LMFDB - Embedded newform 10000.2.a.k.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +3.00000 q^{11} -5.47214 q^{13} -1.14590 q^{17} +7.23607 q^{19} -1.61803 q^{21} +1.85410 q^{23} -5.00000 q^{27} +6.70820 q^{29} +5.76393 q^{31} +3.00000 q^{33} +6.09017 q^{37} -5.47214 q^{39} -9.70820 q^{41} -12.0902 q^{43} -3.00000 q^{47} -4.38197 q^{49} -1.14590 q^{51} -12.7082 q^{53} +7.23607 q^{57} +6.70820 q^{59} -5.76393 q^{61} +3.23607 q^{63} +2.52786 q^{67} +1.85410 q^{69} +5.56231 q^{71} -3.23607 q^{73} -4.85410 q^{77} -3.94427 q^{79} +1.00000 q^{81} -4.85410 q^{83} +6.70820 q^{87} +13.4164 q^{89} +8.85410 q^{91} +5.76393 q^{93} +5.76393 q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^7 - 4 * q^9	$ 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 9 q^{17} + 10 q^{19} - q^{21} - 3 q^{23} - 10 q^{27} + 16 q^{31} + 6 q^{33} + q^{37} - 2 q^{39} - 6 q^{41} - 13 q^{43} - 6 q^{47} - 11 q^{49} - 9 q^{51} - 12 q^{53} + 10 q^{57} - 16 q^{61} + 2 q^{63} + 14 q^{67} - 3 q^{69} - 9 q^{71} - 2 q^{73} - 3 q^{77} + 10 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{91} + 16 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 - 2 * q^13 - 9 * q^17 + 10 * q^19 - q^21 - 3 * q^23 - 10 * q^27 + 16 * q^31 + 6 * q^33 + q^37 - 2 * q^39 - 6 * q^41 - 13 * q^43 - 6 * q^47 - 11 * q^49 - 9 * q^51 - 12 * q^53 + 10 * q^57 - 16 * q^61 + 2 * q^63 + 14 * q^67 - 3 * q^69 - 9 * q^71 - 2 * q^73 - 3 * q^77 + 10 * q^79 + 2 * q^81 - 3 * q^83 + 11 * q^91 + 16 * q^93 + 16 * q^97 - 12 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.14590	−0.277921	−0.138961	−	0.990298i	$-0.544376\pi$
$17$	−1.14590	−0.277921	−0.138961	+	0.990298i	$0.544376\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	−1.61803	−0.353084
$22$	0	0
$23$	1.85410	0.386607	0.193303	−	0.981139i	$-0.438080\pi$
$23$	1.85410	0.386607	0.193303	+	0.981139i	$0.438080\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	6.70820	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$29$	6.70820	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$30$	0	0
$31$	5.76393	1.03523	0.517616	−	0.855613i	$-0.326819\pi$
$31$	5.76393	1.03523	0.517616	+	0.855613i	$0.326819\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.09017	1.00122	0.500609	−	0.865674i	$-0.333109\pi$
$37$	6.09017	1.00122	0.500609	+	0.865674i	$0.333109\pi$
$38$	0	0
$39$	−5.47214	−0.876243
$40$	0	0
$41$	−9.70820	−1.51617	−0.758083	−	0.652158i	$-0.773864\pi$
$41$	−9.70820	−1.51617	−0.758083	+	0.652158i	$0.773864\pi$
$42$	0	0
$43$	−12.0902	−1.84373	−0.921867	−	0.387507i	$-0.873336\pi$
$43$	−12.0902	−1.84373	−0.921867	+	0.387507i	$0.873336\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	0	0
$51$	−1.14590	−0.160458
$52$	0	0
$53$	−12.7082	−1.74561	−0.872803	−	0.488073i	$-0.837700\pi$
$53$	−12.7082	−1.74561	−0.872803	+	0.488073i	$0.837700\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.23607	0.958441
$58$	0	0
$59$	6.70820	0.873334	0.436667	−	0.899623i	$-0.356159\pi$
$59$	6.70820	0.873334	0.436667	+	0.899623i	$0.356159\pi$
$60$	0	0
$61$	−5.76393	−0.737996	−0.368998	−	0.929430i	$-0.620299\pi$
$61$	−5.76393	−0.737996	−0.368998	+	0.929430i	$0.620299\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.52786	0.308828	0.154414	−	0.988006i	$-0.450651\pi$
$67$	2.52786	0.308828	0.154414	+	0.988006i	$0.450651\pi$
$68$	0	0
$69$	1.85410	0.223208
$70$	0	0
$71$	5.56231	0.660124	0.330062	−	0.943959i	$-0.392930\pi$
$71$	5.56231	0.660124	0.330062	+	0.943959i	$0.392930\pi$
$72$	0	0
$73$	−3.23607	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$73$	−3.23607	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.85410	−0.553176
$78$	0	0
$79$	−3.94427	−0.443765	−0.221883	−	0.975073i	$-0.571220\pi$
$79$	−3.94427	−0.443765	−0.221883	+	0.975073i	$0.571220\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.85410	−0.532807	−0.266403	−	0.963862i	$-0.585835\pi$
$83$	−4.85410	−0.532807	−0.266403	+	0.963862i	$0.585835\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.70820	0.719195
$88$	0	0
$89$	13.4164	1.42214	0.711068	−	0.703123i	$-0.248212\pi$
$89$	13.4164	1.42214	0.711068	+	0.703123i	$0.248212\pi$
$90$	0	0
$91$	8.85410	0.928162
$92$	0	0
$93$	5.76393	0.597692
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.76393	0.585239	0.292619	−	0.956229i	$-0.405473\pi$
$97$	5.76393	0.585239	0.292619	+	0.956229i	$0.405473\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	5.47214	0.539186	0.269593	−	0.962974i	$-0.413111\pi$
$103$	5.47214	0.539186	0.269593	+	0.962974i	$0.413111\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.4164	1.00699	0.503496	−	0.863998i	$-0.332047\pi$
$107$	10.4164	1.00699	0.503496	+	0.863998i	$0.332047\pi$
$108$	0	0
$109$	−13.9443	−1.33562	−0.667810	−	0.744332i	$-0.732768\pi$
$109$	−13.9443	−1.33562	−0.667810	+	0.744332i	$0.732768\pi$
$110$	0	0
$111$	6.09017	0.578053
$112$	0	0
$113$	11.5623	1.08769	0.543845	−	0.839186i	$-0.316968\pi$
$113$	11.5623	1.08769	0.543845	+	0.839186i	$0.316968\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.9443	1.01180
$118$	0	0
$119$	1.85410	0.169965
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−9.70820	−0.875359
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.61803	−0.143577	−0.0717886	−	0.997420i	$-0.522871\pi$
$127$	−1.61803	−0.143577	−0.0717886	+	0.997420i	$0.522871\pi$
$128$	0	0
$129$	−12.0902	−1.06448
$130$	0	0
$131$	−18.7082	−1.63454	−0.817272	−	0.576253i	$-0.804514\pi$
$131$	−18.7082	−1.63454	−0.817272	+	0.576253i	$0.804514\pi$
$132$	0	0
$133$	−11.7082	−1.01523
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.70820	0.829428	0.414714	−	0.909952i	$-0.363882\pi$
$137$	9.70820	0.829428	0.414714	+	0.909952i	$0.363882\pi$
$138$	0	0
$139$	16.1803	1.37240	0.686199	−	0.727414i	$-0.259278\pi$
$139$	16.1803	1.37240	0.686199	+	0.727414i	$0.259278\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−16.4164	−1.37281
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.38197	−0.361418
$148$	0	0
$149$	−2.56231	−0.209912	−0.104956	−	0.994477i	$-0.533470\pi$
$149$	−2.56231	−0.209912	−0.104956	+	0.994477i	$0.533470\pi$
$150$	0	0
$151$	−18.5066	−1.50604	−0.753022	−	0.657995i	$-0.771405\pi$
$151$	−18.5066	−1.50604	−0.753022	+	0.657995i	$0.771405\pi$
$152$	0	0
$153$	2.29180	0.185281
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.562306	0.0448769	0.0224384	−	0.999748i	$-0.492857\pi$
$157$	0.562306	0.0448769	0.0224384	+	0.999748i	$0.492857\pi$
$158$	0	0
$159$	−12.7082	−1.00783
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−1.43769	−0.112609	−0.0563044	−	0.998414i	$-0.517932\pi$
$163$	−1.43769	−0.112609	−0.0563044	+	0.998414i	$0.517932\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.7082	−1.91198	−0.955989	−	0.293402i	$-0.905213\pi$
$167$	−24.7082	−1.91198	−0.955989	+	0.293402i	$0.905213\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	−14.4721	−1.10671
$172$	0	0
$173$	0.708204	0.0538437	0.0269219	−	0.999638i	$-0.491429\pi$
$173$	0.708204	0.0538437	0.0269219	+	0.999638i	$0.491429\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.70820	0.504219
$178$	0	0
$179$	−4.14590	−0.309879	−0.154939	−	0.987924i	$-0.549518\pi$
$179$	−4.14590	−0.309879	−0.154939	+	0.987924i	$0.549518\pi$
$180$	0	0
$181$	−12.7984	−0.951296	−0.475648	−	0.879636i	$-0.657786\pi$
$181$	−12.7984	−0.951296	−0.475648	+	0.879636i	$0.657786\pi$
$182$	0	0
$183$	−5.76393	−0.426082
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.43769	−0.251389
$188$	0	0
$189$	8.09017	0.588473
$190$	0	0
$191$	−14.5623	−1.05369	−0.526846	−	0.849961i	$-0.676625\pi$
$191$	−14.5623	−1.05369	−0.526846	+	0.849961i	$0.676625\pi$
$192$	0	0
$193$	−16.1246	−1.16067	−0.580337	−	0.814376i	$-0.697079\pi$
$193$	−16.1246	−1.16067	−0.580337	+	0.814376i	$0.697079\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.4164	−1.81084	−0.905422	−	0.424513i	$-0.860445\pi$
$197$	−25.4164	−1.81084	−0.905422	+	0.424513i	$0.860445\pi$
$198$	0	0
$199$	−8.09017	−0.573497	−0.286748	−	0.958006i	$-0.592574\pi$
$199$	−8.09017	−0.573497	−0.286748	+	0.958006i	$0.592574\pi$
$200$	0	0
$201$	2.52786	0.178302
$202$	0	0
$203$	−10.8541	−0.761809
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.70820	−0.257738
$208$	0	0
$209$	21.7082	1.50159
$210$	0	0
$211$	−8.90983	−0.613378	−0.306689	−	0.951810i	$-0.599221\pi$
$211$	−8.90983	−0.613378	−0.306689	+	0.951810i	$0.599221\pi$
$212$	0	0
$213$	5.56231	0.381123
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.32624	−0.633106
$218$	0	0
$219$	−3.23607	−0.218673
$220$	0	0
$221$	6.27051	0.421800
$222$	0	0
$223$	8.03444	0.538026	0.269013	−	0.963137i	$-0.413303\pi$
$223$	8.03444	0.538026	0.269013	+	0.963137i	$0.413303\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.56231	−0.369183	−0.184592	−	0.982815i	$-0.559096\pi$
$227$	−5.56231	−0.369183	−0.184592	+	0.982815i	$0.559096\pi$
$228$	0	0
$229$	−24.7984	−1.63872	−0.819361	−	0.573277i	$-0.805672\pi$
$229$	−24.7984	−1.63872	−0.819361	+	0.573277i	$0.805672\pi$
$230$	0	0
$231$	−4.85410	−0.319376
$232$	0	0
$233$	−26.1246	−1.71148	−0.855740	−	0.517406i	$-0.826898\pi$
$233$	−26.1246	−1.71148	−0.855740	+	0.517406i	$0.826898\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.94427	−0.256208
$238$	0	0
$239$	21.7082	1.40419	0.702093	−	0.712085i	$-0.252249\pi$
$239$	21.7082	1.40419	0.702093	+	0.712085i	$0.252249\pi$
$240$	0	0
$241$	4.76393	0.306872	0.153436	−	0.988159i	$-0.450966\pi$
$241$	4.76393	0.306872	0.153436	+	0.988159i	$0.450966\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−39.5967	−2.51948
$248$	0	0
$249$	−4.85410	−0.307616
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	5.56231	0.349699
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.4164	1.02403	0.512014	−	0.858977i	$-0.328900\pi$
$257$	16.4164	1.02403	0.512014	+	0.858977i	$0.328900\pi$
$258$	0	0
$259$	−9.85410	−0.612304
$260$	0	0
$261$	−13.4164	−0.830455
$262$	0	0
$263$	−26.5623	−1.63790	−0.818951	−	0.573863i	$-0.805444\pi$
$263$	−26.5623	−1.63790	−0.818951	+	0.573863i	$0.805444\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.4164	0.821071
$268$	0	0
$269$	4.14590	0.252780	0.126390	−	0.991981i	$-0.459661\pi$
$269$	4.14590	0.252780	0.126390	+	0.991981i	$0.459661\pi$
$270$	0	0
$271$	−14.0344	−0.852532	−0.426266	−	0.904598i	$-0.640171\pi$
$271$	−14.0344	−0.852532	−0.426266	+	0.904598i	$0.640171\pi$
$272$	0	0
$273$	8.85410	0.535875
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.52786	−0.151885	−0.0759423	−	0.997112i	$-0.524196\pi$
$277$	−2.52786	−0.151885	−0.0759423	+	0.997112i	$0.524196\pi$
$278$	0	0
$279$	−11.5279	−0.690155
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	8.43769	0.501569	0.250784	−	0.968043i	$-0.419311\pi$
$283$	8.43769	0.501569	0.250784	+	0.968043i	$0.419311\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.7082	0.927226
$288$	0	0
$289$	−15.6869	−0.922760
$290$	0	0
$291$	5.76393	0.337888
$292$	0	0
$293$	−10.1459	−0.592730	−0.296365	−	0.955075i	$-0.595774\pi$
$293$	−10.1459	−0.592730	−0.296365	+	0.955075i	$0.595774\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	−10.1459	−0.586752
$300$	0	0
$301$	19.5623	1.12755
$302$	0	0
$303$	−3.00000	−0.172345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.618034	0.0352731	0.0176365	−	0.999844i	$-0.494386\pi$
$307$	0.618034	0.0352731	0.0176365	+	0.999844i	$0.494386\pi$
$308$	0	0
$309$	5.47214	0.311299
$310$	0	0
$311$	−22.8541	−1.29594	−0.647969	−	0.761667i	$-0.724381\pi$
$311$	−22.8541	−1.29594	−0.647969	+	0.761667i	$0.724381\pi$
$312$	0	0
$313$	−7.38197	−0.417253	−0.208627	−	0.977995i	$-0.566899\pi$
$313$	−7.38197	−0.417253	−0.208627	+	0.977995i	$0.566899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.5623	−0.817901	−0.408950	−	0.912557i	$-0.634105\pi$
$317$	−14.5623	−0.817901	−0.408950	+	0.912557i	$0.634105\pi$
$318$	0	0
$319$	20.1246	1.12676
$320$	0	0
$321$	10.4164	0.581387
$322$	0	0
$323$	−8.29180	−0.461368
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.9443	−0.771120
$328$	0	0
$329$	4.85410	0.267615
$330$	0	0
$331$	7.67376	0.421788	0.210894	−	0.977509i	$-0.432362\pi$
$331$	7.67376	0.421788	0.210894	+	0.977509i	$0.432362\pi$
$332$	0	0
$333$	−12.1803	−0.667479
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.7426	1.18440	0.592199	−	0.805792i	$-0.298260\pi$
$337$	21.7426	1.18440	0.592199	+	0.805792i	$0.298260\pi$
$338$	0	0
$339$	11.5623	0.627978
$340$	0	0
$341$	17.2918	0.936403
$342$	0	0
$343$	18.4164	0.994393
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.1246	0.919297	0.459649	−	0.888101i	$-0.347975\pi$
$347$	17.1246	0.919297	0.459649	+	0.888101i	$0.347975\pi$
$348$	0	0
$349$	−28.9443	−1.54935	−0.774676	−	0.632359i	$-0.782087\pi$
$349$	−28.9443	−1.54935	−0.774676	+	0.632359i	$0.782087\pi$
$350$	0	0
$351$	27.3607	1.46041
$352$	0	0
$353$	−16.8541	−0.897053	−0.448527	−	0.893769i	$-0.648051\pi$
$353$	−16.8541	−0.897053	−0.448527	+	0.893769i	$0.648051\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.85410	0.0981295
$358$	0	0
$359$	−25.8541	−1.36453	−0.682264	−	0.731106i	$-0.739004\pi$
$359$	−25.8541	−1.36453	−0.682264	+	0.731106i	$0.739004\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.8885	1.29917	0.649586	−	0.760288i	$-0.274942\pi$
$367$	24.8885	1.29917	0.649586	+	0.760288i	$0.274942\pi$
$368$	0	0
$369$	19.4164	1.01078
$370$	0	0
$371$	20.5623	1.06754
$372$	0	0
$373$	−8.43769	−0.436887	−0.218444	−	0.975850i	$-0.570098\pi$
$373$	−8.43769	−0.436887	−0.218444	+	0.975850i	$0.570098\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−36.7082	−1.89057
$378$	0	0
$379$	−8.09017	−0.415564	−0.207782	−	0.978175i	$-0.566624\pi$
$379$	−8.09017	−0.415564	−0.207782	+	0.978175i	$0.566624\pi$
$380$	0	0
$381$	−1.61803	−0.0828944
$382$	0	0
$383$	−4.85410	−0.248033	−0.124017	−	0.992280i	$-0.539578\pi$
$383$	−4.85410	−0.248033	−0.124017	+	0.992280i	$0.539578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.1803	1.22916
$388$	0	0
$389$	9.27051	0.470034	0.235017	−	0.971991i	$-0.424485\pi$
$389$	9.27051	0.470034	0.235017	+	0.971991i	$0.424485\pi$
$390$	0	0
$391$	−2.12461	−0.107446
$392$	0	0
$393$	−18.7082	−0.943704
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.0557	−0.655248	−0.327624	−	0.944808i	$-0.606248\pi$
$397$	−13.0557	−0.655248	−0.327624	+	0.944808i	$0.606248\pi$
$398$	0	0
$399$	−11.7082	−0.586143
$400$	0	0
$401$	−7.14590	−0.356849	−0.178425	−	0.983954i	$-0.557100\pi$
$401$	−7.14590	−0.356849	−0.178425	+	0.983954i	$0.557100\pi$
$402$	0	0
$403$	−31.5410	−1.57117
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.2705	0.905636
$408$	0	0
$409$	−16.1803	−0.800066	−0.400033	−	0.916501i	$-0.631001\pi$
$409$	−16.1803	−0.800066	−0.400033	+	0.916501i	$0.631001\pi$
$410$	0	0
$411$	9.70820	0.478870
$412$	0	0
$413$	−10.8541	−0.534095
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.1803	0.792355
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	−34.1803	−1.66585	−0.832924	−	0.553388i	$-0.813335\pi$
$421$	−34.1803	−1.66585	−0.832924	+	0.553388i	$0.813335\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.32624	0.451328
$428$	0	0
$429$	−16.4164	−0.792592
$430$	0	0
$431$	16.4164	0.790751	0.395375	−	0.918520i	$-0.370615\pi$
$431$	16.4164	0.790751	0.395375	+	0.918520i	$0.370615\pi$
$432$	0	0
$433$	12.7426	0.612372	0.306186	−	0.951972i	$-0.400947\pi$
$433$	12.7426	0.612372	0.306186	+	0.951972i	$0.400947\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.4164	0.641794
$438$	0	0
$439$	3.09017	0.147486	0.0737429	−	0.997277i	$-0.476506\pi$
$439$	3.09017	0.147486	0.0737429	+	0.997277i	$0.476506\pi$
$440$	0	0
$441$	8.76393	0.417330
$442$	0	0
$443$	−27.5410	−1.30851	−0.654257	−	0.756273i	$-0.727018\pi$
$443$	−27.5410	−1.30851	−0.654257	+	0.756273i	$0.727018\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.56231	−0.121193
$448$	0	0
$449$	9.27051	0.437502	0.218751	−	0.975781i	$-0.429802\pi$
$449$	9.27051	0.437502	0.218751	+	0.975781i	$0.429802\pi$
$450$	0	0
$451$	−29.1246	−1.37142
$452$	0	0
$453$	−18.5066	−0.869515
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.4721	−0.536644	−0.268322	−	0.963329i	$-0.586469\pi$
$457$	−11.4721	−0.536644	−0.268322	+	0.963329i	$0.586469\pi$
$458$	0	0
$459$	5.72949	0.267430
$460$	0	0
$461$	6.27051	0.292047	0.146023	−	0.989281i	$-0.453353\pi$
$461$	6.27051	0.292047	0.146023	+	0.989281i	$0.453353\pi$
$462$	0	0
$463$	−20.3820	−0.947230	−0.473615	−	0.880732i	$-0.657051\pi$
$463$	−20.3820	−0.947230	−0.473615	+	0.880732i	$0.657051\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.58359	−0.212103	−0.106052	−	0.994361i	$-0.533821\pi$
$467$	−4.58359	−0.212103	−0.106052	+	0.994361i	$0.533821\pi$
$468$	0	0
$469$	−4.09017	−0.188866
$470$	0	0
$471$	0.562306	0.0259097
$472$	0	0
$473$	−36.2705	−1.66772
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.4164	1.16374
$478$	0	0
$479$	−10.8541	−0.495937	−0.247968	−	0.968768i	$-0.579763\pi$
$479$	−10.8541	−0.495937	−0.247968	+	0.968768i	$0.579763\pi$
$480$	0	0
$481$	−33.3262	−1.51955
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−1.43769	−0.0650148
$490$	0	0
$491$	9.70820	0.438125	0.219063	−	0.975711i	$-0.429700\pi$
$491$	9.70820	0.438125	0.219063	+	0.975711i	$0.429700\pi$
$492$	0	0
$493$	−7.68692	−0.346201
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	7.43769	0.332957	0.166478	−	0.986045i	$-0.446760\pi$
$499$	7.43769	0.332957	0.166478	+	0.986045i	$0.446760\pi$
$500$	0	0
$501$	−24.7082	−1.10388
$502$	0	0
$503$	−18.2705	−0.814642	−0.407321	−	0.913285i	$-0.633537\pi$
$503$	−18.2705	−0.814642	−0.407321	+	0.913285i	$0.633537\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.9443	0.752522
$508$	0	0
$509$	−12.4377	−0.551291	−0.275646	−	0.961259i	$-0.588892\pi$
$509$	−12.4377	−0.551291	−0.275646	+	0.961259i	$0.588892\pi$
$510$	0	0
$511$	5.23607	0.231630
$512$	0	0
$513$	−36.1803	−1.59740
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	0.708204	0.0310867
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−41.1591	−1.79976	−0.899880	−	0.436138i	$-0.856346\pi$
$523$	−41.1591	−1.79976	−0.899880	+	0.436138i	$0.856346\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.60488	−0.287713
$528$	0	0
$529$	−19.5623	−0.850535
$530$	0	0
$531$	−13.4164	−0.582223
$532$	0	0
$533$	53.1246	2.30108
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.14590	−0.178909
$538$	0	0
$539$	−13.1459	−0.566234
$540$	0	0
$541$	22.1246	0.951211	0.475606	−	0.879659i	$-0.342229\pi$
$541$	22.1246	0.951211	0.475606	+	0.879659i	$0.342229\pi$
$542$	0	0
$543$	−12.7984	−0.549231
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−40.2361	−1.72037	−0.860185	−	0.509982i	$-0.829652\pi$
$547$	−40.2361	−1.72037	−0.860185	+	0.509982i	$0.829652\pi$
$548$	0	0
$549$	11.5279	0.491997
$550$	0	0
$551$	48.5410	2.06792
$552$	0	0
$553$	6.38197	0.271389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.2918	0.478449	0.239224	−	0.970964i	$-0.423107\pi$
$557$	11.2918	0.478449	0.239224	+	0.970964i	$0.423107\pi$
$558$	0	0
$559$	66.1591	2.79823
$560$	0	0
$561$	−3.43769	−0.145140
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.61803	−0.0679510
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	21.7426	0.909901	0.454951	−	0.890517i	$-0.349657\pi$
$571$	21.7426	0.909901	0.454951	+	0.890517i	$0.349657\pi$
$572$	0	0
$573$	−14.5623	−0.608349
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.1591	−1.42206	−0.711030	−	0.703162i	$-0.751771\pi$
$577$	−34.1591	−1.42206	−0.711030	+	0.703162i	$0.751771\pi$
$578$	0	0
$579$	−16.1246	−0.670116
$580$	0	0
$581$	7.85410	0.325843
$582$	0	0
$583$	−38.1246	−1.57896
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.4164	1.66816	0.834082	−	0.551641i	$-0.185998\pi$
$587$	40.4164	1.66816	0.834082	+	0.551641i	$0.185998\pi$
$588$	0	0
$589$	41.7082	1.71856
$590$	0	0
$591$	−25.4164	−1.04549
$592$	0	0
$593$	2.29180	0.0941128	0.0470564	−	0.998892i	$-0.485016\pi$
$593$	2.29180	0.0941128	0.0470564	+	0.998892i	$0.485016\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.09017	−0.331109
$598$	0	0
$599$	32.5623	1.33046	0.665230	−	0.746639i	$-0.268334\pi$
$599$	32.5623	1.33046	0.665230	+	0.746639i	$0.268334\pi$
$600$	0	0
$601$	−15.5623	−0.634800	−0.317400	−	0.948292i	$-0.602810\pi$
$601$	−15.5623	−0.634800	−0.317400	+	0.948292i	$0.602810\pi$
$602$	0	0
$603$	−5.05573	−0.205885
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.5066	0.751159	0.375579	−	0.926790i	$-0.377444\pi$
$607$	18.5066	0.751159	0.375579	+	0.926790i	$0.377444\pi$
$608$	0	0
$609$	−10.8541	−0.439830
$610$	0	0
$611$	16.4164	0.664137
$612$	0	0
$613$	−31.6525	−1.27843	−0.639216	−	0.769027i	$-0.720741\pi$
$613$	−31.6525	−1.27843	−0.639216	+	0.769027i	$0.720741\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.2705	0.493992	0.246996	−	0.969016i	$-0.420557\pi$
$617$	12.2705	0.493992	0.246996	+	0.969016i	$0.420557\pi$
$618$	0	0
$619$	2.11146	0.0848666	0.0424333	−	0.999099i	$-0.486489\pi$
$619$	2.11146	0.0848666	0.0424333	+	0.999099i	$0.486489\pi$
$620$	0	0
$621$	−9.27051	−0.372013
$622$	0	0
$623$	−21.7082	−0.869721
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.7082	0.866942
$628$	0	0
$629$	−6.97871	−0.278260
$630$	0	0
$631$	38.6525	1.53873	0.769365	−	0.638809i	$-0.220573\pi$
$631$	38.6525	1.53873	0.769365	+	0.638809i	$0.220573\pi$
$632$	0	0
$633$	−8.90983	−0.354134
$634$	0	0
$635$	0	0
$636$	0	0
$637$	23.9787	0.950071
$638$	0	0
$639$	−11.1246	−0.440083
$640$	0	0
$641$	−36.5410	−1.44328	−0.721642	−	0.692267i	$-0.756612\pi$
$641$	−36.5410	−1.44328	−0.721642	+	0.692267i	$0.756612\pi$
$642$	0	0
$643$	3.23607	0.127618	0.0638090	−	0.997962i	$-0.479675\pi$
$643$	3.23607	0.127618	0.0638090	+	0.997962i	$0.479675\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.1459	0.634761	0.317380	−	0.948298i	$-0.397197\pi$
$647$	16.1459	0.634761	0.317380	+	0.948298i	$0.397197\pi$
$648$	0	0
$649$	20.1246	0.789960
$650$	0	0
$651$	−9.32624	−0.365524
$652$	0	0
$653$	46.6869	1.82700	0.913500	−	0.406838i	$-0.133369\pi$
$653$	46.6869	1.82700	0.913500	+	0.406838i	$0.133369\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.47214	0.252502
$658$	0	0
$659$	−18.5410	−0.722256	−0.361128	−	0.932516i	$-0.617608\pi$
$659$	−18.5410	−0.722256	−0.361128	+	0.932516i	$0.617608\pi$
$660$	0	0
$661$	10.2148	0.397309	0.198654	−	0.980070i	$-0.436343\pi$
$661$	10.2148	0.397309	0.198654	+	0.980070i	$0.436343\pi$
$662$	0	0
$663$	6.27051	0.243526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.4377	0.481589
$668$	0	0
$669$	8.03444	0.310629
$670$	0	0
$671$	−17.2918	−0.667542
$672$	0	0
$673$	6.56231	0.252958	0.126479	−	0.991969i	$-0.459632\pi$
$673$	6.56231	0.252958	0.126479	+	0.991969i	$0.459632\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.5623	−0.559675	−0.279837	−	0.960047i	$-0.590281\pi$
$677$	−14.5623	−0.559675	−0.279837	+	0.960047i	$0.590281\pi$
$678$	0	0
$679$	−9.32624	−0.357908
$680$	0	0
$681$	−5.56231	−0.213148
$682$	0	0
$683$	−22.4164	−0.857740	−0.428870	−	0.903366i	$-0.641088\pi$
$683$	−22.4164	−0.857740	−0.428870	+	0.903366i	$0.641088\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.7984	−0.946117
$688$	0	0
$689$	69.5410	2.64930
$690$	0	0
$691$	9.90983	0.376988	0.188494	−	0.982074i	$-0.439639\pi$
$691$	9.90983	0.376988	0.188494	+	0.982074i	$0.439639\pi$
$692$	0	0
$693$	9.70820	0.368784
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.1246	0.421375
$698$	0	0
$699$	−26.1246	−0.988124
$700$	0	0
$701$	5.29180	0.199868	0.0999342	−	0.994994i	$-0.468137\pi$
$701$	5.29180	0.199868	0.0999342	+	0.994994i	$0.468137\pi$
$702$	0	0
$703$	44.0689	1.66209
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.85410	0.182557
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	7.88854	0.295844
$712$	0	0
$713$	10.6869	0.400228
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.7082	0.810708
$718$	0	0
$719$	8.29180	0.309232	0.154616	−	0.987975i	$-0.450586\pi$
$719$	8.29180	0.309232	0.154616	+	0.987975i	$0.450586\pi$
$720$	0	0
$721$	−8.85410	−0.329744
$722$	0	0
$723$	4.76393	0.177173
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.4721	−1.57520	−0.787602	−	0.616184i	$-0.788678\pi$
$727$	−42.4721	−1.57520	−0.787602	+	0.616184i	$0.788678\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	13.8541	0.512412
$732$	0	0
$733$	−16.1246	−0.595576	−0.297788	−	0.954632i	$-0.596249\pi$
$733$	−16.1246	−0.595576	−0.297788	+	0.954632i	$0.596249\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.58359	0.279345
$738$	0	0
$739$	13.9443	0.512948	0.256474	−	0.966551i	$-0.417439\pi$
$739$	13.9443	0.512948	0.256474	+	0.966551i	$0.417439\pi$
$740$	0	0
$741$	−39.5967	−1.45462
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.70820	0.355205
$748$	0	0
$749$	−16.8541	−0.615835
$750$	0	0
$751$	23.6525	0.863091	0.431546	−	0.902091i	$-0.357968\pi$
$751$	23.6525	0.863091	0.431546	+	0.902091i	$0.357968\pi$
$752$	0	0
$753$	3.00000	0.109326
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.90983	0.360179	0.180089	−	0.983650i	$-0.442361\pi$
$757$	9.90983	0.360179	0.180089	+	0.983650i	$0.442361\pi$
$758$	0	0
$759$	5.56231	0.201899
$760$	0	0
$761$	−35.5623	−1.28913	−0.644566	−	0.764548i	$-0.722962\pi$
$761$	−35.5623	−1.28913	−0.644566	+	0.764548i	$0.722962\pi$
$762$	0	0
$763$	22.5623	0.816810
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.7082	−1.32546
$768$	0	0
$769$	−16.1803	−0.583478	−0.291739	−	0.956498i	$-0.594234\pi$
$769$	−16.1803	−0.583478	−0.291739	+	0.956498i	$0.594234\pi$
$770$	0	0
$771$	16.4164	0.591222
$772$	0	0
$773$	7.41641	0.266750	0.133375	−	0.991066i	$-0.457419\pi$
$773$	7.41641	0.266750	0.133375	+	0.991066i	$0.457419\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.85410	−0.353514
$778$	0	0
$779$	−70.2492	−2.51694
$780$	0	0
$781$	16.6869	0.597105
$782$	0	0
$783$	−33.5410	−1.19866
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.1246	1.85804	0.929021	−	0.370027i	$-0.120652\pi$
$787$	52.1246	1.85804	0.929021	+	0.370027i	$0.120652\pi$
$788$	0	0
$789$	−26.5623	−0.945643
$790$	0	0
$791$	−18.7082	−0.665187
$792$	0	0
$793$	31.5410	1.12005
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.8328	−0.844202	−0.422101	−	0.906549i	$-0.638707\pi$
$797$	−23.8328	−0.844202	−0.422101	+	0.906549i	$0.638707\pi$
$798$	0	0
$799$	3.43769	0.121617
$800$	0	0
$801$	−26.8328	−0.948091
$802$	0	0
$803$	−9.70820	−0.342595
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.14590	0.145943
$808$	0	0
$809$	10.8541	0.381610	0.190805	−	0.981628i	$-0.438890\pi$
$809$	10.8541	0.381610	0.190805	+	0.981628i	$0.438890\pi$
$810$	0	0
$811$	−22.1246	−0.776900	−0.388450	−	0.921470i	$-0.626989\pi$
$811$	−22.1246	−0.776900	−0.388450	+	0.921470i	$0.626989\pi$
$812$	0	0
$813$	−14.0344	−0.492209
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−87.4853	−3.06072
$818$	0	0
$819$	−17.7082	−0.618775
$820$	0	0
$821$	−5.56231	−0.194126	−0.0970629	−	0.995278i	$-0.530945\pi$
$821$	−5.56231	−0.194126	−0.0970629	+	0.995278i	$0.530945\pi$
$822$	0	0
$823$	52.4296	1.82758	0.913790	−	0.406187i	$-0.133142\pi$
$823$	52.4296	1.82758	0.913790	+	0.406187i	$0.133142\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.29180	0.184014	0.0920069	−	0.995758i	$-0.470672\pi$
$827$	5.29180	0.184014	0.0920069	+	0.995758i	$0.470672\pi$
$828$	0	0
$829$	−19.6738	−0.683298	−0.341649	−	0.939828i	$-0.610985\pi$
$829$	−19.6738	−0.683298	−0.341649	+	0.939828i	$0.610985\pi$
$830$	0	0
$831$	−2.52786	−0.0876906
$832$	0	0
$833$	5.02129	0.173977
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−28.8197	−0.996153
$838$	0	0
$839$	38.2918	1.32198	0.660990	−	0.750395i	$-0.270137\pi$
$839$	38.2918	1.32198	0.660990	+	0.750395i	$0.270137\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	0	0
$843$	27.0000	0.929929
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.23607	0.111193
$848$	0	0
$849$	8.43769	0.289581
$850$	0	0
$851$	11.2918	0.387078
$852$	0	0
$853$	−35.7984	−1.22571	−0.612856	−	0.790194i	$-0.709980\pi$
$853$	−35.7984	−1.22571	−0.612856	+	0.790194i	$0.709980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.29180	−0.180764	−0.0903822	−	0.995907i	$-0.528809\pi$
$857$	−5.29180	−0.180764	−0.0903822	+	0.995907i	$0.528809\pi$
$858$	0	0
$859$	12.0344	0.410610	0.205305	−	0.978698i	$-0.434181\pi$
$859$	12.0344	0.410610	0.205305	+	0.978698i	$0.434181\pi$
$860$	0	0
$861$	15.7082	0.535334
$862$	0	0
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.6869	−0.532756
$868$	0	0
$869$	−11.8328	−0.401401
$870$	0	0
$871$	−13.8328	−0.468707
$872$	0	0
$873$	−11.5279	−0.390159
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.6869	0.698548	0.349274	−	0.937021i	$-0.386428\pi$
$877$	20.6869	0.698548	0.349274	+	0.937021i	$0.386428\pi$
$878$	0	0
$879$	−10.1459	−0.342213
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	35.4721	1.19373	0.596866	−	0.802341i	$-0.296412\pi$
$883$	35.4721	1.19373	0.596866	+	0.802341i	$0.296412\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.5410	−1.73058	−0.865289	−	0.501273i	$-0.832865\pi$
$887$	−51.5410	−1.73058	−0.865289	+	0.501273i	$0.832865\pi$
$888$	0	0
$889$	2.61803	0.0878060
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	−21.7082	−0.726437
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.1459	−0.338762
$898$	0	0
$899$	38.6656	1.28957
$900$	0	0
$901$	14.5623	0.485141
$902$	0	0
$903$	19.5623	0.650993
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.1803	0.603668	0.301834	−	0.953360i	$-0.402401\pi$
$907$	18.1803	0.603668	0.301834	+	0.953360i	$0.402401\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	0.437694	0.0145015	0.00725073	−	0.999974i	$-0.497692\pi$
$911$	0.437694	0.0145015	0.00725073	+	0.999974i	$0.497692\pi$
$912$	0	0
$913$	−14.5623	−0.481942
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.2705	0.999620
$918$	0	0
$919$	3.74265	0.123458	0.0617292	−	0.998093i	$-0.480338\pi$
$919$	3.74265	0.123458	0.0617292	+	0.998093i	$0.480338\pi$
$920$	0	0
$921$	0.618034	0.0203649
$922$	0	0
$923$	−30.4377	−1.00187
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.9443	−0.359457
$928$	0	0
$929$	−25.8541	−0.848246	−0.424123	−	0.905605i	$-0.639418\pi$
$929$	−25.8541	−0.848246	−0.424123	+	0.905605i	$0.639418\pi$
$930$	0	0
$931$	−31.7082	−1.03919
$932$	0	0
$933$	−22.8541	−0.748210
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.32624	−0.239338	−0.119669	−	0.992814i	$-0.538183\pi$
$937$	−7.32624	−0.239338	−0.119669	+	0.992814i	$0.538183\pi$
$938$	0	0
$939$	−7.38197	−0.240901
$940$	0	0
$941$	33.7082	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$941$	33.7082	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.85410	0.255224	0.127612	−	0.991824i	$-0.459269\pi$
$947$	7.85410	0.255224	0.127612	+	0.991824i	$0.459269\pi$
$948$	0	0
$949$	17.7082	0.574833
$950$	0	0
$951$	−14.5623	−0.472215
$952$	0	0
$953$	−23.5623	−0.763258	−0.381629	−	0.924316i	$-0.624637\pi$
$953$	−23.5623	−0.763258	−0.381629	+	0.924316i	$0.624637\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.1246	0.650536
$958$	0	0
$959$	−15.7082	−0.507244
$960$	0	0
$961$	2.22291	0.0717069
$962$	0	0
$963$	−20.8328	−0.671328
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.06888	0.195162	0.0975811	−	0.995228i	$-0.468889\pi$
$967$	6.06888	0.195162	0.0975811	+	0.995228i	$0.468889\pi$
$968$	0	0
$969$	−8.29180	−0.266371
$970$	0	0
$971$	37.1459	1.19207	0.596034	−	0.802959i	$-0.296742\pi$
$971$	37.1459	1.19207	0.596034	+	0.802959i	$0.296742\pi$
$972$	0	0
$973$	−26.1803	−0.839303
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.5623	1.13774	0.568869	−	0.822428i	$-0.307381\pi$
$977$	35.5623	1.13774	0.568869	+	0.822428i	$0.307381\pi$
$978$	0	0
$979$	40.2492	1.28637
$980$	0	0
$981$	27.8885	0.890413
$982$	0	0
$983$	48.8115	1.55685	0.778423	−	0.627740i	$-0.216020\pi$
$983$	48.8115	1.55685	0.778423	+	0.627740i	$0.216020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.85410	0.154508
$988$	0	0
$989$	−22.4164	−0.712800
$990$	0	0
$991$	6.09017	0.193461	0.0967303	−	0.995311i	$-0.469162\pi$
$991$	6.09017	0.193461	0.0967303	+	0.995311i	$0.469162\pi$
$992$	0	0
$993$	7.67376	0.243519
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.65248	0.274027	0.137013	−	0.990569i	$-0.456250\pi$
$997$	8.65248	0.274027	0.137013	+	0.990569i	$0.456250\pi$
$998$	0	0
$999$	−30.4508	−0.963422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.k.1.1		2
4.3	odd	2		1250.2.a.c.1.2	yes	2
5.4	even	2		10000.2.a.d.1.2		2
20.3	even	4		1250.2.b.a.1249.1		4
20.7	even	4		1250.2.b.a.1249.4		4
20.19	odd	2		1250.2.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.2.a.b.1.1	✓	2	20.19	odd	2
1250.2.a.c.1.2	yes	2	4.3	odd	2
1250.2.b.a.1249.1		4	20.3	even	4
1250.2.b.a.1249.4		4	20.7	even	4
10000.2.a.d.1.2		2	5.4	even	2
10000.2.a.k.1.1		2	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.61803 q^{7} -2.00000 q^{9} +3.00000 q^{11} -5.47214 q^{13} -1.14590 q^{17} +7.23607 q^{19} -1.61803 q^{21} +1.85410 q^{23} -5.00000 q^{27} +6.70820 q^{29} +5.76393 q^{31} +3.00000 q^{33} +6.09017 q^{37} -5.47214 q^{39} -9.70820 q^{41} -12.0902 q^{43} -3.00000 q^{47} -4.38197 q^{49} -1.14590 q^{51} -12.7082 q^{53} +7.23607 q^{57} +6.70820 q^{59} -5.76393 q^{61} +3.23607 q^{63} +2.52786 q^{67} +1.85410 q^{69} +5.56231 q^{71} -3.23607 q^{73} -4.85410 q^{77} -3.94427 q^{79} +1.00000 q^{81} -4.85410 q^{83} +6.70820 q^{87} +13.4164 q^{89} +8.85410 q^{91} +5.76393 q^{93} +5.76393 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} - 2 q^{13} - 9 q^{17} + 10 q^{19} - q^{21} - 3 q^{23} - 10 q^{27} + 16 q^{31} + 6 q^{33} + q^{37} - 2 q^{39} - 6 q^{41} - 13 q^{43} - 6 q^{47} - 11 q^{49} - 9 q^{51} - 12 q^{53} + 10 q^{57} - 16 q^{61} + 2 q^{63} + 14 q^{67} - 3 q^{69} - 9 q^{71} - 2 q^{73} - 3 q^{77} + 10 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{91} + 16 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 - 2 * q^13 - 9 * q^17 + 10 * q^19 - q^21 - 3 * q^23 - 10 * q^27 + 16 * q^31 + 6 * q^33 + q^37 - 2 * q^39 - 6 * q^41 - 13 * q^43 - 6 * q^47 - 11 * q^49 - 9 * q^51 - 12 * q^53 + 10 * q^57 - 16 * q^61 + 2 * q^63 + 14 * q^67 - 3 * q^69 - 9 * q^71 - 2 * q^73 - 3 * q^77 + 10 * q^79 + 2 * q^81 - 3 * q^83 + 11 * q^91 + 16 * q^93 + 16 * q^97 - 12 * q^99

Introduction

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