LMFDB - Embedded newform 5780.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.14414517.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 12x^{4} + 27x^{3} + 21x^{2} - 48x + 17 $ x^6 - 3x^5 - 12x^4 + 27x^3 + 21x^2 - 48*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.18216$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{3} -1.00000 q^{5} +1.53209 q^{7} +2.30278 q^{9} +2.44842 q^{11} -6.18216 q^{13} -2.30278 q^{15} -0.345896 q^{19} +3.52806 q^{21} -9.04065 q^{23} +1.00000 q^{25} -1.60555 q^{27} -5.05397 q^{29} -2.71070 q^{31} +5.63816 q^{33} -1.53209 q^{35} -4.08706 q^{37} -14.2361 q^{39} +1.14564 q^{41} +0.730943 q^{43} -2.30278 q^{45} +0.594059 q^{47} -4.65270 q^{49} -9.59749 q^{53} -2.44842 q^{55} -0.796521 q^{57} +6.74937 q^{59} -10.5856 q^{61} +3.52806 q^{63} +6.18216 q^{65} +12.3030 q^{67} -20.8186 q^{69} +1.08582 q^{71} +9.34420 q^{73} +2.30278 q^{75} +3.75119 q^{77} -11.0401 q^{79} -10.6056 q^{81} -12.2524 q^{83} -11.6382 q^{87} -10.8303 q^{89} -9.47162 q^{91} -6.24214 q^{93} +0.345896 q^{95} +6.97153 q^{97} +5.63816 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} - 6 q^{5} + 3 q^{9} - 15 q^{13} - 3 q^{15} - 3 q^{19} + 6 q^{25} + 12 q^{27} + 6 q^{29} + 6 q^{31} + 3 q^{37} - 27 q^{39} + 3 q^{41} + 3 q^{43} - 3 q^{45} - 15 q^{47} - 30 q^{49} - 12 q^{53}+ \cdots + 18 q^{97}+O(q^{100}) $ 6 * q + 3 * q^3 - 6 * q^5 + 3 * q^9 - 15 * q^13 - 3 * q^15 - 3 * q^19 + 6 * q^25 + 12 * q^27 + 6 * q^29 + 6 * q^31 + 3 * q^37 - 27 * q^39 + 3 * q^41 + 3 * q^43 - 3 * q^45 - 15 * q^47 - 30 * q^49 - 12 * q^53 + 18 * q^57 - 6 * q^59 + 3 * q^61 + 15 * q^65 - 6 * q^67 + 6 * q^71 + 9 * q^73 + 3 * q^75 - 3 * q^77 - 39 * q^79 - 42 * q^81 - 15 * q^83 - 36 * q^87 - 6 * q^89 - 6 * q^91 - 36 * q^93 + 3 * q^95 + 18 * 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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.53209	0.579075	0.289538	−	0.957167i	$-0.406498\pi$
$7$	1.53209	0.579075	0.289538	+	0.957167i	$0.406498\pi$
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	2.44842	0.738226	0.369113	−	0.929385i	$-0.379662\pi$
$11$	2.44842	0.738226	0.369113	+	0.929385i	$0.379662\pi$
$12$	0	0
$13$	−6.18216	−1.71462	−0.857311	−	0.514798i	$-0.827867\pi$
$13$	−6.18216	−1.71462	−0.857311	+	0.514798i	$0.827867\pi$
$14$	0	0
$15$	−2.30278	−0.594574
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.345896	−0.0793540	−0.0396770	−	0.999213i	$-0.512633\pi$
$19$	−0.345896	−0.0793540	−0.0396770	+	0.999213i	$0.512633\pi$
$20$	0	0
$21$	3.52806	0.769885
$22$	0	0
$23$	−9.04065	−1.88511	−0.942553	−	0.334056i	$-0.891582\pi$
$23$	−9.04065	−1.88511	−0.942553	+	0.334056i	$0.891582\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	−5.05397	−0.938498	−0.469249	−	0.883066i	$-0.655475\pi$
$29$	−5.05397	−0.938498	−0.469249	+	0.883066i	$0.655475\pi$
$30$	0	0
$31$	−2.71070	−0.486857	−0.243428	−	0.969919i	$-0.578272\pi$
$31$	−2.71070	−0.486857	−0.243428	+	0.969919i	$0.578272\pi$
$32$	0	0
$33$	5.63816	0.981477
$34$	0	0
$35$	−1.53209	−0.258970
$36$	0	0
$37$	−4.08706	−0.671908	−0.335954	−	0.941878i	$-0.609059\pi$
$37$	−4.08706	−0.671908	−0.335954	+	0.941878i	$0.609059\pi$
$38$	0	0
$39$	−14.2361	−2.27961
$40$	0	0
$41$	1.14564	0.178919	0.0894596	−	0.995990i	$-0.471486\pi$
$41$	1.14564	0.178919	0.0894596	+	0.995990i	$0.471486\pi$
$42$	0	0
$43$	0.730943	0.111468	0.0557339	−	0.998446i	$-0.482250\pi$
$43$	0.730943	0.111468	0.0557339	+	0.998446i	$0.482250\pi$
$44$	0	0
$45$	−2.30278	−0.343278
$46$	0	0
$47$	0.594059	0.0866524	0.0433262	−	0.999061i	$-0.486205\pi$
$47$	0.594059	0.0866524	0.0433262	+	0.999061i	$0.486205\pi$
$48$	0	0
$49$	−4.65270	−0.664672
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.59749	−1.31832	−0.659158	−	0.752004i	$-0.729087\pi$
$53$	−9.59749	−1.31832	−0.659158	+	0.752004i	$0.729087\pi$
$54$	0	0
$55$	−2.44842	−0.330145
$56$	0	0
$57$	−0.796521	−0.105502
$58$	0	0
$59$	6.74937	0.878693	0.439346	−	0.898318i	$-0.355210\pi$
$59$	6.74937	0.878693	0.439346	+	0.898318i	$0.355210\pi$
$60$	0	0
$61$	−10.5856	−1.35534	−0.677672	−	0.735365i	$-0.737011\pi$
$61$	−10.5856	−1.35534	−0.677672	+	0.735365i	$0.737011\pi$
$62$	0	0
$63$	3.52806	0.444493
$64$	0	0
$65$	6.18216	0.766803
$66$	0	0
$67$	12.3030	1.50306	0.751528	−	0.659701i	$-0.229317\pi$
$67$	12.3030	1.50306	0.751528	+	0.659701i	$0.229317\pi$
$68$	0	0
$69$	−20.8186	−2.50626
$70$	0	0
$71$	1.08582	0.128863	0.0644314	−	0.997922i	$-0.479477\pi$
$71$	1.08582	0.128863	0.0644314	+	0.997922i	$0.479477\pi$
$72$	0	0
$73$	9.34420	1.09366	0.546828	−	0.837245i	$-0.315835\pi$
$73$	9.34420	1.09366	0.546828	+	0.837245i	$0.315835\pi$
$74$	0	0
$75$	2.30278	0.265902
$76$	0	0
$77$	3.75119	0.427488
$78$	0	0
$79$	−11.0401	−1.24210	−0.621052	−	0.783769i	$-0.713295\pi$
$79$	−11.0401	−1.24210	−0.621052	+	0.783769i	$0.713295\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−12.2524	−1.34487	−0.672436	−	0.740156i	$-0.734752\pi$
$83$	−12.2524	−1.34487	−0.672436	+	0.740156i	$0.734752\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.6382	−1.24774
$88$	0	0
$89$	−10.8303	−1.14801	−0.574005	−	0.818852i	$-0.694611\pi$
$89$	−10.8303	−1.14801	−0.574005	+	0.818852i	$0.694611\pi$
$90$	0	0
$91$	−9.47162	−0.992896
$92$	0	0
$93$	−6.24214	−0.647280
$94$	0	0
$95$	0.345896	0.0354882
$96$	0	0
$97$	6.97153	0.707851	0.353926	−	0.935274i	$-0.384847\pi$
$97$	6.97153	0.707851	0.353926	+	0.935274i	$0.384847\pi$
$98$	0	0
$99$	5.63816	0.566656
$100$	0	0
$101$	16.9191	1.68351	0.841756	−	0.539858i	$-0.181522\pi$
$101$	16.9191	1.68351	0.841756	+	0.539858i	$0.181522\pi$
$102$	0	0
$103$	9.06890	0.893586	0.446793	−	0.894637i	$-0.352566\pi$
$103$	9.06890	0.893586	0.446793	+	0.894637i	$0.352566\pi$
$104$	0	0
$105$	−3.52806	−0.344303
$106$	0	0
$107$	5.38006	0.520110	0.260055	−	0.965594i	$-0.416259\pi$
$107$	5.38006	0.520110	0.260055	+	0.965594i	$0.416259\pi$
$108$	0	0
$109$	7.14640	0.684501	0.342250	−	0.939609i	$-0.388811\pi$
$109$	7.14640	0.684501	0.342250	+	0.939609i	$0.388811\pi$
$110$	0	0
$111$	−9.41158	−0.893308
$112$	0	0
$113$	18.8990	1.77787	0.888935	−	0.458033i	$-0.151446\pi$
$113$	18.8990	1.77787	0.888935	+	0.458033i	$0.151446\pi$
$114$	0	0
$115$	9.04065	0.843045
$116$	0	0
$117$	−14.2361	−1.31613
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.00525	−0.455023
$122$	0	0
$123$	2.63816	0.237874
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.6094	−1.65131	−0.825657	−	0.564172i	$-0.809195\pi$
$127$	−18.6094	−1.65131	−0.825657	+	0.564172i	$0.809195\pi$
$128$	0	0
$129$	1.68320	0.148197
$130$	0	0
$131$	−3.69739	−0.323043	−0.161521	−	0.986869i	$-0.551640\pi$
$131$	−3.69739	−0.323043	−0.161521	+	0.986869i	$0.551640\pi$
$132$	0	0
$133$	−0.529944	−0.0459519
$134$	0	0
$135$	1.60555	0.138184
$136$	0	0
$137$	−6.28824	−0.537240	−0.268620	−	0.963246i	$-0.586568\pi$
$137$	−6.28824	−0.537240	−0.268620	+	0.963246i	$0.586568\pi$
$138$	0	0
$139$	−10.0556	−0.852903	−0.426452	−	0.904510i	$-0.640237\pi$
$139$	−10.0556	−0.852903	−0.426452	+	0.904510i	$0.640237\pi$
$140$	0	0
$141$	1.36798	0.115205
$142$	0	0
$143$	−15.1365	−1.26578
$144$	0	0
$145$	5.05397	0.419709
$146$	0	0
$147$	−10.7141	−0.883687
$148$	0	0
$149$	1.27054	0.104086	0.0520432	−	0.998645i	$-0.483427\pi$
$149$	1.27054	0.104086	0.0520432	+	0.998645i	$0.483427\pi$
$150$	0	0
$151$	−22.4948	−1.83060	−0.915300	−	0.402773i	$-0.868046\pi$
$151$	−22.4948	−1.83060	−0.915300	+	0.402773i	$0.868046\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.71070	0.217729
$156$	0	0
$157$	17.0193	1.35829	0.679143	−	0.734006i	$-0.262352\pi$
$157$	17.0193	1.35829	0.679143	+	0.734006i	$0.262352\pi$
$158$	0	0
$159$	−22.1009	−1.75271
$160$	0	0
$161$	−13.8511	−1.09162
$162$	0	0
$163$	0.799925	0.0626550	0.0313275	−	0.999509i	$-0.490027\pi$
$163$	0.799925	0.0626550	0.0313275	+	0.999509i	$0.490027\pi$
$164$	0	0
$165$	−5.63816	−0.438930
$166$	0	0
$167$	17.8105	1.37822	0.689111	−	0.724656i	$-0.258001\pi$
$167$	17.8105	1.37822	0.689111	+	0.724656i	$0.258001\pi$
$168$	0	0
$169$	25.2191	1.93993
$170$	0	0
$171$	−0.796521	−0.0609115
$172$	0	0
$173$	6.13517	0.466448	0.233224	−	0.972423i	$-0.425072\pi$
$173$	6.13517	0.466448	0.233224	+	0.972423i	$0.425072\pi$
$174$	0	0
$175$	1.53209	0.115815
$176$	0	0
$177$	15.5423	1.16823
$178$	0	0
$179$	−7.55270	−0.564516	−0.282258	−	0.959339i	$-0.591083\pi$
$179$	−7.55270	−0.564516	−0.282258	+	0.959339i	$0.591083\pi$
$180$	0	0
$181$	−16.2037	−1.20442	−0.602208	−	0.798340i	$-0.705712\pi$
$181$	−16.2037	−1.20442	−0.602208	+	0.798340i	$0.705712\pi$
$182$	0	0
$183$	−24.3762	−1.80194
$184$	0	0
$185$	4.08706	0.300487
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.45985	−0.178928
$190$	0	0
$191$	21.8640	1.58203	0.791014	−	0.611799i	$-0.209554\pi$
$191$	21.8640	1.58203	0.791014	+	0.611799i	$0.209554\pi$
$192$	0	0
$193$	2.35506	0.169521	0.0847606	−	0.996401i	$-0.472987\pi$
$193$	2.35506	0.169521	0.0847606	+	0.996401i	$0.472987\pi$
$194$	0	0
$195$	14.2361	1.01947
$196$	0	0
$197$	9.36359	0.667128	0.333564	−	0.942727i	$-0.391749\pi$
$197$	9.36359	0.667128	0.333564	+	0.942727i	$0.391749\pi$
$198$	0	0
$199$	−12.3432	−0.874987	−0.437494	−	0.899222i	$-0.644134\pi$
$199$	−12.3432	−0.874987	−0.437494	+	0.899222i	$0.644134\pi$
$200$	0	0
$201$	28.3312	1.99833
$202$	0	0
$203$	−7.74313	−0.543461
$204$	0	0
$205$	−1.14564	−0.0800151
$206$	0	0
$207$	−20.8186	−1.44699
$208$	0	0
$209$	−0.846898	−0.0585812
$210$	0	0
$211$	5.16250	0.355401	0.177701	−	0.984085i	$-0.443134\pi$
$211$	5.16250	0.355401	0.177701	+	0.984085i	$0.443134\pi$
$212$	0	0
$213$	2.50039	0.171324
$214$	0	0
$215$	−0.730943	−0.0498499
$216$	0	0
$217$	−4.15304	−0.281927
$218$	0	0
$219$	21.5176	1.45402
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.9115	0.998547	0.499274	−	0.866444i	$-0.333600\pi$
$223$	14.9115	0.998547	0.499274	+	0.866444i	$0.333600\pi$
$224$	0	0
$225$	2.30278	0.153518
$226$	0	0
$227$	−19.9006	−1.32085	−0.660425	−	0.750892i	$-0.729624\pi$
$227$	−19.9006	−1.32085	−0.660425	+	0.750892i	$0.729624\pi$
$228$	0	0
$229$	25.4356	1.68083	0.840417	−	0.541941i	$-0.182310\pi$
$229$	25.4356	1.68083	0.840417	+	0.541941i	$0.182310\pi$
$230$	0	0
$231$	8.63816	0.568349
$232$	0	0
$233$	18.0696	1.18378	0.591890	−	0.806019i	$-0.298382\pi$
$233$	18.0696	1.18378	0.591890	+	0.806019i	$0.298382\pi$
$234$	0	0
$235$	−0.594059	−0.0387521
$236$	0	0
$237$	−25.4228	−1.65139
$238$	0	0
$239$	−3.46127	−0.223891	−0.111945	−	0.993714i	$-0.535708\pi$
$239$	−3.46127	−0.223891	−0.111945	+	0.993714i	$0.535708\pi$
$240$	0	0
$241$	14.0380	0.904268	0.452134	−	0.891950i	$-0.350663\pi$
$241$	14.0380	0.904268	0.452134	+	0.891950i	$0.350663\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	4.65270	0.297250
$246$	0	0
$247$	2.13839	0.136062
$248$	0	0
$249$	−28.2144	−1.78802
$250$	0	0
$251$	−3.61602	−0.228241	−0.114121	−	0.993467i	$-0.536405\pi$
$251$	−3.61602	−0.228241	−0.114121	+	0.993467i	$0.536405\pi$
$252$	0	0
$253$	−22.1353	−1.39163
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.91274	−0.618340	−0.309170	−	0.951007i	$-0.600051\pi$
$257$	−9.91274	−0.618340	−0.309170	+	0.951007i	$0.600051\pi$
$258$	0	0
$259$	−6.26174	−0.389085
$260$	0	0
$261$	−11.6382	−0.720384
$262$	0	0
$263$	−22.8485	−1.40890	−0.704449	−	0.709755i	$-0.748806\pi$
$263$	−22.8485	−1.40890	−0.704449	+	0.709755i	$0.748806\pi$
$264$	0	0
$265$	9.59749	0.589569
$266$	0	0
$267$	−24.9398	−1.52629
$268$	0	0
$269$	−16.3450	−0.996571	−0.498286	−	0.867013i	$-0.666037\pi$
$269$	−16.3450	−0.996571	−0.498286	+	0.867013i	$0.666037\pi$
$270$	0	0
$271$	−24.3015	−1.47621	−0.738105	−	0.674686i	$-0.764279\pi$
$271$	−24.3015	−1.47621	−0.738105	+	0.674686i	$0.764279\pi$
$272$	0	0
$273$	−21.8110	−1.32006
$274$	0	0
$275$	2.44842	0.147645
$276$	0	0
$277$	22.9348	1.37802	0.689010	−	0.724752i	$-0.258046\pi$
$277$	22.9348	1.37802	0.689010	+	0.724752i	$0.258046\pi$
$278$	0	0
$279$	−6.24214	−0.373707
$280$	0	0
$281$	−16.4468	−0.981132	−0.490566	−	0.871404i	$-0.663210\pi$
$281$	−16.4468	−0.981132	−0.490566	+	0.871404i	$0.663210\pi$
$282$	0	0
$283$	−4.05544	−0.241071	−0.120535	−	0.992709i	$-0.538461\pi$
$283$	−4.05544	−0.241071	−0.120535	+	0.992709i	$0.538461\pi$
$284$	0	0
$285$	0.796521	0.0471818
$286$	0	0
$287$	1.75522	0.103608
$288$	0	0
$289$	0	0
$290$	0	0
$291$	16.0539	0.941094
$292$	0	0
$293$	3.85188	0.225029	0.112515	−	0.993650i	$-0.464109\pi$
$293$	3.85188	0.225029	0.112515	+	0.993650i	$0.464109\pi$
$294$	0	0
$295$	−6.74937	−0.392963
$296$	0	0
$297$	−3.93106	−0.228103
$298$	0	0
$299$	55.8908	3.23225
$300$	0	0
$301$	1.11987	0.0645482
$302$	0	0
$303$	38.9609	2.23824
$304$	0	0
$305$	10.5856	0.606128
$306$	0	0
$307$	−11.6595	−0.665444	−0.332722	−	0.943025i	$-0.607967\pi$
$307$	−11.6595	−0.665444	−0.332722	+	0.943025i	$0.607967\pi$
$308$	0	0
$309$	20.8836	1.18803
$310$	0	0
$311$	−12.6065	−0.714850	−0.357425	−	0.933942i	$-0.616345\pi$
$311$	−12.6065	−0.714850	−0.357425	+	0.933942i	$0.616345\pi$
$312$	0	0
$313$	−17.1453	−0.969112	−0.484556	−	0.874760i	$-0.661019\pi$
$313$	−17.1453	−0.969112	−0.484556	+	0.874760i	$0.661019\pi$
$314$	0	0
$315$	−3.52806	−0.198783
$316$	0	0
$317$	2.62353	0.147352	0.0736762	−	0.997282i	$-0.476527\pi$
$317$	2.62353	0.147352	0.0736762	+	0.997282i	$0.476527\pi$
$318$	0	0
$319$	−12.3742	−0.692824
$320$	0	0
$321$	12.3891	0.691491
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6.18216	−0.342925
$326$	0	0
$327$	16.4566	0.910049
$328$	0	0
$329$	0.910151	0.0501783
$330$	0	0
$331$	−12.6609	−0.695906	−0.347953	−	0.937512i	$-0.613123\pi$
$331$	−12.6609	−0.695906	−0.347953	+	0.937512i	$0.613123\pi$
$332$	0	0
$333$	−9.41158	−0.515751
$334$	0	0
$335$	−12.3030	−0.672187
$336$	0	0
$337$	4.77135	0.259912	0.129956	−	0.991520i	$-0.458516\pi$
$337$	4.77135	0.259912	0.129956	+	0.991520i	$0.458516\pi$
$338$	0	0
$339$	43.5202	2.36369
$340$	0	0
$341$	−6.63693	−0.359410
$342$	0	0
$343$	−17.8530	−0.963970
$344$	0	0
$345$	20.8186	1.12084
$346$	0	0
$347$	−8.54464	−0.458700	−0.229350	−	0.973344i	$-0.573660\pi$
$347$	−8.54464	−0.458700	−0.229350	+	0.973344i	$0.573660\pi$
$348$	0	0
$349$	−17.7864	−0.952083	−0.476041	−	0.879423i	$-0.657929\pi$
$349$	−17.7864	−0.952083	−0.476041	+	0.879423i	$0.657929\pi$
$350$	0	0
$351$	9.92578	0.529799
$352$	0	0
$353$	−32.1311	−1.71016	−0.855082	−	0.518492i	$-0.826493\pi$
$353$	−32.1311	−1.71016	−0.855082	+	0.518492i	$0.826493\pi$
$354$	0	0
$355$	−1.08582	−0.0576292
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.0091	1.63660	0.818299	−	0.574793i	$-0.194918\pi$
$359$	31.0091	1.63660	0.818299	+	0.574793i	$0.194918\pi$
$360$	0	0
$361$	−18.8804	−0.993703
$362$	0	0
$363$	−11.5260	−0.604957
$364$	0	0
$365$	−9.34420	−0.489098
$366$	0	0
$367$	23.4378	1.22344	0.611722	−	0.791073i	$-0.290477\pi$
$367$	23.4378	1.22344	0.611722	+	0.791073i	$0.290477\pi$
$368$	0	0
$369$	2.63816	0.137337
$370$	0	0
$371$	−14.7042	−0.763404
$372$	0	0
$373$	−22.4192	−1.16082	−0.580412	−	0.814323i	$-0.697108\pi$
$373$	−22.4192	−1.16082	−0.580412	+	0.814323i	$0.697108\pi$
$374$	0	0
$375$	−2.30278	−0.118915
$376$	0	0
$377$	31.2444	1.60917
$378$	0	0
$379$	29.9129	1.53652	0.768262	−	0.640136i	$-0.221122\pi$
$379$	29.9129	1.53652	0.768262	+	0.640136i	$0.221122\pi$
$380$	0	0
$381$	−42.8532	−2.19544
$382$	0	0
$383$	−28.1941	−1.44065	−0.720325	−	0.693637i	$-0.756007\pi$
$383$	−28.1941	−1.44065	−0.720325	+	0.693637i	$0.756007\pi$
$384$	0	0
$385$	−3.75119	−0.191178
$386$	0	0
$387$	1.68320	0.0855618
$388$	0	0
$389$	−5.87615	−0.297933	−0.148966	−	0.988842i	$-0.547595\pi$
$389$	−5.87615	−0.297933	−0.148966	+	0.988842i	$0.547595\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−8.51427	−0.429488
$394$	0	0
$395$	11.0401	0.555486
$396$	0	0
$397$	39.1164	1.96319	0.981597	−	0.190966i	$-0.0611620\pi$
$397$	39.1164	1.96319	0.981597	+	0.190966i	$0.0611620\pi$
$398$	0	0
$399$	−1.22034	−0.0610935
$400$	0	0
$401$	−27.9328	−1.39490	−0.697448	−	0.716635i	$-0.745681\pi$
$401$	−27.9328	−1.39490	−0.697448	+	0.716635i	$0.745681\pi$
$402$	0	0
$403$	16.7580	0.834776
$404$	0	0
$405$	10.6056	0.526994
$406$	0	0
$407$	−10.0068	−0.496020
$408$	0	0
$409$	−3.35372	−0.165831	−0.0829154	−	0.996557i	$-0.526423\pi$
$409$	−3.35372	−0.165831	−0.0829154	+	0.996557i	$0.526423\pi$
$410$	0	0
$411$	−14.4804	−0.714266
$412$	0	0
$413$	10.3406	0.508829
$414$	0	0
$415$	12.2524	0.601445
$416$	0	0
$417$	−23.1558	−1.13394
$418$	0	0
$419$	−17.7106	−0.865219	−0.432610	−	0.901581i	$-0.642407\pi$
$419$	−17.7106	−0.865219	−0.432610	+	0.901581i	$0.642407\pi$
$420$	0	0
$421$	−39.1191	−1.90655	−0.953274	−	0.302106i	$-0.902310\pi$
$421$	−39.1191	−1.90655	−0.953274	+	0.302106i	$0.902310\pi$
$422$	0	0
$423$	1.36798	0.0665137
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.2180	−0.784846
$428$	0	0
$429$	−34.8560	−1.68286
$430$	0	0
$431$	34.4019	1.65708	0.828542	−	0.559927i	$-0.189171\pi$
$431$	34.4019	1.65708	0.828542	+	0.559927i	$0.189171\pi$
$432$	0	0
$433$	−6.72002	−0.322943	−0.161472	−	0.986877i	$-0.551624\pi$
$433$	−6.72002	−0.322943	−0.161472	+	0.986877i	$0.551624\pi$
$434$	0	0
$435$	11.6382	0.558007
$436$	0	0
$437$	3.12713	0.149591
$438$	0	0
$439$	−28.4568	−1.35817	−0.679085	−	0.734060i	$-0.737623\pi$
$439$	−28.4568	−1.35817	−0.679085	+	0.734060i	$0.737623\pi$
$440$	0	0
$441$	−10.7141	−0.510197
$442$	0	0
$443$	−6.39254	−0.303719	−0.151859	−	0.988402i	$-0.548526\pi$
$443$	−6.39254	−0.303719	−0.151859	+	0.988402i	$0.548526\pi$
$444$	0	0
$445$	10.8303	0.513406
$446$	0	0
$447$	2.92576	0.138384
$448$	0	0
$449$	24.8875	1.17451	0.587257	−	0.809400i	$-0.300208\pi$
$449$	24.8875	1.17451	0.587257	+	0.809400i	$0.300208\pi$
$450$	0	0
$451$	2.80501	0.132083
$452$	0	0
$453$	−51.8004	−2.43380
$454$	0	0
$455$	9.47162	0.444036
$456$	0	0
$457$	15.0624	0.704589	0.352294	−	0.935889i	$-0.385402\pi$
$457$	15.0624	0.704589	0.352294	+	0.935889i	$0.385402\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.44077	0.160253	0.0801264	−	0.996785i	$-0.474468\pi$
$461$	3.44077	0.160253	0.0801264	+	0.996785i	$0.474468\pi$
$462$	0	0
$463$	13.7212	0.637680	0.318840	−	0.947809i	$-0.396707\pi$
$463$	13.7212	0.637680	0.318840	+	0.947809i	$0.396707\pi$
$464$	0	0
$465$	6.24214	0.289472
$466$	0	0
$467$	−3.18218	−0.147254	−0.0736269	−	0.997286i	$-0.523457\pi$
$467$	−3.18218	−0.147254	−0.0736269	+	0.997286i	$0.523457\pi$
$468$	0	0
$469$	18.8494	0.870383
$470$	0	0
$471$	39.1916	1.80585
$472$	0	0
$473$	1.78965	0.0822884
$474$	0	0
$475$	−0.345896	−0.0158708
$476$	0	0
$477$	−22.1009	−1.01193
$478$	0	0
$479$	34.1428	1.56002	0.780012	−	0.625765i	$-0.215213\pi$
$479$	34.1428	1.56002	0.780012	+	0.625765i	$0.215213\pi$
$480$	0	0
$481$	25.2669	1.15207
$482$	0	0
$483$	−31.8959	−1.45132
$484$	0	0
$485$	−6.97153	−0.316561
$486$	0	0
$487$	10.0187	0.453989	0.226995	−	0.973896i	$-0.427110\pi$
$487$	10.0187	0.453989	0.226995	+	0.973896i	$0.427110\pi$
$488$	0	0
$489$	1.84205	0.0833003
$490$	0	0
$491$	0.623364	0.0281320	0.0140660	−	0.999901i	$-0.495522\pi$
$491$	0.623364	0.0281320	0.0140660	+	0.999901i	$0.495522\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.63816	−0.253416
$496$	0	0
$497$	1.66357	0.0746212
$498$	0	0
$499$	1.53671	0.0687925	0.0343962	−	0.999408i	$-0.489049\pi$
$499$	1.53671	0.0687925	0.0343962	+	0.999408i	$0.489049\pi$
$500$	0	0
$501$	41.0137	1.83236
$502$	0	0
$503$	−11.6041	−0.517401	−0.258700	−	0.965958i	$-0.583294\pi$
$503$	−11.6041	−0.517401	−0.258700	+	0.965958i	$0.583294\pi$
$504$	0	0
$505$	−16.9191	−0.752889
$506$	0	0
$507$	58.0740	2.57916
$508$	0	0
$509$	38.2148	1.69384	0.846921	−	0.531719i	$-0.178454\pi$
$509$	38.2148	1.69384	0.846921	+	0.531719i	$0.178454\pi$
$510$	0	0
$511$	14.3161	0.633309
$512$	0	0
$513$	0.555354	0.0245195
$514$	0	0
$515$	−9.06890	−0.399624
$516$	0	0
$517$	1.45450	0.0639690
$518$	0	0
$519$	14.1279	0.620146
$520$	0	0
$521$	−27.4152	−1.20108	−0.600540	−	0.799595i	$-0.705048\pi$
$521$	−27.4152	−1.20108	−0.600540	+	0.799595i	$0.705048\pi$
$522$	0	0
$523$	−12.8711	−0.562815	−0.281408	−	0.959588i	$-0.590801\pi$
$523$	−12.8711	−0.562815	−0.281408	+	0.959588i	$0.590801\pi$
$524$	0	0
$525$	3.52806	0.153977
$526$	0	0
$527$	0	0
$528$	0	0
$529$	58.7334	2.55363
$530$	0	0
$531$	15.5423	0.674478
$532$	0	0
$533$	−7.08254	−0.306779
$534$	0	0
$535$	−5.38006	−0.232600
$536$	0	0
$537$	−17.3922	−0.750528
$538$	0	0
$539$	−11.3918	−0.490678
$540$	0	0
$541$	38.7809	1.66732	0.833660	−	0.552278i	$-0.186241\pi$
$541$	38.7809	1.66732	0.833660	+	0.552278i	$0.186241\pi$
$542$	0	0
$543$	−37.3136	−1.60128
$544$	0	0
$545$	−7.14640	−0.306118
$546$	0	0
$547$	−34.1248	−1.45907	−0.729536	−	0.683942i	$-0.760264\pi$
$547$	−34.1248	−1.45907	−0.729536	+	0.683942i	$0.760264\pi$
$548$	0	0
$549$	−24.3762	−1.04035
$550$	0	0
$551$	1.74815	0.0744736
$552$	0	0
$553$	−16.9144	−0.719272
$554$	0	0
$555$	9.41158	0.399499
$556$	0	0
$557$	33.8036	1.43231	0.716153	−	0.697943i	$-0.245901\pi$
$557$	33.8036	1.43231	0.716153	+	0.697943i	$0.245901\pi$
$558$	0	0
$559$	−4.51881	−0.191125
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.6208	0.869063	0.434532	−	0.900657i	$-0.356914\pi$
$563$	20.6208	0.869063	0.434532	+	0.900657i	$0.356914\pi$
$564$	0	0
$565$	−18.8990	−0.795088
$566$	0	0
$567$	−16.2486	−0.682379
$568$	0	0
$569$	−15.7173	−0.658906	−0.329453	−	0.944172i	$-0.606864\pi$
$569$	−15.7173	−0.658906	−0.329453	+	0.944172i	$0.606864\pi$
$570$	0	0
$571$	39.4858	1.65243	0.826214	−	0.563357i	$-0.190490\pi$
$571$	39.4858	1.65243	0.826214	+	0.563357i	$0.190490\pi$
$572$	0	0
$573$	50.3480	2.10332
$574$	0	0
$575$	−9.04065	−0.377021
$576$	0	0
$577$	−10.9057	−0.454008	−0.227004	−	0.973894i	$-0.572893\pi$
$577$	−10.9057	−0.454008	−0.227004	+	0.973894i	$0.572893\pi$
$578$	0	0
$579$	5.42318	0.225380
$580$	0	0
$581$	−18.7717	−0.778781
$582$	0	0
$583$	−23.4987	−0.973215
$584$	0	0
$585$	14.2361	0.588592
$586$	0	0
$587$	15.2969	0.631371	0.315686	−	0.948864i	$-0.397766\pi$
$587$	15.2969	0.631371	0.315686	+	0.948864i	$0.397766\pi$
$588$	0	0
$589$	0.937622	0.0386340
$590$	0	0
$591$	21.5622	0.886952
$592$	0	0
$593$	25.4750	1.04613	0.523067	−	0.852292i	$-0.324788\pi$
$593$	25.4750	1.04613	0.523067	+	0.852292i	$0.324788\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−28.4237	−1.16330
$598$	0	0
$599$	−19.8734	−0.812006	−0.406003	−	0.913872i	$-0.633078\pi$
$599$	−19.8734	−0.812006	−0.406003	+	0.913872i	$0.633078\pi$
$600$	0	0
$601$	−36.9027	−1.50529	−0.752647	−	0.658424i	$-0.771223\pi$
$601$	−36.9027	−1.50529	−0.752647	+	0.658424i	$0.771223\pi$
$602$	0	0
$603$	28.3312	1.15373
$604$	0	0
$605$	5.00525	0.203492
$606$	0	0
$607$	−27.0762	−1.09899	−0.549494	−	0.835497i	$-0.685180\pi$
$607$	−27.0762	−1.09899	−0.549494	+	0.835497i	$0.685180\pi$
$608$	0	0
$609$	−17.8307	−0.722536
$610$	0	0
$611$	−3.67257	−0.148576
$612$	0	0
$613$	34.9934	1.41337	0.706685	−	0.707529i	$-0.250190\pi$
$613$	34.9934	1.41337	0.706685	+	0.707529i	$0.250190\pi$
$614$	0	0
$615$	−2.63816	−0.106381
$616$	0	0
$617$	34.4944	1.38869	0.694346	−	0.719641i	$-0.255694\pi$
$617$	34.4944	1.38869	0.694346	+	0.719641i	$0.255694\pi$
$618$	0	0
$619$	13.2602	0.532971	0.266485	−	0.963839i	$-0.414138\pi$
$619$	13.2602	0.532971	0.266485	+	0.963839i	$0.414138\pi$
$620$	0	0
$621$	14.5152	0.582476
$622$	0	0
$623$	−16.5930	−0.664784
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.95022	−0.0778841
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−10.2715	−0.408900	−0.204450	−	0.978877i	$-0.565541\pi$
$631$	−10.2715	−0.408900	−0.204450	+	0.978877i	$0.565541\pi$
$632$	0	0
$633$	11.8881	0.472509
$634$	0	0
$635$	18.6094	0.738490
$636$	0	0
$637$	28.7638	1.13966
$638$	0	0
$639$	2.50039	0.0989140
$640$	0	0
$641$	1.97743	0.0781037	0.0390519	−	0.999237i	$-0.487566\pi$
$641$	1.97743	0.0781037	0.0390519	+	0.999237i	$0.487566\pi$
$642$	0	0
$643$	−43.7321	−1.72462	−0.862312	−	0.506377i	$-0.830984\pi$
$643$	−43.7321	−1.72462	−0.862312	+	0.506377i	$0.830984\pi$
$644$	0	0
$645$	−1.68320	−0.0662758
$646$	0	0
$647$	44.5243	1.75043	0.875216	−	0.483732i	$-0.160719\pi$
$647$	44.5243	1.75043	0.875216	+	0.483732i	$0.160719\pi$
$648$	0	0
$649$	16.5253	0.648674
$650$	0	0
$651$	−9.56352	−0.374824
$652$	0	0
$653$	−10.5300	−0.412071	−0.206035	−	0.978545i	$-0.566056\pi$
$653$	−10.5300	−0.412071	−0.206035	+	0.978545i	$0.566056\pi$
$654$	0	0
$655$	3.69739	0.144469
$656$	0	0
$657$	21.5176	0.839481
$658$	0	0
$659$	−19.1763	−0.747004	−0.373502	−	0.927629i	$-0.621843\pi$
$659$	−19.1763	−0.747004	−0.373502	+	0.927629i	$0.621843\pi$
$660$	0	0
$661$	−38.9860	−1.51638	−0.758190	−	0.652034i	$-0.773916\pi$
$661$	−38.9860	−1.51638	−0.758190	+	0.652034i	$0.773916\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.529944	0.0205503
$666$	0	0
$667$	45.6912	1.76917
$668$	0	0
$669$	34.3378	1.32758
$670$	0	0
$671$	−25.9179	−1.00055
$672$	0	0
$673$	−45.3288	−1.74730	−0.873648	−	0.486558i	$-0.838252\pi$
$673$	−45.3288	−1.74730	−0.873648	+	0.486558i	$0.838252\pi$
$674$	0	0
$675$	−1.60555	−0.0617977
$676$	0	0
$677$	33.3408	1.28139	0.640696	−	0.767794i	$-0.278646\pi$
$677$	33.3408	1.28139	0.640696	+	0.767794i	$0.278646\pi$
$678$	0	0
$679$	10.6810	0.409899
$680$	0	0
$681$	−45.8266	−1.75608
$682$	0	0
$683$	10.9924	0.420613	0.210306	−	0.977636i	$-0.432554\pi$
$683$	10.9924	0.420613	0.210306	+	0.977636i	$0.432554\pi$
$684$	0	0
$685$	6.28824	0.240261
$686$	0	0
$687$	58.5725	2.23468
$688$	0	0
$689$	59.3332	2.26042
$690$	0	0
$691$	37.6221	1.43121	0.715607	−	0.698503i	$-0.246150\pi$
$691$	37.6221	1.43121	0.715607	+	0.698503i	$0.246150\pi$
$692$	0	0
$693$	8.63816	0.328136
$694$	0	0
$695$	10.0556	0.381430
$696$	0	0
$697$	0	0
$698$	0	0
$699$	41.6103	1.57384
$700$	0	0
$701$	21.0184	0.793855	0.396927	−	0.917850i	$-0.370076\pi$
$701$	21.0184	0.793855	0.396927	+	0.917850i	$0.370076\pi$
$702$	0	0
$703$	1.41370	0.0533186
$704$	0	0
$705$	−1.36798	−0.0515213
$706$	0	0
$707$	25.9215	0.974880
$708$	0	0
$709$	−35.0263	−1.31544	−0.657720	−	0.753263i	$-0.728479\pi$
$709$	−35.0263	−1.31544	−0.657720	+	0.753263i	$0.728479\pi$
$710$	0	0
$711$	−25.4228	−0.953429
$712$	0	0
$713$	24.5065	0.917777
$714$	0	0
$715$	15.1365	0.566073
$716$	0	0
$717$	−7.97053	−0.297665
$718$	0	0
$719$	−12.7653	−0.476066	−0.238033	−	0.971257i	$-0.576503\pi$
$719$	−12.7653	−0.476066	−0.238033	+	0.971257i	$0.576503\pi$
$720$	0	0
$721$	13.8944	0.517453
$722$	0	0
$723$	32.3264	1.20223
$724$	0	0
$725$	−5.05397	−0.187700
$726$	0	0
$727$	−40.6908	−1.50914	−0.754569	−	0.656221i	$-0.772154\pi$
$727$	−40.6908	−1.50914	−0.754569	+	0.656221i	$0.772154\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−15.1041	−0.557881	−0.278941	−	0.960308i	$-0.589983\pi$
$733$	−15.1041	−0.557881	−0.278941	+	0.960308i	$0.589983\pi$
$734$	0	0
$735$	10.7141	0.395197
$736$	0	0
$737$	30.1230	1.10959
$738$	0	0
$739$	10.4340	0.383822	0.191911	−	0.981412i	$-0.438532\pi$
$739$	10.4340	0.383822	0.191911	+	0.981412i	$0.438532\pi$
$740$	0	0
$741$	4.92422	0.180896
$742$	0	0
$743$	21.8006	0.799786	0.399893	−	0.916562i	$-0.369047\pi$
$743$	21.8006	0.799786	0.399893	+	0.916562i	$0.369047\pi$
$744$	0	0
$745$	−1.27054	−0.0465489
$746$	0	0
$747$	−28.2144	−1.03231
$748$	0	0
$749$	8.24274	0.301183
$750$	0	0
$751$	26.5462	0.968683	0.484342	−	0.874879i	$-0.339059\pi$
$751$	26.5462	0.968683	0.484342	+	0.874879i	$0.339059\pi$
$752$	0	0
$753$	−8.32688	−0.303448
$754$	0	0
$755$	22.4948	0.818669
$756$	0	0
$757$	−25.5323	−0.927988	−0.463994	−	0.885838i	$-0.653584\pi$
$757$	−25.5323	−0.927988	−0.463994	+	0.885838i	$0.653584\pi$
$758$	0	0
$759$	−50.9726	−1.85019
$760$	0	0
$761$	−13.4659	−0.488139	−0.244070	−	0.969758i	$-0.578483\pi$
$761$	−13.4659	−0.488139	−0.244070	+	0.969758i	$0.578483\pi$
$762$	0	0
$763$	10.9489	0.396377
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.7257	−1.50663
$768$	0	0
$769$	−30.5735	−1.10251	−0.551255	−	0.834337i	$-0.685851\pi$
$769$	−30.5735	−1.10251	−0.551255	+	0.834337i	$0.685851\pi$
$770$	0	0
$771$	−22.8268	−0.822088
$772$	0	0
$773$	−19.6308	−0.706073	−0.353036	−	0.935610i	$-0.614851\pi$
$773$	−19.6308	−0.706073	−0.353036	+	0.935610i	$0.614851\pi$
$774$	0	0
$775$	−2.71070	−0.0973714
$776$	0	0
$777$	−14.4194	−0.517292
$778$	0	0
$779$	−0.396273	−0.0141980
$780$	0	0
$781$	2.65853	0.0951298
$782$	0	0
$783$	8.11441	0.289985
$784$	0	0
$785$	−17.0193	−0.607444
$786$	0	0
$787$	12.7767	0.455439	0.227720	−	0.973727i	$-0.426873\pi$
$787$	12.7767	0.455439	0.227720	+	0.973727i	$0.426873\pi$
$788$	0	0
$789$	−52.6149	−1.87314
$790$	0	0
$791$	28.9550	1.02952
$792$	0	0
$793$	65.4417	2.32390
$794$	0	0
$795$	22.1009	0.783837
$796$	0	0
$797$	−9.76822	−0.346008	−0.173004	−	0.984921i	$-0.555347\pi$
$797$	−9.76822	−0.346008	−0.173004	+	0.984921i	$0.555347\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−24.9398	−0.881203
$802$	0	0
$803$	22.8785	0.807365
$804$	0	0
$805$	13.8511	0.488187
$806$	0	0
$807$	−37.6388	−1.32495
$808$	0	0
$809$	−39.9796	−1.40561	−0.702803	−	0.711384i	$-0.748069\pi$
$809$	−39.9796	−1.40561	−0.702803	+	0.711384i	$0.748069\pi$
$810$	0	0
$811$	6.22771	0.218684	0.109342	−	0.994004i	$-0.465126\pi$
$811$	6.22771	0.218684	0.109342	+	0.994004i	$0.465126\pi$
$812$	0	0
$813$	−55.9609	−1.96263
$814$	0	0
$815$	−0.799925	−0.0280202
$816$	0	0
$817$	−0.252830	−0.00884541
$818$	0	0
$819$	−21.8110	−0.762139
$820$	0	0
$821$	3.43848	0.120004	0.0600019	−	0.998198i	$-0.480889\pi$
$821$	3.43848	0.120004	0.0600019	+	0.998198i	$0.480889\pi$
$822$	0	0
$823$	29.2112	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$823$	29.2112	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$824$	0	0
$825$	5.63816	0.196295
$826$	0	0
$827$	54.9557	1.91100	0.955498	−	0.294999i	$-0.0953192\pi$
$827$	54.9557	1.91100	0.955498	+	0.294999i	$0.0953192\pi$
$828$	0	0
$829$	11.5572	0.401397	0.200698	−	0.979653i	$-0.435679\pi$
$829$	11.5572	0.401397	0.200698	+	0.979653i	$0.435679\pi$
$830$	0	0
$831$	52.8137	1.83209
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.8105	−0.616359
$836$	0	0
$837$	4.35217	0.150433
$838$	0	0
$839$	−49.5915	−1.71209	−0.856045	−	0.516901i	$-0.827085\pi$
$839$	−49.5915	−1.71209	−0.856045	+	0.516901i	$0.827085\pi$
$840$	0	0
$841$	−3.45740	−0.119221
$842$	0	0
$843$	−37.8732	−1.30442
$844$	0	0
$845$	−25.2191	−0.867564
$846$	0	0
$847$	−7.66849	−0.263493
$848$	0	0
$849$	−9.33877	−0.320506
$850$	0	0
$851$	36.9497	1.26662
$852$	0	0
$853$	1.12515	0.0385242	0.0192621	−	0.999814i	$-0.493868\pi$
$853$	1.12515	0.0385242	0.0192621	+	0.999814i	$0.493868\pi$
$854$	0	0
$855$	0.796521	0.0272404
$856$	0	0
$857$	−16.8681	−0.576202	−0.288101	−	0.957600i	$-0.593024\pi$
$857$	−16.8681	−0.576202	−0.288101	+	0.957600i	$0.593024\pi$
$858$	0	0
$859$	−26.9983	−0.921168	−0.460584	−	0.887616i	$-0.652360\pi$
$859$	−26.9983	−0.921168	−0.460584	+	0.887616i	$0.652360\pi$
$860$	0	0
$861$	4.04189	0.137747
$862$	0	0
$863$	27.1761	0.925086	0.462543	−	0.886597i	$-0.346937\pi$
$863$	27.1761	0.925086	0.462543	+	0.886597i	$0.346937\pi$
$864$	0	0
$865$	−6.13517	−0.208602
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−27.0307	−0.916953
$870$	0	0
$871$	−76.0594	−2.57717
$872$	0	0
$873$	16.0539	0.543341
$874$	0	0
$875$	−1.53209	−0.0517941
$876$	0	0
$877$	−30.5465	−1.03148	−0.515741	−	0.856745i	$-0.672483\pi$
$877$	−30.5465	−1.03148	−0.515741	+	0.856745i	$0.672483\pi$
$878$	0	0
$879$	8.87002	0.299178
$880$	0	0
$881$	−31.8061	−1.07157	−0.535787	−	0.844353i	$-0.679985\pi$
$881$	−31.8061	−1.07157	−0.535787	+	0.844353i	$0.679985\pi$
$882$	0	0
$883$	−4.30075	−0.144732	−0.0723658	−	0.997378i	$-0.523055\pi$
$883$	−4.30075	−0.144732	−0.0723658	+	0.997378i	$0.523055\pi$
$884$	0	0
$885$	−15.5423	−0.522448
$886$	0	0
$887$	19.7877	0.664405	0.332202	−	0.943208i	$-0.392208\pi$
$887$	19.7877	0.664405	0.332202	+	0.943208i	$0.392208\pi$
$888$	0	0
$889$	−28.5112	−0.956235
$890$	0	0
$891$	−25.9668	−0.869921
$892$	0	0
$893$	−0.205483	−0.00687622
$894$	0	0
$895$	7.55270	0.252459
$896$	0	0
$897$	128.704	4.29730
$898$	0	0
$899$	13.6998	0.456914
$900$	0	0
$901$	0	0
$902$	0	0
$903$	2.57881	0.0858174
$904$	0	0
$905$	16.2037	0.538631
$906$	0	0
$907$	−36.4938	−1.21176	−0.605878	−	0.795558i	$-0.707178\pi$
$907$	−36.4938	−1.21176	−0.605878	+	0.795558i	$0.707178\pi$
$908$	0	0
$909$	38.9609	1.29225
$910$	0	0
$911$	−29.9058	−0.990825	−0.495412	−	0.868658i	$-0.664983\pi$
$911$	−29.9058	−0.990825	−0.495412	+	0.868658i	$0.664983\pi$
$912$	0	0
$913$	−29.9989	−0.992818
$914$	0	0
$915$	24.3762	0.805852
$916$	0	0
$917$	−5.66474	−0.187066
$918$	0	0
$919$	−20.3390	−0.670923	−0.335462	−	0.942054i	$-0.608892\pi$
$919$	−20.3390	−0.670923	−0.335462	+	0.942054i	$0.608892\pi$
$920$	0	0
$921$	−26.8493	−0.884713
$922$	0	0
$923$	−6.71269	−0.220951
$924$	0	0
$925$	−4.08706	−0.134382
$926$	0	0
$927$	20.8836	0.685909
$928$	0	0
$929$	30.0855	0.987074	0.493537	−	0.869725i	$-0.335704\pi$
$929$	30.0855	0.987074	0.493537	+	0.869725i	$0.335704\pi$
$930$	0	0
$931$	1.60935	0.0527444
$932$	0	0
$933$	−29.0300	−0.950399
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.4044	−0.666583	−0.333292	−	0.942824i	$-0.608159\pi$
$937$	−20.4044	−0.666583	−0.333292	+	0.942824i	$0.608159\pi$
$938$	0	0
$939$	−39.4819	−1.28844
$940$	0	0
$941$	34.5644	1.12677	0.563383	−	0.826196i	$-0.309499\pi$
$941$	34.5644	1.12677	0.563383	+	0.826196i	$0.309499\pi$
$942$	0	0
$943$	−10.3573	−0.337282
$944$	0	0
$945$	2.45985	0.0800188
$946$	0	0
$947$	−0.166858	−0.00542215	−0.00271107	−	0.999996i	$-0.500863\pi$
$947$	−0.166858	−0.00542215	−0.00271107	+	0.999996i	$0.500863\pi$
$948$	0	0
$949$	−57.7673	−1.87521
$950$	0	0
$951$	6.04141	0.195906
$952$	0	0
$953$	−26.0938	−0.845261	−0.422631	−	0.906302i	$-0.638893\pi$
$953$	−26.0938	−0.845261	−0.422631	+	0.906302i	$0.638893\pi$
$954$	0	0
$955$	−21.8640	−0.707504
$956$	0	0
$957$	−28.4951	−0.921115
$958$	0	0
$959$	−9.63414	−0.311103
$960$	0	0
$961$	−23.6521	−0.762970
$962$	0	0
$963$	12.3891	0.399233
$964$	0	0
$965$	−2.35506	−0.0758122
$966$	0	0
$967$	−28.5476	−0.918030	−0.459015	−	0.888428i	$-0.651798\pi$
$967$	−28.5476	−0.918030	−0.459015	+	0.888428i	$0.651798\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.54834	−0.145963	−0.0729815	−	0.997333i	$-0.523251\pi$
$971$	−4.54834	−0.145963	−0.0729815	+	0.997333i	$0.523251\pi$
$972$	0	0
$973$	−15.4060	−0.493895
$974$	0	0
$975$	−14.2361	−0.455921
$976$	0	0
$977$	25.7740	0.824583	0.412292	−	0.911052i	$-0.364728\pi$
$977$	25.7740	0.824583	0.412292	+	0.911052i	$0.364728\pi$
$978$	0	0
$979$	−26.5171	−0.847490
$980$	0	0
$981$	16.4566	0.525417
$982$	0	0
$983$	18.1671	0.579441	0.289720	−	0.957111i	$-0.406438\pi$
$983$	18.1671	0.579441	0.289720	+	0.957111i	$0.406438\pi$
$984$	0	0
$985$	−9.36359	−0.298349
$986$	0	0
$987$	2.09587	0.0667124
$988$	0	0
$989$	−6.60820	−0.210129
$990$	0	0
$991$	−30.3752	−0.964900	−0.482450	−	0.875923i	$-0.660253\pi$
$991$	−30.3752	−0.964900	−0.482450	+	0.875923i	$0.660253\pi$
$992$	0	0
$993$	−29.1552	−0.925212
$994$	0	0
$995$	12.3432	0.391306
$996$	0	0
$997$	29.8884	0.946574	0.473287	−	0.880908i	$-0.343067\pi$
$997$	29.8884	0.946574	0.473287	+	0.880908i	$0.343067\pi$
$998$	0	0
$999$	6.56198	0.207612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.o.1.6	yes	6
17.4	even	4		5780.2.c.g.5201.3		12
17.13	even	4		5780.2.c.g.5201.10		12
17.16	even	2		5780.2.a.l.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5780.2.a.l.1.1	✓	6	17.16	even	2
5780.2.a.o.1.6	yes	6	1.1	even	1	trivial
5780.2.c.g.5201.3		12	17.4	even	4
5780.2.c.g.5201.10		12	17.13	even	4

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{3} -1.00000 q^{5} +1.53209 q^{7} +2.30278 q^{9} +2.44842 q^{11} -6.18216 q^{13} -2.30278 q^{15} -0.345896 q^{19} +3.52806 q^{21} -9.04065 q^{23} +1.00000 q^{25} -1.60555 q^{27} -5.05397 q^{29} -2.71070 q^{31} +5.63816 q^{33} -1.53209 q^{35} -4.08706 q^{37} -14.2361 q^{39} +1.14564 q^{41} +0.730943 q^{43} -2.30278 q^{45} +0.594059 q^{47} -4.65270 q^{49} -9.59749 q^{53} -2.44842 q^{55} -0.796521 q^{57} +6.74937 q^{59} -10.5856 q^{61} +3.52806 q^{63} +6.18216 q^{65} +12.3030 q^{67} -20.8186 q^{69} +1.08582 q^{71} +9.34420 q^{73} +2.30278 q^{75} +3.75119 q^{77} -11.0401 q^{79} -10.6056 q^{81} -12.2524 q^{83} -11.6382 q^{87} -10.8303 q^{89} -9.47162 q^{91} -6.24214 q^{93} +0.345896 q^{95} +6.97153 q^{97} +5.63816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} - 6 q^{5} + 3 q^{9} - 15 q^{13} - 3 q^{15} - 3 q^{19} + 6 q^{25} + 12 q^{27} + 6 q^{29} + 6 q^{31} + 3 q^{37} - 27 q^{39} + 3 q^{41} + 3 q^{43} - 3 q^{45} - 15 q^{47} - 30 q^{49} - 12 q^{53}+ \cdots + 18 q^{97}+O(q^{100}) \) 6 * q + 3 * q^3 - 6 * q^5 + 3 * q^9 - 15 * q^13 - 3 * q^15 - 3 * q^19 + 6 * q^25 + 12 * q^27 + 6 * q^29 + 6 * q^31 + 3 * q^37 - 27 * q^39 + 3 * q^41 + 3 * q^43 - 3 * q^45 - 15 * q^47 - 30 * q^49 - 12 * q^53 + 18 * q^57 - 6 * q^59 + 3 * q^61 + 15 * q^65 - 6 * q^67 + 6 * q^71 + 9 * q^73 + 3 * q^75 - 3 * q^77 - 39 * q^79 - 42 * q^81 - 15 * q^83 - 36 * q^87 - 6 * q^89 - 6 * q^91 - 36 * q^93 + 3 * q^95 + 18 * q^97

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