LMFDB - Embedded newform 10000.2.a.l.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:	$0$
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 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +5.23607 q^{11} -1.85410 q^{13} +5.23607 q^{17} -0.854102 q^{19} +0.618034 q^{21} +3.76393 q^{23} -5.00000 q^{27} -3.61803 q^{29} +3.00000 q^{31} +5.23607 q^{33} +0.236068 q^{37} -1.85410 q^{39} -0.763932 q^{41} -4.85410 q^{43} +0.618034 q^{47} -6.61803 q^{49} +5.23607 q^{51} +3.47214 q^{53} -0.854102 q^{57} +10.8541 q^{59} +8.70820 q^{61} -1.23607 q^{63} +4.76393 q^{67} +3.76393 q^{69} +6.61803 q^{71} +9.00000 q^{73} +3.23607 q^{77} +8.09017 q^{79} +1.00000 q^{81} -6.23607 q^{83} -3.61803 q^{87} -8.94427 q^{89} -1.14590 q^{91} +3.00000 q^{93} +3.85410 q^{97} -10.4721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 3 q^{13} + 6 q^{17} + 5 q^{19} - q^{21} + 12 q^{23} - 10 q^{27} - 5 q^{29} + 6 q^{31} + 6 q^{33} - 4 q^{37} + 3 q^{39} - 6 q^{41} - 3 q^{43} - q^{47} - 11 q^{49} + 6 q^{51} - 2 q^{53} + 5 q^{57} + 15 q^{59} + 4 q^{61} + 2 q^{63} + 14 q^{67} + 12 q^{69} + 11 q^{71} + 18 q^{73} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{87} - 9 q^{91} + 6 q^{93} + q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 + 3 * q^13 + 6 * q^17 + 5 * q^19 - q^21 + 12 * q^23 - 10 * q^27 - 5 * q^29 + 6 * q^31 + 6 * q^33 - 4 * q^37 + 3 * q^39 - 6 * q^41 - 3 * q^43 - q^47 - 11 * q^49 + 6 * q^51 - 2 * q^53 + 5 * q^57 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 14 * q^67 + 12 * q^69 + 11 * q^71 + 18 * q^73 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 8 * q^83 - 5 * q^87 - 9 * q^91 + 6 * q^93 + 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
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	−1.85410	−0.514235	−0.257118	−	0.966380i	$-0.582773\pi$
$13$	−1.85410	−0.514235	−0.257118	+	0.966380i	$0.582773\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	−0.854102	−0.195944	−0.0979722	−	0.995189i	$-0.531236\pi$
$19$	−0.854102	−0.195944	−0.0979722	+	0.995189i	$0.531236\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	0	0
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−3.61803	−0.671852	−0.335926	−	0.941888i	$-0.609049\pi$
$29$	−3.61803	−0.671852	−0.335926	+	0.941888i	$0.609049\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	5.23607	0.911482
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−1.85410	−0.296894
$40$	0	0
$41$	−0.763932	−0.119306	−0.0596531	−	0.998219i	$-0.518999\pi$
$41$	−0.763932	−0.119306	−0.0596531	+	0.998219i	$0.518999\pi$
$42$	0	0
$43$	−4.85410	−0.740244	−0.370122	−	0.928983i	$-0.620684\pi$
$43$	−4.85410	−0.740244	−0.370122	+	0.928983i	$0.620684\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.618034	0.0901495	0.0450748	−	0.998984i	$-0.485647\pi$
$47$	0.618034	0.0901495	0.0450748	+	0.998984i	$0.485647\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	5.23607	0.733196
$52$	0	0
$53$	3.47214	0.476935	0.238467	−	0.971151i	$-0.423355\pi$
$53$	3.47214	0.476935	0.238467	+	0.971151i	$0.423355\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.854102	−0.113129
$58$	0	0
$59$	10.8541	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$59$	10.8541	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$60$	0	0
$61$	8.70820	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$61$	8.70820	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.76393	0.582007	0.291003	−	0.956722i	$-0.406011\pi$
$67$	4.76393	0.582007	0.291003	+	0.956722i	$0.406011\pi$
$68$	0	0
$69$	3.76393	0.453124
$70$	0	0
$71$	6.61803	0.785416	0.392708	−	0.919663i	$-0.371538\pi$
$71$	6.61803	0.785416	0.392708	+	0.919663i	$0.371538\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	8.09017	0.910215	0.455108	−	0.890436i	$-0.349601\pi$
$79$	8.09017	0.910215	0.455108	+	0.890436i	$0.349601\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.23607	−0.684497	−0.342249	−	0.939609i	$-0.611189\pi$
$83$	−6.23607	−0.684497	−0.342249	+	0.939609i	$0.611189\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.61803	−0.387894
$88$	0	0
$89$	−8.94427	−0.948091	−0.474045	−	0.880500i	$-0.657207\pi$
$89$	−8.94427	−0.948091	−0.474045	+	0.880500i	$0.657207\pi$
$90$	0	0
$91$	−1.14590	−0.120123
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.85410	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$97$	3.85410	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$98$	0	0
$99$	−10.4721	−1.05249
$100$	0	0
$101$	1.47214	0.146483	0.0732415	−	0.997314i	$-0.476666\pi$
$101$	1.47214	0.146483	0.0732415	+	0.997314i	$0.476666\pi$
$102$	0	0
$103$	8.56231	0.843669	0.421835	−	0.906673i	$-0.361386\pi$
$103$	8.56231	0.843669	0.421835	+	0.906673i	$0.361386\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4164	−1.58703	−0.793517	−	0.608548i	$-0.791752\pi$
$107$	−16.4164	−1.58703	−0.793517	+	0.608548i	$0.791752\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0.236068	0.0224066
$112$	0	0
$113$	−16.8541	−1.58550	−0.792750	−	0.609547i	$-0.791352\pi$
$113$	−16.8541	−1.58550	−0.792750	+	0.609547i	$0.791352\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.70820	0.342824
$118$	0	0
$119$	3.23607	0.296650
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	−0.763932	−0.0688814
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.8885	1.76482	0.882411	−	0.470479i	$-0.155919\pi$
$127$	19.8885	1.76482	0.882411	+	0.470479i	$0.155919\pi$
$128$	0	0
$129$	−4.85410	−0.427380
$130$	0	0
$131$	−6.79837	−0.593977	−0.296988	−	0.954881i	$-0.595982\pi$
$131$	−6.79837	−0.593977	−0.296988	+	0.954881i	$0.595982\pi$
$132$	0	0
$133$	−0.527864	−0.0457716
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.9443	1.02047	0.510234	−	0.860036i	$-0.329559\pi$
$137$	11.9443	1.02047	0.510234	+	0.860036i	$0.329559\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0.618034	0.0520479
$142$	0	0
$143$	−9.70820	−0.811841
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.61803	−0.545846
$148$	0	0
$149$	−3.94427	−0.323127	−0.161564	−	0.986862i	$-0.551654\pi$
$149$	−3.94427	−0.323127	−0.161564	+	0.986862i	$0.551654\pi$
$150$	0	0
$151$	−14.5623	−1.18506	−0.592532	−	0.805547i	$-0.701872\pi$
$151$	−14.5623	−1.18506	−0.592532	+	0.805547i	$0.701872\pi$
$152$	0	0
$153$	−10.4721	−0.846622
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.1803	−1.05191	−0.525953	−	0.850514i	$-0.676291\pi$
$157$	−13.1803	−1.05191	−0.525953	+	0.850514i	$0.676291\pi$
$158$	0	0
$159$	3.47214	0.275358
$160$	0	0
$161$	2.32624	0.183333
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.5623	1.12687	0.563433	−	0.826162i	$-0.309480\pi$
$167$	14.5623	1.12687	0.563433	+	0.826162i	$0.309480\pi$
$168$	0	0
$169$	−9.56231	−0.735562
$170$	0	0
$171$	1.70820	0.130630
$172$	0	0
$173$	−18.8885	−1.43607	−0.718035	−	0.696007i	$-0.754958\pi$
$173$	−18.8885	−1.43607	−0.718035	+	0.696007i	$0.754958\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.8541	0.815844
$178$	0	0
$179$	0.527864	0.0394544	0.0197272	−	0.999805i	$-0.493720\pi$
$179$	0.527864	0.0394544	0.0197272	+	0.999805i	$0.493720\pi$
$180$	0	0
$181$	0.291796	0.0216890	0.0108445	−	0.999941i	$-0.496548\pi$
$181$	0.291796	0.0216890	0.0108445	+	0.999941i	$0.496548\pi$
$182$	0	0
$183$	8.70820	0.643729
$184$	0	0
$185$	0	0
$186$	0	0
$187$	27.4164	2.00489
$188$	0	0
$189$	−3.09017	−0.224777
$190$	0	0
$191$	1.81966	0.131666	0.0658330	−	0.997831i	$-0.479030\pi$
$191$	1.81966	0.131666	0.0658330	+	0.997831i	$0.479030\pi$
$192$	0	0
$193$	−7.70820	−0.554849	−0.277424	−	0.960747i	$-0.589481\pi$
$193$	−7.70820	−0.554849	−0.277424	+	0.960747i	$0.589481\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.70820	−0.264199	−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820	−0.264199	−0.132099	+	0.991236i	$0.542172\pi$
$198$	0	0
$199$	17.5623	1.24496	0.622479	−	0.782636i	$-0.286125\pi$
$199$	17.5623	1.24496	0.622479	+	0.782636i	$0.286125\pi$
$200$	0	0
$201$	4.76393	0.336022
$202$	0	0
$203$	−2.23607	−0.156941
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.52786	−0.523223
$208$	0	0
$209$	−4.47214	−0.309344
$210$	0	0
$211$	9.18034	0.632001	0.316000	−	0.948759i	$-0.397660\pi$
$211$	9.18034	0.632001	0.316000	+	0.948759i	$0.397660\pi$
$212$	0	0
$213$	6.61803	0.453460
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.85410	0.125865
$218$	0	0
$219$	9.00000	0.608164
$220$	0	0
$221$	−9.70820	−0.653044
$222$	0	0
$223$	−0.180340	−0.0120765	−0.00603823	−	0.999982i	$-0.501922\pi$
$223$	−0.180340	−0.0120765	−0.00603823	+	0.999982i	$0.501922\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.7639	0.979917	0.489958	−	0.871746i	$-0.337012\pi$
$227$	14.7639	0.979917	0.489958	+	0.871746i	$0.337012\pi$
$228$	0	0
$229$	21.7082	1.43452	0.717259	−	0.696806i	$-0.245396\pi$
$229$	21.7082	1.43452	0.717259	+	0.696806i	$0.245396\pi$
$230$	0	0
$231$	3.23607	0.212918
$232$	0	0
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.09017	0.525513
$238$	0	0
$239$	20.5279	1.32784	0.663919	−	0.747805i	$-0.268892\pi$
$239$	20.5279	1.32784	0.663919	+	0.747805i	$0.268892\pi$
$240$	0	0
$241$	2.52786	0.162834	0.0814170	−	0.996680i	$-0.474055\pi$
$241$	2.52786	0.162834	0.0814170	+	0.996680i	$0.474055\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.58359	0.100762
$248$	0	0
$249$	−6.23607	−0.395195
$250$	0	0
$251$	29.1803	1.84185	0.920923	−	0.389744i	$-0.127436\pi$
$251$	29.1803	1.84185	0.920923	+	0.389744i	$0.127436\pi$
$252$	0	0
$253$	19.7082	1.23904
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.8541	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$257$	−22.8541	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$258$	0	0
$259$	0.145898	0.00906566
$260$	0	0
$261$	7.23607	0.447901
$262$	0	0
$263$	−10.9098	−0.672729	−0.336364	−	0.941732i	$-0.609197\pi$
$263$	−10.9098	−0.672729	−0.336364	+	0.941732i	$0.609197\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.94427	−0.547381
$268$	0	0
$269$	−12.7639	−0.778231	−0.389115	−	0.921189i	$-0.627219\pi$
$269$	−12.7639	−0.778231	−0.389115	+	0.921189i	$0.627219\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−1.14590	−0.0693529
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.7082	1.48457	0.742286	−	0.670083i	$-0.233742\pi$
$277$	24.7082	1.48457	0.742286	+	0.670083i	$0.233742\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	10.0902	0.601929	0.300965	−	0.953635i	$-0.402691\pi$
$281$	10.0902	0.601929	0.300965	+	0.953635i	$0.402691\pi$
$282$	0	0
$283$	−29.8541	−1.77464	−0.887321	−	0.461152i	$-0.847436\pi$
$283$	−29.8541	−1.77464	−0.887321	+	0.461152i	$0.847436\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.472136	−0.0278693
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	3.85410	0.225931
$292$	0	0
$293$	19.5279	1.14083	0.570415	−	0.821357i	$-0.306782\pi$
$293$	19.5279	1.14083	0.570415	+	0.821357i	$0.306782\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−26.1803	−1.51914
$298$	0	0
$299$	−6.97871	−0.403589
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	1.47214	0.0845720
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.23607	0.527130	0.263565	−	0.964642i	$-0.415102\pi$
$307$	9.23607	0.527130	0.263565	+	0.964642i	$0.415102\pi$
$308$	0	0
$309$	8.56231	0.487093
$310$	0	0
$311$	−8.50658	−0.482364	−0.241182	−	0.970480i	$-0.577535\pi$
$311$	−8.50658	−0.482364	−0.241182	+	0.970480i	$0.577535\pi$
$312$	0	0
$313$	16.7639	0.947553	0.473777	−	0.880645i	$-0.342890\pi$
$313$	16.7639	0.947553	0.473777	+	0.880645i	$0.342890\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.65248	−0.429806	−0.214903	−	0.976635i	$-0.568944\pi$
$317$	−7.65248	−0.429806	−0.214903	+	0.976635i	$0.568944\pi$
$318$	0	0
$319$	−18.9443	−1.06068
$320$	0	0
$321$	−16.4164	−0.916275
$322$	0	0
$323$	−4.47214	−0.248836
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0.381966	0.0210585
$330$	0	0
$331$	23.1246	1.27104	0.635522	−	0.772083i	$-0.280785\pi$
$331$	23.1246	1.27104	0.635522	+	0.772083i	$0.280785\pi$
$332$	0	0
$333$	−0.472136	−0.0258729
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.85410	−0.427840	−0.213920	−	0.976851i	$-0.568623\pi$
$337$	−7.85410	−0.427840	−0.213920	+	0.976851i	$0.568623\pi$
$338$	0	0
$339$	−16.8541	−0.915389
$340$	0	0
$341$	15.7082	0.850647
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.9098	−1.06882	−0.534408	−	0.845227i	$-0.679465\pi$
$347$	−19.9098	−1.06882	−0.534408	+	0.845227i	$0.679465\pi$
$348$	0	0
$349$	21.7082	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$349$	21.7082	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$350$	0	0
$351$	9.27051	0.494823
$352$	0	0
$353$	−12.9098	−0.687121	−0.343560	−	0.939131i	$-0.611633\pi$
$353$	−12.9098	−0.687121	−0.343560	+	0.939131i	$0.611633\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.23607	0.171271
$358$	0	0
$359$	−13.7426	−0.725309	−0.362655	−	0.931924i	$-0.618130\pi$
$359$	−13.7426	−0.725309	−0.362655	+	0.931924i	$0.618130\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0	0
$363$	16.4164	0.861638
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.5623	−1.33434	−0.667171	−	0.744905i	$-0.732495\pi$
$367$	−25.5623	−1.33434	−0.667171	+	0.744905i	$0.732495\pi$
$368$	0	0
$369$	1.52786	0.0795374
$370$	0	0
$371$	2.14590	0.111409
$372$	0	0
$373$	28.2705	1.46379	0.731896	−	0.681417i	$-0.238636\pi$
$373$	28.2705	1.46379	0.731896	+	0.681417i	$0.238636\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.70820	0.345490
$378$	0	0
$379$	−14.5967	−0.749785	−0.374892	−	0.927068i	$-0.622320\pi$
$379$	−14.5967	−0.749785	−0.374892	+	0.927068i	$0.622320\pi$
$380$	0	0
$381$	19.8885	1.01892
$382$	0	0
$383$	33.3607	1.70465	0.852326	−	0.523012i	$-0.175192\pi$
$383$	33.3607	1.70465	0.852326	+	0.523012i	$0.175192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.70820	0.493496
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	19.7082	0.996687
$392$	0	0
$393$	−6.79837	−0.342933
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−29.0344	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$397$	−29.0344	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$398$	0	0
$399$	−0.527864	−0.0264263
$400$	0	0
$401$	26.5967	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$401$	26.5967	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$402$	0	0
$403$	−5.56231	−0.277078
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.23607	0.0612696
$408$	0	0
$409$	1.58359	0.0783036	0.0391518	−	0.999233i	$-0.487534\pi$
$409$	1.58359	0.0783036	0.0391518	+	0.999233i	$0.487534\pi$
$410$	0	0
$411$	11.9443	0.589167
$412$	0	0
$413$	6.70820	0.330089
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	9.47214	0.462744	0.231372	−	0.972865i	$-0.425679\pi$
$419$	9.47214	0.462744	0.231372	+	0.972865i	$0.425679\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	−1.23607	−0.0600997
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.38197	0.260452
$428$	0	0
$429$	−9.70820	−0.468717
$430$	0	0
$431$	29.8328	1.43700	0.718498	−	0.695529i	$-0.244830\pi$
$431$	29.8328	1.43700	0.718498	+	0.695529i	$0.244830\pi$
$432$	0	0
$433$	−26.8541	−1.29053	−0.645263	−	0.763961i	$-0.723252\pi$
$433$	−26.8541	−1.29053	−0.645263	+	0.763961i	$0.723252\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.21478	−0.153784
$438$	0	0
$439$	40.9787	1.95581	0.977904	−	0.209056i	$-0.0670391\pi$
$439$	40.9787	1.95581	0.977904	+	0.209056i	$0.0670391\pi$
$440$	0	0
$441$	13.2361	0.630289
$442$	0	0
$443$	29.9443	1.42270	0.711348	−	0.702840i	$-0.248085\pi$
$443$	29.9443	1.42270	0.711348	+	0.702840i	$0.248085\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.94427	−0.186558
$448$	0	0
$449$	4.67376	0.220568	0.110284	−	0.993900i	$-0.464824\pi$
$449$	4.67376	0.220568	0.110284	+	0.993900i	$0.464824\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	0	0
$453$	−14.5623	−0.684197
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.4164	1.00182	0.500909	−	0.865500i	$-0.332999\pi$
$457$	21.4164	1.00182	0.500909	+	0.865500i	$0.332999\pi$
$458$	0	0
$459$	−26.1803	−1.22199
$460$	0	0
$461$	0.819660	0.0381754	0.0190877	−	0.999818i	$-0.493924\pi$
$461$	0.819660	0.0381754	0.0190877	+	0.999818i	$0.493924\pi$
$462$	0	0
$463$	−24.1246	−1.12117	−0.560583	−	0.828098i	$-0.689423\pi$
$463$	−24.1246	−1.12117	−0.560583	+	0.828098i	$0.689423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.4508	1.27027	0.635137	−	0.772400i	$-0.280944\pi$
$467$	27.4508	1.27027	0.635137	+	0.772400i	$0.280944\pi$
$468$	0	0
$469$	2.94427	0.135954
$470$	0	0
$471$	−13.1803	−0.607318
$472$	0	0
$473$	−25.4164	−1.16865
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.94427	−0.317956
$478$	0	0
$479$	10.8541	0.495937	0.247968	−	0.968768i	$-0.420237\pi$
$479$	10.8541	0.495937	0.247968	+	0.968768i	$0.420237\pi$
$480$	0	0
$481$	−0.437694	−0.0199571
$482$	0	0
$483$	2.32624	0.105847
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−36.4164	−1.65018	−0.825092	−	0.564998i	$-0.808877\pi$
$487$	−36.4164	−1.65018	−0.825092	+	0.564998i	$0.808877\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	43.2492	1.95181	0.975905	−	0.218196i	$-0.0700171\pi$
$491$	43.2492	1.95181	0.975905	+	0.218196i	$0.0700171\pi$
$492$	0	0
$493$	−18.9443	−0.853207
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.09017	0.183469
$498$	0	0
$499$	−7.56231	−0.338535	−0.169268	−	0.985570i	$-0.554140\pi$
$499$	−7.56231	−0.338535	−0.169268	+	0.985570i	$0.554140\pi$
$500$	0	0
$501$	14.5623	0.650596
$502$	0	0
$503$	−37.4164	−1.66832	−0.834158	−	0.551526i	$-0.814046\pi$
$503$	−37.4164	−1.66832	−0.834158	+	0.551526i	$0.814046\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.56231	−0.424677
$508$	0	0
$509$	−20.3262	−0.900945	−0.450472	−	0.892790i	$-0.648744\pi$
$509$	−20.3262	−0.900945	−0.450472	+	0.892790i	$0.648744\pi$
$510$	0	0
$511$	5.56231	0.246062
$512$	0	0
$513$	4.27051	0.188548
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.23607	0.142322
$518$	0	0
$519$	−18.8885	−0.829115
$520$	0	0
$521$	29.3607	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$521$	29.3607	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$522$	0	0
$523$	−13.1459	−0.574830	−0.287415	−	0.957806i	$-0.592796\pi$
$523$	−13.1459	−0.574830	−0.287415	+	0.957806i	$0.592796\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.7082	0.684260
$528$	0	0
$529$	−8.83282	−0.384035
$530$	0	0
$531$	−21.7082	−0.942056
$532$	0	0
$533$	1.41641	0.0613514
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.527864	0.0227790
$538$	0	0
$539$	−34.6525	−1.49259
$540$	0	0
$541$	27.1246	1.16618	0.583089	−	0.812408i	$-0.301844\pi$
$541$	27.1246	1.16618	0.583089	+	0.812408i	$0.301844\pi$
$542$	0	0
$543$	0.291796	0.0125222
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−21.2918	−0.910371	−0.455186	−	0.890397i	$-0.650427\pi$
$547$	−21.2918	−0.910371	−0.455186	+	0.890397i	$0.650427\pi$
$548$	0	0
$549$	−17.4164	−0.743314
$550$	0	0
$551$	3.09017	0.131646
$552$	0	0
$553$	5.00000	0.212622
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.76393	−0.201854	−0.100927	−	0.994894i	$-0.532181\pi$
$557$	−4.76393	−0.201854	−0.100927	+	0.994894i	$0.532181\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	27.4164	1.15752
$562$	0	0
$563$	7.38197	0.311113	0.155556	−	0.987827i	$-0.450283\pi$
$563$	7.38197	0.311113	0.155556	+	0.987827i	$0.450283\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.618034	0.0259550
$568$	0	0
$569$	20.5279	0.860573	0.430286	−	0.902692i	$-0.358413\pi$
$569$	20.5279	0.860573	0.430286	+	0.902692i	$0.358413\pi$
$570$	0	0
$571$	8.12461	0.340004	0.170002	−	0.985444i	$-0.445623\pi$
$571$	8.12461	0.340004	0.170002	+	0.985444i	$0.445623\pi$
$572$	0	0
$573$	1.81966	0.0760174
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.7771	1.40616	0.703079	−	0.711111i	$-0.251808\pi$
$577$	33.7771	1.40616	0.703079	+	0.711111i	$0.251808\pi$
$578$	0	0
$579$	−7.70820	−0.320342
$580$	0	0
$581$	−3.85410	−0.159895
$582$	0	0
$583$	18.1803	0.752953
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.29180	0.218416	0.109208	−	0.994019i	$-0.465169\pi$
$587$	5.29180	0.218416	0.109208	+	0.994019i	$0.465169\pi$
$588$	0	0
$589$	−2.56231	−0.105578
$590$	0	0
$591$	−3.70820	−0.152535
$592$	0	0
$593$	10.9098	0.448013	0.224007	−	0.974588i	$-0.428086\pi$
$593$	10.9098	0.448013	0.224007	+	0.974588i	$0.428086\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17.5623	0.718777
$598$	0	0
$599$	9.47214	0.387021	0.193510	−	0.981098i	$-0.438013\pi$
$599$	9.47214	0.387021	0.193510	+	0.981098i	$0.438013\pi$
$600$	0	0
$601$	2.72949	0.111338	0.0556691	−	0.998449i	$-0.482271\pi$
$601$	2.72949	0.111338	0.0556691	+	0.998449i	$0.482271\pi$
$602$	0	0
$603$	−9.52786	−0.388005
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.5623	−1.44343	−0.721715	−	0.692191i	$-0.756646\pi$
$607$	−35.5623	−1.44343	−0.721715	+	0.692191i	$0.756646\pi$
$608$	0	0
$609$	−2.23607	−0.0906100
$610$	0	0
$611$	−1.14590	−0.0463581
$612$	0	0
$613$	14.9787	0.604985	0.302492	−	0.953152i	$-0.402181\pi$
$613$	14.9787	0.604985	0.302492	+	0.953152i	$0.402181\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.2361	−0.573123	−0.286561	−	0.958062i	$-0.592512\pi$
$617$	−14.2361	−0.573123	−0.286561	+	0.958062i	$0.592512\pi$
$618$	0	0
$619$	−30.5279	−1.22702	−0.613509	−	0.789688i	$-0.710243\pi$
$619$	−30.5279	−1.22702	−0.613509	+	0.789688i	$0.710243\pi$
$620$	0	0
$621$	−18.8197	−0.755207
$622$	0	0
$623$	−5.52786	−0.221469
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.47214	−0.178600
$628$	0	0
$629$	1.23607	0.0492853
$630$	0	0
$631$	10.2361	0.407491	0.203746	−	0.979024i	$-0.434688\pi$
$631$	10.2361	0.407491	0.203746	+	0.979024i	$0.434688\pi$
$632$	0	0
$633$	9.18034	0.364886
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.2705	0.486175
$638$	0	0
$639$	−13.2361	−0.523611
$640$	0	0
$641$	−1.09017	−0.0430591	−0.0215296	−	0.999768i	$-0.506854\pi$
$641$	−1.09017	−0.0430591	−0.0215296	+	0.999768i	$0.506854\pi$
$642$	0	0
$643$	−30.8328	−1.21593	−0.607964	−	0.793965i	$-0.708013\pi$
$643$	−30.8328	−1.21593	−0.607964	+	0.793965i	$0.708013\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.5410	−1.43658	−0.718288	−	0.695746i	$-0.755074\pi$
$647$	−36.5410	−1.43658	−0.718288	+	0.695746i	$0.755074\pi$
$648$	0	0
$649$	56.8328	2.23088
$650$	0	0
$651$	1.85410	0.0726680
$652$	0	0
$653$	−19.0902	−0.747056	−0.373528	−	0.927619i	$-0.621852\pi$
$653$	−19.0902	−0.747056	−0.373528	+	0.927619i	$0.621852\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	−15.5279	−0.604880	−0.302440	−	0.953168i	$-0.597801\pi$
$659$	−15.5279	−0.604880	−0.302440	+	0.953168i	$0.597801\pi$
$660$	0	0
$661$	19.6869	0.765732	0.382866	−	0.923804i	$-0.374937\pi$
$661$	19.6869	0.765732	0.382866	+	0.923804i	$0.374937\pi$
$662$	0	0
$663$	−9.70820	−0.377035
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.6180	−0.527292
$668$	0	0
$669$	−0.180340	−0.00697234
$670$	0	0
$671$	45.5967	1.76024
$672$	0	0
$673$	−12.1803	−0.469518	−0.234759	−	0.972054i	$-0.575430\pi$
$673$	−12.1803	−0.469518	−0.234759	+	0.972054i	$0.575430\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.6180	−0.408084	−0.204042	−	0.978962i	$-0.565408\pi$
$677$	−10.6180	−0.408084	−0.204042	+	0.978962i	$0.565408\pi$
$678$	0	0
$679$	2.38197	0.0914115
$680$	0	0
$681$	14.7639	0.565755
$682$	0	0
$683$	−13.4721	−0.515497	−0.257748	−	0.966212i	$-0.582981\pi$
$683$	−13.4721	−0.515497	−0.257748	+	0.966212i	$0.582981\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.7082	0.828220
$688$	0	0
$689$	−6.43769	−0.245257
$690$	0	0
$691$	−36.2705	−1.37980	−0.689898	−	0.723907i	$-0.742344\pi$
$691$	−36.2705	−1.37980	−0.689898	+	0.723907i	$0.742344\pi$
$692$	0	0
$693$	−6.47214	−0.245856
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	2.94427	0.111363
$700$	0	0
$701$	−41.0132	−1.54905	−0.774523	−	0.632546i	$-0.782010\pi$
$701$	−41.0132	−1.54905	−0.774523	+	0.632546i	$0.782010\pi$
$702$	0	0
$703$	−0.201626	−0.00760447
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.909830	0.0342177
$708$	0	0
$709$	−33.5410	−1.25966	−0.629830	−	0.776733i	$-0.716875\pi$
$709$	−33.5410	−1.25966	−0.629830	+	0.776733i	$0.716875\pi$
$710$	0	0
$711$	−16.1803	−0.606810
$712$	0	0
$713$	11.2918	0.422881
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.5279	0.766627
$718$	0	0
$719$	23.2918	0.868637	0.434319	−	0.900759i	$-0.356989\pi$
$719$	23.2918	0.868637	0.434319	+	0.900759i	$0.356989\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	0	0
$723$	2.52786	0.0940123
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.5623	0.910966	0.455483	−	0.890245i	$-0.349467\pi$
$727$	24.5623	0.910966	0.455483	+	0.890245i	$0.349467\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−25.4164	−0.940060
$732$	0	0
$733$	19.9787	0.737931	0.368965	−	0.929443i	$-0.379712\pi$
$733$	19.9787	0.737931	0.368965	+	0.929443i	$0.379712\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.9443	0.918834
$738$	0	0
$739$	−15.9787	−0.587786	−0.293893	−	0.955838i	$-0.594951\pi$
$739$	−15.9787	−0.587786	−0.293893	+	0.955838i	$0.594951\pi$
$740$	0	0
$741$	1.58359	0.0581747
$742$	0	0
$743$	28.3607	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$743$	28.3607	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.4721	0.456332
$748$	0	0
$749$	−10.1459	−0.370723
$750$	0	0
$751$	5.11146	0.186520	0.0932598	−	0.995642i	$-0.470271\pi$
$751$	5.11146	0.186520	0.0932598	+	0.995642i	$0.470271\pi$
$752$	0	0
$753$	29.1803	1.06339
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.4164	−1.10550	−0.552752	−	0.833346i	$-0.686422\pi$
$757$	−30.4164	−1.10550	−0.552752	+	0.833346i	$0.686422\pi$
$758$	0	0
$759$	19.7082	0.715362
$760$	0	0
$761$	−18.4508	−0.668843	−0.334421	−	0.942424i	$-0.608541\pi$
$761$	−18.4508	−0.668843	−0.334421	+	0.942424i	$0.608541\pi$
$762$	0	0
$763$	6.18034	0.223743
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.1246	−0.726658
$768$	0	0
$769$	13.4164	0.483808	0.241904	−	0.970300i	$-0.422228\pi$
$769$	13.4164	0.483808	0.241904	+	0.970300i	$0.422228\pi$
$770$	0	0
$771$	−22.8541	−0.823070
$772$	0	0
$773$	36.1591	1.30055	0.650275	−	0.759699i	$-0.274654\pi$
$773$	36.1591	1.30055	0.650275	+	0.759699i	$0.274654\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.145898	0.00523406
$778$	0	0
$779$	0.652476	0.0233774
$780$	0	0
$781$	34.6525	1.23996
$782$	0	0
$783$	18.0902	0.646490
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−11.8197	−0.421325	−0.210663	−	0.977559i	$-0.567562\pi$
$787$	−11.8197	−0.421325	−0.210663	+	0.977559i	$0.567562\pi$
$788$	0	0
$789$	−10.9098	−0.388400
$790$	0	0
$791$	−10.4164	−0.370365
$792$	0	0
$793$	−16.1459	−0.573358
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.76393	−0.345856	−0.172928	−	0.984934i	$-0.555323\pi$
$797$	−9.76393	−0.345856	−0.172928	+	0.984934i	$0.555323\pi$
$798$	0	0
$799$	3.23607	0.114484
$800$	0	0
$801$	17.8885	0.632061
$802$	0	0
$803$	47.1246	1.66299
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.7639	−0.449312
$808$	0	0
$809$	30.9787	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$809$	30.9787	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$810$	0	0
$811$	14.7082	0.516475	0.258237	−	0.966081i	$-0.416858\pi$
$811$	14.7082	0.516475	0.258237	+	0.966081i	$0.416858\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.14590	0.145047
$818$	0	0
$819$	2.29180	0.0800818
$820$	0	0
$821$	−40.6869	−1.41998	−0.709992	−	0.704210i	$-0.751301\pi$
$821$	−40.6869	−1.41998	−0.709992	+	0.704210i	$0.751301\pi$
$822$	0	0
$823$	47.7082	1.66300	0.831502	−	0.555522i	$-0.187482\pi$
$823$	47.7082	1.66300	0.831502	+	0.555522i	$0.187482\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.965558	−0.0335757	−0.0167879	−	0.999859i	$-0.505344\pi$
$827$	−0.965558	−0.0335757	−0.0167879	+	0.999859i	$0.505344\pi$
$828$	0	0
$829$	−35.8541	−1.24526	−0.622632	−	0.782515i	$-0.713937\pi$
$829$	−35.8541	−1.24526	−0.622632	+	0.782515i	$0.713937\pi$
$830$	0	0
$831$	24.7082	0.857118
$832$	0	0
$833$	−34.6525	−1.20064
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−15.0000	−0.518476
$838$	0	0
$839$	−10.8541	−0.374725	−0.187363	−	0.982291i	$-0.559994\pi$
$839$	−10.8541	−0.374725	−0.187363	+	0.982291i	$0.559994\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	0	0
$843$	10.0902	0.347524
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.1459	0.348617
$848$	0	0
$849$	−29.8541	−1.02459
$850$	0	0
$851$	0.888544	0.0304589
$852$	0	0
$853$	15.3050	0.524032	0.262016	−	0.965064i	$-0.415613\pi$
$853$	15.3050	0.524032	0.262016	+	0.965064i	$0.415613\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.6869	−0.672492	−0.336246	−	0.941774i	$-0.609157\pi$
$857$	−19.6869	−0.672492	−0.336246	+	0.941774i	$0.609157\pi$
$858$	0	0
$859$	1.58359	0.0540315	0.0270157	−	0.999635i	$-0.491400\pi$
$859$	1.58359	0.0540315	0.0270157	+	0.999635i	$0.491400\pi$
$860$	0	0
$861$	−0.472136	−0.0160904
$862$	0	0
$863$	−21.4377	−0.729748	−0.364874	−	0.931057i	$-0.618888\pi$
$863$	−21.4377	−0.729748	−0.364874	+	0.931057i	$0.618888\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.4164	0.353760
$868$	0	0
$869$	42.3607	1.43699
$870$	0	0
$871$	−8.83282	−0.299289
$872$	0	0
$873$	−7.70820	−0.260883
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.5410	1.23390	0.616951	−	0.787001i	$-0.288368\pi$
$877$	36.5410	1.23390	0.616951	+	0.787001i	$0.288368\pi$
$878$	0	0
$879$	19.5279	0.658659
$880$	0	0
$881$	−40.3607	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$881$	−40.3607	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$882$	0	0
$883$	−20.5836	−0.692693	−0.346347	−	0.938107i	$-0.612578\pi$
$883$	−20.5836	−0.692693	−0.346347	+	0.938107i	$0.612578\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.8885	1.00356	0.501780	−	0.864996i	$-0.332679\pi$
$887$	29.8885	1.00356	0.501780	+	0.864996i	$0.332679\pi$
$888$	0	0
$889$	12.2918	0.412254
$890$	0	0
$891$	5.23607	0.175415
$892$	0	0
$893$	−0.527864	−0.0176643
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.97871	−0.233012
$898$	0	0
$899$	−10.8541	−0.362005
$900$	0	0
$901$	18.1803	0.605675
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.2492	−1.10402	−0.552011	−	0.833837i	$-0.686139\pi$
$907$	−33.2492	−1.10402	−0.552011	+	0.833837i	$0.686139\pi$
$908$	0	0
$909$	−2.94427	−0.0976553
$910$	0	0
$911$	40.2361	1.33308	0.666540	−	0.745469i	$-0.267774\pi$
$911$	40.2361	1.33308	0.666540	+	0.745469i	$0.267774\pi$
$912$	0	0
$913$	−32.6525	−1.08064
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.20163	−0.138750
$918$	0	0
$919$	53.2148	1.75539	0.877697	−	0.479216i	$-0.159079\pi$
$919$	53.2148	1.75539	0.877697	+	0.479216i	$0.159079\pi$
$920$	0	0
$921$	9.23607	0.304339
$922$	0	0
$923$	−12.2705	−0.403889
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.1246	−0.562446
$928$	0	0
$929$	41.6312	1.36588	0.682938	−	0.730477i	$-0.260702\pi$
$929$	41.6312	1.36588	0.682938	+	0.730477i	$0.260702\pi$
$930$	0	0
$931$	5.65248	0.185252
$932$	0	0
$933$	−8.50658	−0.278493
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.7295	−0.579197	−0.289599	−	0.957148i	$-0.593522\pi$
$937$	−17.7295	−0.579197	−0.289599	+	0.957148i	$0.593522\pi$
$938$	0	0
$939$	16.7639	0.547070
$940$	0	0
$941$	−46.4164	−1.51313	−0.756566	−	0.653918i	$-0.773124\pi$
$941$	−46.4164	−1.51313	−0.756566	+	0.653918i	$0.773124\pi$
$942$	0	0
$943$	−2.87539	−0.0936355
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.65248	0.0861939	0.0430969	−	0.999071i	$-0.486278\pi$
$947$	2.65248	0.0861939	0.0430969	+	0.999071i	$0.486278\pi$
$948$	0	0
$949$	−16.6869	−0.541680
$950$	0	0
$951$	−7.65248	−0.248149
$952$	0	0
$953$	7.74265	0.250809	0.125404	−	0.992106i	$-0.459977\pi$
$953$	7.74265	0.250809	0.125404	+	0.992106i	$0.459977\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.9443	−0.612381
$958$	0	0
$959$	7.38197	0.238376
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	32.8328	1.05802
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.11146	0.132216	0.0661078	−	0.997812i	$-0.478942\pi$
$967$	4.11146	0.132216	0.0661078	+	0.997812i	$0.478942\pi$
$968$	0	0
$969$	−4.47214	−0.143666
$970$	0	0
$971$	−5.61803	−0.180291	−0.0901456	−	0.995929i	$-0.528733\pi$
$971$	−5.61803	−0.180291	−0.0901456	+	0.995929i	$0.528733\pi$
$972$	0	0
$973$	−3.09017	−0.0990663
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.34752	0.0751040	0.0375520	−	0.999295i	$-0.488044\pi$
$977$	2.34752	0.0751040	0.0375520	+	0.999295i	$0.488044\pi$
$978$	0	0
$979$	−46.8328	−1.49678
$980$	0	0
$981$	−20.0000	−0.638551
$982$	0	0
$983$	9.61803	0.306768	0.153384	−	0.988167i	$-0.450983\pi$
$983$	9.61803	0.306768	0.153384	+	0.988167i	$0.450983\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.381966	0.0121581
$988$	0	0
$989$	−18.2705	−0.580968
$990$	0	0
$991$	15.3607	0.487948	0.243974	−	0.969782i	$-0.421549\pi$
$991$	15.3607	0.487948	0.243974	+	0.969782i	$0.421549\pi$
$992$	0	0
$993$	23.1246	0.733837
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.8885	−0.788228	−0.394114	−	0.919062i	$-0.628949\pi$
$997$	−24.8885	−0.788228	−0.394114	+	0.919062i	$0.628949\pi$
$998$	0	0
$999$	−1.18034	−0.0373443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.l.1.2		2
4.3	odd	2		625.2.a.c.1.2		2
5.4	even	2		10000.2.a.c.1.1		2
12.11	even	2		5625.2.a.d.1.1		2
20.3	even	4		625.2.b.a.624.1		4
20.7	even	4		625.2.b.a.624.4		4
20.19	odd	2		625.2.a.b.1.1		2
25.9	even	10		400.2.u.b.81.1		4
25.14	even	10		400.2.u.b.321.1		4
60.59	even	2		5625.2.a.f.1.2		2
100.3	even	20		625.2.e.c.249.1		8
100.11	odd	10		125.2.d.a.101.1		4
100.19	odd	10		625.2.d.h.251.1		4
100.23	even	20		125.2.e.a.24.1		8
100.27	even	20		125.2.e.a.24.2		8
100.31	odd	10		625.2.d.b.251.1		4
100.39	odd	10		25.2.d.a.21.1	yes	4
100.47	even	20		625.2.e.c.249.2		8
100.59	odd	10		25.2.d.a.6.1	✓	4
100.63	even	20		125.2.e.a.99.2		8
100.67	even	20		625.2.e.c.374.1		8
100.71	odd	10		625.2.d.b.376.1		4
100.79	odd	10		625.2.d.h.376.1		4
100.83	even	20		625.2.e.c.374.2		8
100.87	even	20		125.2.e.a.99.1		8
100.91	odd	10		125.2.d.a.26.1		4
300.59	even	10		225.2.h.b.181.1		4
300.239	even	10		225.2.h.b.46.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.6.1	✓	4	100.59	odd	10
25.2.d.a.21.1	yes	4	100.39	odd	10
125.2.d.a.26.1		4	100.91	odd	10
125.2.d.a.101.1		4	100.11	odd	10
125.2.e.a.24.1		8	100.23	even	20
125.2.e.a.24.2		8	100.27	even	20
125.2.e.a.99.1		8	100.87	even	20
125.2.e.a.99.2		8	100.63	even	20
225.2.h.b.46.1		4	300.239	even	10
225.2.h.b.181.1		4	300.59	even	10
400.2.u.b.81.1		4	25.9	even	10
400.2.u.b.321.1		4	25.14	even	10
625.2.a.b.1.1		2	20.19	odd	2
625.2.a.c.1.2		2	4.3	odd	2
625.2.b.a.624.1		4	20.3	even	4
625.2.b.a.624.4		4	20.7	even	4
625.2.d.b.251.1		4	100.31	odd	10
625.2.d.b.376.1		4	100.71	odd	10
625.2.d.h.251.1		4	100.19	odd	10
625.2.d.h.376.1		4	100.79	odd	10
625.2.e.c.249.1		8	100.3	even	20
625.2.e.c.249.2		8	100.47	even	20
625.2.e.c.374.1		8	100.67	even	20
625.2.e.c.374.2		8	100.83	even	20
5625.2.a.d.1.1		2	12.11	even	2
5625.2.a.f.1.2		2	60.59	even	2
10000.2.a.c.1.1		2	5.4	even	2
10000.2.a.l.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +5.23607 q^{11} -1.85410 q^{13} +5.23607 q^{17} -0.854102 q^{19} +0.618034 q^{21} +3.76393 q^{23} -5.00000 q^{27} -3.61803 q^{29} +3.00000 q^{31} +5.23607 q^{33} +0.236068 q^{37} -1.85410 q^{39} -0.763932 q^{41} -4.85410 q^{43} +0.618034 q^{47} -6.61803 q^{49} +5.23607 q^{51} +3.47214 q^{53} -0.854102 q^{57} +10.8541 q^{59} +8.70820 q^{61} -1.23607 q^{63} +4.76393 q^{67} +3.76393 q^{69} +6.61803 q^{71} +9.00000 q^{73} +3.23607 q^{77} +8.09017 q^{79} +1.00000 q^{81} -6.23607 q^{83} -3.61803 q^{87} -8.94427 q^{89} -1.14590 q^{91} +3.00000 q^{93} +3.85410 q^{97} -10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9	\( 2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} + 3 q^{13} + 6 q^{17} + 5 q^{19} - q^{21} + 12 q^{23} - 10 q^{27} - 5 q^{29} + 6 q^{31} + 6 q^{33} - 4 q^{37} + 3 q^{39} - 6 q^{41} - 3 q^{43} - q^{47} - 11 q^{49} + 6 q^{51} - 2 q^{53} + 5 q^{57} + 15 q^{59} + 4 q^{61} + 2 q^{63} + 14 q^{67} + 12 q^{69} + 11 q^{71} + 18 q^{73} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{87} - 9 q^{91} + 6 q^{93} + q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 + 3 * q^13 + 6 * q^17 + 5 * q^19 - q^21 + 12 * q^23 - 10 * q^27 - 5 * q^29 + 6 * q^31 + 6 * q^33 - 4 * q^37 + 3 * q^39 - 6 * q^41 - 3 * q^43 - q^47 - 11 * q^49 + 6 * q^51 - 2 * q^53 + 5 * q^57 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 14 * q^67 + 12 * q^69 + 11 * q^71 + 18 * q^73 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 8 * q^83 - 5 * q^87 - 9 * q^91 + 6 * q^93 + q^97 - 12 * q^99

Introduction

L-functions

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Database

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