$	18.7846	1.03250	0.516248	−	0.856439i	$-0.327328\pi$
$331$	18.7846	1.03250	0.516248	+	0.856439i	$0.327328\pi$
$332$	0	0
$333$	3.26795	0.179083
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	22.7846	1.24116	0.620578	−	0.784144i	$-0.286898\pi$
$337$	22.7846	1.24116	0.620578	+	0.784144i	$0.286898\pi$
$338$	0	0
$339$	−21.4641	−1.16577
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	22.3923	1.20556
$346$	0	0
$347$	23.0718	1.23856	0.619279	−	0.785171i	$-0.287425\pi$
$347$	23.0718	1.23856	0.619279	+	0.785171i	$0.287425\pi$
$348$	0	0
$349$	−5.60770	−0.300173	−0.150087	−	0.988673i	$-0.547955\pi$
$349$	−5.60770	−0.300173	−0.150087	+	0.988673i	$0.547955\pi$
$350$	0	0
$351$	5.85641	0.312592
$352$	0	0
$353$	14.1962	0.755585	0.377792	−	0.925890i	$-0.376683\pi$
$353$	14.1962	0.755585	0.377792	+	0.925890i	$0.376683\pi$
$354$	0	0
$355$	9.46410	0.502302
$356$	0	0
$357$	−9.46410	−0.500893
$358$	0	0
$359$	34.0526	1.79723	0.898613	−	0.438743i	$-0.144576\pi$
$359$	34.0526	1.79723	0.898613	+	0.438743i	$0.144576\pi$
$360$	0	0
$361$	26.3205	1.38529
$362$	0	0
$363$	−2.73205	−0.143395
$364$	0	0
$365$	14.3923	0.753328
$366$	0	0
$367$	−12.3923	−0.646873	−0.323437	−	0.946250i	$-0.604838\pi$
$367$	−12.3923	−0.646873	−0.323437	+	0.946250i	$0.604838\pi$
$368$	0	0
$369$	−9.80385	−0.510368
$370$	0	0
$371$	7.26795	0.377333
$372$	0	0
$373$	18.3923	0.952317	0.476159	−	0.879359i	$-0.342029\pi$
$373$	18.3923	0.952317	0.476159	+	0.879359i	$0.342029\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	6.92820	0.356821
$378$	0	0
$379$	−6.14359	−0.315575	−0.157788	−	0.987473i	$-0.550436\pi$
$379$	−6.14359	−0.315575	−0.157788	+	0.987473i	$0.550436\pi$
$380$	0	0
$381$	2.92820	0.150016
$382$	0	0
$383$	33.4641	1.70994	0.854968	−	0.518681i	$-0.173577\pi$
$383$	33.4641	1.70994	0.854968	+	0.518681i	$0.173577\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−8.92820	−0.453846
$388$	0	0
$389$	−1.60770	−0.0815134	−0.0407567	−	0.999169i	$-0.512977\pi$
$389$	−1.60770	−0.0815134	−0.0407567	+	0.999169i	$0.512977\pi$
$390$	0	0
$391$	−28.3923	−1.43586
$392$	0	0
$393$	15.4641	0.780061
$394$	0	0
$395$	−12.1962	−0.613655
$396$	0	0
$397$	30.3923	1.52535	0.762673	−	0.646784i	$-0.223887\pi$
$397$	30.3923	1.52535	0.762673	+	0.646784i	$0.223887\pi$
$398$	0	0
$399$	−18.3923	−0.920767
$400$	0	0
$401$	2.53590	0.126637	0.0633184	−	0.997993i	$-0.479832\pi$
$401$	2.53590	0.126637	0.0633184	+	0.997993i	$0.479832\pi$
$402$	0	0
$403$	2.92820	0.145864
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	−0.732051	−0.0362864
$408$	0	0
$409$	10.8756	0.537766	0.268883	−	0.963173i	$-0.413346\pi$
$409$	10.8756	0.537766	0.268883	+	0.963173i	$0.413346\pi$
$410$	0	0
$411$	2.53590	0.125087
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	−16.3923	−0.804667
$416$	0	0
$417$	37.3205	1.82759
$418$	0	0
$419$	−30.9282	−1.51094	−0.755471	−	0.655182i	$-0.772592\pi$
$419$	−30.9282	−1.51094	−0.755471	+	0.655182i	$0.772592\pi$
$420$	0	0
$421$	−35.8564	−1.74753	−0.873767	−	0.486344i	$-0.838330\pi$
$421$	−35.8564	−1.74753	−0.873767	+	0.486344i	$0.838330\pi$
$422$	0	0
$423$	−30.9282	−1.50378
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	4.92820	0.238492
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−3.12436	−0.150495	−0.0752475	−	0.997165i	$-0.523975\pi$
$431$	−3.12436	−0.150495	−0.0752475	+	0.997165i	$0.523975\pi$
$432$	0	0
$433$	−18.1962	−0.874451	−0.437226	−	0.899352i	$-0.644039\pi$
$433$	−18.1962	−0.874451	−0.437226	+	0.899352i	$0.644039\pi$
$434$	0	0
$435$	−12.9282	−0.619860
$436$	0	0
$437$	−55.1769	−2.63947
$438$	0	0
$439$	−26.2487	−1.25278	−0.626391	−	0.779509i	$-0.715469\pi$
$439$	−26.2487	−1.25278	−0.626391	+	0.779509i	$0.715469\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−3.46410	−0.164214
$446$	0	0
$447$	19.8564	0.939176
$448$	0	0
$449$	40.3923	1.90623	0.953115	−	0.302607i	$-0.0978570\pi$
$449$	40.3923	1.90623	0.953115	+	0.302607i	$0.0978570\pi$
$450$	0	0
$451$	2.19615	0.103413
$452$	0	0
$453$	30.3923	1.42796
$454$	0	0
$455$	−1.46410	−0.0686381
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	22.3923	1.04291	0.521457	−	0.853278i	$-0.325389\pi$
$461$	22.3923	1.04291	0.521457	+	0.853278i	$0.325389\pi$
$462$	0	0
$463$	−27.5167	−1.27881	−0.639404	−	0.768871i	$-0.720819\pi$
$463$	−27.5167	−1.27881	−0.639404	+	0.768871i	$0.720819\pi$
$464$	0	0
$465$	−5.46410	−0.253392
$466$	0	0
$467$	34.0526	1.57576	0.787882	−	0.615826i	$-0.211178\pi$
$467$	34.0526	1.57576	0.787882	+	0.615826i	$0.211178\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−12.3923	−0.571007
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−6.73205	−0.308888
$476$	0	0
$477$	−32.4449	−1.48555
$478$	0	0
$479$	32.7846	1.49797	0.748984	−	0.662589i	$-0.230542\pi$
$479$	32.7846	1.49797	0.748984	+	0.662589i	$0.230542\pi$
$480$	0	0
$481$	−1.07180	−0.0488697
$482$	0	0
$483$	22.3923	1.01889
$484$	0	0
$485$	−14.5885	−0.662428
$486$	0	0
$487$	−4.19615	−0.190146	−0.0950729	−	0.995470i	$-0.530308\pi$
$487$	−4.19615	−0.190146	−0.0950729	+	0.995470i	$0.530308\pi$
$488$	0	0
$489$	−10.9282	−0.494190
$490$	0	0
$491$	27.7128	1.25066	0.625331	−	0.780360i	$-0.284964\pi$
$491$	27.7128	1.25066	0.625331	+	0.780360i	$0.284964\pi$
$492$	0	0
$493$	16.3923	0.738272
$494$	0	0
$495$	4.46410	0.200646
$496$	0	0
$497$	9.46410	0.424523
$498$	0	0
$499$	−12.1436	−0.543622	−0.271811	−	0.962351i	$-0.587623\pi$
$499$	−12.1436	−0.543622	−0.271811	+	0.962351i	$0.587623\pi$
$500$	0	0
$501$	37.8564	1.69130
$502$	0	0
$503$	−8.78461	−0.391686	−0.195843	−	0.980635i	$-0.562744\pi$
$503$	−8.78461	−0.391686	−0.195843	+	0.980635i	$0.562744\pi$
$504$	0	0
$505$	7.85641	0.349605
$506$	0	0
$507$	29.6603	1.31726
$508$	0	0
$509$	11.0718	0.490749	0.245374	−	0.969428i	$-0.421089\pi$
$509$	11.0718	0.490749	0.245374	+	0.969428i	$0.421089\pi$
$510$	0	0
$511$	14.3923	0.636678
$512$	0	0
$513$	26.9282	1.18891
$514$	0	0
$515$	12.3923	0.546070
$516$	0	0
$517$	6.92820	0.304702
$518$	0	0
$519$	−2.53590	−0.111314
$520$	0	0
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	0	0
$523$	29.1769	1.27582	0.637909	−	0.770112i	$-0.279800\pi$
$523$	29.1769	1.27582	0.637909	+	0.770112i	$0.279800\pi$
$524$	0	0
$525$	2.73205	0.119236
$526$	0	0
$527$	6.92820	0.301797
$528$	0	0
$529$	44.1769	1.92074
$530$	0	0
$531$	−30.9282	−1.34217
$532$	0	0
$533$	3.21539	0.139274
$534$	0	0
$535$	7.85641	0.339662
$536$	0	0
$537$	−16.3923	−0.707380
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−20.7321	−0.891340	−0.445670	−	0.895197i	$-0.647035\pi$
$541$	−20.7321	−0.891340	−0.445670	+	0.895197i	$0.647035\pi$
$542$	0	0
$543$	−38.2487	−1.64141
$544$	0	0
$545$	15.6603	0.670812
$546$	0	0
$547$	−28.7846	−1.23074	−0.615371	−	0.788238i	$-0.710994\pi$
$547$	−28.7846	−1.23074	−0.615371	+	0.788238i	$0.710994\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	31.8564	1.35713
$552$	0	0
$553$	−12.1962	−0.518633
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	−25.6077	−1.08503	−0.542516	−	0.840045i	$-0.682528\pi$
$557$	−25.6077	−1.08503	−0.542516	+	0.840045i	$0.682528\pi$
$558$	0	0
$559$	2.92820	0.123850
$560$	0	0
$561$	−9.46410	−0.399575
$562$	0	0
$563$	−5.07180	−0.213751	−0.106875	−	0.994272i	$-0.534085\pi$
$563$	−5.07180	−0.213751	−0.106875	+	0.994272i	$0.534085\pi$
$564$	0	0
$565$	−7.85641	−0.330522
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−3.60770	−0.150977	−0.0754887	−	0.997147i	$-0.524052\pi$
$571$	−3.60770	−0.150977	−0.0754887	+	0.997147i	$0.524052\pi$
$572$	0	0
$573$	−32.7846	−1.36960
$574$	0	0
$575$	8.19615	0.341803
$576$	0	0
$577$	−34.5885	−1.43994	−0.719968	−	0.694007i	$-0.755844\pi$
$577$	−34.5885	−1.43994	−0.719968	+	0.694007i	$0.755844\pi$
$578$	0	0
$579$	−33.8564	−1.40702
$580$	0	0
$581$	−16.3923	−0.680067
$582$	0	0
$583$	7.26795	0.301008
$584$	0	0
$585$	6.53590	0.270226
$586$	0	0
$587$	11.4115	0.471005	0.235502	−	0.971874i	$-0.424326\pi$
$587$	11.4115	0.471005	0.235502	+	0.971874i	$0.424326\pi$
$588$	0	0
$589$	13.4641	0.554779
$590$	0	0
$591$	66.2487	2.72511
$592$	0	0
$593$	0.248711	0.0102133	0.00510667	−	0.999987i	$-0.498374\pi$
$593$	0.248711	0.0102133	0.00510667	+	0.999987i	$0.498374\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	−37.1769	−1.51901	−0.759504	−	0.650503i	$-0.774558\pi$
$599$	−37.1769	−1.51901	−0.759504	+	0.650503i	$0.774558\pi$
$600$	0	0
$601$	−5.51666	−0.225029	−0.112515	−	0.993650i	$-0.535891\pi$
$601$	−5.51666	−0.225029	−0.112515	+	0.993650i	$0.535891\pi$
$602$	0	0
$603$	17.8564	0.727169
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−20.9282	−0.849450	−0.424725	−	0.905323i	$-0.639629\pi$
$607$	−20.9282	−0.849450	−0.424725	+	0.905323i	$0.639629\pi$
$608$	0	0
$609$	−12.9282	−0.523877
$610$	0	0
$611$	10.1436	0.410366
$612$	0	0
$613$	−35.1769	−1.42078	−0.710391	−	0.703807i	$-0.751482\pi$
$613$	−35.1769	−1.42078	−0.710391	+	0.703807i	$0.751482\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	17.3205	0.697297	0.348649	−	0.937253i	$-0.386641\pi$
$617$	17.3205	0.697297	0.348649	+	0.937253i	$0.386641\pi$
$618$	0	0
$619$	−28.7846	−1.15695	−0.578476	−	0.815700i	$-0.696352\pi$
$619$	−28.7846	−1.15695	−0.578476	+	0.815700i	$0.696352\pi$
$620$	0	0
$621$	−32.7846	−1.31560
$622$	0	0
$623$	−3.46410	−0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.3923	−0.734518
$628$	0	0
$629$	−2.53590	−0.101113
$630$	0	0
$631$	46.9282	1.86818	0.934091	−	0.357035i	$-0.116212\pi$
$631$	46.9282	1.86818	0.934091	+	0.357035i	$0.116212\pi$
$632$	0	0
$633$	73.5692	2.92411
$634$	0	0
$635$	1.07180	0.0425330
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	0	0
$639$	−42.2487	−1.67133
$640$	0	0
$641$	−35.5692	−1.40490	−0.702450	−	0.711733i	$-0.747910\pi$
$641$	−35.5692	−1.40490	−0.702450	+	0.711733i	$0.747910\pi$
$642$	0	0
$643$	−33.2679	−1.31196	−0.655980	−	0.754778i	$-0.727744\pi$
$643$	−33.2679	−1.31196	−0.655980	+	0.754778i	$0.727744\pi$
$644$	0	0
$645$	−5.46410	−0.215149
$646$	0	0
$647$	26.5359	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$647$	26.5359	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$648$	0	0
$649$	6.92820	0.271956
$650$	0	0
$651$	−5.46410	−0.214155
$652$	0	0
$653$	11.6603	0.456301	0.228151	−	0.973626i	$-0.426732\pi$
$653$	11.6603	0.456301	0.228151	+	0.973626i	$0.426732\pi$
$654$	0	0
$655$	5.66025	0.221164
$656$	0	0
$657$	−64.2487	−2.50658
$658$	0	0
$659$	−30.2487	−1.17832	−0.589161	−	0.808015i	$-0.700542\pi$
$659$	−30.2487	−1.17832	−0.589161	+	0.808015i	$0.700542\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	0	0
$663$	−13.8564	−0.538138
$664$	0	0
$665$	−6.73205	−0.261058
$666$	0	0
$667$	−38.7846	−1.50175
$668$	0	0
$669$	−69.5692	−2.68970
$670$	0	0
$671$	4.92820	0.190251
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−4.14359	−0.159251	−0.0796256	−	0.996825i	$-0.525372\pi$
$677$	−4.14359	−0.159251	−0.0796256	+	0.996825i	$0.525372\pi$
$678$	0	0
$679$	−14.5885	−0.559854
$680$	0	0
$681$	18.9282	0.725330
$682$	0	0
$683$	−6.24871	−0.239100	−0.119550	−	0.992828i	$-0.538145\pi$
$683$	−6.24871	−0.239100	−0.119550	+	0.992828i	$0.538145\pi$
$684$	0	0
$685$	0.928203	0.0354648
$686$	0	0
$687$	−66.6410	−2.54251
$688$	0	0
$689$	10.6410	0.405390
$690$	0	0
$691$	13.4641	0.512199	0.256099	−	0.966650i	$-0.417563\pi$
$691$	13.4641	0.512199	0.256099	+	0.966650i	$0.417563\pi$
$692$	0	0
$693$	4.46410	0.169577
$694$	0	0
$695$	13.6603	0.518163
$696$	0	0
$697$	7.60770	0.288162
$698$	0	0
$699$	21.4641	0.811847
$700$	0	0
$701$	41.9090	1.58288	0.791440	−	0.611247i	$-0.209332\pi$
$701$	41.9090	1.58288	0.791440	+	0.611247i	$0.209332\pi$
$702$	0	0
$703$	−4.92820	−0.185871
$704$	0	0
$705$	−18.9282	−0.712877
$706$	0	0
$707$	7.85641	0.295471
$708$	0	0
$709$	25.3205	0.950932	0.475466	−	0.879734i	$-0.342280\pi$
$709$	25.3205	0.950932	0.475466	+	0.879734i	$0.342280\pi$
$710$	0	0
$711$	54.4449	2.04184
$712$	0	0
$713$	−16.3923	−0.613897
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	0	0
$717$	3.46410	0.129369
$718$	0	0
$719$	27.7128	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$720$	0	0
$721$	12.3923	0.461514
$722$	0	0
$723$	−8.92820	−0.332043
$724$	0	0
$725$	−4.73205	−0.175744
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	6.92820	0.256249
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	2.73205	0.100773
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−11.7128	−0.430863	−0.215431	−	0.976519i	$-0.569116\pi$
$739$	−11.7128	−0.430863	−0.215431	+	0.976519i	$0.569116\pi$
$740$	0	0
$741$	−26.9282	−0.989232
$742$	0	0
$743$	32.7846	1.20275	0.601375	−	0.798967i	$-0.294620\pi$
$743$	32.7846	1.20275	0.601375	+	0.798967i	$0.294620\pi$
$744$	0	0
$745$	7.26795	0.266277
$746$	0	0
$747$	73.1769	2.67740
$748$	0	0
$749$	7.85641	0.287067
$750$	0	0
$751$	11.6077	0.423571	0.211785	−	0.977316i	$-0.432072\pi$
$751$	11.6077	0.423571	0.211785	+	0.977316i	$0.432072\pi$
$752$	0	0
$753$	32.7846	1.19474
$754$	0	0
$755$	11.1244	0.404857
$756$	0	0
$757$	−43.3731	−1.57642	−0.788210	−	0.615406i	$-0.788992\pi$
$757$	−43.3731	−1.57642	−0.788210	+	0.615406i	$0.788992\pi$
$758$	0	0
$759$	22.3923	0.812789
$760$	0	0
$761$	−12.3397	−0.447315	−0.223658	−	0.974668i	$-0.571800\pi$
$761$	−12.3397	−0.447315	−0.223658	+	0.974668i	$0.571800\pi$
$762$	0	0
$763$	15.6603	0.566939
$764$	0	0
$765$	15.4641	0.559106
$766$	0	0
$767$	10.1436	0.366264
$768$	0	0
$769$	−18.1962	−0.656170	−0.328085	−	0.944648i	$-0.606403\pi$
$769$	−18.1962	−0.656170	−0.328085	+	0.944648i	$0.606403\pi$
$770$	0	0
$771$	64.6410	2.32799
$772$	0	0
$773$	−0.928203	−0.0333851	−0.0166926	−	0.999861i	$-0.505314\pi$
$773$	−0.928203	−0.0333851	−0.0166926	+	0.999861i	$0.505314\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	14.7846	0.529714
$780$	0	0
$781$	9.46410	0.338652
$782$	0	0
$783$	18.9282	0.676439
$784$	0	0
$785$	−4.53590	−0.161893
$786$	0	0
$787$	−18.1436	−0.646749	−0.323375	−	0.946271i	$-0.604817\pi$
$787$	−18.1436	−0.646749	−0.323375	+	0.946271i	$0.604817\pi$
$788$	0	0
$789$	−65.5692	−2.33433
$790$	0	0
$791$	−7.85641	−0.279342
$792$	0	0
$793$	7.21539	0.256226
$794$	0	0
$795$	−19.8564	−0.704234
$796$	0	0
$797$	−25.6077	−0.907071	−0.453536	−	0.891238i	$-0.649837\pi$
$797$	−25.6077	−0.907071	−0.453536	+	0.891238i	$0.649837\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	15.4641	0.546397
$802$	0	0
$803$	14.3923	0.507893
$804$	0	0
$805$	8.19615	0.288876
$806$	0	0
$807$	−77.5692	−2.73057
$808$	0	0
$809$	−31.1769	−1.09612	−0.548061	−	0.836438i	$-0.684634\pi$
$809$	−31.1769	−1.09612	−0.548061	+	0.836438i	$0.684634\pi$
$810$	0	0
$811$	43.1244	1.51430	0.757150	−	0.653241i	$-0.226591\pi$
$811$	43.1244	1.51430	0.757150	+	0.653241i	$0.226591\pi$
$812$	0	0
$813$	1.07180	0.0375896
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	13.4641	0.471049
$818$	0	0
$819$	6.53590	0.228383
$820$	0	0
$821$	17.9090	0.625027	0.312514	−	0.949913i	$-0.398829\pi$
$821$	17.9090	0.625027	0.312514	+	0.949913i	$0.398829\pi$
$822$	0	0
$823$	0.875644	0.0305230	0.0152615	−	0.999884i	$-0.495142\pi$
$823$	0.875644	0.0305230	0.0152615	+	0.999884i	$0.495142\pi$
$824$	0	0
$825$	2.73205	0.0951178
$826$	0	0
$827$	25.8564	0.899115	0.449558	−	0.893251i	$-0.351582\pi$
$827$	25.8564	0.899115	0.449558	+	0.893251i	$0.351582\pi$
$828$	0	0
$829$	2.24871	0.0781010	0.0390505	−	0.999237i	$-0.487567\pi$
$829$	2.24871	0.0781010	0.0390505	+	0.999237i	$0.487567\pi$
$830$	0	0
$831$	−19.3205	−0.670221
$832$	0	0
$833$	−3.46410	−0.120024
$834$	0	0
$835$	13.8564	0.479521
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	31.8564	1.09981	0.549903	−	0.835229i	$-0.314665\pi$
$839$	31.8564	1.09981	0.549903	+	0.835229i	$0.314665\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	0	0
$843$	−61.1769	−2.10704
$844$	0	0
$845$	10.8564	0.373472
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−86.6410	−2.97351
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−54.7846	−1.87579	−0.937895	−	0.346920i	$-0.887227\pi$
$853$	−54.7846	−1.87579	−0.937895	+	0.346920i	$0.887227\pi$
$854$	0	0
$855$	30.0526	1.02778
$856$	0	0
$857$	−27.4641	−0.938156	−0.469078	−	0.883157i	$-0.655414\pi$
$857$	−27.4641	−0.938156	−0.469078	+	0.883157i	$0.655414\pi$
$858$	0	0
$859$	12.7846	0.436205	0.218103	−	0.975926i	$-0.430013\pi$
$859$	12.7846	0.436205	0.218103	+	0.975926i	$0.430013\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	7.51666	0.255870	0.127935	−	0.991783i	$-0.459165\pi$
$863$	7.51666	0.255870	0.127935	+	0.991783i	$0.459165\pi$
$864$	0	0
$865$	−0.928203	−0.0315599
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	−12.1962	−0.413726
$870$	0	0
$871$	−5.85641	−0.198437
$872$	0	0
$873$	65.1244	2.20413
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−21.3205	−0.719942	−0.359971	−	0.932963i	$-0.617213\pi$
$877$	−21.3205	−0.719942	−0.359971	+	0.932963i	$0.617213\pi$
$878$	0	0
$879$	−58.6410	−1.97791
$880$	0	0
$881$	11.0718	0.373018	0.186509	−	0.982453i	$-0.440283\pi$
$881$	11.0718	0.373018	0.186509	+	0.982453i	$0.440283\pi$
$882$	0	0
$883$	35.6077	1.19829	0.599147	−	0.800639i	$-0.295506\pi$
$883$	35.6077	1.19829	0.599147	+	0.800639i	$0.295506\pi$
$884$	0	0
$885$	−18.9282	−0.636265
$886$	0	0
$887$	−13.8564	−0.465253	−0.232626	−	0.972566i	$-0.574732\pi$
$887$	−13.8564	−0.465253	−0.232626	+	0.972566i	$0.574732\pi$
$888$	0	0
$889$	1.07180	0.0359469
$890$	0	0
$891$	2.46410	0.0825505
$892$	0	0
$893$	46.6410	1.56078
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	32.7846	1.09465
$898$	0	0
$899$	9.46410	0.315645
$900$	0	0
$901$	25.1769	0.838765
$902$	0	0
$903$	−5.46410	−0.181834
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	31.0333	1.03044	0.515222	−	0.857057i	$-0.327709\pi$
$907$	31.0333	1.03044	0.515222	+	0.857057i	$0.327709\pi$
$908$	0	0
$909$	−35.0718	−1.16326
$910$	0	0
$911$	−9.46410	−0.313560	−0.156780	−	0.987634i	$-0.550111\pi$
$911$	−9.46410	−0.313560	−0.156780	+	0.987634i	$0.550111\pi$
$912$	0	0
$913$	−16.3923	−0.542506
$914$	0	0
$915$	−13.4641	−0.445109
$916$	0	0
$917$	5.66025	0.186918
$918$	0	0
$919$	25.3731	0.836980	0.418490	−	0.908221i	$-0.362559\pi$
$919$	25.3731	0.836980	0.418490	+	0.908221i	$0.362559\pi$
$920$	0	0
$921$	66.6410	2.19590
$922$	0	0
$923$	13.8564	0.456089
$924$	0	0
$925$	0.732051	0.0240697
$926$	0	0
$927$	−55.3205	−1.81696
$928$	0	0
$929$	−25.6077	−0.840161	−0.420081	−	0.907487i	$-0.637998\pi$
$929$	−25.6077	−0.840161	−0.420081	+	0.907487i	$0.637998\pi$
$930$	0	0
$931$	−6.73205	−0.220634
$932$	0	0
$933$	68.1051	2.22966
$934$	0	0
$935$	−3.46410	−0.113288
$936$	0	0
$937$	−1.21539	−0.0397051	−0.0198525	−	0.999803i	$-0.506320\pi$
$937$	−1.21539	−0.0397051	−0.0198525	+	0.999803i	$0.506320\pi$
$938$	0	0
$939$	−60.6410	−1.97894
$940$	0	0
$941$	−14.7846	−0.481965	−0.240982	−	0.970530i	$-0.577470\pi$
$941$	−14.7846	−0.481965	−0.240982	+	0.970530i	$0.577470\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−47.3205	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$947$	−47.3205	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$948$	0	0
$949$	21.0718	0.684019
$950$	0	0
$951$	−83.5692	−2.70992
$952$	0	0
$953$	26.5359	0.859582	0.429791	−	0.902928i	$-0.358587\pi$
$953$	26.5359	0.859582	0.429791	+	0.902928i	$0.358587\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	−12.9282	−0.417909
$958$	0	0
$959$	0.928203	0.0299732
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−35.0718	−1.13017
$964$	0	0
$965$	−12.3923	−0.398922
$966$	0	0
$967$	−26.9282	−0.865953	−0.432976	−	0.901405i	$-0.642537\pi$
$967$	−26.9282	−0.865953	−0.432976	+	0.901405i	$0.642537\pi$
$968$	0	0
$969$	−63.7128	−2.04675
$970$	0	0
$971$	−42.9282	−1.37763	−0.688816	−	0.724936i	$-0.741869\pi$
$971$	−42.9282	−1.37763	−0.688816	+	0.724936i	$0.741869\pi$
$972$	0	0
$973$	13.6603	0.437928
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−33.7128	−1.07857	−0.539284	−	0.842124i	$-0.681305\pi$
$977$	−33.7128	−1.07857	−0.539284	+	0.842124i	$0.681305\pi$
$978$	0	0
$979$	−3.46410	−0.110713
$980$	0	0
$981$	−69.9090	−2.23202
$982$	0	0
$983$	37.1769	1.18576	0.592880	−	0.805291i	$-0.297991\pi$
$983$	37.1769	1.18576	0.592880	+	0.805291i	$0.297991\pi$
$984$	0	0
$985$	24.2487	0.772628
$986$	0	0
$987$	−18.9282	−0.602491
$988$	0	0
$989$	−16.3923	−0.521245
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	−51.3205	−1.62861
$994$	0	0
$995$	2.92820	0.0928303
$996$	0	0
$997$	17.7128	0.560970	0.280485	−	0.959858i	$-0.409505\pi$
$997$	17.7128	0.560970	0.280485	+	0.959858i	$0.409505\pi$
$998$	0	0
$999$	−2.92820	−0.0926443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.t.1.1		2
4.3	odd	2		770.2.a.j.1.2	✓	2
12.11	even	2		6930.2.a.bv.1.1		2
20.3	even	4		3850.2.c.x.1849.2		4
20.7	even	4		3850.2.c.x.1849.3		4
20.19	odd	2		3850.2.a.bd.1.1		2
28.27	even	2		5390.2.a.bs.1.1		2
44.43	even	2		8470.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.j.1.2	✓	2	4.3	odd	2
3850.2.a.bd.1.1		2	20.19	odd	2
3850.2.c.x.1849.2		4	20.3	even	4
3850.2.c.x.1849.3		4	20.7	even	4
5390.2.a.bs.1.1		2	28.27	even	2
6160.2.a.t.1.1		2	1.1	even	1	trivial
6930.2.a.bv.1.1		2	12.11	even	2
8470.2.a.br.1.2		2	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} -1.46410 q^{13} +2.73205 q^{15} -3.46410 q^{17} -6.73205 q^{19} +2.73205 q^{21} +8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -4.73205 q^{29} -2.00000 q^{31} +2.73205 q^{33} +1.00000 q^{35} +0.732051 q^{37} +4.00000 q^{39} -2.19615 q^{41} -2.00000 q^{43} -4.46410 q^{45} -6.92820 q^{47} +1.00000 q^{49} +9.46410 q^{51} -7.26795 q^{53} +1.00000 q^{55} +18.3923 q^{57} -6.92820 q^{59} -4.92820 q^{61} -4.46410 q^{63} +1.46410 q^{65} +4.00000 q^{67} -22.3923 q^{69} -9.46410 q^{71} -14.3923 q^{73} -2.73205 q^{75} +1.00000 q^{77} +12.1962 q^{79} -2.46410 q^{81} +16.3923 q^{83} +3.46410 q^{85} +12.9282 q^{87} +3.46410 q^{89} +1.46410 q^{91} +5.46410 q^{93} +6.73205 q^{95} +14.5885 q^{97} -4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 - 10 * q^19 + 2 * q^21 + 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 2 * q^35 - 2 * q^37 + 8 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^49 + 12 * q^51 - 18 * q^53 + 2 * q^55 + 16 * q^57 + 4 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 - 24 * q^69 - 12 * q^71 - 8 * q^73 - 2 * q^75 + 2 * q^77 + 14 * q^79 + 2 * q^81 + 12 * q^83 + 12 * q^87 - 4 * q^91 + 4 * q^93 + 10 * q^95 - 2 * q^97 - 2 * q^99

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