LMFDB - Embedded newform 6480.2.a.ca.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.62352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 24 $ x^4 - 2x^3 - 11x^2 + 12*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.16910$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.562950 q^{7} -6.36314 q^{11} +3.75699 q^{13} +6.65815 q^{17} +3.33820 q^{19} -4.16910 q^{23} +1.00000 q^{25} -9.82725 q^{29} -5.16910 q^{31} -0.562950 q^{35} +3.14416 q^{37} -11.5842 q^{41} -8.60826 q^{43} +12.2913 q^{47} -6.68309 q^{49} -0.434941 q^{53} -6.36314 q^{55} +1.73205 q^{59} +4.92399 q^{61} +3.75699 q^{65} +11.6545 q^{67} -11.7774 q^{71} +4.31994 q^{73} +3.58213 q^{77} -7.46410 q^{79} +8.99635 q^{83} +6.65815 q^{85} +6.36736 q^{89} -2.11500 q^{91} +3.33820 q^{95} -16.4604 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{17} - 10 q^{23} + 4 q^{25} - 4 q^{29} - 14 q^{31} - 2 q^{35} + 14 q^{37} + 4 q^{41} - 22 q^{43} + 10 q^{49} - 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} - 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^17 - 10 * q^23 + 4 * q^25 - 4 * q^29 - 14 * q^31 - 2 * q^35 + 14 * q^37 + 4 * q^41 - 22 * q^43 + 10 * q^49 - 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 - 28 * q^71 + 2 * q^73 - 30 * q^77 - 16 * q^79 - 6 * q^83 - 2 * q^85 - 8 * q^89 - 30 * q^91 - 10 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.562950	−0.212775	−0.106388	−	0.994325i	$-0.533928\pi$
$7$	−0.562950	−0.212775	−0.106388	+	0.994325i	$0.533928\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.36314	−1.91856	−0.959280	−	0.282456i	$-0.908851\pi$
$11$	−6.36314	−1.91856	−0.959280	+	0.282456i	$0.908851\pi$
$12$	0	0
$13$	3.75699	1.04200	0.521001	−	0.853556i	$-0.325559\pi$
$13$	3.75699	1.04200	0.521001	+	0.853556i	$0.325559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.65815	1.61484	0.807419	−	0.589979i	$-0.200864\pi$
$17$	6.65815	1.61484	0.807419	+	0.589979i	$0.200864\pi$
$18$	0	0
$19$	3.33820	0.765836	0.382918	−	0.923782i	$-0.374919\pi$
$19$	3.33820	0.765836	0.382918	+	0.923782i	$0.374919\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.16910	−0.869318	−0.434659	−	0.900595i	$-0.643131\pi$
$23$	−4.16910	−0.869318	−0.434659	+	0.900595i	$0.643131\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.82725	−1.82487	−0.912437	−	0.409217i	$-0.865802\pi$
$29$	−9.82725	−1.82487	−0.912437	+	0.409217i	$0.865802\pi$
$30$	0	0
$31$	−5.16910	−0.928398	−0.464199	−	0.885731i	$-0.653658\pi$
$31$	−5.16910	−0.928398	−0.464199	+	0.885731i	$0.653658\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.562950	−0.0951559
$36$	0	0
$37$	3.14416	0.516896	0.258448	−	0.966025i	$-0.416789\pi$
$37$	3.14416	0.516896	0.258448	+	0.966025i	$0.416789\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.5842	−1.80915	−0.904577	−	0.426310i	$-0.859813\pi$
$41$	−11.5842	−1.80915	−0.904577	+	0.426310i	$0.859813\pi$
$42$	0	0
$43$	−8.60826	−1.31275	−0.656374	−	0.754436i	$-0.727911\pi$
$43$	−8.60826	−1.31275	−0.656374	+	0.754436i	$0.727911\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.2913	1.79288	0.896439	−	0.443168i	$-0.146145\pi$
$47$	12.2913	1.79288	0.896439	+	0.443168i	$0.146145\pi$
$48$	0	0
$49$	−6.68309	−0.954727
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.434941	−0.0597438	−0.0298719	−	0.999554i	$-0.509510\pi$
$53$	−0.434941	−0.0597438	−0.0298719	+	0.999554i	$0.509510\pi$
$54$	0	0
$55$	−6.36314	−0.858006
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	4.92399	0.630452	0.315226	−	0.949017i	$-0.397920\pi$
$61$	4.92399	0.630452	0.315226	+	0.949017i	$0.397920\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.75699	0.465998
$66$	0	0
$67$	11.6545	1.42382	0.711911	−	0.702269i	$-0.247830\pi$
$67$	11.6545	1.42382	0.711911	+	0.702269i	$0.247830\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.7774	−1.39772	−0.698858	−	0.715261i	$-0.746308\pi$
$71$	−11.7774	−1.39772	−0.698858	+	0.715261i	$0.746308\pi$
$72$	0	0
$73$	4.31994	0.505611	0.252806	−	0.967517i	$-0.418647\pi$
$73$	4.31994	0.505611	0.252806	+	0.967517i	$0.418647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.58213	0.408222
$78$	0	0
$79$	−7.46410	−0.839777	−0.419889	−	0.907576i	$-0.637931\pi$
$79$	−7.46410	−0.839777	−0.419889	+	0.907576i	$0.637931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.99635	0.987477	0.493739	−	0.869610i	$-0.335630\pi$
$83$	8.99635	0.987477	0.493739	+	0.869610i	$0.335630\pi$
$84$	0	0
$85$	6.65815	0.722177
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.36736	0.674939	0.337470	−	0.941336i	$-0.390429\pi$
$89$	6.36736	0.674939	0.337470	+	0.941336i	$0.390429\pi$
$90$	0	0
$91$	−2.11500	−0.221712
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.33820	0.342492
$96$	0	0
$97$	−16.4604	−1.67131	−0.835653	−	0.549258i	$-0.814910\pi$
$97$	−16.4604	−1.67131	−0.835653	+	0.549258i	$0.814910\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.226856	−0.0225731	−0.0112865	−	0.999936i	$-0.503593\pi$
$101$	−0.226856	−0.0225731	−0.0112865	+	0.999936i	$0.503593\pi$
$102$	0	0
$103$	0.833008	0.0820787	0.0410394	−	0.999158i	$-0.486933\pi$
$103$	0.833008	0.0820787	0.0410394	+	0.999158i	$0.486933\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.39230	−0.617967	−0.308984	−	0.951067i	$-0.599989\pi$
$107$	−6.39230	−0.617967	−0.308984	+	0.951067i	$0.599989\pi$
$108$	0	0
$109$	10.6332	1.01848	0.509238	−	0.860626i	$-0.329927\pi$
$109$	10.6332	1.01848	0.509238	+	0.860626i	$0.329927\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.48236	−0.703881	−0.351941	−	0.936022i	$-0.614478\pi$
$113$	−7.48236	−0.703881	−0.351941	+	0.936022i	$0.614478\pi$
$114$	0	0
$115$	−4.16910	−0.388771
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.74820	−0.343597
$120$	0	0
$121$	29.4896	2.68087
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.4912	−1.01967	−0.509837	−	0.860271i	$-0.670294\pi$
$127$	−11.4912	−1.01967	−0.509837	+	0.860271i	$0.670294\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.19193	−0.453621	−0.226811	−	0.973939i	$-0.572830\pi$
$131$	−5.19193	−0.453621	−0.226811	+	0.973939i	$0.572830\pi$
$132$	0	0
$133$	−1.87924	−0.162951
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.12590	−0.437935	−0.218968	−	0.975732i	$-0.570269\pi$
$137$	−5.12590	−0.437935	−0.218968	+	0.975732i	$0.570269\pi$
$138$	0	0
$139$	−19.7762	−1.67739	−0.838697	−	0.544599i	$-0.816682\pi$
$139$	−19.7762	−1.67739	−0.838697	+	0.544599i	$0.816682\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.9063	−1.99914
$144$	0	0
$145$	−9.82725	−0.816108
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.5900	−1.19526	−0.597630	−	0.801772i	$-0.703891\pi$
$149$	−14.5900	−1.19526	−0.597630	+	0.801772i	$0.703891\pi$
$150$	0	0
$151$	−3.21899	−0.261957	−0.130979	−	0.991385i	$-0.541812\pi$
$151$	−3.21899	−0.261957	−0.130979	+	0.991385i	$0.541812\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.16910	−0.415192
$156$	0	0
$157$	8.16334	0.651505	0.325753	−	0.945455i	$-0.394382\pi$
$157$	8.16334	0.651505	0.325753	+	0.945455i	$0.394382\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.34699	0.184969
$162$	0	0
$163$	6.12225	0.479531	0.239766	−	0.970831i	$-0.422929\pi$
$163$	6.12225	0.479531	0.239766	+	0.970831i	$0.422929\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.1721	0.787143	0.393572	−	0.919294i	$-0.371239\pi$
$167$	10.1721	0.787143	0.393572	+	0.919294i	$0.371239\pi$
$168$	0	0
$169$	1.11500	0.0857691
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.0973	−1.14783	−0.573913	−	0.818916i	$-0.694575\pi$
$173$	−15.0973	−1.14783	−0.573913	+	0.818916i	$0.694575\pi$
$174$	0	0
$175$	−0.562950	−0.0425550
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5842	−0.865847	−0.432923	−	0.901431i	$-0.642518\pi$
$179$	−11.5842	−0.865847	−0.432923	+	0.901431i	$0.642518\pi$
$180$	0	0
$181$	−1.50730	−0.112037	−0.0560185	−	0.998430i	$-0.517841\pi$
$181$	−1.50730	−0.112037	−0.0560185	+	0.998430i	$0.517841\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.14416	0.231163
$186$	0	0
$187$	−42.3667	−3.09816
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.62955	0.117910	0.0589550	−	0.998261i	$-0.481223\pi$
$191$	1.62955	0.117910	0.0589550	+	0.998261i	$0.481223\pi$
$192$	0	0
$193$	−7.92033	−0.570118	−0.285059	−	0.958510i	$-0.592013\pi$
$193$	−7.92033	−0.570118	−0.285059	+	0.958510i	$0.592013\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.4708	−1.03100	−0.515500	−	0.856889i	$-0.672394\pi$
$197$	−14.4708	−1.03100	−0.515500	+	0.856889i	$0.672394\pi$
$198$	0	0
$199$	−5.41422	−0.383804	−0.191902	−	0.981414i	$-0.561466\pi$
$199$	−5.41422	−0.383804	−0.191902	+	0.981414i	$0.561466\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.53225	0.388288
$204$	0	0
$205$	−11.5842	−0.809078
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−21.2415	−1.46930
$210$	0	0
$211$	0.121682	0.00837691	0.00418846	−	0.999991i	$-0.498667\pi$
$211$	0.121682	0.00837691	0.00418846	+	0.999991i	$0.498667\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.60826	−0.587078
$216$	0	0
$217$	2.90994	0.197540
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25.0146	1.68266
$222$	0	0
$223$	9.26219	0.620242	0.310121	−	0.950697i	$-0.399630\pi$
$223$	9.26219	0.620242	0.310121	+	0.950697i	$0.399630\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.6083	0.836840	0.418420	−	0.908254i	$-0.362584\pi$
$227$	12.6083	0.836840	0.418420	+	0.908254i	$0.362584\pi$
$228$	0	0
$229$	3.32360	0.219629	0.109815	−	0.993952i	$-0.464974\pi$
$229$	3.32360	0.219629	0.109815	+	0.993952i	$0.464974\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.12225	0.401082	0.200541	−	0.979685i	$-0.435730\pi$
$233$	6.12225	0.401082	0.200541	+	0.979685i	$0.435730\pi$
$234$	0	0
$235$	12.2913	0.801799
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.4641	−1.00029	−0.500145	−	0.865942i	$-0.666720\pi$
$239$	−15.4641	−1.00029	−0.500145	+	0.865942i	$0.666720\pi$
$240$	0	0
$241$	−18.6259	−1.19980	−0.599900	−	0.800075i	$-0.704793\pi$
$241$	−18.6259	−1.19980	−0.599900	+	0.800075i	$0.704793\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.68309	−0.426967
$246$	0	0
$247$	12.5416	0.798003
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.49481	−0.536187	−0.268094	−	0.963393i	$-0.586394\pi$
$251$	−8.49481	−0.536187	−0.268094	+	0.963393i	$0.586394\pi$
$252$	0	0
$253$	26.5286	1.66784
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.3881	−1.27177	−0.635887	−	0.771782i	$-0.719366\pi$
$257$	−20.3881	−1.27177	−0.635887	+	0.771782i	$0.719366\pi$
$258$	0	0
$259$	−1.77000	−0.109983
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.5353	−1.69790	−0.848949	−	0.528475i	$-0.822764\pi$
$263$	−27.5353	−1.69790	−0.848949	+	0.528475i	$0.822764\pi$
$264$	0	0
$265$	−0.434941	−0.0267182
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.417247	0.0254400	0.0127200	−	0.999919i	$-0.495951\pi$
$269$	0.417247	0.0254400	0.0127200	+	0.999919i	$0.495951\pi$
$270$	0	0
$271$	15.2664	0.927368	0.463684	−	0.886001i	$-0.346527\pi$
$271$	15.2664	0.927368	0.463684	+	0.886001i	$0.346527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.36314	−0.383712
$276$	0	0
$277$	16.8397	1.01180	0.505901	−	0.862592i	$-0.331160\pi$
$277$	16.8397	1.01180	0.505901	+	0.862592i	$0.331160\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.6238	−1.11100	−0.555501	−	0.831516i	$-0.687474\pi$
$281$	−18.6238	−1.11100	−0.555501	+	0.831516i	$0.687474\pi$
$282$	0	0
$283$	3.80230	0.226023	0.113012	−	0.993594i	$-0.463950\pi$
$283$	3.80230	0.226023	0.113012	+	0.993594i	$0.463950\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.52134	0.384943
$288$	0	0
$289$	27.3309	1.60770
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.7293	−1.91207	−0.956034	−	0.293257i	$-0.905261\pi$
$293$	−32.7293	−1.91207	−0.956034	+	0.293257i	$0.905261\pi$
$294$	0	0
$295$	1.73205	0.100844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−15.6633	−0.905831
$300$	0	0
$301$	4.84602	0.279320
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.92399	0.281947
$306$	0	0
$307$	−24.7080	−1.41016	−0.705081	−	0.709127i	$-0.749089\pi$
$307$	−24.7080	−1.41016	−0.705081	+	0.709127i	$0.749089\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0030	−1.53120	−0.765601	−	0.643316i	$-0.777558\pi$
$311$	−27.0030	−1.53120	−0.765601	+	0.643316i	$0.777558\pi$
$312$	0	0
$313$	20.6284	1.16598	0.582992	−	0.812478i	$-0.301882\pi$
$313$	20.6284	1.16598	0.582992	+	0.812478i	$0.301882\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.629550	0.0353590	0.0176795	−	0.999844i	$-0.494372\pi$
$317$	0.629550	0.0353590	0.0176795	+	0.999844i	$0.494372\pi$
$318$	0	0
$319$	62.5322	3.50113
$320$	0	0
$321$	0	0
$322$	0	0
$323$	22.2262	1.23670
$324$	0	0
$325$	3.75699	0.208400
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.91941	−0.381479
$330$	0	0
$331$	1.46657	0.0806098	0.0403049	−	0.999187i	$-0.487167\pi$
$331$	1.46657	0.0806098	0.0403049	+	0.999187i	$0.487167\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.6545	0.636753
$336$	0	0
$337$	−3.13012	−0.170508	−0.0852542	−	0.996359i	$-0.527170\pi$
$337$	−3.13012	−0.170508	−0.0852542	+	0.996359i	$0.527170\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.8917	1.78119
$342$	0	0
$343$	7.70289	0.415917
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.4385	1.58034	0.790172	−	0.612885i	$-0.209991\pi$
$347$	29.4385	1.58034	0.790172	+	0.612885i	$0.209991\pi$
$348$	0	0
$349$	−1.30961	−0.0701017	−0.0350508	−	0.999386i	$-0.511159\pi$
$349$	−1.30961	−0.0701017	−0.0350508	+	0.999386i	$0.511159\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.35646	0.444770	0.222385	−	0.974959i	$-0.428616\pi$
$353$	8.35646	0.444770	0.222385	+	0.974959i	$0.428616\pi$
$354$	0	0
$355$	−11.7774	−0.625077
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.5578	0.768329	0.384164	−	0.923265i	$-0.374490\pi$
$359$	14.5578	0.768329	0.384164	+	0.923265i	$0.374490\pi$
$360$	0	0
$361$	−7.85641	−0.413495
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.31994	0.226116
$366$	0	0
$367$	0.635668	0.0331816	0.0165908	−	0.999862i	$-0.494719\pi$
$367$	0.635668	0.0331816	0.0165908	+	0.999862i	$0.494719\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.244850	0.0127120
$372$	0	0
$373$	22.9349	1.18753	0.593763	−	0.804640i	$-0.297642\pi$
$373$	22.9349	1.18753	0.593763	+	0.804640i	$0.297642\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−36.9209	−1.90152
$378$	0	0
$379$	13.8351	0.710662	0.355331	−	0.934741i	$-0.384368\pi$
$379$	13.8351	0.710662	0.355331	+	0.934741i	$0.384368\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.9453	−1.17245	−0.586224	−	0.810149i	$-0.699386\pi$
$383$	−22.9453	−1.17245	−0.586224	+	0.810149i	$0.699386\pi$
$384$	0	0
$385$	3.58213	0.182562
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.55050	0.484230	0.242115	−	0.970248i	$-0.422159\pi$
$389$	9.55050	0.484230	0.242115	+	0.970248i	$0.422159\pi$
$390$	0	0
$391$	−27.7585	−1.40381
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.46410	−0.375560
$396$	0	0
$397$	−13.1217	−0.658558	−0.329279	−	0.944233i	$-0.606806\pi$
$397$	−13.1217	−0.658558	−0.329279	+	0.944233i	$0.606806\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.3251	0.865173	0.432587	−	0.901592i	$-0.357601\pi$
$401$	17.3251	0.865173	0.432587	+	0.901592i	$0.357601\pi$
$402$	0	0
$403$	−19.4203	−0.967393
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.0067	−0.991697
$408$	0	0
$409$	30.8340	1.52464	0.762321	−	0.647199i	$-0.224060\pi$
$409$	30.8340	1.52464	0.762321	+	0.647199i	$0.224060\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.975058	−0.0479794
$414$	0	0
$415$	8.99635	0.441613
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.19770	0.302777	0.151389	−	0.988474i	$-0.451626\pi$
$419$	6.19770	0.302777	0.151389	+	0.988474i	$0.451626\pi$
$420$	0	0
$421$	−29.0754	−1.41705	−0.708524	−	0.705687i	$-0.750639\pi$
$421$	−29.0754	−1.41705	−0.708524	+	0.705687i	$0.750639\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.65815	0.322967
$426$	0	0
$427$	−2.77196	−0.134144
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.1612	−1.83816	−0.919081	−	0.394069i	$-0.871067\pi$
$431$	−38.1612	−1.83816	−0.919081	+	0.394069i	$0.871067\pi$
$432$	0	0
$433$	−16.8261	−0.808609	−0.404304	−	0.914624i	$-0.632486\pi$
$433$	−16.8261	−0.808609	−0.404304	+	0.914624i	$0.632486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.9173	−0.665755
$438$	0	0
$439$	−12.5718	−0.600019	−0.300010	−	0.953936i	$-0.596990\pi$
$439$	−12.5718	−0.600019	−0.300010	+	0.953936i	$0.596990\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.17944	0.293594	0.146797	−	0.989167i	$-0.453104\pi$
$443$	6.17944	0.293594	0.146797	+	0.989167i	$0.453104\pi$
$444$	0	0
$445$	6.36736	0.301842
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.9723	−0.848167	−0.424083	−	0.905623i	$-0.639404\pi$
$449$	−17.9723	−0.848167	−0.424083	+	0.905623i	$0.639404\pi$
$450$	0	0
$451$	73.7122	3.47097
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.11500	−0.0991527
$456$	0	0
$457$	−2.31994	−0.108522	−0.0542612	−	0.998527i	$-0.517280\pi$
$457$	−2.31994	−0.108522	−0.0542612	+	0.998527i	$0.517280\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.1070	1.82140	0.910698	−	0.413073i	$-0.135545\pi$
$461$	39.1070	1.82140	0.910698	+	0.413073i	$0.135545\pi$
$462$	0	0
$463$	17.0234	0.791144	0.395572	−	0.918435i	$-0.370546\pi$
$463$	17.0234	0.791144	0.395572	+	0.918435i	$0.370546\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.5395	−1.50575	−0.752875	−	0.658163i	$-0.771334\pi$
$467$	−32.5395	−1.50575	−0.752875	+	0.658163i	$0.771334\pi$
$468$	0	0
$469$	−6.56089	−0.302954
$470$	0	0
$471$	0	0
$472$	0	0
$473$	54.7756	2.51858
$474$	0	0
$475$	3.33820	0.153167
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.94276	−0.454296	−0.227148	−	0.973860i	$-0.572940\pi$
$479$	−9.94276	−0.454296	−0.227148	+	0.973860i	$0.572940\pi$
$480$	0	0
$481$	11.8126	0.538607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.4604	−0.747430
$486$	0	0
$487$	−25.7040	−1.16476	−0.582380	−	0.812917i	$-0.697878\pi$
$487$	−25.7040	−1.16476	−0.582380	+	0.812917i	$0.697878\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.9723	0.811080	0.405540	−	0.914077i	$-0.367084\pi$
$491$	17.9723	0.811080	0.405540	+	0.914077i	$0.367084\pi$
$492$	0	0
$493$	−65.4312	−2.94687
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.63006	0.297399
$498$	0	0
$499$	18.6180	0.833455	0.416727	−	0.909031i	$-0.363177\pi$
$499$	18.6180	0.833455	0.416727	+	0.909031i	$0.363177\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.4963	1.18141	0.590706	−	0.806887i	$-0.298849\pi$
$503$	26.4963	1.18141	0.590706	+	0.806887i	$0.298849\pi$
$504$	0	0
$505$	−0.226856	−0.0100950
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.3528	−1.03510	−0.517548	−	0.855654i	$-0.673155\pi$
$509$	−23.3528	−1.03510	−0.517548	+	0.855654i	$0.673155\pi$
$510$	0	0
$511$	−2.43191	−0.107581
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.833008	0.0367067
$516$	0	0
$517$	−78.2116	−3.43974
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.1378	−0.969874	−0.484937	−	0.874549i	$-0.661157\pi$
$521$	−22.1378	−0.969874	−0.484937	+	0.874549i	$0.661157\pi$
$522$	0	0
$523$	−28.5743	−1.24947	−0.624733	−	0.780839i	$-0.714792\pi$
$523$	−28.5743	−1.24947	−0.624733	+	0.780839i	$0.714792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.4166	−1.49921
$528$	0	0
$529$	−5.61860	−0.244287
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−43.5219	−1.88514
$534$	0	0
$535$	−6.39230	−0.276363
$536$	0	0
$537$	0	0
$538$	0	0
$539$	42.5255	1.83170
$540$	0	0
$541$	−33.1295	−1.42435	−0.712174	−	0.702003i	$-0.752289\pi$
$541$	−33.1295	−1.42435	−0.712174	+	0.702003i	$0.752289\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.6332	0.455476
$546$	0	0
$547$	4.42039	0.189002	0.0945011	−	0.995525i	$-0.469874\pi$
$547$	4.42039	0.189002	0.0945011	+	0.995525i	$0.469874\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−32.8053	−1.39755
$552$	0	0
$553$	4.20191	0.178684
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.5583	−1.80325	−0.901627	−	0.432515i	$-0.857626\pi$
$557$	−42.5583	−1.80325	−0.901627	+	0.432515i	$0.857626\pi$
$558$	0	0
$559$	−32.3412	−1.36789
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.97752	0.251923	0.125961	−	0.992035i	$-0.459798\pi$
$563$	5.97752	0.251923	0.125961	+	0.992035i	$0.459798\pi$
$564$	0	0
$565$	−7.48236	−0.314785
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.46986	−0.355075	−0.177538	−	0.984114i	$-0.556813\pi$
$569$	−8.46986	−0.355075	−0.177538	+	0.984114i	$0.556813\pi$
$570$	0	0
$571$	−13.0297	−0.545277	−0.272639	−	0.962117i	$-0.587896\pi$
$571$	−13.0297	−0.545277	−0.272639	+	0.962117i	$0.587896\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.16910	−0.173864
$576$	0	0
$577$	37.9713	1.58077	0.790384	−	0.612612i	$-0.209881\pi$
$577$	37.9713	1.58077	0.790384	+	0.612612i	$0.209881\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.06449	−0.210111
$582$	0	0
$583$	2.76759	0.114622
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.6728	1.22473	0.612363	−	0.790577i	$-0.290219\pi$
$587$	29.6728	1.22473	0.612363	+	0.790577i	$0.290219\pi$
$588$	0	0
$589$	−17.2555	−0.711001
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.34859	0.219640	0.109820	−	0.993951i	$-0.464973\pi$
$593$	5.34859	0.219640	0.109820	+	0.993951i	$0.464973\pi$
$594$	0	0
$595$	−3.74820	−0.153661
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.9988	0.531117	0.265559	−	0.964095i	$-0.414444\pi$
$599$	12.9988	0.531117	0.265559	+	0.964095i	$0.414444\pi$
$600$	0	0
$601$	−40.9489	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$601$	−40.9489	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.4896	1.19892
$606$	0	0
$607$	17.0547	0.692228	0.346114	−	0.938193i	$-0.387501\pi$
$607$	17.0547	0.692228	0.346114	+	0.938193i	$0.387501\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46.1785	1.86818
$612$	0	0
$613$	−29.0192	−1.17207	−0.586037	−	0.810284i	$-0.699313\pi$
$613$	−29.0192	−1.17207	−0.586037	+	0.810284i	$0.699313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.65758	−0.388800	−0.194400	−	0.980922i	$-0.562276\pi$
$617$	−9.65758	−0.388800	−0.194400	+	0.980922i	$0.562276\pi$
$618$	0	0
$619$	2.12836	0.0855462	0.0427731	−	0.999085i	$-0.486381\pi$
$619$	2.12836	0.0855462	0.0427731	+	0.999085i	$0.486381\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.58450	−0.143610
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.9343	0.834704
$630$	0	0
$631$	−37.7111	−1.50125	−0.750627	−	0.660726i	$-0.770248\pi$
$631$	−37.7111	−1.50125	−0.750627	+	0.660726i	$0.770248\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.4912	−0.456012
$636$	0	0
$637$	−25.1083	−0.994828
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.3820	0.844537	0.422268	−	0.906471i	$-0.361234\pi$
$641$	21.3820	0.844537	0.422268	+	0.906471i	$0.361234\pi$
$642$	0	0
$643$	22.4562	0.885587	0.442794	−	0.896624i	$-0.353987\pi$
$643$	22.4562	0.885587	0.442794	+	0.896624i	$0.353987\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.0061	−1.02240	−0.511202	−	0.859461i	$-0.670800\pi$
$647$	−26.0061	−1.02240	−0.511202	+	0.859461i	$0.670800\pi$
$648$	0	0
$649$	−11.0213	−0.432623
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.82606	−0.110592	−0.0552961	−	0.998470i	$-0.517610\pi$
$653$	−2.82606	−0.110592	−0.0552961	+	0.998470i	$0.517610\pi$
$654$	0	0
$655$	−5.19193	−0.202866
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.9432	1.47806	0.739028	−	0.673675i	$-0.235285\pi$
$659$	37.9432	1.47806	0.739028	+	0.673675i	$0.235285\pi$
$660$	0	0
$661$	16.6739	0.648541	0.324271	−	0.945964i	$-0.394881\pi$
$661$	16.6739	0.648541	0.324271	+	0.945964i	$0.394881\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.87924	−0.0728738
$666$	0	0
$667$	40.9708	1.58640
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−31.3320	−1.20956
$672$	0	0
$673$	43.8449	1.69010	0.845048	−	0.534690i	$-0.179572\pi$
$673$	43.8449	1.69010	0.845048	+	0.534690i	$0.179572\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.3772	−0.667860	−0.333930	−	0.942598i	$-0.608375\pi$
$677$	−17.3772	−0.667860	−0.333930	+	0.942598i	$0.608375\pi$
$678$	0	0
$679$	9.26641	0.355612
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.0875	1.53391	0.766953	−	0.641703i	$-0.221772\pi$
$683$	40.0875	1.53391	0.766953	+	0.641703i	$0.221772\pi$
$684$	0	0
$685$	−5.12590	−0.195851
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.63407	−0.0622532
$690$	0	0
$691$	1.92883	0.0733760	0.0366880	−	0.999327i	$-0.488319\pi$
$691$	1.92883	0.0733760	0.0366880	+	0.999327i	$0.488319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.7762	−0.750153
$696$	0	0
$697$	−77.1295	−2.92149
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.7585	0.406344	0.203172	−	0.979143i	$-0.434875\pi$
$701$	10.7585	0.406344	0.203172	+	0.979143i	$0.434875\pi$
$702$	0	0
$703$	10.4958	0.395858
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.127709	0.00480298
$708$	0	0
$709$	38.0103	1.42751	0.713753	−	0.700398i	$-0.246994\pi$
$709$	38.0103	1.42751	0.713753	+	0.700398i	$0.246994\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.5505	0.807073
$714$	0	0
$715$	−23.9063	−0.894045
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.2019	1.23822	0.619110	−	0.785304i	$-0.287493\pi$
$719$	33.2019	1.23822	0.619110	+	0.785304i	$0.287493\pi$
$720$	0	0
$721$	−0.468942	−0.0174643
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.82725	−0.364975
$726$	0	0
$727$	6.28980	0.233276	0.116638	−	0.993174i	$-0.462788\pi$
$727$	6.28980	0.233276	0.116638	+	0.993174i	$0.462788\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−57.3150	−2.11987
$732$	0	0
$733$	13.2664	0.490006	0.245003	−	0.969522i	$-0.421211\pi$
$733$	13.2664	0.490006	0.245003	+	0.969522i	$0.421211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−74.1592	−2.73169
$738$	0	0
$739$	49.4520	1.81912	0.909560	−	0.415573i	$-0.136419\pi$
$739$	49.4520	1.81912	0.909560	+	0.415573i	$0.136419\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.2293	1.73268	0.866338	−	0.499459i	$-0.166468\pi$
$743$	47.2293	1.73268	0.866338	+	0.499459i	$0.166468\pi$
$744$	0	0
$745$	−14.5900	−0.534536
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.59855	0.131488
$750$	0	0
$751$	−28.1508	−1.02724	−0.513618	−	0.858019i	$-0.671695\pi$
$751$	−28.1508	−1.02724	−0.513618	+	0.858019i	$0.671695\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.21899	−0.117151
$756$	0	0
$757$	17.3178	0.629425	0.314713	−	0.949187i	$-0.398092\pi$
$757$	17.3178	0.629425	0.314713	+	0.949187i	$0.398092\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.48756	0.0539239	0.0269620	−	0.999636i	$-0.491417\pi$
$761$	1.48756	0.0539239	0.0269620	+	0.999636i	$0.491417\pi$
$762$	0	0
$763$	−5.98596	−0.216706
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.50730	0.234965
$768$	0	0
$769$	−1.66355	−0.0599892	−0.0299946	−	0.999550i	$-0.509549\pi$
$769$	−1.66355	−0.0599892	−0.0299946	+	0.999550i	$0.509549\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.9483	−1.18507	−0.592534	−	0.805545i	$-0.701872\pi$
$773$	−32.9483	−1.18507	−0.592534	+	0.805545i	$0.701872\pi$
$774$	0	0
$775$	−5.16910	−0.185680
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−38.6705	−1.38552
$780$	0	0
$781$	74.9411	2.68160
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.16334	0.291362
$786$	0	0
$787$	−17.2439	−0.614680	−0.307340	−	0.951600i	$-0.599439\pi$
$787$	−17.2439	−0.614680	−0.307340	+	0.951600i	$0.599439\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.21219	0.149768
$792$	0	0
$793$	18.4994	0.656932
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0583	0.781346	0.390673	−	0.920530i	$-0.372242\pi$
$797$	22.0583	0.781346	0.390673	+	0.920530i	$0.372242\pi$
$798$	0	0
$799$	81.8376	2.89521
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−27.4884	−0.970045
$804$	0	0
$805$	2.34699	0.0827207
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.7394	0.940106	0.470053	−	0.882638i	$-0.344235\pi$
$809$	26.7394	0.940106	0.470053	+	0.882638i	$0.344235\pi$
$810$	0	0
$811$	20.3898	0.715984	0.357992	−	0.933725i	$-0.383461\pi$
$811$	20.3898	0.715984	0.357992	+	0.933725i	$0.383461\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.12225	0.214453
$816$	0	0
$817$	−28.7361	−1.00535
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.7013	0.722482	0.361241	−	0.932472i	$-0.382353\pi$
$821$	20.7013	0.722482	0.361241	+	0.932472i	$0.382353\pi$
$822$	0	0
$823$	9.67938	0.337402	0.168701	−	0.985667i	$-0.446043\pi$
$823$	9.67938	0.337402	0.168701	+	0.985667i	$0.446043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.1321	1.04779	0.523897	−	0.851781i	$-0.324478\pi$
$827$	30.1321	1.04779	0.523897	+	0.851781i	$0.324478\pi$
$828$	0	0
$829$	−18.6332	−0.647158	−0.323579	−	0.946201i	$-0.604886\pi$
$829$	−18.6332	−0.647158	−0.323579	+	0.946201i	$0.604886\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−44.4970	−1.54173
$834$	0	0
$835$	10.1721	0.352021
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.8188	0.477078	0.238539	−	0.971133i	$-0.423331\pi$
$839$	13.8188	0.477078	0.238539	+	0.971133i	$0.423331\pi$
$840$	0	0
$841$	67.5748	2.33016
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.11500	0.0383571
$846$	0	0
$847$	−16.6012	−0.570423
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.1083	−0.449347
$852$	0	0
$853$	−19.9702	−0.683767	−0.341884	−	0.939742i	$-0.611065\pi$
$853$	−19.9702	−0.683767	−0.341884	+	0.939742i	$0.611065\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.5098	−0.461485	−0.230742	−	0.973015i	$-0.574115\pi$
$857$	−13.5098	−0.461485	−0.230742	+	0.973015i	$0.574115\pi$
$858$	0	0
$859$	51.3595	1.75236	0.876182	−	0.481981i	$-0.160083\pi$
$859$	51.3595	1.75236	0.876182	+	0.481981i	$0.160083\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0967	−0.820263	−0.410131	−	0.912027i	$-0.634517\pi$
$863$	−24.0967	−0.820263	−0.410131	+	0.912027i	$0.634517\pi$
$864$	0	0
$865$	−15.0973	−0.513324
$866$	0	0
$867$	0	0
$868$	0	0
$869$	47.4952	1.61116
$870$	0	0
$871$	43.7858	1.48363
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.562950	−0.0190312
$876$	0	0
$877$	−13.6134	−0.459692	−0.229846	−	0.973227i	$-0.573822\pi$
$877$	−13.6134	−0.459692	−0.229846	+	0.973227i	$0.573822\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.3090	0.751611	0.375805	−	0.926699i	$-0.377366\pi$
$881$	22.3090	0.751611	0.375805	+	0.926699i	$0.377366\pi$
$882$	0	0
$883$	41.7444	1.40481	0.702406	−	0.711776i	$-0.252109\pi$
$883$	41.7444	1.40481	0.702406	+	0.711776i	$0.252109\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.2975	−0.446487	−0.223243	−	0.974763i	$-0.571665\pi$
$887$	−13.2975	−0.446487	−0.223243	+	0.974763i	$0.571665\pi$
$888$	0	0
$889$	6.46894	0.216961
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41.0310	1.37305
$894$	0	0
$895$	−11.5842	−0.387218
$896$	0	0
$897$	0	0
$898$	0	0
$899$	50.7980	1.69421
$900$	0	0
$901$	−2.89590	−0.0964765
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.50730	−0.0501045
$906$	0	0
$907$	−12.0292	−0.399423	−0.199712	−	0.979855i	$-0.564001\pi$
$907$	−12.0292	−0.399423	−0.199712	+	0.979855i	$0.564001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.5930	−1.17925	−0.589625	−	0.807677i	$-0.700724\pi$
$911$	−35.5930	−1.17925	−0.589625	+	0.807677i	$0.700724\pi$
$912$	0	0
$913$	−57.2451	−1.89453
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.92280	0.0965193
$918$	0	0
$919$	6.53836	0.215681	0.107840	−	0.994168i	$-0.465606\pi$
$919$	6.53836	0.215681	0.107840	+	0.994168i	$0.465606\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−44.2475	−1.45642
$924$	0	0
$925$	3.14416	0.103379
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.21771	−0.0727606	−0.0363803	−	0.999338i	$-0.511583\pi$
$929$	−2.21771	−0.0727606	−0.0363803	+	0.999338i	$0.511583\pi$
$930$	0	0
$931$	−22.3095	−0.731164
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−42.3667	−1.38554
$936$	0	0
$937$	−41.3662	−1.35137	−0.675687	−	0.737189i	$-0.736153\pi$
$937$	−41.3662	−1.35137	−0.675687	+	0.737189i	$0.736153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.0603	0.360555	0.180277	−	0.983616i	$-0.442300\pi$
$941$	11.0603	0.360555	0.180277	+	0.983616i	$0.442300\pi$
$942$	0	0
$943$	48.2959	1.57273
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.5504	1.57768	0.788838	−	0.614601i	$-0.210683\pi$
$947$	48.5504	1.57768	0.788838	+	0.614601i	$0.210683\pi$
$948$	0	0
$949$	16.2300	0.526848
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.1965	1.56124	0.780618	−	0.625008i	$-0.214904\pi$
$953$	48.1965	1.56124	0.780618	+	0.625008i	$0.214904\pi$
$954$	0	0
$955$	1.62955	0.0527310
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.88562	0.0931817
$960$	0	0
$961$	−4.28039	−0.138077
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.92033	−0.254965
$966$	0	0
$967$	33.2043	1.06778	0.533890	−	0.845554i	$-0.320730\pi$
$967$	33.2043	1.06778	0.533890	+	0.845554i	$0.320730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.27217	−0.0729173	−0.0364587	−	0.999335i	$-0.511608\pi$
$971$	−2.27217	−0.0729173	−0.0364587	+	0.999335i	$0.511608\pi$
$972$	0	0
$973$	11.1330	0.356907
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.3309	0.970371	0.485186	−	0.874411i	$-0.338752\pi$
$977$	30.3309	0.970371	0.485186	+	0.874411i	$0.338752\pi$
$978$	0	0
$979$	−40.5164	−1.29491
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.42404	0.173000	0.0865000	−	0.996252i	$-0.472432\pi$
$983$	5.42404	0.173000	0.0865000	+	0.996252i	$0.472432\pi$
$984$	0	0
$985$	−14.4708	−0.461078
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.8887	1.14119
$990$	0	0
$991$	−29.2025	−0.927649	−0.463824	−	0.885927i	$-0.653523\pi$
$991$	−29.2025	−0.927649	−0.463824	+	0.885927i	$0.653523\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.41422	−0.171642
$996$	0	0
$997$	19.0958	0.604769	0.302384	−	0.953186i	$-0.402217\pi$
$997$	19.0958	0.604769	0.302384	+	0.953186i	$0.402217\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.ca.1.3		4
3.2	odd	2		6480.2.a.by.1.3		4
4.3	odd	2		3240.2.a.v.1.2	yes	4
12.11	even	2		3240.2.a.t.1.2	✓	4
36.7	odd	6		3240.2.q.bg.1081.3		8
36.11	even	6		3240.2.q.bh.1081.3		8
36.23	even	6		3240.2.q.bh.2161.3		8
36.31	odd	6		3240.2.q.bg.2161.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.2	✓	4	12.11	even	2
3240.2.a.v.1.2	yes	4	4.3	odd	2
3240.2.q.bg.1081.3		8	36.7	odd	6
3240.2.q.bg.2161.3		8	36.31	odd	6
3240.2.q.bh.1081.3		8	36.11	even	6
3240.2.q.bh.2161.3		8	36.23	even	6
6480.2.a.by.1.3		4	3.2	odd	2
6480.2.a.ca.1.3		4	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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.562950 q^{7} -6.36314 q^{11} +3.75699 q^{13} +6.65815 q^{17} +3.33820 q^{19} -4.16910 q^{23} +1.00000 q^{25} -9.82725 q^{29} -5.16910 q^{31} -0.562950 q^{35} +3.14416 q^{37} -11.5842 q^{41} -8.60826 q^{43} +12.2913 q^{47} -6.68309 q^{49} -0.434941 q^{53} -6.36314 q^{55} +1.73205 q^{59} +4.92399 q^{61} +3.75699 q^{65} +11.6545 q^{67} -11.7774 q^{71} +4.31994 q^{73} +3.58213 q^{77} -7.46410 q^{79} +8.99635 q^{83} +6.65815 q^{85} +6.36736 q^{89} -2.11500 q^{91} +3.33820 q^{95} -16.4604 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{17} - 10 q^{23} + 4 q^{25} - 4 q^{29} - 14 q^{31} - 2 q^{35} + 14 q^{37} + 4 q^{41} - 22 q^{43} + 10 q^{49} - 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} - 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 4 * q^11 - 2 * q^17 - 10 * q^23 + 4 * q^25 - 4 * q^29 - 14 * q^31 - 2 * q^35 + 14 * q^37 + 4 * q^41 - 22 * q^43 + 10 * q^49 - 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 - 28 * q^71 + 2 * q^73 - 30 * q^77 - 16 * q^79 - 6 * q^83 - 2 * q^85 - 8 * q^89 - 30 * q^91 - 10 * q^97

Introduction

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