LMFDB - Embedded newform 1840.3.k.d.321.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.8
Root			$-2.26343i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43837 q^{3} +2.23607i q^{5} -10.1866i q^{7} -6.93108 q^{9} +O(q^{10})$	$q-1.43837 q^{3} +2.23607i q^{5} -10.1866i q^{7} -6.93108 q^{9} -13.0237i q^{11} +23.2154 q^{13} -3.21630i q^{15} -28.2228i q^{17} +11.6665i q^{19} +14.6521i q^{21} +(17.3999 + 15.0414i) q^{23} -5.00000 q^{25} +22.9149 q^{27} +42.4794 q^{29} -18.7683 q^{31} +18.7330i q^{33} +22.7778 q^{35} -1.14094i q^{37} -33.3924 q^{39} -72.8198 q^{41} +4.96573i q^{43} -15.4984i q^{45} +0.813360 q^{47} -54.7661 q^{49} +40.5950i q^{51} +26.7286i q^{53} +29.1219 q^{55} -16.7808i q^{57} -94.0845 q^{59} -74.5293i q^{61} +70.6039i q^{63} +51.9112i q^{65} -80.0906i q^{67} +(-25.0275 - 21.6352i) q^{69} +83.5303 q^{71} -8.98897 q^{73} +7.19187 q^{75} -132.667 q^{77} +80.2841i q^{79} +29.4195 q^{81} +94.6451i q^{83} +63.1081 q^{85} -61.1014 q^{87} -136.812i q^{89} -236.485i q^{91} +26.9958 q^{93} -26.0871 q^{95} -2.32666i q^{97} +90.2684i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.43837			−0.479458			−0.239729	−	0.970840i	$-0.577059\pi$
$3$	−1.43837			−0.479458			−0.239729	+	0.970840i	$0.577059\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	10.1866i		−	1.45522i	−0.685989	−	0.727612i	$-0.740630\pi$
$7$		−	10.1866i		−	1.45522i	0.685989	−	0.727612i	$-0.259370\pi$
$8$		0			0
$9$	−6.93108			−0.770120
$10$		0			0
$11$		−	13.0237i		−	1.18397i	−0.805947	−	0.591987i	$-0.798343\pi$
$11$		−	13.0237i		−	1.18397i	0.805947	−	0.591987i	$-0.201657\pi$
$12$		0			0
$13$	23.2154			1.78580			0.892900	−	0.450255i	$-0.148667\pi$
$13$	23.2154			1.78580			0.892900	+	0.450255i	$0.148667\pi$
$14$		0			0
$15$		−	3.21630i		−	0.214420i
$16$		0			0
$17$		−	28.2228i		−	1.66016i	−0.557641	−	0.830082i	$-0.688293\pi$
$17$		−	28.2228i		−	1.66016i	0.557641	−	0.830082i	$-0.311707\pi$
$18$		0			0
$19$			11.6665i			0.614027i	0.951705	+	0.307013i	$0.0993297\pi$
$19$			11.6665i			0.614027i	−0.951705	+	0.307013i	$0.900670\pi$
$20$		0			0
$21$			14.6521i			0.697719i
$22$		0			0
$23$	17.3999	+	15.0414i	0.756516	+	0.653976i
$23$	17.3999	+	15.0414i	0.756516	+	0.653976i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	22.9149			0.848699
$28$		0			0
$29$	42.4794			1.46481			0.732404	−	0.680870i	$-0.238398\pi$
$29$	42.4794			1.46481			0.732404	+	0.680870i	$0.238398\pi$
$30$		0			0
$31$	−18.7683			−0.605428			−0.302714	−	0.953082i	$-0.597893\pi$
$31$	−18.7683			−0.605428			−0.302714	+	0.953082i	$0.597893\pi$
$32$		0			0
$33$			18.7330i			0.567667i
$34$		0			0
$35$	22.7778			0.650796
$36$		0			0
$37$		−	1.14094i		−	0.0308363i	−0.999881	−	0.0154182i	$-0.995092\pi$
$37$		−	1.14094i		−	0.0308363i	0.999881	−	0.0154182i	$-0.00490795\pi$
$38$		0			0
$39$	−33.3924			−0.856217
$40$		0			0
$41$	−72.8198			−1.77609			−0.888046	−	0.459755i	$-0.847937\pi$
$41$	−72.8198			−1.77609			−0.888046	+	0.459755i	$0.847937\pi$
$42$		0			0
$43$			4.96573i			0.115482i	0.998332	+	0.0577411i	$0.0183898\pi$
$43$			4.96573i			0.115482i	−0.998332	+	0.0577411i	$0.981610\pi$
$44$		0			0
$45$		−	15.4984i		−	0.344408i
$46$		0			0
$47$	0.813360			0.0173055			0.00865277	−	0.999963i	$-0.497246\pi$
$47$	0.813360			0.0173055			0.00865277	+	0.999963i	$0.497246\pi$
$48$		0			0
$49$	−54.7661			−1.11768
$50$		0			0
$51$			40.5950i			0.795980i
$52$		0			0
$53$			26.7286i			0.504313i	0.967686	+	0.252157i	$0.0811398\pi$
$53$			26.7286i			0.504313i	−0.967686	+	0.252157i	$0.918860\pi$
$54$		0			0
$55$	29.1219			0.529490
$56$		0			0
$57$		−	16.7808i		−	0.294400i
$58$		0			0
$59$	−94.0845			−1.59465			−0.797326	−	0.603549i	$-0.793753\pi$
$59$	−94.0845			−1.59465			−0.797326	+	0.603549i	$0.793753\pi$
$60$		0			0
$61$		−	74.5293i		−	1.22179i	−0.791711	−	0.610896i	$-0.790809\pi$
$61$		−	74.5293i		−	1.22179i	0.791711	−	0.610896i	$-0.209191\pi$
$62$		0			0
$63$			70.6039i			1.12070i
$64$		0			0
$65$			51.9112i			0.798634i
$66$		0			0
$67$		−	80.0906i		−	1.19538i	−0.801727	−	0.597691i	$-0.796085\pi$
$67$		−	80.0906i		−	1.19538i	0.801727	−	0.597691i	$-0.203915\pi$
$68$		0			0
$69$	−25.0275	−	21.6352i	−0.362718	−	0.313554i
$70$		0			0
$71$	83.5303			1.17648			0.588241	−	0.808686i	$-0.299821\pi$
$71$	83.5303			1.17648			0.588241	+	0.808686i	$0.299821\pi$
$72$		0			0
$73$	−8.98897			−0.123137			−0.0615683	−	0.998103i	$-0.519610\pi$
$73$	−8.98897			−0.123137			−0.0615683	+	0.998103i	$0.519610\pi$
$74$		0			0
$75$	7.19187			0.0958917
$76$		0			0
$77$	−132.667			−1.72295
$78$		0			0
$79$			80.2841i			1.01626i	0.861282	+	0.508128i	$0.169662\pi$
$79$			80.2841i			1.01626i	−0.861282	+	0.508128i	$0.830338\pi$
$80$		0			0
$81$	29.4195			0.363204
$82$		0			0
$83$			94.6451i			1.14030i	0.821540	+	0.570151i	$0.193115\pi$
$83$			94.6451i			1.14030i	−0.821540	+	0.570151i	$0.806885\pi$
$84$		0			0
$85$	63.1081			0.742448
$86$		0			0
$87$	−61.1014			−0.702315
$88$		0			0
$89$		−	136.812i		−	1.53722i	−0.639720	−	0.768608i	$-0.720950\pi$
$89$		−	136.812i		−	1.53722i	0.639720	−	0.768608i	$-0.279050\pi$
$90$		0			0
$91$		−	236.485i		−	2.59874i
$92$		0			0
$93$	26.9958			0.290277
$94$		0			0
$95$	−26.0871			−0.274601
$96$		0			0
$97$		−	2.32666i		−	0.0239862i	−0.999928	−	0.0119931i	$-0.996182\pi$
$97$		−	2.32666i		−	0.0239862i	0.999928	−	0.0119931i	$-0.00381761\pi$
$98$		0			0
$99$			90.2684i			0.911802i
$100$		0			0
$101$	12.9918			0.128632			0.0643161	−	0.997930i	$-0.479513\pi$
$101$	12.9918			0.128632			0.0643161	+	0.997930i	$0.479513\pi$
$102$		0			0
$103$		−	90.3165i		−	0.876859i	−0.898766	−	0.438430i	$-0.855535\pi$
$103$		−	90.3165i		−	0.876859i	0.898766	−	0.438430i	$-0.144465\pi$
$104$		0			0
$105$	−32.7631			−0.312029
$106$		0			0
$107$		−	185.173i		−	1.73059i	−0.501264	−	0.865294i	$-0.667131\pi$
$107$		−	185.173i		−	1.73059i	0.501264	−	0.865294i	$-0.332869\pi$
$108$		0			0
$109$			13.8248i			0.126833i	0.997987	+	0.0634167i	$0.0201997\pi$
$109$			13.8248i			0.126833i	−0.997987	+	0.0634167i	$0.979800\pi$
$110$		0			0
$111$			1.64110i			0.0147847i
$112$		0			0
$113$		−	178.530i		−	1.57991i	−0.613164	−	0.789956i	$-0.710103\pi$
$113$		−	178.530i		−	1.57991i	0.613164	−	0.789956i	$-0.289897\pi$
$114$		0			0
$115$	−33.6337	+	38.9073i	−0.292467	+	0.338324i
$116$		0			0
$117$	−160.908			−1.37528
$118$		0			0
$119$	−287.493			−2.41591
$120$		0			0
$121$	−48.6174			−0.401797
$122$		0			0
$123$	104.742			0.851562
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	112.959			0.889442			0.444721	−	0.895669i	$-0.353303\pi$
$127$	112.959			0.889442			0.444721	+	0.895669i	$0.353303\pi$
$128$		0			0
$129$		−	7.14258i		−	0.0553689i
$130$		0			0
$131$	12.1419			0.0926860			0.0463430	−	0.998926i	$-0.485243\pi$
$131$	12.1419			0.0926860			0.0463430	+	0.998926i	$0.485243\pi$
$132$		0			0
$133$	118.842			0.893546
$134$		0			0
$135$			51.2392i			0.379550i
$136$		0			0
$137$			170.543i			1.24484i	0.782685	+	0.622418i	$0.213850\pi$
$137$			170.543i			1.24484i	−0.782685	+	0.622418i	$0.786150\pi$
$138$		0			0
$139$	−38.5478			−0.277322			−0.138661	−	0.990340i	$-0.544280\pi$
$139$	−38.5478			−0.277322			−0.138661	+	0.990340i	$0.544280\pi$
$140$		0			0
$141$	−1.16992			−0.00829728
$142$		0			0
$143$		−	302.351i		−	2.11434i
$144$		0			0
$145$			94.9869i			0.655082i
$146$		0			0
$147$	78.7742			0.535879
$148$		0			0
$149$		−	81.4125i		−	0.546393i	−0.961958	−	0.273196i	$-0.911919\pi$
$149$		−	81.4125i		−	0.546393i	0.961958	−	0.273196i	$-0.0880808\pi$
$150$		0			0
$151$	−159.680			−1.05748			−0.528742	−	0.848782i	$-0.677336\pi$
$151$	−159.680			−1.05748			−0.528742	+	0.848782i	$0.677336\pi$
$152$		0			0
$153$			195.614i			1.27853i
$154$		0			0
$155$		−	41.9671i		−	0.270755i
$156$		0			0
$157$			38.5953i			0.245830i	0.992417	+	0.122915i	$0.0392242\pi$
$157$			38.5953i			0.245830i	−0.992417	+	0.122915i	$0.960776\pi$
$158$		0			0
$159$		−	38.4457i		−	0.241797i
$160$		0			0
$161$	153.221	−	177.245i	0.951681	−	1.10090i
$162$		0			0
$163$	−150.291			−0.922028			−0.461014	−	0.887393i	$-0.652514\pi$
$163$	−150.291			−0.922028			−0.461014	+	0.887393i	$0.652514\pi$
$164$		0			0
$165$	−41.8883			−0.253868
$166$		0			0
$167$	−27.7604			−0.166230			−0.0831149	−	0.996540i	$-0.526487\pi$
$167$	−27.7604			−0.166230			−0.0831149	+	0.996540i	$0.526487\pi$
$168$		0			0
$169$	369.955			2.18908
$170$		0			0
$171$		−	80.8615i		−	0.472874i
$172$		0			0
$173$	−24.0064			−0.138765			−0.0693826	−	0.997590i	$-0.522103\pi$
$173$	−24.0064			−0.138765			−0.0693826	+	0.997590i	$0.522103\pi$
$174$		0			0
$175$			50.9328i			0.291045i
$176$		0			0
$177$	135.329			0.764569
$178$		0			0
$179$	201.178			1.12390			0.561949	−	0.827172i	$-0.310052\pi$
$179$	201.178			1.12390			0.561949	+	0.827172i	$0.310052\pi$
$180$		0			0
$181$			202.358i			1.11800i	0.829168	+	0.558999i	$0.188814\pi$
$181$			202.358i			1.11800i	−0.829168	+	0.558999i	$0.811186\pi$
$182$		0			0
$183$			107.201i			0.585798i
$184$		0			0
$185$	2.55123			0.0137904
$186$		0			0
$187$	−367.566			−1.96559
$188$		0			0
$189$		−	233.424i		−	1.23505i
$190$		0			0
$191$			111.302i			0.582735i	0.956611	+	0.291368i	$0.0941103\pi$
$191$			111.302i			0.582735i	−0.956611	+	0.291368i	$0.905890\pi$
$192$		0			0
$193$	−164.081			−0.850163			−0.425081	−	0.905155i	$-0.639754\pi$
$193$	−164.081			−0.850163			−0.425081	+	0.905155i	$0.639754\pi$
$194$		0			0
$195$		−	74.6678i		−	0.382912i
$196$		0			0
$197$	−329.562			−1.67290			−0.836451	−	0.548042i	$-0.815373\pi$
$197$	−329.562			−1.67290			−0.836451	+	0.548042i	$0.815373\pi$
$198$		0			0
$199$		−	337.886i		−	1.69792i	−0.528455	−	0.848961i	$-0.677229\pi$
$199$		−	337.886i		−	1.69792i	0.528455	−	0.848961i	$-0.322771\pi$
$200$		0			0
$201$			115.200i			0.573136i
$202$		0			0
$203$		−	432.720i		−	2.13162i
$204$		0			0
$205$		−	162.830i		−	0.794292i
$206$		0			0
$207$	−120.600	−	104.253i	−0.582608	−	0.503640i
$208$		0			0
$209$	151.941			0.726992
$210$		0			0
$211$	−111.656			−0.529173			−0.264587	−	0.964362i	$-0.585235\pi$
$211$	−111.656			−0.529173			−0.264587	+	0.964362i	$0.585235\pi$
$212$		0			0
$213$	−120.148			−0.564074
$214$		0			0
$215$	−11.1037			−0.0516452
$216$		0			0
$217$			191.184i			0.881032i
$218$		0			0
$219$	12.9295			0.0590389
$220$		0			0
$221$		−	655.204i		−	2.96472i
$222$		0			0
$223$	363.545			1.63025			0.815123	−	0.579288i	$-0.196669\pi$
$223$	363.545			1.63025			0.815123	+	0.579288i	$0.196669\pi$
$224$		0			0
$225$	34.6554			0.154024
$226$		0			0
$227$			11.3368i			0.0499418i	0.999688	+	0.0249709i	$0.00794931\pi$
$227$			11.3368i			0.0499418i	−0.999688	+	0.0249709i	$0.992051\pi$
$228$		0			0
$229$			112.509i			0.491306i	0.969358	+	0.245653i	$0.0790024\pi$
$229$			112.509i			0.491306i	−0.969358	+	0.245653i	$0.920998\pi$
$230$		0			0
$231$	190.825			0.826082
$232$		0			0
$233$	−381.634			−1.63791			−0.818957	−	0.573855i	$-0.805447\pi$
$233$	−381.634			−1.63791			−0.818957	+	0.573855i	$0.805447\pi$
$234$		0			0
$235$			1.81873i			0.00773927i
$236$		0			0
$237$		−	115.479i		−	0.487252i
$238$		0			0
$239$	−139.296			−0.582828			−0.291414	−	0.956597i	$-0.594126\pi$
$239$	−139.296			−0.582828			−0.291414	+	0.956597i	$0.594126\pi$
$240$		0			0
$241$			82.3347i			0.341638i	0.985302	+	0.170819i	$0.0546413\pi$
$241$			82.3347i			0.341638i	−0.985302	+	0.170819i	$0.945359\pi$
$242$		0			0
$243$	−248.550			−1.02284
$244$		0			0
$245$		−	122.461i		−	0.499840i
$246$		0			0
$247$			270.843i			1.09653i
$248$		0			0
$249$		−	136.135i		−	0.546728i
$250$		0			0
$251$		−	323.924i		−	1.29054i	−0.763957	−	0.645268i	$-0.776746\pi$
$251$		−	323.924i		−	1.29054i	0.763957	−	0.645268i	$-0.223254\pi$
$252$		0			0
$253$	195.896	−	226.611i	0.774291	−	0.895695i
$254$		0			0
$255$	−90.7731			−0.355973
$256$		0			0
$257$	−250.796			−0.975858			−0.487929	−	0.872883i	$-0.662248\pi$
$257$	−250.796			−0.975858			−0.487929	+	0.872883i	$0.662248\pi$
$258$		0			0
$259$	−11.6223			−0.0448737
$260$		0			0
$261$	−294.428			−1.12808
$262$		0			0
$263$			276.878i			1.05277i	0.850247	+	0.526384i	$0.176452\pi$
$263$			276.878i			1.05277i	−0.850247	+	0.526384i	$0.823548\pi$
$264$		0			0
$265$	−59.7670			−0.225536
$266$		0			0
$267$			196.787i			0.737031i
$268$		0			0
$269$	−78.1002			−0.290335			−0.145168	−	0.989407i	$-0.546372\pi$
$269$	−78.1002			−0.290335			−0.145168	+	0.989407i	$0.546372\pi$
$270$		0			0
$271$	−331.551			−1.22343			−0.611717	−	0.791076i	$-0.709521\pi$
$271$	−331.551			−1.22343			−0.611717	+	0.791076i	$0.709521\pi$
$272$		0			0
$273$			340.154i			1.24599i
$274$		0			0
$275$			65.1186i			0.236795i
$276$		0			0
$277$	−355.974			−1.28511			−0.642553	−	0.766241i	$-0.722125\pi$
$277$	−355.974			−1.28511			−0.642553	+	0.766241i	$0.722125\pi$
$278$		0			0
$279$	130.084			0.466252
$280$		0			0
$281$		−	391.437i		−	1.39301i	−0.717550	−	0.696507i	$-0.754736\pi$
$281$		−	391.437i		−	1.39301i	0.717550	−	0.696507i	$-0.245264\pi$
$282$		0			0
$283$			133.616i			0.472143i	0.971736	+	0.236071i	$0.0758599\pi$
$283$			133.616i			0.472143i	−0.971736	+	0.236071i	$0.924140\pi$
$284$		0			0
$285$	37.5230			0.131660
$286$		0			0
$287$			741.783i			2.58461i
$288$		0			0
$289$	−507.527			−1.75615
$290$		0			0
$291$			3.34661i			0.0115004i
$292$		0			0
$293$		−	533.454i		−	1.82066i	−0.413880	−	0.910331i	$-0.635827\pi$
$293$		−	533.454i		−	1.82066i	0.413880	−	0.910331i	$-0.364173\pi$
$294$		0			0
$295$		−	210.379i		−	0.713150i
$296$		0			0
$297$		−	298.437i		−	1.00484i
$298$		0			0
$299$	403.945	+	349.193i	1.35099	+	1.16787i
$300$		0			0
$301$	50.5837			0.168052
$302$		0			0
$303$	−18.6871			−0.0616737
$304$		0			0
$305$	166.653			0.546402
$306$		0			0
$307$	−282.656			−0.920705			−0.460352	−	0.887736i	$-0.652277\pi$
$307$	−282.656			−0.920705			−0.460352	+	0.887736i	$0.652277\pi$
$308$		0			0
$309$			129.909i			0.420417i
$310$		0			0
$311$	193.206			0.621241			0.310620	−	0.950534i	$-0.399463\pi$
$311$	193.206			0.621241			0.310620	+	0.950534i	$0.399463\pi$
$312$		0			0
$313$			399.279i			1.27565i	0.770181	+	0.637826i	$0.220166\pi$
$313$			399.279i			1.27565i	−0.770181	+	0.637826i	$0.779834\pi$
$314$		0			0
$315$	−157.875			−0.501191
$316$		0			0
$317$	−36.5297			−0.115236			−0.0576178	−	0.998339i	$-0.518350\pi$
$317$	−36.5297			−0.115236			−0.0576178	+	0.998339i	$0.518350\pi$
$318$		0			0
$319$		−	553.241i		−	1.73430i
$320$		0			0
$321$			266.348i			0.829745i
$322$		0			0
$323$	329.262			1.01939
$324$		0			0
$325$	−116.077			−0.357160
$326$		0			0
$327$		−	19.8853i		−	0.0608113i
$328$		0			0
$329$		−	8.28534i		−	0.0251834i
$330$		0			0
$331$	−85.9406			−0.259639			−0.129820	−	0.991538i	$-0.541440\pi$
$331$	−85.9406			−0.259639			−0.129820	+	0.991538i	$0.541440\pi$
$332$		0			0
$333$			7.90797i			0.0237476i
$334$		0			0
$335$	179.088			0.534591
$336$		0			0
$337$			548.713i			1.62823i	0.580706	+	0.814114i	$0.302777\pi$
$337$			548.713i			1.62823i	−0.580706	+	0.814114i	$0.697223\pi$
$338$		0			0
$339$			256.793i			0.757502i
$340$		0			0
$341$			244.433i			0.716811i
$342$		0			0
$343$			58.7366i			0.171244i
$344$		0			0
$345$	48.3778	−	55.9632i	0.140226	−	0.162212i
$346$		0			0
$347$	−62.0036			−0.178685			−0.0893424	−	0.996001i	$-0.528477\pi$
$347$	−62.0036			−0.178685			−0.0893424	+	0.996001i	$0.528477\pi$
$348$		0			0
$349$	−131.393			−0.376485			−0.188242	−	0.982123i	$-0.560279\pi$
$349$	−131.393			−0.376485			−0.188242	+	0.982123i	$0.560279\pi$
$350$		0			0
$351$	531.978			1.51561
$352$		0			0
$353$	−176.990			−0.501389			−0.250694	−	0.968066i	$-0.580659\pi$
$353$	−176.990			−0.501389			−0.250694	+	0.968066i	$0.580659\pi$
$354$		0			0
$355$			186.779i			0.526139i
$356$		0			0
$357$	413.523			1.15833
$358$		0			0
$359$			187.851i			0.523260i	0.965168	+	0.261630i	$0.0842601\pi$
$359$			187.851i			0.523260i	−0.965168	+	0.261630i	$0.915740\pi$
$360$		0			0
$361$	224.893			0.622971
$362$		0			0
$363$	69.9300			0.192645
$364$		0			0
$365$		−	20.0999i		−	0.0550684i
$366$		0			0
$367$			128.510i			0.350162i	0.984554	+	0.175081i	$0.0560188\pi$
$367$			128.510i			0.350162i	−0.984554	+	0.175081i	$0.943981\pi$
$368$		0			0
$369$	504.719			1.36780
$370$		0			0
$371$	272.273			0.733888
$372$		0			0
$373$		−	79.2928i		−	0.212581i	−0.994335	−	0.106291i	$-0.966103\pi$
$373$		−	79.2928i		−	0.212581i	0.994335	−	0.106291i	$-0.0338974\pi$
$374$		0			0
$375$			16.0815i			0.0428841i
$376$		0			0
$377$	986.177			2.61585
$378$		0			0
$379$			402.569i			1.06219i	0.847313	+	0.531094i	$0.178219\pi$
$379$			402.569i			1.06219i	−0.847313	+	0.531094i	$0.821781\pi$
$380$		0			0
$381$	−162.478			−0.426450
$382$		0			0
$383$		−	370.198i		−	0.966575i	−0.875462	−	0.483288i	$-0.839443\pi$
$383$		−	370.198i		−	0.966575i	0.875462	−	0.483288i	$-0.160557\pi$
$384$		0			0
$385$		−	296.652i		−	0.770526i
$386$		0			0
$387$		−	34.4179i		−	0.0889351i
$388$		0			0
$389$		−	12.0598i		−	0.0310021i	−0.999880	−	0.0155011i	$-0.995066\pi$
$389$		−	12.0598i		−	0.0310021i	0.999880	−	0.0155011i	$-0.00493434\pi$
$390$		0			0
$391$	424.512	−	491.073i	1.08571	−	1.25594i
$392$		0			0
$393$	−17.4645			−0.0444391
$394$		0			0
$395$	−179.521			−0.454483
$396$		0			0
$397$	233.543			0.588270			0.294135	−	0.955764i	$-0.404968\pi$
$397$	233.543			0.588270			0.294135	+	0.955764i	$0.404968\pi$
$398$		0			0
$399$	−170.939			−0.428418
$400$		0			0
$401$		−	114.072i		−	0.284469i	−0.989833	−	0.142235i	$-0.954571\pi$
$401$		−	114.072i		−	0.284469i	0.989833	−	0.142235i	$-0.0454287\pi$
$402$		0			0
$403$	−435.713			−1.08117
$404$		0			0
$405$			65.7841i			0.162430i
$406$		0			0
$407$	−14.8593			−0.0365094
$408$		0			0
$409$	125.952			0.307950			0.153975	−	0.988075i	$-0.450792\pi$
$409$	125.952			0.307950			0.153975	+	0.988075i	$0.450792\pi$
$410$		0			0
$411$		−	245.304i		−	0.596847i
$412$		0			0
$413$			958.397i			2.32057i
$414$		0			0
$415$	−211.633			−0.509959
$416$		0			0
$417$	55.4462			0.132965
$418$		0			0
$419$		−	135.868i		−	0.324268i	−0.986769	−	0.162134i	$-0.948162\pi$
$419$		−	135.868i		−	0.324268i	0.986769	−	0.162134i	$-0.0518377\pi$
$420$		0			0
$421$			194.111i			0.461070i	0.973064	+	0.230535i	$0.0740477\pi$
$421$			194.111i			0.461070i	−0.973064	+	0.230535i	$0.925952\pi$
$422$		0			0
$423$	−5.63746			−0.0133273
$424$		0			0
$425$			141.114i			0.332033i
$426$		0			0
$427$	−759.198			−1.77798
$428$		0			0
$429$			434.894i			1.01374i
$430$		0			0
$431$		−	421.699i		−	0.978420i	−0.872166	−	0.489210i	$-0.837285\pi$
$431$		−	421.699i		−	0.978420i	0.872166	−	0.489210i	$-0.162715\pi$
$432$		0			0
$433$			3.78505i			0.00874144i	0.999990	+	0.00437072i	$0.00139125\pi$
$433$			3.78505i			0.00874144i	−0.999990	+	0.00437072i	$0.998609\pi$
$434$		0			0
$435$		−	136.627i		−	0.314085i
$436$		0			0
$437$	−175.481	+	202.996i	−0.401559	+	0.464521i
$438$		0			0
$439$	6.17744			0.0140716			0.00703581	−	0.999975i	$-0.497760\pi$
$439$	6.17744			0.0140716			0.00703581	+	0.999975i	$0.497760\pi$
$440$		0			0
$441$	379.588			0.860744
$442$		0			0
$443$	406.821			0.918331			0.459166	−	0.888351i	$-0.348148\pi$
$443$	406.821			0.918331			0.459166	+	0.888351i	$0.348148\pi$
$444$		0			0
$445$	305.921			0.687464
$446$		0			0
$447$			117.102i			0.261972i
$448$		0			0
$449$	−289.981			−0.645837			−0.322918	−	0.946427i	$-0.604664\pi$
$449$	−289.981			−0.645837			−0.322918	+	0.946427i	$0.604664\pi$
$450$		0			0
$451$			948.385i			2.10285i
$452$		0			0
$453$	229.680			0.507020
$454$		0			0
$455$	528.797			1.16219
$456$		0			0
$457$			224.430i			0.491094i	0.969385	+	0.245547i	$0.0789676\pi$
$457$			224.430i			0.491094i	−0.969385	+	0.245547i	$0.921032\pi$
$458$		0			0
$459$		−	646.722i		−	1.40898i
$460$		0			0
$461$	534.660			1.15978			0.579891	−	0.814694i	$-0.303095\pi$
$461$	534.660			1.15978			0.579891	+	0.814694i	$0.303095\pi$
$462$		0			0
$463$	−166.226			−0.359019			−0.179509	−	0.983756i	$-0.557451\pi$
$463$	−166.226			−0.359019			−0.179509	+	0.983756i	$0.557451\pi$
$464$		0			0
$465$			60.3644i			0.129816i
$466$		0			0
$467$			113.755i			0.243587i	0.992555	+	0.121794i	$0.0388646\pi$
$467$			113.755i			0.243587i	−0.992555	+	0.121794i	$0.961135\pi$
$468$		0			0
$469$	−815.848			−1.73955
$470$		0			0
$471$		−	55.5145i		−	0.117865i
$472$		0			0
$473$	64.6723			0.136728
$474$		0			0
$475$		−	58.3326i		−	0.122805i
$476$		0			0
$477$		−	185.258i		−	0.388382i
$478$		0			0
$479$		−	156.314i		−	0.326334i	−0.986598	−	0.163167i	$-0.947829\pi$
$479$		−	156.314i		−	0.326334i	0.986598	−	0.163167i	$-0.0521710\pi$
$480$		0			0
$481$		−	26.4875i		−	0.0550675i
$482$		0			0
$483$	−220.389	+	254.944i	−0.456291	+	0.527835i
$484$		0			0
$485$	5.20257			0.0107270
$486$		0			0
$487$	207.919			0.426938			0.213469	−	0.976950i	$-0.431524\pi$
$487$	207.919			0.426938			0.213469	+	0.976950i	$0.431524\pi$
$488$		0			0
$489$	216.174			0.442074
$490$		0			0
$491$	463.345			0.943676			0.471838	−	0.881685i	$-0.343591\pi$
$491$	463.345			0.943676			0.471838	+	0.881685i	$0.343591\pi$
$492$		0			0
$493$		−	1198.89i		−	2.43182i
$494$		0			0
$495$	−201.846			−0.407770
$496$		0			0
$497$		−	850.886i		−	1.71204i
$498$		0			0
$499$	50.7777			0.101759			0.0508794	−	0.998705i	$-0.483798\pi$
$499$	50.7777			0.101759			0.0508794	+	0.998705i	$0.483798\pi$
$500$		0			0
$501$	39.9298			0.0797003
$502$		0			0
$503$		−	91.1857i		−	0.181284i	−0.995884	−	0.0906419i	$-0.971108\pi$
$503$		−	91.1857i		−	0.181284i	0.995884	−	0.0906419i	$-0.0288919\pi$
$504$		0			0
$505$			29.0506i			0.0575260i
$506$		0			0
$507$	−532.134			−1.04957
$508$		0			0
$509$	317.681			0.624128			0.312064	−	0.950061i	$-0.398980\pi$
$509$	317.681			0.624128			0.312064	+	0.950061i	$0.398980\pi$
$510$		0			0
$511$			91.5667i			0.179191i
$512$		0			0
$513$			267.337i			0.521124i
$514$		0			0
$515$	201.954			0.392143
$516$		0			0
$517$		−	10.5930i		−	0.0204893i
$518$		0			0
$519$	34.5302			0.0665322
$520$		0			0
$521$			522.346i			1.00258i	0.865279	+	0.501291i	$0.167142\pi$
$521$			522.346i			1.00258i	−0.865279	+	0.501291i	$0.832858\pi$
$522$		0			0
$523$			245.926i			0.470221i	0.971969	+	0.235111i	$0.0755452\pi$
$523$			245.926i			0.470221i	−0.971969	+	0.235111i	$0.924455\pi$
$524$		0			0
$525$		−	73.2605i		−	0.139544i
$526$		0			0
$527$			529.693i			1.00511i
$528$		0			0
$529$	76.5101	+	523.438i	0.144632	+	0.989486i
$530$		0			0
$531$	652.107			1.22807
$532$		0			0
$533$	−1690.54			−3.17174
$534$		0			0
$535$	414.059			0.773943
$536$		0			0
$537$	−289.369			−0.538862
$538$		0			0
$539$			713.258i			1.32330i
$540$		0			0
$541$	−307.138			−0.567723			−0.283861	−	0.958865i	$-0.591616\pi$
$541$	−307.138			−0.567723			−0.283861	+	0.958865i	$0.591616\pi$
$542$		0			0
$543$		−	291.066i		−	0.536033i
$544$		0			0
$545$	−30.9133			−0.0567216
$546$		0			0
$547$	712.547			1.30265			0.651323	−	0.758801i	$-0.274214\pi$
$547$	712.547			1.30265			0.651323	+	0.758801i	$0.274214\pi$
$548$		0			0
$549$			516.569i			0.940926i
$550$		0			0
$551$			495.587i			0.899432i
$552$		0			0
$553$	817.820			1.47888
$554$		0			0
$555$	−3.66962			−0.00661193
$556$		0			0
$557$			1.17109i			0.00210250i	0.999999	+	0.00105125i	$0.000334623\pi$
$557$			1.17109i			0.00210250i	−0.999999	+	0.00105125i	$0.999665\pi$
$558$		0			0
$559$			115.281i			0.206228i
$560$		0			0
$561$	528.698			0.942420
$562$		0			0
$563$		−	900.058i		−	1.59868i	−0.600878	−	0.799341i	$-0.705182\pi$
$563$		−	900.058i		−	1.59868i	0.600878	−	0.799341i	$-0.294818\pi$
$564$		0			0
$565$	399.205			0.706558
$566$		0			0
$567$		−	299.684i		−	0.528543i
$568$		0			0
$569$		−	589.836i		−	1.03662i	−0.855193	−	0.518309i	$-0.826562\pi$
$569$		−	589.836i		−	1.03662i	0.855193	−	0.518309i	$-0.173438\pi$
$570$		0			0
$571$		−	117.755i		−	0.206225i	−0.994670	−	0.103113i	$-0.967120\pi$
$571$		−	117.755i		−	0.206225i	0.994670	−	0.103113i	$-0.0328802\pi$
$572$		0			0
$573$		−	160.095i		−	0.279397i
$574$		0			0
$575$	−86.9993	−	75.2072i	−0.151303	−	0.130795i
$576$		0			0
$577$	−403.533			−0.699365			−0.349682	−	0.936868i	$-0.613710\pi$
$577$	−403.533			−0.699365			−0.349682	+	0.936868i	$0.613710\pi$
$578$		0			0
$579$	236.011			0.407618
$580$		0			0
$581$	964.109			1.65940
$582$		0			0
$583$	348.106			0.597094
$584$		0			0
$585$		−	359.801i		−	0.615044i
$586$		0			0
$587$	−607.731			−1.03532			−0.517659	−	0.855587i	$-0.673196\pi$
$587$	−607.731			−1.03532			−0.517659	+	0.855587i	$0.673196\pi$
$588$		0			0
$589$		−	218.960i		−	0.371749i
$590$		0			0
$591$	474.033			0.802087
$592$		0			0
$593$	428.555			0.722690			0.361345	−	0.932432i	$-0.382318\pi$
$593$	428.555			0.722690			0.361345	+	0.932432i	$0.382318\pi$
$594$		0			0
$595$		−	642.855i		−	1.08043i
$596$		0			0
$597$			486.007i			0.814083i
$598$		0			0
$599$	162.171			0.270736			0.135368	−	0.990795i	$-0.456778\pi$
$599$	162.171			0.270736			0.135368	+	0.990795i	$0.456778\pi$
$600$		0			0
$601$	957.407			1.59302			0.796512	−	0.604623i	$-0.206676\pi$
$601$	957.407			1.59302			0.796512	+	0.604623i	$0.206676\pi$
$602$		0			0
$603$			555.114i			0.920587i
$604$		0			0
$605$		−	108.712i		−	0.179689i
$606$		0			0
$607$	989.893			1.63080			0.815398	−	0.578901i	$-0.196518\pi$
$607$	989.893			1.63080			0.815398	+	0.578901i	$0.196518\pi$
$608$		0			0
$609$			622.413i			1.02202i
$610$		0			0
$611$	18.8825			0.0309042
$612$		0			0
$613$		−	134.181i		−	0.218892i	−0.993993	−	0.109446i	$-0.965092\pi$
$613$		−	134.181i		−	0.218892i	0.993993	−	0.109446i	$-0.0349076\pi$
$614$		0			0
$615$			234.211i			0.380830i
$616$		0			0
$617$			151.269i			0.245168i	0.992458	+	0.122584i	$0.0391181\pi$
$617$			151.269i			0.245168i	−0.992458	+	0.122584i	$0.960882\pi$
$618$		0			0
$619$			115.852i			0.187159i	0.995612	+	0.0935797i	$0.0298310\pi$
$619$			115.852i			0.187159i	−0.995612	+	0.0935797i	$0.970169\pi$
$620$		0			0
$621$	398.715	+	344.673i	0.642054	+	0.555028i
$622$		0			0
$623$	−1393.65			−2.23699
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−218.549			−0.348563
$628$		0			0
$629$	−32.2006			−0.0511934
$630$		0			0
$631$			730.187i			1.15719i	0.815615	+	0.578595i	$0.196399\pi$
$631$			730.187i			1.15719i	−0.815615	+	0.578595i	$0.803601\pi$
$632$		0			0
$633$	160.602			0.253716
$634$		0			0
$635$			252.584i			0.397771i
$636$		0			0
$637$	−1271.42			−1.99594
$638$		0			0
$639$	−578.955			−0.906032
$640$		0			0
$641$			725.551i			1.13190i	0.824438	+	0.565952i	$0.191491\pi$
$641$			725.551i			1.13190i	−0.824438	+	0.565952i	$0.808509\pi$
$642$		0			0
$643$			505.908i			0.786792i	0.919369	+	0.393396i	$0.128700\pi$
$643$			505.908i			0.786792i	−0.919369	+	0.393396i	$0.871300\pi$
$644$		0			0
$645$	15.9713			0.0247617
$646$		0			0
$647$	324.943			0.502230			0.251115	−	0.967957i	$-0.419203\pi$
$647$	324.943			0.502230			0.251115	+	0.967957i	$0.419203\pi$
$648$		0			0
$649$			1225.33i			1.88803i
$650$		0			0
$651$		−	274.994i		−	0.422418i
$652$		0			0
$653$	−146.801			−0.224810			−0.112405	−	0.993662i	$-0.535855\pi$
$653$	−146.801			−0.224810			−0.112405	+	0.993662i	$0.535855\pi$
$654$		0			0
$655$			27.1500i			0.0414504i
$656$		0			0
$657$	62.3033			0.0948299
$658$		0			0
$659$			366.667i			0.556399i	0.960523	+	0.278200i	$0.0897377\pi$
$659$			366.667i			0.556399i	−0.960523	+	0.278200i	$0.910262\pi$
$660$		0			0
$661$		−	1241.87i		−	1.87878i	−0.342852	−	0.939389i	$-0.611393\pi$
$661$		−	1241.87i		−	1.87878i	0.342852	−	0.939389i	$-0.388607\pi$
$662$		0			0
$663$			942.428i			1.42146i
$664$		0			0
$665$			265.738i			0.399606i
$666$		0			0
$667$	739.136	+	638.952i	1.10815	+	0.957949i
$668$		0			0
$669$	−522.914			−0.781635
$670$		0			0
$671$	−970.649			−1.44657
$672$		0			0
$673$	812.966			1.20797			0.603987	−	0.796994i	$-0.293578\pi$
$673$	812.966			1.20797			0.603987	+	0.796994i	$0.293578\pi$
$674$		0			0
$675$	−114.574			−0.169740
$676$		0			0
$677$			1053.03i			1.55544i	0.628610	+	0.777721i	$0.283624\pi$
$677$			1053.03i			1.55544i	−0.628610	+	0.777721i	$0.716376\pi$
$678$		0			0
$679$	−23.7007			−0.0349053
$680$		0			0
$681$		−	16.3066i		−	0.0239450i
$682$		0			0
$683$	−947.718			−1.38758			−0.693791	−	0.720177i	$-0.744061\pi$
$683$	−947.718			−1.38758			−0.693791	+	0.720177i	$0.744061\pi$
$684$		0			0
$685$	−381.345			−0.556708
$686$		0			0
$687$		−	161.830i		−	0.235561i
$688$		0			0
$689$			620.515i			0.900602i
$690$		0			0
$691$	600.541			0.869090			0.434545	−	0.900650i	$-0.356909\pi$
$691$	600.541			0.869090			0.434545	+	0.900650i	$0.356909\pi$
$692$		0			0
$693$	919.525			1.32688
$694$		0			0
$695$		−	86.1955i		−	0.124022i
$696$		0			0
$697$			2055.18i			2.94861i
$698$		0			0
$699$	548.932			0.785311
$700$		0			0
$701$		−	907.376i		−	1.29440i	−0.762319	−	0.647201i	$-0.775939\pi$
$701$		−	907.376i		−	1.29440i	0.762319	−	0.647201i	$-0.224061\pi$
$702$		0			0
$703$	13.3108			0.0189343
$704$		0			0
$705$		−	2.61601i		−	0.00371066i
$706$		0			0
$707$		−	132.342i		−	0.187188i
$708$		0			0
$709$		−	657.300i		−	0.927081i	−0.886076	−	0.463540i	$-0.846579\pi$
$709$		−	657.300i		−	0.927081i	0.886076	−	0.463540i	$-0.153421\pi$
$710$		0			0
$711$		−	556.456i		−	0.782638i
$712$		0			0
$713$	−326.565	−	282.302i	−0.458015	−	0.395935i
$714$		0			0
$715$	676.077			0.945563
$716$		0			0
$717$	200.360			0.279442
$718$		0			0
$719$	−1095.72			−1.52395			−0.761975	−	0.647606i	$-0.775770\pi$
$719$	−1095.72			−1.52395			−0.761975	+	0.647606i	$0.775770\pi$
$720$		0			0
$721$	−920.015			−1.27603
$722$		0			0
$723$		−	118.428i		−	0.163801i
$724$		0			0
$725$	−212.397			−0.292962
$726$		0			0
$727$			814.942i			1.12097i	0.828166	+	0.560483i	$0.189384\pi$
$727$			814.942i			1.12097i	−0.828166	+	0.560483i	$0.810616\pi$
$728$		0			0
$729$	92.7324			0.127205
$730$		0			0
$731$	140.147			0.191719
$732$		0			0
$733$			742.293i			1.01268i	0.862335	+	0.506339i	$0.169001\pi$
$733$			742.293i			1.01268i	−0.862335	+	0.506339i	$0.830999\pi$
$734$		0			0
$735$			176.144i			0.239652i
$736$		0			0
$737$	−1043.08			−1.41530
$738$		0			0
$739$	288.013			0.389733			0.194866	−	0.980830i	$-0.437573\pi$
$739$	288.013			0.389733			0.194866	+	0.980830i	$0.437573\pi$
$740$		0			0
$741$		−	389.573i		−	0.525740i
$742$		0			0
$743$			199.651i			0.268709i	0.990933	+	0.134354i	$0.0428961\pi$
$743$			199.651i			0.268709i	−0.990933	+	0.134354i	$0.957104\pi$
$744$		0			0
$745$	182.044			0.244354
$746$		0			0
$747$		−	655.993i		−	0.878170i
$748$		0			0
$749$	−1886.28			−2.51839
$750$		0			0
$751$			610.667i			0.813138i	0.913620	+	0.406569i	$0.133275\pi$
$751$			610.667i			0.813138i	−0.913620	+	0.406569i	$0.866725\pi$
$752$		0			0
$753$			465.925i			0.618758i
$754$		0			0
$755$		−	357.056i		−	0.472922i
$756$		0			0
$757$		−	241.235i		−	0.318672i	−0.987224	−	0.159336i	$-0.949065\pi$
$757$		−	241.235i		−	0.318672i	0.987224	−	0.159336i	$-0.0509354\pi$
$758$		0			0
$759$	−281.771	+	325.952i	−0.371240	+	0.429449i
$760$		0			0
$761$	568.126			0.746552			0.373276	−	0.927720i	$-0.378235\pi$
$761$	568.126			0.746552			0.373276	+	0.927720i	$0.378235\pi$
$762$		0			0
$763$	140.828			0.184571
$764$		0			0
$765$	−437.407			−0.571774
$766$		0			0
$767$	−2184.21			−2.84773
$768$		0			0
$769$			425.966i			0.553922i	0.960881	+	0.276961i	$0.0893273\pi$
$769$			425.966i			0.553922i	−0.960881	+	0.276961i	$0.910673\pi$
$770$		0			0
$771$	360.738			0.467883
$772$		0			0
$773$		−	849.417i		−	1.09886i	−0.835540	−	0.549429i	$-0.814845\pi$
$773$		−	849.417i		−	1.09886i	0.835540	−	0.549429i	$-0.185155\pi$
$774$		0			0
$775$	93.8413			0.121086
$776$		0			0
$777$	16.7172			0.0215151
$778$		0			0
$779$		−	849.553i		−	1.09057i
$780$		0			0
$781$		−	1087.87i		−	1.39293i
$782$		0			0
$783$	973.411			1.24318
$784$		0			0
$785$	−86.3017			−0.109938
$786$		0			0
$787$		−	1448.81i		−	1.84093i	−0.390828	−	0.920464i	$-0.627811\pi$
$787$		−	1448.81i		−	1.84093i	0.390828	−	0.920464i	$-0.372189\pi$
$788$		0			0
$789$		−	398.254i		−	0.504758i
$790$		0			0
$791$	−1818.61			−2.29912
$792$		0			0
$793$		−	1730.23i		−	2.18188i
$794$		0			0
$795$	85.9673			0.108135
$796$		0			0
$797$		−	405.786i		−	0.509141i	−0.967054	−	0.254571i	$-0.918066\pi$
$797$		−	405.786i		−	0.509141i	0.967054	−	0.254571i	$-0.0819342\pi$
$798$		0			0
$799$		−	22.9553i		−	0.0287300i
$800$		0			0
$801$			948.256i			1.18384i
$802$		0			0
$803$			117.070i			0.145791i
$804$		0			0
$805$	396.331	+	342.612i	0.492337	+	0.425605i
$806$		0			0
$807$	112.337			0.139204
$808$		0			0
$809$	755.183			0.933477			0.466739	−	0.884395i	$-0.345429\pi$
$809$	755.183			0.933477			0.466739	+	0.884395i	$0.345429\pi$
$810$		0			0
$811$	1132.66			1.39662			0.698311	−	0.715795i	$-0.253935\pi$
$811$	1132.66			1.39662			0.698311	+	0.715795i	$0.253935\pi$
$812$		0			0
$813$	476.894			0.586586
$814$		0			0
$815$		−	336.060i		−	0.412344i
$816$		0			0
$817$	−57.9328			−0.0709091
$818$		0			0
$819$			1639.10i			2.00134i
$820$		0			0
$821$	167.684			0.204244			0.102122	−	0.994772i	$-0.467437\pi$
$821$	167.684			0.204244			0.102122	+	0.994772i	$0.467437\pi$
$822$		0			0
$823$	403.176			0.489886			0.244943	−	0.969538i	$-0.421231\pi$
$823$	403.176			0.489886			0.244943	+	0.969538i	$0.421231\pi$
$824$		0			0
$825$		−	93.6650i		−	0.113533i
$826$		0			0
$827$		−	901.234i		−	1.08976i	−0.838513	−	0.544881i	$-0.816575\pi$
$827$		−	901.234i		−	1.08976i	0.838513	−	0.544881i	$-0.183425\pi$
$828$		0			0
$829$	−455.150			−0.549034			−0.274517	−	0.961582i	$-0.588518\pi$
$829$	−455.150			−0.549034			−0.274517	+	0.961582i	$0.588518\pi$
$830$		0			0
$831$	512.024			0.616155
$832$		0			0
$833$			1545.65i			1.85553i
$834$		0			0
$835$		−	62.0741i		−	0.0743402i
$836$		0			0
$837$	−430.072			−0.513826
$838$		0			0
$839$			150.136i			0.178946i	0.995989	+	0.0894730i	$0.0285183\pi$
$839$			150.136i			0.178946i	−0.995989	+	0.0894730i	$0.971482\pi$
$840$		0			0
$841$	963.503			1.14566
$842$		0			0
$843$			563.033i			0.667892i
$844$		0			0
$845$			827.244i			0.978987i
$846$		0			0
$847$			495.244i			0.584704i
$848$		0			0
$849$		−	192.190i		−	0.226373i
$850$		0			0
$851$	17.1614	−	19.8523i	0.0201662	−	0.0233281i
$852$		0			0
$853$	582.826			0.683266			0.341633	−	0.939833i	$-0.389020\pi$
$853$	582.826			0.683266			0.341633	+	0.939833i	$0.389020\pi$
$854$		0			0
$855$	180.812			0.211476
$856$		0			0
$857$	−435.647			−0.508339			−0.254170	−	0.967160i	$-0.581802\pi$
$857$	−435.647			−0.508339			−0.254170	+	0.967160i	$0.581802\pi$
$858$		0			0
$859$	427.848			0.498076			0.249038	−	0.968494i	$-0.419886\pi$
$859$	427.848			0.498076			0.249038	+	0.968494i	$0.419886\pi$
$860$		0			0
$861$		−	1066.96i		−	1.23921i
$862$		0			0
$863$	698.719			0.809640			0.404820	−	0.914396i	$-0.367334\pi$
$863$	698.719			0.809640			0.404820	+	0.914396i	$0.367334\pi$
$864$		0			0
$865$		−	53.6799i		−	0.0620577i
$866$		0			0
$867$	730.013			0.841999
$868$		0			0
$869$	1045.60			1.20322
$870$		0			0
$871$		−	1859.34i		−	2.13471i
$872$		0			0
$873$			16.1263i			0.0184722i
$874$		0			0
$875$	−113.889			−0.130159
$876$		0			0
$877$	779.272			0.888566			0.444283	−	0.895887i	$-0.353459\pi$
$877$	779.272			0.888566			0.444283	+	0.895887i	$0.353459\pi$
$878$		0			0
$879$			767.307i			0.872932i
$880$		0			0
$881$		−	1612.85i		−	1.83070i	−0.402659	−	0.915350i	$-0.631914\pi$
$881$		−	1612.85i		−	1.83070i	0.402659	−	0.915350i	$-0.368086\pi$
$882$		0			0
$883$	336.489			0.381075			0.190537	−	0.981680i	$-0.438977\pi$
$883$	336.489			0.381075			0.190537	+	0.981680i	$0.438977\pi$
$884$		0			0
$885$			302.604i			0.341926i
$886$		0			0
$887$	12.5177			0.0141124			0.00705619	−	0.999975i	$-0.497754\pi$
$887$	12.5177			0.0141124			0.00705619	+	0.999975i	$0.497754\pi$
$888$		0			0
$889$		−	1150.67i		−	1.29434i
$890$		0			0
$891$		−	383.152i		−	0.430025i
$892$		0			0
$893$			9.48907i			0.0106261i
$894$		0			0
$895$			449.847i			0.502623i
$896$		0			0
$897$	−581.024	−	502.271i	−0.647741	−	0.559945i
$898$		0			0
$899$	−797.265			−0.886835
$900$		0			0
$901$	754.356			0.837243
$902$		0			0
$903$	−72.7584			−0.0805741
$904$		0			0
$905$	−452.485			−0.499984
$906$		0			0
$907$			1320.74i			1.45616i	0.685492	+	0.728080i	$0.259587\pi$
$907$			1320.74i			1.45616i	−0.685492	+	0.728080i	$0.740413\pi$
$908$		0			0
$909$	−90.0475			−0.0990621
$910$		0			0
$911$		−	1388.64i		−	1.52430i	−0.647398	−	0.762152i	$-0.724143\pi$
$911$		−	1388.64i		−	1.52430i	0.647398	−	0.762152i	$-0.275857\pi$
$912$		0			0
$913$	1232.63			1.35009
$914$		0			0
$915$	−239.709			−0.261977
$916$		0			0
$917$		−	123.684i		−	0.134879i
$918$		0			0
$919$		−	814.108i		−	0.885863i	−0.896555	−	0.442932i	$-0.853938\pi$
$919$		−	814.108i		−	0.885863i	0.896555	−	0.442932i	$-0.146062\pi$
$920$		0			0
$921$	406.566			0.441440
$922$		0			0
$923$	1939.19			2.10096
$924$		0			0
$925$			5.70472i			0.00616726i
$926$		0			0
$927$			625.991i			0.675286i
$928$		0			0
$929$	951.041			1.02373			0.511863	−	0.859067i	$-0.328956\pi$
$929$	951.041			1.02373			0.511863	+	0.859067i	$0.328956\pi$
$930$		0			0
$931$		−	638.929i		−	0.686283i
$932$		0			0
$933$	−277.903			−0.297859
$934$		0			0
$935$		−	821.903i		−	0.879040i
$936$		0			0
$937$			25.4596i			0.0271713i	0.999908	+	0.0135857i	$0.00432459\pi$
$937$			25.4596i			0.0271713i	−0.999908	+	0.0135857i	$0.995675\pi$
$938$		0			0
$939$		−	574.313i		−	0.611622i
$940$		0			0
$941$		−	1304.69i		−	1.38650i	−0.720698	−	0.693249i	$-0.756179\pi$
$941$		−	1304.69i		−	1.38650i	0.720698	−	0.693249i	$-0.243821\pi$
$942$		0			0
$943$	−1267.05	−	1095.31i	−1.34364	−	1.16152i
$944$		0			0
$945$	521.951			0.552329
$946$		0			0
$947$	25.1178			0.0265236			0.0132618	−	0.999912i	$-0.495779\pi$
$947$	25.1178			0.0265236			0.0132618	+	0.999912i	$0.495779\pi$
$948$		0			0
$949$	−208.683			−0.219897
$950$		0			0
$951$	52.5434			0.0552507
$952$		0			0
$953$			663.560i			0.696286i	0.937442	+	0.348143i	$0.113188\pi$
$953$			663.560i			0.696286i	−0.937442	+	0.348143i	$0.886812\pi$
$954$		0			0
$955$	−248.880			−0.260607
$956$		0			0
$957$			795.767i			0.831523i
$958$		0			0
$959$	1737.24			1.81152
$960$		0			0
$961$	−608.753			−0.633457
$962$		0			0
$963$			1283.45i			1.33276i
$964$		0			0
$965$		−	366.897i		−	0.380204i
$966$		0			0
$967$	661.312			0.683880			0.341940	−	0.939722i	$-0.388916\pi$
$967$	661.312			0.683880			0.341940	+	0.939722i	$0.388916\pi$
$968$		0			0
$969$	−473.602			−0.488753
$970$		0			0
$971$		−	1198.53i		−	1.23433i	−0.786834	−	0.617165i	$-0.788281\pi$
$971$		−	1198.53i		−	1.23433i	0.786834	−	0.617165i	$-0.211719\pi$
$972$		0			0
$973$			392.670i			0.403566i
$974$		0			0
$975$	166.962			0.171243
$976$		0			0
$977$		−	1156.31i		−	1.18353i	−0.806110	−	0.591765i	$-0.798431\pi$
$977$		−	1156.31i		−	1.18353i	0.806110	−	0.591765i	$-0.201569\pi$
$978$		0			0
$979$	−1781.80			−1.82002
$980$		0			0
$981$		−	95.8210i		−	0.0976768i
$982$		0			0
$983$			534.831i			0.544080i	0.962286	+	0.272040i	$0.0876983\pi$
$983$			534.831i			0.544080i	−0.962286	+	0.272040i	$0.912302\pi$
$984$		0			0
$985$		−	736.922i		−	0.748144i
$986$		0			0
$987$			11.9174i			0.0120744i
$988$		0			0
$989$	−74.6918	+	86.4030i	−0.0755225	+	0.0873640i
$990$		0			0
$991$	217.485			0.219461			0.109730	−	0.993961i	$-0.465001\pi$
$991$	217.485			0.219461			0.109730	+	0.993961i	$0.465001\pi$
$992$		0			0
$993$	123.615			0.124486
$994$		0			0
$995$	755.537			0.759334
$996$		0			0
$997$	−248.055			−0.248801			−0.124400	−	0.992232i	$-0.539701\pi$
$997$	−248.055			−0.248801			−0.124400	+	0.992232i	$0.539701\pi$
$998$		0			0
$999$		−	26.1446i		−	0.0261707i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.8		16
4.3	odd	2		230.3.d.a.91.14	yes	16
12.11	even	2		2070.3.c.a.91.4		16
20.3	even	4		1150.3.c.c.1149.31		32
20.7	even	4		1150.3.c.c.1149.2		32
20.19	odd	2		1150.3.d.b.551.3		16
23.22	odd	2	inner	1840.3.k.d.321.7		16
92.91	even	2		230.3.d.a.91.13	✓	16
276.275	odd	2		2070.3.c.a.91.5		16
460.183	odd	4		1150.3.c.c.1149.1		32
460.367	odd	4		1150.3.c.c.1149.32		32
460.459	even	2		1150.3.d.b.551.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.13	✓	16	92.91	even	2
230.3.d.a.91.14	yes	16	4.3	odd	2
1150.3.c.c.1149.1		32	460.183	odd	4
1150.3.c.c.1149.2		32	20.7	even	4
1150.3.c.c.1149.31		32	20.3	even	4
1150.3.c.c.1149.32		32	460.367	odd	4
1150.3.d.b.551.3		16	20.19	odd	2
1150.3.d.b.551.4		16	460.459	even	2
1840.3.k.d.321.7		16	23.22	odd	2	inner
1840.3.k.d.321.8		16	1.1	even	1	trivial
2070.3.c.a.91.4		16	12.11	even	2
2070.3.c.a.91.5		16	276.275	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.43837 q^{3} +2.23607i q^{5} -10.1866i q^{7} -6.93108 q^{9} +O(q^{10})\)	\(q-1.43837 q^{3} +2.23607i q^{5} -10.1866i q^{7} -6.93108 q^{9} -13.0237i q^{11} +23.2154 q^{13} -3.21630i q^{15} -28.2228i q^{17} +11.6665i q^{19} +14.6521i q^{21} +(17.3999 + 15.0414i) q^{23} -5.00000 q^{25} +22.9149 q^{27} +42.4794 q^{29} -18.7683 q^{31} +18.7330i q^{33} +22.7778 q^{35} -1.14094i q^{37} -33.3924 q^{39} -72.8198 q^{41} +4.96573i q^{43} -15.4984i q^{45} +0.813360 q^{47} -54.7661 q^{49} +40.5950i q^{51} +26.7286i q^{53} +29.1219 q^{55} -16.7808i q^{57} -94.0845 q^{59} -74.5293i q^{61} +70.6039i q^{63} +51.9112i q^{65} -80.0906i q^{67} +(-25.0275 - 21.6352i) q^{69} +83.5303 q^{71} -8.98897 q^{73} +7.19187 q^{75} -132.667 q^{77} +80.2841i q^{79} +29.4195 q^{81} +94.6451i q^{83} +63.1081 q^{85} -61.1014 q^{87} -136.812i q^{89} -236.485i q^{91} +26.9958 q^{93} -26.0871 q^{95} -2.32666i q^{97} +90.2684i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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