LMFDB - Embedded newform 1104.2.e.h.47.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,2,Mod(47,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1104.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.e (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:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.4
Root			$0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1104.47
Dual form			1104.2.e.h.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.618034 + 1.61803i) q^{3} +3.18448i q^{5} +3.68850i q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})$	\(q+(-0.618034 + 1.61803i) q^{3} +3.18448i q^{5} +3.68850i q^{7} +(-2.23607 - 2.00000i) q^{9} -2.73205 q^{11} +5.46410 q^{13} +(-5.15260 - 1.96812i) q^{15} +6.38867i q^{17} +3.18448i q^{19} +(-5.96812 - 2.27962i) q^{21} +1.00000 q^{23} -5.14093 q^{25} +(4.61803 - 2.38197i) q^{27} -4.47214i q^{29} +2.14093i q^{31} +(1.68850 - 4.42055i) q^{33} -11.7460 q^{35} +6.87298 q^{37} +(-3.37700 + 8.84110i) q^{39} -8.70017i q^{41} +0.0893476i q^{43} +(6.36897 - 7.12072i) q^{45} -6.14093 q^{47} -6.60503 q^{49} +(-10.3371 - 3.94842i) q^{51} -11.5695i q^{53} -8.70017i q^{55} +(-5.15260 - 1.96812i) q^{57} +5.70820 q^{59} +4.05522 q^{61} +(7.37700 - 8.24774i) q^{63} +17.4003i q^{65} -11.1207i q^{67} +(-0.618034 + 1.61803i) q^{69} -7.84914 q^{71} +2.81776 q^{73} +(3.17727 - 8.31820i) q^{75} -10.0772i q^{77} +13.7118i q^{79} +(1.00000 + 8.94427i) q^{81} -5.66025 q^{83} -20.3446 q^{85} +(7.23607 + 2.76393i) q^{87} +14.8447i q^{89} +20.1543i q^{91} +(-3.46410 - 1.32317i) q^{93} -10.1409 q^{95} +8.92093 q^{97} +(6.10905 + 5.46410i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3}+O(q^{10}) $ 8 * q + 4 * q^3	$ 8 q + 4 q^{3} - 8 q^{11} + 16 q^{13} + 4 q^{15} - 16 q^{21} + 8 q^{23} - 24 q^{25} + 28 q^{27} - 4 q^{33} - 32 q^{35} + 24 q^{37} + 8 q^{39} + 24 q^{45} - 32 q^{47} - 8 q^{49} - 24 q^{51} + 4 q^{57} - 8 q^{59} + 8 q^{61} + 24 q^{63} + 4 q^{69} + 8 q^{71} + 16 q^{73} - 12 q^{75} + 8 q^{81} + 24 q^{83} + 40 q^{87} - 64 q^{95} + 56 q^{97}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^11 + 16 * q^13 + 4 * q^15 - 16 * q^21 + 8 * q^23 - 24 * q^25 + 28 * q^27 - 4 * q^33 - 32 * q^35 + 24 * q^37 + 8 * q^39 + 24 * q^45 - 32 * q^47 - 8 * q^49 - 24 * q^51 + 4 * q^57 - 8 * q^59 + 8 * q^61 + 24 * q^63 + 4 * q^69 + 8 * q^71 + 16 * q^73 - 12 * q^75 + 8 * q^81 + 24 * q^83 + 40 * q^87 - 64 * q^95 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−0.618034	+	1.61803i	−0.356822	+	0.934172i
$3$	−0.618034	+	1.61803i	−0.356822	+	0.934172i
$4$		0			0
$5$			3.18448i			1.42414i	0.702106	+	0.712072i	$0.252243\pi$
$5$			3.18448i			1.42414i	−0.702106	+	0.712072i	$0.747757\pi$
$6$		0			0
$7$			3.68850i			1.39412i	0.717012	+	0.697061i	$0.245509\pi$
$7$			3.68850i			1.39412i	−0.717012	+	0.697061i	$0.754491\pi$
$8$		0			0
$9$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$10$		0			0
$11$	−2.73205			−0.823744			−0.411872	−	0.911242i	$-0.635125\pi$
$11$	−2.73205			−0.823744			−0.411872	+	0.911242i	$0.635125\pi$
$12$		0			0
$13$	5.46410			1.51547			0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410			1.51547			0.757735	+	0.652563i	$0.226306\pi$
$14$		0			0
$15$	−5.15260	−	1.96812i	−1.33040	−	0.508166i
$16$		0			0
$17$			6.38867i			1.54948i	0.632280	+	0.774740i	$0.282119\pi$
$17$			6.38867i			1.54948i	−0.632280	+	0.774740i	$0.717881\pi$
$18$		0			0
$19$			3.18448i			0.730571i	0.930896	+	0.365285i	$0.119029\pi$
$19$			3.18448i			0.730571i	−0.930896	+	0.365285i	$0.880971\pi$
$20$		0			0
$21$	−5.96812	−	2.27962i	−1.30235	−	0.497454i
$22$		0			0
$23$	1.00000			0.208514
$23$	1.00000			0.208514
$24$		0			0
$25$	−5.14093			−1.02819
$26$		0			0
$27$	4.61803	−	2.38197i	0.888741	−	0.458410i
$28$		0			0
$29$		−	4.47214i		−	0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$		−	4.47214i		−	0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$		0			0
$31$			2.14093i			0.384523i	0.981344	+	0.192261i	$0.0615822\pi$
$31$			2.14093i			0.384523i	−0.981344	+	0.192261i	$0.938418\pi$
$32$		0			0
$33$	1.68850	−	4.42055i	0.293930	−	0.769519i
$34$		0			0
$35$	−11.7460			−1.98543
$36$		0			0
$37$	6.87298			1.12991			0.564956	−	0.825121i	$-0.308893\pi$
$37$	6.87298			1.12991			0.564956	+	0.825121i	$0.308893\pi$
$38$		0			0
$39$	−3.37700	+	8.84110i	−0.540753	+	1.41571i
$40$		0			0
$41$		−	8.70017i		−	1.35874i	−0.733797	−	0.679369i	$-0.762254\pi$
$41$		−	8.70017i		−	1.35874i	0.733797	−	0.679369i	$-0.237746\pi$
$42$		0			0
$43$			0.0893476i			0.0136254i	0.999977	+	0.00681269i	$0.00216856\pi$
$43$			0.0893476i			0.0136254i	−0.999977	+	0.00681269i	$0.997831\pi$
$44$		0			0
$45$	6.36897	−	7.12072i	0.949429	−	1.06149i
$46$		0			0
$47$	−6.14093			−0.895747			−0.447874	−	0.894097i	$-0.647819\pi$
$47$	−6.14093			−0.895747			−0.447874	+	0.894097i	$0.647819\pi$
$48$		0			0
$49$	−6.60503			−0.943576
$50$		0			0
$51$	−10.3371	−	3.94842i	−1.44748	−	0.552889i
$52$		0			0
$53$		−	11.5695i		−	1.58920i	−0.607136	−	0.794598i	$-0.707682\pi$
$53$		−	11.5695i		−	1.58920i	0.607136	−	0.794598i	$-0.292318\pi$
$54$		0			0
$55$		−	8.70017i		−	1.17313i
$56$		0			0
$57$	−5.15260	−	1.96812i	−0.682479	−	0.260684i
$58$		0			0
$59$	5.70820			0.743145			0.371572	−	0.928404i	$-0.378819\pi$
$59$	5.70820			0.743145			0.371572	+	0.928404i	$0.378819\pi$
$60$		0			0
$61$	4.05522			0.519218			0.259609	−	0.965714i	$-0.416406\pi$
$61$	4.05522			0.519218			0.259609	+	0.965714i	$0.416406\pi$
$62$		0			0
$63$	7.37700	−	8.24774i	0.929415	−	1.03912i
$64$		0			0
$65$			17.4003i			2.15825i
$66$		0			0
$67$		−	11.1207i		−	1.35861i	−0.733855	−	0.679306i	$-0.762281\pi$
$67$		−	11.1207i		−	1.35861i	0.733855	−	0.679306i	$-0.237719\pi$
$68$		0			0
$69$	−0.618034	+	1.61803i	−0.0744025	+	0.194788i
$70$		0			0
$71$	−7.84914			−0.931521			−0.465761	−	0.884911i	$-0.654219\pi$
$71$	−7.84914			−0.931521			−0.465761	+	0.884911i	$0.654219\pi$
$72$		0			0
$73$	2.81776			0.329794			0.164897	−	0.986311i	$-0.447271\pi$
$73$	2.81776			0.329794			0.164897	+	0.986311i	$0.447271\pi$
$74$		0			0
$75$	3.17727	−	8.31820i	0.366880	−	0.960503i
$76$		0			0
$77$		−	10.0772i		−	1.14840i
$78$		0			0
$79$			13.7118i			1.54270i	0.636410	+	0.771351i	$0.280419\pi$
$79$			13.7118i			1.54270i	−0.636410	+	0.771351i	$0.719581\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	−5.66025			−0.621294			−0.310647	−	0.950525i	$-0.600546\pi$
$83$	−5.66025			−0.621294			−0.310647	+	0.950525i	$0.600546\pi$
$84$		0			0
$85$	−20.3446			−2.20668
$86$		0			0
$87$	7.23607	+	2.76393i	0.775788	+	0.296325i
$88$		0			0
$89$			14.8447i			1.57354i	0.617247	+	0.786769i	$0.288248\pi$
$89$			14.8447i			1.57354i	−0.617247	+	0.786769i	$0.711752\pi$
$90$		0			0
$91$			20.1543i			2.11275i
$92$		0			0
$93$	−3.46410	−	1.32317i	−0.359211	−	0.137206i
$94$		0			0
$95$	−10.1409			−1.04044
$96$		0			0
$97$	8.92093			0.905784			0.452892	−	0.891566i	$-0.350392\pi$
$97$	8.92093			0.905784			0.452892	+	0.891566i	$0.350392\pi$
$98$		0			0
$99$	6.10905	+	5.46410i	0.613983	+	0.549163i
$100$		0			0
$101$		−	4.30786i		−	0.428649i	−0.976763	−	0.214324i	$-0.931245\pi$
$101$		−	4.30786i		−	0.428649i	0.976763	−	0.214324i	$-0.0687549\pi$
$102$		0			0
$103$		−	2.68047i		−	0.264114i	−0.991242	−	0.132057i	$-0.957842\pi$
$103$		−	2.68047i		−	0.264114i	0.991242	−	0.132057i	$-0.0421582\pi$
$104$		0			0
$105$	7.25941	−	19.0054i	0.708446	−	1.85473i
$106$		0			0
$107$	11.1171			1.07473			0.537365	−	0.843350i	$-0.319420\pi$
$107$	11.1171			1.07473			0.537365	+	0.843350i	$0.319420\pi$
$108$		0			0
$109$	−13.9044			−1.33180			−0.665898	−	0.746043i	$-0.731951\pi$
$109$	−13.9044			−1.33180			−0.665898	+	0.746043i	$0.731951\pi$
$110$		0			0
$111$	−4.24774	+	11.1207i	−0.403177	+	1.05553i
$112$		0			0
$113$			0.0197037i			0.00185357i	1.00000		0.000926783i	$0.000295004\pi$
$113$			0.0197037i			0.00185357i	−1.00000		0.000926783i	$0.999705\pi$
$114$		0			0
$115$			3.18448i			0.296955i
$116$		0			0
$117$	−12.2181	−	10.9282i	−1.12956	−	1.01031i
$118$		0			0
$119$	−23.5646			−2.16016
$120$		0			0
$121$	−3.53590			−0.321445
$122$		0			0
$123$	14.0772	+	5.37700i	1.26930	+	0.484828i
$124$		0			0
$125$		−	0.448797i		−	0.0401416i
$126$		0			0
$127$		−	1.89683i		−	0.168316i	−0.996452	−	0.0841582i	$-0.973180\pi$
$127$		−	1.89683i		−	0.168316i	0.996452	−	0.0841582i	$-0.0268201\pi$
$128$		0			0
$129$	−0.144568	−	0.0552199i	−0.0127285	−	0.00486184i
$130$		0			0
$131$	12.9660			1.13284			0.566421	−	0.824116i	$-0.308328\pi$
$131$	12.9660			1.13284			0.566421	+	0.824116i	$0.308328\pi$
$132$		0			0
$133$	−11.7460			−1.01850
$134$		0			0
$135$	7.58533	+	14.7061i	0.652841	+	1.26570i
$136$		0			0
$137$			15.1148i			1.29135i	0.763613	+	0.645674i	$0.223423\pi$
$137$			15.1148i			1.29135i	−0.763613	+	0.645674i	$0.776577\pi$
$138$		0			0
$139$		−	15.4641i		−	1.31165i	−0.754914	−	0.655824i	$-0.772321\pi$
$139$		−	15.4641i		−	1.31165i	0.754914	−	0.655824i	$-0.227679\pi$
$140$		0			0
$141$	3.79531	−	9.93624i	0.319622	−	0.836782i
$142$		0			0
$143$	−14.9282			−1.24836
$144$		0			0
$145$	14.2414			1.18269
$146$		0			0
$147$	4.08214	−	10.6872i	0.336689	−	0.881463i
$148$		0			0
$149$			13.4259i			1.09989i	0.835199	+	0.549947i	$0.185352\pi$
$149$			13.4259i			1.09989i	−0.835199	+	0.549947i	$0.814648\pi$
$150$		0			0
$151$		−	20.3186i		−	1.65351i	−0.562566	−	0.826753i	$-0.690186\pi$
$151$		−	20.3186i		−	1.65351i	0.562566	−	0.826753i	$-0.309814\pi$
$152$		0			0
$153$	12.7773	−	14.2855i	1.03299	−	1.15491i
$154$		0			0
$155$	−6.81776			−0.547616
$156$		0			0
$157$	5.69075			0.454171			0.227086	−	0.973875i	$-0.427080\pi$
$157$	5.69075			0.454171			0.227086	+	0.973875i	$0.427080\pi$
$158$		0			0
$159$	18.7199	+	7.15036i	1.48458	+	0.567060i
$160$		0			0
$161$			3.68850i			0.290695i
$162$		0			0
$163$			21.8331i			1.71010i	0.518547	+	0.855049i	$0.326473\pi$
$163$			21.8331i			1.71010i	−0.518547	+	0.855049i	$0.673527\pi$
$164$		0			0
$165$	14.0772	+	5.37700i	1.09591	+	0.418599i
$166$		0			0
$167$	17.9874			1.39191			0.695954	−	0.718087i	$-0.254982\pi$
$167$	17.9874			1.39191			0.695954	+	0.718087i	$0.254982\pi$
$168$		0			0
$169$	16.8564			1.29665
$170$		0			0
$171$	6.36897	−	7.12072i	0.487047	−	0.544535i
$172$		0			0
$173$			7.52786i			0.572333i	0.958180	+	0.286166i	$0.0923810\pi$
$173$			7.52786i			0.572333i	−0.958180	+	0.286166i	$0.907619\pi$
$174$		0			0
$175$		−	18.9623i		−	1.43342i
$176$		0			0
$177$	−3.52786	+	9.23607i	−0.265170	+	0.694225i
$178$		0			0
$179$	4.03776			0.301797			0.150898	−	0.988549i	$-0.451783\pi$
$179$	4.03776			0.301797			0.150898	+	0.988549i	$0.451783\pi$
$180$		0			0
$181$	−19.6503			−1.46060			−0.730299	−	0.683128i	$-0.760619\pi$
$181$	−19.6503			−1.46060			−0.730299	+	0.683128i	$0.760619\pi$
$182$		0			0
$183$	−2.50626	+	6.56148i	−0.185268	+	0.485039i
$184$		0			0
$185$			21.8869i			1.60916i
$186$		0			0
$187$		−	17.4542i		−	1.27638i
$188$		0			0
$189$	8.78588	+	17.0336i	0.639079	+	1.23901i
$190$		0			0
$191$	11.2101			0.811132			0.405566	−	0.914066i	$-0.367074\pi$
$191$	11.2101			0.811132			0.405566	+	0.914066i	$0.367074\pi$
$192$		0			0
$193$	−19.8869			−1.43149			−0.715745	−	0.698362i	$-0.753913\pi$
$193$	−19.8869			−1.43149			−0.715745	+	0.698362i	$0.753913\pi$
$194$		0			0
$195$	−28.1543	−	10.7540i	−2.01617	−	0.770110i
$196$		0			0
$197$		−	4.86710i		−	0.346767i	−0.984854	−	0.173383i	$-0.944530\pi$
$197$		−	4.86710i		−	0.346767i	0.984854	−	0.173383i	$-0.0554700\pi$
$198$		0			0
$199$		−	19.1804i		−	1.35966i	−0.733367	−	0.679832i	$-0.762052\pi$
$199$		−	19.1804i		−	1.35966i	0.733367	−	0.679832i	$-0.237948\pi$
$200$		0			0
$201$	17.9937	+	6.87298i	1.26918	+	0.484783i
$202$		0			0
$203$	16.4955			1.15776
$204$		0			0
$205$	27.7055			1.93504
$206$		0			0
$207$	−2.23607	−	2.00000i	−0.155417	−	0.139010i
$208$		0			0
$209$		−	8.70017i		−	0.601803i
$210$		0			0
$211$			7.51344i			0.517247i	0.965978	+	0.258623i	$0.0832688\pi$
$211$			7.51344i			0.517247i	−0.965978	+	0.258623i	$0.916731\pi$
$212$		0			0
$213$	4.85103	−	12.7002i	0.332387	−	0.870201i
$214$		0			0
$215$	−0.284526			−0.0194045
$216$		0			0
$217$	−7.89683			−0.536072
$218$		0			0
$219$	−1.74147	+	4.55924i	−0.117678	+	0.308085i
$220$		0			0
$221$			34.9083i			2.34819i
$222$		0			0
$223$			17.1841i			1.15073i	0.817897	+	0.575365i	$0.195140\pi$
$223$			17.1841i			1.15073i	−0.817897	+	0.575365i	$0.804860\pi$
$224$		0			0
$225$	11.4955	+	10.2819i	0.766365	+	0.685458i
$226$		0			0
$227$	20.4853			1.35966			0.679828	−	0.733371i	$-0.262054\pi$
$227$	20.4853			1.35966			0.679828	+	0.733371i	$0.262054\pi$
$228$		0			0
$229$	16.5158			1.09139			0.545697	−	0.837983i	$-0.316265\pi$
$229$	16.5158			1.09139			0.545697	+	0.837983i	$0.316265\pi$
$230$		0			0
$231$	16.3052	+	6.22803i	1.07280	+	0.409775i
$232$		0			0
$233$			0.737932i			0.0483436i	0.999708	+	0.0241718i	$0.00769487\pi$
$233$			0.737932i			0.0483436i	−0.999708	+	0.0241718i	$0.992305\pi$
$234$		0			0
$235$		−	19.5557i		−	1.27567i
$236$		0			0
$237$	−22.1862	−	8.47438i	−1.44115	−	0.550470i
$238$		0			0
$239$	−7.71813			−0.499245			−0.249622	−	0.968343i	$-0.580306\pi$
$239$	−7.71813			−0.499245			−0.249622	+	0.968343i	$0.580306\pi$
$240$		0			0
$241$	14.8805			0.958538			0.479269	−	0.877668i	$-0.340902\pi$
$241$	14.8805			0.958538			0.479269	+	0.877668i	$0.340902\pi$
$242$		0			0
$243$	−15.0902	−	3.90983i	−0.968035	−	0.250816i
$244$		0			0
$245$		−	21.0336i		−	1.34379i
$246$		0			0
$247$			17.4003i			1.10716i
$248$		0			0
$249$	3.49823	−	9.15848i	0.221691	−	0.580395i
$250$		0			0
$251$	−23.2275			−1.46611			−0.733054	−	0.680170i	$-0.761906\pi$
$251$	−23.2275			−1.46611			−0.733054	+	0.680170i	$0.761906\pi$
$252$		0			0
$253$	−2.73205			−0.171763
$254$		0			0
$255$	12.5737	−	32.9183i	0.787393	−	2.06142i
$256$		0			0
$257$			23.8347i			1.48677i	0.668865	+	0.743384i	$0.266781\pi$
$257$			23.8347i			1.48677i	−0.668865	+	0.743384i	$0.733219\pi$
$258$		0			0
$259$			25.3510i			1.57523i
$260$		0			0
$261$	−8.94427	+	10.0000i	−0.553637	+	0.618984i
$262$		0			0
$263$	−5.57454			−0.343741			−0.171870	−	0.985120i	$-0.554981\pi$
$263$	−5.57454			−0.343741			−0.171870	+	0.985120i	$0.554981\pi$
$264$		0			0
$265$	36.8429			2.26324
$266$		0			0
$267$	−24.0193	−	9.17455i	−1.46996	−	0.561473i
$268$		0			0
$269$			27.6678i			1.68693i	0.537181	+	0.843467i	$0.319489\pi$
$269$			27.6678i			1.68693i	−0.537181	+	0.843467i	$0.680511\pi$
$270$		0			0
$271$		−	24.0672i		−	1.46198i	−0.682388	−	0.730990i	$-0.739058\pi$
$271$		−	24.0672i		−	1.46198i	0.682388	−	0.730990i	$-0.260942\pi$
$272$		0			0
$273$	−32.6104	−	12.4561i	−1.97367	−	0.753875i
$274$		0			0
$275$	14.0453			0.846963
$276$		0			0
$277$	−6.95603			−0.417948			−0.208974	−	0.977921i	$-0.567012\pi$
$277$	−6.95603			−0.417948			−0.208974	+	0.977921i	$0.567012\pi$
$278$		0			0
$279$	4.28187	−	4.78727i	0.256349	−	0.286606i
$280$		0			0
$281$			19.0556i			1.13676i	0.822766	+	0.568380i	$0.192430\pi$
$281$			19.0556i			1.13676i	−0.822766	+	0.568380i	$0.807570\pi$
$282$		0			0
$283$		−	6.04892i		−	0.359571i	−0.983706	−	0.179786i	$-0.942460\pi$
$283$		−	6.04892i		−	0.359571i	0.983706	−	0.179786i	$-0.0575404\pi$
$284$		0			0
$285$	6.26744	−	16.4084i	0.371251	−	0.971948i
$286$		0			0
$287$	32.0906			1.89425
$288$		0			0
$289$	−23.8151			−1.40089
$290$		0			0
$291$	−5.51344	+	14.4344i	−0.323204	+	0.846158i
$292$		0			0
$293$		−	26.6521i		−	1.55703i	−0.627626	−	0.778515i	$-0.715973\pi$
$293$		−	26.6521i		−	1.55703i	0.627626	−	0.778515i	$-0.284027\pi$
$294$		0			0
$295$			18.1777i			1.05835i
$296$		0			0
$297$	−12.6167	+	6.50765i	−0.732095	+	0.377612i
$298$		0			0
$299$	5.46410			0.315997
$300$		0			0
$301$	−0.329559			−0.0189955
$302$		0			0
$303$	6.97027	+	2.66241i	0.400432	+	0.152951i
$304$		0			0
$305$			12.9138i			0.739441i
$306$		0			0
$307$			21.0431i			1.20100i	0.799627	+	0.600498i	$0.205031\pi$
$307$			21.0431i			1.20100i	−0.799627	+	0.600498i	$0.794969\pi$
$308$		0			0
$309$	4.33708	+	1.65662i	0.246728	+	0.0942418i
$310$		0			0
$311$	28.0528			1.59073			0.795365	−	0.606131i	$-0.207279\pi$
$311$	28.0528			1.59073			0.795365	+	0.606131i	$0.207279\pi$
$312$		0			0
$313$	4.05548			0.229229			0.114615	−	0.993410i	$-0.463437\pi$
$313$	4.05548			0.229229			0.114615	+	0.993410i	$0.463437\pi$
$314$		0			0
$315$	26.2648	+	23.4919i	1.47985	+	1.32362i
$316$		0			0
$317$		−	14.4561i		−	0.811934i	−0.913888	−	0.405967i	$-0.866935\pi$
$317$		−	14.4561i		−	0.811934i	0.913888	−	0.405967i	$-0.133065\pi$
$318$		0			0
$319$			12.2181i			0.684082i
$320$		0			0
$321$	−6.87074	+	17.9878i	−0.383487	+	1.00398i
$322$		0			0
$323$	−20.3446			−1.13200
$324$		0			0
$325$	−28.0906			−1.55818
$326$		0			0
$327$	8.59336	−	22.4977i	0.475214	−	1.24413i
$328$		0			0
$329$		−	22.6508i		−	1.24878i
$330$		0			0
$331$			9.04130i			0.496955i	0.968638	+	0.248478i	$0.0799302\pi$
$331$			9.04130i			0.496955i	−0.968638	+	0.248478i	$0.920070\pi$
$332$		0			0
$333$	−15.3685	−	13.7460i	−0.842186	−	0.753274i
$334$		0			0
$335$	35.4137			1.93486
$336$		0			0
$337$	0.920933			0.0501664			0.0250832	−	0.999685i	$-0.492015\pi$
$337$	0.920933			0.0501664			0.0250832	+	0.999685i	$0.492015\pi$
$338$		0			0
$339$	−0.0318812	−	0.0121775i	−0.00173155	−	0.000661393i
$340$		0			0
$341$		−	5.84914i		−	0.316748i
$342$		0			0
$343$			1.45683i			0.0786615i
$344$		0			0
$345$	−5.15260	−	1.96812i	−0.277407	−	0.105960i
$346$		0			0
$347$	23.7360			1.27422			0.637109	−	0.770774i	$-0.280130\pi$
$347$	23.7360			1.27422			0.637109	+	0.770774i	$0.280130\pi$
$348$		0			0
$349$	12.4882			0.668478			0.334239	−	0.942488i	$-0.391521\pi$
$349$	12.4882			0.668478			0.334239	+	0.942488i	$0.391521\pi$
$350$		0			0
$351$	25.2334	−	13.0153i	1.34686	−	0.694706i
$352$		0			0
$353$		−	3.83307i		−	0.204014i	−0.994784	−	0.102007i	$-0.967474\pi$
$353$		−	3.83307i		−	0.204014i	0.994784	−	0.102007i	$-0.0325264\pi$
$354$		0			0
$355$		−	24.9954i		−	1.32662i
$356$		0			0
$357$	14.5637	−	38.1283i	0.770794	−	2.01797i
$358$		0			0
$359$	−31.8087			−1.67880			−0.839400	−	0.543514i	$-0.817093\pi$
$359$	−31.8087			−1.67880			−0.839400	+	0.543514i	$0.817093\pi$
$360$		0			0
$361$	8.85907			0.466267
$362$		0			0
$363$	2.18531	−	5.72120i	0.114699	−	0.300285i
$364$		0			0
$365$			8.97312i			0.469675i
$366$		0			0
$367$			7.97037i			0.416050i	0.978124	+	0.208025i	$0.0667035\pi$
$367$			7.97037i			0.416050i	−0.978124	+	0.208025i	$0.933297\pi$
$368$		0			0
$369$	−17.4003	+	19.4542i	−0.905825	+	1.01274i
$370$		0			0
$371$	42.6742			2.21553
$372$		0			0
$373$	−19.6503			−1.01746			−0.508728	−	0.860928i	$-0.669884\pi$
$373$	−19.6503			−1.01746			−0.508728	+	0.860928i	$0.669884\pi$
$374$		0			0
$375$	0.726169	+	0.277372i	0.0374992	+	0.0143234i
$376$		0			0
$377$		−	24.4362i		−	1.25853i
$378$		0			0
$379$		−	28.5211i		−	1.46503i	−0.680752	−	0.732514i	$-0.738347\pi$
$379$		−	28.5211i		−	1.46503i	0.680752	−	0.732514i	$-0.261653\pi$
$380$		0			0
$381$	3.06914	+	1.17231i	0.157237	+	0.0600590i
$382$		0			0
$383$	−23.3888			−1.19511			−0.597555	−	0.801828i	$-0.703861\pi$
$383$	−23.3888			−1.19511			−0.597555	+	0.801828i	$0.703861\pi$
$384$		0			0
$385$	32.0906			1.63549
$386$		0			0
$387$	0.178695	−	0.199787i	0.00908359	−	0.0101558i
$388$		0			0
$389$		−	18.7158i		−	0.948930i	−0.880274	−	0.474465i	$-0.842642\pi$
$389$		−	18.7158i		−	0.948930i	0.880274	−	0.474465i	$-0.157358\pi$
$390$		0			0
$391$			6.38867i			0.323089i
$392$		0			0
$393$	−8.01341	+	20.9794i	−0.404223	+	1.05827i
$394$		0			0
$395$	−43.6651			−2.19703
$396$		0			0
$397$	3.85641			0.193547			0.0967737	−	0.995306i	$-0.469148\pi$
$397$	3.85641			0.193547			0.0967737	+	0.995306i	$0.469148\pi$
$398$		0			0
$399$	7.25941	−	19.0054i	0.363425	−	0.951459i
$400$		0			0
$401$		−	0.948923i		−	0.0473870i	−0.999719	−	0.0236935i	$-0.992457\pi$
$401$		−	0.948923i		−	0.0473870i	0.999719	−	0.0236935i	$-0.00754258\pi$
$402$		0			0
$403$			11.6983i			0.582732i
$404$		0			0
$405$	−28.4829	+	3.18448i	−1.41533	+	0.158238i
$406$		0			0
$407$	−18.7773			−0.930758
$408$		0			0
$409$	−29.4416			−1.45579			−0.727896	−	0.685687i	$-0.759502\pi$
$409$	−29.4416			−1.45579			−0.727896	+	0.685687i	$0.759502\pi$
$410$		0			0
$411$	−24.4563	−	9.34148i	−1.20634	−	0.460781i
$412$		0			0
$413$			21.0547i			1.03603i
$414$		0			0
$415$		−	18.0250i		−	0.884812i
$416$		0			0
$417$	25.0214	+	9.55734i	1.22531	+	0.468025i
$418$		0			0
$419$	−24.4654			−1.19521			−0.597607	−	0.801789i	$-0.703882\pi$
$419$	−24.4654			−1.19521			−0.597607	+	0.801789i	$0.703882\pi$
$420$		0			0
$421$	25.4036			1.23809			0.619047	−	0.785354i	$-0.287519\pi$
$421$	25.4036			1.23809			0.619047	+	0.785354i	$0.287519\pi$
$422$		0			0
$423$	13.7315	+	12.2819i	0.667650	+	0.597165i
$424$		0			0
$425$		−	32.8437i		−	1.59315i
$426$		0			0
$427$			14.9577i			0.723853i
$428$		0			0
$429$	9.22614	−	24.1543i	0.445442	−	1.16618i
$430$		0			0
$431$	14.1383			0.681017			0.340508	−	0.940242i	$-0.389401\pi$
$431$	14.1383			0.681017			0.340508	+	0.940242i	$0.389401\pi$
$432$		0			0
$433$	−6.59137			−0.316761			−0.158381	−	0.987378i	$-0.550627\pi$
$433$	−6.59137			−0.316761			−0.158381	+	0.987378i	$0.550627\pi$
$434$		0			0
$435$	−8.80169	+	23.0431i	−0.422009	+	1.10483i
$436$		0			0
$437$			3.18448i			0.152334i
$438$		0			0
$439$		−	22.8805i		−	1.09203i	−0.837776	−	0.546014i	$-0.816145\pi$
$439$		−	22.8805i		−	1.09203i	0.837776	−	0.546014i	$-0.183855\pi$
$440$		0			0
$441$	14.7693	+	13.2101i	0.703300	+	0.629051i
$442$		0			0
$443$	−19.2001			−0.912226			−0.456113	−	0.889922i	$-0.650759\pi$
$443$	−19.2001			−0.912226			−0.456113	+	0.889922i	$0.650759\pi$
$444$		0			0
$445$	−47.2728			−2.24095
$446$		0			0
$447$	−21.7236	−	8.29768i	−1.02749	−	0.392467i
$448$		0			0
$449$			7.06464i			0.333401i	0.986008	+	0.166701i	$0.0533113\pi$
$449$			7.06464i			0.333401i	−0.986008	+	0.166701i	$0.946689\pi$
$450$		0			0
$451$			23.7693i			1.11925i
$452$		0			0
$453$	32.8762	+	12.5576i	1.54466	+	0.590007i
$454$		0			0
$455$	−64.1812			−3.00886
$456$		0			0
$457$	−41.5825			−1.94515			−0.972574	−	0.232594i	$-0.925279\pi$
$457$	−41.5825			−1.94515			−0.972574	+	0.232594i	$0.925279\pi$
$458$		0			0
$459$	15.2176	+	29.5031i	0.710296	+	1.37709i
$460$		0			0
$461$		−	5.49737i		−	0.256038i	−0.991772	−	0.128019i	$-0.959138\pi$
$461$		−	5.49737i		−	0.256038i	0.991772	−	0.128019i	$-0.0408619\pi$
$462$		0			0
$463$		−	9.02410i		−	0.419386i	−0.977767	−	0.209693i	$-0.932754\pi$
$463$		−	9.02410i		−	0.419386i	0.977767	−	0.209693i	$-0.0672464\pi$
$464$		0			0
$465$	4.21361	−	11.0314i	0.195401	−	0.511568i
$466$		0			0
$467$	−7.59751			−0.351571			−0.175785	−	0.984429i	$-0.556246\pi$
$467$	−7.59751			−0.351571			−0.175785	+	0.984429i	$0.556246\pi$
$468$		0			0
$469$	41.0188			1.89407
$470$		0			0
$471$	−3.51707	+	9.20782i	−0.162058	+	0.424274i
$472$		0			0
$473$		−	0.244102i		−	0.0112238i
$474$		0			0
$475$		−	16.3712i		−	0.751163i
$476$		0			0
$477$	−23.1390	+	25.8702i	−1.05946	+	1.18452i
$478$		0			0
$479$	−9.64280			−0.440591			−0.220295	−	0.975433i	$-0.570702\pi$
$479$	−9.64280			−0.440591			−0.220295	+	0.975433i	$0.570702\pi$
$480$		0			0
$481$	37.5547			1.71235
$482$		0			0
$483$	−5.96812	−	2.27962i	−0.271559	−	0.103726i
$484$		0			0
$485$			28.4086i			1.28997i
$486$		0			0
$487$		−	3.08356i		−	0.139729i	−0.997556	−	0.0698647i	$-0.977743\pi$
$487$		−	3.08356i		−	0.139729i	0.997556	−	0.0698647i	$-0.0222568\pi$
$488$		0			0
$489$	−35.3266	−	13.4936i	−1.59753	−	0.610201i
$490$		0			0
$491$	18.6841			0.843202			0.421601	−	0.906782i	$-0.361468\pi$
$491$	18.6841			0.843202			0.421601	+	0.906782i	$0.361468\pi$
$492$		0			0
$493$	28.5710			1.28677
$494$		0			0
$495$	−17.4003	+	19.4542i	−0.782087	+	0.874400i
$496$		0			0
$497$		−	28.9515i		−	1.29865i
$498$		0			0
$499$		−	14.4955i		−	0.648907i	−0.945902	−	0.324453i	$-0.894820\pi$
$499$		−	14.4955i		−	0.648907i	0.945902	−	0.324453i	$-0.105180\pi$
$500$		0			0
$501$	−11.1168	+	29.1042i	−0.496663	+	1.30028i
$502$		0			0
$503$	−4.74243			−0.211454			−0.105727	−	0.994395i	$-0.533717\pi$
$503$	−4.74243			−0.211454			−0.105727	+	0.994395i	$0.533717\pi$
$504$		0			0
$505$	13.7183			0.610457
$506$		0			0
$507$	−10.4178	+	27.2742i	−0.462672	+	1.21129i
$508$		0			0
$509$		−	7.11847i		−	0.315521i	−0.987477	−	0.157760i	$-0.949573\pi$
$509$		−	7.11847i		−	0.315521i	0.987477	−	0.157760i	$-0.0504274\pi$
$510$		0			0
$511$			10.3933i			0.459773i
$512$		0			0
$513$	7.58533	+	14.7061i	0.334901	+	0.649288i
$514$		0			0
$515$	8.53590			0.376137
$516$		0			0
$517$	16.7773			0.737867
$518$		0			0
$519$	−12.1803	−	4.65248i	−0.534658	−	0.204221i
$520$		0			0
$521$			24.0387i			1.05316i	0.850127	+	0.526578i	$0.176525\pi$
$521$			24.0387i			1.05316i	−0.850127	+	0.526578i	$0.823475\pi$
$522$		0			0
$523$			22.2203i			0.971628i	0.874062	+	0.485814i	$0.161477\pi$
$523$			22.2203i			0.971628i	−0.874062	+	0.485814i	$0.838523\pi$
$524$		0			0
$525$	30.6817	+	11.7194i	1.33906	+	0.511475i
$526$		0			0
$527$	−13.6777			−0.595810
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	−12.7639	−	11.4164i	−0.553907	−	0.495430i
$532$		0			0
$533$		−	47.5386i		−	2.05913i
$534$		0			0
$535$			35.4022i			1.53057i
$536$		0			0
$537$	−2.49547	+	6.53324i	−0.107688	+	0.281930i
$538$		0			0
$539$	18.0453			0.777266
$540$		0			0
$541$	40.7647			1.75261			0.876307	−	0.481754i	$-0.160000\pi$
$541$	40.7647			1.75261			0.876307	+	0.481754i	$0.160000\pi$
$542$		0			0
$543$	12.1446	−	31.7949i	0.521173	−	1.36445i
$544$		0			0
$545$		−	44.2782i		−	1.89667i
$546$		0			0
$547$		−	20.2514i		−	0.865886i	−0.901421	−	0.432943i	$-0.857475\pi$
$547$		−	20.2514i		−	0.865886i	0.901421	−	0.432943i	$-0.142525\pi$
$548$		0			0
$549$	−9.06775	−	8.11044i	−0.387002	−	0.346145i
$550$		0			0
$551$	14.2414			0.606706
$552$		0			0
$553$	−50.5761			−2.15072
$554$		0			0
$555$	−35.4137	−	13.5268i	−1.50323	−	0.574183i
$556$		0			0
$557$		−	11.6406i		−	0.493226i	−0.969114	−	0.246613i	$-0.920682\pi$
$557$		−	11.6406i		−	0.493226i	0.969114	−	0.246613i	$-0.0793176\pi$
$558$		0			0
$559$			0.488205i			0.0206489i
$560$		0			0
$561$	28.2414	+	10.7873i	1.19235	+	0.455439i
$562$		0			0
$563$	−1.25536			−0.0529070			−0.0264535	−	0.999650i	$-0.508421\pi$
$563$	−1.25536			−0.0529070			−0.0264535	+	0.999650i	$0.508421\pi$
$564$		0			0
$565$	−0.0627460			−0.00263974
$566$		0			0
$567$	−32.9909	+	3.68850i	−1.38549	+	0.154902i
$568$		0			0
$569$		−	22.5629i		−	0.945885i	−0.881093	−	0.472942i	$-0.843192\pi$
$569$		−	22.5629i		−	0.945885i	0.881093	−	0.472942i	$-0.156808\pi$
$570$		0			0
$571$			12.9187i			0.540630i	0.962772	+	0.270315i	$0.0871279\pi$
$571$			12.9187i			0.540630i	−0.962772	+	0.270315i	$0.912872\pi$
$572$		0			0
$573$	−6.92820	+	18.1383i	−0.289430	+	0.757737i
$574$		0			0
$575$	−5.14093			−0.214392
$576$		0			0
$577$	−21.7910			−0.907171			−0.453586	−	0.891213i	$-0.649855\pi$
$577$	−21.7910			−0.907171			−0.453586	+	0.891213i	$0.649855\pi$
$578$		0			0
$579$	12.2908	−	32.1777i	0.510787	−	1.33726i
$580$		0			0
$581$		−	20.8778i		−	0.866159i
$582$		0			0
$583$			31.6085i			1.30909i
$584$		0			0
$585$	34.8007	−	38.9083i	1.43883	−	1.60866i
$586$		0			0
$587$	17.0968			0.705660			0.352830	−	0.935688i	$-0.385219\pi$
$587$	17.0968			0.705660			0.352830	+	0.935688i	$0.385219\pi$
$588$		0			0
$589$	−6.81776			−0.280921
$590$		0			0
$591$	7.87514	+	3.00803i	0.323940	+	0.123734i
$592$		0			0
$593$			31.2280i			1.28238i	0.767383	+	0.641189i	$0.221559\pi$
$593$			31.2280i			1.28238i	−0.767383	+	0.641189i	$0.778441\pi$
$594$		0			0
$595$		−	75.0411i		−	3.07639i
$596$		0			0
$597$	31.0346	+	11.8542i	1.27016	+	0.485158i
$598$		0			0
$599$	36.8601			1.50606			0.753032	−	0.657984i	$-0.228590\pi$
$599$	36.8601			1.50606			0.753032	+	0.657984i	$0.228590\pi$
$600$		0			0
$601$	−34.5611			−1.40978			−0.704888	−	0.709319i	$-0.749003\pi$
$601$	−34.5611			−1.40978			−0.704888	+	0.709319i	$0.749003\pi$
$602$		0			0
$603$	−22.2414	+	24.8667i	−0.905742	+	1.01265i
$604$		0			0
$605$		−	11.2600i		−	0.457784i
$606$		0			0
$607$			19.0925i			0.774940i	0.921882	+	0.387470i	$0.126651\pi$
$607$			19.0925i			0.774940i	−0.921882	+	0.387470i	$0.873349\pi$
$608$		0			0
$609$	−10.1948	+	26.6902i	−0.413113	+	1.08154i
$610$		0			0
$611$	−33.5547			−1.35748
$612$		0			0
$613$	−0.158390			−0.00639730			−0.00319865	−	0.999995i	$-0.501018\pi$
$613$	−0.158390			−0.00639730			−0.00319865	+	0.999995i	$0.501018\pi$
$614$		0			0
$615$	−17.1230	+	44.8285i	−0.690465	+	1.80766i
$616$		0			0
$617$		−	17.6381i		−	0.710085i	−0.934850	−	0.355043i	$-0.884466\pi$
$617$		−	17.6381i		−	0.710085i	0.934850	−	0.355043i	$-0.115534\pi$
$618$		0			0
$619$			1.17620i			0.0472753i	0.999721	+	0.0236377i	$0.00752480\pi$
$619$			1.17620i			0.0472753i	−0.999721	+	0.0236377i	$0.992475\pi$
$620$		0			0
$621$	4.61803	−	2.38197i	0.185315	−	0.0955850i
$622$		0			0
$623$	−54.7548			−2.19371
$624$		0			0
$625$	−24.2755			−0.971019
$626$		0			0
$627$	14.0772	+	5.37700i	0.562188	+	0.214737i
$628$		0			0
$629$			43.9092i			1.75078i
$630$		0			0
$631$			8.83911i			0.351879i	0.984401	+	0.175940i	$0.0562964\pi$
$631$			8.83911i			0.351879i	−0.984401	+	0.175940i	$0.943704\pi$
$632$		0			0
$633$	−12.1570	−	4.64356i	−0.483197	−	0.184565i
$634$		0			0
$635$	6.04042			0.239707
$636$		0			0
$637$	−36.0906			−1.42996
$638$		0			0
$639$	17.5512	+	15.6983i	0.694315	+	0.621014i
$640$		0			0
$641$			22.7782i			0.899685i	0.893108	+	0.449842i	$0.148520\pi$
$641$			22.7782i			0.899685i	−0.893108	+	0.449842i	$0.851480\pi$
$642$		0			0
$643$			12.0183i			0.473956i	0.971515	+	0.236978i	$0.0761569\pi$
$643$			12.0183i			0.473956i	−0.971515	+	0.236978i	$0.923843\pi$
$644$		0			0
$645$	0.175847	−	0.460373i	0.00692396	−	0.0181272i
$646$		0			0
$647$	−13.2200			−0.519732			−0.259866	−	0.965645i	$-0.583678\pi$
$647$	−13.2200			−0.519732			−0.259866	+	0.965645i	$0.583678\pi$
$648$		0			0
$649$	−15.5951			−0.612161
$650$		0			0
$651$	4.88051	−	12.7773i	0.191282	−	0.500783i
$652$		0			0
$653$		−	14.1321i		−	0.553033i	−0.961009	−	0.276517i	$-0.910820\pi$
$653$		−	14.1321i		−	0.553033i	0.961009	−	0.276517i	$-0.0891801\pi$
$654$		0			0
$655$			41.2899i			1.61333i
$656$		0			0
$657$	−6.30071	−	5.63553i	−0.245814	−	0.219863i
$658$		0			0
$659$	26.3548			1.02664			0.513319	−	0.858198i	$-0.328416\pi$
$659$	26.3548			1.02664			0.513319	+	0.858198i	$0.328416\pi$
$660$		0			0
$661$	−30.7849			−1.19739			−0.598696	−	0.800976i	$-0.704314\pi$
$661$	−30.7849			−1.19739			−0.598696	+	0.800976i	$0.704314\pi$
$662$		0			0
$663$	−56.4829	−	21.5745i	−2.19361	−	0.837886i
$664$		0			0
$665$		−	37.4048i		−	1.45050i
$666$		0			0
$667$		−	4.47214i		−	0.173162i
$668$		0			0
$669$	−27.8044	−	10.6203i	−1.07498	−	0.410606i
$670$		0			0
$671$	−11.0791			−0.427703
$672$		0			0
$673$	4.21646			0.162533			0.0812663	−	0.996692i	$-0.474104\pi$
$673$	4.21646			0.162533			0.0812663	+	0.996692i	$0.474104\pi$
$674$		0			0
$675$	−23.7410	+	12.2455i	−0.913792	+	0.471331i
$676$		0			0
$677$		−	49.5238i		−	1.90335i	−0.307102	−	0.951677i	$-0.599359\pi$
$677$		−	49.5238i		−	1.90335i	0.307102	−	0.951677i	$-0.400641\pi$
$678$		0			0
$679$			32.9049i			1.26277i
$680$		0			0
$681$	−12.6606	+	33.1459i	−0.485155	+	1.27015i
$682$		0			0
$683$	26.7270			1.02268			0.511340	−	0.859379i	$-0.329149\pi$
$683$	26.7270			1.02268			0.511340	+	0.859379i	$0.329149\pi$
$684$		0			0
$685$	−48.1329			−1.83907
$686$		0			0
$687$	−10.2073	+	26.7231i	−0.389433	+	1.01955i
$688$		0			0
$689$		−	63.2170i		−	2.40838i
$690$		0			0
$691$			3.17958i			0.120957i	0.998170	+	0.0604784i	$0.0192626\pi$
$691$			3.17958i			0.120957i	−0.998170	+	0.0604784i	$0.980737\pi$
$692$		0			0
$693$	−20.1543	+	22.5332i	−0.765600	+	0.855967i
$694$		0			0
$695$	49.2452			1.86798
$696$		0			0
$697$	55.5825			2.10534
$698$		0			0
$699$	−1.19400	−	0.456067i	−0.0451612	−	0.0172500i
$700$		0			0
$701$			5.27158i			0.199105i	0.995032	+	0.0995525i	$0.0317411\pi$
$701$			5.27158i			0.199105i	−0.995032	+	0.0995525i	$0.968259\pi$
$702$		0			0
$703$			21.8869i			0.825480i
$704$		0			0
$705$	31.6418	+	12.0861i	1.19170	+	0.455188i
$706$		0			0
$707$	15.8896			0.597588
$708$		0			0
$709$	25.7131			0.965675			0.482837	−	0.875710i	$-0.339606\pi$
$709$	25.7131			0.965675			0.482837	+	0.875710i	$0.339606\pi$
$710$		0			0
$711$	27.4237	−	30.6606i	1.02847	−	1.14986i
$712$		0			0
$713$			2.14093i			0.0801786i
$714$		0			0
$715$		−	47.5386i		−	1.77784i
$716$		0			0
$717$	4.77007	−	12.4882i	0.178142	−	0.466381i
$718$		0			0
$719$	−33.4065			−1.24585			−0.622926	−	0.782281i	$-0.714056\pi$
$719$	−33.4065			−1.24585			−0.622926	+	0.782281i	$0.714056\pi$
$720$		0			0
$721$	9.88690			0.368207
$722$		0			0
$723$	−9.19666	+	24.0772i	−0.342027	+	0.895440i
$724$		0			0
$725$			22.9909i			0.853862i
$726$		0			0
$727$		−	11.8238i		−	0.438521i	−0.975666	−	0.219260i	$-0.929636\pi$
$727$		−	11.8238i		−	0.438521i	0.975666	−	0.219260i	$-0.0703644\pi$
$728$		0			0
$729$	15.6525	−	22.0000i	0.579721	−	0.814815i
$730$		0			0
$731$	−0.570813			−0.0211123
$732$		0			0
$733$	21.5470			0.795855			0.397928	−	0.917417i	$-0.369729\pi$
$733$	21.5470			0.795855			0.397928	+	0.917417i	$0.369729\pi$
$734$		0			0
$735$	34.0331	+	12.9995i	1.25533	+	0.479493i
$736$		0			0
$737$			30.3824i			1.11915i
$738$		0			0
$739$			26.9489i			0.991331i	0.868514	+	0.495665i	$0.165076\pi$
$739$			26.9489i			0.991331i	−0.868514	+	0.495665i	$0.834924\pi$
$740$		0			0
$741$	−28.1543	−	10.7540i	−1.03428	−	0.395058i
$742$		0			0
$743$	−17.1769			−0.630160			−0.315080	−	0.949065i	$-0.602031\pi$
$743$	−17.1769			−0.630160			−0.315080	+	0.949065i	$0.602031\pi$
$744$		0			0
$745$	−42.7546			−1.56641
$746$		0			0
$747$	12.6567	+	11.3205i	0.463085	+	0.414196i
$748$		0			0
$749$			41.0054i			1.49830i
$750$		0			0
$751$		−	10.0575i		−	0.367002i	−0.983020	−	0.183501i	$-0.941257\pi$
$751$		−	10.0575i		−	0.367002i	0.983020	−	0.183501i	$-0.0587431\pi$
$752$		0			0
$753$	14.3554	−	37.5829i	0.523140	−	1.36960i
$754$		0			0
$755$	64.7043			2.35483
$756$		0			0
$757$	−26.1180			−0.949274			−0.474637	−	0.880182i	$-0.657421\pi$
$757$	−26.1180			−0.949274			−0.474637	+	0.880182i	$0.657421\pi$
$758$		0			0
$759$	1.68850	−	4.42055i	0.0612887	−	0.160456i
$760$		0			0
$761$			16.4289i			0.595548i	0.954636	+	0.297774i	$0.0962443\pi$
$761$			16.4289i			0.595548i	−0.954636	+	0.297774i	$0.903756\pi$
$762$		0			0
$763$		−	51.2862i		−	1.85669i
$764$		0			0
$765$	45.4919	+	40.6892i	1.64476	+	1.47112i
$766$		0			0
$767$	31.1902			1.12621
$768$		0			0
$769$	19.4442			0.701177			0.350589	−	0.936530i	$-0.385982\pi$
$769$	19.4442			0.701177			0.350589	+	0.936530i	$0.385982\pi$
$770$		0			0
$771$	−38.5654	−	14.7307i	−1.38890	−	0.530512i
$772$		0			0
$773$		−	23.1995i		−	0.834429i	−0.908808	−	0.417215i	$-0.863006\pi$
$773$		−	23.1995i		−	0.834429i	0.908808	−	0.417215i	$-0.136994\pi$
$774$		0			0
$775$		−	11.0064i		−	0.395361i
$776$		0			0
$777$	−41.0188	−	15.6678i	−1.47154	−	0.562078i
$778$		0			0
$779$	27.7055			0.992654
$780$		0			0
$781$	21.4442			0.767335
$782$		0			0
$783$	−10.6525	−	20.6525i	−0.380688	−	0.738059i
$784$		0			0
$785$			18.1221i			0.646805i
$786$		0			0
$787$			11.6800i			0.416346i	0.978092	+	0.208173i	$0.0667516\pi$
$787$			11.6800i			0.416346i	−0.978092	+	0.208173i	$0.933248\pi$
$788$		0			0
$789$	3.44526	−	9.01980i	0.122654	−	0.321113i
$790$		0			0
$791$	−0.0726770			−0.00258410
$792$		0			0
$793$	22.1581			0.786858
$794$		0			0
$795$	−22.7702	+	59.6131i	−0.807575	+	2.11426i
$796$		0			0
$797$			39.8225i			1.41059i	0.708916	+	0.705293i	$0.249185\pi$
$797$			39.8225i			1.41059i	−0.708916	+	0.705293i	$0.750815\pi$
$798$		0			0
$799$		−	39.2324i		−	1.38794i
$800$		0			0
$801$	29.6895	−	33.1938i	1.04903	−	1.17285i
$802$		0			0
$803$	−7.69827			−0.271666
$804$		0			0
$805$	−11.7460			−0.413991
$806$		0			0
$807$	−44.7674	−	17.0996i	−1.57589	−	0.601935i
$808$		0			0
$809$		−	43.0753i		−	1.51445i	−0.653156	−	0.757223i	$-0.726556\pi$
$809$		−	43.0753i		−	1.51445i	0.653156	−	0.757223i	$-0.273444\pi$
$810$		0			0
$811$			49.6785i			1.74445i	0.489106	+	0.872225i	$0.337323\pi$
$811$			49.6785i			1.74445i	−0.489106	+	0.872225i	$0.662677\pi$
$812$		0			0
$813$	38.9416	+	14.8744i	1.36574	+	0.521667i
$814$		0			0
$815$	−69.5270			−2.43543
$816$		0			0
$817$	−0.284526			−0.00995431
$818$		0			0
$819$	40.3087	−	45.0665i	1.40850	−	1.57475i
$820$		0			0
$821$		−	23.8143i		−	0.831126i	−0.909564	−	0.415563i	$-0.863585\pi$
$821$		−	23.8143i		−	0.831126i	0.909564	−	0.415563i	$-0.136415\pi$
$822$		0			0
$823$		−	53.7117i		−	1.87227i	−0.351639	−	0.936136i	$-0.614375\pi$
$823$		−	53.7117i		−	1.87227i	0.351639	−	0.936136i	$-0.385625\pi$
$824$		0			0
$825$	−8.68047	+	22.7258i	−0.302215	+	0.791209i
$826$		0			0
$827$	−13.0543			−0.453944			−0.226972	−	0.973901i	$-0.572883\pi$
$827$	−13.0543			−0.453944			−0.226972	+	0.973901i	$0.572883\pi$
$828$		0			0
$829$	−37.1420			−1.28999			−0.644997	−	0.764185i	$-0.723142\pi$
$829$	−37.1420			−1.28999			−0.644997	+	0.764185i	$0.723142\pi$
$830$		0			0
$831$	4.29907	−	11.2551i	0.149133	−	0.390435i
$832$		0			0
$833$		−	42.1974i		−	1.46205i
$834$		0			0
$835$			57.2806i			1.98228i
$836$		0			0
$837$	5.09963	+	9.88690i	0.176269	+	0.341741i
$838$		0			0
$839$	−16.7424			−0.578013			−0.289006	−	0.957327i	$-0.593325\pi$
$839$	−16.7424			−0.578013			−0.289006	+	0.957327i	$0.593325\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$	−30.8326	−	11.7770i	−1.06193	−	0.405621i
$844$		0			0
$845$			53.6789i			1.84661i
$846$		0			0
$847$		−	13.0422i		−	0.448134i
$848$		0			0
$849$	9.78736	+	3.73844i	0.335901	+	0.128303i
$850$		0			0
$851$	6.87298			0.235603
$852$		0			0
$853$	6.78993			0.232483			0.116241	−	0.993221i	$-0.462915\pi$
$853$	6.78993			0.232483			0.116241	+	0.993221i	$0.462915\pi$
$854$		0			0
$855$	22.6758	+	20.2819i	0.775496	+	0.693625i
$856$		0			0
$857$		−	4.53159i		−	0.154796i	−0.997000	−	0.0773981i	$-0.975339\pi$
$857$		−	4.53159i		−	0.154796i	0.997000	−	0.0773981i	$-0.0246613\pi$
$858$		0			0
$859$		−	3.67760i		−	0.125478i	−0.998030	−	0.0627390i	$-0.980016\pi$
$859$		−	3.67760i		−	0.125478i	0.998030	−	0.0627390i	$-0.0199836\pi$
$860$		0			0
$861$	−19.8331	+	51.9236i	−0.675909	+	1.76955i
$862$		0			0
$863$	42.4730			1.44580			0.722898	−	0.690955i	$-0.242810\pi$
$863$	42.4730			1.44580			0.722898	+	0.690955i	$0.242810\pi$
$864$		0			0
$865$	−23.9724			−0.815085
$866$		0			0
$867$	14.7185	−	38.5336i	0.499868	−	1.30867i
$868$		0			0
$869$		−	37.4614i		−	1.27079i
$870$		0			0
$871$		−	60.7647i		−	2.05893i
$872$		0			0
$873$	−19.9478	−	17.8419i	−0.675131	−	0.603856i
$874$		0			0
$875$	1.65539			0.0559624
$876$		0			0
$877$	−22.7497			−0.768203			−0.384101	−	0.923291i	$-0.625489\pi$
$877$	−22.7497			−0.768203			−0.384101	+	0.923291i	$0.625489\pi$
$878$		0			0
$879$	43.1239	+	16.4719i	1.45453	+	0.555583i
$880$		0			0
$881$		−	13.7531i		−	0.463353i	−0.972793	−	0.231676i	$-0.925579\pi$
$881$		−	13.7531i		−	0.463353i	0.972793	−	0.231676i	$-0.0744211\pi$
$882$		0			0
$883$		−	23.0574i		−	0.775942i	−0.921672	−	0.387971i	$-0.873176\pi$
$883$		−	23.0574i		−	0.775942i	0.921672	−	0.387971i	$-0.126824\pi$
$884$		0			0
$885$	−29.4121	−	11.2344i	−0.988677	−	0.377641i
$886$		0			0
$887$	33.2900			1.11777			0.558885	−	0.829245i	$-0.311229\pi$
$887$	33.2900			1.11777			0.558885	+	0.829245i	$0.311229\pi$
$888$		0			0
$889$	6.99646			0.234654
$890$		0			0
$891$	−2.73205	−	24.4362i	−0.0915271	−	0.818644i
$892$		0			0
$893$		−	19.5557i		−	0.654406i
$894$		0			0
$895$			12.8582i			0.429802i
$896$		0			0
$897$	−3.37700	+	8.84110i	−0.112755	+	0.295196i
$898$		0			0
$899$	9.57454			0.319329
$900$		0			0
$901$	73.9138			2.46243
$902$		0			0
$903$	0.203679	−	0.533237i	0.00677800	−	0.0177450i
$904$		0			0
$905$		−	62.5761i		−	2.08010i
$906$		0			0
$907$		−	54.1440i		−	1.79782i	−0.438131	−	0.898911i	$-0.644360\pi$
$907$		−	54.1440i		−	1.79782i	0.438131	−	0.898911i	$-0.355640\pi$
$908$		0			0
$909$	−8.61573	+	9.63268i	−0.285766	+	0.319496i
$910$		0			0
$911$	−0.199070			−0.00659547			−0.00329773	−	0.999995i	$-0.501050\pi$
$911$	−0.199070			−0.00659547			−0.00329773	+	0.999995i	$0.501050\pi$
$912$		0			0
$913$	15.4641			0.511787
$914$		0			0
$915$	−20.8949	−	7.98115i	−0.690765	−	0.263849i
$916$		0			0
$917$			47.8250i			1.57932i
$918$		0			0
$919$			16.5178i			0.544873i	0.962174	+	0.272437i	$0.0878295\pi$
$919$			16.5178i			0.544873i	−0.962174	+	0.272437i	$0.912170\pi$
$920$		0			0
$921$	−34.0485	−	13.0054i	−1.12194	−	0.428542i
$922$		0			0
$923$	−42.8885			−1.41169
$924$		0			0
$925$	−35.3335			−1.16176
$926$		0			0
$927$	−5.36093	+	5.99370i	−0.176076	+	0.196859i
$928$		0			0
$929$			33.5978i			1.10231i	0.834404	+	0.551153i	$0.185812\pi$
$929$			33.5978i			1.10231i	−0.834404	+	0.551153i	$0.814188\pi$
$930$		0			0
$931$		−	21.0336i		−	0.689349i
$932$		0			0
$933$	−17.3376	+	45.3904i	−0.567607	+	1.48602i
$934$		0			0
$935$	55.5825			1.81774
$936$		0			0
$937$	30.8878			1.00906			0.504530	−	0.863394i	$-0.331666\pi$
$937$	30.8878			1.00906			0.504530	+	0.863394i	$0.331666\pi$
$938$		0			0
$939$	−2.50642	+	6.56190i	−0.0817940	+	0.214139i
$940$		0			0
$941$			18.4661i			0.601978i	0.953628	+	0.300989i	$0.0973167\pi$
$941$			18.4661i			0.601978i	−0.953628	+	0.300989i	$0.902683\pi$
$942$		0			0
$943$		−	8.70017i		−	0.283317i
$944$		0			0
$945$	−54.2433	+	27.9785i	−1.76453	+	0.910140i
$946$		0			0
$947$	53.9498			1.75313			0.876567	−	0.481280i	$-0.159828\pi$
$947$	53.9498			1.75313			0.876567	+	0.481280i	$0.159828\pi$
$948$		0			0
$949$	15.3965			0.499793
$950$		0			0
$951$	23.3904	+	8.93434i	0.758486	+	0.289716i
$952$		0			0
$953$			7.42833i			0.240627i	0.992736	+	0.120314i	$0.0383900\pi$
$953$			7.42833i			0.240627i	−0.992736	+	0.120314i	$0.961610\pi$
$954$		0			0
$955$			35.6983i			1.15517i
$956$		0			0
$957$	−19.7693	−	7.55120i	−0.639051	−	0.244096i
$958$		0			0
$959$	−55.7511			−1.80030
$960$		0			0
$961$	26.4164			0.852142
$962$		0			0
$963$	−24.8586	−	22.2342i	−0.801056	−	0.716486i
$964$		0			0
$965$		−	63.3295i		−	2.03865i
$966$		0			0
$967$		−	8.35802i		−	0.268776i	−0.990929	−	0.134388i	$-0.957093\pi$
$967$		−	8.35802i		−	0.268776i	0.990929	−	0.134388i	$-0.0429068\pi$
$968$		0			0
$969$	12.5737	−	32.9183i	0.403924	−	1.05749i
$970$		0			0
$971$	22.4027			0.718936			0.359468	−	0.933157i	$-0.382958\pi$
$971$	22.4027			0.718936			0.359468	+	0.933157i	$0.382958\pi$
$972$		0			0
$973$	57.0393			1.82860
$974$		0			0
$975$	17.3609	−	45.4515i	0.555995	−	1.45561i
$976$		0			0
$977$			26.8131i			0.857827i	0.903345	+	0.428914i	$0.141104\pi$
$977$			26.8131i			0.857827i	−0.903345	+	0.428914i	$0.858896\pi$
$978$		0			0
$979$		−	40.5566i		−	1.29619i
$980$		0			0
$981$	31.0911	+	27.8087i	0.992662	+	0.887864i
$982$		0			0
$983$	31.4164			1.00203			0.501014	−	0.865439i	$-0.332961\pi$
$983$	31.4164			1.00203			0.501014	+	0.865439i	$0.332961\pi$
$984$		0			0
$985$	15.4992			0.493846
$986$		0			0
$987$	36.6498	+	13.9990i	1.16658	+	0.445593i
$988$		0			0
$989$			0.0893476i			0.00284109i
$990$		0			0
$991$			6.64368i			0.211043i	0.994417	+	0.105522i	$0.0336512\pi$
$991$			6.64368i			0.211043i	−0.994417	+	0.105522i	$0.966349\pi$
$992$		0			0
$993$	−14.6291	−	5.58783i	−0.464242	−	0.177325i
$994$		0			0
$995$	61.0798			1.93636
$996$		0			0
$997$	22.9839			0.727906			0.363953	−	0.931417i	$-0.381427\pi$
$997$	22.9839			0.727906			0.363953	+	0.931417i	$0.381427\pi$
$998$		0			0
$999$	31.7397	−	16.3712i	1.00420	−	0.517962i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.e.h.47.4	yes	8
3.2	odd	2		1104.2.e.e.47.7	yes	8
4.3	odd	2		1104.2.e.e.47.6	✓	8
12.11	even	2	inner	1104.2.e.h.47.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1104.2.e.e.47.6	✓	8	4.3	odd	2
1104.2.e.e.47.7	yes	8	3.2	odd	2
1104.2.e.h.47.1	yes	8	12.11	even	2	inner
1104.2.e.h.47.4	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.618034 + 1.61803i) q^{3} +3.18448i q^{5} +3.68850i q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(-0.618034 + 1.61803i) q^{3} +3.18448i q^{5} +3.68850i q^{7} +(-2.23607 - 2.00000i) q^{9} -2.73205 q^{11} +5.46410 q^{13} +(-5.15260 - 1.96812i) q^{15} +6.38867i q^{17} +3.18448i q^{19} +(-5.96812 - 2.27962i) q^{21} +1.00000 q^{23} -5.14093 q^{25} +(4.61803 - 2.38197i) q^{27} -4.47214i q^{29} +2.14093i q^{31} +(1.68850 - 4.42055i) q^{33} -11.7460 q^{35} +6.87298 q^{37} +(-3.37700 + 8.84110i) q^{39} -8.70017i q^{41} +0.0893476i q^{43} +(6.36897 - 7.12072i) q^{45} -6.14093 q^{47} -6.60503 q^{49} +(-10.3371 - 3.94842i) q^{51} -11.5695i q^{53} -8.70017i q^{55} +(-5.15260 - 1.96812i) q^{57} +5.70820 q^{59} +4.05522 q^{61} +(7.37700 - 8.24774i) q^{63} +17.4003i q^{65} -11.1207i q^{67} +(-0.618034 + 1.61803i) q^{69} -7.84914 q^{71} +2.81776 q^{73} +(3.17727 - 8.31820i) q^{75} -10.0772i q^{77} +13.7118i q^{79} +(1.00000 + 8.94427i) q^{81} -5.66025 q^{83} -20.3446 q^{85} +(7.23607 + 2.76393i) q^{87} +14.8447i q^{89} +20.1543i q^{91} +(-3.46410 - 1.32317i) q^{93} -10.1409 q^{95} +8.92093 q^{97} +(6.10905 + 5.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3}+O(q^{10}) \) 8 * q + 4 * q^3	\( 8 q + 4 q^{3} - 8 q^{11} + 16 q^{13} + 4 q^{15} - 16 q^{21} + 8 q^{23} - 24 q^{25} + 28 q^{27} - 4 q^{33} - 32 q^{35} + 24 q^{37} + 8 q^{39} + 24 q^{45} - 32 q^{47} - 8 q^{49} - 24 q^{51} + 4 q^{57} - 8 q^{59} + 8 q^{61} + 24 q^{63} + 4 q^{69} + 8 q^{71} + 16 q^{73} - 12 q^{75} + 8 q^{81} + 24 q^{83} + 40 q^{87} - 64 q^{95} + 56 q^{97}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^11 + 16 * q^13 + 4 * q^15 - 16 * q^21 + 8 * q^23 - 24 * q^25 + 28 * q^27 - 4 * q^33 - 32 * q^35 + 24 * q^37 + 8 * q^39 + 24 * q^45 - 32 * q^47 - 8 * q^49 - 24 * q^51 + 4 * q^57 - 8 * q^59 + 8 * q^61 + 24 * q^63 + 4 * q^69 + 8 * q^71 + 16 * q^73 - 12 * q^75 + 8 * q^81 + 24 * q^83 + 40 * q^87 - 64 * q^95 + 56 * q^97

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