LMFDB - Embedded newform 2217.1.p.a.692.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2217,1,Mod(20,2217)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2217, base_ring=CyclotomicField(82))

chi = DirichletCharacter(H, H._module([41, 6]))

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

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

chi := DirichletCharacter("2217.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2217 = 3 \cdot 739 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2217.p (of order $82$, degree $40$, 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:	$1.10642713301$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{82})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} + \cdots + 1 $ x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{41}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{41} - \cdots)$

Embedding invariants

Embedding label			692.1
Root			$0.114683 - 0.993402i$ of defining polynomial
Character	$\chi$	$=$	2217.692
Dual form			2217.1.p.a.1022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.953396 + 0.301721i) q^{3} +(-0.409069 - 0.912504i) q^{4} +(-0.519346 + 0.801963i) q^{7} +(0.817929 + 0.575319i) q^{9} +O(q^{10})$	\(q+(0.953396 + 0.301721i) q^{3} +(-0.409069 - 0.912504i) q^{4} +(-0.519346 + 0.801963i) q^{7} +(0.817929 + 0.575319i) q^{9} +(-0.114683 - 0.993402i) q^{12} +(1.03830 - 0.999266i) q^{13} +(-0.665326 + 0.746553i) q^{16} +(0.455540 - 0.270821i) q^{19} +(-0.737111 + 0.607891i) q^{21} +(0.817929 - 0.575319i) q^{25} +(0.606225 + 0.795293i) q^{27} +(0.944242 + 0.145847i) q^{28} +(1.17867 - 1.13436i) q^{31} +(0.190391 - 0.981708i) q^{36} +(-1.19248 + 1.33807i) q^{37} +(1.29141 - 0.639419i) q^{39} +(-0.321277 + 0.421476i) q^{43} +(-0.859570 + 0.511019i) q^{48} +(0.0356442 + 0.0795111i) q^{49} +(-1.33657 - 0.538687i) q^{52} +(0.516023 - 0.120754i) q^{57} +(1.02658 + 0.846617i) q^{61} +(-0.886173 + 0.357160i) q^{63} +(0.953396 + 0.301721i) q^{64} +(-0.669178 - 0.470690i) q^{67} +(0.552948 - 1.53957i) q^{73} +(0.953396 - 0.301721i) q^{75} +(-0.433472 - 0.304898i) q^{76} +(-0.974253 + 0.482383i) q^{79} +(0.338017 + 0.941140i) q^{81} +(0.856232 + 0.423948i) q^{84} +(0.262136 + 1.35165i) q^{91} +(1.46600 - 0.725863i) q^{93} +(0.124676 - 0.192523i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) $ 40 * q - q^3 - q^4 - 2 * q^7 - q^9	$ 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 39 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 4 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) $ 40 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 39 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - 2 * q^39 - 2 * q^43 - q^48 - 3 * q^49 - 2 * q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 4 * q^91 - 2 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2217\mathbb{Z}\right)^\times$.

$n$	$740$	$742$
$\chi(n)$	$-1$	$e\left(\frac{4}{41}\right)$

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 $

$2$		0			0		0.543568	−	0.839365i	$-0.317073\pi$
$2$		0			0		−0.543568	+	0.839365i	$0.682927\pi$
$3$	0.953396	+	0.301721i	0.953396	+	0.301721i
$3$	0.953396	+	0.301721i	0.953396	+	0.301721i
$4$	−0.409069	−	0.912504i	−0.409069	−	0.912504i
$5$		0			0		0.953396	−	0.301721i	$-0.0975610\pi$
$5$		0			0		−0.953396	+	0.301721i	$0.902439\pi$
$6$		0			0
$7$	−0.519346	+	0.801963i	−0.519346	+	0.801963i	−0.997066	−	0.0765493i	$-0.975610\pi$
$7$	−0.519346	+	0.801963i	−0.519346	+	0.801963i	0.477720	+	0.878512i	$0.341463\pi$
$8$		0			0
$9$	0.817929	+	0.575319i	0.817929	+	0.575319i
$10$		0			0
$11$		0			0		−0.0383027	−	0.999266i	$-0.512195\pi$
$11$		0			0		0.0383027	+	0.999266i	$0.487805\pi$
$12$	−0.114683	−	0.993402i	−0.114683	−	0.993402i
$13$	1.03830	−	0.999266i	1.03830	−	0.999266i	0.0383027	−	0.999266i	$-0.487805\pi$
$13$	1.03830	−	0.999266i	1.03830	−	0.999266i	1.00000			$0$
$14$		0			0
$15$		0			0
$16$	−0.665326	+	0.746553i	−0.665326	+	0.746553i
$17$		0			0		−0.477720	−	0.878512i	$-0.658537\pi$
$17$		0			0		0.477720	+	0.878512i	$0.341463\pi$
$18$		0			0
$19$	0.455540	−	0.270821i	0.455540	−	0.270821i	−0.264982	−	0.964253i	$-0.585366\pi$
$19$	0.455540	−	0.270821i	0.455540	−	0.270821i	0.720522	+	0.693433i	$0.243902\pi$
$20$		0			0
$21$	−0.737111	+	0.607891i	−0.737111	+	0.607891i
$22$		0			0
$23$		0			0		0.543568	−	0.839365i	$-0.317073\pi$
$23$		0			0		−0.543568	+	0.839365i	$0.682927\pi$
$24$		0			0
$25$	0.817929	−	0.575319i	0.817929	−	0.575319i
$26$		0			0
$27$	0.606225	+	0.795293i	0.606225	+	0.795293i
$28$	0.944242	+	0.145847i	0.944242	+	0.145847i
$29$		0			0		0.190391	−	0.981708i	$-0.439024\pi$
$29$		0			0		−0.190391	+	0.981708i	$0.560976\pi$
$30$		0			0
$31$	1.17867	−	1.13436i	1.17867	−	1.13436i	0.190391	−	0.981708i	$-0.439024\pi$
$31$	1.17867	−	1.13436i	1.17867	−	1.13436i	0.988280	−	0.152649i	$-0.0487805\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.190391	−	0.981708i	0.190391	−	0.981708i
$37$	−1.19248	+	1.33807i	−1.19248	+	1.33807i	−0.264982	+	0.964253i	$0.585366\pi$
$37$	−1.19248	+	1.33807i	−1.19248	+	1.33807i	−0.927502	+	0.373817i	$0.878049\pi$
$38$		0			0
$39$	1.29141	−	0.639419i	1.29141	−	0.639419i
$40$		0			0
$41$		0			0		−0.927502	−	0.373817i	$-0.878049\pi$
$41$		0			0		0.927502	+	0.373817i	$0.121951\pi$
$42$		0			0
$43$	−0.321277	+	0.421476i	−0.321277	+	0.421476i	−0.927502	−	0.373817i	$-0.878049\pi$
$43$	−0.321277	+	0.421476i	−0.321277	+	0.421476i	0.606225	+	0.795293i	$0.292683\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.997066	−	0.0765493i	$-0.975610\pi$
$47$		0			0		0.997066	+	0.0765493i	$0.0243902\pi$
$48$	−0.859570	+	0.511019i	−0.859570	+	0.511019i
$49$	0.0356442	+	0.0795111i	0.0356442	+	0.0795111i
$50$		0			0
$51$		0			0
$52$	−1.33657	−	0.538687i	−1.33657	−	0.538687i
$53$		0			0		0.665326	−	0.746553i	$-0.268293\pi$
$53$		0			0		−0.665326	+	0.746553i	$0.731707\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.516023	−	0.120754i	0.516023	−	0.120754i
$58$		0			0
$59$		0			0		−0.606225	−	0.795293i	$-0.707317\pi$
$59$		0			0		0.606225	+	0.795293i	$0.292683\pi$
$60$		0			0
$61$	1.02658	+	0.846617i	1.02658	+	0.846617i	0.988280	−	0.152649i	$-0.0487805\pi$
$61$	1.02658	+	0.846617i	1.02658	+	0.846617i	0.0383027	+	0.999266i	$0.487805\pi$
$62$		0			0
$63$	−0.886173	+	0.357160i	−0.886173	+	0.357160i
$64$	0.953396	+	0.301721i	0.953396	+	0.301721i
$65$		0			0
$66$		0			0
$67$	−0.669178	−	0.470690i	−0.669178	−	0.470690i	0.190391	−	0.981708i	$-0.439024\pi$
$67$	−0.669178	−	0.470690i	−0.669178	−	0.470690i	−0.859570	+	0.511019i	$0.829268\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.997066	−	0.0765493i	$-0.0243902\pi$
$71$		0			0		−0.997066	+	0.0765493i	$0.975610\pi$
$72$		0			0
$73$	0.552948	−	1.53957i	0.552948	−	1.53957i	−0.264982	−	0.964253i	$-0.585366\pi$
$73$	0.552948	−	1.53957i	0.552948	−	1.53957i	0.817929	−	0.575319i	$-0.195122\pi$
$74$		0			0
$75$	0.953396	−	0.301721i	0.953396	−	0.301721i
$76$	−0.433472	−	0.304898i	−0.433472	−	0.304898i
$77$		0			0
$78$		0			0
$79$	−0.974253	+	0.482383i	−0.974253	+	0.482383i	−0.859570	−	0.511019i	$-0.829268\pi$
$79$	−0.974253	+	0.482383i	−0.974253	+	0.482383i	−0.114683	+	0.993402i	$0.536585\pi$
$80$		0			0
$81$	0.338017	+	0.941140i	0.338017	+	0.941140i
$82$		0			0
$83$		0			0		0.896166	−	0.443720i	$-0.146341\pi$
$83$		0			0		−0.896166	+	0.443720i	$0.853659\pi$
$84$	0.856232	+	0.423948i	0.856232	+	0.423948i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.720522	−	0.693433i	$-0.756098\pi$
$89$		0			0		0.720522	+	0.693433i	$0.243902\pi$
$90$		0			0
$91$	0.262136	+	1.35165i	0.262136	+	1.35165i
$92$		0			0
$93$	1.46600	−	0.725863i	1.46600	−	0.725863i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.124676	−	0.192523i	0.124676	−	0.192523i	−0.771489	−	0.636242i	$-0.780488\pi$
$97$	0.124676	−	0.192523i	0.124676	−	0.192523i	0.896166	+	0.443720i	$0.146341\pi$
$98$		0			0
$99$		0			0
$100$	−0.859570	−	0.511019i	−0.859570	−	0.511019i
$101$		0			0		0.665326	−	0.746553i	$-0.268293\pi$
$101$		0			0		−0.665326	+	0.746553i	$0.731707\pi$
$102$		0			0
$103$	−1.51726	−	1.06722i	−1.51726	−	1.06722i	−0.973695	−	0.227854i	$-0.926829\pi$
$103$	−1.51726	−	1.06722i	−1.51726	−	1.06722i	−0.543568	−	0.839365i	$-0.682927\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.953396	−	0.301721i	$-0.0975610\pi$
$107$		0			0		−0.953396	+	0.301721i	$0.902439\pi$
$108$	0.477720	−	0.878512i	0.477720	−	0.878512i
$109$	−0.139048	−	1.20445i	−0.139048	−	1.20445i	−0.859570	−	0.511019i	$-0.829268\pi$
$109$	−0.139048	−	1.20445i	−0.139048	−	1.20445i	0.720522	−	0.693433i	$-0.243902\pi$
$110$		0			0
$111$	−1.54063	+	0.915915i	−1.54063	+	0.915915i
$112$	−0.253174	−	0.921286i	−0.253174	−	0.921286i
$113$		0			0		−0.190391	−	0.981708i	$-0.560976\pi$
$113$		0			0		0.190391	+	0.981708i	$0.439024\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.42415	−	0.219974i	1.42415	−	0.219974i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.997066	+	0.0765493i	−0.997066	+	0.0765493i
$122$		0			0
$123$		0			0
$124$	−1.51726	−	0.611512i	−1.51726	−	0.611512i
$125$		0			0
$126$		0			0
$127$	−1.47107	+	0.465548i	−1.47107	+	0.465548i	−0.927502	−	0.373817i	$-0.878049\pi$
$127$	−1.47107	+	0.465548i	−1.47107	+	0.465548i	−0.543568	+	0.839365i	$0.682927\pi$
$128$		0			0
$129$	−0.433472	+	0.304898i	−0.433472	+	0.304898i
$130$		0			0
$131$		0			0		−0.720522	−	0.693433i	$-0.756098\pi$
$131$		0			0		0.720522	+	0.693433i	$0.243902\pi$
$132$		0			0
$133$	−0.0193945	+	0.505976i	−0.0193945	+	0.505976i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.973695	−	0.227854i	$-0.926829\pi$
$137$		0			0		0.973695	+	0.227854i	$0.0731707\pi$
$138$		0			0
$139$	−1.07439	+	0.165950i	−1.07439	+	0.165950i	−0.665326	−	0.746553i	$-0.731707\pi$
$139$	−1.07439	+	0.165950i	−1.07439	+	0.165950i	−0.409069	+	0.912504i	$0.634146\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.973695	+	0.227854i	−0.973695	+	0.227854i
$145$		0			0
$146$		0			0
$147$	0.00999295	+	0.0865602i	0.00999295	+	0.0865602i
$148$	1.70880	+	0.540783i	1.70880	+	0.540783i
$149$		0			0		0.927502	−	0.373817i	$-0.121951\pi$
$149$		0			0		−0.927502	+	0.373817i	$0.878049\pi$
$150$		0			0
$151$	−1.69899	+	1.01006i	−1.69899	+	1.01006i	−0.771489	+	0.636242i	$0.780488\pi$
$151$	−1.69899	+	1.01006i	−1.69899	+	1.01006i	−0.927502	+	0.373817i	$0.878049\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.11175	−	0.916853i	−1.11175	−	0.916853i
$157$	−0.0775299	−	0.671573i	−0.0775299	−	0.671573i	−0.973695	−	0.227854i	$-0.926829\pi$
$157$	−0.0775299	−	0.671573i	−0.0775299	−	0.671573i	0.896166	−	0.443720i	$-0.146341\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.230840	+	0.302833i	0.230840	+	0.302833i	0.896166	−	0.443720i	$-0.146341\pi$
$163$	0.230840	+	0.302833i	0.230840	+	0.302833i	−0.665326	+	0.746553i	$0.731707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.338017	−	0.941140i	$-0.390244\pi$
$167$		0			0		−0.338017	+	0.941140i	$0.609756\pi$
$168$		0			0
$169$	0.0412369	−	1.07582i	0.0412369	−	1.07582i
$170$		0			0
$171$	0.528408	+	0.0405683i	0.528408	+	0.0405683i
$172$	0.516023	+	0.120754i	0.516023	+	0.120754i
$173$		0			0		0.477720	−	0.878512i	$-0.341463\pi$
$173$		0			0		−0.477720	+	0.878512i	$0.658537\pi$
$174$		0			0
$175$	0.0365959	+	0.954739i	0.0365959	+	0.954739i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.543568	−	0.839365i	$-0.682927\pi$
$179$		0			0		0.543568	+	0.839365i	$0.317073\pi$
$180$		0			0
$181$	−0.818137	+	1.82501i	−0.818137	+	1.82501i	−0.409069	+	0.912504i	$0.634146\pi$
$181$	−0.818137	+	1.82501i	−0.818137	+	1.82501i	−0.409069	+	0.912504i	$0.634146\pi$
$182$		0			0
$183$	0.723299	+	1.11690i	0.723299	+	1.11690i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.952636	+	0.0731382i	−0.952636	+	0.0731382i
$190$		0			0
$191$		0			0		−0.190391	−	0.981708i	$-0.560976\pi$
$191$		0			0		0.190391	+	0.981708i	$0.439024\pi$
$192$	0.817929	+	0.575319i	0.817929	+	0.575319i
$193$	−0.0591003	−	0.0487396i	−0.0591003	−	0.0487396i	0.606225	−	0.795293i	$-0.292683\pi$
$193$	−0.0591003	−	0.0487396i	−0.0591003	−	0.0487396i	−0.665326	+	0.746553i	$0.731707\pi$
$194$		0			0
$195$		0			0
$196$	0.0579732	−	0.0650510i	0.0579732	−	0.0650510i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−0.253344	+	1.30631i	−0.253344	+	1.30631i	0.606225	+	0.795293i	$0.292683\pi$
$199$	−0.253344	+	1.30631i	−0.253344	+	1.30631i	−0.859570	+	0.511019i	$0.829268\pi$
$200$		0			0
$201$	−0.495976	−	0.650659i	−0.495976	−	0.650659i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.0551959	+	1.43999i	0.0551959	+	1.43999i
$209$		0			0
$210$		0			0
$211$	1.00832	+	1.55703i	1.00832	+	1.55703i	0.817929	+	0.575319i	$0.195122\pi$
$211$	1.00832	+	1.55703i	1.00832	+	1.55703i	0.190391	+	0.981708i	$0.439024\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.297574	+	1.53438i	0.297574	+	1.53438i
$218$		0			0
$219$	0.991699	−	1.30099i	0.991699	−	1.30099i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.63902	−	0.974407i	−1.63902	−	0.974407i	−0.973695	−	0.227854i	$-0.926829\pi$
$223$	−1.63902	−	0.974407i	−1.63902	−	0.974407i	−0.665326	−	0.746553i	$-0.731707\pi$
$224$		0			0
$225$	1.00000			1.00000
$226$		0			0
$227$		0			0		−0.896166	−	0.443720i	$-0.853659\pi$
$227$		0			0		0.896166	+	0.443720i	$0.146341\pi$
$228$	−0.321277	−	0.421476i	−0.321277	−	0.421476i
$229$	1.84956	+	0.141999i	1.84956	+	0.141999i	0.953396	−	0.301721i	$-0.0975610\pi$
$229$	1.84956	+	0.141999i	1.84956	+	0.141999i	0.896166	+	0.443720i	$0.146341\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.896166	−	0.443720i	$-0.146341\pi$
$233$		0			0		−0.896166	+	0.443720i	$0.853659\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.07439	+	0.165950i	−1.07439	+	0.165950i
$238$		0			0
$239$		0			0		−0.973695	−	0.227854i	$-0.926829\pi$
$239$		0			0		0.973695	+	0.227854i	$0.0731707\pi$
$240$		0			0
$241$	−0.276544	+	0.616883i	−0.276544	+	0.616883i	−0.997066	−	0.0765493i	$-0.975610\pi$
$241$	−0.276544	+	0.616883i	−0.276544	+	0.616883i	0.720522	+	0.693433i	$0.243902\pi$
$242$		0			0
$243$	0.0383027	+	0.999266i	0.0383027	+	0.999266i
$244$	0.352598	−	1.28309i	0.352598	−	1.28309i
$245$		0			0
$246$		0			0
$247$	0.202366	−	0.736400i	0.202366	−	0.736400i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.665326	−	0.746553i	$-0.731707\pi$
$251$		0			0		0.665326	+	0.746553i	$0.268293\pi$
$252$	0.688415	+	0.662533i	0.688415	+	0.662533i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.114683	−	0.993402i	−0.114683	−	0.993402i
$257$		0			0		−0.543568	−	0.839365i	$-0.682927\pi$
$257$		0			0		0.543568	+	0.839365i	$0.317073\pi$
$258$		0			0
$259$	−0.453771	−	1.65125i	−0.453771	−	1.65125i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.338017	−	0.941140i	$-0.609756\pi$
$263$		0			0		0.338017	+	0.941140i	$0.390244\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.155766	+	0.803172i	−0.155766	+	0.803172i
$269$		0			0		0.338017	−	0.941140i	$-0.390244\pi$
$269$		0			0		−0.338017	+	0.941140i	$0.609756\pi$
$270$		0			0
$271$	0.124676	+	0.192523i	0.124676	+	0.192523i	0.896166	−	0.443720i	$-0.146341\pi$
$271$	0.124676	+	0.192523i	0.124676	+	0.192523i	−0.771489	+	0.636242i	$0.780488\pi$
$272$		0			0
$273$	−0.157900	+	1.36775i	−0.157900	+	1.36775i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.97076	+	0.304403i	−1.97076	+	0.304403i	−0.973695	+	0.227854i	$0.926829\pi$
$277$	−1.97076	+	0.304403i	−1.97076	+	0.304403i	−0.997066	+	0.0765493i	$0.975610\pi$
$278$		0			0
$279$	1.61669	−	0.249713i	1.61669	−	0.249713i
$280$		0			0
$281$		0			0		0.606225	−	0.795293i	$-0.292683\pi$
$281$		0			0		−0.606225	+	0.795293i	$0.707317\pi$
$282$		0			0
$283$	−1.20889	+	0.0928122i	−1.20889	+	0.0928122i	−0.665326	−	0.746553i	$-0.731707\pi$
$283$	−1.20889	+	0.0928122i	−1.20889	+	0.0928122i	−0.543568	+	0.839365i	$0.682927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.543568	+	0.839365i	−0.543568	+	0.839365i
$290$		0			0
$291$	0.176954	−	0.145933i	0.176954	−	0.145933i
$292$	−1.63106	+	0.125224i	−1.63106	+	0.125224i
$293$		0			0		0.896166	−	0.443720i	$-0.146341\pi$
$293$		0			0		−0.896166	+	0.443720i	$0.853659\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.665326	−	0.746553i	−0.665326	−	0.746553i
$301$	−0.171154	−	0.476544i	−0.171154	−	0.476544i
$302$		0			0
$303$		0			0
$304$	−0.100900	+	0.520269i	−0.100900	+	0.520269i
$305$		0			0
$306$		0			0
$307$	0.815737	+	0.0626278i	0.815737	+	0.0626278i	0.477720	−	0.878512i	$-0.341463\pi$
$307$	0.815737	+	0.0626278i	0.815737	+	0.0626278i	0.338017	+	0.941140i	$0.390244\pi$
$308$		0			0
$309$	−1.12455	−	1.47527i	−1.12455	−	1.47527i
$310$		0			0
$311$		0			0		−0.665326	−	0.746553i	$-0.731707\pi$
$311$		0			0		0.665326	+	0.746553i	$0.268293\pi$
$312$		0			0
$313$	0.873597	−	0.840753i	0.873597	−	0.840753i	−0.114683	−	0.993402i	$-0.536585\pi$
$313$	0.873597	−	0.840753i	0.873597	−	0.840753i	0.988280	+	0.152649i	$0.0487805\pi$
$314$		0			0
$315$		0			0
$316$	0.838713	+	0.691681i	0.838713	+	0.691681i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.720522	−	0.693433i	0.720522	−	0.693433i
$325$	0.274362	−	1.41468i	0.274362	−	1.41468i
$326$		0			0
$327$	0.230840	−	1.19027i	0.230840	−	1.19027i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0591003	−	1.54185i	−0.0591003	−	1.54185i	−0.665326	−	0.746553i	$-0.731707\pi$
$331$	−0.0591003	−	1.54185i	−0.0591003	−	1.54185i	0.606225	−	0.795293i	$-0.292683\pi$
$332$		0			0
$333$	−1.74518	+	0.408389i	−1.74518	+	0.408389i
$334$		0			0
$335$		0			0
$336$	0.0365959	−	0.954739i	0.0365959	−	0.954739i
$337$	0.815737	−	0.0626278i	0.815737	−	0.0626278i	0.338017	−	0.941140i	$-0.390244\pi$
$337$	0.815737	−	0.0626278i	0.815737	−	0.0626278i	0.477720	+	0.878512i	$0.341463\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.02652	−	0.158556i	−1.02652	−	0.158556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.264982	−	0.964253i	$-0.585366\pi$
$347$		0			0		0.264982	+	0.964253i	$0.414634\pi$
$348$		0			0
$349$	−0.327309	−	1.68769i	−0.327309	−	1.68769i	−0.665326	−	0.746553i	$-0.731707\pi$
$349$	−0.327309	−	1.68769i	−0.327309	−	1.68769i	0.338017	−	0.941140i	$-0.390244\pi$
$350$		0			0
$351$	1.42415	+	0.219974i	1.42415	+	0.219974i
$352$		0			0
$353$		0			0		0.953396	−	0.301721i	$-0.0975610\pi$
$353$		0			0		−0.953396	+	0.301721i	$0.902439\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.953396	−	0.301721i	$-0.902439\pi$
$359$		0			0		0.953396	+	0.301721i	$0.0975610\pi$
$360$		0			0
$361$	−0.343547	+	0.631773i	−0.343547	+	0.631773i
$362$		0			0
$363$	−0.973695	−	0.227854i	−0.973695	−	0.227854i
$364$	1.12615	−	0.792116i	1.12615	−	0.792116i
$365$		0			0
$366$		0			0
$367$	1.02658	+	0.846617i	1.02658	+	0.846617i	0.988280	−	0.152649i	$-0.0487805\pi$
$367$	1.02658	+	0.846617i	1.02658	+	0.846617i	0.0383027	+	0.999266i	$0.487805\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.26205	−	1.04080i	−1.26205	−	1.04080i
$373$	−1.54298			−1.54298			−0.771489	−	0.636242i	$-0.780488\pi$
$373$	−1.54298			−1.54298			−0.771489	+	0.636242i	$0.780488\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.77133	+	0.877039i	1.77133	+	0.877039i	0.953396	+	0.301721i	$0.0975610\pi$
$379$	1.77133	+	0.877039i	1.77133	+	0.877039i	0.817929	+	0.575319i	$0.195122\pi$
$380$		0			0
$381$	−1.54298			−1.54298
$382$		0			0
$383$		0			0		0.338017	−	0.941140i	$-0.390244\pi$
$383$		0			0		−0.338017	+	0.941140i	$0.609756\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.505265	+	0.159901i	−0.505265	+	0.159901i
$388$	−0.226679	−	0.0350127i	−0.226679	−	0.0350127i
$389$		0			0		−0.859570	−	0.511019i	$-0.829268\pi$
$389$		0			0		0.859570	+	0.511019i	$0.170732\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.19824	−	1.57194i	1.19824	−	1.57194i	0.477720	−	0.878512i	$-0.341463\pi$
$397$	1.19824	−	1.57194i	1.19824	−	1.57194i	0.720522	−	0.693433i	$-0.243902\pi$
$398$		0			0
$399$	−0.171154	+	0.476544i	−0.171154	+	0.476544i
$400$	−0.114683	+	0.993402i	−0.114683	+	0.993402i
$401$		0			0		−0.817929	−	0.575319i	$-0.804878\pi$
$401$		0			0		0.817929	+	0.575319i	$0.195122\pi$
$402$		0			0
$403$	0.0902927	−	2.35561i	0.0902927	−	2.35561i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.47772	+	0.878512i	1.47772	+	0.878512i	1.00000			$0$
$409$	1.47772	+	0.878512i	1.47772	+	0.878512i	0.477720	+	0.878512i	$0.341463\pi$
$410$		0			0
$411$		0			0
$412$	−0.353176	+	1.82107i	−0.353176	+	1.82107i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.07439	−	0.165950i	−1.07439	−	0.165950i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−0.0763807	+	1.99267i	−0.0763807	+	1.99267i	0.0383027	+	0.999266i	$0.487805\pi$
$421$	−0.0763807	+	1.99267i	−0.0763807	+	1.99267i	−0.114683	+	0.993402i	$0.536585\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.21211	+	0.383595i	−1.21211	+	0.383595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.114683	−	0.993402i	$-0.463415\pi$
$431$		0			0		−0.114683	+	0.993402i	$0.536585\pi$
$432$	−0.997066	−	0.0765493i	−0.997066	−	0.0765493i
$433$	−0.293769	−	0.242270i	−0.293769	−	0.242270i	0.477720	−	0.878512i	$-0.341463\pi$
$433$	−0.293769	−	0.242270i	−0.293769	−	0.242270i	−0.771489	+	0.636242i	$0.780488\pi$
$434$		0			0
$435$		0			0
$436$	−1.04219	+	0.619585i	−1.04219	+	0.619585i
$437$		0			0
$438$		0			0
$439$	−0.635679	−	0.713287i	−0.635679	−	0.713287i	0.338017	−	0.941140i	$-0.390244\pi$
$439$	−0.635679	−	0.713287i	−0.635679	−	0.713287i	−0.973695	+	0.227854i	$0.926829\pi$
$440$		0			0
$441$	−0.0165897	+	0.0855413i	−0.0165897	+	0.0855413i
$442$		0			0
$443$		0			0		0.771489	−	0.636242i	$-0.219512\pi$
$443$		0			0		−0.771489	+	0.636242i	$0.780488\pi$
$444$	1.46600	+	1.03116i	1.46600	+	1.03116i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.737111	+	0.607891i	−0.737111	+	0.607891i
$449$		0			0		−0.896166	−	0.443720i	$-0.853659\pi$
$449$		0			0		0.896166	+	0.443720i	$0.146341\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.92457	+	0.450366i	−1.92457	+	0.450366i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.17867	−	0.829059i	1.17867	−	0.829059i	0.190391	−	0.981708i	$-0.439024\pi$
$457$	1.17867	−	0.829059i	1.17867	−	0.829059i	0.988280	+	0.152649i	$0.0487805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.409069	−	0.912504i	$-0.365854\pi$
$461$		0			0		−0.409069	+	0.912504i	$0.634146\pi$
$462$		0			0
$463$	−1.19248	−	0.590436i	−1.19248	−	0.590436i	−0.264982	−	0.964253i	$-0.585366\pi$
$463$	−1.19248	−	0.590436i	−1.19248	−	0.590436i	−0.927502	+	0.373817i	$0.878049\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.973695	−	0.227854i	$-0.926829\pi$
$467$		0			0		0.973695	+	0.227854i	$0.0731707\pi$
$468$	−0.783304	−	1.20956i	−0.783304	−	1.20956i
$469$	0.725011	−	0.292206i	0.725011	−	0.292206i
$470$		0			0
$471$	0.128711	−	0.663668i	0.128711	−	0.663668i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.216791	−	0.483593i	0.216791	−	0.483593i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.997066	−	0.0765493i	$-0.975610\pi$
$479$		0			0		0.997066	+	0.0765493i	$0.0243902\pi$
$480$		0			0
$481$	0.0989293	+	2.58093i	0.0989293	+	2.58093i
$482$		0			0
$483$		0			0
$484$	0.477720	+	0.878512i	0.477720	+	0.878512i
$485$		0			0
$486$		0			0
$487$	1.29141	+	1.24286i	1.29141	+	1.24286i	0.953396	+	0.301721i	$0.0975610\pi$
$487$	1.29141	+	1.24286i	1.29141	+	1.24286i	0.338017	+	0.941140i	$0.390244\pi$
$488$		0			0
$489$	0.128711	+	0.358369i	0.128711	+	0.358369i
$490$		0			0
$491$		0			0		0.817929	−	0.575319i	$-0.195122\pi$
$491$		0			0		−0.817929	+	0.575319i	$0.804878\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.0626579	+	1.63466i	0.0626579	+	1.63466i
$497$		0			0
$498$		0			0
$499$	1.29565	−	1.45383i	1.29565	−	1.45383i	0.477720	−	0.878512i	$-0.341463\pi$
$499$	1.29565	−	1.45383i	1.29565	−	1.45383i	0.817929	−	0.575319i	$-0.195122\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.477720	−	0.878512i	$-0.341463\pi$
$503$		0			0		−0.477720	+	0.878512i	$0.658537\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.363911	−	1.01324i	0.363911	−	1.01324i
$508$	1.02658	+	1.15192i	1.02658	+	1.15192i
$509$		0			0		−0.665326	−	0.746553i	$-0.731707\pi$
$509$		0			0		0.665326	+	0.746553i	$0.268293\pi$
$510$		0			0
$511$	0.947509	+	1.24301i	0.947509	+	1.24301i
$512$		0			0
$513$	0.491542	+	0.198109i	0.491542	+	0.198109i
$514$		0			0
$515$		0			0
$516$	0.455540	+	0.270821i	0.455540	+	0.270821i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.896166	−	0.443720i	$-0.853659\pi$
$521$		0			0		0.896166	+	0.443720i	$0.146341\pi$
$522$		0			0
$523$	−0.390840	−	0.718744i	−0.390840	−	0.718744i	0.606225	−	0.795293i	$-0.292683\pi$
$523$	−0.390840	−	0.718744i	−0.390840	−	0.718744i	−0.997066	+	0.0765493i	$0.975610\pi$
$524$		0			0
$525$	−0.253174	+	0.921286i	−0.253174	+	0.921286i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.409069	−	0.912504i	−0.409069	−	0.912504i
$530$		0			0
$531$		0			0
$532$	0.469639	−	0.189281i	0.469639	−	0.189281i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.808549	−	1.80362i	−0.808549	−	1.80362i	−0.543568	−	0.839365i	$-0.682927\pi$
$541$	−0.808549	−	1.80362i	−0.808549	−	1.80362i	−0.264982	−	0.964253i	$-0.585366\pi$
$542$		0			0
$543$	−1.33065	+	1.49311i	−1.33065	+	1.49311i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.63106	−	1.14726i	−1.63106	−	1.14726i	−0.859570	−	0.511019i	$-0.829268\pi$
$547$	−1.63106	−	1.14726i	−1.63106	−	1.14726i	−0.771489	−	0.636242i	$-0.780488\pi$
$548$		0			0
$549$	0.352598	+	1.28309i	0.352598	+	1.28309i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.119121	−	1.03184i	0.119121	−	1.03184i
$554$		0			0
$555$		0			0
$556$	0.590931	+	0.912504i	0.590931	+	0.912504i
$557$		0			0		−0.927502	−	0.373817i	$-0.878049\pi$
$557$		0			0		0.927502	+	0.373817i	$0.121951\pi$
$558$		0			0
$559$	0.0875837	+	0.758661i	0.0875837	+	0.758661i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.771489	−	0.636242i	$-0.780488\pi$
$563$		0			0		0.771489	+	0.636242i	$0.219512\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.930307	−	0.217700i	−0.930307	−	0.217700i
$568$		0			0
$569$		0			0		0.973695	−	0.227854i	$-0.0731707\pi$
$569$		0			0		−0.973695	+	0.227854i	$0.926829\pi$
$570$		0			0
$571$	0.703246	−	0.418083i	0.703246	−	0.418083i	−0.114683	−	0.993402i	$-0.536585\pi$
$571$	0.703246	−	0.418083i	0.703246	−	0.418083i	0.817929	+	0.575319i	$0.195122\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.606225	+	0.795293i	0.606225	+	0.795293i
$577$	−0.370766	−	0.0867626i	−0.370766	−	0.0867626i	0.0383027	−	0.999266i	$-0.487805\pi$
$577$	−0.370766	−	0.0867626i	−0.370766	−	0.0867626i	−0.409069	+	0.912504i	$0.634146\pi$
$578$		0			0
$579$	−0.0416402	−	0.0643000i	−0.0416402	−	0.0643000i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.927502	−	0.373817i	$-0.121951\pi$
$587$		0			0		−0.927502	+	0.373817i	$0.878049\pi$
$588$	0.0748987	−	0.0445277i	0.0748987	−	0.0445277i
$589$	0.229724	−	0.835954i	0.229724	−	0.835954i
$590$		0			0
$591$		0			0
$592$	−0.205551	−	1.78051i	−0.205551	−	1.78051i
$593$		0			0		−0.606225	−	0.795293i	$-0.707317\pi$
$593$		0			0		0.606225	+	0.795293i	$0.292683\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.635679	+	1.16899i	−0.635679	+	1.16899i
$598$		0			0
$599$		0			0		0.988280	−	0.152649i	$-0.0487805\pi$
$599$		0			0		−0.988280	+	0.152649i	$0.951220\pi$
$600$		0			0
$601$	−0.218678	−	1.89421i	−0.218678	−	1.89421i	−0.409069	−	0.912504i	$-0.634146\pi$
$601$	−0.218678	−	1.89421i	−0.218678	−	1.89421i	0.190391	−	0.981708i	$-0.439024\pi$
$602$		0			0
$603$	−0.276544	−	0.769982i	−0.276544	−	0.769982i
$604$	1.61669	+	1.13715i	1.61669	+	1.13715i
$605$		0			0
$606$		0			0
$607$	−1.52490	+	0.235535i	−1.52490	+	0.235535i	−0.859570	−	0.511019i	$-0.829268\pi$
$607$	−1.52490	+	0.235535i	−1.52490	+	0.235535i	−0.665326	+	0.746553i	$0.731707\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.363036	−	1.87191i	0.363036	−	1.87191i	−0.114683	−	0.993402i	$-0.536585\pi$
$613$	0.363036	−	1.87191i	0.363036	−	1.87191i	0.477720	−	0.878512i	$-0.341463\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.953396	−	0.301721i	$-0.0975610\pi$
$617$		0			0		−0.953396	+	0.301721i	$0.902439\pi$
$618$		0			0
$619$	1.19824	+	0.185080i	1.19824	+	0.185080i	0.720522	−	0.693433i	$-0.243902\pi$
$619$	1.19824	+	0.185080i	1.19824	+	0.185080i	0.477720	+	0.878512i	$0.341463\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.381850	+	1.38953i	−0.381850	+	1.38953i
$625$	0.338017	−	0.941140i	0.338017	−	0.941140i
$626$		0			0
$627$		0			0
$628$	−0.581098	+	0.345466i	−0.581098	+	0.345466i
$629$		0			0
$630$		0			0
$631$	0.856232	−	1.57458i	0.856232	−	1.57458i	0.0383027	−	0.999266i	$-0.487805\pi$
$631$	0.856232	−	1.57458i	0.856232	−	1.57458i	0.817929	−	0.575319i	$-0.195122\pi$
$632$		0			0
$633$	0.491542	+	1.78869i	0.491542	+	1.78869i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.116462	+	0.0469385i	0.116462	+	0.0469385i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.988280	−	0.152649i	$-0.951220\pi$
$641$		0			0		0.988280	+	0.152649i	$0.0487805\pi$
$642$		0			0
$643$	−0.0745904	+	1.94596i	−0.0745904	+	1.94596i	0.190391	+	0.981708i	$0.439024\pi$
$643$	−0.0745904	+	1.94596i	−0.0745904	+	1.94596i	−0.264982	+	0.964253i	$0.585366\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.988280	−	0.152649i	$-0.951220\pi$
$647$		0			0		0.988280	+	0.152649i	$0.0487805\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.179246	+	1.55265i	−0.179246	+	1.55265i
$652$	0.181907	−	0.334522i	0.181907	−	0.334522i
$653$		0			0		−0.338017	−	0.941140i	$-0.609756\pi$
$653$		0			0		0.338017	+	0.941140i	$0.390244\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.33802	−	0.941140i	1.33802	−	0.941140i
$658$		0			0
$659$		0			0		0.409069	−	0.912504i	$-0.365854\pi$
$659$		0			0		−0.409069	+	0.912504i	$0.634146\pi$
$660$		0			0
$661$	−0.00878538	+	0.0761000i	−0.00878538	+	0.0761000i	−0.997066	−	0.0765493i	$-0.975610\pi$
$661$	−0.00878538	+	0.0761000i	−0.00878538	+	0.0761000i	0.988280	+	0.152649i	$0.0487805\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.26864	−	1.42352i	−1.26864	−	1.42352i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.14379	+	1.28343i	1.14379	+	1.28343i	0.953396	+	0.301721i	$0.0975610\pi$
$673$	1.14379	+	1.28343i	1.14379	+	1.28343i	0.190391	+	0.981708i	$0.439024\pi$
$674$		0			0
$675$	0.953396	+	0.301721i	0.953396	+	0.301721i
$676$	−0.998554	+	0.402454i	−0.998554	+	0.402454i
$677$		0			0		−0.190391	−	0.981708i	$-0.560976\pi$
$677$		0			0		0.190391	+	0.981708i	$0.439024\pi$
$678$		0			0
$679$	0.0896458	+	0.199972i	0.0896458	+	0.199972i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.665326	−	0.746553i	$-0.268293\pi$
$683$		0			0		−0.665326	+	0.746553i	$0.731707\pi$
$684$	−0.179136	−	0.498769i	−0.179136	−	0.498769i
$685$		0			0
$686$		0			0
$687$	1.72052	+	0.693433i	1.72052	+	0.693433i
$688$	−0.100900	−	0.520269i	−0.100900	−	0.520269i
$689$		0			0
$690$		0			0
$691$	−1.63902	+	0.974407i	−1.63902	+	0.974407i	−0.665326	+	0.746553i	$0.731707\pi$
$691$	−1.63902	+	0.974407i	−1.63902	+	0.974407i	−0.973695	+	0.227854i	$0.926829\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.856232	−	0.423948i	0.856232	−	0.423948i
$701$		0			0		0.0383027	−	0.999266i	$-0.487805\pi$
$701$		0			0		−0.0383027	+	0.999266i	$0.512195\pi$
$702$		0			0
$703$	−0.180847	+	0.932494i	−0.180847	+	0.932494i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.838713	−	1.29512i	0.838713	−	1.29512i	−0.114683	−	0.993402i	$-0.536585\pi$
$709$	0.838713	−	1.29512i	0.838713	−	1.29512i	0.953396	−	0.301721i	$-0.0975610\pi$
$710$		0			0
$711$	−1.07439	−	0.165950i	−1.07439	−	0.165950i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.665326	−	0.746553i	$-0.731707\pi$
$719$		0			0		0.665326	+	0.746553i	$0.268293\pi$
$720$		0			0
$721$	1.64385	−	0.662533i	1.64385	−	0.662533i
$722$		0			0
$723$	−0.449783	+	0.504695i	−0.449783	+	0.504695i
$724$	2.00000			2.00000
$725$		0			0
$726$		0			0
$727$	−0.00878538	−	0.0761000i	−0.00878538	−	0.0761000i	0.988280	−	0.152649i	$-0.0487805\pi$
$727$	−0.00878538	−	0.0761000i	−0.00878538	−	0.0761000i	−0.997066	+	0.0765493i	$0.975610\pi$
$728$		0			0
$729$	−0.264982	+	0.964253i	−0.264982	+	0.964253i
$730$		0			0
$731$		0			0
$732$	0.723299	−	1.11690i	0.723299	−	1.11690i
$733$	−0.737111	+	0.607891i	−0.737111	+	0.607891i	−0.927502	−	0.373817i	$-0.878049\pi$
$733$	−0.737111	+	0.607891i	−0.737111	+	0.607891i	0.190391	+	0.981708i	$0.439024\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.114683	−	0.993402i	−0.114683	−	0.993402i
$739$	−0.114683	−	0.993402i	−0.114683	−	0.993402i
$740$		0			0
$741$	0.415122	−	0.641023i	0.415122	−	0.641023i
$742$		0			0
$743$		0			0		−0.409069	−	0.912504i	$-0.634146\pi$
$743$		0			0		0.409069	+	0.912504i	$0.365854\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.212738	+	1.84277i	0.212738	+	1.84277i	0.477720	+	0.878512i	$0.341463\pi$
$751$	0.212738	+	1.84277i	0.212738	+	1.84277i	−0.264982	+	0.964253i	$0.585366\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.456432	+	0.839365i	0.456432	+	0.839365i
$757$	1.59451	−	0.642644i	1.59451	−	0.642644i	0.606225	−	0.795293i	$-0.292683\pi$
$757$	1.59451	−	0.642644i	1.59451	−	0.642644i	0.988280	+	0.152649i	$0.0487805\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.859570	−	0.511019i	$-0.829268\pi$
$761$		0			0		0.859570	+	0.511019i	$0.170732\pi$
$762$		0			0
$763$	1.03814	+	0.514016i	1.03814	+	0.514016i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.190391	−	0.981708i	0.190391	−	0.981708i
$769$	−0.889200	+	1.37308i	−0.889200	+	1.37308i	0.0383027	+	0.999266i	$0.487805\pi$
$769$	−0.889200	+	1.37308i	−0.889200	+	1.37308i	−0.927502	+	0.373817i	$0.878049\pi$
$770$		0			0
$771$		0			0
$772$	−0.0202990	+	0.0738671i	−0.0202990	+	0.0738671i
$773$		0			0		−0.997066	−	0.0765493i	$-0.975610\pi$
$773$		0			0		0.997066	+	0.0765493i	$0.0243902\pi$
$774$		0			0
$775$	0.311453	−	1.60594i	0.311453	−	1.60594i
$776$		0			0
$777$	0.0655921	−	1.71121i	0.0655921	−	1.71121i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.0830743	−	0.0262905i	−0.0830743	−	0.0262905i
$785$		0			0
$786$		0			0
$787$	−1.04219	+	0.619585i	−1.04219	+	0.619585i	−0.927502	−	0.373817i	$-0.878049\pi$
$787$	−1.04219	+	0.619585i	−1.04219	+	0.619585i	−0.114683	+	0.993402i	$0.536585\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.91190	−	0.146785i	1.91190	−	0.146785i
$794$		0			0
$795$		0			0
$796$	1.29565	−	0.303194i	1.29565	−	0.303194i
$797$		0			0		−0.720522	−	0.693433i	$-0.756098\pi$
$797$		0			0		0.720522	+	0.693433i	$0.243902\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.390840	+	0.718744i	−0.390840	+	0.718744i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.665326	−	0.746553i	$-0.731707\pi$
$809$		0			0		0.665326	+	0.746553i	$0.268293\pi$
$810$		0			0
$811$	1.29141	+	1.24286i	1.29141	+	1.24286i	0.953396	+	0.301721i	$0.0975610\pi$
$811$	1.29141	+	1.24286i	1.29141	+	1.24286i	0.338017	+	0.941140i	$0.390244\pi$
$812$		0			0
$813$	0.0607780	+	0.221168i	0.0607780	+	0.221168i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.0322101	+	0.279008i	−0.0322101	+	0.279008i
$818$		0			0
$819$	−0.563218	+	1.25636i	−0.563218	+	1.25636i
$820$		0			0
$821$		0			0		0.817929	−	0.575319i	$-0.195122\pi$
$821$		0			0		−0.817929	+	0.575319i	$0.804878\pi$
$822$		0			0
$823$	0.341244	+	0.168961i	0.341244	+	0.168961i	0.606225	−	0.795293i	$-0.292683\pi$
$823$	0.341244	+	0.168961i	0.341244	+	0.168961i	−0.264982	+	0.964253i	$0.585366\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.114683	−	0.993402i	$-0.463415\pi$
$827$		0			0		−0.114683	+	0.993402i	$0.536585\pi$
$828$		0			0
$829$	−0.0591003	+	0.0487396i	−0.0591003	+	0.0487396i	−0.665326	−	0.746553i	$-0.731707\pi$
$829$	−0.0591003	+	0.0487396i	−0.0591003	+	0.0487396i	0.606225	+	0.795293i	$0.292683\pi$
$830$		0			0
$831$	−1.97076	−	0.304403i	−1.97076	−	0.304403i
$832$	1.29141	−	0.639419i	1.29141	−	0.639419i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.61669	+	0.249713i	1.61669	+	0.249713i
$838$		0			0
$839$		0			0		−0.859570	−	0.511019i	$-0.829268\pi$
$839$		0			0		0.859570	+	0.511019i	$0.170732\pi$
$840$		0			0
$841$	−0.927502	−	0.373817i	−0.927502	−	0.373817i
$842$		0			0
$843$		0			0
$844$	1.00832	−	1.55703i	1.00832	−	1.55703i
$845$		0			0
$846$		0			0
$847$	0.456432	−	0.839365i	0.456432	−	0.839365i
$848$		0			0
$849$	−1.18056	−	0.276261i	−1.18056	−	0.276261i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.228511	−	0.636242i	0.228511	−	0.636242i	−0.771489	−	0.636242i	$-0.780488\pi$
$853$	0.228511	−	0.636242i	0.228511	−	0.636242i	1.00000			$0$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.997066	−	0.0765493i	$-0.0243902\pi$
$857$		0			0		−0.997066	+	0.0765493i	$0.975610\pi$
$858$		0			0
$859$	1.77133	+	0.273598i	1.77133	+	0.273598i	0.953396	−	0.301721i	$-0.0975610\pi$
$859$	1.77133	+	0.273598i	1.77133	+	0.273598i	0.817929	+	0.575319i	$0.195122\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.927502	−	0.373817i	$-0.878049\pi$
$863$		0			0		0.927502	+	0.373817i	$0.121951\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.771489	+	0.636242i	−0.771489	+	0.636242i
$868$	1.27839	−	0.899203i	1.27839	−	0.899203i
$869$		0			0
$870$		0			0
$871$	−1.16515	+	0.179969i	−1.16515	+	0.179969i
$872$		0			0
$873$	0.212738	−	0.0857412i	0.212738	−	0.0857412i
$874$		0			0
$875$		0			0
$876$	−1.59283	−	0.372736i	−1.59283	−	0.372736i
$877$	0.228694	+	1.98097i	0.228694	+	1.98097i	0.190391	+	0.981708i	$0.439024\pi$
$877$	0.228694	+	1.98097i	0.228694	+	1.98097i	0.0383027	+	0.999266i	$0.487805\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.477720	−	0.878512i	$-0.341463\pi$
$881$		0			0		−0.477720	+	0.878512i	$0.658537\pi$
$882$		0			0
$883$	0.223333	−	0.0522620i	0.223333	−	0.0522620i	−0.114683	−	0.993402i	$-0.536585\pi$
$883$	0.223333	−	0.0522620i	0.223333	−	0.0522620i	0.338017	+	0.941140i	$0.390244\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.953396	−	0.301721i	$-0.902439\pi$
$887$		0			0		0.953396	+	0.301721i	$0.0975610\pi$
$888$		0			0
$889$	0.390642	−	1.42152i	0.390642	−	1.42152i
$890$		0			0
$891$		0			0
$892$	−0.218678	+	1.89421i	−0.218678	+	1.89421i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.409069	−	0.912504i	−0.409069	−	0.912504i
$901$		0			0
$902$		0			0
$903$	−0.0193945	−	0.505976i	−0.0193945	−	0.505976i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.703246	−	0.418083i	0.703246	−	0.418083i	−0.114683	−	0.993402i	$-0.536585\pi$
$907$	0.703246	−	0.418083i	0.703246	−	0.418083i	0.817929	+	0.575319i	$0.195122\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.973695	−	0.227854i	$-0.926829\pi$
$911$		0			0		0.973695	+	0.227854i	$0.0731707\pi$
$912$	−0.253174	+	0.465579i	−0.253174	+	0.465579i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.627023	−	1.74582i	−0.627023	−	1.74582i
$917$		0			0
$918$		0			0
$919$	−0.0436694	−	0.378270i	−0.0436694	−	0.378270i	−0.997066	−	0.0765493i	$-0.975610\pi$
$919$	−0.0436694	−	0.378270i	−0.0436694	−	0.378270i	0.953396	−	0.301721i	$-0.0975610\pi$
$920$		0			0
$921$	0.758824	+	0.305834i	0.758824	+	0.305834i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.205551	+	1.78051i	−0.205551	+	1.78051i
$926$		0			0
$927$	−0.627023	−	1.74582i	−0.627023	−	1.74582i
$928$		0			0
$929$		0			0		−0.264982	−	0.964253i	$-0.585366\pi$
$929$		0			0		0.264982	+	0.964253i	$0.414634\pi$
$930$		0			0
$931$	0.0377707	+	0.0265673i	0.0377707	+	0.0265673i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.815737	+	1.81965i	0.815737	+	1.81965i	0.477720	+	0.878512i	$0.341463\pi$
$937$	0.815737	+	1.81965i	0.815737	+	1.81965i	0.338017	+	0.941140i	$0.390244\pi$
$938$		0			0
$939$	1.08656	−	0.537988i	1.08656	−	0.537988i
$940$		0			0
$941$		0			0		−0.264982	−	0.964253i	$-0.585366\pi$
$941$		0			0		0.264982	+	0.964253i	$0.414634\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0383027	−	0.999266i	$-0.512195\pi$
$947$		0			0		0.0383027	+	0.999266i	$0.487805\pi$
$948$	0.590931	+	0.912504i	0.590931	+	0.912504i
$949$	−0.964315	−	2.15108i	−0.964315	−	2.15108i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.264982	−	0.964253i	$-0.414634\pi$
$953$		0			0		−0.264982	+	0.964253i	$0.585366\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0641967	−	1.67480i	0.0641967	−	1.67480i
$962$		0			0
$963$		0			0
$964$	0.676034			0.676034
$965$		0			0
$966$		0			0
$967$	0.230840	+	0.302833i	0.230840	+	0.302833i	0.896166	−	0.443720i	$-0.146341\pi$
$967$	0.230840	+	0.302833i	0.230840	+	0.302833i	−0.665326	+	0.746553i	$0.731707\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.338017	−	0.941140i	$-0.390244\pi$
$971$		0			0		−0.338017	+	0.941140i	$0.609756\pi$
$972$	0.896166	−	0.443720i	0.896166	−	0.443720i
$973$	0.424896	−	0.947810i	0.424896	−	0.947810i
$974$		0			0
$975$	0.688415	−	1.26597i	0.688415	−	1.26597i
$976$	−1.31506	+	0.203123i	−1.31506	+	0.203123i
$977$		0			0		0.606225	−	0.795293i	$-0.292683\pi$
$977$		0			0		−0.606225	+	0.795293i	$0.707317\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.579212	−	1.06515i	0.579212	−	1.06515i
$982$		0			0
$983$		0			0		0.264982	−	0.964253i	$-0.414634\pi$
$983$		0			0		−0.264982	+	0.964253i	$0.585366\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.754749	+	0.116578i	−0.754749	+	0.116578i
$989$		0			0
$990$		0			0
$991$	1.44104	+	1.38687i	1.44104	+	1.38687i	0.720522	+	0.693433i	$0.243902\pi$
$991$	1.44104	+	1.38687i	1.44104	+	1.38687i	0.720522	+	0.693433i	$0.243902\pi$
$992$		0			0
$993$	0.408861	−	1.48782i	0.408861	−	1.48782i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0745904	−	1.94596i	−0.0745904	−	1.94596i	−0.264982	−	0.964253i	$-0.585366\pi$
$997$	−0.0745904	−	1.94596i	−0.0745904	−	1.94596i	0.190391	−	0.981708i	$-0.439024\pi$
$998$		0			0
$999$	−1.78707	−	0.137202i	−1.78707	−	0.137202i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2217.1.p.a.692.1	✓	40
3.2	odd	2	CM	2217.1.p.a.692.1	✓	40
739.283	even	41	inner	2217.1.p.a.1022.1	yes	40
2217.1022	odd	82	inner	2217.1.p.a.1022.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2217.1.p.a.692.1	✓	40	1.1	even	1	trivial
2217.1.p.a.692.1	✓	40	3.2	odd	2	CM
2217.1.p.a.1022.1	yes	40	739.283	even	41	inner
2217.1.p.a.1022.1	yes	40	2217.1022	odd	82	inner

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:	\( N \)	\(=\)	\( 2217 = 3 \cdot 739 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2217.p (of order \(82\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(0.953396 + 0.301721i) q^{3} +(-0.409069 - 0.912504i) q^{4} +(-0.519346 + 0.801963i) q^{7} +(0.817929 + 0.575319i) q^{9} +O(q^{10})\)	\(q+(0.953396 + 0.301721i) q^{3} +(-0.409069 - 0.912504i) q^{4} +(-0.519346 + 0.801963i) q^{7} +(0.817929 + 0.575319i) q^{9} +(-0.114683 - 0.993402i) q^{12} +(1.03830 - 0.999266i) q^{13} +(-0.665326 + 0.746553i) q^{16} +(0.455540 - 0.270821i) q^{19} +(-0.737111 + 0.607891i) q^{21} +(0.817929 - 0.575319i) q^{25} +(0.606225 + 0.795293i) q^{27} +(0.944242 + 0.145847i) q^{28} +(1.17867 - 1.13436i) q^{31} +(0.190391 - 0.981708i) q^{36} +(-1.19248 + 1.33807i) q^{37} +(1.29141 - 0.639419i) q^{39} +(-0.321277 + 0.421476i) q^{43} +(-0.859570 + 0.511019i) q^{48} +(0.0356442 + 0.0795111i) q^{49} +(-1.33657 - 0.538687i) q^{52} +(0.516023 - 0.120754i) q^{57} +(1.02658 + 0.846617i) q^{61} +(-0.886173 + 0.357160i) q^{63} +(0.953396 + 0.301721i) q^{64} +(-0.669178 - 0.470690i) q^{67} +(0.552948 - 1.53957i) q^{73} +(0.953396 - 0.301721i) q^{75} +(-0.433472 - 0.304898i) q^{76} +(-0.974253 + 0.482383i) q^{79} +(0.338017 + 0.941140i) q^{81} +(0.856232 + 0.423948i) q^{84} +(0.262136 + 1.35165i) q^{91} +(1.46600 - 0.725863i) q^{93} +(0.124676 - 0.192523i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) 40 * q - q^3 - q^4 - 2 * q^7 - q^9	\( 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 39 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 4 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) 40 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 39 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - 2 * q^39 - 2 * q^43 - q^48 - 3 * q^49 - 2 * q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 4 * q^91 - 2 * q^93 - 2 * q^97

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