LMFDB - Embedded newform 168.2.i.d.125.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(125,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("168.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	168.i (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:	$1.34148675396$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			125.5
Root			$1.68014 - 0.420861i$ of defining polynomial
Character	$\chi$	$=$	168.125
Dual form			168.2.i.d.125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(-1.68014 + 0.420861i) q^{3} -2.00000 q^{4} -3.91044i q^{5} +(-0.595188 - 2.37608i) q^{6} -2.64575 q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(-1.68014 + 0.420861i) q^{3} -2.00000 q^{4} -3.91044i q^{5} +(-0.595188 - 2.37608i) q^{6} -2.64575 q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +5.53019 q^{10} +(3.36028 - 0.841723i) q^{12} -4.55066 q^{13} -3.74166i q^{14} +(1.64575 + 6.57008i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -0.979531 q^{19} +7.82087i q^{20} +(4.44524 - 1.11349i) q^{21} -7.48331i q^{23} +(1.19038 + 4.75216i) q^{24} -10.2915 q^{25} -6.43560i q^{26} +(-3.85005 + 3.48957i) q^{27} +5.29150 q^{28} +(-9.29150 + 2.32744i) q^{30} +5.65685i q^{32} +10.3460i q^{35} +(-5.29150 + 2.82843i) q^{36} -1.38527i q^{38} +(7.64575 - 1.91520i) q^{39} -11.0604 q^{40} +(1.57472 + 6.28651i) q^{42} +(-5.53019 - 10.3460i) q^{45} +10.5830 q^{46} +(-6.72057 + 1.68345i) q^{48} +7.00000 q^{49} -14.5544i q^{50} +9.10132 q^{52} +(-4.93500 - 5.44479i) q^{54} +7.48331i q^{56} +(1.64575 - 0.412247i) q^{57} +5.29570i q^{59} +(-3.29150 - 13.1402i) q^{60} +15.6110 q^{61} +(-7.00000 + 3.74166i) q^{63} -8.00000 q^{64} +17.7951i q^{65} +(3.14944 + 12.5730i) q^{69} -14.6315 q^{70} -5.65685i q^{71} +(-4.00000 - 7.48331i) q^{72} +(17.2912 - 4.33130i) q^{75} +1.95906 q^{76} +(2.70850 + 10.8127i) q^{78} -5.29150 q^{79} -15.6417i q^{80} +(5.00000 - 7.48331i) q^{81} -18.1669i q^{83} +(-8.89047 + 2.22699i) q^{84} +(14.6315 - 7.82087i) q^{90} +12.0399 q^{91} +14.9666i q^{92} +3.83039i q^{95} +(-2.38075 - 9.50432i) q^{96} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} - 8 q^{15} + 32 q^{16} + 16 q^{18} - 40 q^{25} - 32 q^{30} + 40 q^{39} + 56 q^{49} - 8 q^{57} + 16 q^{60} - 56 q^{63} - 64 q^{64} - 32 q^{72} + 64 q^{78} + 40 q^{81}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^15 + 32 * q^16 + 16 * q^18 - 40 * q^25 - 32 * q^30 + 40 * q^39 + 56 * q^49 - 8 * q^57 + 16 * q^60 - 56 * q^63 - 64 * q^64 - 32 * q^72 + 64 * q^78 + 40 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	−1.68014	+	0.420861i	−0.970030	+	0.242984i
$3$	−1.68014	+	0.420861i	−0.970030	+	0.242984i
$4$	−2.00000			−1.00000
$5$		−	3.91044i		−	1.74880i	−0.485206	−	0.874400i	$-0.661255\pi$
$5$		−	3.91044i		−	1.74880i	0.485206	−	0.874400i	$-0.338745\pi$
$6$	−0.595188	−	2.37608i	−0.242984	−	0.970030i
$7$	−2.64575			−1.00000
$7$	−2.64575			−1.00000
$8$		−	2.82843i		−	1.00000i
$9$	2.64575	−	1.41421i	0.881917	−	0.471405i
$10$	5.53019			1.74880
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	3.36028	−	0.841723i	0.970030	−	0.242984i
$13$	−4.55066			−1.26213			−0.631063	−	0.775732i	$-0.717381\pi$
$13$	−4.55066			−1.26213			−0.631063	+	0.775732i	$0.717381\pi$
$14$		−	3.74166i		−	1.00000i
$15$	1.64575	+	6.57008i	0.424931	+	1.69639i
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	2.00000	+	3.74166i	0.471405	+	0.881917i
$19$	−0.979531			−0.224720			−0.112360	−	0.993668i	$-0.535841\pi$
$19$	−0.979531			−0.224720			−0.112360	+	0.993668i	$0.535841\pi$
$20$			7.82087i			1.74880i
$21$	4.44524	−	1.11349i	0.970030	−	0.242984i
$22$		0			0
$23$		−	7.48331i		−	1.56038i	−0.625543	−	0.780189i	$-0.715123\pi$
$23$		−	7.48331i		−	1.56038i	0.625543	−	0.780189i	$-0.284877\pi$
$24$	1.19038	+	4.75216i	0.242984	+	0.970030i
$25$	−10.2915			−2.05830
$26$		−	6.43560i		−	1.26213i
$27$	−3.85005	+	3.48957i	−0.740942	+	0.671569i
$28$	5.29150			1.00000
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−9.29150	+	2.32744i	−1.69639	+	0.424931i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			5.65685i			1.00000i
$33$		0			0
$34$		0			0
$35$			10.3460i			1.74880i
$36$	−5.29150	+	2.82843i	−0.881917	+	0.471405i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	1.38527i		−	0.224720i
$39$	7.64575	−	1.91520i	1.22430	−	0.306677i
$40$	−11.0604			−1.74880
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	1.57472	+	6.28651i	0.242984	+	0.970030i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−5.53019	−	10.3460i	−0.824392	−	1.54230i
$46$	10.5830			1.56038
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−6.72057	+	1.68345i	−0.970030	+	0.242984i
$49$	7.00000			1.00000
$50$		−	14.5544i		−	2.05830i
$51$		0			0
$52$	9.10132			1.26213
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−4.93500	−	5.44479i	−0.671569	−	0.740942i
$55$		0			0
$56$			7.48331i			1.00000i
$57$	1.64575	−	0.412247i	0.217985	−	0.0546034i
$58$		0			0
$59$			5.29570i			0.689442i	0.938705	+	0.344721i	$0.112026\pi$
$59$			5.29570i			0.689442i	−0.938705	+	0.344721i	$0.887974\pi$
$60$	−3.29150	−	13.1402i	−0.424931	−	1.69639i
$61$	15.6110			1.99879			0.999394	−	0.0347968i	$-0.0110784\pi$
$61$	15.6110			1.99879			0.999394	+	0.0347968i	$0.0110784\pi$
$62$		0			0
$63$	−7.00000	+	3.74166i	−0.881917	+	0.471405i
$64$	−8.00000			−1.00000
$65$			17.7951i			2.20721i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	3.14944	+	12.5730i	0.379148	+	1.51361i
$70$	−14.6315			−1.74880
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$	−4.00000	−	7.48331i	−0.471405	−	0.881917i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	17.2912	−	4.33130i	1.99661	−	0.500135i
$76$	1.95906			0.224720
$77$		0			0
$78$	2.70850	+	10.8127i	0.306677	+	1.22430i
$79$	−5.29150			−0.595341			−0.297670	−	0.954669i	$-0.596210\pi$
$79$	−5.29150			−0.595341			−0.297670	+	0.954669i	$0.596210\pi$
$80$		−	15.6417i		−	1.74880i
$81$	5.00000	−	7.48331i	0.555556	−	0.831479i
$82$		0			0
$83$		−	18.1669i		−	1.99408i	−0.0769020	−	0.997039i	$-0.524503\pi$
$83$		−	18.1669i		−	1.99408i	0.0769020	−	0.997039i	$-0.475497\pi$
$84$	−8.89047	+	2.22699i	−0.970030	+	0.242984i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	14.6315	−	7.82087i	1.54230	−	0.824392i
$91$	12.0399			1.26213
$92$			14.9666i			1.56038i
$93$		0			0
$94$		0			0
$95$			3.83039i			0.392990i
$96$	−2.38075	−	9.50432i	−0.242984	−	0.970030i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	20.5830			2.05830
$101$		−	8.96077i		−	0.891630i	−0.895125	−	0.445815i	$-0.852914\pi$
$101$		−	8.96077i		−	0.891630i	0.895125	−	0.445815i	$-0.147086\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			12.8712i			1.26213i
$105$	−4.35425	−	17.3828i	−0.424931	−	1.69639i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	7.70010	−	6.97915i	0.740942	−	0.671569i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−10.5830			−1.00000
$113$		−	14.1421i		−	1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$		−	14.1421i		−	1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$	0.583005	+	2.32744i	0.0546034	+	0.217985i
$115$	−29.2630			−2.72879
$116$		0			0
$117$	−12.0399	+	6.43560i	−1.11309	+	0.594972i
$118$	−7.48925			−0.689442
$119$		0			0
$120$	18.5830	−	4.65489i	1.69639	−	0.424931i
$121$	−11.0000			−1.00000
$122$			22.0773i			1.99879i
$123$		0			0
$124$		0			0
$125$			20.6921i			1.85076i
$126$	−5.29150	−	9.89949i	−0.471405	−	0.881917i
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		−	11.3137i		−	1.00000i
$129$		0			0
$130$	−25.1660			−2.20721
$131$		−	13.1166i		−	1.14600i	−0.819555	−	0.573000i	$-0.805779\pi$
$131$		−	13.1166i		−	1.14600i	0.819555	−	0.573000i	$-0.194221\pi$
$132$		0			0
$133$	2.59160			0.224720
$134$		0			0
$135$	13.6458	+	15.0554i	1.17444	+	1.29576i
$136$		0			0
$137$			14.9666i			1.27869i	0.768922	+	0.639343i	$0.220793\pi$
$137$			14.9666i			1.27869i	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−17.7809	+	4.45398i	−1.51361	+	0.379148i
$139$	19.1822			1.62701			0.813505	−	0.581558i	$-0.197557\pi$
$139$	19.1822			1.62701			0.813505	+	0.581558i	$0.197557\pi$
$140$		−	20.6921i		−	1.74880i
$141$		0			0
$142$	8.00000			0.671345
$143$		0			0
$144$	10.5830	−	5.65685i	0.881917	−	0.471405i
$145$		0			0
$146$		0			0
$147$	−11.7610	+	2.94603i	−0.970030	+	0.242984i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	6.12538	+	24.4534i	0.500135	+	1.99661i
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$			2.77053i			0.224720i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−15.2915	+	3.83039i	−1.22430	+	0.306677i
$157$	−17.5701			−1.40225			−0.701123	−	0.713040i	$-0.747318\pi$
$157$	−17.5701			−1.40225			−0.701123	+	0.713040i	$0.747318\pi$
$158$		−	7.48331i		−	0.595341i
$159$		0			0
$160$	22.1208			1.74880
$161$			19.7990i			1.56038i
$162$	10.5830	+	7.07107i	0.831479	+	0.555556i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	25.6919			1.99408
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−3.14944	−	12.5730i	−0.242984	−	0.970030i
$169$	7.70850			0.592961
$170$		0			0
$171$	−2.59160	+	1.38527i	−0.198184	+	0.105934i
$172$		0			0
$173$			24.6025i			1.87049i	0.353995	+	0.935247i	$0.384823\pi$
$173$			24.6025i			1.87049i	−0.353995	+	0.935247i	$0.615177\pi$
$174$		0			0
$175$	27.2288			2.05830
$176$		0			0
$177$	−2.22876	−	8.89753i	−0.167524	−	0.668779i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	11.0604	+	20.6921i	0.824392	+	1.54230i
$181$	−24.7124			−1.83686			−0.918428	−	0.395589i	$-0.870540\pi$
$181$	−24.7124			−1.83686			−0.918428	+	0.395589i	$0.870540\pi$
$182$			17.0270i			1.26213i
$183$	−26.2288	+	6.57008i	−1.93889	+	0.485675i
$184$	−21.1660			−1.56038
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.1863	−	9.23254i	0.740942	−	0.671569i
$190$	−5.41699			−0.392990
$191$		−	22.6274i		−	1.63726i	−0.574320	−	0.818631i	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$	13.4411	−	3.36689i	0.970030	−	0.242984i
$193$	26.4575			1.90445			0.952227	−	0.305392i	$-0.0987875\pi$
$193$	26.4575			1.90445			0.952227	+	0.305392i	$0.0987875\pi$
$194$		0			0
$195$	−7.48925	−	29.8982i	−0.536317	−	2.14106i
$196$	−14.0000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			29.1088i			2.05830i
$201$		0			0
$202$	12.6724			0.891630
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.5830	−	19.7990i	−0.735570	−	1.37612i
$208$	−18.2026			−1.26213
$209$		0			0
$210$	24.5830	−	6.15784i	1.69639	−	0.424931i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	2.38075	+	9.50432i	0.163126	+	0.651225i
$214$		0			0
$215$		0			0
$216$	9.87000	+	10.8896i	0.671569	+	0.740942i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		−	14.9666i		−	1.00000i
$225$	−27.2288	+	14.5544i	−1.81525	+	0.970292i
$226$	20.0000			1.33038
$227$			15.3964i			1.02189i	0.859612	+	0.510947i	$0.170705\pi$
$227$			15.3964i			1.02189i	−0.859612	+	0.510947i	$0.829295\pi$
$228$	−3.29150	+	0.824494i	−0.217985	+	0.0546034i
$229$	8.46878			0.559633			0.279817	−	0.960053i	$-0.409726\pi$
$229$	8.46878			0.559633			0.279817	+	0.960053i	$0.409726\pi$
$230$		−	41.3842i		−	2.72879i
$231$		0			0
$232$		0			0
$233$		−	29.9333i		−	1.96099i	−0.196537	−	0.980497i	$-0.562969\pi$
$233$		−	29.9333i		−	1.96099i	0.196537	−	0.980497i	$-0.437031\pi$
$234$	−9.10132	−	17.0270i	−0.594972	−	1.11309i
$235$		0			0
$236$		−	10.5914i		−	0.689442i
$237$	8.89047	−	2.22699i	0.577498	−	0.144659i
$238$		0			0
$239$		−	7.48331i		−	0.484055i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$		−	7.48331i		−	0.484055i	0.970269	−	0.242028i	$-0.0778125\pi$
$240$	6.58301	+	26.2803i	0.424931	+	1.69639i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	15.5563i		−	1.00000i
$243$	−5.25127	+	14.6773i	−0.336869	+	0.941551i
$244$	−31.2221			−1.99879
$245$		−	27.3730i		−	1.74880i
$246$		0			0
$247$	4.45751			0.283625
$248$		0			0
$249$	7.64575	+	30.5230i	0.484530	+	1.93432i
$250$	−29.2630			−1.85076
$251$		−	23.2172i		−	1.46546i	−0.680520	−	0.732730i	$-0.738246\pi$
$251$		−	23.2172i		−	1.46546i	0.680520	−	0.732730i	$-0.261754\pi$
$252$	14.0000	−	7.48331i	0.881917	−	0.471405i
$253$		0			0
$254$			2.82843i			0.177471i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		−	35.5901i		−	2.20721i
$261$		0			0
$262$	18.5496			1.14600
$263$			28.2843i			1.74408i	0.489432	+	0.872041i	$0.337204\pi$
$263$			28.2843i			1.74408i	−0.489432	+	0.872041i	$0.662796\pi$
$264$		0			0
$265$		0			0
$266$			3.66507i			0.224720i
$267$		0			0
$268$		0			0
$269$		−	32.4234i		−	1.97689i	−0.151585	−	0.988444i	$-0.548438\pi$
$269$		−	32.4234i		−	1.97689i	0.151585	−	0.988444i	$-0.451562\pi$
$270$	−21.2915	+	19.2980i	−1.29576	+	1.17444i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−20.2288	+	5.06713i	−1.22430	+	0.306677i
$274$	−21.1660			−1.27869
$275$		0			0
$276$	−6.29888	−	25.1461i	−0.379148	−	1.51361i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			27.1277i			1.62701i
$279$		0			0
$280$	29.2630			1.74880
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	32.2016			1.91419			0.957094	−	0.289779i	$-0.0935819\pi$
$283$	32.2016			1.91419			0.957094	+	0.289779i	$0.0935819\pi$
$284$			11.3137i			0.671345i
$285$	−1.61206	−	6.43560i	−0.0954905	−	0.381212i
$286$		0			0
$287$		0			0
$288$	8.00000	+	14.9666i	0.471405	+	0.881917i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.5018i			0.847206i	0.905848	+	0.423603i	$0.139235\pi$
$293$			14.5018i			0.847206i	−0.905848	+	0.423603i	$0.860765\pi$
$294$	−4.16632	−	16.6326i	−0.242984	−	0.970030i
$295$	20.7085			1.20570
$296$		0			0
$297$		0			0
$298$		0			0
$299$			34.0540i			1.96939i
$300$	−34.5824	+	8.66259i	−1.99661	+	0.500135i
$301$		0			0
$302$		−	14.1421i		−	0.813788i
$303$	3.77124	+	15.0554i	0.216652	+	0.864908i
$304$	−3.91813			−0.224720
$305$		−	61.0460i		−	3.49548i
$306$		0			0
$307$	−13.9990			−0.798964			−0.399482	−	0.916741i	$-0.630810\pi$
$307$	−13.9990			−0.798964			−0.399482	+	0.916741i	$0.630810\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−5.41699	−	21.6255i	−0.306677	−	1.22430i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		−	24.8479i		−	1.40225i
$315$	14.6315	+	27.3730i	0.824392	+	1.54230i
$316$	10.5830			0.595341
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$			31.2835i			1.74880i
$321$		0			0
$322$	−28.0000			−1.56038
$323$		0			0
$324$	−10.0000	+	14.9666i	−0.555556	+	0.831479i
$325$	46.8331			2.59783
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			36.3338i			1.99408i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	17.7809	−	4.45398i	0.970030	−	0.242984i
$337$	26.4575			1.44123			0.720616	−	0.693334i	$-0.243859\pi$
$337$	26.4575			1.44123			0.720616	+	0.693334i	$0.243859\pi$
$338$			10.9015i			0.592961i
$339$	5.95188	+	23.7608i	0.323262	+	1.29051i
$340$		0			0
$341$		0			0
$342$	−1.95906	−	3.66507i	−0.105934	−	0.198184i
$343$	−18.5203			−1.00000
$344$		0			0
$345$	49.1660	−	12.3157i	2.64701	−	0.663054i
$346$	−34.7932			−1.87049
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	28.6305			1.53255			0.766277	−	0.642510i	$-0.222107\pi$
$349$	28.6305			1.53255			0.766277	+	0.642510i	$0.222107\pi$
$350$			38.5073i			2.05830i
$351$	17.5203	−	15.8799i	0.935162	−	0.847604i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	12.5830	−	3.15194i	0.668779	−	0.167524i
$355$	−22.1208			−1.17405
$356$		0			0
$357$		0			0
$358$		0			0
$359$			37.4166i			1.97477i	0.158334	+	0.987386i	$0.449388\pi$
$359$			37.4166i			1.97477i	−0.158334	+	0.987386i	$0.550612\pi$
$360$	−29.2630	+	15.6417i	−1.54230	+	0.824392i
$361$	−18.0405			−0.949501
$362$		−	34.9486i		−	1.83686i
$363$	18.4816	−	4.62948i	0.970030	−	0.242984i
$364$	−24.0798			−1.26213
$365$		0			0
$366$	−9.29150	−	37.0931i	−0.485675	−	1.93889i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	29.9333i		−	1.56038i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−8.70850	−	34.7656i	−0.449705	−	1.79529i
$376$		0			0
$377$		0			0
$378$	13.0568	+	14.4056i	0.671569	+	0.740942i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		−	7.66079i		−	0.392990i
$381$	−3.36028	+	0.841723i	−0.172153	+	0.0431228i
$382$	32.0000			1.63726
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	4.76150	+	19.0086i	0.242984	+	0.970030i
$385$		0			0
$386$			37.4166i			1.90445i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	42.2825	−	10.5914i	2.14106	−	0.536317i
$391$		0			0
$392$		−	19.7990i		−	1.00000i
$393$	5.52026	+	22.0377i	0.278460	+	1.11165i
$394$		0			0
$395$			20.6921i			1.04113i
$396$		0			0
$397$	−37.7318			−1.89370			−0.946852	−	0.321668i	$-0.895756\pi$
$397$	−37.7318			−1.89370			−0.946852	+	0.321668i	$0.895756\pi$
$398$		0			0
$399$	−4.35425	+	1.09070i	−0.217985	+	0.0546034i
$400$	−41.1660			−2.05830
$401$			36.7696i			1.83618i	0.396368	+	0.918092i	$0.370271\pi$
$401$			36.7696i			1.83618i	−0.396368	+	0.918092i	$0.629729\pi$
$402$		0			0
$403$		0			0
$404$			17.9215i			0.891630i
$405$	−29.2630	−	19.5522i	−1.45409	−	0.971555i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−6.29888	−	25.1461i	−0.310701	−	1.24036i
$412$		0			0
$413$		−	14.0111i		−	0.689442i
$414$	28.0000	−	14.9666i	1.37612	−	0.735570i
$415$	−71.0405			−3.48724
$416$		−	25.7424i		−	1.26213i
$417$	−32.2288	+	8.07303i	−1.57825	+	0.395338i
$418$		0			0
$419$			38.8590i			1.89839i	0.314695	+	0.949193i	$0.398098\pi$
$419$			38.8590i			1.89839i	−0.314695	+	0.949193i	$0.601902\pi$
$420$	8.70850	+	34.7656i	0.424931	+	1.69639i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−13.4411	+	3.36689i	−0.651225	+	0.163126i
$427$	−41.3029			−1.99879
$428$		0			0
$429$		0			0
$430$		0			0
$431$			37.4166i			1.80229i	0.433515	+	0.901146i	$0.357273\pi$
$431$			37.4166i			1.80229i	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−15.4002	+	13.9583i	−0.740942	+	0.671569i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			7.33014i			0.350648i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	18.5203	−	9.89949i	0.881917	−	0.471405i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	21.1660			1.00000
$449$		−	29.9333i		−	1.41264i	−0.707894	−	0.706319i	$-0.750354\pi$
$449$		−	29.9333i		−	1.41264i	0.707894	−	0.706319i	$-0.249646\pi$
$450$	−20.5830	−	38.5073i	−0.970292	−	1.81525i
$451$		0			0
$452$			28.2843i			1.33038i
$453$	16.8014	−	4.20861i	0.789399	−	0.197738i
$454$	−21.7738			−1.02189
$455$		−	47.0813i		−	2.20721i
$456$	−1.16601	−	4.65489i	−0.0546034	−	0.217985i
$457$	−5.29150			−0.247526			−0.123763	−	0.992312i	$-0.539496\pi$
$457$	−5.29150			−0.247526			−0.123763	+	0.992312i	$0.539496\pi$
$458$			11.9767i			0.559633i
$459$		0			0
$460$	58.5260			2.72879
$461$			6.19024i			0.288308i	0.989555	+	0.144154i	$0.0460461\pi$
$461$			6.19024i			0.288308i	−0.989555	+	0.144154i	$0.953954\pi$
$462$		0			0
$463$	26.4575			1.22958			0.614792	−	0.788689i	$-0.289240\pi$
$463$	26.4575			1.22958			0.614792	+	0.788689i	$0.289240\pi$
$464$		0			0
$465$		0			0
$466$	42.3320			1.96099
$467$		−	41.6295i		−	1.92638i	−0.268814	−	0.963192i	$-0.586632\pi$
$467$		−	41.6295i		−	1.92638i	0.268814	−	0.963192i	$-0.413368\pi$
$468$	24.0798	−	12.8712i	1.11309	−	0.594972i
$469$		0			0
$470$		0			0
$471$	29.5203	−	7.39458i	1.36022	−	0.340724i
$472$	14.9785			0.689442
$473$		0			0
$474$	3.14944	+	12.5730i	0.144659	+	0.577498i
$475$	10.0808			0.462541
$476$		0			0
$477$		0			0
$478$	10.5830			0.484055
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−37.1660	+	9.30978i	−1.69639	+	0.424931i
$481$		0			0
$482$		0			0
$483$	−8.33263	−	33.2651i	−0.379148	−	1.51361i
$484$	22.0000			1.00000
$485$		0			0
$486$	−20.7569	−	7.42642i	−0.941551	−	0.336869i
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$		−	44.1547i		−	1.99879i
$489$		0			0
$490$	38.7113			1.74880
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$			6.30388i			0.283625i
$495$		0			0
$496$		0			0
$497$			14.9666i			0.671345i
$498$	−43.1660	+	10.8127i	−1.93432	+	0.484530i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		−	41.3842i		−	1.85076i
$501$		0			0
$502$	32.8341			1.46546
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	10.5830	+	19.7990i	0.471405	+	0.881917i
$505$	−35.0405			−1.55928
$506$		0			0
$507$	−12.9514	+	3.24421i	−0.575190	+	0.144080i
$508$	−4.00000			−0.177471
$509$		−	37.4737i		−	1.66099i	−0.557024	−	0.830497i	$-0.688057\pi$
$509$		−	37.4737i		−	1.66099i	0.557024	−	0.830497i	$-0.311943\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$	3.77124	−	3.41815i	0.166504	−	0.150915i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−10.3542	−	41.3357i	−0.454501	−	1.81444i
$520$	50.3320			2.20721
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	25.0594			1.09577			0.547885	−	0.836554i	$-0.315433\pi$
$523$	25.0594			1.09577			0.547885	+	0.836554i	$0.315433\pi$
$524$			26.2331i			1.14600i
$525$	−45.7482	+	11.4595i	−1.99661	+	0.500135i
$526$	−40.0000			−1.74408
$527$		0			0
$528$		0			0
$529$	−33.0000			−1.43478
$530$		0			0
$531$	7.48925	+	14.0111i	0.325006	+	0.608030i
$532$	−5.18319			−0.224720
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	45.8536			1.97689
$539$		0			0
$540$	−27.2915	−	30.1107i	−1.17444	−	1.29576i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	41.5203	−	10.4005i	1.78180	−	0.446327i
$544$		0			0
$545$		0			0
$546$	−7.16601	−	28.6078i	−0.306677	−	1.22430i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		−	29.9333i		−	1.27869i
$549$	41.3029	−	22.0773i	1.76277	−	0.942238i
$550$		0			0
$551$		0			0
$552$	35.5619	−	8.90796i	1.51361	−	0.379148i
$553$	14.0000			0.595341
$554$		0			0
$555$		0			0
$556$	−38.3643			−1.62701
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$			41.3842i			1.74880i
$561$		0			0
$562$	−21.1660			−0.892834
$563$		−	46.6799i		−	1.96732i	−0.180032	−	0.983661i	$-0.557620\pi$
$563$		−	46.6799i		−	1.96732i	0.180032	−	0.983661i	$-0.442380\pi$
$564$		0			0
$565$	−55.3019			−2.32657
$566$			45.5400i			1.91419i
$567$	−13.2288	+	19.7990i	−0.555556	+	0.831479i
$568$	−16.0000			−0.671345
$569$			2.82843i			0.118574i	0.998241	+	0.0592869i	$0.0188827\pi$
$569$			2.82843i			0.118574i	−0.998241	+	0.0592869i	$0.981117\pi$
$570$	9.10132	−	2.27980i	0.381212	−	0.0954905i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	9.52301	+	38.0173i	0.397829	+	1.58819i
$574$		0			0
$575$			77.0146i			3.21173i
$576$	−21.1660	+	11.3137i	−0.881917	+	0.471405i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	24.0416i		−	1.00000i
$579$	−44.4524	+	11.1349i	−1.84738	+	0.462753i
$580$		0			0
$581$			48.0651i			1.99408i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	25.1660	+	47.0813i	1.04049	+	1.94657i
$586$	−20.5087			−0.847206
$587$		−	36.5792i		−	1.50978i	−0.655849	−	0.754892i	$-0.727689\pi$
$587$		−	36.5792i		−	1.50978i	0.655849	−	0.754892i	$-0.272311\pi$
$588$	23.5220	−	5.89206i	0.970030	−	0.242984i
$589$		0			0
$590$			29.2862i			1.20570i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−48.1596			−1.96939
$599$			11.3137i			0.462266i	0.972922	+	0.231133i	$0.0742432\pi$
$599$			11.3137i			0.462266i	−0.972922	+	0.231133i	$0.925757\pi$
$600$	−12.2508	−	48.9068i	−0.500135	−	1.99661i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	20.0000			0.813788
$605$			43.0148i			1.74880i
$606$	−21.2915	+	5.33334i	−0.864908	+	0.216652i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	5.54107i		−	0.224720i
$609$		0			0
$610$	86.3320			3.49548
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	19.7975i		−	0.798964i
$615$		0			0
$616$		0			0
$617$		−	14.1421i		−	0.569341i	−0.958625	−	0.284670i	$-0.908116\pi$
$617$		−	14.1421i		−	0.569341i	0.958625	−	0.284670i	$-0.0918842\pi$
$618$		0			0
$619$	26.3244			1.05807			0.529034	−	0.848601i	$-0.322554\pi$
$619$	26.3244			1.05807			0.529034	+	0.848601i	$0.322554\pi$
$620$		0			0
$621$	26.1136	+	28.8111i	1.04790	+	1.15615i
$622$		0			0
$623$		0			0
$624$	30.5830	−	7.66079i	1.22430	−	0.306677i
$625$	29.4575			1.17830
$626$		0			0
$627$		0			0
$628$	35.1402			1.40225
$629$		0			0
$630$	−38.7113	+	20.6921i	−1.54230	+	0.824392i
$631$	−37.0405			−1.47456			−0.737280	−	0.675587i	$-0.763890\pi$
$631$	−37.0405			−1.47456			−0.737280	+	0.675587i	$0.763890\pi$
$632$			14.9666i			0.595341i
$633$		0			0
$634$		0			0
$635$		−	7.82087i		−	0.310362i
$636$		0			0
$637$	−31.8546			−1.26213
$638$		0			0
$639$	−8.00000	−	14.9666i	−0.316475	−	0.592071i
$640$	−44.2415			−1.74880
$641$		−	48.0833i		−	1.89917i	−0.313503	−	0.949587i	$-0.601502\pi$
$641$		−	48.0833i		−	1.89917i	0.313503	−	0.949587i	$-0.398498\pi$
$642$		0			0
$643$	−15.2640			−0.601955			−0.300978	−	0.953631i	$-0.597313\pi$
$643$	−15.2640			−0.601955			−0.300978	+	0.953631i	$0.597313\pi$
$644$		−	39.5980i		−	1.56038i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−21.1660	−	14.1421i	−0.831479	−	0.555556i
$649$		0			0
$650$			66.2320i			2.59783i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	−51.2915			−2.00412
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30.5895			−1.18980			−0.594898	−	0.803801i	$-0.702807\pi$
$661$	−30.5895			−1.18980			−0.594898	+	0.803801i	$0.702807\pi$
$662$		0			0
$663$		0			0
$664$	−51.3838			−1.99408
$665$		−	10.1343i		−	0.392990i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	6.29888	+	25.1461i	0.242984	+	0.970030i
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$			37.4166i			1.44123i
$675$	39.6228	−	35.9130i	1.52508	−	1.38229i
$676$	−15.4170			−0.592961
$677$			9.45150i			0.363251i	0.983368	+	0.181625i	$0.0581358\pi$
$677$			9.45150i			0.363251i	−0.983368	+	0.181625i	$0.941864\pi$
$678$	−33.6028	+	8.41723i	−1.29051	+	0.323262i
$679$		0			0
$680$		0			0
$681$	−6.47974	−	25.8681i	−0.248304	−	0.991267i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	5.18319	−	2.77053i	0.198184	−	0.105934i
$685$	58.5260			2.23616
$686$		−	26.1916i		−	1.00000i
$687$	−14.2288	+	3.56418i	−0.542861	+	0.135982i
$688$		0			0
$689$		0			0
$690$	17.4170	+	69.5312i	0.663054	+	2.64701i
$691$	52.3633			1.99199			0.995997	−	0.0893857i	$-0.0284904\pi$
$691$	52.3633			1.99199			0.995997	+	0.0893857i	$0.0284904\pi$
$692$		−	49.2050i		−	1.87049i
$693$		0			0
$694$		0			0
$695$		−	75.0106i		−	2.84532i
$696$		0			0
$697$		0			0
$698$			40.4896i			1.53255i
$699$	12.5978	+	50.2921i	0.476491	+	1.90222i
$700$	−54.4575			−2.05830
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	22.4575	+	24.7774i	0.847604	+	0.935162i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			23.7080i			0.891630i
$708$	4.45751	+	17.7951i	0.167524	+	0.668779i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		−	31.2835i		−	1.17405i
$711$	−14.0000	+	7.48331i	−0.525041	+	0.280646i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	3.14944	+	12.5730i	0.117618	+	0.469548i
$718$	−52.9150			−1.97477
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−22.1208	−	41.3842i	−0.824392	−	1.54230i
$721$		0			0
$722$		−	25.5131i		−	0.949501i
$723$		0			0
$724$	49.4247			1.83686
$725$		0			0
$726$	6.54707	+	26.1369i	0.242984	+	0.970030i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		−	34.0540i		−	1.26213i
$729$	2.64575	−	26.8701i	0.0979908	−	0.995187i
$730$		0			0
$731$		0			0
$732$	52.4575	−	13.1402i	1.93889	−	0.485675i
$733$	42.9150			1.58510			0.792551	−	0.609806i	$-0.208753\pi$
$733$	42.9150			1.58510			0.792551	+	0.609806i	$0.208753\pi$
$734$		0			0
$735$	11.5203	+	45.9906i	0.424931	+	1.69639i
$736$	42.3320			1.56038
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	−7.48925	+	1.87600i	−0.275125	+	0.0689164i
$742$		0			0
$743$		−	7.48331i		−	0.274536i	−0.990534	−	0.137268i	$-0.956168\pi$
$743$		−	7.48331i		−	0.274536i	0.990534	−	0.137268i	$-0.0438322\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−25.6919	−	48.0651i	−0.940017	−	1.75861i
$748$		0			0
$749$		0			0
$750$	49.1660	−	12.3157i	1.79529	−	0.449705i
$751$	50.0000			1.82453			0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000			1.82453			0.912263	+	0.409605i	$0.134333\pi$
$752$		0			0
$753$	9.77124	+	39.0083i	0.356084	+	1.42154i
$754$		0			0
$755$			39.1044i			1.42315i
$756$	−20.3725	+	18.4651i	−0.740942	+	0.671569i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	10.8340			0.392990
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	−1.19038	−	4.75216i	−0.0431228	−	0.172153i
$763$		0			0
$764$			45.2548i			1.63726i
$765$		0			0
$766$		0			0
$767$		−	24.0989i		−	0.870162i
$768$	−26.8823	+	6.73378i	−0.970030	+	0.242984i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−52.9150			−1.90445
$773$			53.1155i			1.91043i	0.295912	+	0.955215i	$0.404376\pi$
$773$			53.1155i			1.91043i	−0.295912	+	0.955215i	$0.595624\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	14.9785	+	59.7964i	0.536317	+	2.14106i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$			68.7067i			2.45225i
$786$	−31.1660	+	7.80683i	−1.11165	+	0.278460i
$787$	45.2211			1.61196			0.805978	−	0.591945i	$-0.201640\pi$
$787$	45.2211			1.61196			0.805978	+	0.591945i	$0.201640\pi$
$788$		0			0
$789$	−11.9038	−	47.5216i	−0.423785	−	1.69181i
$790$	−29.2630			−1.04113
$791$			37.4166i			1.33038i
$792$		0			0
$793$	−71.0405			−2.52272
$794$		−	53.3608i		−	1.89370i
$795$		0			0
$796$		0			0
$797$		−	55.8860i		−	1.97958i	−0.142521	−	0.989792i	$-0.545521\pi$
$797$		−	55.8860i		−	1.97958i	0.142521	−	0.989792i	$-0.454479\pi$
$798$	−1.54249	−	6.15784i	−0.0546034	−	0.217985i
$799$		0			0
$800$		−	58.2175i		−	2.05830i
$801$		0			0
$802$	−52.0000			−1.83618
$803$		0			0
$804$		0			0
$805$	77.4227			2.72879
$806$		0			0
$807$	13.6458	+	54.4759i	0.480353	+	1.91764i
$808$	−25.3449			−0.891630
$809$		−	31.1127i		−	1.09386i	−0.837177	−	0.546932i	$-0.815796\pi$
$809$		−	31.1127i		−	1.09386i	0.837177	−	0.546932i	$-0.184204\pi$
$810$	27.6510	−	41.3842i	0.971555	−	1.45409i
$811$	−48.4452			−1.70114			−0.850570	−	0.525861i	$-0.823743\pi$
$811$	−48.4452			−1.70114			−0.850570	+	0.525861i	$0.823743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	31.8546	−	17.0270i	1.11309	−	0.594972i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	35.5619	−	8.90796i	1.24036	−	0.310701i
$823$	26.4575			0.922251			0.461125	−	0.887335i	$-0.347446\pi$
$823$	26.4575			0.922251			0.461125	+	0.887335i	$0.347446\pi$
$824$		0			0
$825$		0			0
$826$	19.8147			0.689442
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	21.1660	+	39.5980i	0.735570	+	1.37612i
$829$	1.32653			0.0460723			0.0230361	−	0.999735i	$-0.492667\pi$
$829$	1.32653			0.0460723			0.0230361	+	0.999735i	$0.492667\pi$
$830$		−	100.466i		−	3.48724i
$831$		0			0
$832$	36.4053			1.26213
$833$		0			0
$834$	−11.4170	−	45.5783i	−0.395338	−	1.57825i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−54.9549			−1.89839
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−49.1660	+	12.3157i	−1.69639	+	0.424931i
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−6.29888	−	25.1461i	−0.216945	−	0.866076i
$844$		0			0
$845$		−	30.1436i		−	1.03697i
$846$		0			0
$847$	29.1033			1.00000
$848$		0			0
$849$	−54.1033	+	13.5524i	−1.85682	+	0.465118i
$850$		0			0
$851$		0			0
$852$	−4.76150	−	19.0086i	−0.163126	−	0.651225i
$853$	−57.8935			−1.98224			−0.991118	−	0.132987i	$-0.957543\pi$
$853$	−57.8935			−1.98224			−0.991118	+	0.132987i	$0.957543\pi$
$854$		−	58.4112i		−	1.99879i
$855$	5.41699	+	10.1343i	0.185257	+	0.346585i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−54.3224			−1.85346			−0.926728	−	0.375734i	$-0.877391\pi$
$859$	−54.3224			−1.85346			−0.926728	+	0.375734i	$0.877391\pi$
$860$		0			0
$861$		0			0
$862$	−52.9150			−1.80229
$863$		−	56.5685i		−	1.92562i	−0.270187	−	0.962808i	$-0.587086\pi$
$863$		−	56.5685i		−	1.92562i	0.270187	−	0.962808i	$-0.412914\pi$
$864$	−19.7400	−	21.7792i	−0.671569	−	0.740942i
$865$	96.2065			3.27112
$866$		0			0
$867$	28.5624	−	7.15464i	0.970030	−	0.242984i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−10.3664			−0.350648
$875$		−	54.7461i		−	1.85076i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−6.10326	−	24.3651i	−0.205858	−	0.821816i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	14.0000	+	26.1916i	0.471405	+	0.881917i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−34.7932	+	8.71541i	−1.16956	+	0.292965i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−5.29150			−0.177471
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$			29.9333i			1.00000i
$897$	−14.3320	−	57.2156i	−0.478532	−	1.91037i
$898$	42.3320			1.41264
$899$		0			0
$900$	54.4575	−	29.1088i	1.81525	−	0.970292i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−40.0000			−1.33038
$905$			96.6361i			3.21229i
$906$	5.95188	+	23.7608i	0.197738	+	0.789399i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		−	30.7928i		−	1.02189i
$909$	−12.6724	−	23.7080i	−0.420318	−	0.786344i
$910$	66.5830			2.20721
$911$		−	52.3832i		−	1.73553i	−0.496972	−	0.867766i	$-0.665555\pi$
$911$		−	52.3832i		−	1.73553i	0.496972	−	0.867766i	$-0.334445\pi$
$912$	6.58301	−	1.64899i	0.217985	−	0.0546034i
$913$		0			0
$914$		−	7.48331i		−	0.247526i
$915$	25.6919	+	102.566i	0.849348	+	3.39072i
$916$	−16.9376			−0.559633
$917$			34.7032i			1.14600i
$918$		0			0
$919$	58.2065			1.92006			0.960028	−	0.279904i	$-0.0903025\pi$
$919$	58.2065			1.92006			0.960028	+	0.279904i	$0.0903025\pi$
$920$			82.7683i			2.72879i
$921$	23.5203	−	5.89163i	0.775019	−	0.194136i
$922$	−8.75432			−0.288308
$923$			25.7424i			0.847322i
$924$		0			0
$925$		0			0
$926$			37.4166i			1.22958i
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−6.85672			−0.224720
$932$			59.8665i			1.96099i
$933$		0			0
$934$	58.8730			1.92638
$935$		0			0
$936$	18.2026	+	34.0540i	0.594972	+	1.11309i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	19.0614i		−	0.621385i	−0.950510	−	0.310693i	$-0.899439\pi$
$941$		−	19.0614i		−	0.621385i	0.950510	−	0.310693i	$-0.100561\pi$
$942$	10.4575	+	41.7480i	0.340724	+	1.36022i
$943$		0			0
$944$			21.1828i			0.689442i
$945$	−36.1033	−	39.8328i	−1.17444	−	1.29576i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$	−17.7809	+	4.45398i	−0.577498	+	0.144659i
$949$		0			0
$950$			14.2565i			0.462541i
$951$		0			0
$952$		0			0
$953$		−	29.9333i		−	0.969633i	−0.874616	−	0.484817i	$-0.838886\pi$
$953$		−	29.9333i		−	0.969633i	0.874616	−	0.484817i	$-0.161114\pi$
$954$		0			0
$955$	−88.4830			−2.86324
$956$			14.9666i			0.484055i
$957$		0			0
$958$		0			0
$959$		−	39.5980i		−	1.27869i
$960$	−13.1660	−	52.5607i	−0.424931	−	1.69639i
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	103.460i		−	3.33051i
$966$	47.0440	−	11.7841i	1.51361	−	0.379148i
$967$	−58.0000			−1.86515			−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000			−1.86515			−0.932577	+	0.360971i	$0.882445\pi$
$968$			31.1127i			1.00000i
$969$		0			0
$970$		0			0
$971$			62.3216i			2.00000i	0.00218468	+	0.999998i	$0.499305\pi$
$971$			62.3216i			2.00000i	−0.00218468	+	0.999998i	$0.500695\pi$
$972$	10.5025	−	29.3547i	0.336869	−	0.941551i
$973$	−50.7512			−1.62701
$974$			53.7401i			1.72194i
$975$	−78.6863	+	19.7103i	−2.51998	+	0.631233i
$976$	62.4442			1.99879
$977$			59.8665i			1.91530i	0.287936	+	0.957650i	$0.407031\pi$
$977$			59.8665i			1.91530i	−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		0			0
$980$			54.7461i			1.74880i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−8.91503			−0.283625
$989$		0			0
$990$		0			0
$991$	−37.0405			−1.17663			−0.588315	−	0.808632i	$-0.700209\pi$
$991$	−37.0405			−1.17663			−0.588315	+	0.808632i	$0.700209\pi$
$992$		0			0
$993$		0			0
$994$	−21.1660			−0.671345
$995$		0			0
$996$	−15.2915	−	61.0460i	−0.484530	−	1.93432i
$997$	41.6499			1.31907			0.659533	−	0.751675i	$-0.270754\pi$
$997$	41.6499			1.31907			0.659533	+	0.751675i	$0.270754\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.2.i.d.125.5	yes	8
3.2	odd	2	inner	168.2.i.d.125.1	✓	8
4.3	odd	2		672.2.i.e.209.7		8
7.6	odd	2	inner	168.2.i.d.125.8	yes	8
8.3	odd	2		672.2.i.e.209.2		8
8.5	even	2	inner	168.2.i.d.125.8	yes	8
12.11	even	2		672.2.i.e.209.8		8
21.20	even	2	inner	168.2.i.d.125.4	yes	8
24.5	odd	2	inner	168.2.i.d.125.4	yes	8
24.11	even	2		672.2.i.e.209.1		8
28.27	even	2		672.2.i.e.209.2		8
56.13	odd	2	CM	168.2.i.d.125.5	yes	8
56.27	even	2		672.2.i.e.209.7		8
84.83	odd	2		672.2.i.e.209.1		8
168.83	odd	2		672.2.i.e.209.8		8
168.125	even	2	inner	168.2.i.d.125.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.d.125.1	✓	8	3.2	odd	2	inner
168.2.i.d.125.1	✓	8	168.125	even	2	inner
168.2.i.d.125.4	yes	8	21.20	even	2	inner
168.2.i.d.125.4	yes	8	24.5	odd	2	inner
168.2.i.d.125.5	yes	8	1.1	even	1	trivial
168.2.i.d.125.5	yes	8	56.13	odd	2	CM
168.2.i.d.125.8	yes	8	7.6	odd	2	inner
168.2.i.d.125.8	yes	8	8.5	even	2	inner
672.2.i.e.209.1		8	24.11	even	2
672.2.i.e.209.1		8	84.83	odd	2
672.2.i.e.209.2		8	8.3	odd	2
672.2.i.e.209.2		8	28.27	even	2
672.2.i.e.209.7		8	4.3	odd	2
672.2.i.e.209.7		8	56.27	even	2
672.2.i.e.209.8		8	12.11	even	2
672.2.i.e.209.8		8	168.83	odd	2

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	168.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-1.68014 + 0.420861i) q^{3} -2.00000 q^{4} -3.91044i q^{5} +(-0.595188 - 2.37608i) q^{6} -2.64575 q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(-1.68014 + 0.420861i) q^{3} -2.00000 q^{4} -3.91044i q^{5} +(-0.595188 - 2.37608i) q^{6} -2.64575 q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +5.53019 q^{10} +(3.36028 - 0.841723i) q^{12} -4.55066 q^{13} -3.74166i q^{14} +(1.64575 + 6.57008i) q^{15} +4.00000 q^{16} +(2.00000 + 3.74166i) q^{18} -0.979531 q^{19} +7.82087i q^{20} +(4.44524 - 1.11349i) q^{21} -7.48331i q^{23} +(1.19038 + 4.75216i) q^{24} -10.2915 q^{25} -6.43560i q^{26} +(-3.85005 + 3.48957i) q^{27} +5.29150 q^{28} +(-9.29150 + 2.32744i) q^{30} +5.65685i q^{32} +10.3460i q^{35} +(-5.29150 + 2.82843i) q^{36} -1.38527i q^{38} +(7.64575 - 1.91520i) q^{39} -11.0604 q^{40} +(1.57472 + 6.28651i) q^{42} +(-5.53019 - 10.3460i) q^{45} +10.5830 q^{46} +(-6.72057 + 1.68345i) q^{48} +7.00000 q^{49} -14.5544i q^{50} +9.10132 q^{52} +(-4.93500 - 5.44479i) q^{54} +7.48331i q^{56} +(1.64575 - 0.412247i) q^{57} +5.29570i q^{59} +(-3.29150 - 13.1402i) q^{60} +15.6110 q^{61} +(-7.00000 + 3.74166i) q^{63} -8.00000 q^{64} +17.7951i q^{65} +(3.14944 + 12.5730i) q^{69} -14.6315 q^{70} -5.65685i q^{71} +(-4.00000 - 7.48331i) q^{72} +(17.2912 - 4.33130i) q^{75} +1.95906 q^{76} +(2.70850 + 10.8127i) q^{78} -5.29150 q^{79} -15.6417i q^{80} +(5.00000 - 7.48331i) q^{81} -18.1669i q^{83} +(-8.89047 + 2.22699i) q^{84} +(14.6315 - 7.82087i) q^{90} +12.0399 q^{91} +14.9666i q^{92} +3.83039i q^{95} +(-2.38075 - 9.50432i) q^{96} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} - 8 q^{15} + 32 q^{16} + 16 q^{18} - 40 q^{25} - 32 q^{30} + 40 q^{39} + 56 q^{49} - 8 q^{57} + 16 q^{60} - 56 q^{63} - 64 q^{64} - 32 q^{72} + 64 q^{78} + 40 q^{81}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^15 + 32 * q^16 + 16 * q^18 - 40 * q^25 - 32 * q^30 + 40 * q^39 + 56 * q^49 - 8 * q^57 + 16 * q^60 - 56 * q^63 - 64 * q^64 - 32 * q^72 + 64 * q^78 + 40 * q^81

Introduction

L-functions

Modular forms

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Database

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