LMFDB - Embedded newform 768.3.l.d.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(319,768)]

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([2, 1, 0]))

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

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

chi := DirichletCharacter("768.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.l (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:	$20.9264843029$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			703.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.703
Dual form			768.3.l.d.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} -5.27792 q^{7} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} -5.27792 q^{7} -3.00000i q^{9} +(-0.757875 - 0.757875i) q^{11} +(0.464102 + 0.464102i) q^{13} +4.89898i q^{15} +15.8564 q^{17} +(-3.96524 + 3.96524i) q^{19} +(6.46410 - 6.46410i) q^{21} -21.8695 q^{23} +17.0000i q^{25} +(3.67423 + 3.67423i) q^{27} +(-14.9282 - 14.9282i) q^{29} +57.2992i q^{31} +1.85641 q^{33} +(-10.5558 + 10.5558i) q^{35} +(14.6077 - 14.6077i) q^{37} -1.13681 q^{39} +79.5692i q^{41} +(-15.2789 - 15.2789i) q^{43} +(-6.00000 - 6.00000i) q^{45} +2.27362i q^{47} -21.1436 q^{49} +(-19.4201 + 19.4201i) q^{51} +(27.2154 - 27.2154i) q^{53} -3.03150 q^{55} -9.71281i q^{57} +(50.5055 + 50.5055i) q^{59} +(-68.8179 - 68.8179i) q^{61} +15.8338i q^{63} +1.85641 q^{65} +(-59.9518 + 59.9518i) q^{67} +(26.7846 - 26.7846i) q^{69} -82.9853 q^{71} +77.8564i q^{73} +(-20.8207 - 20.8207i) q^{75} +(4.00000 + 4.00000i) q^{77} +18.0530i q^{79} -9.00000 q^{81} +(-99.9015 + 99.9015i) q^{83} +(31.7128 - 31.7128i) q^{85} +36.5665 q^{87} +74.0000i q^{89} +(-2.44949 - 2.44949i) q^{91} +(-70.1769 - 70.1769i) q^{93} +15.8610i q^{95} +49.2820 q^{97} +(-2.27362 + 2.27362i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5}+O(q^{10}) $ 8 * q + 16 * q^5	$ 8 q + 16 q^{5} - 24 q^{13} + 16 q^{17} + 24 q^{21} - 64 q^{29} - 96 q^{33} + 200 q^{37} - 48 q^{45} - 280 q^{49} + 384 q^{53} - 24 q^{61} - 96 q^{65} + 48 q^{69} + 32 q^{77} - 72 q^{81} + 32 q^{85} - 312 q^{93} - 160 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 24 * q^13 + 16 * q^17 + 24 * q^21 - 64 * q^29 - 96 * q^33 + 200 * q^37 - 48 * q^45 - 280 * q^49 + 384 * q^53 - 24 * q^61 - 96 * q^65 + 48 * q^69 + 32 * q^77 - 72 * q^81 + 32 * q^85 - 312 * q^93 - 160 * 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
$3$	−1.22474	+	1.22474i	−0.408248	+	0.408248i
$3$	−1.22474	+	1.22474i	−0.408248	+	0.408248i
$4$		0			0
$5$	2.00000	−	2.00000i	0.400000	−	0.400000i	−0.478233	−	0.878233i	$-0.658723\pi$
$5$	2.00000	−	2.00000i	0.400000	−	0.400000i	0.878233	+	0.478233i	$0.158723\pi$
$6$		0			0
$7$	−5.27792			−0.753988			−0.376994	−	0.926216i	$-0.623042\pi$
$7$	−5.27792			−0.753988			−0.376994	+	0.926216i	$0.623042\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−0.757875	−	0.757875i	−0.0688977	−	0.0688977i	0.671818	−	0.740716i	$-0.265514\pi$
$11$	−0.757875	−	0.757875i	−0.0688977	−	0.0688977i	−0.740716	+	0.671818i	$0.765514\pi$
$12$		0			0
$13$	0.464102	+	0.464102i	0.0357001	+	0.0357001i	0.724732	−	0.689031i	$-0.241964\pi$
$13$	0.464102	+	0.464102i	0.0357001	+	0.0357001i	−0.689031	+	0.724732i	$0.741964\pi$
$14$		0			0
$15$			4.89898i			0.326599i
$16$		0			0
$17$	15.8564			0.932730			0.466365	−	0.884592i	$-0.345563\pi$
$17$	15.8564			0.932730			0.466365	+	0.884592i	$0.345563\pi$
$18$		0			0
$19$	−3.96524	+	3.96524i	−0.208697	+	0.208697i	−0.803713	−	0.595017i	$-0.797145\pi$
$19$	−3.96524	+	3.96524i	−0.208697	+	0.208697i	0.595017	+	0.803713i	$0.297145\pi$
$20$		0			0
$21$	6.46410	−	6.46410i	0.307814	−	0.307814i
$22$		0			0
$23$	−21.8695			−0.950850			−0.475425	−	0.879756i	$-0.657706\pi$
$23$	−21.8695			−0.950850			−0.475425	+	0.879756i	$0.657706\pi$
$24$		0			0
$25$			17.0000i			0.680000i
$26$		0			0
$27$	3.67423	+	3.67423i	0.136083	+	0.136083i
$28$		0			0
$29$	−14.9282	−	14.9282i	−0.514766	−	0.514766i	0.401217	−	0.915983i	$-0.368587\pi$
$29$	−14.9282	−	14.9282i	−0.514766	−	0.514766i	−0.915983	+	0.401217i	$0.868587\pi$
$30$		0			0
$31$			57.2992i			1.84836i	0.381955	+	0.924181i	$0.375251\pi$
$31$			57.2992i			1.84836i	−0.381955	+	0.924181i	$0.624749\pi$
$32$		0			0
$33$	1.85641			0.0562547
$34$		0			0
$35$	−10.5558	+	10.5558i	−0.301595	+	0.301595i
$36$		0			0
$37$	14.6077	−	14.6077i	0.394803	−	0.394803i	−0.481593	−	0.876395i	$-0.659942\pi$
$37$	14.6077	−	14.6077i	0.394803	−	0.394803i	0.876395	+	0.481593i	$0.159942\pi$
$38$		0			0
$39$	−1.13681			−0.0291490
$40$		0			0
$41$			79.5692i			1.94071i	0.241679	+	0.970356i	$0.422302\pi$
$41$			79.5692i			1.94071i	−0.241679	+	0.970356i	$0.577698\pi$
$42$		0			0
$43$	−15.2789	−	15.2789i	−0.355324	−	0.355324i	0.506762	−	0.862086i	$-0.330842\pi$
$43$	−15.2789	−	15.2789i	−0.355324	−	0.355324i	−0.862086	+	0.506762i	$0.830842\pi$
$44$		0			0
$45$	−6.00000	−	6.00000i	−0.133333	−	0.133333i
$46$		0			0
$47$			2.27362i			0.0483750i	0.999707	+	0.0241875i	$0.00769987\pi$
$47$			2.27362i			0.0483750i	−0.999707	+	0.0241875i	$0.992300\pi$
$48$		0			0
$49$	−21.1436			−0.431502
$50$		0			0
$51$	−19.4201	+	19.4201i	−0.380785	+	0.380785i
$52$		0			0
$53$	27.2154	−	27.2154i	0.513498	−	0.513498i	−0.402098	−	0.915596i	$-0.631719\pi$
$53$	27.2154	−	27.2154i	0.513498	−	0.513498i	0.915596	+	0.402098i	$0.131719\pi$
$54$		0			0
$55$	−3.03150			−0.0551182
$56$		0			0
$57$		−	9.71281i		−	0.170400i
$58$		0			0
$59$	50.5055	+	50.5055i	0.856026	+	0.856026i	0.990867	−	0.134841i	$-0.0430524\pi$
$59$	50.5055	+	50.5055i	0.856026	+	0.856026i	−0.134841	+	0.990867i	$0.543052\pi$
$60$		0			0
$61$	−68.8179	−	68.8179i	−1.12816	−	1.12816i	−0.990476	−	0.137687i	$-0.956033\pi$
$61$	−68.8179	−	68.8179i	−1.12816	−	1.12816i	−0.137687	−	0.990476i	$-0.543967\pi$
$62$		0			0
$63$			15.8338i			0.251329i
$64$		0			0
$65$	1.85641			0.0285601
$66$		0			0
$67$	−59.9518	+	59.9518i	−0.894803	+	0.894803i	−0.994971	−	0.100168i	$-0.968062\pi$
$67$	−59.9518	+	59.9518i	−0.894803	+	0.894803i	0.100168	+	0.994971i	$0.468062\pi$
$68$		0			0
$69$	26.7846	−	26.7846i	0.388183	−	0.388183i
$70$		0			0
$71$	−82.9853			−1.16881			−0.584404	−	0.811463i	$-0.698672\pi$
$71$	−82.9853			−1.16881			−0.584404	+	0.811463i	$0.698672\pi$
$72$		0			0
$73$			77.8564i			1.06653i	0.845949	+	0.533263i	$0.179034\pi$
$73$			77.8564i			1.06653i	−0.845949	+	0.533263i	$0.820966\pi$
$74$		0			0
$75$	−20.8207	−	20.8207i	−0.277609	−	0.277609i
$76$		0			0
$77$	4.00000	+	4.00000i	0.0519481	+	0.0519481i
$78$		0			0
$79$			18.0530i			0.228519i	0.993451	+	0.114259i	$0.0364495\pi$
$79$			18.0530i			0.228519i	−0.993451	+	0.114259i	$0.963551\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−99.9015	+	99.9015i	−1.20363	+	1.20363i	−0.230579	+	0.973054i	$0.574062\pi$
$83$	−99.9015	+	99.9015i	−1.20363	+	1.20363i	−0.973054	+	0.230579i	$0.925938\pi$
$84$		0			0
$85$	31.7128	−	31.7128i	0.373092	−	0.373092i
$86$		0			0
$87$	36.5665			0.420304
$88$		0			0
$89$			74.0000i			0.831461i	0.909488	+	0.415730i	$0.136474\pi$
$89$			74.0000i			0.831461i	−0.909488	+	0.415730i	$0.863526\pi$
$90$		0			0
$91$	−2.44949	−	2.44949i	−0.0269175	−	0.0269175i
$92$		0			0
$93$	−70.1769	−	70.1769i	−0.754590	−	0.754590i
$94$		0			0
$95$			15.8610i			0.166957i
$96$		0			0
$97$	49.2820			0.508062			0.254031	−	0.967196i	$-0.418243\pi$
$97$	49.2820			0.508062			0.254031	+	0.967196i	$0.418243\pi$
$98$		0			0
$99$	−2.27362	+	2.27362i	−0.0229659	+	0.0229659i
$100$		0			0
$101$	−83.9230	+	83.9230i	−0.830921	+	0.830921i	−0.987643	−	0.156722i	$-0.949907\pi$
$101$	−83.9230	+	83.9230i	−0.830921	+	0.830921i	0.156722	+	0.987643i	$0.449907\pi$
$102$		0			0
$103$	−22.6546			−0.219948			−0.109974	−	0.993934i	$-0.535077\pi$
$103$	−22.6546			−0.219948			−0.109974	+	0.993934i	$0.535077\pi$
$104$		0			0
$105$		−	25.8564i		−	0.246251i
$106$		0			0
$107$	70.4532	+	70.4532i	0.658441	+	0.658441i	0.955011	−	0.296570i	$-0.0958428\pi$
$107$	70.4532	+	70.4532i	0.658441	+	0.658441i	−0.296570	+	0.955011i	$0.595843\pi$
$108$		0			0
$109$	−32.7513	−	32.7513i	−0.300471	−	0.300471i	0.540727	−	0.841198i	$-0.318149\pi$
$109$	−32.7513	−	32.7513i	−0.300471	−	0.300471i	−0.841198	+	0.540727i	$0.818149\pi$
$110$		0			0
$111$			35.7814i			0.322355i
$112$		0			0
$113$	−56.8513			−0.503108			−0.251554	−	0.967843i	$-0.580942\pi$
$113$	−56.8513			−0.503108			−0.251554	+	0.967843i	$0.580942\pi$
$114$		0			0
$115$	−43.7391	+	43.7391i	−0.380340	+	0.380340i
$116$		0			0
$117$	1.39230	−	1.39230i	0.0119000	−	0.0119000i
$118$		0			0
$119$	−83.6888			−0.703267
$120$		0			0
$121$		−	119.851i		−	0.990506i
$122$		0			0
$123$	−97.4520	−	97.4520i	−0.792293	−	0.792293i
$124$		0			0
$125$	84.0000	+	84.0000i	0.672000	+	0.672000i
$126$		0			0
$127$			52.0485i			0.409831i	0.978780	+	0.204915i	$0.0656919\pi$
$127$			52.0485i			0.409831i	−0.978780	+	0.204915i	$0.934308\pi$
$128$		0			0
$129$	37.4256			0.290121
$130$		0			0
$131$	67.5305	−	67.5305i	0.515500	−	0.515500i	−0.400706	−	0.916207i	$-0.631235\pi$
$131$	67.5305	−	67.5305i	0.515500	−	0.515500i	0.916207	+	0.400706i	$0.131235\pi$
$132$		0			0
$133$	20.9282	−	20.9282i	0.157355	−	0.157355i
$134$		0			0
$135$	14.6969			0.108866
$136$		0			0
$137$			245.559i			1.79240i	0.443649	+	0.896200i	$0.353684\pi$
$137$			245.559i			1.79240i	−0.443649	+	0.896200i	$0.646316\pi$
$138$		0			0
$139$	−86.3142	−	86.3142i	−0.620965	−	0.620965i	0.324813	−	0.945778i	$-0.394699\pi$
$139$	−86.3142	−	86.3142i	−0.620965	−	0.620965i	−0.945778	+	0.324813i	$0.894699\pi$
$140$		0			0
$141$	−2.78461	−	2.78461i	−0.0197490	−	0.0197490i
$142$		0			0
$143$		−	0.703462i		−	0.00491931i
$144$		0			0
$145$	−59.7128			−0.411813
$146$		0			0
$147$	25.8955	−	25.8955i	0.176160	−	0.176160i
$148$		0			0
$149$	−4.14359	+	4.14359i	−0.0278094	+	0.0278094i	−0.720875	−	0.693065i	$-0.756260\pi$
$149$	−4.14359	+	4.14359i	−0.0278094	+	0.0278094i	0.693065	+	0.720875i	$0.256260\pi$
$150$		0			0
$151$	91.2403			0.604241			0.302120	−	0.953270i	$-0.402306\pi$
$151$	91.2403			0.604241			0.302120	+	0.953270i	$0.402306\pi$
$152$		0			0
$153$		−	47.5692i		−	0.310910i
$154$		0			0
$155$	114.598	+	114.598i	0.739345	+	0.739345i
$156$		0			0
$157$	−72.4641	−	72.4641i	−0.461555	−	0.461555i	0.437610	−	0.899165i	$-0.355825\pi$
$157$	−72.4641	−	72.4641i	−0.461555	−	0.461555i	−0.899165	+	0.437610i	$0.855825\pi$
$158$		0			0
$159$			66.6638i			0.419269i
$160$		0			0
$161$	115.426			0.716929
$162$		0			0
$163$	80.8332	−	80.8332i	0.495909	−	0.495909i	−0.414253	−	0.910162i	$-0.635957\pi$
$163$	80.8332	−	80.8332i	0.495909	−	0.495909i	0.910162	+	0.414253i	$0.135957\pi$
$164$		0			0
$165$	3.71281	−	3.71281i	0.0225019	−	0.0225019i
$166$		0			0
$167$	−139.391			−0.834674			−0.417337	−	0.908752i	$-0.637037\pi$
$167$	−139.391			−0.834674			−0.417337	+	0.908752i	$0.637037\pi$
$168$		0			0
$169$		−	168.569i		−	0.997451i
$170$		0			0
$171$	11.8957	+	11.8957i	0.0695656	+	0.0695656i
$172$		0			0
$173$	214.420	+	214.420i	1.23942	+	1.23942i	0.960236	+	0.279188i	$0.0900654\pi$
$173$	214.420	+	214.420i	1.23942	+	1.23942i	0.279188	+	0.960236i	$0.409935\pi$
$174$		0			0
$175$		−	89.7246i		−	0.512712i
$176$		0			0
$177$	−123.713			−0.698942
$178$		0			0
$179$	126.670	−	126.670i	0.707654	−	0.707654i	−0.258388	−	0.966041i	$-0.583191\pi$
$179$	126.670	−	126.670i	0.707654	−	0.707654i	0.966041	+	0.258388i	$0.0831912\pi$
$180$		0			0
$181$	208.310	−	208.310i	1.15089	−	1.15089i	0.164510	−	0.986375i	$-0.447396\pi$
$181$	208.310	−	208.310i	1.15089	−	1.15089i	0.986375	−	0.164510i	$-0.0526041\pi$
$182$		0			0
$183$	168.569			0.921141
$184$		0			0
$185$		−	58.4308i		−	0.315842i
$186$		0			0
$187$	−12.0172	−	12.0172i	−0.0642629	−	0.0642629i
$188$		0			0
$189$	−19.3923	−	19.3923i	−0.102605	−	0.102605i
$190$		0			0
$191$		−	340.006i		−	1.78014i	−0.455828	−	0.890068i	$-0.650657\pi$
$191$		−	340.006i		−	1.78014i	0.455828	−	0.890068i	$-0.349343\pi$
$192$		0			0
$193$	195.990			1.01549			0.507745	−	0.861507i	$-0.330479\pi$
$193$	195.990			1.01549			0.507745	+	0.861507i	$0.330479\pi$
$194$		0			0
$195$	−2.27362	+	2.27362i	−0.0116596	+	0.0116596i
$196$		0			0
$197$	−204.631	+	204.631i	−1.03873	+	1.03873i	−0.0395156	+	0.999219i	$0.512581\pi$
$197$	−204.631	+	204.631i	−1.03873	+	1.03873i	−0.999219	+	0.0395156i	$0.987419\pi$
$198$		0			0
$199$	234.584			1.17881			0.589406	−	0.807837i	$-0.299362\pi$
$199$	234.584			1.17881			0.589406	+	0.807837i	$0.299362\pi$
$200$		0			0
$201$		−	146.851i		−	0.730603i
$202$		0			0
$203$	78.7898	+	78.7898i	0.388127	+	0.388127i
$204$		0			0
$205$	159.138	+	159.138i	0.776285	+	0.776285i
$206$		0			0
$207$			65.6086i			0.316950i
$208$		0			0
$209$	6.01031			0.0287575
$210$		0			0
$211$	130.866	−	130.866i	0.620216	−	0.620216i	−0.325371	−	0.945587i	$-0.605489\pi$
$211$	130.866	−	130.866i	0.620216	−	0.620216i	0.945587	+	0.325371i	$0.105489\pi$
$212$		0			0
$213$	101.636	−	101.636i	0.477164	−	0.477164i
$214$		0			0
$215$	−61.1158			−0.284259
$216$		0			0
$217$		−	302.420i		−	1.39364i
$218$		0			0
$219$	−95.3542	−	95.3542i	−0.435407	−	0.435407i
$220$		0			0
$221$	7.35898	+	7.35898i	0.0332986	+	0.0332986i
$222$		0			0
$223$			16.5372i			0.0741579i	0.999312	+	0.0370790i	$0.0118053\pi$
$223$			16.5372i			0.0741579i	−0.999312	+	0.0370790i	$0.988195\pi$
$224$		0			0
$225$	51.0000			0.226667
$226$		0			0
$227$	199.506	−	199.506i	0.878880	−	0.878880i	−0.114539	−	0.993419i	$-0.536539\pi$
$227$	199.506	−	199.506i	0.878880	−	0.878880i	0.993419	+	0.114539i	$0.0365392\pi$
$228$		0			0
$229$	−237.105	+	237.105i	−1.03539	+	1.03539i	−0.0360433	+	0.999350i	$0.511475\pi$
$229$	−237.105	+	237.105i	−1.03539	+	1.03539i	−0.999350	+	0.0360433i	$0.988525\pi$
$230$		0			0
$231$	−9.79796			−0.0424154
$232$		0			0
$233$			155.990i			0.669484i	0.942310	+	0.334742i	$0.108649\pi$
$233$			155.990i			0.669484i	−0.942310	+	0.334742i	$0.891351\pi$
$234$		0			0
$235$	4.54725	+	4.54725i	0.0193500	+	0.0193500i
$236$		0			0
$237$	−22.1103	−	22.1103i	−0.0932923	−	0.0932923i
$238$		0			0
$239$		−	250.903i		−	1.04980i	−0.851163	−	0.524902i	$-0.824102\pi$
$239$		−	250.903i		−	1.04980i	0.851163	−	0.524902i	$-0.175898\pi$
$240$		0			0
$241$	−334.554			−1.38819			−0.694095	−	0.719883i	$-0.744195\pi$
$241$	−334.554			−1.38819			−0.694095	+	0.719883i	$0.744195\pi$
$242$		0			0
$243$	11.0227	−	11.0227i	0.0453609	−	0.0453609i
$244$		0			0
$245$	−42.2872	+	42.2872i	−0.172601	+	0.172601i
$246$		0			0
$247$	−3.68055			−0.0149010
$248$		0			0
$249$		−	244.708i		−	0.982762i
$250$		0			0
$251$	54.2949	+	54.2949i	0.216314	+	0.216314i	0.806943	−	0.590629i	$-0.201120\pi$
$251$	54.2949	+	54.2949i	0.216314	+	0.216314i	−0.590629	+	0.806943i	$0.701120\pi$
$252$		0			0
$253$	16.5744	+	16.5744i	0.0655114	+	0.0655114i
$254$		0			0
$255$			77.6802i			0.304628i
$256$		0			0
$257$	110.862			0.431368			0.215684	−	0.976463i	$-0.430802\pi$
$257$	110.862			0.431368			0.215684	+	0.976463i	$0.430802\pi$
$258$		0			0
$259$	−77.0982	+	77.0982i	−0.297676	+	0.297676i
$260$		0			0
$261$	−44.7846	+	44.7846i	−0.171589	+	0.171589i
$262$		0			0
$263$	197.584			0.751269			0.375635	−	0.926768i	$-0.377425\pi$
$263$	197.584			0.751269			0.375635	+	0.926768i	$0.377425\pi$
$264$		0			0
$265$		−	108.862i		−	0.410798i
$266$		0			0
$267$	−90.6311	−	90.6311i	−0.339442	−	0.339442i
$268$		0			0
$269$	223.061	+	223.061i	0.829225	+	0.829225i	0.987410	−	0.158185i	$-0.0505641\pi$
$269$	223.061	+	223.061i	0.829225	+	0.829225i	−0.158185	+	0.987410i	$0.550564\pi$
$270$		0			0
$271$			5.87255i			0.0216699i	0.999941	+	0.0108350i	$0.00344894\pi$
$271$			5.87255i			0.0216699i	−0.999941	+	0.0108350i	$0.996551\pi$
$272$		0			0
$273$	6.00000			0.0219780
$274$		0			0
$275$	12.8839	−	12.8839i	0.0468504	−	0.0468504i
$276$		0			0
$277$	−164.751	+	164.751i	−0.594770	+	0.594770i	−0.938916	−	0.344146i	$-0.888168\pi$
$277$	−164.751	+	164.751i	−0.594770	+	0.594770i	0.344146	+	0.938916i	$0.388168\pi$
$278$		0			0
$279$	171.898			0.616121
$280$		0			0
$281$		−	78.0000i		−	0.277580i	−0.990322	−	0.138790i	$-0.955679\pi$
$281$		−	78.0000i		−	0.277580i	0.990322	−	0.138790i	$-0.0443213\pi$
$282$		0			0
$283$	−222.782	−	222.782i	−0.787216	−	0.787216i	0.193821	−	0.981037i	$-0.437912\pi$
$283$	−222.782	−	222.782i	−0.787216	−	0.787216i	−0.981037	+	0.193821i	$0.937912\pi$
$284$		0			0
$285$	−19.4256	−	19.4256i	−0.0681601	−	0.0681601i
$286$		0			0
$287$		−	419.960i		−	1.46327i
$288$		0			0
$289$	−37.5744			−0.130015
$290$		0			0
$291$	−60.3579	+	60.3579i	−0.207416	+	0.207416i
$292$		0			0
$293$	−322.067	+	322.067i	−1.09920	+	1.09920i	−0.104700	+	0.994504i	$0.533388\pi$
$293$	−322.067	+	322.067i	−1.09920	+	1.09920i	−0.994504	+	0.104700i	$0.966612\pi$
$294$		0			0
$295$	202.022			0.684821
$296$		0			0
$297$		−	5.56922i		−	0.0187516i
$298$		0			0
$299$	−10.1497	−	10.1497i	−0.0339455	−	0.0339455i
$300$		0			0
$301$	80.6410	+	80.6410i	0.267910	+	0.267910i
$302$		0			0
$303$		−	205.569i		−	0.678444i
$304$		0			0
$305$	−275.272			−0.902530
$306$		0			0
$307$	300.462	−	300.462i	0.978705	−	0.978705i	−0.0210733	−	0.999778i	$-0.506708\pi$
$307$	300.462	−	300.462i	0.978705	−	0.978705i	0.999778	+	0.0210733i	$0.00670833\pi$
$308$		0			0
$309$	27.7461	−	27.7461i	0.0897933	−	0.0897933i
$310$		0			0
$311$	−496.070			−1.59508			−0.797540	−	0.603266i	$-0.793866\pi$
$311$	−496.070			−1.59508			−0.797540	+	0.603266i	$0.793866\pi$
$312$		0			0
$313$		−	218.267i		−	0.697337i	−0.937246	−	0.348669i	$-0.886634\pi$
$313$		−	218.267i		−	0.697337i	0.937246	−	0.348669i	$-0.113366\pi$
$314$		0			0
$315$	31.6675	+	31.6675i	0.100532	+	0.100532i
$316$		0			0
$317$	99.2154	+	99.2154i	0.312982	+	0.312982i	0.846064	−	0.533082i	$-0.178966\pi$
$317$	99.2154	+	99.2154i	0.312982	+	0.312982i	−0.533082	+	0.846064i	$0.678966\pi$
$318$		0			0
$319$			22.6274i			0.0709323i
$320$		0			0
$321$	−172.574			−0.537615
$322$		0			0
$323$	−62.8744	+	62.8744i	−0.194658	+	0.194658i
$324$		0			0
$325$	−7.88973	+	7.88973i	−0.0242761	+	0.0242761i
$326$		0			0
$327$	80.2239			0.245333
$328$		0			0
$329$		−	12.0000i		−	0.0364742i
$330$		0			0
$331$	−99.0348	−	99.0348i	−0.299199	−	0.299199i	0.541501	−	0.840700i	$-0.317856\pi$
$331$	−99.0348	−	99.0348i	−0.299199	−	0.299199i	−0.840700	+	0.541501i	$0.817856\pi$
$332$		0			0
$333$	−43.8231	−	43.8231i	−0.131601	−	0.131601i
$334$		0			0
$335$			239.807i			0.715842i
$336$		0			0
$337$	−35.5589			−0.105516			−0.0527580	−	0.998607i	$-0.516801\pi$
$337$	−35.5589			−0.105516			−0.0527580	+	0.998607i	$0.516801\pi$
$338$		0			0
$339$	69.6283	−	69.6283i	0.205393	−	0.205393i
$340$		0			0
$341$	43.4256	−	43.4256i	0.127348	−	0.127348i
$342$		0			0
$343$	370.212			1.07934
$344$		0			0
$345$		−	107.138i		−	0.310546i
$346$		0			0
$347$	410.513	+	410.513i	1.18304	+	1.18304i	0.978954	+	0.204082i	$0.0654210\pi$
$347$	410.513	+	410.513i	1.18304	+	1.18304i	0.204082	+	0.978954i	$0.434579\pi$
$348$		0			0
$349$	−338.454	−	338.454i	−0.969782	−	0.969782i	0.0297750	−	0.999557i	$-0.490521\pi$
$349$	−338.454	−	338.454i	−0.969782	−	0.969782i	−0.999557	+	0.0297750i	$0.990521\pi$
$350$		0			0
$351$			3.41044i			0.00971634i
$352$		0			0
$353$	299.990			0.849829			0.424915	−	0.905233i	$-0.360304\pi$
$353$	299.990			0.849829			0.424915	+	0.905233i	$0.360304\pi$
$354$		0			0
$355$	−165.971	+	165.971i	−0.467523	+	0.467523i
$356$		0			0
$357$	102.497	−	102.497i	0.287108	−	0.287108i
$358$		0			0
$359$	340.873			0.949506			0.474753	−	0.880119i	$-0.342537\pi$
$359$	340.873			0.949506			0.474753	+	0.880119i	$0.342537\pi$
$360$		0			0
$361$			329.554i			0.912891i
$362$		0			0
$363$	146.787	+	146.787i	0.404372	+	0.404372i
$364$		0			0
$365$	155.713	+	155.713i	0.426610	+	0.426610i
$366$		0			0
$367$		−	271.230i		−	0.739046i	−0.929222	−	0.369523i	$-0.879521\pi$
$367$		−	271.230i		−	0.739046i	0.929222	−	0.369523i	$-0.120479\pi$
$368$		0			0
$369$	238.708			0.646904
$370$		0			0
$371$	−143.641	+	143.641i	−0.387171	+	0.387171i
$372$		0			0
$373$	197.105	−	197.105i	0.528432	−	0.528432i	−0.391673	−	0.920105i	$-0.628103\pi$
$373$	197.105	−	197.105i	0.528432	−	0.528432i	0.920105	+	0.391673i	$0.128103\pi$
$374$		0			0
$375$	−205.757			−0.548686
$376$		0			0
$377$		−	13.8564i		−	0.0367544i
$378$		0			0
$379$	−115.235	−	115.235i	−0.304050	−	0.304050i	0.538546	−	0.842596i	$-0.318974\pi$
$379$	−115.235	−	115.235i	−0.304050	−	0.304050i	−0.842596	+	0.538546i	$0.818974\pi$
$380$		0			0
$381$	−63.7461	−	63.7461i	−0.167313	−	0.167313i
$382$		0			0
$383$		−	119.904i		−	0.313064i	−0.987673	−	0.156532i	$-0.949969\pi$
$383$		−	119.904i		−	0.313064i	0.987673	−	0.156532i	$-0.0500314\pi$
$384$		0			0
$385$	16.0000			0.0415584
$386$		0			0
$387$	−45.8368	+	45.8368i	−0.118441	+	0.118441i
$388$		0			0
$389$	316.431	−	316.431i	0.813447	−	0.813447i	−0.171702	−	0.985149i	$-0.554927\pi$
$389$	316.431	−	316.431i	0.813447	−	0.813447i	0.985149	+	0.171702i	$0.0549267\pi$
$390$		0			0
$391$	−346.772			−0.886886
$392$		0			0
$393$			165.415i			0.420904i
$394$		0			0
$395$	36.1059	+	36.1059i	0.0914074	+	0.0914074i
$396$		0			0
$397$	56.5974	+	56.5974i	0.142563	+	0.142563i	0.774786	−	0.632223i	$-0.217858\pi$
$397$	56.5974	+	56.5974i	0.142563	+	0.142563i	−0.632223	+	0.774786i	$0.717858\pi$
$398$		0			0
$399$			51.2634i			0.128480i
$400$		0			0
$401$	−420.144			−1.04774			−0.523870	−	0.851798i	$-0.675512\pi$
$401$	−420.144			−1.04774			−0.523870	+	0.851798i	$0.675512\pi$
$402$		0			0
$403$	−26.5927	+	26.5927i	−0.0659867	+	0.0659867i
$404$		0			0
$405$	−18.0000	+	18.0000i	−0.0444444	+	0.0444444i
$406$		0			0
$407$	−22.1416			−0.0544020
$408$		0			0
$409$		−	152.133i		−	0.371964i	−0.982553	−	0.185982i	$-0.940453\pi$
$409$		−	152.133i		−	0.371964i	0.982553	−	0.185982i	$-0.0595466\pi$
$410$		0			0
$411$	−300.747	−	300.747i	−0.731745	−	0.731745i
$412$		0			0
$413$	−266.564	−	266.564i	−0.645434	−	0.645434i
$414$		0			0
$415$			399.606i			0.962906i
$416$		0			0
$417$	211.426			0.507016
$418$		0			0
$419$	203.132	−	203.132i	0.484801	−	0.484801i	−0.421860	−	0.906661i	$-0.638623\pi$
$419$	203.132	−	203.132i	0.484801	−	0.484801i	0.906661	+	0.421860i	$0.138623\pi$
$420$		0			0
$421$	−38.9615	+	38.9615i	−0.0925452	+	0.0925452i	−0.751864	−	0.659319i	$-0.770845\pi$
$421$	−38.9615	+	38.9615i	−0.0925452	+	0.0925452i	0.659319	+	0.751864i	$0.270845\pi$
$422$		0			0
$423$	6.82087			0.0161250
$424$		0			0
$425$			269.559i			0.634256i
$426$		0			0
$427$	363.215	+	363.215i	0.850621	+	0.850621i
$428$		0			0
$429$	0.861561	+	0.861561i	0.00200830	+	0.00200830i
$430$		0			0
$431$			427.538i			0.991969i	0.868331	+	0.495984i	$0.165193\pi$
$431$			427.538i			0.991969i	−0.868331	+	0.495984i	$0.834807\pi$
$432$		0			0
$433$	325.549			0.751844			0.375922	−	0.926651i	$-0.377326\pi$
$433$	325.549			0.751844			0.375922	+	0.926651i	$0.377326\pi$
$434$		0			0
$435$	73.1330	−	73.1330i	0.168122	−	0.168122i
$436$		0			0
$437$	86.7180	−	86.7180i	0.198439	−	0.198439i
$438$		0			0
$439$	−227.005			−0.517095			−0.258548	−	0.965999i	$-0.583244\pi$
$439$	−227.005			−0.517095			−0.258548	+	0.965999i	$0.583244\pi$
$440$		0			0
$441$			63.4308i			0.143834i
$442$		0			0
$443$	−114.138	−	114.138i	−0.257648	−	0.257648i	0.566449	−	0.824097i	$-0.308317\pi$
$443$	−114.138	−	114.138i	−0.257648	−	0.257648i	−0.824097	+	0.566449i	$0.808317\pi$
$444$		0			0
$445$	148.000	+	148.000i	0.332584	+	0.332584i
$446$		0			0
$447$		−	10.1497i		−	0.0227062i
$448$		0			0
$449$	137.559			0.306367			0.153184	−	0.988198i	$-0.451047\pi$
$449$	137.559			0.306367			0.153184	+	0.988198i	$0.451047\pi$
$450$		0			0
$451$	60.3035	−	60.3035i	0.133711	−	0.133711i
$452$		0			0
$453$	−111.746	+	111.746i	−0.246680	+	0.246680i
$454$		0			0
$455$	−9.79796			−0.0215340
$456$		0			0
$457$			680.708i			1.48951i	0.667336	+	0.744757i	$0.267434\pi$
$457$			680.708i			1.48951i	−0.667336	+	0.744757i	$0.732566\pi$
$458$		0			0
$459$	58.2602	+	58.2602i	0.126928	+	0.126928i
$460$		0			0
$461$	100.277	+	100.277i	0.217520	+	0.217520i	0.807453	−	0.589932i	$-0.200846\pi$
$461$	100.277	+	100.277i	0.217520	+	0.217520i	−0.589932	+	0.807453i	$0.700846\pi$
$462$		0			0
$463$			465.024i			1.00437i	0.864760	+	0.502186i	$0.167471\pi$
$463$			465.024i			1.00437i	−0.864760	+	0.502186i	$0.832529\pi$
$464$		0			0
$465$	−280.708			−0.603672
$466$		0			0
$467$	−412.950	+	412.950i	−0.884262	+	0.884262i	−0.993964	−	0.109702i	$-0.965010\pi$
$467$	−412.950	+	412.950i	−0.884262	+	0.884262i	0.109702	+	0.993964i	$0.465010\pi$
$468$		0			0
$469$	316.420	−	316.420i	0.674671	−	0.674671i
$470$		0			0
$471$	177.500			0.376858
$472$		0			0
$473$			23.1591i			0.0489621i
$474$		0			0
$475$	−67.4091	−	67.4091i	−0.141914	−	0.141914i
$476$		0			0
$477$	−81.6462	−	81.6462i	−0.171166	−	0.171166i
$478$		0			0
$479$			519.941i			1.08547i	0.839903	+	0.542736i	$0.182611\pi$
$479$			519.941i			1.08547i	−0.839903	+	0.542736i	$0.817389\pi$
$480$		0			0
$481$	13.5589			0.0281890
$482$		0			0
$483$	−141.367	+	141.367i	−0.292685	+	0.292685i
$484$		0			0
$485$	98.5641	−	98.5641i	0.203225	−	0.203225i
$486$		0			0
$487$	−763.133			−1.56701			−0.783504	−	0.621386i	$-0.786570\pi$
$487$	−763.133			−1.56701			−0.783504	+	0.621386i	$0.786570\pi$
$488$		0			0
$489$			198.000i			0.404908i
$490$		0			0
$491$	−592.860	−	592.860i	−1.20745	−	1.20745i	−0.971849	−	0.235606i	$-0.924293\pi$
$491$	−592.860	−	592.860i	−1.20745	−	1.20745i	−0.235606	−	0.971849i	$-0.575707\pi$
$492$		0			0
$493$	−236.708	−	236.708i	−0.480137	−	0.480137i
$494$		0			0
$495$			9.09450i			0.0183727i
$496$		0			0
$497$	437.990			0.881267
$498$		0			0
$499$	−323.090	+	323.090i	−0.647474	+	0.647474i	−0.952382	−	0.304908i	$-0.901374\pi$
$499$	−323.090	+	323.090i	−0.647474	+	0.647474i	0.304908	+	0.952382i	$0.401374\pi$
$500$		0			0
$501$	170.718	−	170.718i	0.340754	−	0.340754i
$502$		0			0
$503$	−646.829			−1.28594			−0.642971	−	0.765891i	$-0.722298\pi$
$503$	−646.829			−1.28594			−0.642971	+	0.765891i	$0.722298\pi$
$504$		0			0
$505$			335.692i			0.664737i
$506$		0			0
$507$	206.454	+	206.454i	0.407208	+	0.407208i
$508$		0			0
$509$	−604.344	−	604.344i	−1.18732	−	1.18732i	−0.977807	−	0.209509i	$-0.932814\pi$
$509$	−604.344	−	604.344i	−1.18732	−	1.18732i	−0.209509	−	0.977807i	$-0.567186\pi$
$510$		0			0
$511$		−	410.920i		−	0.804148i
$512$		0			0
$513$	−29.1384			−0.0568001
$514$		0			0
$515$	−45.3092	+	45.3092i	−0.0879791	+	0.0879791i
$516$		0			0
$517$	1.72312	−	1.72312i	0.00333293	−	0.00333293i
$518$		0			0
$519$	−525.221			−1.01199
$520$		0			0
$521$		−	16.7180i		−	0.0320882i	−0.999871	−	0.0160441i	$-0.994893\pi$
$521$		−	16.7180i		−	0.0320882i	0.999871	−	0.0160441i	$-0.00510722\pi$
$522$		0			0
$523$	297.443	+	297.443i	0.568726	+	0.568726i	0.931771	−	0.363046i	$-0.118263\pi$
$523$	297.443	+	297.443i	0.568726	+	0.568726i	−0.363046	+	0.931771i	$0.618263\pi$
$524$		0			0
$525$	109.890	+	109.890i	0.209314	+	0.209314i
$526$		0			0
$527$			908.560i			1.72402i
$528$		0			0
$529$	−50.7231			−0.0958849
$530$		0			0
$531$	151.517	−	151.517i	0.285342	−	0.285342i
$532$		0			0
$533$	−36.9282	+	36.9282i	−0.0692837	+	0.0692837i
$534$		0			0
$535$	281.813			0.526753
$536$		0			0
$537$			310.277i			0.577797i
$538$		0			0
$539$	16.0242	+	16.0242i	0.0297295	+	0.0297295i
$540$		0			0
$541$	−8.46410	−	8.46410i	−0.0156453	−	0.0156453i	0.699241	−	0.714886i	$-0.253521\pi$
$541$	−8.46410	−	8.46410i	−0.0156453	−	0.0156453i	−0.714886	+	0.699241i	$0.753521\pi$
$542$		0			0
$543$			510.254i			0.939694i
$544$		0			0
$545$	−131.005			−0.240376
$546$		0			0
$547$	−547.400	+	547.400i	−1.00073	+	1.00073i	−0.000731952		1.00000i	$0.500233\pi$
$547$	−547.400	+	547.400i	−1.00073	+	1.00073i	−1.00000		0.000731952i	$0.999767\pi$
$548$		0			0
$549$	−206.454	+	206.454i	−0.376054	+	0.376054i
$550$		0			0
$551$	118.388			0.214860
$552$		0			0
$553$		−	95.2820i		−	0.172300i
$554$		0			0
$555$	71.5628	+	71.5628i	0.128942	+	0.128942i
$556$		0			0
$557$	−370.133	−	370.133i	−0.664512	−	0.664512i	0.291928	−	0.956440i	$-0.405703\pi$
$557$	−370.133	−	370.133i	−0.664512	−	0.664512i	−0.956440	+	0.291928i	$0.905703\pi$
$558$		0			0
$559$		−	14.1820i		−	0.0253702i
$560$		0			0
$561$	29.4359			0.0524705
$562$		0			0
$563$	−269.364	+	269.364i	−0.478444	+	0.478444i	−0.904634	−	0.426189i	$-0.859856\pi$
$563$	−269.364	+	269.364i	−0.478444	+	0.478444i	0.426189	+	0.904634i	$0.359856\pi$
$564$		0			0
$565$	−113.703	+	113.703i	−0.201243	+	0.201243i
$566$		0			0
$567$	47.5013			0.0837765
$568$		0			0
$569$			258.154i			0.453698i	0.973930	+	0.226849i	$0.0728423\pi$
$569$			258.154i			0.453698i	−0.973930	+	0.226849i	$0.927158\pi$
$570$		0			0
$571$	−29.9633	−	29.9633i	−0.0524751	−	0.0524751i	0.680382	−	0.732857i	$-0.261814\pi$
$571$	−29.9633	−	29.9633i	−0.0524751	−	0.0524751i	−0.732857	+	0.680382i	$0.761814\pi$
$572$		0			0
$573$	416.420	+	416.420i	0.726737	+	0.726737i
$574$		0			0
$575$		−	371.782i		−	0.646578i
$576$		0			0
$577$	596.995			1.03465			0.517327	−	0.855788i	$-0.326927\pi$
$577$	596.995			1.03465			0.517327	+	0.855788i	$0.326927\pi$
$578$		0			0
$579$	−240.037	+	240.037i	−0.414572	+	0.414572i
$580$		0			0
$581$	527.272	−	527.272i	0.907524	−	0.907524i
$582$		0			0
$583$	−41.2517			−0.0707577
$584$		0			0
$585$		−	5.56922i		−	0.00952003i
$586$		0			0
$587$	−294.860	−	294.860i	−0.502317	−	0.502317i	0.409841	−	0.912157i	$-0.365584\pi$
$587$	−294.860	−	294.860i	−0.502317	−	0.502317i	−0.912157	+	0.409841i	$0.865584\pi$
$588$		0			0
$589$	−227.205	−	227.205i	−0.385747	−	0.385747i
$590$		0			0
$591$		−	501.241i		−	0.848123i
$592$		0			0
$593$	30.0000			0.0505902			0.0252951	−	0.999680i	$-0.491947\pi$
$593$	30.0000			0.0505902			0.0252951	+	0.999680i	$0.491947\pi$
$594$		0			0
$595$	−167.378	+	167.378i	−0.281307	+	0.281307i
$596$		0			0
$597$	−287.305	+	287.305i	−0.481248	+	0.481248i
$598$		0			0
$599$	653.269			1.09060			0.545299	−	0.838241i	$-0.316416\pi$
$599$	653.269			1.09060			0.545299	+	0.838241i	$0.316416\pi$
$600$		0			0
$601$			143.559i			0.238867i	0.992842	+	0.119433i	$0.0381078\pi$
$601$			143.559i			0.238867i	−0.992842	+	0.119433i	$0.961892\pi$
$602$		0			0
$603$	179.855	+	179.855i	0.298268	+	0.298268i
$604$		0			0
$605$	−239.703	−	239.703i	−0.396202	−	0.396202i
$606$		0			0
$607$			313.076i			0.515776i	0.966175	+	0.257888i	$0.0830266\pi$
$607$			313.076i			0.515776i	−0.966175	+	0.257888i	$0.916973\pi$
$608$		0			0
$609$	−192.995			−0.316905
$610$		0			0
$611$	−1.05519	+	1.05519i	−0.00172699	+	0.00172699i
$612$		0			0
$613$	386.895	−	386.895i	0.631150	−	0.631150i	−0.317207	−	0.948356i	$-0.602745\pi$
$613$	386.895	−	386.895i	0.631150	−	0.631150i	0.948356	+	0.317207i	$0.102745\pi$
$614$		0			0
$615$	−389.808			−0.633834
$616$		0			0
$617$		−	159.395i		−	0.258338i	−0.991623	−	0.129169i	$-0.958769\pi$
$617$		−	159.395i		−	0.258338i	0.991623	−	0.129169i	$-0.0412310\pi$
$618$		0			0
$619$	404.723	+	404.723i	0.653833	+	0.653833i	0.953914	−	0.300081i	$-0.0970137\pi$
$619$	404.723	+	404.723i	0.653833	+	0.653833i	−0.300081	+	0.953914i	$0.597014\pi$
$620$		0			0
$621$	−80.3538	−	80.3538i	−0.129394	−	0.129394i
$622$		0			0
$623$		−	390.566i		−	0.626911i
$624$		0			0
$625$	−89.0000			−0.142400
$626$		0			0
$627$	−7.36110	+	7.36110i	−0.0117402	+	0.0117402i
$628$		0			0
$629$	231.626	−	231.626i	0.368244	−	0.368244i
$630$		0			0
$631$	−684.151			−1.08423			−0.542116	−	0.840303i	$-0.682377\pi$
$631$	−684.151			−1.08423			−0.542116	+	0.840303i	$0.682377\pi$
$632$		0			0
$633$			320.554i			0.506404i
$634$		0			0
$635$	104.097	+	104.097i	0.163932	+	0.163932i
$636$		0			0
$637$	−9.81278	−	9.81278i	−0.0154047	−	0.0154047i
$638$		0			0
$639$			248.956i			0.389603i
$640$		0			0
$641$	−459.590			−0.716989			−0.358494	−	0.933532i	$-0.616710\pi$
$641$	−459.590			−0.716989			−0.358494	+	0.933532i	$0.616710\pi$
$642$		0			0
$643$	268.158	−	268.158i	0.417043	−	0.417043i	−0.467140	−	0.884183i	$-0.654716\pi$
$643$	268.158	−	268.158i	0.417043	−	0.417043i	0.884183	+	0.467140i	$0.154716\pi$
$644$		0			0
$645$	74.8513	−	74.8513i	0.116048	−	0.116048i
$646$		0			0
$647$	910.184			1.40678			0.703388	−	0.710806i	$-0.251670\pi$
$647$	910.184			1.40678			0.703388	+	0.710806i	$0.251670\pi$
$648$		0			0
$649$		−	76.5538i		−	0.117956i
$650$		0			0
$651$	370.388	+	370.388i	0.568952	+	0.568952i
$652$		0			0
$653$	−805.538	−	805.538i	−1.23360	−	1.23360i	−0.962573	−	0.271023i	$-0.912638\pi$
$653$	−805.538	−	805.538i	−1.23360	−	1.23360i	−0.271023	−	0.962573i	$-0.587362\pi$
$654$		0			0
$655$		−	270.122i		−	0.412400i
$656$		0			0
$657$	233.569			0.355509
$658$		0			0
$659$	−436.796	+	436.796i	−0.662817	+	0.662817i	−0.956043	−	0.293226i	$-0.905271\pi$
$659$	−436.796	+	436.796i	−0.662817	+	0.662817i	0.293226	+	0.956043i	$0.405271\pi$
$660$		0			0
$661$	74.0333	−	74.0333i	0.112002	−	0.112002i	−0.648885	−	0.760887i	$-0.724764\pi$
$661$	74.0333	−	74.0333i	0.112002	−	0.112002i	0.760887	+	0.648885i	$0.224764\pi$
$662$		0			0
$663$	−18.0258			−0.0271882
$664$		0			0
$665$		−	83.7128i		−	0.125884i
$666$		0			0
$667$	326.473	+	326.473i	0.489465	+	0.489465i
$668$		0			0
$669$	−20.2539	−	20.2539i	−0.0302748	−	0.0302748i
$670$		0			0
$671$			104.311i			0.155456i
$672$		0			0
$673$	−1232.95			−1.83203			−0.916013	−	0.401148i	$-0.868611\pi$
$673$	−1232.95			−1.83203			−0.916013	+	0.401148i	$0.868611\pi$
$674$		0			0
$675$	−62.4620	+	62.4620i	−0.0925363	+	0.0925363i
$676$		0			0
$677$	−282.000	+	282.000i	−0.416544	+	0.416544i	−0.884011	−	0.467467i	$-0.845167\pi$
$677$	−282.000	+	282.000i	−0.416544	+	0.416544i	0.467467	+	0.884011i	$0.345167\pi$
$678$		0			0
$679$	−260.106			−0.383073
$680$		0			0
$681$			488.687i			0.717602i
$682$		0			0
$683$	−907.287	−	907.287i	−1.32838	−	1.32838i	−0.906780	−	0.421605i	$-0.861467\pi$
$683$	−907.287	−	907.287i	−1.32838	−	1.32838i	−0.421605	−	0.906780i	$-0.638533\pi$
$684$		0			0
$685$	491.118	+	491.118i	0.716960	+	0.716960i
$686$		0			0
$687$		−	580.787i		−	0.845395i
$688$		0			0
$689$	25.2614			0.0366639
$690$		0			0
$691$	−463.657	+	463.657i	−0.670994	+	0.670994i	−0.957945	−	0.286951i	$-0.907358\pi$
$691$	−463.657	+	463.657i	−0.670994	+	0.670994i	0.286951	+	0.957945i	$0.407358\pi$
$692$		0			0
$693$	12.0000	−	12.0000i	0.0173160	−	0.0173160i
$694$		0			0
$695$	−345.257			−0.496772
$696$		0			0
$697$			1261.68i			1.81016i
$698$		0			0
$699$	−191.048	−	191.048i	−0.273316	−	0.273316i
$700$		0			0
$701$	255.349	+	255.349i	0.364263	+	0.364263i	0.865380	−	0.501116i	$-0.167077\pi$
$701$	255.349	+	255.349i	0.364263	+	0.364263i	−0.501116	+	0.865380i	$0.667077\pi$
$702$		0			0
$703$			115.846i			0.164788i
$704$		0			0
$705$	−11.1384			−0.0157992
$706$		0			0
$707$	442.939	−	442.939i	0.626505	−	0.626505i
$708$		0			0
$709$	−586.741	+	586.741i	−0.827561	+	0.827561i	−0.987179	−	0.159618i	$-0.948974\pi$
$709$	−586.741	+	586.741i	−0.827561	+	0.827561i	0.159618	+	0.987179i	$0.448974\pi$
$710$		0			0
$711$	54.1589			0.0761728
$712$		0			0
$713$		−	1253.11i		−	1.75751i
$714$		0			0
$715$	−1.40692	−	1.40692i	−0.00196773	−	0.00196773i
$716$		0			0
$717$	307.292	+	307.292i	0.428581	+	0.428581i
$718$		0			0
$719$			1213.59i			1.68789i	0.536428	+	0.843946i	$0.319773\pi$
$719$			1213.59i			1.68789i	−0.536428	+	0.843946i	$0.680227\pi$
$720$		0			0
$721$	119.569			0.165838
$722$		0			0
$723$	409.743	−	409.743i	0.566726	−	0.566726i
$724$		0			0
$725$	253.779	−	253.779i	0.350041	−	0.350041i
$726$		0			0
$727$	1363.03			1.87487			0.937433	−	0.348165i	$-0.113195\pi$
$727$	1363.03			1.87487			0.937433	+	0.348165i	$0.113195\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	−242.269	−	242.269i	−0.331422	−	0.331422i
$732$		0			0
$733$	296.443	+	296.443i	0.404425	+	0.404425i	0.879789	−	0.475364i	$-0.157684\pi$
$733$	296.443	+	296.443i	0.404425	+	0.404425i	−0.475364	+	0.879789i	$0.657684\pi$
$734$		0			0
$735$		−	103.582i		−	0.140928i
$736$		0			0
$737$	90.8719			0.123300
$738$		0			0
$739$	525.707	−	525.707i	0.711376	−	0.711376i	−0.255447	−	0.966823i	$-0.582223\pi$
$739$	525.707	−	525.707i	0.711376	−	0.711376i	0.966823	+	0.255447i	$0.0822228\pi$
$740$		0			0
$741$	4.50773	−	4.50773i	0.00608331	−	0.00608331i
$742$		0			0
$743$	702.045			0.944878			0.472439	−	0.881363i	$-0.343374\pi$
$743$	702.045			0.944878			0.472439	+	0.881363i	$0.343374\pi$
$744$		0			0
$745$			16.5744i			0.0222475i
$746$		0			0
$747$	299.704	+	299.704i	0.401211	+	0.401211i
$748$		0			0
$749$	−371.846	−	371.846i	−0.496457	−	0.496457i
$750$		0			0
$751$		−	1292.93i		−	1.72161i	−0.508938	−	0.860803i	$-0.669962\pi$
$751$		−	1292.93i		−	1.72161i	0.508938	−	0.860803i	$-0.330038\pi$
$752$		0			0
$753$	−132.995			−0.176620
$754$		0			0
$755$	182.481	−	182.481i	0.241696	−	0.241696i
$756$		0			0
$757$	−869.792	+	869.792i	−1.14900	+	1.14900i	−0.162249	+	0.986750i	$0.551875\pi$
$757$	−869.792	+	869.792i	−1.14900	+	1.14900i	−0.986750	+	0.162249i	$0.948125\pi$
$758$		0			0
$759$	−40.5988			−0.0534898
$760$		0			0
$761$		−	845.005i		−	1.11039i	−0.831721	−	0.555194i	$-0.812644\pi$
$761$		−	845.005i		−	1.11039i	0.831721	−	0.555194i	$-0.187356\pi$
$762$		0			0
$763$	172.859	+	172.859i	0.226551	+	0.226551i
$764$		0			0
$765$	−95.1384	−	95.1384i	−0.124364	−	0.124364i
$766$		0			0
$767$			46.8794i			0.0611205i
$768$		0			0
$769$	548.585			0.713374			0.356687	−	0.934224i	$-0.383906\pi$
$769$	548.585			0.713374			0.356687	+	0.934224i	$0.383906\pi$
$770$		0			0
$771$	−135.777	+	135.777i	−0.176105	+	0.176105i
$772$		0			0
$773$	287.041	−	287.041i	0.371334	−	0.371334i	−0.496629	−	0.867963i	$-0.665429\pi$
$773$	287.041	−	287.041i	0.371334	−	0.371334i	0.867963	+	0.496629i	$0.165429\pi$
$774$		0			0
$775$	−974.087			−1.25689
$776$		0			0
$777$		−	188.851i		−	0.243052i
$778$		0			0
$779$	−315.511	−	315.511i	−0.405021	−	0.405021i
$780$		0			0
$781$	62.8925	+	62.8925i	0.0805282	+	0.0805282i
$782$		0			0
$783$		−	109.699i		−	0.140101i
$784$		0			0
$785$	−289.856			−0.369244
$786$		0			0
$787$	1079.85	−	1079.85i	1.37211	−	1.37211i	0.514791	−	0.857316i	$-0.327870\pi$
$787$	1079.85	−	1079.85i	1.37211	−	1.37211i	0.857316	−	0.514791i	$-0.172130\pi$
$788$		0			0
$789$	−241.990	+	241.990i	−0.306704	+	0.306704i
$790$		0			0
$791$	300.056			0.379338
$792$		0			0
$793$		−	63.8770i		−	0.0805511i
$794$		0			0
$795$	133.328	+	133.328i	0.167708	+	0.167708i
$796$		0			0
$797$	858.379	+	858.379i	1.07701	+	1.07701i	0.996776	+	0.0802370i	$0.0255677\pi$
$797$	858.379	+	858.379i	1.07701	+	1.07701i	0.0802370	+	0.996776i	$0.474432\pi$
$798$		0			0
$799$			36.0515i			0.0451208i
$800$		0			0
$801$	222.000			0.277154
$802$		0			0
$803$	59.0054	−	59.0054i	0.0734812	−	0.0734812i
$804$		0			0
$805$	230.851	−	230.851i	0.286772	−	0.286772i
$806$		0			0
$807$	−546.387			−0.677059
$808$		0			0
$809$		−	1068.39i		−	1.32063i	−0.750989	−	0.660315i	$-0.770423\pi$
$809$		−	1068.39i		−	1.32063i	0.750989	−	0.660315i	$-0.229577\pi$
$810$		0			0
$811$	−695.157	−	695.157i	−0.857160	−	0.857160i	0.133843	−	0.991003i	$-0.457268\pi$
$811$	−695.157	−	695.157i	−0.857160	−	0.857160i	−0.991003	+	0.133843i	$0.957268\pi$
$812$		0			0
$813$	−7.19238	−	7.19238i	−0.00884671	−	0.00884671i
$814$		0			0
$815$		−	323.333i		−	0.396727i
$816$		0			0
$817$	121.169			0.148310
$818$		0			0
$819$	−7.34847	+	7.34847i	−0.00897249	+	0.00897249i
$820$		0			0
$821$	276.477	−	276.477i	0.336756	−	0.336756i	−0.518389	−	0.855145i	$-0.673468\pi$
$821$	276.477	−	276.477i	0.336756	−	0.336756i	0.855145	+	0.518389i	$0.173468\pi$
$822$		0			0
$823$	−466.862			−0.567269			−0.283635	−	0.958932i	$-0.591540\pi$
$823$	−466.862			−0.567269			−0.283635	+	0.958932i	$0.591540\pi$
$824$		0			0
$825$			31.5589i			0.0382532i
$826$		0			0
$827$	550.394	+	550.394i	0.665531	+	0.665531i	0.956678	−	0.291148i	$-0.0940370\pi$
$827$	550.394	+	550.394i	0.665531	+	0.665531i	−0.291148	+	0.956678i	$0.594037\pi$
$828$		0			0
$829$	791.561	+	791.561i	0.954839	+	0.954839i	0.999023	−	0.0441847i	$-0.0140690\pi$
$829$	791.561	+	791.561i	0.954839	+	0.954839i	−0.0441847	+	0.999023i	$0.514069\pi$
$830$		0			0
$831$		−	403.557i		−	0.485628i
$832$		0			0
$833$	−335.261			−0.402475
$834$		0			0
$835$	−278.781	+	278.781i	−0.333870	+	0.333870i
$836$		0			0
$837$	−210.531	+	210.531i	−0.251530	+	0.251530i
$838$		0			0
$839$	1476.79			1.76018			0.880090	−	0.474807i	$-0.157482\pi$
$839$	1476.79			1.76018			0.880090	+	0.474807i	$0.157482\pi$
$840$		0			0
$841$		−	395.297i		−	0.470033i
$842$		0			0
$843$	95.5301	+	95.5301i	0.113322	+	0.113322i
$844$		0			0
$845$	−337.138	−	337.138i	−0.398980	−	0.398980i
$846$		0			0
$847$			632.565i			0.746830i
$848$		0			0
$849$	545.703			0.642759
$850$		0			0
$851$	−319.464	+	319.464i	−0.375398	+	0.375398i
$852$		0			0
$853$	−851.869	+	851.869i	−0.998674	+	0.998674i	−0.999999	−	0.00132490i	$-0.999578\pi$
$853$	−851.869	+	851.869i	−0.998674	+	0.998674i	0.00132490	+	0.999999i	$0.499578\pi$
$854$		0			0
$855$	47.5829			0.0556525
$856$		0			0
$857$		−	889.559i		−	1.03799i	−0.854777	−	0.518996i	$-0.826306\pi$
$857$		−	889.559i		−	1.03799i	0.854777	−	0.518996i	$-0.173694\pi$
$858$		0			0
$859$	−179.734	−	179.734i	−0.209236	−	0.209236i	0.594707	−	0.803943i	$-0.297268\pi$
$859$	−179.734	−	179.734i	−0.209236	−	0.209236i	−0.803943	+	0.594707i	$0.797268\pi$
$860$		0			0
$861$	514.344	+	514.344i	0.597379	+	0.597379i
$862$		0			0
$863$			128.563i			0.148972i	0.997222	+	0.0744859i	$0.0237316\pi$
$863$			128.563i			0.148972i	−0.997222	+	0.0744859i	$0.976268\pi$
$864$		0			0
$865$	857.682			0.991540
$866$		0			0
$867$	46.0190	−	46.0190i	0.0530785	−	0.0530785i
$868$		0			0
$869$	13.6819	−	13.6819i	0.0157444	−	0.0157444i
$870$		0			0
$871$	−55.6474			−0.0638891
$872$		0			0
$873$		−	147.846i		−	0.169354i
$874$		0			0
$875$	−443.345	−	443.345i	−0.506680	−	0.506680i
$876$		0			0
$877$	−261.746	−	261.746i	−0.298456	−	0.298456i	0.541953	−	0.840409i	$-0.317685\pi$
$877$	−261.746	−	261.746i	−0.298456	−	0.298456i	−0.840409	+	0.541953i	$0.817685\pi$
$878$		0			0
$879$		−	788.899i		−	0.897496i
$880$		0			0
$881$	547.395			0.621333			0.310667	−	0.950519i	$-0.399448\pi$
$881$	547.395			0.621333			0.310667	+	0.950519i	$0.399448\pi$
$882$		0			0
$883$	949.795	−	949.795i	1.07565	−	1.07565i	0.0787509	−	0.996894i	$-0.474907\pi$
$883$	949.795	−	949.795i	1.07565	−	1.07565i	0.996894	−	0.0787509i	$-0.0250932\pi$
$884$		0			0
$885$	−247.426	+	247.426i	−0.279577	+	0.279577i
$886$		0			0
$887$	−589.393			−0.664479			−0.332240	−	0.943195i	$-0.607804\pi$
$887$	−589.393			−0.664479			−0.332240	+	0.943195i	$0.607804\pi$
$888$		0			0
$889$		−	274.708i		−	0.309007i
$890$		0			0
$891$	6.82087	+	6.82087i	0.00765530	+	0.00765530i
$892$		0			0
$893$	−9.01546	−	9.01546i	−0.0100957	−	0.0100957i
$894$		0			0
$895$		−	506.680i		−	0.566123i
$896$		0			0
$897$	24.8616			0.0277163
$898$		0			0
$899$	855.374	−	855.374i	0.951473	−	0.951473i
$900$		0			0
$901$	431.538	−	431.538i	0.478955	−	0.478955i
$902$		0			0
$903$	−197.529			−0.218748
$904$		0			0
$905$		−	833.241i		−	0.920708i
$906$		0			0
$907$	871.872	+	871.872i	0.961270	+	0.961270i	0.999277	−	0.0380077i	$-0.0121011\pi$
$907$	871.872	+	871.872i	0.961270	+	0.961270i	−0.0380077	+	0.999277i	$0.512101\pi$
$908$		0			0
$909$	251.769	+	251.769i	0.276974	+	0.276974i
$910$		0			0
$911$			273.313i			0.300014i	0.988685	+	0.150007i	$0.0479297\pi$
$911$			273.313i			0.300014i	−0.988685	+	0.150007i	$0.952070\pi$
$912$		0			0
$913$	151.426			0.165855
$914$		0			0
$915$	337.138	−	337.138i	0.368456	−	0.368456i
$916$		0			0
$917$	−356.420	+	356.420i	−0.388681	+	0.388681i
$918$		0			0
$919$	432.925			0.471083			0.235541	−	0.971864i	$-0.424314\pi$
$919$	432.925			0.471083			0.235541	+	0.971864i	$0.424314\pi$
$920$		0			0
$921$			735.979i			0.799109i
$922$		0			0
$923$	−38.5136	−	38.5136i	−0.0417266	−	0.0417266i
$924$		0			0
$925$	248.331	+	248.331i	0.268466	+	0.268466i
$926$		0			0
$927$			67.9639i			0.0733159i
$928$		0			0
$929$	−657.251			−0.707482			−0.353741	−	0.935343i	$-0.615091\pi$
$929$	−657.251			−0.707482			−0.353741	+	0.935343i	$0.615091\pi$
$930$		0			0
$931$	83.8394	−	83.8394i	0.0900531	−	0.0900531i
$932$		0			0
$933$	607.559	−	607.559i	0.651189	−	0.651189i
$934$		0			0
$935$	−48.0687			−0.0514104
$936$		0			0
$937$		−	1612.28i		−	1.72068i	−0.509721	−	0.860340i	$-0.670251\pi$
$937$		−	1612.28i		−	1.72068i	0.509721	−	0.860340i	$-0.329749\pi$
$938$		0			0
$939$	267.321	+	267.321i	0.284687	+	0.284687i
$940$		0			0
$941$	823.856	+	823.856i	0.875512	+	0.875512i	0.993066	−	0.117555i	$-0.0375056\pi$
$941$	823.856	+	823.856i	0.875512	+	0.875512i	−0.117555	+	0.993066i	$0.537506\pi$
$942$		0			0
$943$		−	1740.14i		−	1.84533i
$944$		0			0
$945$	−77.5692			−0.0820838
$946$		0			0
$947$	−1.84223	+	1.84223i	−0.00194533	+	0.00194533i	−0.708079	−	0.706133i	$-0.750438\pi$
$947$	−1.84223	+	1.84223i	−0.00194533	+	0.00194533i	0.706133	+	0.708079i	$0.250438\pi$
$948$		0			0
$949$	−36.1333	+	36.1333i	−0.0380751	+	0.0380751i
$950$		0			0
$951$	−243.027			−0.255549
$952$		0			0
$953$		−	738.687i		−	0.775118i	−0.921845	−	0.387559i	$-0.873318\pi$
$953$		−	738.687i		−	0.775118i	0.921845	−	0.387559i	$-0.126682\pi$
$954$		0			0
$955$	−680.012	−	680.012i	−0.712054	−	0.712054i
$956$		0			0
$957$	−27.7128	−	27.7128i	−0.0289580	−	0.0289580i
$958$		0			0
$959$		−	1296.04i		−	1.35145i
$960$		0			0
$961$	−2322.20			−2.41644
$962$		0			0
$963$	211.360	−	211.360i	0.219480	−	0.219480i
$964$		0			0
$965$	391.979	−	391.979i	0.406196	−	0.406196i
$966$		0			0
$967$	1108.39			1.14621			0.573107	−	0.819480i	$-0.305738\pi$
$967$	1108.39			1.14621			0.573107	+	0.819480i	$0.305738\pi$
$968$		0			0
$969$		−	154.010i		−	0.158937i
$970$		0			0
$971$	−958.931	−	958.931i	−0.987571	−	0.987571i	0.0123531	−	0.999924i	$-0.496068\pi$
$971$	−958.931	−	958.931i	−0.987571	−	0.987571i	−0.999924	+	0.0123531i	$0.996068\pi$
$972$		0			0
$973$	455.559	+	455.559i	0.468200	+	0.468200i
$974$		0			0
$975$		−	19.3258i		−	0.0198213i
$976$		0			0
$977$	1557.23			1.59389			0.796945	−	0.604052i	$-0.206448\pi$
$977$	1557.23			1.59389			0.796945	+	0.604052i	$0.206448\pi$
$978$		0			0
$979$	56.0827	−	56.0827i	0.0572857	−	0.0572857i
$980$		0			0
$981$	−98.2539	+	98.2539i	−0.100157	+	0.100157i
$982$		0			0
$983$	771.711			0.785057			0.392528	−	0.919740i	$-0.371600\pi$
$983$	771.711			0.785057			0.392528	+	0.919740i	$0.371600\pi$
$984$		0			0
$985$			818.523i			0.830988i
$986$		0			0
$987$	14.6969	+	14.6969i	0.0148905	+	0.0148905i
$988$		0			0
$989$	334.144	+	334.144i	0.337860	+	0.337860i
$990$		0			0
$991$			1009.27i			1.01844i	0.860637	+	0.509219i	$0.170066\pi$
$991$			1009.27i			1.01844i	−0.860637	+	0.509219i	$0.829934\pi$
$992$		0			0
$993$	242.585			0.244295
$994$		0			0
$995$	469.167	−	469.167i	0.471525	−	0.471525i
$996$		0			0
$997$	−307.823	+	307.823i	−0.308749	+	0.308749i	−0.844424	−	0.535675i	$-0.820057\pi$
$997$	−307.823	+	307.823i	−0.308749	+	0.308749i	0.535675	+	0.844424i	$0.320057\pi$
$998$		0			0
$999$	107.344			0.107452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.l.d.703.1	yes	8
4.3	odd	2	inner	768.3.l.d.703.4	yes	8
8.3	odd	2		768.3.l.a.703.2	yes	8
8.5	even	2		768.3.l.a.703.3	yes	8
16.3	odd	4		768.3.l.a.319.3	yes	8
16.5	even	4	inner	768.3.l.d.319.4	yes	8
16.11	odd	4	inner	768.3.l.d.319.1	yes	8
16.13	even	4		768.3.l.a.319.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.3.l.a.319.2	✓	8	16.13	even	4
768.3.l.a.319.3	yes	8	16.3	odd	4
768.3.l.a.703.2	yes	8	8.3	odd	2
768.3.l.a.703.3	yes	8	8.5	even	2
768.3.l.d.319.1	yes	8	16.11	odd	4	inner
768.3.l.d.319.4	yes	8	16.5	even	4	inner
768.3.l.d.703.1	yes	8	1.1	even	1	trivial
768.3.l.d.703.4	yes	8	4.3	odd	2	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} -5.27792 q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{3} +(2.00000 - 2.00000i) q^{5} -5.27792 q^{7} -3.00000i q^{9} +(-0.757875 - 0.757875i) q^{11} +(0.464102 + 0.464102i) q^{13} +4.89898i q^{15} +15.8564 q^{17} +(-3.96524 + 3.96524i) q^{19} +(6.46410 - 6.46410i) q^{21} -21.8695 q^{23} +17.0000i q^{25} +(3.67423 + 3.67423i) q^{27} +(-14.9282 - 14.9282i) q^{29} +57.2992i q^{31} +1.85641 q^{33} +(-10.5558 + 10.5558i) q^{35} +(14.6077 - 14.6077i) q^{37} -1.13681 q^{39} +79.5692i q^{41} +(-15.2789 - 15.2789i) q^{43} +(-6.00000 - 6.00000i) q^{45} +2.27362i q^{47} -21.1436 q^{49} +(-19.4201 + 19.4201i) q^{51} +(27.2154 - 27.2154i) q^{53} -3.03150 q^{55} -9.71281i q^{57} +(50.5055 + 50.5055i) q^{59} +(-68.8179 - 68.8179i) q^{61} +15.8338i q^{63} +1.85641 q^{65} +(-59.9518 + 59.9518i) q^{67} +(26.7846 - 26.7846i) q^{69} -82.9853 q^{71} +77.8564i q^{73} +(-20.8207 - 20.8207i) q^{75} +(4.00000 + 4.00000i) q^{77} +18.0530i q^{79} -9.00000 q^{81} +(-99.9015 + 99.9015i) q^{83} +(31.7128 - 31.7128i) q^{85} +36.5665 q^{87} +74.0000i q^{89} +(-2.44949 - 2.44949i) q^{91} +(-70.1769 - 70.1769i) q^{93} +15.8610i q^{95} +49.2820 q^{97} +(-2.27362 + 2.27362i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5}+O(q^{10}) \) 8 * q + 16 * q^5	\( 8 q + 16 q^{5} - 24 q^{13} + 16 q^{17} + 24 q^{21} - 64 q^{29} - 96 q^{33} + 200 q^{37} - 48 q^{45} - 280 q^{49} + 384 q^{53} - 24 q^{61} - 96 q^{65} + 48 q^{69} + 32 q^{77} - 72 q^{81} + 32 q^{85} - 312 q^{93} - 160 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 24 * q^13 + 16 * q^17 + 24 * q^21 - 64 * q^29 - 96 * q^33 + 200 * q^37 - 48 * q^45 - 280 * q^49 + 384 * q^53 - 24 * q^61 - 96 * q^65 + 48 * q^69 + 32 * q^77 - 72 * q^81 + 32 * q^85 - 312 * q^93 - 160 * q^97

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