LMFDB - Embedded newform 2872.1.b.e.717.4

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			717.4
Root			$0.401695 + 0.915773i$ of defining polynomial
Character	$\chi$	$=$	2872.717
Dual form			2872.1.b.e.717.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.789141 + 0.614213i) q^{2} -0.329189i q^{3} +(0.245485 - 0.969400i) q^{4} +1.67433i q^{5} +(0.202192 + 0.259777i) q^{6} +(0.401695 + 0.915773i) q^{8} +0.891634 q^{9} +O(q^{10})$	\(q+(-0.789141 + 0.614213i) q^{2} -0.329189i q^{3} +(0.245485 - 0.969400i) q^{4} +1.67433i q^{5} +(0.202192 + 0.259777i) q^{6} +(0.401695 + 0.915773i) q^{8} +0.891634 q^{9} +(-1.02840 - 1.32128i) q^{10} +1.99317i q^{11} +(-0.319116 - 0.0808112i) q^{12} +0.551172 q^{15} +(-0.879474 - 0.475947i) q^{16} +1.75895 q^{17} +(-0.703625 + 0.547653i) q^{18} +(1.62310 + 0.411024i) q^{20} +(-1.22423 - 1.57289i) q^{22} +0.490971 q^{23} +(0.301463 - 0.132234i) q^{24} -1.80339 q^{25} -0.622706i q^{27} +(-0.434952 + 0.338537i) q^{30} +(0.986361 - 0.164595i) q^{32} +0.656130 q^{33} +(-1.38806 + 1.08037i) q^{34} +(0.218883 - 0.864351i) q^{36} -1.47145i q^{37} +(-1.53331 + 0.672572i) q^{40} -1.35456 q^{41} +(1.93218 + 0.489294i) q^{44} +1.49289i q^{45} +(-0.387445 + 0.301561i) q^{46} -1.89163 q^{47} +(-0.156677 + 0.289513i) q^{48} +1.00000 q^{49} +(1.42313 - 1.10767i) q^{50} -0.579026i q^{51} +(0.382474 + 0.491402i) q^{54} -3.33723 q^{55} +(0.135305 - 0.534307i) q^{60} +(-0.677282 + 0.735724i) q^{64} +(-0.517778 + 0.403003i) q^{66} +(0.431796 - 1.70512i) q^{68} -0.161622i q^{69} +(0.358165 + 0.816535i) q^{72} +1.57828 q^{73} +(0.903782 + 1.16118i) q^{74} +0.593657i q^{75} -1.09390 q^{79} +(0.796894 - 1.47253i) q^{80} +0.686647 q^{81} +(1.06894 - 0.831990i) q^{82} +2.94506i q^{85} +(-1.82529 + 0.800647i) q^{88} +(-0.916954 - 1.17810i) q^{90} +(0.120526 - 0.475947i) q^{92} +(1.49277 - 1.16187i) q^{94} +(-0.0541828 - 0.324699i) q^{96} +(-0.789141 + 0.614213i) q^{98} +1.77718i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.789141	+	0.614213i	−0.789141	+	0.614213i
$2$	−0.789141	+	0.614213i	−0.789141	+	0.614213i
$3$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$3$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$4$	0.245485	−	0.969400i	0.245485	−	0.969400i
$5$			1.67433i			1.67433i	0.546948	+	0.837166i	$0.315789\pi$
$5$			1.67433i			1.67433i	−0.546948	+	0.837166i	$0.684211\pi$
$6$	0.202192	+	0.259777i	0.202192	+	0.259777i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	0.401695	+	0.915773i	0.401695	+	0.915773i
$9$	0.891634			0.891634
$10$	−1.02840	−	1.32128i	−1.02840	−	1.32128i
$11$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$11$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$12$	−0.319116	−	0.0808112i	−0.319116	−	0.0808112i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	0.551172			0.551172
$16$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$17$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$17$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$18$	−0.703625	+	0.547653i	−0.703625	+	0.547653i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	1.62310	+	0.411024i	1.62310	+	0.411024i
$21$		0			0
$22$	−1.22423	−	1.57289i	−1.22423	−	1.57289i
$23$	0.490971			0.490971			0.245485	−	0.969400i	$-0.421053\pi$
$23$	0.490971			0.490971			0.245485	+	0.969400i	$0.421053\pi$
$24$	0.301463	−	0.132234i	0.301463	−	0.132234i
$25$	−1.80339			−1.80339
$26$		0			0
$27$		−	0.622706i		−	0.622706i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−0.434952	+	0.338537i	−0.434952	+	0.338537i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.986361	−	0.164595i	0.986361	−	0.164595i
$33$	0.656130			0.656130
$34$	−1.38806	+	1.08037i	−1.38806	+	1.08037i
$35$		0			0
$36$	0.218883	−	0.864351i	0.218883	−	0.864351i
$37$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$37$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$38$		0			0
$39$		0			0
$40$	−1.53331	+	0.672572i	−1.53331	+	0.672572i
$41$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$41$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	1.93218	+	0.489294i	1.93218	+	0.489294i
$45$			1.49289i			1.49289i
$46$	−0.387445	+	0.301561i	−0.387445	+	0.301561i
$47$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$47$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$48$	−0.156677	+	0.289513i	−0.156677	+	0.289513i
$49$	1.00000			1.00000
$50$	1.42313	−	1.10767i	1.42313	−	1.10767i
$51$		−	0.579026i		−	0.579026i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	0.382474	+	0.491402i	0.382474	+	0.491402i
$55$	−3.33723			−3.33723
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	0.135305	−	0.534307i	0.135305	−	0.534307i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.677282	+	0.735724i	−0.677282	+	0.735724i
$65$		0			0
$66$	−0.517778	+	0.403003i	−0.517778	+	0.403003i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	0.431796	−	1.70512i	0.431796	−	1.70512i
$69$		−	0.161622i		−	0.161622i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.358165	+	0.816535i	0.358165	+	0.816535i
$73$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$73$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$74$	0.903782	+	1.16118i	0.903782	+	1.16118i
$75$			0.593657i			0.593657i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$79$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$80$	0.796894	−	1.47253i	0.796894	−	1.47253i
$81$	0.686647			0.686647
$82$	1.06894	−	0.831990i	1.06894	−	0.831990i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$			2.94506i			2.94506i
$86$		0			0
$87$		0			0
$88$	−1.82529	+	0.800647i	−1.82529	+	0.800647i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.916954	−	1.17810i	−0.916954	−	1.17810i
$91$		0			0
$92$	0.120526	−	0.475947i	0.120526	−	0.475947i
$93$		0			0
$94$	1.49277	−	1.16187i	1.49277	−	1.16187i
$95$		0			0
$96$	−0.0541828	−	0.324699i	−0.0541828	−	0.324699i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.789141	+	0.614213i	−0.789141	+	0.614213i
$99$			1.77718i			1.77718i
$100$	−0.442706	+	1.74821i	−0.442706	+	1.74821i
$101$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$101$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$102$	0.355645	+	0.456933i	0.355645	+	0.456933i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$107$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$108$	−0.603651	−	0.152865i	−0.603651	−	0.152865i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	2.63354	−	2.04977i	2.63354	−	2.04977i
$111$	−0.484385			−0.484385
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			0.822049i			0.822049i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.221403	+	0.504749i	0.221403	+	0.504749i
$121$	−2.97272			−2.97272
$122$		0			0
$123$			0.445908i			0.445908i
$124$		0			0
$125$		−	1.34514i		−	1.34514i
$126$		0			0
$127$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$127$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$128$	0.0825793	−	0.996584i	0.0825793	−	0.996584i
$129$		0			0
$130$		0			0
$131$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$131$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$132$	0.161070	−	0.636052i	0.161070	−	0.636052i
$133$		0			0
$134$		0			0
$135$	1.04262			1.04262
$136$	0.706561	+	1.61080i	0.706561	+	1.61080i
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	0.0992705	+	0.127543i	0.0992705	+	0.127543i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$			0.622706i			0.622706i
$142$		0			0
$143$		0			0
$144$	−0.784169	−	0.424371i	−0.784169	−	0.424371i
$145$		0			0
$146$	−1.24549	+	0.969400i	−1.24549	+	0.969400i
$147$		−	0.329189i		−	0.329189i
$148$	−1.42642	−	0.361219i	−1.42642	−	0.361219i
$149$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$149$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$150$	−0.364632	−	0.468479i	−0.364632	−	0.468479i
$151$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$151$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$152$		0			0
$153$	1.56834			1.56834
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	0.863238	−	0.671885i	0.863238	−	0.671885i
$159$		0			0
$160$	0.275586	+	1.65150i	0.275586	+	1.65150i
$161$		0			0
$162$	−0.541861	+	0.421747i	−0.541861	+	0.421747i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−0.332526	+	1.31311i	−0.332526	+	1.31311i
$165$			1.09858i			1.09858i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$	−1.80890	−	2.32407i	−1.80890	−	2.32407i
$171$		0			0
$172$		0			0
$173$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$173$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$174$		0			0
$175$		0			0
$176$	0.948644	−	1.75294i	0.948644	−	1.75294i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	1.44721	+	0.366484i	1.44721	+	0.366484i
$181$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$181$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$182$		0			0
$183$		0			0
$184$	0.197221	+	0.449618i	0.197221	+	0.449618i
$185$	2.46369			2.46369
$186$		0			0
$187$			3.50588i			3.50588i
$188$	−0.464369	+	1.83375i	−0.464369	+	1.83375i
$189$		0			0
$190$		0			0
$191$	−1.75895			−1.75895			−0.879474	−	0.475947i	$-0.842105\pi$
$191$	−1.75895			−1.75895			−0.879474	+	0.475947i	$0.842105\pi$
$192$	0.242192	+	0.222954i	0.242192	+	0.222954i
$193$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$193$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$194$		0			0
$195$		0			0
$196$	0.245485	−	0.969400i	0.245485	−	0.969400i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−1.09157	−	1.40244i	−1.09157	−	1.40244i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.724414	−	1.65150i	−0.724414	−	1.65150i
$201$		0			0
$202$	1.12496	+	1.44535i	1.12496	+	1.44535i
$203$		0			0
$204$	−0.561308	−	0.142143i	−0.561308	−	0.142143i
$205$		−	2.26799i		−	2.26799i
$206$		0			0
$207$	0.437767			0.437767
$208$		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
$214$	−0.903782	−	1.16118i	−0.903782	−	1.16118i
$215$		0			0
$216$	0.570257	−	0.250138i	0.570257	−	0.250138i
$217$		0			0
$218$		0			0
$219$		−	0.519553i		−	0.519553i
$220$	−0.819241	+	3.23511i	−0.819241	+	3.23511i
$221$		0			0
$222$	0.382248	−	0.297515i	0.382248	−	0.297515i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.60797			−1.60797
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$229$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$230$	−0.504913	−	0.648712i	−0.504913	−	0.648712i
$231$		0			0
$232$		0			0
$233$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$233$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$234$		0			0
$235$		−	3.16723i		−	3.16723i
$236$		0			0
$237$			0.360099i			0.360099i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.484742	−	0.262329i	−0.484742	−	0.262329i
$241$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$241$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$242$	2.34590	−	1.82588i	2.34590	−	1.82588i
$243$		−	0.848742i		−	0.848742i
$244$		0			0
$245$			1.67433i			1.67433i
$246$	−0.273882	−	0.351884i	−0.273882	−	0.351884i
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.826204	+	1.06151i	0.826204	+	1.06151i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			0.978588i			0.978588i
$254$	−1.49277	+	1.16187i	−1.49277	+	1.16187i
$255$	0.969483			0.969483
$256$	0.546948	+	0.837166i	0.546948	+	0.837166i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.12496	+	1.44535i	1.12496	+	1.44535i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$	0.263564	+	0.600866i	0.263564	+	0.600866i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−0.822771	+	0.640388i	−0.822771	+	0.640388i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−1.54695	−	0.837166i	−1.54695	−	0.837166i
$273$		0			0
$274$		0			0
$275$		−	3.59446i		−	3.59446i
$276$	−0.156677	−	0.0396759i	−0.156677	−	0.0396759i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$281$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$282$	−0.382474	−	0.491402i	−0.382474	−	0.491402i
$283$		−	1.83155i		−	1.83155i	−0.401695	−	0.915773i	$-0.631579\pi$
$283$		−	1.83155i		−	1.83155i	0.401695	−	0.915773i	$-0.368421\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.879474	−	0.146758i	0.879474	−	0.146758i
$289$	2.09390			2.09390
$290$		0			0
$291$		0			0
$292$	0.387445	−	1.52999i	0.387445	−	1.52999i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.202192	+	0.259777i	0.202192	+	0.259777i
$295$		0			0
$296$	1.34751	−	0.591074i	1.34751	−	0.591074i
$297$	1.24116			1.24116
$298$	−1.19084	−	1.52999i	−1.19084	−	1.52999i
$299$		0			0
$300$	0.575491	+	0.145734i	0.575491	+	0.145734i
$301$		0			0
$302$	−0.130333	+	0.101443i	−0.130333	+	0.101443i
$303$	−0.602925			−0.602925
$304$		0			0
$305$		0			0
$306$	−1.23764	+	0.963293i	−1.23764	+	0.963293i
$307$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$307$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$308$		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
$316$	−0.268536	+	1.06042i	−0.268536	+	1.06042i
$317$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$317$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$318$		0			0
$319$		0			0
$320$	−1.23185	−	1.13399i	−1.23185	−	1.13399i
$321$	0.484385			0.484385
$322$		0			0
$323$		0			0
$324$	0.168562	−	0.665635i	0.168562	−	0.665635i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.544122	−	1.24047i	−0.544122	−	1.24047i
$329$		0			0
$330$	−0.674762	−	0.866934i	−0.674762	−	0.866934i
$331$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$331$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$332$		0			0
$333$		−	1.31199i		−	1.31199i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.789141	−	0.614213i	0.789141	−	0.614213i
$339$		0			0
$340$	2.85495	+	0.722970i	2.85495	+	0.722970i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.270610			0.270610
$346$	0.398869	+	0.512467i	0.398869	+	0.512467i
$347$		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
$352$	0.328065	+	1.96598i	0.328065	+	1.96598i
$353$		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
$360$	−1.36715	+	0.599688i	−1.36715	+	0.599688i
$361$	−1.00000			−1.00000
$362$	−0.754515	−	0.969400i	−0.754515	−	0.969400i
$363$			0.978588i			0.978588i
$364$		0			0
$365$			2.64257i			2.64257i
$366$		0			0
$367$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$367$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$368$	−0.431796	−	0.233676i	−0.431796	−	0.233676i
$369$	−1.20778			−1.20778
$370$	−1.94420	+	1.51323i	−1.94420	+	1.51323i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−2.15336	−	2.76663i	−2.15336	−	2.76663i
$375$	−0.442807			−0.442807
$376$	−0.759861	−	1.73231i	−0.759861	−	1.73231i
$377$		0			0
$378$		0			0
$379$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$379$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$380$		0			0
$381$		−	0.622706i		−	0.622706i
$382$	1.38806	−	1.08037i	1.38806	−	1.08037i
$383$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$383$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$384$	−0.328065	−	0.0271842i	−0.328065	−	0.0271842i
$385$		0			0
$386$	0.130333	−	0.101443i	0.130333	−	0.101443i
$387$		0			0
$388$		0			0
$389$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$389$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$390$		0			0
$391$	0.863592			0.863592
$392$	0.401695	+	0.915773i	0.401695	+	0.915773i
$393$	−0.602925			−0.602925
$394$		0			0
$395$		−	1.83155i		−	1.83155i
$396$	1.72280	+	0.436271i	1.72280	+	0.436271i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	1.58603	+	0.858319i	1.58603	+	0.858319i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−1.77550	−	0.449618i	−1.77550	−	0.449618i
$405$			1.14967i			1.14967i
$406$		0			0
$407$	2.93284			2.93284
$408$	0.530257	−	0.232592i	0.530257	−	0.232592i
$409$	1.35456			1.35456			0.677282	−	0.735724i	$-0.263158\pi$
$409$	1.35456			1.35456			0.677282	+	0.735724i	$0.263158\pi$
$410$	1.39303	+	1.78976i	1.39303	+	1.78976i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.345459	+	0.268882i	−0.345459	+	0.268882i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$419$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−1.68665			−1.68665
$424$		0			0
$425$	−3.17207			−3.17207
$426$		0			0
$427$		0			0
$428$	1.42642	+	0.361219i	1.42642	+	0.361219i
$429$		0			0
$430$		0			0
$431$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$431$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$432$	−0.296375	+	0.547653i	−0.296375	+	0.547653i
$433$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$433$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.319116	+	0.410000i	0.319116	+	0.410000i
$439$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$439$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$440$	−1.34055	−	3.05614i	−1.34055	−	3.05614i
$441$	0.891634			0.891634
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−0.118909	+	0.469563i	−0.118909	+	0.469563i
$445$		0			0
$446$		0			0
$447$	0.638232			0.638232
$448$		0			0
$449$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$449$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$450$	1.26891	−	0.987633i	1.26891	−	0.987633i
$451$		−	2.69987i		−	2.69987i
$452$		0			0
$453$		−	0.0543685i		−	0.0543685i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.75895			1.75895			0.879474	−	0.475947i	$-0.157895\pi$
$457$	1.75895			1.75895			0.879474	+	0.475947i	$0.157895\pi$
$458$	−0.398869	−	0.512467i	−0.398869	−	0.512467i
$459$		−	1.09531i		−	1.09531i
$460$	0.796894	+	0.201801i	0.796894	+	0.201801i
$461$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$461$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.633988	+	0.493453i	−0.633988	+	0.493453i
$467$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$467$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$468$		0			0
$469$		0			0
$470$	1.94535	+	2.49939i	1.94535	+	2.49939i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.221177	−	0.284169i	−0.221177	−	0.284169i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$479$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$480$	0.543655	−	0.0907200i	0.543655	−	0.0907200i
$481$		0			0
$482$	0.387445	−	0.301561i	0.387445	−	0.301561i
$483$		0			0
$484$	−0.729760	+	2.88176i	−0.729760	+	2.88176i
$485$		0			0
$486$	0.521308	+	0.669777i	0.521308	+	0.669777i
$487$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$487$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$488$		0			0
$489$		0			0
$490$	−1.02840	−	1.32128i	−1.02840	−	1.32128i
$491$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$491$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$492$	0.432263	+	0.109464i	0.432263	+	0.109464i
$493$		0			0
$494$		0			0
$495$	−2.97559			−2.97559
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−1.30398	−	0.330213i	−1.30398	−	0.330213i
$501$		0			0
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$	3.06662			3.06662
$506$	−0.601061	−	0.772244i	−0.601061	−	0.772244i
$507$			0.329189i			0.329189i
$508$	0.464369	−	1.83375i	0.464369	−	1.83375i
$509$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$509$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$510$	−0.765058	+	0.595469i	−0.765058	+	0.595469i
$511$		0			0
$512$	−0.945817	−	0.324699i	−0.945817	−	0.324699i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	3.77035i		−	3.77035i
$518$		0			0
$519$	−0.213775			−0.213775
$520$		0			0
$521$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$521$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$522$		0			0
$523$		−	1.99317i		−	1.99317i	−0.0825793	−	0.996584i	$-0.526316\pi$
$523$		−	1.99317i		−	1.99317i	0.0825793	−	0.996584i	$-0.473684\pi$
$524$	−1.77550	−	0.449618i	−1.77550	−	0.449618i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.577049	−	0.312283i	−0.577049	−	0.312283i
$529$	−0.758948			−0.758948
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.46369			−2.46369
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.99317i			1.99317i
$540$	0.255947	−	1.01071i	0.255947	−	1.01071i
$541$			1.22843i			1.22843i	0.789141	+	0.614213i	$0.210526\pi$
$541$			1.22843i			1.22843i	−0.789141	+	0.614213i	$0.789474\pi$
$542$		0			0
$543$	0.404384			0.404384
$544$	1.73496	−	0.289513i	1.73496	−	0.289513i
$545$		0			0
$546$		0			0
$547$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$547$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$548$		0			0
$549$		0			0
$550$	2.20776	+	2.83654i	2.20776	+	2.83654i
$551$		0			0
$552$	0.148009	−	0.0649230i	0.148009	−	0.0649230i
$553$		0			0
$554$		0			0
$555$		−	0.811021i		−	0.811021i
$556$		0			0
$557$		−	1.22843i		−	1.22843i	−0.789141	−	0.614213i	$-0.789474\pi$
$557$		−	1.22843i		−	1.22843i	0.789141	−	0.614213i	$-0.210526\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.15410			1.15410
$562$	1.55676	−	1.21167i	1.55676	−	1.21167i
$563$			1.93880i			1.93880i	0.245485	+	0.969400i	$0.421053\pi$
$563$			1.93880i			1.93880i	−0.245485	+	0.969400i	$0.578947\pi$
$564$	0.603651	+	0.152865i	0.603651	+	0.152865i
$565$		0			0
$566$	1.12496	+	1.44535i	1.12496	+	1.44535i
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$			0.579026i			0.579026i
$574$		0			0
$575$	−0.885413			−0.885413
$576$	−0.603888	+	0.655997i	−0.603888	+	0.655997i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−1.65238	+	1.28610i	−1.65238	+	1.28610i
$579$			0.0543685i			0.0543685i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.633988	+	1.44535i	0.633988	+	1.44535i
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.319116	−	0.0808112i	−0.319116	−	0.0808112i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.700332	+	1.29410i	−0.700332	+	1.29410i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.979448	+	0.762335i	−0.979448	+	0.762335i
$595$		0			0
$596$	1.87947	+	0.475947i	1.87947	+	0.475947i
$597$		0			0
$598$		0			0
$599$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$599$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$600$	−0.543655	+	0.238469i	−0.543655	+	0.238469i
$601$	1.97272			1.97272			0.986361	−	0.164595i	$-0.0526316\pi$
$601$	1.97272			1.97272			0.986361	+	0.164595i	$0.0526316\pi$
$602$		0			0
$603$		0			0
$604$	0.0405441	−	0.160105i	0.0405441	−	0.160105i
$605$		−	4.97733i		−	4.97733i
$606$	0.475793	−	0.370324i	0.475793	−	0.370324i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.385004	−	1.52035i	0.385004	−	1.52035i
$613$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$613$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$614$	1.19084	+	1.52999i	1.19084	+	1.52999i
$615$	−0.746598			−0.746598
$616$		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
$621$		−	0.305730i		−	0.305730i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.448828			0.448828
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	2.58820i		−	2.58820i
$630$		0			0
$631$	−0.803391			−0.803391			−0.401695	−	0.915773i	$-0.631579\pi$
$631$	−0.803391			−0.803391			−0.401695	+	0.915773i	$0.631579\pi$
$632$	−0.439413	−	1.00176i	−0.439413	−	1.00176i
$633$		0			0
$634$	1.22423	+	1.57289i	1.22423	+	1.57289i
$635$			3.16723i			3.16723i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.66861	+	0.138265i	1.66861	+	0.138265i
$641$	1.89163			1.89163			0.945817	−	0.324699i	$-0.105263\pi$
$641$	1.89163			1.89163			0.945817	+	0.324699i	$0.105263\pi$
$642$	−0.382248	+	0.297515i	−0.382248	+	0.297515i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$647$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$648$	0.275823	+	0.628813i	0.275823	+	0.628813i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	0.329189i		−	0.329189i	−0.986361	−	0.164595i	$-0.947368\pi$
$653$		−	0.329189i		−	0.329189i	0.986361	−	0.164595i	$-0.0526316\pi$
$654$		0			0
$655$	3.06662			3.06662
$656$	1.19130	+	0.644701i	1.19130	+	0.644701i
$657$	1.40725			1.40725
$658$		0			0
$659$		−	1.67433i		−	1.67433i	−0.546948	−	0.837166i	$-0.684211\pi$
$659$		−	1.67433i		−	1.67433i	0.546948	−	0.837166i	$-0.315789\pi$
$660$	1.06496	+	0.269685i	1.06496	+	0.269685i
$661$		−	0.951895i		−	0.951895i	−0.879474	−	0.475947i	$-0.842105\pi$
$661$		−	0.951895i		−	0.951895i	0.879474	−	0.475947i	$-0.157895\pi$
$662$	1.02840	+	1.32128i	1.02840	+	1.32128i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.805843	+	1.03535i	0.805843	+	1.03535i
$667$		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
$675$			1.12298i			1.12298i
$676$	−0.245485	+	0.969400i	−0.245485	+	0.969400i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$	−2.69701	+	1.18302i	−2.69701	+	1.18302i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.213775			0.213775
$688$		0			0
$689$		0			0
$690$	−0.213549	+	0.166212i	−0.213549	+	0.166212i
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−0.629528	−	0.159418i	−0.629528	−	0.159418i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.38261			−2.38261
$698$		0			0
$699$		−	0.264468i		−	0.264468i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−1.46642	−	1.34994i	−1.46642	−	1.34994i
$705$	−1.04262			−1.04262
$706$		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
$711$	−0.975356			−0.975356
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.789141	−	0.614213i	0.789141	−	0.614213i
$719$	−1.35456			−1.35456			−0.677282	−	0.735724i	$-0.736842\pi$
$719$	−1.35456			−1.35456			−0.677282	+	0.735724i	$0.736842\pi$
$720$	0.710539	−	1.31296i	0.710539	−	1.31296i
$721$		0			0
$722$	0.789141	−	0.614213i	0.789141	−	0.614213i
$723$			0.161622i			0.161622i
$724$	1.19084	+	0.301561i	1.19084	+	0.301561i
$725$		0			0
$726$	−0.601061	−	0.772244i	−0.601061	−	0.772244i
$727$	−0.165159			−0.165159			−0.0825793	−	0.996584i	$-0.526316\pi$
$727$	−0.165159			−0.165159			−0.0825793	+	0.996584i	$0.526316\pi$
$728$		0			0
$729$	0.407250			0.407250
$730$	−1.62310	−	2.08536i	−1.62310	−	2.08536i
$731$		0			0
$732$		0			0
$733$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$733$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$734$	−0.130333	+	0.101443i	−0.130333	+	0.101443i
$735$	0.551172			0.551172
$736$	0.484275	−	0.0808112i	0.484275	−	0.0808112i
$737$		0			0
$738$	0.953104	−	0.741831i	0.953104	−	0.741831i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	0.604801	−	2.38831i	0.604801	−	2.38831i
$741$		0			0
$742$		0			0
$743$	1.09390			1.09390			0.546948	−	0.837166i	$-0.315789\pi$
$743$	1.09390			1.09390			0.546948	+	0.837166i	$0.315789\pi$
$744$		0			0
$745$	−3.24620			−3.24620
$746$		0			0
$747$		0			0
$748$	3.39860	+	0.860643i	3.39860	+	0.860643i
$749$		0			0
$750$	0.349437	−	0.271978i	0.349437	−	0.271978i
$751$	−1.89163			−1.89163			−0.945817	−	0.324699i	$-0.894737\pi$
$751$	−1.89163			−1.89163			−0.945817	+	0.324699i	$0.894737\pi$
$752$	1.66364	+	0.900319i	1.66364	+	0.900319i
$753$		0			0
$754$		0			0
$755$			0.276531i			0.276531i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	−0.584666	−	0.751179i	−0.584666	−	0.751179i
$759$	0.322141			0.322141
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	0.382474	+	0.491402i	0.382474	+	0.491402i
$763$		0			0
$764$	−0.431796	+	1.70512i	−0.431796	+	1.70512i
$765$			2.62592i			2.62592i
$766$	−0.633988	+	0.493453i	−0.633988	+	0.493453i
$767$		0			0
$768$	0.275586	−	0.180049i	0.275586	−	0.180049i
$769$	0.165159			0.165159			0.0825793	−	0.996584i	$-0.473684\pi$
$769$	0.165159			0.165159			0.0825793	+	0.996584i	$0.473684\pi$
$770$		0			0
$771$		0			0
$772$	−0.0405441	+	0.160105i	−0.0405441	+	0.160105i
$773$		−	1.47145i		−	1.47145i	−0.677282	−	0.735724i	$-0.736842\pi$
$773$		−	1.47145i		−	1.47145i	0.677282	−	0.735724i	$-0.263158\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.903782	−	1.16118i	−0.903782	−	1.16118i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.681496	+	0.530429i	−0.681496	+	0.530429i
$783$		0			0
$784$	−0.879474	−	0.475947i	−0.879474	−	0.475947i
$785$		0			0
$786$	0.475793	−	0.370324i	0.475793	−	0.370324i
$787$			0.649399i			0.649399i	0.945817	+	0.324699i	$0.105263\pi$
$787$			0.649399i			0.649399i	−0.945817	+	0.324699i	$0.894737\pi$
$788$		0			0
$789$		0			0
$790$	1.12496	+	1.44535i	1.12496	+	1.44535i
$791$		0			0
$792$	−1.62749	+	0.713884i	−1.62749	+	0.713884i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			0.951895i			0.951895i	0.879474	+	0.475947i	$0.157895\pi$
$797$			0.951895i			0.951895i	−0.879474	+	0.475947i	$0.842105\pi$
$798$		0			0
$799$	−3.32729			−3.32729
$800$	−1.77879	+	0.296828i	−1.77879	+	0.296828i
$801$		0			0
$802$		0			0
$803$			3.14578i			3.14578i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.67728	−	0.735724i	1.67728	−	0.735724i
$809$	−0.490971			−0.490971			−0.245485	−	0.969400i	$-0.578947\pi$
$809$	−0.490971			−0.490971			−0.245485	+	0.969400i	$0.578947\pi$
$810$	−0.706145	−	0.907255i	−0.706145	−	0.907255i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$	−2.31443	+	1.80139i	−2.31443	+	1.80139i
$815$		0			0
$816$	−0.275586	+	0.509239i	−0.275586	+	0.509239i
$817$		0			0
$818$	−1.06894	+	0.831990i	−1.06894	+	0.831990i
$819$		0			0
$820$	−2.19859	−	0.556759i	−2.19859	−	0.556759i
$821$		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
$825$	−1.18326			−1.18326
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.107465	−	0.424371i	0.107465	−	0.424371i
$829$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$829$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.75895			1.75895
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.754515	−	0.969400i	−0.754515	−	0.969400i
$839$	1.57828			1.57828			0.789141	−	0.614213i	$-0.210526\pi$
$839$	1.57828			1.57828			0.789141	+	0.614213i	$0.210526\pi$
$840$		0			0
$841$	−1.00000			−1.00000
$842$		0			0
$843$			0.649399i			0.649399i
$844$		0			0
$845$		−	1.67433i		−	1.67433i
$846$	1.33100	−	1.03596i	1.33100	−	1.03596i
$847$		0			0
$848$		0			0
$849$	−0.602925			−0.602925
$850$	2.50321	−	1.94833i	2.50321	−	1.94833i
$851$		−	0.722438i		−	0.722438i
$852$		0			0
$853$		−	0.649399i		−	0.649399i	−0.945817	−	0.324699i	$-0.894737\pi$
$853$		−	0.649399i		−	0.649399i	0.945817	−	0.324699i	$-0.105263\pi$
$854$		0			0
$855$		0			0
$856$	−1.34751	+	0.591074i	−1.34751	+	0.591074i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$			1.47145i			1.47145i	0.677282	+	0.735724i	$0.263158\pi$
$859$			1.47145i			1.47145i	−0.677282	+	0.735724i	$0.736842\pi$
$860$		0			0
$861$		0			0
$862$	−1.38806	+	1.08037i	−1.38806	+	1.08037i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.102494	−	0.614213i	−0.102494	−	0.614213i
$865$	1.08731			1.08731
$866$	0.863238	−	0.671885i	0.863238	−	0.671885i
$867$		−	0.689288i		−	0.689288i
$868$		0			0
$869$		−	2.18032i		−	2.18032i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.503655	−	0.127543i	−0.503655	−	0.127543i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	0.633988	−	0.493453i	0.633988	−	0.493453i
$879$		0			0
$880$	2.93500	+	1.58835i	2.93500	+	1.58835i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.703625	+	0.547653i	−0.703625	+	0.547653i
$883$			1.99317i			1.99317i	0.0825793	+	0.996584i	$0.473684\pi$
$883$			1.99317i			1.99317i	−0.0825793	+	0.996584i	$0.526316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$887$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$888$	−0.194575	−	0.443587i	−0.194575	−	0.443587i
$889$		0			0
$890$		0			0
$891$			1.36860i			1.36860i
$892$		0			0
$893$		0			0
$894$	−0.503655	+	0.392010i	−0.503655	+	0.392010i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.24549	−	0.969400i	1.24549	−	0.969400i
$899$		0			0
$900$	−0.394732	+	1.55876i	−0.394732	+	1.55876i
$901$		0			0
$902$	1.65830	+	2.13058i	1.65830	+	2.13058i
$903$		0			0
$904$		0			0
$905$	−2.05679			−2.05679
$906$	0.0333938	+	0.0429043i	0.0333938	+	0.0429043i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		−	1.63307i		−	1.63307i
$910$		0			0
$911$	0.803391			0.803391			0.401695	−	0.915773i	$-0.368421\pi$
$911$	0.803391			0.803391			0.401695	+	0.915773i	$0.368421\pi$
$912$		0			0
$913$		0			0
$914$	−1.38806	+	1.08037i	−1.38806	+	1.08037i
$915$		0			0
$916$	0.629528	+	0.159418i	0.629528	+	0.159418i
$917$		0			0
$918$	0.672751	+	0.864351i	0.672751	+	0.864351i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	−0.752810	+	0.330213i	−0.752810	+	0.330213i
$921$	−0.638232			−0.638232
$922$	0.202192	+	0.259777i	0.202192	+	0.259777i
$923$		0			0
$924$		0			0
$925$			2.65360i			2.65360i
$926$		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
$932$	0.197221	−	0.778807i	0.197221	−	0.778807i
$933$		0			0
$934$	−0.398869	−	0.512467i	−0.398869	−	0.512467i
$935$	−5.87001			−5.87001
$936$		0			0
$937$	−1.09390			−1.09390			−0.546948	−	0.837166i	$-0.684211\pi$
$937$	−1.09390			−1.09390			−0.546948	+	0.837166i	$0.684211\pi$
$938$		0			0
$939$		0			0
$940$	−3.07031	−	0.777508i	−3.07031	−	0.777508i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−0.665051			−0.665051
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.83155i			1.83155i	0.401695	+	0.915773i	$0.368421\pi$
$947$			1.83155i			1.83155i	−0.401695	+	0.915773i	$0.631579\pi$
$948$	0.349080	+	0.0883990i	0.349080	+	0.0883990i
$949$		0			0
$950$		0			0
$951$	−0.656130			−0.656130
$952$		0			0
$953$	−1.97272			−1.97272			−0.986361	−	0.164595i	$-0.947368\pi$
$953$	−1.97272			−1.97272			−0.986361	+	0.164595i	$0.947368\pi$
$954$		0			0
$955$		−	2.94506i		−	2.94506i
$956$		0			0
$957$		0			0
$958$	−1.55676	+	1.21167i	−1.55676	+	1.21167i
$959$		0			0
$960$	−0.373299	+	0.405511i	−0.373299	+	0.405511i
$961$	1.00000			1.00000
$962$		0			0
$963$			1.31199i			1.31199i
$964$	−0.120526	+	0.475947i	−0.120526	+	0.475947i
$965$		−	0.276531i		−	0.276531i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−1.19413	−	2.72234i	−1.19413	−	2.72234i
$969$		0			0
$970$		0			0
$971$		−	1.93880i		−	1.93880i	−0.245485	−	0.969400i	$-0.578947\pi$
$971$		−	1.93880i		−	1.93880i	0.245485	−	0.969400i	$-0.421053\pi$
$972$	−0.822771	−	0.208354i	−0.822771	−	0.208354i
$973$		0			0
$974$	−1.55676	+	1.21167i	−1.55676	+	1.21167i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	1.62310	+	0.411024i	1.62310	+	0.411024i
$981$		0			0
$982$	0.903782	+	1.16118i	0.903782	+	1.16118i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	−0.408350	+	0.179119i	−0.408350	+	0.179119i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	2.34816	−	1.82764i	2.34816	−	1.82764i
$991$	−1.57828			−1.57828			−0.789141	−	0.614213i	$-0.789474\pi$
$991$	−1.57828			−1.57828			−0.789141	+	0.614213i	$0.789474\pi$
$992$		0			0
$993$	−0.551172			−0.551172
$994$		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
$999$	−0.916279			−0.916279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2872.1.b.e.717.4	yes	18
8.5	even	2	inner	2872.1.b.e.717.3	✓	18
359.358	odd	2	CM	2872.1.b.e.717.4	yes	18
2872.717	odd	2	inner	2872.1.b.e.717.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2872.1.b.e.717.3	✓	18	8.5	even	2	inner
2872.1.b.e.717.3	✓	18	2872.717	odd	2	inner
2872.1.b.e.717.4	yes	18	1.1	even	1	trivial
2872.1.b.e.717.4	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)

\(f(q)\)	\(=\)	\(q+(-0.789141 + 0.614213i) q^{2} -0.329189i q^{3} +(0.245485 - 0.969400i) q^{4} +1.67433i q^{5} +(0.202192 + 0.259777i) q^{6} +(0.401695 + 0.915773i) q^{8} +0.891634 q^{9} +O(q^{10})\)	\(q+(-0.789141 + 0.614213i) q^{2} -0.329189i q^{3} +(0.245485 - 0.969400i) q^{4} +1.67433i q^{5} +(0.202192 + 0.259777i) q^{6} +(0.401695 + 0.915773i) q^{8} +0.891634 q^{9} +(-1.02840 - 1.32128i) q^{10} +1.99317i q^{11} +(-0.319116 - 0.0808112i) q^{12} +0.551172 q^{15} +(-0.879474 - 0.475947i) q^{16} +1.75895 q^{17} +(-0.703625 + 0.547653i) q^{18} +(1.62310 + 0.411024i) q^{20} +(-1.22423 - 1.57289i) q^{22} +0.490971 q^{23} +(0.301463 - 0.132234i) q^{24} -1.80339 q^{25} -0.622706i q^{27} +(-0.434952 + 0.338537i) q^{30} +(0.986361 - 0.164595i) q^{32} +0.656130 q^{33} +(-1.38806 + 1.08037i) q^{34} +(0.218883 - 0.864351i) q^{36} -1.47145i q^{37} +(-1.53331 + 0.672572i) q^{40} -1.35456 q^{41} +(1.93218 + 0.489294i) q^{44} +1.49289i q^{45} +(-0.387445 + 0.301561i) q^{46} -1.89163 q^{47} +(-0.156677 + 0.289513i) q^{48} +1.00000 q^{49} +(1.42313 - 1.10767i) q^{50} -0.579026i q^{51} +(0.382474 + 0.491402i) q^{54} -3.33723 q^{55} +(0.135305 - 0.534307i) q^{60} +(-0.677282 + 0.735724i) q^{64} +(-0.517778 + 0.403003i) q^{66} +(0.431796 - 1.70512i) q^{68} -0.161622i q^{69} +(0.358165 + 0.816535i) q^{72} +1.57828 q^{73} +(0.903782 + 1.16118i) q^{74} +0.593657i q^{75} -1.09390 q^{79} +(0.796894 - 1.47253i) q^{80} +0.686647 q^{81} +(1.06894 - 0.831990i) q^{82} +2.94506i q^{85} +(-1.82529 + 0.800647i) q^{88} +(-0.916954 - 1.17810i) q^{90} +(0.120526 - 0.475947i) q^{92} +(1.49277 - 1.16187i) q^{94} +(-0.0541828 - 0.324699i) q^{96} +(-0.789141 + 0.614213i) q^{98} +1.77718i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - q^{4} + q^{8} - 20 q^{9}+O(q^{10}) \) 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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