LMFDB - Embedded newform 448.2.q.c.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(31,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("448.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.q (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 16x^{8} + 8x^{7} + 8x^{6} + 32x^{5} + 240x^{4} + 120x^{3} + 32x^{2} + 16x + 4 $ x^12 - 4x^9 - 16x^8 + 8x^7 + 8x^6 + 32x^5 + 240x^4 + 120x^3 + 32x^2 + 16*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Root			$0.463767 - 1.73080i$ of defining polynomial
Character	$\chi$	$=$	448.31
Dual form			448.2.q.c.159.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.182520 - 0.105378i) q^{3} +(-0.767035 - 1.32854i) q^{5} +(2.19457 + 1.47779i) q^{7} +(-1.47779 - 2.55961i) q^{9} +O(q^{10})$	\(q+(-0.182520 - 0.105378i) q^{3} +(-0.767035 - 1.32854i) q^{5} +(2.19457 + 1.47779i) q^{7} +(-1.47779 - 2.55961i) q^{9} +(2.37709 - 4.11724i) q^{11} -4.95558 q^{13} +0.323314i q^{15} +(2.30111 + 1.32854i) q^{17} +(3.74410 - 2.16166i) q^{19} +(-0.244826 - 0.500986i) q^{21} +(0.547560 - 0.316134i) q^{23} +(1.32331 - 2.29205i) q^{25} +1.25517i q^{27} -7.85324i q^{29} +(0.645038 - 1.11724i) q^{31} +(-0.867732 + 0.500986i) q^{33} +(0.279998 - 4.04910i) q^{35} +(-1.50000 + 0.866025i) q^{37} +(0.904493 + 0.522209i) q^{39} -7.85324i q^{41} -6.92820 q^{43} +(-2.26704 + 3.92662i) q^{45} +(4.47418 + 7.74951i) q^{47} +(2.63227 + 6.48623i) q^{49} +(-0.279998 - 0.484971i) q^{51} +(0.867732 + 0.500986i) q^{53} -7.29324 q^{55} -0.911164 q^{57} +(8.40080 + 4.85020i) q^{59} +(6.45558 + 11.1814i) q^{61} +(0.539453 - 7.80111i) q^{63} +(3.80111 + 6.58371i) q^{65} +(-2.83961 + 4.91834i) q^{67} -0.133254 q^{69} -6.00000i q^{71} +(-9.73448 - 5.62020i) q^{73} +(-0.483063 + 0.278896i) q^{75} +(11.3011 - 5.52273i) q^{77} +(-4.57166 + 2.63945i) q^{79} +(-4.30111 + 7.44973i) q^{81} +15.0681i q^{83} -4.07616i q^{85} +(-0.827558 + 1.43337i) q^{87} +(-4.50000 + 2.59808i) q^{89} +(-10.8754 - 7.32331i) q^{91} +(-0.235465 + 0.135946i) q^{93} +(-5.74371 - 3.31613i) q^{95} +6.96929i q^{97} -14.0514 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{5} + 4 q^{9}+O(q^{10}) $ 12 * q + 6 * q^5 + 4 * q^9	$ 12 q + 6 q^{5} + 4 q^{9} - 16 q^{13} - 18 q^{17} + 34 q^{21} - 8 q^{25} - 30 q^{33} - 18 q^{37} - 12 q^{45} + 12 q^{49} + 30 q^{53} + 76 q^{57} + 34 q^{61} - 132 q^{69} - 6 q^{73} + 90 q^{77} - 6 q^{81} - 54 q^{89} - 42 q^{93}+O(q^{100}) $ 12 * q + 6 * q^5 + 4 * q^9 - 16 * q^13 - 18 * q^17 + 34 * q^21 - 8 * q^25 - 30 * q^33 - 18 * q^37 - 12 * q^45 + 12 * q^49 + 30 * q^53 + 76 * q^57 + 34 * q^61 - 132 * q^69 - 6 * q^73 + 90 * q^77 - 6 * q^81 - 54 * q^89 - 42 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$

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
$3$	−0.182520	−	0.105378i	−0.105378	−	0.0608400i	0.446385	−	0.894841i	$-0.352711\pi$
$3$	−0.182520	−	0.105378i	−0.105378	−	0.0608400i	−0.551763	+	0.834001i	$0.686045\pi$
$4$		0			0
$5$	−0.767035	−	1.32854i	−0.343029	−	0.594143i	0.641965	−	0.766734i	$-0.278119\pi$
$5$	−0.767035	−	1.32854i	−0.343029	−	0.594143i	−0.984994	+	0.172591i	$0.944786\pi$
$6$		0			0
$7$	2.19457	+	1.47779i	0.829469	+	0.558552i
$7$	2.19457	+	1.47779i	0.829469	+	0.558552i
$8$		0			0
$9$	−1.47779	−	2.55961i	−0.492597	−	0.853203i
$10$		0			0
$11$	2.37709	−	4.11724i	0.716719	−	1.24139i	−0.245573	−	0.969378i	$-0.578976\pi$
$11$	2.37709	−	4.11724i	0.716719	−	1.24139i	0.962293	−	0.272016i	$-0.0876905\pi$
$12$		0			0
$13$	−4.95558			−1.37443			−0.687216	−	0.726454i	$-0.741167\pi$
$13$	−4.95558			−1.37443			−0.687216	+	0.726454i	$0.741167\pi$
$14$		0			0
$15$			0.323314i			0.0834794i
$16$		0			0
$17$	2.30111	+	1.32854i	0.558100	+	0.322219i	0.752383	−	0.658726i	$-0.228905\pi$
$17$	2.30111	+	1.32854i	0.558100	+	0.322219i	−0.194283	+	0.980946i	$0.562238\pi$
$18$		0			0
$19$	3.74410	−	2.16166i	0.858955	−	0.495918i	−0.00470690	−	0.999989i	$-0.501498\pi$
$19$	3.74410	−	2.16166i	0.858955	−	0.495918i	0.863662	+	0.504071i	$0.168165\pi$
$20$		0			0
$21$	−0.244826	−	0.500986i	−0.0534254	−	0.109324i
$22$		0			0
$23$	0.547560	−	0.316134i	0.114174	−	0.0659185i	−0.441825	−	0.897101i	$-0.645669\pi$
$23$	0.547560	−	0.316134i	0.114174	−	0.0659185i	0.556000	+	0.831183i	$0.312336\pi$
$24$		0			0
$25$	1.32331	−	2.29205i	0.264663	−	0.458410i
$26$		0			0
$27$			1.25517i			0.241558i
$28$		0			0
$29$		−	7.85324i		−	1.45831i	−0.684349	−	0.729155i	$-0.739913\pi$
$29$		−	7.85324i		−	1.45831i	0.684349	−	0.729155i	$-0.260087\pi$
$30$		0			0
$31$	0.645038	−	1.11724i	0.115852	−	0.200662i	−0.802268	−	0.596964i	$-0.796373\pi$
$31$	0.645038	−	1.11724i	0.115852	−	0.200662i	0.918120	+	0.396302i	$0.129707\pi$
$32$		0			0
$33$	−0.867732	+	0.500986i	−0.151053	+	0.0872104i
$34$		0			0
$35$	0.279998	−	4.04910i	0.0473284	−	0.684423i
$36$		0			0
$37$	−1.50000	+	0.866025i	−0.246598	+	0.142374i	−0.618206	−	0.786016i	$-0.712140\pi$
$37$	−1.50000	+	0.866025i	−0.246598	+	0.142374i	0.371607	+	0.928390i	$0.378807\pi$
$38$		0			0
$39$	0.904493	+	0.522209i	0.144835	+	0.0836204i
$40$		0			0
$41$		−	7.85324i		−	1.22647i	−0.789901	−	0.613235i	$-0.789868\pi$
$41$		−	7.85324i		−	1.22647i	0.789901	−	0.613235i	$-0.210132\pi$
$42$		0			0
$43$	−6.92820			−1.05654			−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820			−1.05654			−0.528271	+	0.849076i	$0.677159\pi$
$44$		0			0
$45$	−2.26704	+	3.92662i	−0.337950	+	0.585346i
$46$		0			0
$47$	4.47418	+	7.74951i	0.652626	+	1.13038i	0.982483	+	0.186351i	$0.0596661\pi$
$47$	4.47418	+	7.74951i	0.652626	+	1.13038i	−0.329857	+	0.944031i	$0.607001\pi$
$48$		0			0
$49$	2.63227	+	6.48623i	0.376038	+	0.926604i
$50$		0			0
$51$	−0.279998	−	0.484971i	−0.0392076	−	0.0679096i
$52$		0			0
$53$	0.867732	+	0.500986i	0.119192	+	0.0688157i	0.558411	−	0.829565i	$-0.311411\pi$
$53$	0.867732	+	0.500986i	0.119192	+	0.0688157i	−0.439219	+	0.898380i	$0.644745\pi$
$54$		0			0
$55$	−7.29324			−0.983421
$56$		0			0
$57$	−0.911164			−0.120687
$58$		0			0
$59$	8.40080	+	4.85020i	1.09369	+	0.631443i	0.934557	−	0.355814i	$-0.115796\pi$
$59$	8.40080	+	4.85020i	1.09369	+	0.631443i	0.159134	+	0.987257i	$0.449130\pi$
$60$		0			0
$61$	6.45558	+	11.1814i	0.826553	+	1.43163i	0.900727	+	0.434386i	$0.143035\pi$
$61$	6.45558	+	11.1814i	0.826553	+	1.43163i	−0.0741744	+	0.997245i	$0.523632\pi$
$62$		0			0
$63$	0.539453	−	7.80111i	0.0679646	−	0.982847i
$64$		0			0
$65$	3.80111	+	6.58371i	0.471469	+	0.816608i
$66$		0			0
$67$	−2.83961	+	4.91834i	−0.346913	+	0.600871i	−0.985699	−	0.168513i	$-0.946104\pi$
$67$	−2.83961	+	4.91834i	−0.346913	+	0.600871i	0.638786	+	0.769384i	$0.279437\pi$
$68$		0			0
$69$	−0.133254			−0.0160419
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$	−9.73448	−	5.62020i	−1.13933	−	0.657795i	−0.193069	−	0.981185i	$-0.561844\pi$
$73$	−9.73448	−	5.62020i	−1.13933	−	0.657795i	−0.946266	+	0.323390i	$0.895177\pi$
$74$		0			0
$75$	−0.483063	+	0.278896i	−0.0557793	+	0.0322042i
$76$		0			0
$77$	11.3011	−	5.52273i	1.28788	−	0.629373i
$78$		0			0
$79$	−4.57166	+	2.63945i	−0.514352	+	0.296961i	−0.734621	−	0.678478i	$-0.762640\pi$
$79$	−4.57166	+	2.63945i	−0.514352	+	0.296961i	0.220269	+	0.975439i	$0.429307\pi$
$80$		0			0
$81$	−4.30111	+	7.44973i	−0.477901	+	0.827748i
$82$		0			0
$83$			15.0681i			1.65394i	0.562244	+	0.826972i	$0.309938\pi$
$83$			15.0681i			1.65394i	−0.562244	+	0.826972i	$0.690062\pi$
$84$		0			0
$85$		−	4.07616i		−	0.442121i
$86$		0			0
$87$	−0.827558	+	1.43337i	−0.0887236	+	0.153674i
$88$		0			0
$89$	−4.50000	+	2.59808i	−0.476999	+	0.275396i	−0.719165	−	0.694839i	$-0.755475\pi$
$89$	−4.50000	+	2.59808i	−0.476999	+	0.275396i	0.242166	+	0.970235i	$0.422142\pi$
$90$		0			0
$91$	−10.8754	−	7.32331i	−1.14005	−	0.767692i
$92$		0			0
$93$	−0.235465	+	0.135946i	−0.0244166	+	0.0140969i
$94$		0			0
$95$	−5.74371	−	3.31613i	−0.589292	−	0.340228i
$96$		0			0
$97$			6.96929i			0.707624i	0.935316	+	0.353812i	$0.115115\pi$
$97$			6.96929i			0.707624i	−0.935316	+	0.353812i	$0.884885\pi$
$98$		0			0
$99$	−14.0514			−1.41222
$100$		0			0
$101$	−2.23296	+	3.86761i	−0.222188	+	0.384841i	−0.955472	−	0.295081i	$-0.904653\pi$
$101$	−2.23296	+	3.86761i	−0.222188	+	0.384841i	0.733284	+	0.679923i	$0.237987\pi$
$102$		0			0
$103$	−2.83961	−	4.91834i	−0.279795	−	0.484619i	0.691539	−	0.722339i	$-0.256933\pi$
$103$	−2.83961	−	4.91834i	−0.279795	−	0.484619i	−0.971334	+	0.237720i	$0.923600\pi$
$104$		0			0
$105$	−0.477791	+	0.709536i	−0.0466276	+	0.0692436i
$106$		0			0
$107$	5.03418	+	8.71945i	0.486672	+	0.842941i	0.999883	−	0.0153218i	$-0.00487728\pi$
$107$	5.03418	+	8.71945i	0.486672	+	0.842941i	−0.513210	+	0.858263i	$0.671544\pi$
$108$		0			0
$109$	8.93337	+	5.15769i	0.855662	+	0.494017i	0.862557	−	0.505960i	$-0.168862\pi$
$109$	8.93337	+	5.15769i	0.855662	+	0.494017i	−0.00689525	+	0.999976i	$0.502195\pi$
$110$		0			0
$111$	0.365040			0.0346481
$112$		0			0
$113$	14.8667			1.39855			0.699273	−	0.714855i	$-0.253507\pi$
$113$	14.8667			1.39855			0.699273	+	0.714855i	$0.253507\pi$
$114$		0			0
$115$	−0.839995	−	0.484971i	−0.0783300	−	0.0452238i
$116$		0			0
$117$	7.32331	+	12.6844i	0.677041	+	1.17267i
$118$		0			0
$119$	3.08662	+	6.31613i	0.282950	+	0.578999i
$120$		0			0
$121$	−5.80111	−	10.0478i	−0.527373	−	0.913437i
$122$		0			0
$123$	−0.827558	+	1.43337i	−0.0746184	+	0.129243i
$124$		0			0
$125$	−11.7305			−1.04920
$126$		0			0
$127$		−	17.2044i		−	1.52665i	−0.646017	−	0.763323i	$-0.723567\pi$
$127$		−	17.2044i		−	1.52665i	0.646017	−	0.763323i	$-0.276433\pi$
$128$		0			0
$129$	1.26454	+	0.730080i	0.111336	+	0.0642799i
$130$		0			0
$131$	11.0579	−	6.38427i	0.966132	−	0.557797i	0.0680772	−	0.997680i	$-0.478314\pi$
$131$	11.0579	−	6.38427i	0.966132	−	0.557797i	0.898055	+	0.439883i	$0.144980\pi$
$132$		0			0
$133$	11.4112	+	0.789091i	0.989473	+	0.0684229i
$134$		0			0
$135$	1.66755	−	0.962762i	0.143520	−	0.0828614i
$136$		0			0
$137$	−2.30111	+	3.98563i	−0.196597	+	0.340515i	−0.947423	−	0.319984i	$-0.896322\pi$
$137$	−2.30111	+	3.98563i	−0.196597	+	0.340515i	0.750826	+	0.660500i	$0.229656\pi$
$138$		0			0
$139$			15.9112i			1.34957i	0.738016	+	0.674784i	$0.235763\pi$
$139$			15.9112i			1.34957i	−0.738016	+	0.674784i	$0.764237\pi$
$140$		0			0
$141$		−	1.88592i		−	0.158823i
$142$		0			0
$143$	−11.7799	+	20.4033i	−0.985081	+	1.70621i
$144$		0			0
$145$	−10.4334	+	6.02371i	−0.866444	+	0.500242i
$146$		0			0
$147$	0.203064	−	1.46125i	0.0167484	−	0.120522i
$148$		0			0
$149$	4.03657	−	2.33051i	0.330689	−	0.190923i	−0.325458	−	0.945556i	$-0.605519\pi$
$149$	4.03657	−	2.33051i	0.330689	−	0.190923i	0.656147	+	0.754633i	$0.272185\pi$
$150$		0			0
$151$	15.7915	+	9.11724i	1.28510	+	0.741950i	0.977775	−	0.209656i	$-0.0672343\pi$
$151$	15.7915	+	9.11724i	1.28510	+	0.741950i	0.307321	+	0.951606i	$0.400568\pi$
$152$		0			0
$153$		−	7.85324i		−	0.634897i
$154$		0			0
$155$	−1.97907			−0.158963
$156$		0			0
$157$	0.455582	−	0.789091i	0.0363594	−	0.0629763i	−0.847273	−	0.531158i	$-0.821757\pi$
$157$	0.455582	−	0.789091i	0.0363594	−	0.0629763i	0.883632	+	0.468181i	$0.155091\pi$
$158$		0			0
$159$	−0.105586	−	0.182880i	−0.00837349	−	0.0145033i
$160$		0			0
$161$	1.66884	+	0.115401i	0.131523	+	0.00909491i
$162$		0			0
$163$	−1.10756	−	1.91834i	−0.0867505	−	0.150256i	0.819385	−	0.573243i	$-0.194315\pi$
$163$	−1.10756	−	1.91834i	−0.0867505	−	0.150256i	−0.906136	+	0.422987i	$0.860982\pi$
$164$		0			0
$165$	1.33116	+	0.768547i	0.103631	+	0.0598313i
$166$		0			0
$167$	−18.2455			−1.41188			−0.705941	−	0.708270i	$-0.749476\pi$
$167$	−18.2455			−1.41188			−0.705941	+	0.708270i	$0.749476\pi$
$168$		0			0
$169$	11.5578			0.889061
$170$		0			0
$171$	−11.0660	−	6.38895i	−0.846238	−	0.488576i
$172$		0			0
$173$	−5.20041	−	9.00737i	−0.395380	−	0.684818i	0.597770	−	0.801668i	$-0.296054\pi$
$173$	−5.20041	−	9.00737i	−0.395380	−	0.684818i	−0.993150	+	0.116850i	$0.962720\pi$
$174$		0			0
$175$	6.29127	−	3.07448i	0.475575	−	0.232408i
$176$		0			0
$177$	−1.02221	−	1.77052i	−0.0768339	−	0.133080i
$178$		0			0
$179$	6.57127	−	11.3818i	0.491160	−	0.850714i	−0.508788	−	0.860892i	$-0.669906\pi$
$179$	6.57127	−	11.3818i	0.491160	−	0.850714i	0.999948	+	0.0101779i	$0.00323977\pi$
$180$		0			0
$181$	−23.8510			−1.77283			−0.886417	−	0.462887i	$-0.846814\pi$
$181$	−23.8510			−1.77283			−0.886417	+	0.462887i	$0.846814\pi$
$182$		0			0
$183$		−	2.72110i		−	0.201150i
$184$		0			0
$185$	2.30111	+	1.32854i	0.169181	+	0.0976765i
$186$		0			0
$187$	10.9399	−	6.31613i	0.800002	−	0.461881i
$188$		0			0
$189$	−1.85488	+	2.75457i	−0.134923	+	0.200365i
$190$		0			0
$191$	11.2323	−	6.48497i	0.812741	−	0.469236i	−0.0351660	−	0.999381i	$-0.511196\pi$
$191$	11.2323	−	6.48497i	0.812741	−	0.469236i	0.847907	+	0.530145i	$0.177863\pi$
$192$		0			0
$193$	−0.0222090	+	0.0384672i	−0.00159864	+	0.00276893i	−0.866824	−	0.498615i	$-0.833842\pi$
$193$	−0.0222090	+	0.0384672i	−0.00159864	+	0.00276893i	0.865225	+	0.501384i	$0.167176\pi$
$194$		0			0
$195$		−	1.60221i		−	0.114737i
$196$		0			0
$197$		−	26.8700i		−	1.91440i	−0.289419	−	0.957202i	$-0.593462\pi$
$197$		−	26.8700i		−	1.91440i	0.289419	−	0.957202i	$-0.406538\pi$
$198$		0			0
$199$	9.67033	−	16.7495i	0.685512	−	1.18734i	−0.287764	−	0.957701i	$-0.592912\pi$
$199$	9.67033	−	16.7495i	0.685512	−	1.18734i	0.973276	−	0.229640i	$-0.0737548\pi$
$200$		0			0
$201$	1.03657	−	0.598464i	0.0731140	−	0.0422124i
$202$		0			0
$203$	11.6054	−	17.2345i	0.814543	−	1.20962i
$204$		0			0
$205$	−10.4334	+	6.02371i	−0.728698	+	0.420714i
$206$		0			0
$207$	−1.61836	−	0.934359i	−0.112484	−	0.0649425i
$208$		0			0
$209$		−	20.5538i		−	1.42174i
$210$		0			0
$211$	−12.5825			−0.866218			−0.433109	−	0.901342i	$-0.642583\pi$
$211$	−12.5825			−0.866218			−0.433109	+	0.901342i	$0.642583\pi$
$212$		0			0
$213$	−0.632268	+	1.09512i	−0.0433223	+	0.0750364i
$214$		0			0
$215$	5.31418	+	9.20442i	0.362424	+	0.627736i
$216$		0			0
$217$	3.06663	−	1.49863i	0.208176	−	0.101733i
$218$		0			0
$219$	1.18449	+	2.05160i	0.0800405	+	0.138634i
$220$		0			0
$221$	−11.4033	−	6.58371i	−0.767070	−	0.442868i
$222$		0			0
$223$	11.5123			0.770921			0.385460	−	0.922724i	$-0.374043\pi$
$223$	11.5123			0.770921			0.385460	+	0.922724i	$0.374043\pi$
$224$		0			0
$225$	−7.82233			−0.521489
$226$		0			0
$227$	19.0855	+	11.0190i	1.26675	+	0.731359i	0.974372	−	0.224943i	$-0.0722195\pi$
$227$	19.0855	+	11.0190i	1.26675	+	0.731359i	0.292380	+	0.956302i	$0.405553\pi$
$228$		0			0
$229$	2.02221	+	3.50257i	0.133631	+	0.231456i	0.925074	−	0.379787i	$-0.124003\pi$
$229$	2.02221	+	3.50257i	0.133631	+	0.231456i	−0.791442	+	0.611244i	$0.790670\pi$
$230$		0			0
$231$	−2.64465	−	0.182880i	−0.174005	−	0.0120326i
$232$		0			0
$233$	8.93337	+	15.4731i	0.585245	+	1.01367i	0.994845	+	0.101408i	$0.0323349\pi$
$233$	8.93337	+	15.4731i	0.585245	+	1.01367i	−0.409600	+	0.912265i	$0.634332\pi$
$234$		0			0
$235$	6.86371	−	11.8883i	0.447739	−	0.775506i
$236$		0			0
$237$	1.11256			0.0722684
$238$		0			0
$239$			24.0712i			1.55703i	0.627623	+	0.778517i	$0.284028\pi$
$239$			24.0712i			1.55703i	−0.627623	+	0.778517i	$0.715972\pi$
$240$		0			0
$241$	−7.36675	−	4.25319i	−0.474534	−	0.273972i	0.243602	−	0.969875i	$-0.421671\pi$
$241$	−7.36675	−	4.25319i	−0.474534	−	0.273972i	−0.718136	+	0.695903i	$0.755004\pi$
$242$		0			0
$243$	4.83111	−	2.78924i	0.309916	−	0.178930i
$244$		0			0
$245$	6.59820	−	8.47225i	0.421543	−	0.541272i
$246$		0			0
$247$	−18.5542	+	10.7123i	−1.18058	+	0.681605i
$248$		0			0
$249$	1.58785	−	2.75024i	0.100626	−	0.174289i
$250$		0			0
$251$		−	5.93489i		−	0.374607i	−0.982302	−	0.187303i	$-0.940025\pi$
$251$		−	5.93489i		−	0.374607i	0.982302	−	0.187303i	$-0.0599747\pi$
$252$		0			0
$253$		−	3.00591i		−	0.188980i
$254$		0			0
$255$	−0.429537	+	0.743980i	−0.0268987	+	0.0465899i
$256$		0			0
$257$	15.9033	−	9.18178i	0.992022	−	0.572744i	0.0861436	−	0.996283i	$-0.472546\pi$
$257$	15.9033	−	9.18178i	0.992022	−	0.572744i	0.905878	+	0.423539i	$0.139212\pi$
$258$		0			0
$259$	−4.57166	−	0.316134i	−0.284069	−	0.0196436i
$260$		0			0
$261$	−20.1012	+	11.6054i	−1.24423	+	0.718359i
$262$		0			0
$263$	16.1360	+	9.31613i	0.994989	+	0.574457i	0.906762	−	0.421643i	$-0.138546\pi$
$263$	16.1360	+	9.31613i	0.994989	+	0.574457i	0.0882275	+	0.996100i	$0.471880\pi$
$264$		0			0
$265$		−	1.53709i		−	0.0944229i
$266$		0			0
$267$	1.09512			0.0670202
$268$		0			0
$269$	0.0340702	−	0.0590113i	0.00207730	−	0.00359798i	−0.864985	−	0.501798i	$-0.832672\pi$
$269$	0.0340702	−	0.0590113i	0.00207730	−	0.00359798i	0.867062	+	0.498200i	$0.166005\pi$
$270$		0			0
$271$	9.40277	+	16.2861i	0.571178	+	0.989309i	0.996445	+	0.0842411i	$0.0268466\pi$
$271$	9.40277	+	16.2861i	0.571178	+	0.989309i	−0.425268	+	0.905068i	$0.639820\pi$
$272$		0			0
$273$	1.21326	+	2.48267i	0.0734296	+	0.150258i
$274$		0			0
$275$	−6.29127	−	10.8968i	−0.379378	−	0.657102i
$276$		0			0
$277$	2.46994	+	1.42602i	0.148404	+	0.0856814i	0.572364	−	0.820000i	$-0.306027\pi$
$277$	2.46994	+	1.42602i	0.148404	+	0.0856814i	−0.423959	+	0.905681i	$0.639360\pi$
$278$		0			0
$279$	−3.81293			−0.228274
$280$		0			0
$281$	−0.337675			−0.0201440			−0.0100720	−	0.999949i	$-0.503206\pi$
$281$	−0.337675			−0.0201440			−0.0100720	+	0.999949i	$0.503206\pi$
$282$		0			0
$283$	5.32228	+	3.07282i	0.316377	+	0.182660i	0.649776	−	0.760125i	$-0.274862\pi$
$283$	5.32228	+	3.07282i	0.316377	+	0.182660i	−0.333400	+	0.942786i	$0.608196\pi$
$284$		0			0
$285$	0.698895	+	1.21052i	0.0413989	+	0.0717051i
$286$		0			0
$287$	11.6054	−	17.2345i	0.685048	−	1.01732i
$288$		0			0
$289$	−4.96994	−	8.60819i	−0.292350	−	0.506364i
$290$		0			0
$291$	0.734410	−	1.27203i	0.0430519	−	0.0745680i
$292$		0			0
$293$	0.136281			0.00796161			0.00398080	−	0.999992i	$-0.498733\pi$
$293$	0.136281			0.00796161			0.00398080	+	0.999992i	$0.498733\pi$
$294$		0			0
$295$		−	14.8811i		−	0.866412i
$296$		0			0
$297$	5.16785	+	2.98366i	0.299869	+	0.173130i
$298$		0			0
$299$	−2.71348	+	1.56663i	−0.156924	+	0.0906004i
$300$		0			0
$301$	−15.2044	−	10.2384i	−0.876368	−	0.590134i
$302$		0			0
$303$	0.815121	−	0.470610i	0.0468275	−	0.0270359i
$304$		0			0
$305$	9.90332	−	17.1530i	0.567062	−	0.982181i
$306$		0			0
$307$		−	3.35337i		−	0.191387i	−0.995411	−	0.0956935i	$-0.969493\pi$
$307$		−	3.35337i		−	0.191387i	0.995411	−	0.0956935i	$-0.0305069\pi$
$308$		0			0
$309$			1.19693i			0.0680909i
$310$		0			0
$311$	8.40080	−	14.5506i	0.476366	−	0.825090i	−0.523268	−	0.852168i	$-0.675287\pi$
$311$	8.40080	−	14.5506i	0.476366	−	0.825090i	0.999633	+	0.0270789i	$0.00862053\pi$
$312$		0			0
$313$	−6.86773	+	3.96509i	−0.388187	+	0.224120i	−0.681374	−	0.731935i	$-0.738617\pi$
$313$	−6.86773	+	3.96509i	−0.388187	+	0.224120i	0.293187	+	0.956055i	$0.405284\pi$
$314$		0			0
$315$	−10.7779	+	5.26704i	−0.607265	+	0.296764i
$316$		0			0
$317$	21.7345	−	12.5484i	1.22073	−	0.704789i	0.255656	−	0.966768i	$-0.417708\pi$
$317$	21.7345	−	12.5484i	1.22073	−	0.704789i	0.965074	+	0.261979i	$0.0843751\pi$
$318$		0			0
$319$	−32.3337	−	18.6679i	−1.81034	−	1.04520i
$320$		0			0
$321$		−	2.12196i		−	0.118437i
$322$		0			0
$323$	11.4874			0.639177
$324$		0			0
$325$	−6.55779	+	11.3584i	−0.363761	+	0.630052i
$326$		0			0
$327$	−1.08701	−	1.88276i	−0.0601119	−	0.104117i
$328$		0			0
$329$	−1.63325	+	23.6187i	−0.0900442	+	1.30214i
$330$		0			0
$331$	−3.39960	−	5.88829i	−0.186859	−	0.323650i	0.757342	−	0.653018i	$-0.226497\pi$
$331$	−3.39960	−	5.88829i	−0.186859	−	0.323650i	−0.944201	+	0.329369i	$0.893164\pi$
$332$		0			0
$333$	4.43337	+	2.55961i	0.242947	+	0.140266i
$334$		0			0
$335$	8.71231			0.476005
$336$		0			0
$337$	−1.69105			−0.0921172			−0.0460586	−	0.998939i	$-0.514666\pi$
$337$	−1.69105			−0.0921172			−0.0460586	+	0.998939i	$0.514666\pi$
$338$		0			0
$339$	−2.71348	−	1.56663i	−0.147376	−	0.0850875i
$340$		0			0
$341$	−3.06663	−	5.31155i	−0.166067	−	0.287637i
$342$		0			0
$343$	−3.80860	+	18.1244i	−0.205645	+	0.978627i
$344$		0			0
$345$	0.102211	+	0.177034i	0.00550283	+	0.00953119i
$346$		0			0
$347$	−12.0599	+	20.8883i	−0.647407	+	1.12134i	0.336333	+	0.941743i	$0.390813\pi$
$347$	−12.0599	+	20.8883i	−0.647407	+	1.12134i	−0.983740	+	0.179599i	$0.942520\pi$
$348$		0			0
$349$	17.5134			0.937469			0.468735	−	0.883339i	$-0.344710\pi$
$349$	17.5134			0.937469			0.468735	+	0.883339i	$0.344710\pi$
$350$		0			0
$351$		−	6.22012i		−	0.332005i
$352$		0			0
$353$	−8.59570	−	4.96273i	−0.457503	−	0.264139i	0.253491	−	0.967338i	$-0.418421\pi$
$353$	−8.59570	−	4.96273i	−0.457503	−	0.264139i	−0.710994	+	0.703198i	$0.751755\pi$
$354$		0			0
$355$	−7.97126	+	4.60221i	−0.423071	+	0.244260i
$356$		0			0
$357$	0.102211	−	1.47808i	0.00540956	−	0.0782284i
$358$		0			0
$359$	−8.22639	+	4.74951i	−0.434172	+	0.250669i	−0.701122	−	0.713041i	$-0.747317\pi$
$359$	−8.22639	+	4.74951i	−0.434172	+	0.250669i	0.266950	+	0.963710i	$0.413984\pi$
$360$		0			0
$361$	−0.154477	+	0.267561i	−0.00813035	+	0.0140822i
$362$		0			0
$363$			2.44523i			0.128342i
$364$		0			0
$365$			17.2436i			0.902570i
$366$		0			0
$367$	−17.9861	+	31.1528i	−0.938866	+	1.62616i	−0.171276	+	0.985223i	$0.554789\pi$
$367$	−17.9861	+	31.1528i	−0.938866	+	1.62616i	−0.767590	+	0.640941i	$0.778544\pi$
$368$		0			0
$369$	−20.1012	+	11.6054i	−1.04643	+	0.604155i
$370$		0			0
$371$	1.16395	+	2.38177i	0.0604291	+	0.123656i
$372$		0			0
$373$	14.2034	−	8.20036i	0.735426	−	0.424598i	−0.0849779	−	0.996383i	$-0.527082\pi$
$373$	14.2034	−	8.20036i	0.735426	−	0.424598i	0.820404	+	0.571784i	$0.193749\pi$
$374$		0			0
$375$	2.14104	+	1.23613i	0.110563	+	0.0638336i
$376$		0			0
$377$			38.9174i			2.00435i
$378$		0			0
$379$	−6.23921			−0.320487			−0.160243	−	0.987078i	$-0.551228\pi$
$379$	−6.23921			−0.320487			−0.160243	+	0.987078i	$0.551228\pi$
$380$		0			0
$381$	−1.81297	+	3.14015i	−0.0928811	+	0.160875i
$382$		0			0
$383$	−14.9845	−	25.9539i	−0.765673	−	1.32618i	−0.939890	−	0.341477i	$-0.889073\pi$
$383$	−14.9845	−	25.9539i	−0.765673	−	1.32618i	0.174218	−	0.984707i	$-0.444260\pi$
$384$		0			0
$385$	−16.0055	−	10.7779i	−0.815717	−	0.549292i
$386$		0			0
$387$	10.2384	+	17.7335i	0.520449	+	0.901444i
$388$		0			0
$389$	4.03657	+	2.33051i	0.204662	+	0.118162i	0.598828	−	0.800877i	$-0.295633\pi$
$389$	4.03657	+	2.33051i	0.204662	+	0.118162i	−0.394166	+	0.919039i	$0.628967\pi$
$390$		0			0
$391$	1.67999			0.0849608
$392$		0			0
$393$	−2.69105			−0.135745
$394$		0			0
$395$	7.01324	+	4.04910i	0.352875	+	0.203732i
$396$		0			0
$397$	8.58000	+	14.8610i	0.430618	+	0.745852i	0.996927	−	0.0783411i	$-0.0249623\pi$
$397$	8.58000	+	14.8610i	0.430618	+	0.745852i	−0.566309	+	0.824193i	$0.691629\pi$
$398$		0			0
$399$	−1.99961	−	1.34651i	−0.100106	−	0.0674098i
$400$		0			0
$401$	−11.1679	−	19.3433i	−0.557696	−	0.965958i	−0.997688	−	0.0679561i	$-0.978352\pi$
$401$	−11.1679	−	19.3433i	−0.557696	−	0.965958i	0.439992	−	0.898001i	$-0.354981\pi$
$402$		0			0
$403$	−3.19654	+	5.53657i	−0.159231	+	0.275796i
$404$		0			0
$405$	13.1964			0.655734
$406$		0			0
$407$			8.23448i			0.408168i
$408$		0			0
$409$	29.1679	+	16.8401i	1.44226	+	0.832688i	0.998000	−	0.0632115i	$-0.0201343\pi$
$409$	29.1679	+	16.8401i	1.44226	+	0.832688i	0.444257	+	0.895899i	$0.353468\pi$
$410$		0			0
$411$	0.839995	−	0.484971i	0.0414339	−	0.0239219i
$412$		0			0
$413$	11.2685	+	23.0587i	0.554489	+	1.13465i
$414$		0			0
$415$	20.0187	−	11.5578i	0.982679	−	0.567350i
$416$		0			0
$417$	1.67669	−	2.90410i	0.0821077	−	0.142215i
$418$		0			0
$419$			10.3327i			0.504784i	0.967625	+	0.252392i	$0.0812173\pi$
$419$			10.3327i			0.504784i	−0.967625	+	0.252392i	$0.918783\pi$
$420$		0			0
$421$			18.9756i			0.924815i	0.886667	+	0.462408i	$0.153014\pi$
$421$			18.9756i			0.924815i	−0.886667	+	0.462408i	$0.846986\pi$
$422$		0			0
$423$	13.2238	−	22.9043i	0.642963	−	1.11365i
$424$		0			0
$425$	6.09017	−	3.51616i	0.295417	−	0.170559i
$426$		0			0
$427$	−2.35654	+	34.0783i	−0.114041	+	1.64917i
$428$		0			0
$429$	4.30012	−	2.48267i	0.207612	−	0.119865i
$430$		0			0
$431$	−14.7485	−	8.51503i	−0.710408	−	0.410154i	0.100804	−	0.994906i	$-0.467859\pi$
$431$	−14.7485	−	8.51503i	−0.710408	−	0.410154i	−0.811212	+	0.584752i	$0.801192\pi$
$432$		0			0
$433$			3.26914i			0.157105i	0.996910	+	0.0785525i	$0.0250298\pi$
$433$			3.26914i			0.157105i	−0.996910	+	0.0785525i	$0.974970\pi$
$434$		0			0
$435$	2.53906			0.121739
$436$		0			0
$437$	1.36675	−	2.36727i	0.0653803	−	0.113242i
$438$		0			0
$439$	−9.13521	−	15.8226i	−0.436000	−	0.755174i	0.561377	−	0.827560i	$-0.310272\pi$
$439$	−9.13521	−	15.8226i	−0.436000	−	0.755174i	−0.997377	+	0.0723865i	$0.976938\pi$
$440$		0			0
$441$	12.7123	−	16.3229i	0.605346	−	0.777279i
$442$		0			0
$443$	−8.57521	−	14.8527i	−0.407421	−	0.705673i	0.587179	−	0.809457i	$-0.300238\pi$
$443$	−8.57521	−	14.8527i	−0.407421	−	0.705673i	−0.994600	+	0.103784i	$0.966905\pi$
$444$		0			0
$445$	6.90332	+	3.98563i	0.327249	+	0.188937i
$446$		0			0
$447$	−0.982339			−0.0464631
$448$		0			0
$449$	−22.1313			−1.04444			−0.522220	−	0.852811i	$-0.674896\pi$
$449$	−22.1313			−1.04444			−0.522220	+	0.852811i	$0.674896\pi$
$450$		0			0
$451$	−32.3337	−	18.6679i	−1.52253	−	0.879035i
$452$		0			0
$453$	−1.92151	−	3.32816i	−0.0902805	−	0.156370i
$454$		0			0
$455$	−1.38756	+	20.0656i	−0.0650496	+	0.940692i
$456$		0			0
$457$	9.82331	+	17.0145i	0.459515	+	0.795904i	0.998935	−	0.0461330i	$-0.0146898\pi$
$457$	9.82331	+	17.0145i	0.459515	+	0.795904i	−0.539420	+	0.842037i	$0.681356\pi$
$458$		0			0
$459$	−1.66755	+	2.88829i	−0.0778347	+	0.134814i
$460$		0			0
$461$	−4.67035			−0.217520			−0.108760	−	0.994068i	$-0.534688\pi$
$461$	−4.67035			−0.217520			−0.108760	+	0.994068i	$0.534688\pi$
$462$		0			0
$463$			19.1757i			0.891170i	0.895240	+	0.445585i	$0.147004\pi$
$463$			19.1757i			0.891170i	−0.895240	+	0.445585i	$0.852996\pi$
$464$		0			0
$465$	0.361219	+	0.208550i	0.0167511	+	0.00967128i
$466$		0			0
$467$	−31.9605	+	18.4524i	−1.47896	+	0.853876i	−0.999717	−	0.0238080i	$-0.992421\pi$
$467$	−31.9605	+	18.4524i	−1.47896	+	0.853876i	−0.479240	+	0.877684i	$0.659088\pi$
$468$		0			0
$469$	−13.5000	+	6.59730i	−0.623372	+	0.304635i
$470$		0			0
$471$	−0.166306	+	0.0960166i	−0.00766296	+	0.00442421i
$472$		0			0
$473$	−16.4690	+	28.5251i	−0.757243	+	1.31158i
$474$		0			0
$475$		−	11.4422i		−	0.525004i
$476$		0			0
$477$		−	2.96141i		−	0.135594i
$478$		0			0
$479$	−1.99151	+	3.44939i	−0.0909942	+	0.157607i	−0.907930	−	0.419122i	$-0.862338\pi$
$479$	−1.99151	+	3.44939i	−0.0909942	+	0.157607i	0.816936	+	0.576729i	$0.195671\pi$
$480$		0			0
$481$	7.43337	−	4.29166i	0.338933	−	0.195683i
$482$		0			0
$483$	−0.292435	−	0.196922i	−0.0133063	−	0.00896025i
$484$		0			0
$485$	9.25901	−	5.34569i	0.420430	−	0.242735i
$486$		0			0
$487$	7.01757	+	4.05160i	0.317997	+	0.183595i	0.650499	−	0.759507i	$-0.274560\pi$
$487$	7.01757	+	4.05160i	0.317997	+	0.183595i	−0.332503	+	0.943102i	$0.607893\pi$
$488$		0			0
$489$			0.466848i			0.0211116i
$490$		0			0
$491$	−0.771168			−0.0348023			−0.0174012	−	0.999849i	$-0.505539\pi$
$491$	−0.771168			−0.0348023			−0.0174012	+	0.999849i	$0.505539\pi$
$492$		0			0
$493$	10.4334	−	18.0711i	0.469895	−	0.813883i
$494$		0			0
$495$	10.7779	+	18.6679i	0.484430	+	0.839058i
$496$		0			0
$497$	8.86675	−	13.1674i	0.397728	−	0.590639i
$498$		0			0
$499$	17.1791	+	29.7550i	0.769041	+	1.33202i	0.938083	+	0.346409i	$0.112599\pi$
$499$	17.1791	+	29.7550i	0.769041	+	1.33202i	−0.169042	+	0.985609i	$0.554067\pi$
$500$		0			0
$501$	3.33018	+	1.92268i	0.148781	+	0.0858989i
$502$		0			0
$503$	−28.2890			−1.26135			−0.630673	−	0.776049i	$-0.717221\pi$
$503$	−28.2890			−1.26135			−0.630673	+	0.776049i	$0.717221\pi$
$504$		0			0
$505$	6.85105			0.304868
$506$		0			0
$507$	−2.10953	−	1.21794i	−0.0936874	−	0.0540905i
$508$		0			0
$509$	−14.2330	−	24.6522i	−0.630865	−	1.09269i	−0.987375	−	0.158399i	$-0.949367\pi$
$509$	−14.2330	−	24.6522i	−0.630865	−	1.09269i	0.356510	−	0.934291i	$-0.383967\pi$
$510$		0			0
$511$	−13.0575	−	26.7194i	−0.577630	−	1.18200i
$512$		0			0
$513$	2.71326	+	4.69950i	0.119793	+	0.207488i
$514$		0			0
$515$	−4.35616	+	7.54509i	−0.191955	+	0.332476i
$516$		0			0
$517$	42.5421			1.87100
$518$		0			0
$519$			2.19203i			0.0962196i
$520$		0			0
$521$	−18.1022	−	10.4513i	−0.793072	−	0.457880i	0.0479708	−	0.998849i	$-0.484725\pi$
$521$	−18.1022	−	10.4513i	−0.793072	−	0.457880i	−0.841043	+	0.540968i	$0.818058\pi$
$522$		0			0
$523$	−8.83844	+	5.10288i	−0.386478	+	0.223133i	−0.680633	−	0.732624i	$-0.738295\pi$
$523$	−8.83844	+	5.10288i	−0.386478	+	0.223133i	0.294155	+	0.955758i	$0.404962\pi$
$524$		0			0
$525$	−1.47226	−	0.101808i	−0.0642549	−	0.00444328i
$526$		0			0
$527$	2.96860	−	1.71392i	0.129314	−	0.0746597i
$528$		0			0
$529$	−11.3001	+	19.5724i	−0.491310	+	0.850973i
$530$		0			0
$531$		−	28.6704i		−	1.24419i
$532$		0			0
$533$			38.9174i			1.68570i
$534$		0			0
$535$	7.72278	−	13.3762i	0.333885	−	0.578306i
$536$		0			0
$537$	−2.39878	+	1.38493i	−0.103515	+	0.0597643i
$538$		0			0
$539$	32.9625	+	4.58067i	1.41979	+	0.197303i
$540$		0			0
$541$	−33.6368	+	19.4202i	−1.44616	+	0.834940i	−0.998250	−	0.0591384i	$-0.981165\pi$
$541$	−33.6368	+	19.4202i	−1.44616	+	0.834940i	−0.447910	+	0.894079i	$0.647831\pi$
$542$		0			0
$543$	4.35329	+	2.51337i	0.186818	+	0.107859i
$544$		0			0
$545$		−	15.8245i		−	0.677847i
$546$		0			0
$547$	40.5725			1.73475			0.867377	−	0.497651i	$-0.165804\pi$
$547$	40.5725			1.73475			0.867377	+	0.497651i	$0.165804\pi$
$548$		0			0
$549$	19.0800	−	33.0475i	0.814315	−	1.41043i
$550$		0			0
$551$	−16.9760	−	29.4033i	−0.723202	−	1.25262i
$552$		0			0
$553$	−13.9334	−	0.963504i	−0.592507	−	0.0409724i
$554$		0			0
$555$	−0.279998	−	0.484971i	−0.0118853	−	0.0205859i
$556$		0			0
$557$	−10.3312	−	5.96470i	−0.437745	−	0.252732i	0.264895	−	0.964277i	$-0.414663\pi$
$557$	−10.3312	−	5.96470i	−0.437745	−	0.252732i	−0.702641	+	0.711545i	$0.747996\pi$
$558$		0			0
$559$	34.3333			1.45214
$560$		0			0
$561$	−2.66232			−0.112403
$562$		0			0
$563$	−1.01441	−	0.585669i	−0.0427522	−	0.0246830i	0.478472	−	0.878103i	$-0.341191\pi$
$563$	−1.01441	−	0.585669i	−0.0427522	−	0.0246830i	−0.521224	+	0.853420i	$0.674524\pi$
$564$		0			0
$565$	−11.4033	−	19.7511i	−0.479741	−	0.830936i
$566$		0			0
$567$	−20.4482	+	9.99282i	−0.858745	+	0.419659i
$568$		0			0
$569$	3.73448	+	6.46831i	0.156557	+	0.271165i	0.933625	−	0.358252i	$-0.116627\pi$
$569$	3.73448	+	6.46831i	0.156557	+	0.271165i	−0.777068	+	0.629417i	$0.783294\pi$
$570$		0			0
$571$	11.6905	−	20.2485i	0.489232	−	0.847374i	−0.510692	−	0.859764i	$-0.670611\pi$
$571$	11.6905	−	20.2485i	0.489232	−	0.847374i	0.999923	+	0.0123898i	$0.00394390\pi$
$572$		0			0
$573$	−2.73349			−0.114193
$574$		0			0
$575$		−	1.67338i		−	0.0697847i
$576$		0			0
$577$	1.50000	+	0.866025i	0.0624458	+	0.0360531i	0.530898	−	0.847436i	$-0.321855\pi$
$577$	1.50000	+	0.866025i	0.0624458	+	0.0360531i	−0.468452	+	0.883489i	$0.655188\pi$
$578$		0			0
$579$	0.00810718	−	0.00468068i	0.000336923	−	0.000194522i
$580$		0			0
$581$	−22.2676	+	33.0681i	−0.923814	+	1.37189i
$582$		0			0
$583$	4.12535	−	2.38177i	0.170855	−	0.0986430i
$584$		0			0
$585$	11.2345	−	19.4587i	0.464489	−	0.804518i
$586$		0			0
$587$		−	16.1964i		−	0.668497i	−0.942485	−	0.334248i	$-0.891518\pi$
$587$		−	16.1964i		−	0.668497i	0.942485	−	0.334248i	$-0.108482\pi$
$588$		0			0
$589$		−	5.57741i		−	0.229813i
$590$		0			0
$591$	−2.83150	+	4.90430i	−0.116472	+	0.201736i
$592$		0			0
$593$	36.0045	−	20.7872i	1.47853	−	0.853629i	0.478824	−	0.877911i	$-0.341063\pi$
$593$	36.0045	−	20.7872i	1.47853	−	0.853629i	0.999705	+	0.0242816i	$0.00772984\pi$
$594$		0			0
$595$	6.02371	−	8.94541i	0.246948	−	0.366726i
$596$		0			0
$597$	−3.53006	+	2.03808i	−0.144476	+	0.0834130i
$598$		0			0
$599$	36.1796	+	20.8883i	1.47826	+	0.853472i	0.999698	−	0.0245828i	$-0.00782574\pi$
$599$	36.1796	+	20.8883i	1.47826	+	0.853472i	0.478560	+	0.878055i	$0.341159\pi$
$600$		0			0
$601$		−	0.0410883i		−	0.00167603i	−1.00000		0.000838014i	$-0.999733\pi$
$601$		−	0.0410883i		−	0.00167603i	1.00000		0.000838014i	$-0.000266748\pi$
$602$		0			0
$603$	16.7854			0.683554
$604$		0			0
$605$	−8.89930	+	15.4140i	−0.361808	+	0.626670i
$606$		0			0
$607$	15.1546	+	26.2485i	0.615106	+	1.06539i	0.990366	+	0.138475i	$0.0442201\pi$
$607$	15.1546	+	26.2485i	0.615106	+	1.06539i	−0.375260	+	0.926920i	$0.622447\pi$
$608$		0			0
$609$	−3.93436	+	1.92268i	−0.159428	+	0.0779108i
$610$		0			0
$611$	−22.1722	−	38.4033i	−0.896990	−	1.55363i
$612$		0			0
$613$	−31.6068	−	18.2482i	−1.27658	−	0.737036i	−0.300366	−	0.953824i	$-0.597109\pi$
$613$	−31.6068	−	18.2482i	−1.27658	−	0.737036i	−0.976219	+	0.216788i	$0.930442\pi$
$614$		0			0
$615$	2.53906			0.102385
$616$		0			0
$617$	−31.6022			−1.27226			−0.636129	−	0.771583i	$-0.719465\pi$
$617$	−31.6022			−1.27226			−0.636129	+	0.771583i	$0.719465\pi$
$618$		0			0
$619$	−11.5519	−	6.66951i	−0.464311	−	0.268070i	0.249544	−	0.968363i	$-0.419719\pi$
$619$	−11.5519	−	6.66951i	−0.464311	−	0.268070i	−0.713855	+	0.700293i	$0.753053\pi$
$620$		0			0
$621$	0.396803	+	0.687283i	0.0159232	+	0.0275797i
$622$		0			0
$623$	−13.7150	−	0.948401i	−0.549479	−	0.0379969i
$624$		0			0
$625$	2.38111	+	4.12420i	0.0952443	+	0.164968i
$626$		0			0
$627$	−2.16592	+	3.75148i	−0.0864984	+	0.149820i
$628$		0			0
$629$	−4.60221			−0.183502
$630$		0			0
$631$			11.9112i			0.474176i	0.971488	+	0.237088i	$0.0761930\pi$
$631$			11.9112i			0.474176i	−0.971488	+	0.237088i	$0.923807\pi$
$632$		0			0
$633$	2.29657	+	1.32592i	0.0912803	+	0.0527007i
$634$		0			0
$635$	−22.8568	+	13.1964i	−0.907045	+	0.523683i
$636$		0			0
$637$	−13.0444	−	32.1430i	−0.516839	−	1.27355i
$638$		0			0
$639$	−15.3577	+	8.86675i	−0.607539	+	0.350763i
$640$		0			0
$641$	12.1022	−	20.9616i	0.478009	−	0.827935i	−0.521674	−	0.853145i	$-0.674692\pi$
$641$	12.1022	−	20.9616i	0.478009	−	0.827935i	0.999682	+	0.0252100i	$0.00802545\pi$
$642$		0			0
$643$		−	23.1757i		−	0.913960i	−0.889477	−	0.456980i	$-0.848931\pi$
$643$		−	23.1757i		−	0.913960i	0.889477	−	0.456980i	$-0.151069\pi$
$644$		0			0
$645$		−	2.23999i		−	0.0881994i
$646$		0			0
$647$	0.547560	−	0.948401i	0.0215268	−	0.0372855i	−0.855061	−	0.518527i	$-0.826481\pi$
$647$	0.547560	−	0.948401i	0.0215268	−	0.0372855i	0.876588	+	0.481241i	$0.159814\pi$
$648$		0			0
$649$	39.9389	−	23.0587i	1.56774	−	0.905134i
$650$		0			0
$651$	−0.717643	−	0.0496256i	−0.0281266	−	0.00194498i
$652$		0			0
$653$	18.8677	−	10.8933i	0.738351	−	0.426287i	−0.0831182	−	0.996540i	$-0.526488\pi$
$653$	18.8677	−	10.8933i	0.738351	−	0.426287i	0.821470	+	0.570252i	$0.193155\pi$
$654$		0			0
$655$	−16.9636	−	9.79392i	−0.662822	−	0.382680i
$656$		0			0
$657$			33.2219i			1.29611i
$658$		0			0
$659$	32.2969			1.25811			0.629054	−	0.777361i	$-0.283442\pi$
$659$	32.2969			1.25811			0.629054	+	0.777361i	$0.283442\pi$
$660$		0			0
$661$	10.1466	−	17.5745i	0.394658	−	0.683568i	−0.598399	−	0.801198i	$-0.704196\pi$
$661$	10.1466	−	17.5745i	0.394658	−	0.683568i	0.993057	+	0.117630i	$0.0375297\pi$
$662$		0			0
$663$	1.38756	+	2.40332i	0.0538882	+	0.0933371i
$664$		0			0
$665$	−7.70442	−	15.7655i	−0.298765	−	0.611360i
$666$		0			0
$667$	−2.48267	−	4.30012i	−0.0961296	−	0.166501i
$668$		0			0
$669$	−2.10122	−	1.21314i	−0.0812380	−	0.0469028i
$670$		0			0
$671$	61.3820			2.36962
$672$		0			0
$673$	6.13128			0.236344			0.118172	−	0.992993i	$-0.462297\pi$
$673$	6.13128			0.236344			0.118172	+	0.992993i	$0.462297\pi$
$674$		0			0
$675$	2.87692	+	1.66099i	0.110733	+	0.0639315i
$676$		0			0
$677$	−10.6007	−	18.3609i	−0.407418	−	0.705668i	0.587182	−	0.809455i	$-0.300237\pi$
$677$	−10.6007	−	18.3609i	−0.407418	−	0.705668i	−0.994600	+	0.103787i	$0.966904\pi$
$678$		0			0
$679$	−10.2992	+	15.2946i	−0.395245	+	0.586953i
$680$		0			0
$681$	−2.32233	−	4.02239i	−0.0889918	−	0.154138i
$682$		0			0
$683$	−10.8835	+	18.8507i	−0.416445	+	0.721303i	−0.995579	−	0.0939290i	$-0.970057\pi$
$683$	−10.8835	+	18.8507i	−0.416445	+	0.721303i	0.579134	+	0.815232i	$0.303391\pi$
$684$		0			0
$685$	7.06011			0.269753
$686$		0			0
$687$		−	0.852385i		−	0.0325205i
$688$		0			0
$689$	−4.30012	−	2.48267i	−0.163821	−	0.0945824i
$690$		0			0
$691$	14.9119	−	8.60939i	0.567276	−	0.327517i	−0.188785	−	0.982018i	$-0.560455\pi$
$691$	14.9119	−	8.60939i	0.567276	−	0.327517i	0.756060	+	0.654502i	$0.227122\pi$
$692$		0			0
$693$	−30.8367	−	20.7650i	−1.17139	−	0.788796i
$694$		0			0
$695$	21.1387	−	12.2044i	0.801836	−	0.462940i
$696$		0			0
$697$	10.4334	−	18.0711i	0.395192	−	0.684493i
$698$		0			0
$699$		−	3.76552i		−	0.142425i
$700$		0			0
$701$			29.1730i			1.10185i	0.834555	+	0.550924i	$0.185725\pi$
$701$			29.1730i			1.10185i	−0.834555	+	0.550924i	$0.814275\pi$
$702$		0			0
$703$	−3.74410	+	6.48497i	−0.141211	+	0.244585i
$704$		0			0
$705$	−2.50553	+	1.44657i	−0.0943636	+	0.0544808i
$706$		0			0
$707$	−10.6159	+	5.18788i	−0.399253	+	0.195110i
$708$		0			0
$709$	7.36675	−	4.25319i	0.276664	−	0.159732i	−0.355248	−	0.934772i	$-0.615604\pi$
$709$	7.36675	−	4.25319i	0.276664	−	0.159732i	0.631912	+	0.775040i	$0.282270\pi$
$710$		0			0
$711$	13.5119	+	7.80111i	0.506736	+	0.292564i
$712$		0			0
$713$		−	0.815674i		−	0.0305472i
$714$		0			0
$715$	36.1423			1.35164
$716$		0			0
$717$	2.53657	−	4.39347i	0.0947299	−	0.164077i
$718$		0			0
$719$	−7.18766	−	12.4494i	−0.268054	−	0.464284i	0.700305	−	0.713844i	$-0.253047\pi$
$719$	−7.18766	−	12.4494i	−0.268054	−	0.464284i	−0.968359	+	0.249560i	$0.919714\pi$
$720$		0			0
$721$	1.03657	−	14.9900i	0.0386039	−	0.558257i
$722$		0			0
$723$	0.896385	+	1.55259i	0.0333369	+	0.0577413i
$724$		0			0
$725$	−18.0000	−	10.3923i	−0.668503	−	0.385961i
$726$		0			0
$727$	10.7325			0.398045			0.199023	−	0.979995i	$-0.436223\pi$
$727$	10.7325			0.398045			0.199023	+	0.979995i	$0.436223\pi$
$728$		0			0
$729$	24.6309			0.912257
$730$		0			0
$731$	−15.9425	−	9.20442i	−0.589656	−	0.340438i
$732$		0			0
$733$	−7.08785	−	12.2765i	−0.261796	−	0.453443i	0.704923	−	0.709283i	$-0.250981\pi$
$733$	−7.08785	−	12.2765i	−0.261796	−	0.453443i	−0.966719	+	0.255840i	$0.917648\pi$
$734$		0			0
$735$	−2.09709	+	0.851050i	−0.0773524	+	0.0313914i
$736$		0			0
$737$	13.5000	+	23.3827i	0.497279	+	0.861312i
$738$		0			0
$739$	5.55309	−	9.61823i	0.204274	−	0.353812i	−0.745627	−	0.666363i	$-0.767850\pi$
$739$	5.55309	−	9.61823i	0.204274	−	0.353812i	0.949901	+	0.312551i	$0.101183\pi$
$740$		0			0
$741$	4.51535			0.165875
$742$		0			0
$743$		−	3.47093i		−	0.127336i	−0.997971	−	0.0636680i	$-0.979720\pi$
$743$		−	3.47093i		−	0.127336i	0.997971	−	0.0636680i	$-0.0202799\pi$
$744$		0			0
$745$	−6.19238	−	3.57517i	−0.226871	−	0.130984i
$746$		0			0
$747$	38.5685	−	22.2676i	1.41115	−	0.814727i
$748$		0			0
$749$	−1.83768	+	26.5749i	−0.0671472	+	0.971026i
$750$		0			0
$751$	−8.86332	+	5.11724i	−0.323427	+	0.186731i	−0.652919	−	0.757428i	$-0.726456\pi$
$751$	−8.86332	+	5.11724i	−0.323427	+	0.186731i	0.329492	+	0.944158i	$0.393123\pi$
$752$		0			0
$753$	−0.625406	+	1.08324i	−0.0227911	+	0.0394753i
$754$		0			0
$755$		−	27.9730i		−	1.01804i
$756$		0			0
$757$			24.6300i			0.895191i	0.894236	+	0.447596i	$0.147720\pi$
$757$			24.6300i			0.895191i	−0.894236	+	0.447596i	$0.852280\pi$
$758$		0			0
$759$	−0.316757	+	0.548639i	−0.0114975	+	0.0199143i
$760$		0			0
$761$	−24.2991	+	14.0291i	−0.880843	+	0.508555i	−0.870936	−	0.491396i	$-0.836487\pi$
$761$	−24.2991	+	14.0291i	−0.880843	+	0.508555i	−0.00990664	+	0.999951i	$0.503153\pi$
$762$		0			0
$763$	11.9829	+	24.5206i	0.433811	+	0.887704i
$764$		0			0
$765$	−10.4334	+	6.02371i	−0.377219	+	0.217788i
$766$		0			0
$767$	−41.6309	−	24.0356i	−1.50320	−	0.867875i
$768$		0			0
$769$			23.9496i			0.863646i	0.901958	+	0.431823i	$0.142129\pi$
$769$			23.9496i			0.863646i	−0.901958	+	0.431823i	$0.857871\pi$
$770$		0			0
$771$	−3.87023			−0.139383
$772$		0			0
$773$	−15.9715	+	27.6634i	−0.574453	+	0.994982i	0.421647	+	0.906760i	$0.361452\pi$
$773$	−15.9715	+	27.6634i	−0.574453	+	0.994982i	−0.996101	+	0.0882225i	$0.971881\pi$
$774$		0			0
$775$	−1.70718	−	2.95692i	−0.0613236	−	0.106216i
$776$		0			0
$777$	0.801105	+	0.539453i	0.0287395	+	0.0193528i
$778$		0			0
$779$	−16.9760	−	29.4033i	−0.608229	−	1.05348i
$780$		0			0
$781$	−24.7034	−	14.2625i	−0.883958	−	0.510354i
$782$		0			0
$783$	9.85718			0.352267
$784$		0			0
$785$	−1.39779			−0.0498892
$786$		0			0
$787$	19.9044	+	11.4918i	0.709517	+	0.409640i	0.810882	−	0.585210i	$-0.198988\pi$
$787$	19.9044	+	11.4918i	0.709517	+	0.409640i	−0.101365	+	0.994849i	$0.532321\pi$
$788$		0			0
$789$	−1.96343	−	3.40076i	−0.0699000	−	0.121070i
$790$		0			0
$791$	32.6261	+	21.9699i	1.16005	+	0.781161i
$792$		0			0
$793$	−31.9912	−	55.4103i	−1.13604	−	1.96768i
$794$		0			0
$795$	−0.161976	+	0.280550i	−0.00574469	+	0.00995009i
$796$		0			0
$797$	33.7285			1.19472			0.597362	−	0.801972i	$-0.296215\pi$
$797$	33.7285			1.19472			0.597362	+	0.801972i	$0.296215\pi$
$798$		0			0
$799$			23.7766i			0.841155i
$800$		0			0
$801$	13.3001	+	7.67883i	0.469937	+	0.271318i
$802$		0			0
$803$	−46.2794	+	26.7194i	−1.63317	+	0.942909i
$804$		0			0
$805$	−1.12674	−	2.30564i	−0.0397124	−	0.0812632i
$806$		0			0
$807$	−0.0124370	+	0.00718049i	−0.000437802	+	0.000252765i
$808$		0			0
$809$	0.867732	−	1.50296i	0.0305078	−	0.0528411i	−0.850368	−	0.526188i	$-0.823621\pi$
$809$	0.867732	−	1.50296i	0.0305078	−	0.0528411i	0.880876	+	0.473347i	$0.156954\pi$
$810$		0			0
$811$		−	41.5845i		−	1.46023i	−0.683324	−	0.730115i	$-0.739466\pi$
$811$		−	41.5845i		−	1.46023i	0.683324	−	0.730115i	$-0.260534\pi$
$812$		0			0
$813$		−	3.96338i		−	0.139002i
$814$		0			0
$815$	−1.69907	+	2.94287i	−0.0595158	+	0.103084i
$816$		0			0
$817$	−25.9399	+	14.9764i	−0.907522	+	0.523958i
$818$		0			0
$819$	−2.67330	+	38.6590i	−0.0934127	+	1.35086i
$820$		0			0
$821$	8.59570	−	4.96273i	0.299992	−	0.173200i	−0.342447	−	0.939537i	$-0.611256\pi$
$821$	8.59570	−	4.96273i	0.299992	−	0.173200i	0.642439	+	0.766337i	$0.277922\pi$
$822$		0			0
$823$	−6.25165	−	3.60939i	−0.217919	−	0.125815i	0.387067	−	0.922051i	$-0.373488\pi$
$823$	−6.25165	−	3.60939i	−0.217919	−	0.125815i	−0.604986	+	0.796236i	$0.706821\pi$
$824$		0			0
$825$			2.65185i			0.0923254i
$826$		0			0
$827$	5.07813			0.176584			0.0882919	−	0.996095i	$-0.471859\pi$
$827$	5.07813			0.176584			0.0882919	+	0.996095i	$0.471859\pi$
$828$		0			0
$829$	14.0222	−	24.2872i	0.487011	−	0.843529i	−0.512877	−	0.858462i	$-0.671420\pi$
$829$	14.0222	−	24.2872i	0.487011	−	0.843529i	0.999888	+	0.0149335i	$0.00475366\pi$
$830$		0			0
$831$	−0.300543	−	0.520555i	−0.0104257	−	0.0180579i
$832$		0			0
$833$	−2.56011	+	18.4226i	−0.0887027	+	0.638305i
$834$		0			0
$835$	13.9950	+	24.2400i	0.484316	+	0.838860i
$836$		0			0
$837$	1.40233	+	0.809635i	0.0484716	+	0.0279851i
$838$		0			0
$839$	−36.4911			−1.25981			−0.629906	−	0.776671i	$-0.716907\pi$
$839$	−36.4911			−1.25981			−0.629906	+	0.776671i	$0.716907\pi$
$840$		0			0
$841$	−32.6734			−1.12667
$842$		0			0
$843$	0.0616325	+	0.0355835i	0.00212274	+	0.00122556i
$844$		0			0
$845$	−8.86523	−	15.3550i	−0.304973	−	0.528229i
$846$		0			0
$847$	2.11763	−	30.6234i	0.0727628	−	1.05223i
$848$		0			0
$849$	−0.647615	−	1.12170i	−0.0222261	−	0.0384967i
$850$		0			0
$851$	−0.547560	+	0.948401i	−0.0187701	+	0.0325108i
$852$		0			0
$853$	14.4265			0.493954			0.246977	−	0.969021i	$-0.420563\pi$
$853$	14.4265			0.493954			0.246977	+	0.969021i	$0.420563\pi$
$854$		0			0
$855$			19.6022i			0.670381i
$856$		0			0
$857$	−7.20541	−	4.16004i	−0.246132	−	0.142104i	0.371860	−	0.928289i	$-0.378720\pi$
$857$	−7.20541	−	4.16004i	−0.246132	−	0.142104i	−0.617992	+	0.786184i	$0.712054\pi$
$858$		0			0
$859$	−9.52512	+	5.49933i	−0.324993	+	0.187635i	−0.653616	−	0.756826i	$-0.726749\pi$
$859$	−9.52512	+	5.49933i	−0.324993	+	0.187635i	0.328623	+	0.944461i	$0.393415\pi$
$860$		0			0
$861$	−3.93436	+	1.92268i	−0.134083	+	0.0655247i
$862$		0			0
$863$	21.3322	−	12.3161i	0.726155	−	0.419246i	−0.0908586	−	0.995864i	$-0.528961\pi$
$863$	21.3322	−	12.3161i	0.726155	−	0.419246i	0.817014	+	0.576618i	$0.195628\pi$
$864$		0			0
$865$	−7.97779	+	13.8179i	−0.271253	+	0.469824i
$866$		0			0
$867$			2.09489i			0.0711462i
$868$		0			0
$869$			25.0968i			0.851351i
$870$		0			0
$871$	14.0719	−	24.3733i	0.476808	−	0.825856i
$872$		0			0
$873$	17.8387	−	10.2992i	0.603747	−	0.348574i
$874$		0			0
$875$	−25.7433	−	17.3352i	−0.870283	−	0.586036i
$876$		0			0
$877$	26.8432	−	15.4979i	0.906431	−	0.523328i	0.0271496	−	0.999631i	$-0.491357\pi$
$877$	26.8432	−	15.4979i	0.906431	−	0.523328i	0.879281	+	0.476303i	$0.158024\pi$
$878$		0			0
$879$	−0.0248740	−	0.0143610i	−0.000838978	−	0.000484384i
$880$		0			0
$881$			18.1328i			0.610908i	0.952207	+	0.305454i	$0.0988083\pi$
$881$			18.1328i			0.610908i	−0.952207	+	0.305454i	$0.901192\pi$
$882$		0			0
$883$	−22.8210			−0.767987			−0.383994	−	0.923336i	$-0.625452\pi$
$883$	−22.8210			−0.767987			−0.383994	+	0.923336i	$0.625452\pi$
$884$		0			0
$885$	−1.56814	+	2.71610i	−0.0527125	+	0.0913007i
$886$		0			0
$887$	−24.0509	−	41.6574i	−0.807550	−	1.39872i	−0.914556	−	0.404459i	$-0.867460\pi$
$887$	−24.0509	−	41.6574i	−0.807550	−	1.39872i	0.107006	−	0.994258i	$-0.465873\pi$
$888$		0			0
$889$	25.4245	−	37.7563i	0.852712	−	1.26631i
$890$		0			0
$891$	20.4482	+	35.4174i	0.685041	+	1.18653i
$892$		0			0
$893$	33.5036	+	19.3433i	1.12115	+	0.647298i
$894$		0			0
$895$	−20.1616			−0.673927
$896$		0			0
$897$	0.660352			0.0220485
$898$		0			0
$899$	−8.77395	−	5.06564i	−0.292628	−	0.168949i
$900$		0			0
$901$	1.33116	+	2.30564i	0.0443474	+	0.0768120i
$902$		0			0
$903$	1.69620	+	3.47093i	0.0564462	+	0.115505i
$904$		0			0
$905$	18.2946	+	31.6872i	0.608133	+	1.05332i
$906$		0			0
$907$	−26.4514	+	45.8151i	−0.878304	+	1.52127i	−0.0251028	+	0.999685i	$0.507991\pi$
$907$	−26.4514	+	45.8151i	−0.878304	+	1.52127i	−0.853201	+	0.521582i	$0.825342\pi$
$908$		0			0
$909$	13.1994			0.437797
$910$		0			0
$911$			10.0601i			0.333306i	0.986016	+	0.166653i	$0.0532960\pi$
$911$			10.0601i			0.333306i	−0.986016	+	0.166653i	$0.946704\pi$
$912$		0			0
$913$	62.0391	+	35.8183i	2.05320	+	1.18541i
$914$		0			0
$915$	−3.61511	+	2.08718i	−0.119512	+	0.0690001i
$916$		0			0
$917$	33.7019	+	2.33051i	1.11294	+	0.0769604i
$918$		0			0
$919$	−10.5705	+	6.10288i	−0.348688	+	0.201315i	−0.664107	−	0.747637i	$-0.731188\pi$
$919$	−10.5705	+	6.10288i	−0.348688	+	0.201315i	0.315419	+	0.948953i	$0.397855\pi$
$920$		0			0
$921$	−0.353371	+	0.612057i	−0.0116440	+	0.0201680i
$922$		0			0
$923$			29.7335i			0.978690i
$924$		0			0
$925$			4.58410i			0.150724i
$926$		0			0
$927$	−8.39269	+	14.5366i	−0.275652	+	0.477444i
$928$		0			0
$929$	7.00099	−	4.04202i	0.229695	−	0.132614i	−0.380736	−	0.924684i	$-0.624329\pi$
$929$	7.00099	−	4.04202i	0.229695	−	0.132614i	0.610431	+	0.792069i	$0.290996\pi$
$930$		0			0
$931$	23.8765	+	18.5950i	0.782520	+	0.609427i
$932$		0			0
$933$	−3.06663	+	1.77052i	−0.100397	+	0.0579642i
$934$		0			0
$935$	−16.7825	−	9.68939i	−0.548847	−	0.316877i
$936$		0			0
$937$		−	28.1849i		−	0.920761i	−0.887722	−	0.460380i	$-0.847713\pi$
$937$		−	28.1849i		−	0.920761i	0.887722	−	0.460380i	$-0.152287\pi$
$938$		0			0
$939$	1.67133			0.0545418
$940$		0			0
$941$	−20.6062	+	35.6910i	−0.671744	+	1.16349i	0.305666	+	0.952139i	$0.401121\pi$
$941$	−20.6062	+	35.6910i	−0.671744	+	1.16349i	−0.977409	+	0.211355i	$0.932212\pi$
$942$		0			0
$943$	−2.48267	−	4.30012i	−0.0808470	−	0.140031i
$944$		0			0
$945$	5.08232	+	0.351447i	0.165328	+	0.0114326i
$946$		0			0
$947$	−15.6009	−	27.0215i	−0.506961	−	0.878082i	−0.999968	−	0.00805654i	$-0.997435\pi$
$947$	−15.6009	−	27.0215i	−0.506961	−	0.878082i	0.493007	−	0.870026i	$-0.335898\pi$
$948$		0			0
$949$	48.2400	+	27.8514i	1.56594	+	0.904094i
$950$		0			0
$951$	−5.28930			−0.171517
$952$		0			0
$953$	−42.7465			−1.38470			−0.692348	−	0.721564i	$-0.743424\pi$
$953$	−42.7465			−1.38470			−0.692348	+	0.721564i	$0.743424\pi$
$954$		0			0
$955$	−17.2311	−	9.94840i	−0.557586	−	0.321923i
$956$		0			0
$957$	3.93436	+	6.81451i	0.127180	+	0.220282i
$958$		0			0
$959$	−10.9399	+	5.34619i	−0.353267	+	0.172638i
$960$		0			0
$961$	14.6679	+	25.4055i	0.473156	+	0.819531i
$962$		0			0
$963$	14.8789	−	25.7710i	0.479467	−	0.830460i
$964$		0			0
$965$	0.0681404			0.00219352
$966$		0			0
$967$			34.9379i			1.12353i	0.827298	+	0.561764i	$0.189877\pi$
$967$			34.9379i			1.12353i	−0.827298	+	0.561764i	$0.810123\pi$
$968$		0			0
$969$	−2.09668	−	1.21052i	−0.0673552	−	0.0388875i
$970$		0			0
$971$	−51.4192	+	29.6869i	−1.65012	+	0.952698i	−0.673101	+	0.739551i	$0.735038\pi$
$971$	−51.4192	+	29.6869i	−1.65012	+	0.952698i	−0.977020	+	0.213147i	$0.931629\pi$
$972$		0			0
$973$	−23.5134	+	34.9181i	−0.753804	+	1.11942i
$974$		0			0
$975$	2.39386	−	1.38209i	0.0766647	−	0.0442624i
$976$		0			0
$977$	1.33116	−	2.30564i	0.0425877	−	0.0737640i	−0.843946	−	0.536428i	$-0.819773\pi$
$977$	1.33116	−	2.30564i	0.0425877	−	0.0737640i	0.886534	+	0.462664i	$0.153106\pi$
$978$		0			0
$979$			24.7034i			0.789525i
$980$		0			0
$981$		−	30.4879i		−	0.973404i
$982$		0			0
$983$	10.8835	−	18.8507i	0.347129	−	0.601245i	−0.638609	−	0.769531i	$-0.720490\pi$
$983$	10.8835	−	18.8507i	0.347129	−	0.601245i	0.985738	+	0.168286i	$0.0538233\pi$
$984$		0			0
$985$	−35.6979	+	20.6102i	−1.13743	+	0.656695i
$986$		0			0
$987$	2.78699	−	4.13878i	0.0887110	−	0.131739i
$988$		0			0
$989$	−3.79361	+	2.19024i	−0.120630	+	0.0696456i
$990$		0			0
$991$	25.0485	+	14.4618i	0.795693	+	0.459393i	0.841963	−	0.539536i	$-0.181400\pi$
$991$	25.0485	+	14.4618i	0.795693	+	0.459393i	−0.0462700	+	0.998929i	$0.514733\pi$
$992$		0			0
$993$			1.43297i			0.0454740i
$994$		0			0
$995$	−29.6699			−0.940600
$996$		0			0
$997$	3.32233	−	5.75444i	0.105219	−	0.182245i	−0.808609	−	0.588347i	$-0.799779\pi$
$997$	3.32233	−	5.75444i	0.105219	−	0.182245i	0.913828	+	0.406102i	$0.133112\pi$
$998$		0			0
$999$	−1.08701	−	1.88276i	−0.0343916	−	0.0595679i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.q.c.31.3	yes	12
4.3	odd	2	inner	448.2.q.c.31.4	yes	12
7.3	odd	6		3136.2.e.e.1567.5		12
7.4	even	3		3136.2.e.d.1567.7		12
7.5	odd	6		448.2.q.b.159.3	yes	12
8.3	odd	2		448.2.q.b.31.3	✓	12
8.5	even	2		448.2.q.b.31.4	yes	12
28.3	even	6		3136.2.e.e.1567.8		12
28.11	odd	6		3136.2.e.d.1567.6		12
28.19	even	6		448.2.q.b.159.4	yes	12
56.3	even	6		3136.2.e.d.1567.5		12
56.5	odd	6	inner	448.2.q.c.159.4	yes	12
56.11	odd	6		3136.2.e.e.1567.7		12
56.19	even	6	inner	448.2.q.c.159.3	yes	12
56.45	odd	6		3136.2.e.d.1567.8		12
56.53	even	6		3136.2.e.e.1567.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.q.b.31.3	✓	12	8.3	odd	2
448.2.q.b.31.4	yes	12	8.5	even	2
448.2.q.b.159.3	yes	12	7.5	odd	6
448.2.q.b.159.4	yes	12	28.19	even	6
448.2.q.c.31.3	yes	12	1.1	even	1	trivial
448.2.q.c.31.4	yes	12	4.3	odd	2	inner
448.2.q.c.159.3	yes	12	56.19	even	6	inner
448.2.q.c.159.4	yes	12	56.5	odd	6	inner
3136.2.e.d.1567.5		12	56.3	even	6
3136.2.e.d.1567.6		12	28.11	odd	6
3136.2.e.d.1567.7		12	7.4	even	3
3136.2.e.d.1567.8		12	56.45	odd	6
3136.2.e.e.1567.5		12	7.3	odd	6
3136.2.e.e.1567.6		12	56.53	even	6
3136.2.e.e.1567.7		12	56.11	odd	6
3136.2.e.e.1567.8		12	28.3	even	6

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.182520 - 0.105378i) q^{3} +(-0.767035 - 1.32854i) q^{5} +(2.19457 + 1.47779i) q^{7} +(-1.47779 - 2.55961i) q^{9} +O(q^{10})\)	\(q+(-0.182520 - 0.105378i) q^{3} +(-0.767035 - 1.32854i) q^{5} +(2.19457 + 1.47779i) q^{7} +(-1.47779 - 2.55961i) q^{9} +(2.37709 - 4.11724i) q^{11} -4.95558 q^{13} +0.323314i q^{15} +(2.30111 + 1.32854i) q^{17} +(3.74410 - 2.16166i) q^{19} +(-0.244826 - 0.500986i) q^{21} +(0.547560 - 0.316134i) q^{23} +(1.32331 - 2.29205i) q^{25} +1.25517i q^{27} -7.85324i q^{29} +(0.645038 - 1.11724i) q^{31} +(-0.867732 + 0.500986i) q^{33} +(0.279998 - 4.04910i) q^{35} +(-1.50000 + 0.866025i) q^{37} +(0.904493 + 0.522209i) q^{39} -7.85324i q^{41} -6.92820 q^{43} +(-2.26704 + 3.92662i) q^{45} +(4.47418 + 7.74951i) q^{47} +(2.63227 + 6.48623i) q^{49} +(-0.279998 - 0.484971i) q^{51} +(0.867732 + 0.500986i) q^{53} -7.29324 q^{55} -0.911164 q^{57} +(8.40080 + 4.85020i) q^{59} +(6.45558 + 11.1814i) q^{61} +(0.539453 - 7.80111i) q^{63} +(3.80111 + 6.58371i) q^{65} +(-2.83961 + 4.91834i) q^{67} -0.133254 q^{69} -6.00000i q^{71} +(-9.73448 - 5.62020i) q^{73} +(-0.483063 + 0.278896i) q^{75} +(11.3011 - 5.52273i) q^{77} +(-4.57166 + 2.63945i) q^{79} +(-4.30111 + 7.44973i) q^{81} +15.0681i q^{83} -4.07616i q^{85} +(-0.827558 + 1.43337i) q^{87} +(-4.50000 + 2.59808i) q^{89} +(-10.8754 - 7.32331i) q^{91} +(-0.235465 + 0.135946i) q^{93} +(-5.74371 - 3.31613i) q^{95} +6.96929i q^{97} -14.0514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{5} + 4 q^{9}+O(q^{10}) \) 12 * q + 6 * q^5 + 4 * q^9	\( 12 q + 6 q^{5} + 4 q^{9} - 16 q^{13} - 18 q^{17} + 34 q^{21} - 8 q^{25} - 30 q^{33} - 18 q^{37} - 12 q^{45} + 12 q^{49} + 30 q^{53} + 76 q^{57} + 34 q^{61} - 132 q^{69} - 6 q^{73} + 90 q^{77} - 6 q^{81} - 54 q^{89} - 42 q^{93}+O(q^{100}) \) 12 * q + 6 * q^5 + 4 * q^9 - 16 * q^13 - 18 * q^17 + 34 * q^21 - 8 * q^25 - 30 * q^33 - 18 * q^37 - 12 * q^45 + 12 * q^49 + 30 * q^53 + 76 * q^57 + 34 * q^61 - 132 * q^69 - 6 * q^73 + 90 * q^77 - 6 * q^81 - 54 * q^89 - 42 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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