LMFDB - Embedded newform 2872.1.b.e.717.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2872,1,Mod(717,2872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2872, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("2872.717");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2872 = 2^{3} \cdot 359 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2872.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.43331471628$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\Q(\zeta_{38})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} + \cdots + 1 $ 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{38}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{38} - \cdots)$

Embedding invariants

Embedding label			717.9
Root			$-0.245485 + 0.969400i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0825793 - 0.996584i) q^{2} -1.83155i q^{3} +(-0.986361 - 0.164595i) q^{4} +1.22843i q^{5} +(-1.82529 - 0.151248i) q^{6} +(-0.245485 + 0.969400i) q^{8} -2.35456 q^{9} +O(q^{10})$	\(q+(0.0825793 - 0.996584i) q^{2} -1.83155i q^{3} +(-0.986361 - 0.164595i) q^{4} +1.22843i q^{5} +(-1.82529 - 0.151248i) q^{6} +(-0.245485 + 0.969400i) q^{8} -2.35456 q^{9} +(1.22423 + 0.101443i) q^{10} -1.67433i q^{11} +(-0.301463 + 1.80657i) q^{12} +2.24992 q^{15} +(0.945817 + 0.324699i) q^{16} -1.89163 q^{17} +(-0.194438 + 2.34652i) q^{18} +(0.202192 - 1.21167i) q^{20} +(-1.66861 - 0.138265i) q^{22} -1.97272 q^{23} +(1.77550 + 0.449618i) q^{24} -0.509029 q^{25} +2.48095i q^{27} +(0.185797 - 2.24223i) q^{30} +(0.401695 - 0.915773i) q^{32} -3.06662 q^{33} +(-0.156210 + 1.88517i) q^{34} +(2.32245 + 0.387548i) q^{36} +0.951895i q^{37} +(-1.19084 - 0.301561i) q^{40} -1.75895 q^{41} +(-0.275586 + 1.65150i) q^{44} -2.89241i q^{45} +(-0.162906 + 1.96598i) q^{46} +1.35456 q^{47} +(0.594702 - 1.73231i) q^{48} +1.00000 q^{49} +(-0.0420353 + 0.507290i) q^{50} +3.46462i q^{51} +(2.47247 + 0.204875i) q^{54} +2.05679 q^{55} +(-2.21923 - 0.370324i) q^{60} +(-0.879474 - 0.475947i) q^{64} +(-0.253239 + 3.05614i) q^{66} +(1.86584 + 0.311353i) q^{68} +3.61313i q^{69} +(0.578011 - 2.28251i) q^{72} -0.165159 q^{73} +(0.948644 + 0.0786068i) q^{74} +0.932310i q^{75} -1.57828 q^{79} +(-0.398869 + 1.16187i) q^{80} +2.18940 q^{81} +(-0.145253 + 1.75294i) q^{82} -2.32373i q^{85} +(1.62310 + 0.411024i) q^{88} +(-2.88253 - 0.238853i) q^{90} +(1.94582 + 0.324699i) q^{92} +(0.111859 - 1.34994i) q^{94} +(-1.67728 - 0.735724i) q^{96} +(0.0825793 - 0.996584i) q^{98} +3.94232i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) $ 18 * q + q^2 - q^4 + q^8 - 20 * q^9	$ 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9} - q^{16} + 2 q^{17} + q^{18} - 2 q^{23} - 20 q^{25} + q^{32} - 2 q^{34} - q^{36} - 2 q^{41} + 2 q^{46} + 2 q^{47} + 18 q^{49} + q^{50} - q^{64} + 2 q^{68} + q^{72} - 2 q^{73} + 2 q^{79} + 18 q^{81} + 2 q^{82} - 19 q^{90} + 17 q^{92} - 2 q^{94} - 19 q^{96} + q^{98}+O(q^{100}) $ 18 * q + q^2 - q^4 + q^8 - 20 * q^9 - q^16 + 2 * q^17 + q^18 - 2 * q^23 - 20 * q^25 + q^32 - 2 * q^34 - q^36 - 2 * q^41 + 2 * q^46 + 2 * q^47 + 18 * q^49 + q^50 - q^64 + 2 * q^68 + q^72 - 2 * q^73 + 2 * q^79 + 18 * q^81 + 2 * q^82 - 19 * q^90 + 17 * q^92 - 2 * q^94 - 19 * q^96 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2872\mathbb{Z}\right)^\times$.

$n$	$719$	$1437$	$2161$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$2$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$3$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$3$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$4$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$5$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$5$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$6$	−1.82529	−	0.151248i	−1.82529	−	0.151248i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$9$	−2.35456			−2.35456
$10$	1.22423	+	0.101443i	1.22423	+	0.101443i
$11$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$11$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$12$	−0.301463	+	1.80657i	−0.301463	+	1.80657i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	2.24992			2.24992
$16$	0.945817	+	0.324699i	0.945817	+	0.324699i
$17$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$17$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$18$	−0.194438	+	2.34652i	−0.194438	+	2.34652i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	0.202192	−	1.21167i	0.202192	−	1.21167i
$21$		0			0
$22$	−1.66861	−	0.138265i	−1.66861	−	0.138265i
$23$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$23$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$24$	1.77550	+	0.449618i	1.77550	+	0.449618i
$25$	−0.509029			−0.509029
$26$		0			0
$27$			2.48095i			2.48095i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	0.185797	−	2.24223i	0.185797	−	2.24223i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.401695	−	0.915773i	0.401695	−	0.915773i
$33$	−3.06662			−3.06662
$34$	−0.156210	+	1.88517i	−0.156210	+	1.88517i
$35$		0			0
$36$	2.32245	+	0.387548i	2.32245	+	0.387548i
$37$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$37$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$38$		0			0
$39$		0			0
$40$	−1.19084	−	0.301561i	−1.19084	−	0.301561i
$41$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$41$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.275586	+	1.65150i	−0.275586	+	1.65150i
$45$		−	2.89241i		−	2.89241i
$46$	−0.162906	+	1.96598i	−0.162906	+	1.96598i
$47$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$47$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$48$	0.594702	−	1.73231i	0.594702	−	1.73231i
$49$	1.00000			1.00000
$50$	−0.0420353	+	0.507290i	−0.0420353	+	0.507290i
$51$			3.46462i			3.46462i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	2.47247	+	0.204875i	2.47247	+	0.204875i
$55$	2.05679			2.05679
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−2.21923	−	0.370324i	−2.21923	−	0.370324i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$65$		0			0
$66$	−0.253239	+	3.05614i	−0.253239	+	3.05614i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	1.86584	+	0.311353i	1.86584	+	0.311353i
$69$			3.61313i			3.61313i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.578011	−	2.28251i	0.578011	−	2.28251i
$73$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$73$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$74$	0.948644	+	0.0786068i	0.948644	+	0.0786068i
$75$			0.932310i			0.932310i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$79$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$80$	−0.398869	+	1.16187i	−0.398869	+	1.16187i
$81$	2.18940			2.18940
$82$	−0.145253	+	1.75294i	−0.145253	+	1.75294i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	2.32373i		−	2.32373i
$86$		0			0
$87$		0			0
$88$	1.62310	+	0.411024i	1.62310	+	0.411024i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−2.88253	−	0.238853i	−2.88253	−	0.238853i
$91$		0			0
$92$	1.94582	+	0.324699i	1.94582	+	0.324699i
$93$		0			0
$94$	0.111859	−	1.34994i	0.111859	−	1.34994i
$95$		0			0
$96$	−1.67728	−	0.735724i	−1.67728	−	0.735724i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$99$			3.94232i			3.94232i
$100$	0.502087	+	0.0837834i	0.502087	+	0.0837834i
$101$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$101$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$102$	3.45278	+	0.286106i	3.45278	+	0.286106i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$107$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$108$	0.408350	−	2.44711i	0.408350	−	2.44711i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.169849	−	2.04977i	0.169849	−	2.04977i
$111$	1.74344			1.74344
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	2.42334i		−	2.42334i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.552322	+	2.18107i	−0.552322	+	2.18107i
$121$	−1.80339			−1.80339
$122$		0			0
$123$			3.22159i			3.22159i
$124$		0			0
$125$			0.603121i			0.603121i
$126$		0			0
$127$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$127$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$128$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$129$		0			0
$130$		0			0
$131$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$131$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$132$	3.02479	+	0.504749i	3.02479	+	0.504749i
$133$		0			0
$134$		0			0
$135$	−3.04766			−3.04766
$136$	0.464369	−	1.83375i	0.464369	−	1.83375i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	3.60079	+	0.298370i	3.60079	+	0.298370i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	2.48095i		−	2.48095i
$142$		0			0
$143$		0			0
$144$	−2.22699	−	0.764525i	−2.22699	−	0.764525i
$145$		0			0
$146$	−0.0136387	+	0.164595i	−0.0136387	+	0.164595i
$147$		−	1.83155i		−	1.83155i
$148$	0.156677	−	0.938912i	0.156677	−	0.938912i
$149$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$149$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$150$	0.929126	+	0.0769896i	0.929126	+	0.0769896i
$151$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$151$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$152$		0			0
$153$	4.45397			4.45397
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−0.130333	+	1.57289i	−0.130333	+	1.57289i
$159$		0			0
$160$	1.12496	+	0.493453i	1.12496	+	0.493453i
$161$		0			0
$162$	0.180800	−	2.18193i	0.180800	−	2.18193i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	1.73496	+	0.289513i	1.73496	+	0.289513i
$165$		−	3.76711i		−	3.76711i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−2.31580	−	0.191892i	−2.31580	−	0.191892i
$171$		0			0
$172$		0			0
$173$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$173$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$174$		0			0
$175$		0			0
$176$	0.543655	−	1.58361i	0.543655	−	1.58361i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−0.476074	+	2.85296i	−0.476074	+	2.85296i
$181$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$181$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$182$		0			0
$183$		0			0
$184$	0.484275	−	1.91236i	0.484275	−	1.91236i
$185$	−1.16933			−1.16933
$186$		0			0
$187$			3.16723i			3.16723i
$188$	−1.33609	−	0.222954i	−1.33609	−	0.222954i
$189$		0			0
$190$		0			0
$191$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$191$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$192$	−0.871720	+	1.61080i	−0.871720	+	1.61080i
$193$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$193$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$194$		0			0
$195$		0			0
$196$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	3.92886	+	0.325554i	3.92886	+	0.325554i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.124959	−	0.493453i	0.124959	−	0.493453i
$201$		0			0
$202$	−1.93218	−	0.160105i	−1.93218	−	0.160105i
$203$		0			0
$204$	0.570257	−	3.41736i	0.570257	−	3.41736i
$205$		−	2.16074i		−	2.16074i
$206$		0			0
$207$	4.64490			4.64490
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$	−0.948644	−	0.0786068i	−0.948644	−	0.0786068i
$215$		0			0
$216$	−2.40503	−	0.609036i	−2.40503	−	0.609036i
$217$		0			0
$218$		0			0
$219$			0.302496i			0.302496i
$220$	−2.02874	−	0.338537i	−2.02874	−	0.338537i
$221$		0			0
$222$	0.143972	−	1.73748i	0.143972	−	1.73748i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.19854			1.19854
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$229$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$230$	−2.41507	−	0.200118i	−2.41507	−	0.200118i
$231$		0			0
$232$		0			0
$233$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$233$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$234$		0			0
$235$			1.66398i			1.66398i
$236$		0			0
$237$			2.89070i			2.89070i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	2.12801	+	0.730547i	2.12801	+	0.730547i
$241$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$241$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$242$	−0.148923	+	1.79723i	−0.148923	+	1.79723i
$243$		−	1.52905i		−	1.52905i
$244$		0			0
$245$			1.22843i			1.22843i
$246$	3.21059	+	0.266037i	3.21059	+	0.266037i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.601061	+	0.0498054i	0.601061	+	0.0498054i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			3.30299i			3.30299i
$254$	−0.111859	+	1.34994i	−0.111859	+	1.34994i
$255$	−4.25602			−4.25602
$256$	0.789141	+	0.614213i	0.789141	+	0.614213i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.93218	−	0.160105i	−1.93218	−	0.160105i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	0.752810	−	2.97278i	0.752810	−	2.97278i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−0.251673	+	3.03725i	−0.251673	+	3.03725i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−1.78914	−	0.614213i	−1.78914	−	0.614213i
$273$		0			0
$274$		0			0
$275$			0.852284i			0.852284i
$276$	0.594702	−	3.56386i	0.594702	−	3.56386i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$281$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$282$	−2.47247	−	0.204875i	−2.47247	−	0.204875i
$283$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$283$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.945817	+	2.15625i	−0.945817	+	2.15625i
$289$	2.57828			2.57828
$290$		0			0
$291$		0			0
$292$	0.162906	+	0.0271842i	0.162906	+	0.0271842i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1.82529	−	0.151248i	−1.82529	−	0.151248i
$295$		0			0
$296$	−0.922767	−	0.233676i	−0.922767	−	0.233676i
$297$	4.15393			4.15393
$298$	0.328065	+	0.0271842i	0.328065	+	0.0271842i
$299$		0			0
$300$	0.153453	−	0.919595i	0.153453	−	0.919595i
$301$		0			0
$302$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i
$303$	−3.55100			−3.55100
$304$		0			0
$305$		0			0
$306$	0.367806	−	4.43876i	0.367806	−	4.43876i
$307$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$307$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	1.55676	+	0.259777i	1.55676	+	0.259777i
$317$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$317$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$318$		0			0
$319$		0			0
$320$	0.584666	−	1.08037i	0.584666	−	1.08037i
$321$	−1.74344			−1.74344
$322$		0			0
$323$		0			0
$324$	−2.15954	−	0.360364i	−2.15954	−	0.360364i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	0.431796	−	1.70512i	0.431796	−	1.70512i
$329$		0			0
$330$	−3.75425	−	0.311086i	−3.75425	−	0.311086i
$331$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$331$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$332$		0			0
$333$		−	2.24130i		−	2.24130i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.0825793	+	0.996584i	−0.0825793	+	0.996584i
$339$		0			0
$340$	−0.382474	+	2.29204i	−0.382474	+	2.29204i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−4.43847			−4.43847
$346$	−1.46642	−	0.121511i	−1.46642	−	0.121511i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−1.53331	−	0.672572i	−1.53331	−	0.672572i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000			−1.00000
$359$	−1.00000			−1.00000
$360$	2.80390	+	0.710044i	2.80390	+	0.710044i
$361$	−1.00000			−1.00000
$362$	−1.98636	−	0.164595i	−1.98636	−	0.164595i
$363$			3.30299i			3.30299i
$364$		0			0
$365$		−	0.202885i		−	0.202885i
$366$		0			0
$367$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$367$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$368$	−1.86584	−	0.640542i	−1.86584	−	0.640542i
$369$	4.14155			4.14155
$370$	−0.0965627	+	1.16534i	−0.0965627	+	1.16534i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	3.15641	+	0.261547i	3.15641	+	0.261547i
$375$	1.10464			1.10464
$376$	−0.332526	+	1.31311i	−0.332526	+	1.31311i
$377$		0			0
$378$		0			0
$379$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$379$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$380$		0			0
$381$			2.48095i			2.48095i
$382$	0.156210	−	1.88517i	0.156210	−	1.88517i
$383$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$383$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$384$	1.53331	+	1.00176i	1.53331	+	1.00176i
$385$		0			0
$386$	0.0903332	−	1.09016i	0.0903332	−	1.09016i
$387$		0			0
$388$		0			0
$389$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$389$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$390$		0			0
$391$	3.73167			3.73167
$392$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$393$	−3.55100			−3.55100
$394$		0			0
$395$		−	1.93880i		−	1.93880i
$396$	0.648885	−	3.88855i	0.648885	−	3.88855i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−0.481448	−	0.165281i	−0.481448	−	0.165281i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−0.319116	+	1.91236i	−0.319116	+	1.91236i
$405$			2.68952i			2.68952i
$406$		0			0
$407$	1.59379			1.59379
$408$	−3.35860	−	0.850513i	−3.35860	−	0.850513i
$409$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$409$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$410$	−2.15336	−	0.178432i	−2.15336	−	0.178432i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.383573	−	4.62904i	0.383573	−	4.62904i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$419$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−3.18940			−3.18940
$424$		0			0
$425$	0.962897			0.962897
$426$		0			0
$427$		0			0
$428$	−0.156677	+	0.938912i	−0.156677	+	0.938912i
$429$		0			0
$430$		0			0
$431$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$431$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$432$	−0.805562	+	2.34652i	−0.805562	+	2.34652i
$433$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$433$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.301463	+	0.0249799i	0.301463	+	0.0249799i
$439$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$439$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$440$	−0.504913	+	1.99386i	−0.504913	+	1.99386i
$441$	−2.35456			−2.35456
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−1.71966	−	0.286961i	−1.71966	−	0.286961i
$445$		0			0
$446$		0			0
$447$	0.602925			0.602925
$448$		0			0
$449$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$449$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$450$	0.0989747	−	1.19445i	0.0989747	−	1.19445i
$451$			2.94506i			2.94506i
$452$		0			0
$453$			2.00352i			2.00352i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$457$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$458$	1.46642	+	0.121511i	1.46642	+	0.121511i
$459$		−	4.69304i		−	4.69304i
$460$	−0.398869	+	2.39029i	−0.398869	+	2.39029i
$461$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$461$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.0405441	+	0.489294i	−0.0405441	+	0.489294i
$467$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$467$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$468$		0			0
$469$		0			0
$470$	1.65830	+	0.137410i	1.65830	+	0.137410i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	2.88082	+	0.238712i	2.88082	+	0.238712i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$479$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$480$	0.903782	−	2.06042i	0.903782	−	2.06042i
$481$		0			0
$482$	0.162906	−	1.96598i	0.162906	−	1.96598i
$483$		0			0
$484$	1.77879	+	0.296828i	1.77879	+	0.296828i
$485$		0			0
$486$	−1.52383	−	0.126268i	−1.52383	−	0.126268i
$487$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$487$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$488$		0			0
$489$		0			0
$490$	1.22423	+	0.101443i	1.22423	+	0.101443i
$491$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$491$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$492$	0.530257	−	3.17766i	0.530257	−	3.17766i
$493$		0			0
$494$		0			0
$495$	−4.84285			−4.84285
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	0.0992705	−	0.594895i	0.0992705	−	0.594895i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	2.38167			2.38167
$506$	3.29171	+	0.272759i	3.29171	+	0.272759i
$507$			1.83155i			1.83155i
$508$	1.33609	+	0.222954i	1.33609	+	0.222954i
$509$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$509$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$510$	−0.351460	+	4.24149i	−0.351460	+	4.24149i
$511$		0			0
$512$	0.677282	−	0.735724i	0.677282	−	0.735724i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	2.26799i		−	2.26799i
$518$		0			0
$519$	−2.69503			−2.69503
$520$		0			0
$521$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$521$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$522$		0			0
$523$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$523$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$524$	−0.319116	+	1.91236i	−0.319116	+	1.91236i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−2.90046	−	0.995730i	−2.90046	−	0.995730i
$529$	2.89163			2.89163
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.16933			1.16933
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.67433i		−	1.67433i
$540$	3.00609	+	0.501628i	3.00609	+	0.501628i
$541$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$541$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$542$		0			0
$543$	−3.65058			−3.65058
$544$	−0.759861	+	1.73231i	−0.759861	+	1.73231i
$545$		0			0
$546$		0			0
$547$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$547$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$548$		0			0
$549$		0			0
$550$	0.849373	+	0.0703811i	0.849373	+	0.0703811i
$551$		0			0
$552$	−3.50257	−	0.886972i	−3.50257	−	0.886972i
$553$		0			0
$554$		0			0
$555$			2.14169i			2.14169i
$556$		0			0
$557$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$557$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.80092			5.80092
$562$	−0.0663435	+	0.800647i	−0.0663435	+	0.800647i
$563$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$563$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$564$	−0.408350	+	2.44711i	−0.408350	+	2.44711i
$565$		0			0
$566$	−1.93218	−	0.160105i	−1.93218	−	0.160105i
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	3.46462i		−	3.46462i
$574$		0			0
$575$	1.00417			1.00417
$576$	2.07078	+	1.12065i	2.07078	+	1.12065i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.212913	−	2.56947i	0.212913	−	2.56947i
$579$		−	2.00352i		−	2.00352i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.0405441	−	0.160105i	0.0405441	−	0.160105i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.301463	+	1.80657i	−0.301463	+	1.80657i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.309080	+	0.900319i	−0.309080	+	0.900319i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.343029	−	4.13974i	0.343029	−	4.13974i
$595$		0			0
$596$	0.0541828	−	0.324699i	0.0541828	−	0.324699i
$597$		0			0
$598$		0			0
$599$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$599$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$600$	−0.903782	−	0.228869i	−0.903782	−	0.228869i
$601$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$601$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$602$		0			0
$603$		0			0
$604$	1.07898	+	0.180049i	1.07898	+	0.180049i
$605$		−	2.21533i		−	2.21533i
$606$	−0.293240	+	3.53888i	−0.293240	+	3.53888i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−4.39323	−	0.733100i	−4.39323	−	0.733100i
$613$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$613$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$614$	−0.328065	−	0.0271842i	−0.328065	−	0.0271842i
$615$	−3.95749			−3.95749
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		−	4.89422i		−	4.89422i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.24992			−1.24992
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.80064i		−	1.80064i
$630$		0			0
$631$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$631$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$632$	0.387445	−	1.52999i	0.387445	−	1.52999i
$633$		0			0
$634$	1.66861	+	0.138265i	1.66861	+	0.138265i
$635$		−	1.66398i		−	1.66398i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−1.02840	−	0.671885i	−1.02840	−	0.671885i
$641$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$641$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$642$	−0.143972	+	1.73748i	−0.143972	+	1.73748i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$647$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$648$	−0.537467	+	2.12241i	−0.537467	+	2.12241i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$653$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$654$		0			0
$655$	2.38167			2.38167
$656$	−1.66364	−	0.571129i	−1.66364	−	0.571129i
$657$	0.388877			0.388877
$658$		0			0
$659$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$659$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$660$	−0.620046	+	3.71573i	−0.620046	+	3.71573i
$661$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$661$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$662$	−1.22423	−	0.101443i	−1.22423	−	0.101443i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.23364	−	0.185085i	−2.23364	−	0.185085i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	1.26287i		−	1.26287i
$676$	0.986361	+	0.164595i	0.986361	+	0.164595i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	2.25263	+	0.570442i	2.25263	+	0.570442i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.69503			2.69503
$688$		0			0
$689$		0			0
$690$	−0.366526	+	4.42331i	−0.366526	+	4.42331i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−0.242192	+	1.45138i	−0.242192	+	1.45138i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.32729			3.32729
$698$		0			0
$699$			0.899236i			0.899236i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.796894	+	1.47253i	−0.796894	+	1.47253i
$705$	3.04766			3.04766
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	3.71616			3.71616
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.0825793	+	0.996584i	−0.0825793	+	0.996584i
$719$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$719$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$720$	0.939162	−	2.73569i	0.939162	−	2.73569i
$721$		0			0
$722$	−0.0825793	+	0.996584i	−0.0825793	+	0.996584i
$723$		−	3.61313i		−	3.61313i
$724$	−0.328065	+	1.96598i	−0.328065	+	1.96598i
$725$		0			0
$726$	3.29171	+	0.272759i	3.29171	+	0.272759i
$727$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$727$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$728$		0			0
$729$	−0.611123			−0.611123
$730$	−0.202192	−	0.0167541i	−0.202192	−	0.0167541i
$731$		0			0
$732$		0			0
$733$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$733$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$734$	−0.0903332	+	1.09016i	−0.0903332	+	1.09016i
$735$	2.24992			2.24992
$736$	−0.792434	+	1.80657i	−0.792434	+	1.80657i
$737$		0			0
$738$	0.342007	−	4.12741i	0.342007	−	4.12741i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	1.15338	+	0.192466i	1.15338	+	0.192466i
$741$		0			0
$742$		0			0
$743$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$743$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$744$		0			0
$745$	−0.404384			−0.404384
$746$		0			0
$747$		0			0
$748$	0.521308	−	3.12403i	0.521308	−	3.12403i
$749$		0			0
$750$	0.0912208	−	1.10087i	0.0912208	−	1.10087i
$751$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$751$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$752$	1.28117	+	0.439826i	1.28117	+	0.439826i
$753$		0			0
$754$		0			0
$755$		−	1.34377i		−	1.34377i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−0.647181	−	0.0536269i	−0.647181	−	0.0536269i
$759$	6.04959			6.04959
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	2.47247	+	0.204875i	2.47247	+	0.204875i
$763$		0			0
$764$	−1.86584	−	0.311353i	−1.86584	−	0.311353i
$765$			5.47137i			5.47137i
$766$	−0.0405441	+	0.489294i	−0.0405441	+	0.489294i
$767$		0			0
$768$	1.12496	−	1.44535i	1.12496	−	1.44535i
$769$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$769$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$770$		0			0
$771$		0			0
$772$	−1.07898	−	0.180049i	−1.07898	−	0.180049i
$773$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$773$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.948644	−	0.0786068i	−0.948644	−	0.0786068i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.308159	−	3.71892i	0.308159	−	3.71892i
$783$		0			0
$784$	0.945817	+	0.324699i	0.945817	+	0.324699i
$785$		0			0
$786$	−0.293240	+	3.53888i	−0.293240	+	3.53888i
$787$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$787$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$788$		0			0
$789$		0			0
$790$	−1.93218	−	0.160105i	−1.93218	−	0.160105i
$791$		0			0
$792$	−3.82169	−	0.967783i	−3.82169	−	0.967783i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$797$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$798$		0			0
$799$	−2.56234			−2.56234
$800$	−0.204475	+	0.466155i	−0.204475	+	0.466155i
$801$		0			0
$802$		0			0
$803$			0.276531i			0.276531i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.87947	+	0.475947i	1.87947	+	0.475947i
$809$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$809$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$810$	2.68033	+	0.222099i	2.68033	+	0.222099i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	0.131614	−	1.58835i	0.131614	−	1.58835i
$815$		0			0
$816$	−1.12496	+	3.27689i	−1.12496	+	3.27689i
$817$		0			0
$818$	0.145253	−	1.75294i	0.145253	−	1.75294i
$819$		0			0
$820$	−0.355645	+	2.13127i	−0.355645	+	2.13127i
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	1.56100			1.56100
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−4.58155	−	0.764525i	−4.58155	−	0.764525i
$829$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$829$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.89163			−1.89163
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.98636	−	0.164595i	−1.98636	−	0.164595i
$839$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$839$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			1.47145i			1.47145i
$844$		0			0
$845$		−	1.22843i		−	1.22843i
$846$	−0.263379	+	3.17851i	−0.263379	+	3.17851i
$847$		0			0
$848$		0			0
$849$	−3.55100			−3.55100
$850$	0.0795154	−	0.959608i	0.0795154	−	0.959608i
$851$		−	1.87782i		−	1.87782i
$852$		0			0
$853$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$853$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$854$		0			0
$855$		0			0
$856$	0.922767	+	0.233676i	0.922767	+	0.233676i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$859$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$860$		0			0
$861$		0			0
$862$	−0.156210	+	1.88517i	−0.156210	+	1.88517i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	2.27198	+	0.996584i	2.27198	+	0.996584i
$865$	1.80756			1.80756
$866$	−0.130333	+	1.57289i	−0.130333	+	1.57289i
$867$		−	4.72224i		−	4.72224i
$868$		0			0
$869$			2.64257i			2.64257i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.0497892	−	0.298370i	0.0497892	−	0.298370i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	0.0405441	−	0.489294i	0.0405441	−	0.489294i
$879$		0			0
$880$	1.94535	+	0.667840i	1.94535	+	0.667840i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.194438	+	2.34652i	−0.194438	+	2.34652i
$883$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$883$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$887$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$888$	−0.427989	+	1.69009i	−0.427989	+	1.69009i
$889$		0			0
$890$		0			0
$891$		−	3.66579i		−	3.66579i
$892$		0			0
$893$		0			0
$894$	0.0497892	−	0.600866i	0.0497892	−	0.600866i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.0136387	−	0.164595i	0.0136387	−	0.164595i
$899$		0			0
$900$	−1.18219	−	0.197273i	−1.18219	−	0.197273i
$901$		0			0
$902$	2.93500	+	0.243201i	2.93500	+	0.243201i
$903$		0			0
$904$		0			0
$905$	2.44846			2.44846
$906$	1.99668	+	0.165450i	1.99668	+	0.165450i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			4.56503i			4.56503i
$910$		0			0
$911$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$911$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$912$		0			0
$913$		0			0
$914$	−0.156210	+	1.88517i	−0.156210	+	1.88517i
$915$		0			0
$916$	0.242192	−	1.45138i	0.242192	−	1.45138i
$917$		0			0
$918$	−4.67701	−	0.387548i	−4.67701	−	0.387548i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	2.34919	+	0.594895i	2.34919	+	0.594895i
$921$	−0.602925			−0.602925
$922$	−1.82529	−	0.151248i	−1.82529	−	0.151248i
$923$		0			0
$924$		0			0
$925$		−	0.484542i		−	0.484542i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	0.484275	+	0.0808112i	0.484275	+	0.0808112i
$933$		0			0
$934$	1.46642	+	0.121511i	1.46642	+	0.121511i
$935$	−3.89070			−3.89070
$936$		0			0
$937$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$937$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$938$		0			0
$939$		0			0
$940$	0.273882	−	1.64129i	0.273882	−	1.64129i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	3.46992			3.46992
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$947$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$948$	0.475793	−	2.85127i	0.475793	−	2.85127i
$949$		0			0
$950$		0			0
$951$	3.06662			3.06662
$952$		0			0
$953$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$953$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$954$		0			0
$955$			2.32373i			2.32373i
$956$		0			0
$957$		0			0
$958$	0.0663435	−	0.800647i	0.0663435	−	0.800647i
$959$		0			0
$960$	−1.97874	−	1.07084i	−1.97874	−	1.07084i
$961$	1.00000			1.00000
$962$		0			0
$963$			2.24130i			2.24130i
$964$	−1.94582	−	0.324699i	−1.94582	−	0.324699i
$965$			1.34377i			1.34377i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.442706	−	1.74821i	0.442706	−	1.74821i
$969$		0			0
$970$		0			0
$971$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$971$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$972$	−0.251673	+	1.50820i	−0.251673	+	1.50820i
$973$		0			0
$974$	0.0663435	−	0.800647i	0.0663435	−	0.800647i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	0.202192	−	1.21167i	0.202192	−	1.21167i
$981$		0			0
$982$	0.948644	+	0.0786068i	0.948644	+	0.0786068i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−3.12301	−	0.790855i	−3.12301	−	0.790855i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.399919	+	4.82631i	−0.399919	+	4.82631i
$991$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$991$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$992$		0			0
$993$	−2.24992			−2.24992
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	−2.36160			−2.36160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.9	✓	18
8.5	even	2	inner	2872.1.b.e.717.10	yes	18
359.358	odd	2	CM	2872.1.b.e.717.9	✓	18
2872.717	odd	2	inner	2872.1.b.e.717.10	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.9	✓	18	1.1	even	1	trivial
2872.1.b.e.717.9	✓	18	359.358	odd	2	CM
2872.1.b.e.717.10	yes	18	8.5	even	2	inner
2872.1.b.e.717.10	yes	18	2872.717	odd	2	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:	\( N \)	\(=\)	\( 2872 = 2^{3} \cdot 359 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2872.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0825793 - 0.996584i) q^{2} -1.83155i q^{3} +(-0.986361 - 0.164595i) q^{4} +1.22843i q^{5} +(-1.82529 - 0.151248i) q^{6} +(-0.245485 + 0.969400i) q^{8} -2.35456 q^{9} +O(q^{10})\)	\(q+(0.0825793 - 0.996584i) q^{2} -1.83155i q^{3} +(-0.986361 - 0.164595i) q^{4} +1.22843i q^{5} +(-1.82529 - 0.151248i) q^{6} +(-0.245485 + 0.969400i) q^{8} -2.35456 q^{9} +(1.22423 + 0.101443i) q^{10} -1.67433i q^{11} +(-0.301463 + 1.80657i) q^{12} +2.24992 q^{15} +(0.945817 + 0.324699i) q^{16} -1.89163 q^{17} +(-0.194438 + 2.34652i) q^{18} +(0.202192 - 1.21167i) q^{20} +(-1.66861 - 0.138265i) q^{22} -1.97272 q^{23} +(1.77550 + 0.449618i) q^{24} -0.509029 q^{25} +2.48095i q^{27} +(0.185797 - 2.24223i) q^{30} +(0.401695 - 0.915773i) q^{32} -3.06662 q^{33} +(-0.156210 + 1.88517i) q^{34} +(2.32245 + 0.387548i) q^{36} +0.951895i q^{37} +(-1.19084 - 0.301561i) q^{40} -1.75895 q^{41} +(-0.275586 + 1.65150i) q^{44} -2.89241i q^{45} +(-0.162906 + 1.96598i) q^{46} +1.35456 q^{47} +(0.594702 - 1.73231i) q^{48} +1.00000 q^{49} +(-0.0420353 + 0.507290i) q^{50} +3.46462i q^{51} +(2.47247 + 0.204875i) q^{54} +2.05679 q^{55} +(-2.21923 - 0.370324i) q^{60} +(-0.879474 - 0.475947i) q^{64} +(-0.253239 + 3.05614i) q^{66} +(1.86584 + 0.311353i) q^{68} +3.61313i q^{69} +(0.578011 - 2.28251i) q^{72} -0.165159 q^{73} +(0.948644 + 0.0786068i) q^{74} +0.932310i q^{75} -1.57828 q^{79} +(-0.398869 + 1.16187i) q^{80} +2.18940 q^{81} +(-0.145253 + 1.75294i) q^{82} -2.32373i q^{85} +(1.62310 + 0.411024i) q^{88} +(-2.88253 - 0.238853i) q^{90} +(1.94582 + 0.324699i) q^{92} +(0.111859 - 1.34994i) q^{94} +(-1.67728 - 0.735724i) q^{96} +(0.0825793 - 0.996584i) q^{98} +3.94232i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 18 * q + q^2 - q^4 + q^8 - 20 * q^9	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9} - q^{16} + 2 q^{17} + q^{18} - 2 q^{23} - 20 q^{25} + q^{32} - 2 q^{34} - q^{36} - 2 q^{41} + 2 q^{46} + 2 q^{47} + 18 q^{49} + q^{50} - q^{64} + 2 q^{68} + q^{72} - 2 q^{73} + 2 q^{79} + 18 q^{81} + 2 q^{82} - 19 q^{90} + 17 q^{92} - 2 q^{94} - 19 q^{96} + q^{98}+O(q^{100}) \) 18 * q + q^2 - q^4 + q^8 - 20 * q^9 - q^16 + 2 * q^17 + q^18 - 2 * q^23 - 20 * q^25 + q^32 - 2 * q^34 - q^36 - 2 * q^41 + 2 * q^46 + 2 * q^47 + 18 * q^49 + q^50 - q^64 + 2 * q^68 + q^72 - 2 * q^73 + 2 * q^79 + 18 * q^81 + 2 * q^82 - 19 * q^90 + 17 * q^92 - 2 * q^94 - 19 * q^96 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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