LMFDB - Embedded newform 368.2.j.a.277.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [368,2,Mod(93,368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("368.93"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(368, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	368.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.93849479438$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			277.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	368.277
Dual form			368.2.j.a.93.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} +(1.00000 - 1.00000i) q^{3} -2.00000i q^{4} -2.00000i q^{6} -4.00000i q^{7} +(-2.00000 - 2.00000i) q^{8} +1.00000i q^{9} +(4.00000 + 4.00000i) q^{11} +(-2.00000 - 2.00000i) q^{12} +(-3.00000 + 3.00000i) q^{13} +(-4.00000 - 4.00000i) q^{14} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +(-4.00000 - 4.00000i) q^{21} +8.00000 q^{22} +1.00000i q^{23} -4.00000 q^{24} -5.00000i q^{25} +6.00000i q^{26} +(4.00000 + 4.00000i) q^{27} -8.00000 q^{28} +(7.00000 - 7.00000i) q^{29} +2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +8.00000 q^{33} +(-2.00000 + 2.00000i) q^{34} +2.00000 q^{36} +(4.00000 + 4.00000i) q^{37} +6.00000i q^{39} +10.0000i q^{41} -8.00000 q^{42} +(-4.00000 - 4.00000i) q^{43} +(8.00000 - 8.00000i) q^{44} +(1.00000 + 1.00000i) q^{46} +(-4.00000 + 4.00000i) q^{48} -9.00000 q^{49} +(-5.00000 - 5.00000i) q^{50} +(-2.00000 + 2.00000i) q^{51} +(6.00000 + 6.00000i) q^{52} +(-8.00000 - 8.00000i) q^{53} +8.00000 q^{54} +(-8.00000 + 8.00000i) q^{56} -14.0000i q^{58} +(7.00000 + 7.00000i) q^{59} +(2.00000 - 2.00000i) q^{61} +(2.00000 - 2.00000i) q^{62} +4.00000 q^{63} +8.00000i q^{64} +(8.00000 - 8.00000i) q^{66} +(-4.00000 + 4.00000i) q^{67} +4.00000i q^{68} +(1.00000 + 1.00000i) q^{69} +14.0000i q^{71} +(2.00000 - 2.00000i) q^{72} +10.0000i q^{73} +8.00000 q^{74} +(-5.00000 - 5.00000i) q^{75} +(16.0000 - 16.0000i) q^{77} +(6.00000 + 6.00000i) q^{78} -4.00000 q^{79} +5.00000 q^{81} +(10.0000 + 10.0000i) q^{82} +(4.00000 - 4.00000i) q^{83} +(-8.00000 + 8.00000i) q^{84} -8.00000 q^{86} -14.0000i q^{87} -16.0000i q^{88} -6.00000i q^{89} +(12.0000 + 12.0000i) q^{91} +2.00000 q^{92} +(2.00000 - 2.00000i) q^{93} +8.00000i q^{96} -14.0000 q^{97} +(-9.00000 + 9.00000i) q^{98} +(-4.00000 + 4.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 4 q^{8} + 8 q^{11} - 4 q^{12} - 6 q^{13} - 8 q^{14} - 8 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{21} + 16 q^{22} - 8 q^{24} + 8 q^{27} - 16 q^{28} + 14 q^{29} + 4 q^{31} - 8 q^{32}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 - 4 * q^8 + 8 * q^11 - 4 * q^12 - 6 * q^13 - 8 * q^14 - 8 * q^16 - 4 * q^17 + 2 * q^18 - 8 * q^21 + 16 * q^22 - 8 * q^24 + 8 * q^27 - 16 * q^28 + 14 * q^29 + 4 * q^31 - 8 * q^32 + 16 * q^33 - 4 * q^34 + 4 * q^36 + 8 * q^37 - 16 * q^42 - 8 * q^43 + 16 * q^44 + 2 * q^46 - 8 * q^48 - 18 * q^49 - 10 * q^50 - 4 * q^51 + 12 * q^52 - 16 * q^53 + 16 * q^54 - 16 * q^56 + 14 * q^59 + 4 * q^61 + 4 * q^62 + 8 * q^63 + 16 * q^66 - 8 * q^67 + 2 * q^69 + 4 * q^72 + 16 * q^74 - 10 * q^75 + 32 * q^77 + 12 * q^78 - 8 * q^79 + 10 * q^81 + 20 * q^82 + 8 * q^83 - 16 * q^84 - 16 * q^86 + 24 * q^91 + 4 * q^92 + 4 * q^93 - 28 * q^97 - 18 * q^98 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/368\mathbb{Z}\right)^\times$.

$n$	$47$	$97$	$277$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$		−	2.00000i		−	1.00000i
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		−	2.00000i		−	0.816497i
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			1.00000i			0.333333i
$10$		0			0
$11$	4.00000	+	4.00000i	1.20605	+	1.20605i	0.972297	+	0.233748i	$0.0750991\pi$
$11$	4.00000	+	4.00000i	1.20605	+	1.20605i	0.233748	+	0.972297i	$0.424901\pi$
$12$	−2.00000	−	2.00000i	−0.577350	−	0.577350i
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$	−4.00000	−	4.00000i	−1.06904	−	1.06904i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	+	1.00000i	0.235702	+	0.235702i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$	−4.00000	−	4.00000i	−0.872872	−	0.872872i
$22$	8.00000			1.70561
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$	−4.00000			−0.816497
$25$		−	5.00000i		−	1.00000i
$26$			6.00000i			1.17670i
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$	−8.00000			−1.51186
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$	8.00000			1.39262
$34$	−2.00000	+	2.00000i	−0.342997	+	0.342997i
$35$		0			0
$36$	2.00000			0.333333
$37$	4.00000	+	4.00000i	0.657596	+	0.657596i	0.954811	−	0.297215i	$-0.0960577\pi$
$37$	4.00000	+	4.00000i	0.657596	+	0.657596i	−0.297215	+	0.954811i	$0.596058\pi$
$38$		0			0
$39$			6.00000i			0.960769i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−8.00000			−1.23443
$43$	−4.00000	−	4.00000i	−0.609994	−	0.609994i	0.332950	−	0.942944i	$-0.391956\pi$
$43$	−4.00000	−	4.00000i	−0.609994	−	0.609994i	−0.942944	+	0.332950i	$0.891956\pi$
$44$	8.00000	−	8.00000i	1.20605	−	1.20605i
$45$		0			0
$46$	1.00000	+	1.00000i	0.147442	+	0.147442i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−4.00000	+	4.00000i	−0.577350	+	0.577350i
$49$	−9.00000			−1.28571
$50$	−5.00000	−	5.00000i	−0.707107	−	0.707107i
$51$	−2.00000	+	2.00000i	−0.280056	+	0.280056i
$52$	6.00000	+	6.00000i	0.832050	+	0.832050i
$53$	−8.00000	−	8.00000i	−1.09888	−	1.09888i	−0.994541	−	0.104343i	$-0.966726\pi$
$53$	−8.00000	−	8.00000i	−1.09888	−	1.09888i	−0.104343	−	0.994541i	$-0.533274\pi$
$54$	8.00000			1.08866
$55$		0			0
$56$	−8.00000	+	8.00000i	−1.06904	+	1.06904i
$57$		0			0
$58$		−	14.0000i		−	1.83829i
$59$	7.00000	+	7.00000i	0.911322	+	0.911322i	0.996376	−	0.0850540i	$-0.0271063\pi$
$59$	7.00000	+	7.00000i	0.911322	+	0.911322i	−0.0850540	+	0.996376i	$0.527106\pi$
$60$		0			0
$61$	2.00000	−	2.00000i	0.256074	−	0.256074i	−0.567381	−	0.823455i	$-0.692043\pi$
$61$	2.00000	−	2.00000i	0.256074	−	0.256074i	0.823455	+	0.567381i	$0.192043\pi$
$62$	2.00000	−	2.00000i	0.254000	−	0.254000i
$63$	4.00000			0.503953
$64$			8.00000i			1.00000i
$65$		0			0
$66$	8.00000	−	8.00000i	0.984732	−	0.984732i
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	−0.907889	−	0.419211i	$-0.862307\pi$
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	0.419211	+	0.907889i	$0.362307\pi$
$68$			4.00000i			0.485071i
$69$	1.00000	+	1.00000i	0.120386	+	0.120386i
$70$		0			0
$71$			14.0000i			1.66149i	0.556650	+	0.830747i	$0.312086\pi$
$71$			14.0000i			1.66149i	−0.556650	+	0.830747i	$0.687914\pi$
$72$	2.00000	−	2.00000i	0.235702	−	0.235702i
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	8.00000			0.929981
$75$	−5.00000	−	5.00000i	−0.577350	−	0.577350i
$76$		0			0
$77$	16.0000	−	16.0000i	1.82337	−	1.82337i
$78$	6.00000	+	6.00000i	0.679366	+	0.679366i
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$	10.0000	+	10.0000i	1.10432	+	1.10432i
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	−0.452638	−	0.891695i	$-0.649517\pi$
$83$	4.00000	−	4.00000i	0.439057	−	0.439057i	0.891695	+	0.452638i	$0.149517\pi$
$84$	−8.00000	+	8.00000i	−0.872872	+	0.872872i
$85$		0			0
$86$	−8.00000			−0.862662
$87$		−	14.0000i		−	1.50096i
$88$		−	16.0000i		−	1.70561i
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	12.0000	+	12.0000i	1.25794	+	1.25794i
$92$	2.00000			0.208514
$93$	2.00000	−	2.00000i	0.207390	−	0.207390i
$94$		0			0
$95$		0			0
$96$			8.00000i			0.816497i
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	−9.00000	+	9.00000i	−0.909137	+	0.909137i
$99$	−4.00000	+	4.00000i	−0.402015	+	0.402015i
$100$	−10.0000			−1.00000
$101$	−11.0000	−	11.0000i	−1.09454	−	1.09454i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−11.0000	−	11.0000i	−1.09454	−	1.09454i	−0.0995037	−	0.995037i	$-0.531726\pi$
$102$			4.00000i			0.396059i
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	12.0000			1.17670
$105$		0			0
$106$	−16.0000			−1.55406
$107$	−6.00000	−	6.00000i	−0.580042	−	0.580042i	0.354873	−	0.934915i	$-0.384524\pi$
$107$	−6.00000	−	6.00000i	−0.580042	−	0.580042i	−0.934915	+	0.354873i	$0.884524\pi$
$108$	8.00000	−	8.00000i	0.769800	−	0.769800i
$109$	2.00000	−	2.00000i	0.191565	−	0.191565i	−0.604807	−	0.796372i	$-0.706750\pi$
$109$	2.00000	−	2.00000i	0.191565	−	0.191565i	0.796372	+	0.604807i	$0.206750\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$			16.0000i			1.51186i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$	−14.0000	−	14.0000i	−1.29987	−	1.29987i
$117$	−3.00000	−	3.00000i	−0.277350	−	0.277350i
$118$	14.0000			1.28880
$119$			8.00000i			0.733359i
$120$		0			0
$121$			21.0000i			1.90909i
$122$		−	4.00000i		−	0.362143i
$123$	10.0000	+	10.0000i	0.901670	+	0.901670i
$124$		−	4.00000i		−	0.359211i
$125$		0			0
$126$	4.00000	−	4.00000i	0.356348	−	0.356348i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$	−8.00000			−0.704361
$130$		0			0
$131$	5.00000	−	5.00000i	0.436852	−	0.436852i	−0.454099	−	0.890951i	$-0.650039\pi$
$131$	5.00000	−	5.00000i	0.436852	−	0.436852i	0.890951	+	0.454099i	$0.150039\pi$
$132$		−	16.0000i		−	1.39262i
$133$		0			0
$134$			8.00000i			0.691095i
$135$		0			0
$136$	4.00000	+	4.00000i	0.342997	+	0.342997i
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	2.00000			0.170251
$139$	15.0000	+	15.0000i	1.27228	+	1.27228i	0.944887	+	0.327396i	$0.106171\pi$
$139$	15.0000	+	15.0000i	1.27228	+	1.27228i	0.327396	+	0.944887i	$0.393829\pi$
$140$		0			0
$141$		0			0
$142$	14.0000	+	14.0000i	1.17485	+	1.17485i
$143$	−24.0000			−2.00698
$144$		−	4.00000i		−	0.333333i
$145$		0			0
$146$	10.0000	+	10.0000i	0.827606	+	0.827606i
$147$	−9.00000	+	9.00000i	−0.742307	+	0.742307i
$148$	8.00000	−	8.00000i	0.657596	−	0.657596i
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	−10.0000			−0.816497
$151$		−	16.0000i		−	1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$		−	16.0000i		−	1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		−	32.0000i		−	2.57863i
$155$		0			0
$156$	12.0000			0.960769
$157$	−12.0000	+	12.0000i	−0.957704	+	0.957704i	−0.999141	−	0.0414369i	$-0.986806\pi$
$157$	−12.0000	+	12.0000i	−0.957704	+	0.957704i	0.0414369	+	0.999141i	$0.486806\pi$
$158$	−4.00000	+	4.00000i	−0.318223	+	0.318223i
$159$	−16.0000			−1.26888
$160$		0			0
$161$	4.00000			0.315244
$162$	5.00000	−	5.00000i	0.392837	−	0.392837i
$163$	−3.00000	+	3.00000i	−0.234978	+	0.234978i	−0.814767	−	0.579789i	$-0.803135\pi$
$163$	−3.00000	+	3.00000i	−0.234978	+	0.234978i	0.579789	+	0.814767i	$0.303135\pi$
$164$	20.0000			1.56174
$165$		0			0
$166$		−	8.00000i		−	0.620920i
$167$		−	14.0000i		−	1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$		−	14.0000i		−	1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$			16.0000i			1.23443i
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$	−8.00000	+	8.00000i	−0.609994	+	0.609994i
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−11.0000	+	11.0000i	−0.836315	+	0.836315i	0.152057	+	0.988372i	$0.451410\pi$
$174$	−14.0000	−	14.0000i	−1.06134	−	1.06134i
$175$	−20.0000			−1.51186
$176$	−16.0000	−	16.0000i	−1.20605	−	1.20605i
$177$	14.0000			1.05230
$178$	−6.00000	−	6.00000i	−0.449719	−	0.449719i
$179$	15.0000	−	15.0000i	1.12115	−	1.12115i	0.129584	−	0.991568i	$-0.458636\pi$
$179$	15.0000	−	15.0000i	1.12115	−	1.12115i	0.991568	−	0.129584i	$-0.0413643\pi$
$180$		0			0
$181$	−6.00000	−	6.00000i	−0.445976	−	0.445976i	0.448038	−	0.894015i	$-0.352123\pi$
$181$	−6.00000	−	6.00000i	−0.445976	−	0.445976i	−0.894015	+	0.448038i	$0.852123\pi$
$182$	24.0000			1.77900
$183$		−	4.00000i		−	0.295689i
$184$	2.00000	−	2.00000i	0.147442	−	0.147442i
$185$		0			0
$186$		−	4.00000i		−	0.293294i
$187$	−8.00000	−	8.00000i	−0.585018	−	0.585018i
$188$		0			0
$189$	16.0000	−	16.0000i	1.16383	−	1.16383i
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	8.00000	+	8.00000i	0.577350	+	0.577350i
$193$	−10.0000			−0.719816			−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000			−0.719816			−0.359908	+	0.932988i	$0.617192\pi$
$194$	−14.0000	+	14.0000i	−1.00514	+	1.00514i
$195$		0			0
$196$			18.0000i			1.28571i
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	0.862423	−	0.506188i	$-0.168946\pi$
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	−0.506188	+	0.862423i	$0.668946\pi$
$198$			8.00000i			0.568535i
$199$			20.0000i			1.41776i	0.705328	+	0.708881i	$0.250800\pi$
$199$			20.0000i			1.41776i	−0.705328	+	0.708881i	$0.749200\pi$
$200$	−10.0000	+	10.0000i	−0.707107	+	0.707107i
$201$			8.00000i			0.564276i
$202$	−22.0000			−1.54791
$203$	−28.0000	−	28.0000i	−1.96521	−	1.96521i
$204$	4.00000	+	4.00000i	0.280056	+	0.280056i
$205$		0			0
$206$	4.00000	+	4.00000i	0.278693	+	0.278693i
$207$	−1.00000			−0.0695048
$208$	12.0000	−	12.0000i	0.832050	−	0.832050i
$209$		0			0
$210$		0			0
$211$	9.00000	−	9.00000i	0.619586	−	0.619586i	−0.325840	−	0.945425i	$-0.605647\pi$
$211$	9.00000	−	9.00000i	0.619586	−	0.619586i	0.945425	+	0.325840i	$0.105647\pi$
$212$	−16.0000	+	16.0000i	−1.09888	+	1.09888i
$213$	14.0000	+	14.0000i	0.959264	+	0.959264i
$214$	−12.0000			−0.820303
$215$		0			0
$216$		−	16.0000i		−	1.08866i
$217$		−	8.00000i		−	0.543075i
$218$		−	4.00000i		−	0.270914i
$219$	10.0000	+	10.0000i	0.675737	+	0.675737i
$220$		0			0
$221$	6.00000	−	6.00000i	0.403604	−	0.403604i
$222$	8.00000	−	8.00000i	0.536925	−	0.536925i
$223$	−18.0000			−1.20537			−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000			−1.20537			−0.602685	+	0.797980i	$0.705902\pi$
$224$	16.0000	+	16.0000i	1.06904	+	1.06904i
$225$	5.00000			0.333333
$226$	−6.00000	+	6.00000i	−0.399114	+	0.399114i
$227$	10.0000	−	10.0000i	0.663723	−	0.663723i	−0.292532	−	0.956256i	$-0.594498\pi$
$227$	10.0000	−	10.0000i	0.663723	−	0.663723i	0.956256	+	0.292532i	$0.0944979\pi$
$228$		0			0
$229$	−6.00000	−	6.00000i	−0.396491	−	0.396491i	0.480502	−	0.876993i	$-0.340454\pi$
$229$	−6.00000	−	6.00000i	−0.396491	−	0.396491i	−0.876993	+	0.480502i	$0.840454\pi$
$230$		0			0
$231$		−	32.0000i		−	2.10545i
$232$	−28.0000			−1.83829
$233$			4.00000i			0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$			4.00000i			0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$	−6.00000			−0.392232
$235$		0			0
$236$	14.0000	−	14.0000i	0.911322	−	0.911322i
$237$	−4.00000	+	4.00000i	−0.259828	+	0.259828i
$238$	8.00000	+	8.00000i	0.518563	+	0.518563i
$239$	−2.00000			−0.129369			−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000			−0.129369			−0.0646846	+	0.997906i	$0.520604\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$	21.0000	+	21.0000i	1.34993	+	1.34993i
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$	−4.00000	−	4.00000i	−0.256074	−	0.256074i
$245$		0			0
$246$	20.0000			1.27515
$247$		0			0
$248$	−4.00000	−	4.00000i	−0.254000	−	0.254000i
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$	6.00000	+	6.00000i	0.378717	+	0.378717i	0.870639	−	0.491922i	$-0.163706\pi$
$251$	6.00000	+	6.00000i	0.378717	+	0.378717i	−0.491922	+	0.870639i	$0.663706\pi$
$252$		−	8.00000i		−	0.503953i
$253$	−4.00000	+	4.00000i	−0.251478	+	0.251478i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−20.0000			−1.24757			−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000			−1.24757			−0.623783	+	0.781598i	$0.714405\pi$
$258$	−8.00000	+	8.00000i	−0.498058	+	0.498058i
$259$	16.0000	−	16.0000i	0.994192	−	0.994192i
$260$		0			0
$261$	7.00000	+	7.00000i	0.433289	+	0.433289i
$262$		−	10.0000i		−	0.617802i
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	−16.0000	−	16.0000i	−0.984732	−	0.984732i
$265$		0			0
$266$		0			0
$267$	−6.00000	−	6.00000i	−0.367194	−	0.367194i
$268$	8.00000	+	8.00000i	0.488678	+	0.488678i
$269$	1.00000	−	1.00000i	0.0609711	−	0.0609711i	−0.675964	−	0.736935i	$-0.736272\pi$
$269$	1.00000	−	1.00000i	0.0609711	−	0.0609711i	0.736935	+	0.675964i	$0.236272\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	8.00000			0.485071
$273$	24.0000			1.45255
$274$	−18.0000	−	18.0000i	−1.08742	−	1.08742i
$275$	20.0000	−	20.0000i	1.20605	−	1.20605i
$276$	2.00000	−	2.00000i	0.120386	−	0.120386i
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	0.611213	−	0.791466i	$-0.290682\pi$
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	−0.791466	+	0.611213i	$0.790682\pi$
$278$	30.0000			1.79928
$279$			2.00000i			0.119737i
$280$		0			0
$281$		−	2.00000i		−	0.119310i	−0.998219	−	0.0596550i	$-0.981000\pi$
$281$		−	2.00000i		−	0.119310i	0.998219	−	0.0596550i	$-0.0190001\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$	28.0000			1.66149
$285$		0			0
$286$	−24.0000	+	24.0000i	−1.41915	+	1.41915i
$287$	40.0000			2.36113
$288$	−4.00000	−	4.00000i	−0.235702	−	0.235702i
$289$	−13.0000			−0.764706
$290$		0			0
$291$	−14.0000	+	14.0000i	−0.820695	+	0.820695i
$292$	20.0000			1.17041
$293$	16.0000	+	16.0000i	0.934730	+	0.934730i	0.997997	−	0.0632667i	$-0.0201519\pi$
$293$	16.0000	+	16.0000i	0.934730	+	0.934730i	−0.0632667	+	0.997997i	$0.520152\pi$
$294$			18.0000i			1.04978i
$295$		0			0
$296$		−	16.0000i		−	0.929981i
$297$			32.0000i			1.85683i
$298$		0			0
$299$	−3.00000	−	3.00000i	−0.173494	−	0.173494i
$300$	−10.0000	+	10.0000i	−0.577350	+	0.577350i
$301$	−16.0000	+	16.0000i	−0.922225	+	0.922225i
$302$	−16.0000	−	16.0000i	−0.920697	−	0.920697i
$303$	−22.0000			−1.26387
$304$		0			0
$305$		0			0
$306$	−2.00000	−	2.00000i	−0.114332	−	0.114332i
$307$	5.00000	−	5.00000i	0.285365	−	0.285365i	−0.549879	−	0.835244i	$-0.685326\pi$
$307$	5.00000	−	5.00000i	0.285365	−	0.285365i	0.835244	+	0.549879i	$0.185326\pi$
$308$	−32.0000	−	32.0000i	−1.82337	−	1.82337i
$309$	4.00000	+	4.00000i	0.227552	+	0.227552i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	12.0000	−	12.0000i	0.679366	−	0.679366i
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$			24.0000i			1.35440i
$315$		0			0
$316$			8.00000i			0.450035i
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	−0.833439	−	0.552611i	$-0.813631\pi$
$317$	−5.00000	+	5.00000i	−0.280828	+	0.280828i	0.552611	+	0.833439i	$0.313631\pi$
$318$	−16.0000	+	16.0000i	−0.897235	+	0.897235i
$319$	56.0000			3.13540
$320$		0			0
$321$	−12.0000			−0.669775
$322$	4.00000	−	4.00000i	0.222911	−	0.222911i
$323$		0			0
$324$		−	10.0000i		−	0.555556i
$325$	15.0000	+	15.0000i	0.832050	+	0.832050i
$326$			6.00000i			0.332309i
$327$		−	4.00000i		−	0.221201i
$328$	20.0000	−	20.0000i	1.10432	−	1.10432i
$329$		0			0
$330$		0			0
$331$	13.0000	+	13.0000i	0.714545	+	0.714545i	0.967483	−	0.252938i	$-0.0813968\pi$
$331$	13.0000	+	13.0000i	0.714545	+	0.714545i	−0.252938	+	0.967483i	$0.581397\pi$
$332$	−8.00000	−	8.00000i	−0.439057	−	0.439057i
$333$	−4.00000	+	4.00000i	−0.219199	+	0.219199i
$334$	−14.0000	−	14.0000i	−0.766046	−	0.766046i
$335$		0			0
$336$	16.0000	+	16.0000i	0.872872	+	0.872872i
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$	−5.00000	−	5.00000i	−0.271964	−	0.271964i
$339$	−6.00000	+	6.00000i	−0.325875	+	0.325875i
$340$		0			0
$341$	8.00000	+	8.00000i	0.433224	+	0.433224i
$342$		0			0
$343$			8.00000i			0.431959i
$344$			16.0000i			0.862662i
$345$		0			0
$346$			22.0000i			1.18273i
$347$	−1.00000	−	1.00000i	−0.0536828	−	0.0536828i	0.679756	−	0.733439i	$-0.262086\pi$
$347$	−1.00000	−	1.00000i	−0.0536828	−	0.0536828i	−0.733439	+	0.679756i	$0.762086\pi$
$348$	−28.0000			−1.50096
$349$	1.00000	−	1.00000i	0.0535288	−	0.0535288i	−0.679836	−	0.733364i	$-0.737949\pi$
$349$	1.00000	−	1.00000i	0.0535288	−	0.0535288i	0.733364	+	0.679836i	$0.237949\pi$
$350$	−20.0000	+	20.0000i	−1.06904	+	1.06904i
$351$	−24.0000			−1.28103
$352$	−32.0000			−1.70561
$353$	30.0000			1.59674			0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000			1.59674			0.798369	+	0.602168i	$0.205696\pi$
$354$	14.0000	−	14.0000i	0.744092	−	0.744092i
$355$		0			0
$356$	−12.0000			−0.635999
$357$	8.00000	+	8.00000i	0.423405	+	0.423405i
$358$		−	30.0000i		−	1.58555i
$359$		−	32.0000i		−	1.68890i	−0.535638	−	0.844448i	$-0.679929\pi$
$359$		−	32.0000i		−	1.68890i	0.535638	−	0.844448i	$-0.320071\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$	−12.0000			−0.630706
$363$	21.0000	+	21.0000i	1.10221	+	1.10221i
$364$	24.0000	−	24.0000i	1.25794	−	1.25794i
$365$		0			0
$366$	−4.00000	−	4.00000i	−0.209083	−	0.209083i
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$		−	4.00000i		−	0.208514i
$369$	−10.0000			−0.520579
$370$		0			0
$371$	−32.0000	+	32.0000i	−1.66136	+	1.66136i
$372$	−4.00000	−	4.00000i	−0.207390	−	0.207390i
$373$	−16.0000	−	16.0000i	−0.828449	−	0.828449i	0.158854	−	0.987302i	$-0.449220\pi$
$373$	−16.0000	−	16.0000i	−0.828449	−	0.828449i	−0.987302	+	0.158854i	$0.949220\pi$
$374$	−16.0000			−0.827340
$375$		0			0
$376$		0			0
$377$			42.0000i			2.16311i
$378$		−	32.0000i		−	1.64590i
$379$	−18.0000	−	18.0000i	−0.924598	−	0.924598i	0.0727522	−	0.997350i	$-0.476822\pi$
$379$	−18.0000	−	18.0000i	−0.924598	−	0.924598i	−0.997350	+	0.0727522i	$0.976822\pi$
$380$		0			0
$381$		0			0
$382$	−8.00000	+	8.00000i	−0.409316	+	0.409316i
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$	16.0000			0.816497
$385$		0			0
$386$	−10.0000	+	10.0000i	−0.508987	+	0.508987i
$387$	4.00000	−	4.00000i	0.203331	−	0.203331i
$388$			28.0000i			1.42148i
$389$	22.0000	+	22.0000i	1.11544	+	1.11544i	0.992401	+	0.123043i	$0.0392653\pi$
$389$	22.0000	+	22.0000i	1.11544	+	1.11544i	0.123043	+	0.992401i	$0.460735\pi$
$390$		0			0
$391$		−	2.00000i		−	0.101144i
$392$	18.0000	+	18.0000i	0.909137	+	0.909137i
$393$		−	10.0000i		−	0.504433i
$394$	10.0000			0.503793
$395$		0			0
$396$	8.00000	+	8.00000i	0.402015	+	0.402015i
$397$	23.0000	−	23.0000i	1.15434	−	1.15434i	0.168663	−	0.985674i	$-0.446055\pi$
$397$	23.0000	−	23.0000i	1.15434	−	1.15434i	0.985674	−	0.168663i	$-0.0539450\pi$
$398$	20.0000	+	20.0000i	1.00251	+	1.00251i
$399$		0			0
$400$			20.0000i			1.00000i
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$	8.00000	+	8.00000i	0.399004	+	0.399004i
$403$	−6.00000	+	6.00000i	−0.298881	+	0.298881i
$404$	−22.0000	+	22.0000i	−1.09454	+	1.09454i
$405$		0			0
$406$	−56.0000			−2.77923
$407$			32.0000i			1.58618i
$408$	8.00000			0.396059
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		0			0
$411$	−18.0000	−	18.0000i	−0.887875	−	0.887875i
$412$	8.00000			0.394132
$413$	28.0000	−	28.0000i	1.37779	−	1.37779i
$414$	−1.00000	+	1.00000i	−0.0491473	+	0.0491473i
$415$		0			0
$416$		−	24.0000i		−	1.17670i
$417$	30.0000			1.46911
$418$		0			0
$419$	8.00000	−	8.00000i	0.390826	−	0.390826i	−0.484156	−	0.874982i	$-0.660873\pi$
$419$	8.00000	−	8.00000i	0.390826	−	0.390826i	0.874982	+	0.484156i	$0.160873\pi$
$420$		0			0
$421$	−16.0000	−	16.0000i	−0.779792	−	0.779792i	0.200003	−	0.979795i	$-0.435905\pi$
$421$	−16.0000	−	16.0000i	−0.779792	−	0.779792i	−0.979795	+	0.200003i	$0.935905\pi$
$422$		−	18.0000i		−	0.876226i
$423$		0			0
$424$			32.0000i			1.55406i
$425$			10.0000i			0.485071i
$426$	28.0000			1.35660
$427$	−8.00000	−	8.00000i	−0.387147	−	0.387147i
$428$	−12.0000	+	12.0000i	−0.580042	+	0.580042i
$429$	−24.0000	+	24.0000i	−1.15873	+	1.15873i
$430$		0			0
$431$	−4.00000			−0.192673			−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000			−0.192673			−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−16.0000	−	16.0000i	−0.769800	−	0.769800i
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$	−8.00000	−	8.00000i	−0.384012	−	0.384012i
$435$		0			0
$436$	−4.00000	−	4.00000i	−0.191565	−	0.191565i
$437$		0			0
$438$	20.0000			0.955637
$439$		−	2.00000i		−	0.0954548i	−0.998860	−	0.0477274i	$-0.984802\pi$
$439$		−	2.00000i		−	0.0954548i	0.998860	−	0.0477274i	$-0.0151979\pi$
$440$		0			0
$441$		−	9.00000i		−	0.428571i
$442$		−	12.0000i		−	0.570782i
$443$	11.0000	+	11.0000i	0.522626	+	0.522626i	0.918364	−	0.395738i	$-0.129511\pi$
$443$	11.0000	+	11.0000i	0.522626	+	0.522626i	−0.395738	+	0.918364i	$0.629511\pi$
$444$		−	16.0000i		−	0.759326i
$445$		0			0
$446$	−18.0000	+	18.0000i	−0.852325	+	0.852325i
$447$		0			0
$448$	32.0000			1.51186
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	5.00000	−	5.00000i	0.235702	−	0.235702i
$451$	−40.0000	+	40.0000i	−1.88353	+	1.88353i
$452$			12.0000i			0.564433i
$453$	−16.0000	−	16.0000i	−0.751746	−	0.751746i
$454$		−	20.0000i		−	0.938647i
$455$		0			0
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	−12.0000			−0.560723
$459$	−8.00000	−	8.00000i	−0.373408	−	0.373408i
$460$		0			0
$461$	−1.00000	+	1.00000i	−0.0465746	+	0.0465746i	−0.730011	−	0.683436i	$-0.760485\pi$
$461$	−1.00000	+	1.00000i	−0.0465746	+	0.0465746i	0.683436	+	0.730011i	$0.260485\pi$
$462$	−32.0000	−	32.0000i	−1.48877	−	1.48877i
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$	−28.0000	+	28.0000i	−1.29987	+	1.29987i
$465$		0			0
$466$	4.00000	+	4.00000i	0.185296	+	0.185296i
$467$	28.0000	−	28.0000i	1.29569	−	1.29569i	0.364471	−	0.931215i	$-0.381250\pi$
$467$	28.0000	−	28.0000i	1.29569	−	1.29569i	0.931215	−	0.364471i	$-0.118750\pi$
$468$	−6.00000	+	6.00000i	−0.277350	+	0.277350i
$469$	16.0000	+	16.0000i	0.738811	+	0.738811i
$470$		0			0
$471$			24.0000i			1.10586i
$472$		−	28.0000i		−	1.28880i
$473$		−	32.0000i		−	1.47136i
$474$			8.00000i			0.367452i
$475$		0			0
$476$	16.0000			0.733359
$477$	8.00000	−	8.00000i	0.366295	−	0.366295i
$478$	−2.00000	+	2.00000i	−0.0914779	+	0.0914779i
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	−24.0000			−1.09431
$482$	−10.0000	+	10.0000i	−0.455488	+	0.455488i
$483$	4.00000	−	4.00000i	0.182006	−	0.182006i
$484$	42.0000			1.90909
$485$		0			0
$486$			14.0000i			0.635053i
$487$		−	14.0000i		−	0.634401i	−0.948359	−	0.317200i	$-0.897257\pi$
$487$		−	14.0000i		−	0.634401i	0.948359	−	0.317200i	$-0.102743\pi$
$488$	−8.00000			−0.362143
$489$			6.00000i			0.271329i
$490$		0			0
$491$	−7.00000	−	7.00000i	−0.315906	−	0.315906i	0.531287	−	0.847192i	$-0.321709\pi$
$491$	−7.00000	−	7.00000i	−0.315906	−	0.315906i	−0.847192	+	0.531287i	$0.821709\pi$
$492$	20.0000	−	20.0000i	0.901670	−	0.901670i
$493$	−14.0000	+	14.0000i	−0.630528	+	0.630528i
$494$		0			0
$495$		0			0
$496$	−8.00000			−0.359211
$497$	56.0000			2.51194
$498$	−8.00000	−	8.00000i	−0.358489	−	0.358489i
$499$	−15.0000	+	15.0000i	−0.671492	+	0.671492i	−0.958060	−	0.286568i	$-0.907486\pi$
$499$	−15.0000	+	15.0000i	−0.671492	+	0.671492i	0.286568	+	0.958060i	$0.407486\pi$
$500$		0			0
$501$	−14.0000	−	14.0000i	−0.625474	−	0.625474i
$502$	12.0000			0.535586
$503$			36.0000i			1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$			36.0000i			1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	−8.00000	−	8.00000i	−0.356348	−	0.356348i
$505$		0			0
$506$			8.00000i			0.355643i
$507$	−5.00000	−	5.00000i	−0.222058	−	0.222058i
$508$		0			0
$509$	−7.00000	+	7.00000i	−0.310270	+	0.310270i	−0.845014	−	0.534744i	$-0.820408\pi$
$509$	−7.00000	+	7.00000i	−0.310270	+	0.310270i	0.534744	+	0.845014i	$0.320408\pi$
$510$		0			0
$511$	40.0000			1.76950
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	−20.0000	+	20.0000i	−0.882162	+	0.882162i
$515$		0			0
$516$			16.0000i			0.704361i
$517$		0			0
$518$		−	32.0000i		−	1.40600i
$519$			22.0000i			0.965693i
$520$		0			0
$521$		−	18.0000i		−	0.788594i	−0.918983	−	0.394297i	$-0.870988\pi$
$521$		−	18.0000i		−	0.788594i	0.918983	−	0.394297i	$-0.129012\pi$
$522$	14.0000			0.612763
$523$	−14.0000	−	14.0000i	−0.612177	−	0.612177i	0.331336	−	0.943513i	$-0.392501\pi$
$523$	−14.0000	−	14.0000i	−0.612177	−	0.612177i	−0.943513	+	0.331336i	$0.892501\pi$
$524$	−10.0000	−	10.0000i	−0.436852	−	0.436852i
$525$	−20.0000	+	20.0000i	−0.872872	+	0.872872i
$526$	4.00000	+	4.00000i	0.174408	+	0.174408i
$527$	−4.00000			−0.174243
$528$	−32.0000			−1.39262
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−7.00000	+	7.00000i	−0.303774	+	0.303774i
$532$		0			0
$533$	−30.0000	−	30.0000i	−1.29944	−	1.29944i
$534$	−12.0000			−0.519291
$535$		0			0
$536$	16.0000			0.691095
$537$		−	30.0000i		−	1.29460i
$538$		−	2.00000i		−	0.0862261i
$539$	−36.0000	−	36.0000i	−1.55063	−	1.55063i
$540$		0			0
$541$	−9.00000	+	9.00000i	−0.386940	+	0.386940i	−0.873595	−	0.486654i	$-0.838217\pi$
$541$	−9.00000	+	9.00000i	−0.386940	+	0.386940i	0.486654	+	0.873595i	$0.338217\pi$
$542$	−8.00000	+	8.00000i	−0.343629	+	0.343629i
$543$	−12.0000			−0.514969
$544$	8.00000	−	8.00000i	0.342997	−	0.342997i
$545$		0			0
$546$	24.0000	−	24.0000i	1.02711	−	1.02711i
$547$	21.0000	−	21.0000i	0.897895	−	0.897895i	−0.0973546	−	0.995250i	$-0.531038\pi$
$547$	21.0000	−	21.0000i	0.897895	−	0.897895i	0.995250	+	0.0973546i	$0.0310381\pi$
$548$	−36.0000			−1.53784
$549$	2.00000	+	2.00000i	0.0853579	+	0.0853579i
$550$		−	40.0000i		−	1.70561i
$551$		0			0
$552$		−	4.00000i		−	0.170251i
$553$			16.0000i			0.680389i
$554$	−6.00000			−0.254916
$555$		0			0
$556$	30.0000	−	30.0000i	1.27228	−	1.27228i
$557$	10.0000	−	10.0000i	0.423714	−	0.423714i	−0.462767	−	0.886480i	$-0.653143\pi$
$557$	10.0000	−	10.0000i	0.423714	−	0.423714i	0.886480	+	0.462767i	$0.153143\pi$
$558$	2.00000	+	2.00000i	0.0846668	+	0.0846668i
$559$	24.0000			1.01509
$560$		0			0
$561$	−16.0000			−0.675521
$562$	−2.00000	−	2.00000i	−0.0843649	−	0.0843649i
$563$	−10.0000	+	10.0000i	−0.421450	+	0.421450i	−0.885703	−	0.464253i	$-0.846323\pi$
$563$	−10.0000	+	10.0000i	−0.421450	+	0.421450i	0.464253	+	0.885703i	$0.346323\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	20.0000i		−	0.839921i
$568$	28.0000	−	28.0000i	1.17485	−	1.17485i
$569$		−	34.0000i		−	1.42535i	−0.701492	−	0.712677i	$-0.747483\pi$
$569$		−	34.0000i		−	1.42535i	0.701492	−	0.712677i	$-0.252517\pi$
$570$		0			0
$571$	−2.00000	−	2.00000i	−0.0836974	−	0.0836974i	0.664019	−	0.747716i	$-0.268850\pi$
$571$	−2.00000	−	2.00000i	−0.0836974	−	0.0836974i	−0.747716	+	0.664019i	$0.768850\pi$
$572$			48.0000i			2.00698i
$573$	−8.00000	+	8.00000i	−0.334205	+	0.334205i
$574$	40.0000	−	40.0000i	1.66957	−	1.66957i
$575$	5.00000			0.208514
$576$	−8.00000			−0.333333
$577$	−8.00000			−0.333044			−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000			−0.333044			−0.166522	+	0.986038i	$0.553254\pi$
$578$	−13.0000	+	13.0000i	−0.540729	+	0.540729i
$579$	−10.0000	+	10.0000i	−0.415586	+	0.415586i
$580$		0			0
$581$	−16.0000	−	16.0000i	−0.663792	−	0.663792i
$582$			28.0000i			1.16064i
$583$		−	64.0000i		−	2.65061i
$584$	20.0000	−	20.0000i	0.827606	−	0.827606i
$585$		0			0
$586$	32.0000			1.32191
$587$	−9.00000	−	9.00000i	−0.371470	−	0.371470i	0.496543	−	0.868012i	$-0.334603\pi$
$587$	−9.00000	−	9.00000i	−0.371470	−	0.371470i	−0.868012	+	0.496543i	$0.834603\pi$
$588$	18.0000	+	18.0000i	0.742307	+	0.742307i
$589$		0			0
$590$		0			0
$591$	10.0000			0.411345
$592$	−16.0000	−	16.0000i	−0.657596	−	0.657596i
$593$	−28.0000			−1.14982			−0.574911	−	0.818216i	$-0.694963\pi$
$593$	−28.0000			−1.14982			−0.574911	+	0.818216i	$0.694963\pi$
$594$	32.0000	+	32.0000i	1.31298	+	1.31298i
$595$		0			0
$596$		0			0
$597$	20.0000	+	20.0000i	0.818546	+	0.818546i
$598$	−6.00000			−0.245358
$599$			16.0000i			0.653742i	0.945069	+	0.326871i	$0.105994\pi$
$599$			16.0000i			0.653742i	−0.945069	+	0.326871i	$0.894006\pi$
$600$			20.0000i			0.816497i
$601$		−	38.0000i		−	1.55005i	−0.631929	−	0.775026i	$-0.717737\pi$
$601$		−	38.0000i		−	1.55005i	0.631929	−	0.775026i	$-0.282263\pi$
$602$			32.0000i			1.30422i
$603$	−4.00000	−	4.00000i	−0.162893	−	0.162893i
$604$	−32.0000			−1.30206
$605$		0			0
$606$	−22.0000	+	22.0000i	−0.893689	+	0.893689i
$607$	10.0000			0.405887			0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000			0.405887			0.202944	+	0.979190i	$0.434949\pi$
$608$		0			0
$609$	−56.0000			−2.26923
$610$		0			0
$611$		0			0
$612$	−4.00000			−0.161690
$613$	6.00000	+	6.00000i	0.242338	+	0.242338i	0.817817	−	0.575479i	$-0.195184\pi$
$613$	6.00000	+	6.00000i	0.242338	+	0.242338i	−0.575479	+	0.817817i	$0.695184\pi$
$614$		−	10.0000i		−	0.403567i
$615$		0			0
$616$	−64.0000			−2.57863
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	8.00000			0.321807
$619$	28.0000	+	28.0000i	1.12542	+	1.12542i	0.990913	+	0.134502i	$0.0429434\pi$
$619$	28.0000	+	28.0000i	1.12542	+	1.12542i	0.134502	+	0.990913i	$0.457057\pi$
$620$		0			0
$621$	−4.00000	+	4.00000i	−0.160514	+	0.160514i
$622$		0			0
$623$	−24.0000			−0.961540
$624$		−	24.0000i		−	0.960769i
$625$	−25.0000			−1.00000
$626$	6.00000	+	6.00000i	0.239808	+	0.239808i
$627$		0			0
$628$	24.0000	+	24.0000i	0.957704	+	0.957704i
$629$	−8.00000	−	8.00000i	−0.318981	−	0.318981i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	8.00000	+	8.00000i	0.318223	+	0.318223i
$633$		−	18.0000i		−	0.715436i
$634$			10.0000i			0.397151i
$635$		0			0
$636$			32.0000i			1.26888i
$637$	27.0000	−	27.0000i	1.06978	−	1.06978i
$638$	56.0000	−	56.0000i	2.21706	−	2.21706i
$639$	−14.0000			−0.553831
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$	−12.0000	+	12.0000i	−0.473602	+	0.473602i
$643$	−10.0000	+	10.0000i	−0.394362	+	0.394362i	−0.876239	−	0.481877i	$-0.839955\pi$
$643$	−10.0000	+	10.0000i	−0.394362	+	0.394362i	0.481877	+	0.876239i	$0.339955\pi$
$644$		−	8.00000i		−	0.315244i
$645$		0			0
$646$		0			0
$647$		−	8.00000i		−	0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$		−	8.00000i		−	0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	−10.0000	−	10.0000i	−0.392837	−	0.392837i
$649$			56.0000i			2.19819i
$650$	30.0000			1.17670
$651$	−8.00000	−	8.00000i	−0.313545	−	0.313545i
$652$	6.00000	+	6.00000i	0.234978	+	0.234978i
$653$	−7.00000	+	7.00000i	−0.273931	+	0.273931i	−0.830681	−	0.556749i	$-0.812048\pi$
$653$	−7.00000	+	7.00000i	−0.273931	+	0.273931i	0.556749	+	0.830681i	$0.312048\pi$
$654$	−4.00000	−	4.00000i	−0.156412	−	0.156412i
$655$		0			0
$656$		−	40.0000i		−	1.56174i
$657$	−10.0000			−0.390137
$658$		0			0
$659$	12.0000	−	12.0000i	0.467454	−	0.467454i	−0.433635	−	0.901089i	$-0.642769\pi$
$659$	12.0000	−	12.0000i	0.467454	−	0.467454i	0.901089	+	0.433635i	$0.142769\pi$
$660$		0			0
$661$	6.00000	+	6.00000i	0.233373	+	0.233373i	0.814099	−	0.580726i	$-0.197231\pi$
$661$	6.00000	+	6.00000i	0.233373	+	0.233373i	−0.580726	+	0.814099i	$0.697231\pi$
$662$	26.0000			1.01052
$663$		−	12.0000i		−	0.466041i
$664$	−16.0000			−0.620920
$665$		0			0
$666$			8.00000i			0.309994i
$667$	7.00000	+	7.00000i	0.271041	+	0.271041i
$668$	−28.0000			−1.08335
$669$	−18.0000	+	18.0000i	−0.695920	+	0.695920i
$670$		0			0
$671$	16.0000			0.617673
$672$	32.0000			1.23443
$673$	32.0000			1.23351			0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000			1.23351			0.616755	+	0.787155i	$0.288447\pi$
$674$	6.00000	−	6.00000i	0.231111	−	0.231111i
$675$	20.0000	−	20.0000i	0.769800	−	0.769800i
$676$	−10.0000			−0.384615
$677$	−4.00000	−	4.00000i	−0.153732	−	0.153732i	0.626050	−	0.779783i	$-0.284670\pi$
$677$	−4.00000	−	4.00000i	−0.153732	−	0.153732i	−0.779783	+	0.626050i	$0.784670\pi$
$678$			12.0000i			0.460857i
$679$			56.0000i			2.14908i
$680$		0			0
$681$		−	20.0000i		−	0.766402i
$682$	16.0000			0.612672
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	0.933232	−	0.359273i	$-0.116975\pi$
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	−0.359273	+	0.933232i	$0.616975\pi$
$684$		0			0
$685$		0			0
$686$	8.00000	+	8.00000i	0.305441	+	0.305441i
$687$	−12.0000			−0.457829
$688$	16.0000	+	16.0000i	0.609994	+	0.609994i
$689$	48.0000			1.82865
$690$		0			0
$691$	15.0000	−	15.0000i	0.570627	−	0.570627i	−0.361677	−	0.932304i	$-0.617796\pi$
$691$	15.0000	−	15.0000i	0.570627	−	0.570627i	0.932304	+	0.361677i	$0.117796\pi$
$692$	22.0000	+	22.0000i	0.836315	+	0.836315i
$693$	16.0000	+	16.0000i	0.607790	+	0.607790i
$694$	−2.00000			−0.0759190
$695$		0			0
$696$	−28.0000	+	28.0000i	−1.06134	+	1.06134i
$697$		−	20.0000i		−	0.757554i
$698$		−	2.00000i		−	0.0757011i
$699$	4.00000	+	4.00000i	0.151294	+	0.151294i
$700$			40.0000i			1.51186i
$701$	−10.0000	+	10.0000i	−0.377695	+	0.377695i	−0.870270	−	0.492575i	$-0.836056\pi$
$701$	−10.0000	+	10.0000i	−0.377695	+	0.377695i	0.492575	+	0.870270i	$0.336056\pi$
$702$	−24.0000	+	24.0000i	−0.905822	+	0.905822i
$703$		0			0
$704$	−32.0000	+	32.0000i	−1.20605	+	1.20605i
$705$		0			0
$706$	30.0000	−	30.0000i	1.12906	−	1.12906i
$707$	−44.0000	+	44.0000i	−1.65479	+	1.65479i
$708$		−	28.0000i		−	1.05230i
$709$	20.0000	+	20.0000i	0.751116	+	0.751116i	0.974687	−	0.223572i	$-0.0717717\pi$
$709$	20.0000	+	20.0000i	0.751116	+	0.751116i	−0.223572	+	0.974687i	$0.571772\pi$
$710$		0			0
$711$		−	4.00000i		−	0.150012i
$712$	−12.0000	+	12.0000i	−0.449719	+	0.449719i
$713$			2.00000i			0.0749006i
$714$	16.0000			0.598785
$715$		0			0
$716$	−30.0000	−	30.0000i	−1.12115	−	1.12115i
$717$	−2.00000	+	2.00000i	−0.0746914	+	0.0746914i
$718$	−32.0000	−	32.0000i	−1.19423	−	1.19423i
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$	19.0000	+	19.0000i	0.707107	+	0.707107i
$723$	−10.0000	+	10.0000i	−0.371904	+	0.371904i
$724$	−12.0000	+	12.0000i	−0.445976	+	0.445976i
$725$	−35.0000	−	35.0000i	−1.29987	−	1.29987i
$726$	42.0000			1.55877
$727$			44.0000i			1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$			44.0000i			1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$		−	48.0000i		−	1.77900i
$729$			29.0000i			1.07407i
$730$		0			0
$731$	8.00000	+	8.00000i	0.295891	+	0.295891i
$732$	−8.00000			−0.295689
$733$	16.0000	−	16.0000i	0.590973	−	0.590973i	−0.346921	−	0.937894i	$-0.612773\pi$
$733$	16.0000	−	16.0000i	0.590973	−	0.590973i	0.937894	+	0.346921i	$0.112773\pi$
$734$	32.0000	−	32.0000i	1.18114	−	1.18114i
$735$		0			0
$736$	−4.00000	−	4.00000i	−0.147442	−	0.147442i
$737$	−32.0000			−1.17874
$738$	−10.0000	+	10.0000i	−0.368105	+	0.368105i
$739$	15.0000	−	15.0000i	0.551784	−	0.551784i	−0.375171	−	0.926955i	$-0.622416\pi$
$739$	15.0000	−	15.0000i	0.551784	−	0.551784i	0.926955	+	0.375171i	$0.122416\pi$
$740$		0			0
$741$		0			0
$742$			64.0000i			2.34951i
$743$			20.0000i			0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$			20.0000i			0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	−8.00000			−0.293294
$745$		0			0
$746$	−32.0000			−1.17160
$747$	4.00000	+	4.00000i	0.146352	+	0.146352i
$748$	−16.0000	+	16.0000i	−0.585018	+	0.585018i
$749$	−24.0000	+	24.0000i	−0.876941	+	0.876941i
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$	12.0000			0.437304
$754$	42.0000	+	42.0000i	1.52955	+	1.52955i
$755$		0			0
$756$	−32.0000	−	32.0000i	−1.16383	−	1.16383i
$757$	−26.0000	−	26.0000i	−0.944986	−	0.944986i	0.0535776	−	0.998564i	$-0.482938\pi$
$757$	−26.0000	−	26.0000i	−0.944986	−	0.944986i	−0.998564	+	0.0535776i	$0.982938\pi$
$758$	−36.0000			−1.30758
$759$			8.00000i			0.290382i
$760$		0			0
$761$		−	12.0000i		−	0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$		−	12.0000i		−	0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$		0			0
$763$	−8.00000	−	8.00000i	−0.289619	−	0.289619i
$764$			16.0000i			0.578860i
$765$		0			0
$766$	−12.0000	+	12.0000i	−0.433578	+	0.433578i
$767$	−42.0000			−1.51653
$768$	16.0000	−	16.0000i	0.577350	−	0.577350i
$769$	−34.0000			−1.22607			−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000			−1.22607			−0.613036	+	0.790055i	$0.710052\pi$
$770$		0			0
$771$	−20.0000	+	20.0000i	−0.720282	+	0.720282i
$772$			20.0000i			0.719816i
$773$	10.0000	+	10.0000i	0.359675	+	0.359675i	0.863693	−	0.504018i	$-0.168146\pi$
$773$	10.0000	+	10.0000i	0.359675	+	0.359675i	−0.504018	+	0.863693i	$0.668146\pi$
$774$		−	8.00000i		−	0.287554i
$775$		−	10.0000i		−	0.359211i
$776$	28.0000	+	28.0000i	1.00514	+	1.00514i
$777$		−	32.0000i		−	1.14799i
$778$	44.0000			1.57748
$779$		0			0
$780$		0			0
$781$	−56.0000	+	56.0000i	−2.00384	+	2.00384i
$782$	−2.00000	−	2.00000i	−0.0715199	−	0.0715199i
$783$	56.0000			2.00128
$784$	36.0000			1.28571
$785$		0			0
$786$	−10.0000	−	10.0000i	−0.356688	−	0.356688i
$787$	14.0000	−	14.0000i	0.499046	−	0.499046i	−0.412095	−	0.911141i	$-0.635203\pi$
$787$	14.0000	−	14.0000i	0.499046	−	0.499046i	0.911141	+	0.412095i	$0.135203\pi$
$788$	10.0000	−	10.0000i	0.356235	−	0.356235i
$789$	4.00000	+	4.00000i	0.142404	+	0.142404i
$790$		0			0
$791$			24.0000i			0.853342i
$792$	16.0000			0.568535
$793$			12.0000i			0.426132i
$794$		−	46.0000i		−	1.63248i
$795$		0			0
$796$	40.0000			1.41776
$797$	−30.0000	+	30.0000i	−1.06265	+	1.06265i	−0.0647532	+	0.997901i	$0.520626\pi$
$797$	−30.0000	+	30.0000i	−1.06265	+	1.06265i	−0.997901	+	0.0647532i	$0.979374\pi$
$798$		0			0
$799$		0			0
$800$	20.0000	+	20.0000i	0.707107	+	0.707107i
$801$	6.00000			0.212000
$802$	30.0000	−	30.0000i	1.05934	−	1.05934i
$803$	−40.0000	+	40.0000i	−1.41157	+	1.41157i
$804$	16.0000			0.564276
$805$		0			0
$806$			12.0000i			0.422682i
$807$		−	2.00000i		−	0.0704033i
$808$			44.0000i			1.54791i
$809$			52.0000i			1.82822i	0.405463	+	0.914111i	$0.367110\pi$
$809$			52.0000i			1.82822i	−0.405463	+	0.914111i	$0.632890\pi$
$810$		0			0
$811$	−1.00000	−	1.00000i	−0.0351147	−	0.0351147i	0.689331	−	0.724446i	$-0.257904\pi$
$811$	−1.00000	−	1.00000i	−0.0351147	−	0.0351147i	−0.724446	+	0.689331i	$0.757904\pi$
$812$	−56.0000	+	56.0000i	−1.96521	+	1.96521i
$813$	−8.00000	+	8.00000i	−0.280572	+	0.280572i
$814$	32.0000	+	32.0000i	1.12160	+	1.12160i
$815$		0			0
$816$	8.00000	−	8.00000i	0.280056	−	0.280056i
$817$		0			0
$818$	4.00000	+	4.00000i	0.139857	+	0.139857i
$819$	−12.0000	+	12.0000i	−0.419314	+	0.419314i
$820$		0			0
$821$	−5.00000	−	5.00000i	−0.174501	−	0.174501i	0.614453	−	0.788954i	$-0.289377\pi$
$821$	−5.00000	−	5.00000i	−0.174501	−	0.174501i	−0.788954	+	0.614453i	$0.789377\pi$
$822$	−36.0000			−1.25564
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	8.00000	−	8.00000i	0.278693	−	0.278693i
$825$		−	40.0000i		−	1.39262i
$826$		−	56.0000i		−	1.94849i
$827$	18.0000	+	18.0000i	0.625921	+	0.625921i	0.947039	−	0.321118i	$-0.104059\pi$
$827$	18.0000	+	18.0000i	0.625921	+	0.625921i	−0.321118	+	0.947039i	$0.604059\pi$
$828$			2.00000i			0.0695048i
$829$	−19.0000	+	19.0000i	−0.659897	+	0.659897i	−0.955356	−	0.295458i	$-0.904528\pi$
$829$	−19.0000	+	19.0000i	−0.659897	+	0.659897i	0.295458	+	0.955356i	$0.404528\pi$
$830$		0			0
$831$	−6.00000			−0.208138
$832$	−24.0000	−	24.0000i	−0.832050	−	0.832050i
$833$	18.0000			0.623663
$834$	30.0000	−	30.0000i	1.03882	−	1.03882i
$835$		0			0
$836$		0			0
$837$	8.00000	+	8.00000i	0.276520	+	0.276520i
$838$		−	16.0000i		−	0.552711i
$839$		−	12.0000i		−	0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$839$		−	12.0000i		−	0.414286i	0.978311	−	0.207143i	$-0.0664165\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$	−32.0000			−1.10279
$843$	−2.00000	−	2.00000i	−0.0688837	−	0.0688837i
$844$	−18.0000	−	18.0000i	−0.619586	−	0.619586i
$845$		0			0
$846$		0			0
$847$	84.0000			2.88627
$848$	32.0000	+	32.0000i	1.09888	+	1.09888i
$849$		0			0
$850$	10.0000	+	10.0000i	0.342997	+	0.342997i
$851$	−4.00000	+	4.00000i	−0.137118	+	0.137118i
$852$	28.0000	−	28.0000i	0.959264	−	0.959264i
$853$	37.0000	+	37.0000i	1.26686	+	1.26686i	0.947703	+	0.319152i	$0.103398\pi$
$853$	37.0000	+	37.0000i	1.26686	+	1.26686i	0.319152	+	0.947703i	$0.396602\pi$
$854$	−16.0000			−0.547509
$855$		0			0
$856$			24.0000i			0.820303i
$857$		−	32.0000i		−	1.09310i	−0.837427	−	0.546550i	$-0.815941\pi$
$857$		−	32.0000i		−	1.09310i	0.837427	−	0.546550i	$-0.184059\pi$
$858$			48.0000i			1.63869i
$859$	−23.0000	−	23.0000i	−0.784750	−	0.784750i	0.195878	−	0.980628i	$-0.437244\pi$
$859$	−23.0000	−	23.0000i	−0.784750	−	0.784750i	−0.980628	+	0.195878i	$0.937244\pi$
$860$		0			0
$861$	40.0000	−	40.0000i	1.36320	−	1.36320i
$862$	−4.00000	+	4.00000i	−0.136241	+	0.136241i
$863$	10.0000			0.340404			0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000			0.340404			0.170202	+	0.985409i	$0.445558\pi$
$864$	−32.0000			−1.08866
$865$		0			0
$866$	6.00000	−	6.00000i	0.203888	−	0.203888i
$867$	−13.0000	+	13.0000i	−0.441503	+	0.441503i
$868$	−16.0000			−0.543075
$869$	−16.0000	−	16.0000i	−0.542763	−	0.542763i
$870$		0			0
$871$		−	24.0000i		−	0.813209i
$872$	−8.00000			−0.270914
$873$		−	14.0000i		−	0.473828i
$874$		0			0
$875$		0			0
$876$	20.0000	−	20.0000i	0.675737	−	0.675737i
$877$	19.0000	−	19.0000i	0.641584	−	0.641584i	−0.309360	−	0.950945i	$-0.600115\pi$
$877$	19.0000	−	19.0000i	0.641584	−	0.641584i	0.950945	+	0.309360i	$0.100115\pi$
$878$	−2.00000	−	2.00000i	−0.0674967	−	0.0674967i
$879$	32.0000			1.07933
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$	−9.00000	−	9.00000i	−0.303046	−	0.303046i
$883$	27.0000	−	27.0000i	0.908622	−	0.908622i	−0.0875388	−	0.996161i	$-0.527900\pi$
$883$	27.0000	−	27.0000i	0.908622	−	0.908622i	0.996161	+	0.0875388i	$0.0279002\pi$
$884$	−12.0000	−	12.0000i	−0.403604	−	0.403604i
$885$		0			0
$886$	22.0000			0.739104
$887$		−	10.0000i		−	0.335767i	−0.985807	−	0.167884i	$-0.946307\pi$
$887$		−	10.0000i		−	0.335767i	0.985807	−	0.167884i	$-0.0536933\pi$
$888$	−16.0000	−	16.0000i	−0.536925	−	0.536925i
$889$		0			0
$890$		0			0
$891$	20.0000	+	20.0000i	0.670025	+	0.670025i
$892$			36.0000i			1.20537i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	32.0000	−	32.0000i	1.06904	−	1.06904i
$897$	−6.00000			−0.200334
$898$		0			0
$899$	14.0000	−	14.0000i	0.466926	−	0.466926i
$900$		−	10.0000i		−	0.333333i
$901$	16.0000	+	16.0000i	0.533037	+	0.533037i
$902$			80.0000i			2.66371i
$903$			32.0000i			1.06489i
$904$	12.0000	+	12.0000i	0.399114	+	0.399114i
$905$		0			0
$906$	−32.0000			−1.06313
$907$	16.0000	+	16.0000i	0.531271	+	0.531271i	0.920951	−	0.389679i	$-0.127414\pi$
$907$	16.0000	+	16.0000i	0.531271	+	0.531271i	−0.389679	+	0.920951i	$0.627414\pi$
$908$	−20.0000	−	20.0000i	−0.663723	−	0.663723i
$909$	11.0000	−	11.0000i	0.364847	−	0.364847i
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$	32.0000			1.05905
$914$	−22.0000	−	22.0000i	−0.727695	−	0.727695i
$915$		0			0
$916$	−12.0000	+	12.0000i	−0.396491	+	0.396491i
$917$	−20.0000	−	20.0000i	−0.660458	−	0.660458i
$918$	−16.0000			−0.528079
$919$		−	56.0000i		−	1.84727i	−0.383274	−	0.923635i	$-0.625203\pi$
$919$		−	56.0000i		−	1.84727i	0.383274	−	0.923635i	$-0.374797\pi$
$920$		0			0
$921$		−	10.0000i		−	0.329511i
$922$			2.00000i			0.0658665i
$923$	−42.0000	−	42.0000i	−1.38245	−	1.38245i
$924$	−64.0000			−2.10545
$925$	20.0000	−	20.0000i	0.657596	−	0.657596i
$926$	8.00000	−	8.00000i	0.262896	−	0.262896i
$927$	−4.00000			−0.131377
$928$			56.0000i			1.83829i
$929$	−32.0000			−1.04989			−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000			−1.04989			−0.524943	+	0.851137i	$0.675913\pi$
$930$		0			0
$931$		0			0
$932$	8.00000			0.262049
$933$		0			0
$934$		−	56.0000i		−	1.83238i
$935$		0			0
$936$			12.0000i			0.392232i
$937$			22.0000i			0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$			22.0000i			0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	32.0000			1.04484
$939$	6.00000	+	6.00000i	0.195803	+	0.195803i
$940$		0			0
$941$	−8.00000	+	8.00000i	−0.260793	+	0.260793i	−0.825376	−	0.564583i	$-0.809037\pi$
$941$	−8.00000	+	8.00000i	−0.260793	+	0.260793i	0.564583	+	0.825376i	$0.309037\pi$
$942$	24.0000	+	24.0000i	0.781962	+	0.781962i
$943$	−10.0000			−0.325645
$944$	−28.0000	−	28.0000i	−0.911322	−	0.911322i
$945$		0			0
$946$	−32.0000	−	32.0000i	−1.04041	−	1.04041i
$947$	29.0000	−	29.0000i	0.942373	−	0.942373i	−0.0560543	−	0.998428i	$-0.517852\pi$
$947$	29.0000	−	29.0000i	0.942373	−	0.942373i	0.998428	+	0.0560543i	$0.0178520\pi$
$948$	8.00000	+	8.00000i	0.259828	+	0.259828i
$949$	−30.0000	−	30.0000i	−0.973841	−	0.973841i
$950$		0			0
$951$			10.0000i			0.324272i
$952$	16.0000	−	16.0000i	0.518563	−	0.518563i
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		−	16.0000i		−	0.518019i
$955$		0			0
$956$			4.00000i			0.129369i
$957$	56.0000	−	56.0000i	1.81022	−	1.81022i
$958$	24.0000	−	24.0000i	0.775405	−	0.775405i
$959$	−72.0000			−2.32500
$960$		0			0
$961$	−27.0000			−0.870968
$962$	−24.0000	+	24.0000i	−0.773791	+	0.773791i
$963$	6.00000	−	6.00000i	0.193347	−	0.193347i
$964$			20.0000i			0.644157i
$965$		0			0
$966$		−	8.00000i		−	0.257396i
$967$			16.0000i			0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$			16.0000i			0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	42.0000	−	42.0000i	1.34993	−	1.34993i
$969$		0			0
$970$		0			0
$971$	−24.0000	−	24.0000i	−0.770197	−	0.770197i	0.207944	−	0.978141i	$-0.433323\pi$
$971$	−24.0000	−	24.0000i	−0.770197	−	0.770197i	−0.978141	+	0.207944i	$0.933323\pi$
$972$	14.0000	+	14.0000i	0.449050	+	0.449050i
$973$	60.0000	−	60.0000i	1.92351	−	1.92351i
$974$	−14.0000	−	14.0000i	−0.448589	−	0.448589i
$975$	30.0000			0.960769
$976$	−8.00000	+	8.00000i	−0.256074	+	0.256074i
$977$	26.0000			0.831814			0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000			0.831814			0.415907	+	0.909407i	$0.363464\pi$
$978$	6.00000	+	6.00000i	0.191859	+	0.191859i
$979$	24.0000	−	24.0000i	0.767043	−	0.767043i
$980$		0			0
$981$	2.00000	+	2.00000i	0.0638551	+	0.0638551i
$982$	−14.0000			−0.446758
$983$			48.0000i			1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$			48.0000i			1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$		−	40.0000i		−	1.27515i
$985$		0			0
$986$			28.0000i			0.891702i
$987$		0			0
$988$		0			0
$989$	4.00000	−	4.00000i	0.127193	−	0.127193i
$990$		0			0
$991$	−2.00000			−0.0635321			−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000			−0.0635321			−0.0317660	+	0.999495i	$0.510113\pi$
$992$	−8.00000	+	8.00000i	−0.254000	+	0.254000i
$993$	26.0000			0.825085
$994$	56.0000	−	56.0000i	1.77621	−	1.77621i
$995$		0			0
$996$	−16.0000			−0.506979
$997$	25.0000	+	25.0000i	0.791758	+	0.791758i	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	25.0000	+	25.0000i	0.791758	+	0.791758i	−0.190022	+	0.981780i	$0.560856\pi$
$998$			30.0000i			0.949633i
$999$			32.0000i			1.01244i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.2.j.a.277.1	yes	2
4.3	odd	2		1472.2.j.a.369.1		2
16.3	odd	4		1472.2.j.a.1105.1		2
16.13	even	4	inner	368.2.j.a.93.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.2.j.a.93.1	✓	2	16.13	even	4	inner
368.2.j.a.277.1	yes	2	1.1	even	1	trivial
1472.2.j.a.369.1		2	4.3	odd	2
1472.2.j.a.1105.1		2	16.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	368.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(1.00000 - 1.00000i) q^{3} -2.00000i q^{4} -2.00000i q^{6} -4.00000i q^{7} +(-2.00000 - 2.00000i) q^{8} +1.00000i q^{9} +(4.00000 + 4.00000i) q^{11} +(-2.00000 - 2.00000i) q^{12} +(-3.00000 + 3.00000i) q^{13} +(-4.00000 - 4.00000i) q^{14} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +(-4.00000 - 4.00000i) q^{21} +8.00000 q^{22} +1.00000i q^{23} -4.00000 q^{24} -5.00000i q^{25} +6.00000i q^{26} +(4.00000 + 4.00000i) q^{27} -8.00000 q^{28} +(7.00000 - 7.00000i) q^{29} +2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +8.00000 q^{33} +(-2.00000 + 2.00000i) q^{34} +2.00000 q^{36} +(4.00000 + 4.00000i) q^{37} +6.00000i q^{39} +10.0000i q^{41} -8.00000 q^{42} +(-4.00000 - 4.00000i) q^{43} +(8.00000 - 8.00000i) q^{44} +(1.00000 + 1.00000i) q^{46} +(-4.00000 + 4.00000i) q^{48} -9.00000 q^{49} +(-5.00000 - 5.00000i) q^{50} +(-2.00000 + 2.00000i) q^{51} +(6.00000 + 6.00000i) q^{52} +(-8.00000 - 8.00000i) q^{53} +8.00000 q^{54} +(-8.00000 + 8.00000i) q^{56} -14.0000i q^{58} +(7.00000 + 7.00000i) q^{59} +(2.00000 - 2.00000i) q^{61} +(2.00000 - 2.00000i) q^{62} +4.00000 q^{63} +8.00000i q^{64} +(8.00000 - 8.00000i) q^{66} +(-4.00000 + 4.00000i) q^{67} +4.00000i q^{68} +(1.00000 + 1.00000i) q^{69} +14.0000i q^{71} +(2.00000 - 2.00000i) q^{72} +10.0000i q^{73} +8.00000 q^{74} +(-5.00000 - 5.00000i) q^{75} +(16.0000 - 16.0000i) q^{77} +(6.00000 + 6.00000i) q^{78} -4.00000 q^{79} +5.00000 q^{81} +(10.0000 + 10.0000i) q^{82} +(4.00000 - 4.00000i) q^{83} +(-8.00000 + 8.00000i) q^{84} -8.00000 q^{86} -14.0000i q^{87} -16.0000i q^{88} -6.00000i q^{89} +(12.0000 + 12.0000i) q^{91} +2.00000 q^{92} +(2.00000 - 2.00000i) q^{93} +8.00000i q^{96} -14.0000 q^{97} +(-9.00000 + 9.00000i) q^{98} +(-4.00000 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 4 q^{8} + 8 q^{11} - 4 q^{12} - 6 q^{13} - 8 q^{14} - 8 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{21} + 16 q^{22} - 8 q^{24} + 8 q^{27} - 16 q^{28} + 14 q^{29} + 4 q^{31} - 8 q^{32}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 4 * q^8 + 8 * q^11 - 4 * q^12 - 6 * q^13 - 8 * q^14 - 8 * q^16 - 4 * q^17 + 2 * q^18 - 8 * q^21 + 16 * q^22 - 8 * q^24 + 8 * q^27 - 16 * q^28 + 14 * q^29 + 4 * q^31 - 8 * q^32 + 16 * q^33 - 4 * q^34 + 4 * q^36 + 8 * q^37 - 16 * q^42 - 8 * q^43 + 16 * q^44 + 2 * q^46 - 8 * q^48 - 18 * q^49 - 10 * q^50 - 4 * q^51 + 12 * q^52 - 16 * q^53 + 16 * q^54 - 16 * q^56 + 14 * q^59 + 4 * q^61 + 4 * q^62 + 8 * q^63 + 16 * q^66 - 8 * q^67 + 2 * q^69 + 4 * q^72 + 16 * q^74 - 10 * q^75 + 32 * q^77 + 12 * q^78 - 8 * q^79 + 10 * q^81 + 20 * q^82 + 8 * q^83 - 16 * q^84 - 16 * q^86 + 24 * q^91 + 4 * q^92 + 4 * q^93 - 28 * q^97 - 18 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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