LMFDB - Embedded newform 1368.3.o.c.721.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,3,Mod(721,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, 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("1368.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1368.o (of order $2$, degree $1$, 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:	$37.2753001645$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.9
Root			$-1.34238 - 2.32507i$ of defining polynomial
Character	$\chi$	$=$	1368.721
Dual form			1368.3.o.c.721.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0118390 q^{5} +5.38136 q^{7} +O(q^{10})$	$q+0.0118390 q^{5} +5.38136 q^{7} +20.1709 q^{11} -18.7435i q^{13} +2.61324 q^{17} +(14.1212 - 12.7119i) q^{19} -12.3236 q^{23} -24.9999 q^{25} +4.07280i q^{29} +14.9617i q^{31} +0.0637101 q^{35} -62.7522i q^{37} +46.8535i q^{41} -72.6553 q^{43} +81.3241 q^{47} -20.0410 q^{49} +32.5146i q^{53} +0.238804 q^{55} -48.9963i q^{59} +81.6937 q^{61} -0.221905i q^{65} -64.8329i q^{67} -18.7083i q^{71} +117.506 q^{73} +108.547 q^{77} +80.1054i q^{79} -61.0368 q^{83} +0.0309383 q^{85} -132.639i q^{89} -100.865i q^{91} +(0.167181 - 0.150497i) q^{95} -91.8294i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 16 q^{7}+O(q^{10}) $ 20 * q - 16 * q^7	$ 20 q - 16 q^{7} - 16 q^{11} - 32 q^{17} + 40 q^{19} - 64 q^{23} + 68 q^{25} + 208 q^{35} + 64 q^{43} - 48 q^{47} + 20 q^{49} - 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} - 224 q^{83} - 136 q^{85} + 320 q^{95}+O(q^{100}) $ 20 * q - 16 * q^7 - 16 * q^11 - 32 * q^17 + 40 * q^19 - 64 * q^23 + 68 * q^25 + 208 * q^35 + 64 * q^43 - 48 * q^47 + 20 * q^49 - 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 - 224 * q^83 - 136 * q^85 + 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0118390			0.00236781			0.00118390	−	0.999999i	$-0.499623\pi$
$5$	0.0118390			0.00236781			0.00118390	+	0.999999i	$0.499623\pi$
$6$		0			0
$7$	5.38136			0.768765			0.384383	−	0.923174i	$-0.374414\pi$
$7$	5.38136			0.768765			0.384383	+	0.923174i	$0.374414\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.1709			1.83372			0.916859	−	0.399211i	$-0.130716\pi$
$11$	20.1709			1.83372			0.916859	+	0.399211i	$0.130716\pi$
$12$		0			0
$13$		−	18.7435i		−	1.44181i	−0.693035	−	0.720904i	$-0.743727\pi$
$13$		−	18.7435i		−	1.44181i	0.693035	−	0.720904i	$-0.256273\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.61324			0.153720			0.0768600	−	0.997042i	$-0.475511\pi$
$17$	2.61324			0.153720			0.0768600	+	0.997042i	$0.475511\pi$
$18$		0			0
$19$	14.1212	−	12.7119i	0.743219	−	0.669049i
$19$	14.1212	−	12.7119i	0.743219	−	0.669049i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.3236			−0.535810			−0.267905	−	0.963445i	$-0.586331\pi$
$23$	−12.3236			−0.535810			−0.267905	+	0.963445i	$0.586331\pi$
$24$		0			0
$25$	−24.9999			−0.999994
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.07280i			0.140441i	0.997531	+	0.0702206i	$0.0223703\pi$
$29$			4.07280i			0.140441i	−0.997531	+	0.0702206i	$0.977630\pi$
$30$		0			0
$31$			14.9617i			0.482635i	0.970446	+	0.241317i	$0.0775795\pi$
$31$			14.9617i			0.482635i	−0.970446	+	0.241317i	$0.922421\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.0637101			0.00182029
$36$		0			0
$37$		−	62.7522i		−	1.69601i	−0.529992	−	0.848003i	$-0.677805\pi$
$37$		−	62.7522i		−	1.69601i	0.529992	−	0.848003i	$-0.322195\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			46.8535i			1.14277i	0.820683	+	0.571384i	$0.193593\pi$
$41$			46.8535i			1.14277i	−0.820683	+	0.571384i	$0.806407\pi$
$42$		0			0
$43$	−72.6553			−1.68966			−0.844829	−	0.535037i	$-0.820298\pi$
$43$	−72.6553			−1.68966			−0.844829	+	0.535037i	$0.820298\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	81.3241			1.73030			0.865150	−	0.501512i	$-0.167223\pi$
$47$	81.3241			1.73030			0.865150	+	0.501512i	$0.167223\pi$
$48$		0			0
$49$	−20.0410			−0.409000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			32.5146i			0.613483i	0.951793	+	0.306741i	$0.0992387\pi$
$53$			32.5146i			0.613483i	−0.951793	+	0.306741i	$0.900761\pi$
$54$		0			0
$55$	0.238804			0.00434189
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	48.9963i		−	0.830446i	−0.909720	−	0.415223i	$-0.863704\pi$
$59$		−	48.9963i		−	0.830446i	0.909720	−	0.415223i	$-0.136296\pi$
$60$		0			0
$61$	81.6937			1.33924			0.669620	−	0.742704i	$-0.266457\pi$
$61$	81.6937			1.33924			0.669620	+	0.742704i	$0.266457\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.221905i		−	0.00341392i
$66$		0			0
$67$		−	64.8329i		−	0.967655i	−0.875164	−	0.483827i	$-0.839246\pi$
$67$		−	64.8329i		−	0.967655i	0.875164	−	0.483827i	$-0.160754\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	18.7083i		−	0.263498i	−0.991283	−	0.131749i	$-0.957941\pi$
$71$		−	18.7083i		−	0.263498i	0.991283	−	0.131749i	$-0.0420592\pi$
$72$		0			0
$73$	117.506			1.60966			0.804832	−	0.593502i	$-0.202255\pi$
$73$	117.506			1.60966			0.804832	+	0.593502i	$0.202255\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	108.547			1.40970
$78$		0			0
$79$			80.1054i			1.01399i	0.861948	+	0.506996i	$0.169244\pi$
$79$			80.1054i			1.01399i	−0.861948	+	0.506996i	$0.830756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−61.0368			−0.735383			−0.367692	−	0.929948i	$-0.619852\pi$
$83$	−61.0368			−0.735383			−0.367692	+	0.929948i	$0.619852\pi$
$84$		0			0
$85$	0.0309383			0.000363979
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	132.639i		−	1.49032i	−0.666884	−	0.745162i	$-0.732372\pi$
$89$		−	132.639i		−	1.49032i	0.666884	−	0.745162i	$-0.267628\pi$
$90$		0			0
$91$		−	100.865i		−	1.10841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.167181	−	0.150497i	0.00175980	−	0.00158418i
$96$		0			0
$97$		−	91.8294i		−	0.946695i	−0.880876	−	0.473347i	$-0.843046\pi$
$97$		−	91.8294i		−	0.946695i	0.880876	−	0.473347i	$-0.156954\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.18062			0.0413923			0.0206961	−	0.999786i	$-0.493412\pi$
$101$	4.18062			0.0413923			0.0206961	+	0.999786i	$0.493412\pi$
$102$		0			0
$103$		−	49.0424i		−	0.476140i	−0.971248	−	0.238070i	$-0.923485\pi$
$103$		−	49.0424i		−	0.476140i	0.971248	−	0.238070i	$-0.0765148\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	123.082i		−	1.15030i	−0.818049	−	0.575149i	$-0.804944\pi$
$107$		−	123.082i		−	1.15030i	0.818049	−	0.575149i	$-0.195056\pi$
$108$		0			0
$109$			40.7625i			0.373968i	0.982363	+	0.186984i	$0.0598712\pi$
$109$			40.7625i			0.373968i	−0.982363	+	0.186984i	$0.940129\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			188.610i			1.66911i	0.550921	+	0.834557i	$0.314277\pi$
$113$			188.610i			1.66911i	−0.550921	+	0.834557i	$0.685723\pi$
$114$		0			0
$115$	−0.145900			−0.00126870
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.0628			0.118175
$120$		0			0
$121$	285.865			2.36252
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.591950			−0.00473560
$126$		0			0
$127$		−	189.176i		−	1.48958i	−0.667300	−	0.744789i	$-0.732550\pi$
$127$		−	189.176i		−	1.48958i	0.667300	−	0.744789i	$-0.267450\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−57.7649			−0.440953			−0.220477	−	0.975392i	$-0.570761\pi$
$131$	−57.7649			−0.440953			−0.220477	+	0.975392i	$0.570761\pi$
$132$		0			0
$133$	75.9910	−	68.4074i	0.571361	−	0.514341i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−190.267			−1.38881			−0.694406	−	0.719584i	$-0.744333\pi$
$137$	−190.267			−1.38881			−0.694406	+	0.719584i	$0.744333\pi$
$138$		0			0
$139$	59.9097			0.431005			0.215502	−	0.976503i	$-0.430861\pi$
$139$	59.9097			0.431005			0.215502	+	0.976503i	$0.430861\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	378.073i		−	2.64387i
$144$		0			0
$145$			0.0482180i			0.000332538i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	126.063			0.846063			0.423032	−	0.906115i	$-0.360966\pi$
$149$	126.063			0.846063			0.423032	+	0.906115i	$0.360966\pi$
$150$		0			0
$151$			255.446i			1.69170i	0.533422	+	0.845849i	$0.320906\pi$
$151$			255.446i			1.69170i	−0.533422	+	0.845849i	$0.679094\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.177132i			0.00114279i
$156$		0			0
$157$	52.9750			0.337421			0.168710	−	0.985666i	$-0.446040\pi$
$157$	52.9750			0.337421			0.168710	+	0.985666i	$0.446040\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−66.3179			−0.411912
$162$		0			0
$163$	260.416			1.59765			0.798823	−	0.601566i	$-0.205456\pi$
$163$	260.416			1.59765			0.798823	+	0.601566i	$0.205456\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	283.157i		−	1.69555i	−0.530357	−	0.847774i	$-0.677942\pi$
$167$		−	283.157i		−	1.69555i	0.530357	−	0.847774i	$-0.322058\pi$
$168$		0			0
$169$	−182.319			−1.07881
$170$		0			0
$171$		0			0
$172$		0			0
$173$			97.1696i			0.561674i	0.959755	+	0.280837i	$0.0906120\pi$
$173$			97.1696i			0.561674i	−0.959755	+	0.280837i	$0.909388\pi$
$174$		0			0
$175$	−134.533			−0.768761
$176$		0			0
$177$		0			0
$178$		0			0
$179$			309.155i			1.72712i	0.504245	+	0.863560i	$0.331771\pi$
$179$			309.155i			1.72712i	−0.504245	+	0.863560i	$0.668229\pi$
$180$		0			0
$181$			62.7290i			0.346569i	0.984872	+	0.173284i	$0.0554380\pi$
$181$			62.7290i			0.346569i	−0.984872	+	0.173284i	$0.944562\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	0.742926i		−	0.00401581i
$186$		0			0
$187$	52.7114			0.281879
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−238.858			−1.25057			−0.625283	−	0.780398i	$-0.715016\pi$
$191$	−238.858			−1.25057			−0.625283	+	0.780398i	$0.715016\pi$
$192$		0			0
$193$			32.4003i			0.167877i	0.996471	+	0.0839385i	$0.0267499\pi$
$193$			32.4003i			0.167877i	−0.996471	+	0.0839385i	$0.973250\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	186.657			0.947495			0.473748	−	0.880661i	$-0.342901\pi$
$197$	186.657			0.947495			0.473748	+	0.880661i	$0.342901\pi$
$198$		0			0
$199$	218.064			1.09580			0.547899	−	0.836545i	$-0.315428\pi$
$199$	218.064			1.09580			0.547899	+	0.836545i	$0.315428\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			21.9172i			0.107966i
$204$		0			0
$205$			0.554700i			0.00270585i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	284.836	−	256.411i	1.36285	−	1.22685i
$210$		0			0
$211$		−	65.8204i		−	0.311945i	−0.987761	−	0.155973i	$-0.950149\pi$
$211$		−	65.8204i		−	0.311945i	0.987761	−	0.155973i	$-0.0498512\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.860168			−0.00400078
$216$		0			0
$217$			80.5142i			0.371033i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	48.9813i		−	0.221635i
$222$		0			0
$223$		−	18.3832i		−	0.0824358i	−0.999150	−	0.0412179i	$-0.986876\pi$
$223$		−	18.3832i		−	0.0824358i	0.999150	−	0.0412179i	$-0.0131238\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			16.7659i			0.0738586i	0.999318	+	0.0369293i	$0.0117576\pi$
$227$			16.7659i			0.0738586i	−0.999318	+	0.0369293i	$0.988242\pi$
$228$		0			0
$229$	126.904			0.554165			0.277083	−	0.960846i	$-0.410632\pi$
$229$	126.904			0.554165			0.277083	+	0.960846i	$0.410632\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	286.202			1.22834			0.614168	−	0.789175i	$-0.289492\pi$
$233$	286.202			1.22834			0.614168	+	0.789175i	$0.289492\pi$
$234$		0			0
$235$	0.962799			0.00409702
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−184.163			−0.770557			−0.385278	−	0.922800i	$-0.625895\pi$
$239$	−184.163			−0.770557			−0.385278	+	0.922800i	$0.625895\pi$
$240$		0			0
$241$		−	13.5417i		−	0.0561895i	−0.999605	−	0.0280948i	$-0.991056\pi$
$241$		−	13.5417i		−	0.0561895i	0.999605	−	0.0280948i	$-0.00894402\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.237266			−0.000968433
$246$		0			0
$247$	−238.266	−	264.680i	−0.964640	−	1.07158i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−35.0962			−0.139825			−0.0699127	−	0.997553i	$-0.522272\pi$
$251$	−35.0962			−0.139825			−0.0699127	+	0.997553i	$0.522272\pi$
$252$		0			0
$253$	−248.579			−0.982525
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	157.231i		−	0.611794i	−0.952065	−	0.305897i	$-0.901044\pi$
$257$		−	157.231i		−	0.611794i	0.952065	−	0.305897i	$-0.0989564\pi$
$258$		0			0
$259$		−	337.692i		−	1.30383i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	268.630			1.02141			0.510704	−	0.859757i	$-0.329385\pi$
$263$	268.630			1.02141			0.510704	+	0.859757i	$0.329385\pi$
$264$		0			0
$265$			0.384941i			0.00145261i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			170.930i			0.635426i	0.948187	+	0.317713i	$0.102915\pi$
$269$			170.930i			0.635426i	−0.948187	+	0.317713i	$0.897085\pi$
$270$		0			0
$271$	−372.857			−1.37585			−0.687927	−	0.725779i	$-0.741479\pi$
$271$	−372.857			−1.37585			−0.687927	+	0.725779i	$0.741479\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−504.270			−1.83371
$276$		0			0
$277$	400.543			1.44600			0.723001	−	0.690847i	$-0.242762\pi$
$277$	400.543			1.44600			0.723001	+	0.690847i	$0.242762\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			149.073i			0.530510i	0.964178	+	0.265255i	$0.0854562\pi$
$281$			149.073i			0.530510i	−0.964178	+	0.265255i	$0.914544\pi$
$282$		0			0
$283$	26.4609			0.0935013			0.0467506	−	0.998907i	$-0.485113\pi$
$283$	26.4609			0.0935013			0.0467506	+	0.998907i	$0.485113\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			252.135i			0.878520i
$288$		0			0
$289$	−282.171			−0.976370
$290$		0			0
$291$		0			0
$292$		0			0
$293$			538.517i			1.83794i	0.394326	+	0.918970i	$0.370978\pi$
$293$			538.517i			1.83794i	−0.394326	+	0.918970i	$0.629022\pi$
$294$		0			0
$295$		−	0.580069i		−	0.00196634i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			230.988i			0.772536i
$300$		0			0
$301$	−390.984			−1.29895
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.967174			0.00317106
$306$		0			0
$307$			284.758i			0.927552i	0.885953	+	0.463776i	$0.153506\pi$
$307$			284.758i			0.927552i	−0.885953	+	0.463776i	$0.846494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−128.577			−0.413431			−0.206715	−	0.978401i	$-0.566277\pi$
$311$	−128.577			−0.413431			−0.206715	+	0.978401i	$0.566277\pi$
$312$		0			0
$313$	13.0647			0.0417402			0.0208701	−	0.999782i	$-0.493356\pi$
$313$	13.0647			0.0417402			0.0208701	+	0.999782i	$0.493356\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	238.719i		−	0.753056i	−0.926405	−	0.376528i	$-0.877118\pi$
$317$		−	238.719i		−	0.753056i	0.926405	−	0.376528i	$-0.122882\pi$
$318$		0			0
$319$			82.1520i			0.257530i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	36.9020	−	33.2193i	0.114248	−	0.102846i
$324$		0			0
$325$			468.585i			1.44180i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	437.634			1.33020
$330$		0			0
$331$		−	483.447i		−	1.46057i	−0.683144	−	0.730283i	$-0.739388\pi$
$331$		−	483.447i		−	1.46057i	0.683144	−	0.730283i	$-0.260612\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	0.767558i		−	0.00229122i
$336$		0			0
$337$			238.148i			0.706670i	0.935497	+	0.353335i	$0.114952\pi$
$337$			238.148i			0.706670i	−0.935497	+	0.353335i	$0.885048\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			301.791i			0.885016i
$342$		0			0
$343$	−371.534			−1.08319
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−283.931			−0.818244			−0.409122	−	0.912480i	$-0.634165\pi$
$347$	−283.931			−0.818244			−0.409122	+	0.912480i	$0.634165\pi$
$348$		0			0
$349$	−125.357			−0.359189			−0.179595	−	0.983741i	$-0.557479\pi$
$349$	−125.357			−0.359189			−0.179595	+	0.983741i	$0.557479\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−134.945			−0.382281			−0.191140	−	0.981563i	$-0.561219\pi$
$353$	−134.945			−0.382281			−0.191140	+	0.981563i	$0.561219\pi$
$354$		0			0
$355$		−	0.221488i		−	0.000623911i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	434.019			1.20897			0.604484	−	0.796617i	$-0.293379\pi$
$359$	434.019			1.20897			0.604484	+	0.796617i	$0.293379\pi$
$360$		0			0
$361$	37.8139	−	359.014i	0.104748	−	0.994499i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.39115			0.00381137
$366$		0			0
$367$	−190.842			−0.520005			−0.260002	−	0.965608i	$-0.583723\pi$
$367$	−190.842			−0.520005			−0.260002	+	0.965608i	$0.583723\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			174.973i			0.471624i
$372$		0			0
$373$			628.473i			1.68491i	0.538763	+	0.842457i	$0.318892\pi$
$373$			628.473i			1.68491i	−0.538763	+	0.842457i	$0.681108\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	76.3385			0.202489
$378$		0			0
$379$		−	296.275i		−	0.781727i	−0.920449	−	0.390864i	$-0.872176\pi$
$379$		−	296.275i		−	0.781727i	0.920449	−	0.390864i	$-0.127824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			442.460i			1.15525i	0.816303	+	0.577623i	$0.196020\pi$
$383$			442.460i			1.15525i	−0.816303	+	0.577623i	$0.803980\pi$
$384$		0			0
$385$	1.28509			0.00333789
$386$		0			0
$387$		0			0
$388$		0			0
$389$	266.272			0.684505			0.342253	−	0.939608i	$-0.388810\pi$
$389$	266.272			0.684505			0.342253	+	0.939608i	$0.388810\pi$
$390$		0			0
$391$	−32.2046			−0.0823648
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.948371i			0.00240094i
$396$		0			0
$397$	−198.584			−0.500211			−0.250106	−	0.968219i	$-0.580465\pi$
$397$	−198.584			−0.500211			−0.250106	+	0.968219i	$0.580465\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			219.350i			0.547007i	0.961871	+	0.273503i	$0.0881825\pi$
$401$			219.350i			0.547007i	−0.961871	+	0.273503i	$0.911818\pi$
$402$		0			0
$403$	280.434			0.695867
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1265.77i		−	3.11000i
$408$		0			0
$409$			73.0707i			0.178657i	0.996002	+	0.0893285i	$0.0284721\pi$
$409$			73.0707i			0.178657i	−0.996002	+	0.0893285i	$0.971528\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	263.667i		−	0.638418i
$414$		0			0
$415$	−0.722617			−0.00174125
$416$		0			0
$417$		0			0
$418$		0			0
$419$	130.721			0.311983			0.155991	−	0.987758i	$-0.450143\pi$
$419$	130.721			0.311983			0.155991	+	0.987758i	$0.450143\pi$
$420$		0			0
$421$			205.731i			0.488673i	0.969690	+	0.244337i	$0.0785702\pi$
$421$			205.731i			0.488673i	−0.969690	+	0.244337i	$0.921430\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−65.3307			−0.153719
$426$		0			0
$427$	439.623			1.02956
$428$		0			0
$429$		0			0
$430$		0			0
$431$			346.636i			0.804259i	0.915583	+	0.402129i	$0.131730\pi$
$431$			346.636i			0.804259i	−0.915583	+	0.402129i	$0.868270\pi$
$432$		0			0
$433$			44.3462i			0.102416i	0.998688	+	0.0512081i	$0.0163072\pi$
$433$			44.3462i			0.102416i	−0.998688	+	0.0512081i	$0.983693\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−174.024	+	156.657i	−0.398224	+	0.358483i
$438$		0			0
$439$			227.421i			0.518043i	0.965872	+	0.259021i	$0.0834000\pi$
$439$			227.421i			0.518043i	−0.965872	+	0.259021i	$0.916600\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	390.532			0.881563			0.440782	−	0.897614i	$-0.354701\pi$
$443$	390.532			0.881563			0.440782	+	0.897614i	$0.354701\pi$
$444$		0			0
$445$		−	1.57032i		−	0.00352880i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	366.375i		−	0.815980i	−0.912986	−	0.407990i	$-0.866230\pi$
$449$		−	366.375i		−	0.815980i	0.912986	−	0.407990i	$-0.133770\pi$
$450$		0			0
$451$			945.077i			2.09551i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.19415i		−	0.00262451i
$456$		0			0
$457$	−152.655			−0.334038			−0.167019	−	0.985954i	$-0.553414\pi$
$457$	−152.655			−0.334038			−0.167019	+	0.985954i	$0.553414\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	295.846			0.641749			0.320874	−	0.947122i	$-0.396023\pi$
$461$	295.846			0.641749			0.320874	+	0.947122i	$0.396023\pi$
$462$		0			0
$463$	−596.448			−1.28822			−0.644112	−	0.764931i	$-0.722773\pi$
$463$	−596.448			−1.28822			−0.644112	+	0.764931i	$0.722773\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−795.094			−1.70256			−0.851278	−	0.524715i	$-0.824172\pi$
$467$	−795.094			−1.70256			−0.851278	+	0.524715i	$0.824172\pi$
$468$		0			0
$469$		−	348.889i		−	0.743899i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1465.52			−3.09836
$474$		0			0
$475$	−353.027	+	317.796i	−0.743214	+	0.669045i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−258.081			−0.538791			−0.269396	−	0.963030i	$-0.586824\pi$
$479$	−258.081			−0.538791			−0.269396	+	0.963030i	$0.586824\pi$
$480$		0			0
$481$	−1176.20			−2.44531
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.08717i		−	0.00224159i
$486$		0			0
$487$			608.037i			1.24854i	0.781210	+	0.624268i	$0.214603\pi$
$487$			608.037i			1.24854i	−0.781210	+	0.624268i	$0.785397\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−503.753			−1.02597			−0.512987	−	0.858396i	$-0.671461\pi$
$491$	−503.753			−1.02597			−0.512987	+	0.858396i	$0.671461\pi$
$492$		0			0
$493$			10.6432i			0.0215886i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	100.676i		−	0.202568i
$498$		0			0
$499$	−485.555			−0.973056			−0.486528	−	0.873665i	$-0.661737\pi$
$499$	−485.555			−0.973056			−0.486528	+	0.873665i	$0.661737\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−389.683			−0.774718			−0.387359	−	0.921929i	$-0.626613\pi$
$503$	−389.683			−0.774718			−0.387359	+	0.921929i	$0.626613\pi$
$504$		0			0
$505$	0.0494945			9.80089e−5
$506$		0			0
$507$		0			0
$508$		0			0
$509$			361.385i			0.709990i	0.934868	+	0.354995i	$0.115517\pi$
$509$			361.385i			0.709990i	−0.934868	+	0.354995i	$0.884483\pi$
$510$		0			0
$511$	632.339			1.23745
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	0.580615i		−	0.00112741i
$516$		0			0
$517$	1640.38			3.17288
$518$		0			0
$519$		0			0
$520$		0			0
$521$			798.564i			1.53275i	0.642392	+	0.766377i	$0.277942\pi$
$521$			798.564i			1.53275i	−0.642392	+	0.766377i	$0.722058\pi$
$522$		0			0
$523$		−	346.310i		−	0.662160i	−0.943603	−	0.331080i	$-0.892587\pi$
$523$		−	346.310i		−	0.662160i	0.943603	−	0.331080i	$-0.107413\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			39.0985i			0.0741907i
$528$		0			0
$529$	−377.128			−0.712907
$530$		0			0
$531$		0			0
$532$		0			0
$533$	878.199			1.64765
$534$		0			0
$535$		−	1.45717i		−	0.00272368i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−404.245			−0.749990
$540$		0			0
$541$	−8.91567			−0.0164800			−0.00823999	−	0.999966i	$-0.502623\pi$
$541$	−8.91567			−0.0164800			−0.00823999	+	0.999966i	$0.502623\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			0.482589i			0.000885484i
$546$		0			0
$547$		−	412.090i		−	0.753363i	−0.926343	−	0.376681i	$-0.877065\pi$
$547$		−	412.090i		−	0.753363i	0.926343	−	0.376681i	$-0.122935\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	51.7731	+	57.5126i	0.0939620	+	0.104379i
$552$		0			0
$553$			431.076i			0.779522i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−164.634			−0.295573			−0.147786	−	0.989019i	$-0.547215\pi$
$557$	−164.634			−0.295573			−0.147786	+	0.989019i	$0.547215\pi$
$558$		0			0
$559$			1361.81i			2.43616i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			863.263i			1.53333i	0.642050	+	0.766663i	$0.278084\pi$
$563$			863.263i			1.53333i	−0.642050	+	0.766663i	$0.721916\pi$
$564$		0			0
$565$			2.23296i			0.00395214i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	635.576i		−	1.11700i	−0.829503	−	0.558502i	$-0.811376\pi$
$569$		−	635.576i		−	1.11700i	0.829503	−	0.558502i	$-0.188624\pi$
$570$		0			0
$571$	−111.520			−0.195306			−0.0976530	−	0.995221i	$-0.531134\pi$
$571$	−111.520			−0.195306			−0.0976530	+	0.995221i	$0.531134\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	308.089			0.535807
$576$		0			0
$577$	658.831			1.14182			0.570911	−	0.821012i	$-0.306590\pi$
$577$	658.831			1.14182			0.570911	+	0.821012i	$0.306590\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−328.461			−0.565337
$582$		0			0
$583$			655.849i			1.12495i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	608.235			1.03618			0.518088	−	0.855327i	$-0.326644\pi$
$587$	608.235			1.03618			0.518088	+	0.855327i	$0.326644\pi$
$588$		0			0
$589$	190.192	+	211.276i	0.322906	+	0.358703i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1086.74			−1.83261			−0.916305	−	0.400481i	$-0.868843\pi$
$593$	−1086.74			−1.83261			−0.916305	+	0.400481i	$0.868843\pi$
$594$		0			0
$595$	0.166490			0.000279815
$596$		0			0
$597$		0			0
$598$		0			0
$599$			420.739i			0.702402i	0.936300	+	0.351201i	$0.114227\pi$
$599$			420.739i			0.702402i	−0.936300	+	0.351201i	$0.885773\pi$
$600$		0			0
$601$			577.036i			0.960126i	0.877234	+	0.480063i	$0.159386\pi$
$601$			577.036i			0.960126i	−0.877234	+	0.480063i	$0.840614\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.38437			0.00559400
$606$		0			0
$607$		−	150.995i		−	0.248756i	−0.992235	−	0.124378i	$-0.960306\pi$
$607$		−	150.995i		−	0.248756i	0.992235	−	0.124378i	$-0.0396935\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1524.30i		−	2.49476i
$612$		0			0
$613$	−451.848			−0.737109			−0.368554	−	0.929606i	$-0.620147\pi$
$613$	−451.848			−0.737109			−0.368554	+	0.929606i	$0.620147\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−528.451			−0.856485			−0.428242	−	0.903664i	$-0.640867\pi$
$617$	−528.451			−0.856485			−0.428242	+	0.903664i	$0.640867\pi$
$618$		0			0
$619$	−63.6940			−0.102898			−0.0514491	−	0.998676i	$-0.516384\pi$
$619$	−63.6940			−0.102898			−0.0514491	+	0.998676i	$0.516384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	713.777i		−	1.14571i
$624$		0			0
$625$	624.989			0.999983
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	163.987i		−	0.260710i
$630$		0			0
$631$	773.865			1.22641			0.613206	−	0.789923i	$-0.289880\pi$
$631$	773.865			1.22641			0.613206	+	0.789923i	$0.289880\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	2.23967i		−	0.00352703i
$636$		0			0
$637$			375.638i			0.589699i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			5.28345i			0.00824252i	0.999992	+	0.00412126i	$0.00131184\pi$
$641$			5.28345i			0.00824252i	−0.999992	+	0.00412126i	$0.998688\pi$
$642$		0			0
$643$	−694.339			−1.07984			−0.539922	−	0.841715i	$-0.681546\pi$
$643$	−694.339			−1.07984			−0.539922	+	0.841715i	$0.681546\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	279.022			0.431256			0.215628	−	0.976476i	$-0.430820\pi$
$647$	279.022			0.431256			0.215628	+	0.976476i	$0.430820\pi$
$648$		0			0
$649$		−	988.300i		−	1.52280i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−421.633			−0.645686			−0.322843	−	0.946453i	$-0.604639\pi$
$653$	−421.633			−0.645686			−0.322843	+	0.946453i	$0.604639\pi$
$654$		0			0
$655$	−0.683880			−0.00104409
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	376.000i		−	0.570561i	−0.958444	−	0.285280i	$-0.907913\pi$
$659$		−	376.000i		−	0.570561i	0.958444	−	0.285280i	$-0.0920867\pi$
$660$		0			0
$661$		−	720.200i		−	1.08956i	−0.838579	−	0.544781i	$-0.816613\pi$
$661$		−	720.200i		−	1.08956i	0.838579	−	0.544781i	$-0.183387\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.899660	−	0.809878i	0.00135287	−	0.00121786i
$666$		0			0
$667$		−	50.1917i		−	0.0752499i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1647.84			2.45579
$672$		0			0
$673$		−	868.848i		−	1.29101i	−0.763757	−	0.645504i	$-0.776647\pi$
$673$		−	868.848i		−	1.29101i	0.763757	−	0.645504i	$-0.223353\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			977.114i			1.44330i	0.692258	+	0.721650i	$0.256616\pi$
$677$			977.114i			1.44330i	−0.692258	+	0.721650i	$0.743384\pi$
$678$		0			0
$679$		−	494.167i		−	0.727786i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	228.420i		−	0.334437i	−0.985920	−	0.167218i	$-0.946521\pi$
$683$		−	228.420i		−	0.334437i	0.985920	−	0.167218i	$-0.0534785\pi$
$684$		0			0
$685$	−2.25258			−0.00328844
$686$		0			0
$687$		0			0
$688$		0			0
$689$	609.437			0.884525
$690$		0			0
$691$	94.8120			0.137210			0.0686049	−	0.997644i	$-0.478145\pi$
$691$	94.8120			0.137210			0.0686049	+	0.997644i	$0.478145\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.709273			0.00102054
$696$		0			0
$697$			122.439i			0.175666i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	828.374			1.18170			0.590852	−	0.806780i	$-0.298792\pi$
$701$	828.374			1.18170			0.590852	+	0.806780i	$0.298792\pi$
$702$		0			0
$703$	−797.701	−	886.134i	−1.13471	−	1.26050i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	22.4974			0.0318209
$708$		0			0
$709$	−475.081			−0.670073			−0.335036	−	0.942205i	$-0.608749\pi$
$709$	−475.081			−0.670073			−0.335036	+	0.942205i	$0.608749\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	184.382i		−	0.258601i
$714$		0			0
$715$		−	4.47602i		−	0.00626017i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1015.22			1.41199			0.705997	−	0.708215i	$-0.250499\pi$
$719$	1015.22			1.41199			0.705997	+	0.708215i	$0.250499\pi$
$720$		0			0
$721$		−	263.915i		−	0.366040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	101.819i		−	0.140440i
$726$		0			0
$727$	−7.82363			−0.0107615			−0.00538076	−	0.999986i	$-0.501713\pi$
$727$	−7.82363			−0.0107615			−0.00538076	+	0.999986i	$0.501713\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−189.866			−0.259734
$732$		0			0
$733$	168.524			0.229910			0.114955	−	0.993371i	$-0.463328\pi$
$733$	168.524			0.229910			0.114955	+	0.993371i	$0.463328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1307.74i		−	1.77441i
$738$		0			0
$739$	256.826			0.347531			0.173766	−	0.984787i	$-0.444406\pi$
$739$	256.826			0.347531			0.173766	+	0.984787i	$0.444406\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			848.220i			1.14161i	0.821084	+	0.570807i	$0.193370\pi$
$743$			848.220i			1.14161i	−0.821084	+	0.570807i	$0.806630\pi$
$744$		0			0
$745$	1.49247			0.00200331
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	662.348i		−	0.884310i
$750$		0			0
$751$			61.3237i			0.0816560i	0.999166	+	0.0408280i	$0.0129996\pi$
$751$			61.3237i			0.0816560i	−0.999166	+	0.0408280i	$0.987000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			3.02424i			0.00400561i
$756$		0			0
$757$	−275.977			−0.364566			−0.182283	−	0.983246i	$-0.558349\pi$
$757$	−275.977			−0.364566			−0.182283	+	0.983246i	$0.558349\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1142.17			−1.50088			−0.750441	−	0.660938i	$-0.770159\pi$
$761$	−1142.17			−1.50088			−0.750441	+	0.660938i	$0.770159\pi$
$762$		0			0
$763$			219.358i			0.287494i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−918.362			−1.19734
$768$		0			0
$769$	−71.7054			−0.0932450			−0.0466225	−	0.998913i	$-0.514846\pi$
$769$	−71.7054			−0.0932450			−0.0466225	+	0.998913i	$0.514846\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			744.760i			0.963467i	0.876318	+	0.481734i	$0.159993\pi$
$773$			744.760i			0.963467i	−0.876318	+	0.481734i	$0.840007\pi$
$774$		0			0
$775$		−	374.040i		−	0.482632i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	595.598	+	661.625i	0.764567	+	0.849326i
$780$		0			0
$781$		−	377.364i		−	0.483180i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.627173			0.000798947
$786$		0			0
$787$			426.395i			0.541797i	0.962608	+	0.270899i	$0.0873209\pi$
$787$			426.395i			0.541797i	−0.962608	+	0.270899i	$0.912679\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1014.98i			1.28316i
$792$		0			0
$793$		−	1531.23i		−	1.93093i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	490.997i		−	0.616056i	−0.951377	−	0.308028i	$-0.900331\pi$
$797$		−	490.997i		−	0.616056i	0.951377	−	0.308028i	$-0.0996691\pi$
$798$		0			0
$799$	212.520			0.265982
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2370.19			2.95167
$804$		0			0
$805$	−0.785140			−0.000975329
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1223.89			−1.51284			−0.756420	−	0.654087i	$-0.773053\pi$
$809$	−1223.89			−1.51284			−0.756420	+	0.654087i	$0.773053\pi$
$810$		0			0
$811$		−	590.727i		−	0.728393i	−0.931322	−	0.364197i	$-0.881344\pi$
$811$		−	590.727i		−	0.728393i	0.931322	−	0.364197i	$-0.118656\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.08308			0.00378292
$816$		0			0
$817$	−1025.98	+	923.588i	−1.25578	+	1.13046i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−188.273			−0.229321			−0.114661	−	0.993405i	$-0.536578\pi$
$821$	−188.273			−0.229321			−0.114661	+	0.993405i	$0.536578\pi$
$822$		0			0
$823$	1419.13			1.72434			0.862168	−	0.506623i	$-0.169106\pi$
$823$	1419.13			1.72434			0.862168	+	0.506623i	$0.169106\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1005.56i			1.21591i	0.793972	+	0.607954i	$0.208009\pi$
$827$			1005.56i			1.21591i	−0.793972	+	0.607954i	$0.791991\pi$
$828$		0			0
$829$			280.910i			0.338854i	0.985543	+	0.169427i	$0.0541917\pi$
$829$			280.910i			0.338854i	−0.985543	+	0.169427i	$0.945808\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−52.3719			−0.0628715
$834$		0			0
$835$		−	3.35230i		−	0.00401473i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			351.194i			0.418586i	0.977853	+	0.209293i	$0.0671163\pi$
$839$			351.194i			0.418586i	−0.977853	+	0.209293i	$0.932884\pi$
$840$		0			0
$841$	824.412			0.980276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.15848			−0.00255441
$846$		0			0
$847$	1538.34			1.81623
$848$		0			0
$849$		0			0
$850$		0			0
$851$			773.336i			0.908737i
$852$		0			0
$853$	−341.736			−0.400628			−0.200314	−	0.979732i	$-0.564196\pi$
$853$	−341.736			−0.400628			−0.200314	+	0.979732i	$0.564196\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	738.972i		−	0.862278i	−0.902286	−	0.431139i	$-0.858112\pi$
$857$		−	738.972i		−	0.862278i	0.902286	−	0.431139i	$-0.141888\pi$
$858$		0			0
$859$	−375.173			−0.436756			−0.218378	−	0.975864i	$-0.570077\pi$
$859$	−375.173			−0.436756			−0.218378	+	0.975864i	$0.570077\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	195.029i		−	0.225989i	−0.993596	−	0.112995i	$-0.963956\pi$
$863$		−	195.029i		−	0.225989i	0.993596	−	0.112995i	$-0.0360443\pi$
$864$		0			0
$865$			1.15039i			0.00132994i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1615.80i			1.85938i
$870$		0			0
$871$	−1215.19			−1.39517
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.18549			−0.00364057
$876$		0			0
$877$		−	1744.13i		−	1.98874i	−0.105952	−	0.994371i	$-0.533789\pi$
$877$		−	1744.13i		−	1.98874i	0.105952	−	0.994371i	$-0.466211\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	387.876			0.440267			0.220134	−	0.975470i	$-0.429351\pi$
$881$	387.876			0.440267			0.220134	+	0.975470i	$0.429351\pi$
$882$		0			0
$883$	348.266			0.394413			0.197206	−	0.980362i	$-0.436813\pi$
$883$	348.266			0.394413			0.197206	+	0.980362i	$0.436813\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1499.32i			1.69032i	0.534510	+	0.845162i	$0.320496\pi$
$887$			1499.32i			1.69032i	−0.534510	+	0.845162i	$0.679504\pi$
$888$		0			0
$889$		−	1018.03i		−	1.14514i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1148.39	−	1033.79i	1.28599	−	1.15766i
$894$		0			0
$895$			3.66009i			0.00408949i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−60.9359			−0.0677818
$900$		0			0
$901$			84.9685i			0.0943046i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.742650i			0.000820608i
$906$		0			0
$907$			1270.98i			1.40130i	0.713505	+	0.700650i	$0.247107\pi$
$907$			1270.98i			1.40130i	−0.713505	+	0.700650i	$0.752893\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1691.31i		−	1.85654i	−0.371910	−	0.928269i	$-0.621297\pi$
$911$		−	1691.31i		−	1.85654i	0.371910	−	0.928269i	$-0.378703\pi$
$912$		0			0
$913$	−1231.17			−1.34849
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−310.854			−0.338990
$918$		0			0
$919$	803.862			0.874713			0.437357	−	0.899288i	$-0.355915\pi$
$919$	803.862			0.874713			0.437357	+	0.899288i	$0.355915\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−350.660			−0.379913
$924$		0			0
$925$			1568.80i			1.69600i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	818.827			0.881407			0.440703	−	0.897653i	$-0.354729\pi$
$929$	818.827			0.881407			0.440703	+	0.897653i	$0.354729\pi$
$930$		0			0
$931$	−283.002	+	254.760i	−0.303976	+	0.273641i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.624052			0.000667436
$936$		0			0
$937$	−298.683			−0.318765			−0.159383	−	0.987217i	$-0.550950\pi$
$937$	−298.683			−0.318765			−0.159383	+	0.987217i	$0.550950\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			416.099i			0.442188i	0.975253	+	0.221094i	$0.0709627\pi$
$941$			416.099i			0.442188i	−0.975253	+	0.221094i	$0.929037\pi$
$942$		0			0
$943$		−	577.405i		−	0.612307i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	700.253			0.739444			0.369722	−	0.929142i	$-0.379453\pi$
$947$	700.253			0.739444			0.369722	+	0.929142i	$0.379453\pi$
$948$		0			0
$949$		−	2202.47i		−	2.32083i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	264.222i		−	0.277253i	−0.990345	−	0.138627i	$-0.955731\pi$
$953$		−	264.222i		−	0.277253i	0.990345	−	0.138627i	$-0.0442688\pi$
$954$		0			0
$955$	−2.82785			−0.00296110
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1023.90			−1.06767
$960$		0			0
$961$	737.148			0.767064
$962$		0			0
$963$		0			0
$964$		0			0
$965$			0.383588i			0.000397500i
$966$		0			0
$967$	−275.678			−0.285086			−0.142543	−	0.989789i	$-0.545528\pi$
$967$	−275.678			−0.285086			−0.142543	+	0.989789i	$0.545528\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1535.48i		−	1.58133i	−0.612246	−	0.790667i	$-0.709734\pi$
$971$		−	1535.48i		−	1.58133i	0.612246	−	0.790667i	$-0.290266\pi$
$972$		0			0
$973$	322.395			0.331342
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1030.08i		−	1.05432i	−0.849765	−	0.527162i	$-0.823256\pi$
$977$		−	1030.08i		−	1.05432i	0.849765	−	0.527162i	$-0.176744\pi$
$978$		0			0
$979$		−	2675.44i		−	2.73283i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	718.441i		−	0.730866i	−0.930838	−	0.365433i	$-0.880921\pi$
$983$		−	718.441i		−	0.730866i	0.930838	−	0.365433i	$-0.119079\pi$
$984$		0			0
$985$	2.20983			0.00224349
$986$		0			0
$987$		0			0
$988$		0			0
$989$	895.377			0.905336
$990$		0			0
$991$		−	183.179i		−	0.184842i	−0.995720	−	0.0924211i	$-0.970539\pi$
$991$		−	183.179i		−	0.184842i	0.995720	−	0.0924211i	$-0.0294606\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.58166			0.00259464
$996$		0			0
$997$	1980.18			1.98613			0.993067	−	0.117549i	$-0.0375037\pi$
$997$	1980.18			1.98613			0.993067	+	0.117549i	$0.0375037\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.3.o.c.721.9		20
3.2	odd	2		456.3.o.a.265.6	✓	20
4.3	odd	2		2736.3.o.r.721.9		20
12.11	even	2		912.3.o.e.721.16		20
19.18	odd	2	inner	1368.3.o.c.721.10		20
57.56	even	2		456.3.o.a.265.16	yes	20
76.75	even	2		2736.3.o.r.721.10		20
228.227	odd	2		912.3.o.e.721.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.6	✓	20	3.2	odd	2
456.3.o.a.265.16	yes	20	57.56	even	2
912.3.o.e.721.6		20	228.227	odd	2
912.3.o.e.721.16		20	12.11	even	2
1368.3.o.c.721.9		20	1.1	even	1	trivial
1368.3.o.c.721.10		20	19.18	odd	2	inner
2736.3.o.r.721.9		20	4.3	odd	2
2736.3.o.r.721.10		20	76.75	even	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1368.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.0118390 q^{5} +5.38136 q^{7} +O(q^{10})\)	\(q+0.0118390 q^{5} +5.38136 q^{7} +20.1709 q^{11} -18.7435i q^{13} +2.61324 q^{17} +(14.1212 - 12.7119i) q^{19} -12.3236 q^{23} -24.9999 q^{25} +4.07280i q^{29} +14.9617i q^{31} +0.0637101 q^{35} -62.7522i q^{37} +46.8535i q^{41} -72.6553 q^{43} +81.3241 q^{47} -20.0410 q^{49} +32.5146i q^{53} +0.238804 q^{55} -48.9963i q^{59} +81.6937 q^{61} -0.221905i q^{65} -64.8329i q^{67} -18.7083i q^{71} +117.506 q^{73} +108.547 q^{77} +80.1054i q^{79} -61.0368 q^{83} +0.0309383 q^{85} -132.639i q^{89} -100.865i q^{91} +(0.167181 - 0.150497i) q^{95} -91.8294i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 16 q^{7}+O(q^{10}) \) 20 * q - 16 * q^7	\( 20 q - 16 q^{7} - 16 q^{11} - 32 q^{17} + 40 q^{19} - 64 q^{23} + 68 q^{25} + 208 q^{35} + 64 q^{43} - 48 q^{47} + 20 q^{49} - 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} - 224 q^{83} - 136 q^{85} + 320 q^{95}+O(q^{100}) \) 20 * q - 16 * q^7 - 16 * q^11 - 32 * q^17 + 40 * q^19 - 64 * q^23 + 68 * q^25 + 208 * q^35 + 64 * q^43 - 48 * q^47 + 20 * q^49 - 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 - 224 * q^83 - 136 * q^85 + 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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