LMFDB - Embedded newform 760.2.d.e.609.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(609,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("760.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.06863055362$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 9 x^{10} - 8 x^{9} - 11 x^{8} + 60 x^{7} - 126 x^{6} + 180 x^{5} - 99 x^{4} + \cdots + 729 $ x^12 - 4x^11 + 9x^10 - 8x^9 - 11x^8 + 60x^7 - 126x^6 + 180x^5 - 99x^4 - 216x^3 + 729x^2 - 972*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.3
Root			$0.588529 - 1.62900i$ of defining polynomial
Character	$\chi$	$=$	760.609
Dual form			760.2.d.e.609.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43031i q^{3} +(-2.06801 - 0.850485i) q^{5} +3.60737i q^{7} -2.90640 q^{9} -2.37997 q^{11} +3.96613i q^{13} +(-2.06694 + 5.02591i) q^{15} +4.28257i q^{17} -1.00000 q^{19} +8.76701 q^{21} +1.07377i q^{23} +(3.55335 + 3.51763i) q^{25} -0.227486i q^{27} -9.47021 q^{29} +6.38211 q^{31} +5.78405i q^{33} +(3.06801 - 7.46008i) q^{35} +2.04540i q^{37} +9.63892 q^{39} -4.38211 q^{41} -7.86284i q^{43} +(6.01046 + 2.47185i) q^{45} +3.83485i q^{47} -6.01310 q^{49} +10.4080 q^{51} +11.7789i q^{53} +(4.92180 + 2.02413i) q^{55} +2.43031i q^{57} -4.59593 q^{59} +6.62819 q^{61} -10.4844i q^{63} +(3.37314 - 8.20201i) q^{65} +7.02837i q^{67} +2.60959 q^{69} +4.99450 q^{71} +2.93216i q^{73} +(8.54892 - 8.63573i) q^{75} -8.58541i q^{77} -0.860616 q^{79} -9.27205 q^{81} -11.9179i q^{83} +(3.64227 - 8.85642i) q^{85} +23.0155i q^{87} -13.6460 q^{89} -14.3073 q^{91} -15.5105i q^{93} +(2.06801 + 0.850485i) q^{95} +18.5757i q^{97} +6.91712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{5} - 16 q^{9} - 4 q^{11} - 12 q^{15} - 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} + 16 q^{31} + 6 q^{35} + 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} - 68 q^{51} - 18 q^{55} + 4 q^{59} + 20 q^{61}+ \cdots + 44 q^{99}+O(q^{100}) $ 12 * q + 6 * q^5 - 16 * q^9 - 4 * q^11 - 12 * q^15 - 12 * q^19 + 36 * q^21 + 18 * q^25 + 4 * q^29 + 16 * q^31 + 6 * q^35 + 36 * q^39 + 8 * q^41 - 2 * q^45 - 4 * q^49 - 68 * q^51 - 18 * q^55 + 4 * q^59 + 20 * q^61 + 20 * q^65 - 36 * q^69 - 16 * q^71 - 16 * q^75 + 40 * q^79 + 12 * q^81 + 6 * q^85 - 64 * q^89 + 20 * q^91 - 6 * q^95 + 44 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	2.43031i		−	1.40314i	−0.712601	−	0.701569i	$-0.752483\pi$
$3$		−	2.43031i		−	1.40314i	0.712601	−	0.701569i	$-0.247517\pi$
$4$		0			0
$5$	−2.06801	−	0.850485i	−0.924843	−	0.380349i
$5$	−2.06801	−	0.850485i	−0.924843	−	0.380349i
$6$		0			0
$7$			3.60737i			1.36346i	0.731605	+	0.681728i	$0.238771\pi$
$7$			3.60737i			1.36346i	−0.731605	+	0.681728i	$0.761229\pi$
$8$		0			0
$9$	−2.90640			−0.968799
$10$		0			0
$11$	−2.37997			−0.717587			−0.358793	−	0.933417i	$-0.616812\pi$
$11$	−2.37997			−0.717587			−0.358793	+	0.933417i	$0.616812\pi$
$12$		0			0
$13$			3.96613i			1.10001i	0.835162	+	0.550003i	$0.185374\pi$
$13$			3.96613i			1.10001i	−0.835162	+	0.550003i	$0.814626\pi$
$14$		0			0
$15$	−2.06694	+	5.02591i	−0.533682	+	1.29768i
$16$		0			0
$17$			4.28257i			1.03868i	0.854569	+	0.519338i	$0.173822\pi$
$17$			4.28257i			1.03868i	−0.854569	+	0.519338i	$0.826178\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	8.76701			1.91312
$22$		0			0
$23$			1.07377i			0.223897i	0.993714	+	0.111948i	$0.0357091\pi$
$23$			1.07377i			0.223897i	−0.993714	+	0.111948i	$0.964291\pi$
$24$		0			0
$25$	3.55335	+	3.51763i	0.710670	+	0.703526i
$26$		0			0
$27$		−	0.227486i		−	0.0437796i
$28$		0			0
$29$	−9.47021			−1.75857			−0.879287	−	0.476293i	$-0.841980\pi$
$29$	−9.47021			−1.75857			−0.879287	+	0.476293i	$0.841980\pi$
$30$		0			0
$31$	6.38211			1.14626			0.573130	−	0.819464i	$-0.305729\pi$
$31$	6.38211			1.14626			0.573130	+	0.819464i	$0.305729\pi$
$32$		0			0
$33$			5.78405i			1.00687i
$34$		0			0
$35$	3.06801	−	7.46008i	0.518589	−	1.26098i
$36$		0			0
$37$			2.04540i			0.336262i	0.985765	+	0.168131i	$0.0537732\pi$
$37$			2.04540i			0.336262i	−0.985765	+	0.168131i	$0.946227\pi$
$38$		0			0
$39$	9.63892			1.54346
$40$		0			0
$41$	−4.38211			−0.684370			−0.342185	−	0.939633i	$-0.611167\pi$
$41$	−4.38211			−0.684370			−0.342185	+	0.939633i	$0.611167\pi$
$42$		0			0
$43$		−	7.86284i		−	1.19907i	−0.800348	−	0.599536i	$-0.795352\pi$
$43$		−	7.86284i		−	1.19907i	0.800348	−	0.599536i	$-0.204648\pi$
$44$		0			0
$45$	6.01046	+	2.47185i	0.895987	+	0.368481i
$46$		0			0
$47$			3.83485i			0.559371i	0.960092	+	0.279685i	$0.0902301\pi$
$47$			3.83485i			0.559371i	−0.960092	+	0.279685i	$0.909770\pi$
$48$		0			0
$49$	−6.01310			−0.859014
$50$		0			0
$51$	10.4080			1.45741
$52$		0			0
$53$			11.7789i			1.61796i	0.587836	+	0.808980i	$0.299980\pi$
$53$			11.7789i			1.61796i	−0.587836	+	0.808980i	$0.700020\pi$
$54$		0			0
$55$	4.92180	+	2.02413i	0.663655	+	0.272933i
$56$		0			0
$57$			2.43031i			0.321902i
$58$		0			0
$59$	−4.59593			−0.598340			−0.299170	−	0.954200i	$-0.596710\pi$
$59$	−4.59593			−0.598340			−0.299170	+	0.954200i	$0.596710\pi$
$60$		0			0
$61$	6.62819			0.848653			0.424327	−	0.905509i	$-0.360511\pi$
$61$	6.62819			0.848653			0.424327	+	0.905509i	$0.360511\pi$
$62$		0			0
$63$		−	10.4844i		−	1.32092i
$64$		0			0
$65$	3.37314	−	8.20201i	0.418386	−	1.01733i
$66$		0			0
$67$			7.02837i			0.858652i	0.903150	+	0.429326i	$0.141249\pi$
$67$			7.02837i			0.858652i	−0.903150	+	0.429326i	$0.858751\pi$
$68$		0			0
$69$	2.60959			0.314158
$70$		0			0
$71$	4.99450			0.592738			0.296369	−	0.955074i	$-0.404224\pi$
$71$	4.99450			0.592738			0.296369	+	0.955074i	$0.404224\pi$
$72$		0			0
$73$			2.93216i			0.343183i	0.985168	+	0.171592i	$0.0548910\pi$
$73$			2.93216i			0.343183i	−0.985168	+	0.171592i	$0.945109\pi$
$74$		0			0
$75$	8.54892	−	8.63573i	0.987144	−	0.997169i
$76$		0			0
$77$		−	8.58541i		−	0.978398i
$78$		0			0
$79$	−0.860616			−0.0968268			−0.0484134	−	0.998827i	$-0.515416\pi$
$79$	−0.860616			−0.0968268			−0.0484134	+	0.998827i	$0.515416\pi$
$80$		0			0
$81$	−9.27205			−1.03023
$82$		0			0
$83$		−	11.9179i		−	1.30816i	−0.756424	−	0.654081i	$-0.773055\pi$
$83$		−	11.9179i		−	1.30816i	0.756424	−	0.654081i	$-0.226945\pi$
$84$		0			0
$85$	3.64227	−	8.85642i	0.395059	−	0.960613i
$86$		0			0
$87$			23.0155i			2.46752i
$88$		0			0
$89$	−13.6460			−1.44647			−0.723237	−	0.690600i	$-0.757346\pi$
$89$	−13.6460			−1.44647			−0.723237	+	0.690600i	$0.757346\pi$
$90$		0			0
$91$	−14.3073			−1.49981
$92$		0			0
$93$		−	15.5105i		−	1.60836i
$94$		0			0
$95$	2.06801	+	0.850485i	0.212174	+	0.0872579i
$96$		0			0
$97$			18.5757i			1.88608i	0.332681	+	0.943040i	$0.392047\pi$
$97$			18.5757i			1.88608i	−0.332681	+	0.943040i	$0.607953\pi$
$98$		0			0
$99$	6.91712			0.695197
$100$		0			0
$101$	−9.82645			−0.977769			−0.488884	−	0.872349i	$-0.662596\pi$
$101$	−9.82645			−0.977769			−0.488884	+	0.872349i	$0.662596\pi$
$102$		0			0
$103$			8.85269i			0.872282i	0.899878	+	0.436141i	$0.143655\pi$
$103$			8.85269i			0.872282i	−0.899878	+	0.436141i	$0.856345\pi$
$104$		0			0
$105$	−18.1303	−	7.45621i	−1.76934	−	0.727652i
$106$		0			0
$107$			6.58269i			0.636372i	0.948028	+	0.318186i	$0.103074\pi$
$107$			6.58269i			0.636372i	−0.948028	+	0.318186i	$0.896926\pi$
$108$		0			0
$109$	−3.40919			−0.326541			−0.163271	−	0.986581i	$-0.552204\pi$
$109$	−3.40919			−0.326541			−0.163271	+	0.986581i	$0.552204\pi$
$110$		0			0
$111$	4.97096			0.471823
$112$		0			0
$113$			6.58065i			0.619055i	0.950890	+	0.309528i	$0.100171\pi$
$113$			6.58065i			0.619055i	−0.950890	+	0.309528i	$0.899829\pi$
$114$		0			0
$115$	0.913226	−	2.22057i	0.0851588	−	0.207069i
$116$		0			0
$117$		−	11.5271i		−	1.06569i
$118$		0			0
$119$	−15.4488			−1.41619
$120$		0			0
$121$	−5.33576			−0.485069
$122$		0			0
$123$			10.6499i			0.960267i
$124$		0			0
$125$	−4.35668	−	10.2966i	−0.389673	−	0.920953i
$126$		0			0
$127$		−	12.2826i		−	1.08991i	−0.838466	−	0.544953i	$-0.816547\pi$
$127$		−	12.2826i		−	1.08991i	0.838466	−	0.544953i	$-0.183453\pi$
$128$		0			0
$129$	−19.1091			−1.68246
$130$		0			0
$131$	−10.0308			−0.876395			−0.438198	−	0.898879i	$-0.644383\pi$
$131$	−10.0308			−0.876395			−0.438198	+	0.898879i	$0.644383\pi$
$132$		0			0
$133$		−	3.60737i		−	0.312798i
$134$		0			0
$135$	−0.193473	+	0.470443i	−0.0166515	+	0.0404893i
$136$		0			0
$137$		−	17.1273i		−	1.46329i	−0.681687	−	0.731644i	$-0.738753\pi$
$137$		−	17.1273i		−	1.46329i	0.681687	−	0.731644i	$-0.261247\pi$
$138$		0			0
$139$	5.50494			0.466923			0.233461	−	0.972366i	$-0.424995\pi$
$139$	5.50494			0.466923			0.233461	+	0.972366i	$0.424995\pi$
$140$		0			0
$141$	9.31987			0.784875
$142$		0			0
$143$		−	9.43926i		−	0.789350i
$144$		0			0
$145$	19.5845	+	8.05427i	1.62641	+	0.668871i
$146$		0			0
$147$			14.6137i			1.20532i
$148$		0			0
$149$	21.4593			1.75801			0.879007	−	0.476808i	$-0.158206\pi$
$149$	21.4593			1.75801			0.879007	+	0.476808i	$0.158206\pi$
$150$		0			0
$151$	−14.1805			−1.15399			−0.576996	−	0.816747i	$-0.695775\pi$
$151$	−14.1805			−1.15399			−0.576996	+	0.816747i	$0.695775\pi$
$152$		0			0
$153$		−	12.4469i		−	1.00627i
$154$		0			0
$155$	−13.1983	−	5.42789i	−1.06011	−	0.435979i
$156$		0			0
$157$			2.69891i			0.215397i	0.994184	+	0.107698i	$0.0343481\pi$
$157$			2.69891i			0.215397i	−0.994184	+	0.107698i	$0.965652\pi$
$158$		0			0
$159$	28.6264			2.27022
$160$		0			0
$161$	−3.87349			−0.305273
$162$		0			0
$163$			11.1955i			0.876898i	0.898756	+	0.438449i	$0.144472\pi$
$163$			11.1955i			0.876898i	−0.898756	+	0.438449i	$0.855528\pi$
$164$		0			0
$165$	4.91925	−	11.9615i	0.382963	−	0.931200i
$166$		0			0
$167$		−	9.71331i		−	0.751639i	−0.926693	−	0.375819i	$-0.877361\pi$
$167$		−	9.71331i		−	0.751639i	0.926693	−	0.375819i	$-0.122639\pi$
$168$		0			0
$169$	−2.73019			−0.210015
$170$		0			0
$171$	2.90640			0.222258
$172$		0			0
$173$			14.2783i			1.08556i	0.839874	+	0.542781i	$0.182629\pi$
$173$			14.2783i			1.08556i	−0.839874	+	0.542781i	$0.817371\pi$
$174$		0			0
$175$	−12.6894	+	12.8182i	−0.959227	+	0.968968i
$176$		0			0
$177$			11.1695i			0.839554i
$178$		0			0
$179$	17.1327			1.28056			0.640278	−	0.768143i	$-0.278819\pi$
$179$	17.1327			1.28056			0.640278	+	0.768143i	$0.278819\pi$
$180$		0			0
$181$	−23.6883			−1.76074			−0.880370	−	0.474288i	$-0.842705\pi$
$181$	−23.6883			−1.76074			−0.880370	+	0.474288i	$0.842705\pi$
$182$		0			0
$183$		−	16.1085i		−	1.19078i
$184$		0			0
$185$	1.73959	−	4.22992i	0.127897	−	0.310990i
$186$		0			0
$187$		−	10.1924i		−	0.745341i
$188$		0			0
$189$	0.820624			0.0596916
$190$		0			0
$191$	25.2506			1.82707			0.913535	−	0.406761i	$-0.133342\pi$
$191$	25.2506			1.82707			0.913535	+	0.406761i	$0.133342\pi$
$192$		0			0
$193$		−	16.6814i		−	1.20076i	−0.799716	−	0.600378i	$-0.795017\pi$
$193$		−	16.6814i		−	1.20076i	0.799716	−	0.600378i	$-0.204983\pi$
$194$		0			0
$195$	−19.9334	−	8.19776i	−1.42746	−	0.587054i
$196$		0			0
$197$			17.1471i			1.22168i	0.791755	+	0.610839i	$0.209168\pi$
$197$			17.1471i			1.22168i	−0.791755	+	0.610839i	$0.790832\pi$
$198$		0			0
$199$	−22.1273			−1.56856			−0.784280	−	0.620407i	$-0.786967\pi$
$199$	−22.1273			−1.56856			−0.784280	+	0.620407i	$0.786967\pi$
$200$		0			0
$201$	17.0811			1.20481
$202$		0			0
$203$		−	34.1625i		−	2.39774i
$204$		0			0
$205$	9.06225	+	3.72692i	0.632935	+	0.260299i
$206$		0			0
$207$		−	3.12080i		−	0.216911i
$208$		0			0
$209$	2.37997			0.164626
$210$		0			0
$211$	−24.6690			−1.69828			−0.849141	−	0.528166i	$-0.822880\pi$
$211$	−24.6690			−1.69828			−0.849141	+	0.528166i	$0.822880\pi$
$212$		0			0
$213$		−	12.1382i		−	0.831694i
$214$		0			0
$215$	−6.68723	+	16.2605i	−0.456065	+	1.10895i
$216$		0			0
$217$			23.0226i			1.56288i
$218$		0			0
$219$	7.12605			0.481534
$220$		0			0
$221$	−16.9852			−1.14255
$222$		0			0
$223$		−	21.4787i		−	1.43832i	−0.694843	−	0.719162i	$-0.744526\pi$
$223$		−	21.4787i		−	1.43832i	0.694843	−	0.719162i	$-0.255474\pi$
$224$		0			0
$225$	−10.3274	−	10.2236i	−0.688496	−	0.681575i
$226$		0			0
$227$		−	20.5568i		−	1.36440i	−0.731164	−	0.682201i	$-0.761023\pi$
$227$		−	20.5568i		−	1.36440i	0.731164	−	0.682201i	$-0.238977\pi$
$228$		0			0
$229$	17.6465			1.16611			0.583057	−	0.812431i	$-0.301856\pi$
$229$	17.6465			1.16611			0.583057	+	0.812431i	$0.301856\pi$
$230$		0			0
$231$	−20.8652			−1.37283
$232$		0			0
$233$		−	15.5021i		−	1.01558i	−0.861482	−	0.507788i	$-0.830463\pi$
$233$		−	15.5021i		−	1.01558i	0.861482	−	0.507788i	$-0.169537\pi$
$234$		0			0
$235$	3.26149	−	7.93052i	0.212756	−	0.517330i
$236$		0			0
$237$			2.09156i			0.135862i
$238$		0			0
$239$	28.2186			1.82531			0.912656	−	0.408730i	$-0.134028\pi$
$239$	28.2186			1.82531			0.912656	+	0.408730i	$0.134028\pi$
$240$		0			0
$241$	23.8653			1.53730			0.768649	−	0.639671i	$-0.220929\pi$
$241$	23.8653			1.53730			0.768649	+	0.639671i	$0.220929\pi$
$242$		0			0
$243$			21.8515i			1.40177i
$244$		0			0
$245$	12.4352	+	5.11405i	0.794453	+	0.326725i
$246$		0			0
$247$		−	3.96613i		−	0.252359i
$248$		0			0
$249$	−28.9642			−1.83553
$250$		0			0
$251$	8.15793			0.514924			0.257462	−	0.966288i	$-0.417114\pi$
$251$	8.15793			0.514924			0.257462	+	0.966288i	$0.417114\pi$
$252$		0			0
$253$		−	2.55554i		−	0.160665i
$254$		0			0
$255$	−21.5238	−	8.85183i	−1.34787	−	0.554323i
$256$		0			0
$257$			6.07899i			0.379197i	0.981862	+	0.189598i	$0.0607186\pi$
$257$			6.07899i			0.379197i	−0.981862	+	0.189598i	$0.939281\pi$
$258$		0			0
$259$	−7.37852			−0.458479
$260$		0			0
$261$	27.5242			1.70370
$262$		0			0
$263$		−	8.47535i		−	0.522612i	−0.965256	−	0.261306i	$-0.915847\pi$
$263$		−	8.47535i		−	0.522612i	0.965256	−	0.261306i	$-0.0841532\pi$
$264$		0			0
$265$	10.0178	−	24.3590i	0.615389	−	1.49636i
$266$		0			0
$267$			33.1640i			2.02960i
$268$		0			0
$269$	16.6667			1.01619			0.508093	−	0.861302i	$-0.330351\pi$
$269$	16.6667			1.01619			0.508093	+	0.861302i	$0.330351\pi$
$270$		0			0
$271$	−11.8151			−0.717718			−0.358859	−	0.933392i	$-0.616834\pi$
$271$	−11.8151			−0.717718			−0.358859	+	0.933392i	$0.616834\pi$
$272$		0			0
$273$			34.7711i			2.10444i
$274$		0			0
$275$	−8.45685	−	8.37183i	−0.509967	−	0.504841i
$276$		0			0
$277$		−	2.10858i		−	0.126692i	−0.997992	−	0.0633462i	$-0.979823\pi$
$277$		−	2.10858i		−	0.126692i	0.997992	−	0.0633462i	$-0.0201772\pi$
$278$		0			0
$279$	−18.5489			−1.11050
$280$		0			0
$281$	−3.76859			−0.224815			−0.112408	−	0.993662i	$-0.535856\pi$
$281$	−3.76859			−0.224815			−0.112408	+	0.993662i	$0.535856\pi$
$282$		0			0
$283$		−	0.0720389i		−	0.00428227i	−0.999998	−	0.00214114i	$-0.999318\pi$
$283$		−	0.0720389i		−	0.00428227i	0.999998	−	0.00214114i	$-0.000681545\pi$
$284$		0			0
$285$	2.06694	−	5.02591i	0.122435	−	0.297709i
$286$		0			0
$287$		−	15.8079i		−	0.933109i
$288$		0			0
$289$	−1.34044			−0.0788494
$290$		0			0
$291$	45.1447			2.64643
$292$		0			0
$293$			9.98245i			0.583181i	0.956543	+	0.291590i	$0.0941844\pi$
$293$			9.98245i			0.583181i	−0.956543	+	0.291590i	$0.905816\pi$
$294$		0			0
$295$	9.50445	+	3.90877i	0.553370	+	0.227578i
$296$		0			0
$297$			0.541408i			0.0314157i
$298$		0			0
$299$	−4.25872			−0.246288
$300$		0			0
$301$	28.3642			1.63488
$302$		0			0
$303$			23.8813i			1.37195i
$304$		0			0
$305$	−13.7072	−	5.63718i	−0.784871	−	0.322784i
$306$		0			0
$307$			21.0679i			1.20241i	0.799096	+	0.601204i	$0.205312\pi$
$307$			21.0679i			1.20241i	−0.799096	+	0.601204i	$0.794688\pi$
$308$		0			0
$309$	21.5148			1.22393
$310$		0			0
$311$	−29.9966			−1.70095			−0.850475	−	0.526015i	$-0.823686\pi$
$311$	−29.9966			−1.70095			−0.850475	+	0.526015i	$0.823686\pi$
$312$		0			0
$313$		−	5.48121i		−	0.309817i	−0.987929	−	0.154908i	$-0.950492\pi$
$313$		−	5.48121i		−	0.309817i	0.987929	−	0.154908i	$-0.0495082\pi$
$314$		0			0
$315$	−8.91686	+	21.6819i	−0.502408	+	1.22164i
$316$		0			0
$317$			0.687909i			0.0386368i	0.999813	+	0.0193184i	$0.00614962\pi$
$317$			0.687909i			0.0386368i	−0.999813	+	0.0193184i	$0.993850\pi$
$318$		0			0
$319$	22.5388			1.26193
$320$		0			0
$321$	15.9980			0.892919
$322$		0			0
$323$		−	4.28257i		−	0.238289i
$324$		0			0
$325$	−13.9514	+	14.0930i	−0.773883	+	0.781742i
$326$		0			0
$327$			8.28538i			0.458183i
$328$		0			0
$329$	−13.8337			−0.762678
$330$		0			0
$331$	23.1679			1.27342			0.636712	−	0.771102i	$-0.280294\pi$
$331$	23.1679			1.27342			0.636712	+	0.771102i	$0.280294\pi$
$332$		0			0
$333$		−	5.94475i		−	0.325771i
$334$		0			0
$335$	5.97752	−	14.5347i	0.326587	−	0.794118i
$336$		0			0
$337$			28.8512i			1.57162i	0.618465	+	0.785812i	$0.287755\pi$
$337$			28.8512i			1.57162i	−0.618465	+	0.785812i	$0.712245\pi$
$338$		0			0
$339$	15.9930			0.868620
$340$		0			0
$341$	−15.1892			−0.822541
$342$		0			0
$343$			3.56013i			0.192229i
$344$		0			0
$345$	−5.39667	−	2.21942i	−0.290547	−	0.119490i
$346$		0			0
$347$			5.51365i			0.295988i	0.988988	+	0.147994i	$0.0472816\pi$
$347$			5.51365i			0.295988i	−0.988988	+	0.147994i	$0.952718\pi$
$348$		0			0
$349$	6.62869			0.354826			0.177413	−	0.984137i	$-0.443227\pi$
$349$	6.62869			0.354826			0.177413	+	0.984137i	$0.443227\pi$
$350$		0			0
$351$	0.902238			0.0481579
$352$		0			0
$353$			10.0657i			0.535745i	0.963454	+	0.267872i	$0.0863206\pi$
$353$			10.0657i			0.535745i	−0.963454	+	0.267872i	$0.913679\pi$
$354$		0			0
$355$	−10.3287	−	4.24775i	−0.548190	−	0.225447i
$356$		0			0
$357$			37.5454i			1.98711i
$358$		0			0
$359$	23.9747			1.26534			0.632668	−	0.774423i	$-0.281960\pi$
$359$	23.9747			1.26534			0.632668	+	0.774423i	$0.281960\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			12.9675i			0.680620i
$364$		0			0
$365$	2.49376	−	6.06374i	0.130529	−	0.317391i
$366$		0			0
$367$			2.10030i			0.109635i	0.998496	+	0.0548174i	$0.0174577\pi$
$367$			2.10030i			0.109635i	−0.998496	+	0.0548174i	$0.982542\pi$
$368$		0			0
$369$	12.7361			0.663017
$370$		0			0
$371$	−42.4909			−2.20602
$372$		0			0
$373$		−	19.2927i		−	0.998939i	−0.866331	−	0.499470i	$-0.833528\pi$
$373$		−	19.2927i		−	0.998939i	0.866331	−	0.499470i	$-0.166472\pi$
$374$		0			0
$375$	−25.0238	+	10.5881i	−1.29223	+	0.546766i
$376$		0			0
$377$		−	37.5601i		−	1.93444i
$378$		0			0
$379$	−4.29783			−0.220765			−0.110382	−	0.993889i	$-0.535208\pi$
$379$	−4.29783			−0.220765			−0.110382	+	0.993889i	$0.535208\pi$
$380$		0			0
$381$	−29.8506			−1.52929
$382$		0			0
$383$			15.6099i			0.797628i	0.917032	+	0.398814i	$0.130578\pi$
$383$			15.6099i			0.797628i	−0.917032	+	0.398814i	$0.869422\pi$
$384$		0			0
$385$	−7.30176	+	17.7547i	−0.372132	+	0.904865i
$386$		0			0
$387$			22.8525i			1.16166i
$388$		0			0
$389$	−15.4509			−0.783390			−0.391695	−	0.920095i	$-0.628111\pi$
$389$	−15.4509			−0.783390			−0.391695	+	0.920095i	$0.628111\pi$
$390$		0			0
$391$	−4.59850			−0.232556
$392$		0			0
$393$			24.3779i			1.22970i
$394$		0			0
$395$	1.77976	+	0.731941i	0.0895496	+	0.0368279i
$396$		0			0
$397$			0.852887i			0.0428051i	0.999771	+	0.0214026i	$0.00681317\pi$
$397$			0.852887i			0.0428051i	−0.999771	+	0.0214026i	$0.993187\pi$
$398$		0			0
$399$	−8.76701			−0.438900
$400$		0			0
$401$	38.8284			1.93900			0.969500	−	0.245091i	$-0.0788180\pi$
$401$	38.8284			1.93900			0.969500	+	0.245091i	$0.0788180\pi$
$402$		0			0
$403$			25.3123i			1.26089i
$404$		0			0
$405$	19.1747	+	7.88574i	0.952799	+	0.391846i
$406$		0			0
$407$		−	4.86799i		−	0.241297i
$408$		0			0
$409$	−6.16871			−0.305023			−0.152512	−	0.988302i	$-0.548736\pi$
$409$	−6.16871			−0.305023			−0.152512	+	0.988302i	$0.548736\pi$
$410$		0			0
$411$	−41.6247			−2.05320
$412$		0			0
$413$		−	16.5792i		−	0.815810i
$414$		0			0
$415$	−10.1360	+	24.6464i	−0.497558	+	1.20985i
$416$		0			0
$417$		−	13.3787i		−	0.655157i
$418$		0			0
$419$	14.0712			0.687422			0.343711	−	0.939075i	$-0.388316\pi$
$419$	14.0712			0.687422			0.343711	+	0.939075i	$0.388316\pi$
$420$		0			0
$421$	4.85298			0.236520			0.118260	−	0.992983i	$-0.462268\pi$
$421$	4.85298			0.236520			0.118260	+	0.992983i	$0.462268\pi$
$422$		0			0
$423$		−	11.1456i		−	0.541918i
$424$		0			0
$425$	−15.0645	+	15.2175i	−0.730736	+	0.738156i
$426$		0			0
$427$			23.9103i			1.15710i
$428$		0			0
$429$	−22.9403			−1.10757
$430$		0			0
$431$	−15.6171			−0.752251			−0.376126	−	0.926569i	$-0.622744\pi$
$431$	−15.6171			−0.752251			−0.376126	+	0.926569i	$0.622744\pi$
$432$		0			0
$433$		−	25.3497i		−	1.21823i	−0.793082	−	0.609114i	$-0.791525\pi$
$433$		−	25.3497i		−	1.21823i	0.793082	−	0.609114i	$-0.208475\pi$
$434$		0			0
$435$	19.5744	−	47.5964i	0.938519	−	2.28207i
$436$		0			0
$437$		−	1.07377i		−	0.0513654i
$438$		0			0
$439$	−23.4184			−1.11770			−0.558849	−	0.829269i	$-0.688757\pi$
$439$	−23.4184			−1.11770			−0.558849	+	0.829269i	$0.688757\pi$
$440$		0			0
$441$	17.4764			0.832211
$442$		0			0
$443$			25.1133i			1.19317i	0.802551	+	0.596584i	$0.203476\pi$
$443$			25.1133i			1.19317i	−0.802551	+	0.596584i	$0.796524\pi$
$444$		0			0
$445$	28.2201	+	11.6057i	1.33776	+	0.550164i
$446$		0			0
$447$		−	52.1527i		−	2.46674i
$448$		0			0
$449$	−6.37270			−0.300746			−0.150373	−	0.988629i	$-0.548047\pi$
$449$	−6.37270			−0.300746			−0.150373	+	0.988629i	$0.548047\pi$
$450$		0			0
$451$	10.4293			0.491095
$452$		0			0
$453$			34.4630i			1.61921i
$454$		0			0
$455$	29.5876	+	12.1681i	1.38709	+	0.570451i
$456$		0			0
$457$		−	40.6893i		−	1.90336i	−0.307085	−	0.951682i	$-0.599354\pi$
$457$		−	40.6893i		−	1.90336i	0.307085	−	0.951682i	$-0.400646\pi$
$458$		0			0
$459$	0.974224			0.0454729
$460$		0			0
$461$	−11.5304			−0.537024			−0.268512	−	0.963276i	$-0.586532\pi$
$461$	−11.5304			−0.537024			−0.268512	+	0.963276i	$0.586532\pi$
$462$		0			0
$463$			32.0882i			1.49127i	0.666357	+	0.745633i	$0.267853\pi$
$463$			32.0882i			1.49127i	−0.666357	+	0.745633i	$0.732147\pi$
$464$		0			0
$465$	−13.1914	+	32.0759i	−0.611738	+	1.48748i
$466$		0			0
$467$			20.6254i			0.954429i	0.878787	+	0.477214i	$0.158353\pi$
$467$			20.6254i			0.954429i	−0.878787	+	0.477214i	$0.841647\pi$
$468$		0			0
$469$	−25.3539			−1.17073
$470$		0			0
$471$	6.55919			0.302231
$472$		0			0
$473$			18.7133i			0.860438i
$474$		0			0
$475$	−3.55335	−	3.51763i	−0.163039	−	0.161400i
$476$		0			0
$477$		−	34.2342i		−	1.56748i
$478$		0			0
$479$	−10.8936			−0.497740			−0.248870	−	0.968537i	$-0.580059\pi$
$479$	−10.8936			−0.497740			−0.248870	+	0.968537i	$0.580059\pi$
$480$		0			0
$481$	−8.11234			−0.369891
$482$		0			0
$483$			9.41376i			0.428341i
$484$		0			0
$485$	15.7984	−	38.4148i	0.717367	−	1.74433i
$486$		0			0
$487$		−	5.05668i		−	0.229140i	−0.993415	−	0.114570i	$-0.963451\pi$
$487$		−	5.05668i		−	0.229140i	0.993415	−	0.114570i	$-0.0365491\pi$
$488$		0			0
$489$	27.2085			1.23041
$490$		0			0
$491$	−25.5833			−1.15456			−0.577279	−	0.816547i	$-0.695886\pi$
$491$	−25.5833			−1.15456			−0.577279	+	0.816547i	$0.695886\pi$
$492$		0			0
$493$		−	40.5569i		−	1.82659i
$494$		0			0
$495$	−14.3047	−	5.88291i	−0.642948	−	0.264417i
$496$		0			0
$497$			18.0170i			0.808172i
$498$		0			0
$499$	34.0476			1.52418			0.762090	−	0.647471i	$-0.224173\pi$
$499$	34.0476			1.52418			0.762090	+	0.647471i	$0.224173\pi$
$500$		0			0
$501$	−23.6063			−1.05465
$502$		0			0
$503$			5.70145i			0.254215i	0.991889	+	0.127108i	$0.0405693\pi$
$503$			5.70145i			0.254215i	−0.991889	+	0.127108i	$0.959431\pi$
$504$		0			0
$505$	20.3212	+	8.35725i	0.904283	+	0.371893i
$506$		0			0
$507$			6.63521i			0.294680i
$508$		0			0
$509$	27.9604			1.23932			0.619661	−	0.784870i	$-0.287270\pi$
$509$	27.9604			1.23932			0.619661	+	0.784870i	$0.287270\pi$
$510$		0			0
$511$	−10.5774			−0.467916
$512$		0			0
$513$			0.227486i			0.0100437i
$514$		0			0
$515$	7.52909	−	18.3075i	0.331771	−	0.806724i
$516$		0			0
$517$		−	9.12682i		−	0.401397i
$518$		0			0
$519$	34.7008			1.52319
$520$		0			0
$521$	−0.419591			−0.0183826			−0.00919131	−	0.999958i	$-0.502926\pi$
$521$	−0.419591			−0.0183826			−0.00919131	+	0.999958i	$0.502926\pi$
$522$		0			0
$523$			2.23302i			0.0976433i	0.998808	+	0.0488217i	$0.0155466\pi$
$523$			2.23302i			0.0976433i	−0.998808	+	0.0488217i	$0.984453\pi$
$524$		0			0
$525$	31.1523	+	30.8391i	1.35960	+	1.34593i
$526$		0			0
$527$			27.3319i			1.19059i
$528$		0			0
$529$	21.8470			0.949870
$530$		0			0
$531$	13.3576			0.579671
$532$		0			0
$533$		−	17.3800i		−	0.752812i
$534$		0			0
$535$	5.59848	−	13.6131i	0.242043	−	0.588545i
$536$		0			0
$537$		−	41.6377i		−	1.79680i
$538$		0			0
$539$	14.3110			0.616417
$540$		0			0
$541$	39.9938			1.71947			0.859734	−	0.510741i	$-0.170629\pi$
$541$	39.9938			1.71947			0.859734	+	0.510741i	$0.170629\pi$
$542$		0			0
$543$			57.5699i			2.47056i
$544$		0			0
$545$	7.05025	+	2.89947i	0.301999	+	0.124199i
$546$		0			0
$547$			15.5651i			0.665516i	0.943012	+	0.332758i	$0.107979\pi$
$547$			15.5651i			0.665516i	−0.943012	+	0.332758i	$0.892021\pi$
$548$		0			0
$549$	−19.2642			−0.822174
$550$		0			0
$551$	9.47021			0.403444
$552$		0			0
$553$		−	3.10456i		−	0.132019i
$554$		0			0
$555$	−10.2800	−	4.22773i	−0.436362	−	0.179457i
$556$		0			0
$557$			2.68789i			0.113890i	0.998377	+	0.0569448i	$0.0181359\pi$
$557$			2.68789i			0.113890i	−0.998377	+	0.0569448i	$0.981864\pi$
$558$		0			0
$559$	31.1851			1.31899
$560$		0			0
$561$	−24.7706			−1.04582
$562$		0			0
$563$		−	12.7089i		−	0.535618i	−0.963472	−	0.267809i	$-0.913700\pi$
$563$		−	12.7089i		−	0.535618i	0.963472	−	0.267809i	$-0.0862996\pi$
$564$		0			0
$565$	5.59674	−	13.6089i	0.235457	−	0.572529i
$566$		0			0
$567$		−	33.4477i		−	1.40467i
$568$		0			0
$569$	−28.9522			−1.21374			−0.606870	−	0.794801i	$-0.707575\pi$
$569$	−28.9522			−1.21374			−0.606870	+	0.794801i	$0.707575\pi$
$570$		0			0
$571$	8.89885			0.372405			0.186203	−	0.982511i	$-0.440382\pi$
$571$	8.89885			0.372405			0.186203	+	0.982511i	$0.440382\pi$
$572$		0			0
$573$		−	61.3667i		−	2.56363i
$574$		0			0
$575$	−3.77713	+	3.81548i	−0.157517	+	0.159117i
$576$		0			0
$577$			25.3754i			1.05639i	0.849123	+	0.528196i	$0.177131\pi$
$577$			25.3754i			1.05639i	−0.849123	+	0.528196i	$0.822869\pi$
$578$		0			0
$579$	−40.5411			−1.68483
$580$		0			0
$581$	42.9924			1.78362
$582$		0			0
$583$		−	28.0334i		−	1.16103i
$584$		0			0
$585$	−9.80367	+	23.8383i	−0.405332	+	0.985592i
$586$		0			0
$587$		−	38.7611i		−	1.59984i	−0.600107	−	0.799920i	$-0.704875\pi$
$587$		−	38.7611i		−	1.59984i	0.600107	−	0.799920i	$-0.295125\pi$
$588$		0			0
$589$	−6.38211			−0.262970
$590$		0			0
$591$	41.6727			1.71418
$592$		0			0
$593$		−	35.5132i		−	1.45835i	−0.684327	−	0.729176i	$-0.739904\pi$
$593$		−	35.5132i		−	1.45835i	0.684327	−	0.729176i	$-0.260096\pi$
$594$		0			0
$595$	31.9483	+	13.1390i	1.30975	+	0.538646i
$596$		0			0
$597$			53.7761i			2.20091i
$598$		0			0
$599$	22.7248			0.928512			0.464256	−	0.885701i	$-0.346322\pi$
$599$	22.7248			0.928512			0.464256	+	0.885701i	$0.346322\pi$
$600$		0			0
$601$	−25.3951			−1.03589			−0.517944	−	0.855415i	$-0.673302\pi$
$601$	−25.3951			−1.03589			−0.517944	+	0.855415i	$0.673302\pi$
$602$		0			0
$603$		−	20.4272i		−	0.831861i
$604$		0			0
$605$	11.0344	+	4.53799i	0.448613	+	0.184495i
$606$		0			0
$607$		−	18.2509i		−	0.740779i	−0.928876	−	0.370390i	$-0.879224\pi$
$607$		−	18.2509i		−	0.740779i	0.928876	−	0.370390i	$-0.120776\pi$
$608$		0			0
$609$	−83.0254			−3.36436
$610$		0			0
$611$	−15.2095			−0.615312
$612$		0			0
$613$			3.43574i			0.138768i	0.997590	+	0.0693842i	$0.0221034\pi$
$613$			3.43574i			0.138768i	−0.997590	+	0.0693842i	$0.977897\pi$
$614$		0			0
$615$	9.05756	−	22.0241i	0.365236	−	0.888096i
$616$		0			0
$617$			11.7976i			0.474952i	0.971393	+	0.237476i	$0.0763201\pi$
$617$			11.7976i			0.474952i	−0.971393	+	0.237476i	$0.923680\pi$
$618$		0			0
$619$	−27.6561			−1.11159			−0.555797	−	0.831318i	$-0.687587\pi$
$619$	−27.6561			−1.11159			−0.555797	+	0.831318i	$0.687587\pi$
$620$		0			0
$621$	0.244267			0.00980211
$622$		0			0
$623$		−	49.2261i		−	1.97220i
$624$		0			0
$625$	0.252588	+	24.9987i	0.0101035	+	0.999949i
$626$		0			0
$627$		−	5.78405i		−	0.230993i
$628$		0			0
$629$	−8.75959			−0.349268
$630$		0			0
$631$	7.23176			0.287892			0.143946	−	0.989586i	$-0.454021\pi$
$631$	7.23176			0.287892			0.143946	+	0.989586i	$0.454021\pi$
$632$		0			0
$633$			59.9532i			2.38293i
$634$		0			0
$635$	−10.4462	+	25.4006i	−0.414544	+	1.00799i
$636$		0			0
$637$		−	23.8487i		−	0.944921i
$638$		0			0
$639$	−14.5160			−0.574244
$640$		0			0
$641$	20.1802			0.797070			0.398535	−	0.917153i	$-0.369519\pi$
$641$	20.1802			0.797070			0.398535	+	0.917153i	$0.369519\pi$
$642$		0			0
$643$		−	4.68579i		−	0.184790i	−0.995722	−	0.0923948i	$-0.970548\pi$
$643$		−	4.68579i		−	0.184790i	0.995722	−	0.0923948i	$-0.0294522\pi$
$644$		0			0
$645$	39.5179	+	16.2520i	1.55602	+	0.639923i
$646$		0			0
$647$		−	2.17831i		−	0.0856383i	−0.999083	−	0.0428191i	$-0.986366\pi$
$647$		−	2.17831i		−	0.0856383i	0.999083	−	0.0428191i	$-0.0136339\pi$
$648$		0			0
$649$	10.9382			0.429361
$650$		0			0
$651$	55.9520			2.19293
$652$		0			0
$653$		−	3.73150i		−	0.146025i	−0.997331	−	0.0730124i	$-0.976739\pi$
$653$		−	3.73150i		−	0.146025i	0.997331	−	0.0730124i	$-0.0232613\pi$
$654$		0			0
$655$	20.7438	+	8.53105i	0.810528	+	0.333336i
$656$		0			0
$657$		−	8.52202i		−	0.332476i
$658$		0			0
$659$	18.1674			0.707700			0.353850	−	0.935302i	$-0.384872\pi$
$659$	18.1674			0.707700			0.353850	+	0.935302i	$0.384872\pi$
$660$		0			0
$661$	8.87307			0.345123			0.172561	−	0.984999i	$-0.444796\pi$
$661$	8.87307			0.345123			0.172561	+	0.984999i	$0.444796\pi$
$662$		0			0
$663$			41.2794i			1.60316i
$664$		0			0
$665$	−3.06801	+	7.46008i	−0.118972	+	0.289289i
$666$		0			0
$667$		−	10.1688i		−	0.393739i
$668$		0			0
$669$	−52.2000			−2.01817
$670$		0			0
$671$	−15.7749			−0.608982
$672$		0			0
$673$		−	35.7318i		−	1.37736i	−0.725065	−	0.688680i	$-0.758191\pi$
$673$		−	35.7318i		−	1.37736i	0.725065	−	0.688680i	$-0.241809\pi$
$674$		0			0
$675$	0.800210	−	0.808336i	0.0308001	−	0.0311129i
$676$		0			0
$677$		−	21.0748i		−	0.809971i	−0.914323	−	0.404985i	$-0.867277\pi$
$677$		−	21.0748i		−	0.809971i	0.914323	−	0.404985i	$-0.132723\pi$
$678$		0			0
$679$	−67.0095			−2.57159
$680$		0			0
$681$	−49.9594			−1.91445
$682$		0			0
$683$			9.90904i			0.379159i	0.981865	+	0.189579i	$0.0607124\pi$
$683$			9.90904i			0.379159i	−0.981865	+	0.189579i	$0.939288\pi$
$684$		0			0
$685$	−14.5666	+	35.4196i	−0.556559	+	1.35331i
$686$		0			0
$687$		−	42.8865i		−	1.63622i
$688$		0			0
$689$	−46.7168			−1.77977
$690$		0			0
$691$	−11.9960			−0.456348			−0.228174	−	0.973620i	$-0.573276\pi$
$691$	−11.9960			−0.456348			−0.228174	+	0.973620i	$0.573276\pi$
$692$		0			0
$693$			24.9526i			0.947871i
$694$		0			0
$695$	−11.3843	−	4.68187i	−0.431830	−	0.177593i
$696$		0			0
$697$		−	18.7667i		−	0.710840i
$698$		0			0
$699$	−37.6748			−1.42499
$700$		0			0
$701$	−42.9782			−1.62326			−0.811632	−	0.584169i	$-0.801420\pi$
$701$	−42.9782			−1.62326			−0.811632	+	0.584169i	$0.801420\pi$
$702$		0			0
$703$		−	2.04540i		−	0.0771439i
$704$		0			0
$705$	−19.2736	−	7.92641i	−0.725886	−	0.298526i
$706$		0			0
$707$		−	35.4476i		−	1.33314i
$708$		0			0
$709$	30.3524			1.13991			0.569954	−	0.821676i	$-0.306961\pi$
$709$	30.3524			1.13991			0.569954	+	0.821676i	$0.306961\pi$
$710$		0			0
$711$	2.50129			0.0938057
$712$		0			0
$713$			6.85292i			0.256644i
$714$		0			0
$715$	−8.02795	+	19.5205i	−0.300228	+	0.730025i
$716$		0			0
$717$		−	68.5799i		−	2.56116i
$718$		0			0
$719$	−38.0282			−1.41821			−0.709106	−	0.705102i	$-0.750901\pi$
$719$	−38.0282			−1.41821			−0.709106	+	0.705102i	$0.750901\pi$
$720$		0			0
$721$	−31.9349			−1.18932
$722$		0			0
$723$		−	58.0000i		−	2.15704i
$724$		0			0
$725$	−33.6510	−	33.3127i	−1.24977	−	1.23720i
$726$		0			0
$727$		−	17.1429i		−	0.635796i	−0.948125	−	0.317898i	$-0.897023\pi$
$727$		−	17.1429i		−	0.635796i	0.948125	−	0.317898i	$-0.102977\pi$
$728$		0			0
$729$	25.2897			0.936654
$730$		0			0
$731$	33.6732			1.24545
$732$		0			0
$733$			14.0860i			0.520277i	0.965571	+	0.260139i	$0.0837683\pi$
$733$			14.0860i			0.520277i	−0.965571	+	0.260139i	$0.916232\pi$
$734$		0			0
$735$	12.4287	−	30.2213i	0.458440	−	1.11473i
$736$		0			0
$737$		−	16.7273i		−	0.616157i
$738$		0			0
$739$	4.08166			0.150146			0.0750731	−	0.997178i	$-0.476081\pi$
$739$	4.08166			0.150146			0.0750731	+	0.997178i	$0.476081\pi$
$740$		0			0
$741$	−9.63892			−0.354095
$742$		0			0
$743$			22.5661i			0.827872i	0.910306	+	0.413936i	$0.135846\pi$
$743$			22.5661i			0.827872i	−0.910306	+	0.413936i	$0.864154\pi$
$744$		0			0
$745$	−44.3781	−	18.2508i	−1.62589	−	0.668658i
$746$		0			0
$747$			34.6382i			1.26735i
$748$		0			0
$749$	−23.7462			−0.867666
$750$		0			0
$751$	39.5626			1.44366			0.721830	−	0.692070i	$-0.243301\pi$
$751$	39.5626			1.44366			0.721830	+	0.692070i	$0.243301\pi$
$752$		0			0
$753$		−	19.8263i		−	0.722510i
$754$		0			0
$755$	29.3254	+	12.0603i	1.06726	+	0.438919i
$756$		0			0
$757$			33.4582i			1.21606i	0.793915	+	0.608029i	$0.208040\pi$
$757$			33.4582i			1.21606i	−0.793915	+	0.608029i	$0.791960\pi$
$758$		0			0
$759$	−6.21074			−0.225436
$760$		0			0
$761$	4.30846			0.156181			0.0780907	−	0.996946i	$-0.475118\pi$
$761$	4.30846			0.156181			0.0780907	+	0.996946i	$0.475118\pi$
$762$		0			0
$763$		−	12.2982i		−	0.445225i
$764$		0			0
$765$	−10.5859	+	25.7403i	−0.382733	+	0.930641i
$766$		0			0
$767$		−	18.2281i		−	0.658178i
$768$		0			0
$769$	29.3601			1.05875			0.529376	−	0.848387i	$-0.322426\pi$
$769$	29.3601			1.05875			0.529376	+	0.848387i	$0.322426\pi$
$770$		0			0
$771$	14.7738			0.532066
$772$		0			0
$773$			43.7346i			1.57303i	0.617574	+	0.786513i	$0.288116\pi$
$773$			43.7346i			1.57303i	−0.617574	+	0.786513i	$0.711884\pi$
$774$		0			0
$775$	22.6779	+	22.4499i	0.814613	+	0.806424i
$776$		0			0
$777$			17.9321i			0.643310i
$778$		0			0
$779$	4.38211			0.157005
$780$		0			0
$781$	−11.8867			−0.425341
$782$		0			0
$783$			2.15434i			0.0769897i
$784$		0			0
$785$	2.29538	−	5.58138i	0.0819258	−	0.199208i
$786$		0			0
$787$		−	29.4309i		−	1.04910i	−0.851380	−	0.524550i	$-0.824234\pi$
$787$		−	29.4309i		−	1.04910i	0.851380	−	0.524550i	$-0.175766\pi$
$788$		0			0
$789$	−20.5977			−0.733298
$790$		0			0
$791$	−23.7388			−0.844055
$792$		0			0
$793$			26.2883i			0.933524i
$794$		0			0
$795$	−59.1998	−	24.3463i	−2.09960	−	0.863476i
$796$		0			0
$797$			21.2121i

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.43031i q^{3} +(-2.06801 - 0.850485i) q^{5} +3.60737i q^{7} -2.90640 q^{9} -2.37997 q^{11} +3.96613i q^{13} +(-2.06694 + 5.02591i) q^{15} +4.28257i q^{17} -1.00000 q^{19} +8.76701 q^{21} +1.07377i q^{23} +(3.55335 + 3.51763i) q^{25} -0.227486i q^{27} -9.47021 q^{29} +6.38211 q^{31} +5.78405i q^{33} +(3.06801 - 7.46008i) q^{35} +2.04540i q^{37} +9.63892 q^{39} -4.38211 q^{41} -7.86284i q^{43} +(6.01046 + 2.47185i) q^{45} +3.83485i q^{47} -6.01310 q^{49} +10.4080 q^{51} +11.7789i q^{53} +(4.92180 + 2.02413i) q^{55} +2.43031i q^{57} -4.59593 q^{59} +6.62819 q^{61} -10.4844i q^{63} +(3.37314 - 8.20201i) q^{65} +7.02837i q^{67} +2.60959 q^{69} +4.99450 q^{71} +2.93216i q^{73} +(8.54892 - 8.63573i) q^{75} -8.58541i q^{77} -0.860616 q^{79} -9.27205 q^{81} -11.9179i q^{83} +(3.64227 - 8.85642i) q^{85} +23.0155i q^{87} -13.6460 q^{89} -14.3073 q^{91} -15.5105i q^{93} +(2.06801 + 0.850485i) q^{95} +18.5757i q^{97} +6.91712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{5} - 16 q^{9} - 4 q^{11} - 12 q^{15} - 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} + 16 q^{31} + 6 q^{35} + 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} - 68 q^{51} - 18 q^{55} + 4 q^{59} + 20 q^{61}+ \cdots + 44 q^{99}+O(q^{100}) \) 12 * q + 6 * q^5 - 16 * q^9 - 4 * q^11 - 12 * q^15 - 12 * q^19 + 36 * q^21 + 18 * q^25 + 4 * q^29 + 16 * q^31 + 6 * q^35 + 36 * q^39 + 8 * q^41 - 2 * q^45 - 4 * q^49 - 68 * q^51 - 18 * q^55 + 4 * q^59 + 20 * q^61 + 20 * q^65 - 36 * q^69 - 16 * q^71 - 16 * q^75 + 40 * q^79 + 12 * q^81 + 6 * q^85 - 64 * q^89 + 20 * q^91 - 6 * q^95 + 44 * q^99

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