LMFDB - Embedded newform 2879.1.b.b.2878.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2879,1,Mod(2878,2879)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2879, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2879.2878");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2879 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2879.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:	yes
Analytic conductor:	$1.43680817137$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			2878.1
Root			$-1.89163$ of defining polynomial
Character	$\chi$	$=$	2879.2878

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.97272 q^{2} -0.165159 q^{3} +2.89163 q^{4} -1.75895 q^{5} +0.325812 q^{6} -3.73167 q^{8} -0.972723 q^{9} +O(q^{10})$	\(q-1.97272 q^{2} -0.165159 q^{3} +2.89163 q^{4} -1.75895 q^{5} +0.325812 q^{6} -3.73167 q^{8} -0.972723 q^{9} +3.46992 q^{10} +1.57828 q^{11} -0.477579 q^{12} +0.290505 q^{15} +4.46992 q^{16} +1.91891 q^{18} -0.165159 q^{19} -5.08623 q^{20} -3.11351 q^{22} +0.616318 q^{24} +2.09390 q^{25} +0.325812 q^{27} -0.573087 q^{30} -1.35456 q^{31} -5.08623 q^{32} -0.260667 q^{33} -2.81276 q^{36} -1.35456 q^{37} +0.325812 q^{38} +6.56381 q^{40} -1.75895 q^{41} +4.56381 q^{44} +1.71097 q^{45} -0.738245 q^{48} +1.00000 q^{49} -4.13068 q^{50} -0.803391 q^{53} -0.642737 q^{54} -2.77611 q^{55} +0.0272774 q^{57} +2.00000 q^{59} +0.840036 q^{60} -0.803391 q^{61} +2.67218 q^{62} +5.56381 q^{64} +0.514223 q^{66} +0.490971 q^{71} +3.62988 q^{72} +0.490971 q^{73} +2.67218 q^{74} -0.345825 q^{75} -0.477579 q^{76} +1.89163 q^{79} -7.86235 q^{80} +0.918912 q^{81} +3.46992 q^{82} +1.89163 q^{83} -5.88962 q^{88} -3.37527 q^{90} +0.223718 q^{93} +0.290505 q^{95} +0.840036 q^{96} -1.75895 q^{97} -1.97272 q^{98} -1.53523 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10}) $ 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9	$ 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{15} + 7 q^{16} + 16 q^{18} - q^{19} - 3 q^{20} - 2 q^{22} - 4 q^{24} + 8 q^{25} - 2 q^{27} - 4 q^{30} - q^{31} - 3 q^{32} - 2 q^{33} + 5 q^{36} - q^{37} - 2 q^{38} + 15 q^{40} - q^{41} - 3 q^{44} - 3 q^{45} - 5 q^{48} + 9 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} + 17 q^{57} + 18 q^{59} - 6 q^{60} - q^{61} - 2 q^{62} + 6 q^{64} - 4 q^{66} - q^{71} + 13 q^{72} - q^{73} - 2 q^{74} - 3 q^{75} - 3 q^{76} - q^{79} - 5 q^{80} + 7 q^{81} - 2 q^{82} - q^{83} - 4 q^{88} - 6 q^{90} - 2 q^{93} - 2 q^{95} - 6 q^{96} - q^{97} - q^{98} - 3 q^{99}+O(q^{100}) $ 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^15 + 7 * q^16 + 16 * q^18 - q^19 - 3 * q^20 - 2 * q^22 - 4 * q^24 + 8 * q^25 - 2 * q^27 - 4 * q^30 - q^31 - 3 * q^32 - 2 * q^33 + 5 * q^36 - q^37 - 2 * q^38 + 15 * q^40 - q^41 - 3 * q^44 - 3 * q^45 - 5 * q^48 + 9 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 + 17 * q^57 + 18 * q^59 - 6 * q^60 - q^61 - 2 * q^62 + 6 * q^64 - 4 * q^66 - q^71 + 13 * q^72 - q^73 - 2 * q^74 - 3 * q^75 - 3 * q^76 - q^79 - 5 * q^80 + 7 * q^81 - 2 * q^82 - q^83 - 4 * q^88 - 6 * q^90 - 2 * q^93 - 2 * q^95 - 6 * q^96 - q^97 - q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2879\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

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 $

$2$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$2$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$3$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$3$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$4$	2.89163	2.89163
$5$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$5$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$6$	0.325812	0.325812
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−3.73167	−3.73167
$9$	−0.972723	−0.972723
$10$	3.46992	3.46992
$11$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$11$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$12$	−0.477579	−0.477579
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0.290505	0.290505
$16$	4.46992	4.46992
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.91891	1.91891
$19$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$19$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$20$	−5.08623	−5.08623
$21$	0	0
$22$	−3.11351	−3.11351
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.616318	0.616318
$25$	2.09390	2.09390
$26$	0	0
$27$	0.325812	0.325812
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.573087	−0.573087
$31$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$31$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$32$	−5.08623	−5.08623
$33$	−0.260667	−0.260667
$34$	0	0
$35$	0	0
$36$	−2.81276	−2.81276
$37$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$37$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$38$	0.325812	0.325812
$39$	0	0
$40$	6.56381	6.56381
$41$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$41$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	4.56381	4.56381
$45$	1.71097	1.71097
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.738245	−0.738245
$49$	1.00000	1.00000
$50$	−4.13068	−4.13068
$51$	0	0
$52$	0	0
$53$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$53$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$54$	−0.642737	−0.642737
$55$	−2.77611	−2.77611
$56$	0	0
$57$	0.0272774	0.0272774
$58$	0	0
$59$	2.00000	2.00000	1.00000			$0$
$59$	2.00000	2.00000	1.00000			$0$
$60$	0.840036	0.840036
$61$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$61$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$62$	2.67218	2.67218
$63$	0	0
$64$	5.56381	5.56381
$65$	0	0
$66$	0.514223	0.514223
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$71$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$72$	3.62988	3.62988
$73$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$73$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$74$	2.67218	2.67218
$75$	−0.345825	−0.345825
$76$	−0.477579	−0.477579
$77$	0	0
$78$	0	0
$79$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$79$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$80$	−7.86235	−7.86235
$81$	0.918912	0.918912
$82$	3.46992	3.46992
$83$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$83$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−5.88962	−5.88962
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.37527	−3.37527
$91$	0	0
$92$	0	0
$93$	0.223718	0.223718
$94$	0	0
$95$	0.290505	0.290505
$96$	0.840036	0.840036
$97$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$97$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$98$	−1.97272	−1.97272
$99$	−1.53523	−1.53523
$100$	6.05478	6.05478
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	1.58487	1.58487
$107$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$107$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$108$	0.942130	0.942130
$109$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$109$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$110$	5.47650	5.47650
$111$	0.223718	0.223718
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−0.0538107	−0.0538107
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−3.94545	−3.94545
$119$	0	0
$120$	−1.08407	−1.08407
$121$	1.49097	1.49097
$122$	1.58487	1.58487
$123$	0.290505	0.290505
$124$	−3.91690	−3.91690
$125$	−1.92411	−1.92411
$126$	0	0
$127$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$127$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$128$	−5.88962	−5.88962
$129$	0	0
$130$	0	0
$131$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$131$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$132$	−0.753753	−0.753753
$133$	0	0
$134$	0	0
$135$	−0.573087	−0.573087
$136$	0	0
$137$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$137$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−0.968550	−0.968550
$143$	0	0
$144$	−4.34799	−4.34799
$145$	0	0
$146$	−0.968550	−0.968550
$147$	−0.165159	−0.165159
$148$	−3.91690	−3.91690
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.682217	0.682217
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.616318	0.616318
$153$	0	0
$154$	0	0
$155$	2.38261	2.38261
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−3.73167	−3.73167
$159$	0.132687	0.132687
$160$	8.94642	8.94642
$161$	0	0
$162$	−1.81276	−1.81276
$163$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$163$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$164$	−5.08623	−5.08623
$165$	0.458499	0.458499
$166$	−3.73167	−3.73167
$167$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$167$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0.160654	0.160654
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	7.05478	7.05478
$177$	−0.330317	−0.330317
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	4.94749	4.94749
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0.132687	0.132687
$184$	0	0
$185$	2.38261	2.38261
$186$	−0.441333	−0.441333
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−0.573087	−0.573087
$191$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$191$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$192$	−0.918912	−0.918912
$193$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$193$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$194$	3.46992	3.46992
$195$	0	0
$196$	2.89163	2.89163
$197$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$197$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$198$	3.02858	3.02858
$199$	2.00000	2.00000	1.00000			$0$
$199$	2.00000	2.00000	1.00000			$0$
$200$	−7.81373	−7.81373
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.09390	3.09390
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.260667	−0.260667
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−2.32311	−2.32311
$213$	−0.0810881	−0.0810881
$214$	−3.11351	−3.11351
$215$	0	0
$216$	−1.21582	−1.21582
$217$	0	0
$218$	−0.968550	−0.968550
$219$	−0.0810881	−0.0810881
$220$	−8.02751	−8.02751
$221$	0	0
$222$	−0.441333	−0.441333
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.03678	−2.03678
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0.0788763	0.0788763
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	5.78327	5.78327
$237$	−0.312420	−0.312420
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.29853	1.29853
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−2.94127	−2.94127
$243$	−0.477579	−0.477579
$244$	−2.32311	−2.32311
$245$	−1.75895	−1.75895
$246$	−0.573087	−0.573087
$247$	0	0
$248$	5.05478	5.05478
$249$	−0.312420	−0.312420
$250$	3.79573	3.79573
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0.325812	0.325812
$255$	0	0
$256$	6.05478	6.05478
$257$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$257$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	2.67218	2.67218
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0.972723	0.972723
$265$	1.41312	1.41312
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$269$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$270$	1.13054	1.13054
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−3.11351	−3.11351
$275$	3.30476	3.30476
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	1.31761	1.31761
$280$	0	0
$281$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$281$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.41971	1.41971
$285$	−0.0479795	−0.0479795
$286$	0	0
$287$	0	0
$288$	4.94749	4.94749
$289$	1.00000	1.00000
$290$	0	0
$291$	0.290505	0.290505
$292$	1.41971	1.41971
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.325812	0.325812
$295$	−3.51790	−3.51790
$296$	5.05478	5.05478
$297$	0.514223	0.514223
$298$	0	0
$299$	0	0
$300$	−1.00000	−1.00000
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−0.738245	−0.738245
$305$	1.41312	1.41312
$306$	0	0
$307$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$307$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$308$	0	0
$309$	0	0
$310$	−4.70022	−4.70022
$311$	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
$316$	5.46992	5.46992
$317$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$317$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$318$	−0.261755	−0.261755
$319$	0	0
$320$	−9.78645	−9.78645
$321$	−0.260667	−0.260667
$322$	0	0
$323$	0	0
$324$	2.65716	2.65716
$325$	0	0
$326$	−2.15795	−2.15795
$327$	−0.0810881	−0.0810881
$328$	6.56381	6.56381
$329$	0	0
$330$	−0.904492	−0.904492
$331$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$331$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$332$	5.46992	5.46992
$333$	1.31761	1.31761
$334$	−2.15795	−2.15795
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.97272	−1.97272
$339$	0	0
$340$	0	0
$341$	−2.13788	−2.13788
$342$	−0.316925	−0.316925
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$347$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$348$	0	0
$349$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$349$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$350$	0	0
$351$	0	0
$352$	−8.02751	−8.02751
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0.651625	0.651625
$355$	−0.863592	−0.863592
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$359$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$360$	−6.38477	−6.38477
$361$	−0.972723	−0.972723
$362$	0	0
$363$	−0.246247	−0.246247
$364$	0	0
$365$	−0.863592	−0.863592
$366$	−0.261755	−0.261755
$367$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$367$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$368$	0	0
$369$	1.71097	1.71097
$370$	−4.70022	−4.70022
$371$	0	0
$372$	0.646910	0.646910
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.317783	0.317783
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0.840036	0.840036
$381$	0.0272774	0.0272774
$382$	−3.11351	−3.11351
$383$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$383$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$384$	0.972723	0.972723
$385$	0	0
$386$	1.58487	1.58487
$387$	0	0
$388$	−5.08623	−5.08623
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−3.73167	−3.73167
$393$	0.223718	0.223718
$394$	−0.968550	−0.968550
$395$	−3.32729	−3.32729
$396$	−4.43932	−4.43932
$397$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$397$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$398$	−3.94545	−3.94545
$399$	0	0
$400$	9.35954	9.35954
$401$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$401$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.61632	−1.61632
$406$	0	0
$407$	−2.13788	−2.13788
$408$	0	0
$409$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$409$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$410$	−6.10340	−6.10340
$411$	−0.260667	−0.260667
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.32729	−3.32729
$416$	0	0
$417$	0	0
$418$	0.514223	0.514223
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$421$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$422$	0	0
$423$	0	0
$424$	2.99799	2.99799
$425$	0	0
$426$	0.159964	0.159964
$427$	0	0
$428$	4.56381	4.56381
$429$	0	0
$430$	0	0
$431$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$431$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$432$	1.45635	1.45635
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	1.41971	1.41971
$437$	0	0
$438$	0.159964	0.159964
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	10.3595	10.3595
$441$	−0.972723	−0.972723
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.646910	0.646910
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	4.01800	4.01800
$451$	−2.77611	−2.77611
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	−0.101790	−0.101790
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$461$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$462$	0	0
$463$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$463$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$464$	0	0
$465$	−0.393508	−0.393508
$466$	0	0
$467$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$467$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−7.46334	−7.46334
$473$	0	0
$474$	0.616318	0.616318
$475$	−0.345825	−0.345825
$476$	0	0
$477$	0.781476	0.781476
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.47758	−1.47758
$481$	0	0
$482$	0	0
$483$	0	0
$484$	4.31134	4.31134
$485$	3.09390	3.09390
$486$	0.942130	0.942130
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	2.99799	2.99799
$489$	−0.180666	−0.180666
$490$	3.46992	3.46992
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.840036	0.840036
$493$	0	0
$494$	0	0
$495$	2.70039	2.70039
$496$	−6.05478	−6.05478
$497$	0	0
$498$	0.616318	0.616318
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−5.56381	−5.56381
$501$	−0.180666	−0.180666
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.165159	−0.165159
$508$	−0.477579	−0.477579
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−6.05478	−6.05478
$513$	−0.0538107	−0.0538107
$514$	−3.73167	−3.73167
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−3.91690	−3.91690
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−1.16516	−1.16516
$529$	1.00000	1.00000
$530$	−2.78770	−2.78770
$531$	−1.94545	−1.94545
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.77611	−2.77611
$536$	0	0
$537$	0	0
$538$	1.58487	1.58487
$539$	1.57828	1.57828
$540$	−1.65716	−1.65716
$541$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$541$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.863592	−0.863592
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	4.56381	4.56381
$549$	0.781476	0.781476
$550$	−6.51937	−6.51937
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.393508	−0.393508
$556$	0	0
$557$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$557$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$558$	−2.59929	−2.59929
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1.58487	1.58487
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−1.83214	−1.83214
$569$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$569$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$570$	0.0946502	0.0946502
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	−0.260667	−0.260667
$574$	0	0
$575$	0	0
$576$	−5.41205	−5.41205
$577$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$577$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$578$	−1.97272	−1.97272
$579$	0.132687	0.132687
$580$	0	0
$581$	0	0
$582$	−0.573087	−0.573087
$583$	−1.26798	−1.26798
$584$	−1.83214	−1.83214
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.477579	−0.477579
$589$	0.223718	0.223718
$590$	6.93983	6.93983
$591$	−0.0810881	−0.0810881
$592$	−6.05478	−6.05478
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−1.01442	−1.01442
$595$	0	0
$596$	0	0
$597$	−0.330317	−0.330317
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.29051	1.29051
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.62254	−2.62254
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0.840036	0.840036
$609$	0	0
$610$	−2.78770	−2.78770
$611$	0	0
$612$	0	0
$613$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$613$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$614$	−0.968550	−0.968550
$615$	−0.510984	−0.510984
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$619$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$620$	6.88962	6.88962
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.29051	1.29051
$626$	0	0
$627$	0.0430514	0.0430514
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−7.05896	−7.05896
$633$	0	0
$634$	−3.73167	−3.73167
$635$	0.290505	0.290505
$636$	0.383682	0.383682
$637$	0	0
$638$	0	0
$639$	−0.477579	−0.477579
$640$	10.3595	10.3595
$641$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$641$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$642$	0.514223	0.514223
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−3.42908	−3.42908
$649$	3.15656	3.15656
$650$	0	0
$651$	0	0
$652$	3.16315	3.16315
$653$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$653$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$654$	0.159964	0.159964
$655$	2.38261	2.38261
$656$	−7.86235	−7.86235
$657$	−0.477579	−0.477579
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	1.32581	1.32581
$661$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$661$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$662$	3.89163	3.89163
$663$	0	0
$664$	−7.05896	−7.05896
$665$	0	0
$666$	−2.59929	−2.59929
$667$	0	0
$668$	3.16315	3.16315
$669$	0	0
$670$	0	0
$671$	−1.26798	−1.26798
$672$	0	0
$673$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$673$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$674$	0	0
$675$	0.682217	0.682217
$676$	2.89163	2.89163
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	4.21745	4.21745
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.464551	0.464551
$685$	−2.77611	−2.77611
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$691$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$692$	0	0
$693$	0	0
$694$	3.89163	3.89163
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−2.15795	−2.15795
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0.223718	0.223718
$704$	8.78126	8.78126
$705$	0	0
$706$	0	0
$707$	0	0
$708$	−0.955157	−0.955157
$709$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$709$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$710$	1.70363	1.70363
$711$	−1.84004	−1.84004
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	2.67218	2.67218
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	7.64788	7.64788
$721$	0	0
$722$	1.91891	1.91891
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.485777	0.485777
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.840036	−0.840036
$730$	1.70363	1.70363
$731$	0	0
$732$	0.383682	0.383682
$733$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$733$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$734$	−0.968550	−0.968550
$735$	0.290505	0.290505
$736$	0	0
$737$	0	0
$738$	−3.37527	−3.37527
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	6.88962	6.88962
$741$	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	$0.526316\pi$
$744$	−0.834841	−0.834841
$745$	0	0
$746$	0	0
$747$	−1.84004	−1.84004
$748$	0	0
$749$	0	0
$750$	−0.626897	−0.626897
$751$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$751$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	−1.08407	−1.08407
$761$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$761$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$762$	−0.0538107	−0.0538107
$763$	0	0
$764$	4.56381	4.56381
$765$	0	0
$766$	−3.73167	−3.73167
$767$	0	0
$768$	−1.00000	−1.00000
$769$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$769$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$770$	0	0
$771$	−0.312420	−0.312420
$772$	−2.32311	−2.32311
$773$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$773$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$774$	0	0
$775$	−2.83631	−2.83631
$776$	6.56381	6.56381
$777$	0	0
$778$	0	0
$779$	0.290505	0.290505
$780$	0	0
$781$	0.774890	0.774890
$782$	0	0
$783$	0	0
$784$	4.46992	4.46992
$785$	0	0
$786$	−0.441333	−0.441333
$787$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$787$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$788$	1.41971	1.41971
$789$	0	0
$790$	6.56381	6.56381
$791$	0	0
$792$	5.72897	5.72897
$793$	0	0
$794$	3.46992	3.46992
$795$	−0.233389	−0.233389
$796$	5.78327	5.78327
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−10.6500	−10.6500
$801$	0	0
$802$	1.58487	1.58487
$803$	0.774890	0.774890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.132687	0.132687
$808$	0	0
$809$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$809$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$810$	3.18855	3.18855
$811$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$811$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$812$	0	0
$813$	0	0
$814$	4.21745	4.21745
$815$	−1.92411	−1.92411
$816$	0	0
$817$	0	0
$818$	−3.73167	−3.73167
$819$	0	0
$820$	8.94642	8.94642
$821$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$821$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$822$	0.514223	0.514223
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−0.545809	−0.545809
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	6.56381	6.56381
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.92411	−1.92411
$836$	−0.753753	−0.753753
$837$	−0.441333	−0.441333
$838$	0	0
$839$	−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
$842$	0.325812	0.325812
$843$	0.132687	0.132687
$844$	0	0
$845$	−1.75895	−1.75895
$846$	0	0
$847$	0	0
$848$	−3.59109	−3.59109
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−0.234477	−0.234477
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−0.282581	−0.282581
$856$	−5.88962	−5.88962
$857$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$857$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$858$	0	0
$859$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$859$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$860$	0	0
$861$	0	0
$862$	−0.968550	−0.968550
$863$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$863$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$864$	−1.65716	−1.65716
$865$	0	0
$866$	0	0
$867$	−0.165159	−0.165159
$868$	0	0
$869$	2.98553	2.98553
$870$	0	0
$871$	0	0
$872$	−1.83214	−1.83214
$873$	1.71097	1.71097
$874$	0	0
$875$	0	0
$876$	−0.234477	−0.234477
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	−12.4090	−12.4090
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.91891	1.91891
$883$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$883$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$884$	0	0
$885$	0.581011	0.581011
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.834841	−0.834841
$889$	0	0
$890$	0	0
$891$	1.45030	1.45030
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−5.88962	−5.88962
$901$	0	0
$902$	5.47650	5.47650
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.121928	0.121928
$913$	2.98553	2.98553
$914$	0	0
$915$	−0.233389	−0.233389
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−0.0810881	−0.0810881
$922$	−2.15795	−2.15795
$923$	0	0
$924$	0	0
$925$	−2.83631	−2.83631
$926$	−3.11351	−3.11351
$927$	0	0
$928$	0	0
$929$	2.00000	2.00000	1.00000			$0$
$929$	2.00000	2.00000	1.00000			$0$
$930$	0.776282	0.776282
$931$	−0.165159	−0.165159
$932$	0	0
$933$	0	0
$934$	2.67218	2.67218
$935$	0	0
$936$	0	0
$937$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$937$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$941$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$942$	0	0
$943$	0	0
$944$	8.93983	8.93983
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−0.903404	−0.903404
$949$	0	0
$950$	0.682217	0.682217
$951$	−0.312420	−0.312420
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.54164	−1.54164
$955$	−2.77611	−2.77611
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	1.61632	1.61632
$961$	0.834841	0.834841
$962$	0	0
$963$	−1.53523	−1.53523
$964$	0	0
$965$	1.41312	1.41312
$966$	0	0
$967$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$967$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$968$	−5.56381	−5.56381
$969$	0	0
$970$	−6.10340	−6.10340
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.38098	−1.38098
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−3.59109	−3.59109
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0.356405	0.356405
$979$	0	0
$980$	−5.08623	−5.08623
$981$	−0.477579	−0.477579
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−1.08407	−1.08407
$985$	−0.863592	−0.863592
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−5.32712	−5.32712
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	6.88962	6.88962
$993$	0.325812	0.325812
$994$	0	0
$995$	−3.51790	−3.51790
$996$	−0.903404	−0.903404
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	−0.441333	−0.441333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2879.1.b.b.2878.1	✓	9
2879.2878	odd	2	CM	2879.1.b.b.2878.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2879.1.b.b.2878.1	✓	9	1.1	even	1	trivial
2879.1.b.b.2878.1	✓	9	2879.2878	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:	\( N \)	\(=\)	\( 2879 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2879.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.97272 q^{2} -0.165159 q^{3} +2.89163 q^{4} -1.75895 q^{5} +0.325812 q^{6} -3.73167 q^{8} -0.972723 q^{9} +O(q^{10})\)	\(q-1.97272 q^{2} -0.165159 q^{3} +2.89163 q^{4} -1.75895 q^{5} +0.325812 q^{6} -3.73167 q^{8} -0.972723 q^{9} +3.46992 q^{10} +1.57828 q^{11} -0.477579 q^{12} +0.290505 q^{15} +4.46992 q^{16} +1.91891 q^{18} -0.165159 q^{19} -5.08623 q^{20} -3.11351 q^{22} +0.616318 q^{24} +2.09390 q^{25} +0.325812 q^{27} -0.573087 q^{30} -1.35456 q^{31} -5.08623 q^{32} -0.260667 q^{33} -2.81276 q^{36} -1.35456 q^{37} +0.325812 q^{38} +6.56381 q^{40} -1.75895 q^{41} +4.56381 q^{44} +1.71097 q^{45} -0.738245 q^{48} +1.00000 q^{49} -4.13068 q^{50} -0.803391 q^{53} -0.642737 q^{54} -2.77611 q^{55} +0.0272774 q^{57} +2.00000 q^{59} +0.840036 q^{60} -0.803391 q^{61} +2.67218 q^{62} +5.56381 q^{64} +0.514223 q^{66} +0.490971 q^{71} +3.62988 q^{72} +0.490971 q^{73} +2.67218 q^{74} -0.345825 q^{75} -0.477579 q^{76} +1.89163 q^{79} -7.86235 q^{80} +0.918912 q^{81} +3.46992 q^{82} +1.89163 q^{83} -5.88962 q^{88} -3.37527 q^{90} +0.223718 q^{93} +0.290505 q^{95} +0.840036 q^{96} -1.75895 q^{97} -1.97272 q^{98} -1.53523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9	\( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{15} + 7 q^{16} + 16 q^{18} - q^{19} - 3 q^{20} - 2 q^{22} - 4 q^{24} + 8 q^{25} - 2 q^{27} - 4 q^{30} - q^{31} - 3 q^{32} - 2 q^{33} + 5 q^{36} - q^{37} - 2 q^{38} + 15 q^{40} - q^{41} - 3 q^{44} - 3 q^{45} - 5 q^{48} + 9 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} + 17 q^{57} + 18 q^{59} - 6 q^{60} - q^{61} - 2 q^{62} + 6 q^{64} - 4 q^{66} - q^{71} + 13 q^{72} - q^{73} - 2 q^{74} - 3 q^{75} - 3 q^{76} - q^{79} - 5 q^{80} + 7 q^{81} - 2 q^{82} - q^{83} - 4 q^{88} - 6 q^{90} - 2 q^{93} - 2 q^{95} - 6 q^{96} - q^{97} - q^{98} - 3 q^{99}+O(q^{100}) \) 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^15 + 7 * q^16 + 16 * q^18 - q^19 - 3 * q^20 - 2 * q^22 - 4 * q^24 + 8 * q^25 - 2 * q^27 - 4 * q^30 - q^31 - 3 * q^32 - 2 * q^33 + 5 * q^36 - q^37 - 2 * q^38 + 15 * q^40 - q^41 - 3 * q^44 - 3 * q^45 - 5 * q^48 + 9 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 + 17 * q^57 + 18 * q^59 - 6 * q^60 - q^61 - 2 * q^62 + 6 * q^64 - 4 * q^66 - q^71 + 13 * q^72 - q^73 - 2 * q^74 - 3 * q^75 - 3 * q^76 - q^79 - 5 * q^80 + 7 * q^81 - 2 * q^82 - q^83 - 4 * q^88 - 6 * q^90 - 2 * q^93 - 2 * q^95 - 6 * q^96 - q^97 - q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more