LMFDB - Embedded newform 2872.1.b.e.717.6

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			717.6
Root			$0.986361 + 0.164595i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.546948 + 0.837166i) q^{2} -1.93880i q^{3} +(-0.401695 - 0.915773i) q^{4} -1.99317i q^{5} +(1.62310 + 1.06042i) q^{6} +(0.986361 + 0.164595i) q^{8} -2.75895 q^{9} +O(q^{10})$	\(q+(-0.546948 + 0.837166i) q^{2} -1.93880i q^{3} +(-0.401695 - 0.915773i) q^{4} -1.99317i q^{5} +(1.62310 + 1.06042i) q^{6} +(0.986361 + 0.164595i) q^{8} -2.75895 q^{9} +(1.66861 + 1.09016i) q^{10} -1.22843i q^{11} +(-1.77550 + 0.778807i) q^{12} -3.86436 q^{15} +(-0.677282 + 0.735724i) q^{16} +1.35456 q^{17} +(1.50900 - 2.30970i) q^{18} +(-1.82529 + 0.800647i) q^{20} +(1.02840 + 0.671885i) q^{22} -0.803391 q^{23} +(0.319116 - 1.91236i) q^{24} -2.97272 q^{25} +3.41025i q^{27} +(2.11360 - 3.23511i) q^{30} +(-0.245485 - 0.969400i) q^{32} -2.38167 q^{33} +(-0.740876 + 1.13399i) q^{34} +(1.10826 + 2.52657i) q^{36} -0.649399i q^{37} +(0.328065 - 1.96598i) q^{40} +1.89163 q^{41} +(-1.12496 + 0.493453i) q^{44} +5.49905i q^{45} +(0.439413 - 0.672572i) q^{46} +1.75895 q^{47} +(1.42642 + 1.31311i) q^{48} +1.00000 q^{49} +(1.62593 - 2.48866i) q^{50} -2.62623i q^{51} +(-2.85495 - 1.86523i) q^{54} -2.44846 q^{55} +(1.55229 + 3.53888i) q^{60} +(0.945817 + 0.324699i) q^{64} +(1.30265 - 1.99386i) q^{66} +(-0.544122 - 1.24047i) q^{68} +1.55761i q^{69} +(-2.72132 - 0.454108i) q^{72} +1.09390 q^{73} +(0.543655 + 0.355188i) q^{74} +5.76352i q^{75} +0.165159 q^{79} +(1.46642 + 1.34994i) q^{80} +3.85284 q^{81} +(-1.03463 + 1.58361i) q^{82} -2.69987i q^{85} +(0.202192 - 1.21167i) q^{88} +(-4.60362 - 3.00769i) q^{90} +(0.322718 + 0.735724i) q^{92} +(-0.962053 + 1.47253i) q^{94} +(-1.87947 + 0.475947i) q^{96} +(-0.546948 + 0.837166i) q^{98} +3.38916i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$2$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$3$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$3$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$4$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$5$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$5$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$6$	1.62310	+	1.06042i	1.62310	+	1.06042i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.986361	+	0.164595i	0.986361	+	0.164595i
$9$	−2.75895			−2.75895
$10$	1.66861	+	1.09016i	1.66861	+	1.09016i
$11$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$11$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$12$	−1.77550	+	0.778807i	−1.77550	+	0.778807i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−3.86436			−3.86436
$16$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$17$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$17$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$18$	1.50900	−	2.30970i	1.50900	−	2.30970i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1.82529	+	0.800647i	−1.82529	+	0.800647i
$21$		0			0
$22$	1.02840	+	0.671885i	1.02840	+	0.671885i
$23$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$23$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$24$	0.319116	−	1.91236i	0.319116	−	1.91236i
$25$	−2.97272			−2.97272
$26$		0			0
$27$			3.41025i			3.41025i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	2.11360	−	3.23511i	2.11360	−	3.23511i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.245485	−	0.969400i	−0.245485	−	0.969400i
$33$	−2.38167			−2.38167
$34$	−0.740876	+	1.13399i	−0.740876	+	1.13399i
$35$		0			0
$36$	1.10826	+	2.52657i	1.10826	+	2.52657i
$37$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$37$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$38$		0			0
$39$		0			0
$40$	0.328065	−	1.96598i	0.328065	−	1.96598i
$41$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$41$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−1.12496	+	0.493453i	−1.12496	+	0.493453i
$45$			5.49905i			5.49905i
$46$	0.439413	−	0.672572i	0.439413	−	0.672572i
$47$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$47$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$48$	1.42642	+	1.31311i	1.42642	+	1.31311i
$49$	1.00000			1.00000
$50$	1.62593	−	2.48866i	1.62593	−	2.48866i
$51$		−	2.62623i		−	2.62623i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−2.85495	−	1.86523i	−2.85495	−	1.86523i
$55$	−2.44846			−2.44846
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.55229	+	3.53888i	1.55229	+	3.53888i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	0.945817	+	0.324699i	0.945817	+	0.324699i
$65$		0			0
$66$	1.30265	−	1.99386i	1.30265	−	1.99386i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−0.544122	−	1.24047i	−0.544122	−	1.24047i
$69$			1.55761i			1.55761i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−2.72132	−	0.454108i	−2.72132	−	0.454108i
$73$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$73$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$74$	0.543655	+	0.355188i	0.543655	+	0.355188i
$75$			5.76352i			5.76352i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$79$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$80$	1.46642	+	1.34994i	1.46642	+	1.34994i
$81$	3.85284			3.85284
$82$	−1.03463	+	1.58361i	−1.03463	+	1.58361i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	2.69987i		−	2.69987i
$86$		0			0
$87$		0			0
$88$	0.202192	−	1.21167i	0.202192	−	1.21167i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−4.60362	−	3.00769i	−4.60362	−	3.00769i
$91$		0			0
$92$	0.322718	+	0.735724i	0.322718	+	0.735724i
$93$		0			0
$94$	−0.962053	+	1.47253i	−0.962053	+	1.47253i
$95$		0			0
$96$	−1.87947	+	0.475947i	−1.87947	+	0.475947i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.546948	+	0.837166i	−0.546948	+	0.837166i
$99$			3.38916i			3.38916i
$100$	1.19413	+	2.72234i	1.19413	+	2.72234i
$101$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$101$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$102$	2.19859	+	1.43641i	2.19859	+	1.43641i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$107$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$108$	3.12301	−	1.36988i	3.12301	−	1.36988i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	1.33918	−	2.04977i	1.33918	−	2.04977i
$111$	−1.25906			−1.25906
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			1.60129i			1.60129i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−3.81165	−	0.636052i	−3.81165	−	0.636052i
$121$	−0.509029			−0.509029
$122$		0			0
$123$		−	3.66750i		−	3.66750i
$124$		0			0
$125$			3.93197i			3.93197i
$126$		0			0
$127$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$127$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$128$	−0.789141	+	0.614213i	−0.789141	+	0.614213i
$129$		0			0
$130$		0			0
$131$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$131$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$132$	0.956707	+	2.18107i	0.956707	+	2.18107i
$133$		0			0
$134$		0			0
$135$	6.79720			6.79720
$136$	1.33609	+	0.222954i	1.33609	+	0.222954i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−1.30398	−	0.851934i	−1.30398	−	0.851934i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	3.41025i		−	3.41025i
$142$		0			0
$143$		0			0
$144$	1.86858	−	2.02982i	1.86858	−	2.02982i
$145$		0			0
$146$	−0.598305	+	0.915773i	−0.598305	+	0.915773i
$147$		−	1.93880i		−	1.93880i
$148$	−0.594702	+	0.260861i	−0.594702	+	0.260861i
$149$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$149$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$150$	−4.82502	−	3.15234i	−4.82502	−	3.15234i
$151$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$151$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$152$		0			0
$153$	−3.73717			−3.73717
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−0.0903332	+	0.138265i	−0.0903332	+	0.138265i
$159$		0			0
$160$	−1.93218	+	0.489294i	−1.93218	+	0.489294i
$161$		0			0
$162$	−2.10731	+	3.22547i	−2.10731	+	3.22547i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−0.759861	−	1.73231i	−0.759861	−	1.73231i
$165$			4.74707i			4.74707i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	2.26024	+	1.47669i	2.26024	+	1.47669i
$171$		0			0
$172$		0			0
$173$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$173$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$174$		0			0
$175$		0			0
$176$	0.903782	+	0.831990i	0.903782	+	0.831990i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	5.03588	−	2.20894i	5.03588	−	2.20894i
$181$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$181$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$182$		0			0
$183$		0			0
$184$	−0.792434	−	0.132234i	−0.792434	−	0.132234i
$185$	−1.29436			−1.29436
$186$		0			0
$187$		−	1.66398i		−	1.66398i
$188$	−0.706561	−	1.61080i	−0.706561	−	1.61080i
$189$		0			0
$190$		0			0
$191$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$191$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$192$	0.629528	−	1.83375i	0.629528	−	1.83375i
$193$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$193$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$194$		0			0
$195$		0			0
$196$	−0.401695	−	0.915773i	−0.401695	−	0.915773i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−2.83729	−	1.85370i	−2.83729	−	1.85370i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2.93218	−	0.489294i	−2.93218	−	0.489294i
$201$		0			0
$202$	0.275586	+	0.180049i	0.275586	+	0.180049i
$203$		0			0
$204$	−2.40503	+	1.05494i	−2.40503	+	1.05494i
$205$		−	3.77035i		−	3.77035i
$206$		0			0
$207$	2.21651			2.21651
$208$		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
$214$	−0.543655	−	0.355188i	−0.543655	−	0.355188i
$215$		0			0
$216$	−0.561308	+	3.36374i	−0.561308	+	3.36374i
$217$		0			0
$218$		0			0
$219$		−	2.12085i		−	2.12085i
$220$	0.983535	+	2.24223i	0.983535	+	2.24223i
$221$		0			0
$222$	0.688638	−	1.05404i	0.688638	−	1.05404i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	8.20159			8.20159
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$229$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$230$	−1.34055	−	0.875825i	−1.34055	−	0.875825i
$231$		0			0
$232$		0			0
$233$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$233$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$234$		0			0
$235$		−	3.50588i		−	3.50588i
$236$		0			0
$237$		−	0.320210i		−	0.320210i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	2.61726	−	2.84310i	2.61726	−	2.84310i
$241$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$241$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$242$	0.278412	−	0.426142i	0.278412	−	0.426142i
$243$		−	4.05965i		−	4.05965i
$244$		0			0
$245$		−	1.99317i		−	1.99317i
$246$	3.07031	+	2.00593i	3.07031	+	2.00593i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−3.29171	−	2.15058i	−3.29171	−	2.15058i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			0.986906i			0.986906i
$254$	0.962053	−	1.47253i	0.962053	−	1.47253i
$255$	−5.23452			−5.23452
$256$	−0.0825793	−	0.996584i	−0.0825793	−	0.996584i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.275586	+	0.180049i	0.275586	+	0.180049i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	−2.34919	−	0.392010i	−2.34919	−	0.392010i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−3.71772	+	5.69039i	−3.71772	+	5.69039i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.917421	+	0.996584i	−0.917421	+	0.996584i
$273$		0			0
$274$		0			0
$275$			3.65177i			3.65177i
$276$	1.42642	−	0.625687i	1.42642	−	0.625687i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$281$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$282$	2.85495	+	1.86523i	2.85495	+	1.86523i
$283$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$283$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.677282	+	2.67452i	0.677282	+	2.67452i
$289$	0.834841			0.834841
$290$		0			0
$291$		0			0
$292$	−0.439413	−	1.00176i	−0.439413	−	1.00176i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	1.62310	+	1.06042i	1.62310	+	1.06042i
$295$		0			0
$296$	0.106888	−	0.640542i	0.106888	−	0.640542i
$297$	4.18924			4.18924
$298$	−1.53331	−	1.00176i	−1.53331	−	1.00176i
$299$		0			0
$300$	5.27807	−	2.31518i	5.27807	−	2.31518i
$301$		0			0
$302$	0.863238	−	1.32128i	0.863238	−	1.32128i
$303$	−0.638232			−0.638232
$304$		0			0
$305$		0			0
$306$	2.04404	−	3.12863i	2.04404	−	3.12863i
$307$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$307$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$308$		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
$316$	−0.0663435	−	0.151248i	−0.0663435	−	0.151248i
$317$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$317$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$318$		0			0
$319$		0			0
$320$	0.647181	−	1.88517i	0.647181	−	1.88517i
$321$	1.25906			1.25906
$322$		0			0
$323$		0			0
$324$	−1.54767	−	3.52833i	−1.54767	−	3.52833i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	1.86584	+	0.311353i	1.86584	+	0.311353i
$329$		0			0
$330$	−3.97409	−	2.59640i	−3.97409	−	2.59640i
$331$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$331$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$332$		0			0
$333$			1.79166i			1.79166i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.546948	−	0.837166i	0.546948	−	0.837166i
$339$		0			0
$340$	−2.47247	+	1.08453i	−2.47247	+	1.08453i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.10459			3.10459
$346$	−0.796894	−	0.520637i	−0.796894	−	0.520637i
$347$		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
$352$	−1.19084	+	0.301561i	−1.19084	+	0.301561i
$353$		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
$360$	−0.905114	+	5.42405i	−0.905114	+	5.42405i
$361$	−1.00000			−1.00000
$362$	−1.40170	−	0.915773i	−1.40170	−	0.915773i
$363$			0.986906i			0.986906i
$364$		0			0
$365$		−	2.18032i		−	2.18032i
$366$		0			0
$367$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$367$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$368$	0.544122	−	0.591074i	0.544122	−	0.591074i
$369$	−5.21892			−5.21892
$370$	0.707949	−	1.08360i	0.707949	−	1.08360i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	1.39303	+	0.910111i	1.39303	+	0.910111i
$375$	7.62330			7.62330
$376$	1.73496	+	0.289513i	1.73496	+	0.289513i
$377$		0			0
$378$		0			0
$379$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$379$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$380$		0			0
$381$			3.41025i			3.41025i
$382$	0.740876	−	1.13399i	0.740876	−	1.13399i
$383$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$383$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$384$	1.19084	+	1.52999i	1.19084	+	1.52999i
$385$		0			0
$386$	−0.863238	+	1.32128i	−0.863238	+	1.32128i
$387$		0			0
$388$		0			0
$389$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$389$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$390$		0			0
$391$	−1.08824			−1.08824
$392$	0.986361	+	0.164595i	0.986361	+	0.164595i
$393$	−0.638232			−0.638232
$394$		0			0
$395$		−	0.329189i		−	0.329189i
$396$	3.10370	−	1.36141i	3.10370	−	1.36141i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	2.01337	−	2.18710i	2.01337	−	2.18710i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−0.301463	+	0.132234i	−0.301463	+	0.132234i
$405$		−	7.67937i		−	7.67937i
$406$		0			0
$407$	−0.797738			−0.797738
$408$	0.432263	−	2.59041i	0.432263	−	2.59041i
$409$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$409$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$410$	3.15641	+	2.06218i	3.15641	+	2.06218i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.21232	+	1.85559i	−1.21232	+	1.85559i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$419$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−4.85284			−4.85284
$424$		0			0
$425$	−4.02674			−4.02674
$426$		0			0
$427$		0			0
$428$	0.594702	−	0.260861i	0.594702	−	0.260861i
$429$		0			0
$430$		0			0
$431$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$431$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$432$	−2.50900	−	2.30970i	−2.50900	−	2.30970i
$433$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$433$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.77550	+	1.15999i	1.77550	+	1.15999i
$439$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$439$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$440$	−2.41507	−	0.403003i	−2.41507	−	0.403003i
$441$	−2.75895			−2.75895
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	0.505757	+	1.15301i	0.505757	+	1.15301i
$445$		0			0
$446$		0			0
$447$	3.55100			3.55100
$448$		0			0
$449$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$449$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$450$	−4.48584	+	6.86609i	−4.48584	+	6.86609i
$451$		−	2.32373i		−	2.32373i
$452$		0			0
$453$			3.05997i			3.05997i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$457$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$458$	0.796894	+	0.520637i	0.796894	+	0.520637i
$459$			4.61940i			4.61940i
$460$	1.46642	−	0.643232i	1.46642	−	0.643232i
$461$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$461$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.07898	+	1.65150i	−1.07898	+	1.65150i
$467$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$467$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$468$		0			0
$469$		0			0
$470$	2.93500	+	1.91753i	2.93500	+	1.91753i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	0.268069	+	0.175138i	0.268069	+	0.175138i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$479$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$480$	0.948644	+	3.74611i	0.948644	+	3.74611i
$481$		0			0
$482$	−0.439413	+	0.672572i	−0.439413	+	0.672572i
$483$		0			0
$484$	0.204475	+	0.466155i	0.204475	+	0.466155i
$485$		0			0
$486$	3.39860	+	2.22042i	3.39860	+	2.22042i
$487$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$487$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$488$		0			0
$489$		0			0
$490$	1.66861	+	1.09016i	1.66861	+	1.09016i
$491$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$491$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$492$	−3.35860	+	1.47322i	−3.35860	+	1.47322i
$493$		0			0
$494$		0			0
$495$	6.75517			6.75517
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	3.60079	−	1.57945i	3.60079	−	1.57945i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	−0.656130			−0.656130
$506$	−0.826204	−	0.539786i	−0.826204	−	0.539786i
$507$			1.93880i			1.93880i
$508$	0.706561	+	1.61080i	0.706561	+	1.61080i
$509$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$509$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$510$	2.86301	−	4.38216i	2.86301	−	4.38216i
$511$		0			0
$512$	0.879474	+	0.475947i	0.879474	+	0.475947i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	2.16074i		−	2.16074i
$518$		0			0
$519$	1.84553			1.84553
$520$		0			0
$521$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$521$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$522$		0			0
$523$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$523$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$524$	−0.301463	+	0.132234i	−0.301463	+	0.132234i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.61306	−	1.75225i	1.61306	−	1.75225i
$529$	−0.354563			−0.354563
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.29436			1.29436
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.22843i		−	1.22843i
$540$	−2.73040	−	6.22470i	−2.73040	−	6.22470i
$541$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$541$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$542$		0			0
$543$	3.24620			3.24620
$544$	−0.332526	−	1.31311i	−0.332526	−	1.31311i
$545$		0			0
$546$		0			0
$547$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$547$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$548$		0			0
$549$		0			0
$550$	−3.05714	−	1.99733i	−3.05714	−	1.99733i
$551$		0			0
$552$	−0.256375	+	1.53637i	−0.256375	+	1.53637i
$553$		0			0
$554$		0			0
$555$			2.50951i			2.50951i
$556$		0			0
$557$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$557$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−3.22612			−3.22612
$562$	−0.268536	+	0.411024i	−0.268536	+	0.411024i
$563$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$563$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$564$	−3.12301	+	1.36988i	−3.12301	+	1.36988i
$565$		0			0
$566$	0.275586	+	0.180049i	0.275586	+	0.180049i
$567$		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
$573$			2.62623i			2.62623i
$574$		0			0
$575$	2.38826			2.38826
$576$	−2.60946	−	0.895829i	−2.60946	−	0.895829i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.456615	+	0.698901i	−0.456615	+	0.698901i
$579$		−	3.05997i		−	3.05997i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.07898	+	0.180049i	1.07898	+	0.180049i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−1.77550	+	0.778807i	−1.77550	+	0.778807i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.477778	+	0.439826i	0.477778	+	0.439826i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−2.29129	+	3.50709i	−2.29129	+	3.50709i
$595$		0			0
$596$	1.67728	−	0.735724i	1.67728	−	0.735724i
$597$		0			0
$598$		0			0
$599$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$599$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$600$	−0.948644	+	5.68491i	−0.948644	+	5.68491i
$601$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$601$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$602$		0			0
$603$		0			0
$604$	0.633988	+	1.44535i	0.633988	+	1.44535i
$605$			1.01458i			1.01458i
$606$	0.349080	−	0.534307i	0.349080	−	0.534307i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.50120	+	3.42240i	1.50120	+	3.42240i
$613$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$613$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$614$	1.53331	+	1.00176i	1.53331	+	1.00176i
$615$	−7.30995			−7.30995
$616$		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
$621$		−	2.73976i		−	2.73976i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.86436			4.86436
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.879652i		−	0.879652i
$630$		0			0
$631$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$631$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$632$	0.162906	+	0.0271842i	0.162906	+	0.0271842i
$633$		0			0
$634$	−1.02840	−	0.671885i	−1.02840	−	0.671885i
$635$			3.50588i			3.50588i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.22423	+	1.57289i	1.22423	+	1.57289i
$641$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$641$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$642$	−0.688638	+	1.05404i	−0.688638	+	1.05404i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$647$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$648$	3.80030	+	0.634157i	3.80030	+	0.634157i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$653$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$654$		0			0
$655$	−0.656130			−0.656130
$656$	−1.28117	+	1.39172i	−1.28117	+	1.39172i
$657$	−3.01800			−3.01800
$658$		0			0
$659$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$659$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$660$	4.34724	−	1.90688i	4.34724	−	1.90688i
$661$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$661$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$662$	−1.66861	−	1.09016i	−1.66861	−	1.09016i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.49992	−	0.979944i	−1.49992	−	0.979944i
$667$		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
$675$		−	10.1377i		−	10.1377i
$676$	0.401695	+	0.915773i	0.401695	+	0.915773i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	0.444385	−	2.66305i	0.444385	−	2.66305i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.84553			−1.84553
$688$		0			0
$689$		0			0
$690$	−1.69805	+	2.59906i	−1.69805	+	2.59906i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	0.871720	−	0.382372i	0.871720	−	0.382372i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.56234			2.56234
$698$		0			0
$699$		−	3.82472i		−	3.82472i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.398869	−	1.16187i	0.398869	−	1.16187i
$705$	−6.79720			−6.79720
$706$		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
$711$	−0.455664			−0.455664
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.546948	−	0.837166i	0.546948	−	0.837166i
$719$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$719$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$720$	−4.04578	−	3.72440i	−4.04578	−	3.72440i
$721$		0			0
$722$	0.546948	−	0.837166i	0.546948	−	0.837166i
$723$		−	1.55761i		−	1.55761i
$724$	1.53331	−	0.672572i	1.53331	−	0.672572i
$725$		0			0
$726$	−0.826204	−	0.539786i	−0.826204	−	0.539786i
$727$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$727$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$728$		0			0
$729$	−4.01800			−4.01800
$730$	1.82529	+	1.19252i	1.82529	+	1.19252i
$731$		0			0
$732$		0			0
$733$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$733$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$734$	0.863238	−	1.32128i	0.863238	−	1.32128i
$735$	−3.86436			−3.86436
$736$	0.197221	+	0.778807i	0.197221	+	0.778807i
$737$		0			0
$738$	2.85448	−	4.36911i	2.85448	−	4.36911i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	0.519939	+	1.18534i	0.519939	+	1.18534i
$741$		0			0
$742$		0			0
$743$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$743$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$744$		0			0
$745$	3.65058			3.65058
$746$		0			0
$747$		0			0
$748$	−1.52383	+	0.668413i	−1.52383	+	0.668413i
$749$		0			0
$750$	−4.16955	+	6.38198i	−4.16955	+	6.38198i
$751$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$751$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$752$	−1.19130	+	1.29410i	−1.19130	+	1.29410i
$753$		0			0
$754$		0			0
$755$			3.14578i			3.14578i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	1.23185	+	0.804806i	1.23185	+	0.804806i
$759$	1.91341			1.91341
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	−2.85495	−	1.86523i	−2.85495	−	1.86523i
$763$		0			0
$764$	0.544122	+	1.24047i	0.544122	+	1.24047i
$765$			7.44881i			7.44881i
$766$	−1.07898	+	1.65150i	−1.07898	+	1.65150i
$767$		0			0
$768$	−1.93218	+	0.160105i	−1.93218	+	0.160105i
$769$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$769$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$770$		0			0
$771$		0			0
$772$	−0.633988	−	1.44535i	−0.633988	−	1.44535i
$773$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$773$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.543655	−	0.355188i	−0.543655	−	0.355188i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.595213	−	0.911041i	0.595213	−	0.911041i
$783$		0			0
$784$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$785$		0			0
$786$	0.349080	−	0.534307i	0.349080	−	0.534307i
$787$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$787$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$788$		0			0
$789$		0			0
$790$	0.275586	+	0.180049i	0.275586	+	0.180049i
$791$		0			0
$792$	−0.557838	+	3.34294i	−0.557838	+	3.34294i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$797$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$798$		0			0
$799$	2.38261			2.38261
$800$	0.729760	+	2.88176i	0.729760	+	2.88176i
$801$		0			0
$802$		0			0
$803$		−	1.34377i		−	1.34377i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.0541828	−	0.324699i	0.0541828	−	0.324699i
$809$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$809$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$810$	6.42891	+	4.20022i	6.42891	+	4.20022i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	0.436321	−	0.667840i	0.436321	−	0.667840i
$815$		0			0
$816$	1.93218	+	1.77870i	1.93218	+	1.77870i
$817$		0			0
$818$	1.03463	−	1.58361i	1.03463	−	1.58361i
$819$		0			0
$820$	−3.45278	+	1.51453i	−3.45278	+	1.51453i
$821$		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
$825$	7.08005			7.08005
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−0.890363	−	2.02982i	−0.890363	−	2.02982i
$829$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$829$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.35456			1.35456
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.40170	−	0.915773i	−1.40170	−	0.915773i
$839$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$839$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$		−	0.951895i		−	0.951895i
$844$		0			0
$845$			1.99317i			1.99317i
$846$	2.65425	−	4.06264i	2.65425	−	4.06264i
$847$		0			0
$848$		0			0
$849$	−0.638232			−0.638232
$850$	2.20242	−	3.37105i	2.20242	−	3.37105i
$851$			0.521721i			0.521721i
$852$		0			0
$853$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$853$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$854$		0			0
$855$		0			0
$856$	−0.106888	+	0.640542i	−0.106888	+	0.640542i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$859$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$860$		0			0
$861$		0			0
$862$	−0.740876	+	1.13399i	−0.740876	+	1.13399i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	3.30590	−	0.837166i	3.30590	−	0.837166i
$865$	1.89729			1.89729
$866$	−0.0903332	+	0.138265i	−0.0903332	+	0.138265i
$867$		−	1.61859i		−	1.61859i
$868$		0			0
$869$		−	0.202885i		−	0.202885i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−1.94221	+	0.851934i	−1.94221	+	0.851934i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	1.07898	−	1.65150i	1.07898	−	1.65150i
$879$		0			0
$880$	1.65830	−	1.80139i	1.65830	−	1.80139i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.50900	−	2.30970i	1.50900	−	2.30970i
$883$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$883$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$887$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$888$	−1.24188	−	0.207234i	−1.24188	−	0.207234i
$889$		0			0
$890$		0			0
$891$		−	4.73293i		−	4.73293i
$892$		0			0
$893$		0			0
$894$	−1.94221	+	2.97278i	−1.94221	+	2.97278i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.598305	−	0.915773i	0.598305	−	0.915773i
$899$		0			0
$900$	−3.29454	−	7.51079i	−3.29454	−	7.51079i
$901$		0			0
$902$	1.94535	+	1.27096i	1.94535	+	1.27096i
$903$		0			0
$904$		0			0
$905$	3.33723			3.33723
$906$	−2.56171	−	1.67365i	−2.56171	−	1.67365i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			0.908216i			0.908216i
$910$		0			0
$911$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$911$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$912$		0			0
$913$		0			0
$914$	−0.740876	+	1.13399i	−0.740876	+	1.13399i
$915$		0			0
$916$	−0.871720	+	0.382372i	−0.871720	+	0.382372i
$917$		0			0
$918$	−3.86720	−	2.52657i	−3.86720	−	2.52657i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−0.263564	+	1.57945i	−0.263564	+	1.57945i
$921$	−3.55100			−3.55100
$922$	1.62310	+	1.06042i	1.62310	+	1.06042i
$923$		0			0
$924$		0			0
$925$			1.93048i			1.93048i
$926$		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
$932$	−0.792434	−	1.80657i	−0.792434	−	1.80657i
$933$		0			0
$934$	0.796894	+	0.520637i	0.796894	+	0.520637i
$935$	−3.31659			−3.31659
$936$		0			0
$937$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$937$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$938$		0			0
$939$		0			0
$940$	−3.21059	+	1.40830i	−3.21059	+	1.40830i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−1.51972			−1.51972
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$947$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$948$	−0.293240	+	0.128627i	−0.293240	+	0.128627i
$949$		0			0
$950$		0			0
$951$	2.38167			2.38167
$952$		0			0
$953$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$953$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$954$		0			0
$955$			2.69987i			2.69987i
$956$		0			0
$957$		0			0
$958$	0.268536	−	0.411024i	0.268536	−	0.411024i
$959$		0			0
$960$	−3.65498	−	1.25475i	−3.65498	−	1.25475i
$961$	1.00000			1.00000
$962$		0			0
$963$		−	1.79166i		−	1.79166i
$964$	−0.322718	−	0.735724i	−0.322718	−	0.735724i
$965$		−	3.14578i		−	3.14578i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.502087	−	0.0837834i	−0.502087	−	0.0837834i
$969$		0			0
$970$		0			0
$971$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$971$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$972$	−3.71772	+	1.63074i	−3.71772	+	1.63074i
$973$		0			0
$974$	0.268536	−	0.411024i	0.268536	−	0.411024i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−1.82529	+	0.800647i	−1.82529	+	0.800647i
$981$		0			0
$982$	0.543655	+	0.355188i	0.543655	+	0.355188i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	0.603651	−	3.61748i	0.603651	−	3.61748i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−3.69473	+	5.65520i	−3.69473	+	5.65520i
$991$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$991$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$992$		0			0
$993$	3.86436			3.86436
$994$		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
$999$	2.21461			2.21461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.6	yes	18
8.5	even	2	inner	2872.1.b.e.717.5	✓	18
359.358	odd	2	CM	2872.1.b.e.717.6	yes	18
2872.717	odd	2	inner	2872.1.b.e.717.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.5	✓	18	8.5	even	2	inner
2872.1.b.e.717.5	✓	18	2872.717	odd	2	inner
2872.1.b.e.717.6	yes	18	1.1	even	1	trivial
2872.1.b.e.717.6	yes	18	359.358	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2872 = 2^{3} \cdot 359 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2872.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.546948 + 0.837166i) q^{2} -1.93880i q^{3} +(-0.401695 - 0.915773i) q^{4} -1.99317i q^{5} +(1.62310 + 1.06042i) q^{6} +(0.986361 + 0.164595i) q^{8} -2.75895 q^{9} +O(q^{10})\)	\(q+(-0.546948 + 0.837166i) q^{2} -1.93880i q^{3} +(-0.401695 - 0.915773i) q^{4} -1.99317i q^{5} +(1.62310 + 1.06042i) q^{6} +(0.986361 + 0.164595i) q^{8} -2.75895 q^{9} +(1.66861 + 1.09016i) q^{10} -1.22843i q^{11} +(-1.77550 + 0.778807i) q^{12} -3.86436 q^{15} +(-0.677282 + 0.735724i) q^{16} +1.35456 q^{17} +(1.50900 - 2.30970i) q^{18} +(-1.82529 + 0.800647i) q^{20} +(1.02840 + 0.671885i) q^{22} -0.803391 q^{23} +(0.319116 - 1.91236i) q^{24} -2.97272 q^{25} +3.41025i q^{27} +(2.11360 - 3.23511i) q^{30} +(-0.245485 - 0.969400i) q^{32} -2.38167 q^{33} +(-0.740876 + 1.13399i) q^{34} +(1.10826 + 2.52657i) q^{36} -0.649399i q^{37} +(0.328065 - 1.96598i) q^{40} +1.89163 q^{41} +(-1.12496 + 0.493453i) q^{44} +5.49905i q^{45} +(0.439413 - 0.672572i) q^{46} +1.75895 q^{47} +(1.42642 + 1.31311i) q^{48} +1.00000 q^{49} +(1.62593 - 2.48866i) q^{50} -2.62623i q^{51} +(-2.85495 - 1.86523i) q^{54} -2.44846 q^{55} +(1.55229 + 3.53888i) q^{60} +(0.945817 + 0.324699i) q^{64} +(1.30265 - 1.99386i) q^{66} +(-0.544122 - 1.24047i) q^{68} +1.55761i q^{69} +(-2.72132 - 0.454108i) q^{72} +1.09390 q^{73} +(0.543655 + 0.355188i) q^{74} +5.76352i q^{75} +0.165159 q^{79} +(1.46642 + 1.34994i) q^{80} +3.85284 q^{81} +(-1.03463 + 1.58361i) q^{82} -2.69987i q^{85} +(0.202192 - 1.21167i) q^{88} +(-4.60362 - 3.00769i) q^{90} +(0.322718 + 0.735724i) q^{92} +(-0.962053 + 1.47253i) q^{94} +(-1.87947 + 0.475947i) q^{96} +(-0.546948 + 0.837166i) q^{98} +3.38916i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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