LMFDB - Embedded newform 2872.1.b.e.717.12

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			717.12
Root			$-0.945817 - 0.324699i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.401695 + 0.915773i) q^{2} -0.951895i q^{3} +(-0.677282 + 0.735724i) q^{4} +0.329189i q^{5} +(0.871720 - 0.382372i) q^{6} +(-0.945817 - 0.324699i) q^{8} +0.0938963 q^{9} +O(q^{10})$	\(q+(0.401695 + 0.915773i) q^{2} -0.951895i q^{3} +(-0.677282 + 0.735724i) q^{4} +0.329189i q^{5} +(0.871720 - 0.382372i) q^{6} +(-0.945817 - 0.324699i) q^{8} +0.0938963 q^{9} +(-0.301463 + 0.132234i) q^{10} -1.93880i q^{11} +(0.700332 + 0.644701i) q^{12} +0.313353 q^{15} +(-0.0825793 - 0.996584i) q^{16} +0.165159 q^{17} +(0.0377177 + 0.0859877i) q^{18} +(-0.242192 - 0.222954i) q^{20} +(1.77550 - 0.778807i) q^{22} -1.35456 q^{23} +(-0.309080 + 0.900319i) q^{24} +0.891634 q^{25} -1.04127i q^{27} +(0.125873 + 0.286961i) q^{30} +(0.879474 - 0.475947i) q^{32} -1.84553 q^{33} +(0.0663435 + 0.151248i) q^{34} +(-0.0635942 + 0.0690818i) q^{36} -1.22843i q^{37} +(0.106888 - 0.311353i) q^{40} +1.57828 q^{41} +(1.42642 + 1.31311i) q^{44} +0.0309097i q^{45} +(-0.544122 - 1.24047i) q^{46} -1.09390 q^{47} +(-0.948644 + 0.0786068i) q^{48} +1.00000 q^{49} +(0.358165 + 0.816535i) q^{50} -0.157214i q^{51} +(0.953571 - 0.418275i) q^{54} +0.638232 q^{55} +(-0.212229 + 0.230542i) q^{60} +(0.789141 + 0.614213i) q^{64} +(-0.741343 - 1.69009i) q^{66} +(-0.111859 + 0.121511i) q^{68} +1.28940i q^{69} +(-0.0888088 - 0.0304881i) q^{72} -0.803391 q^{73} +(1.12496 - 0.493453i) q^{74} -0.848742i q^{75} +1.97272 q^{79} +(0.328065 - 0.0271842i) q^{80} -0.897287 q^{81} +(0.633988 + 1.44535i) q^{82} +0.0543685i q^{85} +(-0.629528 + 1.83375i) q^{88} +(-0.0283062 + 0.0124163i) q^{90} +(0.917421 - 0.996584i) q^{92} +(-0.439413 - 1.00176i) q^{94} +(-0.453052 - 0.837166i) q^{96} +(0.401695 + 0.915773i) q^{98} -0.182046i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.401695	+	0.915773i	0.401695	+	0.915773i
$2$	0.401695	+	0.915773i	0.401695	+	0.915773i
$3$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$3$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$4$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$5$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$5$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$6$	0.871720	−	0.382372i	0.871720	−	0.382372i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$9$	0.0938963			0.0938963
$10$	−0.301463	+	0.132234i	−0.301463	+	0.132234i
$11$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$11$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$12$	0.700332	+	0.644701i	0.700332	+	0.644701i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	0.313353			0.313353
$16$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$17$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$17$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$18$	0.0377177	+	0.0859877i	0.0377177	+	0.0859877i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.242192	−	0.222954i	−0.242192	−	0.222954i
$21$		0			0
$22$	1.77550	−	0.778807i	1.77550	−	0.778807i
$23$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$23$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$24$	−0.309080	+	0.900319i	−0.309080	+	0.900319i
$25$	0.891634			0.891634
$26$		0			0
$27$		−	1.04127i		−	1.04127i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	0.125873	+	0.286961i	0.125873	+	0.286961i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.879474	−	0.475947i	0.879474	−	0.475947i
$33$	−1.84553			−1.84553
$34$	0.0663435	+	0.151248i	0.0663435	+	0.151248i
$35$		0			0
$36$	−0.0635942	+	0.0690818i	−0.0635942	+	0.0690818i
$37$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$37$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$38$		0			0
$39$		0			0
$40$	0.106888	−	0.311353i	0.106888	−	0.311353i
$41$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$41$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	1.42642	+	1.31311i	1.42642	+	1.31311i
$45$			0.0309097i			0.0309097i
$46$	−0.544122	−	1.24047i	−0.544122	−	1.24047i
$47$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$47$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$48$	−0.948644	+	0.0786068i	−0.948644	+	0.0786068i
$49$	1.00000			1.00000
$50$	0.358165	+	0.816535i	0.358165	+	0.816535i
$51$		−	0.157214i		−	0.157214i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	0.953571	−	0.418275i	0.953571	−	0.418275i
$55$	0.638232			0.638232
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−0.212229	+	0.230542i	−0.212229	+	0.230542i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	0.789141	+	0.614213i	0.789141	+	0.614213i
$65$		0			0
$66$	−0.741343	−	1.69009i	−0.741343	−	1.69009i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.111859	+	0.121511i	−0.111859	+	0.121511i
$69$			1.28940i			1.28940i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.0888088	−	0.0304881i	−0.0888088	−	0.0304881i
$73$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$73$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$74$	1.12496	−	0.493453i	1.12496	−	0.493453i
$75$		−	0.848742i		−	0.848742i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$79$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$80$	0.328065	−	0.0271842i	0.328065	−	0.0271842i
$81$	−0.897287			−0.897287
$82$	0.633988	+	1.44535i	0.633988	+	1.44535i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$			0.0543685i			0.0543685i
$86$		0			0
$87$		0			0
$88$	−0.629528	+	1.83375i	−0.629528	+	1.83375i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.0283062	+	0.0124163i	−0.0283062	+	0.0124163i
$91$		0			0
$92$	0.917421	−	0.996584i	0.917421	−	0.996584i
$93$		0			0
$94$	−0.439413	−	1.00176i	−0.439413	−	1.00176i
$95$		0			0
$96$	−0.453052	−	0.837166i	−0.453052	−	0.837166i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.401695	+	0.915773i	0.401695	+	0.915773i
$99$		−	0.182046i		−	0.182046i
$100$	−0.603888	+	0.655997i	−0.603888	+	0.655997i
$101$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$101$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$102$	0.143972	−	0.0631520i	0.143972	−	0.0631520i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$107$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$108$	0.766090	+	0.705236i	0.766090	+	0.705236i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.256375	+	0.584476i	0.256375	+	0.584476i
$111$	−1.16933			−1.16933
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	0.445908i		−	0.445908i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.296375	−	0.101746i	−0.296375	−	0.101746i
$121$	−2.75895			−2.75895
$122$		0			0
$123$		−	1.50236i		−	1.50236i
$124$		0			0
$125$			0.622706i			0.622706i
$126$		0			0
$127$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$127$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$128$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$129$		0			0
$130$		0			0
$131$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$131$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$132$	1.24995	−	1.35780i	1.24995	−	1.35780i
$133$		0			0
$134$		0			0
$135$	0.342776			0.342776
$136$	−0.156210	−	0.0536269i	−0.156210	−	0.0536269i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−1.18080	+	0.517947i	−1.18080	+	0.517947i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			1.04127i			1.04127i
$142$		0			0
$143$		0			0
$144$	−0.00775390	−	0.0935756i	−0.00775390	−	0.0935756i
$145$		0			0
$146$	−0.322718	−	0.735724i	−0.322718	−	0.735724i
$147$		−	0.951895i		−	0.951895i
$148$	0.903782	+	0.831990i	0.903782	+	0.831990i
$149$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$149$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$150$	0.777255	−	0.340936i	0.777255	−	0.340936i
$151$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$151$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$152$		0			0
$153$	0.0155078			0.0155078
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	0.792434	+	1.80657i	0.792434	+	1.80657i
$159$		0			0
$160$	0.156677	+	0.289513i	0.156677	+	0.289513i
$161$		0			0
$162$	−0.360436	−	0.821712i	−0.360436	−	0.821712i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−1.06894	+	1.16118i	−1.06894	+	1.16118i
$165$		−	0.607530i		−	0.607530i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−0.0497892	+	0.0218396i	−0.0497892	+	0.0218396i
$171$		0			0
$172$		0			0
$173$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$173$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$174$		0			0
$175$		0			0
$176$	−1.93218	+	0.160105i	−1.93218	+	0.160105i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−0.0227410	−	0.0209345i	−0.0227410	−	0.0209345i
$181$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$181$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$182$		0			0
$183$		0			0
$184$	1.28117	+	0.439826i	1.28117	+	0.439826i
$185$	0.404384			0.404384
$186$		0			0
$187$		−	0.320210i		−	0.320210i
$188$	0.740876	−	0.804806i	0.740876	−	0.804806i
$189$		0			0
$190$		0			0
$191$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$191$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$192$	0.584666	−	0.751179i	0.584666	−	0.751179i
$193$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$193$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$194$		0			0
$195$		0			0
$196$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	0.166713	−	0.0731271i	0.166713	−	0.0731271i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.843323	−	0.289513i	−0.843323	−	0.289513i
$201$		0			0
$202$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$203$		0			0
$204$	0.115666	+	0.106478i	0.115666	+	0.106478i
$205$			0.519553i			0.519553i
$206$		0			0
$207$	−0.127188			−0.127188
$208$		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
$214$	−1.12496	+	0.493453i	−1.12496	+	0.493453i
$215$		0			0
$216$	−0.338101	+	0.984855i	−0.338101	+	0.984855i
$217$		0			0
$218$		0			0
$219$			0.764744i			0.764744i
$220$	−0.432263	+	0.469563i	−0.432263	+	0.469563i
$221$		0			0
$222$	−0.469715	−	1.07084i	−0.469715	−	1.07084i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	0.0837212			0.0837212
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$229$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$230$	0.408350	−	0.179119i	0.408350	−	0.179119i
$231$		0			0
$232$		0			0
$233$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$233$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$234$		0			0
$235$		−	0.360099i		−	0.360099i
$236$		0			0
$237$		−	1.87782i		−	1.87782i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.0258765	−	0.312283i	−0.0258765	−	0.312283i
$241$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$241$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$242$	−1.10826	−	2.52657i	−1.10826	−	2.52657i
$243$		−	0.187151i		−	0.187151i
$244$		0			0
$245$			0.329189i			0.329189i
$246$	1.37582	−	0.603490i	1.37582	−	0.603490i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.570257	+	0.250138i	−0.570257	+	0.250138i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			2.62623i			2.62623i
$254$	0.439413	+	1.00176i	0.439413	+	1.00176i
$255$	0.0517530			0.0517530
$256$	−0.986361	+	0.164595i	−0.986361	+	0.164595i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	1.74554	+	0.599244i	1.74554	+	0.599244i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	0.137692	+	0.313905i	0.137692	+	0.313905i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.0136387	−	0.164595i	−0.0136387	−	0.164595i
$273$		0			0
$274$		0			0
$275$		−	1.72870i		−	1.72870i
$276$	−0.948644	−	0.873288i	−0.948644	−	0.873288i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$281$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$282$	−0.953571	+	0.418275i	−0.953571	+	0.418275i
$283$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$283$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.0825793	−	0.0446897i	0.0825793	−	0.0446897i
$289$	−0.972723			−0.972723
$290$		0			0
$291$		0			0
$292$	0.544122	−	0.591074i	0.544122	−	0.591074i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.871720	−	0.382372i	0.871720	−	0.382372i
$295$		0			0
$296$	−0.398869	+	1.16187i	−0.398869	+	1.16187i
$297$	−2.01882			−2.01882
$298$	1.34751	−	0.591074i	1.34751	−	0.591074i
$299$		0			0
$300$	0.624440	+	0.574837i	0.624440	+	0.574837i
$301$		0			0
$302$	−0.197221	−	0.449618i	−0.197221	−	0.449618i
$303$	0.618159			0.618159
$304$		0			0
$305$		0			0
$306$	0.00622941	+	0.0142016i	0.00622941	+	0.0142016i
$307$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$307$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$308$		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
$316$	−1.33609	+	1.45138i	−1.33609	+	1.45138i
$317$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$317$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$318$		0			0
$319$		0			0
$320$	−0.202192	+	0.259777i	−0.202192	+	0.259777i
$321$	1.16933			1.16933
$322$		0			0
$323$		0			0
$324$	0.607716	−	0.660156i	0.607716	−	0.660156i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−1.49277	−	0.512467i	−1.49277	−	0.512467i
$329$		0			0
$330$	0.556360	−	0.244042i	0.556360	−	0.244042i
$331$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$331$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$332$		0			0
$333$		−	0.115345i		−	0.115345i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$339$		0			0
$340$	−0.0400002	−	0.0368228i	−0.0400002	−	0.0368228i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.424457			−0.424457
$346$	1.53331	−	0.672572i	1.53331	−	0.672572i
$347$		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
$352$	−0.922767	−	1.70512i	−0.922767	−	1.70512i
$353$		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
$360$	0.0100363	−	0.0292349i	0.0100363	−	0.0292349i
$361$	−1.00000			−1.00000
$362$	−1.67728	+	0.735724i	−1.67728	+	0.735724i
$363$			2.62623i			2.62623i
$364$		0			0
$365$		−	0.264468i		−	0.264468i
$366$		0			0
$367$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$367$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$368$	0.111859	+	1.34994i	0.111859	+	1.34994i
$369$	0.148195			0.148195
$370$	0.162439	+	0.370324i	0.162439	+	0.370324i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	0.293240	−	0.128627i	0.293240	−	0.128627i
$375$	0.592750			0.592750
$376$	1.03463	+	0.355188i	1.03463	+	0.355188i
$377$		0			0
$378$		0			0
$379$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$379$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$380$		0			0
$381$		−	1.04127i		−	1.04127i
$382$	−0.0663435	−	0.151248i	−0.0663435	−	0.151248i
$383$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$383$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$384$	0.922767	+	0.233676i	0.922767	+	0.233676i
$385$		0			0
$386$	0.197221	+	0.449618i	0.197221	+	0.449618i
$387$		0			0
$388$		0			0
$389$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$389$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$390$		0			0
$391$	−0.223718			−0.223718
$392$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$393$	0.618159			0.618159
$394$		0			0
$395$			0.649399i			0.649399i
$396$	0.133936	+	0.123297i	0.133936	+	0.123297i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−0.0736306	−	0.888589i	−0.0736306	−	0.888589i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−0.477778	−	0.439826i	−0.477778	−	0.439826i
$405$		−	0.295377i		−	0.295377i
$406$		0			0
$407$	−2.38167			−2.38167
$408$	−0.0510472	+	0.148695i	−0.0510472	+	0.148695i
$409$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$409$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$410$	−0.475793	+	0.208702i	−0.475793	+	0.208702i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.0510910	−	0.116476i	−0.0510910	−	0.116476i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$419$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−0.102713			−0.102713
$424$		0			0
$425$	0.147261			0.147261
$426$		0			0
$427$		0			0
$428$	−0.903782	−	0.831990i	−0.903782	−	0.831990i
$429$		0			0
$430$		0			0
$431$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$431$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$432$	−1.03772	+	0.0859877i	−1.03772	+	0.0859877i
$433$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$433$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.700332	+	0.307194i	−0.700332	+	0.307194i
$439$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$439$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$440$	−0.603651	−	0.207234i	−0.603651	−	0.207234i
$441$	0.0938963			0.0938963
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	0.791967	−	0.860305i	0.791967	−	0.860305i
$445$		0			0
$446$		0			0
$447$	−1.40066			−1.40066
$448$		0			0
$449$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$449$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$450$	0.0336304	+	0.0766696i	0.0336304	+	0.0766696i
$451$		−	3.05997i		−	3.05997i
$452$		0			0
$453$			0.467353i			0.467353i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$457$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$458$	−1.53331	+	0.672572i	−1.53331	+	0.672572i
$459$		−	0.171975i		−	0.171975i
$460$	0.328065	+	0.302005i	0.328065	+	0.302005i
$461$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$461$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.759861	−	1.73231i	−0.759861	−	1.73231i
$467$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$467$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$468$		0			0
$469$		0			0
$470$	0.329769	−	0.144650i	0.329769	−	0.144650i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.71966	−	0.754313i	1.71966	−	0.754313i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$479$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$480$	0.275586	−	0.149140i	0.275586	−	0.149140i
$481$		0			0
$482$	0.544122	+	1.24047i	0.544122	+	1.24047i
$483$		0			0
$484$	1.86858	−	2.02982i	1.86858	−	2.02982i
$485$		0			0
$486$	0.171388	−	0.0751778i	0.171388	−	0.0751778i
$487$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$487$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$488$		0			0
$489$		0			0
$490$	−0.301463	+	0.132234i	−0.301463	+	0.132234i
$491$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$491$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$492$	1.10532	+	1.01752i	1.10532	+	1.01752i
$493$		0			0
$494$		0			0
$495$	0.0599276			0.0599276
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.458139	−	0.421747i	−0.458139	−	0.421747i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	−0.213775			−0.213775
$506$	−2.40503	+	1.05494i	−2.40503	+	1.05494i
$507$			0.951895i			0.951895i
$508$	−0.740876	+	0.804806i	−0.740876	+	0.804806i
$509$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$509$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$510$	0.0207890	+	0.0473941i	0.0207890	+	0.0473941i
$511$		0			0
$512$	−0.546948	−	0.837166i	−0.546948	−	0.837166i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			2.12085i			2.12085i
$518$		0			0
$519$	−1.59379			−1.59379
$520$		0			0
$521$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$521$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$522$		0			0
$523$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$523$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$524$	−0.477778	−	0.439826i	−0.477778	−	0.439826i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.152403	+	1.83923i	0.152403	+	1.83923i
$529$	0.834841			0.834841
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.404384			−0.404384
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.93880i		−	1.93880i
$540$	−0.232156	+	0.252189i	−0.232156	+	0.252189i
$541$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$541$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$542$		0			0
$543$	1.74344			1.74344
$544$	0.145253	−	0.0786068i	0.145253	−	0.0786068i
$545$		0			0
$546$		0			0
$547$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$547$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$548$		0			0
$549$		0			0
$550$	1.58310	−	0.694411i	1.58310	−	0.694411i
$551$		0			0
$552$	0.418668	−	1.21954i	0.418668	−	1.21954i
$553$		0			0
$554$		0			0
$555$		−	0.384931i		−	0.384931i
$556$		0			0
$557$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$557$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.304806			−0.304806
$562$	−0.706561	−	1.61080i	−0.706561	−	1.61080i
$563$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$563$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$564$	−0.766090	−	0.705236i	−0.766090	−	0.705236i
$565$		0			0
$566$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$567$		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
$573$			0.157214i			0.157214i
$574$		0			0
$575$	−1.20778			−1.20778
$576$	0.0740974	+	0.0576723i	0.0740974	+	0.0576723i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.390738	−	0.890793i	−0.390738	−	0.890793i
$579$		−	0.467353i		−	0.467353i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.759861	+	0.260861i	0.759861	+	0.260861i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.700332	+	0.644701i	0.700332	+	0.644701i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.22423	+	0.101443i	−1.22423	+	0.101443i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.810952	−	1.84878i	−0.810952	−	1.84878i
$595$		0			0
$596$	1.08258	+	0.996584i	1.08258	+	0.996584i
$597$		0			0
$598$		0			0
$599$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$599$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$600$	−0.275586	+	0.802755i	−0.275586	+	0.802755i
$601$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$601$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$602$		0			0
$603$		0			0
$604$	0.332526	−	0.361219i	0.332526	−	0.361219i
$605$		−	0.908216i		−	0.908216i
$606$	0.248312	+	0.566094i	0.248312	+	0.566094i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.0105031	+	0.0114095i	−0.0105031	+	0.0114095i
$613$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$613$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$614$	−1.34751	+	0.591074i	−1.34751	+	0.591074i
$615$	0.494560			0.494560
$616$		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
$621$			1.41047i			1.41047i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.686647			0.686647
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.202885i		−	0.202885i
$630$		0			0
$631$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$631$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$632$	−1.86584	−	0.640542i	−1.86584	−	0.640542i
$633$		0			0
$634$	−1.77550	+	0.778807i	−1.77550	+	0.778807i
$635$			0.360099i			0.360099i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.319116	−	0.0808112i	−0.319116	−	0.0808112i
$641$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$641$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$642$	0.469715	+	1.07084i	0.469715	+	1.07084i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$647$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$648$	0.848670	+	0.291349i	0.848670	+	0.291349i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$653$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$654$		0			0
$655$	−0.213775			−0.213775
$656$	−0.130333	−	1.57289i	−0.130333	−	1.57289i
$657$	−0.0754354			−0.0754354
$658$		0			0
$659$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$659$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$660$	0.446974	+	0.411469i	0.446974	+	0.411469i
$661$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$661$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$662$	0.301463	−	0.132234i	0.301463	−	0.132234i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.105630	−	0.0463334i	0.105630	−	0.0463334i
$667$		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
$675$		−	0.928436i		−	0.928436i
$676$	0.677282	−	0.735724i	0.677282	−	0.735724i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	0.0176534	−	0.0514226i	0.0176534	−	0.0514226i
$681$		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
$687$	1.59379			1.59379
$688$		0			0
$689$		0			0
$690$	−0.170502	−	0.388706i	−0.170502	−	0.388706i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	1.23185	+	1.13399i	1.23185	+	1.13399i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.260667			0.260667
$698$		0			0
$699$			1.80064i			1.80064i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	1.19084	−	1.52999i	1.19084	−	1.52999i
$705$	−0.342776			−0.342776
$706$		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
$711$	0.185231			0.185231
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$719$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$719$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$720$	0.0308041	−	0.00255250i	0.0308041	−	0.00255250i
$721$		0			0
$722$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$723$		−	1.28940i		−	1.28940i
$724$	−1.34751	−	1.24047i	−1.34751	−	1.24047i
$725$		0			0
$726$	−2.40503	+	1.05494i	−2.40503	+	1.05494i
$727$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$727$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$728$		0			0
$729$	−1.07544			−1.07544
$730$	0.242192	−	0.106235i	0.242192	−	0.106235i
$731$		0			0
$732$		0			0
$733$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$733$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$734$	−0.197221	−	0.449618i	−0.197221	−	0.449618i
$735$	0.313353			0.313353
$736$	−1.19130	+	0.644701i	−1.19130	+	0.644701i
$737$		0			0
$738$	0.0595292	+	0.135713i	0.0595292	+	0.135713i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−0.273882	+	0.297515i	−0.273882	+	0.297515i
$741$		0			0
$742$		0			0
$743$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$743$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$744$		0			0
$745$	0.484385			0.484385
$746$		0			0
$747$		0			0
$748$	0.235586	+	0.216872i	0.235586	+	0.216872i
$749$		0			0
$750$	0.238105	+	0.542825i	0.238105	+	0.542825i
$751$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$751$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$752$	0.0903332	+	1.09016i	0.0903332	+	1.09016i
$753$		0			0
$754$		0			0
$755$		−	0.161622i		−	0.161622i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−1.82529	+	0.800647i	−1.82529	+	0.800647i
$759$	2.49989			2.49989
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	0.953571	−	0.418275i	0.953571	−	0.418275i
$763$		0			0
$764$	0.111859	−	0.121511i	0.111859	−	0.121511i
$765$			0.00510500i			0.00510500i
$766$	−0.759861	−	1.73231i	−0.759861	−	1.73231i
$767$		0			0
$768$	0.156677	+	0.938912i	0.156677	+	0.938912i
$769$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$769$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$770$		0			0
$771$		0			0
$772$	−0.332526	+	0.361219i	−0.332526	+	0.361219i
$773$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$773$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.12496	+	0.493453i	−1.12496	+	0.493453i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.0898664	−	0.204875i	−0.0898664	−	0.204875i
$783$		0			0
$784$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$785$		0			0
$786$	0.248312	+	0.566094i	0.248312	+	0.566094i
$787$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$787$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$788$		0			0
$789$		0			0
$790$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$791$		0			0
$792$	−0.0591103	+	0.172182i	−0.0591103	+	0.172182i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$797$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$798$		0			0
$799$	−0.180666			−0.180666
$800$	0.784169	−	0.424371i	0.784169	−	0.424371i
$801$		0			0
$802$		0			0
$803$			1.55761i			1.55761i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.210859	−	0.614213i	0.210859	−	0.614213i
$809$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$809$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$810$	0.270499	−	0.118652i	0.270499	−	0.118652i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−0.956707	−	2.18107i	−0.956707	−	2.18107i
$815$		0			0
$816$	−0.156677	+	0.0129826i	−0.156677	+	0.0129826i
$817$		0			0
$818$	−0.633988	−	1.44535i	−0.633988	−	1.44535i
$819$		0			0
$820$	−0.382248	−	0.351884i	−0.382248	−	0.351884i
$821$		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
$825$	−1.64554			−1.64554
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.0861424	−	0.0935756i	0.0861424	−	0.0935756i
$829$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$829$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.165159			0.165159
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.67728	+	0.735724i	−1.67728	+	0.735724i
$839$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$839$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			1.67433i			1.67433i
$844$		0			0
$845$		−	0.329189i		−	0.329189i
$846$	−0.0412593	−	0.0940617i	−0.0412593	−	0.0940617i
$847$		0			0
$848$		0			0
$849$	0.618159			0.618159
$850$	0.0591541	+	0.134858i	0.0591541	+	0.134858i
$851$			1.66398i			1.66398i
$852$		0			0
$853$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$853$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$854$		0			0
$855$		0			0
$856$	0.398869	−	1.16187i	0.398869	−	1.16187i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$859$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$860$		0			0
$861$		0			0
$862$	0.0663435	+	0.151248i	0.0663435	+	0.151248i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.495592	−	0.915773i	−0.495592	−	0.915773i
$865$	0.551172			0.551172
$866$	0.792434	+	1.80657i	0.792434	+	1.80657i
$867$			0.925930i			0.925930i
$868$		0			0
$869$		−	3.82472i		−	3.82472i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.562640	−	0.517947i	−0.562640	−	0.517947i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	0.759861	+	1.73231i	0.759861	+	1.73231i
$879$		0			0
$880$	−0.0527048	−	0.636052i	−0.0527048	−	0.636052i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.0377177	+	0.0859877i	0.0377177	+	0.0859877i
$883$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$883$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$887$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$888$	1.10597	+	0.379681i	1.10597	+	0.379681i
$889$		0			0
$890$		0			0
$891$			1.73966i			1.73966i
$892$		0			0
$893$		0			0
$894$	−0.562640	−	1.28269i	−0.562640	−	1.28269i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.322718	+	0.735724i	0.322718	+	0.735724i
$899$		0			0
$900$	−0.0567028	+	0.0615957i	−0.0567028	+	0.0615957i
$901$		0			0
$902$	2.80224	−	1.22918i	2.80224	−	1.22918i
$903$		0			0
$904$		0			0
$905$	−0.602925			−0.602925
$906$	−0.427989	+	0.187733i	−0.427989	+	0.187733i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			0.0609762i			0.0609762i
$910$		0			0
$911$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$911$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$912$		0			0
$913$		0			0
$914$	0.0663435	+	0.151248i	0.0663435	+	0.151248i
$915$		0			0
$916$	−1.23185	−	1.13399i	−1.23185	−	1.13399i
$917$		0			0
$918$	0.157491	−	0.0690818i	0.157491	−	0.0690818i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−0.144786	+	0.421747i	−0.144786	+	0.421747i
$921$	1.40066			1.40066
$922$	0.871720	−	0.382372i	0.871720	−	0.382372i
$923$		0			0
$924$		0			0
$925$		−	1.09531i		−	1.09531i
$926$		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
$932$	1.28117	−	1.39172i	1.28117	−	1.39172i
$933$		0			0
$934$	−1.53331	+	0.672572i	−1.53331	+	0.672572i
$935$	0.105410			0.105410
$936$		0			0
$937$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$937$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$938$		0			0
$939$		0			0
$940$	0.264933	+	0.243888i	0.264933	+	0.243888i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−2.13788			−2.13788
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$947$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$948$	1.38156	+	1.27182i	1.38156	+	1.27182i
$949$		0			0
$950$		0			0
$951$	1.84553			1.84553
$952$		0			0
$953$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$953$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$954$		0			0
$955$		−	0.0543685i		−	0.0543685i
$956$		0			0
$957$		0			0
$958$	0.706561	+	1.61080i	0.706561	+	1.61080i
$959$		0			0
$960$	0.247280	+	0.192466i	0.247280	+	0.192466i
$961$	1.00000			1.00000
$962$		0			0
$963$			0.115345i			0.115345i
$964$	−0.917421	+	0.996584i	−0.917421	+	0.996584i
$965$			0.161622i			0.161622i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	2.60946	+	0.895829i	2.60946	+	0.895829i
$969$		0			0
$970$		0			0
$971$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$971$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$972$	0.137692	+	0.126754i	0.137692	+	0.126754i
$973$		0			0
$974$	0.706561	+	1.61080i	0.706561	+	1.61080i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−0.242192	−	0.222954i	−0.242192	−	0.222954i
$981$		0			0
$982$	1.12496	−	0.493453i	1.12496	−	0.493453i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−0.487815	+	1.42096i	−0.487815	+	1.42096i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.0240727	+	0.0548801i	0.0240727	+	0.0548801i
$991$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$991$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$992$		0			0
$993$	−0.313353			−0.313353
$994$		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
$999$	−1.27913			−1.27913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.12	yes	18
8.5	even	2	inner	2872.1.b.e.717.11	✓	18
359.358	odd	2	CM	2872.1.b.e.717.12	yes	18
2872.717	odd	2	inner	2872.1.b.e.717.11	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.11	✓	18	8.5	even	2	inner
2872.1.b.e.717.11	✓	18	2872.717	odd	2	inner
2872.1.b.e.717.12	yes	18	1.1	even	1	trivial
2872.1.b.e.717.12	yes	18	359.358	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.401695 + 0.915773i) q^{2} -0.951895i q^{3} +(-0.677282 + 0.735724i) q^{4} +0.329189i q^{5} +(0.871720 - 0.382372i) q^{6} +(-0.945817 - 0.324699i) q^{8} +0.0938963 q^{9} +O(q^{10})\)	\(q+(0.401695 + 0.915773i) q^{2} -0.951895i q^{3} +(-0.677282 + 0.735724i) q^{4} +0.329189i q^{5} +(0.871720 - 0.382372i) q^{6} +(-0.945817 - 0.324699i) q^{8} +0.0938963 q^{9} +(-0.301463 + 0.132234i) q^{10} -1.93880i q^{11} +(0.700332 + 0.644701i) q^{12} +0.313353 q^{15} +(-0.0825793 - 0.996584i) q^{16} +0.165159 q^{17} +(0.0377177 + 0.0859877i) q^{18} +(-0.242192 - 0.222954i) q^{20} +(1.77550 - 0.778807i) q^{22} -1.35456 q^{23} +(-0.309080 + 0.900319i) q^{24} +0.891634 q^{25} -1.04127i q^{27} +(0.125873 + 0.286961i) q^{30} +(0.879474 - 0.475947i) q^{32} -1.84553 q^{33} +(0.0663435 + 0.151248i) q^{34} +(-0.0635942 + 0.0690818i) q^{36} -1.22843i q^{37} +(0.106888 - 0.311353i) q^{40} +1.57828 q^{41} +(1.42642 + 1.31311i) q^{44} +0.0309097i q^{45} +(-0.544122 - 1.24047i) q^{46} -1.09390 q^{47} +(-0.948644 + 0.0786068i) q^{48} +1.00000 q^{49} +(0.358165 + 0.816535i) q^{50} -0.157214i q^{51} +(0.953571 - 0.418275i) q^{54} +0.638232 q^{55} +(-0.212229 + 0.230542i) q^{60} +(0.789141 + 0.614213i) q^{64} +(-0.741343 - 1.69009i) q^{66} +(-0.111859 + 0.121511i) q^{68} +1.28940i q^{69} +(-0.0888088 - 0.0304881i) q^{72} -0.803391 q^{73} +(1.12496 - 0.493453i) q^{74} -0.848742i q^{75} +1.97272 q^{79} +(0.328065 - 0.0271842i) q^{80} -0.897287 q^{81} +(0.633988 + 1.44535i) q^{82} +0.0543685i q^{85} +(-0.629528 + 1.83375i) q^{88} +(-0.0283062 + 0.0124163i) q^{90} +(0.917421 - 0.996584i) q^{92} +(-0.439413 - 1.00176i) q^{94} +(-0.453052 - 0.837166i) q^{96} +(0.401695 + 0.915773i) q^{98} -0.182046i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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