LMFDB - Embedded newform 768.2.j.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(193,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.j (of order $4$, degree $2$, 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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.1
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.577
Dual form			768.2.j.e.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.44949i q^{7} +1.00000i q^{9} +O(q^{10})$	\(q+(-0.707107 - 0.707107i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.44949i q^{7} +1.00000i q^{9} +(1.03528 - 1.03528i) q^{11} +(-3.00000 - 3.00000i) q^{13} +3.86370 q^{15} +3.46410 q^{17} +(5.27792 + 5.27792i) q^{19} +(-1.73205 + 1.73205i) q^{21} +2.82843i q^{23} -9.92820i q^{25} +(0.707107 - 0.707107i) q^{27} +(6.19615 + 6.19615i) q^{29} +7.34847 q^{31} -1.46410 q^{33} +(6.69213 + 6.69213i) q^{35} +(-0.464102 + 0.464102i) q^{37} +4.24264i q^{39} +4.53590i q^{41} +(2.44949 - 2.44949i) q^{43} +(-2.73205 - 2.73205i) q^{45} +2.82843 q^{47} +1.00000 q^{49} +(-2.44949 - 2.44949i) q^{51} +(-3.26795 + 3.26795i) q^{53} +5.65685i q^{55} -7.46410i q^{57} +(9.79796 - 9.79796i) q^{59} +(-0.464102 - 0.464102i) q^{61} +2.44949 q^{63} +16.3923 q^{65} +(-10.5558 - 10.5558i) q^{67} +(2.00000 - 2.00000i) q^{69} -6.41473i q^{71} +4.00000i q^{73} +(-7.02030 + 7.02030i) q^{75} +(-2.53590 - 2.53590i) q^{77} +2.44949 q^{79} -1.00000 q^{81} +(3.86370 + 3.86370i) q^{83} +(-9.46410 + 9.46410i) q^{85} -8.76268i q^{87} +4.92820i q^{89} +(-7.34847 + 7.34847i) q^{91} +(-5.19615 - 5.19615i) q^{93} -28.8391 q^{95} +1.07180 q^{97} +(1.03528 + 1.03528i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 24 q^{13} + 8 q^{29} + 16 q^{33} + 24 q^{37} - 8 q^{45} + 8 q^{49} - 40 q^{53} + 24 q^{61} + 48 q^{65} + 16 q^{69} - 48 q^{77} - 8 q^{81} - 48 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 24 * q^13 + 8 * q^29 + 16 * q^33 + 24 * q^37 - 8 * q^45 + 8 * q^49 - 40 * q^53 + 24 * q^61 + 48 * q^65 + 16 * q^69 - 48 * q^77 - 8 * q^81 - 48 * q^85 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$	−2.73205	+	2.73205i	−1.22181	+	1.22181i	−0.254822	+	0.966988i	$0.582017\pi$
$5$	−2.73205	+	2.73205i	−1.22181	+	1.22181i	−0.966988	+	0.254822i	$0.917983\pi$
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	1.03528	−	1.03528i	0.312148	−	0.312148i	−0.533594	−	0.845741i	$-0.679159\pi$
$11$	1.03528	−	1.03528i	0.312148	−	0.312148i	0.845741	+	0.533594i	$0.179159\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$	3.86370			0.997604
$16$		0			0
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		0			0
$19$	5.27792	+	5.27792i	1.21084	+	1.21084i	0.970752	+	0.240085i	$0.0771754\pi$
$19$	5.27792	+	5.27792i	1.21084	+	1.21084i	0.240085	+	0.970752i	$0.422825\pi$
$20$		0			0
$21$	−1.73205	+	1.73205i	−0.377964	+	0.377964i
$22$		0			0
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$		−	9.92820i		−	1.98564i
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$	6.19615	+	6.19615i	1.15060	+	1.15060i	0.986433	+	0.164163i	$0.0524925\pi$
$29$	6.19615	+	6.19615i	1.15060	+	1.15060i	0.164163	+	0.986433i	$0.447508\pi$
$30$		0			0
$31$	7.34847			1.31982			0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847			1.31982			0.659912	+	0.751343i	$0.270594\pi$
$32$		0			0
$33$	−1.46410			−0.254867
$34$		0			0
$35$	6.69213	+	6.69213i	1.13118	+	1.13118i
$36$		0			0
$37$	−0.464102	+	0.464102i	−0.0762978	+	0.0762978i	−0.744226	−	0.667928i	$-0.767181\pi$
$37$	−0.464102	+	0.464102i	−0.0762978	+	0.0762978i	0.667928	+	0.744226i	$0.267181\pi$
$38$		0			0
$39$			4.24264i			0.679366i
$40$		0			0
$41$			4.53590i			0.708388i	0.935172	+	0.354194i	$0.115245\pi$
$41$			4.53590i			0.708388i	−0.935172	+	0.354194i	$0.884755\pi$
$42$		0			0
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	−0.495222	−	0.868766i	$-0.664913\pi$
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	0.868766	+	0.495222i	$0.164913\pi$
$44$		0			0
$45$	−2.73205	−	2.73205i	−0.407270	−	0.407270i
$46$		0			0
$47$	2.82843			0.412568			0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843			0.412568			0.206284	+	0.978492i	$0.433863\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−2.44949	−	2.44949i	−0.342997	−	0.342997i
$52$		0			0
$53$	−3.26795	+	3.26795i	−0.448887	+	0.448887i	−0.894985	−	0.446097i	$-0.852814\pi$
$53$	−3.26795	+	3.26795i	−0.448887	+	0.448887i	0.446097	+	0.894985i	$0.352814\pi$
$54$		0			0
$55$			5.65685i			0.762770i
$56$		0			0
$57$		−	7.46410i		−	0.988644i
$58$		0			0
$59$	9.79796	−	9.79796i	1.27559	−	1.27559i	0.332473	−	0.943113i	$-0.392117\pi$
$59$	9.79796	−	9.79796i	1.27559	−	1.27559i	0.943113	−	0.332473i	$-0.107883\pi$
$60$		0			0
$61$	−0.464102	−	0.464102i	−0.0594221	−	0.0594221i	0.676771	−	0.736193i	$-0.263379\pi$
$61$	−0.464102	−	0.464102i	−0.0594221	−	0.0594221i	−0.736193	+	0.676771i	$0.763379\pi$
$62$		0			0
$63$	2.44949			0.308607
$64$		0			0
$65$	16.3923			2.03322
$66$		0			0
$67$	−10.5558	−	10.5558i	−1.28960	−	1.28960i	−0.935029	−	0.354571i	$-0.884627\pi$
$67$	−10.5558	−	10.5558i	−1.28960	−	1.28960i	−0.354571	−	0.935029i	$-0.615373\pi$
$68$		0			0
$69$	2.00000	−	2.00000i	0.240772	−	0.240772i
$70$		0			0
$71$		−	6.41473i		−	0.761288i	−0.924722	−	0.380644i	$-0.875702\pi$
$71$		−	6.41473i		−	0.761288i	0.924722	−	0.380644i	$-0.124298\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	−7.02030	+	7.02030i	−0.810634	+	0.810634i
$76$		0			0
$77$	−2.53590	−	2.53590i	−0.288992	−	0.288992i
$78$		0			0
$79$	2.44949			0.275589			0.137795	−	0.990461i	$-0.455999\pi$
$79$	2.44949			0.275589			0.137795	+	0.990461i	$0.455999\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	3.86370	+	3.86370i	0.424097	+	0.424097i	0.886611	−	0.462515i	$-0.153053\pi$
$83$	3.86370	+	3.86370i	0.424097	+	0.424097i	−0.462515	+	0.886611i	$0.653053\pi$
$84$		0			0
$85$	−9.46410	+	9.46410i	−1.02653	+	1.02653i
$86$		0			0
$87$		−	8.76268i		−	0.939458i
$88$		0			0
$89$			4.92820i			0.522388i	0.965286	+	0.261194i	$0.0841163\pi$
$89$			4.92820i			0.522388i	−0.965286	+	0.261194i	$0.915884\pi$
$90$		0			0
$91$	−7.34847	+	7.34847i	−0.770329	+	0.770329i
$92$		0			0
$93$	−5.19615	−	5.19615i	−0.538816	−	0.538816i
$94$		0			0
$95$	−28.8391			−2.95883
$96$		0			0
$97$	1.07180			0.108824			0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180			0.108824			0.0544122	+	0.998519i	$0.482671\pi$
$98$		0			0
$99$	1.03528	+	1.03528i	0.104049	+	0.104049i
$100$		0			0
$101$	10.1962	−	10.1962i	1.01456	−	1.01456i	0.0146626	−	0.999892i	$-0.495333\pi$
$101$	10.1962	−	10.1962i	1.01456	−	1.01456i	0.999892	−	0.0146626i	$-0.00466741\pi$
$102$		0			0
$103$			3.20736i			0.316031i	0.987437	+	0.158016i	$0.0505096\pi$
$103$			3.20736i			0.316031i	−0.987437	+	0.158016i	$0.949490\pi$
$104$		0			0
$105$		−	9.46410i		−	0.923602i
$106$		0			0
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	−0.557047	−	0.830481i	$-0.688066\pi$
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	0.830481	+	0.557047i	$0.188066\pi$
$108$		0			0
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	0.657592	−	0.753374i	$-0.271575\pi$
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	−0.753374	+	0.657592i	$0.771575\pi$
$110$		0			0
$111$	0.656339			0.0622969
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	−7.72741	−	7.72741i	−0.720584	−	0.720584i
$116$		0			0
$117$	3.00000	−	3.00000i	0.277350	−	0.277350i
$118$		0			0
$119$		−	8.48528i		−	0.777844i
$120$		0			0
$121$			8.85641i			0.805128i
$122$		0			0
$123$	3.20736	−	3.20736i	0.289198	−	0.289198i
$124$		0			0
$125$	13.4641	+	13.4641i	1.20427	+	1.20427i
$126$		0			0
$127$	14.5211			1.28854			0.644268	−	0.764799i	$-0.277162\pi$
$127$	14.5211			1.28854			0.644268	+	0.764799i	$0.277162\pi$
$128$		0			0
$129$	−3.46410			−0.304997
$130$		0			0
$131$	10.5558	+	10.5558i	0.922267	+	0.922267i	0.997189	−	0.0749219i	$-0.0238708\pi$
$131$	10.5558	+	10.5558i	0.922267	+	0.922267i	−0.0749219	+	0.997189i	$0.523871\pi$
$132$		0			0
$133$	12.9282	−	12.9282i	1.12102	−	1.12102i
$134$		0			0
$135$			3.86370i			0.332535i
$136$		0			0
$137$			14.3923i			1.22962i	0.788676	+	0.614809i	$0.210767\pi$
$137$			14.3923i			1.22962i	−0.788676	+	0.614809i	$0.789233\pi$
$138$		0			0
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	−0.883658	−	0.468132i	$-0.844927\pi$
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	0.468132	+	0.883658i	$0.344927\pi$
$140$		0			0
$141$	−2.00000	−	2.00000i	−0.168430	−	0.168430i
$142$		0			0
$143$	−6.21166			−0.519445
$144$		0			0
$145$	−33.8564			−2.81162
$146$		0			0
$147$	−0.707107	−	0.707107i	−0.0583212	−	0.0583212i
$148$		0			0
$149$	−3.80385	+	3.80385i	−0.311623	+	0.311623i	−0.845538	−	0.533915i	$-0.820720\pi$
$149$	−3.80385	+	3.80385i	−0.311623	+	0.311623i	0.533915	+	0.845538i	$0.320720\pi$
$150$		0			0
$151$			6.59059i			0.536335i	0.963372	+	0.268167i	$0.0864181\pi$
$151$			6.59059i			0.536335i	−0.963372	+	0.268167i	$0.913582\pi$
$152$		0			0
$153$			3.46410i			0.280056i
$154$		0			0
$155$	−20.0764	+	20.0764i	−1.61257	+	1.61257i
$156$		0			0
$157$	1.53590	+	1.53590i	0.122578	+	0.122578i	0.765735	−	0.643157i	$-0.222376\pi$
$157$	1.53590	+	1.53590i	0.122578	+	0.122578i	−0.643157	+	0.765735i	$0.722376\pi$
$158$		0			0
$159$	4.62158			0.366515
$160$		0			0
$161$	6.92820			0.546019
$162$		0			0
$163$	−3.20736	−	3.20736i	−0.251220	−	0.251220i	0.570251	−	0.821471i	$-0.306846\pi$
$163$	−3.20736	−	3.20736i	−0.251220	−	0.251220i	−0.821471	+	0.570251i	$0.806846\pi$
$164$		0			0
$165$	4.00000	−	4.00000i	0.311400	−	0.311400i
$166$		0			0
$167$			7.72741i			0.597965i	0.954259	+	0.298982i	$0.0966472\pi$
$167$			7.72741i			0.597965i	−0.954259	+	0.298982i	$0.903353\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−5.27792	+	5.27792i	−0.403612	+	0.403612i
$172$		0			0
$173$	12.1962	+	12.1962i	0.927256	+	0.927256i	0.997528	−	0.0702716i	$-0.0223866\pi$
$173$	12.1962	+	12.1962i	0.927256	+	0.927256i	−0.0702716	+	0.997528i	$0.522387\pi$
$174$		0			0
$175$	−24.3190			−1.83835
$176$		0			0
$177$	−13.8564			−1.04151
$178$		0			0
$179$	−13.3843	−	13.3843i	−1.00039	−	1.00039i	−1.00000		0.000386700i	$-0.999877\pi$
$179$	−13.3843	−	13.3843i	−1.00039	−	1.00039i	−0.000386700	−	1.00000i	$-0.500123\pi$
$180$		0			0
$181$	5.00000	−	5.00000i	0.371647	−	0.371647i	−0.496430	−	0.868077i	$-0.665356\pi$
$181$	5.00000	−	5.00000i	0.371647	−	0.371647i	0.868077	+	0.496430i	$0.165356\pi$
$182$		0			0
$183$			0.656339i			0.0485180i
$184$		0			0
$185$		−	2.53590i		−	0.186443i
$186$		0			0
$187$	3.58630	−	3.58630i	0.262256	−	0.262256i
$188$		0			0
$189$	−1.73205	−	1.73205i	−0.125988	−	0.125988i
$190$		0			0
$191$	−7.72741			−0.559136			−0.279568	−	0.960126i	$-0.590191\pi$
$191$	−7.72741			−0.559136			−0.279568	+	0.960126i	$0.590191\pi$
$192$		0			0
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$		0			0
$195$	−11.5911	−	11.5911i	−0.830057	−	0.830057i
$196$		0			0
$197$	6.19615	−	6.19615i	0.441458	−	0.441458i	−0.451044	−	0.892502i	$-0.648948\pi$
$197$	6.19615	−	6.19615i	0.441458	−	0.441458i	0.892502	+	0.451044i	$0.148948\pi$
$198$		0			0
$199$		−	18.6622i		−	1.32293i	−0.749977	−	0.661463i	$-0.769936\pi$
$199$		−	18.6622i		−	1.32293i	0.749977	−	0.661463i	$-0.230064\pi$
$200$		0			0
$201$			14.9282i			1.05295i
$202$		0			0
$203$	15.1774	−	15.1774i	1.06525	−	1.06525i
$204$		0			0
$205$	−12.3923	−	12.3923i	−0.865516	−	0.865516i
$206$		0			0
$207$	−2.82843			−0.196589
$208$		0			0
$209$	10.9282			0.755920
$210$		0			0
$211$	4.89898	+	4.89898i	0.337260	+	0.337260i	0.855335	−	0.518075i	$-0.173351\pi$
$211$	4.89898	+	4.89898i	0.337260	+	0.337260i	−0.518075	+	0.855335i	$0.673351\pi$
$212$		0			0
$213$	−4.53590	+	4.53590i	−0.310795	+	0.310795i
$214$		0			0
$215$			13.3843i			0.912799i
$216$		0			0
$217$		−	18.0000i		−	1.22192i
$218$		0			0
$219$	2.82843	−	2.82843i	0.191127	−	0.191127i
$220$		0			0
$221$	−10.3923	−	10.3923i	−0.699062	−	0.699062i
$222$		0			0
$223$	−14.5211			−0.972403			−0.486201	−	0.873847i	$-0.661618\pi$
$223$	−14.5211			−0.972403			−0.486201	+	0.873847i	$0.661618\pi$
$224$		0			0
$225$	9.92820			0.661880
$226$		0			0
$227$	−8.00481	−	8.00481i	−0.531298	−	0.531298i	0.389661	−	0.920958i	$-0.372592\pi$
$227$	−8.00481	−	8.00481i	−0.531298	−	0.531298i	−0.920958	+	0.389661i	$0.872592\pi$
$228$		0			0
$229$	11.9282	−	11.9282i	0.788238	−	0.788238i	−0.192967	−	0.981205i	$-0.561811\pi$
$229$	11.9282	−	11.9282i	0.788238	−	0.788238i	0.981205	+	0.192967i	$0.0618111\pi$
$230$		0			0
$231$			3.58630i			0.235961i
$232$		0			0
$233$			12.9282i			0.846955i	0.905907	+	0.423477i	$0.139191\pi$
$233$			12.9282i			0.846955i	−0.905907	+	0.423477i	$0.860809\pi$
$234$		0			0
$235$	−7.72741	+	7.72741i	−0.504080	+	0.504080i
$236$		0			0
$237$	−1.73205	−	1.73205i	−0.112509	−	0.112509i
$238$		0			0
$239$	−7.72741			−0.499844			−0.249922	−	0.968266i	$-0.580405\pi$
$239$	−7.72741			−0.499844			−0.249922	+	0.968266i	$0.580405\pi$
$240$		0			0
$241$	26.7846			1.72535			0.862674	−	0.505760i	$-0.168788\pi$
$241$	26.7846			1.72535			0.862674	+	0.505760i	$0.168788\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$	−2.73205	+	2.73205i	−0.174544	+	0.174544i
$246$		0			0
$247$		−	31.6675i		−	2.01495i
$248$		0			0
$249$		−	5.46410i		−	0.346273i
$250$		0			0
$251$	−4.62158	+	4.62158i	−0.291711	+	0.291711i	−0.837756	−	0.546045i	$-0.816133\pi$
$251$	−4.62158	+	4.62158i	−0.291711	+	0.291711i	0.546045	+	0.837756i	$0.316133\pi$
$252$		0			0
$253$	2.92820	+	2.92820i	0.184095	+	0.184095i
$254$		0			0
$255$	13.3843			0.838155
$256$		0			0
$257$	−22.7846			−1.42126			−0.710632	−	0.703563i	$-0.751591\pi$
$257$	−22.7846			−1.42126			−0.710632	+	0.703563i	$0.751591\pi$
$258$		0			0
$259$	1.13681	+	1.13681i	0.0706381	+	0.0706381i
$260$		0			0
$261$	−6.19615	+	6.19615i	−0.383532	+	0.383532i
$262$		0			0
$263$		−	22.6274i		−	1.39527i	−0.716455	−	0.697633i	$-0.754237\pi$
$263$		−	22.6274i		−	1.39527i	0.716455	−	0.697633i	$-0.245763\pi$
$264$		0			0
$265$		−	17.8564i		−	1.09691i
$266$		0			0
$267$	3.48477	−	3.48477i	0.213264	−	0.213264i
$268$		0			0
$269$	−7.26795	−	7.26795i	−0.443135	−	0.443135i	0.449929	−	0.893064i	$-0.351449\pi$
$269$	−7.26795	−	7.26795i	−0.443135	−	0.443135i	−0.893064	+	0.449929i	$0.851449\pi$
$270$		0			0
$271$	3.20736			0.194834			0.0974168	−	0.995244i	$-0.468942\pi$
$271$	3.20736			0.194834			0.0974168	+	0.995244i	$0.468942\pi$
$272$		0			0
$273$	10.3923			0.628971
$274$		0			0
$275$	−10.2784	−	10.2784i	−0.619813	−	0.619813i
$276$		0			0
$277$	8.85641	−	8.85641i	0.532130	−	0.532130i	−0.389076	−	0.921206i	$-0.627206\pi$
$277$	8.85641	−	8.85641i	0.532130	−	0.532130i	0.921206	+	0.389076i	$0.127206\pi$
$278$		0			0
$279$			7.34847i			0.439941i
$280$		0			0
$281$		−	0.928203i		−	0.0553720i	−0.999617	−	0.0276860i	$-0.991186\pi$
$281$		−	0.928203i		−	0.0553720i	0.999617	−	0.0276860i	$-0.00881385\pi$
$282$		0			0
$283$	6.96953	−	6.96953i	0.414296	−	0.414296i	−0.468936	−	0.883232i	$-0.655363\pi$
$283$	6.96953	−	6.96953i	0.414296	−	0.414296i	0.883232	+	0.468936i	$0.155363\pi$
$284$		0			0
$285$	20.3923	+	20.3923i	1.20794	+	1.20794i
$286$		0			0
$287$	11.1106			0.655840
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$	−0.757875	−	0.757875i	−0.0444274	−	0.0444274i
$292$		0			0
$293$	−7.66025	+	7.66025i	−0.447517	+	0.447517i	−0.894528	−	0.447011i	$-0.852488\pi$
$293$	−7.66025	+	7.66025i	−0.447517	+	0.447517i	0.447011	+	0.894528i	$0.352488\pi$
$294$		0			0
$295$			53.5370i			3.11705i
$296$		0			0
$297$		−	1.46410i		−	0.0849558i
$298$		0			0
$299$	8.48528	−	8.48528i	0.490716	−	0.490716i
$300$		0			0
$301$	−6.00000	−	6.00000i	−0.345834	−	0.345834i
$302$		0			0
$303$	−14.4195			−0.828381
$304$		0			0
$305$	2.53590			0.145205
$306$		0			0
$307$	16.2127	+	16.2127i	0.925307	+	0.925307i	0.997398	−	0.0720911i	$-0.0229672\pi$
$307$	16.2127	+	16.2127i	0.925307	+	0.925307i	−0.0720911	+	0.997398i	$0.522967\pi$
$308$		0			0
$309$	2.26795	−	2.26795i	0.129019	−	0.129019i
$310$		0			0
$311$			32.4254i			1.83867i	0.393471	+	0.919337i	$0.371274\pi$
$311$			32.4254i			1.83867i	−0.393471	+	0.919337i	$0.628726\pi$
$312$		0			0
$313$		−	3.85641i		−	0.217977i	−0.994043	−	0.108988i	$-0.965239\pi$
$313$		−	3.85641i		−	0.217977i	0.994043	−	0.108988i	$-0.0347612\pi$
$314$		0			0
$315$	−6.69213	+	6.69213i	−0.377059	+	0.377059i
$316$		0			0
$317$	4.73205	+	4.73205i	0.265778	+	0.265778i	0.827397	−	0.561618i	$-0.189821\pi$
$317$	4.73205	+	4.73205i	0.265778	+	0.265778i	−0.561618	+	0.827397i	$0.689821\pi$
$318$		0			0
$319$	12.8295			0.718312
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$	18.2832	+	18.2832i	1.01731	+	1.01731i
$324$		0			0
$325$	−29.7846	+	29.7846i	−1.65215	+	1.65215i
$326$		0			0
$327$			1.41421i			0.0782062i
$328$		0			0
$329$		−	6.92820i		−	0.381964i
$330$		0			0
$331$	−0.757875	+	0.757875i	−0.0416566	+	0.0416566i	−0.727628	−	0.685972i	$-0.759377\pi$
$331$	−0.757875	+	0.757875i	−0.0416566	+	0.0416566i	0.685972	+	0.727628i	$0.259377\pi$
$332$		0			0
$333$	−0.464102	−	0.464102i	−0.0254326	−	0.0254326i
$334$		0			0
$335$	57.6781			3.15129
$336$		0			0
$337$	−9.85641			−0.536913			−0.268456	−	0.963292i	$-0.586514\pi$
$337$	−9.85641			−0.536913			−0.268456	+	0.963292i	$0.586514\pi$
$338$		0			0
$339$	12.7279	+	12.7279i	0.691286	+	0.691286i
$340$		0			0
$341$	7.60770	−	7.60770i	0.411980	−	0.411980i
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$			10.9282i			0.588355i
$346$		0			0
$347$	1.79315	−	1.79315i	0.0962614	−	0.0962614i	−0.657336	−	0.753598i	$-0.728317\pi$
$347$	1.79315	−	1.79315i	0.0962614	−	0.0962614i	0.753598	+	0.657336i	$0.228317\pi$
$348$		0			0
$349$	19.3923	+	19.3923i	1.03805	+	1.03805i	0.999247	+	0.0387993i	$0.0123533\pi$
$349$	19.3923	+	19.3923i	1.03805	+	1.03805i	0.0387993	+	0.999247i	$0.487647\pi$
$350$		0			0
$351$	−4.24264			−0.226455
$352$		0			0
$353$	−0.928203			−0.0494033			−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203			−0.0494033			−0.0247016	+	0.999695i	$0.507864\pi$
$354$		0			0
$355$	17.5254	+	17.5254i	0.930150	+	0.930150i
$356$		0			0
$357$	−6.00000	+	6.00000i	−0.317554	+	0.317554i
$358$		0			0
$359$		−	18.8380i		−	0.994234i	−0.867684	−	0.497117i	$-0.834392\pi$
$359$		−	18.8380i		−	0.994234i	0.867684	−	0.497117i	$-0.165608\pi$
$360$		0			0
$361$			36.7128i			1.93225i
$362$		0			0
$363$	6.26243	−	6.26243i	0.328692	−	0.328692i
$364$		0			0
$365$	−10.9282	−	10.9282i	−0.572008	−	0.572008i
$366$		0			0
$367$	16.3886			0.855476			0.427738	−	0.903903i	$-0.359311\pi$
$367$	16.3886			0.855476			0.427738	+	0.903903i	$0.359311\pi$
$368$		0			0
$369$	−4.53590			−0.236129
$370$		0			0
$371$	8.00481	+	8.00481i	0.415589	+	0.415589i
$372$		0			0
$373$	−9.39230	+	9.39230i	−0.486315	+	0.486315i	−0.907141	−	0.420826i	$-0.861740\pi$
$373$	−9.39230	+	9.39230i	−0.486315	+	0.486315i	0.420826	+	0.907141i	$0.361740\pi$
$374$		0			0
$375$		−	19.0411i		−	0.983279i
$376$		0			0
$377$		−	37.1769i		−	1.91471i
$378$		0			0
$379$	9.41902	−	9.41902i	0.483823	−	0.483823i	−0.422527	−	0.906350i	$-0.638857\pi$
$379$	9.41902	−	9.41902i	0.483823	−	0.483823i	0.906350	+	0.422527i	$0.138857\pi$
$380$		0			0
$381$	−10.2679	−	10.2679i	−0.526043	−	0.526043i
$382$		0			0
$383$	28.8391			1.47361			0.736804	−	0.676106i	$-0.236334\pi$
$383$	28.8391			1.47361			0.736804	+	0.676106i	$0.236334\pi$
$384$		0			0
$385$	13.8564			0.706188
$386$		0			0
$387$	2.44949	+	2.44949i	0.124515	+	0.124515i
$388$		0			0
$389$	−6.73205	+	6.73205i	−0.341329	+	0.341329i	−0.856867	−	0.515538i	$-0.827592\pi$
$389$	−6.73205	+	6.73205i	−0.341329	+	0.341329i	0.515538	+	0.856867i	$0.327592\pi$
$390$		0			0
$391$			9.79796i			0.495504i
$392$		0			0
$393$		−	14.9282i		−	0.753028i
$394$		0			0
$395$	−6.69213	+	6.69213i	−0.336718	+	0.336718i
$396$		0			0
$397$	16.3205	+	16.3205i	0.819103	+	0.819103i	0.985978	−	0.166875i	$-0.0533677\pi$
$397$	16.3205	+	16.3205i	0.819103	+	0.819103i	−0.166875	+	0.985978i	$0.553368\pi$
$398$		0			0
$399$	−18.2832			−0.915307
$400$		0			0
$401$	18.3923			0.918468			0.459234	−	0.888315i	$-0.348124\pi$
$401$	18.3923			0.918468			0.459234	+	0.888315i	$0.348124\pi$
$402$		0			0
$403$	−22.0454	−	22.0454i	−1.09816	−	1.09816i
$404$		0			0
$405$	2.73205	−	2.73205i	0.135757	−	0.135757i
$406$		0			0
$407$			0.960947i			0.0476324i
$408$		0			0
$409$			9.07180i			0.448571i	0.974523	+	0.224286i	$0.0720049\pi$
$409$			9.07180i			0.448571i	−0.974523	+	0.224286i	$0.927995\pi$
$410$		0			0
$411$	10.1769	−	10.1769i	0.501989	−	0.501989i
$412$		0			0
$413$	−24.0000	−	24.0000i	−1.18096	−	1.18096i
$414$		0			0
$415$	−21.1117			−1.03633
$416$		0			0
$417$	6.92820			0.339276
$418$		0			0
$419$	−0.277401	−	0.277401i	−0.0135519	−	0.0135519i	0.700298	−	0.713850i	$-0.253050\pi$
$419$	−0.277401	−	0.277401i	−0.0135519	−	0.0135519i	−0.713850	+	0.700298i	$0.753050\pi$
$420$		0			0
$421$	−13.9282	+	13.9282i	−0.678819	+	0.678819i	−0.959733	−	0.280914i	$-0.909362\pi$
$421$	−13.9282	+	13.9282i	−0.678819	+	0.678819i	0.280914	+	0.959733i	$0.409362\pi$
$422$		0			0
$423$			2.82843i			0.137523i
$424$		0			0
$425$		−	34.3923i		−	1.66827i
$426$		0			0
$427$	−1.13681	+	1.13681i	−0.0550142	+	0.0550142i
$428$		0			0
$429$	4.39230	+	4.39230i	0.212062	+	0.212062i
$430$		0			0
$431$	−4.34418			−0.209252			−0.104626	−	0.994512i	$-0.533364\pi$
$431$	−4.34418			−0.209252			−0.104626	+	0.994512i	$0.533364\pi$
$432$		0			0
$433$	29.8564			1.43481			0.717404	−	0.696658i	$-0.245330\pi$
$433$	29.8564			1.43481			0.717404	+	0.696658i	$0.245330\pi$
$434$		0			0
$435$	23.9401	+	23.9401i	1.14784	+	1.14784i
$436$		0			0
$437$	−14.9282	+	14.9282i	−0.714113	+	0.714113i
$438$		0			0
$439$			5.83272i			0.278381i	0.990266	+	0.139190i	$0.0444500\pi$
$439$			5.83272i			0.278381i	−0.990266	+	0.139190i	$0.955550\pi$
$440$		0			0
$441$			1.00000i			0.0476190i
$442$		0			0
$443$	−24.9754	+	24.9754i	−1.18662	+	1.18662i	−0.208619	+	0.977997i	$0.566897\pi$
$443$	−24.9754	+	24.9754i	−1.18662	+	1.18662i	−0.977997	+	0.208619i	$0.933103\pi$
$444$		0			0
$445$	−13.4641	−	13.4641i	−0.638260	−	0.638260i
$446$		0			0
$447$	5.37945			0.254439
$448$		0			0
$449$	−6.39230			−0.301672			−0.150836	−	0.988559i	$-0.548196\pi$
$449$	−6.39230			−0.301672			−0.150836	+	0.988559i	$0.548196\pi$
$450$		0			0
$451$	4.69591	+	4.69591i	0.221122	+	0.221122i
$452$		0			0
$453$	4.66025	−	4.66025i	0.218958	−	0.218958i
$454$		0			0
$455$		−	40.1528i		−	1.88239i
$456$		0			0
$457$		−	20.7846i		−	0.972263i	−0.873886	−	0.486132i	$-0.838408\pi$
$457$		−	20.7846i		−	0.972263i	0.873886	−	0.486132i	$-0.161592\pi$
$458$		0			0
$459$	2.44949	−	2.44949i	0.114332	−	0.114332i
$460$		0			0
$461$	−22.0526	−	22.0526i	−1.02709	−	1.02709i	−0.999623	−	0.0274674i	$-0.991256\pi$
$461$	−22.0526	−	22.0526i	−1.02709	−	1.02709i	−0.0274674	−	0.999623i	$-0.508744\pi$
$462$		0			0
$463$	29.2180			1.35788			0.678938	−	0.734196i	$-0.262440\pi$
$463$	29.2180			1.35788			0.678938	+	0.734196i	$0.262440\pi$
$464$		0			0
$465$	28.3923			1.31666
$466$		0			0
$467$	−21.3891	−	21.3891i	−0.989768	−	0.989768i	0.0101801	−	0.999948i	$-0.496760\pi$
$467$	−21.3891	−	21.3891i	−0.989768	−	0.989768i	−0.999948	+	0.0101801i	$0.996760\pi$
$468$		0			0
$469$	−25.8564	+	25.8564i	−1.19394	+	1.19394i
$470$		0			0
$471$		−	2.17209i		−	0.100085i
$472$		0			0
$473$		−	5.07180i		−	0.233201i
$474$		0			0
$475$	52.4002	−	52.4002i	2.40429	−	2.40429i
$476$		0			0
$477$	−3.26795	−	3.26795i	−0.149629	−	0.149629i
$478$		0			0
$479$	−29.0421			−1.32697			−0.663485	−	0.748190i	$-0.730923\pi$
$479$	−29.0421			−1.32697			−0.663485	+	0.748190i	$0.730923\pi$
$480$		0			0
$481$	2.78461			0.126967
$482$		0			0
$483$	−4.89898	−	4.89898i	−0.222911	−	0.222911i
$484$		0			0
$485$	−2.92820	+	2.92820i	−0.132963	+	0.132963i
$486$		0			0
$487$		−	11.4896i		−	0.520642i	−0.965522	−	0.260321i	$-0.916172\pi$
$487$		−	11.4896i		−	0.520642i	0.965522	−	0.260321i	$-0.0838284\pi$
$488$		0			0
$489$			4.53590i			0.205120i
$490$		0			0
$491$	−4.14110	+	4.14110i	−0.186885	+	0.186885i	−0.794348	−	0.607463i	$-0.792187\pi$
$491$	−4.14110	+	4.14110i	−0.186885	+	0.186885i	0.607463	+	0.794348i	$0.292187\pi$
$492$		0			0
$493$	21.4641	+	21.4641i	0.966695	+	0.966695i
$494$		0			0
$495$	−5.65685			−0.254257
$496$		0			0
$497$	−15.7128			−0.704816
$498$		0			0
$499$	2.82843	+	2.82843i	0.126618	+	0.126618i	0.767576	−	0.640958i	$-0.221463\pi$
$499$	2.82843	+	2.82843i	0.126618	+	0.126618i	−0.640958	+	0.767576i	$0.721463\pi$
$500$		0			0
$501$	5.46410	−	5.46410i	0.244118	−	0.244118i
$502$		0			0
$503$			23.3853i			1.04270i	0.853343	+	0.521349i	$0.174571\pi$
$503$			23.3853i			1.04270i	−0.853343	+	0.521349i	$0.825429\pi$
$504$		0			0
$505$			55.7128i			2.47919i
$506$		0			0
$507$	3.53553	−	3.53553i	0.157019	−	0.157019i
$508$		0			0
$509$	−5.12436	−	5.12436i	−0.227133	−	0.227133i	0.584361	−	0.811494i	$-0.301345\pi$
$509$	−5.12436	−	5.12436i	−0.227133	−	0.227133i	−0.811494	+	0.584361i	$0.801345\pi$
$510$		0			0
$511$	9.79796			0.433436
$512$		0			0
$513$	7.46410			0.329548
$514$		0			0
$515$	−8.76268	−	8.76268i	−0.386130	−	0.386130i
$516$		0			0
$517$	2.92820	−	2.92820i	0.128782	−	0.128782i
$518$		0			0
$519$		−	17.2480i		−	0.757102i
$520$		0			0
$521$		−	21.3205i		−	0.934068i	−0.884239	−	0.467034i	$-0.845322\pi$
$521$		−	21.3205i		−	0.934068i	0.884239	−	0.467034i	$-0.154678\pi$
$522$		0			0
$523$	−22.2485	+	22.2485i	−0.972858	+	0.972858i	−0.999641	−	0.0267832i	$-0.991474\pi$
$523$	−22.2485	+	22.2485i	−0.972858	+	0.972858i	0.0267832	+	0.999641i	$0.491474\pi$
$524$		0			0
$525$	17.1962	+	17.1962i	0.750502	+	0.750502i
$526$		0			0
$527$	25.4558			1.10887
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$	9.79796	+	9.79796i	0.425195	+	0.425195i
$532$		0			0
$533$	13.6077	−	13.6077i	0.589415	−	0.589415i
$534$		0			0
$535$			15.4548i			0.668170i
$536$		0			0
$537$			18.9282i			0.816812i
$538$		0			0
$539$	1.03528	−	1.03528i	0.0445925	−	0.0445925i
$540$		0			0
$541$	−19.9282	−	19.9282i	−0.856780	−	0.856780i	0.134177	−	0.990957i	$-0.457161\pi$
$541$	−19.9282	−	19.9282i	−0.856780	−	0.856780i	−0.990957	+	0.134177i	$0.957161\pi$
$542$		0			0
$543$	−7.07107			−0.303449
$544$		0			0
$545$	5.46410			0.234056
$546$		0			0
$547$	−16.3886	−	16.3886i	−0.700724	−	0.700724i	0.263842	−	0.964566i	$-0.415010\pi$
$547$	−16.3886	−	16.3886i	−0.700724	−	0.700724i	−0.964566	+	0.263842i	$0.915010\pi$
$548$		0			0
$549$	0.464102	−	0.464102i	0.0198074	−	0.0198074i
$550$		0			0
$551$			65.4056i			2.78637i
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$		0			0
$555$	−1.79315	+	1.79315i	−0.0761150	+	0.0761150i
$556$		0			0
$557$	19.8038	+	19.8038i	0.839116	+	0.839116i	0.988743	−	0.149627i	$-0.0478072\pi$
$557$	19.8038	+	19.8038i	0.839116	+	0.839116i	−0.149627	+	0.988743i	$0.547807\pi$
$558$		0			0
$559$	−14.6969			−0.621614
$560$		0			0
$561$	−5.07180			−0.214131
$562$		0			0
$563$	1.03528	+	1.03528i	0.0436317	+	0.0436317i	0.728586	−	0.684954i	$-0.240178\pi$
$563$	1.03528	+	1.03528i	0.0436317	+	0.0436317i	−0.684954	+	0.728586i	$0.740178\pi$
$564$		0			0
$565$	49.1769	−	49.1769i	2.06889	−	2.06889i
$566$		0			0
$567$			2.44949i			0.102869i
$568$		0			0
$569$			3.46410i			0.145223i	0.997360	+	0.0726113i	$0.0231333\pi$
$569$			3.46410i			0.145223i	−0.997360	+	0.0726113i	$0.976867\pi$
$570$		0			0
$571$	−22.4243	+	22.4243i	−0.938430	+	0.938430i	−0.998211	−	0.0597818i	$-0.980959\pi$
$571$	−22.4243	+	22.4243i	−0.938430	+	0.938430i	0.0597818	+	0.998211i	$0.480959\pi$
$572$		0			0
$573$	5.46410	+	5.46410i	0.228266	+	0.228266i
$574$		0			0
$575$	28.0812			1.17107
$576$		0			0
$577$	−21.8564			−0.909894			−0.454947	−	0.890518i	$-0.650342\pi$
$577$	−21.8564			−0.909894			−0.454947	+	0.890518i	$0.650342\pi$
$578$		0			0
$579$	−4.24264	−	4.24264i	−0.176318	−	0.176318i
$580$		0			0
$581$	9.46410	−	9.46410i	0.392637	−	0.392637i
$582$		0			0
$583$			6.76646i			0.280238i
$584$		0			0
$585$			16.3923i			0.677738i
$586$		0			0
$587$	30.3548	−	30.3548i	1.25288	−	1.25288i	0.298453	−	0.954424i	$-0.403529\pi$
$587$	30.3548	−	30.3548i	1.25288	−	1.25288i	0.954424	−	0.298453i	$-0.0964708\pi$
$588$		0			0
$589$	38.7846	+	38.7846i	1.59809	+	1.59809i
$590$		0			0
$591$	−8.76268			−0.360449
$592$		0			0
$593$	−37.7128			−1.54868			−0.774340	−	0.632770i	$-0.781918\pi$
$593$	−37.7128			−1.54868			−0.774340	+	0.632770i	$0.781918\pi$
$594$		0			0
$595$	23.1822	+	23.1822i	0.950378	+	0.950378i
$596$		0			0
$597$	−13.1962	+	13.1962i	−0.540083	+	0.540083i
$598$		0			0
$599$			6.96953i			0.284767i	0.989812	+	0.142384i	$0.0454767\pi$
$599$			6.96953i			0.284767i	−0.989812	+	0.142384i	$0.954523\pi$
$600$		0			0
$601$		−	19.7128i		−	0.804102i	−0.915617	−	0.402051i	$-0.868297\pi$
$601$		−	19.7128i		−	0.804102i	0.915617	−	0.402051i	$-0.131703\pi$
$602$		0			0
$603$	10.5558	−	10.5558i	0.429867	−	0.429867i
$604$		0			0
$605$	−24.1962	−	24.1962i	−0.983713	−	0.983713i
$606$		0			0
$607$	−31.8434			−1.29248			−0.646241	−	0.763133i	$-0.723660\pi$
$607$	−31.8434			−1.29248			−0.646241	+	0.763133i	$0.723660\pi$
$608$		0			0
$609$	−21.4641			−0.869769
$610$		0			0
$611$	−8.48528	−	8.48528i	−0.343278	−	0.343278i
$612$		0			0
$613$	4.32051	−	4.32051i	0.174504	−	0.174504i	−0.614451	−	0.788955i	$-0.710623\pi$
$613$	4.32051	−	4.32051i	0.174504	−	0.174504i	0.788955	+	0.614451i	$0.210623\pi$
$614$		0			0
$615$			17.5254i			0.706691i
$616$		0			0
$617$		−	39.8564i		−	1.60456i	−0.596949	−	0.802279i	$-0.703621\pi$
$617$		−	39.8564i		−	1.60456i	0.596949	−	0.802279i	$-0.296379\pi$
$618$		0			0
$619$	8.48528	−	8.48528i	0.341052	−	0.341052i	−0.515711	−	0.856763i	$-0.672472\pi$
$619$	8.48528	−	8.48528i	0.341052	−	0.341052i	0.856763	+	0.515711i	$0.172472\pi$
$620$		0			0
$621$	2.00000	+	2.00000i	0.0802572	+	0.0802572i
$622$		0			0
$623$	12.0716			0.483638
$624$		0			0
$625$	−23.9282			−0.957128
$626$		0			0
$627$	−7.72741	−	7.72741i	−0.308603	−	0.308603i
$628$		0			0
$629$	−1.60770	+	1.60770i	−0.0641030	+	0.0641030i
$630$		0			0
$631$		−	23.5612i		−	0.937955i	−0.883210	−	0.468977i	$-0.844623\pi$
$631$		−	23.5612i		−	0.937955i	0.883210	−	0.468977i	$-0.155377\pi$
$632$		0			0
$633$		−	6.92820i		−	0.275371i
$634$		0			0
$635$	−39.6723	+	39.6723i	−1.57435	+	1.57435i
$636$		0			0
$637$	−3.00000	−	3.00000i	−0.118864	−	0.118864i
$638$		0			0
$639$	6.41473			0.253763
$640$		0			0
$641$	−25.6077			−1.01144			−0.505722	−	0.862697i	$-0.668774\pi$
$641$	−25.6077			−1.01144			−0.505722	+	0.862697i	$0.668774\pi$
$642$		0			0
$643$	20.1779	+	20.1779i	0.795740	+	0.795740i	0.982421	−	0.186681i	$-0.0597730\pi$
$643$	20.1779	+	20.1779i	0.795740	+	0.795740i	−0.186681	+	0.982421i	$0.559773\pi$
$644$		0			0
$645$	9.46410	−	9.46410i	0.372649	−	0.372649i
$646$		0			0
$647$		−	40.9107i		−	1.60836i	−0.594383	−	0.804182i	$-0.702604\pi$
$647$		−	40.9107i		−	1.60836i	0.594383	−	0.804182i	$-0.297396\pi$
$648$		0			0
$649$		−	20.2872i		−	0.796342i
$650$		0			0
$651$	−12.7279	+	12.7279i	−0.498847	+	0.498847i
$652$		0			0
$653$	17.6603	+	17.6603i	0.691099	+	0.691099i	0.962474	−	0.271375i	$-0.0874782\pi$
$653$	17.6603	+	17.6603i	0.691099	+	0.691099i	−0.271375	+	0.962474i	$0.587478\pi$
$654$		0			0
$655$	−57.6781			−2.25367
$656$		0			0
$657$	−4.00000			−0.156055
$658$		0			0
$659$	−16.9706	−	16.9706i	−0.661079	−	0.661079i	0.294555	−	0.955634i	$-0.404829\pi$
$659$	−16.9706	−	16.9706i	−0.661079	−	0.661079i	−0.955634	+	0.294555i	$0.904829\pi$
$660$		0			0
$661$	4.60770	−	4.60770i	0.179219	−	0.179219i	−0.611797	−	0.791015i	$-0.709553\pi$
$661$	4.60770	−	4.60770i	0.179219	−	0.179219i	0.791015	+	0.611797i	$0.209553\pi$
$662$		0			0
$663$			14.6969i			0.570782i
$664$		0			0
$665$			70.6410i			2.73934i
$666$		0			0
$667$	−17.5254	+	17.5254i	−0.678585	+	0.678585i
$668$		0			0
$669$	10.2679	+	10.2679i	0.396982	+	0.396982i
$670$		0			0
$671$	−0.960947			−0.0370969
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$	−7.02030	−	7.02030i	−0.270211	−	0.270211i
$676$		0			0
$677$	−2.33975	+	2.33975i	−0.0899237	+	0.0899237i	−0.750638	−	0.660714i	$-0.770254\pi$
$677$	−2.33975	+	2.33975i	−0.0899237	+	0.0899237i	0.660714	+	0.750638i	$0.270254\pi$
$678$		0			0
$679$		−	2.62536i		−	0.100752i
$680$		0			0
$681$			11.3205i			0.433803i
$682$		0			0
$683$	−17.8028	+	17.8028i	−0.681204	+	0.681204i	−0.960271	−	0.279068i	$-0.909975\pi$
$683$	−17.8028	+	17.8028i	−0.681204	+	0.681204i	0.279068	+	0.960271i	$0.409975\pi$
$684$		0			0
$685$	−39.3205	−	39.3205i	−1.50236	−	1.50236i
$686$		0			0
$687$	−16.8690			−0.643594
$688$		0			0
$689$	19.6077			0.746994
$690$		0			0
$691$	18.6622	+	18.6622i	0.709943	+	0.709943i	0.966523	−	0.256580i	$-0.0825958\pi$
$691$	18.6622	+	18.6622i	0.709943	+	0.709943i	−0.256580	+	0.966523i	$0.582596\pi$
$692$		0			0
$693$	2.53590	−	2.53590i	0.0963308	−	0.0963308i
$694$		0			0
$695$		−	26.7685i		−	1.01539i
$696$		0			0
$697$			15.7128i			0.595165i
$698$		0			0
$699$	9.14162	−	9.14162i	0.345768	−	0.345768i
$700$		0			0
$701$	−10.5885	−	10.5885i	−0.399921	−	0.399921i	0.478284	−	0.878205i	$-0.341259\pi$
$701$	−10.5885	−	10.5885i	−0.399921	−	0.399921i	−0.878205	+	0.478284i	$0.841259\pi$
$702$		0			0
$703$	−4.89898			−0.184769
$704$		0			0
$705$	10.9282			0.411580
$706$		0			0
$707$	−24.9754	−	24.9754i	−0.939295	−	0.939295i
$708$		0			0
$709$	34.7128	−	34.7128i	1.30367	−	1.30367i	0.377766	−	0.925901i	$-0.376692\pi$
$709$	34.7128	−	34.7128i	1.30367	−	1.30367i	0.925901	−	0.377766i	$-0.123308\pi$
$710$		0			0
$711$			2.44949i			0.0918630i
$712$		0			0
$713$			20.7846i			0.778390i
$714$		0			0
$715$	16.9706	−	16.9706i	0.634663	−	0.634663i
$716$		0			0
$717$	5.46410	+	5.46410i	0.204061	+	0.204061i
$718$		0			0
$719$	2.27362			0.0847919			0.0423959	−	0.999101i	$-0.486501\pi$
$719$	2.27362			0.0847919			0.0423959	+	0.999101i	$0.486501\pi$
$720$		0			0
$721$	7.85641			0.292588
$722$		0			0
$723$	−18.9396	−	18.9396i	−0.704371	−	0.704371i
$724$		0			0
$725$	61.5167	−	61.5167i	2.28467	−	2.28467i
$726$		0			0
$727$		−	8.10634i		−	0.300648i	−0.988637	−	0.150324i	$-0.951968\pi$
$727$		−	8.10634i		−	0.300648i	0.988637	−	0.150324i	$-0.0480316\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$	8.48528	−	8.48528i	0.313839	−	0.313839i
$732$		0			0
$733$	5.00000	+	5.00000i	0.184679	+	0.184679i	0.793391	−	0.608712i	$-0.208314\pi$
$733$	5.00000	+	5.00000i	0.184679	+	0.184679i	−0.608712	+	0.793391i	$0.708314\pi$
$734$		0			0
$735$	3.86370			0.142515
$736$		0			0
$737$	−21.8564			−0.805091
$738$		0			0
$739$	−25.4558	−	25.4558i	−0.936408	−	0.936408i	0.0616872	−	0.998096i	$-0.480352\pi$
$739$	−25.4558	−	25.4558i	−0.936408	−	0.936408i	−0.998096	+	0.0616872i	$0.980352\pi$
$740$		0			0
$741$	−22.3923	+	22.3923i	−0.822602	+	0.822602i
$742$		0			0
$743$		−	27.3233i		−	1.00240i	−0.865333	−	0.501198i	$-0.832893\pi$
$743$		−	27.3233i		−	1.00240i	0.865333	−	0.501198i	$-0.167107\pi$
$744$		0			0
$745$		−	20.7846i		−	0.761489i
$746$		0			0
$747$	−3.86370	+	3.86370i	−0.141366	+	0.141366i
$748$		0			0
$749$	−6.92820	−	6.92820i	−0.253151	−	0.253151i
$750$		0			0
$751$	−46.1886			−1.68545			−0.842723	−	0.538348i	$-0.819049\pi$
$751$	−46.1886			−1.68545			−0.842723	+	0.538348i	$0.819049\pi$
$752$		0			0
$753$	6.53590			0.238181
$754$		0			0
$755$	−18.0058	−	18.0058i	−0.655300	−	0.655300i
$756$		0			0
$757$	−23.7846	+	23.7846i	−0.864466	+	0.864466i	−0.991853	−	0.127387i	$-0.959341\pi$
$757$	−23.7846	+	23.7846i	−0.864466	+	0.864466i	0.127387	+	0.991853i	$0.459341\pi$
$758$		0			0
$759$		−	4.14110i		−	0.150313i
$760$		0			0
$761$			52.2487i			1.89401i	0.321212	+	0.947007i	$0.395910\pi$
$761$			52.2487i			1.89401i	−0.321212	+	0.947007i	$0.604090\pi$
$762$		0			0
$763$	−2.44949	+	2.44949i	−0.0886775	+	0.0886775i
$764$		0			0
$765$	−9.46410	−	9.46410i	−0.342175	−	0.342175i
$766$		0			0
$767$	−58.7878			−2.12270
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	16.1112	+	16.1112i	0.580229	+	0.580229i
$772$		0			0
$773$	−34.5885	+	34.5885i	−1.24406	+	1.24406i	−0.285759	+	0.958302i	$0.592246\pi$
$773$	−34.5885	+	34.5885i	−1.24406	+	1.24406i	−0.958302	+	0.285759i	$0.907754\pi$
$774$		0			0
$775$		−	72.9571i		−	2.62070i
$776$		0			0
$777$		−	1.60770i		−	0.0576757i
$778$		0			0
$779$	−23.9401	+	23.9401i	−0.857743	+	0.857743i
$780$		0			0
$781$	−6.64102	−	6.64102i	−0.237634	−	0.237634i
$782$		0			0
$783$	8.76268			0.313153
$784$		0			0
$785$	−8.39230			−0.299534
$786$		0			0
$787$	−16.5916	−	16.5916i	−0.591428	−	0.591428i	0.346589	−	0.938017i	$-0.387340\pi$
$787$	−16.5916	−	16.5916i	−0.591428	−	0.591428i	−0.938017	+	0.346589i	$0.887340\pi$
$788$		0			0
$789$	−16.0000	+	16.0000i	−0.569615	+	0.569615i
$790$		0			0
$791$			44.0908i			1.56769i
$792$		0			0
$793$			2.78461i			0.0988844i
$794$		0			0
$795$	−12.6264	+	12.6264i	−0.447812	+	0.447812i
$796$		0			0
$797$	15.5167	+	15.5167i	0.549628	+	0.549628i	0.926333	−	0.376705i	$-0.122943\pi$
$797$	15.5167	+	15.5167i	0.549628	+	0.549628i	−0.376705	+	0.926333i	$0.622943\pi$
$798$		0			0
$799$	9.79796			0.346627
$800$		0			0
$801$	−4.92820			−0.174129
$802$		0			0
$803$	4.14110	+	4.14110i	0.146136	+	0.146136i
$804$		0			0
$805$	−18.9282	+	18.9282i	−0.667132	+	0.667132i
$806$		0			0
$807$			10.2784i			0.361818i
$808$		0			0
$809$		−	6.67949i		−	0.234838i	−0.993082	−	0.117419i	$-0.962538\pi$
$809$		−	6.67949i		−	0.234838i	0.993082	−	0.117419i	$-0.0374621\pi$
$810$		0			0
$811$	−38.4612	+	38.4612i	−1.35055	+	1.35055i	−0.465513	+	0.885041i	$0.654130\pi$
$811$	−38.4612	+	38.4612i	−1.35055	+	1.35055i	−0.885041	+	0.465513i	$0.845870\pi$
$812$		0			0
$813$	−2.26795	−	2.26795i	−0.0795405	−	0.0795405i
$814$		0			0
$815$	17.5254			0.613887
$816$		0			0
$817$	25.8564			0.904601
$818$		0			0
$819$	−7.34847	−	7.34847i	−0.256776	−	0.256776i
$820$		0			0
$821$	−17.8038	+	17.8038i	−0.621359	+	0.621359i	−0.945879	−	0.324520i	$-0.894797\pi$
$821$	−17.8038	+	17.8038i	−0.621359	+	0.621359i	0.324520	+	0.945879i	$0.394797\pi$
$822$		0			0
$823$			9.97382i			0.347666i	0.984775	+	0.173833i	$0.0556152\pi$
$823$			9.97382i			0.347666i	−0.984775	+	0.173833i	$0.944385\pi$
$824$		0			0
$825$			14.5359i			0.506075i
$826$		0			0
$827$	31.6675	−	31.6675i	1.10119	−	1.10119i	0.106919	−	0.994268i	$-0.465901\pi$
$827$	31.6675	−	31.6675i	1.10119	−	1.10119i	0.994268	−	0.106919i	$-0.0340986\pi$
$828$		0			0
$829$	−1.14359	−	1.14359i	−0.0397186	−	0.0397186i	0.686969	−	0.726687i	$-0.258941\pi$
$829$	−1.14359	−	1.14359i	−0.0397186	−	0.0397186i	−0.726687	+	0.686969i	$0.758941\pi$
$830$		0			0
$831$	−12.5249			−0.434482
$832$		0			0
$833$	3.46410			0.120024
$834$		0			0
$835$	−21.1117	−	21.1117i	−0.730599	−	0.730599i
$836$		0			0
$837$	5.19615	−	5.19615i	0.179605	−	0.179605i
$838$		0			0
$839$			26.0106i			0.897987i	0.893535	+	0.448994i	$0.148217\pi$
$839$			26.0106i			0.897987i	−0.893535	+	0.448994i	$0.851783\pi$
$840$		0			0
$841$			47.7846i			1.64775i
$842$		0			0
$843$	−0.656339	+	0.656339i	−0.0226055	+	0.0226055i
$844$		0			0
$845$	−13.6603	−	13.6603i	−0.469927	−	0.469927i
$846$		0			0
$847$	21.6937			0.745404
$848$		0			0
$849$	−9.85641			−0.338271
$850$		0			0
$851$	−1.31268	−	1.31268i	−0.0449980	−	0.0449980i
$852$		0			0
$853$	33.2487	−	33.2487i	1.13841	−	1.13841i	0.149680	−	0.988735i	$-0.452176\pi$
$853$	33.2487	−	33.2487i	1.13841	−	1.13841i	0.988735	−	0.149680i	$-0.0478242\pi$
$854$		0			0
$855$		−	28.8391i		−	0.986276i
$856$		0			0
$857$		−	43.1769i		−	1.47490i	−0.675404	−	0.737448i	$-0.736031\pi$
$857$		−	43.1769i		−	1.47490i	0.675404	−	0.737448i	$-0.263969\pi$
$858$		0			0
$859$	−22.8033	+	22.8033i	−0.778038	+	0.778038i	−0.979497	−	0.201459i	$-0.935432\pi$
$859$	−22.8033	+	22.8033i	−0.778038	+	0.778038i	0.201459	+	0.979497i	$0.435432\pi$
$860$		0			0
$861$	−7.85641	−	7.85641i	−0.267746	−	0.267746i
$862$		0			0
$863$	−37.1213			−1.26362			−0.631812	−	0.775122i	$-0.717688\pi$
$863$	−37.1213			−1.26362			−0.631812	+	0.775122i	$0.717688\pi$
$864$		0			0
$865$	−66.6410			−2.26586
$866$		0			0
$867$	3.53553	+	3.53553i	0.120073	+	0.120073i
$868$		0			0
$869$	2.53590	−	2.53590i	0.0860245	−	0.0860245i
$870$		0			0
$871$			63.3350i			2.14602i
$872$		0			0
$873$			1.07180i			0.0362748i
$874$		0			0
$875$	32.9802	−	32.9802i	1.11493	−	1.11493i
$876$		0			0
$877$	25.2487	+	25.2487i	0.852588	+	0.852588i	0.990451	−	0.137863i	$-0.0440233\pi$
$877$	25.2487	+	25.2487i	0.852588	+	0.852588i	−0.137863	+	0.990451i	$0.544023\pi$
$878$		0			0
$879$	10.8332			0.365396
$880$		0			0
$881$	36.6410			1.23447			0.617234	−	0.786780i	$-0.288253\pi$
$881$	36.6410			1.23447			0.617234	+	0.786780i	$0.288253\pi$
$882$		0			0
$883$	−3.96524	−	3.96524i	−0.133441	−	0.133441i	0.637232	−	0.770672i	$-0.280080\pi$
$883$	−3.96524	−	3.96524i	−0.133441	−	0.133441i	−0.770672	+	0.637232i	$0.780080\pi$
$884$		0			0
$885$	37.8564	−	37.8564i	1.27253	−	1.27253i
$886$		0			0
$887$		−	36.5665i		−	1.22778i	−0.789391	−	0.613891i	$-0.789603\pi$
$887$		−	36.5665i		−	1.22778i	0.789391	−	0.613891i	$-0.210397\pi$
$888$		0			0
$889$		−	35.5692i		−	1.19295i
$890$		0			0
$891$	−1.03528	+	1.03528i	−0.0346831	+	0.0346831i
$892$		0			0
$893$	14.9282	+	14.9282i	0.499553	+	0.499553i
$894$		0			0
$895$	73.1330			2.44457
$896$		0			0
$897$	−12.0000			−0.400668
$898$		0			0
$899$	45.5322	+	45.5322i	1.51859	+	1.51859i
$900$		0			0
$901$	−11.3205	+	11.3205i	−0.377141	+	0.377141i
$902$		0			0
$903$			8.48528i			0.282372i
$904$		0			0
$905$			27.3205i			0.908164i
$906$		0			0
$907$	29.9759	−	29.9759i	0.995333	−	0.995333i	−0.00465619	−	0.999989i	$-0.501482\pi$
$907$	29.9759	−	29.9759i	0.995333	−	0.995333i	0.999989	+	0.00465619i	$0.00148212\pi$
$908$		0			0
$909$	10.1962	+	10.1962i	0.338185	+	0.338185i
$910$		0			0
$911$	−22.0726			−0.731298			−0.365649	−	0.930753i	$-0.619153\pi$
$911$	−22.0726			−0.731298			−0.365649	+	0.930753i	$0.619153\pi$
$912$		0			0
$913$	8.00000			0.264761
$914$		0			0
$915$	−1.79315	−	1.79315i	−0.0592797	−	0.0592797i
$916$		0			0
$917$	25.8564	−	25.8564i	0.853854	−	0.853854i
$918$		0			0
$919$			0.582009i			0.0191987i	0.999954	+	0.00959936i	$0.00305562\pi$
$919$			0.582009i			0.0191987i	−0.999954	+	0.00959936i	$0.996944\pi$
$920$		0			0
$921$		−	22.9282i		−	0.755510i
$922$		0			0
$923$	−19.2442	+	19.2442i	−0.633430	+	0.633430i
$924$		0			0
$925$	4.60770	+	4.60770i	0.151500	+	0.151500i
$926$		0			0
$927$	−3.20736			−0.105344
$928$		0			0
$929$	−45.3205			−1.48692			−0.743459	−	0.668782i	$-0.766816\pi$
$929$	−45.3205			−1.48692			−0.743459	+	0.668782i	$0.766816\pi$
$930$		0			0
$931$	5.27792	+	5.27792i	0.172977	+	0.172977i
$932$		0			0
$933$	22.9282	−	22.9282i	0.750636	−	0.750636i
$934$		0			0
$935$			19.5959i			0.640855i
$936$		0			0
$937$			2.00000i			0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$			2.00000i			0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$		0			0
$939$	−2.72689	+	2.72689i	−0.0889887	+	0.0889887i
$940$		0			0
$941$	25.6603	+	25.6603i	0.836500	+	0.836500i	0.988396	−	0.151896i	$-0.0485379\pi$
$941$	25.6603	+	25.6603i	0.836500	+	0.836500i	−0.151896	+	0.988396i	$0.548538\pi$
$942$		0			0
$943$	−12.8295			−0.417785
$944$		0			0
$945$	9.46410			0.307867
$946$		0			0
$947$	20.5569	+	20.5569i	0.668008	+	0.668008i	0.957255	−	0.289246i	$-0.0934047\pi$
$947$	20.5569	+	20.5569i	0.668008	+	0.668008i	−0.289246	+	0.957255i	$0.593405\pi$
$948$		0			0
$949$	12.0000	−	12.0000i	0.389536	−	0.389536i
$950$		0			0
$951$		−	6.69213i		−	0.217007i
$952$		0			0
$953$		−	46.3923i		−	1.50279i	−0.659850	−	0.751397i	$-0.729380\pi$
$953$		−	46.3923i		−	1.50279i	0.659850	−	0.751397i	$-0.270620\pi$
$954$		0			0
$955$	21.1117	−	21.1117i	0.683158	−	0.683158i
$956$		0			0
$957$	−9.07180	−	9.07180i	−0.293250	−	0.293250i
$958$		0			0
$959$	35.2538			1.13840
$960$		0			0
$961$	23.0000			0.741935
$962$		0			0
$963$	2.82843	+	2.82843i	0.0911448	+	0.0911448i
$964$		0			0
$965$	−16.3923	+	16.3923i	−0.527687	+	0.527687i
$966$		0			0
$967$			10.3800i			0.333797i	0.985974	+	0.166899i	$0.0533753\pi$
$967$			10.3800i			0.333797i	−0.985974	+	0.166899i	$0.946625\pi$
$968$		0			0
$969$		−	25.8564i		−	0.830627i
$970$		0			0
$971$	−15.7322	+	15.7322i	−0.504871	+	0.504871i	−0.912948	−	0.408077i	$-0.866200\pi$
$971$	−15.7322	+	15.7322i	−0.504871	+	0.504871i	0.408077	+	0.912948i	$0.366200\pi$
$972$		0			0
$973$	12.0000	+	12.0000i	0.384702	+	0.384702i
$974$		0			0
$975$	42.1218			1.34898
$976$		0			0
$977$	35.1769			1.12541			0.562705	−	0.826658i	$-0.309761\pi$
$977$	35.1769			1.12541			0.562705	+	0.826658i	$0.309761\pi$
$978$		0			0
$979$	5.10205	+	5.10205i	0.163062	+	0.163062i
$980$		0			0
$981$	1.00000	−	1.00000i	0.0319275	−	0.0319275i
$982$		0			0
$983$		−	23.7370i		−	0.757093i	−0.925582	−	0.378547i	$-0.876424\pi$
$983$		−	23.7370i		−	0.757093i	0.925582	−	0.378547i	$-0.123576\pi$
$984$		0			0
$985$			33.8564i			1.07875i
$986$		0			0
$987$	−4.89898	+	4.89898i	−0.155936	+	0.155936i
$988$		0			0
$989$	6.92820	+	6.92820i	0.220304	+	0.220304i
$990$		0			0
$991$	12.2474			0.389053			0.194527	−	0.980897i	$-0.437683\pi$
$991$	12.2474			0.389053			0.194527	+	0.980897i	$0.437683\pi$
$992$		0			0
$993$	1.07180			0.0340124
$994$		0			0
$995$	50.9860	+	50.9860i	1.61637	+	1.61637i
$996$		0			0
$997$	−12.4641	+	12.4641i	−0.394742	+	0.394742i	−0.876374	−	0.481632i	$-0.840044\pi$
$997$	−12.4641	+	12.4641i	−0.394742	+	0.394742i	0.481632	+	0.876374i	$0.340044\pi$
$998$		0			0
$999$			0.656339i			0.0207656i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.j.e.577.1	yes	8
3.2	odd	2		2304.2.k.l.577.3		8
4.3	odd	2	inner	768.2.j.e.577.3	yes	8
8.3	odd	2		768.2.j.f.577.2	yes	8
8.5	even	2		768.2.j.f.577.4	yes	8
12.11	even	2		2304.2.k.l.577.4		8
16.3	odd	4		768.2.j.f.193.2	yes	8
16.5	even	4	inner	768.2.j.e.193.1	✓	8
16.11	odd	4	inner	768.2.j.e.193.3	yes	8
16.13	even	4		768.2.j.f.193.4	yes	8
24.5	odd	2		2304.2.k.e.577.1		8
24.11	even	2		2304.2.k.e.577.2		8
32.3	odd	8		3072.2.d.h.1537.5		8
32.5	even	8		3072.2.a.r.1.1		4
32.11	odd	8		3072.2.a.r.1.4		4
32.13	even	8		3072.2.d.h.1537.8		8
32.19	odd	8		3072.2.d.h.1537.4		8
32.21	even	8		3072.2.a.l.1.4		4
32.27	odd	8		3072.2.a.l.1.1		4
32.29	even	8		3072.2.d.h.1537.1		8
48.5	odd	4		2304.2.k.l.1729.4		8
48.11	even	4		2304.2.k.l.1729.3		8
48.29	odd	4		2304.2.k.e.1729.2		8
48.35	even	4		2304.2.k.e.1729.1		8
96.5	odd	8		9216.2.a.bc.1.4		4
96.11	even	8		9216.2.a.bc.1.1		4
96.53	odd	8		9216.2.a.bi.1.1		4
96.59	even	8		9216.2.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.e.193.1	✓	8	16.5	even	4	inner
768.2.j.e.193.3	yes	8	16.11	odd	4	inner
768.2.j.e.577.1	yes	8	1.1	even	1	trivial
768.2.j.e.577.3	yes	8	4.3	odd	2	inner
768.2.j.f.193.2	yes	8	16.3	odd	4
768.2.j.f.193.4	yes	8	16.13	even	4
768.2.j.f.577.2	yes	8	8.3	odd	2
768.2.j.f.577.4	yes	8	8.5	even	2
2304.2.k.e.577.1		8	24.5	odd	2
2304.2.k.e.577.2		8	24.11	even	2
2304.2.k.e.1729.1		8	48.35	even	4
2304.2.k.e.1729.2		8	48.29	odd	4
2304.2.k.l.577.3		8	3.2	odd	2
2304.2.k.l.577.4		8	12.11	even	2
2304.2.k.l.1729.3		8	48.11	even	4
2304.2.k.l.1729.4		8	48.5	odd	4
3072.2.a.l.1.1		4	32.27	odd	8
3072.2.a.l.1.4		4	32.21	even	8
3072.2.a.r.1.1		4	32.5	even	8
3072.2.a.r.1.4		4	32.11	odd	8
3072.2.d.h.1537.1		8	32.29	even	8
3072.2.d.h.1537.4		8	32.19	odd	8
3072.2.d.h.1537.5		8	32.3	odd	8
3072.2.d.h.1537.8		8	32.13	even	8
9216.2.a.bc.1.1		4	96.11	even	8
9216.2.a.bc.1.4		4	96.5	odd	8
9216.2.a.bi.1.1		4	96.53	odd	8
9216.2.a.bi.1.4		4	96.59	even	8

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.44949i q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.44949i q^{7} +1.00000i q^{9} +(1.03528 - 1.03528i) q^{11} +(-3.00000 - 3.00000i) q^{13} +3.86370 q^{15} +3.46410 q^{17} +(5.27792 + 5.27792i) q^{19} +(-1.73205 + 1.73205i) q^{21} +2.82843i q^{23} -9.92820i q^{25} +(0.707107 - 0.707107i) q^{27} +(6.19615 + 6.19615i) q^{29} +7.34847 q^{31} -1.46410 q^{33} +(6.69213 + 6.69213i) q^{35} +(-0.464102 + 0.464102i) q^{37} +4.24264i q^{39} +4.53590i q^{41} +(2.44949 - 2.44949i) q^{43} +(-2.73205 - 2.73205i) q^{45} +2.82843 q^{47} +1.00000 q^{49} +(-2.44949 - 2.44949i) q^{51} +(-3.26795 + 3.26795i) q^{53} +5.65685i q^{55} -7.46410i q^{57} +(9.79796 - 9.79796i) q^{59} +(-0.464102 - 0.464102i) q^{61} +2.44949 q^{63} +16.3923 q^{65} +(-10.5558 - 10.5558i) q^{67} +(2.00000 - 2.00000i) q^{69} -6.41473i q^{71} +4.00000i q^{73} +(-7.02030 + 7.02030i) q^{75} +(-2.53590 - 2.53590i) q^{77} +2.44949 q^{79} -1.00000 q^{81} +(3.86370 + 3.86370i) q^{83} +(-9.46410 + 9.46410i) q^{85} -8.76268i q^{87} +4.92820i q^{89} +(-7.34847 + 7.34847i) q^{91} +(-5.19615 - 5.19615i) q^{93} -28.8391 q^{95} +1.07180 q^{97} +(1.03528 + 1.03528i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 24 q^{13} + 8 q^{29} + 16 q^{33} + 24 q^{37} - 8 q^{45} + 8 q^{49} - 40 q^{53} + 24 q^{61} + 48 q^{65} + 16 q^{69} - 48 q^{77} - 8 q^{81} - 48 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 24 * q^13 + 8 * q^29 + 16 * q^33 + 24 * q^37 - 8 * q^45 + 8 * q^49 - 40 * q^53 + 24 * q^61 + 48 * q^65 + 16 * q^69 - 48 * q^77 - 8 * q^81 - 48 * q^85 + 64 * q^97

Introduction

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