LMFDB - Embedded newform 2872.1.b.e.717.15

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			717.15
Root			$0.0825793 - 0.996584i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.879474 - 0.475947i) q^{2} -1.22843i q^{3} +(0.546948 - 0.837166i) q^{4} +1.47145i q^{5} +(-0.584666 - 1.08037i) q^{6} +(0.0825793 - 0.996584i) q^{8} -0.509029 q^{9} +O(q^{10})$	\(q+(0.879474 - 0.475947i) q^{2} -1.22843i q^{3} +(0.546948 - 0.837166i) q^{4} +1.47145i q^{5} +(-0.584666 - 1.08037i) q^{6} +(0.0825793 - 0.996584i) q^{8} -0.509029 q^{9} +(0.700332 + 1.29410i) q^{10} -0.649399i q^{11} +(-1.02840 - 0.671885i) q^{12} +1.80756 q^{15} +(-0.401695 - 0.915773i) q^{16} +0.803391 q^{17} +(-0.447678 + 0.242271i) q^{18} +(1.23185 + 0.804806i) q^{20} +(-0.309080 - 0.571129i) q^{22} +1.09390 q^{23} +(-1.22423 - 0.101443i) q^{24} -1.16516 q^{25} -0.603121i q^{27} +(1.58971 - 0.860305i) q^{30} +(-0.789141 - 0.614213i) q^{32} -0.797738 q^{33} +(0.706561 - 0.382372i) q^{34} +(-0.278412 + 0.426142i) q^{36} +0.329189i q^{37} +(1.46642 + 0.121511i) q^{40} -1.97272 q^{41} +(-0.543655 - 0.355188i) q^{44} -0.749010i q^{45} +(0.962053 - 0.520637i) q^{46} -0.490971 q^{47} +(-1.12496 + 0.493453i) q^{48} +1.00000 q^{49} +(-1.02473 + 0.554554i) q^{50} -0.986906i q^{51} +(-0.287054 - 0.530429i) q^{54} +0.955557 q^{55} +(0.988644 - 1.51323i) q^{60} +(-0.986361 - 0.164595i) q^{64} +(-0.701590 + 0.379681i) q^{66} +(0.439413 - 0.672572i) q^{68} -1.34377i q^{69} +(-0.0420353 + 0.507290i) q^{72} -1.75895 q^{73} +(0.156677 + 0.289513i) q^{74} +1.43131i q^{75} +1.35456 q^{79} +(1.34751 - 0.591074i) q^{80} -1.24992 q^{81} +(-1.73496 + 0.938912i) q^{82} +1.18215i q^{85} +(-0.647181 - 0.0536269i) q^{88} +(-0.356489 - 0.658734i) q^{90} +(0.598305 - 0.915773i) q^{92} +(-0.431796 + 0.233676i) q^{94} +(-0.754515 + 0.969400i) q^{96} +(0.879474 - 0.475947i) q^{98} +0.330563i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.879474	−	0.475947i	0.879474	−	0.475947i
$2$	0.879474	−	0.475947i	0.879474	−	0.475947i
$3$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$3$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$4$	0.546948	−	0.837166i	0.546948	−	0.837166i
$5$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$5$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$6$	−0.584666	−	1.08037i	−0.584666	−	1.08037i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$9$	−0.509029			−0.509029
$10$	0.700332	+	1.29410i	0.700332	+	1.29410i
$11$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$11$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$12$	−1.02840	−	0.671885i	−1.02840	−	0.671885i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	1.80756			1.80756
$16$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$17$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$17$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$18$	−0.447678	+	0.242271i	−0.447678	+	0.242271i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	1.23185	+	0.804806i	1.23185	+	0.804806i
$21$		0			0
$22$	−0.309080	−	0.571129i	−0.309080	−	0.571129i
$23$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$23$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$24$	−1.22423	−	0.101443i	−1.22423	−	0.101443i
$25$	−1.16516			−1.16516
$26$		0			0
$27$		−	0.603121i		−	0.603121i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	1.58971	−	0.860305i	1.58971	−	0.860305i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.789141	−	0.614213i	−0.789141	−	0.614213i
$33$	−0.797738			−0.797738
$34$	0.706561	−	0.382372i	0.706561	−	0.382372i
$35$		0			0
$36$	−0.278412	+	0.426142i	−0.278412	+	0.426142i
$37$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$37$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$38$		0			0
$39$		0			0
$40$	1.46642	+	0.121511i	1.46642	+	0.121511i
$41$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$41$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.543655	−	0.355188i	−0.543655	−	0.355188i
$45$		−	0.749010i		−	0.749010i
$46$	0.962053	−	0.520637i	0.962053	−	0.520637i
$47$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$47$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$48$	−1.12496	+	0.493453i	−1.12496	+	0.493453i
$49$	1.00000			1.00000
$50$	−1.02473	+	0.554554i	−1.02473	+	0.554554i
$51$		−	0.986906i		−	0.986906i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.287054	−	0.530429i	−0.287054	−	0.530429i
$55$	0.955557			0.955557
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	0.988644	−	1.51323i	0.988644	−	1.51323i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$65$		0			0
$66$	−0.701590	+	0.379681i	−0.701590	+	0.379681i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	0.439413	−	0.672572i	0.439413	−	0.672572i
$69$		−	1.34377i		−	1.34377i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.0420353	+	0.507290i	−0.0420353	+	0.507290i
$73$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$73$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$74$	0.156677	+	0.289513i	0.156677	+	0.289513i
$75$			1.43131i			1.43131i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$79$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$80$	1.34751	−	0.591074i	1.34751	−	0.591074i
$81$	−1.24992			−1.24992
$82$	−1.73496	+	0.938912i	−1.73496	+	0.938912i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$			1.18215i			1.18215i
$86$		0			0
$87$		0			0
$88$	−0.647181	−	0.0536269i	−0.647181	−	0.0536269i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.356489	−	0.658734i	−0.356489	−	0.658734i
$91$		0			0
$92$	0.598305	−	0.915773i	0.598305	−	0.915773i
$93$		0			0
$94$	−0.431796	+	0.233676i	−0.431796	+	0.233676i
$95$		0			0
$96$	−0.754515	+	0.969400i	−0.754515	+	0.969400i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.879474	−	0.475947i	0.879474	−	0.475947i
$99$			0.330563i			0.330563i
$100$	−0.637281	+	0.975432i	−0.637281	+	0.975432i
$101$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$101$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$102$	−0.469715	−	0.867958i	−0.469715	−	0.867958i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$107$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$108$	−0.504913	−	0.329876i	−0.504913	−	0.329876i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	0.840387	−	0.454795i	0.840387	−	0.454795i
$111$	0.404384			0.404384
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			1.60961i			1.60961i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.149267	−	1.80139i	0.149267	−	1.80139i
$121$	0.578281			0.578281
$122$		0			0
$123$			2.42334i			2.42334i
$124$		0			0
$125$		−	0.243022i		−	0.243022i
$126$		0			0
$127$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$127$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$128$	−0.945817	+	0.324699i	−0.945817	+	0.324699i
$129$		0			0
$130$		0			0
$131$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$131$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$132$	−0.436321	+	0.667840i	−0.436321	+	0.667840i
$133$		0			0
$134$		0			0
$135$	0.887461			0.887461
$136$	0.0663435	−	0.800647i	0.0663435	−	0.800647i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−0.639564	−	1.18181i	−0.639564	−	1.18181i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			0.603121i			0.603121i
$142$		0			0
$143$		0			0
$144$	0.204475	+	0.466155i	0.204475	+	0.466155i
$145$		0			0
$146$	−1.54695	+	0.837166i	−1.54695	+	0.837166i
$147$		−	1.22843i		−	1.22843i
$148$	0.275586	+	0.180049i	0.275586	+	0.180049i
$149$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$149$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$150$	0.681229	+	1.25880i	0.681229	+	1.25880i
$151$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$151$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$152$		0			0
$153$	−0.408949			−0.408949
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	1.19130	−	0.644701i	1.19130	−	0.644701i
$159$		0			0
$160$	0.903782	−	1.16118i	0.903782	−	1.16118i
$161$		0			0
$162$	−1.09927	+	0.594895i	−1.09927	+	0.594895i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−1.07898	+	1.65150i	−1.07898	+	1.65150i
$165$		−	1.17383i		−	1.17383i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	0.562640	+	1.03967i	0.562640	+	1.03967i
$171$		0			0
$172$		0			0
$173$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$173$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$174$		0			0
$175$		0			0
$176$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−0.627046	−	0.409669i	−0.627046	−	0.409669i
$181$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$181$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$182$		0			0
$183$		0			0
$184$	0.0903332	−	1.09016i	0.0903332	−	1.09016i
$185$	−0.484385			−0.484385
$186$		0			0
$187$		−	0.521721i		−	0.521721i
$188$	−0.268536	+	0.411024i	−0.268536	+	0.411024i
$189$		0			0
$190$		0			0
$191$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$191$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$192$	−0.202192	+	1.21167i	−0.202192	+	1.21167i
$193$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$193$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$194$		0			0
$195$		0			0
$196$	0.546948	−	0.837166i	0.546948	−	0.837166i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	0.157331	+	0.290721i	0.157331	+	0.290721i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.0962180	+	1.16118i	−0.0962180	+	1.16118i
$201$		0			0
$202$	0.948644	+	1.75294i	0.948644	+	1.75294i
$203$		0			0
$204$	−0.826204	−	0.539786i	−0.826204	−	0.539786i
$205$		−	2.90276i		−	2.90276i
$206$		0			0
$207$	−0.556825			−0.556825
$208$		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
$214$	−0.156677	−	0.289513i	−0.156677	−	0.289513i
$215$		0			0
$216$	−0.601061	−	0.0498054i	−0.601061	−	0.0498054i
$217$		0			0
$218$		0			0
$219$			2.16074i			2.16074i
$220$	0.522640	−	0.799960i	0.522640	−	0.799960i
$221$		0			0
$222$	0.355645	−	0.192466i	0.355645	−	0.192466i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	0.593100			0.593100
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$229$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$230$	0.766090	+	1.41561i	0.766090	+	1.41561i
$231$		0			0
$232$		0			0
$233$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$233$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$234$		0			0
$235$		−	0.722438i		−	0.722438i
$236$		0			0
$237$		−	1.66398i		−	1.66398i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.726090	−	1.65532i	−0.726090	−	1.65532i
$241$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$241$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$242$	0.508583	−	0.275231i	0.508583	−	0.275231i
$243$			0.932310i			0.932310i
$244$		0			0
$245$			1.47145i			1.47145i
$246$	1.15338	+	2.13127i	1.15338	+	2.13127i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.115666	−	0.213732i	−0.115666	−	0.213732i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		−	0.710375i		−	0.710375i
$254$	0.431796	−	0.233676i	0.431796	−	0.233676i
$255$	1.45218			1.45218
$256$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.948644	+	1.75294i	0.948644	+	1.75294i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−0.0658767	+	0.795013i	−0.0658767	+	0.795013i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	0.780499	−	0.422385i	0.780499	−	0.422385i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.322718	−	0.735724i	−0.322718	−	0.735724i
$273$		0			0
$274$		0			0
$275$			0.756653i			0.756653i
$276$	−1.12496	−	0.734973i	−1.12496	−	0.734973i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$281$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$282$	0.287054	+	0.530429i	0.287054	+	0.530429i
$283$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$283$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.401695	+	0.312652i	0.401695	+	0.312652i
$289$	−0.354563			−0.354563
$290$		0			0
$291$		0			0
$292$	−0.962053	+	1.47253i	−0.962053	+	1.47253i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.584666	−	1.08037i	−0.584666	−	1.08037i
$295$		0			0
$296$	0.328065	+	0.0271842i	0.328065	+	0.0271842i
$297$	−0.391666			−0.391666
$298$	0.796894	+	1.47253i	0.796894	+	1.47253i
$299$		0			0
$300$	1.19825	+	0.782853i	1.19825	+	0.782853i
$301$		0			0
$302$	−1.66364	+	0.900319i	−1.66364	+	0.900319i
$303$	2.44846			2.44846
$304$		0			0
$305$		0			0
$306$	−0.359660	+	0.194638i	−0.359660	+	0.194638i
$307$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$307$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$308$		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
$316$	0.740876	−	1.13399i	0.740876	−	1.13399i
$317$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$317$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$318$		0			0
$319$		0			0
$320$	0.242192	−	1.45138i	0.242192	−	1.45138i
$321$	−0.404384			−0.404384
$322$		0			0
$323$		0			0
$324$	−0.683641	+	1.04639i	−0.683641	+	1.04639i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.162906	+	1.96598i	−0.162906	+	1.96598i
$329$		0			0
$330$	−0.558681	−	1.03235i	−0.558681	−	1.03235i
$331$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$331$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$332$		0			0
$333$		−	0.167567i		−	0.167567i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.879474	+	0.475947i	−0.879474	+	0.475947i
$339$		0			0
$340$	0.989654	+	0.646574i	0.989654	+	0.646574i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.97729			1.97729
$346$	0.922767	+	1.70512i	0.922767	+	1.70512i
$347$		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
$352$	−0.398869	+	0.512467i	−0.398869	+	0.512467i
$353$		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
$360$	−0.746451	−	0.0618527i	−0.746451	−	0.0618527i
$361$	−1.00000			−1.00000
$362$	−0.453052	−	0.837166i	−0.453052	−	0.837166i
$363$		−	0.710375i		−	0.710375i
$364$		0			0
$365$		−	2.58820i		−	2.58820i
$366$		0			0
$367$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$367$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$368$	−0.439413	−	1.00176i	−0.439413	−	1.00176i
$369$	1.00417			1.00417
$370$	−0.426004	+	0.230542i	−0.426004	+	0.230542i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−0.248312	−	0.458840i	−0.248312	−	0.458840i
$375$	−0.298535			−0.298535
$376$	−0.0405441	+	0.489294i	−0.0405441	+	0.489294i
$377$		0			0
$378$		0			0
$379$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$379$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$380$		0			0
$381$		−	0.603121i		−	0.603121i
$382$	−0.706561	+	0.382372i	−0.706561	+	0.382372i
$383$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$383$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$384$	0.398869	+	1.16187i	0.398869	+	1.16187i
$385$		0			0
$386$	1.66364	−	0.900319i	1.66364	−	0.900319i
$387$		0			0
$388$		0			0
$389$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$389$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$390$		0			0
$391$	0.878826			0.878826
$392$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$393$	2.44846			2.44846
$394$		0			0
$395$			1.99317i			1.99317i
$396$	0.276736	+	0.180801i	0.276736	+	0.180801i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	0.468039	+	1.06702i	0.468039	+	1.06702i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	1.66861	+	1.09016i	1.66861	+	1.09016i
$405$		−	1.83919i		−	1.83919i
$406$		0			0
$407$	0.213775			0.213775
$408$	−0.983535	−	0.0814980i	−0.983535	−	0.0814980i
$409$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$409$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$410$	−1.38156	−	2.55290i	−1.38156	−	2.55290i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.489713	+	0.265019i	−0.489713	+	0.265019i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$419$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	0.249918			0.249918
$424$		0			0
$425$	−0.936078			−0.936078
$426$		0			0
$427$		0			0
$428$	−0.275586	−	0.180049i	−0.275586	−	0.180049i
$429$		0			0
$430$		0			0
$431$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$431$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$432$	−0.552322	+	0.242271i	−0.552322	+	0.242271i
$433$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$433$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.02840	+	1.90031i	1.02840	+	1.90031i
$439$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$439$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$440$	0.0789092	−	0.952293i	0.0789092	−	0.952293i
$441$	−0.509029			−0.509029
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	0.221177	−	0.338537i	0.221177	−	0.338537i
$445$		0			0
$446$		0			0
$447$	2.05679			2.05679
$448$		0			0
$449$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$449$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$450$	0.521616	−	0.282284i	0.521616	−	0.282284i
$451$			1.28108i			1.28108i
$452$		0			0
$453$			2.32373i			2.32373i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$457$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$458$	−0.922767	−	1.70512i	−0.922767	−	1.70512i
$459$		−	0.484542i		−	0.484542i
$460$	1.34751	+	0.880374i	1.34751	+	0.880374i
$461$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$461$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	0.145253	−	0.0786068i	0.145253	−	0.0786068i
$467$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$467$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$468$		0			0
$469$		0			0
$470$	−0.343843	−	0.635365i	−0.343843	−	0.635365i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.791967	−	1.46343i	−0.791967	−	1.46343i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$479$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$480$	−1.42642	−	1.11023i	−1.42642	−	1.11023i
$481$		0			0
$482$	−0.962053	+	0.520637i	−0.962053	+	0.520637i
$483$		0			0
$484$	0.316290	−	0.484117i	0.316290	−	0.484117i
$485$		0			0
$486$	0.443731	+	0.819943i	0.443731	+	0.819943i
$487$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$487$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$488$		0			0
$489$		0			0
$490$	0.700332	+	1.29410i	0.700332	+	1.29410i
$491$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$491$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$492$	2.02874	+	1.32544i	2.02874	+	1.32544i
$493$		0			0
$494$		0			0
$495$	−0.486406			−0.486406
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.203450	−	0.132921i	−0.203450	−	0.132921i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	−2.93284			−2.93284
$506$	−0.338101	−	0.624756i	−0.338101	−	0.624756i
$507$			1.22843i			1.22843i
$508$	0.268536	−	0.411024i	0.268536	−	0.411024i
$509$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$509$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$510$	1.27715	−	0.691161i	1.27715	−	0.691161i
$511$		0			0
$512$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			0.318836i			0.318836i
$518$		0			0
$519$	2.38167			2.38167
$520$		0			0
$521$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$521$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$522$		0			0
$523$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$523$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$524$	1.66861	+	1.09016i	1.66861	+	1.09016i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.320448	+	0.730547i	0.320448	+	0.730547i
$529$	0.196609			0.196609
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.484385			0.484385
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.649399i		−	0.649399i
$540$	0.485395	−	0.742953i	0.485395	−	0.742953i
$541$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$541$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$542$		0			0
$543$	−1.16933			−1.16933
$544$	−0.633988	−	0.493453i	−0.633988	−	0.493453i
$545$		0			0
$546$		0			0
$547$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$547$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$548$		0			0
$549$		0			0
$550$	0.360127	+	0.665456i	0.360127	+	0.665456i
$551$		0			0
$552$	−1.33918	−	0.110968i	−1.33918	−	0.110968i
$553$		0			0
$554$		0			0
$555$			0.595030i			0.595030i
$556$		0			0
$557$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$557$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.640896			−0.640896
$562$	1.38806	−	0.751179i	1.38806	−	0.751179i
$563$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$563$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$564$	0.504913	+	0.329876i	0.504913	+	0.329876i
$565$		0			0
$566$	0.948644	+	1.75294i	0.948644	+	1.75294i
$567$		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
$573$			0.986906i			0.986906i
$574$		0			0
$575$	−1.27456			−1.27456
$576$	0.502087	+	0.0837834i	0.502087	+	0.0837834i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.311829	+	0.168753i	−0.311829	+	0.168753i
$579$		−	2.32373i		−	2.32373i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.145253	+	1.75294i	−0.145253	+	1.75294i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−1.02840	−	0.671885i	−1.02840	−	0.671885i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.301463	−	0.132234i	0.301463	−	0.132234i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.344460	+	0.186413i	−0.344460	+	0.186413i
$595$		0			0
$596$	1.40170	+	0.915773i	1.40170	+	0.915773i
$597$		0			0
$598$		0			0
$599$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$599$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$600$	1.42642	+	0.118197i	1.42642	+	0.118197i
$601$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$601$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$602$		0			0
$603$		0			0
$604$	−1.03463	+	1.58361i	−1.03463	+	1.58361i
$605$			0.850910i			0.850910i
$606$	2.15336	−	1.16534i	2.15336	−	1.16534i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.223674	+	0.342359i	−0.223674	+	0.342359i
$613$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$613$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$614$	−0.796894	−	1.47253i	−0.796894	−	1.47253i
$615$	−3.56582			−3.56582
$616$		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
$621$		−	0.659752i		−	0.659752i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.807564			−0.807564
$626$		0			0
$627$		0			0
$628$		0			0
$629$			0.264468i			0.264468i
$630$		0			0
$631$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$631$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$632$	0.111859	−	1.34994i	0.111859	−	1.34994i
$633$		0			0
$634$	0.309080	+	0.571129i	0.309080	+	0.571129i
$635$			0.722438i			0.722438i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.477778	−	1.39172i	−0.477778	−	1.39172i
$641$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$641$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$642$	−0.355645	+	0.192466i	−0.355645	+	0.192466i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$647$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$648$	−0.103217	+	1.24565i	−0.103217	+	1.24565i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$653$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$654$		0			0
$655$	−2.93284			−2.93284
$656$	0.792434	+	1.80657i	0.792434	+	1.80657i
$657$	0.895355			0.895355
$658$		0			0
$659$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$659$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$660$	−0.982691	−	0.642024i	−0.982691	−	0.642024i
$661$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$661$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$662$	−0.700332	−	1.29410i	−0.700332	−	1.29410i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0797530	−	0.147371i	−0.0797530	−	0.147371i
$667$		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
$675$			0.702732i			0.702732i
$676$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	1.17811	+	0.0976210i	1.17811	+	0.0976210i
$681$		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
$687$	−2.38167			−2.38167
$688$		0			0
$689$		0			0
$690$	1.73897	−	0.941085i	1.73897	−	0.941085i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	1.62310	+	1.06042i	1.62310	+	1.06042i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.58487			−1.58487
$698$		0			0
$699$		−	0.202885i		−	0.202885i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.106888	+	0.640542i	−0.106888	+	0.640542i
$705$	−0.887461			−0.887461
$706$		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
$711$	−0.689512			−0.689512
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.879474	+	0.475947i	−0.879474	+	0.475947i
$719$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$719$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$720$	−0.685923	+	0.300874i	−0.685923	+	0.300874i
$721$		0			0
$722$	−0.879474	+	0.475947i	−0.879474	+	0.475947i
$723$			1.34377i			1.34377i
$724$	−0.796894	−	0.520637i	−0.796894	−	0.520637i
$725$		0			0
$726$	−0.338101	−	0.624756i	−0.338101	−	0.624756i
$727$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$727$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$728$		0			0
$729$	−0.104645			−0.104645
$730$	−1.23185	−	2.27625i	−1.23185	−	2.27625i
$731$		0			0
$732$		0			0
$733$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$733$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$734$	−1.66364	+	0.900319i	−1.66364	+	0.900319i
$735$	1.80756			1.80756
$736$	−0.863238	−	0.671885i	−0.863238	−	0.671885i
$737$		0			0
$738$	0.883144	−	0.477934i	0.883144	−	0.477934i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−0.264933	+	0.405511i	−0.264933	+	0.405511i
$741$		0			0
$742$		0			0
$743$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$743$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$744$		0			0
$745$	−2.46369			−2.46369
$746$		0			0
$747$		0			0
$748$	−0.436767	−	0.285354i	−0.436767	−	0.285354i
$749$		0			0
$750$	−0.262554	+	0.142087i	−0.262554	+	0.142087i
$751$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$751$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$752$	0.197221	+	0.449618i	0.197221	+	0.449618i
$753$		0			0
$754$		0			0
$755$		−	2.78344i		−	2.78344i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	0.871720	+	1.61080i	0.871720	+	1.61080i
$759$	−0.872643			−0.872643
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−0.287054	−	0.530429i	−0.287054	−	0.530429i
$763$		0			0
$764$	−0.439413	+	0.672572i	−0.439413	+	0.672572i
$765$		−	0.601747i		−	0.601747i
$766$	0.145253	−	0.0786068i	0.145253	−	0.0786068i
$767$		0			0
$768$	0.903782	+	0.831990i	0.903782	+	0.831990i
$769$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$769$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$770$		0			0
$771$		0			0
$772$	1.03463	−	1.58361i	1.03463	−	1.58361i
$773$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$773$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.156677	−	0.289513i	−0.156677	−	0.289513i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.772905	−	0.418275i	0.772905	−	0.418275i
$783$		0			0
$784$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$785$		0			0
$786$	2.15336	−	1.16534i	2.15336	−	1.16534i
$787$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$787$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$788$		0			0
$789$		0			0
$790$	0.948644	+	1.75294i	0.948644	+	1.75294i
$791$		0			0
$792$	0.329434	+	0.0272977i	0.329434	+	0.0272977i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$797$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$798$		0			0
$799$	−0.394442			−0.394442
$800$	0.919474	+	0.715655i	0.919474	+	0.715655i
$801$		0			0
$802$		0			0
$803$			1.14226i			1.14226i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.98636	+	0.164595i	1.98636	+	0.164595i
$809$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$809$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$810$	−0.875358	−	1.61752i	−0.875358	−	1.61752i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	0.188010	−	0.101746i	0.188010	−	0.101746i
$815$		0			0
$816$	−0.903782	+	0.396436i	−0.903782	+	0.396436i
$817$		0			0
$818$	1.73496	−	0.938912i	1.73496	−	0.938912i
$819$		0			0
$820$	−2.43009	−	1.58766i	−2.43009	−	1.58766i
$821$		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
$825$	0.929492			0.929492
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−0.304554	+	0.466155i	−0.304554	+	0.466155i
$829$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$829$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.803391			0.803391
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.453052	−	0.837166i	−0.453052	−	0.837166i
$839$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$839$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$		−	1.93880i		−	1.93880i
$844$		0			0
$845$		−	1.47145i		−	1.47145i
$846$	0.219797	−	0.118948i	0.219797	−	0.118948i
$847$		0			0
$848$		0			0
$849$	2.44846			2.44846
$850$	−0.823256	+	0.445524i	−0.823256	+	0.445524i
$851$			0.360099i			0.360099i
$852$		0			0
$853$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$853$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$854$		0			0
$855$		0			0
$856$	−0.328065	−	0.0271842i	−0.328065	−	0.0271842i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$859$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$860$		0			0
$861$		0			0
$862$	0.706561	−	0.382372i	0.706561	−	0.382372i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.370445	+	0.475947i	−0.370445	+	0.475947i
$865$	−2.85284			−2.85284
$866$	1.19130	−	0.644701i	1.19130	−	0.644701i
$867$			0.435554i			0.435554i
$868$		0			0
$869$		−	0.879652i		−	0.879652i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.80890	+	1.18181i	1.80890	+	1.18181i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−0.145253	+	0.0786068i	−0.145253	+	0.0786068i
$879$		0			0
$880$	−0.383843	−	0.875073i	−0.383843	−	0.875073i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.447678	+	0.242271i	−0.447678	+	0.242271i
$883$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$883$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$887$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$888$	0.0333938	−	0.403003i	0.0333938	−	0.403003i
$889$		0			0
$890$		0			0
$891$			0.811696i			0.811696i
$892$		0			0
$893$		0			0
$894$	1.80890	−	0.978925i	1.80890	−	0.978925i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.54695	−	0.837166i	1.54695	−	0.837166i
$899$		0			0
$900$	0.324395	−	0.496523i	0.324395	−	0.496523i
$901$		0			0
$902$	0.609729	+	1.12668i	0.609729	+	1.12668i
$903$		0			0
$904$		0			0
$905$	1.40066			1.40066
$906$	1.10597	+	2.04366i	1.10597	+	2.04366i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		−	1.01458i		−	1.01458i
$910$		0			0
$911$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$911$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$912$		0			0
$913$		0			0
$914$	0.706561	−	0.382372i	0.706561	−	0.382372i
$915$		0			0
$916$	−1.62310	−	1.06042i	−1.62310	−	1.06042i
$917$		0			0
$918$	−0.230617	−	0.426142i	−0.230617	−	0.426142i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	1.60411	+	0.132921i	1.60411	+	0.132921i
$921$	−2.05679			−2.05679
$922$	−0.584666	−	1.08037i	−0.584666	−	1.08037i
$923$		0			0
$924$		0			0
$925$		−	0.383558i		−	0.383558i
$926$		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
$932$	0.0903332	−	0.138265i	0.0903332	−	0.138265i
$933$		0			0
$934$	−0.922767	−	1.70512i	−0.922767	−	1.70512i
$935$	0.767685			0.767685
$936$		0			0
$937$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$937$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$938$		0			0
$939$		0			0
$940$	−0.604801	−	0.395136i	−0.604801	−	0.395136i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−2.15795			−2.15795
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$947$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$948$	−1.39303	−	0.910111i	−1.39303	−	0.910111i
$949$		0			0
$950$		0			0
$951$	0.797738			0.797738
$952$		0			0
$953$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$953$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$954$		0			0
$955$		−	1.18215i		−	1.18215i
$956$		0			0
$957$		0			0
$958$	−1.38806	+	0.751179i	−1.38806	+	0.751179i
$959$		0			0
$960$	−1.78291	−	0.297515i	−1.78291	−	0.297515i
$961$	1.00000			1.00000
$962$		0			0
$963$			0.167567i			0.167567i
$964$	−0.598305	+	0.915773i	−0.598305	+	0.915773i
$965$			2.78344i			2.78344i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.0477541	−	0.576306i	0.0477541	−	0.576306i
$969$		0			0
$970$		0			0
$971$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$971$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$972$	0.780499	+	0.509925i	0.780499	+	0.509925i
$973$		0			0
$974$	−1.38806	+	0.751179i	−1.38806	+	0.751179i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	1.23185	+	0.804806i	1.23185	+	0.804806i
$981$		0			0
$982$	0.156677	+	0.289513i	0.156677	+	0.289513i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	2.41507	+	0.200118i	2.41507	+	0.200118i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.427781	+	0.231504i	−0.427781	+	0.231504i
$991$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$991$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$992$		0			0
$993$	−1.80756			−1.80756
$994$		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
$999$	0.198541			0.198541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.15	✓	18
8.5	even	2	inner	2872.1.b.e.717.16	yes	18
359.358	odd	2	CM	2872.1.b.e.717.15	✓	18
2872.717	odd	2	inner	2872.1.b.e.717.16	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.15	✓	18	1.1	even	1	trivial
2872.1.b.e.717.15	✓	18	359.358	odd	2	CM
2872.1.b.e.717.16	yes	18	8.5	even	2	inner
2872.1.b.e.717.16	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)

\(f(q)\)	\(=\)	\(q+(0.879474 - 0.475947i) q^{2} -1.22843i q^{3} +(0.546948 - 0.837166i) q^{4} +1.47145i q^{5} +(-0.584666 - 1.08037i) q^{6} +(0.0825793 - 0.996584i) q^{8} -0.509029 q^{9} +O(q^{10})\)	\(q+(0.879474 - 0.475947i) q^{2} -1.22843i q^{3} +(0.546948 - 0.837166i) q^{4} +1.47145i q^{5} +(-0.584666 - 1.08037i) q^{6} +(0.0825793 - 0.996584i) q^{8} -0.509029 q^{9} +(0.700332 + 1.29410i) q^{10} -0.649399i q^{11} +(-1.02840 - 0.671885i) q^{12} +1.80756 q^{15} +(-0.401695 - 0.915773i) q^{16} +0.803391 q^{17} +(-0.447678 + 0.242271i) q^{18} +(1.23185 + 0.804806i) q^{20} +(-0.309080 - 0.571129i) q^{22} +1.09390 q^{23} +(-1.22423 - 0.101443i) q^{24} -1.16516 q^{25} -0.603121i q^{27} +(1.58971 - 0.860305i) q^{30} +(-0.789141 - 0.614213i) q^{32} -0.797738 q^{33} +(0.706561 - 0.382372i) q^{34} +(-0.278412 + 0.426142i) q^{36} +0.329189i q^{37} +(1.46642 + 0.121511i) q^{40} -1.97272 q^{41} +(-0.543655 - 0.355188i) q^{44} -0.749010i q^{45} +(0.962053 - 0.520637i) q^{46} -0.490971 q^{47} +(-1.12496 + 0.493453i) q^{48} +1.00000 q^{49} +(-1.02473 + 0.554554i) q^{50} -0.986906i q^{51} +(-0.287054 - 0.530429i) q^{54} +0.955557 q^{55} +(0.988644 - 1.51323i) q^{60} +(-0.986361 - 0.164595i) q^{64} +(-0.701590 + 0.379681i) q^{66} +(0.439413 - 0.672572i) q^{68} -1.34377i q^{69} +(-0.0420353 + 0.507290i) q^{72} -1.75895 q^{73} +(0.156677 + 0.289513i) q^{74} +1.43131i q^{75} +1.35456 q^{79} +(1.34751 - 0.591074i) q^{80} -1.24992 q^{81} +(-1.73496 + 0.938912i) q^{82} +1.18215i q^{85} +(-0.647181 - 0.0536269i) q^{88} +(-0.356489 - 0.658734i) q^{90} +(0.598305 - 0.915773i) q^{92} +(-0.431796 + 0.233676i) q^{94} +(-0.754515 + 0.969400i) q^{96} +(0.879474 - 0.475947i) q^{98} +0.330563i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more