LMFDB - Embedded newform 6160.2.a.v.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:	$1$
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)

$f(q)$	$=$	$q-2.73205 q^{3} +1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} +1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} -1.46410 q^{13} -2.73205 q^{15} +3.46410 q^{17} +2.73205 q^{19} +2.73205 q^{21} -1.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -4.73205 q^{29} -8.92820 q^{31} +2.73205 q^{33} -1.00000 q^{35} +3.26795 q^{37} +4.00000 q^{39} -4.73205 q^{41} +4.92820 q^{43} +4.46410 q^{45} +1.00000 q^{49} -9.46410 q^{51} -4.73205 q^{53} -1.00000 q^{55} -7.46410 q^{57} +13.8564 q^{59} +2.00000 q^{61} -4.46410 q^{63} -1.46410 q^{65} +10.9282 q^{67} +3.46410 q^{69} -9.46410 q^{71} +11.4641 q^{73} -2.73205 q^{75} +1.00000 q^{77} -6.73205 q^{79} -2.46410 q^{81} +4.39230 q^{83} +3.46410 q^{85} +12.9282 q^{87} -15.4641 q^{89} +1.46410 q^{91} +24.3923 q^{93} +2.73205 q^{95} +5.80385 q^{97} -4.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 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} + 10 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} + 2 q^{49} - 12 q^{51} - 6 q^{53} - 2 q^{55} - 8 q^{57} + 4 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 12 q^{71} + 16 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} - 24 q^{89} - 4 q^{91} + 28 q^{93} + 2 q^{95} + 22 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 + 2 * 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 + 10 * q^37 + 8 * q^39 - 6 * q^41 - 4 * q^43 + 2 * q^45 + 2 * q^49 - 12 * q^51 - 6 * q^53 - 2 * q^55 - 8 * q^57 + 4 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 12 * q^71 + 16 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 - 24 * q^89 - 4 * q^91 + 28 * q^93 + 2 * q^95 + 22 * 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	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	2.73205	0.596182
$22$	0	0
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	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
$31$	−8.92820	−1.60355	−0.801776	−	0.597624i	$-0.796111\pi$
$31$	−8.92820	−1.60355	−0.801776	+	0.597624i	$0.796111\pi$
$32$	0	0
$33$	2.73205	0.475589
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.26795	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$37$	3.26795	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−4.73205	−0.739022	−0.369511	−	0.929226i	$-0.620475\pi$
$41$	−4.73205	−0.739022	−0.369511	+	0.929226i	$0.620475\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	4.46410	0.665469
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.46410	−1.32524
$52$	0	0
$53$	−4.73205	−0.649997	−0.324999	−	0.945715i	$-0.605364\pi$
$53$	−4.73205	−0.649997	−0.324999	+	0.945715i	$0.605364\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−7.46410	−0.988644
$58$	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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	−1.46410	−0.181599
$66$	0	0
$67$	10.9282	1.33509	0.667546	−	0.744568i	$-0.267345\pi$
$67$	10.9282	1.33509	0.667546	+	0.744568i	$0.267345\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	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
$73$	11.4641	1.34177	0.670886	−	0.741561i	$-0.265914\pi$
$73$	11.4641	1.34177	0.670886	+	0.741561i	$0.265914\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−6.73205	−0.757415	−0.378707	−	0.925516i	$-0.623631\pi$
$79$	−6.73205	−0.757415	−0.378707	+	0.925516i	$0.623631\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	4.39230	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$83$	4.39230	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	12.9282	1.38605
$88$	0	0
$89$	−15.4641	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$89$	−15.4641	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	24.3923	2.52936
$94$	0	0
$95$	2.73205	0.280302
$96$	0	0
$97$	5.80385	0.589291	0.294646	−	0.955607i	$-0.404798\pi$
$97$	5.80385	0.589291	0.294646	+	0.955607i	$0.404798\pi$
$98$	0	0
$99$	−4.46410	−0.448659
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−10.5359	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$103$	−10.5359	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$104$	0	0
$105$	2.73205	0.266621
$106$	0	0
$107$	−0.928203	−0.0897328	−0.0448664	−	0.998993i	$-0.514286\pi$
$107$	−0.928203	−0.0897328	−0.0448664	+	0.998993i	$0.514286\pi$
$108$	0	0
$109$	−1.80385	−0.172777	−0.0863886	−	0.996262i	$-0.527533\pi$
$109$	−1.80385	−0.172777	−0.0863886	+	0.996262i	$0.527533\pi$
$110$	0	0
$111$	−8.92820	−0.847428
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	−1.26795	−0.118237
$116$	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
$123$	12.9282	1.16570
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.85641	−0.874615	−0.437307	−	0.899312i	$-0.644068\pi$
$127$	−9.85641	−0.874615	−0.437307	+	0.899312i	$0.644068\pi$
$128$	0	0
$129$	−13.4641	−1.18545
$130$	0	0
$131$	17.6603	1.54298	0.771492	−	0.636239i	$-0.219511\pi$
$131$	17.6603	1.54298	0.771492	+	0.636239i	$0.219511\pi$
$132$	0	0
$133$	−2.73205	−0.236899
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	7.85641	0.671218	0.335609	−	0.942001i	$-0.391058\pi$
$137$	7.85641	0.671218	0.335609	+	0.942001i	$0.391058\pi$
$138$	0	0
$139$	−4.19615	−0.355913	−0.177957	−	0.984038i	$-0.556949\pi$
$139$	−4.19615	−0.355913	−0.177957	+	0.984038i	$0.556949\pi$
$140$	0	0
$141$	0	0
$142$	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
$151$	7.80385	0.635068	0.317534	−	0.948247i	$-0.397145\pi$
$151$	7.80385	0.635068	0.317534	+	0.948247i	$0.397145\pi$
$152$	0	0
$153$	15.4641	1.25020
$154$	0	0
$155$	−8.92820	−0.717131
$156$	0	0
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	0	0
$159$	12.9282	1.02527
$160$	0	0
$161$	1.26795	0.0999284
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	2.73205	0.212690
$166$	0	0
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	12.1962	0.932663
$172$	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
$177$	−37.8564	−2.84546
$178$	0	0
$179$	19.8564	1.48414	0.742069	−	0.670324i	$-0.233845\pi$
$179$	19.8564	1.48414	0.742069	+	0.670324i	$0.233845\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	0	0
$183$	−5.46410	−0.403918
$184$	0	0
$185$	3.26795	0.240264
$186$	0	0
$187$	−3.46410	−0.253320
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	−25.4641	−1.83295	−0.916473	−	0.400096i	$-0.868977\pi$
$193$	−25.4641	−1.83295	−0.916473	+	0.400096i	$0.868977\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−22.3923	−1.59539	−0.797693	−	0.603064i	$-0.793946\pi$
$197$	−22.3923	−1.59539	−0.797693	+	0.603064i	$0.793946\pi$
$198$	0	0
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	−29.8564	−2.10591
$202$	0	0
$203$	4.73205	0.332125
$204$	0	0
$205$	−4.73205	−0.330501
$206$	0	0
$207$	−5.66025	−0.393415
$208$	0	0
$209$	−2.73205	−0.188980
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	25.8564	1.77165
$214$	0	0
$215$	4.92820	0.336101
$216$	0	0
$217$	8.92820	0.606086
$218$	0	0
$219$	−31.3205	−2.11644
$220$	0	0
$221$	−5.07180	−0.341166
$222$	0	0
$223$	8.39230	0.561990	0.280995	−	0.959709i	$-0.409336\pi$
$223$	8.39230	0.561990	0.280995	+	0.959709i	$0.409336\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	−13.4641	−0.889733	−0.444866	−	0.895597i	$-0.646749\pi$
$229$	−13.4641	−0.889733	−0.444866	+	0.895597i	$0.646749\pi$
$230$	0	0
$231$	−2.73205	−0.179756
$232$	0	0
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	18.3923	1.19471
$238$	0	0
$239$	3.80385	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$239$	3.80385	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$240$	0	0
$241$	−18.1962	−1.17212	−0.586059	−	0.810269i	$-0.699321\pi$
$241$	−18.1962	−1.17212	−0.586059	+	0.810269i	$0.699321\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	1.26795	0.0797153
$254$	0	0
$255$	−9.46410	−0.592665
$256$	0	0
$257$	22.9808	1.43350	0.716750	−	0.697330i	$-0.245629\pi$
$257$	22.9808	1.43350	0.716750	+	0.697330i	$0.245629\pi$
$258$	0	0
$259$	−3.26795	−0.203060
$260$	0	0
$261$	−21.1244	−1.30756
$262$	0	0
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	0	0
$265$	−4.73205	−0.290688
$266$	0	0
$267$	42.2487	2.58558
$268$	0	0
$269$	4.39230	0.267804	0.133902	−	0.990995i	$-0.457249\pi$
$269$	4.39230	0.267804	0.133902	+	0.990995i	$0.457249\pi$
$270$	0	0
$271$	−24.3923	−1.48173	−0.740863	−	0.671656i	$-0.765583\pi$
$271$	−24.3923	−1.48173	−0.740863	+	0.671656i	$0.765583\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	27.8564	1.67373	0.836865	−	0.547410i	$-0.184386\pi$
$277$	27.8564	1.67373	0.836865	+	0.547410i	$0.184386\pi$
$278$	0	0
$279$	−39.8564	−2.38614
$280$	0	0
$281$	−10.3923	−0.619953	−0.309976	−	0.950744i	$-0.600321\pi$
$281$	−10.3923	−0.619953	−0.309976	+	0.950744i	$0.600321\pi$
$282$	0	0
$283$	10.9282	0.649614	0.324807	−	0.945780i	$-0.394701\pi$
$283$	10.9282	0.649614	0.324807	+	0.945780i	$0.394701\pi$
$284$	0	0
$285$	−7.46410	−0.442135
$286$	0	0
$287$	4.73205	0.279324
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−15.8564	−0.929519
$292$	0	0
$293$	−2.53590	−0.148149	−0.0740744	−	0.997253i	$-0.523600\pi$
$293$	−2.53590	−0.148149	−0.0740744	+	0.997253i	$0.523600\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	1.85641	0.107359
$300$	0	0
$301$	−4.92820	−0.284057
$302$	0	0
$303$	16.3923	0.941713
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	−31.3205	−1.78756	−0.893778	−	0.448510i	$-0.851955\pi$
$307$	−31.3205	−1.78756	−0.893778	+	0.448510i	$0.851955\pi$
$308$	0	0
$309$	28.7846	1.63750
$310$	0	0
$311$	−7.85641	−0.445496	−0.222748	−	0.974876i	$-0.571503\pi$
$311$	−7.85641	−0.445496	−0.222748	+	0.974876i	$0.571503\pi$
$312$	0	0
$313$	27.2679	1.54128	0.770638	−	0.637273i	$-0.219938\pi$
$313$	27.2679	1.54128	0.770638	+	0.637273i	$0.219938\pi$
$314$	0	0
$315$	−4.46410	−0.251524
$316$	0	0
$317$	−32.4449	−1.82229	−0.911143	−	0.412091i	$-0.864798\pi$
$317$	−32.4449	−1.82229	−0.911143	+	0.412091i	$0.864798\pi$
$318$	0	0
$319$	4.73205	0.264944
$320$	0	0
$321$	2.53590	0.141540
$322$	0	0
$323$	9.46410	0.526597
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	0	0
$327$	4.92820	0.272530
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.92820	0.270879	0.135439	−	0.990786i	$-0.456755\pi$
$331$	4.92820	0.270879	0.135439	+	0.990786i	$0.456755\pi$
$332$	0	0
$333$	14.5885	0.799443
$334$	0	0
$335$	10.9282	0.597072
$336$	0	0
$337$	−30.7846	−1.67694	−0.838472	−	0.544944i	$-0.816551\pi$
$337$	−30.7846	−1.67694	−0.838472	+	0.544944i	$0.816551\pi$
$338$	0	0
$339$	−35.3205	−1.91835
$340$	0	0
$341$	8.92820	0.483489
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.46410	0.186501
$346$	0	0
$347$	−7.85641	−0.421754	−0.210877	−	0.977513i	$-0.567632\pi$
$347$	−7.85641	−0.421754	−0.210877	+	0.977513i	$0.567632\pi$
$348$	0	0
$349$	30.3923	1.62686	0.813431	−	0.581661i	$-0.197597\pi$
$349$	30.3923	1.62686	0.813431	+	0.581661i	$0.197597\pi$
$350$	0	0
$351$	5.85641	0.312592
$352$	0	0
$353$	−32.4449	−1.72687	−0.863433	−	0.504464i	$-0.831690\pi$
$353$	−32.4449	−1.72687	−0.863433	+	0.504464i	$0.831690\pi$
$354$	0	0
$355$	−9.46410	−0.502302
$356$	0	0
$357$	9.46410	0.500893
$358$	0	0
$359$	1.26795	0.0669198	0.0334599	−	0.999440i	$-0.489347\pi$
$359$	1.26795	0.0669198	0.0334599	+	0.999440i	$0.489347\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	−2.73205	−0.143395
$364$	0	0
$365$	11.4641	0.600059
$366$	0	0
$367$	−29.4641	−1.53801	−0.769007	−	0.639241i	$-0.779249\pi$
$367$	−29.4641	−1.53801	−0.769007	+	0.639241i	$0.779249\pi$
$368$	0	0
$369$	−21.1244	−1.09969
$370$	0	0
$371$	4.73205	0.245676
$372$	0	0
$373$	30.3923	1.57365	0.786827	−	0.617174i	$-0.211722\pi$
$373$	30.3923	1.57365	0.786827	+	0.617174i	$0.211722\pi$
$374$	0	0
$375$	−2.73205	−0.141082
$376$	0	0
$377$	6.92820	0.356821
$378$	0	0
$379$	−23.7128	−1.21805	−0.609023	−	0.793153i	$-0.708438\pi$
$379$	−23.7128	−1.21805	−0.609023	+	0.793153i	$0.708438\pi$
$380$	0	0
$381$	26.9282	1.37957
$382$	0	0
$383$	−16.3923	−0.837608	−0.418804	−	0.908077i	$-0.637551\pi$
$383$	−16.3923	−0.837608	−0.418804	+	0.908077i	$0.637551\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	22.0000	1.11832
$388$	0	0
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	−4.39230	−0.222128
$392$	0	0
$393$	−48.2487	−2.43383
$394$	0	0
$395$	−6.73205	−0.338726
$396$	0	0
$397$	−38.3923	−1.92685	−0.963427	−	0.267970i	$-0.913647\pi$
$397$	−38.3923	−1.92685	−0.963427	+	0.267970i	$0.913647\pi$
$398$	0	0
$399$	7.46410	0.373672
$400$	0	0
$401$	−25.1769	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$401$	−25.1769	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$402$	0	0
$403$	13.0718	0.651153
$404$	0	0
$405$	−2.46410	−0.122442
$406$	0	0
$407$	−3.26795	−0.161986
$408$	0	0
$409$	22.1962	1.09753	0.548765	−	0.835977i	$-0.315098\pi$
$409$	22.1962	1.09753	0.548765	+	0.835977i	$0.315098\pi$
$410$	0	0
$411$	−21.4641	−1.05875
$412$	0	0
$413$	−13.8564	−0.681829
$414$	0	0
$415$	4.39230	0.215610
$416$	0	0
$417$	11.4641	0.561399
$418$	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
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−18.3397	−0.883394	−0.441697	−	0.897164i	$-0.645623\pi$
$431$	−18.3397	−0.883394	−0.441697	+	0.897164i	$0.645623\pi$
$432$	0	0
$433$	0.732051	0.0351801	0.0175901	−	0.999845i	$-0.494401\pi$
$433$	0.732051	0.0351801	0.0175901	+	0.999845i	$0.494401\pi$
$434$	0	0
$435$	12.9282	0.619860
$436$	0	0
$437$	−3.46410	−0.165710
$438$	0	0
$439$	−36.3923	−1.73691	−0.868455	−	0.495768i	$-0.834887\pi$
$439$	−36.3923	−1.73691	−0.868455	+	0.495768i	$0.834887\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	−20.7846	−0.987507	−0.493753	−	0.869602i	$-0.664375\pi$
$443$	−20.7846	−0.987507	−0.493753	+	0.869602i	$0.664375\pi$
$444$	0	0
$445$	−15.4641	−0.733069
$446$	0	0
$447$	19.8564	0.939176
$448$	0	0
$449$	−35.3205	−1.66688	−0.833439	−	0.552612i	$-0.813631\pi$
$449$	−35.3205	−1.66688	−0.833439	+	0.552612i	$0.813631\pi$
$450$	0	0
$451$	4.73205	0.222824
$452$	0	0
$453$	−21.3205	−1.00172
$454$	0	0
$455$	1.46410	0.0686381
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	−13.8564	−0.646762
$460$	0	0
$461$	29.3205	1.36559	0.682796	−	0.730609i	$-0.260764\pi$
$461$	29.3205	1.36559	0.682796	+	0.730609i	$0.260764\pi$
$462$	0	0
$463$	−12.9808	−0.603267	−0.301634	−	0.953424i	$-0.597532\pi$
$463$	−12.9808	−0.603267	−0.301634	+	0.953424i	$0.597532\pi$
$464$	0	0
$465$	24.3923	1.13117
$466$	0	0
$467$	−17.6603	−0.817219	−0.408610	−	0.912709i	$-0.633986\pi$
$467$	−17.6603	−0.817219	−0.408610	+	0.912709i	$0.633986\pi$
$468$	0	0
$469$	−10.9282	−0.504618
$470$	0	0
$471$	−17.4641	−0.804703
$472$	0	0
$473$	−4.92820	−0.226599
$474$	0	0
$475$	2.73205	0.125355
$476$	0	0
$477$	−21.1244	−0.967218
$478$	0	0
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	−4.78461	−0.218159
$482$	0	0
$483$	−3.46410	−0.157622
$484$	0	0
$485$	5.80385	0.263539
$486$	0	0
$487$	−4.87564	−0.220937	−0.110468	−	0.993880i	$-0.535235\pi$
$487$	−4.87564	−0.220937	−0.110468	+	0.993880i	$0.535235\pi$
$488$	0	0
$489$	21.8564	0.988381
$490$	0	0
$491$	−42.9282	−1.93732	−0.968661	−	0.248385i	$-0.920100\pi$
$491$	−42.9282	−1.93732	−0.968661	+	0.248385i	$0.920100\pi$
$492$	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
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	0	0
$501$	−37.8564	−1.69130
$502$	0	0
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	29.6603	1.31726
$508$	0	0
$509$	2.78461	0.123426	0.0617128	−	0.998094i	$-0.480344\pi$
$509$	2.78461	0.123426	0.0617128	+	0.998094i	$0.480344\pi$
$510$	0	0
$511$	−11.4641	−0.507142
$512$	0	0
$513$	−10.9282	−0.482492
$514$	0	0
$515$	−10.5359	−0.464267
$516$	0	0
$517$	0	0
$518$	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
$523$	−19.3205	−0.844827	−0.422413	−	0.906403i	$-0.638817\pi$
$523$	−19.3205	−0.844827	−0.422413	+	0.906403i	$0.638817\pi$
$524$	0	0
$525$	2.73205	0.119236
$526$	0	0
$527$	−30.9282	−1.34725
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	61.8564	2.68434
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	−0.928203	−0.0401297
$536$	0	0
$537$	−54.2487	−2.34100
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	17.1244	0.736234	0.368117	−	0.929780i	$-0.380003\pi$
$541$	17.1244	0.736234	0.368117	+	0.929780i	$0.380003\pi$
$542$	0	0
$543$	32.3923	1.39009
$544$	0	0
$545$	−1.80385	−0.0772683
$546$	0	0
$547$	−4.78461	−0.204575	−0.102288	−	0.994755i	$-0.532616\pi$
$547$	−4.78461	−0.204575	−0.102288	+	0.994755i	$0.532616\pi$
$548$	0	0
$549$	8.92820	0.381046
$550$	0	0
$551$	−12.9282	−0.550760
$552$	0	0
$553$	6.73205	0.286276
$554$	0	0
$555$	−8.92820	−0.378981
$556$	0	0
$557$	−12.2487	−0.518995	−0.259497	−	0.965744i	$-0.583557\pi$
$557$	−12.2487	−0.518995	−0.259497	+	0.965744i	$0.583557\pi$
$558$	0	0
$559$	−7.21539	−0.305178
$560$	0	0
$561$	9.46410	0.399575
$562$	0	0
$563$	20.7846	0.875967	0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846	0.875967	0.437983	+	0.898983i	$0.355693\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	20.3923	0.853391	0.426696	−	0.904395i	$-0.359678\pi$
$571$	20.3923	0.853391	0.426696	+	0.904395i	$0.359678\pi$
$572$	0	0
$573$	18.9282	0.790737
$574$	0	0
$575$	−1.26795	−0.0528771
$576$	0	0
$577$	8.33975	0.347188	0.173594	−	0.984817i	$-0.444462\pi$
$577$	8.33975	0.347188	0.173594	+	0.984817i	$0.444462\pi$
$578$	0	0
$579$	69.5692	2.89120
$580$	0	0
$581$	−4.39230	−0.182224
$582$	0	0
$583$	4.73205	0.195982
$584$	0	0
$585$	−6.53590	−0.270226
$586$	0	0
$587$	28.9808	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$587$	28.9808	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$588$	0	0
$589$	−24.3923	−1.00507
$590$	0	0
$591$	61.1769	2.51648
$592$	0	0
$593$	32.5359	1.33609	0.668045	−	0.744121i	$-0.267132\pi$
$593$	32.5359	1.33609	0.668045	+	0.744121i	$0.267132\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	0	0
$597$	−67.7128	−2.77130
$598$	0	0
$599$	32.1051	1.31178	0.655890	−	0.754857i	$-0.272294\pi$
$599$	32.1051	1.31178	0.655890	+	0.754857i	$0.272294\pi$
$600$	0	0
$601$	28.4449	1.16029	0.580145	−	0.814513i	$-0.302996\pi$
$601$	28.4449	1.16029	0.580145	+	0.814513i	$0.302996\pi$
$602$	0	0
$603$	48.7846	1.98666
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−34.7846	−1.41186	−0.705932	−	0.708280i	$-0.749472\pi$
$607$	−34.7846	−1.41186	−0.705932	+	0.708280i	$0.749472\pi$
$608$	0	0
$609$	−12.9282	−0.523877
$610$	0	0
$611$	0	0
$612$	0	0
$613$	27.1769	1.09767	0.548833	−	0.835932i	$-0.315072\pi$
$613$	27.1769	1.09767	0.548833	+	0.835932i	$0.315072\pi$
$614$	0	0
$615$	12.9282	0.521315
$616$	0	0
$617$	8.53590	0.343642	0.171821	−	0.985128i	$-0.445035\pi$
$617$	8.53590	0.343642	0.171821	+	0.985128i	$0.445035\pi$
$618$	0	0
$619$	10.9282	0.439242	0.219621	−	0.975585i	$-0.429518\pi$
$619$	10.9282	0.439242	0.219621	+	0.975585i	$0.429518\pi$
$620$	0	0
$621$	5.07180	0.203524
$622$	0	0
$623$	15.4641	0.619556
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.46410	0.298088
$628$	0	0
$629$	11.3205	0.451378
$630$	0	0
$631$	12.7846	0.508947	0.254474	−	0.967080i	$-0.418098\pi$
$631$	12.7846	0.508947	0.254474	+	0.967080i	$0.418098\pi$
$632$	0	0
$633$	21.8564	0.868714
$634$	0	0
$635$	−9.85641	−0.391140
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	0	0
$639$	−42.2487	−1.67133
$640$	0	0
$641$	19.8564	0.784281	0.392140	−	0.919905i	$-0.371735\pi$
$641$	19.8564	0.784281	0.392140	+	0.919905i	$0.371735\pi$
$642$	0	0
$643$	14.7321	0.580975	0.290488	−	0.956879i	$-0.406182\pi$
$643$	14.7321	0.580975	0.290488	+	0.956879i	$0.406182\pi$
$644$	0	0
$645$	−13.4641	−0.530148
$646$	0	0
$647$	−37.1769	−1.46158	−0.730788	−	0.682605i	$-0.760847\pi$
$647$	−37.1769	−1.46158	−0.730788	+	0.682605i	$0.760847\pi$
$648$	0	0
$649$	−13.8564	−0.543912
$650$	0	0
$651$	−24.3923	−0.956010
$652$	0	0
$653$	−9.80385	−0.383654	−0.191827	−	0.981429i	$-0.561441\pi$
$653$	−9.80385	−0.383654	−0.191827	+	0.981429i	$0.561441\pi$
$654$	0	0
$655$	17.6603	0.690043
$656$	0	0
$657$	51.1769	1.99660
$658$	0	0
$659$	−6.24871	−0.243415	−0.121708	−	0.992566i	$-0.538837\pi$
$659$	−6.24871	−0.243415	−0.121708	+	0.992566i	$0.538837\pi$
$660$	0	0
$661$	3.85641	0.149997	0.0749984	−	0.997184i	$-0.476105\pi$
$661$	3.85641	0.149997	0.0749984	+	0.997184i	$0.476105\pi$
$662$	0	0
$663$	13.8564	0.538138
$664$	0	0
$665$	−2.73205	−0.105944
$666$	0	0
$667$	6.00000	0.232321
$668$	0	0
$669$	−22.9282	−0.886456
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−20.1436	−0.776478	−0.388239	−	0.921559i	$-0.626917\pi$
$673$	−20.1436	−0.776478	−0.388239	+	0.921559i	$0.626917\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	19.8564	0.763144	0.381572	−	0.924339i	$-0.375383\pi$
$677$	19.8564	0.763144	0.381572	+	0.924339i	$0.375383\pi$
$678$	0	0
$679$	−5.80385	−0.222731
$680$	0	0
$681$	37.8564	1.45066
$682$	0	0
$683$	−42.2487	−1.61660	−0.808301	−	0.588769i	$-0.799613\pi$
$683$	−42.2487	−1.61660	−0.808301	+	0.588769i	$0.799613\pi$
$684$	0	0
$685$	7.85641	0.300178
$686$	0	0
$687$	36.7846	1.40342
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	−31.3205	−1.19149	−0.595744	−	0.803174i	$-0.703143\pi$
$691$	−31.3205	−1.19149	−0.595744	+	0.803174i	$0.703143\pi$
$692$	0	0
$693$	4.46410	0.169577
$694$	0	0
$695$	−4.19615	−0.159169
$696$	0	0
$697$	−16.3923	−0.620903
$698$	0	0
$699$	−21.4641	−0.811847
$700$	0	0
$701$	14.1962	0.536181	0.268091	−	0.963394i	$-0.413607\pi$
$701$	14.1962	0.536181	0.268091	+	0.963394i	$0.413607\pi$
$702$	0	0
$703$	8.92820	0.336734
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−2.39230	−0.0898449	−0.0449224	−	0.998990i	$-0.514304\pi$
$709$	−2.39230	−0.0898449	−0.0449224	+	0.998990i	$0.514304\pi$
$710$	0	0
$711$	−30.0526	−1.12706
$712$	0	0
$713$	11.3205	0.423956
$714$	0	0
$715$	1.46410	0.0547543
$716$	0	0
$717$	−10.3923	−0.388108
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	10.5359	0.392377
$722$	0	0
$723$	49.7128	1.84884
$724$	0	0
$725$	−4.73205	−0.175744
$726$	0	0
$727$	38.6410	1.43312	0.716558	−	0.697528i	$-0.245717\pi$
$727$	38.6410	1.43312	0.716558	+	0.697528i	$0.245717\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	17.0718	0.631423
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	−2.73205	−0.100773
$736$	0	0
$737$	−10.9282	−0.402546
$738$	0	0
$739$	−18.1436	−0.667423	−0.333711	−	0.942675i	$-0.608301\pi$
$739$	−18.1436	−0.667423	−0.333711	+	0.942675i	$0.608301\pi$
$740$	0	0
$741$	10.9282	0.401458
$742$	0	0
$743$	−3.71281	−0.136210	−0.0681049	−	0.997678i	$-0.521695\pi$
$743$	−3.71281	−0.136210	−0.0681049	+	0.997678i	$0.521695\pi$
$744$	0	0
$745$	−7.26795	−0.266277
$746$	0	0
$747$	19.6077	0.717408
$748$	0	0
$749$	0.928203	0.0339158
$750$	0	0
$751$	−2.24871	−0.0820566	−0.0410283	−	0.999158i	$-0.513063\pi$
$751$	−2.24871	−0.0820566	−0.0410283	+	0.999158i	$0.513063\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.80385	0.284011
$756$	0	0
$757$	23.3731	0.849509	0.424754	−	0.905309i	$-0.360360\pi$
$757$	23.3731	0.849509	0.424754	+	0.905309i	$0.360360\pi$
$758$	0	0
$759$	−3.46410	−0.125739
$760$	0	0
$761$	19.2679	0.698463	0.349231	−	0.937037i	$-0.386443\pi$
$761$	19.2679	0.698463	0.349231	+	0.937037i	$0.386443\pi$
$762$	0	0
$763$	1.80385	0.0653037
$764$	0	0
$765$	15.4641	0.559106
$766$	0	0
$767$	−20.2872	−0.732528
$768$	0	0
$769$	−24.4449	−0.881504	−0.440752	−	0.897629i	$-0.645288\pi$
$769$	−24.4449	−0.881504	−0.440752	+	0.897629i	$0.645288\pi$
$770$	0	0
$771$	−62.7846	−2.26113
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	8.92820	0.320298
$778$	0	0
$779$	−12.9282	−0.463201
$780$	0	0
$781$	9.46410	0.338652
$782$	0	0
$783$	18.9282	0.676439
$784$	0	0
$785$	6.39230	0.228151
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	51.7128	1.84102
$790$	0	0
$791$	−12.9282	−0.459674
$792$	0	0
$793$	−2.92820	−0.103984
$794$	0	0
$795$	12.9282	0.458516
$796$	0	0
$797$	6.67949	0.236600	0.118300	−	0.992978i	$-0.462256\pi$
$797$	6.67949	0.236600	0.118300	+	0.992978i	$0.462256\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−69.0333	−2.43917
$802$	0	0
$803$	−11.4641	−0.404559
$804$	0	0
$805$	1.26795	0.0446893
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	5.32051	0.187059	0.0935296	−	0.995617i	$-0.470185\pi$
$809$	5.32051	0.187059	0.0935296	+	0.995617i	$0.470185\pi$
$810$	0	0
$811$	4.58846	0.161123	0.0805613	−	0.996750i	$-0.474329\pi$
$811$	4.58846	0.161123	0.0805613	+	0.996750i	$0.474329\pi$
$812$	0	0
$813$	66.6410	2.33720
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	13.4641	0.471049
$818$	0	0
$819$	6.53590	0.228383
$820$	0	0
$821$	52.0526	1.81665	0.908323	−	0.418269i	$-0.137363\pi$
$821$	52.0526	1.81665	0.908323	+	0.418269i	$0.137363\pi$
$822$	0	0
$823$	24.1962	0.843425	0.421712	−	0.906730i	$-0.361429\pi$
$823$	24.1962	0.843425	0.421712	+	0.906730i	$0.361429\pi$
$824$	0	0
$825$	2.73205	0.0951178
$826$	0	0
$827$	−53.5692	−1.86278	−0.931392	−	0.364017i	$-0.881405\pi$
$827$	−53.5692	−1.86278	−0.931392	+	0.364017i	$0.881405\pi$
$828$	0	0
$829$	40.1051	1.39291	0.696454	−	0.717601i	$-0.254760\pi$
$829$	40.1051	1.39291	0.696454	+	0.717601i	$0.254760\pi$
$830$	0	0
$831$	−76.1051	−2.64006
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	13.8564	0.479521
$836$	0	0
$837$	35.7128	1.23442
$838$	0	0
$839$	38.7846	1.33899	0.669497	−	0.742815i	$-0.266510\pi$
$839$	38.7846	1.33899	0.669497	+	0.742815i	$0.266510\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	0	0
$843$	28.3923	0.977883
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−29.8564	−1.02467
$850$	0	0
$851$	−4.14359	−0.142041
$852$	0	0
$853$	20.9282	0.716568	0.358284	−	0.933613i	$-0.383362\pi$
$853$	20.9282	0.716568	0.358284	+	0.933613i	$0.383362\pi$
$854$	0	0
$855$	12.1962	0.417100
$856$	0	0
$857$	−19.1769	−0.655071	−0.327535	−	0.944839i	$-0.606218\pi$
$857$	−19.1769	−0.655071	−0.327535	+	0.944839i	$0.606218\pi$
$858$	0	0
$859$	−40.7846	−1.39155	−0.695776	−	0.718258i	$-0.744940\pi$
$859$	−40.7846	−1.39155	−0.695776	+	0.718258i	$0.744940\pi$
$860$	0	0
$861$	−12.9282	−0.440592
$862$	0	0
$863$	−24.5885	−0.837001	−0.418500	−	0.908217i	$-0.637444\pi$
$863$	−24.5885	−0.837001	−0.418500	+	0.908217i	$0.637444\pi$
$864$	0	0
$865$	0.928203	0.0315599
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	6.73205	0.228369
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	25.9090	0.876886
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−23.1769	−0.782629	−0.391314	−	0.920257i	$-0.627979\pi$
$877$	−23.1769	−0.782629	−0.391314	+	0.920257i	$0.627979\pi$
$878$	0	0
$879$	6.92820	0.233682
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	0	0
$883$	41.1769	1.38571	0.692857	−	0.721075i	$-0.256352\pi$
$883$	41.1769	1.38571	0.692857	+	0.721075i	$0.256352\pi$
$884$	0	0
$885$	−37.8564	−1.27253
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	9.85641	0.330573
$890$	0	0
$891$	2.46410	0.0825505
$892$	0	0
$893$	0	0
$894$	0	0
$895$	19.8564	0.663726
$896$	0	0
$897$	−5.07180	−0.169342
$898$	0	0
$899$	42.2487	1.40907
$900$	0	0
$901$	−16.3923	−0.546107
$902$	0	0
$903$	13.4641	0.448057
$904$	0	0
$905$	−11.8564	−0.394120
$906$	0	0
$907$	15.3205	0.508709	0.254355	−	0.967111i	$-0.418137\pi$
$907$	15.3205	0.508709	0.254355	+	0.967111i	$0.418137\pi$
$908$	0	0
$909$	−26.7846	−0.888389
$910$	0	0
$911$	−14.5359	−0.481596	−0.240798	−	0.970575i	$-0.577409\pi$
$911$	−14.5359	−0.481596	−0.240798	+	0.970575i	$0.577409\pi$
$912$	0	0
$913$	−4.39230	−0.145364
$914$	0	0
$915$	−5.46410	−0.180638
$916$	0	0
$917$	−17.6603	−0.583193
$918$	0	0
$919$	−24.9808	−0.824039	−0.412020	−	0.911175i	$-0.635176\pi$
$919$	−24.9808	−0.824039	−0.412020	+	0.911175i	$0.635176\pi$
$920$	0	0
$921$	85.5692	2.81960
$922$	0	0
$923$	13.8564	0.456089
$924$	0	0
$925$	3.26795	0.107450
$926$	0	0
$927$	−47.0333	−1.54478
$928$	0	0
$929$	26.1051	0.856481	0.428241	−	0.903665i	$-0.359134\pi$
$929$	26.1051	0.856481	0.428241	+	0.903665i	$0.359134\pi$
$930$	0	0
$931$	2.73205	0.0895393
$932$	0	0
$933$	21.4641	0.702703
$934$	0	0
$935$	−3.46410	−0.113288
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−74.4974	−2.43113
$940$	0	0
$941$	−35.5692	−1.15952	−0.579762	−	0.814786i	$-0.696854\pi$
$941$	−35.5692	−1.15952	−0.579762	+	0.814786i	$0.696854\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	25.1769	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$947$	25.1769	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$948$	0	0
$949$	−16.7846	−0.544851
$950$	0	0
$951$	88.6410	2.87438
$952$	0	0
$953$	−11.3205	−0.366707	−0.183354	−	0.983047i	$-0.558695\pi$
$953$	−11.3205	−0.366707	−0.183354	+	0.983047i	$0.558695\pi$
$954$	0	0
$955$	−6.92820	−0.224191
$956$	0	0
$957$	−12.9282	−0.417909
$958$	0	0
$959$	−7.85641	−0.253697
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	−4.14359	−0.133525
$964$	0	0
$965$	−25.4641	−0.819718
$966$	0	0
$967$	26.1436	0.840721	0.420361	−	0.907357i	$-0.361904\pi$
$967$	26.1436	0.840721	0.420361	+	0.907357i	$0.361904\pi$
$968$	0	0
$969$	−25.8564	−0.830627
$970$	0	0
$971$	−49.8564	−1.59997	−0.799984	−	0.600021i	$-0.795159\pi$
$971$	−49.8564	−1.59997	−0.799984	+	0.600021i	$0.795159\pi$
$972$	0	0
$973$	4.19615	0.134522
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−0.928203	−0.0296959	−0.0148479	−	0.999890i	$-0.504726\pi$
$977$	−0.928203	−0.0296959	−0.0148479	+	0.999890i	$0.504726\pi$
$978$	0	0
$979$	15.4641	0.494235
$980$	0	0
$981$	−8.05256	−0.257098
$982$	0	0
$983$	7.60770	0.242648	0.121324	−	0.992613i	$-0.461286\pi$
$983$	7.60770	0.242648	0.121324	+	0.992613i	$0.461286\pi$
$984$	0	0
$985$	−22.3923	−0.713478
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.24871	−0.198697
$990$	0	0
$991$	22.9282	0.728338	0.364169	−	0.931333i	$-0.381353\pi$
$991$	22.9282	0.728338	0.364169	+	0.931333i	$0.381353\pi$
$992$	0	0
$993$	−13.4641	−0.427270
$994$	0	0
$995$	24.7846	0.785725
$996$	0	0
$997$	−23.8564	−0.755540	−0.377770	−	0.925899i	$-0.623309\pi$
$997$	−23.8564	−0.755540	−0.377770	+	0.925899i	$0.623309\pi$
$998$	0	0
$999$	−13.0718	−0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.v.1.1		2
4.3	odd	2		770.2.a.h.1.2	✓	2
12.11	even	2		6930.2.a.ca.1.1		2
20.3	even	4		3850.2.c.s.1849.4		4
20.7	even	4		3850.2.c.s.1849.1		4
20.19	odd	2		3850.2.a.bm.1.1		2
28.27	even	2		5390.2.a.bk.1.1		2
44.43	even	2		8470.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.h.1.2	✓	2	4.3	odd	2
3850.2.a.bm.1.1		2	20.19	odd	2
3850.2.c.s.1849.1		4	20.7	even	4
3850.2.c.s.1849.4		4	20.3	even	4
5390.2.a.bk.1.1		2	28.27	even	2
6160.2.a.v.1.1		2	1.1	even	1	trivial
6930.2.a.ca.1.1		2	12.11	even	2
8470.2.a.ce.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)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} +1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} +1.00000 q^{5} -1.00000 q^{7} +4.46410 q^{9} -1.00000 q^{11} -1.46410 q^{13} -2.73205 q^{15} +3.46410 q^{17} +2.73205 q^{19} +2.73205 q^{21} -1.26795 q^{23} +1.00000 q^{25} -4.00000 q^{27} -4.73205 q^{29} -8.92820 q^{31} +2.73205 q^{33} -1.00000 q^{35} +3.26795 q^{37} +4.00000 q^{39} -4.73205 q^{41} +4.92820 q^{43} +4.46410 q^{45} +1.00000 q^{49} -9.46410 q^{51} -4.73205 q^{53} -1.00000 q^{55} -7.46410 q^{57} +13.8564 q^{59} +2.00000 q^{61} -4.46410 q^{63} -1.46410 q^{65} +10.9282 q^{67} +3.46410 q^{69} -9.46410 q^{71} +11.4641 q^{73} -2.73205 q^{75} +1.00000 q^{77} -6.73205 q^{79} -2.46410 q^{81} +4.39230 q^{83} +3.46410 q^{85} +12.9282 q^{87} -15.4641 q^{89} +1.46410 q^{91} +24.3923 q^{93} +2.73205 q^{95} +5.80385 q^{97} -4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{19} + 2 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} + 2 q^{49} - 12 q^{51} - 6 q^{53} - 2 q^{55} - 8 q^{57} + 4 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 12 q^{71} + 16 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} - 24 q^{89} - 4 q^{91} + 28 q^{93} + 2 q^{95} + 22 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 - 2 * q^15 + 2 * 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 + 10 * q^37 + 8 * q^39 - 6 * q^41 - 4 * q^43 + 2 * q^45 + 2 * q^49 - 12 * q^51 - 6 * q^53 - 2 * q^55 - 8 * q^57 + 4 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 12 * q^71 + 16 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 - 24 * q^89 - 4 * q^91 + 28 * q^93 + 2 * q^95 + 22 * q^97 - 2 * q^99

Introduction

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