LMFDB - Embedded newform 2009.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49086 q^{2} +1.49086 q^{3} +4.20440 q^{4} +3.71354 q^{6} +5.49086 q^{8} -0.777326 q^{9} +O(q^{10})$	\(q+2.49086 q^{2} +1.49086 q^{3} +4.20440 q^{4} +3.71354 q^{6} +5.49086 q^{8} -0.777326 q^{9} -3.42708 q^{11} +6.26819 q^{12} +6.26819 q^{13} +5.26819 q^{16} +5.20440 q^{17} -1.93621 q^{18} +2.22267 q^{19} -8.53638 q^{22} +2.91794 q^{23} +8.18613 q^{24} -5.00000 q^{25} +15.6132 q^{26} -5.63148 q^{27} +4.98173 q^{29} +0.854152 q^{31} +2.14061 q^{32} -5.10930 q^{33} +12.9635 q^{34} -3.26819 q^{36} +0.268189 q^{37} +5.53638 q^{38} +9.34502 q^{39} -1.00000 q^{41} -3.20440 q^{43} -14.4088 q^{44} +7.26819 q^{46} -4.14061 q^{47} +7.85415 q^{48} -12.4543 q^{50} +7.75905 q^{51} +26.3540 q^{52} -3.42708 q^{53} -14.0272 q^{54} +3.31370 q^{57} +12.4088 q^{58} +2.12758 q^{59} -14.2447 q^{61} +2.12758 q^{62} -5.20440 q^{64} -12.7266 q^{66} +6.12758 q^{67} +21.8814 q^{68} +4.35025 q^{69} -7.55465 q^{71} -4.26819 q^{72} -12.5364 q^{73} +0.668023 q^{74} -7.45432 q^{75} +9.34502 q^{76} +23.2772 q^{78} -6.00000 q^{79} -6.06379 q^{81} -2.49086 q^{82} +10.4088 q^{83} -7.98173 q^{86} +7.42708 q^{87} -18.8176 q^{88} +6.26819 q^{89} +12.2682 q^{92} +1.27342 q^{93} -10.3137 q^{94} +3.19136 q^{96} +8.06379 q^{97} +2.66395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{4} + 10 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^4 + 10 * q^6 + 9 * q^8 + 4 * q^9	$ 3 q - 3 q^{3} + 4 q^{4} + 10 q^{6} + 9 q^{8} + 4 q^{9} - 8 q^{11} + 5 q^{12} + 5 q^{13} + 2 q^{16} + 7 q^{17} - 11 q^{18} + 13 q^{19} + 2 q^{22} - q^{23} + q^{24} - 15 q^{25} + 21 q^{26} - 6 q^{27} - 2 q^{31} + 3 q^{32} + 10 q^{33} + 9 q^{34} + 4 q^{36} - 13 q^{37} - 11 q^{38} + 16 q^{39} - 3 q^{41} - q^{43} - 26 q^{44} + 8 q^{46} - 9 q^{47} + 19 q^{48} + 2 q^{51} + 17 q^{52} - 8 q^{53} - 7 q^{54} - 24 q^{57} + 20 q^{58} - 4 q^{59} - 6 q^{61} - 4 q^{62} - 7 q^{64} - 44 q^{66} + 8 q^{67} + 26 q^{68} + 9 q^{69} - 10 q^{71} + q^{72} - 10 q^{73} + 21 q^{74} + 15 q^{75} + 16 q^{76} + 6 q^{78} - 18 q^{79} - 13 q^{81} + 14 q^{83} - 9 q^{86} + 20 q^{87} - 22 q^{88} + 5 q^{89} + 23 q^{92} - 2 q^{93} + 3 q^{94} - 6 q^{96} + 19 q^{97} - 30 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^4 + 10 * q^6 + 9 * q^8 + 4 * q^9 - 8 * q^11 + 5 * q^12 + 5 * q^13 + 2 * q^16 + 7 * q^17 - 11 * q^18 + 13 * q^19 + 2 * q^22 - q^23 + q^24 - 15 * q^25 + 21 * q^26 - 6 * q^27 - 2 * q^31 + 3 * q^32 + 10 * q^33 + 9 * q^34 + 4 * q^36 - 13 * q^37 - 11 * q^38 + 16 * q^39 - 3 * q^41 - q^43 - 26 * q^44 + 8 * q^46 - 9 * q^47 + 19 * q^48 + 2 * q^51 + 17 * q^52 - 8 * q^53 - 7 * q^54 - 24 * q^57 + 20 * q^58 - 4 * q^59 - 6 * q^61 - 4 * q^62 - 7 * q^64 - 44 * q^66 + 8 * q^67 + 26 * q^68 + 9 * q^69 - 10 * q^71 + q^72 - 10 * q^73 + 21 * q^74 + 15 * q^75 + 16 * q^76 + 6 * q^78 - 18 * q^79 - 13 * q^81 + 14 * q^83 - 9 * q^86 + 20 * q^87 - 22 * q^88 + 5 * q^89 + 23 * q^92 - 2 * q^93 + 3 * q^94 - 6 * q^96 + 19 * q^97 - 30 * 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 $

$2$	2.49086	1.76131	0.880653	−	0.473761i	$-0.157104\pi$
$2$	2.49086	1.76131	0.880653	+	0.473761i	$0.157104\pi$
$3$	1.49086	0.860751	0.430375	−	0.902650i	$-0.358381\pi$
$3$	1.49086	0.860751	0.430375	+	0.902650i	$0.358381\pi$
$4$	4.20440	2.10220
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	3.71354	1.51605
$7$	0	0
$7$	0	0
$8$	5.49086	1.94131
$9$	−0.777326	−0.259109
$10$	0	0
$11$	−3.42708	−1.03330	−0.516651	−	0.856196i	$-0.672822\pi$
$11$	−3.42708	−1.03330	−0.516651	+	0.856196i	$0.672822\pi$
$12$	6.26819	1.80947
$13$	6.26819	1.73848	0.869241	−	0.494388i	$-0.164608\pi$
$13$	6.26819	1.73848	0.869241	+	0.494388i	$0.164608\pi$
$14$	0	0
$15$	0	0
$16$	5.26819	1.31705
$17$	5.20440	1.26225	0.631126	−	0.775680i	$-0.282593\pi$
$17$	5.20440	1.26225	0.631126	+	0.775680i	$0.282593\pi$
$18$	−1.93621	−0.456370
$19$	2.22267	0.509916	0.254958	−	0.966952i	$-0.417938\pi$
$19$	2.22267	0.509916	0.254958	+	0.966952i	$0.417938\pi$
$20$	0	0
$21$	0	0
$22$	−8.53638	−1.81996
$23$	2.91794	0.608432	0.304216	−	0.952603i	$-0.401605\pi$
$23$	2.91794	0.608432	0.304216	+	0.952603i	$0.401605\pi$
$24$	8.18613	1.67099
$25$	−5.00000	−1.00000
$26$	15.6132	3.06200
$27$	−5.63148	−1.08378
$28$	0	0
$29$	4.98173	0.925084	0.462542	−	0.886597i	$-0.346938\pi$
$29$	4.98173	0.925084	0.462542	+	0.886597i	$0.346938\pi$
$30$	0	0
$31$	0.854152	0.153410	0.0767051	−	0.997054i	$-0.475560\pi$
$31$	0.854152	0.153410	0.0767051	+	0.997054i	$0.475560\pi$
$32$	2.14061	0.378411
$33$	−5.10930	−0.889415
$34$	12.9635	2.22321
$35$	0	0
$36$	−3.26819	−0.544698
$37$	0.268189	0.0440900	0.0220450	−	0.999757i	$-0.492982\pi$
$37$	0.268189	0.0440900	0.0220450	+	0.999757i	$0.492982\pi$
$38$	5.53638	0.898119
$39$	9.34502	1.49640
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.20440	−0.488667	−0.244333	−	0.969691i	$-0.578569\pi$
$43$	−3.20440	−0.488667	−0.244333	+	0.969691i	$0.578569\pi$
$44$	−14.4088	−2.17221
$45$	0	0
$46$	7.26819	1.07164
$47$	−4.14061	−0.603971	−0.301985	−	0.953313i	$-0.597649\pi$
$47$	−4.14061	−0.603971	−0.301985	+	0.953313i	$0.597649\pi$
$48$	7.85415	1.13365
$49$	0	0
$50$	−12.4543	−1.76131
$51$	7.75905	1.08648
$52$	26.3540	3.65464
$53$	−3.42708	−0.470745	−0.235373	−	0.971905i	$-0.575631\pi$
$53$	−3.42708	−0.470745	−0.235373	+	0.971905i	$0.575631\pi$
$54$	−14.0272	−1.90887
$55$	0	0
$56$	0	0
$57$	3.31370	0.438911
$58$	12.4088	1.62936
$59$	2.12758	0.276987	0.138493	−	0.990363i	$-0.455774\pi$
$59$	2.12758	0.276987	0.138493	+	0.990363i	$0.455774\pi$
$60$	0	0
$61$	−14.2447	−1.82384	−0.911922	−	0.410363i	$-0.865402\pi$
$61$	−14.2447	−1.82384	−0.911922	+	0.410363i	$0.865402\pi$
$62$	2.12758	0.270202
$63$	0	0
$64$	−5.20440	−0.650550
$65$	0	0
$66$	−12.7266	−1.56653
$67$	6.12758	0.748602	0.374301	−	0.927307i	$-0.377883\pi$
$67$	6.12758	0.748602	0.374301	+	0.927307i	$0.377883\pi$
$68$	21.8814	2.65351
$69$	4.35025	0.523709
$70$	0	0
$71$	−7.55465	−0.896572	−0.448286	−	0.893890i	$-0.647965\pi$
$71$	−7.55465	−0.896572	−0.448286	+	0.893890i	$0.647965\pi$
$72$	−4.26819	−0.503011
$73$	−12.5364	−1.46727	−0.733636	−	0.679543i	$-0.762178\pi$
$73$	−12.5364	−1.46727	−0.733636	+	0.679543i	$0.762178\pi$
$74$	0.668023	0.0776561
$75$	−7.45432	−0.860751
$76$	9.34502	1.07195
$77$	0	0
$78$	23.2772	2.63562
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	−6.06379	−0.673754
$82$	−2.49086	−0.275070
$83$	10.4088	1.14251	0.571257	−	0.820771i	$-0.306456\pi$
$83$	10.4088	1.14251	0.571257	+	0.820771i	$0.306456\pi$
$84$	0	0
$85$	0	0
$86$	−7.98173	−0.860692
$87$	7.42708	0.796266
$88$	−18.8176	−2.00596
$89$	6.26819	0.664427	0.332213	−	0.943204i	$-0.392205\pi$
$89$	6.26819	0.664427	0.332213	+	0.943204i	$0.392205\pi$
$90$	0	0
$91$	0	0
$92$	12.2682	1.27905
$93$	1.27342	0.132048
$94$	−10.3137	−1.06378
$95$	0	0
$96$	3.19136	0.325717
$97$	8.06379	0.818754	0.409377	−	0.912365i	$-0.365746\pi$
$97$	8.06379	0.818754	0.409377	+	0.912365i	$0.365746\pi$
$98$	0	0
$99$	2.66395	0.267737
$100$	−21.0220	−2.10220
$101$	−12.7538	−1.26905	−0.634526	−	0.772901i	$-0.718805\pi$
$101$	−12.7538	−1.26905	−0.634526	+	0.772901i	$0.718805\pi$
$102$	19.3267	1.91363
$103$	−13.3905	−1.31941	−0.659704	−	0.751525i	$-0.729318\pi$
$103$	−13.3905	−1.31941	−0.659704	+	0.751525i	$0.729318\pi$
$104$	34.4178	3.37494
$105$	0	0
$106$	−8.53638	−0.829126
$107$	9.61320	0.929344	0.464672	−	0.885483i	$-0.346172\pi$
$107$	9.61320	0.929344	0.464672	+	0.885483i	$0.346172\pi$
$108$	−23.6770	−2.27832
$109$	−8.53638	−0.817637	−0.408818	−	0.912616i	$-0.634059\pi$
$109$	−8.53638	−0.817637	−0.408818	+	0.912616i	$0.634059\pi$
$110$	0	0
$111$	0.399834	0.0379505
$112$	0	0
$113$	−16.4726	−1.54961	−0.774805	−	0.632200i	$-0.782152\pi$
$113$	−16.4726	−1.54961	−0.774805	+	0.632200i	$0.782152\pi$
$114$	8.25399	0.773057
$115$	0	0
$116$	20.9452	1.94471
$117$	−4.87242	−0.450456
$118$	5.29950	0.487859
$119$	0	0
$120$	0	0
$121$	0.744849	0.0677135
$122$	−35.4816	−3.21235
$123$	−1.49086	−0.134427
$124$	3.59120	0.322499
$125$	0	0
$126$	0	0
$127$	6.14061	0.544891	0.272446	−	0.962171i	$-0.412167\pi$
$127$	6.14061	0.544891	0.272446	+	0.962171i	$0.412167\pi$
$128$	−17.2447	−1.52423
$129$	−4.77733	−0.420620
$130$	0	0
$131$	3.59120	0.313764	0.156882	−	0.987617i	$-0.449856\pi$
$131$	3.59120	0.313764	0.156882	+	0.987617i	$0.449856\pi$
$132$	−21.4816	−1.86973
$133$	0	0
$134$	15.2630	1.31852
$135$	0	0
$136$	28.5767	2.45043
$137$	−16.8176	−1.43683	−0.718413	−	0.695617i	$-0.755131\pi$
$137$	−16.8176	−1.43683	−0.718413	+	0.695617i	$0.755131\pi$
$138$	10.8359	0.922411
$139$	17.8359	1.51282	0.756410	−	0.654098i	$-0.226952\pi$
$139$	17.8359	1.51282	0.756410	+	0.654098i	$0.226952\pi$
$140$	0	0
$141$	−6.17309	−0.519868
$142$	−18.8176	−1.57914
$143$	−21.4816	−1.79638
$144$	−4.09510	−0.341258
$145$	0	0
$146$	−31.2264	−2.58432
$147$	0	0
$148$	1.12758	0.0926861
$149$	−14.9452	−1.22436	−0.612178	−	0.790720i	$-0.709707\pi$
$149$	−14.9452	−1.22436	−0.612178	+	0.790720i	$0.709707\pi$
$150$	−18.5677	−1.51605
$151$	−4.94518	−0.402433	−0.201217	−	0.979547i	$-0.564490\pi$
$151$	−4.94518	−0.402433	−0.201217	+	0.979547i	$0.564490\pi$
$152$	12.2044	0.989908
$153$	−4.04551	−0.327061
$154$	0	0
$155$	0	0
$156$	39.2902	3.14573
$157$	9.45955	0.754954	0.377477	−	0.926019i	$-0.376792\pi$
$157$	9.45955	0.754954	0.377477	+	0.926019i	$0.376792\pi$
$158$	−14.9452	−1.18897
$159$	−5.10930	−0.405194
$160$	0	0
$161$	0	0
$162$	−15.1041	−1.18669
$163$	4.19136	0.328293	0.164146	−	0.986436i	$-0.447513\pi$
$163$	4.19136	0.328293	0.164146	+	0.986436i	$0.447513\pi$
$164$	−4.20440	−0.328309
$165$	0	0
$166$	25.9269	2.01232
$167$	10.9232	0.845261	0.422630	−	0.906302i	$-0.361107\pi$
$167$	10.9232	0.845261	0.422630	+	0.906302i	$0.361107\pi$
$168$	0	0
$169$	26.2902	2.02232
$170$	0	0
$171$	−1.72774	−0.132124
$172$	−13.4726	−1.02728
$173$	−3.59120	−0.273034	−0.136517	−	0.990638i	$-0.543591\pi$
$173$	−3.59120	−0.273034	−0.136517	+	0.990638i	$0.543591\pi$
$174$	18.4998	1.40247
$175$	0	0
$176$	−18.0545	−1.36091
$177$	3.17192	0.238416
$178$	15.6132	1.17026
$179$	−24.9086	−1.86176	−0.930879	−	0.365327i	$-0.880957\pi$
$179$	−24.9086	−1.86176	−0.930879	+	0.365327i	$0.880957\pi$
$180$	0	0
$181$	8.06379	0.599377	0.299688	−	0.954037i	$-0.403117\pi$
$181$	8.06379	0.599377	0.299688	+	0.954037i	$0.403117\pi$
$182$	0	0
$183$	−21.2369	−1.56988
$184$	16.0220	1.18116
$185$	0	0
$186$	3.17192	0.232577
$187$	−17.8359	−1.30429
$188$	−17.4088	−1.26967
$189$	0	0
$190$	0	0
$191$	22.6640	1.63991	0.819953	−	0.572431i	$-0.193999\pi$
$191$	22.6640	1.63991	0.819953	+	0.572431i	$0.193999\pi$
$192$	−7.75905	−0.559961
$193$	12.9452	0.931815	0.465907	−	0.884834i	$-0.345728\pi$
$193$	12.9452	0.931815	0.465907	+	0.884834i	$0.345728\pi$
$194$	20.0858	1.44208
$195$	0	0
$196$	0	0
$197$	18.8866	1.34562	0.672808	−	0.739817i	$-0.265088\pi$
$197$	18.8866	1.34562	0.672808	+	0.739817i	$0.265088\pi$
$198$	6.63555	0.471568
$199$	23.2499	1.64814	0.824071	−	0.566486i	$-0.191697\pi$
$199$	23.2499	1.64814	0.824071	+	0.566486i	$0.191697\pi$
$200$	−27.4543	−1.94131
$201$	9.13538	0.644360
$202$	−31.7680	−2.23519
$203$	0	0
$204$	32.6222	2.28401
$205$	0	0
$206$	−33.3540	−2.32388
$207$	−2.26819	−0.157650
$208$	33.0220	2.28966
$209$	−7.61727	−0.526898
$210$	0	0
$211$	28.5364	1.96453	0.982263	−	0.187510i	$-0.0600418\pi$
$211$	28.5364	1.96453	0.982263	+	0.187510i	$0.0600418\pi$
$212$	−14.4088	−0.989601
$213$	−11.2630	−0.771725
$214$	23.9452	1.63686
$215$	0	0
$216$	−30.9217	−2.10395
$217$	0	0
$218$	−21.2630	−1.44011
$219$	−18.6900	−1.26296
$220$	0	0
$221$	32.6222	2.19440
$222$	0.995931	0.0668425
$223$	5.29950	0.354881	0.177440	−	0.984132i	$-0.443218\pi$
$223$	5.29950	0.354881	0.177440	+	0.984132i	$0.443218\pi$
$224$	0	0
$225$	3.88663	0.259109
$226$	−41.0310	−2.72934
$227$	25.0728	1.66414	0.832069	−	0.554673i	$-0.187156\pi$
$227$	25.0728	1.66414	0.832069	+	0.554673i	$0.187156\pi$
$228$	13.9321	0.922679
$229$	−17.0090	−1.12398	−0.561992	−	0.827143i	$-0.689965\pi$
$229$	−17.0090	−1.12398	−0.561992	+	0.827143i	$0.689965\pi$
$230$	0	0
$231$	0	0
$232$	27.3540	1.79588
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−12.1365	−0.793391
$235$	0	0
$236$	8.94518	0.582282
$237$	−8.94518	−0.581052
$238$	0	0
$239$	−9.26295	−0.599171	−0.299585	−	0.954069i	$-0.596848\pi$
$239$	−9.26295	−0.599171	−0.299585	+	0.954069i	$0.596848\pi$
$240$	0	0
$241$	5.29950	0.341371	0.170685	−	0.985326i	$-0.445402\pi$
$241$	5.29950	0.341371	0.170685	+	0.985326i	$0.445402\pi$
$242$	1.85532	0.119264
$243$	7.85415	0.503844
$244$	−59.8904	−3.83409
$245$	0	0
$246$	−3.71354	−0.236767
$247$	13.9321	0.886481
$248$	4.69003	0.297817
$249$	15.5181	0.983420
$250$	0	0
$251$	−7.84635	−0.495257	−0.247629	−	0.968855i	$-0.579651\pi$
$251$	−7.84635	−0.495257	−0.247629	+	0.968855i	$0.579651\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	15.2954	0.959721
$255$	0	0
$256$	−32.5453	−2.03408
$257$	−19.9362	−1.24359	−0.621793	−	0.783181i	$-0.713596\pi$
$257$	−19.9362	−1.24359	−0.621793	+	0.783181i	$0.713596\pi$
$258$	−11.8997	−0.740841
$259$	0	0
$260$	0	0
$261$	−3.87242	−0.239697
$262$	8.94518	0.552635
$263$	27.7993	1.71418	0.857090	−	0.515166i	$-0.172270\pi$
$263$	27.7993	1.71418	0.857090	+	0.515166i	$0.172270\pi$
$264$	−28.0545	−1.72663
$265$	0	0
$266$	0	0
$267$	9.34502	0.571906
$268$	25.7628	1.57371
$269$	−31.8904	−1.94439	−0.972195	−	0.234173i	$-0.924762\pi$
$269$	−31.8904	−1.94439	−0.972195	+	0.234173i	$0.924762\pi$
$270$	0	0
$271$	18.6900	1.13534	0.567669	−	0.823257i	$-0.307845\pi$
$271$	18.6900	1.13534	0.567669	+	0.823257i	$0.307845\pi$
$272$	27.4178	1.66245
$273$	0	0
$274$	−41.8904	−2.53069
$275$	17.1354	1.03330
$276$	18.2902	1.10094
$277$	20.3450	1.22241	0.611207	−	0.791471i	$-0.290684\pi$
$277$	20.3450	1.22241	0.611207	+	0.791471i	$0.290684\pi$
$278$	44.4267	2.66454
$279$	−0.663954	−0.0397499
$280$	0	0
$281$	21.9635	1.31023	0.655115	−	0.755529i	$-0.272620\pi$
$281$	21.9635	1.31023	0.655115	+	0.755529i	$0.272620\pi$
$282$	−15.3763	−0.915647
$283$	20.5729	1.22293	0.611467	−	0.791270i	$-0.290580\pi$
$283$	20.5729	1.22293	0.611467	+	0.791270i	$0.290580\pi$
$284$	−31.7628	−1.88477
$285$	0	0
$286$	−53.5076	−3.16397
$287$	0	0
$288$	−1.66395	−0.0980494
$289$	10.0858	0.593282
$290$	0	0
$291$	12.0220	0.704743
$292$	−52.7080	−3.08450
$293$	−11.0728	−0.646877	−0.323439	−	0.946249i	$-0.604839\pi$
$293$	−11.0728	−0.646877	−0.323439	+	0.946249i	$0.604839\pi$
$294$	0	0
$295$	0	0
$296$	1.47259	0.0855926
$297$	19.2995	1.11987
$298$	−37.2264	−2.15647
$299$	18.2902	1.05775
$300$	−31.3409	−1.80947
$301$	0	0
$302$	−12.3178	−0.708808
$303$	−19.0142	−1.09234
$304$	11.7095	0.671584
$305$	0	0
$306$	−10.0768	−0.576054
$307$	−12.5364	−0.715489	−0.357744	−	0.933820i	$-0.616454\pi$
$307$	−12.5364	−0.715489	−0.357744	+	0.933820i	$0.616454\pi$
$308$	0	0
$309$	−19.9635	−1.13568
$310$	0	0
$311$	−24.5584	−1.39258	−0.696289	−	0.717761i	$-0.745167\pi$
$311$	−24.5584	−1.39258	−0.696289	+	0.717761i	$0.745167\pi$
$312$	51.3122	2.90498
$313$	13.9049	0.785951	0.392976	−	0.919549i	$-0.371446\pi$
$313$	13.9049	0.785951	0.392976	+	0.919549i	$0.371446\pi$
$314$	23.5625	1.32971
$315$	0	0
$316$	−25.2264	−1.41910
$317$	−5.14585	−0.289020	−0.144510	−	0.989503i	$-0.546161\pi$
$317$	−5.14585	−0.289020	−0.144510	+	0.989503i	$0.546161\pi$
$318$	−12.7266	−0.713671
$319$	−17.0728	−0.955891
$320$	0	0
$321$	14.3320	0.799933
$322$	0	0
$323$	11.5677	0.643643
$324$	−25.4946	−1.41637
$325$	−31.3409	−1.73848
$326$	10.4401	0.578224
$327$	−12.7266	−0.703781
$328$	−5.49086	−0.303182
$329$	0	0
$330$	0	0
$331$	−26.8176	−1.47403	−0.737014	−	0.675877i	$-0.763765\pi$
$331$	−26.8176	−1.47403	−0.737014	+	0.675877i	$0.763765\pi$
$332$	43.7628	2.40180
$333$	−0.208470	−0.0114241
$334$	27.2081	1.48876
$335$	0	0
$336$	0	0
$337$	4.34502	0.236688	0.118344	−	0.992973i	$-0.462241\pi$
$337$	4.34502	0.236688	0.118344	+	0.992973i	$0.462241\pi$
$338$	65.4853	3.56193
$339$	−24.5584	−1.33383
$340$	0	0
$341$	−2.92724	−0.158519
$342$	−4.30357	−0.232710
$343$	0	0
$344$	−17.5949	−0.948655
$345$	0	0
$346$	−8.94518	−0.480896
$347$	−21.3174	−1.14438	−0.572190	−	0.820121i	$-0.693906\pi$
$347$	−21.3174	−1.14438	−0.572190	+	0.820121i	$0.693906\pi$
$348$	31.2264	1.67391
$349$	34.8721	1.86666	0.933330	−	0.359019i	$-0.116889\pi$
$349$	34.8721	1.86666	0.933330	+	0.359019i	$0.116889\pi$
$350$	0	0
$351$	−35.2992	−1.88413
$352$	−7.33605	−0.391013
$353$	15.0988	0.803630	0.401815	−	0.915721i	$-0.368380\pi$
$353$	15.0988	0.803630	0.401815	+	0.915721i	$0.368380\pi$
$354$	7.90083	0.419924
$355$	0	0
$356$	26.3540	1.39676
$357$	0	0
$358$	−62.0440	−3.27913
$359$	−29.2499	−1.54375	−0.771876	−	0.635773i	$-0.780681\pi$
$359$	−29.2499	−1.54375	−0.771876	+	0.635773i	$0.780681\pi$
$360$	0	0
$361$	−14.0597	−0.739985
$362$	20.0858	1.05569
$363$	1.11047	0.0582845
$364$	0	0
$365$	0	0
$366$	−52.8982	−2.76503
$367$	21.4271	1.11848	0.559242	−	0.829004i	$-0.311092\pi$
$367$	21.4271	1.11848	0.559242	+	0.829004i	$0.311092\pi$
$368$	15.3723	0.801334
$369$	0.777326	0.0404660
$370$	0	0
$371$	0	0
$372$	5.35398	0.277591
$373$	31.8944	1.65143	0.825716	−	0.564087i	$-0.190772\pi$
$373$	31.8944	1.65143	0.825716	+	0.564087i	$0.190772\pi$
$374$	−44.4267	−2.29725
$375$	0	0
$376$	−22.7355	−1.17250
$377$	31.2264	1.60824
$378$	0	0
$379$	3.24618	0.166745	0.0833726	−	0.996518i	$-0.473431\pi$
$379$	3.24618	0.166745	0.0833726	+	0.996518i	$0.473431\pi$
$380$	0	0
$381$	9.15482	0.469016
$382$	56.4528	2.88838
$383$	−11.6550	−0.595542	−0.297771	−	0.954637i	$-0.596243\pi$
$383$	−11.6550	−0.595542	−0.297771	+	0.954637i	$0.596243\pi$
$384$	−25.7095	−1.31198
$385$	0	0
$386$	32.2447	1.64121
$387$	2.49086	0.126618
$388$	33.9034	1.72118
$389$	18.7941	0.952899	0.476449	−	0.879202i	$-0.341924\pi$
$389$	18.7941	0.952899	0.476449	+	0.879202i	$0.341924\pi$
$390$	0	0
$391$	15.1861	0.767996
$392$	0	0
$393$	5.35398	0.270073
$394$	47.0440	2.37004
$395$	0	0
$396$	11.2003	0.562838
$397$	−18.7445	−0.940760	−0.470380	−	0.882464i	$-0.655883\pi$
$397$	−18.7445	−0.940760	−0.470380	+	0.882464i	$0.655883\pi$
$398$	57.9124	2.90288
$399$	0	0
$400$	−26.3409	−1.31705
$401$	−15.6848	−0.783261	−0.391631	−	0.920123i	$-0.628089\pi$
$401$	−15.6848	−0.783261	−0.391631	+	0.920123i	$0.628089\pi$
$402$	22.7550	1.13492
$403$	5.35398	0.266701
$404$	−53.6222	−2.66780
$405$	0	0
$406$	0	0
$407$	−0.919105	−0.0455583
$408$	42.6039	2.10921
$409$	29.5181	1.45958	0.729788	−	0.683673i	$-0.239619\pi$
$409$	29.5181	1.45958	0.729788	+	0.683673i	$0.239619\pi$
$410$	0	0
$411$	−25.0728	−1.23675
$412$	−56.2992	−2.77366
$413$	0	0
$414$	−5.64975	−0.277670
$415$	0	0
$416$	13.4178	0.657860
$417$	26.5909	1.30216
$418$	−18.9736	−0.928029
$419$	19.3540	0.945504	0.472752	−	0.881196i	$-0.343261\pi$
$419$	19.3540	0.945504	0.472752	+	0.881196i	$0.343261\pi$
$420$	0	0
$421$	20.2552	0.987176	0.493588	−	0.869696i	$-0.335685\pi$
$421$	20.2552	0.987176	0.493588	+	0.869696i	$0.335685\pi$
$422$	71.0802	3.46013
$423$	3.21861	0.156494
$424$	−18.8176	−0.913864
$425$	−26.0220	−1.26225
$426$	−28.0545	−1.35924
$427$	0	0
$428$	40.4178	1.95367
$429$	−32.0261	−1.54623
$430$	0	0
$431$	−18.5259	−0.892362	−0.446181	−	0.894943i	$-0.647216\pi$
$431$	−18.5259	−0.892362	−0.446181	+	0.894943i	$0.647216\pi$
$432$	−29.6677	−1.42739
$433$	−7.18239	−0.345164	−0.172582	−	0.984995i	$-0.555211\pi$
$433$	−7.18239	−0.345164	−0.172582	+	0.984995i	$0.555211\pi$
$434$	0	0
$435$	0	0
$436$	−35.8904	−1.71884
$437$	6.48563	0.310250
$438$	−46.5543	−2.22445
$439$	2.70980	0.129332	0.0646659	−	0.997907i	$-0.479402\pi$
$439$	2.70980	0.129332	0.0646659	+	0.997907i	$0.479402\pi$
$440$	0	0
$441$	0	0
$442$	81.2574	3.86502
$443$	31.4946	1.49635	0.748177	−	0.663499i	$-0.230930\pi$
$443$	31.4946	1.49635	0.748177	+	0.663499i	$0.230930\pi$
$444$	1.68106	0.0797796
$445$	0	0
$446$	13.2003	0.625054
$447$	−22.2812	−1.05387
$448$	0	0
$449$	−16.4726	−0.777390	−0.388695	−	0.921367i	$-0.627074\pi$
$449$	−16.4726	−0.777390	−0.388695	+	0.921367i	$0.627074\pi$
$450$	9.68106	0.456370
$451$	3.42708	0.161375
$452$	−69.2574	−3.25759
$453$	−7.37259	−0.346395
$454$	62.4528	2.93106
$455$	0	0
$456$	18.1951	0.852064
$457$	−7.76279	−0.363128	−0.181564	−	0.983379i	$-0.558116\pi$
$457$	−7.76279	−0.363128	−0.181564	+	0.983379i	$0.558116\pi$
$458$	−42.3670	−1.97968
$459$	−29.3085	−1.36800
$460$	0	0
$461$	−22.2812	−1.03774	−0.518870	−	0.854853i	$-0.673647\pi$
$461$	−22.2812	−1.03774	−0.518870	+	0.854853i	$0.673647\pi$
$462$	0	0
$463$	−13.0988	−0.608754	−0.304377	−	0.952552i	$-0.598448\pi$
$463$	−13.0988	−0.608754	−0.304377	+	0.952552i	$0.598448\pi$
$464$	26.2447	1.21838
$465$	0	0
$466$	−14.9452	−0.692322
$467$	15.0988	0.698691	0.349345	−	0.936994i	$-0.386404\pi$
$467$	15.0988	0.698691	0.349345	+	0.936994i	$0.386404\pi$
$468$	−20.4856	−0.946949
$469$	0	0
$470$	0	0
$471$	14.1029	0.649827
$472$	11.6822	0.537718
$473$	10.9817	0.504940
$474$	−22.2812	−1.02341
$475$	−11.1134	−0.509916
$476$	0	0
$477$	2.66395	0.121974
$478$	−23.0728	−1.05532
$479$	2.70980	0.123814	0.0619070	−	0.998082i	$-0.480282\pi$
$479$	2.70980	0.123814	0.0619070	+	0.998082i	$0.480282\pi$
$480$	0	0
$481$	1.68106	0.0766498
$482$	13.2003	0.601259
$483$	0	0
$484$	3.13164	0.142347
$485$	0	0
$486$	19.5636	0.887424
$487$	−24.5845	−1.11403	−0.557014	−	0.830503i	$-0.688053\pi$
$487$	−24.5845	−1.11403	−0.557014	+	0.830503i	$0.688053\pi$
$488$	−78.2156	−3.54065
$489$	6.24875	0.282578
$490$	0	0
$491$	4.95449	0.223593	0.111796	−	0.993731i	$-0.464340\pi$
$491$	4.95449	0.223593	0.111796	+	0.993731i	$0.464340\pi$
$492$	−6.26819	−0.282592
$493$	25.9269	1.16769
$494$	34.7031	1.56136
$495$	0	0
$496$	4.49983	0.202048
$497$	0	0
$498$	38.6535	1.73210
$499$	−25.3540	−1.13500	−0.567500	−	0.823373i	$-0.692089\pi$
$499$	−25.3540	−1.13500	−0.567500	+	0.823373i	$0.692089\pi$
$500$	0	0
$501$	16.2850	0.727558
$502$	−19.5442	−0.872300
$503$	18.1078	0.807387	0.403694	−	0.914894i	$-0.367726\pi$
$503$	18.1078	0.807387	0.403694	+	0.914894i	$0.367726\pi$
$504$	0	0
$505$	0	0
$506$	−24.9086	−1.10732
$507$	39.1951	1.74072
$508$	25.8176	1.14547
$509$	−38.5233	−1.70752	−0.853759	−	0.520669i	$-0.825683\pi$
$509$	−38.5233	−1.70752	−0.853759	+	0.520669i	$0.825683\pi$
$510$	0	0
$511$	0	0
$512$	−46.5767	−2.05842
$513$	−12.5169	−0.552636
$514$	−49.6584	−2.19034
$515$	0	0
$516$	−20.0858	−0.884228
$517$	14.1902	0.624084
$518$	0	0
$519$	−5.35398	−0.235014
$520$	0	0
$521$	−35.3670	−1.54946	−0.774729	−	0.632294i	$-0.782114\pi$
$521$	−35.3670	−1.54946	−0.774729	+	0.632294i	$0.782114\pi$
$522$	−9.64568	−0.422180
$523$	6.38273	0.279097	0.139549	−	0.990215i	$-0.455435\pi$
$523$	6.38273	0.279097	0.139549	+	0.990215i	$0.455435\pi$
$524$	15.0988	0.659596
$525$	0	0
$526$	69.2443	3.01920
$527$	4.44535	0.193642
$528$	−26.9168	−1.17140
$529$	−14.4856	−0.629810
$530$	0	0
$531$	−1.65382	−0.0717696
$532$	0	0
$533$	−6.26819	−0.271505
$534$	23.2772	1.00730
$535$	0	0
$536$	33.6457	1.45327
$537$	−37.1354	−1.60251
$538$	−79.4345	−3.42467
$539$	0	0
$540$	0	0
$541$	−2.55839	−0.109994	−0.0549968	−	0.998487i	$-0.517515\pi$
$541$	−2.55839	−0.109994	−0.0549968	+	0.998487i	$0.517515\pi$
$542$	46.5543	1.99968
$543$	12.0220	0.515914
$544$	11.1406	0.477650
$545$	0	0
$546$	0	0
$547$	8.81761	0.377014	0.188507	−	0.982072i	$-0.439635\pi$
$547$	8.81761	0.377014	0.188507	+	0.982072i	$0.439635\pi$
$548$	−70.7080	−3.02049
$549$	11.0728	0.472574
$550$	42.6819	1.81996
$551$	11.0728	0.471715
$552$	23.8866	1.01668
$553$	0	0
$554$	50.6767	2.15304
$555$	0	0
$556$	74.9892	3.18025
$557$	35.8904	1.52072	0.760362	−	0.649500i	$-0.225022\pi$
$557$	35.8904	1.52072	0.760362	+	0.649500i	$0.225022\pi$
$558$	−1.65382	−0.0700117
$559$	−20.0858	−0.849539
$560$	0	0
$561$	−26.5909	−1.12267
$562$	54.7080	2.30772
$563$	12.6509	0.533173	0.266586	−	0.963811i	$-0.414104\pi$
$563$	12.6509	0.533173	0.266586	+	0.963811i	$0.414104\pi$
$564$	−25.9542	−1.09287
$565$	0	0
$566$	51.2443	2.15396
$567$	0	0
$568$	−41.4816	−1.74053
$569$	−18.9306	−0.793614	−0.396807	−	0.917902i	$-0.629882\pi$
$569$	−18.9306	−0.793614	−0.396807	+	0.917902i	$0.629882\pi$
$570$	0	0
$571$	−29.7889	−1.24663	−0.623313	−	0.781973i	$-0.714214\pi$
$571$	−29.7889	−1.24663	−0.623313	+	0.781973i	$0.714214\pi$
$572$	−90.3171	−3.77635
$573$	33.7889	1.41155
$574$	0	0
$575$	−14.5897	−0.608432
$576$	4.04551	0.168563
$577$	−0.826910	−0.0344247	−0.0172123	−	0.999852i	$-0.505479\pi$
$577$	−0.826910	−0.0344247	−0.0172123	+	0.999852i	$0.505479\pi$
$578$	25.1223	1.04495
$579$	19.2995	0.802060
$580$	0	0
$581$	0	0
$582$	29.9452	1.24127
$583$	11.7448	0.486422
$584$	−68.8355	−2.84844
$585$	0	0
$586$	−27.5807	−1.13935
$587$	−34.9672	−1.44325	−0.721625	−	0.692284i	$-0.756605\pi$
$587$	−34.9672	−1.44325	−0.721625	+	0.692284i	$0.756605\pi$
$588$	0	0
$589$	1.89850	0.0782264
$590$	0	0
$591$	28.1574	1.15824
$592$	1.41287	0.0580687
$593$	28.1846	1.15740	0.578702	−	0.815539i	$-0.303559\pi$
$593$	28.1846	1.15740	0.578702	+	0.815539i	$0.303559\pi$
$594$	48.0724	1.97244
$595$	0	0
$596$	−62.8355	−2.57384
$597$	34.6625	1.41864
$598$	45.5584	1.86302
$599$	8.99477	0.367516	0.183758	−	0.982971i	$-0.441174\pi$
$599$	8.99477	0.367516	0.183758	+	0.982971i	$0.441174\pi$
$600$	−40.9306	−1.67099
$601$	22.7956	0.929852	0.464926	−	0.885350i	$-0.346081\pi$
$601$	22.7956	0.929852	0.464926	+	0.885350i	$0.346081\pi$
$602$	0	0
$603$	−4.76312	−0.193969
$604$	−20.7915	−0.845995
$605$	0	0
$606$	−47.3618	−1.92394
$607$	38.4088	1.55897	0.779483	−	0.626424i	$-0.215482\pi$
$607$	38.4088	1.55897	0.779483	+	0.626424i	$0.215482\pi$
$608$	4.75789	0.192958
$609$	0	0
$610$	0	0
$611$	−25.9542	−1.04999
$612$	−17.0090	−0.687547
$613$	−21.2902	−0.859903	−0.429951	−	0.902852i	$-0.641469\pi$
$613$	−21.2902	−0.859903	−0.429951	+	0.902852i	$0.641469\pi$
$614$	−31.2264	−1.26020
$615$	0	0
$616$	0	0
$617$	17.8579	0.718931	0.359466	−	0.933158i	$-0.382959\pi$
$617$	17.8579	0.718931	0.359466	+	0.933158i	$0.382959\pi$
$618$	−49.7262	−2.00028
$619$	−15.9530	−0.641205	−0.320602	−	0.947214i	$-0.603885\pi$
$619$	−15.9530	−0.641205	−0.320602	+	0.947214i	$0.603885\pi$
$620$	0	0
$621$	−16.4323	−0.659406
$622$	−61.1716	−2.45276
$623$	0	0
$624$	49.2313	1.97083
$625$	25.0000	1.00000
$626$	34.6352	1.38430
$627$	−11.3563	−0.453528
$628$	39.7718	1.58707
$629$	1.39576	0.0556528
$630$	0	0
$631$	−27.2134	−1.08335	−0.541674	−	0.840589i	$-0.682209\pi$
$631$	−27.2134	−1.08335	−0.541674	+	0.840589i	$0.682209\pi$
$632$	−32.9452	−1.31049
$633$	42.5438	1.69097
$634$	−12.8176	−0.509052
$635$	0	0
$636$	−21.4816	−0.851799
$637$	0	0
$638$	−42.5259	−1.68362
$639$	5.87242	0.232310
$640$	0	0
$641$	−12.5468	−0.495571	−0.247785	−	0.968815i	$-0.579703\pi$
$641$	−12.5468	−0.495571	−0.247785	+	0.968815i	$0.579703\pi$
$642$	35.6990	1.40893
$643$	17.7758	0.701010	0.350505	−	0.936561i	$-0.386010\pi$
$643$	17.7758	0.701010	0.350505	+	0.936561i	$0.386010\pi$
$644$	0	0
$645$	0	0
$646$	28.8135	1.13365
$647$	−11.4375	−0.449656	−0.224828	−	0.974398i	$-0.572182\pi$
$647$	−11.4375	−0.449656	−0.224828	+	0.974398i	$0.572182\pi$
$648$	−33.2954	−1.30797
$649$	−7.29136	−0.286211
$650$	−78.0660	−3.06200
$651$	0	0
$652$	17.6222	0.690138
$653$	4.82808	0.188937	0.0944686	−	0.995528i	$-0.469885\pi$
$653$	4.82808	0.188937	0.0944686	+	0.995528i	$0.469885\pi$
$654$	−31.7002	−1.23957
$655$	0	0
$656$	−5.26819	−0.205688
$657$	9.74485	0.380183
$658$	0	0
$659$	−36.2186	−1.41088	−0.705438	−	0.708771i	$-0.749250\pi$
$659$	−36.2186	−1.41088	−0.705438	+	0.708771i	$0.749250\pi$
$660$	0	0
$661$	23.3100	0.906653	0.453326	−	0.891345i	$-0.350237\pi$
$661$	23.3100	0.906653	0.453326	+	0.891345i	$0.350237\pi$
$662$	−66.7990	−2.59622
$663$	48.6352	1.88884
$664$	57.1533	2.21798
$665$	0	0
$666$	−0.519271	−0.0201214
$667$	14.5364	0.562851
$668$	45.9254	1.77691
$669$	7.90083	0.305464
$670$	0	0
$671$	48.8176	1.88458
$672$	0	0
$673$	−36.4088	−1.40346	−0.701728	−	0.712445i	$-0.747588\pi$
$673$	−36.4088	−1.40346	−0.701728	+	0.712445i	$0.747588\pi$
$674$	10.8228	0.416880
$675$	28.1574	1.08378
$676$	110.535	4.25133
$677$	0.663954	0.0255178	0.0127589	−	0.999919i	$-0.495939\pi$
$677$	0.663954	0.0255178	0.0127589	+	0.999919i	$0.495939\pi$
$678$	−61.1716	−2.34928
$679$	0	0
$680$	0	0
$681$	37.3801	1.43241
$682$	−7.29136	−0.279201
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−7.26412	−0.277751
$685$	0	0
$686$	0	0
$687$	−25.3581	−0.967470
$688$	−16.8814	−0.643597
$689$	−21.4816	−0.818382
$690$	0	0
$691$	42.8486	1.63004	0.815018	−	0.579435i	$-0.196727\pi$
$691$	42.8486	1.63004	0.815018	+	0.579435i	$0.196727\pi$
$692$	−15.0988	−0.573972
$693$	0	0
$694$	−53.0988	−2.01560
$695$	0	0
$696$	40.7811	1.54580
$697$	−5.20440	−0.197131
$698$	86.8616	3.28776
$699$	−8.94518	−0.338338
$700$	0	0
$701$	0.596431	0.0225269	0.0112635	−	0.999937i	$-0.496415\pi$
$701$	0.596431	0.0225269	0.0112635	+	0.999937i	$0.496415\pi$
$702$	−87.9254	−3.31853
$703$	0.596097	0.0224822
$704$	17.8359	0.672215
$705$	0	0
$706$	37.6091	1.41544
$707$	0	0
$708$	13.3360	0.501199
$709$	42.4528	1.59435	0.797175	−	0.603749i	$-0.206327\pi$
$709$	42.4528	1.59435	0.797175	+	0.603749i	$0.206327\pi$
$710$	0	0
$711$	4.66395	0.174912
$712$	34.4178	1.28986
$713$	2.49236	0.0933397
$714$	0	0
$715$	0	0
$716$	−104.726	−3.91379
$717$	−13.8098	−0.515737
$718$	−72.8576	−2.71902
$719$	7.18239	0.267858	0.133929	−	0.990991i	$-0.457241\pi$
$719$	7.18239	0.267858	0.133929	+	0.990991i	$0.457241\pi$
$720$	0	0
$721$	0	0
$722$	−35.0208	−1.30334
$723$	7.90083	0.293835
$724$	33.9034	1.26001
$725$	−24.9086	−0.925084
$726$	2.76603	0.102657
$727$	−24.5584	−0.910820	−0.455410	−	0.890282i	$-0.650507\pi$
$727$	−24.5584	−0.910820	−0.455410	+	0.890282i	$0.650507\pi$
$728$	0	0
$729$	29.9008	1.10744
$730$	0	0
$731$	−16.6770	−0.616821
$732$	−89.2884	−3.30019
$733$	49.0623	1.81216	0.906078	−	0.423110i	$-0.139061\pi$
$733$	49.0623	1.81216	0.906078	+	0.423110i	$0.139061\pi$
$734$	53.3719	1.96999
$735$	0	0
$736$	6.24618	0.230237
$737$	−20.9997	−0.773533
$738$	1.93621	0.0712730
$739$	17.8855	0.657927	0.328964	−	0.944343i	$-0.393301\pi$
$739$	17.8855	0.657927	0.328964	+	0.944343i	$0.393301\pi$
$740$	0	0
$741$	20.7709	0.763039
$742$	0	0
$743$	47.4305	1.74006	0.870028	−	0.493003i	$-0.164101\pi$
$743$	47.4305	1.74006	0.870028	+	0.493003i	$0.164101\pi$
$744$	6.99220	0.256346
$745$	0	0
$746$	79.4447	2.90868
$747$	−8.09103	−0.296035
$748$	−74.9892	−2.74188
$749$	0	0
$750$	0	0
$751$	−1.74485	−0.0636704	−0.0318352	−	0.999493i	$-0.510135\pi$
$751$	−1.74485	−0.0636704	−0.0318352	+	0.999493i	$0.510135\pi$
$752$	−21.8135	−0.795458
$753$	−11.6978	−0.426293
$754$	77.7807	2.83261
$755$	0	0
$756$	0	0
$757$	−26.6640	−0.969118	−0.484559	−	0.874759i	$-0.661020\pi$
$757$	−26.6640	−0.969118	−0.484559	+	0.874759i	$0.661020\pi$
$758$	8.08580	0.293689
$759$	−14.9086	−0.541149
$760$	0	0
$761$	14.2992	0.518344	0.259172	−	0.965831i	$-0.416550\pi$
$761$	14.2992	0.518344	0.259172	+	0.965831i	$0.416550\pi$
$762$	22.8034	0.826080
$763$	0	0
$764$	95.2884	3.44741
$765$	0	0
$766$	−29.0310	−1.04893
$767$	13.3360	0.481537
$768$	−48.5207	−1.75084
$769$	−43.7628	−1.57813	−0.789063	−	0.614312i	$-0.789434\pi$
$769$	−43.7628	−1.57813	−0.789063	+	0.614312i	$0.789434\pi$
$770$	0	0
$771$	−29.7222	−1.07042
$772$	54.4267	1.95886
$773$	14.5845	0.524567	0.262283	−	0.964991i	$-0.415524\pi$
$773$	14.5845	0.524567	0.262283	+	0.964991i	$0.415524\pi$
$774$	6.20440	0.223013
$775$	−4.27076	−0.153410
$776$	44.2772	1.58946
$777$	0	0
$778$	46.8135	1.67835
$779$	−2.22267	−0.0796356
$780$	0	0
$781$	25.8904	0.926430
$782$	37.8266	1.35268
$783$	−28.0545	−1.00259
$784$	0	0
$785$	0	0
$786$	13.3360	0.475681
$787$	20.5729	0.733346	0.366673	−	0.930350i	$-0.380497\pi$
$787$	20.5729	0.733346	0.366673	+	0.930350i	$0.380497\pi$
$788$	79.4070	2.82876
$789$	41.4450	1.47548
$790$	0	0
$791$	0	0
$792$	14.6274	0.519762
$793$	−89.2884	−3.17072
$794$	−46.6900	−1.65697
$795$	0	0
$796$	97.7520	3.46473
$797$	−35.9164	−1.27223	−0.636113	−	0.771596i	$-0.719459\pi$
$797$	−35.9164	−1.27223	−0.636113	+	0.771596i	$0.719459\pi$
$798$	0	0
$799$	−21.5494	−0.762364
$800$	−10.7031	−0.378411
$801$	−4.87242	−0.172159
$802$	−39.0687	−1.37956
$803$	42.9631	1.51614
$804$	38.4088	1.35457
$805$	0	0
$806$	13.3360	0.469742
$807$	−47.5442	−1.67363
$808$	−70.0295	−2.46363
$809$	15.4375	0.542755	0.271378	−	0.962473i	$-0.412521\pi$
$809$	15.4375	0.542755	0.271378	+	0.962473i	$0.412521\pi$
$810$	0	0
$811$	−33.7889	−1.18649	−0.593244	−	0.805023i	$-0.702153\pi$
$811$	−33.7889	−1.18649	−0.593244	+	0.805023i	$0.702153\pi$
$812$	0	0
$813$	27.8643	0.977243
$814$	−2.28937	−0.0802422
$815$	0	0
$816$	40.8762	1.43095
$817$	−7.12234	−0.249179
$818$	73.5256	2.57076
$819$	0	0
$820$	0	0
$821$	4.08463	0.142555	0.0712773	−	0.997457i	$-0.477292\pi$
$821$	4.08463	0.142555	0.0712773	+	0.997457i	$0.477292\pi$
$822$	−62.4528	−2.17829
$823$	8.69003	0.302915	0.151458	−	0.988464i	$-0.451603\pi$
$823$	8.69003	0.302915	0.151458	+	0.988464i	$0.451603\pi$
$824$	−73.5256	−2.56138
$825$	25.5465	0.889415
$826$	0	0
$827$	−4.81761	−0.167525	−0.0837623	−	0.996486i	$-0.526694\pi$
$827$	−4.81761	−0.167525	−0.0837623	+	0.996486i	$0.526694\pi$
$828$	−9.53638	−0.331412
$829$	−12.4819	−0.433514	−0.216757	−	0.976226i	$-0.569548\pi$
$829$	−12.4819	−0.433514	−0.216757	+	0.976226i	$0.569548\pi$
$830$	0	0
$831$	30.3316	1.05219
$832$	−32.6222	−1.13097
$833$	0	0
$834$	66.2342	2.29350
$835$	0	0
$836$	−32.0261	−1.10765
$837$	−4.81014	−0.166263
$838$	48.2081	1.66532
$839$	−29.5453	−1.02002	−0.510009	−	0.860169i	$-0.670358\pi$
$839$	−29.5453	−1.02002	−0.510009	+	0.860169i	$0.670358\pi$
$840$	0	0
$841$	−4.18239	−0.144220
$842$	50.4528	1.73872
$843$	32.7445	1.12778
$844$	119.978	4.12983
$845$	0	0
$846$	8.01711	0.275634
$847$	0	0
$848$	−18.0545	−0.619994
$849$	30.6714	1.05264
$850$	−64.8173	−2.22321
$851$	0.782560	0.0268258
$852$	−47.3540	−1.62232
$853$	16.0731	0.550332	0.275166	−	0.961397i	$-0.411267\pi$
$853$	16.0731	0.550332	0.275166	+	0.961397i	$0.411267\pi$
$854$	0	0
$855$	0	0
$856$	52.7848	1.80415
$857$	9.74485	0.332878	0.166439	−	0.986052i	$-0.446773\pi$
$857$	9.74485	0.332878	0.166439	+	0.986052i	$0.446773\pi$
$858$	−79.7726	−2.72339
$859$	−50.8251	−1.73413	−0.867065	−	0.498196i	$-0.833996\pi$
$859$	−50.8251	−1.73413	−0.867065	+	0.498196i	$0.833996\pi$
$860$	0	0
$861$	0	0
$862$	−46.1455	−1.57172
$863$	3.74485	0.127476	0.0637381	−	0.997967i	$-0.479698\pi$
$863$	3.74485	0.127476	0.0637381	+	0.997967i	$0.479698\pi$
$864$	−12.0548	−0.410113
$865$	0	0
$866$	−17.8904	−0.607939
$867$	15.0365	0.510668
$868$	0	0
$869$	20.5625	0.697534
$870$	0	0
$871$	38.4088	1.30143
$872$	−46.8721	−1.58729
$873$	−6.26819	−0.212146
$874$	16.1548	0.546445
$875$	0	0
$876$	−78.5804	−2.65499
$877$	12.9075	0.435854	0.217927	−	0.975965i	$-0.430070\pi$
$877$	12.9075	0.435854	0.217927	+	0.975965i	$0.430070\pi$
$878$	6.74975	0.227793
$879$	−16.5080	−0.556800
$880$	0	0
$881$	−15.3279	−0.516410	−0.258205	−	0.966090i	$-0.583131\pi$
$881$	−15.3279	−0.516410	−0.258205	+	0.966090i	$0.583131\pi$
$882$	0	0
$883$	24.5103	0.824837	0.412419	−	0.910994i	$-0.364684\pi$
$883$	24.5103	0.824837	0.412419	+	0.910994i	$0.364684\pi$
$884$	137.157	4.61308
$885$	0	0
$886$	78.4487	2.63554
$887$	58.3122	1.95793	0.978966	−	0.204023i	$-0.0654017\pi$
$887$	58.3122	1.95793	0.978966	+	0.204023i	$0.0654017\pi$
$888$	2.19543	0.0736739
$889$	0	0
$890$	0	0
$891$	20.7811	0.696192
$892$	22.2812	0.746031
$893$	−9.20324	−0.307975
$894$	−55.4995	−1.85618
$895$	0	0
$896$	0	0
$897$	27.2682	0.910458
$898$	−41.0310	−1.36922
$899$	4.25515	0.141917
$900$	16.3409	0.544698
$901$	−17.8359	−0.594199
$902$	8.53638	0.284230
$903$	0	0
$904$	−90.4487	−3.00828
$905$	0	0
$906$	−18.3641	−0.610107
$907$	27.4946	0.912943	0.456472	−	0.889738i	$-0.349113\pi$
$907$	27.4946	0.912943	0.456472	+	0.889738i	$0.349113\pi$
$908$	105.416	3.49835
$909$	9.91387	0.328822
$910$	0	0
$911$	27.8318	0.922109	0.461055	−	0.887372i	$-0.347471\pi$
$911$	27.8318	0.922109	0.461055	+	0.887372i	$0.347471\pi$
$912$	17.4572	0.578066
$913$	−35.6718	−1.18056
$914$	−19.3360	−0.639580
$915$	0	0
$916$	−71.5125	−2.36284
$917$	0	0
$918$	−73.0034	−2.40947
$919$	19.5076	0.643498	0.321749	−	0.946825i	$-0.395729\pi$
$919$	19.5076	0.643498	0.321749	+	0.946825i	$0.395729\pi$
$920$	0	0
$921$	−18.6900	−0.615857
$922$	−55.4995	−1.82778
$923$	−47.3540	−1.55868
$924$	0	0
$925$	−1.34095	−0.0440900
$926$	−32.6274	−1.07220
$927$	10.4088	0.341870
$928$	10.6640	0.350061
$929$	0.881394	0.0289176	0.0144588	−	0.999895i	$-0.495397\pi$
$929$	0.881394	0.0289176	0.0144588	+	0.999895i	$0.495397\pi$
$930$	0	0
$931$	0	0
$932$	−25.2264	−0.826319
$933$	−36.6132	−1.19866
$934$	37.6091	1.23061
$935$	0	0
$936$	−26.7538	−0.874476
$937$	−8.20557	−0.268064	−0.134032	−	0.990977i	$-0.542793\pi$
$937$	−8.20557	−0.268064	−0.134032	+	0.990977i	$0.542793\pi$
$938$	0	0
$939$	20.7303	0.676508
$940$	0	0
$941$	9.30997	0.303496	0.151748	−	0.988419i	$-0.451510\pi$
$941$	9.30997	0.303496	0.151748	+	0.988419i	$0.451510\pi$
$942$	35.1284	1.14455
$943$	−2.91794	−0.0950212
$944$	11.2085	0.364805
$945$	0	0
$946$	27.3540	0.889355
$947$	10.1548	0.329987	0.164994	−	0.986295i	$-0.447240\pi$
$947$	10.1548	0.329987	0.164994	+	0.986295i	$0.447240\pi$
$948$	−37.6091	−1.22149
$949$	−78.5804	−2.55083
$950$	−27.6819	−0.898119
$951$	−7.67176	−0.248774
$952$	0	0
$953$	−4.75755	−0.154112	−0.0770561	−	0.997027i	$-0.524552\pi$
$953$	−4.75755	−0.154112	−0.0770561	+	0.997027i	$0.524552\pi$
$954$	6.63555	0.214834
$955$	0	0
$956$	−38.9452	−1.25958
$957$	−25.4532	−0.822784
$958$	6.74975	0.218075
$959$	0	0
$960$	0	0
$961$	−30.2704	−0.976465
$962$	4.18729	0.135004
$963$	−7.47259	−0.240801
$964$	22.2812	0.717630
$965$	0	0
$966$	0	0
$967$	34.9892	1.12518	0.562588	−	0.826737i	$-0.309806\pi$
$967$	34.9892	1.12518	0.562588	+	0.826737i	$0.309806\pi$
$968$	4.08986	0.131453
$969$	17.2458	0.554016
$970$	0	0
$971$	−6.96719	−0.223588	−0.111794	−	0.993731i	$-0.535660\pi$
$971$	−6.96719	−0.223588	−0.111794	+	0.993731i	$0.535660\pi$
$972$	33.0220	1.05918
$973$	0	0
$974$	−61.2365	−1.96215
$975$	−46.7251	−1.49640
$976$	−75.0437	−2.40209
$977$	33.1533	1.06067	0.530334	−	0.847789i	$-0.322066\pi$
$977$	33.1533	1.06067	0.530334	+	0.847789i	$0.322066\pi$
$978$	15.5648	0.497707
$979$	−21.4816	−0.686554
$980$	0	0
$981$	6.63555	0.211857
$982$	12.3409	0.393815
$983$	18.0261	0.574943	0.287471	−	0.957789i	$-0.407185\pi$
$983$	18.0261	0.574943	0.287471	+	0.957789i	$0.407185\pi$
$984$	−8.18613	−0.260964
$985$	0	0
$986$	64.5804	2.05666
$987$	0	0
$988$	58.5763	1.86356
$989$	−9.35025	−0.297321
$990$	0	0
$991$	48.0620	1.52674	0.763369	−	0.645962i	$-0.223544\pi$
$991$	48.0620	1.52674	0.763369	+	0.645962i	$0.223544\pi$
$992$	1.82841	0.0580520
$993$	−39.9814	−1.26877
$994$	0	0
$995$	0	0
$996$	65.2443	2.06735
$997$	−59.8631	−1.89588	−0.947942	−	0.318443i	$-0.896840\pi$
$997$	−59.8631	−1.89588	−0.947942	+	0.318443i	$0.896840\pi$
$998$	−63.1533	−1.99908
$999$	−1.51030	−0.0477838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.h.1.3	✓	3
7.6	odd	2		2009.2.a.i.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.h.1.3	✓	3	1.1	even	1	trivial
2009.2.a.i.1.3	yes	3	7.6	odd	2

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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49086 q^{2} +1.49086 q^{3} +4.20440 q^{4} +3.71354 q^{6} +5.49086 q^{8} -0.777326 q^{9} +O(q^{10})\)	\(q+2.49086 q^{2} +1.49086 q^{3} +4.20440 q^{4} +3.71354 q^{6} +5.49086 q^{8} -0.777326 q^{9} -3.42708 q^{11} +6.26819 q^{12} +6.26819 q^{13} +5.26819 q^{16} +5.20440 q^{17} -1.93621 q^{18} +2.22267 q^{19} -8.53638 q^{22} +2.91794 q^{23} +8.18613 q^{24} -5.00000 q^{25} +15.6132 q^{26} -5.63148 q^{27} +4.98173 q^{29} +0.854152 q^{31} +2.14061 q^{32} -5.10930 q^{33} +12.9635 q^{34} -3.26819 q^{36} +0.268189 q^{37} +5.53638 q^{38} +9.34502 q^{39} -1.00000 q^{41} -3.20440 q^{43} -14.4088 q^{44} +7.26819 q^{46} -4.14061 q^{47} +7.85415 q^{48} -12.4543 q^{50} +7.75905 q^{51} +26.3540 q^{52} -3.42708 q^{53} -14.0272 q^{54} +3.31370 q^{57} +12.4088 q^{58} +2.12758 q^{59} -14.2447 q^{61} +2.12758 q^{62} -5.20440 q^{64} -12.7266 q^{66} +6.12758 q^{67} +21.8814 q^{68} +4.35025 q^{69} -7.55465 q^{71} -4.26819 q^{72} -12.5364 q^{73} +0.668023 q^{74} -7.45432 q^{75} +9.34502 q^{76} +23.2772 q^{78} -6.00000 q^{79} -6.06379 q^{81} -2.49086 q^{82} +10.4088 q^{83} -7.98173 q^{86} +7.42708 q^{87} -18.8176 q^{88} +6.26819 q^{89} +12.2682 q^{92} +1.27342 q^{93} -10.3137 q^{94} +3.19136 q^{96} +8.06379 q^{97} +2.66395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{4} + 10 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^4 + 10 * q^6 + 9 * q^8 + 4 * q^9	\( 3 q - 3 q^{3} + 4 q^{4} + 10 q^{6} + 9 q^{8} + 4 q^{9} - 8 q^{11} + 5 q^{12} + 5 q^{13} + 2 q^{16} + 7 q^{17} - 11 q^{18} + 13 q^{19} + 2 q^{22} - q^{23} + q^{24} - 15 q^{25} + 21 q^{26} - 6 q^{27} - 2 q^{31} + 3 q^{32} + 10 q^{33} + 9 q^{34} + 4 q^{36} - 13 q^{37} - 11 q^{38} + 16 q^{39} - 3 q^{41} - q^{43} - 26 q^{44} + 8 q^{46} - 9 q^{47} + 19 q^{48} + 2 q^{51} + 17 q^{52} - 8 q^{53} - 7 q^{54} - 24 q^{57} + 20 q^{58} - 4 q^{59} - 6 q^{61} - 4 q^{62} - 7 q^{64} - 44 q^{66} + 8 q^{67} + 26 q^{68} + 9 q^{69} - 10 q^{71} + q^{72} - 10 q^{73} + 21 q^{74} + 15 q^{75} + 16 q^{76} + 6 q^{78} - 18 q^{79} - 13 q^{81} + 14 q^{83} - 9 q^{86} + 20 q^{87} - 22 q^{88} + 5 q^{89} + 23 q^{92} - 2 q^{93} + 3 q^{94} - 6 q^{96} + 19 q^{97} - 30 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^4 + 10 * q^6 + 9 * q^8 + 4 * q^9 - 8 * q^11 + 5 * q^12 + 5 * q^13 + 2 * q^16 + 7 * q^17 - 11 * q^18 + 13 * q^19 + 2 * q^22 - q^23 + q^24 - 15 * q^25 + 21 * q^26 - 6 * q^27 - 2 * q^31 + 3 * q^32 + 10 * q^33 + 9 * q^34 + 4 * q^36 - 13 * q^37 - 11 * q^38 + 16 * q^39 - 3 * q^41 - q^43 - 26 * q^44 + 8 * q^46 - 9 * q^47 + 19 * q^48 + 2 * q^51 + 17 * q^52 - 8 * q^53 - 7 * q^54 - 24 * q^57 + 20 * q^58 - 4 * q^59 - 6 * q^61 - 4 * q^62 - 7 * q^64 - 44 * q^66 + 8 * q^67 + 26 * q^68 + 9 * q^69 - 10 * q^71 + q^72 - 10 * q^73 + 21 * q^74 + 15 * q^75 + 16 * q^76 + 6 * q^78 - 18 * q^79 - 13 * q^81 + 14 * q^83 - 9 * q^86 + 20 * q^87 - 22 * q^88 + 5 * q^89 + 23 * q^92 - 2 * q^93 + 3 * q^94 - 6 * q^96 + 19 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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