LMFDB - Embedded newform 2872.1.b.e.717.1

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			717.1
Root			$-0.546948 - 0.837166i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.945817 - 0.324699i) q^{2} -1.99317i q^{3} +(0.789141 + 0.614213i) q^{4} +0.951895i q^{5} +(-0.647181 + 1.88517i) q^{6} +(-0.546948 - 0.837166i) q^{8} -2.97272 q^{9} +O(q^{10})$	\(q+(-0.945817 - 0.324699i) q^{2} -1.99317i q^{3} +(0.789141 + 0.614213i) q^{4} +0.951895i q^{5} +(-0.647181 + 1.88517i) q^{6} +(-0.546948 - 0.837166i) q^{8} -2.97272 q^{9} +(0.309080 - 0.900319i) q^{10} +1.47145i q^{11} +(1.22423 - 1.57289i) q^{12} +1.89729 q^{15} +(0.245485 + 0.969400i) q^{16} -0.490971 q^{17} +(2.81165 + 0.965241i) q^{18} +(-0.584666 + 0.751179i) q^{20} +(0.477778 - 1.39172i) q^{22} +1.57828 q^{23} +(-1.66861 + 1.09016i) q^{24} +0.0938963 q^{25} +3.93197i q^{27} +(-1.79449 - 0.616048i) q^{30} +(0.0825793 - 0.996584i) q^{32} +2.93284 q^{33} +(0.464369 + 0.159418i) q^{34} +(-2.34590 - 1.82588i) q^{36} -1.83155i q^{37} +(0.796894 - 0.520637i) q^{40} -0.803391 q^{41} +(-0.903782 + 1.16118i) q^{44} -2.82972i q^{45} +(-1.49277 - 0.512467i) q^{46} +1.97272 q^{47} +(1.93218 - 0.489294i) q^{48} +1.00000 q^{49} +(-0.0888088 - 0.0304881i) q^{50} +0.978588i q^{51} +(1.27671 - 3.71892i) q^{54} -1.40066 q^{55} +(1.49723 + 1.16534i) q^{60} +(-0.401695 + 0.915773i) q^{64} +(-2.77393 - 0.952293i) q^{66} +(-0.387445 - 0.301561i) q^{68} -3.14578i q^{69} +(1.62593 + 2.48866i) q^{72} +1.89163 q^{73} +(-0.594702 + 1.73231i) q^{74} -0.187151i q^{75} +1.75895 q^{79} +(-0.922767 + 0.233676i) q^{80} +4.86436 q^{81} +(0.759861 + 0.260861i) q^{82} -0.467353i q^{85} +(1.23185 - 0.804806i) q^{88} +(-0.918808 + 2.67640i) q^{90} +(1.24549 + 0.969400i) q^{92} +(-1.86584 - 0.640542i) q^{94} +(-1.98636 - 0.164595i) q^{96} +(-0.945817 - 0.324699i) q^{98} -4.37421i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$2$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$3$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$3$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$4$	0.789141	+	0.614213i	0.789141	+	0.614213i
$5$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$5$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$6$	−0.647181	+	1.88517i	−0.647181	+	1.88517i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.546948	−	0.837166i	−0.546948	−	0.837166i
$9$	−2.97272			−2.97272
$10$	0.309080	−	0.900319i	0.309080	−	0.900319i
$11$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$11$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$12$	1.22423	−	1.57289i	1.22423	−	1.57289i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	1.89729			1.89729
$16$	0.245485	+	0.969400i	0.245485	+	0.969400i
$17$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$17$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$18$	2.81165	+	0.965241i	2.81165	+	0.965241i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.584666	+	0.751179i	−0.584666	+	0.751179i
$21$		0			0
$22$	0.477778	−	1.39172i	0.477778	−	1.39172i
$23$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$23$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$24$	−1.66861	+	1.09016i	−1.66861	+	1.09016i
$25$	0.0938963			0.0938963
$26$		0			0
$27$			3.93197i			3.93197i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−1.79449	−	0.616048i	−1.79449	−	0.616048i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$33$	2.93284			2.93284
$34$	0.464369	+	0.159418i	0.464369	+	0.159418i
$35$		0			0
$36$	−2.34590	−	1.82588i	−2.34590	−	1.82588i
$37$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$37$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$38$		0			0
$39$		0			0
$40$	0.796894	−	0.520637i	0.796894	−	0.520637i
$41$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$41$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.903782	+	1.16118i	−0.903782	+	1.16118i
$45$		−	2.82972i		−	2.82972i
$46$	−1.49277	−	0.512467i	−1.49277	−	0.512467i
$47$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$47$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$48$	1.93218	−	0.489294i	1.93218	−	0.489294i
$49$	1.00000			1.00000
$50$	−0.0888088	−	0.0304881i	−0.0888088	−	0.0304881i
$51$			0.978588i			0.978588i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	1.27671	−	3.71892i	1.27671	−	3.71892i
$55$	−1.40066			−1.40066
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.49723	+	1.16534i	1.49723	+	1.16534i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.401695	+	0.915773i	−0.401695	+	0.915773i
$65$		0			0
$66$	−2.77393	−	0.952293i	−2.77393	−	0.952293i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.387445	−	0.301561i	−0.387445	−	0.301561i
$69$		−	3.14578i		−	3.14578i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1.62593	+	2.48866i	1.62593	+	2.48866i
$73$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$73$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$74$	−0.594702	+	1.73231i	−0.594702	+	1.73231i
$75$		−	0.187151i		−	0.187151i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$79$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$80$	−0.922767	+	0.233676i	−0.922767	+	0.233676i
$81$	4.86436			4.86436
$82$	0.759861	+	0.260861i	0.759861	+	0.260861i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	0.467353i		−	0.467353i
$86$		0			0
$87$		0			0
$88$	1.23185	−	0.804806i	1.23185	−	0.804806i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.918808	+	2.67640i	−0.918808	+	2.67640i
$91$		0			0
$92$	1.24549	+	0.969400i	1.24549	+	0.969400i
$93$		0			0
$94$	−1.86584	−	0.640542i	−1.86584	−	0.640542i
$95$		0			0
$96$	−1.98636	−	0.164595i	−1.98636	−	0.164595i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$99$		−	4.37421i		−	4.37421i
$100$	0.0740974	+	0.0576723i	0.0740974	+	0.0576723i
$101$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$101$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$102$	0.317747	−	0.925566i	0.317747	−	0.925566i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$107$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$108$	−2.41507	+	3.10288i	−2.41507	+	3.10288i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	1.32477	+	0.454795i	1.32477	+	0.454795i
$111$	−3.65058			−3.65058
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			1.50236i			1.50236i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−1.03772	−	1.58835i	−1.03772	−	1.58835i
$121$	−1.16516			−1.16516
$122$		0			0
$123$			1.60129i			1.60129i
$124$		0			0
$125$			1.04127i			1.04127i
$126$		0			0
$127$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$127$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$128$	0.677282	−	0.735724i	0.677282	−	0.735724i
$129$		0			0
$130$		0			0
$131$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$131$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$132$	2.31443	+	1.80139i	2.31443	+	1.80139i
$133$		0			0
$134$		0			0
$135$	−3.74282			−3.74282
$136$	0.268536	+	0.411024i	0.268536	+	0.411024i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−1.02143	+	2.97533i	−1.02143	+	2.97533i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	3.93197i		−	3.93197i
$142$		0			0
$143$		0			0
$144$	−0.729760	−	2.88176i	−0.729760	−	2.88176i
$145$		0			0
$146$	−1.78914	−	0.614213i	−1.78914	−	0.614213i
$147$		−	1.99317i		−	1.99317i
$148$	1.12496	−	1.44535i	1.12496	−	1.44535i
$149$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$149$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$150$	−0.0607679	+	0.177011i	−0.0607679	+	0.177011i
$151$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$151$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$152$		0			0
$153$	1.45952			1.45952
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.66364	−	0.571129i	−1.66364	−	0.571129i
$159$		0			0
$160$	0.948644	+	0.0786068i	0.948644	+	0.0786068i
$161$		0			0
$162$	−4.60079	−	1.57945i	−4.60079	−	1.57945i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−0.633988	−	0.493453i	−0.633988	−	0.493453i
$165$			2.79176i			2.79176i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−0.151749	+	0.442030i	−0.151749	+	0.442030i
$171$		0			0
$172$		0			0
$173$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$173$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$174$		0			0
$175$		0			0
$176$	−1.42642	+	0.361219i	−1.42642	+	0.361219i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	1.73805	−	2.23305i	1.73805	−	2.23305i
$181$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$181$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$182$		0			0
$183$		0			0
$184$	−0.863238	−	1.32128i	−0.863238	−	1.32128i
$185$	1.74344			1.74344
$186$		0			0
$187$		−	0.722438i		−	0.722438i
$188$	1.55676	+	1.21167i	1.55676	+	1.21167i
$189$		0			0
$190$		0			0
$191$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$191$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$192$	1.82529	+	0.800647i	1.82529	+	0.800647i
$193$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$193$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$194$		0			0
$195$		0			0
$196$	0.789141	+	0.614213i	0.789141	+	0.614213i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−1.42030	+	4.13720i	−1.42030	+	4.13720i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.0513564	−	0.0786068i	−0.0513564	−	0.0786068i
$201$		0			0
$202$	0.543655	−	1.58361i	0.543655	−	1.58361i
$203$		0			0
$204$	−0.601061	+	0.772244i	−0.601061	+	0.772244i
$205$		−	0.764744i		−	0.764744i
$206$		0			0
$207$	−4.69179			−4.69179
$208$		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
$214$	0.594702	−	1.73231i	0.594702	−	1.73231i
$215$		0			0
$216$	3.29171	−	2.15058i	3.29171	−	2.15058i
$217$		0			0
$218$		0			0
$219$		−	3.77035i		−	3.77035i
$220$	−1.10532	−	0.860305i	−1.10532	−	0.860305i
$221$		0			0
$222$	3.45278	+	1.18534i	3.45278	+	1.18534i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−0.279128			−0.279128
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$229$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$230$	0.487815	−	1.42096i	0.487815	−	1.42096i
$231$		0			0
$232$		0			0
$233$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$233$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$234$		0			0
$235$			1.87782i			1.87782i
$236$		0			0
$237$		−	3.50588i		−	3.50588i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	0.465756	+	1.83923i	0.465756	+	1.83923i
$241$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$241$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$242$	1.10203	+	0.378326i	1.10203	+	0.378326i
$243$		−	5.76352i		−	5.76352i
$244$		0			0
$245$			0.951895i			0.951895i
$246$	0.519939	−	1.51453i	0.519939	−	1.51453i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.338101	−	0.984855i	0.338101	−	0.984855i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			2.32236i			2.32236i
$254$	1.86584	+	0.640542i	1.86584	+	0.640542i
$255$	−0.931513			−0.931513
$256$	−0.879474	+	0.475947i	−0.879474	+	0.475947i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.543655	−	1.58361i	0.543655	−	1.58361i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−1.60411	−	2.45528i	−1.60411	−	2.45528i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	3.54002	+	1.21529i	3.54002	+	1.21529i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.120526	−	0.475947i	−0.120526	−	0.475947i
$273$		0			0
$274$		0			0
$275$			0.138164i			0.138164i
$276$	1.93218	−	2.48246i	1.93218	−	2.48246i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$281$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$282$	−1.27671	+	3.71892i	−1.27671	+	3.71892i
$283$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$283$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.245485	+	2.96257i	−0.245485	+	2.96257i
$289$	−0.758948			−0.758948
$290$		0			0
$291$		0			0
$292$	1.49277	+	1.16187i	1.49277	+	1.16187i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.647181	+	1.88517i	−0.647181	+	1.88517i
$295$		0			0
$296$	−1.53331	+	1.00176i	−1.53331	+	1.00176i
$297$	−5.78569			−5.78569
$298$	−0.398869	+	1.16187i	−0.398869	+	1.16187i
$299$		0			0
$300$	0.114951	−	0.147689i	0.114951	−	0.147689i
$301$		0			0
$302$	−1.28117	−	0.439826i	−1.28117	−	0.439826i
$303$	3.33723			3.33723
$304$		0			0
$305$		0			0
$306$	−1.38044	−	0.473906i	−1.38044	−	0.473906i
$307$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$307$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$308$		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
$316$	1.38806	+	1.08037i	1.38806	+	1.08037i
$317$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$317$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$318$		0			0
$319$		0			0
$320$	−0.871720	−	0.382372i	−0.871720	−	0.382372i
$321$	3.65058			3.65058
$322$		0			0
$323$		0			0
$324$	3.83866	+	2.98775i	3.83866	+	2.98775i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	0.439413	+	0.672572i	0.439413	+	0.672572i
$329$		0			0
$330$	0.906483	−	2.64049i	0.906483	−	2.64049i
$331$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$331$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$332$		0			0
$333$			5.44468i			5.44468i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.945817	+	0.324699i	0.945817	+	0.324699i
$339$		0			0
$340$	0.287054	−	0.368807i	0.287054	−	0.368807i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.99445			2.99445
$346$	−0.106888	+	0.311353i	−0.106888	+	0.311353i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	1.46642	+	0.121511i	1.46642	+	0.121511i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000			−1.00000
$359$	−1.00000			−1.00000
$360$	−2.36895	+	1.54771i	−2.36895	+	1.54771i
$361$	−1.00000			−1.00000
$362$	−0.210859	+	0.614213i	−0.210859	+	0.614213i
$363$			2.32236i			2.32236i
$364$		0			0
$365$			1.80064i			1.80064i
$366$		0			0
$367$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$367$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$368$	0.387445	+	1.52999i	0.387445	+	1.52999i
$369$	2.38826			2.38826
$370$	−1.64898	−	0.566094i	−1.64898	−	0.566094i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−0.234575	+	0.683294i	−0.234575	+	0.683294i
$375$	2.07544			2.07544
$376$	−1.07898	−	1.65150i	−1.07898	−	1.65150i
$377$		0			0
$378$		0			0
$379$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$379$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$380$		0			0
$381$			3.93197i			3.93197i
$382$	−0.464369	−	0.159418i	−0.464369	−	0.159418i
$383$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$383$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$384$	−1.46642	−	1.34994i	−1.46642	−	1.34994i
$385$		0			0
$386$	1.28117	+	0.439826i	1.28117	+	0.439826i
$387$		0			0
$388$		0			0
$389$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$389$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$390$		0			0
$391$	−0.774890			−0.774890
$392$	−0.546948	−	0.837166i	−0.546948	−	0.837166i
$393$	3.33723			3.33723
$394$		0			0
$395$			1.67433i			1.67433i
$396$	2.68669	−	3.45186i	2.68669	−	3.45186i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	0.0230502	+	0.0910231i	0.0230502	+	0.0910231i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−1.02840	+	1.32128i	−1.02840	+	1.32128i
$405$			4.63036i			4.63036i
$406$		0			0
$407$	2.69503			2.69503
$408$	0.819241	−	0.535237i	0.819241	−	0.535237i
$409$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$409$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$410$	−0.248312	+	0.723308i	−0.248312	+	0.723308i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	4.43758	+	1.52342i	4.43758	+	1.52342i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$419$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−5.86436			−5.86436
$424$		0			0
$425$	−0.0461004			−0.0461004
$426$		0			0
$427$		0			0
$428$	−1.12496	+	1.44535i	−1.12496	+	1.44535i
$429$		0			0
$430$		0			0
$431$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$431$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$432$	−3.81165	+	0.965241i	−3.81165	+	0.965241i
$433$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$433$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.22423	+	3.56606i	−1.22423	+	3.56606i
$439$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$439$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$440$	0.766090	+	1.17259i	0.766090	+	1.17259i
$441$	−2.97272			−2.97272
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−2.88082	−	2.24223i	−2.88082	−	2.24223i
$445$		0			0
$446$		0			0
$447$	−2.44846			−2.44846
$448$		0			0
$449$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$449$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$450$	0.264004	+	0.0906326i	0.264004	+	0.0906326i
$451$		−	1.18215i		−	1.18215i
$452$		0			0
$453$		−	2.69987i		−	2.69987i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$457$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$458$	0.106888	−	0.311353i	0.106888	−	0.311353i
$459$		−	1.93048i		−	1.93048i
$460$	−0.922767	+	1.18557i	−0.922767	+	1.18557i
$461$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$461$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	1.03463	+	0.355188i	1.03463	+	0.355188i
$467$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$467$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$468$		0			0
$469$		0			0
$470$	0.609729	−	1.77608i	0.609729	−	1.77608i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−1.13836	+	3.31592i	−1.13836	+	3.31592i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$479$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$480$	0.156677	−	1.89081i	0.156677	−	1.89081i
$481$		0			0
$482$	1.49277	+	0.512467i	1.49277	+	0.512467i
$483$		0			0
$484$	−0.919474	−	0.715655i	−0.919474	−	0.715655i
$485$		0			0
$486$	−1.87141	+	5.45123i	−1.87141	+	5.45123i
$487$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$487$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$488$		0			0
$489$		0			0
$490$	0.309080	−	0.900319i	0.309080	−	0.900319i
$491$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$491$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$492$	−0.983535	+	1.26365i	−0.983535	+	1.26365i
$493$		0			0
$494$		0			0
$495$	4.16378			4.16378
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.639564	+	0.821712i	−0.639564	+	0.821712i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	−1.59379			−1.59379
$506$	0.754068	−	2.19653i	0.754068	−	2.19653i
$507$			1.99317i			1.99317i
$508$	−1.55676	−	1.21167i	−1.55676	−	1.21167i
$509$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$509$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$510$	0.881041	+	0.302462i	0.881041	+	0.302462i
$511$		0			0
$512$	0.986361	−	0.164595i	0.986361	−	0.164595i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			2.90276i			2.90276i
$518$		0			0
$519$	−0.656130			−0.656130
$520$		0			0
$521$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$521$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$522$		0			0
$523$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$523$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$524$	−1.02840	+	1.32128i	−1.02840	+	1.32128i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.719971	+	2.84310i	0.719971	+	2.84310i
$529$	1.49097			1.49097
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.74344			−1.74344
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.47145i			1.47145i
$540$	−2.95361	−	2.29889i	−2.95361	−	2.29889i
$541$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$541$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$542$		0			0
$543$	−1.29436			−1.29436
$544$	−0.0405441	+	0.489294i	−0.0405441	+	0.489294i
$545$		0			0
$546$		0			0
$547$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$547$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$548$		0			0
$549$		0			0
$550$	0.0448616	−	0.130677i	0.0448616	−	0.130677i
$551$		0			0
$552$	−2.63354	+	1.72058i	−2.63354	+	1.72058i
$553$		0			0
$554$		0			0
$555$		−	3.47497i		−	3.47497i
$556$		0			0
$557$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$557$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.43994			−1.43994
$562$	0.156210	+	0.0536269i	0.156210	+	0.0536269i
$563$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$563$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$564$	2.41507	−	3.10288i	2.41507	−	3.10288i
$565$		0			0
$566$	0.543655	−	1.58361i	0.543655	−	1.58361i
$567$		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
$573$		−	0.978588i		−	0.978588i
$574$		0			0
$575$	0.148195			0.148195
$576$	1.19413	−	2.72234i	1.19413	−	2.72234i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.717826	+	0.246430i	0.717826	+	0.246430i
$579$			2.69987i			2.69987i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.03463	−	1.58361i	−1.03463	−	1.58361i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	1.22423	−	1.57289i	1.22423	−	1.57289i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.77550	−	0.449618i	1.77550	−	0.449618i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	5.47220	+	1.87861i	5.47220	+	1.87861i
$595$		0			0
$596$	0.754515	−	0.969400i	0.754515	−	0.969400i
$597$		0			0
$598$		0			0
$599$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$599$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$600$	−0.156677	+	0.102362i	−0.156677	+	0.102362i
$601$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$601$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$602$		0			0
$603$		0			0
$604$	1.06894	+	0.831990i	1.06894	+	0.831990i
$605$		−	1.10911i		−	1.10911i
$606$	−3.15641	−	1.08360i	−3.15641	−	1.08360i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.15177	+	0.896456i	1.15177	+	0.896456i
$613$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$613$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$614$	0.398869	−	1.16187i	0.398869	−	1.16187i
$615$	−1.52426			−1.52426
$616$		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
$621$			6.20575i			6.20575i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.897287			−0.897287
$626$		0			0
$627$		0			0
$628$		0			0
$629$			0.899236i			0.899236i
$630$		0			0
$631$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$631$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$632$	−0.962053	−	1.47253i	−0.962053	−	1.47253i
$633$		0			0
$634$	−0.477778	+	1.39172i	−0.477778	+	1.39172i
$635$		−	1.87782i		−	1.87782i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.700332	+	0.644701i	0.700332	+	0.644701i
$641$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$641$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$642$	−3.45278	−	1.18534i	−3.45278	−	1.18534i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$647$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$648$	−2.66055	−	4.07228i	−2.66055	−	4.07228i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$653$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$654$		0			0
$655$	−1.59379			−1.59379
$656$	−0.197221	−	0.778807i	−0.197221	−	0.778807i
$657$	−5.62330			−5.62330
$658$		0			0
$659$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$659$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$660$	−1.71473	+	2.20309i	−1.71473	+	2.20309i
$661$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$661$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$662$	−0.309080	+	0.900319i	−0.309080	+	0.900319i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.76788	−	5.14967i	1.76788	−	5.14967i
$667$		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
$675$			0.369197i			0.369197i
$676$	−0.789141	−	0.614213i	−0.789141	−	0.614213i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	−0.391252	+	0.255618i	−0.391252	+	0.255618i
$681$		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
$687$	0.656130			0.656130
$688$		0			0
$689$		0			0
$690$	−2.83220	−	0.972297i	−2.83220	−	0.972297i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	0.202192	−	0.259777i	0.202192	−	0.259777i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.394442			0.394442
$698$		0			0
$699$			2.18032i			2.18032i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−1.34751	−	0.591074i	−1.34751	−	0.591074i
$705$	3.74282			3.74282
$706$		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
$711$	−5.22886			−5.22886
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.945817	+	0.324699i	0.945817	+	0.324699i
$719$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$719$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$720$	2.74313	−	0.694655i	2.74313	−	0.694655i
$721$		0			0
$722$	0.945817	+	0.324699i	0.945817	+	0.324699i
$723$			3.14578i			3.14578i
$724$	0.398869	−	0.512467i	0.398869	−	0.512467i
$725$		0			0
$726$	0.754068	−	2.19653i	0.754068	−	2.19653i
$727$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$727$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$728$		0			0
$729$	−6.62330			−6.62330
$730$	0.584666	−	1.70307i	0.584666	−	1.70307i
$731$		0			0
$732$		0			0
$733$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$733$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$734$	−1.28117	−	0.439826i	−1.28117	−	0.439826i
$735$	1.89729			1.89729
$736$	0.130333	−	1.57289i	0.130333	−	1.57289i
$737$		0			0
$738$	−2.25886	−	0.775466i	−2.25886	−	0.775466i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	1.37582	+	1.07084i	1.37582	+	1.07084i
$741$		0			0
$742$		0			0
$743$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$743$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$744$		0			0
$745$	1.16933			1.16933
$746$		0			0
$747$		0			0
$748$	0.443731	−	0.570105i	0.443731	−	0.570105i
$749$		0			0
$750$	−1.96298	−	0.673893i	−1.96298	−	0.673893i
$751$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$751$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$752$	0.484275	+	1.91236i	0.484275	+	1.91236i
$753$		0			0
$754$		0			0
$755$			1.28940i			1.28940i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−0.629528	+	1.83375i	−0.629528	+	1.83375i
$759$	4.62885			4.62885
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	1.27671	−	3.71892i	1.27671	−	3.71892i
$763$		0			0
$764$	0.387445	+	0.301561i	0.387445	+	0.301561i
$765$			1.38931i			1.38931i
$766$	1.03463	+	0.355188i	1.03463	+	0.355188i
$767$		0			0
$768$	0.948644	+	1.75294i	0.948644	+	1.75294i
$769$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$769$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$770$		0			0
$771$		0			0
$772$	−1.06894	−	0.831990i	−1.06894	−	0.831990i
$773$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$773$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.594702	−	1.73231i	0.594702	−	1.73231i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.732904	+	0.251606i	0.732904	+	0.251606i
$783$		0			0
$784$	0.245485	+	0.969400i	0.245485	+	0.969400i
$785$		0			0
$786$	−3.15641	−	1.08360i	−3.15641	−	1.08360i
$787$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$787$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$788$		0			0
$789$		0			0
$790$	0.543655	−	1.58361i	0.543655	−	1.58361i
$791$		0			0
$792$	−3.66194	+	2.39246i	−3.66194	+	2.39246i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$797$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$798$		0			0
$799$	−0.968550			−0.968550
$800$	0.00775390	−	0.0935756i	0.00775390	−	0.0935756i
$801$		0			0
$802$		0			0
$803$			2.78344i			2.78344i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.40170	−	0.915773i	1.40170	−	0.915773i
$809$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$809$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$810$	1.50347	−	4.37947i	1.50347	−	4.37947i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−2.54900	−	0.875073i	−2.54900	−	0.875073i
$815$		0			0
$816$	−0.948644	+	0.240229i	−0.948644	+	0.240229i
$817$		0			0
$818$	−0.759861	−	0.260861i	−0.759861	−	0.260861i
$819$		0			0
$820$	0.469715	−	0.603490i	0.469715	−	0.603490i
$821$		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
$825$	0.275383			0.275383
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−3.70248	−	2.88176i	−3.70248	−	2.88176i
$829$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$829$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.490971			−0.490971
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.210859	+	0.614213i	−0.210859	+	0.614213i
$839$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$839$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			0.329189i			0.329189i
$844$		0			0
$845$		−	0.951895i		−	0.951895i
$846$	5.54661	+	1.90415i	5.54661	+	1.90415i
$847$		0			0
$848$		0			0
$849$	3.33723			3.33723
$850$	0.0436025	+	0.0149688i	0.0436025	+	0.0149688i
$851$		−	2.89070i		−	2.89070i
$852$		0			0
$853$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$853$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$854$		0			0
$855$		0			0
$856$	1.53331	−	1.00176i	1.53331	−	1.00176i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$859$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$860$		0			0
$861$		0			0
$862$	0.464369	+	0.159418i	0.464369	+	0.159418i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	3.91854	+	0.324699i	3.91854	+	0.324699i
$865$	0.313353			0.313353
$866$	−1.66364	−	0.571129i	−1.66364	−	0.571129i
$867$			1.51271i			1.51271i
$868$		0			0
$869$			2.58820i			2.58820i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	2.31580	−	2.97533i	2.31580	−	2.97533i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−1.03463	−	0.355188i	−1.03463	−	0.355188i
$879$		0			0
$880$	−0.343843	−	1.35780i	−0.343843	−	1.35780i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	2.81165	+	0.965241i	2.81165	+	0.965241i
$883$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$883$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$887$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$888$	1.99668	+	3.05614i	1.99668	+	3.05614i
$889$		0			0
$890$		0			0
$891$			7.15765i			7.15765i
$892$		0			0
$893$		0			0
$894$	2.31580	+	0.795013i	2.31580	+	0.795013i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.78914	+	0.614213i	1.78914	+	0.614213i
$899$		0			0
$900$	−0.220271	−	0.171444i	−0.220271	−	0.171444i
$901$		0			0
$902$	−0.383843	+	1.11810i	−0.383843	+	1.11810i
$903$		0			0
$904$		0			0
$905$	0.618159			0.618159
$906$	−0.876647	+	2.55359i	−0.876647	+	2.55359i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		−	4.97733i		−	4.97733i
$910$		0			0
$911$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$911$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$912$		0			0
$913$		0			0
$914$	0.464369	+	0.159418i	0.464369	+	0.159418i
$915$		0			0
$916$	−0.202192	+	0.259777i	−0.202192	+	0.259777i
$917$		0			0
$918$	−0.626827	+	1.82588i	−0.626827	+	1.82588i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	1.25772	−	0.821712i	1.25772	−	0.821712i
$921$	2.44846			2.44846
$922$	−0.647181	+	1.88517i	−0.647181	+	1.88517i
$923$		0			0
$924$		0			0
$925$		−	0.171975i		−	0.171975i
$926$		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
$932$	−0.863238	−	0.671885i	−0.863238	−	0.671885i
$933$		0			0
$934$	0.106888	−	0.311353i	0.106888	−	0.311353i
$935$	0.687685			0.687685
$936$		0			0
$937$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$937$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$938$		0			0
$939$		0			0
$940$	−1.15338	+	1.48187i	−1.15338	+	1.48187i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−1.26798			−1.26798
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$947$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$948$	2.15336	−	2.76663i	2.15336	−	2.76663i
$949$		0			0
$950$		0			0
$951$	−2.93284			−2.93284
$952$		0			0
$953$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$953$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$954$		0			0
$955$			0.467353i			0.467353i
$956$		0			0
$957$		0			0
$958$	−0.156210	−	0.0536269i	−0.156210	−	0.0536269i
$959$		0			0
$960$	−0.762132	+	1.73748i	−0.762132	+	1.73748i
$961$	1.00000			1.00000
$962$		0			0
$963$		−	5.44468i		−	5.44468i
$964$	−1.24549	−	0.969400i	−1.24549	−	0.969400i
$965$		−	1.28940i		−	1.28940i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.637281	+	0.975432i	0.637281	+	0.975432i
$969$		0			0
$970$		0			0
$971$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$971$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$972$	3.54002	−	4.54822i	3.54002	−	4.54822i
$973$		0			0
$974$	−0.156210	−	0.0536269i	−0.156210	−	0.0536269i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−0.584666	+	0.751179i	−0.584666	+	0.751179i
$981$		0			0
$982$	−0.594702	+	1.73231i	−0.594702	+	1.73231i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	1.34055	−	0.875825i	1.34055	−	0.875825i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−3.93818	−	1.35198i	−3.93818	−	1.35198i
$991$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$991$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$992$		0			0
$993$	−1.89729			−1.89729
$994$		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
$999$	7.20159			7.20159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.1	✓	18
8.5	even	2	inner	2872.1.b.e.717.2	yes	18
359.358	odd	2	CM	2872.1.b.e.717.1	✓	18
2872.717	odd	2	inner	2872.1.b.e.717.2	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.1	✓	18	1.1	even	1	trivial
2872.1.b.e.717.1	✓	18	359.358	odd	2	CM
2872.1.b.e.717.2	yes	18	8.5	even	2	inner
2872.1.b.e.717.2	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)

\(f(q)\)	\(=\)	\(q+(-0.945817 - 0.324699i) q^{2} -1.99317i q^{3} +(0.789141 + 0.614213i) q^{4} +0.951895i q^{5} +(-0.647181 + 1.88517i) q^{6} +(-0.546948 - 0.837166i) q^{8} -2.97272 q^{9} +O(q^{10})\)	\(q+(-0.945817 - 0.324699i) q^{2} -1.99317i q^{3} +(0.789141 + 0.614213i) q^{4} +0.951895i q^{5} +(-0.647181 + 1.88517i) q^{6} +(-0.546948 - 0.837166i) q^{8} -2.97272 q^{9} +(0.309080 - 0.900319i) q^{10} +1.47145i q^{11} +(1.22423 - 1.57289i) q^{12} +1.89729 q^{15} +(0.245485 + 0.969400i) q^{16} -0.490971 q^{17} +(2.81165 + 0.965241i) q^{18} +(-0.584666 + 0.751179i) q^{20} +(0.477778 - 1.39172i) q^{22} +1.57828 q^{23} +(-1.66861 + 1.09016i) q^{24} +0.0938963 q^{25} +3.93197i q^{27} +(-1.79449 - 0.616048i) q^{30} +(0.0825793 - 0.996584i) q^{32} +2.93284 q^{33} +(0.464369 + 0.159418i) q^{34} +(-2.34590 - 1.82588i) q^{36} -1.83155i q^{37} +(0.796894 - 0.520637i) q^{40} -0.803391 q^{41} +(-0.903782 + 1.16118i) q^{44} -2.82972i q^{45} +(-1.49277 - 0.512467i) q^{46} +1.97272 q^{47} +(1.93218 - 0.489294i) q^{48} +1.00000 q^{49} +(-0.0888088 - 0.0304881i) q^{50} +0.978588i q^{51} +(1.27671 - 3.71892i) q^{54} -1.40066 q^{55} +(1.49723 + 1.16534i) q^{60} +(-0.401695 + 0.915773i) q^{64} +(-2.77393 - 0.952293i) q^{66} +(-0.387445 - 0.301561i) q^{68} -3.14578i q^{69} +(1.62593 + 2.48866i) q^{72} +1.89163 q^{73} +(-0.594702 + 1.73231i) q^{74} -0.187151i q^{75} +1.75895 q^{79} +(-0.922767 + 0.233676i) q^{80} +4.86436 q^{81} +(0.759861 + 0.260861i) q^{82} -0.467353i q^{85} +(1.23185 - 0.804806i) q^{88} +(-0.918808 + 2.67640i) q^{90} +(1.24549 + 0.969400i) q^{92} +(-1.86584 - 0.640542i) q^{94} +(-1.98636 - 0.164595i) q^{96} +(-0.945817 - 0.324699i) q^{98} -4.37421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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