LMFDB - Embedded newform 1764.4.t.c.521.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (of order $6$, degree $2$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.20
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.c.1097.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.78515 + 11.7522i) q^{5} +O(q^{10})$	$q+(6.78515 + 11.7522i) q^{5} +(-9.62379 - 5.55630i) q^{11} -25.5525i q^{13} +(-11.6574 + 20.1912i) q^{17} +(-46.0440 + 26.5835i) q^{19} +(105.687 - 61.0186i) q^{23} +(-29.5764 + 51.2279i) q^{25} +43.3847i q^{29} +(-114.082 - 65.8655i) q^{31} +(178.531 + 309.225i) q^{37} -91.5050 q^{41} -395.464 q^{43} +(47.2356 + 81.8144i) q^{47} +(-25.2641 - 14.5863i) q^{53} -150.801i q^{55} +(-363.793 + 630.107i) q^{59} +(-630.866 + 364.230i) q^{61} +(300.299 - 173.378i) q^{65} +(18.6321 - 32.2717i) q^{67} +179.807i q^{71} +(-84.7874 - 48.9520i) q^{73} +(61.5145 + 106.546i) q^{79} -217.139 q^{83} -316.388 q^{85} +(-79.2711 - 137.302i) q^{89} +(-624.830 - 360.746i) q^{95} +597.401i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 888 q^{25} - 864 q^{37} - 2496 q^{43} - 1056 q^{67} - 16128 q^{85}+O(q^{100}) $ 48 * q - 888 * q^25 - 864 * q^37 - 2496 * q^43 - 1056 * q^67 - 16128 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	6.78515	+	11.7522i	0.606882	+	1.05115i	0.991751	+	0.128179i	$0.0409133\pi$
$5$	6.78515	+	11.7522i	0.606882	+	1.05115i	−0.384869	+	0.922971i	$0.625753\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.62379	−	5.55630i	−0.263789	−	0.152299i	0.362273	−	0.932072i	$-0.382001\pi$
$11$	−9.62379	−	5.55630i	−0.263789	−	0.152299i	−0.626062	+	0.779773i	$0.715334\pi$
$12$		0			0
$13$		−	25.5525i		−	0.545154i	−0.962134	−	0.272577i	$-0.912124\pi$
$13$		−	25.5525i		−	0.545154i	0.962134	−	0.272577i	$-0.0878759\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−11.6574	+	20.1912i	−0.166313	+	0.288063i	−0.937121	−	0.349005i	$-0.886520\pi$
$17$	−11.6574	+	20.1912i	−0.166313	+	0.288063i	0.770807	+	0.637068i	$0.219853\pi$
$18$		0			0
$19$	−46.0440	+	26.5835i	−0.555959	+	0.320983i	−0.751522	−	0.659708i	$-0.770680\pi$
$19$	−46.0440	+	26.5835i	−0.555959	+	0.320983i	0.195563	+	0.980691i	$0.437347\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	105.687	−	61.0186i	0.958144	−	0.553185i	0.0625430	−	0.998042i	$-0.480079\pi$
$23$	105.687	−	61.0186i	0.958144	−	0.553185i	0.895601	+	0.444857i	$0.146746\pi$
$24$		0			0
$25$	−29.5764	+	51.2279i	−0.236612	+	0.409823i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			43.3847i			0.277804i	0.990306	+	0.138902i	$0.0443574\pi$
$29$			43.3847i			0.277804i	−0.990306	+	0.138902i	$0.955643\pi$
$30$		0			0
$31$	−114.082	−	65.8655i	−0.660961	−	0.381606i	0.131682	−	0.991292i	$-0.457962\pi$
$31$	−114.082	−	65.8655i	−0.660961	−	0.381606i	−0.792643	+	0.609686i	$0.791296\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	178.531	+	309.225i	0.793252	+	1.37395i	0.923943	+	0.382529i	$0.124947\pi$
$37$	178.531	+	309.225i	0.793252	+	1.37395i	−0.130692	+	0.991423i	$0.541720\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−91.5050			−0.348553			−0.174277	−	0.984697i	$-0.555759\pi$
$41$	−91.5050			−0.348553			−0.174277	+	0.984697i	$0.555759\pi$
$42$		0			0
$43$	−395.464			−1.40250			−0.701252	−	0.712914i	$-0.747375\pi$
$43$	−395.464			−1.40250			−0.701252	+	0.712914i	$0.747375\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	47.2356	+	81.8144i	0.146596	+	0.253912i	0.929967	−	0.367642i	$-0.119835\pi$
$47$	47.2356	+	81.8144i	0.146596	+	0.253912i	−0.783371	+	0.621554i	$0.786502\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−25.2641	−	14.5863i	−0.0654773	−	0.0378033i	0.466904	−	0.884308i	$-0.345369\pi$
$53$	−25.2641	−	14.5863i	−0.0654773	−	0.0378033i	−0.532381	+	0.846505i	$0.678703\pi$
$54$		0			0
$55$		−	150.801i		−	0.369709i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−363.793	+	630.107i	−0.802742	+	1.39039i	0.115063	+	0.993358i	$0.463293\pi$
$59$	−363.793	+	630.107i	−0.802742	+	1.39039i	−0.917805	+	0.397031i	$0.870040\pi$
$60$		0			0
$61$	−630.866	+	364.230i	−1.32416	+	0.764507i	−0.984390	−	0.176000i	$-0.943684\pi$
$61$	−630.866	+	364.230i	−1.32416	+	0.764507i	−0.339775	+	0.940507i	$0.610351\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	300.299	−	173.378i	0.573039	−	0.330844i
$66$		0			0
$67$	18.6321	−	32.2717i	0.0339742	−	0.0588451i	−0.848538	−	0.529134i	$-0.822517\pi$
$67$	18.6321	−	32.2717i	0.0339742	−	0.0588451i	0.882513	+	0.470289i	$0.155850\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			179.807i			0.300552i	0.988644	+	0.150276i	$0.0480162\pi$
$71$			179.807i			0.300552i	−0.988644	+	0.150276i	$0.951984\pi$
$72$		0			0
$73$	−84.7874	−	48.9520i	−0.135940	−	0.0784850i	0.430488	−	0.902596i	$-0.358342\pi$
$73$	−84.7874	−	48.9520i	−0.135940	−	0.0784850i	−0.566428	+	0.824111i	$0.691675\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	61.5145	+	106.546i	0.0876066	+	0.151739i	0.906499	−	0.422208i	$-0.138745\pi$
$79$	61.5145	+	106.546i	0.0876066	+	0.151739i	−0.818892	+	0.573947i	$0.805411\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−217.139			−0.287158			−0.143579	−	0.989639i	$-0.545861\pi$
$83$	−217.139			−0.287158			−0.143579	+	0.989639i	$0.545861\pi$
$84$		0			0
$85$	−316.388			−0.403731
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−79.2711	−	137.302i	−0.0944126	−	0.163527i	0.814951	−	0.579530i	$-0.196764\pi$
$89$	−79.2711	−	137.302i	−0.0944126	−	0.163527i	−0.909363	+	0.416003i	$0.863431\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−624.830	−	360.746i	−0.674802	−	0.389597i
$96$		0			0
$97$			597.401i			0.625329i	0.949864	+	0.312664i	$0.101221\pi$
$97$			597.401i			0.625329i	−0.949864	+	0.312664i	$0.898779\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−348.594	+	603.783i	−0.343430	+	0.594838i	−0.985067	−	0.172170i	$-0.944922\pi$
$101$	−348.594	+	603.783i	−0.343430	+	0.594838i	0.641637	+	0.767008i	$0.278255\pi$
$102$		0			0
$103$	303.976	−	175.501i	0.290793	−	0.167889i	−0.347507	−	0.937678i	$-0.612972\pi$
$103$	303.976	−	175.501i	0.290793	−	0.167889i	0.638299	+	0.769788i	$0.279638\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	363.845	−	210.066i	0.328731	−	0.189793i	−0.326546	−	0.945181i	$-0.605885\pi$
$107$	363.845	−	210.066i	0.328731	−	0.189793i	0.655278	+	0.755388i	$0.272552\pi$
$108$		0			0
$109$	−713.900	+	1236.51i	−0.627333	+	1.08657i	0.360752	+	0.932662i	$0.382520\pi$
$109$	−713.900	+	1236.51i	−0.627333	+	1.08657i	−0.988085	+	0.153910i	$0.950813\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2036.67i		−	1.69552i	−0.530377	−	0.847762i	$-0.677950\pi$
$113$		−	2036.67i		−	1.69552i	0.530377	−	0.847762i	$-0.322050\pi$
$114$		0			0
$115$	1434.21	+	828.040i	1.16296	+	0.671436i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−603.755	−	1045.73i	−0.453610	−	0.785676i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	893.565			0.639383
$126$		0			0
$127$	−509.829			−0.356220			−0.178110	−	0.984011i	$-0.556998\pi$
$127$	−509.829			−0.356220			−0.178110	+	0.984011i	$0.556998\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1461.14	−	2530.76i	−0.974505	−	1.68789i	−0.681559	−	0.731763i	$-0.738698\pi$
$131$	−1461.14	−	2530.76i	−0.974505	−	1.68789i	−0.292946	−	0.956129i	$-0.594635\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1371.34	−	791.742i	−0.855191	−	0.493745i	0.00720775	−	0.999974i	$-0.497706\pi$
$137$	−1371.34	−	791.742i	−0.855191	−	0.493745i	−0.862399	+	0.506229i	$0.831039\pi$
$138$		0			0
$139$		−	1130.29i		−	0.689714i	−0.938655	−	0.344857i	$-0.887927\pi$
$139$		−	1130.29i		−	0.689714i	0.938655	−	0.344857i	$-0.112073\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−141.977	+	245.912i	−0.0830262	+	0.143806i
$144$		0			0
$145$	−509.866	+	294.371i	−0.292014	+	0.168595i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1404.83	+	811.077i	−0.772401	+	0.445946i	−0.833731	−	0.552171i	$-0.813799\pi$
$149$	−1404.83	+	811.077i	−0.772401	+	0.445946i	0.0613292	+	0.998118i	$0.480466\pi$
$150$		0			0
$151$	178.370	−	308.946i	0.0961294	−	0.166501i	−0.813950	−	0.580935i	$-0.802687\pi$
$151$	178.370	−	308.946i	0.0961294	−	0.166501i	0.910079	+	0.414434i	$0.136020\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1787.63i		−	0.926360i
$156$		0			0
$157$	2942.23	+	1698.70i	1.49564	+	0.863507i	0.999987	−	0.00501451i	$-0.00159618\pi$
$157$	2942.23	+	1698.70i	1.49564	+	0.863507i	0.495651	+	0.868522i	$0.334930\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1436.68	−	2488.41i	−0.690366	−	1.19575i	−0.971718	−	0.236144i	$-0.924116\pi$
$163$	−1436.68	−	2488.41i	−0.690366	−	1.19575i	0.281352	−	0.959605i	$-0.409217\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1733.75			−0.803364			−0.401682	−	0.915779i	$-0.631574\pi$
$167$	−1733.75			−0.803364			−0.401682	+	0.915779i	$0.631574\pi$
$168$		0			0
$169$	1544.07			0.702807
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−619.304	−	1072.67i	−0.272167	−	0.471406i	0.697250	−	0.716828i	$-0.254407\pi$
$173$	−619.304	−	1072.67i	−0.272167	−	0.471406i	−0.969416	+	0.245422i	$0.921074\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2696.31	−	1556.71i	−1.12587	−	0.650023i	−0.182980	−	0.983117i	$-0.558574\pi$
$179$	−2696.31	−	1556.71i	−1.12587	−	0.650023i	−0.942894	+	0.333093i	$0.891908\pi$
$180$		0			0
$181$			2749.21i			1.12899i	0.825437	+	0.564495i	$0.190929\pi$
$181$			2749.21i			1.12899i	−0.825437	+	0.564495i	$0.809071\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2422.72	+	4196.27i	−0.962820	+	1.66765i
$186$		0			0
$187$	224.376	−	129.544i	0.0877434	−	0.0506587i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	368.401	−	212.696i	0.139563	−	0.0805767i	−0.428593	−	0.903498i	$-0.640990\pi$
$191$	368.401	−	212.696i	0.139563	−	0.0805767i	0.568156	+	0.822921i	$0.307657\pi$
$192$		0			0
$193$	−1413.38	+	2448.04i	−0.527135	+	0.913024i	0.472365	+	0.881403i	$0.343400\pi$
$193$	−1413.38	+	2448.04i	−0.527135	+	0.913024i	−0.999500	+	0.0316212i	$0.989933\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4763.08i		−	1.72262i	−0.508083	−	0.861308i	$-0.669646\pi$
$197$		−	4763.08i		−	1.72262i	0.508083	−	0.861308i	$-0.330354\pi$
$198$		0			0
$199$	3735.13	+	2156.48i	1.33053	+	0.768185i	0.985382	−	0.170362i	$-0.0544937\pi$
$199$	3735.13	+	2156.48i	1.33053	+	0.768185i	0.345153	+	0.938546i	$0.387827\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−620.875	−	1075.39i	−0.211531	−	0.366382i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	590.823			0.195541
$210$		0			0
$211$	−4948.35			−1.61450			−0.807248	−	0.590213i	$-0.799044\pi$
$211$	−4948.35			−1.61450			−0.807248	+	0.590213i	$0.799044\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2683.28	−	4647.58i	−0.851154	−	1.47424i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	515.935	+	297.875i	0.157039	+	0.0906664i
$222$		0			0
$223$			314.818i			0.0945370i	0.998882	+	0.0472685i	$0.0150516\pi$
$223$			314.818i			0.0945370i	−0.998882	+	0.0472685i	$0.984948\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2160.94	+	3742.86i	−0.631836	+	1.09437i	0.355340	+	0.934737i	$0.384365\pi$
$227$	−2160.94	+	3742.86i	−0.631836	+	1.09437i	−0.987176	+	0.159635i	$0.948968\pi$
$228$		0			0
$229$	−2666.96	+	1539.77i	−0.769596	+	0.444327i	−0.832731	−	0.553678i	$-0.813224\pi$
$229$	−2666.96	+	1539.77i	−0.769596	+	0.444327i	0.0631343	+	0.998005i	$0.479890\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3116.93	−	1799.56i	0.876382	−	0.505979i	0.00691826	−	0.999976i	$-0.497798\pi$
$233$	3116.93	−	1799.56i	0.876382	−	0.505979i	0.869464	+	0.493997i	$0.164464\pi$
$234$		0			0
$235$	−641.001	+	1110.25i	−0.177933	+	0.308189i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	242.461i		−	0.0656213i	−0.999462	−	0.0328107i	$-0.989554\pi$
$239$		−	242.461i		−	0.0656213i	0.999462	−	0.0328107i	$-0.0104458\pi$
$240$		0			0
$241$	269.633	+	155.673i	0.0720689	+	0.0416090i	0.535601	−	0.844471i	$-0.320085\pi$
$241$	269.633	+	155.673i	0.0720689	+	0.0416090i	−0.463533	+	0.886080i	$0.653418\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	679.276	+	1176.54i	0.174985	+	0.303083i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1863.72			0.468672			0.234336	−	0.972156i	$-0.424708\pi$
$251$	1863.72			0.468672			0.234336	+	0.972156i	$0.424708\pi$
$252$		0			0
$253$	−1356.15			−0.336998
$254$		0			0
$255$		0			0
$256$		0			0
$257$	166.962	+	289.186i	0.0405245	+	0.0701905i	0.885576	−	0.464494i	$-0.153764\pi$
$257$	166.962	+	289.186i	0.0405245	+	0.0701905i	−0.845052	+	0.534685i	$0.820430\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6283.46	−	3627.76i	−1.47321	−	0.850560i	−0.473667	−	0.880704i	$-0.657070\pi$
$263$	−6283.46	−	3627.76i	−1.47321	−	0.850560i	−0.999546	+	0.0301441i	$0.990403\pi$
$264$		0			0
$265$		−	395.880i		−	0.0917687i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2856.72	+	4947.99i	−0.647499	+	1.12150i	0.336219	+	0.941784i	$0.390852\pi$
$269$	−2856.72	+	4947.99i	−0.647499	+	1.12150i	−0.983718	+	0.179718i	$0.942482\pi$
$270$		0			0
$271$	−5636.53	+	3254.25i	−1.26345	+	0.729453i	−0.973740	−	0.227662i	$-0.926892\pi$
$271$	−5636.53	+	3254.25i	−1.26345	+	0.729453i	−0.289709	+	0.957115i	$0.593559\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	569.275	−	328.671i	0.124831	−	0.0720713i
$276$		0			0
$277$	−2985.26	+	5170.62i	−0.647533	+	1.12156i	0.336177	+	0.941799i	$0.390866\pi$
$277$	−2985.26	+	5170.62i	−0.647533	+	1.12156i	−0.983710	+	0.179762i	$0.942467\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4469.94i		−	0.948947i	−0.880270	−	0.474474i	$-0.842638\pi$
$281$		−	4469.94i		−	0.948947i	0.880270	−	0.474474i	$-0.157362\pi$
$282$		0			0
$283$	478.258	+	276.122i	0.100458	+	0.0579992i	0.549387	−	0.835568i	$-0.314861\pi$
$283$	478.258	+	276.122i	0.100458	+	0.0579992i	−0.448930	+	0.893567i	$0.648195\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2184.71	+	3784.03i	0.444680	+	0.770208i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5834.87			−1.16340			−0.581701	−	0.813403i	$-0.697613\pi$
$293$	−5834.87			−1.16340			−0.581701	+	0.813403i	$0.697613\pi$
$294$		0			0
$295$	−9873.54			−1.94868
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1559.18	−	2700.58i	−0.301571	−	0.522336i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8561.03	−	4942.71i	−1.60722	−	0.927931i
$306$		0			0
$307$		−	6605.19i		−	1.22794i	−0.789329	−	0.613971i	$-0.789571\pi$
$307$		−	6605.19i		−	1.22794i	0.789329	−	0.613971i	$-0.210429\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4457.10	+	7719.92i	−0.812665	+	1.40758i	0.0983278	+	0.995154i	$0.468651\pi$
$311$	−4457.10	+	7719.92i	−0.812665	+	1.40758i	−0.910993	+	0.412423i	$0.864683\pi$
$312$		0			0
$313$	−4758.92	+	2747.56i	−0.859392	+	0.496170i	−0.863809	−	0.503820i	$-0.831928\pi$
$313$	−4758.92	+	2747.56i	−0.859392	+	0.496170i	0.00441640	+	0.999990i	$0.498594\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5597.12	−	3231.50i	0.991689	−	0.572552i	0.0859101	−	0.996303i	$-0.472620\pi$
$317$	5597.12	−	3231.50i	0.991689	−	0.572552i	0.905779	+	0.423751i	$0.139287\pi$
$318$		0			0
$319$	241.058	−	417.525i	0.0423093	−	0.0732818i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1239.58i		−	0.213535i
$324$		0			0
$325$	1309.00	+	755.753i	0.223417	+	0.128990i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−357.428	−	619.084i	−0.0593536	−	0.102803i	0.834822	−	0.550520i	$-0.185571\pi$
$331$	−357.428	−	619.084i	−0.0593536	−	0.102803i	−0.894175	+	0.447717i	$0.852237\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	505.686			0.0824734
$336$		0			0
$337$	−2098.15			−0.339151			−0.169575	−	0.985517i	$-0.554240\pi$
$337$	−2098.15			−0.339151			−0.169575	+	0.985517i	$0.554240\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	731.936	+	1267.75i	0.116236	+	0.201327i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9355.15	+	5401.20i	1.44729	+	0.835596i	0.998320	−	0.0579491i	$-0.0184561\pi$
$347$	9355.15	+	5401.20i	1.44729	+	0.835596i	0.448974	+	0.893545i	$0.351789\pi$
$348$		0			0
$349$			4256.02i			0.652778i	0.945236	+	0.326389i	$0.105832\pi$
$349$			4256.02i			0.652778i	−0.945236	+	0.326389i	$0.894168\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2852.24	+	4940.22i	−0.430055	+	0.744877i	−0.996878	−	0.0789629i	$-0.974839\pi$
$353$	−2852.24	+	4940.22i	−0.430055	+	0.744877i	0.566823	+	0.823840i	$0.308172\pi$
$354$		0			0
$355$	−2113.13	+	1220.02i	−0.315925	+	0.182399i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10133.7	−	5850.67i	1.48979	−	0.860130i	0.489857	−	0.871803i	$-0.337049\pi$
$359$	10133.7	−	5850.67i	1.48979	−	0.860130i	0.999932	+	0.0116725i	$0.00371554\pi$
$360$		0			0
$361$	−2016.13	+	3492.05i	−0.293940	+	0.509119i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1328.59i		−	0.190524i
$366$		0			0
$367$	3200.64	+	1847.89i	0.455237	+	0.262831i	0.710040	−	0.704162i	$-0.248677\pi$
$367$	3200.64	+	1847.89i	0.455237	+	0.262831i	−0.254802	+	0.966993i	$0.582010\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2488.62	+	4310.42i	0.345458	+	0.598351i	0.985437	−	0.170041i	$-0.0543901\pi$
$373$	2488.62	+	4310.42i	0.345458	+	0.598351i	−0.639979	+	0.768393i	$0.721057\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1108.59			0.151446
$378$		0			0
$379$	263.270			0.0356815			0.0178408	−	0.999841i	$-0.494321\pi$
$379$	263.270			0.0356815			0.0178408	+	0.999841i	$0.494321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3779.88	−	6546.95i	−0.504290	−	0.873455i	−0.999988	−	0.00496048i	$-0.998421\pi$
$383$	−3779.88	−	6546.95i	−0.504290	−	0.873455i	0.495698	−	0.868495i	$-0.334912\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6204.83	−	3582.36i	−0.808734	−	0.466923i	0.0377822	−	0.999286i	$-0.487971\pi$
$389$	−6204.83	−	3582.36i	−0.808734	−	0.466923i	−0.846516	+	0.532363i	$0.821304\pi$
$390$		0			0
$391$			2845.27i			0.368008i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−834.770	+	1445.86i	−0.106334	+	0.184176i
$396$		0			0
$397$	2535.14	−	1463.67i	0.320492	−	0.185036i	−0.331120	−	0.943589i	$-0.607426\pi$
$397$	2535.14	−	1463.67i	0.320492	−	0.185036i	0.651612	+	0.758553i	$0.274093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11987.2	+	6920.84i	−1.49280	+	0.861871i	−0.999966	−	0.00825081i	$-0.997374\pi$
$401$	−11987.2	+	6920.84i	−1.49280	+	0.861871i	−0.492838	+	0.870121i	$0.664040\pi$
$402$		0			0
$403$	−1683.03	+	2915.09i	−0.208034	+	0.360326i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3967.88i		−	0.483245i
$408$		0			0
$409$	5442.19	+	3142.05i	0.657944	+	0.379864i	0.791493	−	0.611178i	$-0.209304\pi$
$409$	5442.19	+	3142.05i	0.657944	+	0.379864i	−0.133549	+	0.991042i	$0.542637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1473.32	−	2551.86i	−0.174271	−	0.301846i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4835.26			−0.563766			−0.281883	−	0.959449i	$-0.590959\pi$
$419$	−4835.26			−0.563766			−0.281883	+	0.959449i	$0.590959\pi$
$420$		0			0
$421$	−4348.01			−0.503346			−0.251673	−	0.967812i	$-0.580981\pi$
$421$	−4348.01			−0.503346			−0.251673	+	0.967812i	$0.580981\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−689.567	−	1194.37i	−0.0787034	−	0.136318i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6054.69	−	3495.68i	−0.676669	−	0.390675i	0.121930	−	0.992539i	$-0.461092\pi$
$431$	−6054.69	−	3495.68i	−0.676669	−	0.390675i	−0.798599	+	0.601864i	$0.794425\pi$
$432$		0			0
$433$			2274.87i			0.252479i	0.992000	+	0.126240i	$0.0402908\pi$
$433$			2274.87i			0.252479i	−0.992000	+	0.126240i	$0.959709\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3244.18	+	5619.08i	−0.355126	+	0.615096i
$438$		0			0
$439$	−4184.38	+	2415.85i	−0.454919	+	0.262648i	−0.709905	−	0.704297i	$-0.751262\pi$
$439$	−4184.38	+	2415.85i	−0.454919	+	0.262648i	0.254986	+	0.966945i	$0.417929\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2918.74	+	1685.13i	−0.313033	+	0.180729i	−0.648283	−	0.761400i	$-0.724513\pi$
$443$	−2918.74	+	1685.13i	−0.313033	+	0.180729i	0.335250	+	0.942129i	$0.391179\pi$
$444$		0			0
$445$	1075.73	−	1863.22i	0.114595	−	0.198484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4674.68i		−	0.491340i	−0.969353	−	0.245670i	$-0.920992\pi$
$449$		−	4674.68i		−	0.491340i	0.969353	−	0.245670i	$-0.0790080\pi$
$450$		0			0
$451$	880.624	+	508.429i	0.0919445	+	0.0530842i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5570.06	−	9647.63i	−0.570146	−	0.987521i	−0.996550	−	0.0829889i	$-0.973553\pi$
$457$	−5570.06	−	9647.63i	−0.570146	−	0.987521i	0.426405	−	0.904532i	$-0.359780\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3295.95			0.332989			0.166494	−	0.986042i	$-0.446755\pi$
$461$	3295.95			0.332989			0.166494	+	0.986042i	$0.446755\pi$
$462$		0			0
$463$	16451.6			1.65134			0.825670	−	0.564154i	$-0.190797\pi$
$463$	16451.6			1.65134			0.825670	+	0.564154i	$0.190797\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1230.98	−	2132.12i	−0.121976	−	0.211269i	0.798571	−	0.601901i	$-0.205590\pi$
$467$	−1230.98	−	2132.12i	−0.121976	−	0.211269i	−0.920547	+	0.390632i	$0.872257\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3805.86	+	2197.31i	0.369965	+	0.213600i
$474$		0			0
$475$		−	3144.98i		−	0.303793i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7135.86	−	12359.7i	0.680680	−	1.17897i	−0.294094	−	0.955777i	$-0.595018\pi$
$479$	7135.86	−	12359.7i	0.680680	−	1.17897i	0.974774	−	0.223196i	$-0.0716489\pi$
$480$		0			0
$481$	7901.48	−	4561.92i	0.749015	−	0.432444i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7020.79	+	4053.45i	−0.657315	+	0.379501i
$486$		0			0
$487$	3996.95	−	6922.92i	0.371908	−	0.644163i	−0.617951	−	0.786216i	$-0.712037\pi$
$487$	3996.95	−	6922.92i	0.371908	−	0.644163i	0.989859	+	0.142053i	$0.0453705\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2326.48i		−	0.213834i	−0.994268	−	0.106917i	$-0.965902\pi$
$491$		−	2326.48i		−	0.213834i	0.994268	−	0.106917i	$-0.0340980\pi$
$492$		0			0
$493$	−875.987	−	505.751i	−0.0800253	−	0.0462026i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7582.10	−	13132.6i	−0.680203	−	1.17815i	−0.974919	−	0.222562i	$-0.928558\pi$
$499$	−7582.10	−	13132.6i	−0.680203	−	1.17815i	0.294715	−	0.955585i	$-0.404775\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17052.7			1.51161			0.755807	−	0.654794i	$-0.227245\pi$
$503$	17052.7			1.51161			0.755807	+	0.654794i	$0.227245\pi$
$504$		0			0
$505$	−9461.05			−0.833686
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−943.473	−	1634.14i	−0.0821585	−	0.142303i	0.822018	−	0.569461i	$-0.192848\pi$
$509$	−943.473	−	1634.14i	−0.0821585	−	0.142303i	−0.904177	+	0.427158i	$0.859515\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4125.04	+	2381.60i	0.352954	+	0.203778i
$516$		0			0
$517$		−	1049.82i		−	0.0893056i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10436.9	−	18077.2i	0.877635	−	1.52011i	0.0237053	−	0.999719i	$-0.492454\pi$
$521$	10436.9	−	18077.2i	0.877635	−	1.52011i	0.853929	−	0.520389i	$-0.174213\pi$
$522$		0			0
$523$	5398.17	−	3116.63i	0.451330	−	0.260575i	−0.257062	−	0.966395i	$-0.582754\pi$
$523$	5398.17	−	3116.63i	0.451330	−	0.260575i	0.708392	+	0.705820i	$0.249421\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2659.80	−	1535.64i	0.219854	−	0.126933i
$528$		0			0
$529$	1363.04	−	2360.85i	0.112027	−	0.194037i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2338.18i			0.190015i
$534$		0			0
$535$	4937.49	+	2850.66i	0.399002	+	0.230364i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	458.279	+	793.762i	0.0364195	+	0.0630804i	0.883661	−	0.468128i	$-0.155071\pi$
$541$	458.279	+	793.762i	0.0364195	+	0.0630804i	−0.847241	+	0.531209i	$0.821738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19375.7			−1.52287
$546$		0			0
$547$	14238.3			1.11295			0.556476	−	0.830864i	$-0.312153\pi$
$547$	14238.3			1.11295			0.556476	+	0.830864i	$0.312153\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1153.32	−	1997.60i	−0.0891705	−	0.154448i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9781.49	−	5647.35i	−0.744084	−	0.429597i	0.0794680	−	0.996837i	$-0.474678\pi$
$557$	−9781.49	−	5647.35i	−0.744084	−	0.429597i	−0.823553	+	0.567240i	$0.808011\pi$
$558$		0			0
$559$			10105.1i			0.764580i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6200.07	+	10738.8i	−0.464124	+	0.803887i	−0.999162	−	0.0409416i	$-0.986964\pi$
$563$	−6200.07	+	10738.8i	−0.464124	+	0.803887i	0.535037	+	0.844828i	$0.320298\pi$
$564$		0			0
$565$	23935.4	−	13819.1i	1.78225	−	1.02898i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2983.12	+	1722.30i	−0.219787	+	0.126894i	−0.605852	−	0.795578i	$-0.707167\pi$
$569$	−2983.12	+	1722.30i	−0.219787	+	0.126894i	0.386065	+	0.922472i	$0.373834\pi$
$570$		0			0
$571$	−6371.21	+	11035.3i	−0.466947	+	0.808776i	−0.999287	−	0.0377547i	$-0.987979\pi$
$571$	−6371.21	+	11035.3i	−0.466947	+	0.808776i	0.532340	+	0.846531i	$0.321313\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7218.85i			0.523560i
$576$		0			0
$577$	4269.52	+	2465.01i	0.308046	+	0.177850i	0.646052	−	0.763294i	$-0.276419\pi$
$577$	4269.52	+	2465.01i	0.308046	+	0.177850i	−0.338006	+	0.941144i	$0.609752\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	162.091	+	280.750i	0.0115148	+	0.0199442i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13674.4			0.961503			0.480751	−	0.876857i	$-0.340364\pi$
$587$	13674.4			0.961503			0.480751	+	0.876857i	$0.340364\pi$
$588$		0			0
$589$	7003.74			0.489956
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10753.4	+	18625.5i	0.744671	+	1.28981i	0.950348	+	0.311189i	$0.100727\pi$
$593$	10753.4	+	18625.5i	0.744671	+	1.28981i	−0.205677	+	0.978620i	$0.565940\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23054.7	+	13310.6i	1.57260	+	0.907942i	0.995849	+	0.0910223i	$0.0290135\pi$
$599$	23054.7	+	13310.6i	1.57260	+	0.907942i	0.576752	+	0.816919i	$0.304320\pi$
$600$		0			0
$601$			1559.13i			0.105821i	0.998599	+	0.0529104i	$0.0168498\pi$
$601$			1559.13i			0.105821i	−0.998599	+	0.0529104i	$0.983150\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8193.14	−	14190.9i	0.550576	−	0.953625i
$606$		0			0
$607$	−15691.4	+	9059.43i	−1.04925	+	0.605784i	−0.922439	−	0.386143i	$-0.873807\pi$
$607$	−15691.4	+	9059.43i	−1.04925	+	0.605784i	−0.126810	+	0.991927i	$0.540474\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2090.57	−	1206.99i	0.138421	−	0.0799174i
$612$		0			0
$613$	1508.96	−	2613.59i	0.0994230	−	0.172206i	−0.812023	−	0.583625i	$-0.801634\pi$
$613$	1508.96	−	2613.59i	0.0994230	−	0.172206i	0.911446	+	0.411420i	$0.134967\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12242.8i			0.798830i	0.916770	+	0.399415i	$0.130787\pi$
$617$			12242.8i			0.798830i	−0.916770	+	0.399415i	$0.869213\pi$
$618$		0			0
$619$	−22254.7	−	12848.8i	−1.44506	−	0.834306i	−0.446879	−	0.894595i	$-0.647464\pi$
$619$	−22254.7	−	12848.8i	−1.44506	−	0.834306i	−0.998181	+	0.0602890i	$0.980798\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9760.02	+	16904.9i	0.624641	+	1.08191i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8324.81			−0.527714
$630$		0			0
$631$	17745.4			1.11954			0.559772	−	0.828647i	$-0.310889\pi$
$631$	17745.4			1.11954			0.559772	+	0.828647i	$0.310889\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3459.26	−	5991.62i	−0.216184	−	0.374441i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3888.48	+	2245.01i	0.239603	+	0.138335i	0.614994	−	0.788532i	$-0.289158\pi$
$641$	3888.48	+	2245.01i	0.239603	+	0.138335i	−0.375391	+	0.926866i	$0.622492\pi$
$642$		0			0
$643$		−	14156.1i		−	0.868217i	−0.900861	−	0.434109i	$-0.857063\pi$
$643$		−	14156.1i		−	0.868217i	0.900861	−	0.434109i	$-0.142937\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10440.7	−	18083.8i	0.634414	−	1.09884i	−0.352225	−	0.935915i	$-0.614575\pi$
$647$	10440.7	−	18083.8i	0.634414	−	1.09884i	0.986639	−	0.162922i	$-0.0520917\pi$
$648$		0			0
$649$	7002.12	−	4042.68i	0.423509	−	0.244513i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15826.4	+	9137.40i	−0.948448	+	0.547587i	−0.892598	−	0.450853i	$-0.851120\pi$
$653$	−15826.4	+	9137.40i	−0.948448	+	0.547587i	−0.0558494	+	0.998439i	$0.517787\pi$
$654$		0			0
$655$	19828.1	−	34343.2i	1.18282	−	2.04870i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			9657.98i			0.570897i	0.958394	+	0.285449i	$0.0921426\pi$
$659$			9657.98i			0.570897i	−0.958394	+	0.285449i	$0.907857\pi$
$660$		0			0
$661$	−11029.7	−	6367.98i	−0.649023	−	0.374713i	0.139059	−	0.990284i	$-0.455592\pi$
$661$	−11029.7	−	6367.98i	−0.649023	−	0.374713i	−0.788082	+	0.615571i	$0.788926\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2647.27	+	4585.21i	0.153677	+	0.266177i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8095.09			0.465734
$672$		0			0
$673$	−13482.5			−0.772232			−0.386116	−	0.922450i	$-0.626184\pi$
$673$	−13482.5			−0.772232			−0.386116	+	0.922450i	$0.626184\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14046.2	+	24328.8i	0.797402	+	1.38114i	0.921303	+	0.388845i	$0.127126\pi$
$677$	14046.2	+	24328.8i	0.797402	+	1.38114i	−0.123901	+	0.992295i	$0.539541\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22854.9	+	13195.3i	1.28041	+	0.739244i	0.976923	−	0.213593i	$-0.0685165\pi$
$683$	22854.9	+	13195.3i	1.28041	+	0.739244i	0.303485	+	0.952836i	$0.401850\pi$
$684$		0			0
$685$		−	21488.3i		−	1.19858i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−372.716	+	645.563i	−0.0206086	+	0.0356952i
$690$		0			0
$691$	−21179.8	+	12228.2i	−1.16602	+	0.673200i	−0.952738	−	0.303792i	$-0.901747\pi$
$691$	−21179.8	+	12228.2i	−1.16602	+	0.673200i	−0.213278	+	0.976992i	$0.568414\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13283.5	−	7669.21i	0.724993	−	0.418575i
$696$		0			0
$697$	1066.71	−	1847.59i	0.0579691	−	0.100405i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			16268.1i			0.876517i	0.898849	+	0.438258i	$0.144405\pi$
$701$			16268.1i			0.876517i	−0.898849	+	0.438258i	$0.855595\pi$
$702$		0			0
$703$	−16440.6	−	9491.96i	−0.882030	−	0.509240i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6694.41	+	11595.1i	0.354603	+	0.614191i	0.987050	−	0.160413i	$-0.0512826\pi$
$709$	6694.41	+	11595.1i	0.354603	+	0.614191i	−0.632447	+	0.774604i	$0.717949\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16076.1			−0.844395
$714$		0			0
$715$	−3853.35			−0.201548
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9215.48	+	15961.7i	0.477996	+	0.827914i	0.999682	−	0.0252241i	$-0.00802994\pi$
$719$	9215.48	+	15961.7i	0.477996	+	0.827914i	−0.521686	+	0.853138i	$0.674697\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2222.50	−	1283.16i	−0.113851	−	0.0657317i
$726$		0			0
$727$			28217.0i			1.43949i	0.694238	+	0.719745i	$0.255741\pi$
$727$			28217.0i			1.43949i	−0.694238	+	0.719745i	$0.744259\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4610.07	−	7984.88i	0.233255	−	0.404010i
$732$		0			0
$733$	14776.6	−	8531.28i	0.744593	−	0.429891i	−0.0791439	−	0.996863i	$-0.525219\pi$
$733$	14776.6	−	8531.28i	0.744593	−	0.429891i	0.823737	+	0.566972i	$0.191885\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−358.623	+	207.051i	−0.0179241	+	0.0103485i
$738$		0			0
$739$	−5363.69	+	9290.18i	−0.266991	+	0.462442i	−0.968083	−	0.250628i	$-0.919363\pi$
$739$	−5363.69	+	9290.18i	−0.266991	+	0.462442i	0.701092	+	0.713071i	$0.252696\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			9465.65i			0.467377i	0.972312	+	0.233688i	$0.0750796\pi$
$743$			9465.65i			0.467377i	−0.972312	+	0.233688i	$0.924920\pi$
$744$		0			0
$745$	−19063.9	−	11006.5i	−0.937513	−	0.541273i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	11891.6	+	20596.9i	0.577804	+	1.00079i	0.995731	+	0.0923054i	$0.0294236\pi$
$751$	11891.6	+	20596.9i	0.577804	+	1.00079i	−0.417927	+	0.908481i	$0.637243\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4841.06			0.233357
$756$		0			0
$757$	−1709.14			−0.0820605			−0.0410303	−	0.999158i	$-0.513064\pi$
$757$	−1709.14			−0.0820605			−0.0410303	+	0.999158i	$0.513064\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11182.1	+	19368.0i	0.532656	+	0.922587i	0.999273	+	0.0381276i	$0.0121393\pi$
$761$	11182.1	+	19368.0i	0.532656	+	0.922587i	−0.466617	+	0.884460i	$0.654527\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16100.8	+	9295.82i	0.757976	+	0.437618i
$768$		0			0
$769$			266.623i			0.0125028i	0.999980	+	0.00625141i	$0.00198990\pi$
$769$			266.623i			0.0125028i	−0.999980	+	0.00625141i	$0.998010\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6659.91	−	11535.3i	0.309884	−	0.536734i	−0.668453	−	0.743754i	$-0.733043\pi$
$773$	6659.91	−	11535.3i	0.309884	−	0.536734i	0.978337	+	0.207020i	$0.0663765\pi$
$774$		0			0
$775$	6748.30	−	3896.13i	0.312782	−	0.180585i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4213.25	−	2432.52i	0.193781	−	0.111880i
$780$		0			0
$781$	999.061	−	1730.42i	0.0457736	−	0.0792823i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			46103.6i			2.09619i
$786$		0			0
$787$	−29289.1	−	16910.1i	−1.32661	−	0.765920i	−0.341838	−	0.939759i	$-0.611050\pi$
$787$	−29289.1	−	16910.1i	−1.32661	−	0.765920i	−0.984774	+	0.173839i	$0.944383\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	9307.01	+	16120.2i	0.416774	+	0.721873i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3193.62			−0.141937			−0.0709685	−	0.997479i	$-0.522609\pi$
$797$	−3193.62			−0.141937			−0.0709685	+	0.997479i	$0.522609\pi$
$798$		0			0
$799$	−2202.57			−0.0975236
$800$		0			0
$801$		0			0
$802$		0			0
$803$	543.984	+	942.208i	0.0239063	+	0.0414070i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5481.07	−	3164.50i	−0.238201	−	0.137525i	0.376149	−	0.926559i	$-0.377248\pi$
$809$	−5481.07	−	3164.50i	−0.238201	−	0.137525i	−0.614349	+	0.789034i	$0.710581\pi$
$810$		0			0
$811$		−	36542.9i		−	1.58224i	−0.611661	−	0.791120i	$-0.709498\pi$
$811$		−	36542.9i		−	1.58224i	0.611661	−	0.791120i	$-0.290502\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	19496.2	−	33768.4i	0.837941	−	1.45136i
$816$		0			0
$817$	18208.7	−	10512.8i	0.779734	−	0.450180i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5836.94	−	3369.96i	0.248125	−	0.143255i	−0.370780	−	0.928721i	$-0.620910\pi$
$821$	5836.94	−	3369.96i	0.248125	−	0.143255i	0.618905	+	0.785465i	$0.287576\pi$
$822$		0			0
$823$	10557.7	−	18286.5i	0.447167	−	0.774515i	−0.551034	−	0.834483i	$-0.685766\pi$
$823$	10557.7	−	18286.5i	0.447167	−	0.774515i	0.998200	+	0.0599677i	$0.0190998\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			27049.4i			1.13736i	0.822558	+	0.568681i	$0.192546\pi$
$827$			27049.4i			1.13736i	−0.822558	+	0.568681i	$0.807454\pi$
$828$		0			0
$829$	−12607.0	−	7278.67i	−0.528179	−	0.304944i	0.212096	−	0.977249i	$-0.431971\pi$
$829$	−12607.0	−	7278.67i	−0.528179	−	0.304944i	−0.740275	+	0.672305i	$0.765304\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−11763.8	−	20375.4i	−0.487547	−	0.844457i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33136.3			1.36352			0.681759	−	0.731577i	$-0.261215\pi$
$839$	33136.3			1.36352			0.681759	+	0.731577i	$0.261215\pi$
$840$		0			0
$841$	22506.8			0.922825
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10476.7	+	18146.2i	0.426521	+	0.738756i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	37736.9	+	21787.4i	1.52010	+	0.877630i
$852$		0			0
$853$			38789.6i			1.55701i	0.627638	+	0.778505i	$0.284022\pi$
$853$			38789.6i			1.55701i	−0.627638	+	0.778505i	$0.715978\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−188.396	+	326.312i	−0.00750932	+	0.0130065i	−0.869756	−	0.493483i	$-0.835724\pi$
$857$	−188.396	+	326.312i	−0.00750932	+	0.0130065i	0.862246	+	0.506489i	$0.169057\pi$
$858$		0			0
$859$	734.421	−	424.018i	0.0291713	−	0.0168420i	−0.485343	−	0.874324i	$-0.661305\pi$
$859$	734.421	−	424.018i	0.0291713	−	0.0168420i	0.514515	+	0.857482i	$0.327972\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	12557.4	−	7250.02i	0.495318	−	0.285972i	−0.231460	−	0.972844i	$-0.574350\pi$
$863$	12557.4	−	7250.02i	0.495318	−	0.285972i	0.726778	+	0.686873i	$0.241017\pi$
$864$		0			0
$865$	8404.14	−	14556.4i	0.330346	−	0.572176i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1367.17i		−	0.0533695i
$870$		0			0
$871$	−824.625	−	476.097i	−0.0320796	−	0.0185212i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3246.25	+	5622.67i	0.124992	+	0.216493i	0.921730	−	0.387832i	$-0.126776\pi$
$877$	3246.25	+	5622.67i	0.124992	+	0.216493i	−0.796738	+	0.604325i	$0.793443\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−5687.73			−0.217508			−0.108754	−	0.994069i	$-0.534686\pi$
$881$	−5687.73			−0.217508			−0.108754	+	0.994069i	$0.534686\pi$
$882$		0			0
$883$	38104.3			1.45222			0.726111	−	0.687578i	$-0.241326\pi$
$883$	38104.3			1.45222			0.726111	+	0.687578i	$0.241326\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4065.20	+	7041.13i	0.153885	+	0.266536i	0.932652	−	0.360776i	$-0.117488\pi$
$887$	4065.20	+	7041.13i	0.153885	+	0.266536i	−0.778768	+	0.627313i	$0.784155\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4349.83	−	2511.37i	−0.163003	−	0.0941097i
$894$		0			0
$895$		−	42250.1i		−	1.57795i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2857.55	−	4949.43i	0.106012	−	0.183618i
$900$		0			0
$901$	589.027	−	340.075i	0.0217795	−	0.0125744i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−32309.3	+	18653.8i	−1.18674	+	0.685164i
$906$		0			0
$907$	−1255.16	+	2174.00i	−0.0459502	+	0.0795882i	−0.888086	−	0.459678i	$-0.847965\pi$
$907$	−1255.16	+	2174.00i	−0.0459502	+	0.0795882i	0.842136	+	0.539266i	$0.181298\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	17140.3i		−	0.623361i	−0.950187	−	0.311681i	$-0.899108\pi$
$911$		−	17140.3i		−	0.623361i	0.950187	−	0.311681i	$-0.100892\pi$
$912$		0			0
$913$	2089.70	+	1206.49i	0.0757491	+	0.0437338i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2939.92	−	5092.10i	−0.105527	−	0.182778i	0.808427	−	0.588597i	$-0.200320\pi$
$919$	−2939.92	−	5092.10i	−0.105527	−	0.182778i	−0.913953	+	0.405819i	$0.866986\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4594.52			0.163847
$924$		0			0
$925$	−21121.2			−0.750770
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16148.0	+	27969.2i	0.570290	+	0.987771i	0.996536	+	0.0831640i	$0.0265025\pi$
$929$	16148.0	+	27969.2i	0.570290	+	0.987771i	−0.426246	+	0.904607i	$0.640164\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3044.85	+	1757.95i	0.106500	+	0.0614877i
$936$		0			0
$937$			24422.5i			0.851494i	0.904842	+	0.425747i	$0.139989\pi$
$937$			24422.5i			0.851494i	−0.904842	+	0.425747i	$0.860011\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21898.5	−	37929.3i	0.758629	−	1.31398i	−0.184921	−	0.982753i	$-0.559203\pi$
$941$	21898.5	−	37929.3i	0.758629	−	1.31398i	0.943550	−	0.331231i	$-0.107464\pi$
$942$		0			0
$943$	−9670.91	+	5583.50i	−0.333964	+	0.192814i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27928.3	+	16124.4i	−0.958340	+	0.553298i	−0.895662	−	0.444736i	$-0.853297\pi$
$947$	−27928.3	+	16124.4i	−0.958340	+	0.553298i	−0.0626781	+	0.998034i	$0.519964\pi$
$948$		0			0
$949$	−1250.85	+	2166.53i	−0.0427864	+	0.0741082i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	37006.9i		−	1.25789i	−0.777449	−	0.628946i	$-0.783487\pi$
$953$		−	37006.9i		−	1.25789i	0.777449	−	0.628946i	$-0.216513\pi$
$954$		0			0
$955$	4999.30	+	2886.35i	0.169397	+	0.0978011i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6218.97	−	10771.6i	−0.208753	−	0.361571i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−38359.8			−1.27963
$966$		0			0
$967$	24660.8			0.820100			0.410050	−	0.912063i	$-0.365511\pi$
$967$	24660.8			0.820100			0.410050	+	0.912063i	$0.365511\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23662.9	−	40985.4i	−0.782059	−	1.35457i	−0.930740	−	0.365681i	$-0.880836\pi$
$971$	−23662.9	−	40985.4i	−0.782059	−	1.35457i	0.148681	−	0.988885i	$-0.452497\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4136.98	−	2388.48i	−0.135469	−	0.0782133i	0.430734	−	0.902479i	$-0.358255\pi$
$977$	−4136.98	−	2388.48i	−0.135469	−	0.0782133i	−0.566203	+	0.824266i	$0.691588\pi$
$978$		0			0
$979$			1761.81i			0.0575157i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17576.5	−	30443.4i	0.570299	−	0.987787i	−0.426236	−	0.904612i	$-0.640161\pi$
$983$	17576.5	−	30443.4i	0.570299	−	0.987787i	0.996535	−	0.0831749i	$-0.0265060\pi$
$984$		0			0
$985$	55976.8	−	32318.2i	1.81073	−	1.04543i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41795.5	+	24130.6i	−1.34380	+	0.775844i
$990$		0			0
$991$	19728.2	−	34170.2i	0.632378	−	1.09531i	−0.354686	−	0.934985i	$-0.615412\pi$
$991$	19728.2	−	34170.2i	0.632378	−	1.09531i	0.987064	−	0.160326i	$-0.0512545\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			58528.1i			1.86479i
$996$		0			0
$997$	−13134.5	−	7583.18i	−0.417224	−	0.240884i	0.276665	−	0.960966i	$-0.410771\pi$
$997$	−13134.5	−	7583.18i	−0.417224	−	0.240884i	−0.693889	+	0.720082i	$0.744104\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.c.521.20		48
3.2	odd	2	inner	1764.4.t.c.521.5		48
7.2	even	3	inner	1764.4.t.c.1097.19		48
7.3	odd	6		1764.4.f.b.881.20	yes	24
7.4	even	3		1764.4.f.b.881.6	yes	24
7.5	odd	6	inner	1764.4.t.c.1097.5		48
7.6	odd	2	inner	1764.4.t.c.521.6		48
21.2	odd	6	inner	1764.4.t.c.1097.6		48
21.5	even	6	inner	1764.4.t.c.1097.20		48
21.11	odd	6		1764.4.f.b.881.19	yes	24
21.17	even	6		1764.4.f.b.881.5	✓	24
21.20	even	2	inner	1764.4.t.c.521.19		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.f.b.881.5	✓	24	21.17	even	6
1764.4.f.b.881.6	yes	24	7.4	even	3
1764.4.f.b.881.19	yes	24	21.11	odd	6
1764.4.f.b.881.20	yes	24	7.3	odd	6
1764.4.t.c.521.5		48	3.2	odd	2	inner
1764.4.t.c.521.6		48	7.6	odd	2	inner
1764.4.t.c.521.19		48	21.20	even	2	inner
1764.4.t.c.521.20		48	1.1	even	1	trivial
1764.4.t.c.1097.5		48	7.5	odd	6	inner
1764.4.t.c.1097.6		48	21.2	odd	6	inner
1764.4.t.c.1097.19		48	7.2	even	3	inner
1764.4.t.c.1097.20		48	21.5	even	6	inner

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.78515 + 11.7522i) q^{5} +O(q^{10})\)	\(q+(6.78515 + 11.7522i) q^{5} +(-9.62379 - 5.55630i) q^{11} -25.5525i q^{13} +(-11.6574 + 20.1912i) q^{17} +(-46.0440 + 26.5835i) q^{19} +(105.687 - 61.0186i) q^{23} +(-29.5764 + 51.2279i) q^{25} +43.3847i q^{29} +(-114.082 - 65.8655i) q^{31} +(178.531 + 309.225i) q^{37} -91.5050 q^{41} -395.464 q^{43} +(47.2356 + 81.8144i) q^{47} +(-25.2641 - 14.5863i) q^{53} -150.801i q^{55} +(-363.793 + 630.107i) q^{59} +(-630.866 + 364.230i) q^{61} +(300.299 - 173.378i) q^{65} +(18.6321 - 32.2717i) q^{67} +179.807i q^{71} +(-84.7874 - 48.9520i) q^{73} +(61.5145 + 106.546i) q^{79} -217.139 q^{83} -316.388 q^{85} +(-79.2711 - 137.302i) q^{89} +(-624.830 - 360.746i) q^{95} +597.401i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 888 q^{25} - 864 q^{37} - 2496 q^{43} - 1056 q^{67} - 16128 q^{85}+O(q^{100}) \) 48 * q - 888 * q^25 - 864 * q^37 - 2496 * q^43 - 1056 * q^67 - 16128 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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