LMFDB - Embedded newform 2339.1.b.a.2338.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2339,1,Mod(2338,2339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2339, 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("2339.2338");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2339 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2339.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:	$1.16731306455$
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:	$\Z[a_1, a_2, a_3]$
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			2338.9
Root			$-1.57828$ of defining polynomial
Character	$\chi$	$=$	2339.2338

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.89163 q^{3} +1.00000 q^{4} -1.75895 q^{5} +2.57828 q^{9} +O(q^{10})$	$q+1.89163 q^{3} +1.00000 q^{4} -1.75895 q^{5} +2.57828 q^{9} +0.490971 q^{11} +1.89163 q^{12} -0.803391 q^{13} -3.32729 q^{15} +1.00000 q^{16} -0.165159 q^{19} -1.75895 q^{20} +2.09390 q^{25} +2.98553 q^{27} +0.928738 q^{33} +2.57828 q^{36} -1.51972 q^{39} -1.97272 q^{41} +0.490971 q^{44} -4.53506 q^{45} +1.89163 q^{48} +1.00000 q^{49} -0.803391 q^{52} -1.35456 q^{53} -0.863592 q^{55} -0.312420 q^{57} -0.803391 q^{59} -3.32729 q^{60} +1.00000 q^{64} +1.41312 q^{65} -1.97272 q^{67} +1.09390 q^{71} +1.57828 q^{73} +3.96089 q^{75} -0.165159 q^{76} -1.97272 q^{79} -1.75895 q^{80} +3.06925 q^{81} -0.165159 q^{83} +1.09390 q^{89} +0.290505 q^{95} -1.35456 q^{97} +1.26586 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 9 q^{4} - q^{5} + 8 q^{9}+O(q^{10}) $ 9 * q - q^3 + 9 * q^4 - q^5 + 8 * q^9	$ 9 q - q^{3} + 9 q^{4} - q^{5} + 8 q^{9} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 9 q^{16} - q^{19} - q^{20} + 8 q^{25} - 2 q^{27} - 2 q^{33} + 8 q^{36} - 2 q^{39} - q^{41} - q^{44} - 3 q^{45} - q^{48} + 9 q^{49} - q^{52} - q^{53} - 2 q^{55} - 2 q^{57} - q^{59} - 2 q^{60} + 9 q^{64} - 2 q^{65} - q^{67} - q^{71} - q^{73} - 3 q^{75} - q^{76} - q^{79} - q^{80} + 7 q^{81} - q^{83} - q^{89} - 2 q^{95} - q^{97} - 3 q^{99}+O(q^{100}) $ 9 * q - q^3 + 9 * q^4 - q^5 + 8 * q^9 - q^11 - q^12 - q^13 - 2 * q^15 + 9 * q^16 - q^19 - q^20 + 8 * q^25 - 2 * q^27 - 2 * q^33 + 8 * q^36 - 2 * q^39 - q^41 - q^44 - 3 * q^45 - q^48 + 9 * q^49 - q^52 - q^53 - 2 * q^55 - 2 * q^57 - q^59 - 2 * q^60 + 9 * q^64 - 2 * q^65 - q^67 - q^71 - q^73 - 3 * q^75 - q^76 - q^79 - q^80 + 7 * q^81 - q^83 - q^89 - 2 * q^95 - q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2339\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$3$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$4$	1.00000	1.00000
$5$	−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	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	2.57828	2.57828
$10$	0	0
$11$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$11$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$12$	1.89163	1.89163
$13$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$13$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$14$	0	0
$15$	−3.32729	−3.32729
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$19$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$20$	−1.75895	−1.75895
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	2.09390	2.09390
$26$	0	0
$27$	2.98553	2.98553
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0.928738	0.928738
$34$	0	0
$35$	0	0
$36$	2.57828	2.57828
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−1.51972	−1.51972
$40$	0	0
$41$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$41$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0.490971	0.490971
$45$	−4.53506	−4.53506
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.89163	1.89163
$49$	1.00000	1.00000
$50$	0	0
$51$	0	0
$52$	−0.803391	−0.803391
$53$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$53$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$54$	0	0
$55$	−0.863592	−0.863592
$56$	0	0
$57$	−0.312420	−0.312420
$58$	0	0
$59$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$59$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$60$	−3.32729	−3.32729
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	1.41312	1.41312
$66$	0	0
$67$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$67$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$71$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$72$	0	0
$73$	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	0
$75$	3.96089	3.96089
$76$	−0.165159	−0.165159
$77$	0	0
$78$	0	0
$79$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$79$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$80$	−1.75895	−1.75895
$81$	3.06925	3.06925
$82$	0	0
$83$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$83$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$89$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.290505	0.290505
$96$	0	0
$97$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$97$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$98$	0	0
$99$	1.26586	1.26586
$100$	2.09390	2.09390
$101$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$101$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$102$	0	0
$103$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$103$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.98553	2.98553
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.07137	−2.07137
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.758948	−0.758948
$122$	0	0
$123$	−3.73167	−3.73167
$124$	0	0
$125$	−1.92411	−1.92411
$126$	0	0
$127$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$127$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0.928738	0.928738
$133$	0	0
$134$	0	0
$135$	−5.25139	−5.25139
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$139$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.394442	−0.394442
$144$	2.57828	2.57828
$145$	0	0
$146$	0	0
$147$	1.89163	1.89163
$148$	0	0
$149$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$149$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.51972	−1.51972
$157$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$157$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$158$	0	0
$159$	−2.56234	−2.56234
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−1.97272	−1.97272
$165$	−1.63360	−1.63360
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.354563	−0.354563
$170$	0	0
$171$	−0.425826	−0.425826
$172$	0	0
$173$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$173$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$174$	0	0
$175$	0	0
$176$	0.490971	0.490971
$177$	−1.51972	−1.51972
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−4.53506	−4.53506
$181$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$181$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$191$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$192$	1.89163	1.89163
$193$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$193$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$194$	0	0
$195$	2.67311	2.67311
$196$	1.00000	1.00000
$197$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$197$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$198$	0	0
$199$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$199$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$200$	0	0
$201$	−3.73167	−3.73167
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.46992	3.46992
$206$	0	0
$207$	0	0
$208$	−0.803391	−0.803391
$209$	−0.0810881	−0.0810881
$210$	0	0
$211$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$211$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$212$	−1.35456	−1.35456
$213$	2.06925	2.06925
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.98553	2.98553
$220$	−0.863592	−0.863592
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	5.39865	5.39865
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−0.312420	−0.312420
$229$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$229$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$233$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$234$	0	0
$235$	0	0
$236$	−0.803391	−0.803391
$237$	−3.73167	−3.73167
$238$	0	0
$239$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$239$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$240$	−3.32729	−3.32729
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	2.82037	2.82037
$244$	0	0
$245$	−1.75895	−1.75895
$246$	0	0
$247$	0.132687	0.132687
$248$	0	0
$249$	−0.312420	−0.312420
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$257$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$258$	0	0
$259$	0	0
$260$	1.41312	1.41312
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	2.38261	2.38261
$266$	0	0
$267$	2.06925	2.06925
$268$	−1.97272	−1.97272
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$271$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.02804	1.02804
$276$	0	0
$277$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$277$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$283$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$284$	1.09390	1.09390
$285$	0.549530	0.549530
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	1.00000
$290$	0	0
$291$	−2.56234	−2.56234
$292$	1.57828	1.57828
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	1.41312	1.41312
$296$	0	0
$297$	1.46581	1.46581
$298$	0	0
$299$	0	0
$300$	3.96089	3.96089
$301$	0	0
$302$	0	0
$303$	−2.56234	−2.56234
$304$	−0.165159	−0.165159
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	2.98553	2.98553
$310$	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
$316$	−1.97272	−1.97272
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−1.75895	−1.75895
$321$	0	0
$322$	0	0
$323$	0	0
$324$	3.06925	3.06925
$325$	−1.68222	−1.68222
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−0.165159	−0.165159
$333$	0	0
$334$	0	0
$335$	3.46992	3.46992
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$347$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$348$	0	0
$349$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$349$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$350$	0	0
$351$	−2.39855	−2.39855
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−1.92411	−1.92411
$356$	1.09390	1.09390
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.972723	−0.972723
$362$	0	0
$363$	−1.43565	−1.43565
$364$	0	0
$365$	−2.77611	−2.77611
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−5.08623	−5.08623
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$373$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$374$	0	0
$375$	−3.63971	−3.63971
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$379$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$380$	0.290505	0.290505
$381$	0.928738	0.928738
$382$	0	0
$383$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$383$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−1.35456	−1.35456
$389$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$389$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.46992	3.46992
$396$	1.26586	1.26586
$397$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$397$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$398$	0	0
$399$	0	0
$400$	2.09390	2.09390
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−1.35456	−1.35456
$405$	−5.39865	−5.39865
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$409$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$410$	0	0
$411$	0	0
$412$	1.57828	1.57828
$413$	0	0
$414$	0	0
$415$	0.290505	0.290505
$416$	0	0
$417$	−3.32729	−3.32729
$418$	0	0
$419$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$419$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$420$	0	0
$421$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$421$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.746139	−0.746139
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	2.98553	2.98553
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	2.57828	2.57828
$442$	0	0
$443$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$443$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$444$	0	0
$445$	−1.92411	−1.92411
$446$	0	0
$447$	2.98553	2.98553
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−0.968550	−0.968550
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$457$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$463$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−2.07137	−2.07137
$469$	0	0
$470$	0	0
$471$	−3.73167	−3.73167
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−0.345825	−0.345825
$476$	0	0
$477$	−3.49244	−3.49244
$478$	0	0
$479$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$479$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.758948	−0.758948
$485$	2.38261	2.38261
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−3.73167	−3.73167
$493$	0	0
$494$	0	0
$495$	−2.22658	−2.22658
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$499$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$500$	−1.92411	−1.92411
$501$	0	0
$502$	0	0
$503$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$503$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$504$	0	0
$505$	2.38261	2.38261
$506$	0	0
$507$	−0.670704	−0.670704
$508$	0.490971	0.490971
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.493086	−0.493086
$514$	0	0
$515$	−2.77611	−2.77611
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.56234	−2.56234
$520$	0	0
$521$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$521$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0.928738	0.928738
$529$	1.00000	1.00000
$530$	0	0
$531$	−2.07137	−2.07137
$532$	0	0
$533$	1.58487	1.58487
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.490971	0.490971
$540$	−5.25139	−5.25139
$541$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$541$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$542$	0	0
$543$	0.928738	0.928738
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$547$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−1.75895	−1.75895
$557$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$557$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$563$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	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	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	−0.394442	−0.394442
$573$	−1.51972	−1.51972
$574$	0	0
$575$	0	0
$576$	2.57828	2.57828
$577$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$577$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$578$	0	0
$579$	−3.32729	−3.32729
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.665051	−0.665051
$584$	0	0
$585$	3.64343	3.64343
$586$	0	0
$587$	2.00000	2.00000	1.00000			$0$
$587$	2.00000	2.00000	1.00000			$0$
$588$	1.89163	1.89163
$589$	0	0
$590$	0	0
$591$	3.57828	3.57828
$592$	0	0
$593$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$593$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$594$	0	0
$595$	0	0
$596$	1.57828	1.57828
$597$	3.57828	3.57828
$598$	0	0
$599$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$599$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$600$	0	0
$601$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$601$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$602$	0	0
$603$	−5.08623	−5.08623
$604$	0	0
$605$	1.33495	1.33495
$606$	0	0
$607$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$607$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	6.56381	6.56381
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−1.51972	−1.51972
$625$	1.29051	1.29051
$626$	0	0
$627$	−0.153389	−0.153389
$628$	−1.97272	−1.97272
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	2.06925	2.06925
$634$	0	0
$635$	−0.863592	−0.863592
$636$	−2.56234	−2.56234
$637$	−0.803391	−0.803391
$638$	0	0
$639$	2.82037	2.82037
$640$	0	0
$641$	2.00000	2.00000	1.00000			$0$
$641$	2.00000	2.00000	1.00000			$0$
$642$	0	0
$643$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$643$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$647$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$648$	0	0
$649$	−0.394442	−0.394442
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−1.97272	−1.97272
$657$	4.06925	4.06925
$658$	0	0
$659$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$659$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$660$	−1.63360	−1.63360
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	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
$675$	6.25139	6.25139
$676$	−0.354563	−0.354563
$677$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$677$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.425826	−0.425826
$685$	0	0
$686$	0	0
$687$	2.06925	2.06925
$688$	0	0
$689$	1.08824	1.08824
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−1.35456	−1.35456
$693$	0	0
$694$	0	0
$695$	3.09390	3.09390
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−3.73167	−3.73167
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.490971	0.490971
$705$	0	0
$706$	0	0
$707$	0	0
$708$	−1.51972	−1.51972
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−5.08623	−5.08623
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.693802	0.693802
$716$	0	0
$717$	3.57828	3.57828
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−4.53506	−4.53506
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0.490971	0.490971
$725$	0	0
$726$	0	0
$727$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$727$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$728$	0	0
$729$	2.26586	2.26586
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$733$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$734$	0	0
$735$	−3.32729	−3.32729
$736$	0	0
$737$	−0.968550	−0.968550
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0.250995	0.250995
$742$	0	0
$743$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$743$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$744$	0	0
$745$	−2.77611	−2.77611
$746$	0	0
$747$	−0.425826	−0.425826
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$757$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−0.803391	−0.803391
$765$	0	0
$766$	0	0
$767$	0.645437	0.645437
$768$	1.89163	1.89163
$769$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$769$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$770$	0	0
$771$	−1.51972	−1.51972
$772$	−1.75895	−1.75895
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.325812	0.325812
$780$	2.67311	2.67311
$781$	0.537071	0.537071
$782$	0	0
$783$	0	0
$784$	1.00000	1.00000
$785$	3.46992	3.46992
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1.89163	1.89163
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	4.50702	4.50702
$796$	1.89163	1.89163
$797$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$797$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	2.82037	2.82037
$802$	0	0
$803$	0.774890	0.774890
$804$	−3.73167	−3.73167
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$811$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$812$	0	0
$813$	2.98553	2.98553
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	3.46992	3.46992
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.94468	1.94468
$826$	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			$\pi$
$830$	0	0
$831$	−0.312420	−0.312420
$832$	−0.803391	−0.803391
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.0810881	−0.0810881
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	1.09390	1.09390
$845$	0.623658	0.623658
$846$	0	0
$847$	0	0
$848$	−1.35456	−1.35456
$849$	−2.56234	−2.56234
$850$	0	0
$851$	0	0
$852$	2.06925	2.06925
$853$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$853$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$854$	0	0
$855$	0.749005	0.749005
$856$	0	0
$857$	2.00000	2.00000	1.00000			$0$
$857$	2.00000	2.00000	1.00000			$0$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	2.38261	2.38261
$866$	0	0
$867$	1.89163	1.89163
$868$	0	0
$869$	−0.968550	−0.968550
$870$	0	0
$871$	1.58487	1.58487
$872$	0	0
$873$	−3.49244	−3.49244
$874$	0	0
$875$	0	0
$876$	2.98553	2.98553
$877$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$877$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$878$	0	0
$879$	0	0
$880$	−0.863592	−0.863592
$881$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$881$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$882$	0	0
$883$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$883$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$884$	0	0
$885$	2.67311	2.67311
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.50691	1.50691
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	5.39865	5.39865
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.863592	−0.863592
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−3.49244	−3.49244
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−0.312420	−0.312420
$913$	−0.0810881	−0.0810881
$914$	0	0
$915$	0	0
$916$	1.09390	1.09390
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.878826	−0.878826
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.06925	4.06925
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.165159	−0.165159
$932$	−1.97272	−1.97272
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$941$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$942$	0	0
$943$	0	0
$944$	−0.803391	−0.803391
$945$	0	0
$946$	0	0
$947$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$947$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$948$	−3.73167	−3.73167
$949$	−1.26798	−1.26798
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1.41312	1.41312
$956$	1.89163	1.89163
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−3.32729	−3.32729
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.09390	3.09390
$966$	0	0
$967$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$967$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$971$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$972$	2.82037	2.82037
$973$	0	0
$974$	0	0
$975$	−3.18214	−3.18214
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0.537071	0.537071
$980$	−1.75895	−1.75895
$981$	0	0
$982$	0	0
$983$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$983$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$984$	0	0
$985$	−3.32729	−3.32729
$986$	0	0
$987$	0	0
$988$	0.132687	0.132687
$989$	0	0
$990$	0	0
$991$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$991$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.32729	−3.32729
$996$	−0.312420	−0.312420
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2339.1.b.a.2338.9	✓	9
2339.2338	odd	2	CM	2339.1.b.a.2338.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2339.1.b.a.2338.9	✓	9	1.1	even	1	trivial
2339.1.b.a.2338.9	✓	9	2339.2338	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:	\( N \)	\(=\)	\( 2339 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2339.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.89163 q^{3} +1.00000 q^{4} -1.75895 q^{5} +2.57828 q^{9} +O(q^{10})\)	\(q+1.89163 q^{3} +1.00000 q^{4} -1.75895 q^{5} +2.57828 q^{9} +0.490971 q^{11} +1.89163 q^{12} -0.803391 q^{13} -3.32729 q^{15} +1.00000 q^{16} -0.165159 q^{19} -1.75895 q^{20} +2.09390 q^{25} +2.98553 q^{27} +0.928738 q^{33} +2.57828 q^{36} -1.51972 q^{39} -1.97272 q^{41} +0.490971 q^{44} -4.53506 q^{45} +1.89163 q^{48} +1.00000 q^{49} -0.803391 q^{52} -1.35456 q^{53} -0.863592 q^{55} -0.312420 q^{57} -0.803391 q^{59} -3.32729 q^{60} +1.00000 q^{64} +1.41312 q^{65} -1.97272 q^{67} +1.09390 q^{71} +1.57828 q^{73} +3.96089 q^{75} -0.165159 q^{76} -1.97272 q^{79} -1.75895 q^{80} +3.06925 q^{81} -0.165159 q^{83} +1.09390 q^{89} +0.290505 q^{95} -1.35456 q^{97} +1.26586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 9 q^{4} - q^{5} + 8 q^{9}+O(q^{10}) \) 9 * q - q^3 + 9 * q^4 - q^5 + 8 * q^9	\( 9 q - q^{3} + 9 q^{4} - q^{5} + 8 q^{9} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 9 q^{16} - q^{19} - q^{20} + 8 q^{25} - 2 q^{27} - 2 q^{33} + 8 q^{36} - 2 q^{39} - q^{41} - q^{44} - 3 q^{45} - q^{48} + 9 q^{49} - q^{52} - q^{53} - 2 q^{55} - 2 q^{57} - q^{59} - 2 q^{60} + 9 q^{64} - 2 q^{65} - q^{67} - q^{71} - q^{73} - 3 q^{75} - q^{76} - q^{79} - q^{80} + 7 q^{81} - q^{83} - q^{89} - 2 q^{95} - q^{97} - 3 q^{99}+O(q^{100}) \) 9 * q - q^3 + 9 * q^4 - q^5 + 8 * q^9 - q^11 - q^12 - q^13 - 2 * q^15 + 9 * q^16 - q^19 - q^20 + 8 * q^25 - 2 * q^27 - 2 * q^33 + 8 * q^36 - 2 * q^39 - q^41 - q^44 - 3 * q^45 - q^48 + 9 * q^49 - q^52 - q^53 - 2 * q^55 - 2 * q^57 - q^59 - 2 * q^60 + 9 * q^64 - 2 * q^65 - q^67 - q^71 - q^73 - 3 * q^75 - q^76 - q^79 - q^80 + 7 * q^81 - q^83 - q^89 - 2 * q^95 - q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more