LMFDB - Embedded newform 2872.1.b.e.717.18

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			717.18
Root			$0.879474 + 0.475947i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.986361 + 0.164595i) q^{2} +1.47145i q^{3} +(0.945817 + 0.324699i) q^{4} +1.93880i q^{5} +(-0.242192 + 1.45138i) q^{6} +(0.879474 + 0.475947i) q^{8} -1.16516 q^{9} +O(q^{10})$	\(q+(0.986361 + 0.164595i) q^{2} +1.47145i q^{3} +(0.945817 + 0.324699i) q^{4} +1.93880i q^{5} +(-0.242192 + 1.45138i) q^{6} +(0.879474 + 0.475947i) q^{8} -1.16516 q^{9} +(-0.319116 + 1.91236i) q^{10} -1.83155i q^{11} +(-0.477778 + 1.39172i) q^{12} -2.85284 q^{15} +(0.789141 + 0.614213i) q^{16} -1.57828 q^{17} +(-1.14927 - 0.191779i) q^{18} +(-0.629528 + 1.83375i) q^{20} +(0.301463 - 1.80657i) q^{22} +1.89163 q^{23} +(-0.700332 + 1.29410i) q^{24} -2.75895 q^{25} -0.243022i q^{27} +(-2.81393 - 0.469563i) q^{30} +(0.677282 + 0.735724i) q^{32} +2.69503 q^{33} +(-1.55676 - 0.259777i) q^{34} +(-1.10203 - 0.378326i) q^{36} -1.67433i q^{37} +(-0.922767 + 1.70512i) q^{40} +1.09390 q^{41} +(0.594702 - 1.73231i) q^{44} -2.25901i q^{45} +(1.86584 + 0.311353i) q^{46} +0.165159 q^{47} +(-0.903782 + 1.16118i) q^{48} +1.00000 q^{49} +(-2.72132 - 0.454108i) q^{50} -2.32236i q^{51} +(0.0400002 - 0.239708i) q^{54} +3.55100 q^{55} +(-2.69827 - 0.926317i) q^{60} +(0.546948 + 0.837166i) q^{64} +(2.65827 + 0.443587i) q^{66} +(-1.49277 - 0.512467i) q^{68} +2.78344i q^{69} +(-1.02473 - 0.554554i) q^{72} -1.97272 q^{73} +(0.275586 - 1.65150i) q^{74} -4.05965i q^{75} -0.490971 q^{79} +(-1.19084 + 1.52999i) q^{80} -0.807564 q^{81} +(1.07898 + 0.180049i) q^{82} -3.05997i q^{85} +(0.871720 - 1.61080i) q^{88} +(0.371821 - 2.22820i) q^{90} +(1.78914 + 0.614213i) q^{92} +(0.162906 + 0.0271842i) q^{94} +(-1.08258 + 0.996584i) q^{96} +(0.986361 + 0.164595i) q^{98} +2.13404i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.986361	+	0.164595i	0.986361	+	0.164595i
$2$	0.986361	+	0.164595i	0.986361	+	0.164595i
$3$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$3$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$4$	0.945817	+	0.324699i	0.945817	+	0.324699i
$5$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$5$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$6$	−0.242192	+	1.45138i	−0.242192	+	1.45138i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.879474	+	0.475947i	0.879474	+	0.475947i
$9$	−1.16516			−1.16516
$10$	−0.319116	+	1.91236i	−0.319116	+	1.91236i
$11$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$11$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$12$	−0.477778	+	1.39172i	−0.477778	+	1.39172i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−2.85284			−2.85284
$16$	0.789141	+	0.614213i	0.789141	+	0.614213i
$17$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$17$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$18$	−1.14927	−	0.191779i	−1.14927	−	0.191779i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.629528	+	1.83375i	−0.629528	+	1.83375i
$21$		0			0
$22$	0.301463	−	1.80657i	0.301463	−	1.80657i
$23$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$23$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$24$	−0.700332	+	1.29410i	−0.700332	+	1.29410i
$25$	−2.75895			−2.75895
$26$		0			0
$27$		−	0.243022i		−	0.243022i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−2.81393	−	0.469563i	−2.81393	−	0.469563i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.677282	+	0.735724i	0.677282	+	0.735724i
$33$	2.69503			2.69503
$34$	−1.55676	−	0.259777i	−1.55676	−	0.259777i
$35$		0			0
$36$	−1.10203	−	0.378326i	−1.10203	−	0.378326i
$37$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$37$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$38$		0			0
$39$		0			0
$40$	−0.922767	+	1.70512i	−0.922767	+	1.70512i
$41$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$41$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	0.594702	−	1.73231i	0.594702	−	1.73231i
$45$		−	2.25901i		−	2.25901i
$46$	1.86584	+	0.311353i	1.86584	+	0.311353i
$47$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$47$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$48$	−0.903782	+	1.16118i	−0.903782	+	1.16118i
$49$	1.00000			1.00000
$50$	−2.72132	−	0.454108i	−2.72132	−	0.454108i
$51$		−	2.32236i		−	2.32236i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	0.0400002	−	0.239708i	0.0400002	−	0.239708i
$55$	3.55100			3.55100
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−2.69827	−	0.926317i	−2.69827	−	0.926317i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	0.546948	+	0.837166i	0.546948	+	0.837166i
$65$		0			0
$66$	2.65827	+	0.443587i	2.65827	+	0.443587i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−1.49277	−	0.512467i	−1.49277	−	0.512467i
$69$			2.78344i			2.78344i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−1.02473	−	0.554554i	−1.02473	−	0.554554i
$73$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$73$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$74$	0.275586	−	1.65150i	0.275586	−	1.65150i
$75$		−	4.05965i		−	4.05965i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$79$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$80$	−1.19084	+	1.52999i	−1.19084	+	1.52999i
$81$	−0.807564			−0.807564
$82$	1.07898	+	0.180049i	1.07898	+	0.180049i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		−	3.05997i		−	3.05997i
$86$		0			0
$87$		0			0
$88$	0.871720	−	1.61080i	0.871720	−	1.61080i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.371821	−	2.22820i	0.371821	−	2.22820i
$91$		0			0
$92$	1.78914	+	0.614213i	1.78914	+	0.614213i
$93$		0			0
$94$	0.162906	+	0.0271842i	0.162906	+	0.0271842i
$95$		0			0
$96$	−1.08258	+	0.996584i	−1.08258	+	0.996584i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.986361	+	0.164595i	0.986361	+	0.164595i
$99$			2.13404i			2.13404i
$100$	−2.60946	−	0.895829i	−2.60946	−	0.895829i
$101$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$101$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$102$	0.382248	−	2.29068i	0.382248	−	2.29068i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$107$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$108$	0.0789092	−	0.229855i	0.0789092	−	0.229855i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	3.50257	+	0.584476i	3.50257	+	0.584476i
$111$	2.46369			2.46369
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			3.66750i			3.66750i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−2.50900	−	1.35780i	−2.50900	−	1.35780i
$121$	−2.35456			−2.35456
$122$		0			0
$123$			1.60961i			1.60961i
$124$		0			0
$125$		−	3.41025i		−	3.41025i
$126$		0			0
$127$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$127$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$128$	0.401695	+	0.915773i	0.401695	+	0.915773i
$129$		0			0
$130$		0			0
$131$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$131$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$132$	2.54900	+	0.875073i	2.54900	+	0.875073i
$133$		0			0
$134$		0			0
$135$	0.471172			0.471172
$136$	−1.38806	−	0.751179i	−1.38806	−	0.751179i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−0.458139	+	2.74548i	−0.458139	+	2.74548i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			0.243022i			0.243022i
$142$		0			0
$143$		0			0
$144$	−0.919474	−	0.715655i	−0.919474	−	0.715655i
$145$		0			0
$146$	−1.94582	−	0.324699i	−1.94582	−	0.324699i
$147$			1.47145i			1.47145i
$148$	0.543655	−	1.58361i	0.543655	−	1.58361i
$149$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$149$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$150$	0.668196	−	4.00428i	0.668196	−	4.00428i
$151$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$151$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$152$		0			0
$153$	1.83895			1.83895
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−0.484275	−	0.0808112i	−0.484275	−	0.0808112i
$159$		0			0
$160$	−1.42642	+	1.31311i	−1.42642	+	1.31311i
$161$		0			0
$162$	−0.796550	−	0.132921i	−0.796550	−	0.132921i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	1.03463	+	0.355188i	1.03463	+	0.355188i
$165$			5.22512i			5.22512i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	0.503655	−	3.01824i	0.503655	−	3.01824i
$171$		0			0
$172$		0			0
$173$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$173$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$174$		0			0
$175$		0			0
$176$	1.12496	−	1.44535i	1.12496	−	1.44535i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.733499	−	2.13661i	0.733499	−	2.13661i
$181$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$181$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$182$		0			0
$183$		0			0
$184$	1.66364	+	0.900319i	1.66364	+	0.900319i
$185$	3.24620			3.24620
$186$		0			0
$187$			2.89070i			2.89070i
$188$	0.156210	+	0.0536269i	0.156210	+	0.0536269i
$189$		0			0
$190$		0			0
$191$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$191$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$192$	−1.23185	+	0.804806i	−1.23185	+	0.804806i
$193$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$193$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$194$		0			0
$195$		0			0
$196$	0.945817	+	0.324699i	0.945817	+	0.324699i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−0.351252	+	2.10494i	−0.351252	+	2.10494i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2.42642	−	1.31311i	−2.42642	−	1.31311i
$201$		0			0
$202$	0.156677	−	0.938912i	0.156677	−	0.938912i
$203$		0			0
$204$	0.754068	−	2.19653i	0.754068	−	2.19653i
$205$			2.12085i			2.12085i
$206$		0			0
$207$	−2.20405			−2.20405
$208$		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
$214$	−0.275586	+	1.65150i	−0.275586	+	1.65150i
$215$		0			0
$216$	0.115666	−	0.213732i	0.115666	−	0.213732i
$217$		0			0
$218$		0			0
$219$		−	2.90276i		−	2.90276i
$220$	3.35860	+	1.15301i	3.35860	+	1.15301i
$221$		0			0
$222$	2.43009	+	0.405511i	2.43009	+	0.405511i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	3.21461			3.21461
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$229$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$230$	−0.603651	+	3.61748i	−0.603651	+	3.61748i
$231$		0			0
$232$		0			0
$233$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$233$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$234$		0			0
$235$			0.320210i			0.320210i
$236$		0			0
$237$		−	0.722438i		−	0.722438i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−2.25129	−	1.75225i	−2.25129	−	1.75225i
$241$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$241$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$242$	−2.32245	−	0.387548i	−2.32245	−	0.387548i
$243$		−	1.43131i		−	1.43131i
$244$		0			0
$245$			1.93880i			1.93880i
$246$	−0.264933	+	1.58766i	−0.264933	+	1.58766i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.561308	−	3.36374i	0.561308	−	3.36374i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		−	3.46462i		−	3.46462i
$254$	−0.162906	−	0.0271842i	−0.162906	−	0.0271842i
$255$	4.50259			4.50259
$256$	0.245485	+	0.969400i	0.245485	+	0.969400i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.156677	−	0.938912i	0.156677	−	0.938912i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	2.37020	+	1.28269i	2.37020	+	1.28269i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	0.464746	+	0.0775524i	0.464746	+	0.0775524i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−1.24549	−	0.969400i	−1.24549	−	0.969400i
$273$		0			0
$274$		0			0
$275$			5.05314i			5.05314i
$276$	−0.903782	+	2.63263i	−0.903782	+	2.63263i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$281$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$282$	−0.0400002	+	0.239708i	−0.0400002	+	0.239708i
$283$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$283$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.789141	−	0.857235i	−0.789141	−	0.857235i
$289$	1.49097			1.49097
$290$		0			0
$291$		0			0
$292$	−1.86584	−	0.640542i	−1.86584	−	0.640542i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−0.242192	+	1.45138i	−0.242192	+	1.45138i
$295$		0			0
$296$	0.796894	−	1.47253i	0.796894	−	1.47253i
$297$	−0.445107			−0.445107
$298$	0.106888	−	0.640542i	0.106888	−	0.640542i
$299$		0			0
$300$	1.31817	−	3.83968i	1.31817	−	3.83968i
$301$		0			0
$302$	0.792434	+	0.132234i	0.792434	+	0.132234i
$303$	1.40066			1.40066
$304$		0			0
$305$		0			0
$306$	1.81387	+	0.302681i	1.81387	+	0.302681i
$307$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$307$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$308$		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
$316$	−0.464369	−	0.159418i	−0.464369	−	0.159418i
$317$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$317$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$318$		0			0
$319$		0			0
$320$	−1.62310	+	1.06042i	−1.62310	+	1.06042i
$321$	−2.46369			−2.46369
$322$		0			0
$323$		0			0
$324$	−0.763808	−	0.262216i	−0.763808	−	0.262216i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	0.962053	+	0.520637i	0.962053	+	0.520637i
$329$		0			0
$330$	−0.860026	+	5.15385i	−0.860026	+	5.15385i
$331$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$331$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$332$		0			0
$333$			1.95086i			1.95086i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$339$		0			0
$340$	0.993571	−	2.89417i	0.993571	−	2.89417i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−5.39654			−5.39654
$346$	−0.328065	+	1.96598i	−0.328065	+	1.96598i
$347$		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
$352$	1.34751	−	1.24047i	1.34751	−	1.24047i
$353$		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
$360$	1.07517	−	1.98674i	1.07517	−	1.98674i
$361$	−1.00000			−1.00000
$362$	−0.0541828	+	0.324699i	−0.0541828	+	0.324699i
$363$		−	3.46462i		−	3.46462i
$364$		0			0
$365$		−	3.82472i		−	3.82472i
$366$		0			0
$367$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$367$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$368$	1.49277	+	1.16187i	1.49277	+	1.16187i
$369$	−1.27456			−1.27456
$370$	3.20192	+	0.534307i	3.20192	+	0.534307i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−0.475793	+	2.85127i	−0.475793	+	2.85127i
$375$	5.01800			5.01800
$376$	0.145253	+	0.0786068i	0.145253	+	0.0786068i
$377$		0			0
$378$		0			0
$379$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$379$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$380$		0			0
$381$		−	0.243022i		−	0.243022i
$382$	1.55676	+	0.259777i	1.55676	+	0.259777i
$383$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$383$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$384$	−1.34751	+	0.591074i	−1.34751	+	0.591074i
$385$		0			0
$386$	−0.792434	−	0.132234i	−0.792434	−	0.132234i
$387$		0			0
$388$		0			0
$389$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$389$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$390$		0			0
$391$	−2.98553			−2.98553
$392$	0.879474	+	0.475947i	0.879474	+	0.475947i
$393$	1.40066			1.40066
$394$		0			0
$395$		−	0.951895i		−	0.951895i
$396$	−0.692922	+	2.01841i	−0.692922	+	2.01841i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−2.17720	−	1.69458i	−2.17720	−	1.69458i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	0.309080	−	0.900319i	0.309080	−	0.900319i
$405$		−	1.56571i		−	1.56571i
$406$		0			0
$407$	−3.06662			−3.06662
$408$	1.10532	−	2.04245i	1.10532	−	2.04245i
$409$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$409$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$410$	−0.349080	+	2.09192i	−0.349080	+	2.09192i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−2.17399	−	0.362775i	−2.17399	−	0.362775i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$419$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−0.192436			−0.192436
$424$		0			0
$425$	4.35439			4.35439
$426$		0			0
$427$		0			0
$428$	−0.543655	+	1.58361i	−0.543655	+	1.58361i
$429$		0			0
$430$		0			0
$431$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$431$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$432$	0.149267	−	0.191779i	0.149267	−	0.191779i
$433$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$433$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.477778	−	2.86317i	0.477778	−	2.86317i
$439$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$439$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$440$	3.12301	+	1.69009i	3.12301	+	1.69009i
$441$	−1.16516			−1.16516
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	2.33020	+	0.799960i	2.33020	+	0.799960i
$445$		0			0
$446$		0			0
$447$	0.955557			0.955557
$448$		0			0
$449$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$449$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$450$	3.17077	+	0.529108i	3.17077	+	0.529108i
$451$		−	2.00352i		−	2.00352i
$452$		0			0
$453$			1.18215i			1.18215i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$457$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$458$	0.328065	−	1.96598i	0.328065	−	1.96598i
$459$			0.383558i			0.383558i
$460$	−1.19084	+	3.46879i	−1.19084	+	3.46879i
$461$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$461$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	1.73496	+	0.289513i	1.73496	+	0.289513i
$467$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$467$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$468$		0			0
$469$		0			0
$470$	−0.0527048	+	0.315843i	−0.0527048	+	0.315843i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	0.118909	−	0.712585i	0.118909	−	0.712585i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$479$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$480$	−1.93218	−	2.09891i	−1.93218	−	2.09891i
$481$		0			0
$482$	−1.86584	−	0.311353i	−1.86584	−	0.311353i
$483$		0			0
$484$	−2.22699	−	0.764525i	−2.22699	−	0.764525i
$485$		0			0
$486$	0.235586	−	1.41179i	0.235586	−	1.41179i
$487$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$487$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$488$		0			0
$489$		0			0
$490$	−0.319116	+	1.91236i	−0.319116	+	1.91236i
$491$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$491$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$492$	−0.522640	+	1.52240i	−0.522640	+	1.52240i
$493$		0			0
$494$		0			0
$495$	−4.13748			−4.13748
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	1.10731	−	3.22547i	1.10731	−	3.22547i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	1.84553			1.84553
$506$	0.570257	−	3.41736i	0.570257	−	3.41736i
$507$		−	1.47145i		−	1.47145i
$508$	−0.156210	−	0.0536269i	−0.156210	−	0.0536269i
$509$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$509$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$510$	4.44118	+	0.741102i	4.44118	+	0.741102i
$511$		0			0
$512$	0.0825793	+	0.996584i	0.0825793	+	0.996584i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	0.302496i		−	0.302496i
$518$		0			0
$519$	−2.93284			−2.93284
$520$		0			0
$521$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$521$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$522$		0			0
$523$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$523$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$524$	0.309080	−	0.900319i	0.309080	−	0.900319i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	2.12675	+	1.65532i	2.12675	+	1.65532i
$529$	2.57828			2.57828
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−3.24620			−3.24620
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.83155i		−	1.83155i
$540$	0.445643	+	0.152989i	0.445643	+	0.152989i
$541$			0.329189i			0.329189i	0.986361	+	0.164595i	$0.0526316\pi$
$541$			0.329189i			0.329189i	−0.986361	+	0.164595i	$0.947368\pi$
$542$		0			0
$543$	−0.484385			−0.484385
$544$	−1.06894	−	1.16118i	−1.06894	−	1.16118i
$545$		0			0
$546$		0			0
$547$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$547$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$548$		0			0
$549$		0			0
$550$	−0.831720	+	4.98422i	−0.831720	+	4.98422i
$551$		0			0
$552$	−1.32477	+	2.44796i	−1.32477	+	2.44796i
$553$		0			0
$554$		0			0
$555$			4.77661i			4.77661i
$556$		0			0
$557$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$557$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−4.25351			−4.25351
$562$	−1.33609	−	0.222954i	−1.33609	−	0.222954i
$563$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$563$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$564$	−0.0789092	+	0.229855i	−0.0789092	+	0.229855i
$565$		0			0
$566$	0.156677	−	0.938912i	0.156677	−	0.938912i
$567$		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
$573$			2.32236i			2.32236i
$574$		0			0
$575$	−5.21892			−5.21892
$576$	−0.637281	−	0.975432i	−0.637281	−	0.975432i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.47064	+	0.245406i	1.47064	+	0.245406i
$579$		−	1.18215i		−	1.18215i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.73496	−	0.938912i	−1.73496	−	0.938912i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.477778	+	1.39172i	−0.477778	+	1.39172i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.02840	−	1.32128i	1.02840	−	1.32128i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.439036	−	0.0732622i	−0.439036	−	0.0732622i
$595$		0			0
$596$	0.210859	−	0.614213i	0.210859	−	0.614213i
$597$		0			0
$598$		0			0
$599$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$599$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$600$	1.93218	−	3.57035i	1.93218	−	3.57035i
$601$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$601$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$602$		0			0
$603$		0			0
$604$	0.759861	+	0.260861i	0.759861	+	0.260861i
$605$		−	4.56503i		−	4.56503i
$606$	1.38156	+	0.230542i	1.38156	+	0.230542i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.73931	+	0.597105i	1.73931	+	0.597105i
$613$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$613$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$614$	−0.106888	+	0.640542i	−0.106888	+	0.640542i
$615$	−3.12072			−3.12072
$616$		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
$621$		−	0.459710i		−	0.459710i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3.85284			3.85284
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2.64257i			2.64257i
$630$		0			0
$631$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$631$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$632$	−0.431796	−	0.233676i	−0.431796	−	0.233676i
$633$		0			0
$634$	−0.301463	+	1.80657i	−0.301463	+	1.80657i
$635$		−	0.320210i		−	0.320210i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−1.77550	+	0.778807i	−1.77550	+	0.778807i
$641$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$641$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$642$	−2.43009	−	0.405511i	−2.43009	−	0.405511i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$647$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$648$	−0.710231	−	0.384358i	−0.710231	−	0.384358i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$653$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$654$		0			0
$655$	1.84553			1.84553
$656$	0.863238	+	0.671885i	0.863238	+	0.671885i
$657$	2.29853			2.29853
$658$		0			0
$659$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$659$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$660$	−1.69659	+	4.94201i	−1.69659	+	4.94201i
$661$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$661$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$662$	0.319116	−	1.91236i	0.319116	−	1.91236i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.321102	+	1.92426i	−0.321102	+	1.92426i
$667$		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
$675$			0.670486i			0.670486i
$676$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	1.45639	−	2.69117i	1.45639	−	2.69117i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.93284			2.93284
$688$		0			0
$689$		0			0
$690$	−5.32294	−	0.888241i	−5.32294	−	0.888241i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−0.647181	+	1.88517i	−0.647181	+	1.88517i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.72648			−1.72648
$698$		0			0
$699$			2.58820i			2.58820i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	1.53331	−	1.00176i	1.53331	−	1.00176i
$705$	−0.471172			−0.471172
$706$		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
$711$	0.572059			0.572059
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$719$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$719$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$720$	1.38751	−	1.78268i	1.38751	−	1.78268i
$721$		0			0
$722$	−0.986361	−	0.164595i	−0.986361	−	0.164595i
$723$		−	2.78344i		−	2.78344i
$724$	−0.106888	+	0.311353i	−0.106888	+	0.311353i
$725$		0			0
$726$	0.570257	−	3.41736i	0.570257	−	3.41736i
$727$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$727$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$728$		0			0
$729$	1.29853			1.29853
$730$	0.629528	−	3.77255i	0.629528	−	3.77255i
$731$		0			0
$732$		0			0
$733$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$733$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$734$	0.792434	+	0.132234i	0.792434	+	0.132234i
$735$	−2.85284			−2.85284
$736$	1.28117	+	1.39172i	1.28117	+	1.39172i
$737$		0			0
$738$	−1.25718	−	0.209786i	−1.25718	−	0.209786i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	3.07031	+	1.05404i	3.07031	+	1.05404i
$741$		0			0
$742$		0			0
$743$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$743$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$744$		0			0
$745$	1.25906			1.25906
$746$		0			0
$747$		0			0
$748$	−0.938607	+	2.73407i	−0.938607	+	2.73407i
$749$		0			0
$750$	4.94956	+	0.825936i	4.94956	+	0.825936i
$751$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$751$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$752$	0.130333	+	0.101443i	0.130333	+	0.101443i
$753$		0			0
$754$		0			0
$755$			1.55761i			1.55761i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	0.202192	−	1.21167i	0.202192	−	1.21167i
$759$	5.09800			5.09800
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	0.0400002	−	0.239708i	0.0400002	−	0.239708i
$763$		0			0
$764$	1.49277	+	0.512467i	1.49277	+	0.512467i
$765$			3.56535i			3.56535i
$766$	1.73496	+	0.289513i	1.73496	+	0.289513i
$767$		0			0
$768$	−1.42642	+	0.361219i	−1.42642	+	0.361219i
$769$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$769$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$770$		0			0
$771$		0			0
$772$	−0.759861	−	0.260861i	−0.759861	−	0.260861i
$773$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$773$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.275586	+	1.65150i	−0.275586	+	1.65150i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−2.94481	−	0.491402i	−2.94481	−	0.491402i
$783$		0			0
$784$	0.789141	+	0.614213i	0.789141	+	0.614213i
$785$		0			0
$786$	1.38156	+	0.230542i	1.38156	+	0.230542i
$787$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$787$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$788$		0			0
$789$		0			0
$790$	0.156677	−	0.938912i	0.156677	−	0.938912i
$791$		0			0
$792$	−1.01569	+	1.87683i	−1.01569	+	1.87683i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$797$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$798$		0			0
$799$	−0.260667			−0.260667
$800$	−1.86858	−	2.02982i	−1.86858	−	2.02982i
$801$		0			0
$802$		0			0
$803$			3.61313i			3.61313i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.453052	−	0.837166i	0.453052	−	0.837166i
$809$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$809$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$810$	0.257707	−	1.54435i	0.257707	−	1.54435i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−3.02479	−	0.504749i	−3.02479	−	0.504749i
$815$		0			0
$816$	1.42642	−	1.83267i	1.42642	−	1.83267i
$817$		0			0
$818$	−1.07898	−	0.180049i	−1.07898	−	0.180049i
$819$		0			0
$820$	−0.688638	+	2.00593i	−0.688638	+	2.00593i
$821$		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
$825$	−7.43543			−7.43543
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−2.08463	−	0.715655i	−2.08463	−	0.715655i
$829$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$829$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.57828			−1.57828
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.0541828	+	0.324699i	−0.0541828	+	0.324699i
$839$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$839$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$		−	1.99317i		−	1.99317i
$844$		0			0
$845$		−	1.93880i		−	1.93880i
$846$	−0.189812	−	0.0316739i	−0.189812	−	0.0316739i
$847$		0			0
$848$		0			0
$849$	1.40066			1.40066
$850$	4.29501	+	0.716710i	4.29501	+	0.716710i
$851$		−	3.16723i		−	3.16723i
$852$		0			0
$853$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$853$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$854$		0			0
$855$		0			0
$856$	−0.796894	+	1.47253i	−0.796894	+	1.47253i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$859$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$860$		0			0
$861$		0			0
$862$	−1.55676	−	0.259777i	−1.55676	−	0.259777i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	0.178797	−	0.164595i	0.178797	−	0.164595i
$865$	−3.86436			−3.86436
$866$	−0.484275	−	0.0808112i	−0.484275	−	0.0808112i
$867$			2.19389i			2.19389i
$868$		0			0
$869$			0.899236i			0.899236i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.942524	−	2.74548i	0.942524	−	2.74548i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−1.73496	−	0.289513i	−1.73496	−	0.289513i
$879$		0			0
$880$	2.80224	+	2.18107i	2.80224	+	2.18107i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.14927	−	0.191779i	−1.14927	−	0.191779i
$883$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$883$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$887$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$888$	2.16675	+	1.17259i	2.16675	+	1.17259i
$889$		0			0
$890$		0			0
$891$			1.47909i			1.47909i
$892$		0			0
$893$		0			0
$894$	0.942524	+	0.157279i	0.942524	+	0.157279i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.94582	+	0.324699i	1.94582	+	0.324699i
$899$		0			0
$900$	3.04044	+	1.04378i	3.04044	+	1.04378i
$901$		0			0
$902$	0.329769	−	1.97620i	0.329769	−	1.97620i
$903$		0			0
$904$		0			0
$905$	−0.638232			−0.638232
$906$	−0.194575	+	1.16602i	−0.194575	+	1.16602i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$			1.10911i			1.10911i
$910$		0			0
$911$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$911$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$912$		0			0
$913$		0			0
$914$	−1.55676	−	0.259777i	−1.55676	−	0.259777i
$915$		0			0
$916$	0.647181	−	1.88517i	0.647181	−	1.88517i
$917$		0			0
$918$	−0.0631315	+	0.378326i	−0.0631315	+	0.378326i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−1.74554	+	3.22547i	−1.74554	+	3.22547i
$921$	−0.955557			−0.955557
$922$	−0.242192	+	1.45138i	−0.242192	+	1.45138i
$923$		0			0
$924$		0			0
$925$			4.61940i			4.61940i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	1.66364	+	0.571129i	1.66364	+	0.571129i
$933$		0			0
$934$	0.328065	−	1.96598i	0.328065	−	1.96598i
$935$	−5.60448			−5.60448
$936$		0			0
$937$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$937$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$938$		0			0
$939$		0			0
$940$	−0.103972	+	0.302860i	−0.103972	+	0.302860i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	2.06925			2.06925
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$947$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$948$	0.234575	−	0.683294i	0.234575	−	0.683294i
$949$		0			0
$950$		0			0
$951$	−2.69503			−2.69503
$952$		0			0
$953$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$953$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$954$		0			0
$955$			3.05997i			3.05997i
$956$		0			0
$957$		0			0
$958$	1.33609	+	0.222954i	1.33609	+	0.222954i
$959$		0			0
$960$	−1.56036	−	2.38831i	−1.56036	−	2.38831i
$961$	1.00000			1.00000
$962$		0			0
$963$		−	1.95086i		−	1.95086i
$964$	−1.78914	−	0.614213i	−1.78914	−	0.614213i
$965$		−	1.55761i		−	1.55761i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−2.07078	−	1.12065i	−2.07078	−	1.12065i
$969$		0			0
$970$		0			0
$971$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$971$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$972$	0.464746	−	1.35376i	0.464746	−	1.35376i
$973$		0			0
$974$	1.33609	+	0.222954i	1.33609	+	0.222954i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−0.629528	+	1.83375i	−0.629528	+	1.83375i
$981$		0			0
$982$	0.275586	−	1.65150i	0.275586	−	1.65150i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−0.766090	+	1.41561i	−0.766090	+	1.41561i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−4.08105	−	0.681007i	−4.08105	−	0.681007i
$991$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$991$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$992$		0			0
$993$	2.85284			2.85284
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	−0.406900			−0.406900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.18	yes	18
8.5	even	2	inner	2872.1.b.e.717.17	✓	18
359.358	odd	2	CM	2872.1.b.e.717.18	yes	18
2872.717	odd	2	inner	2872.1.b.e.717.17	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.17	✓	18	8.5	even	2	inner
2872.1.b.e.717.17	✓	18	2872.717	odd	2	inner
2872.1.b.e.717.18	yes	18	1.1	even	1	trivial
2872.1.b.e.717.18	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)

\(f(q)\)	\(=\)	\(q+(0.986361 + 0.164595i) q^{2} +1.47145i q^{3} +(0.945817 + 0.324699i) q^{4} +1.93880i q^{5} +(-0.242192 + 1.45138i) q^{6} +(0.879474 + 0.475947i) q^{8} -1.16516 q^{9} +O(q^{10})\)	\(q+(0.986361 + 0.164595i) q^{2} +1.47145i q^{3} +(0.945817 + 0.324699i) q^{4} +1.93880i q^{5} +(-0.242192 + 1.45138i) q^{6} +(0.879474 + 0.475947i) q^{8} -1.16516 q^{9} +(-0.319116 + 1.91236i) q^{10} -1.83155i q^{11} +(-0.477778 + 1.39172i) q^{12} -2.85284 q^{15} +(0.789141 + 0.614213i) q^{16} -1.57828 q^{17} +(-1.14927 - 0.191779i) q^{18} +(-0.629528 + 1.83375i) q^{20} +(0.301463 - 1.80657i) q^{22} +1.89163 q^{23} +(-0.700332 + 1.29410i) q^{24} -2.75895 q^{25} -0.243022i q^{27} +(-2.81393 - 0.469563i) q^{30} +(0.677282 + 0.735724i) q^{32} +2.69503 q^{33} +(-1.55676 - 0.259777i) q^{34} +(-1.10203 - 0.378326i) q^{36} -1.67433i q^{37} +(-0.922767 + 1.70512i) q^{40} +1.09390 q^{41} +(0.594702 - 1.73231i) q^{44} -2.25901i q^{45} +(1.86584 + 0.311353i) q^{46} +0.165159 q^{47} +(-0.903782 + 1.16118i) q^{48} +1.00000 q^{49} +(-2.72132 - 0.454108i) q^{50} -2.32236i q^{51} +(0.0400002 - 0.239708i) q^{54} +3.55100 q^{55} +(-2.69827 - 0.926317i) q^{60} +(0.546948 + 0.837166i) q^{64} +(2.65827 + 0.443587i) q^{66} +(-1.49277 - 0.512467i) q^{68} +2.78344i q^{69} +(-1.02473 - 0.554554i) q^{72} -1.97272 q^{73} +(0.275586 - 1.65150i) q^{74} -4.05965i q^{75} -0.490971 q^{79} +(-1.19084 + 1.52999i) q^{80} -0.807564 q^{81} +(1.07898 + 0.180049i) q^{82} -3.05997i q^{85} +(0.871720 - 1.61080i) q^{88} +(0.371821 - 2.22820i) q^{90} +(1.78914 + 0.614213i) q^{92} +(0.162906 + 0.0271842i) q^{94} +(-1.08258 + 0.996584i) q^{96} +(0.986361 + 0.164595i) q^{98} +2.13404i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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