LMFDB - Embedded newform 344.1.be.a.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [344,1,Mod(67,344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(344, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 21, 40]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("344.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 344 = 2^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	344.be (of order $42$, degree $12$, 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:	$0.171678364346$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			67.1
Root			$-0.988831 - 0.149042i$ of defining polynomial
Character	$\chi$	$=$	344.67
Dual form			344.1.be.a.267.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.623490 - 0.781831i) q^{2} +(0.440071 - 0.0663300i) q^{3} +(-0.222521 - 0.974928i) q^{4} +(0.222521 - 0.385418i) q^{6} +(-0.900969 - 0.433884i) q^{8} +(-0.766310 + 0.236375i) q^{9} +O(q^{10})$	\(q+(0.623490 - 0.781831i) q^{2} +(0.440071 - 0.0663300i) q^{3} +(-0.222521 - 0.974928i) q^{4} +(0.222521 - 0.385418i) q^{6} +(-0.900969 - 0.433884i) q^{8} +(-0.766310 + 0.236375i) q^{9} +(-0.162592 + 0.712362i) q^{11} +(-0.162592 - 0.414278i) q^{12} +(-0.900969 + 0.433884i) q^{16} +(1.57906 + 1.07659i) q^{17} +(-0.292981 + 0.746503i) q^{18} +(-1.40097 - 0.432142i) q^{19} +(0.455573 + 0.571270i) q^{22} +(-0.425270 - 0.131178i) q^{24} +(0.365341 - 0.930874i) q^{25} +(-0.722521 + 0.347948i) q^{27} +(-0.222521 + 0.974928i) q^{32} +(-0.0243010 + 0.324275i) q^{33} +(1.82624 - 0.563320i) q^{34} +(0.400969 + 0.694498i) q^{36} +(-1.21135 + 0.825886i) q^{38} +(0.0931869 - 0.116853i) q^{41} +(-0.988831 - 0.149042i) q^{43} +0.730682 q^{44} +(-0.367711 + 0.250701i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(0.766310 + 0.369035i) q^{51} +(-0.178448 + 0.781831i) q^{54} +(-0.645190 - 0.0972467i) q^{57} +(1.78181 - 0.858075i) q^{59} +(0.623490 + 0.781831i) q^{64} +(0.238377 + 0.221181i) q^{66} +(-0.955573 - 0.294755i) q^{67} +(0.698220 - 1.77904i) q^{68} +(0.792981 + 0.119523i) q^{72} +(-0.0747301 - 0.997204i) q^{73} +(0.0990311 - 0.433884i) q^{75} +(-0.109562 + 1.46200i) q^{76} +(0.367711 - 0.250701i) q^{81} +(-0.0332580 - 0.145713i) q^{82} +(-1.23305 + 0.185853i) q^{83} +(-0.733052 + 0.680173i) q^{86} +(0.455573 - 0.571270i) q^{88} +(-0.147791 + 0.0222759i) q^{89} +(-0.0332580 + 0.443797i) q^{96} +(0.440071 - 1.92808i) q^{97} +(0.365341 + 0.930874i) q^{98} +(-0.0437890 - 0.584323i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) $ 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9	$ 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{16} - q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{22} - 5 q^{24} + q^{25} - 8 q^{27} - 2 q^{32} - 9 q^{33} + 13 q^{34} - 4 q^{36} - q^{38} + 2 q^{41} + q^{43} + 2 q^{44} + 2 q^{48} - 6 q^{49} - 6 q^{50} - 3 q^{51} + 6 q^{54} - 2 q^{57} + 2 q^{59} - 2 q^{64} - 2 q^{66} - q^{67} - q^{68} + 3 q^{72} - q^{73} + 10 q^{75} - q^{76} - 2 q^{81} + 2 q^{82} - 5 q^{83} + q^{86} - 5 q^{88} - q^{89} + 2 q^{96} + 2 q^{97} + q^{98} + 11 q^{99}+O(q^{100}) $ 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9 + 2 * q^11 + 2 * q^12 - 2 * q^16 - q^17 + 3 * q^18 - 8 * q^19 - 5 * q^22 - 5 * q^24 + q^25 - 8 * q^27 - 2 * q^32 - 9 * q^33 + 13 * q^34 - 4 * q^36 - q^38 + 2 * q^41 + q^43 + 2 * q^44 + 2 * q^48 - 6 * q^49 - 6 * q^50 - 3 * q^51 + 6 * q^54 - 2 * q^57 + 2 * q^59 - 2 * q^64 - 2 * q^66 - q^67 - q^68 + 3 * q^72 - q^73 + 10 * q^75 - q^76 - 2 * q^81 + 2 * q^82 - 5 * q^83 + q^86 - 5 * q^88 - q^89 + 2 * q^96 + 2 * q^97 + q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/344\mathbb{Z}\right)^\times$.

$n$	$87$	$89$	$173$
$\chi(n)$	$-1$	$e\left(\frac{20}{21}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.623490	−	0.781831i	0.623490	−	0.781831i
$2$	0.623490	−	0.781831i	0.623490	−	0.781831i
$3$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.0747301	−	0.997204i	$-0.476190\pi$
$3$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.365341	+	0.930874i	$0.380952\pi$
$4$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$5$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$5$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$6$	0.222521	−	0.385418i	0.222521	−	0.385418i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$9$	−0.766310	+	0.236375i	−0.766310	+	0.236375i
$10$		0			0
$11$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$11$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$12$	−0.162592	−	0.414278i	−0.162592	−	0.414278i
$13$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$13$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$14$		0			0
$15$		0			0
$16$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$17$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.955573	+	0.294755i	$0.0952381\pi$
$17$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.623490	+	0.781831i	$0.285714\pi$
$18$	−0.292981	+	0.746503i	−0.292981	+	0.746503i
$19$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.900969	+	0.433884i	$0.857143\pi$
$20$		0			0
$21$		0			0
$22$	0.455573	+	0.571270i	0.455573	+	0.571270i
$23$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$23$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$24$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$25$	0.365341	−	0.930874i	0.365341	−	0.930874i
$26$		0			0
$27$	−0.722521	+	0.347948i	−0.722521	+	0.347948i
$28$		0			0
$29$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$29$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$30$		0			0
$31$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$31$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$32$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$33$	−0.0243010	+	0.324275i	−0.0243010	+	0.324275i
$34$	1.82624	−	0.563320i	1.82624	−	0.563320i
$35$		0			0
$36$	0.400969	+	0.694498i	0.400969	+	0.694498i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$39$		0			0
$40$		0			0
$41$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	−0.733052	−	0.680173i	$-0.761905\pi$
$41$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	0.826239	+	0.563320i	$0.190476\pi$
$42$		0			0
$43$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$43$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$44$	0.730682			0.730682
$45$		0			0
$46$		0			0
$47$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$47$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$48$	−0.367711	+	0.250701i	−0.367711	+	0.250701i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$	0.766310	+	0.369035i	0.766310	+	0.369035i
$52$		0			0
$53$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$53$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$54$	−0.178448	+	0.781831i	−0.178448	+	0.781831i
$55$		0			0
$56$		0			0
$57$	−0.645190	−	0.0972467i	−0.645190	−	0.0972467i
$58$		0			0
$59$	1.78181	−	0.858075i	1.78181	−	0.858075i	0.826239	−	0.563320i	$-0.190476\pi$
$59$	1.78181	−	0.858075i	1.78181	−	0.858075i	0.955573	−	0.294755i	$-0.0952381\pi$
$60$		0			0
$61$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$61$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$62$		0			0
$63$		0			0
$64$	0.623490	+	0.781831i	0.623490	+	0.781831i
$65$		0			0
$66$	0.238377	+	0.221181i	0.238377	+	0.221181i
$67$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.222521	−	0.974928i	$-0.571429\pi$
$67$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.733052	+	0.680173i	$0.761905\pi$
$68$	0.698220	−	1.77904i	0.698220	−	1.77904i
$69$		0			0
$70$		0			0
$71$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$71$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$72$	0.792981	+	0.119523i	0.792981	+	0.119523i
$73$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	−0.900969	−	0.433884i	$-0.857143\pi$
$73$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	0.826239	−	0.563320i	$-0.190476\pi$
$74$		0			0
$75$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$76$	−0.109562	+	1.46200i	−0.109562	+	1.46200i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	0.367711	−	0.250701i	0.367711	−	0.250701i
$82$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$83$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.733052	−	0.680173i	$-0.761905\pi$
$83$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$87$		0			0
$88$	0.455573	−	0.571270i	0.455573	−	0.571270i
$89$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i	−0.222521	−	0.974928i	$-0.571429\pi$
$89$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i	0.0747301	+	0.997204i	$0.476190\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i
$97$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.0747301	−	0.997204i	$-0.476190\pi$
$97$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.365341	−	0.930874i	$-0.380952\pi$
$98$	0.365341	+	0.930874i	0.365341	+	0.930874i
$99$	−0.0437890	−	0.584323i	−0.0437890	−	0.584323i
$100$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$101$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$101$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$102$	0.766310	−	0.369035i	0.766310	−	0.369035i
$103$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$103$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.955573	+	0.294755i	$0.0952381\pi$
$107$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.0747301	+	0.997204i	$0.476190\pi$
$108$	0.500000	+	0.626980i	0.500000	+	0.626980i
$109$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$109$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.733052	+	0.680173i	$0.761905\pi$
$113$	−1.72188	+	0.829215i	−1.72188	+	0.829215i	−0.988831	+	0.149042i	$0.952381\pi$
$114$	−0.478300	+	0.443797i	−0.478300	+	0.443797i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.440071	−	1.92808i	0.440071	−	1.92808i
$119$		0			0
$120$		0			0
$121$	0.419945	+	0.202235i	0.419945	+	0.202235i
$122$		0			0
$123$	0.0332580	−	0.0576046i	0.0332580	−	0.0576046i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$127$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$128$	1.00000			1.00000
$129$	−0.445042			−0.445042
$130$		0			0
$131$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.0747301	−	0.997204i	$-0.476190\pi$
$131$	1.03030	−	1.29196i	1.03030	−	1.29196i	0.955573	−	0.294755i	$-0.0952381\pi$
$132$	0.321552	−	0.0484662i	0.321552	−	0.0484662i
$133$		0			0
$134$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$135$		0			0
$136$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$137$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$137$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$138$		0			0
$139$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.587862	−	0.545456i	0.587862	−	0.545456i
$145$		0			0
$146$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$147$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$148$		0			0
$149$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$149$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$150$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$151$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$151$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$152$	1.07473	+	0.997204i	1.07473	+	0.997204i
$153$	−1.46453	−	0.451748i	−1.46453	−	0.451748i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$157$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.0332580	−	0.443797i	0.0332580	−	0.443797i
$163$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.623490	−	0.781831i	$-0.285714\pi$
$163$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.955573	+	0.294755i	$0.0952381\pi$
$164$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i
$165$		0			0
$166$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$167$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$167$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$168$		0			0
$169$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$170$		0			0
$171$	1.17572			1.17572
$172$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	−0.162592	−	0.712362i	−0.162592	−	0.712362i
$177$	0.727208	−	0.495802i	0.727208	−	0.495802i
$178$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i
$179$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.623490	+	0.781831i	$0.285714\pi$
$179$	0.988831	+	1.71271i	0.988831	+	1.71271i	0.365341	+	0.930874i	$0.380952\pi$
$180$		0			0
$181$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$181$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.02366	+	0.949820i	−1.02366	+	0.949820i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$191$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$192$	0.326239	+	0.302705i	0.326239	+	0.302705i
$193$	0.777479	+	0.974928i	0.777479	+	0.974928i	1.00000			$0$
$193$	0.777479	+	0.974928i	0.777479	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$194$	−1.23305	−	1.54620i	−1.23305	−	1.54620i
$195$		0			0
$196$	0.955573	+	0.294755i	0.955573	+	0.294755i
$197$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$197$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$198$	−0.484144	−	0.330084i	−0.484144	−	0.330084i
$199$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$199$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$200$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$201$	−0.440071	−	0.0663300i	−0.440071	−	0.0663300i
$202$		0			0
$203$		0			0
$204$	0.189263	−	0.829215i	0.189263	−	0.829215i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.535628	−	0.927735i	0.535628	−	0.927735i
$210$		0			0
$211$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$211$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$212$		0			0
$213$		0			0
$214$	1.65248			1.65248
$215$		0			0
$216$	0.801938			0.801938
$217$		0			0
$218$		0			0
$219$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$223$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$224$		0			0
$225$	−0.0599289	+	0.799695i	−0.0599289	+	0.799695i
$226$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$227$	0.266948	+	0.680173i	0.266948	+	0.680173i	1.00000			$0$
$227$	0.266948	+	0.680173i	0.266948	+	0.680173i	−0.733052	+	0.680173i	$0.761905\pi$
$228$	0.0487597	+	0.650653i	0.0487597	+	0.650653i
$229$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$229$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	0.365341	+	0.930874i	$0.380952\pi$
$233$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$234$		0			0
$235$		0			0
$236$	−1.23305	−	1.54620i	−1.23305	−	1.54620i
$237$		0			0
$238$		0			0
$239$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$239$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$240$		0			0
$241$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.955573	−	0.294755i	$-0.0952381\pi$
$241$	1.03030	+	0.702449i	1.03030	+	0.702449i	0.0747301	+	0.997204i	$0.476190\pi$
$242$	0.419945	−	0.202235i	0.419945	−	0.202235i
$243$	0.733052	−	0.680173i	0.733052	−	0.680173i
$244$		0			0
$245$		0			0
$246$	−0.0243010	−	0.0619180i	−0.0243010	−	0.0619180i
$247$		0			0
$248$		0			0
$249$	−0.530303	+	0.163577i	−0.530303	+	0.163577i
$250$		0			0
$251$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$251$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.623490	−	0.781831i	0.623490	−	0.781831i
$257$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$257$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$258$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$263$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$263$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$264$	0.162592	−	0.281618i	0.162592	−	0.281618i
$265$		0			0
$266$		0			0
$267$	−0.0635609	+	0.0196059i	−0.0635609	+	0.0196059i
$268$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$269$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$269$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$270$		0			0
$271$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$271$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$272$	−1.88980	−	0.284841i	−1.88980	−	0.284841i
$273$		0			0
$274$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$275$	0.603718	+	0.411608i	0.603718	+	0.411608i
$276$		0			0
$277$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$277$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$278$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$279$		0			0
$280$		0			0
$281$	0.733052	+	0.680173i	0.733052	+	0.680173i	0.955573	−	0.294755i	$-0.0952381\pi$
$281$	0.733052	+	0.680173i	0.733052	+	0.680173i	−0.222521	+	0.974928i	$0.571429\pi$
$282$		0			0
$283$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	0.365341	+	0.930874i	$0.380952\pi$
$283$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.0599289	−	0.799695i	−0.0599289	−	0.799695i
$289$	0.969059	+	2.46912i	0.969059	+	2.46912i
$290$		0			0
$291$	0.0657731	−	0.877681i	0.0657731	−	0.877681i
$292$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$293$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$293$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$294$	0.222521	+	0.385418i	0.222521	+	0.385418i
$295$		0			0
$296$		0			0
$297$	−0.130389	−	0.571270i	−0.130389	−	0.571270i
$298$		0			0
$299$		0			0
$300$	−0.445042			−0.445042
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.44973	−	0.218511i	1.44973	−	0.218511i
$305$		0			0
$306$	−1.26631	+	0.863355i	−1.26631	+	0.863355i
$307$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.222521	+	0.974928i	$0.571429\pi$
$307$	−0.955573	+	1.65510i	−0.955573	+	1.65510i	−0.733052	+	0.680173i	$0.761905\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$311$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$312$		0			0
$313$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	−0.733052	−	0.680173i	$-0.761905\pi$
$313$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	0.0747301	−	0.997204i	$-0.476190\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$317$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.539102	+	0.500214i	0.539102	+	0.500214i
$322$		0			0
$323$	−1.74698	−	2.19064i	−1.74698	−	2.19064i
$324$	−0.326239	−	0.302705i	−0.326239	−	0.302705i
$325$		0			0
$326$	0.603718	−	1.53825i	0.603718	−	1.53825i
$327$		0			0
$328$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i
$329$		0			0
$330$		0			0
$331$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	−0.222521	−	0.974928i	$-0.571429\pi$
$331$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	0.0747301	−	0.997204i	$-0.476190\pi$
$332$	0.455573	+	1.16078i	0.455573	+	1.16078i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$337$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$338$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$339$	−0.702749	+	0.479126i	−0.702749	+	0.479126i
$340$		0			0
$341$		0			0
$342$	0.733052	−	0.919218i	0.733052	−	0.919218i
$343$		0			0
$344$	0.826239	+	0.563320i	0.826239	+	0.563320i
$345$		0			0
$346$		0			0
$347$	−1.88980	+	0.284841i	−1.88980	+	0.284841i	−0.988831	−	0.149042i	$-0.952381\pi$
$347$	−1.88980	+	0.284841i	−1.88980	+	0.284841i	−0.900969	+	0.433884i	$0.857143\pi$
$348$		0			0
$349$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$349$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$350$		0			0
$351$		0			0
$352$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$353$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.623490	−	0.781831i	$-0.285714\pi$
$353$	1.57906	−	0.487076i	1.57906	−	0.487076i	0.955573	+	0.294755i	$0.0952381\pi$
$354$	0.0657731	−	0.877681i	0.0657731	−	0.877681i
$355$		0			0
$356$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$357$		0			0
$358$	1.95557	+	0.294755i	1.95557	+	0.294755i
$359$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$359$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$360$		0			0
$361$	0.949729	+	0.647514i	0.949729	+	0.647514i
$362$		0			0
$363$	0.198220	+	0.0611427i	0.198220	+	0.0611427i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$367$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$368$		0			0
$369$	−0.0437890	+	0.111572i	−0.0437890	+	0.111572i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$373$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$374$	0.104356	+	1.39254i	0.104356	+	1.39254i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$379$	1.78181	+	0.858075i	1.78181	+	0.858075i	0.826239	+	0.563320i	$0.190476\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$383$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$384$	0.440071	−	0.0663300i	0.440071	−	0.0663300i
$385$		0			0
$386$	1.24698			1.24698
$387$	0.792981	−	0.119523i	0.792981	−	0.119523i
$388$	−1.97766			−1.97766
$389$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$389$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$390$		0			0
$391$		0			0
$392$	0.826239	−	0.563320i	0.826239	−	0.563320i
$393$	0.367711	−	0.636894i	0.367711	−	0.636894i
$394$		0			0
$395$		0			0
$396$	−0.559929	+	0.172715i	−0.559929	+	0.172715i
$397$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$397$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$398$		0			0
$399$		0			0
$400$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$401$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.733052	+	0.680173i	$0.761905\pi$
$402$	−0.326239	+	0.302705i	−0.326239	+	0.302705i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.530303	−	0.664979i	−0.530303	−	0.664979i
$409$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$409$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$410$		0			0
$411$	−0.530303	−	0.163577i	−0.530303	−	0.163577i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.0549581	−	0.733365i	−0.0549581	−	0.733365i
$418$	−0.391374	−	0.997204i	−0.391374	−	0.997204i
$419$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$419$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$420$		0			0
$421$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$421$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$422$	1.62349	+	0.781831i	1.62349	+	0.781831i
$423$		0			0
$424$		0			0
$425$	1.57906	−	1.07659i	1.57906	−	1.07659i
$426$		0			0
$427$		0			0
$428$	1.03030	−	1.29196i	1.03030	−	1.29196i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.500000	−	0.626980i	0.500000	−	0.626980i
$433$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.826239	−	0.563320i	$-0.190476\pi$
$433$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.955573	+	0.294755i	$0.0952381\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.400969	−	0.193096i	−0.400969	−	0.193096i
$439$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$439$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$440$		0			0
$441$	0.178448	−	0.781831i	0.178448	−	0.781831i
$442$		0			0
$443$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	0.365341	−	0.930874i	$-0.380952\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.88980	−	0.582926i	−1.88980	−	0.582926i	−0.988831	−	0.149042i	$-0.952381\pi$
$449$	−1.88980	−	0.582926i	−1.88980	−	0.582926i	−0.900969	−	0.433884i	$-0.857143\pi$
$450$	0.587862	+	0.545456i	0.587862	+	0.545456i
$451$	0.0680900	+	0.0853822i	0.0680900	+	0.0853822i
$452$	1.19158	+	1.49419i	1.19158	+	1.49419i
$453$		0			0
$454$	0.698220	+	0.215372i	0.698220	+	0.215372i
$455$		0			0
$456$	0.539102	+	0.367554i	0.539102	+	0.367554i
$457$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.988831	−	0.149042i	$-0.952381\pi$
$457$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	−1.51550	−	0.228425i	−1.51550	−	0.228425i
$460$		0			0
$461$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$461$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$462$		0			0
$463$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$463$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$464$		0			0
$465$		0			0
$466$	0.733052	+	1.26968i	0.733052	+	1.26968i
$467$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	−0.900969	−	0.433884i	$-0.857143\pi$
$467$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i	0.826239	+	0.563320i	$0.190476\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.97766			−1.97766
$473$	0.266948	−	0.680173i	0.266948	−	0.680173i
$474$		0			0
$475$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	1.19158	−	0.367554i	1.19158	−	0.367554i
$483$		0			0
$484$	0.103718	−	0.454418i	0.103718	−	0.454418i
$485$		0			0
$486$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$487$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$487$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$488$		0			0
$489$	0.662592	−	0.319088i	0.662592	−	0.319088i
$490$		0			0
$491$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.0747301	−	0.997204i	$-0.476190\pi$
$491$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.623490	−	0.781831i	$-0.285714\pi$
$492$	−0.0635609	−	0.0196059i	−0.0635609	−	0.0196059i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.202749	+	0.516596i	−0.202749	+	0.516596i
$499$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$499$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$500$		0			0
$501$		0			0
$502$	0.988831	+	0.149042i	0.988831	+	0.149042i
$503$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$503$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.425270	+	0.131178i	−0.425270	+	0.131178i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$513$	1.16259	−	0.175233i	1.16259	−	0.175233i
$514$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$515$		0			0
$516$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.36534	−	0.930874i	1.36534	−	0.930874i	0.365341	−	0.930874i	$-0.380952\pi$
$521$	1.36534	−	0.930874i	1.36534	−	0.930874i	1.00000			$0$
$522$		0			0
$523$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	0.826239	−	0.563320i	$-0.190476\pi$
$523$	−0.0747301	−	0.129436i	−0.0747301	−	0.129436i	−0.900969	+	0.433884i	$0.857143\pi$
$524$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.118803	−	0.302705i	−0.118803	−	0.302705i
$529$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$530$		0			0
$531$	−1.16259	+	1.07873i	−1.16259	+	1.07873i
$532$		0			0
$533$		0			0
$534$	−0.0243010	+	0.0619180i	−0.0243010	+	0.0619180i
$535$		0			0
$536$	0.733052	+	0.680173i	0.733052	+	0.680173i
$537$	0.548760	+	0.688123i	0.548760	+	0.688123i
$538$		0			0
$539$	−0.535628	−	0.496990i	−0.535628	−	0.496990i
$540$		0			0
$541$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$541$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$542$		0			0
$543$		0			0
$544$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$545$		0			0
$546$		0			0
$547$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	−0.733052	−	0.680173i	$-0.761905\pi$
$547$	−0.658322	−	1.67738i	−0.658322	−	1.67738i	0.0747301	−	0.997204i	$-0.476190\pi$
$548$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$549$		0			0
$550$	0.698220	−	0.215372i	0.698220	−	0.215372i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$557$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$557$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.387483	+	0.485888i	−0.387483	+	0.485888i
$562$	0.988831	−	0.149042i	0.988831	−	0.149042i
$563$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$563$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$564$		0			0
$565$		0			0
$566$	0.733052	+	1.26968i	0.733052	+	1.26968i
$567$		0			0
$568$		0			0
$569$	−0.147791	+	1.97213i	−0.147791	+	1.97213i	0.0747301	+	0.997204i	$0.476190\pi$
$569$	−0.147791	+	1.97213i	−0.147791	+	1.97213i	−0.222521	+	0.974928i	$0.571429\pi$
$570$		0			0
$571$	0.455573	+	1.16078i	0.455573	+	1.16078i	0.955573	+	0.294755i	$0.0952381\pi$
$571$	0.455573	+	1.16078i	0.455573	+	1.16078i	−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.662592	−	0.451748i	−0.662592	−	0.451748i
$577$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$578$	2.53464	+	0.781831i	2.53464	+	0.781831i
$579$	0.406813	+	0.377467i	0.406813	+	0.377467i
$580$		0			0
$581$		0			0
$582$	−0.645190	−	0.598649i	−0.645190	−	0.598649i
$583$		0			0
$584$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$585$		0			0
$586$		0			0
$587$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	−0.900969	−	0.433884i	$-0.857143\pi$
$587$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	0.365341	+	0.930874i	$0.380952\pi$
$588$	0.440071	+	0.0663300i	0.440071	+	0.0663300i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.900969	−	0.433884i	$-0.857143\pi$
$593$	−1.40097	+	0.432142i	−1.40097	+	0.432142i	−0.500000	+	0.866025i	$0.666667\pi$
$594$	−0.527933	−	0.254239i	−0.527933	−	0.254239i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$599$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$600$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$601$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$601$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$602$		0			0
$603$	0.801938			0.801938
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$607$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$608$	0.733052	−	1.26968i	0.733052	−	1.26968i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.114533	+	1.52833i	−0.114533	+	1.52833i
$613$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$613$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$614$	0.698220	+	1.77904i	0.698220	+	1.77904i
$615$		0			0
$616$		0			0
$617$	0.326239	−	0.302705i	0.326239	−	0.302705i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	0.326239	−	0.302705i	0.326239	−	0.302705i	0.826239	+	0.563320i	$0.190476\pi$
$618$		0			0
$619$	−0.367711	−	0.250701i	−0.367711	−	0.250701i	0.365341	−	0.930874i	$-0.380952\pi$
$619$	−0.367711	−	0.250701i	−0.367711	−	0.250701i	−0.733052	+	0.680173i	$0.761905\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$626$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$627$	0.174178	−	0.443797i	0.174178	−	0.443797i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$631$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$632$		0			0
$633$	0.292981	+	0.746503i	0.292981	+	0.746503i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	−0.988831	−	0.149042i	$-0.952381\pi$
$641$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	0.826239	−	0.563320i	$-0.190476\pi$
$642$	0.727208	−	0.109609i	0.727208	−	0.109609i
$643$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	0.0747301	+	0.997204i	$0.476190\pi$
$643$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	−0.988831	+	0.149042i	$0.952381\pi$
$644$		0			0
$645$		0			0
$646$	−2.80194			−2.80194
$647$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$647$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$648$	−0.440071	+	0.0663300i	−0.440071	+	0.0663300i
$649$	0.321552	+	1.40881i	0.321552	+	1.40881i
$650$		0			0
$651$		0			0
$652$	−0.826239	−	1.43109i	−0.826239	−	1.43109i
$653$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$653$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$654$		0			0
$655$		0			0
$656$	−0.0332580	+	0.145713i	−0.0332580	+	0.145713i
$657$	0.292981	+	0.746503i	0.292981	+	0.746503i
$658$		0			0
$659$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.365341	−	0.930874i	$-0.380952\pi$
$659$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.0747301	+	0.997204i	$0.476190\pi$
$660$		0			0
$661$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$661$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$662$	−1.63402	−	1.11406i	−1.63402	−	1.11406i
$663$		0			0
$664$	1.19158	+	0.367554i	1.19158	+	0.367554i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	−0.733052	−	0.680173i	$-0.761905\pi$
$673$	−0.109562	+	0.101659i	−0.109562	+	0.101659i	0.623490	+	0.781831i	$0.285714\pi$
$674$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$675$	0.0599289	+	0.799695i	0.0599289	+	0.799695i
$676$	0.365341	+	0.930874i	0.365341	+	0.930874i
$677$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$677$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$678$	−0.0635609	+	0.848162i	−0.0635609	+	0.848162i
$679$		0			0
$680$		0			0
$681$	0.162592	+	0.281618i	0.162592	+	0.281618i
$682$		0			0
$683$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	−0.733052	−	0.680173i	$-0.761905\pi$
$683$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	0.365341	+	0.930874i	$0.380952\pi$
$684$	−0.261623	−	1.14625i	−0.261623	−	1.14625i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.955573	−	0.294755i	0.955573	−	0.294755i
$689$		0			0
$690$		0			0
$691$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.365341	−	0.930874i	$-0.380952\pi$
$691$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.623490	+	0.781831i	$0.285714\pi$
$692$		0			0
$693$		0			0
$694$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$695$		0			0
$696$		0			0
$697$	0.272950	−	0.0841939i	0.272950	−	0.0841939i
$698$		0			0
$699$	−0.145190	+	0.636119i	−0.145190	+	0.636119i
$700$		0			0
$701$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$701$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$702$		0			0
$703$		0			0
$704$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$705$		0			0
$706$	0.603718	−	1.53825i	0.603718	−	1.53825i
$707$		0			0
$708$	−0.645190	−	0.598649i	−0.645190	−	0.598649i
$709$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$709$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$710$		0			0
$711$		0			0
$712$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.44973	−	1.34515i	1.44973	−	1.34515i
$717$		0			0
$718$		0			0
$719$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$719$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$720$		0			0
$721$		0			0
$722$	1.09839	−	0.338809i	1.09839	−	0.338809i
$723$	0.500000	+	0.240787i	0.500000	+	0.240787i
$724$		0			0
$725$		0			0
$726$	0.171391	−	0.116853i	0.171391	−	0.116853i
$727$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$727$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$732$		0			0
$733$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$733$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.365341	−	0.632789i	0.365341	−	0.632789i
$738$	0.0599289	+	0.103800i	0.0599289	+	0.103800i
$739$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$739$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$743$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.900969	−	0.433884i	0.900969	−	0.433884i
$748$	1.15379	+	0.786643i	1.15379	+	0.786643i
$749$		0			0
$750$		0			0
$751$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$751$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$752$		0			0
$753$	0.277479	+	0.347948i	0.277479	+	0.347948i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$757$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$758$	1.78181	−	0.858075i	1.78181	−	0.858075i
$759$		0			0
$760$		0			0
$761$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−0.134659	−	1.79690i	−0.134659	−	1.79690i	0.365341	−	0.930874i	$-0.380952\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.222521	−	0.385418i	0.222521	−	0.385418i
$769$	0.603718	−	0.411608i	0.603718	−	0.411608i	−0.222521	−	0.974928i	$-0.571429\pi$
$769$	0.603718	−	0.411608i	0.603718	−	0.411608i	0.826239	+	0.563320i	$0.190476\pi$
$770$		0			0
$771$	−0.440071	+	0.0663300i	−0.440071	+	0.0663300i
$772$	0.777479	−	0.974928i	0.777479	−	0.974928i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	0.400969	−	0.694498i	0.400969	−	0.694498i
$775$		0			0
$776$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$777$		0			0
$778$		0			0
$779$	−0.181049	+	0.123437i	−0.181049	+	0.123437i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$785$		0			0
$786$	−0.268680	−	0.684585i	−0.268680	−	0.684585i
$787$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	−0.988831	−	0.149042i	$-0.952381\pi$
$787$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	0.955573	−	0.294755i	$-0.0952381\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.214076	+	0.545456i	−0.214076	+	0.545456i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$797$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$798$		0			0
$799$		0			0
$800$	0.826239	+	0.563320i	0.826239	+	0.563320i
$801$	0.107988	−	0.0520043i	0.107988	−	0.0520043i
$802$	−0.914101	+	0.848162i	−0.914101	+	0.848162i
$803$	0.722521	+	0.108903i	0.722521	+	0.108903i
$804$	0.0332580	+	0.443797i	0.0332580	+	0.443797i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$809$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$810$		0			0
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.850540			−0.850540
$817$	1.32091	+	0.636119i	1.32091	+	0.636119i
$818$	0.149460			0.149460
$819$		0			0
$820$		0			0
$821$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$821$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$822$	−0.458528	+	0.312619i	−0.458528	+	0.312619i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	0.292981	+	0.141092i	0.292981	+	0.141092i
$826$		0			0
$827$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$827$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	1.00000			$0$
$828$		0			0
$829$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$829$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$834$	−0.607634	−	0.414278i	−0.607634	−	0.414278i
$835$		0			0
$836$	−1.02366	−	0.315758i	−1.02366	−	0.315758i
$837$		0			0
$838$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$839$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$839$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$840$		0			0
$841$	0.955573	+	0.294755i	0.955573	+	0.294755i
$842$		0			0
$843$	0.367711	+	0.250701i	0.367711	+	0.250701i
$844$	1.62349	−	0.781831i	1.62349	−	0.781831i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.145190	+	0.636119i	−0.145190	+	0.636119i
$850$	0.142820	−	1.90580i	0.142820	−	1.90580i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$857$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.0747301	−	0.997204i	$-0.476190\pi$
$857$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.365341	+	0.930874i	$0.380952\pi$
$858$		0			0
$859$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$859$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$863$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$864$	−0.178448	−	0.781831i	−0.178448	−	0.781831i
$865$		0			0
$866$	0.900969	−	1.56052i	0.900969	−	1.56052i
$867$	0.590232	+	1.02231i	0.590232	+	1.02231i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.118519	+	1.58153i	0.118519	+	1.58153i
$874$		0			0
$875$		0			0
$876$	−0.400969	+	0.193096i	−0.400969	+	0.193096i
$877$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$877$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$881$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$882$	−0.500000	−	0.626980i	−0.500000	−	0.626980i
$883$	0.326239	+	0.302705i	0.326239	+	0.302705i	0.826239	−	0.563320i	$-0.190476\pi$
$883$	0.326239	+	0.302705i	0.326239	+	0.302705i	−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$887$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$887$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.118803	+	0.302705i	0.118803	+	0.302705i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.63402	+	1.11406i	−1.63402	+	1.11406i
$899$		0			0
$900$	0.792981	−	0.119523i	0.792981	−	0.119523i
$901$		0			0
$902$	0.109208			0.109208
$903$		0			0
$904$	1.91115			1.91115
$905$		0			0
$906$		0			0
$907$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	0.955573	−	0.294755i	$-0.0952381\pi$
$907$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i	−0.988831	+	0.149042i	$0.952381\pi$
$908$	0.603718	−	0.411608i	0.603718	−	0.411608i
$909$		0			0
$910$		0			0
$911$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$911$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$912$	0.623490	−	0.192321i	0.623490	−	0.192321i
$913$	0.0680900	−	0.908598i	0.0680900	−	0.908598i
$914$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.12349	+	1.04245i	−1.12349	+	1.04245i
$919$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$919$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$920$		0			0
$921$	−0.310737	+	0.791745i	−0.310737	+	0.791745i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$929$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$930$		0			0
$931$	1.07473	−	0.997204i	1.07473	−	0.997204i
$932$	1.44973	+	0.218511i	1.44973	+	0.218511i
$933$		0			0
$934$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$935$		0			0
$936$		0			0
$937$	−1.88980	+	0.582926i	−1.88980	+	0.582926i	−0.900969	+	0.433884i	$0.857143\pi$
$937$	−1.88980	+	0.582926i	−1.88980	+	0.582926i	−0.988831	+	0.149042i	$0.952381\pi$
$938$		0			0
$939$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$940$		0			0
$941$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$941$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$942$		0			0
$943$		0			0
$944$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$945$		0			0
$946$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$947$	−1.46610			−1.46610			−0.733052	−	0.680173i	$-0.761905\pi$
$947$	−1.46610			−1.46610			−0.733052	+	0.680173i	$0.761905\pi$
$948$		0			0
$949$		0			0
$950$	0.326239	+	1.42935i	0.326239	+	1.42935i
$951$		0			0
$952$		0			0
$953$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$953$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$962$		0			0
$963$	−1.09492	−	0.746503i	−1.09492	−	0.746503i
$964$	0.455573	−	1.16078i	0.455573	−	1.16078i
$965$		0			0
$966$		0			0
$967$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$967$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$968$	−0.290611	−	0.364415i	−0.290611	−	0.364415i
$969$	−0.914101	−	0.848162i	−0.914101	−	0.848162i
$970$		0			0
$971$	−0.658322	+	1.67738i	−0.658322	+	1.67738i	0.0747301	+	0.997204i	$0.476190\pi$
$971$	−0.658322	+	1.67738i	−0.658322	+	1.67738i	−0.733052	+	0.680173i	$0.761905\pi$
$972$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.535628	−	1.36476i	−0.535628	−	1.36476i	−0.900969	−	0.433884i	$-0.857143\pi$
$977$	−0.535628	−	1.36476i	−0.535628	−	1.36476i	0.365341	−	0.930874i	$-0.380952\pi$
$978$	0.163647	−	0.716983i	0.163647	−	0.716983i
$979$	0.00816111	−	0.108903i	0.00816111	−	0.108903i
$980$		0			0
$981$		0			0
$982$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	−0.0549581	+	0.0374698i	−0.0549581	+	0.0374698i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$991$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$992$		0			0
$993$	−0.195850	−	0.858075i	−0.195850	−	0.858075i
$994$		0			0
$995$		0			0
$996$	0.277479	+	0.480608i	0.277479	+	0.480608i
$997$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$997$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$998$	−1.88980	+	0.582926i	−1.88980	+	0.582926i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	344.1.be.a.67.1	✓	12
3.2	odd	2		3096.1.eq.a.1099.1		12
4.3	odd	2		1376.1.by.a.239.1		12
8.3	odd	2	CM	344.1.be.a.67.1	✓	12
8.5	even	2		1376.1.by.a.239.1		12
24.11	even	2		3096.1.eq.a.1099.1		12
43.9	even	21	inner	344.1.be.a.267.1	yes	12
129.95	odd	42		3096.1.eq.a.955.1		12
172.95	odd	42		1376.1.by.a.783.1		12
344.181	even	42		1376.1.by.a.783.1		12
344.267	odd	42	inner	344.1.be.a.267.1	yes	12
1032.611	even	42		3096.1.eq.a.955.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.1.be.a.67.1	✓	12	1.1	even	1	trivial
344.1.be.a.67.1	✓	12	8.3	odd	2	CM
344.1.be.a.267.1	yes	12	43.9	even	21	inner
344.1.be.a.267.1	yes	12	344.267	odd	42	inner
1376.1.by.a.239.1		12	4.3	odd	2
1376.1.by.a.239.1		12	8.5	even	2
1376.1.by.a.783.1		12	172.95	odd	42
1376.1.by.a.783.1		12	344.181	even	42
3096.1.eq.a.955.1		12	129.95	odd	42
3096.1.eq.a.955.1		12	1032.611	even	42
3096.1.eq.a.1099.1		12	3.2	odd	2
3096.1.eq.a.1099.1		12	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	344.be (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.623490 - 0.781831i) q^{2} +(0.440071 - 0.0663300i) q^{3} +(-0.222521 - 0.974928i) q^{4} +(0.222521 - 0.385418i) q^{6} +(-0.900969 - 0.433884i) q^{8} +(-0.766310 + 0.236375i) q^{9} +O(q^{10})\)	\(q+(0.623490 - 0.781831i) q^{2} +(0.440071 - 0.0663300i) q^{3} +(-0.222521 - 0.974928i) q^{4} +(0.222521 - 0.385418i) q^{6} +(-0.900969 - 0.433884i) q^{8} +(-0.766310 + 0.236375i) q^{9} +(-0.162592 + 0.712362i) q^{11} +(-0.162592 - 0.414278i) q^{12} +(-0.900969 + 0.433884i) q^{16} +(1.57906 + 1.07659i) q^{17} +(-0.292981 + 0.746503i) q^{18} +(-1.40097 - 0.432142i) q^{19} +(0.455573 + 0.571270i) q^{22} +(-0.425270 - 0.131178i) q^{24} +(0.365341 - 0.930874i) q^{25} +(-0.722521 + 0.347948i) q^{27} +(-0.222521 + 0.974928i) q^{32} +(-0.0243010 + 0.324275i) q^{33} +(1.82624 - 0.563320i) q^{34} +(0.400969 + 0.694498i) q^{36} +(-1.21135 + 0.825886i) q^{38} +(0.0931869 - 0.116853i) q^{41} +(-0.988831 - 0.149042i) q^{43} +0.730682 q^{44} +(-0.367711 + 0.250701i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(0.766310 + 0.369035i) q^{51} +(-0.178448 + 0.781831i) q^{54} +(-0.645190 - 0.0972467i) q^{57} +(1.78181 - 0.858075i) q^{59} +(0.623490 + 0.781831i) q^{64} +(0.238377 + 0.221181i) q^{66} +(-0.955573 - 0.294755i) q^{67} +(0.698220 - 1.77904i) q^{68} +(0.792981 + 0.119523i) q^{72} +(-0.0747301 - 0.997204i) q^{73} +(0.0990311 - 0.433884i) q^{75} +(-0.109562 + 1.46200i) q^{76} +(0.367711 - 0.250701i) q^{81} +(-0.0332580 - 0.145713i) q^{82} +(-1.23305 + 0.185853i) q^{83} +(-0.733052 + 0.680173i) q^{86} +(0.455573 - 0.571270i) q^{88} +(-0.147791 + 0.0222759i) q^{89} +(-0.0332580 + 0.443797i) q^{96} +(0.440071 - 1.92808i) q^{97} +(0.365341 + 0.930874i) q^{98} +(-0.0437890 - 0.584323i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9	\( 12 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 3 q^{9} + 2 q^{11} + 2 q^{12} - 2 q^{16} - q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{22} - 5 q^{24} + q^{25} - 8 q^{27} - 2 q^{32} - 9 q^{33} + 13 q^{34} - 4 q^{36} - q^{38} + 2 q^{41} + q^{43} + 2 q^{44} + 2 q^{48} - 6 q^{49} - 6 q^{50} - 3 q^{51} + 6 q^{54} - 2 q^{57} + 2 q^{59} - 2 q^{64} - 2 q^{66} - q^{67} - q^{68} + 3 q^{72} - q^{73} + 10 q^{75} - q^{76} - 2 q^{81} + 2 q^{82} - 5 q^{83} + q^{86} - 5 q^{88} - q^{89} + 2 q^{96} + 2 q^{97} + q^{98} + 11 q^{99}+O(q^{100}) \) 12 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 2 * q^8 + 3 * q^9 + 2 * q^11 + 2 * q^12 - 2 * q^16 - q^17 + 3 * q^18 - 8 * q^19 - 5 * q^22 - 5 * q^24 + q^25 - 8 * q^27 - 2 * q^32 - 9 * q^33 + 13 * q^34 - 4 * q^36 - q^38 + 2 * q^41 + q^43 + 2 * q^44 + 2 * q^48 - 6 * q^49 - 6 * q^50 - 3 * q^51 + 6 * q^54 - 2 * q^57 + 2 * q^59 - 2 * q^64 - 2 * q^66 - q^67 - q^68 + 3 * q^72 - q^73 + 10 * q^75 - q^76 - 2 * q^81 + 2 * q^82 - 5 * q^83 + q^86 - 5 * q^88 - q^89 + 2 * q^96 + 2 * q^97 + q^98 + 11 * q^99

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