LMFDB - Embedded newform 6003.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.312617.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 $ x^5 - 2x^4 - 5x^3 + 11*x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.26093$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31874 q^{2} -0.260930 q^{4} -2.51371 q^{5} -2.25278 q^{7} -2.98157 q^{8} +O(q^{10})$	\(q+1.31874 q^{2} -0.260930 q^{4} -2.51371 q^{5} -2.25278 q^{7} -2.98157 q^{8} -3.31493 q^{10} +0.206934 q^{11} -0.720644 q^{13} -2.97083 q^{14} -3.41006 q^{16} -2.26093 q^{17} -4.58613 q^{19} +0.655902 q^{20} +0.272892 q^{22} -1.00000 q^{23} +1.31874 q^{25} -0.950341 q^{26} +0.587818 q^{28} -1.00000 q^{29} -0.615304 q^{31} +1.46618 q^{32} -2.98157 q^{34} +5.66284 q^{35} +1.84272 q^{37} -6.04791 q^{38} +7.49481 q^{40} +3.39931 q^{41} -8.50175 q^{43} -0.0539954 q^{44} -1.31874 q^{46} -5.89878 q^{47} -1.92498 q^{49} +1.73907 q^{50} +0.188038 q^{52} -5.61905 q^{53} -0.520173 q^{55} +6.71683 q^{56} -1.31874 q^{58} -1.23398 q^{59} +9.37442 q^{61} -0.811425 q^{62} +8.75362 q^{64} +1.81149 q^{65} -9.43319 q^{67} +0.589944 q^{68} +7.46780 q^{70} -1.35812 q^{71} -1.27374 q^{73} +2.43007 q^{74} +1.19666 q^{76} -0.466178 q^{77} +9.36077 q^{79} +8.57189 q^{80} +4.48280 q^{82} +13.4640 q^{83} +5.68332 q^{85} -11.2116 q^{86} -0.616990 q^{88} +3.61318 q^{89} +1.62345 q^{91} +0.260930 q^{92} -7.77895 q^{94} +11.5282 q^{95} +1.72542 q^{97} -2.53855 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * q^98

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 $

$2$	1.31874	0.932489	0.466244	−	0.884656i	$-0.345607\pi$
$2$	1.31874	0.932489	0.466244	+	0.884656i	$0.345607\pi$
$3$	0	0
$3$	0	0
$4$	−0.260930	−0.130465
$5$	−2.51371	−1.12417	−0.562083	−	0.827081i	$-0.690000\pi$
$5$	−2.51371	−1.12417	−0.562083	+	0.827081i	$0.690000\pi$
$6$	0	0
$7$	−2.25278	−0.851471	−0.425735	−	0.904848i	$-0.639985\pi$
$7$	−2.25278	−0.851471	−0.425735	+	0.904848i	$0.639985\pi$
$8$	−2.98157	−1.05415
$9$	0	0
$10$	−3.31493	−1.04827
$11$	0.206934	0.0623931	0.0311965	−	0.999513i	$-0.490068\pi$
$11$	0.206934	0.0623931	0.0311965	+	0.999513i	$0.490068\pi$
$12$	0	0
$13$	−0.720644	−0.199871	−0.0999354	−	0.994994i	$-0.531864\pi$
$13$	−0.720644	−0.199871	−0.0999354	+	0.994994i	$0.531864\pi$
$14$	−2.97083	−0.793987
$15$	0	0
$16$	−3.41006	−0.852514
$17$	−2.26093	−0.548356	−0.274178	−	0.961679i	$-0.588406\pi$
$17$	−2.26093	−0.548356	−0.274178	+	0.961679i	$0.588406\pi$
$18$	0	0
$19$	−4.58613	−1.05213	−0.526065	−	0.850444i	$-0.676333\pi$
$19$	−4.58613	−1.05213	−0.526065	+	0.850444i	$0.676333\pi$
$20$	0.655902	0.146664
$21$	0	0
$22$	0.272892	0.0581808
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.31874	0.263748
$26$	−0.950341	−0.186377
$27$	0	0
$28$	0.587818	0.111087
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.615304	−0.110512	−0.0552559	−	0.998472i	$-0.517597\pi$
$31$	−0.615304	−0.110512	−0.0552559	+	0.998472i	$0.517597\pi$
$32$	1.46618	0.259186
$33$	0	0
$34$	−2.98157	−0.511336
$35$	5.66284	0.957194
$36$	0	0
$37$	1.84272	0.302942	0.151471	−	0.988462i	$-0.451599\pi$
$37$	1.84272	0.302942	0.151471	+	0.988462i	$0.451599\pi$
$38$	−6.04791	−0.981100
$39$	0	0
$40$	7.49481	1.18503
$41$	3.39931	0.530883	0.265441	−	0.964127i	$-0.414482\pi$
$41$	3.39931	0.530883	0.265441	+	0.964127i	$0.414482\pi$
$42$	0	0
$43$	−8.50175	−1.29650	−0.648252	−	0.761426i	$-0.724500\pi$
$43$	−8.50175	−1.29650	−0.648252	+	0.761426i	$0.724500\pi$
$44$	−0.0539954	−0.00814011
$45$	0	0
$46$	−1.31874	−0.194437
$47$	−5.89878	−0.860425	−0.430213	−	0.902728i	$-0.641561\pi$
$47$	−5.89878	−0.860425	−0.430213	+	0.902728i	$0.641561\pi$
$48$	0	0
$49$	−1.92498	−0.274997
$50$	1.73907	0.245942
$51$	0	0
$52$	0.188038	0.0260761
$53$	−5.61905	−0.771836	−0.385918	−	0.922533i	$-0.626115\pi$
$53$	−5.61905	−0.771836	−0.385918	+	0.922533i	$0.626115\pi$
$54$	0	0
$55$	−0.520173	−0.0701401
$56$	6.71683	0.897574
$57$	0	0
$58$	−1.31874	−0.173159
$59$	−1.23398	−0.160651	−0.0803253	−	0.996769i	$-0.525596\pi$
$59$	−1.23398	−0.160651	−0.0803253	+	0.996769i	$0.525596\pi$
$60$	0	0
$61$	9.37442	1.20027	0.600136	−	0.799898i	$-0.295113\pi$
$61$	9.37442	1.20027	0.600136	+	0.799898i	$0.295113\pi$
$62$	−0.811425	−0.103051
$63$	0	0
$64$	8.75362	1.09420
$65$	1.81149	0.224688
$66$	0	0
$67$	−9.43319	−1.15245	−0.576224	−	0.817292i	$-0.695474\pi$
$67$	−9.43319	−1.15245	−0.576224	+	0.817292i	$0.695474\pi$
$68$	0.589944	0.0715412
$69$	0	0
$70$	7.46780	0.892573
$71$	−1.35812	−0.161179	−0.0805896	−	0.996747i	$-0.525680\pi$
$71$	−1.35812	−0.161179	−0.0805896	+	0.996747i	$0.525680\pi$
$72$	0	0
$73$	−1.27374	−0.149080	−0.0745398	−	0.997218i	$-0.523749\pi$
$73$	−1.27374	−0.149080	−0.0745398	+	0.997218i	$0.523749\pi$
$74$	2.43007	0.282490
$75$	0	0
$76$	1.19666	0.137266
$77$	−0.466178	−0.0531259
$78$	0	0
$79$	9.36077	1.05317	0.526585	−	0.850123i	$-0.323472\pi$
$79$	9.36077	1.05317	0.526585	+	0.850123i	$0.323472\pi$
$80$	8.57189	0.958367
$81$	0	0
$82$	4.48280	0.495042
$83$	13.4640	1.47786	0.738932	−	0.673780i	$-0.235330\pi$
$83$	13.4640	1.47786	0.738932	+	0.673780i	$0.235330\pi$
$84$	0	0
$85$	5.68332	0.616443
$86$	−11.2116	−1.20898
$87$	0	0
$88$	−0.616990	−0.0657714
$89$	3.61318	0.382996	0.191498	−	0.981493i	$-0.438665\pi$
$89$	3.61318	0.382996	0.191498	+	0.981493i	$0.438665\pi$
$90$	0	0
$91$	1.62345	0.170184
$92$	0.260930	0.0272038
$93$	0	0
$94$	−7.77895	−0.802337
$95$	11.5282	1.18277
$96$	0	0
$97$	1.72542	0.175190	0.0875950	−	0.996156i	$-0.472082\pi$
$97$	1.72542	0.175190	0.0875950	+	0.996156i	$0.472082\pi$
$98$	−2.53855	−0.256432
$99$	0	0
$100$	−0.344098	−0.0344098
$101$	5.34150	0.531499	0.265750	−	0.964042i	$-0.414381\pi$
$101$	5.34150	0.531499	0.265750	+	0.964042i	$0.414381\pi$
$102$	0	0
$103$	−8.03732	−0.791941	−0.395970	−	0.918263i	$-0.629592\pi$
$103$	−8.03732	−0.791941	−0.395970	+	0.918263i	$0.629592\pi$
$104$	2.14866	0.210693
$105$	0	0
$106$	−7.41006	−0.719728
$107$	3.51280	0.339595	0.169798	−	0.985479i	$-0.445689\pi$
$107$	3.51280	0.339595	0.169798	+	0.985479i	$0.445689\pi$
$108$	0	0
$109$	18.5175	1.77366	0.886828	−	0.462100i	$-0.152904\pi$
$109$	18.5175	1.77366	0.886828	+	0.462100i	$0.152904\pi$
$110$	−0.685972	−0.0654049
$111$	0	0
$112$	7.68211	0.725891
$113$	17.2446	1.62224	0.811119	−	0.584882i	$-0.198859\pi$
$113$	17.2446	1.62224	0.811119	+	0.584882i	$0.198859\pi$
$114$	0	0
$115$	2.51371	0.234405
$116$	0.260930	0.0242267
$117$	0	0
$118$	−1.62730	−0.149805
$119$	5.09338	0.466909
$120$	0	0
$121$	−10.9572	−0.996107
$122$	12.3624	1.11924
$123$	0	0
$124$	0.160551	0.0144179
$125$	9.25362	0.827669
$126$	0	0
$127$	14.4331	1.28073	0.640364	−	0.768072i	$-0.278783\pi$
$127$	14.4331	1.28073	0.640364	+	0.768072i	$0.278783\pi$
$128$	8.61137	0.761145
$129$	0	0
$130$	2.38888	0.209519
$131$	9.95272	0.869573	0.434787	−	0.900533i	$-0.356824\pi$
$131$	9.95272	0.869573	0.434787	+	0.900533i	$0.356824\pi$
$132$	0	0
$133$	10.3315	0.895859
$134$	−12.4399	−1.07464
$135$	0	0
$136$	6.74113	0.578047
$137$	21.7145	1.85519	0.927597	−	0.373582i	$-0.121871\pi$
$137$	21.7145	1.85519	0.927597	+	0.373582i	$0.121871\pi$
$138$	0	0
$139$	−9.64007	−0.817660	−0.408830	−	0.912610i	$-0.634063\pi$
$139$	−9.64007	−0.817660	−0.408830	+	0.912610i	$0.634063\pi$
$140$	−1.47760	−0.124880
$141$	0	0
$142$	−1.79101	−0.150298
$143$	−0.149126	−0.0124706
$144$	0	0
$145$	2.51371	0.208752
$146$	−1.67973	−0.139015
$147$	0	0
$148$	−0.480822	−0.0395233
$149$	−3.81234	−0.312319	−0.156159	−	0.987732i	$-0.549911\pi$
$149$	−3.81234	−0.312319	−0.156159	+	0.987732i	$0.549911\pi$
$150$	0	0
$151$	−24.1476	−1.96511	−0.982553	−	0.185983i	$-0.940453\pi$
$151$	−24.1476	−1.96511	−0.982553	+	0.185983i	$0.940453\pi$
$152$	13.6739	1.10910
$153$	0	0
$154$	−0.614766	−0.0495393
$155$	1.54670	0.124234
$156$	0	0
$157$	13.7130	1.09442	0.547209	−	0.836996i	$-0.315690\pi$
$157$	13.7130	1.09442	0.547209	+	0.836996i	$0.315690\pi$
$158$	12.3444	0.982068
$159$	0	0
$160$	−3.68555	−0.291368
$161$	2.25278	0.177544
$162$	0	0
$163$	1.12483	0.0881033	0.0440516	−	0.999029i	$-0.485973\pi$
$163$	1.12483	0.0881033	0.0440516	+	0.999029i	$0.485973\pi$
$164$	−0.886981	−0.0692616
$165$	0	0
$166$	17.7555	1.37809
$167$	−7.17952	−0.555568	−0.277784	−	0.960644i	$-0.589600\pi$
$167$	−7.17952	−0.555568	−0.277784	+	0.960644i	$0.589600\pi$
$168$	0	0
$169$	−12.4807	−0.960052
$170$	7.49481	0.574826
$171$	0	0
$172$	2.21836	0.169148
$173$	−3.39322	−0.257982	−0.128991	−	0.991646i	$-0.541174\pi$
$173$	−3.39322	−0.257982	−0.128991	+	0.991646i	$0.541174\pi$
$174$	0	0
$175$	−2.97083	−0.224573
$176$	−0.705658	−0.0531910
$177$	0	0
$178$	4.76483	0.357139
$179$	12.3767	0.925078	0.462539	−	0.886599i	$-0.346939\pi$
$179$	12.3767	0.925078	0.462539	+	0.886599i	$0.346939\pi$
$180$	0	0
$181$	−21.5325	−1.60050	−0.800251	−	0.599666i	$-0.795300\pi$
$181$	−21.5325	−1.60050	−0.800251	+	0.599666i	$0.795300\pi$
$182$	2.14091	0.158695
$183$	0	0
$184$	2.98157	0.219805
$185$	−4.63207	−0.340557
$186$	0	0
$187$	−0.467864	−0.0342136
$188$	1.53917	0.112255
$189$	0	0
$190$	15.2027	1.10292
$191$	18.6497	1.34944	0.674722	−	0.738072i	$-0.264263\pi$
$191$	18.6497	1.34944	0.674722	+	0.738072i	$0.264263\pi$
$192$	0	0
$193$	−8.16030	−0.587391	−0.293696	−	0.955899i	$-0.594885\pi$
$193$	−8.16030	−0.587391	−0.293696	+	0.955899i	$0.594885\pi$
$194$	2.27538	0.163363
$195$	0	0
$196$	0.502285	0.0358775
$197$	−13.5299	−0.963969	−0.481984	−	0.876180i	$-0.660084\pi$
$197$	−13.5299	−0.963969	−0.481984	+	0.876180i	$0.660084\pi$
$198$	0	0
$199$	−3.83619	−0.271941	−0.135970	−	0.990713i	$-0.543415\pi$
$199$	−3.83619	−0.271941	−0.135970	+	0.990713i	$0.543415\pi$
$200$	−3.93192	−0.278028
$201$	0	0
$202$	7.04404	0.495617
$203$	2.25278	0.158114
$204$	0	0
$205$	−8.54488	−0.596800
$206$	−10.5991	−0.738476
$207$	0	0
$208$	2.45744	0.170393
$209$	−0.949029	−0.0656457
$210$	0	0
$211$	−18.0641	−1.24359	−0.621793	−	0.783181i	$-0.713596\pi$
$211$	−18.0641	−1.24359	−0.621793	+	0.783181i	$0.713596\pi$
$212$	1.46618	0.100698
$213$	0	0
$214$	4.63246	0.316669
$215$	21.3709	1.45749
$216$	0	0
$217$	1.38614	0.0940976
$218$	24.4197	1.65391
$219$	0	0
$220$	0.135729	0.00915083
$221$	1.62933	0.109600
$222$	0	0
$223$	5.81999	0.389736	0.194868	−	0.980830i	$-0.437572\pi$
$223$	5.81999	0.389736	0.194868	+	0.980830i	$0.437572\pi$
$224$	−3.30298	−0.220689
$225$	0	0
$226$	22.7411	1.51272
$227$	15.3483	1.01870	0.509352	−	0.860558i	$-0.329885\pi$
$227$	15.3483	1.01870	0.509352	+	0.860558i	$0.329885\pi$
$228$	0	0
$229$	−15.8178	−1.04527	−0.522634	−	0.852557i	$-0.675050\pi$
$229$	−15.8178	−1.04527	−0.522634	+	0.852557i	$0.675050\pi$
$230$	3.31493	0.218580
$231$	0	0
$232$	2.98157	0.195750
$233$	−10.8064	−0.707951	−0.353975	−	0.935255i	$-0.615170\pi$
$233$	−10.8064	−0.707951	−0.353975	+	0.935255i	$0.615170\pi$
$234$	0	0
$235$	14.8278	0.967260
$236$	0.321982	0.0209593
$237$	0	0
$238$	6.71683	0.435388
$239$	12.4863	0.807670	0.403835	−	0.914832i	$-0.367677\pi$
$239$	12.4863	0.807670	0.403835	+	0.914832i	$0.367677\pi$
$240$	0	0
$241$	−3.94214	−0.253935	−0.126968	−	0.991907i	$-0.540524\pi$
$241$	−3.94214	−0.253935	−0.126968	+	0.991907i	$0.540524\pi$
$242$	−14.4496	−0.928859
$243$	0	0
$244$	−2.44607	−0.156593
$245$	4.83884	0.309142
$246$	0	0
$247$	3.30497	0.210290
$248$	1.83457	0.116496
$249$	0	0
$250$	12.2031	0.771792
$251$	−6.79990	−0.429206	−0.214603	−	0.976701i	$-0.568846\pi$
$251$	−6.79990	−0.429206	−0.214603	+	0.976701i	$0.568846\pi$
$252$	0	0
$253$	−0.206934	−0.0130099
$254$	19.0334	1.19426
$255$	0	0
$256$	−6.15109	−0.384443
$257$	25.4934	1.59023	0.795116	−	0.606457i	$-0.207410\pi$
$257$	25.4934	1.59023	0.795116	+	0.606457i	$0.207410\pi$
$258$	0	0
$259$	−4.15125	−0.257946
$260$	−0.472672	−0.0293139
$261$	0	0
$262$	13.1250	0.810867
$263$	−5.87654	−0.362363	−0.181181	−	0.983450i	$-0.557992\pi$
$263$	−5.87654	−0.362363	−0.181181	+	0.983450i	$0.557992\pi$
$264$	0	0
$265$	14.1247	0.867671
$266$	13.6246	0.835378
$267$	0	0
$268$	2.46140	0.150354
$269$	3.61181	0.220216	0.110108	−	0.993920i	$-0.464880\pi$
$269$	3.61181	0.220216	0.110108	+	0.993920i	$0.464880\pi$
$270$	0	0
$271$	−23.1450	−1.40596	−0.702980	−	0.711209i	$-0.748148\pi$
$271$	−23.1450	−1.40596	−0.702980	+	0.711209i	$0.748148\pi$
$272$	7.70990	0.467481
$273$	0	0
$274$	28.6357	1.72995
$275$	0.272892	0.0164560
$276$	0	0
$277$	15.9558	0.958693	0.479346	−	0.877626i	$-0.340874\pi$
$277$	15.9558	0.958693	0.479346	+	0.877626i	$0.340874\pi$
$278$	−12.7127	−0.762459
$279$	0	0
$280$	−16.8842	−1.00902
$281$	3.25850	0.194386	0.0971928	−	0.995266i	$-0.469014\pi$
$281$	3.25850	0.194386	0.0971928	+	0.995266i	$0.469014\pi$
$282$	0	0
$283$	7.75465	0.460966	0.230483	−	0.973076i	$-0.425969\pi$
$283$	7.75465	0.460966	0.230483	+	0.973076i	$0.425969\pi$
$284$	0.354374	0.0210282
$285$	0	0
$286$	−0.196658	−0.0116287
$287$	−7.65790	−0.452031
$288$	0	0
$289$	−11.8882	−0.699306
$290$	3.31493	0.194659
$291$	0	0
$292$	0.332356	0.0194497
$293$	−23.5594	−1.37636	−0.688178	−	0.725541i	$-0.741589\pi$
$293$	−23.5594	−1.37636	−0.688178	+	0.725541i	$0.741589\pi$
$294$	0	0
$295$	3.10187	0.180598
$296$	−5.49422	−0.319345
$297$	0	0
$298$	−5.02747	−0.291234
$299$	0.720644	0.0416759
$300$	0	0
$301$	19.1526	1.10394
$302$	−31.8444	−1.83244
$303$	0	0
$304$	15.6390	0.896956
$305$	−23.5646	−1.34930
$306$	0	0
$307$	11.0124	0.628509	0.314254	−	0.949339i	$-0.398245\pi$
$307$	11.0124	0.628509	0.314254	+	0.949339i	$0.398245\pi$
$308$	0.121640	0.00693107
$309$	0	0
$310$	2.03969	0.115846
$311$	4.46225	0.253031	0.126515	−	0.991965i	$-0.459621\pi$
$311$	4.46225	0.253031	0.126515	+	0.991965i	$0.459621\pi$
$312$	0	0
$313$	11.9157	0.673516	0.336758	−	0.941591i	$-0.390670\pi$
$313$	11.9157	0.673516	0.336758	+	0.941591i	$0.390670\pi$
$314$	18.0839	1.02053
$315$	0	0
$316$	−2.44250	−0.137402
$317$	2.70900	0.152153	0.0760764	−	0.997102i	$-0.475761\pi$
$317$	2.70900	0.152153	0.0760764	+	0.997102i	$0.475761\pi$
$318$	0	0
$319$	−0.206934	−0.0115861
$320$	−22.0041	−1.23006
$321$	0	0
$322$	2.97083	0.165558
$323$	10.3689	0.576942
$324$	0	0
$325$	−0.950341	−0.0527155
$326$	1.48335	0.0821553
$327$	0	0
$328$	−10.1353	−0.559628
$329$	13.2887	0.732627
$330$	0	0
$331$	−10.9224	−0.600349	−0.300175	−	0.953884i	$-0.597045\pi$
$331$	−10.9224	−0.600349	−0.300175	+	0.953884i	$0.597045\pi$
$332$	−3.51316	−0.192809
$333$	0	0
$334$	−9.46790	−0.518061
$335$	23.7123	1.29554
$336$	0	0
$337$	35.6784	1.94353	0.971763	−	0.235960i	$-0.0758235\pi$
$337$	35.6784	1.94353	0.971763	+	0.235960i	$0.0758235\pi$
$338$	−16.4587	−0.895237
$339$	0	0
$340$	−1.48295	−0.0804242
$341$	−0.127328	−0.00689518
$342$	0	0
$343$	20.1060	1.08562
$344$	25.3486	1.36670
$345$	0	0
$346$	−4.47477	−0.240565
$347$	−11.2710	−0.605060	−0.302530	−	0.953140i	$-0.597831\pi$
$347$	−11.2710	−0.605060	−0.302530	+	0.953140i	$0.597831\pi$
$348$	0	0
$349$	−12.9056	−0.690823	−0.345412	−	0.938451i	$-0.612261\pi$
$349$	−12.9056	−0.690823	−0.345412	+	0.938451i	$0.612261\pi$
$350$	−3.91774	−0.209412
$351$	0	0
$352$	0.303403	0.0161714
$353$	−6.75930	−0.359761	−0.179881	−	0.983688i	$-0.557571\pi$
$353$	−6.75930	−0.359761	−0.179881	+	0.983688i	$0.557571\pi$
$354$	0	0
$355$	3.41392	0.181192
$356$	−0.942786	−0.0499675
$357$	0	0
$358$	16.3216	0.862624
$359$	−33.2370	−1.75418	−0.877091	−	0.480325i	$-0.840519\pi$
$359$	−33.2370	−1.75418	−0.877091	+	0.480325i	$0.840519\pi$
$360$	0	0
$361$	2.03260	0.106979
$362$	−28.3958	−1.49245
$363$	0	0
$364$	−0.423607	−0.0222031
$365$	3.20181	0.167590
$366$	0	0
$367$	12.7646	0.666308	0.333154	−	0.942872i	$-0.391887\pi$
$367$	12.7646	0.666308	0.333154	+	0.942872i	$0.391887\pi$
$368$	3.41006	0.177761
$369$	0	0
$370$	−6.10849	−0.317565
$371$	12.6585	0.657196
$372$	0	0
$373$	−18.6085	−0.963511	−0.481756	−	0.876306i	$-0.660001\pi$
$373$	−18.6085	−0.963511	−0.481756	+	0.876306i	$0.660001\pi$
$374$	−0.616990	−0.0319038
$375$	0	0
$376$	17.5876	0.907014
$377$	0.720644	0.0371151
$378$	0	0
$379$	−7.61409	−0.391109	−0.195555	−	0.980693i	$-0.562651\pi$
$379$	−7.61409	−0.391109	−0.195555	+	0.980693i	$0.562651\pi$
$380$	−3.00805	−0.154310
$381$	0	0
$382$	24.5941	1.25834
$383$	6.30303	0.322070	0.161035	−	0.986949i	$-0.448517\pi$
$383$	6.30303	0.322070	0.161035	+	0.986949i	$0.448517\pi$
$384$	0	0
$385$	1.17184	0.0597223
$386$	−10.7613	−0.547736
$387$	0	0
$388$	−0.450214	−0.0228561
$389$	19.1940	0.973175	0.486588	−	0.873632i	$-0.338241\pi$
$389$	19.1940	0.973175	0.486588	+	0.873632i	$0.338241\pi$
$390$	0	0
$391$	2.26093	0.114340
$392$	5.73947	0.289887
$393$	0	0
$394$	−17.8425	−0.898890
$395$	−23.5303	−1.18394
$396$	0	0
$397$	3.97724	0.199612	0.0998059	−	0.995007i	$-0.468178\pi$
$397$	3.97724	0.199612	0.0998059	+	0.995007i	$0.468178\pi$
$398$	−5.05894	−0.253582
$399$	0	0
$400$	−4.49697	−0.224849
$401$	7.76968	0.387999	0.194000	−	0.981002i	$-0.437854\pi$
$401$	7.76968	0.387999	0.194000	+	0.981002i	$0.437854\pi$
$402$	0	0
$403$	0.443415	0.0220881
$404$	−1.39376	−0.0693420
$405$	0	0
$406$	2.97083	0.147440
$407$	0.381323	0.0189015
$408$	0	0
$409$	24.8374	1.22813	0.614066	−	0.789255i	$-0.289533\pi$
$409$	24.8374	1.22813	0.614066	+	0.789255i	$0.289533\pi$
$410$	−11.2685	−0.556509
$411$	0	0
$412$	2.09718	0.103321
$413$	2.77989	0.136789
$414$	0	0
$415$	−33.8446	−1.66136
$416$	−1.05659	−0.0518037
$417$	0	0
$418$	−1.25152	−0.0612138
$419$	−2.53323	−0.123756	−0.0618782	−	0.998084i	$-0.519709\pi$
$419$	−2.53323	−0.123756	−0.0618782	+	0.998084i	$0.519709\pi$
$420$	0	0
$421$	−0.486597	−0.0237153	−0.0118576	−	0.999930i	$-0.503774\pi$
$421$	−0.486597	−0.0237153	−0.0118576	+	0.999930i	$0.503774\pi$
$422$	−23.8219	−1.15963
$423$	0	0
$424$	16.7536	0.813628
$425$	−2.98157	−0.144628
$426$	0	0
$427$	−21.1185	−1.02200
$428$	−0.916594	−0.0443052
$429$	0	0
$430$	28.1827	1.35909
$431$	−17.1122	−0.824267	−0.412133	−	0.911123i	$-0.635216\pi$
$431$	−17.1122	−0.824267	−0.412133	+	0.911123i	$0.635216\pi$
$432$	0	0
$433$	−25.5910	−1.22982	−0.614912	−	0.788596i	$-0.710808\pi$
$433$	−25.5910	−1.22982	−0.614912	+	0.788596i	$0.710808\pi$
$434$	1.82796	0.0877450
$435$	0	0
$436$	−4.83177	−0.231400
$437$	4.58613	0.219384
$438$	0	0
$439$	−7.57936	−0.361743	−0.180872	−	0.983507i	$-0.557892\pi$
$439$	−7.57936	−0.361743	−0.180872	+	0.983507i	$0.557892\pi$
$440$	1.55094	0.0739379
$441$	0	0
$442$	2.14866	0.102201
$443$	−8.39708	−0.398958	−0.199479	−	0.979902i	$-0.563925\pi$
$443$	−8.39708	−0.398958	−0.199479	+	0.979902i	$0.563925\pi$
$444$	0	0
$445$	−9.08248	−0.430551
$446$	7.67505	0.363424
$447$	0	0
$448$	−19.7200	−0.931681
$449$	−30.7310	−1.45029	−0.725144	−	0.688597i	$-0.758227\pi$
$449$	−30.7310	−1.45029	−0.725144	+	0.688597i	$0.758227\pi$
$450$	0	0
$451$	0.703434	0.0331234
$452$	−4.49963	−0.211645
$453$	0	0
$454$	20.2404	0.949931
$455$	−4.08089	−0.191315
$456$	0	0
$457$	1.32954	0.0621932	0.0310966	−	0.999516i	$-0.490100\pi$
$457$	1.32954	0.0621932	0.0310966	+	0.999516i	$0.490100\pi$
$458$	−20.8595	−0.974700
$459$	0	0
$460$	−0.655902	−0.0305816
$461$	7.33804	0.341767	0.170883	−	0.985291i	$-0.445338\pi$
$461$	7.33804	0.341767	0.170883	+	0.985291i	$0.445338\pi$
$462$	0	0
$463$	−2.68755	−0.124901	−0.0624506	−	0.998048i	$-0.519892\pi$
$463$	−2.68755	−0.124901	−0.0624506	+	0.998048i	$0.519892\pi$
$464$	3.41006	0.158308
$465$	0	0
$466$	−14.2508	−0.660156
$467$	−6.84657	−0.316821	−0.158411	−	0.987373i	$-0.550637\pi$
$467$	−6.84657	−0.316821	−0.158411	+	0.987373i	$0.550637\pi$
$468$	0	0
$469$	21.2509	0.981276
$470$	19.5540	0.901959
$471$	0	0
$472$	3.67921	0.169349
$473$	−1.75930	−0.0808929
$474$	0	0
$475$	−6.04791	−0.277497
$476$	−1.32901	−0.0609153
$477$	0	0
$478$	16.4661	0.753143
$479$	−15.1503	−0.692236	−0.346118	−	0.938191i	$-0.612500\pi$
$479$	−15.1503	−0.692236	−0.346118	+	0.938191i	$0.612500\pi$
$480$	0	0
$481$	−1.32795	−0.0605493
$482$	−5.19865	−0.236792
$483$	0	0
$484$	2.85905	0.129957
$485$	−4.33721	−0.196942
$486$	0	0
$487$	−2.30603	−0.104496	−0.0522481	−	0.998634i	$-0.516639\pi$
$487$	−2.30603	−0.104496	−0.0522481	+	0.998634i	$0.516639\pi$
$488$	−27.9505	−1.26526
$489$	0	0
$490$	6.38117	0.288272
$491$	30.1262	1.35958	0.679789	−	0.733408i	$-0.262071\pi$
$491$	30.1262	1.35958	0.679789	+	0.733408i	$0.262071\pi$
$492$	0	0
$493$	2.26093	0.101827
$494$	4.35839	0.196093
$495$	0	0
$496$	2.09822	0.0942129
$497$	3.05955	0.137239
$498$	0	0
$499$	14.1251	0.632328	0.316164	−	0.948705i	$-0.397605\pi$
$499$	14.1251	0.632328	0.316164	+	0.948705i	$0.397605\pi$
$500$	−2.41455	−0.107982
$501$	0	0
$502$	−8.96729	−0.400230
$503$	28.0963	1.25275	0.626377	−	0.779521i	$-0.284537\pi$
$503$	28.0963	1.25275	0.626377	+	0.779521i	$0.284537\pi$
$504$	0	0
$505$	−13.4270	−0.597493
$506$	−0.272892	−0.0121315
$507$	0	0
$508$	−3.76602	−0.167090
$509$	5.49940	0.243757	0.121878	−	0.992545i	$-0.461108\pi$
$509$	5.49940	0.243757	0.121878	+	0.992545i	$0.461108\pi$
$510$	0	0
$511$	2.86945	0.126937
$512$	−25.3344	−1.11963
$513$	0	0
$514$	33.6191	1.48287
$515$	20.2035	0.890273
$516$	0	0
$517$	−1.22066	−0.0536846
$518$	−5.47442	−0.240532
$519$	0	0
$520$	−5.40110	−0.236854
$521$	−19.0771	−0.835783	−0.417892	−	0.908497i	$-0.637231\pi$
$521$	−19.0771	−0.835783	−0.417892	+	0.908497i	$0.637231\pi$
$522$	0	0
$523$	−5.59256	−0.244545	−0.122273	−	0.992497i	$-0.539018\pi$
$523$	−5.59256	−0.244545	−0.122273	+	0.992497i	$0.539018\pi$
$524$	−2.59696	−0.113449
$525$	0	0
$526$	−7.74961	−0.337899
$527$	1.39116	0.0605998
$528$	0	0
$529$	1.00000	0.0434783
$530$	18.6267	0.809094
$531$	0	0
$532$	−2.69581	−0.116878
$533$	−2.44969	−0.106108
$534$	0	0
$535$	−8.83016	−0.381761
$536$	28.1258	1.21485
$537$	0	0
$538$	4.76303	0.205349
$539$	−0.398345	−0.0171579
$540$	0	0
$541$	12.7925	0.549994	0.274997	−	0.961445i	$-0.411323\pi$
$541$	12.7925	0.549994	0.274997	+	0.961445i	$0.411323\pi$
$542$	−30.5222	−1.31104
$543$	0	0
$544$	−3.31493	−0.142126
$545$	−46.5476	−1.99388
$546$	0	0
$547$	−28.1352	−1.20298	−0.601488	−	0.798882i	$-0.705425\pi$
$547$	−28.1352	−1.20298	−0.601488	+	0.798882i	$0.705425\pi$
$548$	−5.66596	−0.242038
$549$	0	0
$550$	0.359874	0.0153451
$551$	4.58613	0.195376
$552$	0	0
$553$	−21.0878	−0.896743
$554$	21.0416	0.893970
$555$	0	0
$556$	2.51538	0.106676
$557$	0.823300	0.0348843	0.0174422	−	0.999848i	$-0.494448\pi$
$557$	0.823300	0.0348843	0.0174422	+	0.999848i	$0.494448\pi$
$558$	0	0
$559$	6.12674	0.259133
$560$	−19.3106	−0.816021
$561$	0	0
$562$	4.29710	0.181262
$563$	−20.8016	−0.876684	−0.438342	−	0.898808i	$-0.644434\pi$
$563$	−20.8016	−0.876684	−0.438342	+	0.898808i	$0.644434\pi$
$564$	0	0
$565$	−43.3480	−1.82366
$566$	10.2263	0.429845
$567$	0	0
$568$	4.04934	0.169906
$569$	8.48857	0.355859	0.177930	−	0.984043i	$-0.443060\pi$
$569$	8.48857	0.355859	0.177930	+	0.984043i	$0.443060\pi$
$570$	0	0
$571$	1.08653	0.0454698	0.0227349	−	0.999742i	$-0.492763\pi$
$571$	1.08653	0.0454698	0.0227349	+	0.999742i	$0.492763\pi$
$572$	0.0389115	0.00162697
$573$	0	0
$574$	−10.0988	−0.421514
$575$	−1.31874	−0.0549952
$576$	0	0
$577$	39.2324	1.63327	0.816634	−	0.577156i	$-0.195838\pi$
$577$	39.2324	1.63327	0.816634	+	0.577156i	$0.195838\pi$
$578$	−15.6774	−0.652095
$579$	0	0
$580$	−0.655902	−0.0272348
$581$	−30.3314	−1.25836
$582$	0	0
$583$	−1.16278	−0.0481572
$584$	3.79774	0.157152
$585$	0	0
$586$	−31.0687	−1.28344
$587$	13.3533	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$587$	13.3533	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$588$	0	0
$589$	2.82186	0.116273
$590$	4.09055	0.168405
$591$	0	0
$592$	−6.28379	−0.258262
$593$	−11.9242	−0.489668	−0.244834	−	0.969565i	$-0.578733\pi$
$593$	−11.9242	−0.489668	−0.244834	+	0.969565i	$0.578733\pi$
$594$	0	0
$595$	−12.8033	−0.524883
$596$	0.994752	0.0407466
$597$	0	0
$598$	0.950341	0.0388623
$599$	−19.6776	−0.804004	−0.402002	−	0.915639i	$-0.631685\pi$
$599$	−19.6776	−0.804004	−0.402002	+	0.915639i	$0.631685\pi$
$600$	0	0
$601$	−2.28447	−0.0931854	−0.0465927	−	0.998914i	$-0.514836\pi$
$601$	−2.28447	−0.0931854	−0.0465927	+	0.998914i	$0.514836\pi$
$602$	25.2572	1.02941
$603$	0	0
$604$	6.30084	0.256377
$605$	27.5432	1.11979
$606$	0	0
$607$	35.5364	1.44238	0.721188	−	0.692739i	$-0.243596\pi$
$607$	35.5364	1.44238	0.721188	+	0.692739i	$0.243596\pi$
$608$	−6.72408	−0.272698
$609$	0	0
$610$	−31.0755	−1.25821
$611$	4.25092	0.171974
$612$	0	0
$613$	−6.04229	−0.244046	−0.122023	−	0.992527i	$-0.538938\pi$
$613$	−6.04229	−0.244046	−0.122023	+	0.992527i	$0.538938\pi$
$614$	14.5224	0.586078
$615$	0	0
$616$	1.38994	0.0560024
$617$	−12.1271	−0.488220	−0.244110	−	0.969748i	$-0.578496\pi$
$617$	−12.1271	−0.488220	−0.244110	+	0.969748i	$0.578496\pi$
$618$	0	0
$619$	40.4628	1.62634	0.813168	−	0.582028i	$-0.197741\pi$
$619$	40.4628	1.62634	0.813168	+	0.582028i	$0.197741\pi$
$620$	−0.403579	−0.0162081
$621$	0	0
$622$	5.88453	0.235948
$623$	−8.13969	−0.326110
$624$	0	0
$625$	−29.8546	−1.19418
$626$	15.7137	0.628046
$627$	0	0
$628$	−3.57814	−0.142783
$629$	−4.16627	−0.166120
$630$	0	0
$631$	−26.9996	−1.07484	−0.537418	−	0.843316i	$-0.680600\pi$
$631$	−26.9996	−1.07484	−0.537418	+	0.843316i	$0.680600\pi$
$632$	−27.9098	−1.11019
$633$	0	0
$634$	3.57247	0.141881
$635$	−36.2806	−1.43975
$636$	0	0
$637$	1.38723	0.0549639
$638$	−0.272892	−0.0108039
$639$	0	0
$640$	−21.6465	−0.855653
$641$	3.52991	0.139423	0.0697116	−	0.997567i	$-0.477792\pi$
$641$	3.52991	0.139423	0.0697116	+	0.997567i	$0.477792\pi$
$642$	0	0
$643$	−24.7084	−0.974405	−0.487203	−	0.873289i	$-0.661983\pi$
$643$	−24.7084	−0.974405	−0.487203	+	0.873289i	$0.661983\pi$
$644$	−0.587818	−0.0231633
$645$	0	0
$646$	13.6739	0.537992
$647$	−9.56460	−0.376023	−0.188012	−	0.982167i	$-0.560204\pi$
$647$	−9.56460	−0.376023	−0.188012	+	0.982167i	$0.560204\pi$
$648$	0	0
$649$	−0.255353	−0.0100235
$650$	−1.25325	−0.0491566
$651$	0	0
$652$	−0.293501	−0.0114944
$653$	9.70004	0.379592	0.189796	−	0.981824i	$-0.439217\pi$
$653$	9.70004	0.379592	0.189796	+	0.981824i	$0.439217\pi$
$654$	0	0
$655$	−25.0183	−0.977544
$656$	−11.5918	−0.452585
$657$	0	0
$658$	17.5243	0.683167
$659$	−16.1697	−0.629881	−0.314940	−	0.949111i	$-0.601985\pi$
$659$	−16.1697	−0.629881	−0.314940	+	0.949111i	$0.601985\pi$
$660$	0	0
$661$	15.2278	0.592295	0.296147	−	0.955142i	$-0.404298\pi$
$661$	15.2278	0.592295	0.296147	+	0.955142i	$0.404298\pi$
$662$	−14.4038	−0.559819
$663$	0	0
$664$	−40.1439	−1.55788
$665$	−25.9705	−1.00709
$666$	0	0
$667$	1.00000	0.0387202
$668$	1.87335	0.0724821
$669$	0	0
$670$	31.2703	1.20808
$671$	1.93989	0.0748886
$672$	0	0
$673$	−30.1373	−1.16171	−0.580854	−	0.814008i	$-0.697281\pi$
$673$	−30.1373	−1.16171	−0.580854	+	0.814008i	$0.697281\pi$
$674$	47.0504	1.81232
$675$	0	0
$676$	3.25658	0.125253
$677$	4.23918	0.162925	0.0814624	−	0.996676i	$-0.474041\pi$
$677$	4.23918	0.162925	0.0814624	+	0.996676i	$0.474041\pi$
$678$	0	0
$679$	−3.88699	−0.149169
$680$	−16.9452	−0.649821
$681$	0	0
$682$	−0.167912	−0.00642967
$683$	45.8651	1.75498	0.877490	−	0.479596i	$-0.159217\pi$
$683$	45.8651	1.75498	0.877490	+	0.479596i	$0.159217\pi$
$684$	0	0
$685$	−54.5839	−2.08554
$686$	26.5146	1.01233
$687$	0	0
$688$	28.9914	1.10529
$689$	4.04934	0.154267
$690$	0	0
$691$	32.6070	1.24043	0.620215	−	0.784432i	$-0.287046\pi$
$691$	32.6070	1.24043	0.620215	+	0.784432i	$0.287046\pi$
$692$	0.885392	0.0336576
$693$	0	0
$694$	−14.8635	−0.564212
$695$	24.2324	0.919185
$696$	0	0
$697$	−7.68560	−0.291113
$698$	−17.0192	−0.644185
$699$	0	0
$700$	0.775177	0.0292990
$701$	35.5331	1.34207	0.671033	−	0.741427i	$-0.265851\pi$
$701$	35.5331	1.34207	0.671033	+	0.741427i	$0.265851\pi$
$702$	0	0
$703$	−8.45097	−0.318735
$704$	1.81142	0.0682706
$705$	0	0
$706$	−8.91375	−0.335473
$707$	−12.0332	−0.452556
$708$	0	0
$709$	−45.4945	−1.70858	−0.854290	−	0.519796i	$-0.826008\pi$
$709$	−45.4945	−1.70858	−0.854290	+	0.519796i	$0.826008\pi$
$710$	4.50207	0.168960
$711$	0	0
$712$	−10.7730	−0.403734
$713$	0.615304	0.0230433
$714$	0	0
$715$	0.374860	0.0140190
$716$	−3.22945	−0.120690
$717$	0	0
$718$	−43.8309	−1.63575
$719$	20.5160	0.765119	0.382560	−	0.923931i	$-0.375043\pi$
$719$	20.5160	0.765119	0.382560	+	0.923931i	$0.375043\pi$
$720$	0	0
$721$	18.1063	0.674315
$722$	2.68047	0.0997566
$723$	0	0
$724$	5.61848	0.208809
$725$	−1.31874	−0.0489767
$726$	0	0
$727$	−28.3740	−1.05233	−0.526167	−	0.850382i	$-0.676371\pi$
$727$	−28.3740	−1.05233	−0.526167	+	0.850382i	$0.676371\pi$
$728$	−4.84045	−0.179399
$729$	0	0
$730$	4.22234	0.156276
$731$	19.2219	0.710946
$732$	0	0
$733$	50.2037	1.85432	0.927158	−	0.374672i	$-0.122245\pi$
$733$	50.2037	1.85432	0.927158	+	0.374672i	$0.122245\pi$
$734$	16.8332	0.621324
$735$	0	0
$736$	−1.46618	−0.0540440
$737$	−1.95205	−0.0719048
$738$	0	0
$739$	−19.7992	−0.728326	−0.364163	−	0.931335i	$-0.618645\pi$
$739$	−19.7992	−0.728326	−0.364163	+	0.931335i	$0.618645\pi$
$740$	1.20865	0.0444307
$741$	0	0
$742$	16.6932	0.612828
$743$	46.4475	1.70399	0.851997	−	0.523546i	$-0.175391\pi$
$743$	46.4475	1.70399	0.851997	+	0.523546i	$0.175391\pi$
$744$	0	0
$745$	9.58311	0.351098
$746$	−24.5397	−0.898463
$747$	0	0
$748$	0.122080	0.00446368
$749$	−7.91356	−0.289155
$750$	0	0
$751$	36.9387	1.34791	0.673956	−	0.738772i	$-0.264594\pi$
$751$	36.9387	1.34791	0.673956	+	0.738772i	$0.264594\pi$
$752$	20.1152	0.733525
$753$	0	0
$754$	0.950341	0.0346094
$755$	60.7001	2.20910
$756$	0	0
$757$	15.5417	0.564872	0.282436	−	0.959286i	$-0.408857\pi$
$757$	15.5417	0.564872	0.282436	+	0.959286i	$0.408857\pi$
$758$	−10.0410	−0.364705
$759$	0	0
$760$	−34.3722	−1.24681
$761$	40.6429	1.47330	0.736652	−	0.676272i	$-0.236406\pi$
$761$	40.6429	1.47330	0.736652	+	0.676272i	$0.236406\pi$
$762$	0	0
$763$	−41.7159	−1.51022
$764$	−4.86626	−0.176055
$765$	0	0
$766$	8.31204	0.300326
$767$	0.889262	0.0321094
$768$	0	0
$769$	−5.13554	−0.185192	−0.0925962	−	0.995704i	$-0.529517\pi$
$769$	−5.13554	−0.185192	−0.0925962	+	0.995704i	$0.529517\pi$
$770$	1.54534	0.0556904
$771$	0	0
$772$	2.12927	0.0766339
$773$	−0.211642	−0.00761222	−0.00380611	−	0.999993i	$-0.501212\pi$
$773$	−0.211642	−0.00761222	−0.00380611	+	0.999993i	$0.501212\pi$
$774$	0	0
$775$	−0.811425	−0.0291472
$776$	−5.14447	−0.184676
$777$	0	0
$778$	25.3119	0.907475
$779$	−15.5897	−0.558558
$780$	0	0
$781$	−0.281042	−0.0100565
$782$	2.98157	0.106621
$783$	0	0
$784$	6.56429	0.234439
$785$	−34.4706	−1.23031
$786$	0	0
$787$	19.8563	0.707799	0.353900	−	0.935283i	$-0.384855\pi$
$787$	19.8563	0.707799	0.353900	+	0.935283i	$0.384855\pi$
$788$	3.53037	0.125764
$789$	0	0
$790$	−31.0303	−1.10401
$791$	−38.8483	−1.38129
$792$	0	0
$793$	−6.75562	−0.239899
$794$	5.24493	0.186136
$795$	0	0
$796$	1.00098	0.0354787
$797$	6.41446	0.227212	0.113606	−	0.993526i	$-0.463760\pi$
$797$	6.41446	0.227212	0.113606	+	0.993526i	$0.463760\pi$
$798$	0	0
$799$	13.3367	0.471819
$800$	1.93350	0.0683597
$801$	0	0
$802$	10.2462	0.361805
$803$	−0.263580	−0.00930154
$804$	0	0
$805$	−5.66284	−0.199589
$806$	0.584749	0.0205969
$807$	0	0
$808$	−15.9261	−0.560278
$809$	−47.7649	−1.67932	−0.839662	−	0.543109i	$-0.817247\pi$
$809$	−47.7649	−1.67932	−0.839662	+	0.543109i	$0.817247\pi$
$810$	0	0
$811$	−12.0735	−0.423959	−0.211979	−	0.977274i	$-0.567991\pi$
$811$	−12.0735	−0.423959	−0.211979	+	0.977274i	$0.567991\pi$
$812$	−0.587818	−0.0206284
$813$	0	0
$814$	0.502865	0.0176254
$815$	−2.82749	−0.0990427
$816$	0	0
$817$	38.9901	1.36409
$818$	32.7540	1.14522
$819$	0	0
$820$	2.22961	0.0778615
$821$	−2.12441	−0.0741424	−0.0370712	−	0.999313i	$-0.511803\pi$
$821$	−2.12441	−0.0741424	−0.0370712	+	0.999313i	$0.511803\pi$
$822$	0	0
$823$	14.3341	0.499656	0.249828	−	0.968290i	$-0.419626\pi$
$823$	14.3341	0.499656	0.249828	+	0.968290i	$0.419626\pi$
$824$	23.9639	0.834821
$825$	0	0
$826$	3.66594	0.127554
$827$	−27.0920	−0.942082	−0.471041	−	0.882111i	$-0.656122\pi$
$827$	−27.0920	−0.942082	−0.471041	+	0.882111i	$0.656122\pi$
$828$	0	0
$829$	14.1735	0.492268	0.246134	−	0.969236i	$-0.420840\pi$
$829$	14.1735	0.492268	0.246134	+	0.969236i	$0.420840\pi$
$830$	−44.6321	−1.54920
$831$	0	0
$832$	−6.30825	−0.218699
$833$	4.35225	0.150796
$834$	0	0
$835$	18.0472	0.624550
$836$	0.247630	0.00856446
$837$	0	0
$838$	−3.34067	−0.115402
$839$	41.6889	1.43926	0.719631	−	0.694357i	$-0.244311\pi$
$839$	41.6889	1.43926	0.719631	+	0.694357i	$0.244311\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−0.641694	−0.0221142
$843$	0	0
$844$	4.71347	0.162244
$845$	31.3728	1.07926
$846$	0	0
$847$	24.6841	0.848156
$848$	19.1613	0.658001
$849$	0	0
$850$	−3.93192	−0.134864
$851$	−1.84272	−0.0631678
$852$	0	0
$853$	24.0066	0.821970	0.410985	−	0.911642i	$-0.365185\pi$
$853$	24.0066	0.821970	0.410985	+	0.911642i	$0.365185\pi$
$854$	−27.8498	−0.953000
$855$	0	0
$856$	−10.4737	−0.357983
$857$	−28.3597	−0.968749	−0.484374	−	0.874861i	$-0.660953\pi$
$857$	−28.3597	−0.968749	−0.484374	+	0.874861i	$0.660953\pi$
$858$	0	0
$859$	−12.0011	−0.409474	−0.204737	−	0.978817i	$-0.565634\pi$
$859$	−12.0011	−0.409474	−0.204737	+	0.978817i	$0.565634\pi$
$860$	−5.57631	−0.190151
$861$	0	0
$862$	−22.5665	−0.768620
$863$	−31.8630	−1.08463	−0.542314	−	0.840176i	$-0.682452\pi$
$863$	−31.8630	−1.08463	−0.542314	+	0.840176i	$0.682452\pi$
$864$	0	0
$865$	8.52957	0.290014
$866$	−33.7478	−1.14680
$867$	0	0
$868$	−0.361686	−0.0122764
$869$	1.93707	0.0657105
$870$	0	0
$871$	6.79798	0.230341
$872$	−55.2113	−1.86969
$873$	0	0
$874$	6.04791	0.204573
$875$	−20.8464	−0.704736
$876$	0	0
$877$	5.74793	0.194094	0.0970469	−	0.995280i	$-0.469060\pi$
$877$	5.74793	0.194094	0.0970469	+	0.995280i	$0.469060\pi$
$878$	−9.99519	−0.337321
$879$	0	0
$880$	1.77382	0.0597955
$881$	11.9952	0.404128	0.202064	−	0.979372i	$-0.435235\pi$
$881$	11.9952	0.404128	0.202064	+	0.979372i	$0.435235\pi$
$882$	0	0
$883$	30.2919	1.01940	0.509701	−	0.860352i	$-0.329756\pi$
$883$	30.2919	1.01940	0.509701	+	0.860352i	$0.329756\pi$
$884$	−0.425140	−0.0142990
$885$	0	0
$886$	−11.0736	−0.372023
$887$	28.8927	0.970123	0.485062	−	0.874480i	$-0.338797\pi$
$887$	28.8927	0.970123	0.485062	+	0.874480i	$0.338797\pi$
$888$	0	0
$889$	−32.5145	−1.09050
$890$	−11.9774	−0.401484
$891$	0	0
$892$	−1.51861	−0.0508468
$893$	27.0526	0.905280
$894$	0	0
$895$	−31.1114	−1.03994
$896$	−19.3995	−0.648093
$897$	0	0
$898$	−40.5262	−1.35238
$899$	0.615304	0.0205215
$900$	0	0
$901$	12.7043	0.423241
$902$	0.927645	0.0308872
$903$	0	0
$904$	−51.4161	−1.71007
$905$	54.1266	1.79923
$906$	0	0
$907$	40.3737	1.34059	0.670293	−	0.742097i	$-0.266168\pi$
$907$	40.3737	1.34059	0.670293	+	0.742097i	$0.266168\pi$
$908$	−4.00484	−0.132905
$909$	0	0
$910$	−5.38163	−0.178399
$911$	8.30349	0.275107	0.137553	−	0.990494i	$-0.456076\pi$
$911$	8.30349	0.275107	0.137553	+	0.990494i	$0.456076\pi$
$912$	0	0
$913$	2.78616	0.0922085
$914$	1.75331	0.0579944
$915$	0	0
$916$	4.12733	0.136371
$917$	−22.4213	−0.740416
$918$	0	0
$919$	−39.2171	−1.29365	−0.646827	−	0.762637i	$-0.723904\pi$
$919$	−39.2171	−1.29365	−0.646827	+	0.762637i	$0.723904\pi$
$920$	−7.49481	−0.247097
$921$	0	0
$922$	9.67695	0.318693
$923$	0.978722	0.0322150
$924$	0	0
$925$	2.43007	0.0799002
$926$	−3.54418	−0.116469
$927$	0	0
$928$	−1.46618	−0.0481296
$929$	−40.1694	−1.31792	−0.658958	−	0.752180i	$-0.729002\pi$
$929$	−40.1694	−1.31792	−0.658958	+	0.752180i	$0.729002\pi$
$930$	0	0
$931$	8.82822	0.289333
$932$	2.81971	0.0923627
$933$	0	0
$934$	−9.02883	−0.295432
$935$	1.17608	0.0384618
$936$	0	0
$937$	−26.5512	−0.867390	−0.433695	−	0.901060i	$-0.642791\pi$
$937$	−26.5512	−0.867390	−0.433695	+	0.901060i	$0.642791\pi$
$938$	28.0244	0.915029
$939$	0	0
$940$	−3.86902	−0.126194
$941$	17.2684	0.562934	0.281467	−	0.959571i	$-0.409179\pi$
$941$	17.2684	0.562934	0.281467	+	0.959571i	$0.409179\pi$
$942$	0	0
$943$	−3.39931	−0.110697
$944$	4.20794	0.136957
$945$	0	0
$946$	−2.32006	−0.0754317
$947$	−44.4634	−1.44487	−0.722434	−	0.691440i	$-0.756977\pi$
$947$	−44.4634	−1.44487	−0.722434	+	0.691440i	$0.756977\pi$
$948$	0	0
$949$	0.917912	0.0297967
$950$	−7.97560	−0.258763
$951$	0	0
$952$	−15.1863	−0.492190
$953$	22.6887	0.734958	0.367479	−	0.930032i	$-0.380221\pi$
$953$	22.6887	0.734958	0.367479	+	0.930032i	$0.380221\pi$
$954$	0	0
$955$	−46.8799	−1.51700
$956$	−3.25804	−0.105373
$957$	0	0
$958$	−19.9793	−0.645502
$959$	−48.9180	−1.57964
$960$	0	0
$961$	−30.6214	−0.987787
$962$	−1.75122	−0.0564615
$963$	0	0
$964$	1.02862	0.0331297
$965$	20.5126	0.660325
$966$	0	0
$967$	4.31252	0.138681	0.0693406	−	0.997593i	$-0.477910\pi$
$967$	4.31252	0.138681	0.0693406	+	0.997593i	$0.477910\pi$
$968$	32.6696	1.05004
$969$	0	0
$970$	−5.71964	−0.183647
$971$	−46.8188	−1.50249	−0.751243	−	0.660025i	$-0.770546\pi$
$971$	−46.8188	−1.50249	−0.751243	+	0.660025i	$0.770546\pi$
$972$	0	0
$973$	21.7170	0.696214
$974$	−3.04105	−0.0974414
$975$	0	0
$976$	−31.9673	−1.02325
$977$	−51.8522	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$977$	−51.8522	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$978$	0	0
$979$	0.747691	0.0238963
$980$	−1.26260	−0.0403322
$981$	0	0
$982$	39.7286	1.26779
$983$	−2.68187	−0.0855382	−0.0427691	−	0.999085i	$-0.513618\pi$
$983$	−2.68187	−0.0855382	−0.0427691	+	0.999085i	$0.513618\pi$
$984$	0	0
$985$	34.0104	1.08366
$986$	2.98157	0.0949527
$987$	0	0
$988$	−0.862365	−0.0274355
$989$	8.50175	0.270340
$990$	0	0
$991$	11.1114	0.352966	0.176483	−	0.984304i	$-0.443528\pi$
$991$	11.1114	0.352966	0.176483	+	0.984304i	$0.443528\pi$
$992$	−0.902145	−0.0286431
$993$	0	0
$994$	4.03474	0.127974
$995$	9.64308	0.305706
$996$	0	0
$997$	35.6328	1.12850	0.564251	−	0.825603i	$-0.309165\pi$
$997$	35.6328	1.12850	0.564251	+	0.825603i	$0.309165\pi$
$998$	18.6274	0.589639
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.h.1.3		5
3.2	odd	2		2001.2.a.h.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.h.1.3	✓	5	3.2	odd	2
6003.2.a.h.1.3		5	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31874 q^{2} -0.260930 q^{4} -2.51371 q^{5} -2.25278 q^{7} -2.98157 q^{8} +O(q^{10})\)	\(q+1.31874 q^{2} -0.260930 q^{4} -2.51371 q^{5} -2.25278 q^{7} -2.98157 q^{8} -3.31493 q^{10} +0.206934 q^{11} -0.720644 q^{13} -2.97083 q^{14} -3.41006 q^{16} -2.26093 q^{17} -4.58613 q^{19} +0.655902 q^{20} +0.272892 q^{22} -1.00000 q^{23} +1.31874 q^{25} -0.950341 q^{26} +0.587818 q^{28} -1.00000 q^{29} -0.615304 q^{31} +1.46618 q^{32} -2.98157 q^{34} +5.66284 q^{35} +1.84272 q^{37} -6.04791 q^{38} +7.49481 q^{40} +3.39931 q^{41} -8.50175 q^{43} -0.0539954 q^{44} -1.31874 q^{46} -5.89878 q^{47} -1.92498 q^{49} +1.73907 q^{50} +0.188038 q^{52} -5.61905 q^{53} -0.520173 q^{55} +6.71683 q^{56} -1.31874 q^{58} -1.23398 q^{59} +9.37442 q^{61} -0.811425 q^{62} +8.75362 q^{64} +1.81149 q^{65} -9.43319 q^{67} +0.589944 q^{68} +7.46780 q^{70} -1.35812 q^{71} -1.27374 q^{73} +2.43007 q^{74} +1.19666 q^{76} -0.466178 q^{77} +9.36077 q^{79} +8.57189 q^{80} +4.48280 q^{82} +13.4640 q^{83} +5.68332 q^{85} -11.2116 q^{86} -0.616990 q^{88} +3.61318 q^{89} +1.62345 q^{91} +0.260930 q^{92} -7.77895 q^{94} +11.5282 q^{95} +1.72542 q^{97} -2.53855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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