LMFDB - Embedded newform 114.2.b.b.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,2,Mod(113,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("114.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	114.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.910294583043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	114.113
Dual form			114.2.b.b.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +(1.50000 - 0.866025i) q^{3} +1.00000 q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{6} +1.00000 q^{7} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} -3.46410i q^{10} -3.46410i q^{11} +(1.50000 - 0.866025i) q^{12} +1.73205i q^{13} -1.00000 q^{14} +(3.00000 + 5.19615i) q^{15} +1.00000 q^{16} +1.73205i q^{17} +(-1.50000 + 2.59808i) q^{18} +(-4.00000 - 1.73205i) q^{19} +3.46410i q^{20} +(1.50000 - 0.866025i) q^{21} +3.46410i q^{22} -5.19615i q^{23} +(-1.50000 + 0.866025i) q^{24} -7.00000 q^{25} -1.73205i q^{26} -5.19615i q^{27} +1.00000 q^{28} -9.00000 q^{29} +(-3.00000 - 5.19615i) q^{30} +10.3923i q^{31} -1.00000 q^{32} +(-3.00000 - 5.19615i) q^{33} -1.73205i q^{34} +3.46410i q^{35} +(1.50000 - 2.59808i) q^{36} -6.92820i q^{37} +(4.00000 + 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{39} -3.46410i q^{40} +(-1.50000 + 0.866025i) q^{42} +2.00000 q^{43} -3.46410i q^{44} +(9.00000 + 5.19615i) q^{45} +5.19615i q^{46} +3.46410i q^{47} +(1.50000 - 0.866025i) q^{48} -6.00000 q^{49} +7.00000 q^{50} +(1.50000 + 2.59808i) q^{51} +1.73205i q^{52} +9.00000 q^{53} +5.19615i q^{54} +12.0000 q^{55} -1.00000 q^{56} +(-7.50000 + 0.866025i) q^{57} +9.00000 q^{58} +3.00000 q^{59} +(3.00000 + 5.19615i) q^{60} -8.00000 q^{61} -10.3923i q^{62} +(1.50000 - 2.59808i) q^{63} +1.00000 q^{64} -6.00000 q^{65} +(3.00000 + 5.19615i) q^{66} -8.66025i q^{67} +1.73205i q^{68} +(-4.50000 - 7.79423i) q^{69} -3.46410i q^{70} +12.0000 q^{71} +(-1.50000 + 2.59808i) q^{72} +11.0000 q^{73} +6.92820i q^{74} +(-10.5000 + 6.06218i) q^{75} +(-4.00000 - 1.73205i) q^{76} -3.46410i q^{77} +(-1.50000 - 2.59808i) q^{78} -6.92820i q^{79} +3.46410i q^{80} +(-4.50000 - 7.79423i) q^{81} +10.3923i q^{83} +(1.50000 - 0.866025i) q^{84} -6.00000 q^{85} -2.00000 q^{86} +(-13.5000 + 7.79423i) q^{87} +3.46410i q^{88} -6.00000 q^{89} +(-9.00000 - 5.19615i) q^{90} +1.73205i q^{91} -5.19615i q^{92} +(9.00000 + 15.5885i) q^{93} -3.46410i q^{94} +(6.00000 - 13.8564i) q^{95} +(-1.50000 + 0.866025i) q^{96} +13.8564i q^{97} +6.00000 q^{98} +(-9.00000 - 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{12} - 2 q^{14} + 6 q^{15} + 2 q^{16} - 3 q^{18} - 8 q^{19} + 3 q^{21} - 3 q^{24} - 14 q^{25} + 2 q^{28} - 18 q^{29} - 6 q^{30}+ \cdots - 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 3 * q^9 + 3 * q^12 - 2 * q^14 + 6 * q^15 + 2 * q^16 - 3 * q^18 - 8 * q^19 + 3 * q^21 - 3 * q^24 - 14 * q^25 + 2 * q^28 - 18 * q^29 - 6 * q^30 - 2 * q^32 - 6 * q^33 + 3 * q^36 + 8 * q^38 + 3 * q^39 - 3 * q^42 + 4 * q^43 + 18 * q^45 + 3 * q^48 - 12 * q^49 + 14 * q^50 + 3 * q^51 + 18 * q^53 + 24 * q^55 - 2 * q^56 - 15 * q^57 + 18 * q^58 + 6 * q^59 + 6 * q^60 - 16 * q^61 + 3 * q^63 + 2 * q^64 - 12 * q^65 + 6 * q^66 - 9 * q^69 + 24 * q^71 - 3 * q^72 + 22 * q^73 - 21 * q^75 - 8 * q^76 - 3 * q^78 - 9 * q^81 + 3 * q^84 - 12 * q^85 - 4 * q^86 - 27 * q^87 - 12 * q^89 - 18 * q^90 + 18 * q^93 + 12 * q^95 - 3 * q^96 + 12 * q^98 - 18 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$.

$n$	$77$	$97$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$	1.00000			0.500000
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	−1.50000	+	0.866025i	−0.612372	+	0.353553i
$7$	1.00000			0.377964			0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000			0.377964			0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000			−0.353553
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		−	3.46410i		−	1.09545i
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	1.50000	−	0.866025i	0.433013	−	0.250000i
$13$			1.73205i			0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$			1.73205i			0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−1.00000			−0.267261
$15$	3.00000	+	5.19615i	0.774597	+	1.34164i
$16$	1.00000			0.250000
$17$			1.73205i			0.420084i	0.977692	+	0.210042i	$0.0673601\pi$
$17$			1.73205i			0.420084i	−0.977692	+	0.210042i	$0.932640\pi$
$18$	−1.50000	+	2.59808i	−0.353553	+	0.612372i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$20$			3.46410i			0.774597i
$21$	1.50000	−	0.866025i	0.327327	−	0.188982i
$22$			3.46410i			0.738549i
$23$		−	5.19615i		−	1.08347i	−0.840548	−	0.541736i	$-0.817767\pi$
$23$		−	5.19615i		−	1.08347i	0.840548	−	0.541736i	$-0.182233\pi$
$24$	−1.50000	+	0.866025i	−0.306186	+	0.176777i
$25$	−7.00000			−1.40000
$26$		−	1.73205i		−	0.339683i
$27$		−	5.19615i		−	1.00000i
$28$	1.00000			0.188982
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$	−3.00000	−	5.19615i	−0.547723	−	0.948683i
$31$			10.3923i			1.86651i	0.359211	+	0.933257i	$0.383046\pi$
$31$			10.3923i			1.86651i	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000			−0.176777
$33$	−3.00000	−	5.19615i	−0.522233	−	0.904534i
$34$		−	1.73205i		−	0.297044i
$35$			3.46410i			0.585540i
$36$	1.50000	−	2.59808i	0.250000	−	0.433013i
$37$		−	6.92820i		−	1.13899i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$		−	6.92820i		−	1.13899i	0.821995	−	0.569495i	$-0.192861\pi$
$38$	4.00000	+	1.73205i	0.648886	+	0.280976i
$39$	1.50000	+	2.59808i	0.240192	+	0.416025i
$40$		−	3.46410i		−	0.547723i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−1.50000	+	0.866025i	−0.231455	+	0.133631i
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		−	3.46410i		−	0.522233i
$45$	9.00000	+	5.19615i	1.34164	+	0.774597i
$46$			5.19615i			0.766131i
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	1.50000	−	0.866025i	0.216506	−	0.125000i
$49$	−6.00000			−0.857143
$50$	7.00000			0.989949
$51$	1.50000	+	2.59808i	0.210042	+	0.363803i
$52$			1.73205i			0.240192i
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$			5.19615i			0.707107i
$55$	12.0000			1.61808
$56$	−1.00000			−0.133631
$57$	−7.50000	+	0.866025i	−0.993399	+	0.114708i
$58$	9.00000			1.18176
$59$	3.00000			0.390567			0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000			0.390567			0.195283	+	0.980747i	$0.437437\pi$
$60$	3.00000	+	5.19615i	0.387298	+	0.670820i
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		−	10.3923i		−	1.31982i
$63$	1.50000	−	2.59808i	0.188982	−	0.327327i
$64$	1.00000			0.125000
$65$	−6.00000			−0.744208
$66$	3.00000	+	5.19615i	0.369274	+	0.639602i
$67$		−	8.66025i		−	1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$		−	8.66025i		−	1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$			1.73205i			0.210042i
$69$	−4.50000	−	7.79423i	−0.541736	−	0.938315i
$70$		−	3.46410i		−	0.414039i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.50000	+	2.59808i	−0.176777	+	0.306186i
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$			6.92820i			0.805387i
$75$	−10.5000	+	6.06218i	−1.21244	+	0.700000i
$76$	−4.00000	−	1.73205i	−0.458831	−	0.198680i
$77$		−	3.46410i		−	0.394771i
$78$	−1.50000	−	2.59808i	−0.169842	−	0.294174i
$79$		−	6.92820i		−	0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$		−	6.92820i		−	0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$			3.46410i			0.387298i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$			10.3923i			1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$			10.3923i			1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$	1.50000	−	0.866025i	0.163663	−	0.0944911i
$85$	−6.00000			−0.650791
$86$	−2.00000			−0.215666
$87$	−13.5000	+	7.79423i	−1.44735	+	0.835629i
$88$			3.46410i			0.369274i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$	−9.00000	−	5.19615i	−0.948683	−	0.547723i
$91$			1.73205i			0.181568i
$92$		−	5.19615i		−	0.541736i
$93$	9.00000	+	15.5885i	0.933257	+	1.61645i
$94$		−	3.46410i		−	0.357295i
$95$	6.00000	−	13.8564i	0.615587	−	1.42164i
$96$	−1.50000	+	0.866025i	−0.153093	+	0.0883883i
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$	6.00000			0.606092
$99$	−9.00000	−	5.19615i	−0.904534	−	0.522233i
$100$	−7.00000			−0.700000
$101$		−	10.3923i		−	1.03407i	−0.855963	−	0.517036i	$-0.827035\pi$
$101$		−	10.3923i		−	1.03407i	0.855963	−	0.517036i	$-0.172965\pi$
$102$	−1.50000	−	2.59808i	−0.148522	−	0.257248i
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		−	1.73205i		−	0.169842i
$105$	3.00000	+	5.19615i	0.292770	+	0.507093i
$106$	−9.00000			−0.874157
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		−	5.19615i		−	0.500000i
$109$			15.5885i			1.49310i	0.665327	+	0.746552i	$0.268292\pi$
$109$			15.5885i			1.49310i	−0.665327	+	0.746552i	$0.731708\pi$
$110$	−12.0000			−1.14416
$111$	−6.00000	−	10.3923i	−0.569495	−	0.986394i
$112$	1.00000			0.0944911
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$	7.50000	−	0.866025i	0.702439	−	0.0811107i
$115$	18.0000			1.67851
$116$	−9.00000			−0.835629
$117$	4.50000	+	2.59808i	0.416025	+	0.240192i
$118$	−3.00000			−0.276172
$119$			1.73205i			0.158777i
$120$	−3.00000	−	5.19615i	−0.273861	−	0.474342i
$121$	−1.00000			−0.0909091
$122$	8.00000			0.724286
$123$		0			0
$124$			10.3923i			0.933257i
$125$		−	6.92820i		−	0.619677i
$126$	−1.50000	+	2.59808i	−0.133631	+	0.231455i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−1.00000			−0.0883883
$129$	3.00000	−	1.73205i	0.264135	−	0.152499i
$130$	6.00000			0.526235
$131$			3.46410i			0.302660i	0.988483	+	0.151330i	$0.0483556\pi$
$131$			3.46410i			0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$132$	−3.00000	−	5.19615i	−0.261116	−	0.452267i
$133$	−4.00000	−	1.73205i	−0.346844	−	0.150188i
$134$			8.66025i			0.748132i
$135$	18.0000			1.54919
$136$		−	1.73205i		−	0.148522i
$137$			8.66025i			0.739895i	0.929053	+	0.369948i	$0.120624\pi$
$137$			8.66025i			0.739895i	−0.929053	+	0.369948i	$0.879376\pi$
$138$	4.50000	+	7.79423i	0.383065	+	0.663489i
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$			3.46410i			0.292770i
$141$	3.00000	+	5.19615i	0.252646	+	0.437595i
$142$	−12.0000			−1.00702
$143$	6.00000			0.501745
$144$	1.50000	−	2.59808i	0.125000	−	0.216506i
$145$		−	31.1769i		−	2.58910i
$146$	−11.0000			−0.910366
$147$	−9.00000	+	5.19615i	−0.742307	+	0.428571i
$148$		−	6.92820i		−	0.569495i
$149$			6.92820i			0.567581i	0.958886	+	0.283790i	$0.0915919\pi$
$149$			6.92820i			0.567581i	−0.958886	+	0.283790i	$0.908408\pi$
$150$	10.5000	−	6.06218i	0.857321	−	0.494975i
$151$			3.46410i			0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$			3.46410i			0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	4.00000	+	1.73205i	0.324443	+	0.140488i
$153$	4.50000	+	2.59808i	0.363803	+	0.210042i
$154$			3.46410i			0.279145i
$155$	−36.0000			−2.89159
$156$	1.50000	+	2.59808i	0.120096	+	0.208013i
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$			6.92820i			0.551178i
$159$	13.5000	−	7.79423i	1.07062	−	0.618123i
$160$		−	3.46410i		−	0.273861i
$161$		−	5.19615i		−	0.409514i
$162$	4.50000	+	7.79423i	0.353553	+	0.612372i
$163$	10.0000			0.783260			0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000			0.783260			0.391630	+	0.920123i	$0.371911\pi$
$164$		0			0
$165$	18.0000	−	10.3923i	1.40130	−	0.809040i
$166$		−	10.3923i		−	0.806599i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−1.50000	+	0.866025i	−0.115728	+	0.0668153i
$169$	10.0000			0.769231
$170$	6.00000			0.460179
$171$	−10.5000	+	7.79423i	−0.802955	+	0.596040i
$172$	2.00000			0.152499
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$	13.5000	−	7.79423i	1.02343	−	0.590879i
$175$	−7.00000			−0.529150
$176$		−	3.46410i		−	0.261116i
$177$	4.50000	−	2.59808i	0.338241	−	0.195283i
$178$	6.00000			0.449719
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$	9.00000	+	5.19615i	0.670820	+	0.387298i
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		−	1.73205i		−	0.128388i
$183$	−12.0000	+	6.92820i	−0.887066	+	0.512148i
$184$			5.19615i			0.383065i
$185$	24.0000			1.76452
$186$	−9.00000	−	15.5885i	−0.659912	−	1.14300i
$187$	6.00000			0.438763
$188$			3.46410i			0.252646i
$189$		−	5.19615i		−	0.377964i
$190$	−6.00000	+	13.8564i	−0.435286	+	1.00525i
$191$			19.0526i			1.37859i	0.724479	+	0.689297i	$0.242081\pi$
$191$			19.0526i			1.37859i	−0.724479	+	0.689297i	$0.757919\pi$
$192$	1.50000	−	0.866025i	0.108253	−	0.0625000i
$193$			3.46410i			0.249351i	0.992198	+	0.124676i	$0.0397891\pi$
$193$			3.46410i			0.249351i	−0.992198	+	0.124676i	$0.960211\pi$
$194$		−	13.8564i		−	0.994832i
$195$	−9.00000	+	5.19615i	−0.644503	+	0.372104i
$196$	−6.00000			−0.428571
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	9.00000	+	5.19615i	0.639602	+	0.369274i
$199$	11.0000			0.779769			0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000			0.779769			0.389885	+	0.920864i	$0.372515\pi$
$200$	7.00000			0.494975
$201$	−7.50000	−	12.9904i	−0.529009	−	0.916271i
$202$			10.3923i			0.731200i
$203$	−9.00000			−0.631676
$204$	1.50000	+	2.59808i	0.105021	+	0.181902i
$205$		0			0
$206$			3.46410i			0.241355i
$207$	−13.5000	−	7.79423i	−0.938315	−	0.541736i
$208$			1.73205i			0.120096i
$209$	−6.00000	+	13.8564i	−0.415029	+	0.958468i
$210$	−3.00000	−	5.19615i	−0.207020	−	0.358569i
$211$		−	12.1244i		−	0.834675i	−0.908752	−	0.417338i	$-0.862963\pi$
$211$		−	12.1244i		−	0.834675i	0.908752	−	0.417338i	$-0.137037\pi$
$212$	9.00000			0.618123
$213$	18.0000	−	10.3923i	1.23334	−	0.712069i
$214$	3.00000			0.205076
$215$			6.92820i			0.472500i
$216$			5.19615i			0.353553i
$217$			10.3923i			0.705476i
$218$		−	15.5885i		−	1.05578i
$219$	16.5000	−	9.52628i	1.11497	−	0.643726i
$220$	12.0000			0.809040
$221$	−3.00000			−0.201802
$222$	6.00000	+	10.3923i	0.402694	+	0.697486i
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	−1.00000			−0.0668153
$225$	−10.5000	+	18.1865i	−0.700000	+	1.21244i
$226$	−12.0000			−0.798228
$227$	3.00000			0.199117			0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000			0.199117			0.0995585	+	0.995032i	$0.468257\pi$
$228$	−7.50000	+	0.866025i	−0.496700	+	0.0573539i
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$	−18.0000			−1.18688
$231$	−3.00000	−	5.19615i	−0.197386	−	0.341882i
$232$	9.00000			0.590879
$233$			6.92820i			0.453882i	0.973909	+	0.226941i	$0.0728724\pi$
$233$			6.92820i			0.453882i	−0.973909	+	0.226941i	$0.927128\pi$
$234$	−4.50000	−	2.59808i	−0.294174	−	0.169842i
$235$	−12.0000			−0.782794
$236$	3.00000			0.195283
$237$	−6.00000	−	10.3923i	−0.389742	−	0.675053i
$238$		−	1.73205i		−	0.112272i
$239$		−	12.1244i		−	0.784259i	−0.919910	−	0.392130i	$-0.871738\pi$
$239$		−	12.1244i		−	0.784259i	0.919910	−	0.392130i	$-0.128262\pi$
$240$	3.00000	+	5.19615i	0.193649	+	0.335410i
$241$		−	13.8564i		−	0.892570i	−0.894891	−	0.446285i	$-0.852747\pi$
$241$		−	13.8564i		−	0.892570i	0.894891	−	0.446285i	$-0.147253\pi$
$242$	1.00000			0.0642824
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$	−8.00000			−0.512148
$245$		−	20.7846i		−	1.32788i
$246$		0			0
$247$	3.00000	−	6.92820i	0.190885	−	0.440831i
$248$		−	10.3923i		−	0.659912i
$249$	9.00000	+	15.5885i	0.570352	+	0.987878i
$250$			6.92820i			0.438178i
$251$		−	13.8564i		−	0.874609i	−0.899314	−	0.437304i	$-0.855933\pi$
$251$		−	13.8564i		−	0.874609i	0.899314	−	0.437304i	$-0.144067\pi$
$252$	1.50000	−	2.59808i	0.0944911	−	0.163663i
$253$	−18.0000			−1.13165
$254$		0			0
$255$	−9.00000	+	5.19615i	−0.563602	+	0.325396i
$256$	1.00000			0.0625000
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$	−3.00000	+	1.73205i	−0.186772	+	0.107833i
$259$		−	6.92820i		−	0.430498i
$260$	−6.00000			−0.372104
$261$	−13.5000	+	23.3827i	−0.835629	+	1.44735i
$262$		−	3.46410i		−	0.214013i
$263$		−	3.46410i		−	0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$263$		−	3.46410i		−	0.213606i	0.994280	−	0.106803i	$-0.0340614\pi$
$264$	3.00000	+	5.19615i	0.184637	+	0.319801i
$265$			31.1769i			1.91518i
$266$	4.00000	+	1.73205i	0.245256	+	0.106199i
$267$	−9.00000	+	5.19615i	−0.550791	+	0.317999i
$268$		−	8.66025i		−	0.529009i
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$	−18.0000			−1.09545
$271$	−1.00000			−0.0607457			−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000			−0.0607457			−0.0303728	+	0.999539i	$0.509669\pi$
$272$			1.73205i			0.105021i
$273$	1.50000	+	2.59808i	0.0907841	+	0.157243i
$274$		−	8.66025i		−	0.523185i
$275$			24.2487i			1.46225i
$276$	−4.50000	−	7.79423i	−0.270868	−	0.469157i
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$	14.0000			0.839664
$279$	27.0000	+	15.5885i	1.61645	+	0.933257i
$280$		−	3.46410i		−	0.207020i
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$	−3.00000	−	5.19615i	−0.178647	−	0.309426i
$283$	−10.0000			−0.594438			−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000			−0.594438			−0.297219	+	0.954809i	$0.596059\pi$
$284$	12.0000			0.712069
$285$	−3.00000	−	25.9808i	−0.177705	−	1.53897i
$286$	−6.00000			−0.354787
$287$		0			0
$288$	−1.50000	+	2.59808i	−0.0883883	+	0.153093i
$289$	14.0000			0.823529
$290$			31.1769i			1.83077i
$291$	12.0000	+	20.7846i	0.703452	+	1.21842i
$292$	11.0000			0.643726
$293$	21.0000			1.22683			0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000			1.22683			0.613417	+	0.789760i	$0.289795\pi$
$294$	9.00000	−	5.19615i	0.524891	−	0.303046i
$295$			10.3923i			0.605063i
$296$			6.92820i			0.402694i
$297$	−18.0000			−1.04447
$298$		−	6.92820i		−	0.401340i
$299$	9.00000			0.520483
$300$	−10.5000	+	6.06218i	−0.606218	+	0.350000i
$301$	2.00000			0.115278
$302$		−	3.46410i		−	0.199337i
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$	−4.00000	−	1.73205i	−0.229416	−	0.0993399i
$305$		−	27.7128i		−	1.58683i
$306$	−4.50000	−	2.59808i	−0.257248	−	0.148522i
$307$		−	10.3923i		−	0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$		−	10.3923i		−	0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$		−	3.46410i		−	0.197386i
$309$	−3.00000	−	5.19615i	−0.170664	−	0.295599i
$310$	36.0000			2.04466
$311$		−	22.5167i		−	1.27680i	−0.769704	−	0.638401i	$-0.779596\pi$
$311$		−	22.5167i		−	1.27680i	0.769704	−	0.638401i	$-0.220404\pi$
$312$	−1.50000	−	2.59808i	−0.0849208	−	0.147087i
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$	−4.00000			−0.225733
$315$	9.00000	+	5.19615i	0.507093	+	0.292770i
$316$		−	6.92820i		−	0.389742i
$317$	−3.00000			−0.168497			−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000			−0.168497			−0.0842484	+	0.996445i	$0.526849\pi$
$318$	−13.5000	+	7.79423i	−0.757042	+	0.437079i
$319$			31.1769i			1.74557i
$320$			3.46410i			0.193649i
$321$	−4.50000	+	2.59808i	−0.251166	+	0.145010i
$322$			5.19615i			0.289570i
$323$	3.00000	−	6.92820i	0.166924	−	0.385496i
$324$	−4.50000	−	7.79423i	−0.250000	−	0.433013i
$325$		−	12.1244i		−	0.672538i
$326$	−10.0000			−0.553849
$327$	13.5000	+	23.3827i	0.746552	+	1.29307i
$328$		0			0
$329$			3.46410i			0.190982i
$330$	−18.0000	+	10.3923i	−0.990867	+	0.572078i
$331$		−	19.0526i		−	1.04722i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$		−	19.0526i		−	1.04722i	0.851957	−	0.523612i	$-0.175416\pi$
$332$			10.3923i			0.570352i
$333$	−18.0000	−	10.3923i	−0.986394	−	0.569495i
$334$		0			0
$335$	30.0000			1.63908
$336$	1.50000	−	0.866025i	0.0818317	−	0.0472456i
$337$		−	24.2487i		−	1.32091i	−0.750865	−	0.660456i	$-0.770363\pi$
$337$		−	24.2487i		−	1.32091i	0.750865	−	0.660456i	$-0.229637\pi$
$338$	−10.0000			−0.543928
$339$	18.0000	−	10.3923i	0.977626	−	0.564433i
$340$	−6.00000			−0.325396
$341$	36.0000			1.94951
$342$	10.5000	−	7.79423i	0.567775	−	0.421464i
$343$	−13.0000			−0.701934
$344$	−2.00000			−0.107833
$345$	27.0000	−	15.5885i	1.45363	−	0.839254i
$346$	6.00000			0.322562
$347$		−	27.7128i		−	1.48770i	−0.668346	−	0.743851i	$-0.732997\pi$
$347$		−	27.7128i		−	1.48770i	0.668346	−	0.743851i	$-0.267003\pi$
$348$	−13.5000	+	7.79423i	−0.723676	+	0.417815i
$349$	−28.0000			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$350$	7.00000			0.374166
$351$	9.00000			0.480384
$352$			3.46410i			0.184637i
$353$			25.9808i			1.38282i	0.722464	+	0.691408i	$0.243009\pi$
$353$			25.9808i			1.38282i	−0.722464	+	0.691408i	$0.756991\pi$
$354$	−4.50000	+	2.59808i	−0.239172	+	0.138086i
$355$			41.5692i			2.20627i
$356$	−6.00000			−0.317999
$357$	1.50000	+	2.59808i	0.0793884	+	0.137505i
$358$	12.0000			0.634220
$359$			19.0526i			1.00556i	0.864416	+	0.502778i	$0.167689\pi$
$359$			19.0526i			1.00556i	−0.864416	+	0.502778i	$0.832311\pi$
$360$	−9.00000	−	5.19615i	−0.474342	−	0.273861i
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		−	13.8564i		−	0.728277i
$363$	−1.50000	+	0.866025i	−0.0787296	+	0.0454545i
$364$			1.73205i			0.0907841i
$365$			38.1051i			1.99451i
$366$	12.0000	−	6.92820i	0.627250	−	0.362143i
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$		−	5.19615i		−	0.270868i
$369$		0			0
$370$	−24.0000			−1.24770
$371$	9.00000			0.467257
$372$	9.00000	+	15.5885i	0.466628	+	0.808224i
$373$			29.4449i			1.52460i	0.647225	+	0.762299i	$0.275929\pi$
$373$			29.4449i			1.52460i	−0.647225	+	0.762299i	$0.724071\pi$
$374$	−6.00000			−0.310253
$375$	−6.00000	−	10.3923i	−0.309839	−	0.536656i
$376$		−	3.46410i		−	0.178647i
$377$		−	15.5885i		−	0.802846i
$378$			5.19615i			0.267261i
$379$		−	12.1244i		−	0.622786i	−0.950281	−	0.311393i	$-0.899204\pi$
$379$		−	12.1244i		−	0.622786i	0.950281	−	0.311393i	$-0.100796\pi$
$380$	6.00000	−	13.8564i	0.307794	−	0.710819i
$381$		0			0
$382$		−	19.0526i		−	0.974814i
$383$	−6.00000			−0.306586			−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000			−0.306586			−0.153293	+	0.988181i	$0.548988\pi$
$384$	−1.50000	+	0.866025i	−0.0765466	+	0.0441942i
$385$	12.0000			0.611577
$386$		−	3.46410i		−	0.176318i
$387$	3.00000	−	5.19615i	0.152499	−	0.264135i
$388$			13.8564i			0.703452i
$389$		−	3.46410i		−	0.175637i	−0.996136	−	0.0878185i	$-0.972010\pi$
$389$		−	3.46410i		−	0.175637i	0.996136	−	0.0878185i	$-0.0279895\pi$
$390$	9.00000	−	5.19615i	0.455733	−	0.263117i
$391$	9.00000			0.455150
$392$	6.00000			0.303046
$393$	3.00000	+	5.19615i	0.151330	+	0.262111i
$394$		0			0
$395$	24.0000			1.20757
$396$	−9.00000	−	5.19615i	−0.452267	−	0.261116i
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$	−11.0000			−0.551380
$399$	−7.50000	+	0.866025i	−0.375470	+	0.0433555i
$400$	−7.00000			−0.350000
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$	7.50000	+	12.9904i	0.374066	+	0.647901i
$403$	−18.0000			−0.896644
$404$		−	10.3923i		−	0.517036i
$405$	27.0000	−	15.5885i	1.34164	−	0.774597i
$406$	9.00000			0.446663
$407$	−24.0000			−1.18964
$408$	−1.50000	−	2.59808i	−0.0742611	−	0.128624i
$409$		−	38.1051i		−	1.88418i	−0.335365	−	0.942088i	$-0.608860\pi$
$409$		−	38.1051i		−	1.88418i	0.335365	−	0.942088i	$-0.391140\pi$
$410$		0			0
$411$	7.50000	+	12.9904i	0.369948	+	0.640768i
$412$		−	3.46410i		−	0.170664i
$413$	3.00000			0.147620
$414$	13.5000	+	7.79423i	0.663489	+	0.383065i
$415$	−36.0000			−1.76717
$416$		−	1.73205i		−	0.0849208i
$417$	−21.0000	+	12.1244i	−1.02837	+	0.593732i
$418$	6.00000	−	13.8564i	0.293470	−	0.677739i
$419$			10.3923i			0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$			10.3923i			0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$	3.00000	+	5.19615i	0.146385	+	0.253546i
$421$			12.1244i			0.590905i	0.955357	+	0.295452i	$0.0954704\pi$
$421$			12.1244i			0.590905i	−0.955357	+	0.295452i	$0.904530\pi$
$422$			12.1244i			0.590204i
$423$	9.00000	+	5.19615i	0.437595	+	0.252646i
$424$	−9.00000			−0.437079
$425$		−	12.1244i		−	0.588118i
$426$	−18.0000	+	10.3923i	−0.872103	+	0.503509i
$427$	−8.00000			−0.387147
$428$	−3.00000			−0.145010
$429$	9.00000	−	5.19615i	0.434524	−	0.250873i
$430$		−	6.92820i		−	0.334108i
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		−	5.19615i		−	0.250000i
$433$		−	10.3923i		−	0.499422i	−0.968320	−	0.249711i	$-0.919664\pi$
$433$		−	10.3923i		−	0.499422i	0.968320	−	0.249711i	$-0.0803357\pi$
$434$		−	10.3923i		−	0.498847i
$435$	−27.0000	−	46.7654i	−1.29455	−	2.24223i
$436$			15.5885i			0.746552i
$437$	−9.00000	+	20.7846i	−0.430528	+	0.994263i
$438$	−16.5000	+	9.52628i	−0.788400	+	0.455183i
$439$			20.7846i			0.991995i	0.868324	+	0.495998i	$0.165198\pi$
$439$			20.7846i			0.991995i	−0.868324	+	0.495998i	$0.834802\pi$
$440$	−12.0000			−0.572078
$441$	−9.00000	+	15.5885i	−0.428571	+	0.742307i
$442$	3.00000			0.142695
$443$		−	17.3205i		−	0.822922i	−0.911427	−	0.411461i	$-0.865019\pi$
$443$		−	17.3205i		−	0.822922i	0.911427	−	0.411461i	$-0.134981\pi$
$444$	−6.00000	−	10.3923i	−0.284747	−	0.493197i
$445$		−	20.7846i		−	0.985285i
$446$			10.3923i			0.492090i
$447$	6.00000	+	10.3923i	0.283790	+	0.491539i
$448$	1.00000			0.0472456
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	10.5000	−	18.1865i	0.494975	−	0.857321i
$451$		0			0
$452$	12.0000			0.564433
$453$	3.00000	+	5.19615i	0.140952	+	0.244137i
$454$	−3.00000			−0.140797
$455$	−6.00000			−0.281284
$456$	7.50000	−	0.866025i	0.351220	−	0.0405554i
$457$	−25.0000			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$458$	8.00000			0.373815
$459$	9.00000			0.420084
$460$	18.0000			0.839254
$461$		−	13.8564i		−	0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$		−	13.8564i		−	0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$	3.00000	+	5.19615i	0.139573	+	0.241747i
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$	−9.00000			−0.417815
$465$	−54.0000	+	31.1769i	−2.50419	+	1.44579i
$466$		−	6.92820i		−	0.320943i
$467$			27.7128i			1.28240i	0.767375	+	0.641198i	$0.221562\pi$
$467$			27.7128i			1.28240i	−0.767375	+	0.641198i	$0.778438\pi$
$468$	4.50000	+	2.59808i	0.208013	+	0.120096i
$469$		−	8.66025i		−	0.399893i
$470$	12.0000			0.553519
$471$	6.00000	−	3.46410i	0.276465	−	0.159617i
$472$	−3.00000			−0.138086
$473$		−	6.92820i		−	0.318559i
$474$	6.00000	+	10.3923i	0.275589	+	0.477334i
$475$	28.0000	+	12.1244i	1.28473	+	0.556304i
$476$			1.73205i			0.0793884i
$477$	13.5000	−	23.3827i	0.618123	−	1.07062i
$478$			12.1244i			0.554555i
$479$		−	10.3923i		−	0.474837i	−0.971408	−	0.237418i	$-0.923699\pi$
$479$		−	10.3923i		−	0.474837i	0.971408	−	0.237418i	$-0.0763012\pi$
$480$	−3.00000	−	5.19615i	−0.136931	−	0.237171i
$481$	12.0000			0.547153
$482$			13.8564i			0.631142i
$483$	−4.50000	−	7.79423i	−0.204757	−	0.354650i
$484$	−1.00000			−0.0454545
$485$	−48.0000			−2.17957
$486$	13.5000	+	7.79423i	0.612372	+	0.353553i
$487$			3.46410i			0.156973i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$			3.46410i			0.156973i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$	8.00000			0.362143
$489$	15.0000	−	8.66025i	0.678323	−	0.391630i
$490$			20.7846i			0.938953i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		−	15.5885i		−	0.702069i
$494$	−3.00000	+	6.92820i	−0.134976	+	0.311715i
$495$	18.0000	−	31.1769i	0.809040	−	1.40130i
$496$			10.3923i			0.466628i
$497$	12.0000			0.538274
$498$	−9.00000	−	15.5885i	−0.403300	−	0.698535i
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$		−	6.92820i		−	0.309839i
$501$		0			0
$502$			13.8564i			0.618442i
$503$		−	25.9808i		−	1.15842i	−0.815177	−	0.579212i	$-0.803360\pi$
$503$		−	25.9808i		−	1.15842i	0.815177	−	0.579212i	$-0.196640\pi$
$504$	−1.50000	+	2.59808i	−0.0668153	+	0.115728i
$505$	36.0000			1.60198
$506$	18.0000			0.800198
$507$	15.0000	−	8.66025i	0.666173	−	0.384615i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$	9.00000	−	5.19615i	0.398527	−	0.230089i
$511$	11.0000			0.486611
$512$	−1.00000			−0.0441942
$513$	−9.00000	+	20.7846i	−0.397360	+	0.917663i
$514$	6.00000			0.264649
$515$	12.0000			0.528783
$516$	3.00000	−	1.73205i	0.132068	−	0.0762493i
$517$	12.0000			0.527759
$518$			6.92820i			0.304408i
$519$	−9.00000	+	5.19615i	−0.395056	+	0.228086i
$520$	6.00000			0.263117
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$	13.5000	−	23.3827i	0.590879	−	1.02343i
$523$			32.9090i			1.43901i	0.694488	+	0.719504i	$0.255631\pi$
$523$			32.9090i			1.43901i	−0.694488	+	0.719504i	$0.744369\pi$
$524$			3.46410i			0.151330i
$525$	−10.5000	+	6.06218i	−0.458258	+	0.264575i
$526$			3.46410i			0.151042i
$527$	−18.0000			−0.784092
$528$	−3.00000	−	5.19615i	−0.130558	−	0.226134i
$529$	−4.00000			−0.173913
$530$		−	31.1769i		−	1.35424i
$531$	4.50000	−	7.79423i	0.195283	−	0.338241i
$532$	−4.00000	−	1.73205i	−0.173422	−	0.0750939i
$533$		0			0
$534$	9.00000	−	5.19615i	0.389468	−	0.224860i
$535$		−	10.3923i		−	0.449299i
$536$			8.66025i			0.374066i
$537$	−18.0000	+	10.3923i	−0.776757	+	0.448461i
$538$	18.0000			0.776035
$539$			20.7846i			0.895257i
$540$	18.0000			0.774597
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$	1.00000			0.0429537
$543$	12.0000	+	20.7846i	0.514969	+	0.891953i
$544$		−	1.73205i		−	0.0742611i
$545$	−54.0000			−2.31311
$546$	−1.50000	−	2.59808i	−0.0641941	−	0.111187i
$547$		−	24.2487i		−	1.03680i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$		−	24.2487i		−	1.03680i	0.855138	−	0.518400i	$-0.173472\pi$
$548$			8.66025i			0.369948i
$549$	−12.0000	+	20.7846i	−0.512148	+	0.887066i
$550$		−	24.2487i		−	1.03397i
$551$	36.0000	+	15.5885i	1.53365	+	0.664091i
$552$	4.50000	+	7.79423i	0.191533	+	0.331744i
$553$		−	6.92820i		−	0.294617i
$554$	2.00000			0.0849719
$555$	36.0000	−	20.7846i	1.52811	−	0.882258i
$556$	−14.0000			−0.593732
$557$			45.0333i			1.90812i	0.299611	+	0.954062i	$0.403143\pi$
$557$			45.0333i			1.90812i	−0.299611	+	0.954062i	$0.596857\pi$
$558$	−27.0000	−	15.5885i	−1.14300	−	0.659912i
$559$			3.46410i			0.146516i
$560$			3.46410i			0.146385i
$561$	9.00000	−	5.19615i	0.379980	−	0.219382i
$562$	−12.0000			−0.506189
$563$	24.0000			1.01148			0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000			1.01148			0.505740	+	0.862686i	$0.331220\pi$
$564$	3.00000	+	5.19615i	0.126323	+	0.218797i
$565$			41.5692i			1.74883i
$566$	10.0000			0.420331
$567$	−4.50000	−	7.79423i	−0.188982	−	0.327327i
$568$	−12.0000			−0.503509
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$	3.00000	+	25.9808i	0.125656	+	1.08821i
$571$	−16.0000			−0.669579			−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000			−0.669579			−0.334790	+	0.942293i	$0.608665\pi$
$572$	6.00000			0.250873
$573$	16.5000	+	28.5788i	0.689297	+	1.19390i
$574$		0			0
$575$			36.3731i			1.51686i
$576$	1.50000	−	2.59808i	0.0625000	−	0.108253i
$577$	25.0000			1.04076			0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000			1.04076			0.520382	+	0.853934i	$0.325790\pi$
$578$	−14.0000			−0.582323
$579$	3.00000	+	5.19615i	0.124676	+	0.215945i
$580$		−	31.1769i		−	1.29455i
$581$			10.3923i			0.431145i
$582$	−12.0000	−	20.7846i	−0.497416	−	0.861550i
$583$		−	31.1769i		−	1.29122i
$584$	−11.0000			−0.455183
$585$	−9.00000	+	15.5885i	−0.372104	+	0.644503i
$586$	−21.0000			−0.867502
$587$		−	13.8564i		−	0.571915i	−0.958242	−	0.285958i	$-0.907688\pi$
$587$		−	13.8564i		−	0.571915i	0.958242	−	0.285958i	$-0.0923116\pi$
$588$	−9.00000	+	5.19615i	−0.371154	+	0.214286i
$589$	18.0000	−	41.5692i	0.741677	−	1.71283i
$590$		−	10.3923i		−	0.427844i
$591$		0			0
$592$		−	6.92820i		−	0.284747i
$593$		−	20.7846i		−	0.853522i	−0.904365	−	0.426761i	$-0.859655\pi$
$593$		−	20.7846i		−	0.853522i	0.904365	−	0.426761i	$-0.140345\pi$
$594$	18.0000			0.738549
$595$	−6.00000			−0.245976
$596$			6.92820i			0.283790i
$597$	16.5000	−	9.52628i	0.675300	−	0.389885i
$598$	−9.00000			−0.368037
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$	10.5000	−	6.06218i	0.428661	−	0.247487i
$601$		−	10.3923i		−	0.423911i	−0.977279	−	0.211955i	$-0.932017\pi$
$601$		−	10.3923i		−	0.423911i	0.977279	−	0.211955i	$-0.0679832\pi$
$602$	−2.00000			−0.0815139
$603$	−22.5000	−	12.9904i	−0.916271	−	0.529009i
$604$			3.46410i			0.140952i
$605$		−	3.46410i		−	0.140836i
$606$	9.00000	+	15.5885i	0.365600	+	0.633238i
$607$		−	27.7128i		−	1.12483i	−0.826856	−	0.562414i	$-0.809873\pi$
$607$		−	27.7128i		−	1.12483i	0.826856	−	0.562414i	$-0.190127\pi$
$608$	4.00000	+	1.73205i	0.162221	+	0.0702439i
$609$	−13.5000	+	7.79423i	−0.547048	+	0.315838i
$610$			27.7128i			1.12206i
$611$	−6.00000			−0.242734
$612$	4.50000	+	2.59808i	0.181902	+	0.105021i
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$			10.3923i			0.419399i
$615$		0			0
$616$			3.46410i			0.139573i
$617$		−	48.4974i		−	1.95243i	−0.216799	−	0.976216i	$-0.569561\pi$
$617$		−	48.4974i		−	1.95243i	0.216799	−	0.976216i	$-0.430439\pi$
$618$	3.00000	+	5.19615i	0.120678	+	0.209020i
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$	−36.0000			−1.44579
$621$	−27.0000			−1.08347
$622$			22.5167i			0.902836i
$623$	−6.00000			−0.240385
$624$	1.50000	+	2.59808i	0.0600481	+	0.104006i
$625$	−11.0000			−0.440000
$626$	−13.0000			−0.519584
$627$	3.00000	+	25.9808i	0.119808	+	1.03757i
$628$	4.00000			0.159617
$629$	12.0000			0.478471
$630$	−9.00000	−	5.19615i	−0.358569	−	0.207020i
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$			6.92820i			0.275589i
$633$	−10.5000	−	18.1865i	−0.417338	−	0.722850i
$634$	3.00000			0.119145
$635$		0			0
$636$	13.5000	−	7.79423i	0.535310	−	0.309061i
$637$		−	10.3923i		−	0.411758i
$638$		−	31.1769i		−	1.23431i
$639$	18.0000	−	31.1769i	0.712069	−	1.23334i
$640$		−	3.46410i		−	0.136931i
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$	4.50000	−	2.59808i	0.177601	−	0.102538i
$643$	−28.0000			−1.10421			−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000			−1.10421			−0.552106	+	0.833774i	$0.686176\pi$
$644$		−	5.19615i		−	0.204757i
$645$	6.00000	+	10.3923i	0.236250	+	0.409197i
$646$	−3.00000	+	6.92820i	−0.118033	+	0.272587i
$647$			1.73205i			0.0680939i	0.999420	+	0.0340470i	$0.0108396\pi$
$647$			1.73205i			0.0680939i	−0.999420	+	0.0340470i	$0.989160\pi$
$648$	4.50000	+	7.79423i	0.176777	+	0.306186i
$649$		−	10.3923i		−	0.407934i
$650$			12.1244i			0.475556i
$651$	9.00000	+	15.5885i	0.352738	+	0.610960i
$652$	10.0000			0.391630
$653$		−	17.3205i		−	0.677804i	−0.940822	−	0.338902i	$-0.889945\pi$
$653$		−	17.3205i		−	0.677804i	0.940822	−	0.338902i	$-0.110055\pi$
$654$	−13.5000	−	23.3827i	−0.527892	−	0.914335i
$655$	−12.0000			−0.468879
$656$		0			0
$657$	16.5000	−	28.5788i	0.643726	−	1.11497i
$658$		−	3.46410i		−	0.135045i
$659$	33.0000			1.28550			0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000			1.28550			0.642749	+	0.766077i	$0.277794\pi$
$660$	18.0000	−	10.3923i	0.700649	−	0.404520i
$661$			22.5167i			0.875797i	0.899025	+	0.437898i	$0.144277\pi$
$661$			22.5167i			0.875797i	−0.899025	+	0.437898i	$0.855723\pi$
$662$			19.0526i			0.740499i
$663$	−4.50000	+	2.59808i	−0.174766	+	0.100901i
$664$		−	10.3923i		−	0.403300i
$665$	6.00000	−	13.8564i	0.232670	−	0.537328i
$666$	18.0000	+	10.3923i	0.697486	+	0.402694i
$667$			46.7654i			1.81076i
$668$		0			0
$669$	−9.00000	−	15.5885i	−0.347960	−	0.602685i
$670$	−30.0000			−1.15900
$671$			27.7128i			1.06984i
$672$	−1.50000	+	0.866025i	−0.0578638	+	0.0334077i
$673$			48.4974i			1.86944i	0.355387	+	0.934719i	$0.384349\pi$
$673$			48.4974i			1.86944i	−0.355387	+	0.934719i	$0.615651\pi$
$674$			24.2487i			0.934025i
$675$			36.3731i			1.40000i
$676$	10.0000			0.384615
$677$	15.0000			0.576497			0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000			0.576497			0.288248	+	0.957556i	$0.406927\pi$
$678$	−18.0000	+	10.3923i	−0.691286	+	0.399114i
$679$			13.8564i			0.531760i
$680$	6.00000			0.230089
$681$	4.50000	−	2.59808i	0.172440	−	0.0995585i
$682$	−36.0000			−1.37851
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$	−10.5000	+	7.79423i	−0.401478	+	0.298020i
$685$	−30.0000			−1.14624
$686$	13.0000			0.496342
$687$	−12.0000	+	6.92820i	−0.457829	+	0.264327i
$688$	2.00000			0.0762493
$689$			15.5885i			0.593873i
$690$	−27.0000	+	15.5885i	−1.02787	+	0.593442i
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$	−6.00000			−0.228086
$693$	−9.00000	−	5.19615i	−0.341882	−	0.197386i
$694$			27.7128i			1.05196i
$695$		−	48.4974i		−	1.83961i
$696$	13.5000	−	7.79423i	0.511716	−	0.295439i
$697$		0			0
$698$	28.0000			1.05982
$699$	6.00000	+	10.3923i	0.226941	+	0.393073i
$700$	−7.00000			−0.264575
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−9.00000			−0.339683
$703$	−12.0000	+	27.7128i	−0.452589	+	1.04521i
$704$		−	3.46410i		−	0.130558i
$705$	−18.0000	+	10.3923i	−0.677919	+	0.391397i
$706$		−	25.9808i		−	0.977799i
$707$		−	10.3923i		−	0.390843i
$708$	4.50000	−	2.59808i	0.169120	−	0.0976417i
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		−	41.5692i		−	1.56007i
$711$	−18.0000	−	10.3923i	−0.675053	−	0.389742i
$712$	6.00000			0.224860
$713$	54.0000			2.02232
$714$	−1.50000	−	2.59808i	−0.0561361	−	0.0972306i
$715$			20.7846i			0.777300i
$716$	−12.0000			−0.448461
$717$	−10.5000	−	18.1865i	−0.392130	−	0.679189i
$718$		−	19.0526i		−	0.711035i
$719$		−	8.66025i		−	0.322973i	−0.986875	−	0.161486i	$-0.948371\pi$
$719$		−	8.66025i		−	0.322973i	0.986875	−	0.161486i	$-0.0516288\pi$
$720$	9.00000	+	5.19615i	0.335410	+	0.193649i
$721$		−	3.46410i		−	0.129010i
$722$	−13.0000	−	13.8564i	−0.483810	−	0.515682i
$723$	−12.0000	−	20.7846i	−0.446285	−	0.772988i
$724$			13.8564i			0.514969i
$725$	63.0000			2.33976
$726$	1.50000	−	0.866025i	0.0556702	−	0.0321412i
$727$	−7.00000			−0.259616			−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000			−0.259616			−0.129808	+	0.991539i	$0.541436\pi$
$728$		−	1.73205i		−	0.0641941i
$729$	−27.0000			−1.00000
$730$		−	38.1051i		−	1.41033i
$731$			3.46410i			0.128124i
$732$	−12.0000	+	6.92820i	−0.443533	+	0.256074i
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$	−32.0000			−1.18114
$735$	−18.0000	−	31.1769i	−0.663940	−	1.14998i
$736$			5.19615i			0.191533i
$737$	−30.0000			−1.10506
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$	24.0000			0.882258
$741$	−1.50000	−	12.9904i	−0.0551039	−	0.477214i
$742$	−9.00000			−0.330400
$743$	−30.0000			−1.10059			−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000			−1.10059			−0.550297	+	0.834969i	$0.685485\pi$
$744$	−9.00000	−	15.5885i	−0.329956	−	0.571501i
$745$	−24.0000			−0.879292
$746$		−	29.4449i		−	1.07805i
$747$	27.0000	+	15.5885i	0.987878	+	0.570352i
$748$	6.00000			0.219382
$749$	−3.00000			−0.109618
$750$	6.00000	+	10.3923i	0.219089	+	0.379473i
$751$		−	6.92820i		−	0.252814i	−0.991978	−	0.126407i	$-0.959656\pi$
$751$		−	6.92820i		−	0.252814i	0.991978	−	0.126407i	$-0.0403445\pi$
$752$			3.46410i			0.126323i
$753$	−12.0000	−	20.7846i	−0.437304	−	0.757433i
$754$			15.5885i			0.567698i
$755$	−12.0000			−0.436725
$756$		−	5.19615i		−	0.188982i
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$			12.1244i			0.440376i
$759$	−27.0000	+	15.5885i	−0.980038	+	0.565825i
$760$	−6.00000	+	13.8564i	−0.217643	+	0.502625i
$761$		−	8.66025i		−	0.313934i	−0.987604	−	0.156967i	$-0.949828\pi$
$761$		−	8.66025i		−	0.313934i	0.987604	−	0.156967i	$-0.0501716\pi$
$762$		0			0
$763$			15.5885i			0.564340i
$764$			19.0526i			0.689297i
$765$	−9.00000	+	15.5885i	−0.325396	+	0.563602i
$766$	6.00000			0.216789
$767$			5.19615i			0.187622i
$768$	1.50000	−	0.866025i	0.0541266	−	0.0312500i
$769$	49.0000			1.76699			0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000			1.76699			0.883493	+	0.468445i	$0.155186\pi$
$770$	−12.0000			−0.432450
$771$	−9.00000	+	5.19615i	−0.324127	+	0.187135i
$772$			3.46410i			0.124676i
$773$	3.00000			0.107903			0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000			0.107903			0.0539513	+	0.998544i	$0.482818\pi$
$774$	−3.00000	+	5.19615i	−0.107833	+	0.186772i
$775$		−	72.7461i		−	2.61312i
$776$		−	13.8564i		−	0.497416i
$777$	−6.00000	−	10.3923i	−0.215249	−	0.372822i
$778$			3.46410i			0.124194i
$779$		0			0
$780$	−9.00000	+	5.19615i	−0.322252	+	0.186052i
$781$		−	41.5692i		−	1.48746i
$782$	−9.00000			−0.321839
$783$			46.7654i			1.67126i
$784$	−6.00000			−0.214286
$785$			13.8564i			0.494556i
$786$	−3.00000	−	5.19615i	−0.107006	−	0.185341i
$787$			22.5167i			0.802632i	0.915940	+	0.401316i	$0.131447\pi$
$787$			22.5167i			0.802632i	−0.915940	+	0.401316i	$0.868553\pi$
$788$		0			0
$789$	−3.00000	−	5.19615i	−0.106803	−	0.184988i
$790$	−24.0000			−0.853882
$791$	12.0000			0.426671
$792$	9.00000	+	5.19615i	0.319801	+	0.184637i
$793$		−	13.8564i		−	0.492055i
$794$	22.0000			0.780751
$795$	27.0000	+	46.7654i	0.957591	+	1.65860i
$796$	11.0000			0.389885
$797$	21.0000			0.743858			0.371929	−	0.928261i	$-0.378696\pi$
$797$	21.0000			0.743858			0.371929	+	0.928261i	$0.378696\pi$
$798$	7.50000	−	0.866025i	0.265497	−	0.0306570i
$799$	−6.00000			−0.212265
$800$	7.00000			0.247487
$801$	−9.00000	+	15.5885i	−0.317999	+	0.550791i
$802$	24.0000			0.847469
$803$		−	38.1051i		−	1.34470i
$804$	−7.50000	−	12.9904i	−0.264505	−	0.458135i
$805$	18.0000			0.634417
$806$	18.0000			0.634023
$807$	−27.0000	+	15.5885i	−0.950445	+	0.548740i
$808$			10.3923i			0.365600i
$809$			32.9090i			1.15702i	0.815676	+	0.578509i	$0.196365\pi$
$809$			32.9090i			1.15702i	−0.815676	+	0.578509i	$0.803635\pi$
$810$	−27.0000	+	15.5885i	−0.948683	+	0.547723i
$811$		−	46.7654i		−	1.64215i	−0.570817	−	0.821077i	$-0.693374\pi$
$811$		−	46.7654i		−	1.64215i	0.570817	−	0.821077i	$-0.306626\pi$
$812$	−9.00000			−0.315838
$813$	−1.50000	+	0.866025i	−0.0526073	+	0.0303728i
$814$	24.0000			0.841200
$815$			34.6410i			1.21342i
$816$	1.50000	+	2.59808i	0.0525105	+	0.0909509i
$817$	−8.00000	−	3.46410i	−0.279885	−	0.121194i
$818$			38.1051i			1.33231i
$819$	4.50000	+	2.59808i	0.157243	+	0.0907841i
$820$		0			0
$821$		−	17.3205i		−	0.604490i	−0.953230	−	0.302245i	$-0.902264\pi$
$821$		−	17.3205i		−	0.604490i	0.953230	−	0.302245i	$-0.0977361\pi$
$822$	−7.50000	−	12.9904i	−0.261593	−	0.453092i
$823$	−23.0000			−0.801730			−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000			−0.801730			−0.400865	+	0.916137i	$0.631290\pi$
$824$			3.46410i			0.120678i
$825$	21.0000	+	36.3731i	0.731126	+	1.26635i
$826$	−3.00000			−0.104383
$827$	33.0000			1.14752			0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000			1.14752			0.573761	+	0.819023i	$0.305484\pi$
$828$	−13.5000	−	7.79423i	−0.469157	−	0.270868i
$829$		−	1.73205i		−	0.0601566i	−0.999548	−	0.0300783i	$-0.990424\pi$
$829$		−	1.73205i		−	0.0601566i	0.999548	−	0.0300783i	$-0.00957567\pi$
$830$	36.0000			1.24958
$831$	−3.00000	+	1.73205i	−0.104069	+	0.0600842i
$832$			1.73205i			0.0600481i
$833$		−	10.3923i		−	0.360072i
$834$	21.0000	−	12.1244i	0.727171	−	0.419832i
$835$		0			0
$836$	−6.00000	+	13.8564i	−0.207514	+	0.479234i
$837$	54.0000			1.86651
$838$		−	10.3923i		−	0.358996i
$839$	−42.0000			−1.45000			−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000			−1.45000			−0.725001	+	0.688748i	$0.758161\pi$
$840$	−3.00000	−	5.19615i	−0.103510	−	0.179284i
$841$	52.0000			1.79310
$842$		−	12.1244i		−	0.417833i
$843$	18.0000	−	10.3923i	0.619953	−	0.357930i
$844$		−	12.1244i		−	0.417338i
$845$			34.6410i			1.19169i
$846$	−9.00000	−	5.19615i	−0.309426	−	0.178647i
$847$	−1.00000			−0.0343604
$848$	9.00000			0.309061
$849$	−15.0000	+	8.66025i	−0.514799	+	0.297219i
$850$			12.1244i			0.415862i
$851$	−36.0000			−1.23406
$852$	18.0000	−	10.3923i	0.616670	−	0.356034i
$853$	14.0000			0.479351			0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000			0.479351			0.239675	+	0.970853i	$0.422959\pi$
$854$	8.00000			0.273754
$855$	−27.0000	−	36.3731i	−0.923381	−	1.24393i
$856$	3.00000			0.102538
$857$	24.0000			0.819824			0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000			0.819824			0.409912	+	0.912125i	$0.365559\pi$
$858$	−9.00000	+	5.19615i	−0.307255	+	0.177394i
$859$	−50.0000			−1.70598			−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000			−1.70598			−0.852989	+	0.521929i	$0.825213\pi$
$860$			6.92820i			0.236250i
$861$		0			0
$862$	12.0000			0.408722
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$			5.19615i			0.176777i
$865$		−	20.7846i		−	0.706698i
$866$			10.3923i			0.353145i
$867$	21.0000	−	12.1244i	0.713197	−	0.411765i
$868$			10.3923i			0.352738i
$869$	−24.0000			−0.814144
$870$	27.0000	+	46.7654i	0.915386	+	1.58549i
$871$	15.0000			0.508256
$872$		−	15.5885i		−	0.527892i
$873$	36.0000	+	20.7846i	1.21842	+	0.703452i
$874$	9.00000	−	20.7846i	0.304430	−	0.703050i
$875$		−	6.92820i		−	0.234216i
$876$	16.5000	−	9.52628i	0.557483	−	0.321863i
$877$			53.6936i			1.81310i	0.422095	+	0.906552i	$0.361295\pi$
$877$			53.6936i			1.81310i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		−	20.7846i		−	0.701447i
$879$	31.5000	−	18.1865i	1.06247	−	0.613417i
$880$	12.0000			0.404520
$881$			34.6410i			1.16709i	0.812082	+	0.583543i	$0.198334\pi$
$881$			34.6410i			1.16709i	−0.812082	+	0.583543i	$0.801666\pi$
$882$	9.00000	−	15.5885i	0.303046	−	0.524891i
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$	−3.00000			−0.100901
$885$	9.00000	+	15.5885i	0.302532	+	0.524000i
$886$			17.3205i			0.581894i
$887$	36.0000			1.20876			0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000			1.20876			0.604381	+	0.796696i	$0.293421\pi$
$888$	6.00000	+	10.3923i	0.201347	+	0.348743i
$889$		0			0
$890$			20.7846i			0.696702i
$891$	−27.0000	+	15.5885i	−0.904534	+	0.522233i
$892$		−	10.3923i		−	0.347960i
$893$	6.00000	−	13.8564i	0.200782	−	0.463687i
$894$	−6.00000	−	10.3923i	−0.200670	−	0.347571i
$895$		−	41.5692i		−	1.38951i
$896$	−1.00000			−0.0334077
$897$	13.5000	−	7.79423i	0.450752	−	0.260242i
$898$	30.0000			1.00111
$899$		−	93.5307i		−	3.11942i
$900$	−10.5000	+	18.1865i	−0.350000	+	0.606218i
$901$			15.5885i			0.519327i
$902$		0			0
$903$	3.00000	−	1.73205i	0.0998337	−	0.0576390i
$904$	−12.0000			−0.399114
$905$	−48.0000			−1.59557
$906$	−3.00000	−	5.19615i	−0.0996683	−	0.172631i
$907$		−	8.66025i		−	0.287559i	−0.989610	−	0.143780i	$-0.954074\pi$
$907$		−	8.66025i		−	0.287559i	0.989610	−	0.143780i	$-0.0459256\pi$
$908$	3.00000			0.0995585
$909$	−27.0000	−	15.5885i	−0.895533	−	0.517036i
$910$	6.00000			0.198898
$911$	−30.0000			−0.993944			−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000			−0.993944			−0.496972	+	0.867766i	$0.665555\pi$
$912$	−7.50000	+	0.866025i	−0.248350	+	0.0286770i
$913$	36.0000			1.19143
$914$	25.0000			0.826927
$915$	−24.0000	−	41.5692i	−0.793416	−	1.37424i
$916$	−8.00000			−0.264327
$917$			3.46410i			0.114395i
$918$	−9.00000			−0.297044
$919$	−11.0000			−0.362857			−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000			−0.362857			−0.181428	+	0.983404i	$0.558072\pi$
$920$	−18.0000			−0.593442
$921$	−9.00000	−	15.5885i	−0.296560	−	0.513657i
$922$			13.8564i			0.456336i
$923$			20.7846i			0.684134i
$924$	−3.00000	−	5.19615i	−0.0986928	−	0.170941i
$925$			48.4974i			1.59459i
$926$	40.0000			1.31448
$927$	−9.00000	−	5.19615i	−0.295599	−	0.170664i
$928$	9.00000			0.295439
$929$			15.5885i			0.511441i	0.966751	+	0.255720i	$0.0823126\pi$
$929$			15.5885i			0.511441i	−0.966751	+	0.255720i	$0.917687\pi$
$930$	54.0000	−	31.1769i	1.77073	−	1.02233i
$931$	24.0000	+	10.3923i	0.786568	+	0.340594i
$932$			6.92820i			0.226941i
$933$	−19.5000	−	33.7750i	−0.638401	−	1.10574i
$934$		−	27.7128i		−	0.906791i
$935$			20.7846i			0.679729i
$936$	−4.50000	−	2.59808i	−0.147087	−	0.0849208i
$937$	−7.00000			−0.228680			−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000			−0.228680			−0.114340	+	0.993442i	$0.536475\pi$
$938$			8.66025i			0.282767i
$939$	19.5000	−	11.2583i	0.636358	−	0.367402i
$940$	−12.0000			−0.391397
$941$	−39.0000			−1.27136			−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000			−1.27136			−0.635682	+	0.771951i	$0.719281\pi$
$942$	−6.00000	+	3.46410i	−0.195491	+	0.112867i
$943$		0			0
$944$	3.00000			0.0976417
$945$	18.0000			0.585540
$946$			6.92820i			0.225255i
$947$			31.1769i			1.01311i	0.862207	+	0.506557i	$0.169082\pi$
$947$			31.1769i			1.01311i	−0.862207	+	0.506557i	$0.830918\pi$
$948$	−6.00000	−	10.3923i	−0.194871	−	0.337526i
$949$			19.0526i			0.618472i
$950$	−28.0000	−	12.1244i	−0.908440	−	0.393366i
$951$	−4.50000	+	2.59808i	−0.145922	+	0.0842484i
$952$		−	1.73205i		−	0.0561361i
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$	−13.5000	+	23.3827i	−0.437079	+	0.757042i
$955$	−66.0000			−2.13571
$956$		−	12.1244i		−	0.392130i
$957$	27.0000	+	46.7654i	0.872786	+	1.51171i
$958$			10.3923i			0.335760i
$959$			8.66025i			0.279654i
$960$	3.00000	+	5.19615i	0.0968246	+	0.167705i
$961$	−77.0000			−2.48387
$962$	−12.0000			−0.386896
$963$	−4.50000	+	7.79423i	−0.145010	+	0.251166i
$964$		−	13.8564i		−	0.446285i
$965$	−12.0000			−0.386294
$966$	4.50000	+	7.79423i	0.144785	+	0.250775i
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$	1.00000			0.0321412
$969$	−1.50000	−	12.9904i	−0.0481869	−	0.417311i
$970$	48.0000			1.54119
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$	−13.5000	−	7.79423i	−0.433013	−	0.250000i
$973$	−14.0000			−0.448819
$974$		−	3.46410i		−	0.110997i
$975$	−10.5000	−	18.1865i	−0.336269	−	0.582435i
$976$	−8.00000			−0.256074
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$	−15.0000	+	8.66025i	−0.479647	+	0.276924i
$979$			20.7846i			0.664279i
$980$		−	20.7846i		−	0.663940i
$981$	40.5000	+	23.3827i	1.29307	+	0.746552i
$982$		0			0
$983$	−60.0000			−1.91370			−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000			−1.91370			−0.956851	+	0.290578i	$0.906153\pi$
$984$		0			0
$985$		0			0
$986$			15.5885i			0.496438i
$987$	3.00000	+	5.19615i	0.0954911	+	0.165395i
$988$	3.00000	−	6.92820i	0.0954427	−	0.220416i
$989$		−	10.3923i		−	0.330456i
$990$	−18.0000	+	31.1769i	−0.572078	+	0.990867i
$991$			3.46410i			0.110041i	0.998485	+	0.0550204i	$0.0175224\pi$
$991$			3.46410i			0.110041i	−0.998485	+	0.0550204i	$0.982478\pi$
$992$		−	10.3923i		−	0.329956i
$993$	−16.5000	−	28.5788i	−0.523612	−	0.906922i
$994$	−12.0000			−0.380617
$995$			38.1051i			1.20801i
$996$	9.00000	+	15.5885i	0.285176	+	0.493939i
$997$	−38.0000			−1.20347			−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000			−1.20347			−0.601736	+	0.798695i	$0.705524\pi$
$998$	−10.0000			−0.316544
$999$	−36.0000			−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.2.b.b.113.1	✓	2
3.2	odd	2		114.2.b.c.113.1	yes	2
4.3	odd	2		912.2.f.a.113.2		2
12.11	even	2		912.2.f.e.113.2		2
19.18	odd	2		114.2.b.c.113.2	yes	2
57.56	even	2	inner	114.2.b.b.113.2	yes	2
76.75	even	2		912.2.f.e.113.1		2
228.227	odd	2		912.2.f.a.113.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.b.b.113.1	✓	2	1.1	even	1	trivial
114.2.b.b.113.2	yes	2	57.56	even	2	inner
114.2.b.c.113.1	yes	2	3.2	odd	2
114.2.b.c.113.2	yes	2	19.18	odd	2
912.2.f.a.113.1		2	228.227	odd	2
912.2.f.a.113.2		2	4.3	odd	2
912.2.f.e.113.1		2	76.75	even	2
912.2.f.e.113.2		2	12.11	even	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:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	114.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +(1.50000 - 0.866025i) q^{3} +1.00000 q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{6} +1.00000 q^{7} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} -3.46410i q^{10} -3.46410i q^{11} +(1.50000 - 0.866025i) q^{12} +1.73205i q^{13} -1.00000 q^{14} +(3.00000 + 5.19615i) q^{15} +1.00000 q^{16} +1.73205i q^{17} +(-1.50000 + 2.59808i) q^{18} +(-4.00000 - 1.73205i) q^{19} +3.46410i q^{20} +(1.50000 - 0.866025i) q^{21} +3.46410i q^{22} -5.19615i q^{23} +(-1.50000 + 0.866025i) q^{24} -7.00000 q^{25} -1.73205i q^{26} -5.19615i q^{27} +1.00000 q^{28} -9.00000 q^{29} +(-3.00000 - 5.19615i) q^{30} +10.3923i q^{31} -1.00000 q^{32} +(-3.00000 - 5.19615i) q^{33} -1.73205i q^{34} +3.46410i q^{35} +(1.50000 - 2.59808i) q^{36} -6.92820i q^{37} +(4.00000 + 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{39} -3.46410i q^{40} +(-1.50000 + 0.866025i) q^{42} +2.00000 q^{43} -3.46410i q^{44} +(9.00000 + 5.19615i) q^{45} +5.19615i q^{46} +3.46410i q^{47} +(1.50000 - 0.866025i) q^{48} -6.00000 q^{49} +7.00000 q^{50} +(1.50000 + 2.59808i) q^{51} +1.73205i q^{52} +9.00000 q^{53} +5.19615i q^{54} +12.0000 q^{55} -1.00000 q^{56} +(-7.50000 + 0.866025i) q^{57} +9.00000 q^{58} +3.00000 q^{59} +(3.00000 + 5.19615i) q^{60} -8.00000 q^{61} -10.3923i q^{62} +(1.50000 - 2.59808i) q^{63} +1.00000 q^{64} -6.00000 q^{65} +(3.00000 + 5.19615i) q^{66} -8.66025i q^{67} +1.73205i q^{68} +(-4.50000 - 7.79423i) q^{69} -3.46410i q^{70} +12.0000 q^{71} +(-1.50000 + 2.59808i) q^{72} +11.0000 q^{73} +6.92820i q^{74} +(-10.5000 + 6.06218i) q^{75} +(-4.00000 - 1.73205i) q^{76} -3.46410i q^{77} +(-1.50000 - 2.59808i) q^{78} -6.92820i q^{79} +3.46410i q^{80} +(-4.50000 - 7.79423i) q^{81} +10.3923i q^{83} +(1.50000 - 0.866025i) q^{84} -6.00000 q^{85} -2.00000 q^{86} +(-13.5000 + 7.79423i) q^{87} +3.46410i q^{88} -6.00000 q^{89} +(-9.00000 - 5.19615i) q^{90} +1.73205i q^{91} -5.19615i q^{92} +(9.00000 + 15.5885i) q^{93} -3.46410i q^{94} +(6.00000 - 13.8564i) q^{95} +(-1.50000 + 0.866025i) q^{96} +13.8564i q^{97} +6.00000 q^{98} +(-9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{12} - 2 q^{14} + 6 q^{15} + 2 q^{16} - 3 q^{18} - 8 q^{19} + 3 q^{21} - 3 q^{24} - 14 q^{25} + 2 q^{28} - 18 q^{29} - 6 q^{30}+ \cdots - 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^7 - 2 * q^8 + 3 * q^9 + 3 * q^12 - 2 * q^14 + 6 * q^15 + 2 * q^16 - 3 * q^18 - 8 * q^19 + 3 * q^21 - 3 * q^24 - 14 * q^25 + 2 * q^28 - 18 * q^29 - 6 * q^30 - 2 * q^32 - 6 * q^33 + 3 * q^36 + 8 * q^38 + 3 * q^39 - 3 * q^42 + 4 * q^43 + 18 * q^45 + 3 * q^48 - 12 * q^49 + 14 * q^50 + 3 * q^51 + 18 * q^53 + 24 * q^55 - 2 * q^56 - 15 * q^57 + 18 * q^58 + 6 * q^59 + 6 * q^60 - 16 * q^61 + 3 * q^63 + 2 * q^64 - 12 * q^65 + 6 * q^66 - 9 * q^69 + 24 * q^71 - 3 * q^72 + 22 * q^73 - 21 * q^75 - 8 * q^76 - 3 * q^78 - 9 * q^81 + 3 * q^84 - 12 * q^85 - 4 * q^86 - 27 * q^87 - 12 * q^89 - 18 * q^90 + 18 * q^93 + 12 * q^95 - 3 * q^96 + 12 * q^98 - 18 * q^99

Introduction

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