LMFDB - Embedded newform 2872.1.b.e.717.13

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			717.13
Root			$-0.789141 - 0.614213i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.677282 - 0.735724i) q^{2} +1.67433i q^{3} +(-0.0825793 - 0.996584i) q^{4} -0.649399i q^{5} +(1.23185 + 1.13399i) q^{6} +(-0.789141 - 0.614213i) q^{8} -1.80339 q^{9} +O(q^{10})$	\(q+(0.677282 - 0.735724i) q^{2} +1.67433i q^{3} +(-0.0825793 - 0.996584i) q^{4} -0.649399i q^{5} +(1.23185 + 1.13399i) q^{6} +(-0.789141 - 0.614213i) q^{8} -1.80339 q^{9} +(-0.477778 - 0.439826i) q^{10} -0.951895i q^{11} +(1.66861 - 0.138265i) q^{12} +1.08731 q^{15} +(-0.986361 + 0.164595i) q^{16} +1.97272 q^{17} +(-1.22140 + 1.32680i) q^{18} +(-0.647181 + 0.0536269i) q^{20} +(-0.700332 - 0.644701i) q^{22} -0.165159 q^{23} +(1.02840 - 1.32128i) q^{24} +0.578281 q^{25} -1.34514i q^{27} +(0.736415 - 0.799960i) q^{30} +(-0.546948 + 0.837166i) q^{32} +1.59379 q^{33} +(1.33609 - 1.45138i) q^{34} +(0.148923 + 1.79723i) q^{36} -1.93880i q^{37} +(-0.398869 + 0.512467i) q^{40} +0.490971 q^{41} +(-0.948644 + 0.0786068i) q^{44} +1.17112i q^{45} +(-0.111859 + 0.121511i) q^{46} +0.803391 q^{47} +(-0.275586 - 1.65150i) q^{48} +1.00000 q^{49} +(0.391659 - 0.425455i) q^{50} +3.30299i q^{51} +(-0.989654 - 0.911041i) q^{54} -0.618159 q^{55} +(-0.0897894 - 1.08360i) q^{60} +(0.245485 + 0.969400i) q^{64} +(1.07944 - 1.17259i) q^{66} +(-0.162906 - 1.96598i) q^{68} -0.276531i q^{69} +(1.42313 + 1.10767i) q^{72} -1.35456 q^{73} +(-1.42642 - 1.31311i) q^{74} +0.968235i q^{75} -1.89163 q^{79} +(0.106888 + 0.640542i) q^{80} +0.448828 q^{81} +(0.332526 - 0.361219i) q^{82} -1.28108i q^{85} +(-0.584666 + 0.751179i) q^{88} +(0.861621 + 0.793178i) q^{90} +(0.0136387 + 0.164595i) q^{92} +(0.544122 - 0.591074i) q^{94} +(-1.40170 - 0.915773i) q^{96} +(0.677282 - 0.735724i) q^{98} +1.71664i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.677282	−	0.735724i	0.677282	−	0.735724i
$2$	0.677282	−	0.735724i	0.677282	−	0.735724i
$3$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$3$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$4$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$5$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$5$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$6$	1.23185	+	1.13399i	1.23185	+	1.13399i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.789141	−	0.614213i	−0.789141	−	0.614213i
$9$	−1.80339			−1.80339
$10$	−0.477778	−	0.439826i	−0.477778	−	0.439826i
$11$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$11$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$12$	1.66861	−	0.138265i	1.66861	−	0.138265i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	1.08731			1.08731
$16$	−0.986361	+	0.164595i	−0.986361	+	0.164595i
$17$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$17$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$18$	−1.22140	+	1.32680i	−1.22140	+	1.32680i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.647181	+	0.0536269i	−0.647181	+	0.0536269i
$21$		0			0
$22$	−0.700332	−	0.644701i	−0.700332	−	0.644701i
$23$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$23$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$24$	1.02840	−	1.32128i	1.02840	−	1.32128i
$25$	0.578281			0.578281
$26$		0			0
$27$		−	1.34514i		−	1.34514i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	0.736415	−	0.799960i	0.736415	−	0.799960i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$33$	1.59379			1.59379
$34$	1.33609	−	1.45138i	1.33609	−	1.45138i
$35$		0			0
$36$	0.148923	+	1.79723i	0.148923	+	1.79723i
$37$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$37$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$38$		0			0
$39$		0			0
$40$	−0.398869	+	0.512467i	−0.398869	+	0.512467i
$41$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$41$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.948644	+	0.0786068i	−0.948644	+	0.0786068i
$45$			1.17112i			1.17112i
$46$	−0.111859	+	0.121511i	−0.111859	+	0.121511i
$47$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$47$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$48$	−0.275586	−	1.65150i	−0.275586	−	1.65150i
$49$	1.00000			1.00000
$50$	0.391659	−	0.425455i	0.391659	−	0.425455i
$51$			3.30299i			3.30299i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.989654	−	0.911041i	−0.989654	−	0.911041i
$55$	−0.618159			−0.618159
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−0.0897894	−	1.08360i	−0.0897894	−	1.08360i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	0.245485	+	0.969400i	0.245485	+	0.969400i
$65$		0			0
$66$	1.07944	−	1.17259i	1.07944	−	1.17259i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.162906	−	1.96598i	−0.162906	−	1.96598i
$69$		−	0.276531i		−	0.276531i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1.42313	+	1.10767i	1.42313	+	1.10767i
$73$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$73$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$74$	−1.42642	−	1.31311i	−1.42642	−	1.31311i
$75$			0.968235i			0.968235i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$79$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$80$	0.106888	+	0.640542i	0.106888	+	0.640542i
$81$	0.448828			0.448828
$82$	0.332526	−	0.361219i	0.332526	−	0.361219i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	1.28108i		−	1.28108i
$86$		0			0
$87$		0			0
$88$	−0.584666	+	0.751179i	−0.584666	+	0.751179i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.861621	+	0.793178i	0.861621	+	0.793178i
$91$		0			0
$92$	0.0136387	+	0.164595i	0.0136387	+	0.164595i
$93$		0			0
$94$	0.544122	−	0.591074i	0.544122	−	0.591074i
$95$		0			0
$96$	−1.40170	−	0.915773i	−1.40170	−	0.915773i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.677282	−	0.735724i	0.677282	−	0.735724i
$99$			1.71664i			1.71664i
$100$	−0.0477541	−	0.576306i	−0.0477541	−	0.576306i
$101$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$101$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$102$	2.43009	+	2.23706i	2.43009	+	2.23706i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$107$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$108$	−1.34055	+	0.111081i	−1.34055	+	0.111081i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	−0.418668	+	0.454795i	−0.418668	+	0.454795i
$111$	3.24620			3.24620
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			0.107254i			0.107254i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.858040	−	0.667840i	−0.858040	−	0.667840i
$121$	0.0938963			0.0938963
$122$		0			0
$123$			0.822049i			0.822049i
$124$		0			0
$125$		−	1.02493i		−	1.02493i
$126$		0			0
$127$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$127$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$128$	0.879474	+	0.475947i	0.879474	+	0.475947i
$129$		0			0
$130$		0			0
$131$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$131$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$132$	−0.131614	−	1.58835i	−0.131614	−	1.58835i
$133$		0			0
$134$		0			0
$135$	−0.873535			−0.873535
$136$	−1.55676	−	1.21167i	−1.55676	−	1.21167i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−0.203450	−	0.187289i	−0.203450	−	0.187289i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			1.34514i			1.34514i
$142$		0			0
$143$		0			0
$144$	1.77879	−	0.296828i	1.77879	−	0.296828i
$145$		0			0
$146$	−0.917421	+	0.996584i	−0.917421	+	0.996584i
$147$			1.67433i			1.67433i
$148$	−1.93218	+	0.160105i	−1.93218	+	0.160105i
$149$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$149$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$150$	0.712354	+	0.655768i	0.712354	+	0.655768i
$151$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$151$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$152$		0			0
$153$	−3.55759			−3.55759
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.28117	+	1.39172i	−1.28117	+	1.39172i
$159$		0			0
$160$	0.543655	+	0.355188i	0.543655	+	0.355188i
$161$		0			0
$162$	0.303983	−	0.330213i	0.303983	−	0.330213i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−0.0405441	−	0.489294i	−0.0405441	−	0.489294i
$165$		−	1.03500i		−	1.03500i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−0.942524	−	0.867655i	−0.942524	−	0.867655i
$171$		0			0
$172$		0			0
$173$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$173$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$174$		0			0
$175$		0			0
$176$	0.156677	+	0.938912i	0.156677	+	0.938912i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	1.16712	−	0.0967103i	1.16712	−	0.0967103i
$181$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$181$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$182$		0			0
$183$		0			0
$184$	0.130333	+	0.101443i	0.130333	+	0.101443i
$185$	−1.25906			−1.25906
$186$		0			0
$187$		−	1.87782i		−	1.87782i
$188$	−0.0663435	−	0.800647i	−0.0663435	−	0.800647i
$189$		0			0
$190$		0			0
$191$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$191$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$192$	−1.62310	+	0.411024i	−1.62310	+	0.411024i
$193$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$193$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$194$		0			0
$195$		0			0
$196$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	1.26297	+	1.16265i	1.26297	+	1.16265i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.456345	−	0.355188i	−0.456345	−	0.355188i
$201$		0			0
$202$	0.903782	+	0.831990i	0.903782	+	0.831990i
$203$		0			0
$204$	3.29171	−	0.272759i	3.29171	−	0.272759i
$205$		−	0.318836i		−	0.318836i
$206$		0			0
$207$	0.297846			0.297846
$208$		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
$214$	1.42642	+	1.31311i	1.42642	+	1.31311i
$215$		0			0
$216$	−0.826204	+	1.06151i	−0.826204	+	1.06151i
$217$		0			0
$218$		0			0
$219$		−	2.26799i		−	2.26799i
$220$	0.0510472	+	0.616048i	0.0510472	+	0.616048i
$221$		0			0
$222$	2.19859	−	2.38831i	2.19859	−	2.38831i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.04287			−1.04287
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$229$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$230$	0.0789092	+	0.0726411i	0.0789092	+	0.0726411i
$231$		0			0
$232$		0			0
$233$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$233$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$234$		0			0
$235$		−	0.521721i		−	0.521721i
$236$		0			0
$237$		−	3.16723i		−	3.16723i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−1.07248	+	0.178965i	−1.07248	+	0.178965i
$241$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$241$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$242$	0.0635942	−	0.0690818i	0.0635942	−	0.0690818i
$243$		−	0.593657i		−	0.593657i
$244$		0			0
$245$		−	0.649399i		−	0.649399i
$246$	0.604801	+	0.556759i	0.604801	+	0.556759i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.754068	−	0.694169i	−0.754068	−	0.694169i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			0.157214i			0.157214i
$254$	−0.544122	+	0.591074i	−0.544122	+	0.591074i
$255$	2.14496			2.14496
$256$	0.945817	−	0.324699i	0.945817	−	0.324699i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.903782	+	0.831990i	0.903782	+	0.831990i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−1.25772	−	0.978925i	−1.25772	−	0.978925i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−0.591629	+	0.642681i	−0.591629	+	0.642681i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−1.94582	+	0.324699i	−1.94582	+	0.324699i
$273$		0			0
$274$		0			0
$275$		−	0.550463i		−	0.550463i
$276$	−0.275586	+	0.0228357i	−0.275586	+	0.0228357i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$281$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$282$	0.989654	+	0.911041i	0.989654	+	0.911041i
$283$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$283$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.986361	−	1.50974i	0.986361	−	1.50974i
$289$	2.89163			2.89163
$290$		0			0
$291$		0			0
$292$	0.111859	+	1.34994i	0.111859	+	1.34994i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	1.23185	+	1.13399i	1.23185	+	1.13399i
$295$		0			0
$296$	−1.19084	+	1.52999i	−1.19084	+	1.52999i
$297$	−1.28044			−1.28044
$298$	1.46642	+	1.34994i	1.46642	+	1.34994i
$299$		0			0
$300$	0.964928	−	0.0799562i	0.964928	−	0.0799562i
$301$		0			0
$302$	1.19130	−	1.29410i	1.19130	−	1.29410i
$303$	−2.05679			−2.05679
$304$		0			0
$305$		0			0
$306$	−2.40949	+	2.61740i	−2.40949	+	2.61740i
$307$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$307$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$308$		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
$316$	0.156210	+	1.88517i	0.156210	+	1.88517i
$317$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$317$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$318$		0			0
$319$		0			0
$320$	0.629528	−	0.159418i	0.629528	−	0.159418i
$321$	−3.24620			−3.24620
$322$		0			0
$323$		0			0
$324$	−0.0370639	−	0.447295i	−0.0370639	−	0.447295i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.387445	−	0.301561i	−0.387445	−	0.301561i
$329$		0			0
$330$	−0.761478	−	0.700990i	−0.761478	−	0.700990i
$331$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$331$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$332$		0			0
$333$			3.49642i			3.49642i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$339$		0			0
$340$	−1.27671	+	0.105791i	−1.27671	+	0.105791i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.179579			−0.179579
$346$	−1.34751	−	1.24047i	−1.34751	−	1.24047i
$347$		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
$352$	0.796894	+	0.520637i	0.796894	+	0.520637i
$353$		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
$360$	0.719317	−	0.924178i	0.719317	−	0.924178i
$361$	−1.00000			−1.00000
$362$	−1.08258	−	0.996584i	−1.08258	−	0.996584i
$363$			0.157214i			0.157214i
$364$		0			0
$365$			0.879652i			0.879652i
$366$		0			0
$367$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$367$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$368$	0.162906	−	0.0271842i	0.162906	−	0.0271842i
$369$	−0.885413			−0.885413
$370$	−0.852735	+	0.926317i	−0.852735	+	0.926317i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−1.38156	−	1.27182i	−1.38156	−	1.27182i
$375$	1.71608			1.71608
$376$	−0.633988	−	0.493453i	−0.633988	−	0.493453i
$377$		0			0
$378$		0			0
$379$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$379$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$380$		0			0
$381$		−	1.34514i		−	1.34514i
$382$	−1.33609	+	1.45138i	−1.33609	+	1.45138i
$383$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$383$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$384$	−0.796894	+	1.47253i	−0.796894	+	1.47253i
$385$		0			0
$386$	−1.19130	+	1.29410i	−1.19130	+	1.29410i
$387$		0			0
$388$		0			0
$389$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$389$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$390$		0			0
$391$	−0.325812			−0.325812
$392$	−0.789141	−	0.614213i	−0.789141	−	0.614213i
$393$	−2.05679			−2.05679
$394$		0			0
$395$			1.22843i			1.22843i
$396$	1.71078	−	0.141759i	1.71078	−	0.141759i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−0.570394	+	0.0951819i	−0.570394	+	0.0951819i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	1.22423	−	0.101443i	1.22423	−	0.101443i
$405$		−	0.291468i		−	0.291468i
$406$		0			0
$407$	−1.84553			−1.84553
$408$	2.02874	−	2.60653i	2.02874	−	2.60653i
$409$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$409$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$410$	−0.234575	−	0.215942i	−0.234575	−	0.215942i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.201725	−	0.219132i	0.201725	−	0.219132i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$419$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−1.44883			−1.44883
$424$		0			0
$425$	1.14079			1.14079
$426$		0			0
$427$		0			0
$428$	1.93218	−	0.160105i	1.93218	−	0.160105i
$429$		0			0
$430$		0			0
$431$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$431$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$432$	0.221403	+	1.32680i	0.221403	+	1.32680i
$433$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$433$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.66861	−	1.53607i	−1.66861	−	1.53607i
$439$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$439$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$440$	0.487815	+	0.379681i	0.487815	+	0.379681i
$441$	−1.80339			−1.80339
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−0.268069	−	3.23511i	−0.268069	−	3.23511i
$445$		0			0
$446$		0			0
$447$	−3.33723			−3.33723
$448$		0			0
$449$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$449$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$450$	−0.706314	+	0.767262i	−0.706314	+	0.767262i
$451$		−	0.467353i		−	0.467353i
$452$		0			0
$453$			2.94506i			2.94506i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$457$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$458$	1.34751	+	1.24047i	1.34751	+	1.24047i
$459$		−	2.65360i		−	2.65360i
$460$	0.106888	−	0.00885696i	0.106888	−	0.00885696i
$461$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$461$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.06894	+	1.16118i	−1.06894	+	1.16118i
$467$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$467$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$468$		0			0
$469$		0			0
$470$	−0.383843	−	0.353352i	−0.383843	−	0.353352i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−2.33020	−	2.14510i	−2.33020	−	2.14510i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$479$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$480$	−0.594702	+	0.910260i	−0.594702	+	0.910260i
$481$		0			0
$482$	0.111859	−	0.121511i	0.111859	−	0.121511i
$483$		0			0
$484$	−0.00775390	−	0.0935756i	−0.00775390	−	0.0935756i
$485$		0			0
$486$	−0.436767	−	0.402073i	−0.436767	−	0.402073i
$487$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$487$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$488$		0			0
$489$		0			0
$490$	−0.477778	−	0.439826i	−0.477778	−	0.439826i
$491$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$491$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$492$	0.819241	−	0.0678843i	0.819241	−	0.0678843i
$493$		0			0
$494$		0			0
$495$	1.11478			1.11478
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−1.02143	+	0.0846384i	−1.02143	+	0.0846384i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	0.797738			0.797738
$506$	0.115666	+	0.106478i	0.115666	+	0.106478i
$507$		−	1.67433i		−	1.67433i
$508$	0.0663435	+	0.800647i	0.0663435	+	0.800647i
$509$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$509$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$510$	1.45274	−	1.57810i	1.45274	−	1.57810i
$511$		0			0
$512$	0.401695	−	0.915773i	0.401695	−	0.915773i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	0.764744i		−	0.764744i
$518$		0			0
$519$	3.06662			3.06662
$520$		0			0
$521$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$521$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$522$		0			0
$523$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$523$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$524$	1.22423	−	0.101443i	1.22423	−	0.101443i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.57205	+	0.262329i	−1.57205	+	0.262329i
$529$	−0.972723			−0.972723
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.25906			1.25906
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.951895i		−	0.951895i
$540$	0.0721359	+	0.870551i	0.0721359	+	0.870551i
$541$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$541$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$542$		0			0
$543$	2.46369			2.46369
$544$	−1.07898	+	1.65150i	−1.07898	+	1.65150i
$545$		0			0
$546$		0			0
$547$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$547$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$548$		0			0
$549$		0			0
$550$	−0.404989	−	0.372818i	−0.404989	−	0.372818i
$551$		0			0
$552$	−0.169849	+	0.218222i	−0.169849	+	0.218222i
$553$		0			0
$554$		0			0
$555$		−	2.10808i		−	2.10808i
$556$		0			0
$557$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$557$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	3.14410			3.14410
$562$	0.740876	−	0.804806i	0.740876	−	0.804806i
$563$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$563$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$564$	1.34055	−	0.111081i	1.34055	−	0.111081i
$565$		0			0
$566$	0.903782	+	0.831990i	0.903782	+	0.831990i
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	3.30299i		−	3.30299i
$574$		0			0
$575$	−0.0955081			−0.0955081
$576$	−0.442706	−	1.74821i	−0.442706	−	1.74821i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.95845	−	2.12744i	1.95845	−	2.12744i
$579$		−	2.94506i		−	2.94506i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.06894	+	0.831990i	1.06894	+	0.831990i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	1.66861	−	0.138265i	1.66861	−	0.138265i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.319116	+	1.91236i	0.319116	+	1.91236i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.867215	+	0.942047i	−0.867215	+	0.942047i
$595$		0			0
$596$	1.98636	−	0.164595i	1.98636	−	0.164595i
$597$		0			0
$598$		0			0
$599$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$599$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$600$	0.594702	−	0.764073i	0.594702	−	0.764073i
$601$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$601$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$602$		0			0
$603$		0			0
$604$	−0.145253	−	1.75294i	−0.145253	−	1.75294i
$605$		−	0.0609762i		−	0.0609762i
$606$	−1.39303	+	1.51323i	−1.39303	+	1.51323i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.293783	+	3.54544i	0.293783	+	3.54544i
$613$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$613$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$614$	−1.46642	−	1.34994i	−1.46642	−	1.34994i
$615$	0.533838			0.533838
$616$		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
$621$			0.222162i			0.222162i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0873100			−0.0873100
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	3.82472i		−	3.82472i
$630$		0			0
$631$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$631$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$632$	1.49277	+	1.16187i	1.49277	+	1.16187i
$633$		0			0
$634$	0.700332	+	0.644701i	0.700332	+	0.644701i
$635$			0.521721i			0.521721i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.309080	−	0.571129i	0.309080	−	0.571129i
$641$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$641$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$642$	−2.19859	+	2.38831i	−2.19859	+	2.38831i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$647$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$648$	−0.354188	−	0.275676i	−0.354188	−	0.275676i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$653$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$654$		0			0
$655$	0.797738			0.797738
$656$	−0.484275	+	0.0808112i	−0.484275	+	0.0808112i
$657$	2.44281			2.44281
$658$		0			0
$659$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$659$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$660$	−1.03147	+	0.0854700i	−1.03147	+	0.0854700i
$661$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$661$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$662$	0.477778	+	0.439826i	0.477778	+	0.439826i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.57240	+	2.36806i	2.57240	+	2.36806i
$667$		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
$675$		−	0.777871i		−	0.777871i
$676$	0.0825793	+	0.996584i	0.0825793	+	0.996584i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	−0.786858	+	1.01096i	−0.786858	+	1.01096i
$681$		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
$687$	−3.06662			−3.06662
$688$		0			0
$689$		0			0
$690$	−0.121625	+	0.132120i	−0.121625	+	0.132120i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−1.82529	+	0.151248i	−1.82529	+	0.151248i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.968550			0.968550
$698$		0			0
$699$		−	2.64257i		−	2.64257i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.922767	−	0.233676i	0.922767	−	0.233676i
$705$	0.873535			0.873535
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	3.41136			3.41136
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$719$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$719$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$720$	−0.192760	−	1.15515i	−0.192760	−	1.15515i
$721$		0			0
$722$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$723$			0.276531i			0.276531i
$724$	−1.46642	+	0.121511i	−1.46642	+	0.121511i
$725$		0			0
$726$	0.115666	+	0.106478i	0.115666	+	0.106478i
$727$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$727$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$728$		0			0
$729$	1.44281			1.44281
$730$	0.647181	+	0.595772i	0.647181	+	0.595772i
$731$		0			0
$732$		0			0
$733$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$733$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$734$	1.19130	−	1.29410i	1.19130	−	1.29410i
$735$	1.08731			1.08731
$736$	0.0903332	−	0.138265i	0.0903332	−	0.138265i
$737$		0			0
$738$	−0.599674	+	0.651419i	−0.599674	+	0.651419i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	0.103972	+	1.25475i	0.103972	+	1.25475i
$741$		0			0
$742$		0			0
$743$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$743$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$744$		0			0
$745$	1.29436			1.29436
$746$		0			0
$747$		0			0
$748$	−1.87141	+	0.155070i	−1.87141	+	0.155070i
$749$		0			0
$750$	1.16227	−	1.26256i	1.16227	−	1.26256i
$751$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$751$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$752$	−0.792434	+	0.132234i	−0.792434	+	0.132234i
$753$		0			0
$754$		0			0
$755$		−	1.14226i		−	1.14226i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−0.242192	−	0.222954i	−0.242192	−	0.222954i
$759$	−0.263228			−0.263228
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−0.989654	−	0.911041i	−0.989654	−	0.911041i
$763$		0			0
$764$	0.162906	+	1.96598i	0.162906	+	1.96598i
$765$			2.31030i			2.31030i
$766$	−1.06894	+	1.16118i	−1.06894	+	1.16118i
$767$		0			0
$768$	0.543655	+	1.58361i	0.543655	+	1.58361i
$769$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$769$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$770$		0			0
$771$		0			0
$772$	0.145253	+	1.75294i	0.145253	+	1.75294i
$773$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$773$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.42642	+	1.31311i	1.42642	+	1.31311i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.220667	+	0.239708i	−0.220667	+	0.239708i
$783$		0			0
$784$	−0.986361	+	0.164595i	−0.986361	+	0.164595i
$785$		0			0
$786$	−1.39303	+	1.51323i	−1.39303	+	1.51323i
$787$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$787$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$788$		0			0
$789$		0			0
$790$	0.903782	+	0.831990i	0.903782	+	0.831990i
$791$		0			0
$792$	1.05438	−	1.35467i	1.05438	−	1.35467i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$797$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$798$		0			0
$799$	1.58487			1.58487
$800$	−0.316290	+	0.484117i	−0.316290	+	0.484117i
$801$		0			0
$802$		0			0
$803$			1.28940i			1.28940i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.754515	−	0.969400i	0.754515	−	0.969400i
$809$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$809$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$810$	−0.214440	−	0.197406i	−0.214440	−	0.197406i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−1.24995	+	1.35780i	−1.24995	+	1.35780i
$815$		0			0
$816$	−0.543655	−	3.25795i	−0.543655	−	3.25795i
$817$		0			0
$818$	−0.332526	+	0.361219i	−0.332526	+	0.361219i
$819$		0			0
$820$	−0.317747	+	0.0263293i	−0.317747	+	0.0263293i
$821$		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
$825$	0.921658			0.921658
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−0.0245959	−	0.296828i	−0.0245959	−	0.296828i
$829$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$829$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.97272			1.97272
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.08258	−	0.996584i	−1.08258	−	0.996584i
$839$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$839$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			1.83155i			1.83155i
$844$		0			0
$845$			0.649399i			0.649399i
$846$	−0.981264	+	1.06594i	−0.981264	+	1.06594i
$847$		0			0
$848$		0			0
$849$	−2.05679			−2.05679
$850$	0.772635	−	0.839305i	0.772635	−	0.839305i
$851$			0.320210i			0.320210i
$852$		0			0
$853$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$853$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$854$		0			0
$855$		0			0
$856$	1.19084	−	1.52999i	1.19084	−	1.52999i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$859$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$860$		0			0
$861$		0			0
$862$	1.33609	−	1.45138i	1.33609	−	1.45138i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	1.12611	+	0.735724i	1.12611	+	0.735724i
$865$	−1.18940			−1.18940
$866$	−1.28117	+	1.39172i	−1.28117	+	1.39172i
$867$			4.84156i			4.84156i
$868$		0			0
$869$			1.80064i			1.80064i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−2.26024	+	0.187289i	−2.26024	+	0.187289i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	1.06894	−	1.16118i	1.06894	−	1.16118i
$879$		0			0
$880$	0.609729	−	0.101746i	0.609729	−	0.101746i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.22140	+	1.32680i	−1.22140	+	1.32680i
$883$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$883$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$887$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$888$	−2.56171	−	1.99386i	−2.56171	−	1.99386i
$889$		0			0
$890$		0			0
$891$		−	0.427237i		−	0.427237i
$892$		0			0
$893$		0			0
$894$	−2.26024	+	2.45528i	−2.26024	+	2.45528i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.917421	−	0.996584i	0.917421	−	0.996584i
$899$		0			0
$900$	0.0861192	+	1.03930i	0.0861192	+	1.03930i
$901$		0			0
$902$	−0.343843	−	0.316529i	−0.343843	−	0.316529i
$903$		0			0
$904$		0			0
$905$	−0.955557			−0.955557
$906$	2.16675	+	1.99464i	2.16675	+	1.99464i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		−	2.21533i		−	2.21533i
$910$		0			0
$911$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$911$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$912$		0			0
$913$		0			0
$914$	1.33609	−	1.45138i	1.33609	−	1.45138i
$915$		0			0
$916$	1.82529	−	0.151248i	1.82529	−	0.151248i
$917$		0			0
$918$	−1.95231	−	1.79723i	−1.95231	−	1.79723i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	0.0658767	−	0.0846384i	0.0658767	−	0.0846384i
$921$	3.33723			3.33723
$922$	1.23185	+	1.13399i	1.23185	+	1.13399i
$923$		0			0
$924$		0			0
$925$		−	1.12117i		−	1.12117i
$926$		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
$932$	0.130333	+	1.57289i	0.130333	+	1.57289i
$933$		0			0
$934$	1.34751	+	1.24047i	1.34751	+	1.24047i
$935$	−1.21946			−1.21946
$936$		0			0
$937$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$937$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$938$		0			0
$939$		0			0
$940$	−0.519939	+	0.0430834i	−0.519939	+	0.0430834i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−0.0810881			−0.0810881
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$947$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$948$	−3.15641	+	0.261547i	−3.15641	+	0.261547i
$949$		0			0
$950$		0			0
$951$	−1.59379			−1.59379
$952$		0			0
$953$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$953$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$954$		0			0
$955$			1.28108i			1.28108i
$956$		0			0
$957$		0			0
$958$	−0.740876	+	0.804806i	−0.740876	+	0.804806i
$959$		0			0
$960$	0.266919	+	1.05404i	0.266919	+	1.05404i
$961$	1.00000			1.00000
$962$		0			0
$963$		−	3.49642i		−	3.49642i
$964$	−0.0136387	−	0.164595i	−0.0136387	−	0.164595i
$965$			1.14226i			1.14226i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.0740974	−	0.0576723i	−0.0740974	−	0.0576723i
$969$		0			0
$970$		0			0
$971$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$971$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$972$	−0.591629	+	0.0490238i	−0.591629	+	0.0490238i
$973$		0			0
$974$	−0.740876	+	0.804806i	−0.740876	+	0.804806i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−0.647181	+	0.0536269i	−0.647181	+	0.0536269i
$981$		0			0
$982$	−1.42642	−	1.31311i	−1.42642	−	1.31311i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	0.504913	−	0.648712i	0.504913	−	0.648712i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.755022	−	0.820173i	0.755022	−	0.820173i
$991$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$991$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$992$		0			0
$993$	−1.08731			−1.08731
$994$		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
$999$	−2.60797			−2.60797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.13	✓	18
8.5	even	2	inner	2872.1.b.e.717.14	yes	18
359.358	odd	2	CM	2872.1.b.e.717.13	✓	18
2872.717	odd	2	inner	2872.1.b.e.717.14	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.13	✓	18	1.1	even	1	trivial
2872.1.b.e.717.13	✓	18	359.358	odd	2	CM
2872.1.b.e.717.14	yes	18	8.5	even	2	inner
2872.1.b.e.717.14	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)

\(f(q)\)	\(=\)	\(q+(0.677282 - 0.735724i) q^{2} +1.67433i q^{3} +(-0.0825793 - 0.996584i) q^{4} -0.649399i q^{5} +(1.23185 + 1.13399i) q^{6} +(-0.789141 - 0.614213i) q^{8} -1.80339 q^{9} +O(q^{10})\)	\(q+(0.677282 - 0.735724i) q^{2} +1.67433i q^{3} +(-0.0825793 - 0.996584i) q^{4} -0.649399i q^{5} +(1.23185 + 1.13399i) q^{6} +(-0.789141 - 0.614213i) q^{8} -1.80339 q^{9} +(-0.477778 - 0.439826i) q^{10} -0.951895i q^{11} +(1.66861 - 0.138265i) q^{12} +1.08731 q^{15} +(-0.986361 + 0.164595i) q^{16} +1.97272 q^{17} +(-1.22140 + 1.32680i) q^{18} +(-0.647181 + 0.0536269i) q^{20} +(-0.700332 - 0.644701i) q^{22} -0.165159 q^{23} +(1.02840 - 1.32128i) q^{24} +0.578281 q^{25} -1.34514i q^{27} +(0.736415 - 0.799960i) q^{30} +(-0.546948 + 0.837166i) q^{32} +1.59379 q^{33} +(1.33609 - 1.45138i) q^{34} +(0.148923 + 1.79723i) q^{36} -1.93880i q^{37} +(-0.398869 + 0.512467i) q^{40} +0.490971 q^{41} +(-0.948644 + 0.0786068i) q^{44} +1.17112i q^{45} +(-0.111859 + 0.121511i) q^{46} +0.803391 q^{47} +(-0.275586 - 1.65150i) q^{48} +1.00000 q^{49} +(0.391659 - 0.425455i) q^{50} +3.30299i q^{51} +(-0.989654 - 0.911041i) q^{54} -0.618159 q^{55} +(-0.0897894 - 1.08360i) q^{60} +(0.245485 + 0.969400i) q^{64} +(1.07944 - 1.17259i) q^{66} +(-0.162906 - 1.96598i) q^{68} -0.276531i q^{69} +(1.42313 + 1.10767i) q^{72} -1.35456 q^{73} +(-1.42642 - 1.31311i) q^{74} +0.968235i q^{75} -1.89163 q^{79} +(0.106888 + 0.640542i) q^{80} +0.448828 q^{81} +(0.332526 - 0.361219i) q^{82} -1.28108i q^{85} +(-0.584666 + 0.751179i) q^{88} +(0.861621 + 0.793178i) q^{90} +(0.0136387 + 0.164595i) q^{92} +(0.544122 - 0.591074i) q^{94} +(-1.40170 - 0.915773i) q^{96} +(0.677282 - 0.735724i) q^{98} +1.71664i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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