LMFDB - Embedded newform 1824.2.d.e.191.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1824,2,Mod(191,1824)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1824.191"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1824 = 2^{5} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1824.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$14.5647133287$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1824.191
Dual form			1824.2.d.e.191.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.70711 - 0.292893i) q^{3} +1.00000i q^{5} -2.41421i q^{7} +(2.82843 - 1.00000i) q^{9} -5.24264 q^{11} +4.24264 q^{13} +(0.292893 + 1.70711i) q^{15} +1.82843i q^{17} -1.00000i q^{19} +(-0.707107 - 4.12132i) q^{21} +5.65685 q^{23} +4.00000 q^{25} +(4.53553 - 2.53553i) q^{27} -8.00000i q^{29} +8.24264i q^{31} +(-8.94975 + 1.53553i) q^{33} +2.41421 q^{35} +7.07107 q^{37} +(7.24264 - 1.24264i) q^{39} -9.41421i q^{41} -11.2426i q^{43} +(1.00000 + 2.82843i) q^{45} +0.414214 q^{47} +1.17157 q^{49} +(0.535534 + 3.12132i) q^{51} +5.07107i q^{53} -5.24264i q^{55} +(-0.292893 - 1.70711i) q^{57} -12.2426 q^{59} +8.65685 q^{61} +(-2.41421 - 6.82843i) q^{63} +4.24264i q^{65} -3.17157i q^{67} +(9.65685 - 1.65685i) q^{69} +9.41421 q^{71} -13.0000 q^{73} +(6.82843 - 1.17157i) q^{75} +12.6569i q^{77} -8.82843i q^{79} +(7.00000 - 5.65685i) q^{81} -11.6569 q^{83} -1.82843 q^{85} +(-2.34315 - 13.6569i) q^{87} +14.2426i q^{89} -10.2426i q^{91} +(2.41421 + 14.0711i) q^{93} +1.00000 q^{95} +3.75736 q^{97} +(-14.8284 + 5.24264i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{11} + 4 q^{15} + 16 q^{25} + 4 q^{27} - 16 q^{33} + 4 q^{35} + 12 q^{39} + 4 q^{45} - 4 q^{47} + 16 q^{49} - 12 q^{51} - 4 q^{57} - 32 q^{59} + 12 q^{61} - 4 q^{63} + 16 q^{69} + 32 q^{71}+ \cdots - 48 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^11 + 4 * q^15 + 16 * q^25 + 4 * q^27 - 16 * q^33 + 4 * q^35 + 12 * q^39 + 4 * q^45 - 4 * q^47 + 16 * q^49 - 12 * q^51 - 4 * q^57 - 32 * q^59 + 12 * q^61 - 4 * q^63 + 16 * q^69 + 32 * q^71 - 52 * q^73 + 16 * q^75 + 28 * q^81 - 24 * q^83 + 4 * q^85 - 32 * q^87 + 4 * q^93 + 4 * q^95 + 32 * q^97 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1824\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$799$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$4$		0			0
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$		−	2.41421i		−	0.912487i	−0.889855	−	0.456243i	$-0.849195\pi$
$7$		−	2.41421i		−	0.912487i	0.889855	−	0.456243i	$-0.150805\pi$
$8$		0			0
$9$	2.82843	−	1.00000i	0.942809	−	0.333333i
$10$		0			0
$11$	−5.24264			−1.58072			−0.790358	−	0.612646i	$-0.790105\pi$
$11$	−5.24264			−1.58072			−0.790358	+	0.612646i	$0.790105\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$	0.292893	+	1.70711i	0.0756247	+	0.440773i
$16$		0			0
$17$			1.82843i			0.443459i	0.975108	+	0.221729i	$0.0711701\pi$
$17$			1.82843i			0.443459i	−0.975108	+	0.221729i	$0.928830\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	−0.707107	−	4.12132i	−0.154303	−	0.899346i
$22$		0			0
$23$	5.65685			1.17954			0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685			1.17954			0.589768	+	0.807573i	$0.299219\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	4.53553	−	2.53553i	0.872864	−	0.487964i
$28$		0			0
$29$		−	8.00000i		−	1.48556i	−0.669534	−	0.742781i	$-0.733506\pi$
$29$		−	8.00000i		−	1.48556i	0.669534	−	0.742781i	$-0.266494\pi$
$30$		0			0
$31$			8.24264i			1.48042i	0.672375	+	0.740211i	$0.265274\pi$
$31$			8.24264i			1.48042i	−0.672375	+	0.740211i	$0.734726\pi$
$32$		0			0
$33$	−8.94975	+	1.53553i	−1.55795	+	0.267302i
$34$		0			0
$35$	2.41421			0.408077
$36$		0			0
$37$	7.07107			1.16248			0.581238	−	0.813733i	$-0.302568\pi$
$37$	7.07107			1.16248			0.581238	+	0.813733i	$0.302568\pi$
$38$		0			0
$39$	7.24264	−	1.24264i	1.15975	−	0.198982i
$40$		0			0
$41$		−	9.41421i		−	1.47025i	−0.677930	−	0.735127i	$-0.737123\pi$
$41$		−	9.41421i		−	1.47025i	0.677930	−	0.735127i	$-0.262877\pi$
$42$		0			0
$43$		−	11.2426i		−	1.71449i	−0.514912	−	0.857243i	$-0.672175\pi$
$43$		−	11.2426i		−	1.71449i	0.514912	−	0.857243i	$-0.327825\pi$
$44$		0			0
$45$	1.00000	+	2.82843i	0.149071	+	0.421637i
$46$		0			0
$47$	0.414214			0.0604193			0.0302096	−	0.999544i	$-0.490383\pi$
$47$	0.414214			0.0604193			0.0302096	+	0.999544i	$0.490383\pi$
$48$		0			0
$49$	1.17157			0.167368
$50$		0			0
$51$	0.535534	+	3.12132i	0.0749897	+	0.437072i
$52$		0			0
$53$			5.07107i			0.696565i	0.937390	+	0.348282i	$0.113235\pi$
$53$			5.07107i			0.696565i	−0.937390	+	0.348282i	$0.886765\pi$
$54$		0			0
$55$		−	5.24264i		−	0.706918i
$56$		0			0
$57$	−0.292893	−	1.70711i	−0.0387947	−	0.226112i
$58$		0			0
$59$	−12.2426			−1.59386			−0.796928	−	0.604074i	$-0.793543\pi$
$59$	−12.2426			−1.59386			−0.796928	+	0.604074i	$0.793543\pi$
$60$		0			0
$61$	8.65685			1.10840			0.554198	−	0.832385i	$-0.313025\pi$
$61$	8.65685			1.10840			0.554198	+	0.832385i	$0.313025\pi$
$62$		0			0
$63$	−2.41421	−	6.82843i	−0.304162	−	0.860301i
$64$		0			0
$65$			4.24264i			0.526235i
$66$		0			0
$67$		−	3.17157i		−	0.387469i	−0.981054	−	0.193735i	$-0.937940\pi$
$67$		−	3.17157i		−	0.387469i	0.981054	−	0.193735i	$-0.0620601\pi$
$68$		0			0
$69$	9.65685	−	1.65685i	1.16255	−	0.199462i
$70$		0			0
$71$	9.41421			1.11726			0.558631	−	0.829416i	$-0.311327\pi$
$71$	9.41421			1.11726			0.558631	+	0.829416i	$0.311327\pi$
$72$		0			0
$73$	−13.0000			−1.52153			−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000			−1.52153			−0.760767	+	0.649025i	$0.775177\pi$
$74$		0			0
$75$	6.82843	−	1.17157i	0.788479	−	0.135282i
$76$		0			0
$77$			12.6569i			1.44238i
$78$		0			0
$79$		−	8.82843i		−	0.993276i	−0.867958	−	0.496638i	$-0.834568\pi$
$79$		−	8.82843i		−	0.993276i	0.867958	−	0.496638i	$-0.165432\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	−11.6569			−1.27951			−0.639753	−	0.768581i	$-0.720963\pi$
$83$	−11.6569			−1.27951			−0.639753	+	0.768581i	$0.720963\pi$
$84$		0			0
$85$	−1.82843			−0.198321
$86$		0			0
$87$	−2.34315	−	13.6569i	−0.251212	−	1.46417i
$88$		0			0
$89$			14.2426i			1.50972i	0.655888	+	0.754858i	$0.272294\pi$
$89$			14.2426i			1.50972i	−0.655888	+	0.754858i	$0.727706\pi$
$90$		0			0
$91$		−	10.2426i		−	1.07372i
$92$		0			0
$93$	2.41421	+	14.0711i	0.250342	+	1.45910i
$94$		0			0
$95$	1.00000			0.102598
$96$		0			0
$97$	3.75736			0.381502			0.190751	−	0.981638i	$-0.438908\pi$
$97$	3.75736			0.381502			0.190751	+	0.981638i	$0.438908\pi$
$98$		0			0
$99$	−14.8284	+	5.24264i	−1.49031	+	0.526905i
$100$		0			0
$101$		−	2.34315i		−	0.233152i	−0.993182	−	0.116576i	$-0.962808\pi$
$101$		−	2.34315i		−	0.233152i	0.993182	−	0.116576i	$-0.0371918\pi$
$102$		0			0
$103$			1.51472i			0.149250i	0.997212	+	0.0746248i	$0.0237759\pi$
$103$			1.51472i			0.149250i	−0.997212	+	0.0746248i	$0.976224\pi$
$104$		0			0
$105$	4.12132	−	0.707107i	0.402200	−	0.0690066i
$106$		0			0
$107$	2.48528			0.240261			0.120131	−	0.992758i	$-0.461669\pi$
$107$	2.48528			0.240261			0.120131	+	0.992758i	$0.461669\pi$
$108$		0			0
$109$	−0.343146			−0.0328674			−0.0164337	−	0.999865i	$-0.505231\pi$
$109$	−0.343146			−0.0328674			−0.0164337	+	0.999865i	$0.505231\pi$
$110$		0			0
$111$	12.0711	−	2.07107i	1.14574	−	0.196577i
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$			5.65685i			0.527504i
$116$		0			0
$117$	12.0000	−	4.24264i	1.10940	−	0.392232i
$118$		0			0
$119$	4.41421			0.404650
$120$		0			0
$121$	16.4853			1.49866
$122$		0			0
$123$	−2.75736	−	16.0711i	−0.248623	−	1.44908i
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$			21.6569i			1.92174i	0.277008	+	0.960868i	$0.410657\pi$
$127$			21.6569i			1.92174i	−0.277008	+	0.960868i	$0.589343\pi$
$128$		0			0
$129$	−3.29289	−	19.1924i	−0.289923	−	1.68980i
$130$		0			0
$131$	9.58579			0.837514			0.418757	−	0.908098i	$-0.362466\pi$
$131$	9.58579			0.837514			0.418757	+	0.908098i	$0.362466\pi$
$132$		0			0
$133$	−2.41421			−0.209339
$134$		0			0
$135$	2.53553	+	4.53553i	0.218224	+	0.390357i
$136$		0			0
$137$			3.82843i			0.327085i	0.986536	+	0.163542i	$0.0522920\pi$
$137$			3.82843i			0.327085i	−0.986536	+	0.163542i	$0.947708\pi$
$138$		0			0
$139$			5.58579i			0.473780i	0.971536	+	0.236890i	$0.0761281\pi$
$139$			5.58579i			0.473780i	−0.971536	+	0.236890i	$0.923872\pi$
$140$		0			0
$141$	0.707107	−	0.121320i	0.0595491	−	0.0102170i
$142$		0			0
$143$	−22.2426			−1.86002
$144$		0			0
$145$	8.00000			0.664364
$146$		0			0
$147$	2.00000	−	0.343146i	0.164957	−	0.0283022i
$148$		0			0
$149$			12.3137i			1.00878i	0.863476	+	0.504389i	$0.168282\pi$
$149$			12.3137i			1.00878i	−0.863476	+	0.504389i	$0.831718\pi$
$150$		0			0
$151$		−	22.3848i		−	1.82165i	−0.412796	−	0.910824i	$-0.635448\pi$
$151$		−	22.3848i		−	1.82165i	0.412796	−	0.910824i	$-0.364552\pi$
$152$		0			0
$153$	1.82843	+	5.17157i	0.147820	+	0.418097i
$154$		0			0
$155$	−8.24264			−0.662065
$156$		0			0
$157$	−17.6569			−1.40917			−0.704585	−	0.709619i	$-0.748867\pi$
$157$	−17.6569			−1.40917			−0.704585	+	0.709619i	$0.748867\pi$
$158$		0			0
$159$	1.48528	+	8.65685i	0.117790	+	0.686533i
$160$		0			0
$161$		−	13.6569i		−	1.07631i
$162$		0			0
$163$			15.3137i			1.19946i	0.800202	+	0.599731i	$0.204726\pi$
$163$			15.3137i			1.19946i	−0.800202	+	0.599731i	$0.795274\pi$
$164$		0			0
$165$	−1.53553	−	8.94975i	−0.119541	−	0.696737i
$166$		0			0
$167$	−12.9706			−1.00369			−0.501846	−	0.864957i	$-0.667346\pi$
$167$	−12.9706			−1.00369			−0.501846	+	0.864957i	$0.667346\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−1.00000	−	2.82843i	−0.0764719	−	0.216295i
$172$		0			0
$173$		−	15.6569i		−	1.19037i	−0.803589	−	0.595184i	$-0.797079\pi$
$173$		−	15.6569i		−	1.19037i	0.803589	−	0.595184i	$-0.202921\pi$
$174$		0			0
$175$		−	9.65685i		−	0.729990i
$176$		0			0
$177$	−20.8995	+	3.58579i	−1.57090	+	0.269524i
$178$		0			0
$179$	−14.3431			−1.07206			−0.536029	−	0.844200i	$-0.680076\pi$
$179$	−14.3431			−1.07206			−0.536029	+	0.844200i	$0.680076\pi$
$180$		0			0
$181$	−18.9706			−1.41007			−0.705035	−	0.709172i	$-0.749069\pi$
$181$	−18.9706			−1.41007			−0.705035	+	0.709172i	$0.749069\pi$
$182$		0			0
$183$	14.7782	−	2.53553i	1.09243	−	0.187432i
$184$		0			0
$185$			7.07107i			0.519875i
$186$		0			0
$187$		−	9.58579i		−	0.700982i
$188$		0			0
$189$	−6.12132	−	10.9497i	−0.445261	−	0.796477i
$190$		0			0
$191$	12.4142			0.898261			0.449130	−	0.893466i	$-0.351734\pi$
$191$	12.4142			0.898261			0.449130	+	0.893466i	$0.351734\pi$
$192$		0			0
$193$	10.9706			0.789678			0.394839	−	0.918750i	$-0.370800\pi$
$193$	10.9706			0.789678			0.394839	+	0.918750i	$0.370800\pi$
$194$		0			0
$195$	1.24264	+	7.24264i	0.0889873	+	0.518656i
$196$		0			0
$197$			5.17157i			0.368459i	0.982883	+	0.184230i	$0.0589790\pi$
$197$			5.17157i			0.368459i	−0.982883	+	0.184230i	$0.941021\pi$
$198$		0			0
$199$			10.0711i			0.713919i	0.934120	+	0.356960i	$0.116187\pi$
$199$			10.0711i			0.713919i	−0.934120	+	0.356960i	$0.883813\pi$
$200$		0			0
$201$	−0.928932	−	5.41421i	−0.0655218	−	0.381889i
$202$		0			0
$203$	−19.3137			−1.35556
$204$		0			0
$205$	9.41421			0.657517
$206$		0			0
$207$	16.0000	−	5.65685i	1.11208	−	0.393179i
$208$		0			0
$209$			5.24264i			0.362641i
$210$		0			0
$211$		−	16.7279i		−	1.15160i	−0.817591	−	0.575799i	$-0.804691\pi$
$211$		−	16.7279i		−	1.15160i	0.817591	−	0.575799i	$-0.195309\pi$
$212$		0			0
$213$	16.0711	−	2.75736i	1.10117	−	0.188931i
$214$		0			0
$215$	11.2426			0.766742
$216$		0			0
$217$	19.8995			1.35087
$218$		0			0
$219$	−22.1924	+	3.80761i	−1.49962	+	0.257295i
$220$		0			0
$221$			7.75736i			0.521816i
$222$		0			0
$223$		−	0.585786i		−	0.0392272i	−0.999808	−	0.0196136i	$-0.993756\pi$
$223$		−	0.585786i		−	0.0392272i	0.999808	−	0.0196136i	$-0.00624360\pi$
$224$		0			0
$225$	11.3137	−	4.00000i	0.754247	−	0.266667i
$226$		0			0
$227$	−13.0711			−0.867557			−0.433779	−	0.901019i	$-0.642820\pi$
$227$	−13.0711			−0.867557			−0.433779	+	0.901019i	$0.642820\pi$
$228$		0			0
$229$	−10.3137			−0.681549			−0.340775	−	0.940145i	$-0.610689\pi$
$229$	−10.3137			−0.681549			−0.340775	+	0.940145i	$0.610689\pi$
$230$		0			0
$231$	3.70711	+	21.6066i	0.243910	+	1.42161i
$232$		0			0
$233$			27.8284i			1.82310i	0.411188	+	0.911550i	$0.365114\pi$
$233$			27.8284i			1.82310i	−0.411188	+	0.911550i	$0.634886\pi$
$234$		0			0
$235$			0.414214i			0.0270203i
$236$		0			0
$237$	−2.58579	−	15.0711i	−0.167965	−	0.978971i
$238$		0			0
$239$	21.7279			1.40546			0.702731	−	0.711455i	$-0.251964\pi$
$239$	21.7279			1.40546			0.702731	+	0.711455i	$0.251964\pi$
$240$		0			0
$241$	−4.24264			−0.273293			−0.136646	−	0.990620i	$-0.543632\pi$
$241$	−4.24264			−0.273293			−0.136646	+	0.990620i	$0.543632\pi$
$242$		0			0
$243$	10.2929	−	11.7071i	0.660289	−	0.751011i
$244$		0			0
$245$			1.17157i			0.0748490i
$246$		0			0
$247$		−	4.24264i		−	0.269953i
$248$		0			0
$249$	−19.8995	+	3.41421i	−1.26108	+	0.216367i
$250$		0			0
$251$	−2.55635			−0.161355			−0.0806777	−	0.996740i	$-0.525708\pi$
$251$	−2.55635			−0.161355			−0.0806777	+	0.996740i	$0.525708\pi$
$252$		0			0
$253$	−29.6569			−1.86451
$254$		0			0
$255$	−3.12132	+	0.535534i	−0.195465	+	0.0335364i
$256$		0			0
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$		0			0
$259$		−	17.0711i		−	1.06074i
$260$		0			0
$261$	−8.00000	−	22.6274i	−0.495188	−	1.40060i
$262$		0			0
$263$	−1.24264			−0.0766245			−0.0383123	−	0.999266i	$-0.512198\pi$
$263$	−1.24264			−0.0766245			−0.0383123	+	0.999266i	$0.512198\pi$
$264$		0			0
$265$	−5.07107			−0.311513
$266$		0			0
$267$	4.17157	+	24.3137i	0.255296	+	1.48797i
$268$		0			0
$269$			4.38478i			0.267345i	0.991026	+	0.133672i	$0.0426769\pi$
$269$			4.38478i			0.267345i	−0.991026	+	0.133672i	$0.957323\pi$
$270$		0			0
$271$		−	1.17157i		−	0.0711680i	−0.999367	−	0.0355840i	$-0.988671\pi$
$271$		−	1.17157i		−	0.0711680i	0.999367	−	0.0355840i	$-0.0113291\pi$
$272$		0			0
$273$	−3.00000	−	17.4853i	−0.181568	−	1.05826i
$274$		0			0
$275$	−20.9706			−1.26457
$276$		0			0
$277$	2.17157			0.130477			0.0652386	−	0.997870i	$-0.479219\pi$
$277$	2.17157			0.130477			0.0652386	+	0.997870i	$0.479219\pi$
$278$		0			0
$279$	8.24264	+	23.3137i	0.493474	+	1.39576i
$280$		0			0
$281$			9.31371i			0.555609i	0.960638	+	0.277805i	$0.0896068\pi$
$281$			9.31371i			0.555609i	−0.960638	+	0.277805i	$0.910393\pi$
$282$		0			0
$283$			12.8995i			0.766795i	0.923583	+	0.383398i	$0.125246\pi$
$283$			12.8995i			0.766795i	−0.923583	+	0.383398i	$0.874754\pi$
$284$		0			0
$285$	1.70711	−	0.292893i	0.101120	−	0.0173495i
$286$		0			0
$287$	−22.7279			−1.34159
$288$		0			0
$289$	13.6569			0.803344
$290$		0			0
$291$	6.41421	−	1.10051i	0.376008	−	0.0645127i
$292$		0			0
$293$		−	2.34315i		−	0.136888i	−0.997655	−	0.0684440i	$-0.978197\pi$
$293$		−	2.34315i		−	0.136888i	0.997655	−	0.0684440i	$-0.0218035\pi$
$294$		0			0
$295$		−	12.2426i		−	0.712794i
$296$		0			0
$297$	−23.7782	+	13.2929i	−1.37975	+	0.771332i
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$	−27.1421			−1.56445
$302$		0			0
$303$	−0.686292	−	4.00000i	−0.0394264	−	0.229794i
$304$		0			0
$305$			8.65685i			0.495690i
$306$		0			0
$307$			10.4853i			0.598427i	0.954186	+	0.299213i	$0.0967242\pi$
$307$			10.4853i			0.598427i	−0.954186	+	0.299213i	$0.903276\pi$
$308$		0			0
$309$	0.443651	+	2.58579i	0.0252384	+	0.147100i
$310$		0			0
$311$	16.7574			0.950223			0.475111	−	0.879926i	$-0.342408\pi$
$311$	16.7574			0.950223			0.475111	+	0.879926i	$0.342408\pi$
$312$		0			0
$313$	−1.65685			−0.0936509			−0.0468255	−	0.998903i	$-0.514910\pi$
$313$	−1.65685			−0.0936509			−0.0468255	+	0.998903i	$0.514910\pi$
$314$		0			0
$315$	6.82843	−	2.41421i	0.384738	−	0.136026i
$316$		0			0
$317$			15.8995i			0.893005i	0.894782	+	0.446502i	$0.147331\pi$
$317$			15.8995i			0.893005i	−0.894782	+	0.446502i	$0.852669\pi$
$318$		0			0
$319$			41.9411i			2.34825i
$320$		0			0
$321$	4.24264	−	0.727922i	0.236801	−	0.0406286i
$322$		0			0
$323$	1.82843			0.101736
$324$		0			0
$325$	16.9706			0.941357
$326$		0			0
$327$	−0.585786	+	0.100505i	−0.0323941	+	0.00555794i
$328$		0			0
$329$		−	1.00000i		−	0.0551318i
$330$		0			0
$331$			12.1421i			0.667392i	0.942681	+	0.333696i	$0.108296\pi$
$331$			12.1421i			0.667392i	−0.942681	+	0.333696i	$0.891704\pi$
$332$		0			0
$333$	20.0000	−	7.07107i	1.09599	−	0.387492i
$334$		0			0
$335$	3.17157			0.173282
$336$		0			0
$337$	−7.55635			−0.411621			−0.205810	−	0.978592i	$-0.565983\pi$
$337$	−7.55635			−0.411621			−0.205810	+	0.978592i	$0.565983\pi$
$338$		0			0
$339$	3.31371	+	19.3137i	0.179976	+	1.04898i
$340$		0			0
$341$		−	43.2132i		−	2.34013i
$342$		0			0
$343$		−	19.7279i		−	1.06521i
$344$		0			0
$345$	1.65685	+	9.65685i	0.0892020	+	0.519908i
$346$		0			0
$347$	12.7574			0.684851			0.342425	−	0.939545i	$-0.388752\pi$
$347$	12.7574			0.684851			0.342425	+	0.939545i	$0.388752\pi$
$348$		0			0
$349$	14.3137			0.766195			0.383098	−	0.923708i	$-0.374857\pi$
$349$	14.3137			0.766195			0.383098	+	0.923708i	$0.374857\pi$
$350$		0			0
$351$	19.2426	−	10.7574i	1.02710	−	0.574185i
$352$		0			0
$353$		−	29.4558i		−	1.56778i	−0.620902	−	0.783888i	$-0.713234\pi$
$353$		−	29.4558i		−	1.56778i	0.620902	−	0.783888i	$-0.286766\pi$
$354$		0			0
$355$			9.41421i			0.499655i
$356$		0			0
$357$	7.53553	−	1.29289i	0.398823	−	0.0684272i
$358$		0			0
$359$	−14.5563			−0.768255			−0.384127	−	0.923280i	$-0.625498\pi$
$359$	−14.5563			−0.768255			−0.384127	+	0.923280i	$0.625498\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	28.1421	−	4.82843i	1.47708	−	0.253427i
$364$		0			0
$365$		−	13.0000i		−	0.680451i
$366$		0			0
$367$			15.5147i			0.809862i	0.914347	+	0.404931i	$0.132704\pi$
$367$			15.5147i			0.809862i	−0.914347	+	0.404931i	$0.867296\pi$
$368$		0			0
$369$	−9.41421	−	26.6274i	−0.490084	−	1.38617i
$370$		0			0
$371$	12.2426			0.635606
$372$		0			0
$373$	−19.3137			−1.00003			−0.500013	−	0.866018i	$-0.666671\pi$
$373$	−19.3137			−1.00003			−0.500013	+	0.866018i	$0.666671\pi$
$374$		0			0
$375$	2.63604	+	15.3640i	0.136124	+	0.793392i
$376$		0			0
$377$		−	33.9411i		−	1.74806i
$378$		0			0
$379$			14.4853i			0.744059i	0.928221	+	0.372029i	$0.121338\pi$
$379$			14.4853i			0.744059i	−0.928221	+	0.372029i	$0.878662\pi$
$380$		0			0
$381$	6.34315	+	36.9706i	0.324969	+	1.89406i
$382$		0			0
$383$	−0.242641			−0.0123984			−0.00619918	−	0.999981i	$-0.501973\pi$
$383$	−0.242641			−0.0123984			−0.00619918	+	0.999981i	$0.501973\pi$
$384$		0			0
$385$	−12.6569			−0.645053
$386$		0			0
$387$	−11.2426	−	31.7990i	−0.571496	−	1.61643i
$388$		0			0
$389$		−	7.82843i		−	0.396917i	−0.980109	−	0.198459i	$-0.936406\pi$
$389$		−	7.82843i		−	0.396917i	0.980109	−	0.198459i	$-0.0635935\pi$
$390$		0			0
$391$			10.3431i			0.523075i
$392$		0			0
$393$	16.3640	−	2.80761i	0.825453	−	0.141625i
$394$		0			0
$395$	8.82843			0.444206
$396$		0			0
$397$	3.00000			0.150566			0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000			0.150566			0.0752828	+	0.997162i	$0.476014\pi$
$398$		0			0
$399$	−4.12132	+	0.707107i	−0.206324	+	0.0353996i
$400$		0			0
$401$			1.31371i			0.0656035i	0.999462	+	0.0328017i	$0.0104430\pi$
$401$			1.31371i			0.0656035i	−0.999462	+	0.0328017i	$0.989557\pi$
$402$		0			0
$403$			34.9706i			1.74201i
$404$		0			0
$405$	5.65685	+	7.00000i	0.281091	+	0.347833i
$406$		0			0
$407$	−37.0711			−1.83754
$408$		0			0
$409$	−27.4142			−1.35555			−0.677773	−	0.735271i	$-0.737055\pi$
$409$	−27.4142			−1.35555			−0.677773	+	0.735271i	$0.737055\pi$
$410$		0			0
$411$	1.12132	+	6.53553i	0.0553107	+	0.322374i
$412$		0			0
$413$			29.5563i			1.45437i
$414$		0			0
$415$		−	11.6569i		−	0.572212i
$416$		0			0
$417$	1.63604	+	9.53553i	0.0801172	+	0.466957i
$418$		0			0
$419$	−24.2843			−1.18636			−0.593182	−	0.805068i	$-0.702129\pi$
$419$	−24.2843			−1.18636			−0.593182	+	0.805068i	$0.702129\pi$
$420$		0			0
$421$	−26.2426			−1.27899			−0.639494	−	0.768796i	$-0.720856\pi$
$421$	−26.2426			−1.27899			−0.639494	+	0.768796i	$0.720856\pi$
$422$		0			0
$423$	1.17157	−	0.414214i	0.0569638	−	0.0201398i
$424$		0			0
$425$			7.31371i			0.354767i
$426$		0			0
$427$		−	20.8995i		−	1.01140i
$428$		0			0
$429$	−37.9706	+	6.51472i	−1.83324	+	0.314534i
$430$		0			0
$431$	3.61522			0.174139			0.0870696	−	0.996202i	$-0.472250\pi$
$431$	3.61522			0.174139			0.0870696	+	0.996202i	$0.472250\pi$
$432$		0			0
$433$	25.6569			1.23299			0.616495	−	0.787359i	$-0.288552\pi$
$433$	25.6569			1.23299			0.616495	+	0.787359i	$0.288552\pi$
$434$		0			0
$435$	13.6569	−	2.34315i	0.654796	−	0.112345i
$436$		0			0
$437$		−	5.65685i		−	0.270604i
$438$		0			0
$439$			32.2843i			1.54084i	0.637534	+	0.770422i	$0.279954\pi$
$439$			32.2843i			1.54084i	−0.637534	+	0.770422i	$0.720046\pi$
$440$		0			0
$441$	3.31371	−	1.17157i	0.157796	−	0.0557892i
$442$		0			0
$443$	−20.4142			−0.969909			−0.484954	−	0.874540i	$-0.661164\pi$
$443$	−20.4142			−0.969909			−0.484954	+	0.874540i	$0.661164\pi$
$444$		0			0
$445$	−14.2426			−0.675166
$446$		0			0
$447$	3.60660	+	21.0208i	0.170586	+	0.994250i
$448$		0			0
$449$			4.68629i			0.221160i	0.993867	+	0.110580i	$0.0352708\pi$
$449$			4.68629i			0.221160i	−0.993867	+	0.110580i	$0.964729\pi$
$450$		0			0
$451$			49.3553i			2.32405i
$452$		0			0
$453$	−6.55635	−	38.2132i	−0.308044	−	1.79541i
$454$		0			0
$455$	10.2426			0.480182
$456$		0			0
$457$	−23.4853			−1.09860			−0.549298	−	0.835627i	$-0.685105\pi$
$457$	−23.4853			−1.09860			−0.549298	+	0.835627i	$0.685105\pi$
$458$		0			0
$459$	4.63604	+	8.29289i	0.216392	+	0.387079i
$460$		0			0
$461$			8.85786i			0.412552i	0.978494	+	0.206276i	$0.0661345\pi$
$461$			8.85786i			0.412552i	−0.978494	+	0.206276i	$0.933866\pi$
$462$		0			0
$463$			7.92893i			0.368489i	0.982880	+	0.184244i	$0.0589838\pi$
$463$			7.92893i			0.368489i	−0.982880	+	0.184244i	$0.941016\pi$
$464$		0			0
$465$	−14.0711	+	2.41421i	−0.652530	+	0.111956i
$466$		0			0
$467$	23.7279			1.09800			0.548999	−	0.835823i	$-0.315009\pi$
$467$	23.7279			1.09800			0.548999	+	0.835823i	$0.315009\pi$
$468$		0			0
$469$	−7.65685			−0.353561
$470$		0			0
$471$	−30.1421	+	5.17157i	−1.38888	+	0.238293i
$472$		0			0
$473$			58.9411i			2.71012i
$474$		0			0
$475$		−	4.00000i		−	0.183533i
$476$		0			0
$477$	5.07107	+	14.3431i	0.232188	+	0.656728i
$478$		0			0
$479$	−20.4853			−0.935996			−0.467998	−	0.883729i	$-0.655025\pi$
$479$	−20.4853			−0.935996			−0.467998	+	0.883729i	$0.655025\pi$
$480$		0			0
$481$	30.0000			1.36788
$482$		0			0
$483$	−4.00000	−	23.3137i	−0.182006	−	1.06081i
$484$		0			0
$485$			3.75736i			0.170613i
$486$		0			0
$487$		−	8.92893i		−	0.404609i	−0.979323	−	0.202304i	$-0.935157\pi$
$487$		−	8.92893i		−	0.404609i	0.979323	−	0.202304i	$-0.0648430\pi$
$488$		0			0
$489$	4.48528	+	26.1421i	0.202831	+	1.18219i
$490$		0			0
$491$	−14.3431			−0.647297			−0.323649	−	0.946177i	$-0.604910\pi$
$491$	−14.3431			−0.647297			−0.323649	+	0.946177i	$0.604910\pi$
$492$		0			0
$493$	14.6274			0.658786
$494$		0			0
$495$	−5.24264	−	14.8284i	−0.235639	−	0.666488i
$496$		0			0
$497$		−	22.7279i		−	1.01949i
$498$		0			0
$499$		−	39.5269i		−	1.76947i	−0.466097	−	0.884734i	$-0.654340\pi$
$499$		−	39.5269i		−	1.76947i	0.466097	−	0.884734i	$-0.345660\pi$
$500$		0			0
$501$	−22.1421	+	3.79899i	−0.989238	+	0.169726i
$502$		0			0
$503$	−17.1716			−0.765643			−0.382821	−	0.923822i	$-0.625048\pi$
$503$	−17.1716			−0.765643			−0.382821	+	0.923822i	$0.625048\pi$
$504$		0			0
$505$	2.34315			0.104269
$506$		0			0
$507$	8.53553	−	1.46447i	0.379076	−	0.0650392i
$508$		0			0
$509$			42.3848i			1.87867i	0.342998	+	0.939336i	$0.388558\pi$
$509$			42.3848i			1.87867i	−0.342998	+	0.939336i	$0.611442\pi$
$510$		0			0
$511$			31.3848i			1.38838i
$512$		0			0
$513$	−2.53553	−	4.53553i	−0.111947	−	0.200249i
$514$		0			0
$515$	−1.51472			−0.0667465
$516$		0			0
$517$	−2.17157			−0.0955057
$518$		0			0
$519$	−4.58579	−	26.7279i	−0.201294	−	1.17323i
$520$		0			0
$521$		−	43.6569i		−	1.91264i	−0.292319	−	0.956321i	$-0.594427\pi$
$521$		−	43.6569i		−	1.91264i	0.292319	−	0.956321i	$-0.405573\pi$
$522$		0			0
$523$		−	13.7990i		−	0.603388i	−0.953405	−	0.301694i	$-0.902448\pi$
$523$		−	13.7990i		−	0.603388i	0.953405	−	0.301694i	$-0.0975520\pi$
$524$		0			0
$525$	−2.82843	−	16.4853i	−0.123443	−	0.719477i
$526$		0			0
$527$	−15.0711			−0.656506
$528$		0			0
$529$	9.00000			0.391304
$530$		0			0
$531$	−34.6274	+	12.2426i	−1.50270	+	0.531285i
$532$		0			0
$533$		−	39.9411i		−	1.73004i
$534$		0			0
$535$			2.48528i			0.107448i
$536$		0			0
$537$	−24.4853	+	4.20101i	−1.05662	+	0.181287i
$538$		0			0
$539$	−6.14214			−0.264561
$540$		0			0
$541$	−19.3431			−0.831627			−0.415813	−	0.909450i	$-0.636503\pi$
$541$	−19.3431			−0.831627			−0.415813	+	0.909450i	$0.636503\pi$
$542$		0			0
$543$	−32.3848	+	5.55635i	−1.38976	+	0.238446i
$544$		0			0
$545$		−	0.343146i		−	0.0146987i
$546$		0			0
$547$			30.9289i			1.32243i	0.750198	+	0.661213i	$0.229958\pi$
$547$			30.9289i			1.32243i	−0.750198	+	0.661213i	$0.770042\pi$
$548$		0			0
$549$	24.4853	−	8.65685i	1.04501	−	0.369466i
$550$		0			0
$551$	−8.00000			−0.340811
$552$		0			0
$553$	−21.3137			−0.906351
$554$		0			0
$555$	2.07107	+	12.0711i	0.0879119	+	0.512388i
$556$		0			0
$557$			22.1716i			0.939440i	0.882816	+	0.469720i	$0.155645\pi$
$557$			22.1716i			0.939440i	−0.882816	+	0.469720i	$0.844355\pi$
$558$		0			0
$559$		−	47.6985i		−	2.01743i
$560$		0			0
$561$	−2.80761	−	16.3640i	−0.118537	−	0.690887i
$562$		0			0
$563$	−11.2132			−0.472580			−0.236290	−	0.971683i	$-0.575932\pi$
$563$	−11.2132			−0.472580			−0.236290	+	0.971683i	$0.575932\pi$
$564$		0			0
$565$	−11.3137			−0.475971
$566$		0			0
$567$	−13.6569	−	16.8995i	−0.573534	−	0.709712i
$568$		0			0
$569$		−	3.07107i		−	0.128746i	−0.997926	−	0.0643729i	$-0.979495\pi$
$569$		−	3.07107i		−	0.128746i	0.997926	−	0.0643729i	$-0.0205047\pi$
$570$		0			0
$571$			29.4558i			1.23269i	0.787477	+	0.616344i	$0.211387\pi$
$571$			29.4558i			1.23269i	−0.787477	+	0.616344i	$0.788613\pi$
$572$		0			0
$573$	21.1924	−	3.63604i	0.885325	−	0.151898i
$574$		0			0
$575$	22.6274			0.943629
$576$		0			0
$577$	−29.7696			−1.23932			−0.619661	−	0.784869i	$-0.712730\pi$
$577$	−29.7696			−1.23932			−0.619661	+	0.784869i	$0.712730\pi$
$578$		0			0
$579$	18.7279	−	3.21320i	0.778306	−	0.133536i
$580$		0			0
$581$			28.1421i			1.16753i
$582$		0			0
$583$		−	26.5858i		−	1.10107i
$584$		0			0
$585$	4.24264	+	12.0000i	0.175412	+	0.496139i
$586$		0			0
$587$	6.07107			0.250580			0.125290	−	0.992120i	$-0.460014\pi$
$587$	6.07107			0.250580			0.125290	+	0.992120i	$0.460014\pi$
$588$		0			0
$589$	8.24264			0.339632
$590$		0			0
$591$	1.51472	+	8.82843i	0.0623072	+	0.363153i
$592$		0			0
$593$		−	18.6863i		−	0.767354i	−0.923467	−	0.383677i	$-0.874658\pi$
$593$		−	18.6863i		−	0.767354i	0.923467	−	0.383677i	$-0.125342\pi$
$594$		0			0
$595$			4.41421i			0.180965i
$596$		0			0
$597$	2.94975	+	17.1924i	0.120725	+	0.703638i
$598$		0			0
$599$	−1.51472			−0.0618897			−0.0309449	−	0.999521i	$-0.509852\pi$
$599$	−1.51472			−0.0618897			−0.0309449	+	0.999521i	$0.509852\pi$
$600$		0			0
$601$	−27.8995			−1.13804			−0.569022	−	0.822322i	$-0.692678\pi$
$601$	−27.8995			−1.13804			−0.569022	+	0.822322i	$0.692678\pi$
$602$		0			0
$603$	−3.17157	−	8.97056i	−0.129156	−	0.365310i
$604$		0			0
$605$			16.4853i			0.670222i
$606$		0			0
$607$		−	4.10051i		−	0.166434i	−0.996531	−	0.0832172i	$-0.973480\pi$
$607$		−	4.10051i		−	0.166434i	0.996531	−	0.0832172i	$-0.0265195\pi$
$608$		0			0
$609$	−32.9706	+	5.65685i	−1.33603	+	0.229227i
$610$		0			0
$611$	1.75736			0.0710951
$612$		0			0
$613$	−33.2843			−1.34434			−0.672170	−	0.740397i	$-0.734637\pi$
$613$	−33.2843			−1.34434			−0.672170	+	0.740397i	$0.734637\pi$
$614$		0			0
$615$	16.0711	−	2.75736i	0.648048	−	0.111187i
$616$		0			0
$617$			19.8284i			0.798262i	0.916894	+	0.399131i	$0.130688\pi$
$617$			19.8284i			0.798262i	−0.916894	+	0.399131i	$0.869312\pi$
$618$		0			0
$619$		−	39.9411i		−	1.60537i	−0.596404	−	0.802685i	$-0.703404\pi$
$619$		−	39.9411i		−	1.60537i	0.596404	−	0.802685i	$-0.296596\pi$
$620$		0			0
$621$	25.6569	−	14.3431i	1.02957	−	0.575571i
$622$		0			0
$623$	34.3848			1.37760
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	1.53553	+	8.94975i	0.0613233	+	0.357418i
$628$		0			0
$629$			12.9289i			0.515510i
$630$		0			0
$631$			17.0416i			0.678417i	0.940711	+	0.339208i	$0.110159\pi$
$631$			17.0416i			0.678417i	−0.940711	+	0.339208i	$0.889841\pi$
$632$		0			0
$633$	−4.89949	−	28.5563i	−0.194737	−	1.13501i
$634$		0			0
$635$	−21.6569			−0.859426
$636$		0			0
$637$	4.97056			0.196941
$638$		0			0
$639$	26.6274	−	9.41421i	1.05336	−	0.372421i
$640$		0			0
$641$		−	20.5269i		−	0.810764i	−0.914147	−	0.405382i	$-0.867138\pi$
$641$		−	20.5269i		−	0.810764i	0.914147	−	0.405382i	$-0.132862\pi$
$642$		0			0
$643$		−	24.0711i		−	0.949270i	−0.880183	−	0.474635i	$-0.842580\pi$
$643$		−	24.0711i		−	0.949270i	0.880183	−	0.474635i	$-0.157420\pi$
$644$		0			0
$645$	19.1924	−	3.29289i	0.755700	−	0.129658i
$646$		0			0
$647$	33.2426			1.30690			0.653452	−	0.756968i	$-0.273320\pi$
$647$	33.2426			1.30690			0.653452	+	0.756968i	$0.273320\pi$
$648$		0			0
$649$	64.1838			2.51943
$650$		0			0
$651$	33.9706	−	5.82843i	1.33141	−	0.228434i
$652$		0			0
$653$			12.6569i			0.495301i	0.968849	+	0.247650i	$0.0796585\pi$
$653$			12.6569i			0.495301i	−0.968849	+	0.247650i	$0.920342\pi$
$654$		0			0
$655$			9.58579i			0.374548i
$656$		0			0
$657$	−36.7696	+	13.0000i	−1.43452	+	0.507178i
$658$		0			0
$659$	−0.142136			−0.00553682			−0.00276841	−	0.999996i	$-0.500881\pi$
$659$	−0.142136			−0.00553682			−0.00276841	+	0.999996i	$0.500881\pi$
$660$		0			0
$661$	36.2843			1.41129			0.705647	−	0.708563i	$-0.250656\pi$
$661$	36.2843			1.41129			0.705647	+	0.708563i	$0.250656\pi$
$662$		0			0
$663$	2.27208	+	13.2426i	0.0882402	+	0.514302i
$664$		0			0
$665$		−	2.41421i		−	0.0936192i
$666$		0			0
$667$		−	45.2548i		−	1.75227i
$668$		0			0
$669$	−0.171573	−	1.00000i	−0.00663339	−	0.0386622i
$670$		0			0
$671$	−45.3848			−1.75206
$672$		0			0
$673$	5.27208			0.203224			0.101612	−	0.994824i	$-0.467600\pi$
$673$	5.27208			0.203224			0.101612	+	0.994824i	$0.467600\pi$
$674$		0			0
$675$	18.1421	−	10.1421i	0.698291	−	0.390371i
$676$		0			0
$677$		−	9.65685i		−	0.371143i	−0.982631	−	0.185572i	$-0.940586\pi$
$677$		−	9.65685i		−	0.371143i	0.982631	−	0.185572i	$-0.0594136\pi$
$678$		0			0
$679$		−	9.07107i		−	0.348116i
$680$		0			0
$681$	−22.3137	+	3.82843i	−0.855063	+	0.146706i
$682$		0			0
$683$	25.3553			0.970195			0.485098	−	0.874460i	$-0.338784\pi$
$683$	25.3553			0.970195			0.485098	+	0.874460i	$0.338784\pi$
$684$		0			0
$685$	−3.82843			−0.146277
$686$		0			0
$687$	−17.6066	+	3.02082i	−0.671734	+	0.115251i
$688$		0			0
$689$			21.5147i			0.819646i
$690$		0			0
$691$			12.7574i			0.485313i	0.970112	+	0.242656i	$0.0780188\pi$
$691$			12.7574i			0.485313i	−0.970112	+	0.242656i	$0.921981\pi$
$692$		0			0
$693$	12.6569	+	35.7990i	0.480794	+	1.35989i
$694$		0			0
$695$	−5.58579			−0.211881
$696$		0			0
$697$	17.2132			0.651997
$698$		0			0
$699$	8.15076	+	47.5061i	0.308290	+	1.79685i
$700$		0			0
$701$		−	10.8284i		−	0.408984i	−0.978868	−	0.204492i	$-0.934446\pi$
$701$		−	10.8284i		−	0.408984i	0.978868	−	0.204492i	$-0.0655542\pi$
$702$		0			0
$703$		−	7.07107i		−	0.266690i
$704$		0			0
$705$	0.121320	+	0.707107i	0.00456919	+	0.0266312i
$706$		0			0
$707$	−5.65685			−0.212748
$708$		0			0
$709$	43.7990			1.64491			0.822453	−	0.568833i	$-0.192605\pi$
$709$	43.7990			1.64491			0.822453	+	0.568833i	$0.192605\pi$
$710$		0			0
$711$	−8.82843	−	24.9706i	−0.331092	−	0.936469i
$712$		0			0
$713$			46.6274i			1.74621i
$714$		0			0
$715$		−	22.2426i		−	0.831828i
$716$		0			0
$717$	37.0919	−	6.36396i	1.38522	−	0.237666i
$718$		0			0
$719$	−4.27208			−0.159322			−0.0796608	−	0.996822i	$-0.525384\pi$
$719$	−4.27208			−0.159322			−0.0796608	+	0.996822i	$0.525384\pi$
$720$		0			0
$721$	3.65685			0.136188
$722$		0			0
$723$	−7.24264	+	1.24264i	−0.269357	+	0.0462143i
$724$		0			0
$725$		−	32.0000i		−	1.18845i
$726$		0			0
$727$		−	13.9289i		−	0.516595i	−0.966065	−	0.258298i	$-0.916838\pi$
$727$		−	13.9289i		−	0.516595i	0.966065	−	0.258298i	$-0.0831616\pi$
$728$		0			0
$729$	14.1421	−	23.0000i	0.523783	−	0.851852i
$730$		0			0
$731$	20.5563			0.760304
$732$		0			0
$733$	20.7696			0.767141			0.383570	−	0.923512i	$-0.374694\pi$
$733$	20.7696			0.767141			0.383570	+	0.923512i	$0.374694\pi$
$734$		0			0
$735$	0.343146	+	2.00000i	0.0126571	+	0.0737711i
$736$		0			0
$737$			16.6274i			0.612479i
$738$		0			0
$739$			32.3553i			1.19021i	0.803648	+	0.595105i	$0.202890\pi$
$739$			32.3553i			1.19021i	−0.803648	+	0.595105i	$0.797110\pi$
$740$		0			0
$741$	−1.24264	−	7.24264i	−0.0456495	−	0.266065i
$742$		0			0
$743$	7.45584			0.273528			0.136764	−	0.990604i	$-0.456330\pi$
$743$	7.45584			0.273528			0.136764	+	0.990604i	$0.456330\pi$
$744$		0			0
$745$	−12.3137			−0.451139
$746$		0			0
$747$	−32.9706	+	11.6569i	−1.20633	+	0.426502i
$748$		0			0
$749$		−	6.00000i		−	0.219235i
$750$		0			0
$751$			16.9706i			0.619265i	0.950856	+	0.309632i	$0.100206\pi$
$751$			16.9706i			0.619265i	−0.950856	+	0.309632i	$0.899794\pi$
$752$		0			0
$753$	−4.36396	+	0.748737i	−0.159032	+	0.0272855i
$754$		0			0
$755$	22.3848			0.814665
$756$		0			0
$757$	38.3137			1.39254			0.696268	−	0.717782i	$-0.254843\pi$
$757$	38.3137			1.39254			0.696268	+	0.717782i	$0.254843\pi$
$758$		0			0
$759$	−50.6274	+	8.68629i	−1.83766	+	0.315292i
$760$		0			0
$761$			9.48528i			0.343841i	0.985111	+	0.171921i	$0.0549973\pi$
$761$			9.48528i			0.343841i	−0.985111	+	0.171921i	$0.945003\pi$
$762$		0			0
$763$			0.828427i			0.0299911i
$764$		0			0
$765$	−5.17157	+	1.82843i	−0.186979	+	0.0661069i
$766$		0			0
$767$	−51.9411			−1.87549
$768$		0			0
$769$	8.17157			0.294674			0.147337	−	0.989086i	$-0.452930\pi$
$769$	8.17157			0.294674			0.147337	+	0.989086i	$0.452930\pi$
$770$		0			0
$771$	−2.34315	−	13.6569i	−0.0843863	−	0.491840i
$772$		0			0
$773$		−	41.0711i		−	1.47722i	−0.674131	−	0.738612i	$-0.735482\pi$
$773$		−	41.0711i		−	1.47722i	0.674131	−	0.738612i	$-0.264518\pi$
$774$		0			0
$775$			32.9706i			1.18434i
$776$		0			0
$777$	−5.00000	−	29.1421i	−0.179374	−	1.04547i
$778$		0			0
$779$	−9.41421			−0.337299
$780$		0			0
$781$	−49.3553			−1.76607
$782$		0			0
$783$	−20.2843	−	36.2843i	−0.724901	−	1.29669i
$784$		0			0
$785$		−	17.6569i		−	0.630200i
$786$		0			0
$787$			39.3137i			1.40138i	0.713465	+	0.700691i	$0.247125\pi$
$787$			39.3137i			1.40138i	−0.713465	+	0.700691i	$0.752875\pi$
$788$		0			0
$789$	−2.12132	+	0.363961i	−0.0755210	+	0.0129574i
$790$		0			0
$791$	27.3137			0.971164
$792$		0			0
$793$	36.7279			1.30425
$794$		0			0
$795$	−8.65685	+	1.48528i	−0.307027	+	0.0526775i
$796$		0			0
$797$			13.3137i			0.471596i	0.971802	+	0.235798i	$0.0757703\pi$
$797$			13.3137i			0.471596i	−0.971802	+	0.235798i	$0.924230\pi$
$798$		0			0
$799$			0.757359i			0.0267934i
$800$		0			0
$801$	14.2426	+	40.2843i	0.503239	+	1.42337i
$802$		0			0
$803$	68.1543			2.40511
$804$		0			0
$805$	13.6569			0.481341
$806$		0			0
$807$	1.28427	+	7.48528i	0.0452085	+	0.263494i
$808$		0			0
$809$		−	6.65685i		−	0.234043i	−0.993129	−	0.117021i	$-0.962665\pi$
$809$		−	6.65685i		−	0.234043i	0.993129	−	0.117021i	$-0.0373346\pi$
$810$		0			0
$811$			10.0416i			0.352609i	0.984336	+	0.176305i	$0.0564144\pi$
$811$			10.0416i			0.352609i	−0.984336	+	0.176305i	$0.943586\pi$
$812$		0			0
$813$	−0.343146	−	2.00000i	−0.0120346	−	0.0701431i
$814$		0			0
$815$	−15.3137			−0.536416
$816$		0			0
$817$	−11.2426			−0.393330
$818$		0			0
$819$	−10.2426	−	28.9706i	−0.357907	−	1.01231i
$820$		0			0
$821$		−	28.1716i		−	0.983195i	−0.870823	−	0.491597i	$-0.836413\pi$
$821$		−	28.1716i		−	0.983195i	0.870823	−	0.491597i	$-0.163587\pi$
$822$		0			0
$823$		−	23.1005i		−	0.805233i	−0.915369	−	0.402616i	$-0.868101\pi$
$823$		−	23.1005i		−	0.805233i	0.915369	−	0.402616i	$-0.131899\pi$
$824$		0			0
$825$	−35.7990	+	6.14214i	−1.24636	+	0.213842i
$826$		0			0
$827$	46.7696			1.62634			0.813168	−	0.582029i	$-0.197741\pi$
$827$	46.7696			1.62634			0.813168	+	0.582029i	$0.197741\pi$
$828$		0			0
$829$	−21.6569			−0.752174			−0.376087	−	0.926584i	$-0.622731\pi$
$829$	−21.6569			−0.752174			−0.376087	+	0.926584i	$0.622731\pi$
$830$		0			0
$831$	3.70711	−	0.636039i	0.128598	−	0.0220639i
$832$		0			0
$833$			2.14214i			0.0742206i
$834$		0			0
$835$		−	12.9706i		−	0.448865i
$836$		0			0
$837$	20.8995	+	37.3848i	0.722392	+	1.29221i
$838$		0			0
$839$	−33.7990			−1.16687			−0.583435	−	0.812160i	$-0.698292\pi$
$839$	−33.7990			−1.16687			−0.583435	+	0.812160i	$0.698292\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$		0			0
$843$	2.72792	+	15.8995i	0.0939546	+	0.547608i
$844$		0			0
$845$			5.00000i			0.172005i
$846$		0			0
$847$		−	39.7990i		−	1.36751i
$848$		0			0
$849$	3.77817	+	22.0208i	0.129667	+	0.755752i
$850$		0			0
$851$	40.0000			1.37118
$852$		0			0
$853$	−19.3137			−0.661289			−0.330644	−	0.943755i	$-0.607266\pi$
$853$	−19.3137			−0.661289			−0.330644	+	0.943755i	$0.607266\pi$
$854$		0			0
$855$	2.82843	−	1.00000i	0.0967302	−	0.0341993i
$856$		0			0
$857$			5.07107i			0.173224i	0.996242	+	0.0866122i	$0.0276041\pi$
$857$			5.07107i			0.173224i	−0.996242	+	0.0866122i	$0.972396\pi$
$858$		0			0
$859$		−	27.3848i		−	0.934357i	−0.884163	−	0.467178i	$-0.845271\pi$
$859$		−	27.3848i		−	0.934357i	0.884163	−	0.467178i	$-0.154729\pi$
$860$		0			0
$861$	−38.7990	+	6.65685i	−1.32227	+	0.226865i
$862$		0			0
$863$	33.3553			1.13543			0.567714	−	0.823226i	$-0.307828\pi$
$863$	33.3553			1.13543			0.567714	+	0.823226i	$0.307828\pi$
$864$		0			0
$865$	15.6569			0.532349
$866$		0			0
$867$	23.3137	−	4.00000i	0.791775	−	0.135847i
$868$		0			0
$869$			46.2843i			1.57009i
$870$		0			0
$871$		−	13.4558i		−	0.455934i
$872$		0			0
$873$	10.6274	−	3.75736i	0.359684	−	0.127167i
$874$		0			0
$875$	21.7279			0.734538
$876$		0			0
$877$	43.6569			1.47419			0.737094	−	0.675791i	$-0.236198\pi$
$877$	43.6569			1.47419			0.737094	+	0.675791i	$0.236198\pi$
$878$		0			0
$879$	−0.686292	−	4.00000i	−0.0231480	−	0.134917i
$880$		0			0
$881$		−	33.6274i		−	1.13294i	−0.824084	−	0.566468i	$-0.808309\pi$
$881$		−	33.6274i		−	1.13294i	0.824084	−	0.566468i	$-0.191691\pi$
$882$		0			0
$883$		−	30.8406i		−	1.03787i	−0.854814	−	0.518935i	$-0.826329\pi$
$883$		−	30.8406i		−	1.03787i	0.854814	−	0.518935i	$-0.173671\pi$
$884$		0			0
$885$	−3.58579	−	20.8995i	−0.120535	−	0.702529i
$886$		0			0
$887$	−24.0416			−0.807239			−0.403619	−	0.914927i	$-0.632248\pi$
$887$	−24.0416			−0.807239			−0.403619	+	0.914927i	$0.632248\pi$
$888$		0			0
$889$	52.2843			1.75356
$890$		0			0
$891$	−36.6985	+	29.6569i	−1.22945	+	0.993542i
$892$		0			0
$893$		−	0.414214i		−	0.0138611i
$894$		0			0
$895$		−	14.3431i		−	0.479438i
$896$		0			0
$897$	40.9706	−	7.02944i	1.36797	−	0.234706i
$898$		0			0
$899$	65.9411			2.19926
$900$		0			0
$901$	−9.27208			−0.308898
$902$		0			0
$903$	−46.3345	+	7.94975i	−1.54192	+	0.264551i
$904$		0			0
$905$		−	18.9706i		−	0.630603i
$906$		0			0
$907$			28.0416i			0.931107i	0.885020	+	0.465554i	$0.154145\pi$
$907$			28.0416i			0.931107i	−0.885020	+	0.465554i	$0.845855\pi$
$908$		0			0
$909$	−2.34315	−	6.62742i	−0.0777172	−	0.219818i
$910$		0			0
$911$	5.51472			0.182711			0.0913554	−	0.995818i	$-0.470880\pi$
$911$	5.51472			0.182711			0.0913554	+	0.995818i	$0.470880\pi$
$912$		0			0
$913$	61.1127			2.02254
$914$		0			0
$915$	2.53553	+	14.7782i	0.0838222	+	0.488551i
$916$		0			0
$917$		−	23.1421i		−	0.764221i
$918$		0			0
$919$			23.5980i			0.778426i	0.921148	+	0.389213i	$0.127253\pi$
$919$			23.5980i			0.778426i	−0.921148	+	0.389213i	$0.872747\pi$
$920$		0			0
$921$	3.07107	+	17.8995i	0.101195	+	0.589808i
$922$		0			0
$923$	39.9411			1.31468
$924$		0			0
$925$	28.2843			0.929981
$926$		0			0
$927$	1.51472	+	4.28427i	0.0497499	+	0.140714i
$928$		0			0
$929$		−	19.5147i		−	0.640257i	−0.947374	−	0.320129i	$-0.896274\pi$
$929$		−	19.5147i		−	0.640257i	0.947374	−	0.320129i	$-0.103726\pi$
$930$		0			0
$931$		−	1.17157i		−	0.0383968i
$932$		0			0
$933$	28.6066	−	4.90812i	0.936538	−	0.160685i
$934$		0			0
$935$	9.58579			0.313489
$936$		0			0
$937$	−11.1421			−0.363998			−0.181999	−	0.983299i	$-0.558257\pi$
$937$	−11.1421			−0.363998			−0.181999	+	0.983299i	$0.558257\pi$
$938$		0			0
$939$	−2.82843	+	0.485281i	−0.0923022	+	0.0158366i
$940$		0			0
$941$		−	47.6569i		−	1.55357i	−0.629766	−	0.776785i	$-0.716849\pi$
$941$		−	47.6569i		−	1.55357i	0.629766	−	0.776785i	$-0.283151\pi$
$942$		0			0
$943$		−	53.2548i		−	1.73422i
$944$		0			0
$945$	10.9497	−	6.12132i	0.356195	−	0.199127i
$946$		0			0
$947$	−32.6863			−1.06216			−0.531081	−	0.847321i	$-0.678214\pi$
$947$	−32.6863			−1.06216			−0.531081	+	0.847321i	$0.678214\pi$
$948$		0			0
$949$	−55.1543			−1.79039
$950$		0			0
$951$	4.65685	+	27.1421i	0.151009	+	0.880144i
$952$		0			0
$953$			17.3137i			0.560846i	0.959877	+	0.280423i	$0.0904747\pi$
$953$			17.3137i			0.560846i	−0.959877	+	0.280423i	$0.909525\pi$
$954$		0			0
$955$			12.4142i			0.401715i
$956$		0			0
$957$	12.2843	+	71.5980i	0.397094	+	2.31443i
$958$		0			0
$959$	9.24264			0.298460
$960$		0			0
$961$	−36.9411			−1.19165
$962$		0			0
$963$	7.02944	−	2.48528i	0.226520	−	0.0800871i
$964$		0			0
$965$			10.9706i			0.353155i
$966$		0			0
$967$		−	27.3137i		−	0.878350i	−0.898402	−	0.439175i	$-0.855271\pi$
$967$		−	27.3137i		−	0.878350i	0.898402	−	0.439175i	$-0.144729\pi$
$968$		0			0
$969$	3.12132	−	0.535534i	0.100271	−	0.0172038i
$970$		0			0
$971$	3.02944			0.0972193			0.0486096	−	0.998818i	$-0.484521\pi$
$971$	3.02944			0.0972193			0.0486096	+	0.998818i	$0.484521\pi$
$972$		0			0
$973$	13.4853			0.432318
$974$		0			0
$975$	28.9706	−	4.97056i	0.927801	−	0.159185i
$976$		0			0
$977$		−	45.3553i		−	1.45105i	−0.688198	−	0.725523i	$-0.741598\pi$
$977$		−	45.3553i		−	1.45105i	0.688198	−	0.725523i	$-0.258402\pi$
$978$		0			0
$979$		−	74.6690i		−	2.38643i
$980$		0			0
$981$	−0.970563	+	0.343146i	−0.0309877	+	0.0109558i
$982$		0			0
$983$	20.9706			0.668857			0.334429	−	0.942421i	$-0.391457\pi$
$983$	20.9706			0.668857			0.334429	+	0.942421i	$0.391457\pi$
$984$		0			0
$985$	−5.17157			−0.164780
$986$		0			0
$987$	−0.292893	−	1.70711i	−0.00932289	−	0.0543378i
$988$		0			0
$989$		−	63.5980i		−	2.02230i
$990$		0			0
$991$			53.7574i			1.70766i	0.520553	+	0.853829i	$0.325726\pi$
$991$			53.7574i			1.70766i	−0.520553	+	0.853829i	$0.674274\pi$
$992$		0			0
$993$	3.55635	+	20.7279i	0.112857	+	0.657781i
$994$		0			0
$995$	−10.0711			−0.319274
$996$		0			0
$997$	−39.6274			−1.25501			−0.627506	−	0.778611i	$-0.715924\pi$
$997$	−39.6274			−1.25501			−0.627506	+	0.778611i	$0.715924\pi$
$998$		0			0
$999$	32.0711	−	17.9289i	1.01468	−	0.567246i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.d.e.191.3	yes	4
3.2	odd	2		1824.2.d.b.191.1	✓	4
4.3	odd	2		1824.2.d.b.191.2	yes	4
12.11	even	2	inner	1824.2.d.e.191.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.d.b.191.1	✓	4	3.2	odd	2
1824.2.d.b.191.2	yes	4	4.3	odd	2
1824.2.d.e.191.3	yes	4	1.1	even	1	trivial
1824.2.d.e.191.4	yes	4	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1824.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.70711 - 0.292893i) q^{3} +1.00000i q^{5} -2.41421i q^{7} +(2.82843 - 1.00000i) q^{9} -5.24264 q^{11} +4.24264 q^{13} +(0.292893 + 1.70711i) q^{15} +1.82843i q^{17} -1.00000i q^{19} +(-0.707107 - 4.12132i) q^{21} +5.65685 q^{23} +4.00000 q^{25} +(4.53553 - 2.53553i) q^{27} -8.00000i q^{29} +8.24264i q^{31} +(-8.94975 + 1.53553i) q^{33} +2.41421 q^{35} +7.07107 q^{37} +(7.24264 - 1.24264i) q^{39} -9.41421i q^{41} -11.2426i q^{43} +(1.00000 + 2.82843i) q^{45} +0.414214 q^{47} +1.17157 q^{49} +(0.535534 + 3.12132i) q^{51} +5.07107i q^{53} -5.24264i q^{55} +(-0.292893 - 1.70711i) q^{57} -12.2426 q^{59} +8.65685 q^{61} +(-2.41421 - 6.82843i) q^{63} +4.24264i q^{65} -3.17157i q^{67} +(9.65685 - 1.65685i) q^{69} +9.41421 q^{71} -13.0000 q^{73} +(6.82843 - 1.17157i) q^{75} +12.6569i q^{77} -8.82843i q^{79} +(7.00000 - 5.65685i) q^{81} -11.6569 q^{83} -1.82843 q^{85} +(-2.34315 - 13.6569i) q^{87} +14.2426i q^{89} -10.2426i q^{91} +(2.41421 + 14.0711i) q^{93} +1.00000 q^{95} +3.75736 q^{97} +(-14.8284 + 5.24264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{11} + 4 q^{15} + 16 q^{25} + 4 q^{27} - 16 q^{33} + 4 q^{35} + 12 q^{39} + 4 q^{45} - 4 q^{47} + 16 q^{49} - 12 q^{51} - 4 q^{57} - 32 q^{59} + 12 q^{61} - 4 q^{63} + 16 q^{69} + 32 q^{71}+ \cdots - 48 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^11 + 4 * q^15 + 16 * q^25 + 4 * q^27 - 16 * q^33 + 4 * q^35 + 12 * q^39 + 4 * q^45 - 4 * q^47 + 16 * q^49 - 12 * q^51 - 4 * q^57 - 32 * q^59 + 12 * q^61 - 4 * q^63 + 16 * q^69 + 32 * q^71 - 52 * q^73 + 16 * q^75 + 28 * q^81 - 24 * q^83 + 4 * q^85 - 32 * q^87 + 4 * q^93 + 4 * q^95 + 32 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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