LMFDB - Embedded newform 2001.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.92022$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} +0.399447 q^{5} -3.44099 q^{7} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} -2.00000 q^{4} +0.399447 q^{5} -3.44099 q^{7} +1.00000 q^{9} -4.17339 q^{11} +2.00000 q^{12} +6.21494 q^{13} -0.399447 q^{15} +4.00000 q^{16} +3.27452 q^{17} +7.38141 q^{19} -0.798895 q^{20} +3.44099 q^{21} +1.00000 q^{23} -4.84044 q^{25} -1.00000 q^{27} +6.88199 q^{28} +1.00000 q^{29} -4.23989 q^{31} +4.17339 q^{33} -1.37450 q^{35} -2.00000 q^{36} -4.80581 q^{37} -6.21494 q^{39} +2.06650 q^{41} -0.409127 q^{43} +8.34678 q^{44} +0.399447 q^{45} -0.399447 q^{47} -4.00000 q^{48} +4.84044 q^{49} -3.27452 q^{51} -12.4299 q^{52} -9.68088 q^{53} -1.66705 q^{55} -7.38141 q^{57} -4.35790 q^{59} +0.798895 q^{60} -8.41604 q^{61} -3.44099 q^{63} -8.00000 q^{64} +2.48254 q^{65} +6.88199 q^{67} -6.54904 q^{68} -1.00000 q^{69} -4.26760 q^{71} -2.55901 q^{73} +4.84044 q^{75} -14.7628 q^{76} +14.3606 q^{77} -6.29947 q^{79} +1.59779 q^{80} +1.00000 q^{81} -17.6282 q^{83} -6.88199 q^{84} +1.30800 q^{85} -1.00000 q^{87} +11.2191 q^{89} -21.3856 q^{91} -2.00000 q^{92} +4.23989 q^{93} +2.94849 q^{95} -10.0138 q^{97} -4.17339 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9 + 8 * q^12 + 2 * q^15 + 16 * q^16 + 2 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^21 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^28 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 8 * q^36 + 4 * q^37 + 6 * q^41 - 2 * q^45 + 2 * q^47 - 16 * q^48 + 4 * q^49 - 2 * q^51 - 8 * q^53 - 8 * q^55 - 4 * q^57 - 22 * q^59 - 4 * q^60 - 16 * q^61 - 2 * q^63 - 32 * q^64 - 10 * q^65 + 4 * q^67 - 4 * q^68 - 4 * q^69 - 22 * q^71 - 22 * q^73 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 20 * q^79 - 8 * q^80 + 4 * q^81 - 10 * q^83 - 4 * q^84 - 2 * q^85 - 4 * q^87 - 14 * q^89 - 26 * q^91 - 8 * q^92 - 2 * q^93 - 14 * q^95 - 8 * q^97

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	0.399447	0.178638	0.0893192	−	0.996003i	$-0.471531\pi$
$5$	0.399447	0.178638	0.0893192	+	0.996003i	$0.471531\pi$
$6$	0	0
$7$	−3.44099	−1.30057	−0.650287	−	0.759689i	$-0.725351\pi$
$7$	−3.44099	−1.30057	−0.650287	+	0.759689i	$0.725351\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.17339	−1.25832	−0.629162	−	0.777274i	$-0.716602\pi$
$11$	−4.17339	−1.25832	−0.629162	+	0.777274i	$0.716602\pi$
$12$	2.00000	0.577350
$13$	6.21494	1.72371	0.861857	−	0.507152i	$-0.169302\pi$
$13$	6.21494	1.72371	0.861857	+	0.507152i	$0.169302\pi$
$14$	0	0
$15$	−0.399447	−0.103137
$16$	4.00000	1.00000
$17$	3.27452	0.794188	0.397094	−	0.917778i	$-0.370019\pi$
$17$	3.27452	0.794188	0.397094	+	0.917778i	$0.370019\pi$
$18$	0	0
$19$	7.38141	1.69341	0.846706	−	0.532061i	$-0.178582\pi$
$19$	7.38141	1.69341	0.846706	+	0.532061i	$0.178582\pi$
$20$	−0.798895	−0.178638
$21$	3.44099	0.750887
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.84044	−0.968088
$26$	0	0
$27$	−1.00000	−0.192450
$28$	6.88199	1.30057
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.23989	−0.761507	−0.380753	−	0.924677i	$-0.624335\pi$
$31$	−4.23989	−0.761507	−0.380753	+	0.924677i	$0.624335\pi$
$32$	0	0
$33$	4.17339	0.726494
$34$	0	0
$35$	−1.37450	−0.232332
$36$	−2.00000	−0.333333
$37$	−4.80581	−0.790071	−0.395035	−	0.918666i	$-0.629268\pi$
$37$	−4.80581	−0.790071	−0.395035	+	0.918666i	$0.629268\pi$
$38$	0	0
$39$	−6.21494	−0.995187
$40$	0	0
$41$	2.06650	0.322733	0.161366	−	0.986895i	$-0.448410\pi$
$41$	2.06650	0.322733	0.161366	+	0.986895i	$0.448410\pi$
$42$	0	0
$43$	−0.409127	−0.0623912	−0.0311956	−	0.999513i	$-0.509931\pi$
$43$	−0.409127	−0.0623912	−0.0311956	+	0.999513i	$0.509931\pi$
$44$	8.34678	1.25832
$45$	0.399447	0.0595461
$46$	0	0
$47$	−0.399447	−0.0582654	−0.0291327	−	0.999576i	$-0.509275\pi$
$47$	−0.399447	−0.0582654	−0.0291327	+	0.999576i	$0.509275\pi$
$48$	−4.00000	−0.577350
$49$	4.84044	0.691492
$50$	0	0
$51$	−3.27452	−0.458524
$52$	−12.4299	−1.72371
$53$	−9.68088	−1.32977	−0.664886	−	0.746945i	$-0.731520\pi$
$53$	−9.68088	−1.32977	−0.664886	+	0.746945i	$0.731520\pi$
$54$	0	0
$55$	−1.66705	−0.224785
$56$	0	0
$57$	−7.38141	−0.977692
$58$	0	0
$59$	−4.35790	−0.567350	−0.283675	−	0.958920i	$-0.591554\pi$
$59$	−4.35790	−0.567350	−0.283675	+	0.958920i	$0.591554\pi$
$60$	0.798895	0.103137
$61$	−8.41604	−1.07756	−0.538782	−	0.842445i	$-0.681115\pi$
$61$	−8.41604	−1.07756	−0.538782	+	0.842445i	$0.681115\pi$
$62$	0	0
$63$	−3.44099	−0.433525
$64$	−8.00000	−1.00000
$65$	2.48254	0.307921
$66$	0	0
$67$	6.88199	0.840769	0.420384	−	0.907346i	$-0.361895\pi$
$67$	6.88199	0.840769	0.420384	+	0.907346i	$0.361895\pi$
$68$	−6.54904	−0.794188
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.26760	−0.506471	−0.253236	−	0.967405i	$-0.581495\pi$
$71$	−4.26760	−0.506471	−0.253236	+	0.967405i	$0.581495\pi$
$72$	0	0
$73$	−2.55901	−0.299509	−0.149754	−	0.988723i	$-0.547848\pi$
$73$	−2.55901	−0.299509	−0.149754	+	0.988723i	$0.547848\pi$
$74$	0	0
$75$	4.84044	0.558926
$76$	−14.7628	−1.69341
$77$	14.3606	1.63654
$78$	0	0
$79$	−6.29947	−0.708746	−0.354373	−	0.935104i	$-0.615306\pi$
$79$	−6.29947	−0.708746	−0.354373	+	0.935104i	$0.615306\pi$
$80$	1.59779	0.178638
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.6282	−1.93495	−0.967474	−	0.252970i	$-0.918593\pi$
$83$	−17.6282	−1.93495	−0.967474	+	0.252970i	$0.918593\pi$
$84$	−6.88199	−0.750887
$85$	1.30800	0.141872
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	11.2191	1.18922	0.594611	−	0.804014i	$-0.297306\pi$
$89$	11.2191	1.18922	0.594611	+	0.804014i	$0.297306\pi$
$90$	0	0
$91$	−21.3856	−2.24182
$92$	−2.00000	−0.208514
$93$	4.23989	0.439656
$94$	0	0
$95$	2.94849	0.302508
$96$	0	0
$97$	−10.0138	−1.01675	−0.508375	−	0.861136i	$-0.669754\pi$
$97$	−10.0138	−1.01675	−0.508375	+	0.861136i	$0.669754\pi$
$98$	0	0
$99$	−4.17339	−0.419442
$100$	9.68088	0.968088
$101$	0.732397	0.0728762	0.0364381	−	0.999336i	$-0.488399\pi$
$101$	0.732397	0.0728762	0.0364381	+	0.999336i	$0.488399\pi$
$102$	0	0
$103$	6.74899	0.664998	0.332499	−	0.943104i	$-0.392108\pi$
$103$	6.74899	0.664998	0.332499	+	0.943104i	$0.392108\pi$
$104$	0	0
$105$	1.37450	0.134137
$106$	0	0
$107$	15.2260	1.47195	0.735977	−	0.677007i	$-0.236723\pi$
$107$	15.2260	1.47195	0.735977	+	0.677007i	$0.236723\pi$
$108$	2.00000	0.192450
$109$	20.3506	1.94924	0.974619	−	0.223869i	$-0.0718687\pi$
$109$	20.3506	1.94924	0.974619	+	0.223869i	$0.0718687\pi$
$110$	0	0
$111$	4.80581	0.456147
$112$	−13.7640	−1.30057
$113$	−2.33295	−0.219465	−0.109733	−	0.993961i	$-0.534999\pi$
$113$	−2.33295	−0.219465	−0.109733	+	0.993961i	$0.534999\pi$
$114$	0	0
$115$	0.399447	0.0372487
$116$	−2.00000	−0.185695
$117$	6.21494	0.574571
$118$	0	0
$119$	−11.2676	−1.03290
$120$	0	0
$121$	6.41719	0.583381
$122$	0	0
$123$	−2.06650	−0.186330
$124$	8.47978	0.761507
$125$	−3.93074	−0.351576
$126$	0	0
$127$	−4.09306	−0.363200	−0.181600	−	0.983372i	$-0.558128\pi$
$127$	−4.09306	−0.363200	−0.181600	+	0.983372i	$0.558128\pi$
$128$	0	0
$129$	0.409127	0.0360216
$130$	0	0
$131$	−12.8266	−1.12066	−0.560331	−	0.828269i	$-0.689326\pi$
$131$	−12.8266	−1.12066	−0.560331	+	0.828269i	$0.689326\pi$
$132$	−8.34678	−0.726494
$133$	−25.3994	−2.20241
$134$	0	0
$135$	−0.399447	−0.0343790
$136$	0	0
$137$	12.8224	1.09549	0.547746	−	0.836645i	$-0.315486\pi$
$137$	12.8224	1.09549	0.547746	+	0.836645i	$0.315486\pi$
$138$	0	0
$139$	−0.506341	−0.0429473	−0.0214736	−	0.999769i	$-0.506836\pi$
$139$	−0.506341	−0.0429473	−0.0214736	+	0.999769i	$0.506836\pi$
$140$	2.74899	0.232332
$141$	0.399447	0.0336395
$142$	0	0
$143$	−25.9374	−2.16899
$144$	4.00000	0.333333
$145$	0.399447	0.0331723
$146$	0	0
$147$	−4.84044	−0.399233
$148$	9.61162	0.790071
$149$	−10.4659	−0.857404	−0.428702	−	0.903446i	$-0.641029\pi$
$149$	−10.4659	−0.857404	−0.428702	+	0.903446i	$0.641029\pi$
$150$	0	0
$151$	−0.0277138	−0.00225532	−0.00112766	−	0.999999i	$-0.500359\pi$
$151$	−0.0277138	−0.00225532	−0.00112766	+	0.999999i	$0.500359\pi$
$152$	0	0
$153$	3.27452	0.264729
$154$	0	0
$155$	−1.69361	−0.136034
$156$	12.4299	0.995187
$157$	7.45788	0.595203	0.297602	−	0.954690i	$-0.403813\pi$
$157$	7.45788	0.595203	0.297602	+	0.954690i	$0.403813\pi$
$158$	0	0
$159$	9.68088	0.767744
$160$	0	0
$161$	−3.44099	−0.271188
$162$	0	0
$163$	−6.95125	−0.544464	−0.272232	−	0.962232i	$-0.587762\pi$
$163$	−6.95125	−0.544464	−0.272232	+	0.962232i	$0.587762\pi$
$164$	−4.13300	−0.322733
$165$	1.66705	0.129780
$166$	0	0
$167$	−10.4620	−0.809576	−0.404788	−	0.914411i	$-0.632655\pi$
$167$	−10.4620	−0.809576	−0.404788	+	0.914411i	$0.632655\pi$
$168$	0	0
$169$	25.6255	1.97119
$170$	0	0
$171$	7.38141	0.564471
$172$	0.818253	0.0623912
$173$	−16.4049	−1.24724	−0.623622	−	0.781726i	$-0.714339\pi$
$173$	−16.4049	−1.24724	−0.623622	+	0.781726i	$0.714339\pi$
$174$	0	0
$175$	16.6559	1.25907
$176$	−16.6936	−1.25832
$177$	4.35790	0.327560
$178$	0	0
$179$	−11.4897	−0.858784	−0.429392	−	0.903118i	$-0.641272\pi$
$179$	−11.4897	−0.858784	−0.429392	+	0.903118i	$0.641272\pi$
$180$	−0.798895	−0.0595461
$181$	−0.0830937	−0.00617631	−0.00308815	−	0.999995i	$-0.500983\pi$
$181$	−0.0830937	−0.00617631	−0.00308815	+	0.999995i	$0.500983\pi$
$182$	0	0
$183$	8.41604	0.622132
$184$	0	0
$185$	−1.91967	−0.141137
$186$	0	0
$187$	−13.6659	−0.999346
$188$	0.798895	0.0582654
$189$	3.44099	0.250296
$190$	0	0
$191$	−4.69892	−0.340002	−0.170001	−	0.985444i	$-0.554377\pi$
$191$	−4.69892	−0.340002	−0.170001	+	0.985444i	$0.554377\pi$
$192$	8.00000	0.577350
$193$	17.3618	1.24973	0.624864	−	0.780734i	$-0.285155\pi$
$193$	17.3618	1.24973	0.624864	+	0.780734i	$0.285155\pi$
$194$	0	0
$195$	−2.48254	−0.177778
$196$	−9.68088	−0.691492
$197$	−13.0376	−0.928893	−0.464446	−	0.885601i	$-0.653747\pi$
$197$	−13.0376	−0.928893	−0.464446	+	0.885601i	$0.653747\pi$
$198$	0	0
$199$	−26.2038	−1.85754	−0.928770	−	0.370657i	$-0.879133\pi$
$199$	−26.2038	−1.85754	−0.928770	+	0.370657i	$0.879133\pi$
$200$	0	0
$201$	−6.88199	−0.485418
$202$	0	0
$203$	−3.44099	−0.241510
$204$	6.54904	0.458524
$205$	0.825457	0.0576524
$206$	0	0
$207$	1.00000	0.0695048
$208$	24.8598	1.72371
$209$	−30.8055	−2.13086
$210$	0	0
$211$	−22.0039	−1.51481	−0.757404	−	0.652946i	$-0.773533\pi$
$211$	−22.0039	−1.51481	−0.757404	+	0.652946i	$0.773533\pi$
$212$	19.3618	1.32977
$213$	4.26760	0.292411
$214$	0	0
$215$	−0.163425	−0.0111455
$216$	0	0
$217$	14.5894	0.990395
$218$	0	0
$219$	2.55901	0.172922
$220$	3.33410	0.224785
$221$	20.3509	1.36895
$222$	0	0
$223$	−14.0692	−0.942144	−0.471072	−	0.882095i	$-0.656133\pi$
$223$	−14.0692	−0.942144	−0.471072	+	0.882095i	$0.656133\pi$
$224$	0	0
$225$	−4.84044	−0.322696
$226$	0	0
$227$	−25.6393	−1.70174	−0.850870	−	0.525377i	$-0.823924\pi$
$227$	−25.6393	−1.70174	−0.850870	+	0.525377i	$0.823924\pi$
$228$	14.7628	0.977692
$229$	−3.94428	−0.260646	−0.130323	−	0.991472i	$-0.541601\pi$
$229$	−3.94428	−0.260646	−0.130323	+	0.991472i	$0.541601\pi$
$230$	0	0
$231$	−14.3606	−0.944859
$232$	0	0
$233$	6.07918	0.398260	0.199130	−	0.979973i	$-0.436188\pi$
$233$	6.07918	0.398260	0.199130	+	0.979973i	$0.436188\pi$
$234$	0	0
$235$	−0.159558	−0.0104084
$236$	8.71580	0.567350
$237$	6.29947	0.409195
$238$	0	0
$239$	−25.4770	−1.64797	−0.823986	−	0.566611i	$-0.808254\pi$
$239$	−25.4770	−1.64797	−0.823986	+	0.566611i	$0.808254\pi$
$240$	−1.59779	−0.103137
$241$	17.6448	1.13660	0.568301	−	0.822821i	$-0.307601\pi$
$241$	17.6448	1.13660	0.568301	+	0.822821i	$0.307601\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	16.8321	1.07756
$245$	1.93350	0.123527
$246$	0	0
$247$	45.8750	2.91896
$248$	0	0
$249$	17.6282	1.11714
$250$	0	0
$251$	−2.21914	−0.140071	−0.0700354	−	0.997545i	$-0.522311\pi$
$251$	−2.21914	−0.140071	−0.0700354	+	0.997545i	$0.522311\pi$
$252$	6.88199	0.433525
$253$	−4.17339	−0.262379
$254$	0	0
$255$	−1.30800	−0.0819100
$256$	16.0000	1.00000
$257$	−26.5783	−1.65791	−0.828955	−	0.559315i	$-0.811064\pi$
$257$	−26.5783	−1.65791	−0.828955	+	0.559315i	$0.811064\pi$
$258$	0	0
$259$	16.5368	1.02754
$260$	−4.96508	−0.307921
$261$	1.00000	0.0618984
$262$	0	0
$263$	−14.5701	−0.898429	−0.449215	−	0.893424i	$-0.648296\pi$
$263$	−14.5701	−0.898429	−0.449215	+	0.893424i	$0.648296\pi$
$264$	0	0
$265$	−3.86700	−0.237548
$266$	0	0
$267$	−11.2191	−0.686597
$268$	−13.7640	−0.840769
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	22.3551	1.35798	0.678988	−	0.734150i	$-0.262419\pi$
$271$	22.3551	1.35798	0.678988	+	0.734150i	$0.262419\pi$
$272$	13.0981	0.794188
$273$	21.3856	1.29431
$274$	0	0
$275$	20.2011	1.21817
$276$	2.00000	0.120386
$277$	11.8255	0.710523	0.355261	−	0.934767i	$-0.384392\pi$
$277$	11.8255	0.710523	0.355261	+	0.934767i	$0.384392\pi$
$278$	0	0
$279$	−4.23989	−0.253836
$280$	0	0
$281$	4.72963	0.282146	0.141073	−	0.989999i	$-0.454945\pi$
$281$	4.72963	0.282146	0.141073	+	0.989999i	$0.454945\pi$
$282$	0	0
$283$	5.12188	0.304464	0.152232	−	0.988345i	$-0.451354\pi$
$283$	5.12188	0.304464	0.152232	+	0.988345i	$0.451354\pi$
$284$	8.53521	0.506471
$285$	−2.94849	−0.174653
$286$	0	0
$287$	−7.11081	−0.419738
$288$	0	0
$289$	−6.27752	−0.369266
$290$	0	0
$291$	10.0138	0.587021
$292$	5.11801	0.299509
$293$	0.190274	0.0111159	0.00555797	−	0.999985i	$-0.498231\pi$
$293$	0.190274	0.0111159	0.00555797	+	0.999985i	$0.498231\pi$
$294$	0	0
$295$	−1.74075	−0.101351
$296$	0	0
$297$	4.17339	0.242165
$298$	0	0
$299$	6.21494	0.359419
$300$	−9.68088	−0.558926
$301$	1.40780	0.0811444
$302$	0	0
$303$	−0.732397	−0.0420751
$304$	29.5257	1.69341
$305$	−3.36177	−0.192494
$306$	0	0
$307$	−10.8083	−0.616862	−0.308431	−	0.951247i	$-0.599804\pi$
$307$	−10.8083	−0.616862	−0.308431	+	0.951247i	$0.599804\pi$
$308$	−28.7212	−1.63654
$309$	−6.74899	−0.383937
$310$	0	0
$311$	15.6116	0.885254	0.442627	−	0.896706i	$-0.354047\pi$
$311$	15.6116	0.885254	0.442627	+	0.896706i	$0.354047\pi$
$312$	0	0
$313$	13.1579	0.743731	0.371866	−	0.928287i	$-0.378718\pi$
$313$	13.1579	0.743731	0.371866	+	0.928287i	$0.378718\pi$
$314$	0	0
$315$	−1.37450	−0.0774441
$316$	12.5989	0.708746
$317$	−23.7451	−1.33366	−0.666828	−	0.745211i	$-0.732349\pi$
$317$	−23.7451	−1.33366	−0.666828	+	0.745211i	$0.732349\pi$
$318$	0	0
$319$	−4.17339	−0.233665
$320$	−3.19558	−0.178638
$321$	−15.2260	−0.849833
$322$	0	0
$323$	24.1706	1.34489
$324$	−2.00000	−0.111111
$325$	−30.0830	−1.66871
$326$	0	0
$327$	−20.3506	−1.12539
$328$	0	0
$329$	1.37450	0.0757784
$330$	0	0
$331$	−3.09808	−0.170286	−0.0851429	−	0.996369i	$-0.527135\pi$
$331$	−3.09808	−0.170286	−0.0851429	+	0.996369i	$0.527135\pi$
$332$	35.2564	1.93495
$333$	−4.80581	−0.263357
$334$	0	0
$335$	2.74899	0.150194
$336$	13.7640	0.750887
$337$	7.75476	0.422429	0.211214	−	0.977440i	$-0.432258\pi$
$337$	7.75476	0.422429	0.211214	+	0.977440i	$0.432258\pi$
$338$	0	0
$339$	2.33295	0.126708
$340$	−2.61600	−0.141872
$341$	17.6947	0.958223
$342$	0	0
$343$	7.43103	0.401238
$344$	0	0
$345$	−0.399447	−0.0215055
$346$	0	0
$347$	−6.44099	−0.345771	−0.172885	−	0.984942i	$-0.555309\pi$
$347$	−6.44099	−0.345771	−0.172885	+	0.984942i	$0.555309\pi$
$348$	2.00000	0.107211
$349$	−17.9651	−0.961649	−0.480824	−	0.876817i	$-0.659663\pi$
$349$	−17.9651	−0.961649	−0.480824	+	0.876817i	$0.659663\pi$
$350$	0	0
$351$	−6.21494	−0.331729
$352$	0	0
$353$	−28.8194	−1.53390	−0.766950	−	0.641707i	$-0.778226\pi$
$353$	−28.8194	−1.53390	−0.766950	+	0.641707i	$0.778226\pi$
$354$	0	0
$355$	−1.70468	−0.0904752
$356$	−22.4382	−1.18922
$357$	11.2676	0.596345
$358$	0	0
$359$	−8.62135	−0.455017	−0.227509	−	0.973776i	$-0.573058\pi$
$359$	−8.62135	−0.455017	−0.227509	+	0.973776i	$0.573058\pi$
$360$	0	0
$361$	35.4853	1.86765
$362$	0	0
$363$	−6.41719	−0.336815
$364$	42.7711	2.24182
$365$	−1.02219	−0.0535038
$366$	0	0
$367$	−9.54789	−0.498396	−0.249198	−	0.968453i	$-0.580167\pi$
$367$	−9.54789	−0.498396	−0.249198	+	0.968453i	$0.580167\pi$
$368$	4.00000	0.208514
$369$	2.06650	0.107578
$370$	0	0
$371$	33.3119	1.72947
$372$	−8.47978	−0.439656
$373$	−3.70468	−0.191821	−0.0959106	−	0.995390i	$-0.530576\pi$
$373$	−3.70468	−0.191821	−0.0959106	+	0.995390i	$0.530576\pi$
$374$	0	0
$375$	3.93074	0.202983
$376$	0	0
$377$	6.21494	0.320086
$378$	0	0
$379$	−11.7504	−0.603579	−0.301790	−	0.953375i	$-0.597584\pi$
$379$	−11.7504	−0.603579	−0.301790	+	0.953375i	$0.597584\pi$
$380$	−5.89697	−0.302508
$381$	4.09306	0.209694
$382$	0	0
$383$	21.3894	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$383$	21.3894	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$384$	0	0
$385$	5.73631	0.292349
$386$	0	0
$387$	−0.409127	−0.0207971
$388$	20.0277	1.01675
$389$	21.5448	1.09237	0.546183	−	0.837666i	$-0.316080\pi$
$389$	21.5448	1.09237	0.546183	+	0.837666i	$0.316080\pi$
$390$	0	0
$391$	3.27452	0.165600
$392$	0	0
$393$	12.8266	0.647014
$394$	0	0
$395$	−2.51631	−0.126609
$396$	8.34678	0.419442
$397$	34.6670	1.73989	0.869943	−	0.493151i	$-0.164155\pi$
$397$	34.6670	1.73989	0.869943	+	0.493151i	$0.164155\pi$
$398$	0	0
$399$	25.3994	1.27156
$400$	−19.3618	−0.968088
$401$	36.6598	1.83070	0.915351	−	0.402656i	$-0.131913\pi$
$401$	36.6598	1.83070	0.915351	+	0.402656i	$0.131913\pi$
$402$	0	0
$403$	−26.3506	−1.31262
$404$	−1.46479	−0.0728762
$405$	0.399447	0.0198487
$406$	0	0
$407$	20.0565	0.994165
$408$	0	0
$409$	−24.9473	−1.23356	−0.616782	−	0.787134i	$-0.711564\pi$
$409$	−24.9473	−1.23356	−0.616782	+	0.787134i	$0.711564\pi$
$410$	0	0
$411$	−12.8224	−0.632483
$412$	−13.4980	−0.664998
$413$	14.9955	0.737881
$414$	0	0
$415$	−7.04155	−0.345656
$416$	0	0
$417$	0.506341	0.0247956
$418$	0	0
$419$	−6.81825	−0.333093	−0.166547	−	0.986034i	$-0.553262\pi$
$419$	−6.81825	−0.333093	−0.166547	+	0.986034i	$0.553262\pi$
$420$	−2.74899	−0.134137
$421$	30.6878	1.49563	0.747815	−	0.663907i	$-0.231103\pi$
$421$	30.6878	1.49563	0.747815	+	0.663907i	$0.231103\pi$
$422$	0	0
$423$	−0.399447	−0.0194218
$424$	0	0
$425$	−15.8501	−0.768844
$426$	0	0
$427$	28.9596	1.40145
$428$	−30.4520	−1.47195
$429$	25.9374	1.25227
$430$	0	0
$431$	−19.5779	−0.943032	−0.471516	−	0.881857i	$-0.656293\pi$
$431$	−19.5779	−0.943032	−0.471516	+	0.881857i	$0.656293\pi$
$432$	−4.00000	−0.192450
$433$	−5.07531	−0.243904	−0.121952	−	0.992536i	$-0.538915\pi$
$433$	−5.07531	−0.243904	−0.121952	+	0.992536i	$0.538915\pi$
$434$	0	0
$435$	−0.399447	−0.0191520
$436$	−40.7013	−1.94924
$437$	7.38141	0.353101
$438$	0	0
$439$	−0.790540	−0.0377304	−0.0188652	−	0.999822i	$-0.506005\pi$
$439$	−0.790540	−0.0377304	−0.0188652	+	0.999822i	$0.506005\pi$
$440$	0	0
$441$	4.84044	0.230497
$442$	0	0
$443$	28.3103	1.34506	0.672530	−	0.740070i	$-0.265207\pi$
$443$	28.3103	1.34506	0.672530	+	0.740070i	$0.265207\pi$
$444$	−9.61162	−0.456147
$445$	4.48144	0.212441
$446$	0	0
$447$	10.4659	0.495022
$448$	27.5280	1.30057
$449$	−15.2343	−0.718951	−0.359475	−	0.933155i	$-0.617044\pi$
$449$	−15.2343	−0.718951	−0.359475	+	0.933155i	$0.617044\pi$
$450$	0	0
$451$	−8.62430	−0.406103
$452$	4.66590	0.219465
$453$	0.0277138	0.00130211
$454$	0	0
$455$	−8.54241	−0.400474
$456$	0	0
$457$	5.77394	0.270094	0.135047	−	0.990839i	$-0.456882\pi$
$457$	5.77394	0.270094	0.135047	+	0.990839i	$0.456882\pi$
$458$	0	0
$459$	−3.27452	−0.152841
$460$	−0.798895	−0.0372487
$461$	10.8765	0.506567	0.253284	−	0.967392i	$-0.418489\pi$
$461$	10.8765	0.506567	0.253284	+	0.967392i	$0.418489\pi$
$462$	0	0
$463$	5.03487	0.233990	0.116995	−	0.993132i	$-0.462674\pi$
$463$	5.03487	0.233990	0.116995	+	0.993132i	$0.462674\pi$
$464$	4.00000	0.185695
$465$	1.69361	0.0785394
$466$	0	0
$467$	28.2246	1.30608	0.653039	−	0.757325i	$-0.273494\pi$
$467$	28.2246	1.30608	0.653039	+	0.757325i	$0.273494\pi$
$468$	−12.4299	−0.574571
$469$	−23.6809	−1.09348
$470$	0	0
$471$	−7.45788	−0.343641
$472$	0	0
$473$	1.70745	0.0785084
$474$	0	0
$475$	−35.7293	−1.63937
$476$	22.5352	1.03290
$477$	−9.68088	−0.443257
$478$	0	0
$479$	−39.9295	−1.82443	−0.912214	−	0.409715i	$-0.865628\pi$
$479$	−39.9295	−1.82443	−0.912214	+	0.409715i	$0.865628\pi$
$480$	0	0
$481$	−29.8678	−1.36186
$482$	0	0
$483$	3.44099	0.156571
$484$	−12.8344	−0.583381
$485$	−4.00000	−0.181631
$486$	0	0
$487$	22.1650	1.00439	0.502196	−	0.864754i	$-0.332526\pi$
$487$	22.1650	1.00439	0.502196	+	0.864754i	$0.332526\pi$
$488$	0	0
$489$	6.95125	0.314346
$490$	0	0
$491$	−3.21218	−0.144963	−0.0724817	−	0.997370i	$-0.523092\pi$
$491$	−3.21218	−0.144963	−0.0724817	+	0.997370i	$0.523092\pi$
$492$	4.13300	0.186330
$493$	3.27452	0.147477
$494$	0	0
$495$	−1.66705	−0.0749283
$496$	−16.9596	−0.761507
$497$	14.6848	0.658703
$498$	0	0
$499$	−20.6737	−0.925481	−0.462741	−	0.886494i	$-0.653134\pi$
$499$	−20.6737	−0.925481	−0.462741	+	0.886494i	$0.653134\pi$
$500$	7.86148	0.351576
$501$	10.4620	0.467409
$502$	0	0
$503$	10.9288	0.487293	0.243646	−	0.969864i	$-0.421656\pi$
$503$	10.9288	0.487293	0.243646	+	0.969864i	$0.421656\pi$
$504$	0	0
$505$	0.292554	0.0130185
$506$	0	0
$507$	−25.6255	−1.13807
$508$	8.18612	0.363200
$509$	−28.7445	−1.27408	−0.637039	−	0.770832i	$-0.719841\pi$
$509$	−28.7445	−1.27408	−0.637039	+	0.770832i	$0.719841\pi$
$510$	0	0
$511$	8.80552	0.389533
$512$	0	0
$513$	−7.38141	−0.325897
$514$	0	0
$515$	2.69587	0.118794
$516$	−0.818253	−0.0360216
$517$	1.66705	0.0733168
$518$	0	0
$519$	16.4049	0.720096
$520$	0	0
$521$	−2.58235	−0.113135	−0.0565673	−	0.998399i	$-0.518016\pi$
$521$	−2.58235	−0.113135	−0.0565673	+	0.998399i	$0.518016\pi$
$522$	0	0
$523$	20.5922	0.900434	0.450217	−	0.892919i	$-0.351347\pi$
$523$	20.5922	0.900434	0.450217	+	0.892919i	$0.351347\pi$
$524$	25.6531	1.12066
$525$	−16.6559	−0.726924
$526$	0	0
$527$	−13.8836	−0.604779
$528$	16.6936	0.726494
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.35790	−0.189117
$532$	50.7988	2.20241
$533$	12.8432	0.556299
$534$	0	0
$535$	6.08199	0.262947
$536$	0	0
$537$	11.4897	0.495819
$538$	0	0
$539$	−20.2011	−0.870121
$540$	0.798895	0.0343790
$541$	−42.7291	−1.83707	−0.918533	−	0.395345i	$-0.870625\pi$
$541$	−42.7291	−1.83707	−0.918533	+	0.395345i	$0.870625\pi$
$542$	0	0
$543$	0.0830937	0.00356589
$544$	0	0
$545$	8.12901	0.348209
$546$	0	0
$547$	15.0011	0.641401	0.320700	−	0.947181i	$-0.396082\pi$
$547$	15.0011	0.641401	0.320700	+	0.947181i	$0.396082\pi$
$548$	−25.6448	−1.09549
$549$	−8.41604	−0.359188
$550$	0	0
$551$	7.38141	0.314459
$552$	0	0
$553$	21.6764	0.921776
$554$	0	0
$555$	1.91967	0.0814854
$556$	1.01268	0.0429473
$557$	−41.4753	−1.75736	−0.878682	−	0.477407i	$-0.841577\pi$
$557$	−41.4753	−1.75736	−0.878682	+	0.477407i	$0.841577\pi$
$558$	0	0
$559$	−2.54270	−0.107545
$560$	−5.49799	−0.232332
$561$	13.6659	0.576973
$562$	0	0
$563$	−44.7067	−1.88416	−0.942080	−	0.335387i	$-0.891133\pi$
$563$	−44.7067	−1.88416	−0.942080	+	0.335387i	$0.891133\pi$
$564$	−0.798895	−0.0336395
$565$	−0.931891	−0.0392049
$566$	0	0
$567$	−3.44099	−0.144508
$568$	0	0
$569$	37.8127	1.58519	0.792595	−	0.609748i	$-0.208730\pi$
$569$	37.8127	1.58519	0.792595	+	0.609748i	$0.208730\pi$
$570$	0	0
$571$	−16.6698	−0.697608	−0.348804	−	0.937196i	$-0.613412\pi$
$571$	−16.6698	−0.697608	−0.348804	+	0.937196i	$0.613412\pi$
$572$	51.8747	2.16899
$573$	4.69892	0.196300
$574$	0	0
$575$	−4.84044	−0.201860
$576$	−8.00000	−0.333333
$577$	−0.619863	−0.0258052	−0.0129026	−	0.999917i	$-0.504107\pi$
$577$	−0.619863	−0.0258052	−0.0129026	+	0.999917i	$0.504107\pi$
$578$	0	0
$579$	−17.3618	−0.721530
$580$	−0.798895	−0.0331723
$581$	60.6586	2.51654
$582$	0	0
$583$	40.4021	1.67328
$584$	0	0
$585$	2.48254	0.102640
$586$	0	0
$587$	31.3550	1.29416	0.647079	−	0.762423i	$-0.275990\pi$
$587$	31.3550	1.29416	0.647079	+	0.762423i	$0.275990\pi$
$588$	9.68088	0.399233
$589$	−31.2964	−1.28954
$590$	0	0
$591$	13.0376	0.536297
$592$	−19.2232	−0.790071
$593$	−25.2200	−1.03566	−0.517830	−	0.855484i	$-0.673260\pi$
$593$	−25.2200	−1.03566	−0.517830	+	0.855484i	$0.673260\pi$
$594$	0	0
$595$	−4.50082	−0.184515
$596$	20.9319	0.857404
$597$	26.2038	1.07245
$598$	0	0
$599$	−29.3313	−1.19845	−0.599223	−	0.800582i	$-0.704524\pi$
$599$	−29.3313	−1.19845	−0.599223	+	0.800582i	$0.704524\pi$
$600$	0	0
$601$	−32.1191	−1.31016	−0.655082	−	0.755558i	$-0.727366\pi$
$601$	−32.1191	−1.31016	−0.655082	+	0.755558i	$0.727366\pi$
$602$	0	0
$603$	6.88199	0.280256
$604$	0.0554277	0.00225532
$605$	2.56333	0.104214
$606$	0	0
$607$	2.45368	0.0995916	0.0497958	−	0.998759i	$-0.484143\pi$
$607$	2.45368	0.0995916	0.0497958	+	0.998759i	$0.484143\pi$
$608$	0	0
$609$	3.44099	0.139436
$610$	0	0
$611$	−2.48254	−0.100433
$612$	−6.54904	−0.264729
$613$	−15.8753	−0.641198	−0.320599	−	0.947215i	$-0.603884\pi$
$613$	−15.8753	−0.641198	−0.320599	+	0.947215i	$0.603884\pi$
$614$	0	0
$615$	−0.825457	−0.0332857
$616$	0	0
$617$	−45.7983	−1.84377	−0.921885	−	0.387465i	$-0.873351\pi$
$617$	−45.7983	−1.84377	−0.921885	+	0.387465i	$0.873351\pi$
$618$	0	0
$619$	−32.3537	−1.30040	−0.650202	−	0.759761i	$-0.725316\pi$
$619$	−32.3537	−1.30040	−0.650202	+	0.759761i	$0.725316\pi$
$620$	3.38723	0.136034
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−38.6048	−1.54667
$624$	−24.8598	−0.995187
$625$	22.6321	0.905283
$626$	0	0
$627$	30.8055	1.23025
$628$	−14.9158	−0.595203
$629$	−15.7367	−0.627464
$630$	0	0
$631$	−13.9363	−0.554794	−0.277397	−	0.960755i	$-0.589472\pi$
$631$	−13.9363	−0.554794	−0.277397	+	0.960755i	$0.589472\pi$
$632$	0	0
$633$	22.0039	0.874575
$634$	0	0
$635$	−1.63496	−0.0648815
$636$	−19.3618	−0.767744
$637$	30.0830	1.19193
$638$	0	0
$639$	−4.26760	−0.168824
$640$	0	0
$641$	−32.2522	−1.27389	−0.636943	−	0.770911i	$-0.719801\pi$
$641$	−32.2522	−1.27389	−0.636943	+	0.770911i	$0.719801\pi$
$642$	0	0
$643$	21.0926	0.831809	0.415905	−	0.909408i	$-0.363465\pi$
$643$	21.0926	0.831809	0.415905	+	0.909408i	$0.363465\pi$
$644$	6.88199	0.271188
$645$	0.163425	0.00643484
$646$	0	0
$647$	17.5296	0.689158	0.344579	−	0.938757i	$-0.388022\pi$
$647$	17.5296	0.689158	0.344579	+	0.938757i	$0.388022\pi$
$648$	0	0
$649$	18.1872	0.713911
$650$	0	0
$651$	−14.5894	−0.571805
$652$	13.9025	0.544464
$653$	−24.3434	−0.952633	−0.476316	−	0.879274i	$-0.658028\pi$
$653$	−24.3434	−0.952633	−0.476316	+	0.879274i	$0.658028\pi$
$654$	0	0
$655$	−5.12354	−0.200193
$656$	8.26599	0.322733
$657$	−2.55901	−0.0998363
$658$	0	0
$659$	33.3492	1.29910	0.649550	−	0.760319i	$-0.274957\pi$
$659$	33.3492	1.29910	0.649550	+	0.760319i	$0.274957\pi$
$660$	−3.33410	−0.129780
$661$	44.2869	1.72256	0.861281	−	0.508130i	$-0.169663\pi$
$661$	44.2869	1.72256	0.861281	+	0.508130i	$0.169663\pi$
$662$	0	0
$663$	−20.3509	−0.790365
$664$	0	0
$665$	−10.1457	−0.393434
$666$	0	0
$667$	1.00000	0.0387202
$668$	20.9241	0.809576
$669$	14.0692	0.543947
$670$	0	0
$671$	35.1234	1.35593
$672$	0	0
$673$	−21.5352	−0.830121	−0.415061	−	0.909794i	$-0.636240\pi$
$673$	−21.5352	−0.830121	−0.415061	+	0.909794i	$0.636240\pi$
$674$	0	0
$675$	4.84044	0.186309
$676$	−51.2509	−1.97119
$677$	−2.85543	−0.109743	−0.0548715	−	0.998493i	$-0.517475\pi$
$677$	−2.85543	−0.109743	−0.0548715	+	0.998493i	$0.517475\pi$
$678$	0	0
$679$	34.4575	1.32236
$680$	0	0
$681$	25.6393	0.982500
$682$	0	0
$683$	6.63501	0.253882	0.126941	−	0.991910i	$-0.459484\pi$
$683$	6.63501	0.253882	0.126941	+	0.991910i	$0.459484\pi$
$684$	−14.7628	−0.564471
$685$	5.12188	0.195697
$686$	0	0
$687$	3.94428	0.150484
$688$	−1.63651	−0.0623912
$689$	−60.1661	−2.29215
$690$	0	0
$691$	−48.0288	−1.82710	−0.913550	−	0.406726i	$-0.866670\pi$
$691$	−48.0288	−1.82710	−0.913550	+	0.406726i	$0.866670\pi$
$692$	32.8099	1.24724
$693$	14.3606	0.545515
$694$	0	0
$695$	−0.202257	−0.00767203
$696$	0	0
$697$	6.76679	0.256310
$698$	0	0
$699$	−6.07918	−0.229936
$700$	−33.3119	−1.25907
$701$	−41.7367	−1.57637	−0.788186	−	0.615437i	$-0.788980\pi$
$701$	−41.7367	−1.57637	−0.788186	+	0.615437i	$0.788980\pi$
$702$	0	0
$703$	−35.4737	−1.33792
$704$	33.3871	1.25832
$705$	0.159558	0.00600931
$706$	0	0
$707$	−2.52017	−0.0947809
$708$	−8.71580	−0.327560
$709$	22.4090	0.841586	0.420793	−	0.907157i	$-0.361752\pi$
$709$	22.4090	0.841586	0.420793	+	0.907157i	$0.361752\pi$
$710$	0	0
$711$	−6.29947	−0.236249
$712$	0	0
$713$	−4.23989	−0.158785
$714$	0	0
$715$	−10.3606	−0.387465
$716$	22.9795	0.858784
$717$	25.4770	0.951457
$718$	0	0
$719$	46.4493	1.73227	0.866133	−	0.499813i	$-0.166598\pi$
$719$	46.4493	1.73227	0.866133	+	0.499813i	$0.166598\pi$
$720$	1.59779	0.0595461
$721$	−23.2232	−0.864879
$722$	0	0
$723$	−17.6448	−0.656218
$724$	0.166187	0.00617631
$725$	−4.84044	−0.179769
$726$	0	0
$727$	28.6315	1.06188	0.530942	−	0.847408i	$-0.321838\pi$
$727$	28.6315	1.06188	0.530942	+	0.847408i	$0.321838\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.33969	−0.0495504
$732$	−16.8321	−0.622132
$733$	16.7628	0.619149	0.309575	−	0.950875i	$-0.399813\pi$
$733$	16.7628	0.619149	0.309575	+	0.950875i	$0.399813\pi$
$734$	0	0
$735$	−1.93350	−0.0713183
$736$	0	0
$737$	−28.7212	−1.05796
$738$	0	0
$739$	33.5008	1.23235	0.616173	−	0.787611i	$-0.288682\pi$
$739$	33.5008	1.23235	0.616173	+	0.787611i	$0.288682\pi$
$740$	3.83934	0.141137
$741$	−45.8750	−1.68526
$742$	0	0
$743$	−17.5215	−0.642801	−0.321401	−	0.946943i	$-0.604154\pi$
$743$	−17.5215	−0.642801	−0.321401	+	0.946943i	$0.604154\pi$
$744$	0	0
$745$	−4.18060	−0.153165
$746$	0	0
$747$	−17.6282	−0.644983
$748$	27.3317	0.999346
$749$	−52.3926	−1.91438
$750$	0	0
$751$	16.2579	0.593258	0.296629	−	0.954993i	$-0.404137\pi$
$751$	16.2579	0.593258	0.296629	+	0.954993i	$0.404137\pi$
$752$	−1.59779	−0.0582654
$753$	2.21914	0.0808699
$754$	0	0
$755$	−0.0110702	−0.000402887 0
$756$	−6.88199	−0.250296
$757$	−21.1165	−0.767493	−0.383747	−	0.923438i	$-0.625366\pi$
$757$	−21.1165	−0.767493	−0.383747	+	0.923438i	$0.625366\pi$
$758$	0	0
$759$	4.17339	0.151485
$760$	0	0
$761$	27.1297	0.983449	0.491725	−	0.870751i	$-0.336367\pi$
$761$	27.1297	0.983449	0.491725	+	0.870751i	$0.336367\pi$
$762$	0	0
$763$	−70.0265	−2.53513
$764$	9.39784	0.340002
$765$	1.30800	0.0472908
$766$	0	0
$767$	−27.0841	−0.977950
$768$	−16.0000	−0.577350
$769$	12.6751	0.457076	0.228538	−	0.973535i	$-0.426605\pi$
$769$	12.6751	0.457076	0.228538	+	0.973535i	$0.426605\pi$
$770$	0	0
$771$	26.5783	0.957195
$772$	−34.7235	−1.24973
$773$	−29.7184	−1.06890	−0.534448	−	0.845201i	$-0.679481\pi$
$773$	−29.7184	−1.06890	−0.534448	+	0.845201i	$0.679481\pi$
$774$	0	0
$775$	20.5229	0.737206
$776$	0	0
$777$	−16.5368	−0.593253
$778$	0	0
$779$	15.2537	0.546520
$780$	4.96508	0.177778
$781$	17.8104	0.637305
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	19.3618	0.691492
$785$	2.97903	0.106326
$786$	0	0
$787$	41.1756	1.46775	0.733877	−	0.679283i	$-0.237709\pi$
$787$	41.1756	1.46775	0.733877	+	0.679283i	$0.237709\pi$
$788$	26.0753	0.928893
$789$	14.5701	0.518708
$790$	0	0
$791$	8.02767	0.285431
$792$	0	0
$793$	−52.3052	−1.85741
$794$	0	0
$795$	3.86700	0.137149
$796$	52.4076	1.85754
$797$	20.6781	0.732454	0.366227	−	0.930525i	$-0.380649\pi$
$797$	20.6781	0.732454	0.366227	+	0.930525i	$0.380649\pi$
$798$	0	0
$799$	−1.30800	−0.0462737
$800$	0	0
$801$	11.2191	0.396407
$802$	0	0
$803$	10.6797	0.376880
$804$	13.7640	0.485418
$805$	−1.37450	−0.0484446
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−4.13069	−0.145227	−0.0726137	−	0.997360i	$-0.523134\pi$
$809$	−4.13069	−0.145227	−0.0726137	+	0.997360i	$0.523134\pi$
$810$	0	0
$811$	−30.5031	−1.07111	−0.535555	−	0.844501i	$-0.679897\pi$
$811$	−30.5031	−1.07111	−0.535555	+	0.844501i	$0.679897\pi$
$812$	6.88199	0.241510
$813$	−22.3551	−0.784027
$814$	0	0
$815$	−2.77666	−0.0972621
$816$	−13.0981	−0.458524
$817$	−3.01993	−0.105654
$818$	0	0
$819$	−21.3856	−0.747272
$820$	−1.65091	−0.0576524
$821$	−36.8431	−1.28583	−0.642917	−	0.765936i	$-0.722276\pi$
$821$	−36.8431	−1.28583	−0.642917	+	0.765936i	$0.722276\pi$
$822$	0	0
$823$	56.8275	1.98088	0.990442	−	0.137930i	$-0.0440450\pi$
$823$	56.8275	1.98088	0.990442	+	0.137930i	$0.0440450\pi$
$824$	0	0
$825$	−20.2011	−0.703311
$826$	0	0
$827$	46.0817	1.60242	0.801209	−	0.598384i	$-0.204190\pi$
$827$	46.0817	1.60242	0.801209	+	0.598384i	$0.204190\pi$
$828$	−2.00000	−0.0695048
$829$	−37.1428	−1.29002	−0.645011	−	0.764173i	$-0.723147\pi$
$829$	−37.1428	−1.29002	−0.645011	+	0.764173i	$0.723147\pi$
$830$	0	0
$831$	−11.8255	−0.410221
$832$	−49.7195	−1.72371
$833$	15.8501	0.549174
$834$	0	0
$835$	−4.17903	−0.144621
$836$	61.6110	2.13086
$837$	4.23989	0.146552
$838$	0	0
$839$	−9.91368	−0.342258	−0.171129	−	0.985249i	$-0.554742\pi$
$839$	−9.91368	−0.342258	−0.171129	+	0.985249i	$0.554742\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−4.72963	−0.162897
$844$	44.0077	1.51481
$845$	10.2360	0.352130
$846$	0	0
$847$	−22.0815	−0.758730
$848$	−38.7235	−1.32977
$849$	−5.12188	−0.175782
$850$	0	0
$851$	−4.80581	−0.164741
$852$	−8.53521	−0.292411
$853$	42.2359	1.44613	0.723065	−	0.690780i	$-0.242733\pi$
$853$	42.2359	1.44613	0.723065	+	0.690780i	$0.242733\pi$
$854$	0	0
$855$	2.94849	0.100836
$856$	0	0
$857$	18.0609	0.616947	0.308474	−	0.951233i	$-0.400182\pi$
$857$	18.0609	0.616947	0.308474	+	0.951233i	$0.400182\pi$
$858$	0	0
$859$	−20.0016	−0.682445	−0.341222	−	0.939983i	$-0.610841\pi$
$859$	−20.0016	−0.682445	−0.341222	+	0.939983i	$0.610841\pi$
$860$	0.326849	0.0111455
$861$	7.11081	0.242336
$862$	0	0
$863$	22.4631	0.764652	0.382326	−	0.924028i	$-0.375123\pi$
$863$	22.4631	0.764652	0.382326	+	0.924028i	$0.375123\pi$
$864$	0	0
$865$	−6.55291	−0.222805
$866$	0	0
$867$	6.27752	0.213196
$868$	−29.1789	−0.990395
$869$	26.2902	0.891832
$870$	0	0
$871$	42.7711	1.44924
$872$	0	0
$873$	−10.0138	−0.338917
$874$	0	0
$875$	13.5257	0.457250
$876$	−5.11801	−0.172922
$877$	17.6876	0.597266	0.298633	−	0.954368i	$-0.403469\pi$
$877$	17.6876	0.597266	0.298633	+	0.954368i	$0.403469\pi$
$878$	0	0
$879$	−0.190274	−0.00641780
$880$	−6.66820	−0.224785
$881$	−58.0246	−1.95490	−0.977449	−	0.211169i	$-0.932273\pi$
$881$	−58.0246	−1.95490	−0.977449	+	0.211169i	$0.932273\pi$
$882$	0	0
$883$	2.84765	0.0958309	0.0479155	−	0.998851i	$-0.484742\pi$
$883$	2.84765	0.0958309	0.0479155	+	0.998851i	$0.484742\pi$
$884$	−40.7019	−1.36895
$885$	1.74075	0.0585147
$886$	0	0
$887$	43.5728	1.46303	0.731516	−	0.681825i	$-0.238813\pi$
$887$	43.5728	1.46303	0.731516	+	0.681825i	$0.238813\pi$
$888$	0	0
$889$	14.0842	0.472369
$890$	0	0
$891$	−4.17339	−0.139814
$892$	28.1384	0.942144
$893$	−2.94849	−0.0986673
$894$	0	0
$895$	−4.58955	−0.153412
$896$	0	0
$897$	−6.21494	−0.207511
$898$	0	0
$899$	−4.23989	−0.141408
$900$	9.68088	0.322696
$901$	−31.7002	−1.05609
$902$	0	0
$903$	−1.40780	−0.0468487
$904$	0	0
$905$	−0.0331916	−0.00110333
$906$	0	0
$907$	52.0167	1.72719	0.863593	−	0.504190i	$-0.168209\pi$
$907$	52.0167	1.72719	0.863593	+	0.504190i	$0.168209\pi$
$908$	51.2786	1.70174
$909$	0.732397	0.0242921
$910$	0	0
$911$	41.4063	1.37185	0.685926	−	0.727671i	$-0.259397\pi$
$911$	41.4063	1.37185	0.685926	+	0.727671i	$0.259397\pi$
$912$	−29.5257	−0.977692
$913$	73.5695	2.43479
$914$	0	0
$915$	3.36177	0.111137
$916$	7.88857	0.260646
$917$	44.1361	1.45750
$918$	0	0
$919$	−6.45211	−0.212836	−0.106418	−	0.994322i	$-0.533938\pi$
$919$	−6.45211	−0.212836	−0.106418	+	0.994322i	$0.533938\pi$
$920$	0	0
$921$	10.8083	0.356145
$922$	0	0
$923$	−26.5229	−0.873012
$924$	28.7212	0.944859
$925$	23.2623	0.764858
$926$	0	0
$927$	6.74899	0.221666
$928$	0	0
$929$	−42.8391	−1.40551	−0.702753	−	0.711434i	$-0.748046\pi$
$929$	−42.8391	−1.40551	−0.702753	+	0.711434i	$0.748046\pi$
$930$	0	0
$931$	35.7293	1.17098
$932$	−12.1584	−0.398260
$933$	−15.6116	−0.511102
$934$	0	0
$935$	−5.45879	−0.178522
$936$	0	0
$937$	−26.6792	−0.871570	−0.435785	−	0.900051i	$-0.643529\pi$
$937$	−26.6792	−0.871570	−0.435785	+	0.900051i	$0.643529\pi$
$938$	0	0
$939$	−13.1579	−0.429393
$940$	0.319117	0.0104084
$941$	5.96451	0.194437	0.0972187	−	0.995263i	$-0.469005\pi$
$941$	5.96451	0.194437	0.0972187	+	0.995263i	$0.469005\pi$
$942$	0	0
$943$	2.06650	0.0672944
$944$	−17.4316	−0.567350
$945$	1.37450	0.0447124
$946$	0	0
$947$	2.74059	0.0890572	0.0445286	−	0.999008i	$-0.485821\pi$
$947$	2.74059	0.0890572	0.0445286	+	0.999008i	$0.485821\pi$
$948$	−12.5989	−0.409195
$949$	−15.9041	−0.516268
$950$	0	0
$951$	23.7451	0.769987
$952$	0	0
$953$	−7.35900	−0.238382	−0.119191	−	0.992871i	$-0.538030\pi$
$953$	−7.35900	−0.238382	−0.119191	+	0.992871i	$0.538030\pi$
$954$	0	0
$955$	−1.87697	−0.0607373
$956$	50.9540	1.64797
$957$	4.17339	0.134907
$958$	0	0
$959$	−44.1218	−1.42477
$960$	3.19558	0.103137
$961$	−13.0233	−0.420108
$962$	0	0
$963$	15.2260	0.490651
$964$	−35.2896	−1.13660
$965$	6.93511	0.223249
$966$	0	0
$967$	8.48975	0.273012	0.136506	−	0.990639i	$-0.456413\pi$
$967$	8.48975	0.273012	0.136506	+	0.990639i	$0.456413\pi$
$968$	0	0
$969$	−24.1706	−0.776471
$970$	0	0
$971$	−45.2082	−1.45080	−0.725400	−	0.688327i	$-0.758345\pi$
$971$	−45.2082	−1.45080	−0.725400	+	0.688327i	$0.758345\pi$
$972$	2.00000	0.0641500
$973$	1.74232	0.0558561
$974$	0	0
$975$	30.0830	0.963429
$976$	−33.6642	−1.07756
$977$	−57.3922	−1.83614	−0.918070	−	0.396419i	$-0.870253\pi$
$977$	−57.3922	−1.83614	−0.918070	+	0.396419i	$0.870253\pi$
$978$	0	0
$979$	−46.8217	−1.49643
$980$	−3.86700	−0.123527
$981$	20.3506	0.649746
$982$	0	0
$983$	56.0950	1.78915	0.894576	−	0.446916i	$-0.147478\pi$
$983$	56.0950	1.78915	0.894576	+	0.446916i	$0.147478\pi$
$984$	0	0
$985$	−5.20785	−0.165936
$986$	0	0
$987$	−1.37450	−0.0437507
$988$	−91.7500	−2.91896
$989$	−0.409127	−0.0130095
$990$	0	0
$991$	−7.76167	−0.246558	−0.123279	−	0.992372i	$-0.539341\pi$
$991$	−7.76167	−0.246558	−0.123279	+	0.992372i	$0.539341\pi$
$992$	0	0
$993$	3.09808	0.0983145
$994$	0	0
$995$	−10.4670	−0.331828
$996$	−35.2564	−1.11714
$997$	15.8776	0.502849	0.251425	−	0.967877i	$-0.419101\pi$
$997$	15.8776	0.502849	0.251425	+	0.967877i	$0.419101\pi$
$998$	0	0
$999$	4.80581	0.152049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.g.1.3	✓	4
3.2	odd	2		6003.2.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.g.1.3	✓	4	1.1	even	1	trivial
6003.2.a.g.1.2		4	3.2	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} +0.399447 q^{5} -3.44099 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} +0.399447 q^{5} -3.44099 q^{7} +1.00000 q^{9} -4.17339 q^{11} +2.00000 q^{12} +6.21494 q^{13} -0.399447 q^{15} +4.00000 q^{16} +3.27452 q^{17} +7.38141 q^{19} -0.798895 q^{20} +3.44099 q^{21} +1.00000 q^{23} -4.84044 q^{25} -1.00000 q^{27} +6.88199 q^{28} +1.00000 q^{29} -4.23989 q^{31} +4.17339 q^{33} -1.37450 q^{35} -2.00000 q^{36} -4.80581 q^{37} -6.21494 q^{39} +2.06650 q^{41} -0.409127 q^{43} +8.34678 q^{44} +0.399447 q^{45} -0.399447 q^{47} -4.00000 q^{48} +4.84044 q^{49} -3.27452 q^{51} -12.4299 q^{52} -9.68088 q^{53} -1.66705 q^{55} -7.38141 q^{57} -4.35790 q^{59} +0.798895 q^{60} -8.41604 q^{61} -3.44099 q^{63} -8.00000 q^{64} +2.48254 q^{65} +6.88199 q^{67} -6.54904 q^{68} -1.00000 q^{69} -4.26760 q^{71} -2.55901 q^{73} +4.84044 q^{75} -14.7628 q^{76} +14.3606 q^{77} -6.29947 q^{79} +1.59779 q^{80} +1.00000 q^{81} -17.6282 q^{83} -6.88199 q^{84} +1.30800 q^{85} -1.00000 q^{87} +11.2191 q^{89} -21.3856 q^{91} -2.00000 q^{92} +4.23989 q^{93} +2.94849 q^{95} -10.0138 q^{97} -4.17339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^9 + 8 * q^12 + 2 * q^15 + 16 * q^16 + 2 * q^17 + 4 * q^19 + 4 * q^20 + 2 * q^21 + 4 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^28 + 4 * q^29 + 2 * q^31 + 4 * q^35 - 8 * q^36 + 4 * q^37 + 6 * q^41 - 2 * q^45 + 2 * q^47 - 16 * q^48 + 4 * q^49 - 2 * q^51 - 8 * q^53 - 8 * q^55 - 4 * q^57 - 22 * q^59 - 4 * q^60 - 16 * q^61 - 2 * q^63 - 32 * q^64 - 10 * q^65 + 4 * q^67 - 4 * q^68 - 4 * q^69 - 22 * q^71 - 22 * q^73 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 20 * q^79 - 8 * q^80 + 4 * q^81 - 10 * q^83 - 4 * q^84 - 2 * q^85 - 4 * q^87 - 14 * q^89 - 26 * q^91 - 8 * q^92 - 2 * q^93 - 14 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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