LMFDB - Embedded newform 10000.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1250)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.209057 q^{3} -0.418114 q^{7} -2.95630 q^{9} +O(q^{10})$	$q-0.209057 q^{3} -0.418114 q^{7} -2.95630 q^{9} -2.65418 q^{11} -2.81795 q^{13} +3.81040 q^{17} -5.26755 q^{19} +0.0874096 q^{21} +9.00723 q^{23} +1.24520 q^{27} +7.77116 q^{29} -5.49448 q^{31} +0.554875 q^{33} +6.80284 q^{37} +0.589113 q^{39} +5.97010 q^{41} -5.47660 q^{43} +0.159705 q^{47} -6.82518 q^{49} -0.796590 q^{51} +5.35304 q^{53} +1.10122 q^{57} +11.5083 q^{59} +13.1040 q^{61} +1.23607 q^{63} +3.39386 q^{67} -1.88302 q^{69} +0.116977 q^{71} -13.1772 q^{73} +1.10975 q^{77} -5.07636 q^{79} +8.60857 q^{81} -13.6431 q^{83} -1.62461 q^{87} +0.191183 q^{89} +1.17823 q^{91} +1.14866 q^{93} +9.56231 q^{97} +7.84655 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^7 - 3 * q^9	$ 4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 6 * q^13 - 7 * q^17 - 15 * q^19 + 18 * q^21 + 6 * q^23 - 5 * q^27 + 10 * q^29 - 8 * q^31 + 13 * q^33 - 12 * q^37 - 24 * q^39 + 3 * q^41 - 14 * q^43 + 2 * q^47 + 8 * q^49 + 7 * q^51 + 4 * q^53 + 10 * q^57 + 20 * q^59 + 18 * q^61 - 4 * q^63 - 23 * q^67 - 16 * q^69 - 8 * q^71 - q^73 + 26 * q^77 - 10 * q^79 - 16 * q^81 - 14 * q^83 - 20 * q^87 + 5 * q^89 - 48 * q^91 - 22 * q^93 - 2 * q^97 + 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
$3$	−0.209057	−0.120699	−0.0603495	−	0.998177i	$-0.519222\pi$
$3$	−0.209057	−0.120699	−0.0603495	+	0.998177i	$0.519222\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.418114	−0.158032	−0.0790161	−	0.996873i	$-0.525178\pi$
$7$	−0.418114	−0.158032	−0.0790161	+	0.996873i	$0.525178\pi$
$8$	0	0
$9$	−2.95630	−0.985432
$10$	0	0
$11$	−2.65418	−0.800266	−0.400133	−	0.916457i	$-0.631036\pi$
$11$	−2.65418	−0.800266	−0.400133	+	0.916457i	$0.631036\pi$
$12$	0	0
$13$	−2.81795	−0.781560	−0.390780	−	0.920484i	$-0.627795\pi$
$13$	−2.81795	−0.781560	−0.390780	+	0.920484i	$0.627795\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.81040	0.924157	0.462079	−	0.886839i	$-0.347104\pi$
$17$	3.81040	0.924157	0.462079	+	0.886839i	$0.347104\pi$
$18$	0	0
$19$	−5.26755	−1.20846	−0.604229	−	0.796811i	$-0.706519\pi$
$19$	−5.26755	−1.20846	−0.604229	+	0.796811i	$0.706519\pi$
$20$	0	0
$21$	0.0874096	0.0190743
$22$	0	0
$23$	9.00723	1.87814	0.939068	−	0.343731i	$-0.111691\pi$
$23$	9.00723	1.87814	0.939068	+	0.343731i	$0.111691\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.24520	0.239640
$28$	0	0
$29$	7.77116	1.44307	0.721534	−	0.692379i	$-0.243437\pi$
$29$	7.77116	1.44307	0.721534	+	0.692379i	$0.243437\pi$
$30$	0	0
$31$	−5.49448	−0.986837	−0.493419	−	0.869792i	$-0.664253\pi$
$31$	−5.49448	−0.986837	−0.493419	+	0.869792i	$0.664253\pi$
$32$	0	0
$33$	0.554875	0.0965914
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.80284	1.11838	0.559190	−	0.829040i	$-0.311112\pi$
$37$	6.80284	1.11838	0.559190	+	0.829040i	$0.311112\pi$
$38$	0	0
$39$	0.589113	0.0943335
$40$	0	0
$41$	5.97010	0.932373	0.466187	−	0.884686i	$-0.345628\pi$
$41$	5.97010	0.932373	0.466187	+	0.884686i	$0.345628\pi$
$42$	0	0
$43$	−5.47660	−0.835174	−0.417587	−	0.908637i	$-0.637124\pi$
$43$	−5.47660	−0.835174	−0.417587	+	0.908637i	$0.637124\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.159705	0.0232954	0.0116477	−	0.999932i	$-0.496292\pi$
$47$	0.159705	0.0232954	0.0116477	+	0.999932i	$0.496292\pi$
$48$	0	0
$49$	−6.82518	−0.975026
$50$	0	0
$51$	−0.796590	−0.111545
$52$	0	0
$53$	5.35304	0.735297	0.367649	−	0.929965i	$-0.380163\pi$
$53$	5.35304	0.735297	0.367649	+	0.929965i	$0.380163\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.10122	0.145860
$58$	0	0
$59$	11.5083	1.49825	0.749125	−	0.662428i	$-0.230474\pi$
$59$	11.5083	1.49825	0.749125	+	0.662428i	$0.230474\pi$
$60$	0	0
$61$	13.1040	1.67779	0.838896	−	0.544291i	$-0.183201\pi$
$61$	13.1040	1.67779	0.838896	+	0.544291i	$0.183201\pi$
$62$	0	0
$63$	1.23607	0.155730
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.39386	0.414626	0.207313	−	0.978275i	$-0.433528\pi$
$67$	3.39386	0.414626	0.207313	+	0.978275i	$0.433528\pi$
$68$	0	0
$69$	−1.88302	−0.226689
$70$	0	0
$71$	0.116977	0.0138826	0.00694130	−	0.999976i	$-0.497790\pi$
$71$	0.116977	0.0138826	0.00694130	+	0.999976i	$0.497790\pi$
$72$	0	0
$73$	−13.1772	−1.54228	−0.771140	−	0.636665i	$-0.780313\pi$
$73$	−13.1772	−1.54228	−0.771140	+	0.636665i	$0.780313\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.10975	0.126468
$78$	0	0
$79$	−5.07636	−0.571135	−0.285568	−	0.958359i	$-0.592182\pi$
$79$	−5.07636	−0.571135	−0.285568	+	0.958359i	$0.592182\pi$
$80$	0	0
$81$	8.60857	0.956507
$82$	0	0
$83$	−13.6431	−1.49753	−0.748764	−	0.662836i	$-0.769353\pi$
$83$	−13.6431	−1.49753	−0.748764	+	0.662836i	$0.769353\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.62461	−0.174177
$88$	0	0
$89$	0.191183	0.0202653	0.0101327	−	0.999949i	$-0.496775\pi$
$89$	0.191183	0.0202653	0.0101327	+	0.999949i	$0.496775\pi$
$90$	0	0
$91$	1.17823	0.123512
$92$	0	0
$93$	1.14866	0.119110
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.56231	0.970905	0.485453	−	0.874263i	$-0.338655\pi$
$97$	9.56231	0.970905	0.485453	+	0.874263i	$0.338655\pi$
$98$	0	0
$99$	7.84655	0.788607
$100$	0	0
$101$	−5.96661	−0.593700	−0.296850	−	0.954924i	$-0.595936\pi$
$101$	−5.96661	−0.593700	−0.296850	+	0.954924i	$0.595936\pi$
$102$	0	0
$103$	−10.4721	−1.03185	−0.515925	−	0.856634i	$-0.672552\pi$
$103$	−10.4721	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.35207	−0.710751	−0.355376	−	0.934724i	$-0.615647\pi$
$107$	−7.35207	−0.710751	−0.355376	+	0.934724i	$0.615647\pi$
$108$	0	0
$109$	12.8475	1.23057	0.615285	−	0.788305i	$-0.289041\pi$
$109$	12.8475	1.23057	0.615285	+	0.788305i	$0.289041\pi$
$110$	0	0
$111$	−1.42218	−0.134987
$112$	0	0
$113$	0.000976391 0	9.18512e−5 0	4.59256e−5	−	1.00000i	$-0.499985\pi$
$113$	0.000976391 0	9.18512e−5 0	4.59256e−5		1.00000i	$0.499985\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.33070	0.770174
$118$	0	0
$119$	−1.59318	−0.146047
$120$	0	0
$121$	−3.95532	−0.359574
$122$	0	0
$123$	−1.24809	−0.112537
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.8324	−1.84858	−0.924289	−	0.381694i	$-0.875341\pi$
$127$	−20.8324	−1.84858	−0.924289	+	0.381694i	$0.875341\pi$
$128$	0	0
$129$	1.14492	0.100805
$130$	0	0
$131$	0.640375	0.0559498	0.0279749	−	0.999609i	$-0.491094\pi$
$131$	0.640375	0.0559498	0.0279749	+	0.999609i	$0.491094\pi$
$132$	0	0
$133$	2.20243	0.190975
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.6167	−1.33423	−0.667114	−	0.744956i	$-0.732470\pi$
$137$	−15.6167	−1.33423	−0.667114	+	0.744956i	$0.732470\pi$
$138$	0	0
$139$	−10.4284	−0.884528	−0.442264	−	0.896885i	$-0.645825\pi$
$139$	−10.4284	−0.884528	−0.442264	+	0.896885i	$0.645825\pi$
$140$	0	0
$141$	−0.0333875	−0.00281173
$142$	0	0
$143$	7.47936	0.625456
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.42685	0.117685
$148$	0	0
$149$	−4.63379	−0.379615	−0.189808	−	0.981821i	$-0.560786\pi$
$149$	−4.63379	−0.379615	−0.189808	+	0.981821i	$0.560786\pi$
$150$	0	0
$151$	−1.42218	−0.115735	−0.0578677	−	0.998324i	$-0.518430\pi$
$151$	−1.42218	−0.115735	−0.0578677	+	0.998324i	$0.518430\pi$
$152$	0	0
$153$	−11.2647	−0.910694
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.07952	−0.565007	−0.282504	−	0.959266i	$-0.591165\pi$
$157$	−7.07952	−0.565007	−0.282504	+	0.959266i	$0.591165\pi$
$158$	0	0
$159$	−1.11909	−0.0887497
$160$	0	0
$161$	−3.76605	−0.296806
$162$	0	0
$163$	−17.9367	−1.40491	−0.702456	−	0.711727i	$-0.747913\pi$
$163$	−17.9367	−1.40491	−0.702456	+	0.711727i	$0.747913\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.96661	0.461710	0.230855	−	0.972988i	$-0.425848\pi$
$167$	5.96661	0.461710	0.230855	+	0.972988i	$0.425848\pi$
$168$	0	0
$169$	−5.05913	−0.389164
$170$	0	0
$171$	15.5724	1.19085
$172$	0	0
$173$	1.36621	0.103871	0.0519354	−	0.998650i	$-0.483461\pi$
$173$	1.36621	0.103871	0.0519354	+	0.998650i	$0.483461\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.40589	−0.180837
$178$	0	0
$179$	16.8132	1.25667	0.628337	−	0.777941i	$-0.283736\pi$
$179$	16.8132	1.25667	0.628337	+	0.777941i	$0.283736\pi$
$180$	0	0
$181$	−7.34411	−0.545884	−0.272942	−	0.962031i	$-0.587997\pi$
$181$	−7.34411	−0.545884	−0.272942	+	0.962031i	$0.587997\pi$
$182$	0	0
$183$	−2.73948	−0.202508
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10.1135	−0.739571
$188$	0	0
$189$	−0.520637	−0.0378708
$190$	0	0
$191$	−6.02957	−0.436284	−0.218142	−	0.975917i	$-0.570000\pi$
$191$	−6.02957	−0.436284	−0.218142	+	0.975917i	$0.570000\pi$
$192$	0	0
$193$	−15.7786	−1.13577	−0.567884	−	0.823109i	$-0.692238\pi$
$193$	−15.7786	−1.13577	−0.567884	+	0.823109i	$0.692238\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.7265	0.835477	0.417738	−	0.908567i	$-0.362823\pi$
$197$	11.7265	0.835477	0.417738	+	0.908567i	$0.362823\pi$
$198$	0	0
$199$	−24.4085	−1.73027	−0.865137	−	0.501535i	$-0.832769\pi$
$199$	−24.4085	−1.73027	−0.865137	+	0.501535i	$0.832769\pi$
$200$	0	0
$201$	−0.709511	−0.0500450
$202$	0	0
$203$	−3.24923	−0.228051
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−26.6280	−1.85078
$208$	0	0
$209$	13.9810	0.967088
$210$	0	0
$211$	−14.9151	−1.02680	−0.513401	−	0.858149i	$-0.671614\pi$
$211$	−14.9151	−1.02680	−0.513401	+	0.858149i	$0.671614\pi$
$212$	0	0
$213$	−0.0244548	−0.00167562
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.29732	0.155952
$218$	0	0
$219$	2.75480	0.186152
$220$	0	0
$221$	−10.7375	−0.722284
$222$	0	0
$223$	−23.0725	−1.54505	−0.772526	−	0.634983i	$-0.781007\pi$
$223$	−23.0725	−1.54505	−0.772526	+	0.634983i	$0.781007\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.9955	0.862544	0.431272	−	0.902222i	$-0.358065\pi$
$227$	12.9955	0.862544	0.431272	+	0.902222i	$0.358065\pi$
$228$	0	0
$229$	−7.16693	−0.473604	−0.236802	−	0.971558i	$-0.576099\pi$
$229$	−7.16693	−0.473604	−0.236802	+	0.971558i	$0.576099\pi$
$230$	0	0
$231$	−0.232001	−0.0152645
$232$	0	0
$233$	6.34069	0.415392	0.207696	−	0.978193i	$-0.433403\pi$
$233$	6.34069	0.415392	0.207696	+	0.978193i	$0.433403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.06125	0.0689355
$238$	0	0
$239$	18.1446	1.17368	0.586838	−	0.809704i	$-0.300373\pi$
$239$	18.1446	1.17368	0.586838	+	0.809704i	$0.300373\pi$
$240$	0	0
$241$	−29.0244	−1.86963	−0.934813	−	0.355141i	$-0.884433\pi$
$241$	−29.0244	−1.86963	−0.934813	+	0.355141i	$0.884433\pi$
$242$	0	0
$243$	−5.53529	−0.355089
$244$	0	0
$245$	0	0
$246$	0	0
$247$	14.8437	0.944482
$248$	0	0
$249$	2.85219	0.180750
$250$	0	0
$251$	26.3222	1.66144	0.830722	−	0.556688i	$-0.187928\pi$
$251$	26.3222	1.66144	0.830722	+	0.556688i	$0.187928\pi$
$252$	0	0
$253$	−23.9068	−1.50301
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.2816	−1.57702	−0.788510	−	0.615022i	$-0.789147\pi$
$257$	−25.2816	−1.57702	−0.788510	+	0.615022i	$0.789147\pi$
$258$	0	0
$259$	−2.84436	−0.176740
$260$	0	0
$261$	−22.9738	−1.42204
$262$	0	0
$263$	−8.08164	−0.498335	−0.249168	−	0.968460i	$-0.580157\pi$
$263$	−8.08164	−0.498335	−0.249168	+	0.968460i	$0.580157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.0399681	−0.00244601
$268$	0	0
$269$	−21.5699	−1.31514	−0.657571	−	0.753393i	$-0.728416\pi$
$269$	−21.5699	−1.31514	−0.657571	+	0.753393i	$0.728416\pi$
$270$	0	0
$271$	−7.71136	−0.468432	−0.234216	−	0.972185i	$-0.575252\pi$
$271$	−7.71136	−0.468432	−0.234216	+	0.972185i	$0.575252\pi$
$272$	0	0
$273$	−0.246316	−0.0149077
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.40155	−0.204379	−0.102190	−	0.994765i	$-0.532585\pi$
$277$	−3.40155	−0.204379	−0.102190	+	0.994765i	$0.532585\pi$
$278$	0	0
$279$	16.2433	0.972461
$280$	0	0
$281$	−0.537604	−0.0320708	−0.0160354	−	0.999871i	$-0.505104\pi$
$281$	−0.537604	−0.0320708	−0.0160354	+	0.999871i	$0.505104\pi$
$282$	0	0
$283$	13.8242	0.821764	0.410882	−	0.911689i	$-0.365221\pi$
$283$	13.8242	0.821764	0.410882	+	0.911689i	$0.365221\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.49618	−0.147345
$288$	0	0
$289$	−2.48087	−0.145934
$290$	0	0
$291$	−1.99907	−0.117187
$292$	0	0
$293$	−32.6280	−1.90615	−0.953075	−	0.302735i	$-0.902100\pi$
$293$	−32.6280	−1.90615	−0.953075	+	0.302735i	$0.902100\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.30500	−0.191776
$298$	0	0
$299$	−25.3820	−1.46788
$300$	0	0
$301$	2.28984	0.131984
$302$	0	0
$303$	1.24736	0.0716591
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.52031	0.372134	0.186067	−	0.982537i	$-0.440426\pi$
$307$	6.52031	0.372134	0.186067	+	0.982537i	$0.440426\pi$
$308$	0	0
$309$	2.18927	0.124543
$310$	0	0
$311$	26.5162	1.50359	0.751797	−	0.659395i	$-0.229188\pi$
$311$	26.5162	1.50359	0.751797	+	0.659395i	$0.229188\pi$
$312$	0	0
$313$	−1.44321	−0.0815753	−0.0407877	−	0.999168i	$-0.512987\pi$
$313$	−1.44321	−0.0815753	−0.0407877	+	0.999168i	$0.512987\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.4102	1.20252	0.601259	−	0.799054i	$-0.294666\pi$
$317$	21.4102	1.20252	0.601259	+	0.799054i	$0.294666\pi$
$318$	0	0
$319$	−20.6261	−1.15484
$320$	0	0
$321$	1.53700	0.0857870
$322$	0	0
$323$	−20.0714	−1.11680
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.68586	−0.148529
$328$	0	0
$329$	−0.0667750	−0.00368142
$330$	0	0
$331$	19.8851	1.09298	0.546491	−	0.837465i	$-0.315963\pi$
$331$	19.8851	1.09298	0.546491	+	0.837465i	$0.315963\pi$
$332$	0	0
$333$	−20.1112	−1.10209
$334$	0	0
$335$	0	0
$336$	0	0
$337$	33.6205	1.83142	0.915712	−	0.401836i	$-0.131628\pi$
$337$	33.6205	1.83142	0.915712	+	0.401836i	$0.131628\pi$
$338$	0	0
$339$	−0.000204121 0	−1.10863e−5 0
$340$	0	0
$341$	14.5833	0.789732
$342$	0	0
$343$	5.78050	0.312118
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.23391	0.442020	0.221010	−	0.975272i	$-0.429065\pi$
$347$	8.23391	0.442020	0.221010	+	0.975272i	$0.429065\pi$
$348$	0	0
$349$	8.24013	0.441084	0.220542	−	0.975377i	$-0.429217\pi$
$349$	8.24013	0.441084	0.220542	+	0.975377i	$0.429217\pi$
$350$	0	0
$351$	−3.50893	−0.187293
$352$	0	0
$353$	2.65378	0.141247	0.0706233	−	0.997503i	$-0.477501\pi$
$353$	2.65378	0.141247	0.0706233	+	0.997503i	$0.477501\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.333065	0.0176277
$358$	0	0
$359$	28.0929	1.48269	0.741344	−	0.671125i	$-0.234189\pi$
$359$	28.0929	1.48269	0.741344	+	0.671125i	$0.234189\pi$
$360$	0	0
$361$	8.74703	0.460370
$362$	0	0
$363$	0.826887	0.0434003
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.3728	1.27225	0.636124	−	0.771587i	$-0.280537\pi$
$367$	24.3728	1.27225	0.636124	+	0.771587i	$0.280537\pi$
$368$	0	0
$369$	−17.6494	−0.918790
$370$	0	0
$371$	−2.23818	−0.116201
$372$	0	0
$373$	−5.66734	−0.293444	−0.146722	−	0.989178i	$-0.546872\pi$
$373$	−5.66734	−0.293444	−0.146722	+	0.989178i	$0.546872\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−21.8988	−1.12784
$378$	0	0
$379$	10.6719	0.548176	0.274088	−	0.961705i	$-0.411624\pi$
$379$	10.6719	0.548176	0.274088	+	0.961705i	$0.411624\pi$
$380$	0	0
$381$	4.35516	0.223122
$382$	0	0
$383$	−17.9390	−0.916640	−0.458320	−	0.888787i	$-0.651549\pi$
$383$	−17.9390	−0.916640	−0.458320	+	0.888787i	$0.651549\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.1905	0.823007
$388$	0	0
$389$	−9.93086	−0.503515	−0.251757	−	0.967790i	$-0.581009\pi$
$389$	−9.93086	−0.503515	−0.251757	+	0.967790i	$0.581009\pi$
$390$	0	0
$391$	34.3211	1.73569
$392$	0	0
$393$	−0.133875	−0.00675309
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.51893	0.377364	0.188682	−	0.982038i	$-0.439578\pi$
$397$	7.51893	0.377364	0.188682	+	0.982038i	$0.439578\pi$
$398$	0	0
$399$	−0.460434	−0.0230505
$400$	0	0
$401$	9.63992	0.481395	0.240697	−	0.970600i	$-0.422624\pi$
$401$	9.63992	0.481395	0.240697	+	0.970600i	$0.422624\pi$
$402$	0	0
$403$	15.4832	0.771272
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.0560	−0.895001
$408$	0	0
$409$	24.0970	1.19152	0.595760	−	0.803163i	$-0.296851\pi$
$409$	24.0970	1.19152	0.595760	+	0.803163i	$0.296851\pi$
$410$	0	0
$411$	3.26479	0.161040
$412$	0	0
$413$	−4.81177	−0.236772
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.18014	0.106762
$418$	0	0
$419$	−28.9257	−1.41311	−0.706556	−	0.707657i	$-0.749752\pi$
$419$	−28.9257	−1.41311	−0.706556	+	0.707657i	$0.749752\pi$
$420$	0	0
$421$	12.7395	0.620884	0.310442	−	0.950592i	$-0.399523\pi$
$421$	12.7395	0.620884	0.310442	+	0.950592i	$0.399523\pi$
$422$	0	0
$423$	−0.472136	−0.0229560
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.47895	−0.265145
$428$	0	0
$429$	−1.56361	−0.0754919
$430$	0	0
$431$	14.0257	0.675596	0.337798	−	0.941219i	$-0.390318\pi$
$431$	14.0257	0.675596	0.337798	+	0.941219i	$0.390318\pi$
$432$	0	0
$433$	−27.7148	−1.33189	−0.665945	−	0.746001i	$-0.731971\pi$
$433$	−27.7148	−1.33189	−0.665945	+	0.746001i	$0.731971\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−47.4460	−2.26965
$438$	0	0
$439$	−25.0072	−1.19353	−0.596765	−	0.802416i	$-0.703548\pi$
$439$	−25.0072	−1.19353	−0.596765	+	0.802416i	$0.703548\pi$
$440$	0	0
$441$	20.1772	0.960821
$442$	0	0
$443$	−28.5509	−1.35650	−0.678248	−	0.734833i	$-0.737260\pi$
$443$	−28.5509	−1.35650	−0.678248	+	0.734833i	$0.737260\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.968727	0.0458192
$448$	0	0
$449$	−4.59694	−0.216943	−0.108472	−	0.994100i	$-0.534596\pi$
$449$	−4.59694	−0.216943	−0.108472	+	0.994100i	$0.534596\pi$
$450$	0	0
$451$	−15.8457	−0.746147
$452$	0	0
$453$	0.297317	0.0139692
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.8037	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$457$	−28.8037	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$458$	0	0
$459$	4.74472	0.221465
$460$	0	0
$461$	−0.326884	−0.0152245	−0.00761225	−	0.999971i	$-0.502423\pi$
$461$	−0.326884	−0.0152245	−0.00761225	+	0.999971i	$0.502423\pi$
$462$	0	0
$463$	30.0740	1.39766	0.698829	−	0.715289i	$-0.253705\pi$
$463$	30.0740	1.39766	0.698829	+	0.715289i	$0.253705\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.1125	0.699324	0.349662	−	0.936876i	$-0.386296\pi$
$467$	15.1125	0.699324	0.349662	+	0.936876i	$0.386296\pi$
$468$	0	0
$469$	−1.41902	−0.0655243
$470$	0	0
$471$	1.48002	0.0681959
$472$	0	0
$473$	14.5359	0.668361
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.8252	−0.724585
$478$	0	0
$479$	−3.50341	−0.160075	−0.0800374	−	0.996792i	$-0.525504\pi$
$479$	−3.50341	−0.160075	−0.0800374	+	0.996792i	$0.525504\pi$
$480$	0	0
$481$	−19.1701	−0.874081
$482$	0	0
$483$	0.787318	0.0358242
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.9423	−0.586472	−0.293236	−	0.956040i	$-0.594732\pi$
$487$	−12.9423	−0.586472	−0.293236	+	0.956040i	$0.594732\pi$
$488$	0	0
$489$	3.74979	0.169572
$490$	0	0
$491$	−26.8717	−1.21270	−0.606352	−	0.795197i	$-0.707368\pi$
$491$	−26.8717	−1.21270	−0.606352	+	0.795197i	$0.707368\pi$
$492$	0	0
$493$	29.6112	1.33362
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.0489097	−0.00219390
$498$	0	0
$499$	10.5468	0.472139	0.236069	−	0.971736i	$-0.424141\pi$
$499$	10.5468	0.472139	0.236069	+	0.971736i	$0.424141\pi$
$500$	0	0
$501$	−1.24736	−0.0557280
$502$	0	0
$503$	−27.3317	−1.21866	−0.609331	−	0.792916i	$-0.708562\pi$
$503$	−27.3317	−1.21866	−0.609331	+	0.792916i	$0.708562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.05765	0.0469718
$508$	0	0
$509$	8.10186	0.359109	0.179554	−	0.983748i	$-0.442534\pi$
$509$	8.10186	0.359109	0.179554	+	0.983748i	$0.442534\pi$
$510$	0	0
$511$	5.50959	0.243730
$512$	0	0
$513$	−6.55917	−0.289595
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.423887	−0.0186425
$518$	0	0
$519$	−0.285615	−0.0125371
$520$	0	0
$521$	22.6230	0.991129	0.495565	−	0.868571i	$-0.334961\pi$
$521$	22.6230	0.991129	0.495565	+	0.868571i	$0.334961\pi$
$522$	0	0
$523$	10.2820	0.449599	0.224799	−	0.974405i	$-0.427827\pi$
$523$	10.2820	0.449599	0.224799	+	0.974405i	$0.427827\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.9361	−0.911992
$528$	0	0
$529$	58.1301	2.52740
$530$	0	0
$531$	−34.0219	−1.47642
$532$	0	0
$533$	−16.8235	−0.728706
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.51491	−0.151679
$538$	0	0
$539$	18.1153	0.780280
$540$	0	0
$541$	−9.89627	−0.425474	−0.212737	−	0.977110i	$-0.568238\pi$
$541$	−9.89627	−0.425474	−0.212737	+	0.977110i	$0.568238\pi$
$542$	0	0
$543$	1.53534	0.0658876
$544$	0	0
$545$	0	0
$546$	0	0
$547$	39.5682	1.69182	0.845908	−	0.533330i	$-0.179059\pi$
$547$	39.5682	1.69182	0.845908	+	0.533330i	$0.179059\pi$
$548$	0	0
$549$	−38.7392	−1.65335
$550$	0	0
$551$	−40.9349	−1.74389
$552$	0	0
$553$	2.12250	0.0902578
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.4813	−0.655964	−0.327982	−	0.944684i	$-0.606369\pi$
$557$	−15.4813	−0.655964	−0.327982	+	0.944684i	$0.606369\pi$
$558$	0	0
$559$	15.4328	0.652738
$560$	0	0
$561$	2.11429	0.0892656
$562$	0	0
$563$	−37.3755	−1.57519	−0.787595	−	0.616193i	$-0.788674\pi$
$563$	−37.3755	−1.57519	−0.787595	+	0.616193i	$0.788674\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.59936	−0.151159
$568$	0	0
$569$	−22.7464	−0.953578	−0.476789	−	0.879018i	$-0.658199\pi$
$569$	−22.7464	−0.953578	−0.476789	+	0.879018i	$0.658199\pi$
$570$	0	0
$571$	−19.9447	−0.834659	−0.417329	−	0.908755i	$-0.637034\pi$
$571$	−19.9447	−0.834659	−0.417329	+	0.908755i	$0.637034\pi$
$572$	0	0
$573$	1.26052	0.0526591
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.9587	0.664370	0.332185	−	0.943214i	$-0.392214\pi$
$577$	15.9587	0.664370	0.332185	+	0.943214i	$0.392214\pi$
$578$	0	0
$579$	3.29862	0.137086
$580$	0	0
$581$	5.70438	0.236658
$582$	0	0
$583$	−14.2080	−0.588433
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.8655	1.15013	0.575067	−	0.818106i	$-0.304976\pi$
$587$	27.8655	1.15013	0.575067	+	0.818106i	$0.304976\pi$
$588$	0	0
$589$	28.9424	1.19255
$590$	0	0
$591$	−2.45150	−0.100841
$592$	0	0
$593$	44.0453	1.80872	0.904361	−	0.426767i	$-0.140348\pi$
$593$	44.0453	1.80872	0.904361	+	0.426767i	$0.140348\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.10277	0.208842
$598$	0	0
$599$	26.2375	1.07204	0.536018	−	0.844207i	$-0.319928\pi$
$599$	26.2375	1.07204	0.536018	+	0.844207i	$0.319928\pi$
$600$	0	0
$601$	−16.4765	−0.672092	−0.336046	−	0.941846i	$-0.609090\pi$
$601$	−16.4765	−0.672092	−0.336046	+	0.941846i	$0.609090\pi$
$602$	0	0
$603$	−10.0333	−0.408586
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.12486	−0.126834	−0.0634172	−	0.997987i	$-0.520200\pi$
$607$	−3.12486	−0.126834	−0.0634172	+	0.997987i	$0.520200\pi$
$608$	0	0
$609$	0.679274	0.0275256
$610$	0	0
$611$	−0.450042	−0.0182068
$612$	0	0
$613$	−31.8458	−1.28624	−0.643120	−	0.765765i	$-0.722360\pi$
$613$	−31.8458	−1.28624	−0.643120	+	0.765765i	$0.722360\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.5134	0.423253	0.211627	−	0.977351i	$-0.432124\pi$
$617$	10.5134	0.423253	0.211627	+	0.977351i	$0.432124\pi$
$618$	0	0
$619$	−19.9853	−0.803276	−0.401638	−	0.915798i	$-0.631559\pi$
$619$	−19.9853	−0.803276	−0.401638	+	0.915798i	$0.631559\pi$
$620$	0	0
$621$	11.2158	0.450076
$622$	0	0
$623$	−0.0799361	−0.00320257
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.92283	−0.116727
$628$	0	0
$629$	25.9215	1.03356
$630$	0	0
$631$	3.29520	0.131180	0.0655900	−	0.997847i	$-0.479107\pi$
$631$	3.29520	0.131180	0.0655900	+	0.997847i	$0.479107\pi$
$632$	0	0
$633$	3.11811	0.123934
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.2330	0.762041
$638$	0	0
$639$	−0.345818	−0.0136804
$640$	0	0
$641$	7.95401	0.314165	0.157082	−	0.987586i	$-0.449791\pi$
$641$	7.95401	0.314165	0.157082	+	0.987586i	$0.449791\pi$
$642$	0	0
$643$	−41.6422	−1.64221	−0.821105	−	0.570778i	$-0.806642\pi$
$643$	−41.6422	−1.64221	−0.821105	+	0.570778i	$0.806642\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.6053	−1.32116	−0.660580	−	0.750756i	$-0.729689\pi$
$647$	−33.6053	−1.32116	−0.660580	+	0.750756i	$0.729689\pi$
$648$	0	0
$649$	−30.5451	−1.19900
$650$	0	0
$651$	−0.480270	−0.0188233
$652$	0	0
$653$	9.91218	0.387894	0.193947	−	0.981012i	$-0.437871\pi$
$653$	9.91218	0.387894	0.193947	+	0.981012i	$0.437871\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	38.9558	1.51981
$658$	0	0
$659$	19.7305	0.768591	0.384295	−	0.923210i	$-0.374444\pi$
$659$	19.7305	0.768591	0.384295	+	0.923210i	$0.374444\pi$
$660$	0	0
$661$	5.81073	0.226011	0.113006	−	0.993594i	$-0.463952\pi$
$661$	5.81073	0.226011	0.113006	+	0.993594i	$0.463952\pi$
$662$	0	0
$663$	2.24475	0.0871790
$664$	0	0
$665$	0	0
$666$	0	0
$667$	69.9966	2.71028
$668$	0	0
$669$	4.82347	0.186486
$670$	0	0
$671$	−34.7803	−1.34268
$672$	0	0
$673$	−8.43217	−0.325036	−0.162518	−	0.986706i	$-0.551962\pi$
$673$	−8.43217	−0.325036	−0.162518	+	0.986706i	$0.551962\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.4885	−0.479970	−0.239985	−	0.970777i	$-0.577143\pi$
$677$	−12.4885	−0.479970	−0.239985	+	0.970777i	$0.577143\pi$
$678$	0	0
$679$	−3.99813	−0.153434
$680$	0	0
$681$	−2.71681	−0.104108
$682$	0	0
$683$	20.6794	0.791273	0.395637	−	0.918407i	$-0.370524\pi$
$683$	20.6794	0.791273	0.395637	+	0.918407i	$0.370524\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.49830	0.0571636
$688$	0	0
$689$	−15.0846	−0.574679
$690$	0	0
$691$	−6.61763	−0.251747	−0.125873	−	0.992046i	$-0.540173\pi$
$691$	−6.61763	−0.251747	−0.125873	+	0.992046i	$0.540173\pi$
$692$	0	0
$693$	−3.28075	−0.124625
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22.7485	0.861659
$698$	0	0
$699$	−1.32557	−0.0501375
$700$	0	0
$701$	26.6731	1.00743	0.503715	−	0.863870i	$-0.331966\pi$
$701$	26.6731	1.00743	0.503715	+	0.863870i	$0.331966\pi$
$702$	0	0
$703$	−35.8343	−1.35152
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.49472	0.0938237
$708$	0	0
$709$	4.14850	0.155800	0.0779000	−	0.996961i	$-0.475179\pi$
$709$	4.14850	0.155800	0.0779000	+	0.996961i	$0.475179\pi$
$710$	0	0
$711$	15.0072	0.562815
$712$	0	0
$713$	−49.4900	−1.85341
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.79325	−0.141662
$718$	0	0
$719$	7.23607	0.269860	0.134930	−	0.990855i	$-0.456919\pi$
$719$	7.23607	0.269860	0.134930	+	0.990855i	$0.456919\pi$
$720$	0	0
$721$	4.37855	0.163066
$722$	0	0
$723$	6.06775	0.225662
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.4054	1.05350	0.526748	−	0.850021i	$-0.323411\pi$
$727$	28.4054	1.05350	0.526748	+	0.850021i	$0.323411\pi$
$728$	0	0
$729$	−24.6685	−0.913648
$730$	0	0
$731$	−20.8680	−0.771832
$732$	0	0
$733$	18.5903	0.686649	0.343325	−	0.939217i	$-0.388447\pi$
$733$	18.5903	0.686649	0.343325	+	0.939217i	$0.388447\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.00793	−0.331811
$738$	0	0
$739$	−6.04233	−0.222271	−0.111135	−	0.993805i	$-0.535449\pi$
$739$	−6.04233	−0.222271	−0.111135	+	0.993805i	$0.535449\pi$
$740$	0	0
$741$	−3.10318	−0.113998
$742$	0	0
$743$	−25.6176	−0.939820	−0.469910	−	0.882714i	$-0.655714\pi$
$743$	−25.6176	−0.939820	−0.469910	+	0.882714i	$0.655714\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	40.3331	1.47571
$748$	0	0
$749$	3.07400	0.112322
$750$	0	0
$751$	−26.1187	−0.953084	−0.476542	−	0.879152i	$-0.658110\pi$
$751$	−26.1187	−0.953084	−0.476542	+	0.879152i	$0.658110\pi$
$752$	0	0
$753$	−5.50284	−0.200535
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−42.7054	−1.55216	−0.776078	−	0.630637i	$-0.782794\pi$
$757$	−42.7054	−1.55216	−0.776078	+	0.630637i	$0.782794\pi$
$758$	0	0
$759$	4.99789	0.181412
$760$	0	0
$761$	−22.4635	−0.814302	−0.407151	−	0.913361i	$-0.633478\pi$
$761$	−22.4635	−0.814302	−0.407151	+	0.913361i	$0.633478\pi$
$762$	0	0
$763$	−5.37173	−0.194470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.4298	−1.17097
$768$	0	0
$769$	9.24269	0.333300	0.166650	−	0.986016i	$-0.446705\pi$
$769$	9.24269	0.333300	0.166650	+	0.986016i	$0.446705\pi$
$770$	0	0
$771$	5.28528	0.190345
$772$	0	0
$773$	−36.9385	−1.32859	−0.664293	−	0.747473i	$-0.731267\pi$
$773$	−36.9385	−1.32859	−0.664293	+	0.747473i	$0.731267\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.594634	0.0213324
$778$	0	0
$779$	−31.4478	−1.12673
$780$	0	0
$781$	−0.310478	−0.0111098
$782$	0	0
$783$	9.67668	0.345816
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.7137	−0.488841	−0.244421	−	0.969669i	$-0.578598\pi$
$787$	−13.7137	−0.488841	−0.244421	+	0.969669i	$0.578598\pi$
$788$	0	0
$789$	1.68952	0.0601486
$790$	0	0
$791$	−0.000408243 0	−1.45154e−5 0
$792$	0	0
$793$	−36.9264	−1.31130
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.144591	0.00512169	0.00256084	−	0.999997i	$-0.499185\pi$
$797$	0.144591	0.00512169	0.00256084	+	0.999997i	$0.499185\pi$
$798$	0	0
$799$	0.608541	0.0215286
$800$	0	0
$801$	−0.565192	−0.0199701
$802$	0	0
$803$	34.9748	1.23423
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.50934	0.158736
$808$	0	0
$809$	−31.9490	−1.12327	−0.561633	−	0.827386i	$-0.689827\pi$
$809$	−31.9490	−1.12327	−0.561633	+	0.827386i	$0.689827\pi$
$810$	0	0
$811$	−51.8201	−1.81965	−0.909824	−	0.414994i	$-0.863784\pi$
$811$	−51.8201	−1.81965	−0.909824	+	0.414994i	$0.863784\pi$
$812$	0	0
$813$	1.61211	0.0565393
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.8483	1.00927
$818$	0	0
$819$	−3.48318	−0.121712
$820$	0	0
$821$	−31.3045	−1.09254	−0.546268	−	0.837610i	$-0.683952\pi$
$821$	−31.3045	−1.09254	−0.546268	+	0.837610i	$0.683952\pi$
$822$	0	0
$823$	−26.1527	−0.911627	−0.455813	−	0.890075i	$-0.650652\pi$
$823$	−26.1527	−0.911627	−0.455813	+	0.890075i	$0.650652\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.9333	−0.588829	−0.294414	−	0.955678i	$-0.595125\pi$
$827$	−16.9333	−0.588829	−0.294414	+	0.955678i	$0.595125\pi$
$828$	0	0
$829$	8.81795	0.306260	0.153130	−	0.988206i	$-0.451065\pi$
$829$	8.81795	0.306260	0.153130	+	0.988206i	$0.451065\pi$
$830$	0	0
$831$	0.711117	0.0246684
$832$	0	0
$833$	−26.0066	−0.901077
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.84175	−0.236485
$838$	0	0
$839$	−2.43323	−0.0840044	−0.0420022	−	0.999118i	$-0.513374\pi$
$839$	−2.43323	−0.0840044	−0.0420022	+	0.999118i	$0.513374\pi$
$840$	0	0
$841$	31.3909	1.08245
$842$	0	0
$843$	0.112390	0.00387091
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.65377	0.0568243
$848$	0	0
$849$	−2.89005	−0.0991861
$850$	0	0
$851$	61.2747	2.10047
$852$	0	0
$853$	−1.87107	−0.0640642	−0.0320321	−	0.999487i	$-0.510198\pi$
$853$	−1.87107	−0.0640642	−0.0320321	+	0.999487i	$0.510198\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.3285	−0.386976	−0.193488	−	0.981103i	$-0.561980\pi$
$857$	−11.3285	−0.386976	−0.193488	+	0.981103i	$0.561980\pi$
$858$	0	0
$859$	8.58301	0.292849	0.146424	−	0.989222i	$-0.453224\pi$
$859$	8.58301	0.292849	0.146424	+	0.989222i	$0.453224\pi$
$860$	0	0
$861$	0.521844	0.0177844
$862$	0	0
$863$	−0.116569	−0.00396804	−0.00198402	−	0.999998i	$-0.500632\pi$
$863$	−0.116569	−0.00396804	−0.00198402	+	0.999998i	$0.500632\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.518644	0.0176141
$868$	0	0
$869$	13.4736	0.457060
$870$	0	0
$871$	−9.56375	−0.324055
$872$	0	0
$873$	−28.2690	−0.956761
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.6777	0.765773	0.382886	−	0.923795i	$-0.374930\pi$
$877$	22.6777	0.765773	0.382886	+	0.923795i	$0.374930\pi$
$878$	0	0
$879$	6.82111	0.230070
$880$	0	0
$881$	25.1472	0.847230	0.423615	−	0.905842i	$-0.360761\pi$
$881$	25.1472	0.847230	0.423615	+	0.905842i	$0.360761\pi$
$882$	0	0
$883$	2.94000	0.0989389	0.0494695	−	0.998776i	$-0.484247\pi$
$883$	2.94000	0.0989389	0.0494695	+	0.998776i	$0.484247\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.2452	−0.478306	−0.239153	−	0.970982i	$-0.576870\pi$
$887$	−14.2452	−0.478306	−0.239153	+	0.970982i	$0.576870\pi$
$888$	0	0
$889$	8.71032	0.292135
$890$	0	0
$891$	−22.8487	−0.765460
$892$	0	0
$893$	−0.841255	−0.0281515
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.30627	0.177171
$898$	0	0
$899$	−42.6984	−1.42407
$900$	0	0
$901$	20.3972	0.679530
$902$	0	0
$903$	−0.478708	−0.0159304
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.5504	−0.981203	−0.490602	−	0.871384i	$-0.663223\pi$
$907$	−29.5504	−0.981203	−0.490602	+	0.871384i	$0.663223\pi$
$908$	0	0
$909$	17.6391	0.585051
$910$	0	0
$911$	20.8638	0.691250	0.345625	−	0.938373i	$-0.387667\pi$
$911$	20.8638	0.691250	0.345625	+	0.938373i	$0.387667\pi$
$912$	0	0
$913$	36.2114	1.19842
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.267749	−0.00884187
$918$	0	0
$919$	−27.7250	−0.914564	−0.457282	−	0.889322i	$-0.651177\pi$
$919$	−27.7250	−0.914564	−0.457282	+	0.889322i	$0.651177\pi$
$920$	0	0
$921$	−1.36312	−0.0449162
$922$	0	0
$923$	−0.329635	−0.0108501
$924$	0	0
$925$	0	0
$926$	0	0
$927$	30.9587	1.01682
$928$	0	0
$929$	−35.3262	−1.15902	−0.579508	−	0.814966i	$-0.696755\pi$
$929$	−35.3262	−1.15902	−0.579508	+	0.814966i	$0.696755\pi$
$930$	0	0
$931$	35.9519	1.17828
$932$	0	0
$933$	−5.54339	−0.181482
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.1912	−0.920966	−0.460483	−	0.887669i	$-0.652324\pi$
$937$	−28.1912	−0.920966	−0.460483	+	0.887669i	$0.652324\pi$
$938$	0	0
$939$	0.301714	0.00984607
$940$	0	0
$941$	−45.3987	−1.47995	−0.739977	−	0.672632i	$-0.765164\pi$
$941$	−45.3987	−1.47995	−0.739977	+	0.672632i	$0.765164\pi$
$942$	0	0
$943$	53.7741	1.75112
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.56206	−0.0507601	−0.0253801	−	0.999678i	$-0.508080\pi$
$947$	−1.56206	−0.0507601	−0.0253801	+	0.999678i	$0.508080\pi$
$948$	0	0
$949$	37.1329	1.20538
$950$	0	0
$951$	−4.47596	−0.145143
$952$	0	0
$953$	11.4102	0.369612	0.184806	−	0.982775i	$-0.440834\pi$
$953$	11.4102	0.369612	0.184806	+	0.982775i	$0.440834\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.31202	0.139388
$958$	0	0
$959$	6.52957	0.210851
$960$	0	0
$961$	−0.810727	−0.0261525
$962$	0	0
$963$	21.7349	0.700397
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.1350	−0.840445	−0.420222	−	0.907421i	$-0.638048\pi$
$967$	−26.1350	−0.840445	−0.420222	+	0.907421i	$0.638048\pi$
$968$	0	0
$969$	4.19607	0.134797
$970$	0	0
$971$	44.8659	1.43981	0.719907	−	0.694070i	$-0.244184\pi$
$971$	44.8659	1.43981	0.719907	+	0.694070i	$0.244184\pi$
$972$	0	0
$973$	4.36027	0.139784
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.0295	−0.992722	−0.496361	−	0.868116i	$-0.665331\pi$
$977$	−31.0295	−0.992722	−0.496361	+	0.868116i	$0.665331\pi$
$978$	0	0
$979$	−0.507434	−0.0162176
$980$	0	0
$981$	−37.9811	−1.21264
$982$	0	0
$983$	−51.8124	−1.65256	−0.826280	−	0.563260i	$-0.809547\pi$
$983$	−51.8124	−1.65256	−0.826280	+	0.563260i	$0.809547\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.0139598	0.000444345 0
$988$	0	0
$989$	−49.3290	−1.56857
$990$	0	0
$991$	37.2396	1.18296	0.591478	−	0.806321i	$-0.298545\pi$
$991$	37.2396	1.18296	0.591478	+	0.806321i	$0.298545\pi$
$992$	0	0
$993$	−4.15711	−0.131922
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.7918	−1.26022	−0.630109	−	0.776506i	$-0.716990\pi$
$997$	−39.7918	−1.26022	−0.630109	+	0.776506i	$0.716990\pi$
$998$	0	0
$999$	8.47093	0.268008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.w.1.2		4
4.3	odd	2		1250.2.a.e.1.3	✓	4
5.4	even	2		10000.2.a.s.1.3		4
20.3	even	4		1250.2.b.f.1249.7		8
20.7	even	4		1250.2.b.f.1249.2		8
20.19	odd	2		1250.2.a.k.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.2.a.e.1.3	✓	4	4.3	odd	2
1250.2.a.k.1.2	yes	4	20.19	odd	2
1250.2.b.f.1249.2		8	20.7	even	4
1250.2.b.f.1249.7		8	20.3	even	4
10000.2.a.s.1.3		4	5.4	even	2
10000.2.a.w.1.2		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.209057 q^{3} -0.418114 q^{7} -2.95630 q^{9} +O(q^{10})\)	\(q-0.209057 q^{3} -0.418114 q^{7} -2.95630 q^{9} -2.65418 q^{11} -2.81795 q^{13} +3.81040 q^{17} -5.26755 q^{19} +0.0874096 q^{21} +9.00723 q^{23} +1.24520 q^{27} +7.77116 q^{29} -5.49448 q^{31} +0.554875 q^{33} +6.80284 q^{37} +0.589113 q^{39} +5.97010 q^{41} -5.47660 q^{43} +0.159705 q^{47} -6.82518 q^{49} -0.796590 q^{51} +5.35304 q^{53} +1.10122 q^{57} +11.5083 q^{59} +13.1040 q^{61} +1.23607 q^{63} +3.39386 q^{67} -1.88302 q^{69} +0.116977 q^{71} -13.1772 q^{73} +1.10975 q^{77} -5.07636 q^{79} +8.60857 q^{81} -13.6431 q^{83} -1.62461 q^{87} +0.191183 q^{89} +1.17823 q^{91} +1.14866 q^{93} +9.56231 q^{97} +7.84655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^7 - 3 * q^9	\( 4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 6 * q^13 - 7 * q^17 - 15 * q^19 + 18 * q^21 + 6 * q^23 - 5 * q^27 + 10 * q^29 - 8 * q^31 + 13 * q^33 - 12 * q^37 - 24 * q^39 + 3 * q^41 - 14 * q^43 + 2 * q^47 + 8 * q^49 + 7 * q^51 + 4 * q^53 + 10 * q^57 + 20 * q^59 + 18 * q^61 - 4 * q^63 - 23 * q^67 - 16 * q^69 - 8 * q^71 - q^73 + 26 * q^77 - 10 * q^79 - 16 * q^81 - 14 * q^83 - 20 * q^87 + 5 * q^89 - 48 * q^91 - 22 * q^93 - 2 * q^97 + q^99

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