LMFDB - Embedded newform 420.2.bb.a.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(109,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))

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

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

chi := DirichletCharacter("420.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.bb (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:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 20 x^{13} - 12 x^{12} + 124 x^{11} - 24 x^{10} + 328 x^{9} + 1132 x^{8} + \cdots + 16 $ x^16 - 2x^15 + 2x^14 - 20x^13 - 12x^12 + 124x^11 - 24x^10 + 328x^9 + 1132x^8 - 760x^7 + 96x^6 - 640x^5 + 96x^4 + 224x^3 + 32x^2 + 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			289.2
Root			$0.786650 - 0.210782i$ of defining polynomial
Character	$\chi$	$=$	420.289
Dual form			420.2.bb.a.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{3} +(-1.58324 - 1.57904i) q^{5} +(0.418148 + 2.61250i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{3} +(-1.58324 - 1.57904i) q^{5} +(0.418148 + 2.61250i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.292387 + 0.506430i) q^{11} +1.75198i q^{13} +(2.16064 + 0.575868i) q^{15} +(-5.39101 + 3.11250i) q^{17} +(-3.48075 + 6.02884i) q^{19} +(-1.66838 - 2.05342i) q^{21} +(2.38328 + 1.37599i) q^{23} +(0.0132825 + 4.99998i) q^{25} +1.00000i q^{27} -3.24047 q^{29} +(1.25198 + 2.16849i) q^{31} +(-0.506430 - 0.292387i) q^{33} +(3.46321 - 4.79648i) q^{35} +(-3.36732 - 1.94412i) q^{37} +(-0.875989 - 1.51726i) q^{39} +4.58477 q^{41} +0.754325i q^{43} +(-2.15911 + 0.581605i) q^{45} +(1.26025 + 0.727603i) q^{47} +(-6.65030 + 2.18482i) q^{49} +(3.11250 - 5.39101i) q^{51} +(-0.456378 + 0.263490i) q^{53} +(0.336753 - 1.26349i) q^{55} -6.96151i q^{57} +(-5.05327 - 8.75253i) q^{59} +(-4.07704 + 7.06164i) q^{61} +(2.47156 + 0.944123i) q^{63} +(2.76644 - 2.77380i) q^{65} +(12.7289 - 7.34901i) q^{67} -2.75198 q^{69} +13.3561 q^{71} +(5.84942 - 3.37716i) q^{73} +(-2.51149 - 4.32347i) q^{75} +(-1.20079 + 0.975625i) q^{77} +(-4.04771 + 7.01085i) q^{79} +(-0.500000 - 0.866025i) q^{81} -5.72895i q^{83} +(13.4500 + 3.58477i) q^{85} +(2.80633 - 1.62023i) q^{87} +(-8.50192 + 14.7258i) q^{89} +(-4.57704 + 0.732586i) q^{91} +(-2.16849 - 1.25198i) q^{93} +(15.0306 - 4.04885i) q^{95} -11.1132i q^{97} +0.584775 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{5} + 8 q^{9}+O(q^{10}) $ 16 * q + 2 * q^5 + 8 * q^9	$ 16 q + 2 q^{5} + 8 q^{9} - 8 q^{11} + 4 q^{15} + 8 q^{19} - 4 q^{21} + 12 q^{25} + 24 q^{29} + 10 q^{35} - 4 q^{39} + 48 q^{41} - 2 q^{45} + 8 q^{49} + 4 q^{51} - 40 q^{55} - 28 q^{59} - 32 q^{61} - 26 q^{65} - 24 q^{69} - 56 q^{71} - 8 q^{75} - 16 q^{79} - 8 q^{81} + 32 q^{85} - 16 q^{89} - 40 q^{91} + 22 q^{95} - 16 q^{99}+O(q^{100}) $ 16 * q + 2 * q^5 + 8 * q^9 - 8 * q^11 + 4 * q^15 + 8 * q^19 - 4 * q^21 + 12 * q^25 + 24 * q^29 + 10 * q^35 - 4 * q^39 + 48 * q^41 - 2 * q^45 + 8 * q^49 + 4 * q^51 - 40 * q^55 - 28 * q^59 - 32 * q^61 - 26 * q^65 - 24 * q^69 - 56 * q^71 - 8 * q^75 - 16 * q^79 - 8 * q^81 + 32 * q^85 - 16 * q^89 - 40 * q^91 + 22 * q^95 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i
$4$		0			0
$5$	−1.58324	−	1.57904i	−0.708045	−	0.706167i
$5$	−1.58324	−	1.57904i	−0.708045	−	0.706167i
$6$		0			0
$7$	0.418148	+	2.61250i	0.158045	+	0.987432i
$7$	0.418148	+	2.61250i	0.158045	+	0.987432i
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	0.292387	+	0.506430i	0.0881581	+	0.152694i	0.906733	−	0.421706i	$-0.138569\pi$
$11$	0.292387	+	0.506430i	0.0881581	+	0.152694i	−0.818574	+	0.574400i	$0.805235\pi$
$12$		0			0
$13$			1.75198i			0.485911i	0.970037	+	0.242956i	$0.0781169\pi$
$13$			1.75198i			0.485911i	−0.970037	+	0.242956i	$0.921883\pi$
$14$		0			0
$15$	2.16064	+	0.575868i	0.557876	+	0.148688i
$16$		0			0
$17$	−5.39101	+	3.11250i	−1.30751	+	0.754892i	−0.981680	−	0.190536i	$-0.938977\pi$
$17$	−5.39101	+	3.11250i	−1.30751	+	0.754892i	−0.325831	+	0.945428i	$0.605644\pi$
$18$		0			0
$19$	−3.48075	+	6.02884i	−0.798540	+	1.38311i	0.122027	+	0.992527i	$0.461060\pi$
$19$	−3.48075	+	6.02884i	−0.798540	+	1.38311i	−0.920567	+	0.390585i	$0.872273\pi$
$20$		0			0
$21$	−1.66838	−	2.05342i	−0.364070	−	0.448092i
$22$		0			0
$23$	2.38328	+	1.37599i	0.496949	+	0.286914i	0.727453	−	0.686158i	$-0.240704\pi$
$23$	2.38328	+	1.37599i	0.496949	+	0.286914i	−0.230504	+	0.973071i	$0.574037\pi$
$24$		0			0
$25$	0.0132825	+	4.99998i	0.00265650	+	0.999996i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−3.24047			−0.601739			−0.300870	−	0.953665i	$-0.597277\pi$
$29$	−3.24047			−0.601739			−0.300870	+	0.953665i	$0.597277\pi$
$30$		0			0
$31$	1.25198	+	2.16849i	0.224862	+	0.389472i	0.956278	−	0.292459i	$-0.0944736\pi$
$31$	1.25198	+	2.16849i	0.224862	+	0.389472i	−0.731416	+	0.681931i	$0.761140\pi$
$32$		0			0
$33$	−0.506430	−	0.292387i	−0.0881581	−	0.0508981i
$34$		0			0
$35$	3.46321	−	4.79648i	0.585389	−	0.810753i
$36$		0			0
$37$	−3.36732	−	1.94412i	−0.553584	−	0.319612i	0.196982	−	0.980407i	$-0.436886\pi$
$37$	−3.36732	−	1.94412i	−0.553584	−	0.319612i	−0.750566	+	0.660795i	$0.770219\pi$
$38$		0			0
$39$	−0.875989	−	1.51726i	−0.140271	−	0.242956i
$40$		0			0
$41$	4.58477			0.716022			0.358011	−	0.933717i	$-0.383455\pi$
$41$	4.58477			0.716022			0.358011	+	0.933717i	$0.383455\pi$
$42$		0			0
$43$			0.754325i			0.115033i	0.998345	+	0.0575167i	$0.0183183\pi$
$43$			0.754325i			0.115033i	−0.998345	+	0.0575167i	$0.981682\pi$
$44$		0			0
$45$	−2.15911	+	0.581605i	−0.321860	+	0.0867006i
$46$		0			0
$47$	1.26025	+	0.727603i	0.183826	+	0.106132i	0.589089	−	0.808068i	$-0.299487\pi$
$47$	1.26025	+	0.727603i	0.183826	+	0.106132i	−0.405263	+	0.914200i	$0.632820\pi$
$48$		0			0
$49$	−6.65030	+	2.18482i	−0.950043	+	0.312118i
$50$		0			0
$51$	3.11250	−	5.39101i	0.435837	−	0.754892i
$52$		0			0
$53$	−0.456378	+	0.263490i	−0.0626883	+	0.0361931i	−0.531017	−	0.847361i	$-0.678190\pi$
$53$	−0.456378	+	0.263490i	−0.0626883	+	0.0361931i	0.468328	+	0.883555i	$0.344856\pi$
$54$		0			0
$55$	0.336753	−	1.26349i	0.0454078	−	0.170369i
$56$		0			0
$57$		−	6.96151i		−	0.922074i
$58$		0			0
$59$	−5.05327	−	8.75253i	−0.657880	−	1.13948i	−0.981163	−	0.193180i	$-0.938120\pi$
$59$	−5.05327	−	8.75253i	−0.657880	−	1.13948i	0.323283	−	0.946302i	$-0.395213\pi$
$60$		0			0
$61$	−4.07704	+	7.06164i	−0.522012	+	0.904151i	0.477660	+	0.878545i	$0.341485\pi$
$61$	−4.07704	+	7.06164i	−0.522012	+	0.904151i	−0.999672	+	0.0256061i	$0.991848\pi$
$62$		0			0
$63$	2.47156	+	0.944123i	0.311388	+	0.118948i
$64$		0			0
$65$	2.76644	−	2.77380i	0.343134	−	0.344047i
$66$		0			0
$67$	12.7289	−	7.34901i	1.55508	−	0.897824i	0.557362	−	0.830270i	$-0.311814\pi$
$67$	12.7289	−	7.34901i	1.55508	−	0.897824i	0.997716	−	0.0675547i	$-0.0215197\pi$
$68$		0			0
$69$	−2.75198			−0.331299
$70$		0			0
$71$	13.3561			1.58507			0.792536	−	0.609825i	$-0.208760\pi$
$71$	13.3561			1.58507			0.792536	+	0.609825i	$0.208760\pi$
$72$		0			0
$73$	5.84942	−	3.37716i	0.684622	−	0.395267i	−0.116972	−	0.993135i	$-0.537319\pi$
$73$	5.84942	−	3.37716i	0.684622	−	0.395267i	0.801594	+	0.597868i	$0.203985\pi$
$74$		0			0
$75$	−2.51149	−	4.32347i	−0.290002	−	0.499231i
$76$		0			0
$77$	−1.20079	+	0.975625i	−0.136842	+	0.111183i
$78$		0			0
$79$	−4.04771	+	7.01085i	−0.455403	+	0.788782i	−0.998711	−	0.0507516i	$-0.983838\pi$
$79$	−4.04771	+	7.01085i	−0.455403	+	0.788782i	0.543308	+	0.839534i	$0.317172\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	5.72895i		−	0.628834i	−0.949285	−	0.314417i	$-0.898191\pi$
$83$		−	5.72895i		−	0.628834i	0.949285	−	0.314417i	$-0.101809\pi$
$84$		0			0
$85$	13.4500	+	3.58477i	1.45886	+	0.388823i
$86$		0			0
$87$	2.80633	−	1.62023i	0.300870	−	0.173707i
$88$		0			0
$89$	−8.50192	+	14.7258i	−0.901202	+	1.56093i	−0.0752651	+	0.997164i	$0.523980\pi$
$89$	−8.50192	+	14.7258i	−0.901202	+	1.56093i	−0.825936	+	0.563763i	$0.809353\pi$
$90$		0			0
$91$	−4.57704	+	0.732586i	−0.479804	+	0.0767959i
$92$		0			0
$93$	−2.16849	−	1.25198i	−0.224862	−	0.129824i
$94$		0			0
$95$	15.0306	−	4.04885i	1.54211	−	0.415403i
$96$		0			0
$97$		−	11.1132i		−	1.12838i	−0.825645	−	0.564189i	$-0.809189\pi$
$97$		−	11.1132i		−	1.12838i	0.825645	−	0.564189i	$-0.190811\pi$
$98$		0			0
$99$	0.584775			0.0587721
$100$		0			0
$101$	−8.67803	−	15.0308i	−0.863496	−	1.49562i	−0.868533	−	0.495632i	$-0.834937\pi$
$101$	−8.67803	−	15.0308i	−0.863496	−	1.49562i	0.00503661	−	0.999987i	$-0.498397\pi$
$102$		0			0
$103$	11.5712	+	6.68063i	1.14014	+	0.658262i	0.946466	−	0.322803i	$-0.104625\pi$
$103$	11.5712	+	6.68063i	1.14014	+	0.658262i	0.193678	+	0.981065i	$0.437958\pi$
$104$		0			0
$105$	−0.600985	+	5.88548i	−0.0586501	+	0.574364i
$106$		0			0
$107$	−1.84223	−	1.06361i	−0.178095	−	0.102823i	0.408302	−	0.912847i	$-0.366121\pi$
$107$	−1.84223	−	1.06361i	−0.178095	−	0.102823i	−0.586397	+	0.810024i	$0.699454\pi$
$108$		0			0
$109$	1.14400	+	1.98147i	0.109575	+	0.189790i	0.915598	−	0.402094i	$-0.131718\pi$
$109$	1.14400	+	1.98147i	0.109575	+	0.189790i	−0.806023	+	0.591884i	$0.798384\pi$
$110$		0			0
$111$	3.88825			0.369056
$112$		0			0
$113$		−	11.7765i		−	1.10784i	−0.832570	−	0.553920i	$-0.813132\pi$
$113$		−	11.7765i		−	1.10784i	0.832570	−	0.553920i	$-0.186868\pi$
$114$		0			0
$115$	−1.60057	−	5.94181i	−0.149253	−	0.554077i
$116$		0			0
$117$	1.51726	+	0.875989i	0.140271	+	0.0809852i
$118$		0			0
$119$	−10.3856	−	12.7825i	−0.952050	−	1.17177i
$120$		0			0
$121$	5.32902	−	9.23013i	0.484456	−	0.839103i
$122$		0			0
$123$	−3.97053	+	2.29239i	−0.358011	+	0.206698i
$124$		0			0
$125$	7.87413	−	7.93713i	0.704284	−	0.709919i
$126$		0			0
$127$			15.2606i			1.35416i	0.735909	+	0.677081i	$0.236755\pi$
$127$			15.2606i			1.35416i	−0.735909	+	0.677081i	$0.763245\pi$
$128$		0			0
$129$	−0.377162	−	0.653265i	−0.0332073	−	0.0575167i
$130$		0			0
$131$	−4.00891	+	6.94363i	−0.350260	+	0.606668i	−0.986295	−	0.164992i	$-0.947240\pi$
$131$	−4.00891	+	6.94363i	−0.350260	+	0.606668i	0.636035	+	0.771660i	$0.280573\pi$
$132$		0			0
$133$	−17.2058	−	6.57252i	−1.49193	−	0.569910i
$134$		0			0
$135$	1.57904	−	1.58324i	0.135902	−	0.136263i
$136$		0			0
$137$	−12.7243	+	7.34641i	−1.08711	+	0.627646i	−0.932807	−	0.360377i	$-0.882648\pi$
$137$	−12.7243	+	7.34641i	−1.08711	+	0.627646i	−0.154308	+	0.988023i	$0.549315\pi$
$138$		0			0
$139$	−19.1080			−1.62072			−0.810361	−	0.585931i	$-0.800729\pi$
$139$	−19.1080			−1.62072			−0.810361	+	0.585931i	$0.800729\pi$
$140$		0			0
$141$	−1.45521			−0.122550
$142$		0			0
$143$	−0.887254	+	0.512256i	−0.0741959	+	0.0428370i
$144$		0			0
$145$	5.13043	+	5.11682i	0.426059	+	0.424929i
$146$		0			0
$147$	4.66692	−	5.21726i	0.384921	−	0.430313i
$148$		0			0
$149$	−0.912621	+	1.58071i	−0.0747648	+	0.129496i	−0.900984	−	0.433852i	$-0.857154\pi$
$149$	−0.912621	+	1.58071i	−0.0747648	+	0.129496i	0.826219	+	0.563349i	$0.190487\pi$
$150$		0			0
$151$	2.41379	+	4.18081i	0.196432	+	0.340230i	0.947369	−	0.320144i	$-0.103731\pi$
$151$	2.41379	+	4.18081i	0.196432	+	0.340230i	−0.750937	+	0.660374i	$0.770398\pi$
$152$		0			0
$153$			6.22500i			0.503261i
$154$		0			0
$155$	1.44195	−	5.41015i	0.115820	−	0.434554i
$156$		0			0
$157$	12.5449	−	7.24281i	1.00119	−	0.578039i	0.0925924	−	0.995704i	$-0.470485\pi$
$157$	12.5449	−	7.24281i	1.00119	−	0.578039i	0.908601	+	0.417665i	$0.137151\pi$
$158$		0			0
$159$	0.263490	−	0.456378i	0.0208961	−	0.0361931i
$160$		0			0
$161$	−2.59820	+	6.80169i	−0.204767	+	0.536048i
$162$		0			0
$163$	−10.0628	−	5.80977i	−0.788181	−	0.455057i	0.0511407	−	0.998691i	$-0.483714\pi$
$163$	−10.0628	−	5.80977i	−0.788181	−	0.455057i	−0.839322	+	0.543635i	$0.817048\pi$
$164$		0			0
$165$	0.340108	+	1.26259i	0.0264774	+	0.0982926i
$166$		0			0
$167$			21.2998i			1.64822i	0.566427	+	0.824112i	$0.308325\pi$
$167$			21.2998i			1.64822i	−0.566427	+	0.824112i	$0.691675\pi$
$168$		0			0
$169$	9.93057			0.763890
$170$		0			0
$171$	3.48075	+	6.02884i	0.266180	+	0.461037i
$172$		0			0
$173$	10.3769	+	5.99109i	0.788939	+	0.455494i	0.839589	−	0.543222i	$-0.182796\pi$
$173$	10.3769	+	5.99109i	0.788939	+	0.455494i	−0.0506497	+	0.998716i	$0.516129\pi$
$174$		0			0
$175$	−13.0569	+	2.12543i	−0.987009	+	0.160668i
$176$		0			0
$177$	8.75253	+	5.05327i	0.657880	+	0.379827i
$178$		0			0
$179$	10.2095	+	17.6834i	0.763096	+	1.32172i	0.941247	+	0.337718i	$0.109655\pi$
$179$	10.2095	+	17.6834i	0.763096	+	1.32172i	−0.178151	+	0.984003i	$0.557012\pi$
$180$		0			0
$181$	19.8276			1.47377			0.736887	−	0.676016i	$-0.236295\pi$
$181$	19.8276			1.47377			0.736887	+	0.676016i	$0.236295\pi$
$182$		0			0
$183$		−	8.15408i		−	0.602767i
$184$		0			0
$185$	2.26142	+	8.39513i	0.166263	+	0.617222i
$186$		0			0
$187$	−3.15253	−	1.82011i	−0.230536	−	0.133100i
$188$		0			0
$189$	−2.61250	+	0.418148i	−0.190031	+	0.0304158i
$190$		0			0
$191$	1.39976	−	2.42445i	0.101283	−	0.175427i	−0.810931	−	0.585142i	$-0.801039\pi$
$191$	1.39976	−	2.42445i	0.101283	−	0.175427i	0.912213	+	0.409715i	$0.134372\pi$
$192$		0			0
$193$	−18.2366	+	10.5289i	−1.31270	+	0.757887i	−0.982542	−	0.186040i	$-0.940435\pi$
$193$	−18.2366	+	10.5289i	−1.31270	+	0.757887i	−0.330156	+	0.943926i	$0.607101\pi$
$194$		0			0
$195$	−1.00891	+	3.78540i	−0.0722494	+	0.271078i
$196$		0			0
$197$			7.18416i			0.511850i	0.966697	+	0.255925i	$0.0823800\pi$
$197$			7.18416i			0.511850i	−0.966697	+	0.255925i	$0.917620\pi$
$198$		0			0
$199$	−5.54771	−	9.60892i	−0.393267	−	0.681159i	0.599611	−	0.800291i	$-0.295322\pi$
$199$	−5.54771	−	9.60892i	−0.393267	−	0.681159i	−0.992878	+	0.119133i	$0.961989\pi$
$200$		0			0
$201$	−7.34901	+	12.7289i	−0.518359	+	0.897824i
$202$		0			0
$203$	−1.35499	−	8.46572i	−0.0951020	−	0.594177i
$204$		0			0
$205$	−7.25879	−	7.23953i	−0.506976	−	0.505631i
$206$		0			0
$207$	2.38328	−	1.37599i	0.165650	−	0.0956378i
$208$		0			0
$209$	−4.07092			−0.281591
$210$		0			0
$211$	12.2340			0.842226			0.421113	−	0.907008i	$-0.361640\pi$
$211$	12.2340			0.842226			0.421113	+	0.907008i	$0.361640\pi$
$212$		0			0
$213$	−11.5667	+	6.67803i	−0.792536	+	0.457571i
$214$		0			0
$215$	1.19111	−	1.19428i	0.0812328	−	0.0814489i
$216$		0			0
$217$	−5.14166	+	4.17754i	−0.349039	+	0.283590i
$218$		0			0
$219$	−3.37716	+	5.84942i	−0.228207	+	0.395267i
$220$		0			0
$221$	−5.45303	−	9.44493i	−0.366810	−	0.635334i
$222$		0			0
$223$			11.8074i			0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$223$			11.8074i			0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$224$		0			0
$225$	4.33675	+	2.48849i	0.289117	+	0.165899i
$226$		0			0
$227$	−22.6871	+	13.0984i	−1.50579	+	0.869370i	−0.505816	+	0.862641i	$0.668809\pi$
$227$	−22.6871	+	13.0984i	−1.50579	+	0.869370i	−0.999977	+	0.00672884i	$0.997858\pi$
$228$		0			0
$229$	4.97067	−	8.60946i	0.328471	−	0.568929i	−0.653737	−	0.756721i	$-0.726800\pi$
$229$	4.97067	−	8.60946i	0.328471	−	0.568929i	0.982209	+	0.187792i	$0.0601332\pi$
$230$		0			0
$231$	0.552099	−	1.44531i	0.0363255	−	0.0950944i
$232$		0			0
$233$	13.3063	+	7.68241i	0.871727	+	0.503292i	0.867922	−	0.496701i	$-0.165455\pi$
$233$	13.3063	+	7.68241i	0.871727	+	0.503292i	0.00380503	+	0.999993i	$0.498789\pi$
$234$		0			0
$235$	−0.846355	−	3.14194i	−0.0552101	−	0.204958i
$236$		0			0
$237$		−	8.09543i		−	0.525855i
$238$		0			0
$239$	15.2183			0.984390			0.492195	−	0.870485i	$-0.336195\pi$
$239$	15.2183			0.984390			0.492195	+	0.870485i	$0.336195\pi$
$240$		0			0
$241$	−13.3730	−	23.1626i	−0.861428	−	1.49204i	−0.870551	−	0.492078i	$-0.836237\pi$
$241$	−13.3730	−	23.1626i	−0.861428	−	1.49204i	0.00912311	−	0.999958i	$-0.497096\pi$
$242$		0			0
$243$	0.866025	+	0.500000i	0.0555556	+	0.0320750i
$244$		0			0
$245$	13.9789	+	7.04198i	0.893081	+	0.449896i
$246$		0			0
$247$	−10.5624	−	6.09820i	−0.672069	−	0.388019i
$248$		0			0
$249$	2.86448	+	4.96142i	0.181529	+	0.314417i
$250$		0			0
$251$	8.05310			0.508307			0.254154	−	0.967164i	$-0.418203\pi$
$251$	8.05310			0.508307			0.254154	+	0.967164i	$0.418203\pi$
$252$		0			0
$253$			1.60929i			0.101175i
$254$		0			0
$255$	−13.4404	+	3.62049i	−0.841672	+	0.226724i
$256$		0			0
$257$	−13.8307	−	7.98514i	−0.862733	−	0.498099i	0.00219336	−	0.999998i	$-0.499302\pi$
$257$	−13.8307	−	7.98514i	−0.862733	−	0.498099i	−0.864927	+	0.501898i	$0.832635\pi$
$258$		0			0
$259$	3.67098	−	9.61005i	0.228104	−	0.597139i
$260$		0			0
$261$	−1.62023	+	2.80633i	−0.100290	+	0.173707i
$262$		0			0
$263$	−20.3437	+	11.7454i	−1.25444	+	0.724253i	−0.971989	−	0.235027i	$-0.924482\pi$
$263$	−20.3437	+	11.7454i	−1.25444	+	0.724253i	−0.282455	+	0.959281i	$0.591149\pi$
$264$		0			0
$265$	1.13861	+	0.303470i	0.0699445	+	0.0186420i
$266$		0			0
$267$		−	17.0038i		−	1.04062i
$268$		0			0
$269$	6.56831	+	11.3766i	0.400477	+	0.693647i	0.993783	−	0.111330i	$-0.0355111\pi$
$269$	6.56831	+	11.3766i	0.400477	+	0.693647i	−0.593306	+	0.804977i	$0.702178\pi$
$270$		0			0
$271$	−2.69275	+	4.66398i	−0.163573	+	0.283317i	−0.936148	−	0.351607i	$-0.885635\pi$
$271$	−2.69275	+	4.66398i	−0.163573	+	0.283317i	0.772575	+	0.634924i	$0.218969\pi$
$272$		0			0
$273$	3.59754	−	2.92296i	0.217733	−	0.176906i
$274$		0			0
$275$	−2.52826	+	1.46866i	−0.152460	+	0.0885635i
$276$		0			0
$277$	8.71716	−	5.03285i	0.523763	−	0.302395i	−0.214710	−	0.976678i	$-0.568881\pi$
$277$	8.71716	−	5.03285i	0.523763	−	0.302395i	0.738473	+	0.674283i	$0.235547\pi$
$278$		0			0
$279$	2.50396			0.149908
$280$		0			0
$281$	−26.8822			−1.60366			−0.801828	−	0.597555i	$-0.796139\pi$
$281$	−26.8822			−1.60366			−0.801828	+	0.597555i	$0.796139\pi$
$282$		0			0
$283$	10.2687	−	5.92865i	0.610413	−	0.352422i	−0.162714	−	0.986673i	$-0.552025\pi$
$283$	10.2687	−	5.92865i	0.610413	−	0.352422i	0.773127	+	0.634251i	$0.218692\pi$
$284$		0			0
$285$	−10.9925	+	11.0217i	−0.651138	+	0.652870i
$286$		0			0
$287$	1.91711	+	11.9777i	0.113164	+	0.707023i
$288$		0			0
$289$	10.8753	−	18.8366i	0.639724	−	1.10803i
$290$		0			0
$291$	5.55662	+	9.62435i	0.325735	+	0.564189i
$292$		0			0
$293$			16.3879i			0.957388i	0.877982	+	0.478694i	$0.158890\pi$
$293$			16.3879i			0.957388i	−0.877982	+	0.478694i	$0.841110\pi$
$294$		0			0
$295$	−5.82003	+	21.8366i	−0.338855	+	1.27138i
$296$		0			0
$297$	−0.506430	+	0.292387i	−0.0293860	+	0.0169660i
$298$		0			0
$299$	−2.41070	+	4.17546i	−0.139415	+	0.241473i
$300$		0			0
$301$	−1.97067	+	0.315420i	−0.113588	+	0.0181805i
$302$		0			0
$303$	15.0308	+	8.67803i	0.863496	+	0.498540i
$304$		0			0
$305$	17.6055	−	4.74246i	1.00809	−	0.271552i
$306$		0			0
$307$		−	20.0141i		−	1.14226i	−0.820858	−	0.571132i	$-0.806504\pi$
$307$		−	20.0141i		−	1.14226i	0.820858	−	0.571132i	$-0.193496\pi$
$308$		0			0
$309$	−13.3613			−0.760096
$310$		0			0
$311$	14.3183	+	24.7999i	0.811914	+	1.40628i	0.911523	+	0.411249i	$0.134907\pi$
$311$	14.3183	+	24.7999i	0.811914	+	1.40628i	−0.0996095	+	0.995027i	$0.531759\pi$
$312$		0			0
$313$	23.6956	+	13.6806i	1.33935	+	0.773275i	0.986711	−	0.162484i	$-0.0519505\pi$
$313$	23.6956	+	13.6806i	1.33935	+	0.773275i	0.352641	+	0.935759i	$0.385284\pi$
$314$		0			0
$315$	−2.42227	−	5.39746i	−0.136479	−	0.304113i
$316$		0			0
$317$	6.94258	+	4.00830i	0.389934	+	0.225129i	0.682132	−	0.731229i	$-0.261053\pi$
$317$	6.94258	+	4.00830i	0.389934	+	0.225129i	−0.292197	+	0.956358i	$0.594386\pi$
$318$		0			0
$319$	−0.947472	−	1.64107i	−0.0530482	−	0.0918822i
$320$		0			0
$321$	2.12722			0.118730
$322$		0			0
$323$		−	43.3354i		−	2.41125i
$324$		0			0
$325$	−8.75986	+	0.0232706i	−0.485910	+	0.00129082i
$326$		0			0
$327$	−1.98147	−	1.14400i	−0.109575	−	0.0632634i
$328$		0			0
$329$	−1.37389	+	3.59664i	−0.0757452	+	0.198289i
$330$		0			0
$331$	−4.98849	+	8.64032i	−0.274192	+	0.474915i	−0.969931	−	0.243380i	$-0.921744\pi$
$331$	−4.98849	+	8.64032i	−0.274192	+	0.474915i	0.695739	+	0.718295i	$0.255077\pi$
$332$		0			0
$333$	−3.36732	+	1.94412i	−0.184528	+	0.106537i
$334$		0			0
$335$	−31.7572	−	8.46411i	−1.73508	−	0.462444i
$336$		0			0
$337$			0.936640i			0.0510220i	0.999675	+	0.0255110i	$0.00812129\pi$
$337$			0.936640i			0.0510220i	−0.999675	+	0.0255110i	$0.991879\pi$
$338$		0			0
$339$	5.88825	+	10.1987i	0.319806	+	0.553920i
$340$		0			0
$341$	−0.732125	+	1.26808i	−0.0396468	+	0.0686703i
$342$		0			0
$343$	−8.48866	−	16.4603i	−0.458345	−	0.888775i
$344$		0			0
$345$	4.35703	+	4.34548i	0.234575	+	0.233953i
$346$		0			0
$347$	−0.153687	+	0.0887310i	−0.00825033	+	0.00476333i	−0.504119	−	0.863634i	$-0.668183\pi$
$347$	−0.153687	+	0.0887310i	−0.00825033	+	0.00476333i	0.495869	+	0.868397i	$0.334850\pi$
$348$		0			0
$349$	31.6891			1.69628			0.848139	−	0.529774i	$-0.177723\pi$
$349$	31.6891			1.69628			0.848139	+	0.529774i	$0.177723\pi$
$350$		0			0
$351$	−1.75198			−0.0935137
$352$		0			0
$353$	21.1870	−	12.2323i	1.12767	−	0.651060i	0.184321	−	0.982866i	$-0.440991\pi$
$353$	21.1870	−	12.2323i	1.12767	−	0.651060i	0.943348	+	0.331806i	$0.107658\pi$
$354$		0			0
$355$	−21.1458	−	21.0897i	−1.12230	−	1.11933i
$356$		0			0
$357$	15.3855	+	5.87716i	0.814286	+	0.311052i
$358$		0			0
$359$	8.01243	−	13.8779i	0.422880	−	0.732450i	−0.573340	−	0.819318i	$-0.694352\pi$
$359$	8.01243	−	13.8779i	0.422880	−	0.732450i	0.996220	+	0.0868679i	$0.0276858\pi$
$360$		0			0
$361$	−14.7313	−	25.5154i	−0.775332	−	1.34291i
$362$		0			0
$363$			10.6580i			0.559402i
$364$		0			0
$365$	−14.5937	−	3.88960i	−0.763868	−	0.203591i
$366$		0			0
$367$	13.2853	−	7.67029i	0.693489	−	0.400386i	−0.111429	−	0.993772i	$-0.535543\pi$
$367$	13.2853	−	7.67029i	0.693489	−	0.400386i	0.804918	+	0.593386i	$0.202209\pi$
$368$		0			0
$369$	2.29239	−	3.97053i	0.119337	−	0.206698i
$370$		0			0
$371$	−0.879200	−	1.08211i	−0.0456458	−	0.0561803i
$372$		0			0
$373$	−3.15327	−	1.82054i	−0.163270	−	0.0942640i	0.416139	−	0.909301i	$-0.363383\pi$
$373$	−3.15327	−	1.82054i	−0.163270	−	0.0942640i	−0.579409	+	0.815037i	$0.696716\pi$
$374$		0			0
$375$	−2.85063	+	10.8108i	−0.147206	+	0.558269i
$376$		0			0
$377$		−	5.67723i		−	0.292392i
$378$		0			0
$379$	−4.08959			−0.210068			−0.105034	−	0.994469i	$-0.533495\pi$
$379$	−4.08959			−0.210068			−0.105034	+	0.994469i	$0.533495\pi$
$380$		0			0
$381$	−7.63031	−	13.2161i	−0.390913	−	0.677081i
$382$		0			0
$383$	6.85895	+	3.96002i	0.350476	+	0.202348i	0.664895	−	0.746937i	$-0.268476\pi$
$383$	6.85895	+	3.96002i	0.350476	+	0.202348i	−0.314419	+	0.949284i	$0.601810\pi$
$384$		0			0
$385$	3.44168	+	0.351441i	0.175404	+	0.0179111i
$386$		0			0
$387$	0.653265	+	0.377162i	0.0332073	+	0.0191722i
$388$		0			0
$389$	−11.3028	−	19.5770i	−0.573074	−	0.992593i	−0.996248	−	0.0865447i	$-0.972417\pi$
$389$	−11.3028	−	19.5770i	−0.573074	−	0.992593i	0.423174	−	0.906048i	$-0.360916\pi$
$390$		0			0
$391$	−17.1311			−0.866355
$392$		0			0
$393$		−	8.01781i		−	0.404445i
$394$		0			0
$395$	17.4789	−	4.70834i	0.879458	−	0.236903i
$396$		0			0
$397$	19.3097	+	11.1484i	0.969125	+	0.559524i	0.898969	−	0.438012i	$-0.144317\pi$
$397$	19.3097	+	11.1484i	0.969125	+	0.559524i	0.0701554	+	0.997536i	$0.477650\pi$
$398$		0			0
$399$	18.1869	−	2.91094i	0.910486	−	0.145729i
$400$		0			0
$401$	3.52177	−	6.09989i	0.175869	−	0.304614i	−0.764593	−	0.644514i	$-0.777060\pi$
$401$	3.52177	−	6.09989i	0.175869	−	0.304614i	0.940462	+	0.339900i	$0.110393\pi$
$402$		0			0
$403$	−3.79915	+	2.19344i	−0.189249	+	0.109263i
$404$		0			0
$405$	−0.575868	+	2.16064i	−0.0286151	+	0.107363i
$406$		0			0
$407$		−	2.27375i		−	0.112706i
$408$		0			0
$409$	4.62957	+	8.01865i	0.228917	+	0.396497i	0.957488	−	0.288475i	$-0.0931481\pi$
$409$	4.62957	+	8.01865i	0.228917	+	0.396497i	−0.728570	+	0.684971i	$0.759815\pi$
$410$		0			0
$411$	7.34641	−	12.7243i	0.362371	−	0.627646i
$412$		0			0
$413$	20.7529	−	16.8615i	1.02119	−	0.829701i
$414$		0			0
$415$	−9.04623	+	9.07030i	−0.444062	+	0.445243i
$416$		0			0
$417$	16.5480	−	9.55402i	0.810361	−	0.467862i
$418$		0			0
$419$	11.9183			0.582248			0.291124	−	0.956685i	$-0.405971\pi$
$419$	11.9183			0.582248			0.291124	+	0.956685i	$0.405971\pi$
$420$		0			0
$421$	−5.68576			−0.277107			−0.138553	−	0.990355i	$-0.544245\pi$
$421$	−5.68576			−0.277107			−0.138553	+	0.990355i	$0.544245\pi$
$422$		0			0
$423$	1.26025	−	0.727603i	0.0612752	−	0.0353773i
$424$		0			0
$425$	−15.6340	−	26.9136i	−0.758363	−	1.30550i
$426$		0			0
$427$	−20.1533	−	7.69845i	−0.975289	−	0.372554i
$428$		0			0
$429$	0.512256	−	0.887254i	0.0247320	−	0.0428370i
$430$		0			0
$431$	6.85866	+	11.8796i	0.330370	+	0.572218i	0.982584	−	0.185817i	$-0.0594931\pi$
$431$	6.85866	+	11.8796i	0.330370	+	0.572218i	−0.652214	+	0.758035i	$0.726160\pi$
$432$		0			0
$433$		−	22.8294i		−	1.09711i	−0.836114	−	0.548556i	$-0.815178\pi$
$433$		−	22.8294i		−	1.09711i	0.836114	−	0.548556i	$-0.184822\pi$
$434$		0			0
$435$	−7.00149	−	1.86608i	−0.335696	−	0.0894717i
$436$		0			0
$437$	−16.5912	+	9.57896i	−0.793667	+	0.458224i
$438$		0			0
$439$	19.6966	−	34.1155i	0.940067	−	1.62824i	0.174727	−	0.984617i	$-0.444096\pi$
$439$	19.6966	−	34.1155i	0.940067	−	1.62824i	0.765340	−	0.643627i	$-0.222571\pi$
$440$		0			0
$441$	−1.43304	+	6.85174i	−0.0682400	+	0.326274i
$442$		0			0
$443$	4.47462	+	2.58342i	0.212596	+	0.122742i	0.602517	−	0.798106i	$-0.294165\pi$
$443$	4.47462	+	2.58342i	0.212596	+	0.122742i	−0.389921	+	0.920848i	$0.627498\pi$
$444$		0			0
$445$	36.7131	−	9.88952i	1.74037	−	0.468808i
$446$		0			0
$447$		−	1.82524i		−	0.0863309i
$448$		0			0
$449$	36.7384			1.73379			0.866895	−	0.498491i	$-0.166112\pi$
$449$	36.7384			1.73379			0.866895	+	0.498491i	$0.166112\pi$
$450$		0			0
$451$	1.34053	+	2.32187i	0.0631231	+	0.109332i
$452$		0			0
$453$	−4.18081	−	2.41379i	−0.196432	−	0.113410i
$454$		0			0
$455$	8.40332	+	6.06746i	0.393954	+	0.284447i
$456$		0			0
$457$	−1.03409	−	0.597031i	−0.0483726	−	0.0279280i	0.475619	−	0.879652i	$-0.342224\pi$
$457$	−1.03409	−	0.597031i	−0.0483726	−	0.0279280i	−0.523991	+	0.851724i	$0.675558\pi$
$458$		0			0
$459$	−3.11250	−	5.39101i	−0.145279	−	0.251631i
$460$		0			0
$461$	−14.5340			−0.676917			−0.338459	−	0.940981i	$-0.609906\pi$
$461$	−14.5340			−0.676917			−0.338459	+	0.940981i	$0.609906\pi$
$462$		0			0
$463$			32.2762i			1.50000i	0.661436	+	0.750001i	$0.269947\pi$
$463$			32.2762i			1.50000i	−0.661436	+	0.750001i	$0.730053\pi$
$464$		0			0
$465$	1.45631	+	5.40630i	0.0675350	+	0.250711i
$466$		0			0
$467$	−10.6551	−	6.15174i	−0.493060	−	0.284668i	0.232783	−	0.972529i	$-0.425217\pi$
$467$	−10.6551	−	6.15174i	−0.493060	−	0.284668i	−0.725843	+	0.687860i	$0.758550\pi$
$468$		0			0
$469$	24.5218	+	30.1812i	1.13231	+	1.39364i
$470$		0			0
$471$	−7.24281	+	12.5449i	−0.333731	+	0.578039i
$472$		0			0
$473$	−0.382013	+	0.220555i	−0.0175650	+	0.0101411i
$474$		0			0
$475$	−30.1903	−	17.3236i	−1.38523	−	0.794863i
$476$		0			0
$477$			0.526980i			0.0241287i
$478$		0			0
$479$	−11.5196	−	19.9525i	−0.526342	−	0.911651i	−0.999529	−	0.0306893i	$-0.990230\pi$
$479$	−11.5196	−	19.9525i	−0.526342	−	0.911651i	0.473187	−	0.880962i	$-0.343104\pi$
$480$		0			0
$481$	3.40606	−	5.89947i	0.155303	−	0.268993i
$482$		0			0
$483$	−1.15073	−	7.18954i	−0.0523602	−	0.327135i
$484$		0			0
$485$	−17.5482	+	17.5949i	−0.796824	+	0.798943i
$486$		0			0
$487$	15.2508	−	8.80507i	0.691081	−	0.398996i	−0.112936	−	0.993602i	$-0.536025\pi$
$487$	15.2508	−	8.80507i	0.691081	−	0.398996i	0.804017	+	0.594606i	$0.202692\pi$
$488$		0			0
$489$	11.6195			0.525454
$490$		0			0
$491$	−18.7403			−0.845740			−0.422870	−	0.906190i	$-0.638977\pi$
$491$	−18.7403			−0.845740			−0.422870	+	0.906190i	$0.638977\pi$
$492$		0			0
$493$	17.4694	−	10.0859i	0.786781	−	0.454248i
$494$		0			0
$495$	−0.925838	−	0.923382i	−0.0416133	−	0.0415029i
$496$		0			0
$497$	5.58481	+	34.8927i	0.250513	+	1.56515i
$498$		0			0
$499$	−14.0063	+	24.2596i	−0.627008	+	1.08601i	0.361141	+	0.932511i	$0.382387\pi$
$499$	−14.0063	+	24.2596i	−0.627008	+	1.08601i	−0.988149	+	0.153499i	$0.950946\pi$
$500$		0			0
$501$	−10.6499	−	18.4461i	−0.475801	−	0.824112i
$502$		0			0
$503$			16.7178i			0.745412i	0.927950	+	0.372706i	$0.121570\pi$
$503$			16.7178i			0.745412i	−0.927950	+	0.372706i	$0.878430\pi$
$504$		0			0
$505$	−9.99479	+	37.5002i	−0.444762	+	1.66874i
$506$		0			0
$507$	−8.60013	+	4.96529i	−0.381945	+	0.220516i
$508$		0			0
$509$	0.131059	−	0.227000i	0.00580907	−	0.0100616i	−0.863106	−	0.505022i	$-0.831484\pi$
$509$	0.131059	−	0.227000i	0.00580907	−	0.0100616i	0.868915	+	0.494961i	$0.164818\pi$
$510$		0			0
$511$	11.2688	+	13.8694i	0.498500	+	0.613548i
$512$		0			0
$513$	−6.02884	−	3.48075i	−0.266180	−	0.153679i
$514$		0			0
$515$	−7.77098	−	28.8484i	−0.342430	−	1.27121i
$516$		0			0
$517$			0.850968i			0.0374255i
$518$		0			0
$519$	−11.9822			−0.525960
$520$		0			0
$521$	19.1710	+	33.2052i	0.839898	+	1.45475i	0.889979	+	0.456002i	$0.150719\pi$
$521$	19.1710	+	33.2052i	0.839898	+	1.45475i	−0.0500804	+	0.998745i	$0.515948\pi$
$522$		0			0
$523$	3.11081	+	1.79603i	0.136026	+	0.0785348i	0.566469	−	0.824083i	$-0.308309\pi$
$523$	3.11081	+	1.79603i	0.136026	+	0.0785348i	−0.430443	+	0.902618i	$0.641642\pi$
$524$		0			0
$525$	10.2449	−	8.36913i	0.447124	−	0.365259i
$526$		0			0
$527$	−13.4988	−	7.79356i	−0.588019	−	0.339493i
$528$		0			0
$529$	−7.71331	−	13.3598i	−0.335361	−	0.580863i
$530$		0			0
$531$	−10.1065			−0.438587
$532$		0			0
$533$			8.03242i			0.347923i
$534$		0			0
$535$	1.23720	+	4.59290i	0.0534890	+	0.198568i
$536$		0			0
$537$	−17.6834	−	10.2095i	−0.763096	−	0.440574i
$538$		0			0
$539$	−3.05093	−	2.72910i	−0.131413	−	0.117551i
$540$		0			0
$541$	−8.48602	+	14.6982i	−0.364843	+	0.631926i	−0.988751	−	0.149572i	$-0.952211\pi$
$541$	−8.48602	+	14.6982i	−0.364843	+	0.631926i	0.623908	+	0.781498i	$0.285544\pi$
$542$		0			0
$543$	−17.1712	+	9.91379i	−0.736887	+	0.425442i
$544$		0			0
$545$	1.31759	−	4.94356i	0.0564392	−	0.211759i
$546$		0			0
$547$			29.0926i			1.24391i	0.783054	+	0.621954i	$0.213661\pi$
$547$			29.0926i			1.24391i	−0.783054	+	0.621954i	$0.786339\pi$
$548$		0			0
$549$	4.07704	+	7.06164i	0.174004	+	0.301384i
$550$		0			0
$551$	11.2793	−	19.5363i	0.480513	−	0.832273i
$552$		0			0
$553$	−20.0084	−	7.64308i	−0.850843	−	0.325017i
$554$		0			0
$555$	−6.15602	−	6.13968i	−0.261308	−	0.260615i
$556$		0			0
$557$	1.22185	−	0.705437i	0.0517716	−	0.0298903i	−0.473891	−	0.880584i	$-0.657151\pi$
$557$	1.22185	−	0.705437i	0.0517716	−	0.0298903i	0.525662	+	0.850693i	$0.323818\pi$
$558$		0			0
$559$	−1.32156			−0.0558961
$560$		0			0
$561$	3.64022			0.153690
$562$		0			0
$563$	−7.61158	+	4.39455i	−0.320790	+	0.185208i	−0.651745	−	0.758439i	$-0.725963\pi$
$563$	−7.61158	+	4.39455i	−0.320790	+	0.185208i	0.330955	+	0.943647i	$0.392629\pi$
$564$		0			0
$565$	−18.5955	+	18.6450i	−0.782319	+	0.784400i
$566$		0			0
$567$	2.05342	−	1.66838i	0.0862354	−	0.0700652i
$568$		0			0
$569$	15.0534	−	26.0733i	0.631072	−	1.09305i	−0.356261	−	0.934386i	$-0.615949\pi$
$569$	15.0534	−	26.0733i	0.631072	−	1.09305i	0.987333	−	0.158662i	$-0.0507180\pi$
$570$		0			0
$571$	13.0025	+	22.5210i	0.544139	+	0.942476i	0.998661	+	0.0517403i	$0.0164768\pi$
$571$	13.0025	+	22.5210i	0.544139	+	0.942476i	−0.454522	+	0.890736i	$0.650190\pi$
$572$		0			0
$573$			2.79951i			0.116951i
$574$		0			0
$575$	−6.84826	+	11.9346i	−0.285592	+	0.497709i
$576$		0			0
$577$	21.8879	−	12.6370i	0.911207	−	0.526086i	0.0303878	−	0.999538i	$-0.490326\pi$
$577$	21.8879	−	12.6370i	0.911207	−	0.526086i	0.880819	+	0.473452i	$0.156992\pi$
$578$		0			0
$579$	10.5289	−	18.2366i	0.437566	−	0.757887i
$580$		0			0
$581$	14.9669	−	2.39555i	0.620931	−	0.0993842i
$582$		0			0
$583$	−0.266878	−	0.154082i	−0.0110530	−	0.00638143i
$584$		0			0
$585$	−1.01896	−	3.78270i	−0.0421288	−	0.156396i
$586$		0			0
$587$			5.75719i			0.237625i	0.992917	+	0.118812i	$0.0379087\pi$
$587$			5.75719i			0.237625i	−0.992917	+	0.118812i	$0.962091\pi$
$588$		0			0
$589$	−17.4313			−0.718245
$590$		0			0
$591$	−3.59208	−	6.22167i	−0.147758	−	0.255925i
$592$		0			0
$593$	13.8307	+	7.98514i	0.567957	+	0.327910i	0.756333	−	0.654187i	$-0.226989\pi$
$593$	13.8307	+	7.98514i	0.567957	+	0.327910i	−0.188376	+	0.982097i	$0.560322\pi$
$594$		0			0
$595$	−3.74113	+	36.6371i	−0.153371	+	1.50197i
$596$		0			0
$597$	9.60892	+	5.54771i	0.393267	+	0.227053i
$598$		0			0
$599$	13.0976	+	22.6858i	0.535155	+	0.926916i	0.999156	+	0.0410809i	$0.0130801\pi$
$599$	13.0976	+	22.6858i	0.535155	+	0.926916i	−0.464001	+	0.885835i	$0.653587\pi$
$600$		0			0
$601$	−21.8615			−0.891749			−0.445875	−	0.895095i	$-0.647107\pi$
$601$	−21.8615			−0.891749			−0.445875	+	0.895095i	$0.647107\pi$
$602$		0			0
$603$		−	14.6980i		−	0.598550i
$604$		0			0
$605$	−23.0118	+	6.19877i	−0.935564	+	0.252016i
$606$		0			0
$607$	−7.44827	−	4.30026i	−0.302316	−	0.174542i	0.341167	−	0.940003i	$-0.389178\pi$
$607$	−7.44827	−	4.30026i	−0.302316	−	0.174542i	−0.643483	+	0.765461i	$0.722511\pi$
$608$		0			0
$609$	5.40632	+	6.65403i	0.219075	+	0.269635i
$610$		0			0
$611$	−1.27474	+	2.20792i	−0.0515706	+	0.0893229i
$612$		0			0
$613$	−38.3217	+	22.1251i	−1.54780	+	0.893623i	−0.549491	+	0.835499i	$0.685179\pi$
$613$	−38.3217	+	22.1251i	−1.54780	+	0.893623i	−0.998309	+	0.0581239i	$0.981488\pi$
$614$		0			0
$615$	9.90606	+	2.64022i	0.399451	+	0.106464i
$616$		0			0
$617$			32.3683i			1.30310i	0.758606	+	0.651550i	$0.225881\pi$
$617$			32.3683i			1.30310i	−0.758606	+	0.651550i	$0.774119\pi$
$618$		0			0
$619$	21.0233	+	36.4134i	0.844996	+	1.46358i	0.885625	+	0.464402i	$0.153731\pi$
$619$	21.0233	+	36.4134i	0.844996	+	1.46358i	−0.0406284	+	0.999174i	$0.512936\pi$
$620$		0			0
$621$	−1.37599	+	2.38328i	−0.0552165	+	0.0956378i
$622$		0			0
$623$	−42.0261	−	16.0537i	−1.68374	−	0.643178i
$624$		0			0
$625$	−24.9996	+	0.132824i	−0.999986	+	0.00531297i
$626$		0			0
$627$	3.52552	−	2.03546i	0.140796	−	0.0812884i
$628$		0			0
$629$	24.2043			0.965090
$630$		0			0
$631$	3.74086			0.148921			0.0744607	−	0.997224i	$-0.476276\pi$
$631$	3.74086			0.148921			0.0744607	+	0.997224i	$0.476276\pi$
$632$		0			0
$633$	−10.5950	+	6.11702i	−0.421113	+	0.243130i
$634$		0			0
$635$	24.0971	−	24.1612i	0.956264	−	0.958808i
$636$		0			0
$637$	−3.82776	−	11.6512i	−0.151661	−	0.461637i
$638$		0			0
$639$	6.67803	−	11.5667i	0.264179	−	0.457571i
$640$		0			0
$641$	−17.4004	−	30.1384i	−0.687275	−	1.19040i	−0.972716	−	0.231999i	$-0.925473\pi$
$641$	−17.4004	−	30.1384i	−0.687275	−	1.19040i	0.285441	−	0.958396i	$-0.407860\pi$
$642$		0			0
$643$			22.9864i			0.906494i	0.891385	+	0.453247i	$0.149734\pi$
$643$			22.9864i			0.906494i	−0.891385	+	0.453247i	$0.850266\pi$
$644$		0			0
$645$	−0.434391	+	1.62983i	−0.0171041	+	0.0641744i
$646$		0			0
$647$	5.73850	−	3.31312i	0.225604	−	0.130252i	−0.382939	−	0.923774i	$-0.625088\pi$
$647$	5.73850	−	3.31312i	0.225604	−	0.130252i	0.608542	+	0.793521i	$0.291755\pi$
$648$		0			0
$649$	2.95503	−	5.11826i	0.115995	−	0.200909i
$650$		0			0
$651$	2.36404	−	6.18869i	0.0926541	−	0.242554i
$652$		0			0
$653$	−9.07953	−	5.24207i	−0.355309	−	0.205138i	0.311712	−	0.950177i	$-0.399098\pi$
$653$	−9.07953	−	5.24207i	−0.355309	−	0.205138i	−0.667021	+	0.745039i	$0.732431\pi$
$654$		0			0
$655$	17.3113	−	4.66320i	0.676409	−	0.182206i
$656$		0			0
$657$		−	6.75432i		−	0.263511i
$658$		0			0
$659$	−13.2344			−0.515539			−0.257770	−	0.966206i	$-0.582988\pi$
$659$	−13.2344			−0.515539			−0.257770	+	0.966206i	$0.582988\pi$
$660$		0			0
$661$	6.75450	+	11.6991i	0.262720	+	0.455044i	0.966964	−	0.254914i	$-0.0820472\pi$
$661$	6.75450	+	11.6991i	0.262720	+	0.455044i	−0.704244	+	0.709958i	$0.748714\pi$
$662$		0			0
$663$	9.44493	+	5.45303i	0.366810	+	0.211778i
$664$		0			0
$665$	16.8626	+	37.5745i	0.653905	+	1.45708i
$666$		0			0
$667$	−7.72295	−	4.45885i	−0.299034	−	0.172647i
$668$		0			0
$669$	−5.90371	−	10.2255i	−0.228251	−	0.395342i
$670$		0			0
$671$	−4.76830			−0.184078
$672$		0			0
$673$		−	14.8059i		−	0.570727i	−0.958419	−	0.285363i	$-0.907886\pi$
$673$		−	14.8059i		−	0.570727i	0.958419	−	0.285363i	$-0.0921143\pi$
$674$		0			0
$675$	−4.99998	+	0.0132825i	−0.192449	+	0.000511243i
$676$		0			0
$677$	−37.1578	−	21.4531i	−1.42809	−	0.824509i	−0.431122	−	0.902294i	$-0.641882\pi$
$677$	−37.1578	−	21.4531i	−1.42809	−	0.824509i	−0.996970	+	0.0777847i	$0.975215\pi$
$678$		0			0
$679$	29.0333	−	4.64698i	1.11420	−	0.178335i
$680$		0			0
$681$	13.0984	−	22.6871i	0.501931	−	0.869370i
$682$		0			0
$683$	−41.3656	+	23.8824i	−1.58281	+	0.913836i	−0.588363	+	0.808597i	$0.700227\pi$
$683$	−41.3656	+	23.8824i	−1.58281	+	0.913836i	−0.994447	+	0.105239i	$0.966439\pi$
$684$		0			0
$685$	31.7459	+	8.46111i	1.21295	+	0.323283i
$686$		0			0
$687$			9.94135i			0.379286i
$688$		0			0
$689$	−0.461628	−	0.799564i	−0.0175866	−	0.0304609i
$690$		0			0
$691$	10.4948	−	18.1775i	0.399241	−	0.691505i	−0.594392	−	0.804176i	$-0.702607\pi$
$691$	10.4948	−	18.1775i	0.399241	−	0.691505i	0.993632	+	0.112671i	$0.0359405\pi$
$692$		0			0
$693$	0.244523	+	1.52772i	0.00928864	+	0.0580334i
$694$		0			0
$695$	30.2526	+	30.1723i	1.14754	+	1.14450i
$696$		0			0
$697$	−24.7166	+	14.2701i	−0.936206	+	0.540519i
$698$		0			0
$699$	−15.3648			−0.581151
$700$		0			0
$701$	−33.1194			−1.25090			−0.625452	−	0.780263i	$-0.715085\pi$
$701$	−33.1194			−1.25090			−0.625452	+	0.780263i	$0.715085\pi$
$702$		0			0
$703$	23.4416	−	13.5340i	0.884118	−	0.510446i
$704$		0			0
$705$	2.30394	+	2.29782i	0.0867713	+	0.0865411i
$706$		0			0
$707$	35.6392	−	28.9564i	1.34035	−	1.08902i
$708$		0			0
$709$	−16.7366	+	28.9886i	−0.628555	+	1.08869i	0.359287	+	0.933227i	$0.383020\pi$
$709$	−16.7366	+	28.9886i	−0.628555	+	1.08869i	−0.987842	+	0.155462i	$0.950313\pi$
$710$		0			0
$711$	4.04771	+	7.01085i	0.151801	+	0.262927i
$712$		0			0
$713$			6.89083i			0.258064i
$714$		0			0
$715$	2.21361	+	0.589984i	0.0827842	+	0.0220641i
$716$		0			0
$717$	−13.1794	+	7.60915i	−0.492195	+	0.284169i
$718$		0			0
$719$	7.43588	−	12.8793i	0.277312	−	0.480318i	−0.693404	−	0.720549i	$-0.743890\pi$
$719$	7.43588	−	12.8793i	0.277312	−	0.480318i	0.970716	+	0.240231i	$0.0772232\pi$
$720$		0			0
$721$	−12.6147	+	33.0232i	−0.469795	+	1.22985i
$722$		0			0
$723$	23.1626	+	13.3730i	0.861428	+	0.497346i
$724$		0			0
$725$	−0.0430414	−	16.2023i	−0.00159852	−	0.601737i
$726$		0			0
$727$			11.9682i			0.443876i	0.975061	+	0.221938i	$0.0712383\pi$
$727$			11.9682i			0.443876i	−0.975061	+	0.221938i	$0.928762\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−2.34784	−	4.06657i	−0.0868378	−	0.150408i
$732$		0			0
$733$	22.2524	+	12.8474i	0.821909	+	0.474530i	0.851074	−	0.525045i	$-0.175952\pi$
$733$	22.2524	+	12.8474i	0.821909	+	0.474530i	−0.0291651	+	0.999575i	$0.509285\pi$
$734$		0			0
$735$	−15.6271	+	0.890927i	−0.576414	+	0.0328623i
$736$		0			0
$737$	7.44352	+	4.29752i	0.274185	+	0.158301i
$738$		0			0
$739$	−2.38298	−	4.12744i	−0.0876593	−	0.151830i	0.818862	−	0.573990i	$-0.194605\pi$
$739$	−2.38298	−	4.12744i	−0.0876593	−	0.151830i	−0.906521	+	0.422160i	$0.861272\pi$
$740$		0			0
$741$	12.1964			0.448046
$742$		0			0
$743$		−	27.9703i		−	1.02613i	−0.858350	−	0.513065i	$-0.828510\pi$
$743$		−	27.9703i		−	1.02613i	0.858350	−	0.513065i	$-0.171490\pi$
$744$		0			0
$745$	3.94089	−	1.06157i	0.144383	−	0.0388929i
$746$		0			0
$747$	−4.96142	−	2.86448i	−0.181529	−	0.104806i
$748$		0			0
$749$	2.00836	−	5.25757i	0.0733838	−	0.192107i
$750$		0			0
$751$	−10.8613	+	18.8123i	−0.396333	+	0.686469i	−0.993270	−	0.115819i	$-0.963051\pi$
$751$	−10.8613	+	18.8123i	−0.396333	+	0.686469i	0.596937	+	0.802288i	$0.296384\pi$
$752$		0			0
$753$	−6.97419	+	4.02655i	−0.254154	+	0.146736i
$754$		0			0
$755$	2.78005	−	10.4307i	0.101176	−	0.379612i
$756$		0			0
$757$			10.2574i			0.372812i	0.982473	+	0.186406i	$0.0596840\pi$
$757$			10.2574i			0.372812i	−0.982473	+	0.186406i	$0.940316\pi$
$758$		0			0
$759$	−0.804644	−	1.39368i	−0.0292067	−	0.0505875i
$760$		0			0
$761$	15.1717	−	26.2782i	0.549975	−	0.952584i	−0.448301	−	0.893883i	$-0.647971\pi$
$761$	15.1717	−	26.2782i	0.549975	−	0.952584i	0.998276	−	0.0587013i	$-0.0186960\pi$
$762$		0			0
$763$	−4.69822	+	3.81725i	−0.170087	+	0.138194i
$764$		0			0
$765$	9.82950	−	9.85565i	0.355386	−	0.356332i
$766$		0			0
$767$	15.3342	−	8.85322i	0.553687	−	0.319671i
$768$		0			0
$769$	−33.8125			−1.21931			−0.609655	−	0.792667i	$-0.708692\pi$
$769$	−33.8125			−1.21931			−0.609655	+	0.792667i	$0.708692\pi$
$770$		0			0
$771$	15.9703			0.575156
$772$		0			0
$773$	−13.0253	+	7.52017i	−0.468488	+	0.270482i	−0.715607	−	0.698504i	$-0.753850\pi$
$773$	−13.0253	+	7.52017i	−0.468488	+	0.270482i	0.247119	+	0.968985i	$0.420516\pi$
$774$		0			0
$775$	−10.8258	+	6.28867i	−0.388873	+	0.225896i
$776$		0			0
$777$	1.62586	+	10.1580i	0.0583275	+	0.364418i
$778$		0			0
$779$	−15.9585	+	27.6409i	−0.571772	+	0.990338i
$780$		0			0
$781$	3.90514	+	6.76391i	0.139737	+	0.242032i
$782$		0			0
$783$		−	3.24047i		−	0.115805i
$784$		0			0
$785$	−31.2983	−	8.34180i	−1.11708	−	0.297732i
$786$		0			0
$787$	−1.57836	+	0.911269i	−0.0562626	+	0.0324832i	−0.527867	−	0.849327i	$-0.677008\pi$
$787$	−1.57836	+	0.911269i	−0.0562626	+	0.0324832i	0.471605	+	0.881810i	$0.343675\pi$
$788$		0			0
$789$	11.7454	−	20.3437i	0.418148	−	0.724253i
$790$		0			0
$791$	30.7661	−	4.92432i	1.09392	−	0.175089i
$792$		0			0
$793$	−12.3718	−	7.14289i	−0.439337	−	0.253651i
$794$		0			0
$795$	−1.13780	+	0.306494i	−0.0403538	+	0.0108702i
$796$		0			0
$797$		−	26.7990i		−	0.949270i	−0.880183	−	0.474635i	$-0.842580\pi$
$797$		−	26.7990i		−	0.949270i	0.880183	−	0.474635i	$-0.157420\pi$
$798$		0			0
$799$	−9.05865			−0.320472
$800$		0			0
$801$	8.50192	+	14.7258i	0.300401	+	0.520309i
$802$		0			0
$803$	3.42059	+	1.97488i	0.120710	+	0.0696920i
$804$		0			0
$805$	14.8537	−	6.66603i	0.523524	−	0.234947i
$806$		0			0
$807$	−11.3766	−	6.56831i	−0.400477	−	0.231216i
$808$		0			0
$809$	−1.01125	−	1.75154i	−0.0355538	−	0.0615810i	0.847701	−	0.530474i	$-0.177986\pi$
$809$	−1.01125	−	1.75154i	−0.0355538	−	0.0615810i	−0.883255	+	0.468893i	$0.844653\pi$
$810$		0			0
$811$	−0.357073			−0.0125385			−0.00626927	−	0.999980i	$-0.501996\pi$
$811$	−0.357073			−0.0125385			−0.00626927	+	0.999980i	$0.501996\pi$
$812$		0			0
$813$		−	5.38550i		−	0.188878i
$814$		0			0
$815$	6.75799	+	25.0878i	0.236722	+	0.878788i
$816$		0			0
$817$	−4.54771	−	2.62562i	−0.159104	−	0.0918588i
$818$		0			0
$819$	−1.65408	+	4.33013i	−0.0577983	+	0.151307i
$820$		0			0
$821$	−2.37963	+	4.12164i	−0.0830496	+	0.143846i	−0.904558	−	0.426350i	$-0.859799\pi$
$821$	−2.37963	+	4.12164i	−0.0830496	+	0.143846i	0.821509	+	0.570196i	$0.193133\pi$
$822$		0			0
$823$	19.6898	−	11.3679i	0.686342	−	0.396260i	−0.115898	−	0.993261i	$-0.536975\pi$
$823$	19.6898	−	11.3679i	0.686342	−	0.396260i	0.802240	+	0.597002i	$0.203641\pi$
$824$		0			0
$825$	1.45521	−	2.53602i	0.0506638	−	0.0882930i
$826$		0			0
$827$		−	6.46781i		−	0.224908i	−0.993657	−	0.112454i	$-0.964129\pi$
$827$		−	6.46781i		−	0.224908i	0.993657	−	0.112454i	$-0.0358711\pi$
$828$		0			0
$829$	1.18015	+	2.04407i	0.0409882	+	0.0709936i	0.885792	−	0.464083i	$-0.153616\pi$
$829$	1.18015	+	2.04407i	0.0409882	+	0.0709936i	−0.844804	+	0.535077i	$0.820283\pi$
$830$		0			0
$831$	−5.03285	+	8.71716i	−0.174588	+	0.302395i
$832$		0			0
$833$	29.0516	−	32.4775i	1.00658	−	1.12528i
$834$		0			0
$835$	33.6331	−	33.7226i	1.16392	−	1.16702i
$836$		0			0
$837$	−2.16849	+	1.25198i	−0.0749540	+	0.0432747i
$838$		0			0
$839$	−31.3590			−1.08263			−0.541317	−	0.840819i	$-0.682074\pi$
$839$	−31.3590			−1.08263			−0.541317	+	0.840819i	$0.682074\pi$
$840$		0			0
$841$	−18.4994			−0.637910
$842$		0			0
$843$	23.2806	−	13.4411i	0.801828	−	0.462936i
$844$		0			0
$845$	−15.7225	−	15.6807i	−0.540869	−	0.539434i
$846$		0			0
$847$	26.3420	+	10.0625i	0.905123	+	0.345751i
$848$		0			0
$849$	−5.92865	+	10.2687i	−0.203471	+	0.352422i
$850$		0			0
$851$	−5.35018	−	9.26679i	−0.183402	−	0.317661i
$852$		0			0
$853$			17.4278i			0.596718i	0.954454	+	0.298359i	$0.0964392\pi$
$853$			17.4278i			0.596718i	−0.954454	+	0.298359i	$0.903561\pi$
$854$		0			0
$855$	4.00891	−	15.0413i	0.137102	−	0.514403i
$856$		0			0
$857$	4.07311	−	2.35161i	0.139135	−	0.0803296i	−0.428817	−	0.903392i	$-0.641069\pi$
$857$	4.07311	−	2.35161i	0.139135	−	0.0803296i	0.567952	+	0.823062i	$0.307736\pi$
$858$		0			0
$859$	3.99708	−	6.92315i	0.136379	−	0.236215i	−0.789745	−	0.613436i	$-0.789787\pi$
$859$	3.99708	−	6.92315i	0.136379	−	0.236215i	0.926123	+	0.377221i	$0.123120\pi$
$860$		0			0
$861$	−7.64913	−	9.41445i	−0.260682	−	0.320844i
$862$		0			0
$863$	17.9349	+	10.3547i	0.610510	+	0.352478i	0.773165	−	0.634205i	$-0.218672\pi$
$863$	17.9349	+	10.3547i	0.610510	+	0.352478i	−0.162655	+	0.986683i	$0.552006\pi$
$864$		0			0
$865$	−6.96890	−	25.8708i	−0.236950	−	0.879634i
$866$		0			0
$867$			21.7506i			0.738689i
$868$		0			0
$869$	−4.73400			−0.160590
$870$		0			0
$871$	12.8753	+	22.3007i	0.436263	+	0.755630i
$872$		0			0
$873$	−9.62435	−	5.55662i	−0.325735	−	0.188063i
$874$		0			0
$875$	24.0283	+	17.2523i	0.812305	+	0.583233i
$876$		0			0
$877$	14.1192	+	8.15174i	0.476772	+	0.275265i	0.719070	−	0.694937i	$-0.244568\pi$
$877$	14.1192	+	8.15174i	0.476772	+	0.275265i	−0.242298	+	0.970202i	$0.577901\pi$
$878$		0			0
$879$	−8.19393	−	14.1923i	−0.276374	−	0.478694i
$880$		0			0
$881$	8.44801			0.284621			0.142310	−	0.989822i	$-0.454547\pi$
$881$	8.44801			0.284621			0.142310	+	0.989822i	$0.454547\pi$
$882$		0			0
$883$		−	40.6622i		−	1.36839i	−0.729298	−	0.684197i	$-0.760153\pi$
$883$		−	40.6622i		−	1.36839i	0.729298	−	0.684197i	$-0.239847\pi$
$884$		0			0
$885$	−5.87802	−	21.8211i	−0.197588	−	0.733508i
$886$		0			0
$887$	18.8414	+	10.8781i	0.632632	+	0.365250i	0.781771	−	0.623566i	$-0.214317\pi$
$887$	18.8414	+	10.8781i	0.632632	+	0.365250i	−0.149139	+	0.988816i	$0.547650\pi$
$888$		0			0
$889$	−39.8684	+	6.38120i	−1.33714	+	0.214019i
$890$		0			0
$891$	0.292387	−	0.506430i	0.00979535	−	0.0169660i
$892$		0			0
$893$	−8.77321	+	5.06521i	−0.293584	+	0.169501i
$894$		0			0
$895$	11.7587	−	44.1183i	0.393049	−	1.47471i
$896$		0			0
$897$		−	4.82140i		−	0.160982i
$898$		0			0
$899$	−4.05699	−	7.02692i	−0.135308	−	0.234361i
$900$		0			0
$901$	1.64022	−	2.84095i	0.0546438	−	0.0946458i
$902$		0			0
$903$	1.54894	−	1.25850i	0.0515456	−	0.0418802i
$904$		0			0
$905$	−31.3918	−	31.3085i	−1.04350	−	1.04073i
$906$		0			0
$907$	21.7543	−	12.5598i	0.722339	−	0.417042i	−0.0932742	−	0.995640i	$-0.529733\pi$
$907$	21.7543	−	12.5598i	0.722339	−	0.417042i	0.815613	+	0.578598i	$0.196400\pi$
$908$		0			0
$909$	−17.3561			−0.575664
$910$		0			0
$911$	−9.33526			−0.309291			−0.154646	−	0.987970i	$-0.549424\pi$
$911$	−9.33526			−0.309291			−0.154646	+	0.987970i	$0.549424\pi$
$912$		0			0
$913$	2.90131	−	1.67507i	0.0960195	−	0.0554369i
$914$		0			0
$915$	−12.8756	+	12.9098i	−0.425654	+	0.426786i
$916$		0			0
$917$	−19.8165	−	7.56980i	−0.654400	−	0.249977i
$918$		0			0
$919$	10.5657	−	18.3003i	0.348531	−	0.603673i	−0.637458	−	0.770485i	$-0.720014\pi$
$919$	10.5657	−	18.3003i	0.348531	−	0.603673i	0.985989	+	0.166812i	$0.0533474\pi$
$920$		0			0
$921$	10.0070	+	17.3327i	0.329743	+	0.571132i
$922$		0			0
$923$			23.3995i			0.770205i
$924$		0			0
$925$	9.67585	−	16.8624i	0.318140	−	0.554431i
$926$		0			0
$927$	11.5712	−	6.68063i	0.380048	−	0.219421i
$928$		0			0
$929$	−28.4611	+	49.2960i	−0.933778	+	1.61735i	−0.156979	+	0.987602i	$0.550175\pi$
$929$	−28.4611	+	49.2960i	−0.933778	+	1.61735i	−0.776799	+	0.629748i	$0.783158\pi$
$930$		0			0
$931$	9.97612	−	47.6985i	0.326954	−	1.56325i
$932$		0			0
$933$	−24.7999	−	14.3183i	−0.811914	−	0.468759i
$934$		0			0
$935$	2.11717	+	7.85963i	0.0692390	+	0.257037i
$936$		0			0
$937$			11.1734i			0.365019i	0.983204	+	0.182509i	$0.0584220\pi$
$937$			11.1734i			0.365019i	−0.983204	+	0.182509i	$0.941578\pi$
$938$		0			0
$939$	−27.3613			−0.892901
$940$		0			0
$941$	−14.9941	−	25.9706i	−0.488794	−	0.846616i	0.511123	−	0.859508i	$-0.329230\pi$
$941$	−14.9941	−	25.9706i	−0.488794	−	0.846616i	−0.999917	+	0.0128918i	$0.995896\pi$
$942$		0			0
$943$	10.9268	+	6.30860i	0.355826	+	0.205436i
$944$		0			0
$945$	4.79648	+	3.46321i	0.156029	+	0.112658i
$946$		0			0
$947$	34.3460	+	19.8297i	1.11610	+	0.644378i	0.940401	−	0.340067i	$-0.110450\pi$
$947$	34.3460	+	19.8297i	1.11610	+	0.644378i	0.175694	+	0.984445i	$0.443783\pi$
$948$		0			0
$949$	5.91671	+	10.2480i	0.192065	+	0.332666i
$950$		0			0
$951$	−8.01660			−0.259956
$952$		0			0
$953$		−	17.9334i		−	0.580921i	−0.956887	−	0.290460i	$-0.906192\pi$
$953$		−	17.9334i		−	0.580921i	0.956887	−	0.290460i	$-0.0938085\pi$
$954$		0			0
$955$	−6.04445	+	1.62821i	−0.195594	+	0.0526877i
$956$		0			0
$957$	1.64107	+	0.947472i	0.0530482	+	0.0306274i
$958$		0			0
$959$	−24.5131	−	30.1705i	−0.791571	−	0.974255i
$960$		0			0
$961$	12.3651	−	21.4170i	0.398874	−	0.690870i
$962$		0			0
$963$	−1.84223	+	1.06361i	−0.0593650	+	0.0342744i
$964$		0			0
$965$	45.4984	+	12.1265i	1.46464	+	0.390366i
$966$		0			0
$967$		−	10.2432i		−	0.329398i	−0.986344	−	0.164699i	$-0.947335\pi$
$967$		−	10.2432i		−	0.329398i	0.986344	−	0.164699i	$-0.0526653\pi$
$968$		0			0
$969$	21.6677	+	37.5295i	0.696066	+	1.20562i
$970$		0			0
$971$	10.3547	−	17.9349i	0.332298	−	0.575558i	−0.650664	−	0.759366i	$-0.725509\pi$
$971$	10.3547	−	17.9349i	0.332298	−	0.575558i	0.982962	+	0.183808i	$0.0588426\pi$
$972$		0			0
$973$	−7.98999	−	49.9197i	−0.256147	−	1.60035i
$974$		0			0
$975$	7.57463	−	4.40008i	0.242582	−	0.140915i
$976$		0			0
$977$	31.4669	−	18.1674i	1.00672	−	0.581228i	0.0964874	−	0.995334i	$-0.469239\pi$
$977$	31.4669	−	18.1674i	1.00672	−	0.581228i	0.910228	+	0.414107i	$0.135906\pi$
$978$		0			0
$979$	−9.94342			−0.317793
$980$		0			0
$981$	2.28800			0.0730503
$982$		0			0
$983$	27.6067	−	15.9387i	0.880517	−	0.508367i	0.00968816	−	0.999953i	$-0.496916\pi$
$983$	27.6067	−	15.9387i	0.880517	−	0.508367i	0.870829	+	0.491586i	$0.163583\pi$
$984$		0			0
$985$	11.3441	−	11.3742i	0.361452	−	0.362413i
$986$		0			0
$987$	−0.608492	−	3.80172i	−0.0193685	−	0.121010i
$988$		0			0
$989$	−1.03794	+	1.79777i	−0.0330047	+	0.0571658i
$990$		0			0
$991$	−5.30365	−	9.18619i	−0.168476	−	0.291809i	0.769408	−	0.638757i	$-0.220551\pi$
$991$	−5.30365	−	9.18619i	−0.168476	−	0.291809i	−0.937884	+	0.346948i	$0.887218\pi$
$992$		0			0
$993$		−	9.97698i		−	0.316610i
$994$		0			0
$995$	−6.38950	+	23.9733i	−0.202561	+	0.760003i
$996$		0			0
$997$	−18.7491	+	10.8248i	−0.593791	+	0.342825i	−0.766595	−	0.642131i	$-0.778051\pi$
$997$	−18.7491	+	10.8248i	−0.593791	+	0.342825i	0.172804	+	0.984956i	$0.444717\pi$
$998$		0			0
$999$	1.94412	−	3.36732i	0.0615093	−	0.106537i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bb.a.289.2	yes	16
3.2	odd	2		1260.2.bm.c.289.6		16
4.3	odd	2		1680.2.di.e.289.6		16
5.2	odd	4		2100.2.q.m.1801.1		8
5.3	odd	4		2100.2.q.l.1801.4		8
5.4	even	2	inner	420.2.bb.a.289.8	yes	16
7.2	even	3		2940.2.k.f.589.2		8
7.3	odd	6		2940.2.bb.i.949.1		16
7.4	even	3	inner	420.2.bb.a.109.8	yes	16
7.5	odd	6		2940.2.k.g.589.7		8
7.6	odd	2		2940.2.bb.i.1549.7		16
15.14	odd	2		1260.2.bm.c.289.1		16
20.19	odd	2		1680.2.di.e.289.4		16
21.11	odd	6		1260.2.bm.c.109.1		16
28.11	odd	6		1680.2.di.e.529.4		16
35.4	even	6	inner	420.2.bb.a.109.2	✓	16
35.9	even	6		2940.2.k.f.589.6		8
35.18	odd	12		2100.2.q.l.1201.4		8
35.19	odd	6		2940.2.k.g.589.3		8
35.24	odd	6		2940.2.bb.i.949.7		16
35.32	odd	12		2100.2.q.m.1201.1		8
35.34	odd	2		2940.2.bb.i.1549.1		16
105.74	odd	6		1260.2.bm.c.109.6		16
140.39	odd	6		1680.2.di.e.529.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bb.a.109.2	✓	16	35.4	even	6	inner
420.2.bb.a.109.8	yes	16	7.4	even	3	inner
420.2.bb.a.289.2	yes	16	1.1	even	1	trivial
420.2.bb.a.289.8	yes	16	5.4	even	2	inner
1260.2.bm.c.109.1		16	21.11	odd	6
1260.2.bm.c.109.6		16	105.74	odd	6
1260.2.bm.c.289.1		16	15.14	odd	2
1260.2.bm.c.289.6		16	3.2	odd	2
1680.2.di.e.289.4		16	20.19	odd	2
1680.2.di.e.289.6		16	4.3	odd	2
1680.2.di.e.529.4		16	28.11	odd	6
1680.2.di.e.529.6		16	140.39	odd	6
2100.2.q.l.1201.4		8	35.18	odd	12
2100.2.q.l.1801.4		8	5.3	odd	4
2100.2.q.m.1201.1		8	35.32	odd	12
2100.2.q.m.1801.1		8	5.2	odd	4
2940.2.k.f.589.2		8	7.2	even	3
2940.2.k.f.589.6		8	35.9	even	6
2940.2.k.g.589.3		8	35.19	odd	6
2940.2.k.g.589.7		8	7.5	odd	6
2940.2.bb.i.949.1		16	7.3	odd	6
2940.2.bb.i.949.7		16	35.24	odd	6
2940.2.bb.i.1549.1		16	35.34	odd	2
2940.2.bb.i.1549.7		16	7.6	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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bb (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{3} +(-1.58324 - 1.57904i) q^{5} +(0.418148 + 2.61250i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{3} +(-1.58324 - 1.57904i) q^{5} +(0.418148 + 2.61250i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.292387 + 0.506430i) q^{11} +1.75198i q^{13} +(2.16064 + 0.575868i) q^{15} +(-5.39101 + 3.11250i) q^{17} +(-3.48075 + 6.02884i) q^{19} +(-1.66838 - 2.05342i) q^{21} +(2.38328 + 1.37599i) q^{23} +(0.0132825 + 4.99998i) q^{25} +1.00000i q^{27} -3.24047 q^{29} +(1.25198 + 2.16849i) q^{31} +(-0.506430 - 0.292387i) q^{33} +(3.46321 - 4.79648i) q^{35} +(-3.36732 - 1.94412i) q^{37} +(-0.875989 - 1.51726i) q^{39} +4.58477 q^{41} +0.754325i q^{43} +(-2.15911 + 0.581605i) q^{45} +(1.26025 + 0.727603i) q^{47} +(-6.65030 + 2.18482i) q^{49} +(3.11250 - 5.39101i) q^{51} +(-0.456378 + 0.263490i) q^{53} +(0.336753 - 1.26349i) q^{55} -6.96151i q^{57} +(-5.05327 - 8.75253i) q^{59} +(-4.07704 + 7.06164i) q^{61} +(2.47156 + 0.944123i) q^{63} +(2.76644 - 2.77380i) q^{65} +(12.7289 - 7.34901i) q^{67} -2.75198 q^{69} +13.3561 q^{71} +(5.84942 - 3.37716i) q^{73} +(-2.51149 - 4.32347i) q^{75} +(-1.20079 + 0.975625i) q^{77} +(-4.04771 + 7.01085i) q^{79} +(-0.500000 - 0.866025i) q^{81} -5.72895i q^{83} +(13.4500 + 3.58477i) q^{85} +(2.80633 - 1.62023i) q^{87} +(-8.50192 + 14.7258i) q^{89} +(-4.57704 + 0.732586i) q^{91} +(-2.16849 - 1.25198i) q^{93} +(15.0306 - 4.04885i) q^{95} -11.1132i q^{97} +0.584775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{5} + 8 q^{9}+O(q^{10}) \) 16 * q + 2 * q^5 + 8 * q^9	\( 16 q + 2 q^{5} + 8 q^{9} - 8 q^{11} + 4 q^{15} + 8 q^{19} - 4 q^{21} + 12 q^{25} + 24 q^{29} + 10 q^{35} - 4 q^{39} + 48 q^{41} - 2 q^{45} + 8 q^{49} + 4 q^{51} - 40 q^{55} - 28 q^{59} - 32 q^{61} - 26 q^{65} - 24 q^{69} - 56 q^{71} - 8 q^{75} - 16 q^{79} - 8 q^{81} + 32 q^{85} - 16 q^{89} - 40 q^{91} + 22 q^{95} - 16 q^{99}+O(q^{100}) \) 16 * q + 2 * q^5 + 8 * q^9 - 8 * q^11 + 4 * q^15 + 8 * q^19 - 4 * q^21 + 12 * q^25 + 24 * q^29 + 10 * q^35 - 4 * q^39 + 48 * q^41 - 2 * q^45 + 8 * q^49 + 4 * q^51 - 40 * q^55 - 28 * q^59 - 32 * q^61 - 26 * q^65 - 24 * q^69 - 56 * q^71 - 8 * q^75 - 16 * q^79 - 8 * q^81 + 32 * q^85 - 16 * q^89 - 40 * q^91 + 22 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more