LMFDB - Embedded newform 768.3.l.d.703.4

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			703.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.703
Dual form			768.3.l.d.319.4

$q$-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	$-0.376958\pi$
$7$	5.27792			0.753988			0.376994	+	0.926216i	$0.376958\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	0.757875	+	0.757875i	0.0688977	+	0.0688977i	0.740716	−	0.671818i	$-0.234486\pi$
$11$	0.757875	+	0.757875i	0.0688977	+	0.0688977i	−0.671818	+	0.740716i	$0.734486\pi$
$12$		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	−0.595017	−	0.803713i	$-0.702855\pi$
$19$	3.96524	−	3.96524i	0.208697	−	0.208697i	0.803713	+	0.595017i	$0.202855\pi$
$20$		0			0
$21$	6.46410	−	6.46410i	0.307814	−	0.307814i
$22$		0			0
$23$	21.8695			0.950850			0.475425	−	0.879756i	$-0.342294\pi$
$23$	21.8695			0.950850			0.475425	+	0.879756i	$0.342294\pi$
$24$		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	$-0.624749\pi$
$31$		−	57.2992i		−	1.84836i	0.381955	−	0.924181i	$-0.375251\pi$
$32$		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	0.862086	−	0.506762i	$-0.169158\pi$
$43$	15.2789	+	15.2789i	0.355324	+	0.355324i	−0.506762	+	0.862086i	$0.669158\pi$
$44$		0			0
$45$	−6.00000	−	6.00000i	−0.133333	−	0.133333i
$46$		0			0
$47$		−	2.27362i		−	0.0483750i	−0.999707	−	0.0241875i	$-0.992300\pi$
$47$		−	2.27362i		−	0.0483750i	0.999707	−	0.0241875i	$-0.00769987\pi$
$48$		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	0.134841	−	0.990867i	$-0.456948\pi$
$59$	−50.5055	−	50.5055i	−0.856026	−	0.856026i	−0.990867	+	0.134841i	$0.956948\pi$
$60$		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	−0.100168	−	0.994971i	$-0.531938\pi$
$67$	59.9518	−	59.9518i	0.894803	−	0.894803i	0.994971	+	0.100168i	$0.0319380\pi$
$68$		0			0
$69$	26.7846	−	26.7846i	0.388183	−	0.388183i
$70$		0			0
$71$	82.9853			1.16881			0.584404	−	0.811463i	$-0.301328\pi$
$71$	82.9853			1.16881			0.584404	+	0.811463i	$0.301328\pi$
$72$		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	$-0.963551\pi$
$79$		−	18.0530i		−	0.228519i	0.993451	−	0.114259i	$-0.0364495\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	99.9015	−	99.9015i	1.20363	−	1.20363i	0.230579	−	0.973054i	$-0.425938\pi$
$83$	99.9015	−	99.9015i	1.20363	−	1.20363i	0.973054	−	0.230579i	$-0.0740619\pi$
$84$		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	$-0.464923\pi$
$103$	22.6546			0.219948			0.109974	+	0.993934i	$0.464923\pi$
$104$		0			0
$105$		−	25.8564i		−	0.246251i
$106$		0			0
$107$	−70.4532	−	70.4532i	−0.658441	−	0.658441i	0.296570	−	0.955011i	$-0.404157\pi$
$107$	−70.4532	−	70.4532i	−0.658441	−	0.658441i	−0.955011	+	0.296570i	$0.904157\pi$
$108$		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	$-0.934308\pi$
$127$		−	52.0485i		−	0.409831i	0.978780	−	0.204915i	$-0.0656919\pi$
$128$		0			0
$129$	37.4256			0.290121
$130$		0			0
$131$	−67.5305	+	67.5305i	−0.515500	+	0.515500i	−0.916207	−	0.400706i	$-0.868765\pi$
$131$	−67.5305	+	67.5305i	−0.515500	+	0.515500i	0.400706	+	0.916207i	$0.368765\pi$
$132$		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	0.945778	−	0.324813i	$-0.105301\pi$
$139$	86.3142	+	86.3142i	0.620965	+	0.620965i	−0.324813	+	0.945778i	$0.605301\pi$
$140$		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	$-0.597694\pi$
$151$	−91.2403			−0.604241			−0.302120	+	0.953270i	$0.597694\pi$
$152$		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	−0.910162	−	0.414253i	$-0.864043\pi$
$163$	−80.8332	+	80.8332i	−0.495909	+	0.495909i	0.414253	+	0.910162i	$0.364043\pi$
$164$		0			0
$165$	3.71281	−	3.71281i	0.0225019	−	0.0225019i
$166$		0			0
$167$	139.391			0.834674			0.417337	−	0.908752i	$-0.362963\pi$
$167$	139.391			0.834674			0.417337	+	0.908752i	$0.362963\pi$
$168$		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	−0.966041	−	0.258388i	$-0.916809\pi$
$179$	−126.670	+	126.670i	−0.707654	+	0.707654i	0.258388	+	0.966041i	$0.416809\pi$
$180$		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	$0.349343\pi$
$191$			340.006i			1.78014i	−0.455828	+	0.890068i	$0.650657\pi$
$192$		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	$-0.700638\pi$
$199$	−234.584			−1.17881			−0.589406	+	0.807837i	$0.700638\pi$
$200$		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	−0.945587	−	0.325371i	$-0.894511\pi$
$211$	−130.866	+	130.866i	−0.620216	+	0.620216i	0.325371	+	0.945587i	$0.394511\pi$
$212$		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	$-0.988195\pi$
$223$		−	16.5372i		−	0.0741579i	0.999312	−	0.0370790i	$-0.0118053\pi$
$224$		0			0
$225$	51.0000			0.226667
$226$		0			0
$227$	−199.506	+	199.506i	−0.878880	+	0.878880i	−0.993419	−	0.114539i	$-0.963461\pi$
$227$	−199.506	+	199.506i	−0.878880	+	0.878880i	0.114539	+	0.993419i	$0.463461\pi$
$228$		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	$0.175898\pi$
$239$			250.903i			1.04980i	−0.851163	+	0.524902i	$0.824102\pi$
$240$		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	0.590629	−	0.806943i	$-0.298880\pi$
$251$	−54.2949	−	54.2949i	−0.216314	−	0.216314i	−0.806943	+	0.590629i	$0.798880\pi$
$252$		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	$-0.622575\pi$
$263$	−197.584			−0.751269			−0.375635	+	0.926768i	$0.622575\pi$
$264$		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	$-0.996551\pi$
$271$		−	5.87255i		−	0.0216699i	0.999941	−	0.0108350i	$-0.00344894\pi$
$272$		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	0.981037	−	0.193821i	$-0.0620881\pi$
$283$	222.782	+	222.782i	0.787216	+	0.787216i	−0.193821	+	0.981037i	$0.562088\pi$
$284$		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	−0.999778	−	0.0210733i	$-0.993292\pi$
$307$	−300.462	+	300.462i	−0.978705	+	0.978705i	0.0210733	+	0.999778i	$0.493292\pi$
$308$		0			0
$309$	27.7461	−	27.7461i	0.0897933	−	0.0897933i
$310$		0			0
$311$	496.070			1.59508			0.797540	−	0.603266i	$-0.206134\pi$
$311$	496.070			1.59508			0.797540	+	0.603266i	$0.206134\pi$
$312$		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	0.840700	−	0.541501i	$-0.182144\pi$
$331$	99.0348	+	99.0348i	0.299199	+	0.299199i	−0.541501	+	0.840700i	$0.682144\pi$
$332$		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	$-0.934579\pi$
$347$	−410.513	−	410.513i	−1.18304	−	1.18304i	−0.204082	−	0.978954i	$-0.565421\pi$
$348$		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	$-0.657463\pi$
$359$	−340.873			−0.949506			−0.474753	+	0.880119i	$0.657463\pi$
$360$		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	$0.120479\pi$
$367$			271.230i			0.739046i	−0.929222	+	0.369523i	$0.879521\pi$
$368$		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	0.842596	−	0.538546i	$-0.181026\pi$
$379$	115.235	+	115.235i	0.304050	+	0.304050i	−0.538546	+	0.842596i	$0.681026\pi$
$380$		0			0
$381$	−63.7461	−	63.7461i	−0.167313	−	0.167313i
$382$		0			0
$383$			119.904i			0.313064i	0.987673	+	0.156532i	$0.0500314\pi$
$383$			119.904i			0.313064i	−0.987673	+	0.156532i	$0.949969\pi$
$384$		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	−0.906661	−	0.421860i	$-0.861377\pi$
$419$	−203.132	+	203.132i	−0.484801	+	0.484801i	0.421860	+	0.906661i	$0.361377\pi$
$420$		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	$-0.834807\pi$
$431$		−	427.538i		−	0.991969i	0.868331	−	0.495984i	$-0.165193\pi$
$432$		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	$-0.416756\pi$
$439$	227.005			0.517095			0.258548	+	0.965999i	$0.416756\pi$
$440$		0			0
$441$			63.4308i			0.143834i
$442$		0			0
$443$	114.138	+	114.138i	0.257648	+	0.257648i	0.824097	−	0.566449i	$-0.191683\pi$
$443$	114.138	+	114.138i	0.257648	+	0.257648i	−0.566449	+	0.824097i	$0.691683\pi$
$444$		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	$-0.832529\pi$
$463$		−	465.024i		−	1.00437i	0.864760	−	0.502186i	$-0.167471\pi$
$464$		0			0
$465$	−280.708			−0.603672
$466$		0			0
$467$	412.950	−	412.950i	0.884262	−	0.884262i	−0.109702	−	0.993964i	$-0.534990\pi$
$467$	412.950	−	412.950i	0.884262	−	0.884262i	0.993964	+	0.109702i	$0.0349898\pi$
$468$		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	$-0.817389\pi$
$479$		−	519.941i		−	1.08547i	0.839903	−	0.542736i	$-0.182611\pi$
$480$		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	$-0.213430\pi$
$487$	763.133			1.56701			0.783504	+	0.621386i	$0.213430\pi$
$488$		0			0
$489$			198.000i			0.404908i
$490$		0			0
$491$	592.860	+	592.860i	1.20745	+	1.20745i	0.971849	+	0.235606i	$0.0757074\pi$
$491$	592.860	+	592.860i	1.20745	+	1.20745i	0.235606	+	0.971849i	$0.424293\pi$
$492$		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	−0.304908	−	0.952382i	$-0.598626\pi$
$499$	323.090	−	323.090i	0.647474	−	0.647474i	0.952382	+	0.304908i	$0.0986256\pi$
$500$		0			0
$501$	170.718	−	170.718i	0.340754	−	0.340754i
$502$		0			0
$503$	646.829			1.28594			0.642971	−	0.765891i	$-0.277702\pi$
$503$	646.829			1.28594			0.642971	+	0.765891i	$0.277702\pi$
$504$		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	0.363046	−	0.931771i	$-0.381737\pi$
$523$	−297.443	−	297.443i	−0.568726	−	0.568726i	−0.931771	+	0.363046i	$0.881737\pi$
$524$		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	$-0.499767\pi$
$547$	547.400	−	547.400i	1.00073	−	1.00073i	1.00000		0.000731952i	$-0.000232988\pi$
$548$		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	−0.426189	−	0.904634i	$-0.640144\pi$
$563$	269.364	−	269.364i	0.478444	−	0.478444i	0.904634	+	0.426189i	$0.140144\pi$
$564$		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	0.732857	−	0.680382i	$-0.238186\pi$
$571$	29.9633	+	29.9633i	0.0524751	+	0.0524751i	−0.680382	+	0.732857i	$0.738186\pi$
$572$		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	0.912157	−	0.409841i	$-0.134416\pi$
$587$	294.860	+	294.860i	0.502317	+	0.502317i	−0.409841	+	0.912157i	$0.634416\pi$
$588$		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	$-0.683584\pi$
$599$	−653.269			−1.09060			−0.545299	+	0.838241i	$0.683584\pi$
$600$		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	$-0.916973\pi$
$607$		−	313.076i		−	0.515776i	0.966175	−	0.257888i	$-0.0830266\pi$
$608$		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	0.300081	−	0.953914i	$-0.402986\pi$
$619$	−404.723	−	404.723i	−0.653833	−	0.653833i	−0.953914	+	0.300081i	$0.902986\pi$
$620$		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	$-0.317623\pi$
$631$	684.151			1.08423			0.542116	+	0.840303i	$0.317623\pi$
$632$		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	−0.884183	−	0.467140i	$-0.845284\pi$
$643$	−268.158	+	268.158i	−0.417043	+	0.417043i	0.467140	+	0.884183i	$0.345284\pi$
$644$		0			0
$645$	74.8513	−	74.8513i	0.116048	−	0.116048i
$646$		0			0
$647$	−910.184			−1.40678			−0.703388	−	0.710806i	$-0.748330\pi$
$647$	−910.184			−1.40678			−0.703388	+	0.710806i	$0.748330\pi$
$648$		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	−0.293226	−	0.956043i	$-0.594729\pi$
$659$	436.796	−	436.796i	0.662817	−	0.662817i	0.956043	+	0.293226i	$0.0947290\pi$
$660$		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	$0.138533\pi$
$683$	907.287	+	907.287i	1.32838	+	1.32838i	0.421605	+	0.906780i	$0.361467\pi$
$684$		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	−0.286951	−	0.957945i	$-0.592642\pi$
$691$	463.657	−	463.657i	0.670994	−	0.670994i	0.957945	+	0.286951i	$0.0926417\pi$
$692$		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	$-0.680227\pi$
$719$		−	1213.59i		−	1.68789i	0.536428	−	0.843946i	$-0.319773\pi$
$720$		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	$-0.886805\pi$
$727$	−1363.03			−1.87487			−0.937433	+	0.348165i	$0.886805\pi$
$728$		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	−0.966823	−	0.255447i	$-0.917777\pi$
$739$	−525.707	+	525.707i	−0.711376	+	0.711376i	0.255447	+	0.966823i	$0.417777\pi$
$740$		0			0
$741$	4.50773	−	4.50773i	0.00608331	−	0.00608331i
$742$		0			0
$743$	−702.045			−0.944878			−0.472439	−	0.881363i	$-0.656626\pi$
$743$	−702.045			−0.944878			−0.472439	+	0.881363i	$0.656626\pi$
$744$		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	$0.330038\pi$
$751$			1292.93i			1.72161i	−0.508938	+	0.860803i	$0.669962\pi$
$752$		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	$0.672130\pi$
$787$	−1079.85	+	1079.85i	−1.37211	+	1.37211i	−0.857316	+	0.514791i	$0.827870\pi$
$788$		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	0.991003	−	0.133843i	$-0.0427317\pi$
$811$	695.157	+	695.157i	0.857160	+	0.857160i	−0.133843	+	0.991003i	$0.542732\pi$
$812$		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	$-0.408460\pi$
$823$	466.862			0.567269			0.283635	+	0.958932i	$0.408460\pi$
$824$		0			0
$825$			31.5589i			0.0382532i
$826$		0			0
$827$	−550.394	−	550.394i	−0.665531	−	0.665531i	0.291148	−	0.956678i	$-0.405963\pi$
$827$	−550.394	−	550.394i	−0.665531	−	0.665531i	−0.956678	+	0.291148i	$0.905963\pi$
$828$		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	$-0.842518\pi$
$839$	−1476.79			−1.76018			−0.880090	+	0.474807i	$0.842518\pi$
$840$		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	0.803943	−	0.594707i	$-0.202732\pi$
$859$	179.734	+	179.734i	0.209236	+	0.209236i	−0.594707	+	0.803943i	$0.702732\pi$
$860$		0			0
$861$	514.344	+	514.344i	0.597379	+	0.597379i
$862$		0			0
$863$		−	128.563i		−	0.148972i	−0.997222	−	0.0744859i	$-0.976268\pi$
$863$		−	128.563i		−	0.148972i	0.997222	−	0.0744859i	$-0.0237316\pi$
$864$		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	$0.525093\pi$
$883$	−949.795	+	949.795i	−1.07565	+	1.07565i	−0.996894	+	0.0787509i	$0.974907\pi$
$884$		0			0
$885$	−247.426	+	247.426i	−0.279577	+	0.279577i
$886$		0			0
$887$	589.393			0.664479			0.332240	−	0.943195i	$-0.392196\pi$
$887$	589.393			0.664479			0.332240	+	0.943195i	$0.392196\pi$
$888$		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	0.0380077	−	0.999277i	$-0.487899\pi$
$907$	−871.872	−	871.872i	−0.961270	−	0.961270i	−0.999277	+	0.0380077i	$0.987899\pi$
$908$		0			0
$909$	251.769	+	251.769i	0.276974	+	0.276974i
$910$		0			0
$911$		−	273.313i		−	0.300014i	−0.988685	−	0.150007i	$-0.952070\pi$
$911$		−	273.313i		−	0.300014i	0.988685	−	0.150007i	$-0.0479297\pi$
$912$		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	$-0.575686\pi$
$919$	−432.925			−0.471083			−0.235541	+	0.971864i	$0.575686\pi$
$920$		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	−0.706133	−	0.708079i	$-0.749562\pi$
$947$	1.84223	−	1.84223i	0.00194533	−	0.00194533i	0.708079	+	0.706133i	$0.249562\pi$
$948$		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	$-0.694262\pi$
$967$	−1108.39			−1.14621			−0.573107	+	0.819480i	$0.694262\pi$
$968$		0			0
$969$		−	154.010i		−	0.158937i
$970$		0			0
$971$	958.931	+	958.931i	0.987571	+	0.987571i	0.999924	−	0.0123531i	$-0.00393222\pi$
$971$	958.931	+	958.931i	0.987571	+	0.987571i	−0.0123531	+	0.999924i	$0.503932\pi$
$972$		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	$-0.628400\pi$
$983$	−771.711			−0.785057			−0.392528	+	0.919740i	$0.628400\pi$
$984$		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	$-0.829934\pi$
$991$		−	1009.27i		−	1.01844i	0.860637	−	0.509219i	$-0.170066\pi$
$992$		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.703.4	yes	8
4.3	odd	2	inner	768.3.l.d.703.1	yes	8
8.3	odd	2		768.3.l.a.703.3	yes	8
8.5	even	2		768.3.l.a.703.2	yes	8
16.3	odd	4		768.3.l.a.319.2	✓	8
16.5	even	4	inner	768.3.l.d.319.1	yes	8
16.11	odd	4	inner	768.3.l.d.319.4	yes	8
16.13	even	4		768.3.l.a.319.3	yes	8

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

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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