LMFDB - Embedded newform 112.2.j.a.27.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,2,Mod(27,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("112.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	112.j (of order $4$, 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:	$0.894324502638$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			27.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	112.27
Dual form			112.2.j.a.83.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.44949 - 2.44949i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.44949 - 2.44949i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +4.89898i q^{10} +(1.00000 - 1.00000i) q^{11} +(2.44949 + 2.44949i) q^{13} +(3.44949 + 1.44949i) q^{14} -4.00000 q^{16} -4.89898i q^{17} +(-3.00000 - 3.00000i) q^{18} +(-4.89898 + 4.89898i) q^{19} +(-4.89898 - 4.89898i) q^{20} +2.00000i q^{22} +4.00000 q^{23} -7.00000i q^{25} -4.89898 q^{26} +(-4.89898 + 2.00000i) q^{28} +(-3.00000 + 3.00000i) q^{29} -4.89898 q^{31} +(4.00000 - 4.00000i) q^{32} +(4.89898 + 4.89898i) q^{34} +(-8.44949 - 3.55051i) q^{35} +6.00000 q^{36} +(5.00000 + 5.00000i) q^{37} -9.79796i q^{38} +9.79796 q^{40} -4.89898 q^{41} +(-5.00000 + 5.00000i) q^{43} +(-2.00000 - 2.00000i) q^{44} +(7.34847 + 7.34847i) q^{45} +(-4.00000 + 4.00000i) q^{46} +4.89898 q^{47} +(-5.00000 + 4.89898i) q^{49} +(7.00000 + 7.00000i) q^{50} +(4.89898 - 4.89898i) q^{52} +(-1.00000 - 1.00000i) q^{53} -4.89898i q^{55} +(2.89898 - 6.89898i) q^{56} -6.00000i q^{58} +(4.89898 + 4.89898i) q^{59} +(-2.44949 - 2.44949i) q^{61} +(4.89898 - 4.89898i) q^{62} +(7.34847 - 3.00000i) q^{63} +8.00000i q^{64} +12.0000 q^{65} +(5.00000 + 5.00000i) q^{67} -9.79796 q^{68} +(12.0000 - 4.89898i) q^{70} +2.00000 q^{71} +(-6.00000 + 6.00000i) q^{72} -9.79796 q^{73} -10.0000 q^{74} +(9.79796 + 9.79796i) q^{76} +(-3.44949 - 1.44949i) q^{77} -4.00000i q^{79} +(-9.79796 + 9.79796i) q^{80} -9.00000 q^{81} +(4.89898 - 4.89898i) q^{82} +(-12.0000 - 12.0000i) q^{85} -10.0000i q^{86} +4.00000 q^{88} -14.6969 q^{90} +(3.55051 - 8.44949i) q^{91} -8.00000i q^{92} +(-4.89898 + 4.89898i) q^{94} +24.0000i q^{95} -4.89898i q^{97} +(0.101021 - 9.89898i) q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^7 + 8 * q^8	$ 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8} + 4 q^{11} + 4 q^{14} - 16 q^{16} - 12 q^{18} + 16 q^{23} - 12 q^{29} + 16 q^{32} - 24 q^{35} + 24 q^{36} + 20 q^{37} - 20 q^{43} - 8 q^{44} - 16 q^{46} - 20 q^{49} + 28 q^{50} - 4 q^{53} - 8 q^{56} + 48 q^{65} + 20 q^{67} + 48 q^{70} + 8 q^{71} - 24 q^{72} - 40 q^{74} - 4 q^{77} - 36 q^{81} - 48 q^{85} + 16 q^{88} + 24 q^{91} + 20 q^{98} + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^7 + 8 * q^8 + 4 * q^11 + 4 * q^14 - 16 * q^16 - 12 * q^18 + 16 * q^23 - 12 * q^29 + 16 * q^32 - 24 * q^35 + 24 * q^36 + 20 * q^37 - 20 * q^43 - 8 * q^44 - 16 * q^46 - 20 * q^49 + 28 * q^50 - 4 * q^53 - 8 * q^56 + 48 * q^65 + 20 * q^67 + 48 * q^70 + 8 * q^71 - 24 * q^72 - 40 * q^74 - 4 * q^77 - 36 * q^81 - 48 * q^85 + 16 * q^88 + 24 * q^91 + 20 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$	2.44949	−	2.44949i	1.09545	−	1.09545i	0.100509	−	0.994936i	$-0.467953\pi$
$5$	2.44949	−	2.44949i	1.09545	−	1.09545i	0.994936	−	0.100509i	$-0.0320471\pi$
$6$		0			0
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$			3.00000i			1.00000i
$10$			4.89898i			1.54919i
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	0.841605	+	0.540094i	$0.181611\pi$
$12$		0			0
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	0.959857	−	0.280491i	$-0.0904971\pi$
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	−0.280491	+	0.959857i	$0.590497\pi$
$14$	3.44949	+	1.44949i	0.921915	+	0.387392i
$15$		0			0
$16$	−4.00000			−1.00000
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$	−4.89898	+	4.89898i	−1.12390	+	1.12390i	−0.132754	+	0.991149i	$0.542382\pi$
$19$	−4.89898	+	4.89898i	−1.12390	+	1.12390i	−0.991149	+	0.132754i	$0.957618\pi$
$20$	−4.89898	−	4.89898i	−1.09545	−	1.09545i
$21$		0			0
$22$			2.00000i			0.426401i
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$		−	7.00000i		−	1.40000i
$26$	−4.89898			−0.960769
$27$		0			0
$28$	−4.89898	+	2.00000i	−0.925820	+	0.377964i
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	−4.89898			−0.879883			−0.439941	−	0.898027i	$-0.645001\pi$
$31$	−4.89898			−0.879883			−0.439941	+	0.898027i	$0.645001\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		0			0
$34$	4.89898	+	4.89898i	0.840168	+	0.840168i
$35$	−8.44949	−	3.55051i	−1.42822	−	0.600146i
$36$	6.00000			1.00000
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		−	9.79796i		−	1.58944i
$39$		0			0
$40$	9.79796			1.54919
$41$	−4.89898			−0.765092			−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898			−0.765092			−0.382546	+	0.923936i	$0.624953\pi$
$42$		0			0
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	−0.976772	−	0.214280i	$-0.931260\pi$
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	0.214280	+	0.976772i	$0.431260\pi$
$44$	−2.00000	−	2.00000i	−0.301511	−	0.301511i
$45$	7.34847	+	7.34847i	1.09545	+	1.09545i
$46$	−4.00000	+	4.00000i	−0.589768	+	0.589768i
$47$	4.89898			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$	7.00000	+	7.00000i	0.989949	+	0.989949i
$51$		0			0
$52$	4.89898	−	4.89898i	0.679366	−	0.679366i
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	0.635083	−	0.772444i	$-0.280966\pi$
$53$	−1.00000	−	1.00000i	−0.137361	−	0.137361i	−0.772444	+	0.635083i	$0.780966\pi$
$54$		0			0
$55$		−	4.89898i		−	0.660578i
$56$	2.89898	−	6.89898i	0.387392	−	0.921915i
$57$		0			0
$58$		−	6.00000i		−	0.787839i
$59$	4.89898	+	4.89898i	0.637793	+	0.637793i	0.950011	−	0.312218i	$-0.101072\pi$
$59$	4.89898	+	4.89898i	0.637793	+	0.637793i	−0.312218	+	0.950011i	$0.601072\pi$
$60$		0			0
$61$	−2.44949	−	2.44949i	−0.313625	−	0.313625i	0.532687	−	0.846312i	$-0.321182\pi$
$61$	−2.44949	−	2.44949i	−0.313625	−	0.313625i	−0.846312	+	0.532687i	$0.821182\pi$
$62$	4.89898	−	4.89898i	0.622171	−	0.622171i
$63$	7.34847	−	3.00000i	0.925820	−	0.377964i
$64$			8.00000i			1.00000i
$65$	12.0000			1.48842
$66$		0			0
$67$	5.00000	+	5.00000i	0.610847	+	0.610847i	0.943167	−	0.332320i	$-0.107831\pi$
$67$	5.00000	+	5.00000i	0.610847	+	0.610847i	−0.332320	+	0.943167i	$0.607831\pi$
$68$	−9.79796			−1.18818
$69$		0			0
$70$	12.0000	−	4.89898i	1.43427	−	0.585540i
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$	−9.79796			−1.14676			−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796			−1.14676			−0.573382	+	0.819288i	$0.694369\pi$
$74$	−10.0000			−1.16248
$75$		0			0
$76$	9.79796	+	9.79796i	1.12390	+	1.12390i
$77$	−3.44949	−	1.44949i	−0.393106	−	0.165185i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$	−9.79796	+	9.79796i	−1.09545	+	1.09545i
$81$	−9.00000			−1.00000
$82$	4.89898	−	4.89898i	0.541002	−	0.541002i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−12.0000	−	12.0000i	−1.30158	−	1.30158i
$86$		−	10.0000i		−	1.07833i
$87$		0			0
$88$	4.00000			0.426401
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−14.6969			−1.54919
$91$	3.55051	−	8.44949i	0.372195	−	0.885747i
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$	−4.89898	+	4.89898i	−0.505291	+	0.505291i
$95$			24.0000i			2.46235i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$	0.101021	−	9.89898i	0.0102046	−	0.999948i
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$	−14.0000			−1.40000
$101$	−2.44949	+	2.44949i	−0.243733	+	0.243733i	−0.818393	−	0.574659i	$-0.805135\pi$
$101$	−2.44949	+	2.44949i	−0.243733	+	0.243733i	0.574659	+	0.818393i	$0.305135\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$			9.79796i			0.960769i
$105$		0			0
$106$	2.00000			0.194257
$107$	3.00000	−	3.00000i	0.290021	−	0.290021i	−0.547068	−	0.837088i	$-0.684256\pi$
$107$	3.00000	−	3.00000i	0.290021	−	0.290021i	0.837088	+	0.547068i	$0.184256\pi$
$108$		0			0
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	7.00000	−	7.00000i	0.670478	−	0.670478i	0.957826	+	0.287348i	$0.0927736\pi$
$110$	4.89898	+	4.89898i	0.467099	+	0.467099i
$111$		0			0
$112$	4.00000	+	9.79796i	0.377964	+	0.925820i
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	9.79796	−	9.79796i	0.913664	−	0.913664i
$116$	6.00000	+	6.00000i	0.557086	+	0.557086i
$117$	−7.34847	+	7.34847i	−0.679366	+	0.679366i
$118$	−9.79796			−0.901975
$119$	−12.0000	+	4.89898i	−1.10004	+	0.449089i
$120$		0			0
$121$			9.00000i			0.818182i
$122$	4.89898			0.443533
$123$		0			0
$124$			9.79796i			0.879883i
$125$	−4.89898	−	4.89898i	−0.438178	−	0.438178i
$126$	−4.34847	+	10.3485i	−0.387392	+	0.921915i
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$		0			0
$130$	−12.0000	+	12.0000i	−1.05247	+	1.05247i
$131$	9.79796	−	9.79796i	0.856052	−	0.856052i	−0.134819	−	0.990870i	$-0.543045\pi$
$131$	9.79796	−	9.79796i	0.856052	−	0.856052i	0.990870	+	0.134819i	$0.0430452\pi$
$132$		0			0
$133$	16.8990	+	7.10102i	1.46533	+	0.615737i
$134$	−10.0000			−0.863868
$135$		0			0
$136$	9.79796	−	9.79796i	0.840168	−	0.840168i
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$	4.89898	+	4.89898i	0.415526	+	0.415526i	0.883658	−	0.468132i	$-0.155073\pi$
$139$	4.89898	+	4.89898i	0.415526	+	0.415526i	−0.468132	+	0.883658i	$0.655073\pi$
$140$	−7.10102	+	16.8990i	−0.600146	+	1.42822i
$141$		0			0
$142$	−2.00000	+	2.00000i	−0.167836	+	0.167836i
$143$	4.89898			0.409673
$144$		−	12.0000i		−	1.00000i
$145$			14.6969i			1.22051i
$146$	9.79796	−	9.79796i	0.810885	−	0.810885i
$147$		0			0
$148$	10.0000	−	10.0000i	0.821995	−	0.821995i
$149$	3.00000	+	3.00000i	0.245770	+	0.245770i	0.819232	−	0.573462i	$-0.194400\pi$
$149$	3.00000	+	3.00000i	0.245770	+	0.245770i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	−19.5959			−1.58944
$153$	14.6969			1.18818
$154$	4.89898	−	2.00000i	0.394771	−	0.161165i
$155$	−12.0000	+	12.0000i	−0.963863	+	0.963863i
$156$		0			0
$157$	−12.2474	−	12.2474i	−0.977453	−	0.977453i	0.0222985	−	0.999751i	$-0.492902\pi$
$157$	−12.2474	−	12.2474i	−0.977453	−	0.977453i	−0.999751	+	0.0222985i	$0.992902\pi$
$158$	4.00000	+	4.00000i	0.318223	+	0.318223i
$159$		0			0
$160$		−	19.5959i		−	1.54919i
$161$	−4.00000	−	9.79796i	−0.315244	−	0.772187i
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$	−1.00000	−	1.00000i	−0.0783260	−	0.0783260i	0.666858	−	0.745184i	$-0.267639\pi$
$163$	−1.00000	−	1.00000i	−0.0783260	−	0.0783260i	−0.745184	+	0.666858i	$0.767639\pi$
$164$			9.79796i			0.765092i
$165$		0			0
$166$		0			0
$167$			19.5959i			1.51638i	0.652035	+	0.758189i	$0.273915\pi$
$167$			19.5959i			1.51638i	−0.652035	+	0.758189i	$0.726085\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$	24.0000			1.84072
$171$	−14.6969	−	14.6969i	−1.12390	−	1.12390i
$172$	10.0000	+	10.0000i	0.762493	+	0.762493i
$173$	2.44949	+	2.44949i	0.186231	+	0.186231i	0.794065	−	0.607833i	$-0.207961\pi$
$173$	2.44949	+	2.44949i	0.186231	+	0.186231i	−0.607833	+	0.794065i	$0.707961\pi$
$174$		0			0
$175$	−17.1464	+	7.00000i	−1.29615	+	0.529150i
$176$	−4.00000	+	4.00000i	−0.301511	+	0.301511i
$177$		0			0
$178$		0			0
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	0.810277	−	0.586047i	$-0.199317\pi$
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	−0.586047	+	0.810277i	$0.699317\pi$
$180$	14.6969	−	14.6969i	1.09545	−	1.09545i
$181$	−2.44949	+	2.44949i	−0.182069	+	0.182069i	−0.792257	−	0.610188i	$-0.791094\pi$
$181$	−2.44949	+	2.44949i	−0.182069	+	0.182069i	0.610188	+	0.792257i	$0.291094\pi$
$182$	4.89898	+	12.0000i	0.363137	+	0.889499i
$183$		0			0
$184$	8.00000	+	8.00000i	0.589768	+	0.589768i
$185$	24.4949			1.80090
$186$		0			0
$187$	−4.89898	−	4.89898i	−0.358249	−	0.358249i
$188$		−	9.79796i		−	0.714590i
$189$		0			0
$190$	−24.0000	−	24.0000i	−1.74114	−	1.74114i
$191$		−	20.0000i		−	1.44715i	−0.690246	−	0.723575i	$-0.742498\pi$
$191$		−	20.0000i		−	1.44715i	0.690246	−	0.723575i	$-0.257502\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$	4.89898	+	4.89898i	0.351726	+	0.351726i
$195$		0			0
$196$	9.79796	+	10.0000i	0.699854	+	0.714286i
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$	−6.00000			−0.426401
$199$			9.79796i			0.694559i	0.937762	+	0.347279i	$0.112894\pi$
$199$			9.79796i			0.694559i	−0.937762	+	0.347279i	$0.887106\pi$
$200$	14.0000	−	14.0000i	0.989949	−	0.989949i
$201$		0			0
$202$		−	4.89898i		−	0.344691i
$203$	10.3485	+	4.34847i	0.726320	+	0.305203i
$204$		0			0
$205$	−12.0000	+	12.0000i	−0.838116	+	0.838116i
$206$	9.79796	+	9.79796i	0.682656	+	0.682656i
$207$			12.0000i			0.834058i
$208$	−9.79796	−	9.79796i	−0.679366	−	0.679366i
$209$			9.79796i			0.677739i
$210$		0			0
$211$	1.00000	+	1.00000i	0.0688428	+	0.0688428i	0.740690	−	0.671847i	$-0.234499\pi$
$211$	1.00000	+	1.00000i	0.0688428	+	0.0688428i	−0.671847	+	0.740690i	$0.734499\pi$
$212$	−2.00000	+	2.00000i	−0.137361	+	0.137361i
$213$		0			0
$214$			6.00000i			0.410152i
$215$			24.4949i			1.67054i
$216$		0			0
$217$	4.89898	+	12.0000i	0.332564	+	0.814613i
$218$			14.0000i			0.948200i
$219$		0			0
$220$	−9.79796			−0.660578
$221$	12.0000	−	12.0000i	0.807207	−	0.807207i
$222$		0			0
$223$	−9.79796			−0.656120			−0.328060	−	0.944657i	$-0.606395\pi$
$223$	−9.79796			−0.656120			−0.328060	+	0.944657i	$0.606395\pi$
$224$	−13.7980	−	5.79796i	−0.921915	−	0.387392i
$225$	21.0000			1.40000
$226$	−4.00000	+	4.00000i	−0.266076	+	0.266076i
$227$	−19.5959	+	19.5959i	−1.30063	+	1.30063i	−0.372658	+	0.927969i	$0.621554\pi$
$227$	−19.5959	+	19.5959i	−1.30063	+	1.30063i	−0.927969	+	0.372658i	$0.878446\pi$
$228$		0			0
$229$	7.34847	−	7.34847i	0.485601	−	0.485601i	−0.421314	−	0.906915i	$-0.638431\pi$
$229$	7.34847	−	7.34847i	0.485601	−	0.485601i	0.906915	+	0.421314i	$0.138431\pi$
$230$			19.5959i			1.29212i
$231$		0			0
$232$	−12.0000			−0.787839
$233$			4.00000i			0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$			4.00000i			0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$		−	14.6969i		−	0.960769i
$235$	12.0000	−	12.0000i	0.782794	−	0.782794i
$236$	9.79796	−	9.79796i	0.637793	−	0.637793i
$237$		0			0
$238$	7.10102	−	16.8990i	0.460291	−	1.09540i
$239$		−	14.0000i		−	0.905585i	−0.891616	−	0.452792i	$-0.850428\pi$
$239$		−	14.0000i		−	0.905585i	0.891616	−	0.452792i	$-0.149572\pi$
$240$		0			0
$241$			24.4949i			1.57786i	0.614486	+	0.788928i	$0.289363\pi$
$241$			24.4949i			1.57786i	−0.614486	+	0.788928i	$0.710637\pi$
$242$	−9.00000	−	9.00000i	−0.578542	−	0.578542i
$243$		0			0
$244$	−4.89898	+	4.89898i	−0.313625	+	0.313625i
$245$	−0.247449	+	24.2474i	−0.0158089	+	1.54911i
$246$		0			0
$247$	−24.0000			−1.52708
$248$	−9.79796	−	9.79796i	−0.622171	−	0.622171i
$249$		0			0
$250$	9.79796			0.619677
$251$	−14.6969	−	14.6969i	−0.927663	−	0.927663i	0.0698920	−	0.997555i	$-0.477735\pi$
$251$	−14.6969	−	14.6969i	−0.927663	−	0.927663i	−0.997555	+	0.0698920i	$0.977735\pi$
$252$	−6.00000	−	14.6969i	−0.377964	−	0.925820i
$253$	4.00000	−	4.00000i	0.251478	−	0.251478i
$254$	18.0000	+	18.0000i	1.12942	+	1.12942i
$255$		0			0
$256$	16.0000			1.00000
$257$		−	29.3939i		−	1.83354i	−0.399416	−	0.916770i	$-0.630787\pi$
$257$		−	29.3939i		−	1.83354i	0.399416	−	0.916770i	$-0.369213\pi$
$258$		0			0
$259$	7.24745	−	17.2474i	0.450335	−	1.07170i
$260$		−	24.0000i		−	1.48842i
$261$	−9.00000	−	9.00000i	−0.557086	−	0.557086i
$262$			19.5959i			1.21064i
$263$	−26.0000			−1.60323			−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000			−1.60323			−0.801614	+	0.597841i	$0.796025\pi$
$264$		0			0
$265$	−4.89898			−0.300942
$266$	−24.0000	+	9.79796i	−1.47153	+	0.600751i
$267$		0			0
$268$	10.0000	−	10.0000i	0.610847	−	0.610847i
$269$	−7.34847	−	7.34847i	−0.448044	−	0.448044i	0.446660	−	0.894704i	$-0.352613\pi$
$269$	−7.34847	−	7.34847i	−0.448044	−	0.448044i	−0.894704	+	0.446660i	$0.852613\pi$
$270$		0			0
$271$	−29.3939			−1.78555			−0.892775	−	0.450502i	$-0.851245\pi$
$271$	−29.3939			−1.78555			−0.892775	+	0.450502i	$0.851245\pi$
$272$			19.5959i			1.18818i
$273$		0			0
$274$	8.00000	+	8.00000i	0.483298	+	0.483298i
$275$	−7.00000	−	7.00000i	−0.422116	−	0.422116i
$276$		0			0
$277$	5.00000	+	5.00000i	0.300421	+	0.300421i	0.841178	−	0.540758i	$-0.181862\pi$
$277$	5.00000	+	5.00000i	0.300421	+	0.300421i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−9.79796			−0.587643
$279$		−	14.6969i		−	0.879883i
$280$	−9.79796	−	24.0000i	−0.585540	−	1.43427i
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	14.6969	+	14.6969i	0.873642	+	0.873642i	0.992867	−	0.119225i	$-0.0380410\pi$
$283$	14.6969	+	14.6969i	0.873642	+	0.873642i	−0.119225	+	0.992867i	$0.538041\pi$
$284$		−	4.00000i		−	0.237356i
$285$		0			0
$286$	−4.89898	+	4.89898i	−0.289683	+	0.289683i
$287$	4.89898	+	12.0000i	0.289178	+	0.708338i
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$	−7.00000			−0.411765
$290$	−14.6969	−	14.6969i	−0.863034	−	0.863034i
$291$		0			0
$292$			19.5959i			1.14676i
$293$	12.2474	−	12.2474i	0.715504	−	0.715504i	−0.252177	−	0.967681i	$-0.581147\pi$
$293$	12.2474	−	12.2474i	0.715504	−	0.715504i	0.967681	+	0.252177i	$0.0811467\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$			20.0000i			1.16248i
$297$		0			0
$298$	−6.00000			−0.347571
$299$	9.79796	+	9.79796i	0.566631	+	0.566631i
$300$		0			0
$301$	17.2474	+	7.24745i	0.994126	+	0.417736i
$302$	−12.0000	+	12.0000i	−0.690522	+	0.690522i
$303$		0			0
$304$	19.5959	−	19.5959i	1.12390	−	1.12390i
$305$	−12.0000			−0.687118
$306$	−14.6969	+	14.6969i	−0.840168	+	0.840168i
$307$	4.89898	−	4.89898i	0.279600	−	0.279600i	−0.553350	−	0.832949i	$-0.686651\pi$
$307$	4.89898	−	4.89898i	0.279600	−	0.279600i	0.832949	+	0.553350i	$0.186651\pi$
$308$	−2.89898	+	6.89898i	−0.165185	+	0.393106i
$309$		0			0
$310$		−	24.0000i		−	1.36311i
$311$		−	24.4949i		−	1.38898i	−0.719503	−	0.694489i	$-0.755630\pi$
$311$		−	24.4949i		−	1.38898i	0.719503	−	0.694489i	$-0.244370\pi$
$312$		0			0
$313$	14.6969			0.830720			0.415360	−	0.909657i	$-0.363656\pi$
$313$	14.6969			0.830720			0.415360	+	0.909657i	$0.363656\pi$
$314$	24.4949			1.38233
$315$	10.6515	−	25.3485i	0.600146	−	1.42822i
$316$	−8.00000			−0.450035
$317$	13.0000	−	13.0000i	0.730153	−	0.730153i	−0.240497	−	0.970650i	$-0.577310\pi$
$317$	13.0000	−	13.0000i	0.730153	−	0.730153i	0.970650	+	0.240497i	$0.0773105\pi$
$318$		0			0
$319$			6.00000i			0.335936i
$320$	19.5959	+	19.5959i	1.09545	+	1.09545i
$321$		0			0
$322$	13.7980	+	5.79796i	0.768930	+	0.323108i
$323$	24.0000	+	24.0000i	1.33540	+	1.33540i
$324$			18.0000i			1.00000i
$325$	17.1464	−	17.1464i	0.951113	−	0.951113i
$326$	2.00000			0.110770
$327$		0			0
$328$	−9.79796	−	9.79796i	−0.541002	−	0.541002i
$329$	−4.89898	−	12.0000i	−0.270089	−	0.661581i
$330$		0			0
$331$	−19.0000	+	19.0000i	−1.04433	+	1.04433i	−0.0453639	+	0.998971i	$0.514445\pi$
$331$	−19.0000	+	19.0000i	−1.04433	+	1.04433i	−0.998971	+	0.0453639i	$0.985555\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$	−19.5959	−	19.5959i	−1.07224	−	1.07224i
$335$	24.4949			1.33830
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$	1.00000	+	1.00000i	0.0543928	+	0.0543928i
$339$		0			0
$340$	−24.0000	+	24.0000i	−1.30158	+	1.30158i
$341$	−4.89898	+	4.89898i	−0.265295	+	0.265295i
$342$	29.3939			1.58944
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$	−20.0000			−1.07833
$345$		0			0
$346$	−4.89898			−0.263371
$347$	3.00000	−	3.00000i	0.161048	−	0.161048i	−0.621983	−	0.783031i	$-0.713673\pi$
$347$	3.00000	−	3.00000i	0.161048	−	0.161048i	0.783031	+	0.621983i	$0.213673\pi$
$348$		0			0
$349$	−7.34847	−	7.34847i	−0.393355	−	0.393355i	0.482527	−	0.875881i	$-0.339719\pi$
$349$	−7.34847	−	7.34847i	−0.393355	−	0.393355i	−0.875881	+	0.482527i	$0.839719\pi$
$350$	10.1464	−	24.1464i	0.542349	−	1.29068i
$351$		0			0
$352$		−	8.00000i		−	0.426401i
$353$		−	9.79796i		−	0.521493i	−0.965407	−	0.260746i	$-0.916031\pi$
$353$		−	9.79796i		−	0.521493i	0.965407	−	0.260746i	$-0.0839686\pi$
$354$		0			0
$355$	4.89898	−	4.89898i	0.260011	−	0.260011i
$356$		0			0
$357$		0			0
$358$	−6.00000			−0.317110
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$			29.3939i			1.54919i
$361$		−	29.0000i		−	1.52632i
$362$		−	4.89898i		−	0.257485i
$363$		0			0
$364$	−16.8990	−	7.10102i	−0.885747	−	0.372195i
$365$	−24.0000	+	24.0000i	−1.25622	+	1.25622i
$366$		0			0
$367$	29.3939			1.53435			0.767174	−	0.641439i	$-0.221662\pi$
$367$	29.3939			1.53435			0.767174	+	0.641439i	$0.221662\pi$
$368$	−16.0000			−0.834058
$369$		−	14.6969i		−	0.765092i
$370$	−24.4949	+	24.4949i	−1.27343	+	1.27343i
$371$	−1.44949	+	3.44949i	−0.0752538	+	0.179089i
$372$		0			0
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	0.362446	−	0.932005i	$-0.381942\pi$
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	9.79796			0.506640
$375$		0			0
$376$	9.79796	+	9.79796i	0.505291	+	0.505291i
$377$	−14.6969			−0.756931
$378$		0			0
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.202057	+	0.979374i	$0.564763\pi$
$379$	−23.0000	+	23.0000i	−1.18143	+	1.18143i	−0.979374	+	0.202057i	$0.935237\pi$
$380$	48.0000			2.46235
$381$		0			0
$382$	20.0000	+	20.0000i	1.02329	+	1.02329i
$383$	14.6969			0.750978			0.375489	−	0.926827i	$-0.377475\pi$
$383$	14.6969			0.750978			0.375489	+	0.926827i	$0.377475\pi$
$384$		0			0
$385$	−12.0000	+	4.89898i	−0.611577	+	0.249675i
$386$	16.0000	−	16.0000i	0.814379	−	0.814379i
$387$	−15.0000	−	15.0000i	−0.762493	−	0.762493i
$388$	−9.79796			−0.497416
$389$	13.0000	+	13.0000i	0.659126	+	0.659126i	0.955173	−	0.296047i	$-0.0956686\pi$
$389$	13.0000	+	13.0000i	0.659126	+	0.659126i	−0.296047	+	0.955173i	$0.595669\pi$
$390$		0			0
$391$		−	19.5959i		−	0.991008i
$392$	−19.7980	−	0.202041i	−0.999948	−	0.0102046i
$393$		0			0
$394$	30.0000			1.51138
$395$	−9.79796	−	9.79796i	−0.492989	−	0.492989i
$396$	6.00000	−	6.00000i	0.301511	−	0.301511i
$397$	12.2474	+	12.2474i	0.614682	+	0.614682i	0.944162	−	0.329481i	$-0.106874\pi$
$397$	12.2474	+	12.2474i	0.614682	+	0.614682i	−0.329481	+	0.944162i	$0.606874\pi$
$398$	−9.79796	−	9.79796i	−0.491127	−	0.491127i
$399$		0			0
$400$			28.0000i			1.40000i
$401$	−8.00000			−0.399501			−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000			−0.399501			−0.199750	+	0.979847i	$0.564013\pi$
$402$		0			0
$403$	−12.0000	−	12.0000i	−0.597763	−	0.597763i
$404$	4.89898	+	4.89898i	0.243733	+	0.243733i
$405$	−22.0454	+	22.0454i	−1.09545	+	1.09545i
$406$	−14.6969	+	6.00000i	−0.729397	+	0.297775i
$407$	10.0000			0.495682
$408$		0			0
$409$	−24.4949			−1.21119			−0.605597	−	0.795771i	$-0.707066\pi$
$409$	−24.4949			−1.21119			−0.605597	+	0.795771i	$0.707066\pi$
$410$		−	24.0000i		−	1.18528i
$411$		0			0
$412$	−19.5959			−0.965422
$413$	7.10102	−	16.8990i	0.349418	−	0.831544i
$414$	−12.0000	−	12.0000i	−0.589768	−	0.589768i
$415$		0			0
$416$	19.5959			0.960769
$417$		0			0
$418$	−9.79796	−	9.79796i	−0.479234	−	0.479234i
$419$	−4.89898	+	4.89898i	−0.239331	+	0.239331i	−0.816573	−	0.577242i	$-0.804129\pi$
$419$	−4.89898	+	4.89898i	−0.239331	+	0.239331i	0.577242	+	0.816573i	$0.304129\pi$
$420$		0			0
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	0.922383	−	0.386276i	$-0.126239\pi$
$421$	11.0000	+	11.0000i	0.536107	+	0.536107i	−0.386276	+	0.922383i	$0.626239\pi$
$422$	−2.00000			−0.0973585
$423$			14.6969i			0.714590i
$424$		−	4.00000i		−	0.194257i
$425$	−34.2929			−1.66345
$426$		0			0
$427$	−3.55051	+	8.44949i	−0.171821	+	0.408899i
$428$	−6.00000	−	6.00000i	−0.290021	−	0.290021i
$429$		0			0
$430$	−24.4949	−	24.4949i	−1.18125	−	1.18125i
$431$		−	10.0000i		−	0.481683i	−0.970564	−	0.240842i	$-0.922577\pi$
$431$		−	10.0000i		−	0.481683i	0.970564	−	0.240842i	$-0.0774234\pi$
$432$		0			0
$433$			14.6969i			0.706290i	0.935569	+	0.353145i	$0.114888\pi$
$433$			14.6969i			0.706290i	−0.935569	+	0.353145i	$0.885112\pi$
$434$	−16.8990	−	7.10102i	−0.811177	−	0.340860i
$435$		0			0
$436$	−14.0000	−	14.0000i	−0.670478	−	0.670478i
$437$	−19.5959	+	19.5959i	−0.937400	+	0.937400i
$438$		0			0
$439$		−	14.6969i		−	0.701447i	−0.936479	−	0.350723i	$-0.885936\pi$
$439$		−	14.6969i		−	0.701447i	0.936479	−	0.350723i	$-0.114064\pi$
$440$	9.79796	−	9.79796i	0.467099	−	0.467099i
$441$	−14.6969	−	15.0000i	−0.699854	−	0.714286i
$442$			24.0000i			1.14156i
$443$	5.00000	−	5.00000i	0.237557	−	0.237557i	−0.578281	−	0.815838i	$-0.696276\pi$
$443$	5.00000	−	5.00000i	0.237557	−	0.237557i	0.815838	+	0.578281i	$0.196276\pi$
$444$		0			0
$445$		0			0
$446$	9.79796	−	9.79796i	0.463947	−	0.463947i
$447$		0			0
$448$	19.5959	−	8.00000i	0.925820	−	0.377964i
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	−21.0000	+	21.0000i	−0.989949	+	0.989949i
$451$	−4.89898	+	4.89898i	−0.230684	+	0.230684i
$452$		−	8.00000i		−	0.376288i
$453$		0			0
$454$		−	39.1918i		−	1.83936i
$455$	−12.0000	−	29.3939i	−0.562569	−	1.37801i
$456$		0			0
$457$		−	18.0000i		−	0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$		−	18.0000i		−	0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$			14.6969i			0.686743i
$459$		0			0
$460$	−19.5959	−	19.5959i	−0.913664	−	0.913664i
$461$	22.0454	+	22.0454i	1.02676	+	1.02676i	0.999632	+	0.0271249i	$0.00863520\pi$
$461$	22.0454	+	22.0454i	1.02676	+	1.02676i	0.0271249	+	0.999632i	$0.491365\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	12.0000	−	12.0000i	0.557086	−	0.557086i
$465$		0			0
$466$	−4.00000	−	4.00000i	−0.185296	−	0.185296i
$467$	−19.5959	+	19.5959i	−0.906791	+	0.906791i	−0.996012	−	0.0892209i	$-0.971562\pi$
$467$	−19.5959	+	19.5959i	−0.906791	+	0.906791i	0.0892209	+	0.996012i	$0.471562\pi$
$468$	14.6969	+	14.6969i	0.679366	+	0.679366i
$469$	7.24745	−	17.2474i	0.334656	−	0.796413i
$470$			24.0000i			1.10704i
$471$		0			0
$472$			19.5959i			0.901975i
$473$			10.0000i			0.459800i
$474$		0			0
$475$	34.2929	+	34.2929i	1.57346	+	1.57346i
$476$	9.79796	+	24.0000i	0.449089	+	1.10004i
$477$	3.00000	−	3.00000i	0.137361	−	0.137361i
$478$	14.0000	+	14.0000i	0.640345	+	0.640345i
$479$	−24.4949			−1.11920			−0.559600	−	0.828763i	$-0.689045\pi$
$479$	−24.4949			−1.11920			−0.559600	+	0.828763i	$0.689045\pi$
$480$		0			0
$481$			24.4949i			1.11687i
$482$	−24.4949	−	24.4949i	−1.11571	−	1.11571i
$483$		0			0
$484$	18.0000			0.818182
$485$	−12.0000	−	12.0000i	−0.544892	−	0.544892i
$486$		0			0
$487$	−12.0000			−0.543772			−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000			−0.543772			−0.271886	+	0.962329i	$0.587647\pi$
$488$		−	9.79796i		−	0.443533i
$489$		0			0
$490$	−24.0000	−	24.4949i	−1.08421	−	1.10657i
$491$	11.0000	−	11.0000i	0.496423	−	0.496423i	−0.413900	−	0.910323i	$-0.635834\pi$
$491$	11.0000	−	11.0000i	0.496423	−	0.496423i	0.910323	+	0.413900i	$0.135834\pi$
$492$		0			0
$493$	14.6969	+	14.6969i	0.661917	+	0.661917i
$494$	24.0000	−	24.0000i	1.07981	−	1.07981i
$495$	14.6969			0.660578
$496$	19.5959			0.879883
$497$	−2.00000	−	4.89898i	−0.0897123	−	0.219749i
$498$		0			0
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.999548	+	0.0300737i	$0.00957421\pi$
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.0300737	+	0.999548i	$0.490426\pi$
$500$	−9.79796	+	9.79796i	−0.438178	+	0.438178i
$501$		0			0
$502$	29.3939			1.31191
$503$			14.6969i			0.655304i	0.944798	+	0.327652i	$0.106257\pi$
$503$			14.6969i			0.655304i	−0.944798	+	0.327652i	$0.893743\pi$
$504$	20.6969	+	8.69694i	0.921915	+	0.387392i
$505$			12.0000i			0.533993i
$506$			8.00000i			0.355643i
$507$		0			0
$508$	−36.0000			−1.59724
$509$	−7.34847	−	7.34847i	−0.325715	−	0.325715i	0.525239	−	0.850955i	$-0.323976\pi$
$509$	−7.34847	−	7.34847i	−0.325715	−	0.325715i	−0.850955	+	0.525239i	$0.823976\pi$
$510$		0			0
$511$	9.79796	+	24.0000i	0.433436	+	1.06170i
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	29.3939	+	29.3939i	1.29651	+	1.29651i
$515$	−24.0000	−	24.0000i	−1.05757	−	1.05757i
$516$		0			0
$517$	4.89898	−	4.89898i	0.215457	−	0.215457i
$518$	10.0000	+	24.4949i	0.439375	+	1.07624i
$519$		0			0
$520$	24.0000	+	24.0000i	1.05247	+	1.05247i
$521$	−4.89898			−0.214628			−0.107314	−	0.994225i	$-0.534225\pi$
$521$	−4.89898			−0.214628			−0.107314	+	0.994225i	$0.534225\pi$
$522$	18.0000			0.787839
$523$	14.6969	+	14.6969i	0.642652	+	0.642652i	0.951207	−	0.308554i	$-0.0998452\pi$
$523$	14.6969	+	14.6969i	0.642652	+	0.642652i	−0.308554	+	0.951207i	$0.599845\pi$
$524$	−19.5959	−	19.5959i	−0.856052	−	0.856052i
$525$		0			0
$526$	26.0000	−	26.0000i	1.13365	−	1.13365i
$527$			24.0000i			1.04546i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	4.89898	−	4.89898i	0.212798	−	0.212798i
$531$	−14.6969	+	14.6969i	−0.637793	+	0.637793i
$532$	14.2020	−	33.7980i	0.615737	−	1.46533i
$533$	−12.0000	−	12.0000i	−0.519778	−	0.519778i
$534$		0			0
$535$		−	14.6969i		−	0.635404i
$536$			20.0000i			0.863868i
$537$		0			0
$538$	14.6969			0.633630
$539$	−0.101021	+	9.89898i	−0.00435126	+	0.426379i
$540$		0			0
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.289685	+	0.957122i	$0.593551\pi$
$541$	−29.0000	+	29.0000i	−1.24681	+	1.24681i	−0.957122	+	0.289685i	$0.906449\pi$
$542$	29.3939	−	29.3939i	1.26258	−	1.26258i
$543$		0			0
$544$	−19.5959	−	19.5959i	−0.840168	−	0.840168i
$545$		−	34.2929i		−	1.46894i
$546$		0			0
$547$	−5.00000	−	5.00000i	−0.213785	−	0.213785i	0.592088	−	0.805873i	$-0.298304\pi$
$547$	−5.00000	−	5.00000i	−0.213785	−	0.213785i	−0.805873	+	0.592088i	$0.798304\pi$
$548$	−16.0000			−0.683486
$549$	7.34847	−	7.34847i	0.313625	−	0.313625i
$550$	14.0000			0.596962
$551$		−	29.3939i		−	1.25222i
$552$		0			0
$553$	−9.79796	+	4.00000i	−0.416652	+	0.170097i
$554$	−10.0000			−0.424859
$555$		0			0
$556$	9.79796	−	9.79796i	0.415526	−	0.415526i
$557$	3.00000	−	3.00000i	0.127114	−	0.127114i	−0.640688	−	0.767802i	$-0.721351\pi$
$557$	3.00000	−	3.00000i	0.127114	−	0.127114i	0.767802	+	0.640688i	$0.221351\pi$
$558$	14.6969	+	14.6969i	0.622171	+	0.622171i
$559$	−24.4949			−1.03602
$560$	33.7980	+	14.2020i	1.42822	+	0.600146i
$561$		0			0
$562$	−20.0000	−	20.0000i	−0.843649	−	0.843649i
$563$	24.4949	−	24.4949i	1.03234	−	1.03234i	0.0328775	−	0.999459i	$-0.489533\pi$
$563$	24.4949	−	24.4949i	1.03234	−	1.03234i	0.999459	−	0.0328775i	$-0.0104671\pi$
$564$		0			0
$565$	9.79796	−	9.79796i	0.412203	−	0.412203i
$566$	−29.3939			−1.23552
$567$	9.00000	+	22.0454i	0.377964	+	0.925820i
$568$	4.00000	+	4.00000i	0.167836	+	0.167836i
$569$		−	14.0000i		−	0.586911i	−0.955973	−	0.293455i	$-0.905195\pi$
$569$		−	14.0000i		−	0.586911i	0.955973	−	0.293455i	$-0.0948052\pi$
$570$		0			0
$571$	31.0000	−	31.0000i	1.29731	−	1.29731i	0.367146	−	0.930163i	$-0.380335\pi$
$571$	31.0000	−	31.0000i	1.29731	−	1.29731i	0.930163	−	0.367146i	$-0.119665\pi$
$572$		−	9.79796i		−	0.409673i
$573$		0			0
$574$	−16.8990	−	7.10102i	−0.705350	−	0.296391i
$575$		−	28.0000i		−	1.16768i
$576$	−24.0000			−1.00000
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$	7.00000	−	7.00000i	0.291162	−	0.291162i
$579$		0			0
$580$	29.3939			1.22051
$581$		0			0
$582$		0			0
$583$	−2.00000			−0.0828315
$584$	−19.5959	−	19.5959i	−0.810885	−	0.810885i
$585$			36.0000i			1.48842i
$586$			24.4949i			1.01187i
$587$	−24.4949	−	24.4949i	−1.01101	−	1.01101i	−0.999939	−	0.0110739i	$-0.996475\pi$
$587$	−24.4949	−	24.4949i	−1.01101	−	1.01101i	−0.0110739	−	0.999939i	$-0.503525\pi$
$588$		0			0
$589$	24.0000	−	24.0000i	0.988903	−	0.988903i
$590$	−24.0000	+	24.0000i	−0.988064	+	0.988064i
$591$		0			0
$592$	−20.0000	−	20.0000i	−0.821995	−	0.821995i
$593$			39.1918i			1.60942i	0.593671	+	0.804708i	$0.297678\pi$
$593$			39.1918i			1.60942i	−0.593671	+	0.804708i	$0.702322\pi$
$594$		0			0
$595$	−17.3939	+	41.3939i	−0.713079	+	1.69698i
$596$	6.00000	−	6.00000i	0.245770	−	0.245770i
$597$		0			0
$598$	−19.5959			−0.801337
$599$	10.0000			0.408589			0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000			0.408589			0.204294	+	0.978909i	$0.434510\pi$
$600$		0			0
$601$	19.5959			0.799334			0.399667	−	0.916660i	$-0.369126\pi$
$601$	19.5959			0.799334			0.399667	+	0.916660i	$0.369126\pi$
$602$	−24.4949	+	10.0000i	−0.998337	+	0.407570i
$603$	−15.0000	+	15.0000i	−0.610847	+	0.610847i
$604$		−	24.0000i		−	0.976546i
$605$	22.0454	+	22.0454i	0.896273	+	0.896273i
$606$		0			0
$607$	29.3939			1.19306			0.596530	−	0.802591i	$-0.296546\pi$
$607$	29.3939			1.19306			0.596530	+	0.802591i	$0.296546\pi$
$608$			39.1918i			1.58944i
$609$		0			0
$610$	12.0000	−	12.0000i	0.485866	−	0.485866i
$611$	12.0000	+	12.0000i	0.485468	+	0.485468i
$612$		−	29.3939i		−	1.18818i
$613$	19.0000	+	19.0000i	0.767403	+	0.767403i	0.977649	−	0.210246i	$-0.0674264\pi$
$613$	19.0000	+	19.0000i	0.767403	+	0.767403i	−0.210246	+	0.977649i	$0.567426\pi$
$614$			9.79796i			0.395413i
$615$		0			0
$616$	−4.00000	−	9.79796i	−0.161165	−	0.394771i
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$	4.89898	+	4.89898i	0.196907	+	0.196907i	0.798673	−	0.601766i	$-0.205536\pi$
$619$	4.89898	+	4.89898i	0.196907	+	0.196907i	−0.601766	+	0.798673i	$0.705536\pi$
$620$	24.0000	+	24.0000i	0.963863	+	0.963863i
$621$		0			0
$622$	24.4949	+	24.4949i	0.982156	+	0.982156i
$623$		0			0
$624$		0			0
$625$	11.0000			0.440000
$626$	−14.6969	+	14.6969i	−0.587408	+	0.587408i
$627$		0			0
$628$	−24.4949	+	24.4949i	−0.977453	+	0.977453i
$629$	24.4949	−	24.4949i	0.976676	−	0.976676i
$630$	14.6969	+	36.0000i	0.585540	+	1.43427i
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$	8.00000	−	8.00000i	0.318223	−	0.318223i
$633$		0			0
$634$			26.0000i			1.03259i
$635$	−44.0908	−	44.0908i	−1.74969	−	1.74969i
$636$		0			0
$637$	−24.2474	−	0.247449i	−0.960719	−	0.00980428i
$638$	−6.00000	−	6.00000i	−0.237542	−	0.237542i
$639$			6.00000i			0.237356i
$640$	−39.1918			−1.54919
$641$	32.0000			1.26392			0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000			1.26392			0.631962	+	0.774999i	$0.282250\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$	−19.5959	+	8.00000i	−0.772187	+	0.315244i
$645$		0			0
$646$	−48.0000			−1.88853
$647$			19.5959i			0.770395i	0.922834	+	0.385198i	$0.125867\pi$
$647$			19.5959i			0.770395i	−0.922834	+	0.385198i	$0.874133\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$	9.79796			0.384604
$650$			34.2929i			1.34508i
$651$		0			0
$652$	−2.00000	+	2.00000i	−0.0783260	+	0.0783260i
$653$	15.0000	−	15.0000i	0.586995	−	0.586995i	−0.349821	−	0.936817i	$-0.613758\pi$
$653$	15.0000	−	15.0000i	0.586995	−	0.586995i	0.936817	+	0.349821i	$0.113758\pi$
$654$		0			0
$655$		−	48.0000i		−	1.87552i
$656$	19.5959			0.765092
$657$		−	29.3939i		−	1.14676i
$658$	16.8990	+	7.10102i	0.658791	+	0.276827i
$659$	−27.0000	−	27.0000i	−1.05177	−	1.05177i	−0.998585	−	0.0531861i	$-0.983062\pi$
$659$	−27.0000	−	27.0000i	−1.05177	−	1.05177i	−0.0531861	−	0.998585i	$-0.516938\pi$
$660$		0			0
$661$	−2.44949	+	2.44949i	−0.0952741	+	0.0952741i	−0.753137	−	0.657863i	$-0.771460\pi$
$661$	−2.44949	+	2.44949i	−0.0952741	+	0.0952741i	0.657863	+	0.753137i	$0.271460\pi$
$662$		−	38.0000i		−	1.47691i
$663$		0			0
$664$		0			0
$665$	58.7878	−	24.0000i	2.27969	−	0.930680i
$666$		−	30.0000i		−	1.16248i
$667$	−12.0000	+	12.0000i	−0.464642	+	0.464642i
$668$	39.1918			1.51638
$669$		0			0
$670$	−24.4949	+	24.4949i	−0.946320	+	0.946320i
$671$	−4.89898			−0.189123
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$	−28.0000	+	28.0000i	−1.07852	+	1.07852i
$675$		0			0
$676$	−2.00000			−0.0769231
$677$	−7.34847	+	7.34847i	−0.282425	+	0.282425i	−0.834075	−	0.551651i	$-0.813998\pi$
$677$	−7.34847	+	7.34847i	−0.282425	+	0.282425i	0.551651	+	0.834075i	$0.313998\pi$
$678$		0			0
$679$	−12.0000	+	4.89898i	−0.460518	+	0.188006i
$680$		−	48.0000i		−	1.84072i
$681$		0			0
$682$		−	9.79796i		−	0.375183i
$683$	−25.0000	+	25.0000i	−0.956598	+	0.956598i	−0.999097	−	0.0424981i	$-0.986468\pi$
$683$	−25.0000	+	25.0000i	−0.956598	+	0.956598i	0.0424981	+	0.999097i	$0.486468\pi$
$684$	−29.3939	+	29.3939i	−1.12390	+	1.12390i
$685$	−19.5959	−	19.5959i	−0.748722	−	0.748722i
$686$	−24.3485	+	9.65153i	−0.929629	+	0.368497i
$687$		0			0
$688$	20.0000	−	20.0000i	0.762493	−	0.762493i
$689$		−	4.89898i		−	0.186636i
$690$		0			0
$691$	9.79796	−	9.79796i	0.372732	−	0.372732i	−0.495739	−	0.868471i	$-0.665103\pi$
$691$	9.79796	−	9.79796i	0.372732	−	0.372732i	0.868471	+	0.495739i	$0.165103\pi$
$692$	4.89898	−	4.89898i	0.186231	−	0.186231i
$693$	4.34847	−	10.3485i	0.165185	−	0.393106i
$694$			6.00000i			0.227757i
$695$	24.0000			0.910372
$696$		0			0
$697$			24.0000i			0.909065i
$698$	14.6969			0.556287
$699$		0			0
$700$	14.0000	+	34.2929i	0.529150	+	1.29615i
$701$	−29.0000	+	29.0000i	−1.09531	+	1.09531i	−0.100364	+	0.994951i	$0.532001\pi$
$701$	−29.0000	+	29.0000i	−1.09531	+	1.09531i	−0.994951	+	0.100364i	$0.967999\pi$
$702$		0			0
$703$	−48.9898			−1.84769
$704$	8.00000	+	8.00000i	0.301511	+	0.301511i
$705$		0			0
$706$	9.79796	+	9.79796i	0.368751	+	0.368751i
$707$	8.44949	+	3.55051i	0.317776	+	0.133531i
$708$		0			0
$709$	23.0000	+	23.0000i	0.863783	+	0.863783i	0.991775	−	0.127992i	$-0.0408532\pi$
$709$	23.0000	+	23.0000i	0.863783	+	0.863783i	−0.127992	+	0.991775i	$0.540853\pi$
$710$			9.79796i			0.367711i
$711$	12.0000			0.450035
$712$		0			0
$713$	−19.5959			−0.733873
$714$		0			0
$715$	12.0000	−	12.0000i	0.448775	−	0.448775i
$716$	6.00000	−	6.00000i	0.224231	−	0.224231i
$717$		0			0
$718$	−20.0000	+	20.0000i	−0.746393	+	0.746393i
$719$	−24.4949			−0.913506			−0.456753	−	0.889594i	$-0.650988\pi$
$719$	−24.4949			−0.913506			−0.456753	+	0.889594i	$0.650988\pi$
$720$	−29.3939	−	29.3939i	−1.09545	−	1.09545i
$721$	−24.0000	+	9.79796i	−0.893807	+	0.364895i
$722$	29.0000	+	29.0000i	1.07927	+	1.07927i
$723$		0			0
$724$	4.89898	+	4.89898i	0.182069	+	0.182069i
$725$	21.0000	+	21.0000i	0.779920	+	0.779920i
$726$		0			0
$727$			19.5959i			0.726772i	0.931639	+	0.363386i	$0.118379\pi$
$727$			19.5959i			0.726772i	−0.931639	+	0.363386i	$0.881621\pi$
$728$	24.0000	−	9.79796i	0.889499	−	0.363137i
$729$		−	27.0000i		−	1.00000i
$730$		−	48.0000i		−	1.77656i
$731$	24.4949	+	24.4949i	0.905977	+	0.905977i
$732$		0			0
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	0.171005	−	0.985270i	$-0.445299\pi$
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	−0.985270	+	0.171005i	$0.945299\pi$
$734$	−29.3939	+	29.3939i	−1.08495	+	1.08495i
$735$		0			0
$736$	16.0000	−	16.0000i	0.589768	−	0.589768i
$737$	10.0000			0.368355
$738$	14.6969	+	14.6969i	0.541002	+	0.541002i
$739$	−17.0000	−	17.0000i	−0.625355	−	0.625355i	0.321541	−	0.946896i	$-0.395799\pi$
$739$	−17.0000	−	17.0000i	−0.625355	−	0.625355i	−0.946896	+	0.321541i	$0.895799\pi$
$740$		−	48.9898i		−	1.80090i
$741$		0			0
$742$	−2.00000	−	4.89898i	−0.0734223	−	0.179847i
$743$	4.00000			0.146746			0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000			0.146746			0.0733729	+	0.997305i	$0.476624\pi$
$744$		0			0
$745$	14.6969			0.538454
$746$	22.0000			0.805477
$747$		0			0
$748$	−9.79796	+	9.79796i	−0.358249	+	0.358249i
$749$	−10.3485	−	4.34847i	−0.378125	−	0.158890i
$750$		0			0
$751$			10.0000i			0.364905i	0.983215	+	0.182453i	$0.0584036\pi$
$751$			10.0000i			0.364905i	−0.983215	+	0.182453i	$0.941596\pi$
$752$	−19.5959			−0.714590
$753$		0			0
$754$	14.6969	−	14.6969i	0.535231	−	0.535231i
$755$	29.3939	−	29.3939i	1.06975	−	1.06975i
$756$		0			0
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	0.610380	−	0.792108i	$-0.291017\pi$
$757$	−5.00000	−	5.00000i	−0.181728	−	0.181728i	−0.792108	+	0.610380i	$0.791017\pi$
$758$		−	46.0000i		−	1.67080i
$759$		0			0
$760$	−48.0000	+	48.0000i	−1.74114	+	1.74114i
$761$	44.0908			1.59829			0.799145	−	0.601138i	$-0.205286\pi$
$761$	44.0908			1.59829			0.799145	+	0.601138i	$0.205286\pi$
$762$		0			0
$763$	−24.1464	−	10.1464i	−0.874159	−	0.367325i
$764$	−40.0000			−1.44715
$765$	36.0000	−	36.0000i	1.30158	−	1.30158i
$766$	−14.6969	+	14.6969i	−0.531022	+	0.531022i
$767$			24.0000i			0.866590i
$768$		0			0
$769$			34.2929i			1.23663i	0.785930	+	0.618316i	$0.212185\pi$
$769$			34.2929i			1.23663i	−0.785930	+	0.618316i	$0.787815\pi$
$770$	7.10102	−	16.8990i	0.255903	−	0.608997i
$771$		0			0
$772$			32.0000i			1.15171i
$773$	−12.2474	+	12.2474i	−0.440510	+	0.440510i	−0.892183	−	0.451673i	$-0.850827\pi$
$773$	−12.2474	+	12.2474i	−0.440510	+	0.440510i	0.451673	+	0.892183i	$0.350827\pi$
$774$	30.0000			1.07833
$775$			34.2929i			1.23184i
$776$	9.79796	−	9.79796i	0.351726	−	0.351726i
$777$		0			0
$778$	−26.0000			−0.932145
$779$	24.0000	−	24.0000i	0.859889	−	0.859889i
$780$		0			0
$781$	2.00000	−	2.00000i	0.0715656	−	0.0715656i
$782$	19.5959	+	19.5959i	0.700749	+	0.700749i
$783$		0			0
$784$	20.0000	−	19.5959i	0.714286	−	0.699854i
$785$	−60.0000			−2.14149
$786$		0			0
$787$	4.89898	−	4.89898i	0.174630	−	0.174630i	−0.614380	−	0.789010i	$-0.710594\pi$
$787$	4.89898	−	4.89898i	0.174630	−	0.174630i	0.789010	+	0.614380i	$0.210594\pi$
$788$	−30.0000	+	30.0000i	−1.06871	+	1.06871i
$789$		0			0
$790$	19.5959			0.697191
$791$	−4.00000	−	9.79796i	−0.142224	−	0.348375i
$792$			12.0000i			0.426401i
$793$		−	12.0000i		−	0.426132i
$794$	−24.4949			−0.869291
$795$		0			0
$796$	19.5959			0.694559
$797$	12.2474	+	12.2474i	0.433827	+	0.433827i	0.889928	−	0.456101i	$-0.150754\pi$
$797$	12.2474	+	12.2474i	0.433827	+	0.433827i	−0.456101	+	0.889928i	$0.650754\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$	−28.0000	−	28.0000i	−0.989949	−	0.989949i
$801$		0			0
$802$	8.00000	−	8.00000i	0.282490	−	0.282490i
$803$	−9.79796	+	9.79796i	−0.345762	+	0.345762i
$804$		0			0
$805$	−33.7980	−	14.2020i	−1.19122	−	0.500556i
$806$	24.0000			0.845364
$807$		0			0
$808$	−9.79796			−0.344691
$809$			46.0000i			1.61727i	0.588308	+	0.808637i	$0.299794\pi$
$809$			46.0000i			1.61727i	−0.588308	+	0.808637i	$0.700206\pi$
$810$		−	44.0908i		−	1.54919i
$811$	−14.6969	−	14.6969i	−0.516079	−	0.516079i	0.400303	−	0.916383i	$-0.368905\pi$
$811$	−14.6969	−	14.6969i	−0.516079	−	0.516079i	−0.916383	+	0.400303i	$0.868905\pi$
$812$	8.69694	−	20.6969i	0.305203	−	0.726320i
$813$		0			0
$814$	−10.0000	+	10.0000i	−0.350500	+	0.350500i
$815$	−4.89898			−0.171604
$816$		0			0
$817$		−	48.9898i		−	1.71394i
$818$	24.4949	−	24.4949i	0.856444	−	0.856444i
$819$	25.3485	+	10.6515i	0.885747	+	0.372195i
$820$	24.0000	+	24.0000i	0.838116	+	0.838116i
$821$	−19.0000	−	19.0000i	−0.663105	−	0.663105i	0.293006	−	0.956111i	$-0.405344\pi$
$821$	−19.0000	−	19.0000i	−0.663105	−	0.663105i	−0.956111	+	0.293006i	$0.905344\pi$
$822$		0			0
$823$	−6.00000			−0.209147			−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000			−0.209147			−0.104573	+	0.994517i	$0.533348\pi$
$824$	19.5959	−	19.5959i	0.682656	−	0.682656i
$825$		0			0
$826$	9.79796	+	24.0000i	0.340915	+	0.835067i
$827$	33.0000	−	33.0000i	1.14752	−	1.14752i	0.160484	−	0.987038i	$-0.448695\pi$
$827$	33.0000	−	33.0000i	1.14752	−	1.14752i	0.987038	−	0.160484i	$-0.0513055\pi$
$828$	24.0000			0.834058
$829$	−7.34847	−	7.34847i	−0.255223	−	0.255223i	0.567885	−	0.823108i	$-0.307762\pi$
$829$	−7.34847	−	7.34847i	−0.255223	−	0.255223i	−0.823108	+	0.567885i	$0.807762\pi$
$830$		0			0
$831$		0			0
$832$	−19.5959	+	19.5959i	−0.679366	+	0.679366i
$833$	24.0000	+	24.4949i	0.831551	+	0.848698i
$834$		0			0
$835$	48.0000	+	48.0000i	1.66111	+	1.66111i
$836$	19.5959			0.677739
$837$		0			0
$838$		−	9.79796i		−	0.338465i
$839$			9.79796i			0.338263i	0.985593	+	0.169132i	$0.0540963\pi$
$839$			9.79796i			0.338263i	−0.985593	+	0.169132i	$0.945904\pi$
$840$		0			0
$841$			11.0000i			0.379310i
$842$	−22.0000			−0.758170
$843$		0			0
$844$	2.00000	−	2.00000i	0.0688428	−	0.0688428i
$845$	−2.44949	−	2.44949i	−0.0842650	−	0.0842650i
$846$	−14.6969	−	14.6969i	−0.505291	−	0.505291i
$847$	22.0454	−	9.00000i	0.757489	−	0.309244i
$848$	4.00000	+	4.00000i	0.137361	+	0.137361i
$849$		0			0
$850$	34.2929	−	34.2929i	1.17624	−	1.17624i
$851$	20.0000	+	20.0000i	0.685591	+	0.685591i
$852$		0			0
$853$	−12.2474	+	12.2474i	−0.419345	+	0.419345i	−0.884978	−	0.465633i	$-0.845827\pi$
$853$	−12.2474	+	12.2474i	−0.419345	+	0.419345i	0.465633	+	0.884978i	$0.345827\pi$
$854$	−4.89898	−	12.0000i	−0.167640	−	0.410632i
$855$	−72.0000			−2.46235
$856$	12.0000			0.410152
$857$	4.89898			0.167346			0.0836730	−	0.996493i	$-0.473335\pi$
$857$	4.89898			0.167346			0.0836730	+	0.996493i	$0.473335\pi$
$858$		0			0
$859$	29.3939	+	29.3939i	1.00291	+	1.00291i	0.999996	+	0.00291037i	$0.000926402\pi$
$859$	29.3939	+	29.3939i	1.00291	+	1.00291i	0.00291037	+	0.999996i	$0.499074\pi$
$860$	48.9898			1.67054
$861$		0			0
$862$	10.0000	+	10.0000i	0.340601	+	0.340601i
$863$			4.00000i			0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$			4.00000i			0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$	−14.6969	−	14.6969i	−0.499422	−	0.499422i
$867$		0			0
$868$	24.0000	−	9.79796i	0.814613	−	0.332564i
$869$	−4.00000	−	4.00000i	−0.135691	−	0.135691i
$870$		0			0
$871$			24.4949i			0.829978i
$872$	28.0000			0.948200
$873$	14.6969			0.497416
$874$		−	39.1918i		−	1.32568i
$875$	−7.10102	+	16.8990i	−0.240058	+	0.571290i
$876$		0			0
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	−0.452689	−	0.891668i	$-0.649535\pi$
$877$	13.0000	−	13.0000i	0.438979	−	0.438979i	0.891668	+	0.452689i	$0.149535\pi$
$878$	14.6969	+	14.6969i	0.495998	+	0.495998i
$879$		0			0
$880$			19.5959i			0.660578i
$881$			48.9898i			1.65051i	0.564762	+	0.825254i	$0.308968\pi$
$881$			48.9898i			1.65051i	−0.564762	+	0.825254i	$0.691032\pi$
$882$	29.6969	+	0.303062i	0.999948	+	0.0102046i
$883$	−31.0000	−	31.0000i	−1.04323	−	1.04323i	−0.999022	−	0.0442108i	$-0.985923\pi$
$883$	−31.0000	−	31.0000i	−1.04323	−	1.04323i	−0.0442108	−	0.999022i	$-0.514077\pi$
$884$	−24.0000	−	24.0000i	−0.807207	−	0.807207i
$885$		0			0
$886$			10.0000i			0.335957i
$887$			19.5959i			0.657967i	0.944336	+	0.328983i	$0.106706\pi$
$887$			19.5959i			0.657967i	−0.944336	+	0.328983i	$0.893294\pi$
$888$		0			0
$889$	−44.0908	+	18.0000i	−1.47876	+	0.603701i
$890$		0			0
$891$	−9.00000	+	9.00000i	−0.301511	+	0.301511i
$892$			19.5959i			0.656120i
$893$	−24.0000	+	24.0000i	−0.803129	+	0.803129i
$894$		0			0
$895$	14.6969			0.491264
$896$	−11.5959	+	27.5959i	−0.387392	+	0.921915i
$897$		0			0
$898$	−10.0000	+	10.0000i	−0.333704	+	0.333704i
$899$	14.6969	−	14.6969i	0.490170	−	0.490170i
$900$		−	42.0000i		−	1.40000i
$901$	−4.89898	+	4.89898i	−0.163209	+	0.163209i
$902$		−	9.79796i		−	0.326236i
$903$		0			0
$904$	8.00000	+	8.00000i	0.266076	+	0.266076i
$905$			12.0000i			0.398893i
$906$		0			0
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	−0.813707	−	0.581276i	$-0.802554\pi$
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	0.581276	+	0.813707i	$0.302554\pi$
$908$	39.1918	+	39.1918i	1.30063	+	1.30063i
$909$	−7.34847	−	7.34847i	−0.243733	−	0.243733i
$910$	41.3939	+	17.3939i	1.37219	+	0.576601i
$911$			50.0000i			1.65657i	0.560304	+	0.828287i	$0.310684\pi$
$911$			50.0000i			1.65657i	−0.560304	+	0.828287i	$0.689316\pi$
$912$		0			0
$913$		0			0
$914$	18.0000	+	18.0000i	0.595387	+	0.595387i
$915$		0			0
$916$	−14.6969	−	14.6969i	−0.485601	−	0.485601i
$917$	−33.7980	−	14.2020i	−1.11611	−	0.468993i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$	39.1918			1.29212
$921$		0			0
$922$	−44.0908			−1.45205
$923$	4.89898	+	4.89898i	0.161252	+	0.161252i
$924$		0			0
$925$	35.0000	−	35.0000i	1.15079	−	1.15079i
$926$	−4.00000	−	4.00000i	−0.131448	−	0.131448i
$927$	29.3939			0.965422
$928$			24.0000i			0.787839i
$929$			34.2929i			1.12511i	0.826759	+	0.562556i	$0.190182\pi$
$929$			34.2929i			1.12511i	−0.826759	+	0.562556i	$0.809818\pi$
$930$		0			0
$931$	0.494897	−	48.4949i	0.0162196	−	1.58936i
$932$	8.00000			0.262049
$933$		0			0
$934$		−	39.1918i		−	1.28240i
$935$	−24.0000			−0.784884
$936$	−29.3939			−0.960769
$937$	29.3939			0.960256			0.480128	−	0.877198i	$-0.340590\pi$
$937$	29.3939			0.960256			0.480128	+	0.877198i	$0.340590\pi$
$938$	10.0000	+	24.4949i	0.326512	+	0.799787i
$939$		0			0
$940$	−24.0000	−	24.0000i	−0.782794	−	0.782794i
$941$	−26.9444	−	26.9444i	−0.878362	−	0.878362i	0.115003	−	0.993365i	$-0.463312\pi$
$941$	−26.9444	−	26.9444i	−0.878362	−	0.878362i	−0.993365	+	0.115003i	$0.963312\pi$
$942$		0			0
$943$	−19.5959			−0.638131
$944$	−19.5959	−	19.5959i	−0.637793	−	0.637793i
$945$		0			0
$946$	−10.0000	−	10.0000i	−0.325128	−	0.325128i
$947$	−15.0000	−	15.0000i	−0.487435	−	0.487435i	0.420061	−	0.907496i	$-0.362009\pi$
$947$	−15.0000	−	15.0000i	−0.487435	−	0.487435i	−0.907496	+	0.420061i	$0.862009\pi$
$948$		0			0
$949$	−24.0000	−	24.0000i	−0.779073	−	0.779073i
$950$	−68.5857			−2.22521
$951$		0			0
$952$	−33.7980	−	14.2020i	−1.09540	−	0.460291i
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$			6.00000i			0.194257i
$955$	−48.9898	−	48.9898i	−1.58527	−	1.58527i
$956$	−28.0000			−0.905585
$957$		0			0
$958$	24.4949	−	24.4949i	0.791394	−	0.791394i
$959$	−19.5959	+	8.00000i	−0.632785	+	0.258333i
$960$		0			0
$961$	−7.00000			−0.225806
$962$	−24.4949	−	24.4949i	−0.789747	−	0.789747i
$963$	9.00000	+	9.00000i	0.290021	+	0.290021i
$964$	48.9898			1.57786
$965$	−39.1918	+	39.1918i	−1.26163	+	1.26163i
$966$		0			0
$967$	28.0000			0.900419			0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000			0.900419			0.450210	+	0.892923i	$0.351349\pi$
$968$	−18.0000	+	18.0000i	−0.578542	+	0.578542i
$969$		0			0
$970$	24.0000			0.770594
$971$	9.79796	+	9.79796i	0.314431	+	0.314431i	0.846624	−	0.532192i	$-0.178632\pi$
$971$	9.79796	+	9.79796i	0.314431	+	0.314431i	−0.532192	+	0.846624i	$0.678632\pi$
$972$		0			0
$973$	7.10102	−	16.8990i	0.227648	−	0.541756i
$974$	12.0000	−	12.0000i	0.384505	−	0.384505i
$975$		0			0
$976$	9.79796	+	9.79796i	0.313625	+	0.313625i
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$		0			0
$979$		0			0
$980$	48.4949	+	0.494897i	1.54911	+	0.0158089i
$981$	21.0000	+	21.0000i	0.670478	+	0.670478i
$982$			22.0000i			0.702048i
$983$		−	9.79796i		−	0.312506i	−0.987717	−	0.156253i	$-0.950058\pi$
$983$		−	9.79796i		−	0.312506i	0.987717	−	0.156253i	$-0.0499416\pi$
$984$		0			0
$985$	−73.4847			−2.34142
$986$	−29.3939			−0.936092
$987$		0			0
$988$			48.0000i			1.52708i
$989$	−20.0000	+	20.0000i	−0.635963	+	0.635963i
$990$	−14.6969	+	14.6969i	−0.467099	+	0.467099i
$991$		−	60.0000i		−	1.90596i	−0.303029	−	0.952981i	$-0.597998\pi$
$991$		−	60.0000i		−	1.90596i	0.303029	−	0.952981i	$-0.402002\pi$
$992$	−19.5959	+	19.5959i	−0.622171	+	0.622171i
$993$		0			0
$994$	6.89898	+	2.89898i	0.218822	+	0.0919500i
$995$	24.0000	+	24.0000i	0.760851	+	0.760851i
$996$		0			0
$997$	−7.34847	+	7.34847i	−0.232728	+	0.232728i	−0.813831	−	0.581102i	$-0.802622\pi$
$997$	−7.34847	+	7.34847i	−0.232728	+	0.232728i	0.581102	+	0.813831i	$0.302622\pi$
$998$	−46.0000			−1.45610
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.2.j.a.27.2	yes	4
4.3	odd	2		448.2.j.b.335.2		4
7.2	even	3		784.2.w.d.619.1		8
7.3	odd	6		784.2.w.d.411.1		8
7.4	even	3		784.2.w.d.411.2		8
7.5	odd	6		784.2.w.d.619.2		8
7.6	odd	2	inner	112.2.j.a.27.1	✓	4
8.3	odd	2		896.2.j.e.671.1		4
8.5	even	2		896.2.j.b.671.1		4
16.3	odd	4	inner	112.2.j.a.83.1	yes	4
16.5	even	4		896.2.j.e.223.2		4
16.11	odd	4		896.2.j.b.223.2		4
16.13	even	4		448.2.j.b.111.1		4
28.27	even	2		448.2.j.b.335.1		4
56.13	odd	2		896.2.j.b.671.2		4
56.27	even	2		896.2.j.e.671.2		4
112.3	even	12		784.2.w.d.19.1		8
112.13	odd	4		448.2.j.b.111.2		4
112.19	even	12		784.2.w.d.227.2		8
112.27	even	4		896.2.j.b.223.1		4
112.51	odd	12		784.2.w.d.227.1		8
112.67	odd	12		784.2.w.d.19.2		8
112.69	odd	4		896.2.j.e.223.1		4
112.83	even	4	inner	112.2.j.a.83.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.a.27.1	✓	4	7.6	odd	2	inner
112.2.j.a.27.2	yes	4	1.1	even	1	trivial
112.2.j.a.83.1	yes	4	16.3	odd	4	inner
112.2.j.a.83.2	yes	4	112.83	even	4	inner
448.2.j.b.111.1		4	16.13	even	4
448.2.j.b.111.2		4	112.13	odd	4
448.2.j.b.335.1		4	28.27	even	2
448.2.j.b.335.2		4	4.3	odd	2
784.2.w.d.19.1		8	112.3	even	12
784.2.w.d.19.2		8	112.67	odd	12
784.2.w.d.227.1		8	112.51	odd	12
784.2.w.d.227.2		8	112.19	even	12
784.2.w.d.411.1		8	7.3	odd	6
784.2.w.d.411.2		8	7.4	even	3
784.2.w.d.619.1		8	7.2	even	3
784.2.w.d.619.2		8	7.5	odd	6
896.2.j.b.223.1		4	112.27	even	4
896.2.j.b.223.2		4	16.11	odd	4
896.2.j.b.671.1		4	8.5	even	2
896.2.j.b.671.2		4	56.13	odd	2
896.2.j.e.223.1		4	112.69	odd	4
896.2.j.e.223.2		4	16.5	even	4
896.2.j.e.671.1		4	8.3	odd	2
896.2.j.e.671.2		4	56.27	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.44949 - 2.44949i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(2.44949 - 2.44949i) q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +4.89898i q^{10} +(1.00000 - 1.00000i) q^{11} +(2.44949 + 2.44949i) q^{13} +(3.44949 + 1.44949i) q^{14} -4.00000 q^{16} -4.89898i q^{17} +(-3.00000 - 3.00000i) q^{18} +(-4.89898 + 4.89898i) q^{19} +(-4.89898 - 4.89898i) q^{20} +2.00000i q^{22} +4.00000 q^{23} -7.00000i q^{25} -4.89898 q^{26} +(-4.89898 + 2.00000i) q^{28} +(-3.00000 + 3.00000i) q^{29} -4.89898 q^{31} +(4.00000 - 4.00000i) q^{32} +(4.89898 + 4.89898i) q^{34} +(-8.44949 - 3.55051i) q^{35} +6.00000 q^{36} +(5.00000 + 5.00000i) q^{37} -9.79796i q^{38} +9.79796 q^{40} -4.89898 q^{41} +(-5.00000 + 5.00000i) q^{43} +(-2.00000 - 2.00000i) q^{44} +(7.34847 + 7.34847i) q^{45} +(-4.00000 + 4.00000i) q^{46} +4.89898 q^{47} +(-5.00000 + 4.89898i) q^{49} +(7.00000 + 7.00000i) q^{50} +(4.89898 - 4.89898i) q^{52} +(-1.00000 - 1.00000i) q^{53} -4.89898i q^{55} +(2.89898 - 6.89898i) q^{56} -6.00000i q^{58} +(4.89898 + 4.89898i) q^{59} +(-2.44949 - 2.44949i) q^{61} +(4.89898 - 4.89898i) q^{62} +(7.34847 - 3.00000i) q^{63} +8.00000i q^{64} +12.0000 q^{65} +(5.00000 + 5.00000i) q^{67} -9.79796 q^{68} +(12.0000 - 4.89898i) q^{70} +2.00000 q^{71} +(-6.00000 + 6.00000i) q^{72} -9.79796 q^{73} -10.0000 q^{74} +(9.79796 + 9.79796i) q^{76} +(-3.44949 - 1.44949i) q^{77} -4.00000i q^{79} +(-9.79796 + 9.79796i) q^{80} -9.00000 q^{81} +(4.89898 - 4.89898i) q^{82} +(-12.0000 - 12.0000i) q^{85} -10.0000i q^{86} +4.00000 q^{88} -14.6969 q^{90} +(3.55051 - 8.44949i) q^{91} -8.00000i q^{92} +(-4.89898 + 4.89898i) q^{94} +24.0000i q^{95} -4.89898i q^{97} +(0.101021 - 9.89898i) q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^7 + 8 * q^8	\( 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8} + 4 q^{11} + 4 q^{14} - 16 q^{16} - 12 q^{18} + 16 q^{23} - 12 q^{29} + 16 q^{32} - 24 q^{35} + 24 q^{36} + 20 q^{37} - 20 q^{43} - 8 q^{44} - 16 q^{46} - 20 q^{49} + 28 q^{50} - 4 q^{53} - 8 q^{56} + 48 q^{65} + 20 q^{67} + 48 q^{70} + 8 q^{71} - 24 q^{72} - 40 q^{74} - 4 q^{77} - 36 q^{81} - 48 q^{85} + 16 q^{88} + 24 q^{91} + 20 q^{98} + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^7 + 8 * q^8 + 4 * q^11 + 4 * q^14 - 16 * q^16 - 12 * q^18 + 16 * q^23 - 12 * q^29 + 16 * q^32 - 24 * q^35 + 24 * q^36 + 20 * q^37 - 20 * q^43 - 8 * q^44 - 16 * q^46 - 20 * q^49 + 28 * q^50 - 4 * q^53 - 8 * q^56 + 48 * q^65 + 20 * q^67 + 48 * q^70 + 8 * q^71 - 24 * q^72 - 40 * q^74 - 4 * q^77 - 36 * q^81 - 48 * q^85 + 16 * q^88 + 24 * q^91 + 20 * q^98 + 12 * q^99

Introduction

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