LMFDB - Embedded newform 1764.4.k.o.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(361,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,18,0,0,0,0,0,36,0,20,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.o.1549.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.00000 + 15.5885i) q^{5} +(18.0000 - 31.1769i) q^{11} +10.0000 q^{13} +(-9.00000 + 15.5885i) q^{17} +(-50.0000 - 86.6025i) q^{19} +(36.0000 + 62.3538i) q^{23} +(-99.5000 + 172.339i) q^{25} +234.000 q^{29} +(-8.00000 + 13.8564i) q^{31} +(113.000 + 195.722i) q^{37} +90.0000 q^{41} +452.000 q^{43} +(-216.000 - 374.123i) q^{47} +(207.000 - 358.535i) q^{53} +648.000 q^{55} +(342.000 - 592.361i) q^{59} +(211.000 + 365.463i) q^{61} +(90.0000 + 155.885i) q^{65} +(-166.000 + 287.520i) q^{67} +360.000 q^{71} +(13.0000 - 22.5167i) q^{73} +(-256.000 - 443.405i) q^{79} -1188.00 q^{83} -324.000 q^{85} +(315.000 + 545.596i) q^{89} +(900.000 - 1558.85i) q^{95} +1054.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{5} + 36 q^{11} + 20 q^{13} - 18 q^{17} - 100 q^{19} + 72 q^{23} - 199 q^{25} + 468 q^{29} - 16 q^{31} + 226 q^{37} + 180 q^{41} + 904 q^{43} - 432 q^{47} + 414 q^{53} + 1296 q^{55} + 684 q^{59}+ \cdots + 2108 q^{97}+O(q^{100}) $ 2 * q + 18 * q^5 + 36 * q^11 + 20 * q^13 - 18 * q^17 - 100 * q^19 + 72 * q^23 - 199 * q^25 + 468 * q^29 - 16 * q^31 + 226 * q^37 + 180 * q^41 + 904 * q^43 - 432 * q^47 + 414 * q^53 + 1296 * q^55 + 684 * q^59 + 422 * q^61 + 180 * q^65 - 332 * q^67 + 720 * q^71 + 26 * q^73 - 512 * q^79 - 2376 * q^83 - 648 * q^85 + 630 * q^89 + 1800 * q^95 + 2108 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	9.00000	+	15.5885i	0.804984	+	1.39427i	0.916302	+	0.400489i	$0.131160\pi$
$5$	9.00000	+	15.5885i	0.804984	+	1.39427i	−0.111317	+	0.993785i	$0.535507\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	−0.506589	−	0.862188i	$-0.669094\pi$
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	0.999971	+	0.00762479i	$0.00242707\pi$
$12$		0			0
$13$	10.0000			0.213346			0.106673	−	0.994294i	$-0.465980\pi$
$13$	10.0000			0.213346			0.106673	+	0.994294i	$0.465980\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.00000	+	15.5885i	−0.128401	+	0.222397i	−0.923057	−	0.384662i	$-0.874318\pi$
$17$	−9.00000	+	15.5885i	−0.128401	+	0.222397i	0.794656	+	0.607060i	$0.207651\pi$
$18$		0			0
$19$	−50.0000	−	86.6025i	−0.603726	−	1.04568i	−0.992251	−	0.124246i	$-0.960349\pi$
$19$	−50.0000	−	86.6025i	−0.603726	−	1.04568i	0.388526	−	0.921438i	$-0.372984\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	36.0000	+	62.3538i	0.326370	+	0.565290i	0.981789	−	0.189976i	$-0.0608410\pi$
$23$	36.0000	+	62.3538i	0.326370	+	0.565290i	−0.655418	+	0.755266i	$0.727508\pi$
$24$		0			0
$25$	−99.5000	+	172.339i	−0.796000	+	1.37871i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	234.000			1.49837			0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000			1.49837			0.749185	+	0.662361i	$0.230446\pi$
$30$		0			0
$31$	−8.00000	+	13.8564i	−0.0463498	+	0.0802801i	−0.888270	−	0.459323i	$-0.848092\pi$
$31$	−8.00000	+	13.8564i	−0.0463498	+	0.0802801i	0.841920	+	0.539603i	$0.181426\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	113.000	+	195.722i	0.502083	+	0.869634i	0.999997	+	0.00240737i	$0.000766290\pi$
$37$	113.000	+	195.722i	0.502083	+	0.869634i	−0.497914	+	0.867227i	$0.665900\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	90.0000			0.342820			0.171410	−	0.985200i	$-0.445168\pi$
$41$	90.0000			0.342820			0.171410	+	0.985200i	$0.445168\pi$
$42$		0			0
$43$	452.000			1.60301			0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000			1.60301			0.801504	+	0.597989i	$0.204033\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−216.000	−	374.123i	−0.670358	−	1.16109i	−0.977803	−	0.209528i	$-0.932807\pi$
$47$	−216.000	−	374.123i	−0.670358	−	1.16109i	0.307444	−	0.951566i	$-0.400526\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	207.000	−	358.535i	0.536484	−	0.929217i	−0.462606	−	0.886564i	$-0.653086\pi$
$53$	207.000	−	358.535i	0.536484	−	0.929217i	0.999090	−	0.0426532i	$-0.0135811\pi$
$54$		0			0
$55$	648.000			1.58866
$56$		0			0
$57$		0			0
$58$		0			0
$59$	342.000	−	592.361i	0.754654	−	1.30710i	−0.190892	−	0.981611i	$-0.561138\pi$
$59$	342.000	−	592.361i	0.754654	−	1.30710i	0.945546	−	0.325489i	$-0.105529\pi$
$60$		0			0
$61$	211.000	+	365.463i	0.442882	+	0.767094i	0.997902	−	0.0647433i	$-0.0206228\pi$
$61$	211.000	+	365.463i	0.442882	+	0.767094i	−0.555020	+	0.831837i	$0.687290\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	90.0000	+	155.885i	0.171740	+	0.297463i
$66$		0			0
$67$	−166.000	+	287.520i	−0.302688	+	0.524272i	−0.976744	−	0.214409i	$-0.931218\pi$
$67$	−166.000	+	287.520i	−0.302688	+	0.524272i	0.674056	+	0.738681i	$0.264551\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	360.000			0.601748			0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000			0.601748			0.300874	+	0.953664i	$0.402722\pi$
$72$		0			0
$73$	13.0000	−	22.5167i	0.0208429	−	0.0361010i	−0.855416	−	0.517942i	$-0.826698\pi$
$73$	13.0000	−	22.5167i	0.0208429	−	0.0361010i	0.876259	+	0.481841i	$0.160032\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−256.000	−	443.405i	−0.364585	−	0.631481i	0.624124	−	0.781325i	$-0.285456\pi$
$79$	−256.000	−	443.405i	−0.364585	−	0.631481i	−0.988710	+	0.149845i	$0.952123\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1188.00			−1.57108			−0.785542	−	0.618809i	$-0.787616\pi$
$83$	−1188.00			−1.57108			−0.785542	+	0.618809i	$0.787616\pi$
$84$		0			0
$85$	−324.000			−0.413444
$86$		0			0
$87$		0			0
$88$		0			0
$89$	315.000	+	545.596i	0.375168	+	0.649810i	0.990352	−	0.138574i	$-0.0442518\pi$
$89$	315.000	+	545.596i	0.375168	+	0.649810i	−0.615184	+	0.788383i	$0.710918\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	900.000	−	1558.85i	0.971979	−	1.68352i
$96$		0			0
$97$	1054.00			1.10327			0.551637	−	0.834085i	$-0.314004\pi$
$97$	1054.00			1.10327			0.551637	+	0.834085i	$0.314004\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−279.000	+	483.242i	−0.274867	+	0.476083i	−0.970101	−	0.242700i	$-0.921967\pi$
$101$	−279.000	+	483.242i	−0.274867	+	0.476083i	0.695235	+	0.718783i	$0.255300\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.00382652	+	0.00662773i	0.867932	−	0.496682i	$-0.165449\pi$
$103$	4.00000	+	6.92820i	0.00382652	+	0.00662773i	−0.864106	+	0.503310i	$0.832115\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	882.000	+	1527.67i	0.796880	+	1.38024i	0.921639	+	0.388049i	$0.126851\pi$
$107$	882.000	+	1527.67i	0.796880	+	1.38024i	−0.124759	+	0.992187i	$0.539816\pi$
$108$		0			0
$109$	−811.000	+	1404.69i	−0.712658	+	1.23436i	0.251198	+	0.967936i	$0.419175\pi$
$109$	−811.000	+	1404.69i	−0.712658	+	1.23436i	−0.963856	+	0.266424i	$0.914158\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1134.00			0.944051			0.472025	−	0.881585i	$-0.343523\pi$
$113$	1134.00			0.944051			0.472025	+	0.881585i	$0.343523\pi$
$114$		0			0
$115$	−648.000	+	1122.37i	−0.525446	+	0.910099i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	17.5000	+	30.3109i	0.0131480	+	0.0227730i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1332.00			−0.953102
$126$		0			0
$127$	−592.000			−0.413634			−0.206817	−	0.978380i	$-0.566310\pi$
$127$	−592.000			−0.413634			−0.206817	+	0.978380i	$0.566310\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	954.000	+	1652.38i	0.636270	+	1.10205i	0.986245	+	0.165293i	$0.0528569\pi$
$131$	954.000	+	1652.38i	0.636270	+	1.10205i	−0.349975	+	0.936759i	$0.613810\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	477.000	−	826.188i	0.297466	−	0.515226i	−0.678089	−	0.734979i	$-0.737192\pi$
$137$	477.000	−	826.188i	0.297466	−	0.515226i	0.975556	+	0.219753i	$0.0705251\pi$
$138$		0			0
$139$	−2564.00			−1.56457			−0.782286	−	0.622919i	$-0.785947\pi$
$139$	−2564.00			−1.56457			−0.782286	+	0.622919i	$0.785947\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	180.000	−	311.769i	0.105261	−	0.182318i
$144$		0			0
$145$	2106.00	+	3647.70i	1.20616	+	2.08914i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−369.000	−	639.127i	−0.202884	−	0.351405i	0.746573	−	0.665304i	$-0.231698\pi$
$149$	−369.000	−	639.127i	−0.202884	−	0.351405i	−0.949456	+	0.313899i	$0.898365\pi$
$150$		0			0
$151$	1220.00	−	2113.10i	0.657498	−	1.13882i	−0.323763	−	0.946138i	$-0.604948\pi$
$151$	1220.00	−	2113.10i	0.657498	−	1.13882i	0.981261	−	0.192682i	$-0.0617185\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−288.000			−0.149243
$156$		0			0
$157$	−1277.00	+	2211.83i	−0.649145	+	1.12435i	0.334183	+	0.942508i	$0.391540\pi$
$157$	−1277.00	+	2211.83i	−0.649145	+	1.12435i	−0.983328	+	0.181843i	$0.941794\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	410.000	+	710.141i	0.197016	+	0.341242i	0.947560	−	0.319579i	$-0.103541\pi$
$163$	410.000	+	710.141i	0.197016	+	0.341242i	−0.750543	+	0.660821i	$0.770208\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1944.00			0.900786			0.450393	−	0.892830i	$-0.351284\pi$
$167$	1944.00			0.900786			0.450393	+	0.892830i	$0.351284\pi$
$168$		0			0
$169$	−2097.00			−0.954483
$170$		0			0
$171$		0			0
$172$		0			0
$173$	621.000	+	1075.60i	0.272912	+	0.472697i	0.969606	−	0.244671i	$-0.0786799\pi$
$173$	621.000	+	1075.60i	0.272912	+	0.472697i	−0.696694	+	0.717368i	$0.745347\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	558.000	−	966.484i	0.232999	−	0.403567i	−0.725690	−	0.688022i	$-0.758479\pi$
$179$	558.000	−	966.484i	0.232999	−	0.403567i	0.958689	+	0.284455i	$0.0918127\pi$
$180$		0			0
$181$	−1070.00			−0.439406			−0.219703	−	0.975567i	$-0.570509\pi$
$181$	−1070.00			−0.439406			−0.219703	+	0.975567i	$0.570509\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2034.00	+	3522.99i	−0.808339	+	1.40008i
$186$		0			0
$187$	324.000	+	561.184i	0.126702	+	0.219454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−288.000	−	498.831i	−0.109104	−	0.188974i	0.806303	−	0.591502i	$-0.201465\pi$
$191$	−288.000	−	498.831i	−0.109104	−	0.188974i	−0.915408	+	0.402528i	$0.868132\pi$
$192$		0			0
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	−0.713339	−	0.700819i	$-0.752818\pi$
$193$	671.000	−	1162.21i	0.250257	−	0.433458i	0.963597	+	0.267361i	$0.0861515\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1422.00			−0.514281			−0.257140	−	0.966374i	$-0.582780\pi$
$197$	−1422.00			−0.514281			−0.257140	+	0.966374i	$0.582780\pi$
$198$		0			0
$199$	436.000	−	755.174i	0.155313	−	0.269009i	−0.777860	−	0.628437i	$-0.783695\pi$
$199$	436.000	−	755.174i	0.155313	−	0.269009i	0.933173	+	0.359428i	$0.117028\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	810.000	+	1402.96i	0.275965	+	0.477986i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3600.00			−1.19147
$210$		0			0
$211$	1340.00			0.437201			0.218600	−	0.975814i	$-0.429851\pi$
$211$	1340.00			0.437201			0.218600	+	0.975814i	$0.429851\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4068.00	+	7045.98i	1.29040	+	2.23503i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−90.0000	+	155.885i	−0.0273939	+	0.0474477i
$222$		0			0
$223$	−4880.00			−1.46542			−0.732711	−	0.680540i	$-0.761745\pi$
$223$	−4880.00			−1.46542			−0.732711	+	0.680540i	$0.761745\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1350.00	+	2338.27i	−0.394725	+	0.683684i	−0.993066	−	0.117557i	$-0.962494\pi$
$227$	−1350.00	+	2338.27i	−0.394725	+	0.683684i	0.598341	+	0.801242i	$0.295827\pi$
$228$		0			0
$229$	127.000	+	219.970i	0.0366480	+	0.0634762i	0.883768	−	0.467926i	$-0.154999\pi$
$229$	127.000	+	219.970i	0.0366480	+	0.0634762i	−0.847120	+	0.531402i	$0.821665\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2205.00	+	3819.17i	0.619976	+	1.07383i	0.989490	+	0.144604i	$0.0461909\pi$
$233$	2205.00	+	3819.17i	0.619976	+	1.07383i	−0.369514	+	0.929225i	$0.620476\pi$
$234$		0			0
$235$	3888.00	−	6734.21i	1.07926	−	1.86933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3888.00			1.05228			0.526138	−	0.850399i	$-0.323640\pi$
$239$	3888.00			1.05228			0.526138	+	0.850399i	$0.323640\pi$
$240$		0			0
$241$	2569.00	−	4449.64i	0.686655	−	1.18932i	−0.286259	−	0.958152i	$-0.592412\pi$
$241$	2569.00	−	4449.64i	0.686655	−	1.18932i	0.972914	−	0.231169i	$-0.0742550\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−500.000	−	866.025i	−0.128803	−	0.223093i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4788.00			1.20405			0.602024	−	0.798478i	$-0.294361\pi$
$251$	4788.00			1.20405			0.602024	+	0.798478i	$0.294361\pi$
$252$		0			0
$253$	2592.00			0.644101
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2943.00	+	5097.43i	0.714316	+	1.23723i	0.963223	+	0.268705i	$0.0865955\pi$
$257$	2943.00	+	5097.43i	0.714316	+	1.23723i	−0.248906	+	0.968528i	$0.580071\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1116.00	−	1932.97i	0.261656	−	0.453201i	−0.705026	−	0.709181i	$-0.749065\pi$
$263$	1116.00	−	1932.97i	0.261656	−	0.453201i	0.966682	+	0.255980i	$0.0823981\pi$
$264$		0			0
$265$	7452.00			1.72744
$266$		0			0
$267$		0			0
$268$		0			0
$269$	333.000	−	576.773i	0.0754772	−	0.130730i	−0.825816	−	0.563939i	$-0.809285\pi$
$269$	333.000	−	576.773i	0.0754772	−	0.130730i	0.901294	+	0.433209i	$0.142619\pi$
$270$		0			0
$271$	−2768.00	−	4794.32i	−0.620458	−	1.07466i	−0.989401	−	0.145212i	$-0.953614\pi$
$271$	−2768.00	−	4794.32i	−0.620458	−	1.07466i	0.368943	−	0.929452i	$-0.379720\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3582.00	+	6204.21i	0.785464	+	1.36046i
$276$		0			0
$277$	−1063.00	+	1841.17i	−0.230576	+	0.399369i	−0.957978	−	0.286843i	$-0.907394\pi$
$277$	−1063.00	+	1841.17i	−0.230576	+	0.399369i	0.727402	+	0.686212i	$0.240728\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2934.00			0.622875			0.311437	−	0.950267i	$-0.399190\pi$
$281$	2934.00			0.622875			0.311437	+	0.950267i	$0.399190\pi$
$282$		0			0
$283$	1018.00	−	1763.23i	0.213830	−	0.370364i	−0.739080	−	0.673617i	$-0.764740\pi$
$283$	1018.00	−	1763.23i	0.213830	−	0.370364i	0.952910	+	0.303253i	$0.0980729\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2294.50	+	3974.19i	0.467026	+	0.808913i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2286.00			0.455800			0.227900	−	0.973684i	$-0.426814\pi$
$293$	2286.00			0.455800			0.227900	+	0.973684i	$0.426814\pi$
$294$		0			0
$295$	12312.0			2.42994
$296$		0			0
$297$		0			0
$298$		0			0
$299$	360.000	+	623.538i	0.0696299	+	0.120603i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3798.00	+	6578.33i	−0.713026	+	1.23500i
$306$		0			0
$307$	−1244.00			−0.231267			−0.115633	−	0.993292i	$-0.536890\pi$
$307$	−1244.00			−0.231267			−0.115633	+	0.993292i	$0.536890\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−612.000	+	1060.02i	−0.111586	+	0.193273i	−0.916410	−	0.400241i	$-0.868926\pi$
$311$	−612.000	+	1060.02i	−0.111586	+	0.193273i	0.804824	+	0.593514i	$0.202260\pi$
$312$		0			0
$313$	949.000	+	1643.72i	0.171376	+	0.296832i	0.938901	−	0.344187i	$-0.111845\pi$
$313$	949.000	+	1643.72i	0.171376	+	0.296832i	−0.767525	+	0.641019i	$0.778512\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4581.00	−	7934.52i	−0.811655	−	1.40583i	−0.911705	−	0.410845i	$-0.865234\pi$
$317$	−4581.00	−	7934.52i	−0.811655	−	1.40583i	0.100050	−	0.994982i	$-0.468100\pi$
$318$		0			0
$319$	4212.00	−	7295.40i	0.739269	−	1.28045i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1800.00			0.310076
$324$		0			0
$325$	−995.000	+	1723.39i	−0.169824	+	0.294143i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2174.00	+	3765.48i	0.361009	+	0.625285i	0.988127	−	0.153639i	$-0.0490992\pi$
$331$	2174.00	+	3765.48i	0.361009	+	0.625285i	−0.627119	+	0.778924i	$0.715766\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5976.00			−0.974638
$336$		0			0
$337$	7154.00			1.15639			0.578195	−	0.815899i	$-0.303757\pi$
$337$	7154.00			1.15639			0.578195	+	0.815899i	$0.303757\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	288.000	+	498.831i	0.0457363	+	0.0792176i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−918.000	+	1590.02i	−0.142020	+	0.245985i	−0.928257	−	0.371939i	$-0.878693\pi$
$347$	−918.000	+	1590.02i	−0.142020	+	0.245985i	0.786237	+	0.617925i	$0.212026\pi$
$348$		0			0
$349$	−5894.00			−0.904007			−0.452004	−	0.892016i	$-0.649291\pi$
$349$	−5894.00			−0.904007			−0.452004	+	0.892016i	$0.649291\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5553.00	+	9618.08i	−0.837270	+	1.45019i	0.0548984	+	0.998492i	$0.482516\pi$
$353$	−5553.00	+	9618.08i	−0.837270	+	1.45019i	−0.892169	+	0.451703i	$0.850817\pi$
$354$		0			0
$355$	3240.00	+	5611.84i	0.484398	+	0.839002i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6588.00	+	11410.8i	0.968527	+	1.67754i	0.699823	+	0.714316i	$0.253262\pi$
$359$	6588.00	+	11410.8i	0.968527	+	1.67754i	0.268704	+	0.963223i	$0.413405\pi$
$360$		0			0
$361$	−1570.50	+	2720.19i	−0.228969	+	0.396586i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	468.000			0.0671130
$366$		0			0
$367$	−3056.00	+	5293.15i	−0.434665	+	0.752861i	−0.997268	−	0.0738656i	$-0.976466\pi$
$367$	−3056.00	+	5293.15i	−0.434665	+	0.752861i	0.562604	+	0.826727i	$0.309800\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6809.00	+	11793.5i	0.945192	+	1.63712i	0.755366	+	0.655303i	$0.227459\pi$
$373$	6809.00	+	11793.5i	0.945192	+	1.63712i	0.189826	+	0.981818i	$0.439208\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2340.00			0.319671
$378$		0			0
$379$	692.000			0.0937880			0.0468940	−	0.998900i	$-0.485068\pi$
$379$	692.000			0.0937880			0.0468940	+	0.998900i	$0.485068\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4032.00	+	6983.63i	0.537926	+	0.931715i	0.999016	+	0.0443613i	$0.0141253\pi$
$383$	4032.00	+	6983.63i	0.537926	+	0.931715i	−0.461090	+	0.887353i	$0.652541\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6327.00	−	10958.7i	0.824657	−	1.42835i	−0.0775239	−	0.996990i	$-0.524701\pi$
$389$	6327.00	−	10958.7i	0.824657	−	1.42835i	0.902181	−	0.431358i	$-0.141965\pi$
$390$		0			0
$391$	−1296.00			−0.167625
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4608.00	−	7981.29i	0.586971	−	1.01666i
$396$		0			0
$397$	−53.0000	−	91.7987i	−0.00670024	−	0.0116051i	0.862656	−	0.505791i	$-0.168799\pi$
$397$	−53.0000	−	91.7987i	−0.00670024	−	0.0116051i	−0.869356	+	0.494186i	$0.835466\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2007.00	−	3476.23i	−0.249937	−	0.432904i	0.713571	−	0.700583i	$-0.247077\pi$
$401$	−2007.00	−	3476.23i	−0.249937	−	0.432904i	−0.963508	+	0.267679i	$0.913743\pi$
$402$		0			0
$403$	−80.0000	+	138.564i	−0.00988855	+	0.0171275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8136.00			0.990876
$408$		0			0
$409$	1957.00	−	3389.62i	0.236595	−	0.409795i	−0.723140	−	0.690702i	$-0.757302\pi$
$409$	1957.00	−	3389.62i	0.236595	−	0.409795i	0.959735	+	0.280907i	$0.0906352\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−10692.0	−	18519.1i	−1.26470	−	2.19052i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4428.00			0.516282			0.258141	−	0.966107i	$-0.416890\pi$
$419$	4428.00			0.516282			0.258141	+	0.966107i	$0.416890\pi$
$420$		0			0
$421$	−15490.0			−1.79320			−0.896599	−	0.442843i	$-0.853970\pi$
$421$	−15490.0			−1.79320			−0.896599	+	0.442843i	$0.853970\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1791.00	−	3102.10i	−0.204415	−	0.354057i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3384.00	−	5861.26i	0.378194	−	0.655051i	−0.612606	−	0.790389i	$-0.709879\pi$
$431$	3384.00	−	5861.26i	0.378194	−	0.655051i	0.990800	+	0.135338i	$0.0432119\pi$
$432$		0			0
$433$	−1298.00			−0.144060			−0.0720299	−	0.997402i	$-0.522948\pi$
$433$	−1298.00			−0.144060			−0.0720299	+	0.997402i	$0.522948\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3600.00	−	6235.38i	0.394076	−	0.682560i
$438$		0			0
$439$	−1124.00	−	1946.83i	−0.122200	−	0.211656i	0.798435	−	0.602081i	$-0.205661\pi$
$439$	−1124.00	−	1946.83i	−0.122200	−	0.211656i	−0.920635	+	0.390425i	$0.872328\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4806.00	−	8324.24i	−0.515440	−	0.892768i	−0.999839	−	0.0179214i	$-0.994295\pi$
$443$	−4806.00	−	8324.24i	−0.515440	−	0.892768i	0.484399	−	0.874847i	$-0.339038\pi$
$444$		0			0
$445$	−5670.00	+	9820.73i	−0.604008	+	1.04617i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−162.000			−0.0170273			−0.00851364	−	0.999964i	$-0.502710\pi$
$449$	−162.000			−0.0170273			−0.00851364	+	0.999964i	$0.502710\pi$
$450$		0			0
$451$	1620.00	−	2805.92i	0.169142	−	0.292962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−685.000	−	1186.45i	−0.0701159	−	0.121444i	0.828836	−	0.559492i	$-0.189004\pi$
$457$	−685.000	−	1186.45i	−0.0701159	−	0.121444i	−0.898952	+	0.438047i	$0.855670\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15354.0			−1.55121			−0.775604	−	0.631220i	$-0.782555\pi$
$461$	−15354.0			−1.55121			−0.775604	+	0.631220i	$0.782555\pi$
$462$		0			0
$463$	−13024.0			−1.30729			−0.653646	−	0.756800i	$-0.726762\pi$
$463$	−13024.0			−1.30729			−0.653646	+	0.756800i	$0.726762\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7218.00	+	12501.9i	0.715223	+	1.23880i	0.962873	+	0.269953i	$0.0870083\pi$
$467$	7218.00	+	12501.9i	0.715223	+	1.23880i	−0.247650	+	0.968849i	$0.579658\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8136.00	−	14092.0i	0.790896	−	1.36987i
$474$		0			0
$475$	19900.0			1.92226
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6048.00	+	10475.4i	−0.576911	+	0.999238i	0.418921	+	0.908023i	$0.362409\pi$
$479$	−6048.00	+	10475.4i	−0.576911	+	0.999238i	−0.995831	+	0.0912156i	$0.970925\pi$
$480$		0			0
$481$	1130.00	+	1957.22i	0.107118	+	0.185533i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9486.00	+	16430.2i	0.888118	+	1.53826i
$486$		0			0
$487$	−3028.00	+	5244.65i	−0.281749	+	0.488004i	−0.971816	−	0.235742i	$-0.924248\pi$
$487$	−3028.00	+	5244.65i	−0.281749	+	0.488004i	0.690067	+	0.723746i	$0.257581\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7524.00			−0.691555			−0.345777	−	0.938317i	$-0.612385\pi$
$491$	−7524.00			−0.691555			−0.345777	+	0.938317i	$0.612385\pi$
$492$		0			0
$493$	−2106.00	+	3647.70i	−0.192392	+	0.333233i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2638.00	−	4569.15i	−0.236660	−	0.409906i	0.723094	−	0.690749i	$-0.242719\pi$
$499$	−2638.00	−	4569.15i	−0.236660	−	0.409906i	−0.959754	+	0.280843i	$0.909386\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4968.00			0.440382			0.220191	−	0.975457i	$-0.429332\pi$
$503$	4968.00			0.440382			0.220191	+	0.975457i	$0.429332\pi$
$504$		0			0
$505$	−10044.0			−0.885054
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5499.00	−	9524.55i	−0.478858	−	0.829407i	0.520848	−	0.853650i	$-0.325616\pi$
$509$	−5499.00	−	9524.55i	−0.478858	−	0.829407i	−0.999706	+	0.0242426i	$0.992283\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−72.0000	+	124.708i	−0.00616058	+	0.0106704i
$516$		0			0
$517$	−15552.0			−1.32297
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4419.00	−	7653.93i	0.371593	−	0.643618i	−0.618218	−	0.786007i	$-0.712145\pi$
$521$	4419.00	−	7653.93i	0.371593	−	0.643618i	0.989811	+	0.142389i	$0.0454784\pi$
$522$		0			0
$523$	11218.0	+	19430.1i	0.937914	+	1.62451i	0.769354	+	0.638823i	$0.220578\pi$
$523$	11218.0	+	19430.1i	0.937914	+	1.62451i	0.168560	+	0.985691i	$0.446088\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−144.000	−	249.415i	−0.0119027	−	0.0206161i
$528$		0			0
$529$	3491.50	−	6047.46i	0.286965	−	0.497038i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	900.000			0.0731395
$534$		0			0
$535$	−15876.0	+	27498.0i	−1.28295	+	2.22214i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2381.00	+	4124.01i	0.189218	+	0.327736i	0.944990	−	0.327100i	$-0.106071\pi$
$541$	2381.00	+	4124.01i	0.189218	+	0.327736i	−0.755771	+	0.654835i	$0.772738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−29196.0			−2.29471
$546$		0			0
$547$	−6004.00			−0.469310			−0.234655	−	0.972079i	$-0.575396\pi$
$547$	−6004.00			−0.469310			−0.234655	+	0.972079i	$0.575396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11700.0	−	20265.0i	−0.904604	−	1.56682i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2637.00	+	4567.42i	−0.200598	+	0.347447i	−0.948721	−	0.316114i	$-0.897622\pi$
$557$	−2637.00	+	4567.42i	−0.200598	+	0.347447i	0.748123	+	0.663560i	$0.230955\pi$
$558$		0			0
$559$	4520.00			0.341996
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6210.00	−	10756.0i	0.464867	−	0.805174i	−0.534328	−	0.845277i	$-0.679435\pi$
$563$	6210.00	−	10756.0i	0.464867	−	0.805174i	0.999196	+	0.0401033i	$0.0127687\pi$
$564$		0			0
$565$	10206.0	+	17677.3i	0.759946	+	1.31627i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10683.0	−	18503.5i	−0.787091	−	1.36328i	−0.927742	−	0.373222i	$-0.878253\pi$
$569$	−10683.0	−	18503.5i	−0.787091	−	1.36328i	0.140651	−	0.990059i	$-0.455080\pi$
$570$		0			0
$571$	−10570.0	+	18307.8i	−0.774677	+	1.34178i	0.160298	+	0.987069i	$0.448754\pi$
$571$	−10570.0	+	18307.8i	−0.774677	+	1.34178i	−0.934976	+	0.354712i	$0.884579\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14328.0			−1.03916
$576$		0			0
$577$	1633.00	−	2828.44i	0.117821	−	0.204072i	−0.801083	−	0.598553i	$-0.795742\pi$
$577$	1633.00	−	2828.44i	0.117821	−	0.204072i	0.918904	+	0.394482i	$0.129076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−7452.00	−	12907.2i	−0.529383	−	0.916918i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17028.0			1.19731			0.598655	−	0.801007i	$-0.295702\pi$
$587$	17028.0			1.19731			0.598655	+	0.801007i	$0.295702\pi$
$588$		0			0
$589$	1600.00			0.111930
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4761.00	−	8246.29i	−0.329698	−	0.571053i	0.652754	−	0.757570i	$-0.273614\pi$
$593$	−4761.00	−	8246.29i	−0.329698	−	0.571053i	−0.982452	+	0.186517i	$0.940280\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5148.00	+	8916.60i	−0.351155	+	0.608218i	−0.986452	−	0.164050i	$-0.947544\pi$
$599$	−5148.00	+	8916.60i	−0.351155	+	0.608218i	0.635297	+	0.772268i	$0.280877\pi$
$600$		0			0
$601$	3382.00			0.229542			0.114771	−	0.993392i	$-0.463387\pi$
$601$	3382.00			0.229542			0.114771	+	0.993392i	$0.463387\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−315.000	+	545.596i	−0.0211679	+	0.0366639i
$606$		0			0
$607$	−10328.0	−	17888.6i	−0.690611	−	1.19617i	−0.971638	−	0.236473i	$-0.924009\pi$
$607$	−10328.0	−	17888.6i	−0.690611	−	1.19617i	0.281028	−	0.959700i	$-0.409325\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2160.00	−	3741.23i	−0.143018	−	0.247715i
$612$		0			0
$613$	11057.0	−	19151.3i	0.728529	−	1.26185i	−0.228976	−	0.973432i	$-0.573538\pi$
$613$	11057.0	−	19151.3i	0.728529	−	1.26185i	0.957505	−	0.288417i	$-0.0931288\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19962.0			−1.30250			−0.651248	−	0.758865i	$-0.725754\pi$
$617$	−19962.0			−1.30250			−0.651248	+	0.758865i	$0.725754\pi$
$618$		0			0
$619$	−302.000	+	523.079i	−0.0196097	+	0.0339650i	−0.875664	−	0.482921i	$-0.839576\pi$
$619$	−302.000	+	523.079i	−0.0196097	+	0.0339650i	0.856054	+	0.516886i	$0.172909\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	449.500	+	778.557i	0.0287680	+	0.0498276i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4068.00			−0.257872
$630$		0			0
$631$	152.000			0.00958958			0.00479479	−	0.999989i	$-0.498474\pi$
$631$	152.000			0.00958958			0.00479479	+	0.999989i	$0.498474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5328.00	−	9228.37i	−0.332969	−	0.576719i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2097.00	−	3632.11i	0.129215	−	0.223806i	−0.794158	−	0.607711i	$-0.792088\pi$
$641$	2097.00	−	3632.11i	0.129215	−	0.223806i	0.923372	+	0.383905i	$0.125421\pi$
$642$		0			0
$643$	7252.00			0.444776			0.222388	−	0.974958i	$-0.428615\pi$
$643$	7252.00			0.444776			0.222388	+	0.974958i	$0.428615\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3348.00	−	5798.91i	0.203437	−	0.352362i	−0.746197	−	0.665725i	$-0.768122\pi$
$647$	3348.00	−	5798.91i	0.203437	−	0.352362i	0.949633	+	0.313363i	$0.101456\pi$
$648$		0			0
$649$	−12312.0	−	21325.0i	−0.744666	−	1.28980i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14211.0	+	24614.2i	0.851638	+	1.47508i	0.879730	+	0.475474i	$0.157724\pi$
$653$	14211.0	+	24614.2i	0.851638	+	1.47508i	−0.0280919	+	0.999605i	$0.508943\pi$
$654$		0			0
$655$	−17172.0	+	29742.8i	−1.02437	+	1.77427i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19908.0			1.17679			0.588396	−	0.808573i	$-0.299760\pi$
$659$	19908.0			1.17679			0.588396	+	0.808573i	$0.299760\pi$
$660$		0			0
$661$	7159.00	−	12399.8i	0.421260	−	0.729644i	−0.574803	−	0.818292i	$-0.694921\pi$
$661$	7159.00	−	12399.8i	0.421260	−	0.729644i	0.996063	+	0.0886482i	$0.0282547\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8424.00	+	14590.8i	0.489023	+	0.847013i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15192.0			0.874040
$672$		0			0
$673$	30050.0			1.72116			0.860581	−	0.509313i	$-0.170101\pi$
$673$	30050.0			1.72116			0.860581	+	0.509313i	$0.170101\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11079.0	−	19189.4i	−0.628952	−	1.08938i	−0.987762	−	0.155966i	$-0.950151\pi$
$677$	−11079.0	−	19189.4i	−0.628952	−	1.08938i	0.358810	−	0.933411i	$-0.383183\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1566.00	+	2712.39i	−0.0877325	+	0.151957i	−0.906552	−	0.422093i	$-0.861295\pi$
$683$	−1566.00	+	2712.39i	−0.0877325	+	0.151957i	0.818820	+	0.574051i	$0.194629\pi$
$684$		0			0
$685$	17172.0			0.957822
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2070.00	−	3585.35i	0.114457	−	0.198245i
$690$		0			0
$691$	−10466.0	−	18127.6i	−0.576187	−	0.997986i	−0.995912	−	0.0903340i	$-0.971207\pi$
$691$	−10466.0	−	18127.6i	−0.576187	−	0.997986i	0.419724	−	0.907652i	$-0.362127\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23076.0	−	39968.8i	−1.25946	−	2.18144i
$696$		0			0
$697$	−810.000	+	1402.96i	−0.0440186	+	0.0762424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21834.0			1.17640			0.588202	−	0.808714i	$-0.299836\pi$
$701$	21834.0			1.17640			0.588202	+	0.808714i	$0.299836\pi$
$702$		0			0
$703$	11300.0	−	19572.2i	0.606241	−	1.05004i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6223.00	−	10778.6i	−0.329633	−	0.570941i	0.652806	−	0.757525i	$-0.273592\pi$
$709$	−6223.00	−	10778.6i	−0.329633	−	0.570941i	−0.982439	+	0.186584i	$0.940258\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1152.00			−0.0605088
$714$		0			0
$715$	6480.00			0.338935
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6264.00	+	10849.6i	0.324907	+	0.562755i	0.981493	−	0.191496i	$-0.0613338\pi$
$719$	6264.00	+	10849.6i	0.324907	+	0.562755i	−0.656587	+	0.754250i	$0.728000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−23283.0	+	40327.3i	−1.19270	+	2.06582i
$726$		0			0
$727$	−11576.0			−0.590550			−0.295275	−	0.955412i	$-0.595411\pi$
$727$	−11576.0			−0.590550			−0.295275	+	0.955412i	$0.595411\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4068.00	+	7045.98i	−0.205828	+	0.356505i
$732$		0			0
$733$	−14669.0	−	25407.5i	−0.739170	−	1.28028i	−0.952869	−	0.303381i	$-0.901884\pi$
$733$	−14669.0	−	25407.5i	−0.739170	−	1.28028i	0.213699	−	0.976900i	$-0.431449\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5976.00	+	10350.7i	0.298682	+	0.517333i
$738$		0			0
$739$	−1270.00	+	2199.70i	−0.0632175	+	0.109496i	−0.895902	−	0.444252i	$-0.853470\pi$
$739$	−1270.00	+	2199.70i	−0.0632175	+	0.109496i	0.832684	+	0.553748i	$0.186803\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18792.0			0.927876			0.463938	−	0.885868i	$-0.346436\pi$
$743$	18792.0			0.927876			0.463938	+	0.885868i	$0.346436\pi$
$744$		0			0
$745$	6642.00	−	11504.3i	0.326636	−	0.565751i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2416.00	−	4184.63i	−0.117392	−	0.203328i	0.801342	−	0.598207i	$-0.204120\pi$
$751$	−2416.00	−	4184.63i	−0.117392	−	0.203328i	−0.918733	+	0.394879i	$0.870787\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	43920.0			2.11710
$756$		0			0
$757$	−20818.0			−0.999529			−0.499764	−	0.866161i	$-0.666580\pi$
$757$	−20818.0			−0.999529			−0.499764	+	0.866161i	$0.666580\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6021.00	−	10428.7i	−0.286808	−	0.496766i	0.686238	−	0.727377i	$-0.259261\pi$
$761$	−6021.00	−	10428.7i	−0.286808	−	0.496766i	−0.973046	+	0.230611i	$0.925928\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3420.00	−	5923.61i	0.161003	−	0.278865i
$768$		0			0
$769$	−13058.0			−0.612332			−0.306166	−	0.951978i	$-0.599046\pi$
$769$	−13058.0			−0.612332			−0.306166	+	0.951978i	$0.599046\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5913.00	−	10241.6i	0.275130	−	0.476540i	−0.695038	−	0.718973i	$-0.744612\pi$
$773$	5913.00	−	10241.6i	0.275130	−	0.476540i	0.970168	+	0.242434i	$0.0779456\pi$
$774$		0			0
$775$	−1592.00	−	2757.42i	−0.0737888	−	0.127806i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4500.00	−	7794.23i	−0.206969	−	0.358482i
$780$		0			0
$781$	6480.00	−	11223.7i	0.296892	−	0.514232i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−45972.0			−2.09021
$786$		0			0
$787$	5998.00	−	10388.8i	0.271672	−	0.470549i	−0.697618	−	0.716470i	$-0.745757\pi$
$787$	5998.00	−	10388.8i	0.271672	−	0.470549i	0.969290	+	0.245920i	$0.0790902\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2110.00	+	3654.63i	0.0944871	+	0.163657i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6966.00			0.309596			0.154798	−	0.987946i	$-0.450527\pi$
$797$	6966.00			0.309596			0.154798	+	0.987946i	$0.450527\pi$
$798$		0			0
$799$	7776.00			0.344299
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−468.000	−	810.600i	−0.0205671	−	0.0356232i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−20403.0	+	35339.0i	−0.886689	+	1.53579i	−0.0429232	+	0.999078i	$0.513667\pi$
$809$	−20403.0	+	35339.0i	−0.886689	+	1.53579i	−0.843766	+	0.536712i	$0.819666\pi$
$810$		0			0
$811$	17980.0			0.778500			0.389250	−	0.921132i	$-0.372734\pi$
$811$	17980.0			0.778500			0.389250	+	0.921132i	$0.372734\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7380.00	+	12782.5i	−0.317190	+	0.549390i
$816$		0			0
$817$	−22600.0	−	39144.3i	−0.967777	−	1.67624i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6417.00	−	11114.6i	−0.272783	−	0.472474i	0.696790	−	0.717275i	$-0.254611\pi$
$821$	−6417.00	−	11114.6i	−0.272783	−	0.472474i	−0.969573	+	0.244801i	$0.921277\pi$
$822$		0			0
$823$	18932.0	−	32791.2i	0.801857	−	1.38886i	−0.116536	−	0.993186i	$-0.537179\pi$
$823$	18932.0	−	32791.2i	0.801857	−	1.38886i	0.918393	−	0.395670i	$-0.129488\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−42516.0			−1.78770			−0.893849	−	0.448368i	$-0.852005\pi$
$827$	−42516.0			−1.78770			−0.893849	+	0.448368i	$0.852005\pi$
$828$		0			0
$829$	22819.0	−	39523.7i	0.956015	−	1.65587i	0.223988	−	0.974592i	$-0.428092\pi$
$829$	22819.0	−	39523.7i	0.956015	−	1.65587i	0.732028	−	0.681275i	$-0.238574\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	17496.0	+	30304.0i	0.725119	+	1.25594i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17496.0			0.719939			0.359970	−	0.932964i	$-0.382787\pi$
$839$	17496.0			0.719939			0.359970	+	0.932964i	$0.382787\pi$
$840$		0			0
$841$	30367.0			1.24511
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18873.0	−	32689.0i	−0.768344	−	1.33081i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8136.00	+	14092.0i	−0.327730	+	0.567646i
$852$		0			0
$853$	−32174.0			−1.29146			−0.645731	−	0.763565i	$-0.723447\pi$
$853$	−32174.0			−1.29146			−0.645731	+	0.763565i	$0.723447\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19467.0	−	33717.8i	0.775939	−	1.34397i	−0.158326	−	0.987387i	$-0.550610\pi$
$857$	19467.0	−	33717.8i	0.775939	−	1.34397i	0.934265	−	0.356579i	$-0.116057\pi$
$858$		0			0
$859$	14890.0	+	25790.2i	0.591432	+	1.02439i	0.994040	+	0.109018i	$0.0347706\pi$
$859$	14890.0	+	25790.2i	0.591432	+	1.02439i	−0.402608	+	0.915373i	$0.631896\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24048.0	−	41652.4i	−0.948556	−	1.64295i	−0.748471	−	0.663168i	$-0.769212\pi$
$863$	−24048.0	−	41652.4i	−0.948556	−	1.64295i	−0.200085	−	0.979779i	$-0.564122\pi$
$864$		0			0
$865$	−11178.0	+	19360.9i	−0.439380	+	0.761028i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18432.0			−0.719520
$870$		0			0
$871$	−1660.00	+	2875.20i	−0.0645774	+	0.111851i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10651.0	−	18448.1i	−0.410101	−	0.710316i	0.584799	−	0.811178i	$-0.301173\pi$
$877$	−10651.0	−	18448.1i	−0.410101	−	0.710316i	−0.994900	+	0.100862i	$0.967840\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7470.00			−0.285665			−0.142832	−	0.989747i	$-0.545621\pi$
$881$	−7470.00			−0.285665			−0.142832	+	0.989747i	$0.545621\pi$
$882$		0			0
$883$	764.000			0.0291174			0.0145587	−	0.999894i	$-0.495366\pi$
$883$	764.000			0.0291174			0.0145587	+	0.999894i	$0.495366\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16164.0	−	27996.9i	−0.611876	−	1.05980i	−0.990924	−	0.134423i	$-0.957082\pi$
$887$	−16164.0	−	27996.9i	−0.611876	−	1.05980i	0.379048	−	0.925377i	$-0.376252\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21600.0	+	37412.3i	−0.809425	+	1.40197i
$894$		0			0
$895$	20088.0			0.750243
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1872.00	+	3242.40i	−0.0694490	+	0.120289i
$900$		0			0
$901$	3726.00	+	6453.62i	0.137770	+	0.238625i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9630.00	−	16679.6i	−0.353715	−	0.612652i
$906$		0			0
$907$	18158.0	−	31450.6i	0.664748	−	1.15138i	−0.314605	−	0.949223i	$-0.601872\pi$
$907$	18158.0	−	31450.6i	0.664748	−	1.15138i	0.979354	−	0.202155i	$-0.0647944\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13392.0			0.487044			0.243522	−	0.969895i	$-0.421697\pi$
$911$	13392.0			0.487044			0.243522	+	0.969895i	$0.421697\pi$
$912$		0			0
$913$	−21384.0	+	37038.2i	−0.775145	+	1.34259i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−19036.0	−	32971.3i	−0.683286	−	1.18349i	−0.973972	−	0.226667i	$-0.927217\pi$
$919$	−19036.0	−	32971.3i	−0.683286	−	1.18349i	0.290686	−	0.956818i	$-0.406116\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3600.00			0.128381
$924$		0			0
$925$	−44974.0			−1.59863
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6399.00	+	11083.4i	0.225990	+	0.391426i	0.956616	−	0.291352i	$-0.0941051\pi$
$929$	6399.00	+	11083.4i	0.225990	+	0.391426i	−0.730626	+	0.682778i	$0.760772\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5832.00	+	10101.3i	−0.203986	+	0.353314i
$936$		0			0
$937$	−34874.0			−1.21588			−0.607942	−	0.793981i	$-0.708005\pi$
$937$	−34874.0			−1.21588			−0.607942	+	0.793981i	$0.708005\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8595.00	+	14887.0i	−0.297757	+	0.515730i	−0.975622	−	0.219456i	$-0.929572\pi$
$941$	−8595.00	+	14887.0i	−0.297757	+	0.515730i	0.677866	+	0.735186i	$0.262905\pi$
$942$		0			0
$943$	3240.00	+	5611.84i	0.111886	+	0.193793i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20142.0	+	34887.0i	0.691158	+	1.19712i	0.971459	+	0.237209i	$0.0762327\pi$
$947$	20142.0	+	34887.0i	0.691158	+	1.19712i	−0.280300	+	0.959912i	$0.590434\pi$
$948$		0			0
$949$	130.000	−	225.167i	0.00444676	−	0.00770202i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15498.0			−0.526789			−0.263394	−	0.964688i	$-0.584842\pi$
$953$	−15498.0			−0.526789			−0.263394	+	0.964688i	$0.584842\pi$
$954$		0			0
$955$	5184.00	−	8978.95i	0.175655	−	0.304243i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	14767.5	+	25578.1i	0.495703	+	0.858583i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24156.0			0.805813
$966$		0			0
$967$	37160.0			1.23577			0.617883	−	0.786270i	$-0.287991\pi$
$967$	37160.0			1.23577			0.617883	+	0.786270i	$0.287991\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9234.00	−	15993.8i	−0.305183	−	0.528593i	0.672119	−	0.740443i	$-0.265384\pi$
$971$	−9234.00	−	15993.8i	−0.305183	−	0.528593i	−0.977302	+	0.211850i	$0.932051\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5193.00	−	8994.54i	0.170050	−	0.294535i	−0.768387	−	0.639985i	$-0.778940\pi$
$977$	5193.00	−	8994.54i	0.170050	−	0.294535i	0.938437	+	0.345450i	$0.112274\pi$
$978$		0			0
$979$	22680.0			0.740404
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22068.0	+	38222.9i	−0.716032	+	1.24020i	0.246528	+	0.969136i	$0.420710\pi$
$983$	−22068.0	+	38222.9i	−0.716032	+	1.24020i	−0.962560	+	0.271069i	$0.912623\pi$
$984$		0			0
$985$	−12798.0	−	22166.8i	−0.413988	−	0.717048i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16272.0	+	28183.9i	0.523174	+	0.906165i
$990$		0			0
$991$	14216.0	−	24622.8i	0.455687	−	0.789274i	−0.543040	−	0.839707i	$-0.682727\pi$
$991$	14216.0	−	24622.8i	0.455687	−	0.789274i	0.998727	+	0.0504332i	$0.0160602\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15696.0			0.500097
$996$		0			0
$997$	−19889.0	+	34448.8i	−0.631786	+	1.09429i	0.355400	+	0.934714i	$0.384345\pi$
$997$	−19889.0	+	34448.8i	−0.631786	+	1.09429i	−0.987186	+	0.159572i	$0.948989\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.o.361.1		2
3.2	odd	2		588.4.i.e.361.1		2
7.2	even	3	inner	1764.4.k.o.1549.1		2
7.3	odd	6		36.4.a.a.1.1		1
7.4	even	3		1764.4.a.b.1.1		1
7.5	odd	6		1764.4.k.b.1549.1		2
7.6	odd	2		1764.4.k.b.361.1		2
21.2	odd	6		588.4.i.e.373.1		2
21.5	even	6		588.4.i.d.373.1		2
21.11	odd	6		588.4.a.c.1.1		1
21.17	even	6		12.4.a.a.1.1	✓	1
21.20	even	2		588.4.i.d.361.1		2
28.3	even	6		144.4.a.g.1.1		1
35.3	even	12		900.4.d.c.649.1		2
35.17	even	12		900.4.d.c.649.2		2
35.24	odd	6		900.4.a.g.1.1		1
56.3	even	6		576.4.a.a.1.1		1
56.45	odd	6		576.4.a.b.1.1		1
63.31	odd	6		324.4.e.a.217.1		2
63.38	even	6		324.4.e.h.109.1		2
63.52	odd	6		324.4.e.a.109.1		2
63.59	even	6		324.4.e.h.217.1		2
84.11	even	6		2352.4.a.bk.1.1		1
84.59	odd	6		48.4.a.a.1.1		1
105.17	odd	12		300.4.d.e.49.1		2
105.38	odd	12		300.4.d.e.49.2		2
105.59	even	6		300.4.a.b.1.1		1
168.59	odd	6		192.4.a.l.1.1		1
168.101	even	6		192.4.a.f.1.1		1
231.164	odd	6		1452.4.a.d.1.1		1
273.38	even	6		2028.4.a.c.1.1		1
273.122	odd	12		2028.4.b.c.337.2		2
273.164	odd	12		2028.4.b.c.337.1		2
336.59	odd	12		768.4.d.j.385.2		2
336.101	even	12		768.4.d.g.385.1		2
336.227	odd	12		768.4.d.j.385.1		2
336.269	even	12		768.4.d.g.385.2		2
420.59	odd	6		1200.4.a.be.1.1		1
420.143	even	12		1200.4.f.d.49.1		2
420.227	even	12		1200.4.f.d.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.a.a.1.1	✓	1	21.17	even	6
36.4.a.a.1.1		1	7.3	odd	6
48.4.a.a.1.1		1	84.59	odd	6
144.4.a.g.1.1		1	28.3	even	6
192.4.a.f.1.1		1	168.101	even	6
192.4.a.l.1.1		1	168.59	odd	6
300.4.a.b.1.1		1	105.59	even	6
300.4.d.e.49.1		2	105.17	odd	12
300.4.d.e.49.2		2	105.38	odd	12
324.4.e.a.109.1		2	63.52	odd	6
324.4.e.a.217.1		2	63.31	odd	6
324.4.e.h.109.1		2	63.38	even	6
324.4.e.h.217.1		2	63.59	even	6
576.4.a.a.1.1		1	56.3	even	6
576.4.a.b.1.1		1	56.45	odd	6
588.4.a.c.1.1		1	21.11	odd	6
588.4.i.d.361.1		2	21.20	even	2
588.4.i.d.373.1		2	21.5	even	6
588.4.i.e.361.1		2	3.2	odd	2
588.4.i.e.373.1		2	21.2	odd	6
768.4.d.g.385.1		2	336.101	even	12
768.4.d.g.385.2		2	336.269	even	12
768.4.d.j.385.1		2	336.227	odd	12
768.4.d.j.385.2		2	336.59	odd	12
900.4.a.g.1.1		1	35.24	odd	6
900.4.d.c.649.1		2	35.3	even	12
900.4.d.c.649.2		2	35.17	even	12
1200.4.a.be.1.1		1	420.59	odd	6
1200.4.f.d.49.1		2	420.143	even	12
1200.4.f.d.49.2		2	420.227	even	12
1452.4.a.d.1.1		1	231.164	odd	6
1764.4.a.b.1.1		1	7.4	even	3
1764.4.k.b.361.1		2	7.6	odd	2
1764.4.k.b.1549.1		2	7.5	odd	6
1764.4.k.o.361.1		2	1.1	even	1	trivial
1764.4.k.o.1549.1		2	7.2	even	3	inner
2028.4.a.c.1.1		1	273.38	even	6
2028.4.b.c.337.1		2	273.164	odd	12
2028.4.b.c.337.2		2	273.122	odd	12
2352.4.a.bk.1.1		1	84.11	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.00000 + 15.5885i) q^{5} +(18.0000 - 31.1769i) q^{11} +10.0000 q^{13} +(-9.00000 + 15.5885i) q^{17} +(-50.0000 - 86.6025i) q^{19} +(36.0000 + 62.3538i) q^{23} +(-99.5000 + 172.339i) q^{25} +234.000 q^{29} +(-8.00000 + 13.8564i) q^{31} +(113.000 + 195.722i) q^{37} +90.0000 q^{41} +452.000 q^{43} +(-216.000 - 374.123i) q^{47} +(207.000 - 358.535i) q^{53} +648.000 q^{55} +(342.000 - 592.361i) q^{59} +(211.000 + 365.463i) q^{61} +(90.0000 + 155.885i) q^{65} +(-166.000 + 287.520i) q^{67} +360.000 q^{71} +(13.0000 - 22.5167i) q^{73} +(-256.000 - 443.405i) q^{79} -1188.00 q^{83} -324.000 q^{85} +(315.000 + 545.596i) q^{89} +(900.000 - 1558.85i) q^{95} +1054.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{5} + 36 q^{11} + 20 q^{13} - 18 q^{17} - 100 q^{19} + 72 q^{23} - 199 q^{25} + 468 q^{29} - 16 q^{31} + 226 q^{37} + 180 q^{41} + 904 q^{43} - 432 q^{47} + 414 q^{53} + 1296 q^{55} + 684 q^{59}+ \cdots + 2108 q^{97}+O(q^{100}) \) 2 * q + 18 * q^5 + 36 * q^11 + 20 * q^13 - 18 * q^17 - 100 * q^19 + 72 * q^23 - 199 * q^25 + 468 * q^29 - 16 * q^31 + 226 * q^37 + 180 * q^41 + 904 * q^43 - 432 * q^47 + 414 * q^53 + 1296 * q^55 + 684 * q^59 + 422 * q^61 + 180 * q^65 - 332 * q^67 + 720 * q^71 + 26 * q^73 - 512 * q^79 - 2376 * q^83 - 648 * q^85 + 630 * q^89 + 1800 * q^95 + 2108 * q^97

Introduction

L-functions

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