LMFDB - Embedded newform 2872.1.b.e.717.8

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			717.8
Root			$0.677282 - 0.735724i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.245485 + 0.969400i) q^{2} +0.649399i q^{3} +(-0.879474 - 0.475947i) q^{4} +1.83155i q^{5} +(-0.629528 - 0.159418i) q^{6} +(0.677282 - 0.735724i) q^{8} +0.578281 q^{9} +O(q^{10})$	\(q+(-0.245485 + 0.969400i) q^{2} +0.649399i q^{3} +(-0.879474 - 0.475947i) q^{4} +1.83155i q^{5} +(-0.629528 - 0.159418i) q^{6} +(0.677282 - 0.735724i) q^{8} +0.578281 q^{9} +(-1.77550 - 0.449618i) q^{10} -0.329189i q^{11} +(0.309080 - 0.571129i) q^{12} -1.18940 q^{15} +(0.546948 + 0.837166i) q^{16} -1.09390 q^{17} +(-0.141960 + 0.560586i) q^{18} +(0.871720 - 1.61080i) q^{20} +(0.319116 + 0.0808112i) q^{22} -1.75895 q^{23} +(0.477778 + 0.439826i) q^{24} -2.35456 q^{25} +1.02493i q^{27} +(0.291982 - 1.15301i) q^{30} +(-0.945817 + 0.324699i) q^{32} +0.213775 q^{33} +(0.268536 - 1.06042i) q^{34} +(-0.508583 - 0.275231i) q^{36} +1.99317i q^{37} +(1.34751 + 1.24047i) q^{40} -0.165159 q^{41} +(-0.156677 + 0.289513i) q^{44} +1.05915i q^{45} +(0.431796 - 1.70512i) q^{46} -1.57828 q^{47} +(-0.543655 + 0.355188i) q^{48} +1.00000 q^{49} +(0.578011 - 2.28251i) q^{50} -0.710375i q^{51} +(-0.993571 - 0.251606i) q^{54} +0.602925 q^{55} +(1.04605 + 0.566094i) q^{60} +(-0.0825793 - 0.996584i) q^{64} +(-0.0524787 + 0.207234i) q^{66} +(0.962053 + 0.520637i) q^{68} -1.14226i q^{69} +(0.391659 - 0.425455i) q^{72} +0.490971 q^{73} +(-1.93218 - 0.489294i) q^{74} -1.52905i q^{75} +0.803391 q^{79} +(-1.53331 + 1.00176i) q^{80} -0.0873100 q^{81} +(0.0405441 - 0.160105i) q^{82} -2.00352i q^{85} +(-0.242192 - 0.222954i) q^{88} +(-1.02674 - 0.260006i) q^{90} +(1.54695 + 0.837166i) q^{92} +(0.387445 - 1.52999i) q^{94} +(-0.210859 - 0.614213i) q^{96} +(-0.245485 + 0.969400i) q^{98} -0.190364i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$2$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$3$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$3$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$4$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$5$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$5$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$6$	−0.629528	−	0.159418i	−0.629528	−	0.159418i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.677282	−	0.735724i	0.677282	−	0.735724i
$9$	0.578281			0.578281
$10$	−1.77550	−	0.449618i	−1.77550	−	0.449618i
$11$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$11$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$12$	0.309080	−	0.571129i	0.309080	−	0.571129i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−1.18940			−1.18940
$16$	0.546948	+	0.837166i	0.546948	+	0.837166i
$17$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$17$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$18$	−0.141960	+	0.560586i	−0.141960	+	0.560586i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	0.871720	−	1.61080i	0.871720	−	1.61080i
$21$		0			0
$22$	0.319116	+	0.0808112i	0.319116	+	0.0808112i
$23$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$23$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$24$	0.477778	+	0.439826i	0.477778	+	0.439826i
$25$	−2.35456			−2.35456
$26$		0			0
$27$			1.02493i			1.02493i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	0.291982	−	1.15301i	0.291982	−	1.15301i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.945817	+	0.324699i	−0.945817	+	0.324699i
$33$	0.213775			0.213775
$34$	0.268536	−	1.06042i	0.268536	−	1.06042i
$35$		0			0
$36$	−0.508583	−	0.275231i	−0.508583	−	0.275231i
$37$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$37$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$38$		0			0
$39$		0			0
$40$	1.34751	+	1.24047i	1.34751	+	1.24047i
$41$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$41$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.156677	+	0.289513i	−0.156677	+	0.289513i
$45$			1.05915i			1.05915i
$46$	0.431796	−	1.70512i	0.431796	−	1.70512i
$47$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$47$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$48$	−0.543655	+	0.355188i	−0.543655	+	0.355188i
$49$	1.00000			1.00000
$50$	0.578011	−	2.28251i	0.578011	−	2.28251i
$51$		−	0.710375i		−	0.710375i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.993571	−	0.251606i	−0.993571	−	0.251606i
$55$	0.602925			0.602925
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.04605	+	0.566094i	1.04605	+	0.566094i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$65$		0			0
$66$	−0.0524787	+	0.207234i	−0.0524787	+	0.207234i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	0.962053	+	0.520637i	0.962053	+	0.520637i
$69$		−	1.14226i		−	1.14226i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.391659	−	0.425455i	0.391659	−	0.425455i
$73$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$73$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$74$	−1.93218	−	0.489294i	−1.93218	−	0.489294i
$75$		−	1.52905i		−	1.52905i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$79$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$80$	−1.53331	+	1.00176i	−1.53331	+	1.00176i
$81$	−0.0873100			−0.0873100
$82$	0.0405441	−	0.160105i	0.0405441	−	0.160105i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	2.00352i		−	2.00352i
$86$		0			0
$87$		0			0
$88$	−0.242192	−	0.222954i	−0.242192	−	0.222954i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−1.02674	−	0.260006i	−1.02674	−	0.260006i
$91$		0			0
$92$	1.54695	+	0.837166i	1.54695	+	0.837166i
$93$		0			0
$94$	0.387445	−	1.52999i	0.387445	−	1.52999i
$95$		0			0
$96$	−0.210859	−	0.614213i	−0.210859	−	0.614213i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$99$		−	0.190364i		−	0.190364i
$100$	2.07078	+	1.12065i	2.07078	+	1.12065i
$101$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$101$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$102$	0.688638	+	0.174387i	0.688638	+	0.174387i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$107$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$108$	0.487815	−	0.901403i	0.487815	−	0.901403i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	−0.148009	+	0.584476i	−0.148009	+	0.584476i
$111$	−1.29436			−1.29436
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	3.22159i		−	3.22159i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.805562	+	0.875073i	−0.805562	+	0.875073i
$121$	0.891634			0.891634
$122$		0			0
$123$		−	0.107254i		−	0.107254i
$124$		0			0
$125$		−	2.48095i		−	2.48095i
$126$		0			0
$127$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$127$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$128$	0.986361	+	0.164595i	0.986361	+	0.164595i
$129$		0			0
$130$		0			0
$131$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$131$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$132$	−0.188010	−	0.101746i	−0.188010	−	0.101746i
$133$		0			0
$134$		0			0
$135$	−1.87721			−1.87721
$136$	−0.740876	+	0.804806i	−0.740876	+	0.804806i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	1.10731	+	0.280408i	1.10731	+	0.280408i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	1.02493i		−	1.02493i
$142$		0			0
$143$		0			0
$144$	0.316290	+	0.484117i	0.316290	+	0.484117i
$145$		0			0
$146$	−0.120526	+	0.475947i	−0.120526	+	0.475947i
$147$			0.649399i			0.649399i
$148$	0.948644	−	1.75294i	0.948644	−	1.75294i
$149$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$149$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$150$	1.48226	+	0.375360i	1.48226	+	0.375360i
$151$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$151$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$152$		0			0
$153$	−0.632579			−0.632579
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−0.197221	+	0.778807i	−0.197221	+	0.778807i
$159$		0			0
$160$	−0.594702	−	1.73231i	−0.594702	−	1.73231i
$161$		0			0
$162$	0.0214333	−	0.0846384i	0.0214333	−	0.0846384i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	0.145253	+	0.0786068i	0.145253	+	0.0786068i
$165$			0.391539i			0.391539i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	1.94221	+	0.491836i	1.94221	+	0.491836i
$171$		0			0
$172$		0			0
$173$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$173$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$174$		0			0
$175$		0			0
$176$	0.275586	−	0.180049i	0.275586	−	0.180049i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.504099	−	0.931493i	0.504099	−	0.931493i
$181$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$181$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$182$		0			0
$183$		0			0
$184$	−1.19130	+	1.29410i	−1.19130	+	1.29410i
$185$	−3.65058			−3.65058
$186$		0			0
$187$			0.360099i			0.360099i
$188$	1.38806	+	0.751179i	1.38806	+	0.751179i
$189$		0			0
$190$		0			0
$191$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$191$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$192$	0.647181	−	0.0536269i	0.647181	−	0.0536269i
$193$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$193$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$194$		0			0
$195$		0			0
$196$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	0.184539	+	0.0467316i	0.184539	+	0.0467316i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−1.59470	+	1.73231i	−1.59470	+	1.73231i
$201$		0			0
$202$	−1.42642	−	0.361219i	−1.42642	−	0.361219i
$203$		0			0
$204$	−0.338101	+	0.624756i	−0.338101	+	0.624756i
$205$		−	0.302496i		−	0.302496i
$206$		0			0
$207$	−1.01717			−1.01717
$208$		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
$214$	1.93218	+	0.489294i	1.93218	+	0.489294i
$215$		0			0
$216$	0.754068	+	0.694169i	0.754068	+	0.694169i
$217$		0			0
$218$		0			0
$219$			0.318836i			0.318836i
$220$	−0.530257	−	0.286961i	−0.530257	−	0.286961i
$221$		0			0
$222$	0.317747	−	1.25475i	0.317747	−	1.25475i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.36160			−1.36160
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$229$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$230$	3.12301	+	0.790855i	3.12301	+	0.790855i
$231$		0			0
$232$		0			0
$233$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$233$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$234$		0			0
$235$		−	2.89070i		−	2.89070i
$236$		0			0
$237$			0.521721i			0.521721i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.650543	−	0.995730i	−0.650543	−	0.995730i
$241$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$241$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$242$	−0.218883	+	0.864351i	−0.218883	+	0.864351i
$243$			0.968235i			0.968235i
$244$		0			0
$245$			1.83155i			1.83155i
$246$	0.103972	+	0.0263293i	0.103972	+	0.0263293i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	2.40503	+	0.609036i	2.40503	+	0.609036i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			0.579026i			0.579026i
$254$	−0.387445	+	1.52999i	−0.387445	+	1.52999i
$255$	1.30109			1.30109
$256$	−0.401695	+	0.915773i	−0.401695	+	0.915773i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.42642	−	0.361219i	−1.42642	−	0.361219i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	0.144786	−	0.157279i	0.144786	−	0.157279i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	0.460829	−	1.81977i	0.460829	−	1.81977i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.598305	−	0.915773i	−0.598305	−	0.915773i
$273$		0			0
$274$		0			0
$275$			0.775097i			0.775097i
$276$	−0.543655	+	1.00459i	−0.543655	+	1.00459i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$281$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$282$	0.993571	+	0.251606i	0.993571	+	0.251606i
$283$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$283$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.546948	+	0.187768i	−0.546948	+	0.187768i
$289$	0.196609			0.196609
$290$		0			0
$291$		0			0
$292$	−0.431796	−	0.233676i	−0.431796	−	0.233676i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.629528	−	0.159418i	−0.629528	−	0.159418i
$295$		0			0
$296$	1.46642	+	1.34994i	1.46642	+	1.34994i
$297$	0.337397			0.337397
$298$	−0.922767	−	0.233676i	−0.922767	−	0.233676i
$299$		0			0
$300$	−0.727748	+	1.34476i	−0.727748	+	1.34476i
$301$		0			0
$302$	−0.484275	+	1.91236i	−0.484275	+	1.91236i
$303$	−0.955557			−0.955557
$304$		0			0
$305$		0			0
$306$	0.155289	−	0.613223i	0.155289	−	0.613223i
$307$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$307$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$308$		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
$316$	−0.706561	−	0.382372i	−0.706561	−	0.382372i
$317$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$317$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$318$		0			0
$319$		0			0
$320$	1.82529	−	0.151248i	1.82529	−	0.151248i
$321$	1.29436			1.29436
$322$		0			0
$323$		0			0
$324$	0.0767869	+	0.0415550i	0.0767869	+	0.0415550i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.111859	+	0.121511i	−0.111859	+	0.121511i
$329$		0			0
$330$	−0.379558	−	0.0961172i	−0.379558	−	0.0961172i
$331$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$331$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$332$		0			0
$333$			1.15261i			1.15261i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.245485	−	0.969400i	0.245485	−	0.969400i
$339$		0			0
$340$	−0.953571	+	1.76205i	−0.953571	+	1.76205i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.09210			2.09210
$346$	1.19084	+	0.301561i	1.19084	+	0.301561i
$347$		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
$352$	0.106888	+	0.311353i	0.106888	+	0.311353i
$353$		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
$360$	0.779241	+	0.717342i	0.779241	+	0.717342i
$361$	−1.00000			−1.00000
$362$	−1.87947	−	0.475947i	−1.87947	−	0.475947i
$363$			0.579026i			0.579026i
$364$		0			0
$365$			0.899236i			0.899236i
$366$		0			0
$367$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$367$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$368$	−0.962053	−	1.47253i	−0.962053	−	1.47253i
$369$	−0.0955081			−0.0955081
$370$	0.896165	−	3.53888i	0.896165	−	3.53888i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−0.349080	−	0.0883990i	−0.349080	−	0.0883990i
$375$	1.61112			1.61112
$376$	−1.06894	+	1.16118i	−1.06894	+	1.16118i
$377$		0			0
$378$		0			0
$379$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$379$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$380$		0			0
$381$			1.02493i			1.02493i
$382$	−0.268536	+	1.06042i	−0.268536	+	1.06042i
$383$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$383$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$384$	−0.106888	+	0.640542i	−0.106888	+	0.640542i
$385$		0			0
$386$	0.484275	−	1.91236i	0.484275	−	1.91236i
$387$		0			0
$388$		0			0
$389$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$389$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$390$		0			0
$391$	1.92411			1.92411
$392$	0.677282	−	0.735724i	0.677282	−	0.735724i
$393$	−0.955557			−0.955557
$394$		0			0
$395$			1.47145i			1.47145i
$396$	−0.0906032	+	0.167420i	−0.0906032	+	0.167420i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−1.28782	−	1.97116i	−1.28782	−	1.97116i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	0.700332	−	1.29410i	0.700332	−	1.29410i
$405$		−	0.159912i		−	0.159912i
$406$		0			0
$407$	0.656130			0.656130
$408$	−0.522640	−	0.481124i	−0.522640	−	0.481124i
$409$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$409$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$410$	0.293240	+	0.0742583i	0.293240	+	0.0742583i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.249699	−	0.986041i	0.249699	−	0.986041i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$419$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−0.912690			−0.912690
$424$		0			0
$425$	2.57565			2.57565
$426$		0			0
$427$		0			0
$428$	−0.948644	+	1.75294i	−0.948644	+	1.75294i
$429$		0			0
$430$		0			0
$431$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$431$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$432$	−0.858040	+	0.560586i	−0.858040	+	0.560586i
$433$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$433$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.309080	−	0.0782696i	−0.309080	−	0.0782696i
$439$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$439$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$440$	0.408350	−	0.443587i	0.408350	−	0.443587i
$441$	0.578281			0.578281
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	1.13836	+	0.616048i	1.13836	+	0.616048i
$445$		0			0
$446$		0			0
$447$	−0.618159			−0.618159
$448$		0			0
$449$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$449$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$450$	0.334253	−	1.31993i	0.334253	−	1.31993i
$451$			0.0543685i			0.0543685i
$452$		0			0
$453$			1.28108i			1.28108i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$457$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$458$	−1.19084	−	0.301561i	−1.19084	−	0.301561i
$459$		−	1.12117i		−	1.12117i
$460$	−1.53331	+	2.83331i	−1.53331	+	2.83331i
$461$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$461$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.332526	+	1.31311i	−0.332526	+	1.31311i
$467$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$467$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$468$		0			0
$469$		0			0
$470$	2.80224	+	0.709624i	2.80224	+	0.709624i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.505757	−	0.128075i	−0.505757	−	0.128075i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$479$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$480$	1.12496	−	0.386199i	1.12496	−	0.386199i
$481$		0			0
$482$	−0.431796	+	1.70512i	−0.431796	+	1.70512i
$483$		0			0
$484$	−0.784169	−	0.424371i	−0.784169	−	0.424371i
$485$		0			0
$486$	−0.938607	−	0.237688i	−0.938607	−	0.237688i
$487$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$487$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$488$		0			0
$489$		0			0
$490$	−1.77550	−	0.449618i	−1.77550	−	0.449618i
$491$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$491$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$492$	−0.0510472	+	0.0943270i	−0.0510472	+	0.0943270i
$493$		0			0
$494$		0			0
$495$	0.348660			0.348660
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−1.18080	+	2.18193i	−1.18080	+	2.18193i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	−2.69503			−2.69503
$506$	−0.561308	−	0.142143i	−0.561308	−	0.142143i
$507$		−	0.649399i		−	0.649399i
$508$	−1.38806	−	0.751179i	−1.38806	−	0.751179i
$509$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$509$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$510$	−0.319398	+	1.26127i	−0.319398	+	1.26127i
$511$		0			0
$512$	−0.789141	−	0.614213i	−0.789141	−	0.614213i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			0.519553i			0.519553i
$518$		0			0
$519$	0.797738			0.797738
$520$		0			0
$521$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$521$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$522$		0			0
$523$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$523$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$524$	0.700332	−	1.29410i	0.700332	−	1.29410i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.116924	+	0.178965i	0.116924	+	0.178965i
$529$	2.09390			2.09390
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	3.65058			3.65058
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.329189i		−	0.329189i
$540$	1.65096	+	0.893455i	1.65096	+	0.893455i
$541$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$541$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$542$		0			0
$543$	−1.25906			−1.25906
$544$	1.03463	−	0.355188i	1.03463	−	0.355188i
$545$		0			0
$546$		0			0
$547$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$547$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$548$		0			0
$549$		0			0
$550$	−0.751379	−	0.190275i	−0.751379	−	0.190275i
$551$		0			0
$552$	−0.840387	−	0.773631i	−0.840387	−	0.773631i
$553$		0			0
$554$		0			0
$555$		−	2.37068i		−	2.37068i
$556$		0			0
$557$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$557$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.233848			−0.233848
$562$	−0.464369	+	1.83375i	−0.464369	+	1.83375i
$563$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$563$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$564$	−0.487815	+	0.901403i	−0.487815	+	0.901403i
$565$		0			0
$566$	−1.42642	−	0.361219i	−1.42642	−	0.361219i
$567$		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
$573$			0.710375i			0.710375i
$574$		0			0
$575$	4.14155			4.14155
$576$	−0.0477541	−	0.576306i	−0.0477541	−	0.576306i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.0482647	+	0.190593i	−0.0482647	+	0.190593i
$579$		−	1.28108i		−	1.28108i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.332526	−	0.361219i	0.332526	−	0.361219i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.309080	−	0.571129i	0.309080	−	0.571129i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.66861	+	1.09016i	−1.66861	+	1.09016i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.0828261	+	0.327073i	−0.0828261	+	0.327073i
$595$		0			0
$596$	0.453052	−	0.837166i	0.453052	−	0.837166i
$597$		0			0
$598$		0			0
$599$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$599$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$600$	−1.12496	−	1.03560i	−1.12496	−	1.03560i
$601$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$601$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$602$		0			0
$603$		0			0
$604$	−1.73496	−	0.938912i	−1.73496	−	0.938912i
$605$			1.63307i			1.63307i
$606$	0.234575	−	0.926317i	0.234575	−	0.926317i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.556337	+	0.301075i	0.556337	+	0.301075i
$613$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$613$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$614$	0.922767	+	0.233676i	0.922767	+	0.233676i
$615$	0.196440			0.196440
$616$		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
$621$		−	1.80281i		−	1.80281i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.18940			2.18940
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	2.18032i		−	2.18032i
$630$		0			0
$631$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$631$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$632$	0.544122	−	0.591074i	0.544122	−	0.591074i
$633$		0			0
$634$	−0.319116	−	0.0808112i	−0.319116	−	0.0808112i
$635$			2.89070i			2.89070i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.301463	+	1.80657i	−0.301463	+	1.80657i
$641$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$641$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$642$	−0.317747	+	1.25475i	−0.317747	+	1.25475i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$647$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$648$	−0.0591335	+	0.0642361i	−0.0591335	+	0.0642361i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$653$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$654$		0			0
$655$	−2.69503			−2.69503
$656$	−0.0903332	−	0.138265i	−0.0903332	−	0.138265i
$657$	0.283919			0.283919
$658$		0			0
$659$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$659$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$660$	0.186352	−	0.344348i	0.186352	−	0.344348i
$661$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$661$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$662$	1.77550	+	0.449618i	1.77550	+	0.449618i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.11734	−	0.282949i	−1.11734	−	0.282949i
$667$		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
$675$		−	2.41327i		−	2.41327i
$676$	0.879474	+	0.475947i	0.879474	+	0.475947i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	−1.47404	−	1.35695i	−1.47404	−	1.35695i
$681$		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
$687$	−0.797738			−0.797738
$688$		0			0
$689$		0			0
$690$	−0.513580	+	2.02808i	−0.513580	+	2.02808i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−0.584666	+	1.08037i	−0.584666	+	1.08037i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.180666			0.180666
$698$		0			0
$699$			0.879652i			0.879652i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.328065	+	0.0271842i	−0.328065	+	0.0271842i
$705$	1.87721			1.87721
$706$		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
$711$	0.464586			0.464586
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.245485	−	0.969400i	0.245485	−	0.969400i
$719$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$719$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$720$	−0.886684	+	0.579299i	−0.886684	+	0.579299i
$721$		0			0
$722$	0.245485	−	0.969400i	0.245485	−	0.969400i
$723$			1.14226i			1.14226i
$724$	0.922767	−	1.70512i	0.922767	−	1.70512i
$725$		0			0
$726$	−0.561308	−	0.142143i	−0.561308	−	0.142143i
$727$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$727$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$728$		0			0
$729$	−0.716081			−0.716081
$730$	−0.871720	−	0.220749i	−0.871720	−	0.220749i
$731$		0			0
$732$		0			0
$733$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$733$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$734$	−0.484275	+	1.91236i	−0.484275	+	1.91236i
$735$	−1.18940			−1.18940
$736$	1.66364	−	0.571129i	1.66364	−	0.571129i
$737$		0			0
$738$	0.0234459	−	0.0925856i	0.0234459	−	0.0925856i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	3.21059	+	1.73748i	3.21059	+	1.73748i
$741$		0			0
$742$		0			0
$743$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$743$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$744$		0			0
$745$	−1.74344			−1.74344
$746$		0			0
$747$		0			0
$748$	0.171388	−	0.316697i	0.171388	−	0.316697i
$749$		0			0
$750$	−0.395507	+	1.56182i	−0.395507	+	1.56182i
$751$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$751$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$752$	−0.863238	−	1.32128i	−0.863238	−	1.32128i
$753$		0			0
$754$		0			0
$755$			3.61313i			3.61313i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	1.62310	+	0.411024i	1.62310	+	0.411024i
$759$	−0.376019			−0.376019
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−0.993571	−	0.251606i	−0.993571	−	0.251606i
$763$		0			0
$764$	−0.962053	−	0.520637i	−0.962053	−	0.520637i
$765$		−	1.15860i		−	1.15860i
$766$	−0.332526	+	1.31311i	−0.332526	+	1.31311i
$767$		0			0
$768$	−0.594702	−	0.260861i	−0.594702	−	0.260861i
$769$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$769$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$770$		0			0
$771$		0			0
$772$	1.73496	+	0.938912i	1.73496	+	0.938912i
$773$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$773$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.93218	+	0.489294i	1.93218	+	0.489294i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.472340	+	1.86523i	−0.472340	+	1.86523i
$783$		0			0
$784$	0.546948	+	0.837166i	0.546948	+	0.837166i
$785$		0			0
$786$	0.234575	−	0.926317i	0.234575	−	0.926317i
$787$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$787$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$788$		0			0
$789$		0			0
$790$	−1.42642	−	0.361219i	−1.42642	−	0.361219i
$791$		0			0
$792$	−0.140055	−	0.128930i	−0.140055	−	0.128930i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$797$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$798$		0			0
$799$	1.72648			1.72648
$800$	2.22699	−	0.764525i	2.22699	−	0.764525i
$801$		0			0
$802$		0			0
$803$		−	0.161622i		−	0.161622i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.08258	+	0.996584i	1.08258	+	0.996584i
$809$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$809$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$810$	0.155019	+	0.0392562i	0.155019	+	0.0392562i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−0.161070	+	0.636052i	−0.161070	+	0.636052i
$815$		0			0
$816$	0.594702	−	0.388538i	0.594702	−	0.388538i
$817$		0			0
$818$	−0.0405441	+	0.160105i	−0.0405441	+	0.160105i
$819$		0			0
$820$	−0.143972	+	0.266037i	−0.143972	+	0.266037i
$821$		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
$825$	−0.503347			−0.503347
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.894571	+	0.484117i	0.894571	+	0.484117i
$829$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$829$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.09390			−1.09390
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.87947	−	0.475947i	−1.87947	−	0.475947i
$839$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$839$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			1.22843i			1.22843i
$844$		0			0
$845$		−	1.83155i		−	1.83155i
$846$	0.224052	−	0.884762i	0.224052	−	0.884762i
$847$		0			0
$848$		0			0
$849$	−0.955557			−0.955557
$850$	−0.632284	+	2.49683i	−0.632284	+	2.49683i
$851$		−	3.50588i		−	3.50588i
$852$		0			0
$853$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$853$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$854$		0			0
$855$		0			0
$856$	−1.46642	−	1.34994i	−1.46642	−	1.34994i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$859$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$860$		0			0
$861$		0			0
$862$	0.268536	−	1.06042i	0.268536	−	1.06042i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.332796	−	0.969400i	−0.332796	−	0.969400i
$865$	2.24992			2.24992
$866$	−0.197221	+	0.778807i	−0.197221	+	0.778807i
$867$			0.127678i			0.127678i
$868$		0			0
$869$		−	0.264468i		−	0.264468i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.151749	−	0.280408i	0.151749	−	0.280408i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	0.332526	−	1.31311i	0.332526	−	1.31311i
$879$		0			0
$880$	0.329769	+	0.504749i	0.329769	+	0.504749i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.141960	+	0.560586i	−0.141960	+	0.560586i
$883$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$883$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$887$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$888$	−0.876647	+	0.952293i	−0.876647	+	0.952293i
$889$		0			0
$890$		0			0
$891$			0.0287415i			0.0287415i
$892$		0			0
$893$		0			0
$894$	0.151749	−	0.599244i	0.151749	−	0.599244i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.120526	−	0.475947i	0.120526	−	0.475947i
$899$		0			0
$900$	1.19749	+	0.648050i	1.19749	+	0.648050i
$901$		0			0
$902$	−0.0527048	−	0.0133467i	−0.0527048	−	0.0133467i
$903$		0			0
$904$		0			0
$905$	−3.55100			−3.55100
$906$	−1.24188	−	0.314488i	−1.24188	−	0.314488i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			0.850910i			0.850910i
$910$		0			0
$911$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$911$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$912$		0			0
$913$		0			0
$914$	0.268536	−	1.06042i	0.268536	−	1.06042i
$915$		0			0
$916$	0.584666	−	1.08037i	0.584666	−	1.08037i
$917$		0			0
$918$	1.08686	+	0.275231i	1.08686	+	0.275231i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−2.37020	−	2.18193i	−2.37020	−	2.18193i
$921$	0.618159			0.618159
$922$	−0.629528	−	0.159418i	−0.629528	−	0.159418i
$923$		0			0
$924$		0			0
$925$		−	4.69304i		−	4.69304i
$926$		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
$932$	−1.19130	−	0.644701i	−1.19130	−	0.644701i
$933$		0			0
$934$	−1.19084	−	0.301561i	−1.19084	−	0.301561i
$935$	−0.659538			−0.659538
$936$		0			0
$937$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$937$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$938$		0			0
$939$		0			0
$940$	−1.37582	+	2.54229i	−1.37582	+	2.54229i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	0.290505			0.290505
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$947$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$948$	0.248312	−	0.458840i	0.248312	−	0.458840i
$949$		0			0
$950$		0			0
$951$	−0.213775			−0.213775
$952$		0			0
$953$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$953$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$954$		0			0
$955$			2.00352i			2.00352i
$956$		0			0
$957$		0			0
$958$	0.464369	−	1.83375i	0.464369	−	1.83375i
$959$		0			0
$960$	0.0982202	+	1.18534i	0.0982202	+	1.18534i
$961$	1.00000			1.00000
$962$		0			0
$963$		−	1.15261i		−	1.15261i
$964$	−1.54695	−	0.837166i	−1.54695	−	0.837166i
$965$		−	3.61313i		−	3.61313i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.603888	−	0.655997i	0.603888	−	0.655997i
$969$		0			0
$970$		0			0
$971$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$971$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$972$	0.460829	−	0.851537i	0.460829	−	0.851537i
$973$		0			0
$974$	0.464369	−	1.83375i	0.464369	−	1.83375i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	0.871720	−	1.61080i	0.871720	−	1.61080i
$981$		0			0
$982$	−1.93218	−	0.489294i	−1.93218	−	0.489294i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−0.0789092	−	0.0726411i	−0.0789092	−	0.0726411i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.0855910	+	0.337991i	−0.0855910	+	0.337991i
$991$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$991$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$992$		0			0
$993$	1.18940			1.18940
$994$		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
$999$	−2.04287			−2.04287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.8	yes	18
8.5	even	2	inner	2872.1.b.e.717.7	✓	18
359.358	odd	2	CM	2872.1.b.e.717.8	yes	18
2872.717	odd	2	inner	2872.1.b.e.717.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.7	✓	18	8.5	even	2	inner
2872.1.b.e.717.7	✓	18	2872.717	odd	2	inner
2872.1.b.e.717.8	yes	18	1.1	even	1	trivial
2872.1.b.e.717.8	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)

\(f(q)\)	\(=\)	\(q+(-0.245485 + 0.969400i) q^{2} +0.649399i q^{3} +(-0.879474 - 0.475947i) q^{4} +1.83155i q^{5} +(-0.629528 - 0.159418i) q^{6} +(0.677282 - 0.735724i) q^{8} +0.578281 q^{9} +O(q^{10})\)	\(q+(-0.245485 + 0.969400i) q^{2} +0.649399i q^{3} +(-0.879474 - 0.475947i) q^{4} +1.83155i q^{5} +(-0.629528 - 0.159418i) q^{6} +(0.677282 - 0.735724i) q^{8} +0.578281 q^{9} +(-1.77550 - 0.449618i) q^{10} -0.329189i q^{11} +(0.309080 - 0.571129i) q^{12} -1.18940 q^{15} +(0.546948 + 0.837166i) q^{16} -1.09390 q^{17} +(-0.141960 + 0.560586i) q^{18} +(0.871720 - 1.61080i) q^{20} +(0.319116 + 0.0808112i) q^{22} -1.75895 q^{23} +(0.477778 + 0.439826i) q^{24} -2.35456 q^{25} +1.02493i q^{27} +(0.291982 - 1.15301i) q^{30} +(-0.945817 + 0.324699i) q^{32} +0.213775 q^{33} +(0.268536 - 1.06042i) q^{34} +(-0.508583 - 0.275231i) q^{36} +1.99317i q^{37} +(1.34751 + 1.24047i) q^{40} -0.165159 q^{41} +(-0.156677 + 0.289513i) q^{44} +1.05915i q^{45} +(0.431796 - 1.70512i) q^{46} -1.57828 q^{47} +(-0.543655 + 0.355188i) q^{48} +1.00000 q^{49} +(0.578011 - 2.28251i) q^{50} -0.710375i q^{51} +(-0.993571 - 0.251606i) q^{54} +0.602925 q^{55} +(1.04605 + 0.566094i) q^{60} +(-0.0825793 - 0.996584i) q^{64} +(-0.0524787 + 0.207234i) q^{66} +(0.962053 + 0.520637i) q^{68} -1.14226i q^{69} +(0.391659 - 0.425455i) q^{72} +0.490971 q^{73} +(-1.93218 - 0.489294i) q^{74} -1.52905i q^{75} +0.803391 q^{79} +(-1.53331 + 1.00176i) q^{80} -0.0873100 q^{81} +(0.0405441 - 0.160105i) q^{82} -2.00352i q^{85} +(-0.242192 - 0.222954i) q^{88} +(-1.02674 - 0.260006i) q^{90} +(1.54695 + 0.837166i) q^{92} +(0.387445 - 1.52999i) q^{94} +(-0.210859 - 0.614213i) q^{96} +(-0.245485 + 0.969400i) q^{98} -0.190364i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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