LMFDB - Embedded newform 2255.1.h.j.2254.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2255,1,Mod(2254,2255)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2255, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2255.2254");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2255 = 5 \cdot 11 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2255.h (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:	$1.12539160349$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.2.129287396253125.1

Embedding invariants

Embedding label			2254.2
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	2255.2254

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.17557 q^{2} -1.90211 q^{3} +0.381966 q^{4} -1.00000 q^{5} +2.23607 q^{6} +0.726543 q^{8} +2.61803 q^{9} +O(q^{10})$	$q-1.17557 q^{2} -1.90211 q^{3} +0.381966 q^{4} -1.00000 q^{5} +2.23607 q^{6} +0.726543 q^{8} +2.61803 q^{9} +1.17557 q^{10} +1.00000 q^{11} -0.726543 q^{12} +1.90211 q^{15} -1.23607 q^{16} -3.07768 q^{18} -0.618034 q^{19} -0.381966 q^{20} -1.17557 q^{22} -1.38197 q^{24} +1.00000 q^{25} -3.07768 q^{27} -1.61803 q^{29} -2.23607 q^{30} +0.618034 q^{31} +0.726543 q^{32} -1.90211 q^{33} +1.00000 q^{36} +0.726543 q^{38} -0.726543 q^{40} -1.00000 q^{41} +0.381966 q^{44} -2.61803 q^{45} +2.35114 q^{48} +1.00000 q^{49} -1.17557 q^{50} +1.17557 q^{53} +3.61803 q^{54} -1.00000 q^{55} +1.17557 q^{57} +1.90211 q^{58} -1.61803 q^{59} +0.726543 q^{60} -0.726543 q^{62} +0.381966 q^{64} +2.23607 q^{66} +1.17557 q^{67} +1.90211 q^{72} +1.90211 q^{73} -1.90211 q^{75} -0.236068 q^{76} -1.61803 q^{79} +1.23607 q^{80} +3.23607 q^{81} +1.17557 q^{82} -1.90211 q^{83} +3.07768 q^{87} +0.726543 q^{88} +3.07768 q^{90} -1.17557 q^{93} +0.618034 q^{95} -1.38197 q^{96} -1.17557 q^{98} +2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} - 4 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^4 - 4 * q^5 + 6 * q^9	$ 4 q + 6 q^{4} - 4 q^{5} + 6 q^{9} + 4 q^{11} + 4 q^{16} + 2 q^{19} - 6 q^{20} - 10 q^{24} + 4 q^{25} - 2 q^{29} - 2 q^{31} + 4 q^{36} - 4 q^{41} + 6 q^{44} - 6 q^{45} + 4 q^{49} + 10 q^{54} - 4 q^{55} - 2 q^{59} + 6 q^{64} + 8 q^{76} - 2 q^{79} - 4 q^{80} + 4 q^{81} - 2 q^{95} - 10 q^{96} + 6 q^{99}+O(q^{100}) $ 4 * q + 6 * q^4 - 4 * q^5 + 6 * q^9 + 4 * q^11 + 4 * q^16 + 2 * q^19 - 6 * q^20 - 10 * q^24 + 4 * q^25 - 2 * q^29 - 2 * q^31 + 4 * q^36 - 4 * q^41 + 6 * q^44 - 6 * q^45 + 4 * q^49 + 10 * q^54 - 4 * q^55 - 2 * q^59 + 6 * q^64 + 8 * q^76 - 2 * q^79 - 4 * q^80 + 4 * q^81 - 2 * q^95 - 10 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2255\mathbb{Z}\right)^\times$.

$n$	$452$	$826$	$1641$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$2$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$3$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$3$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$4$	0.381966	0.381966
$5$	−1.00000	−1.00000
$5$	−1.00000	−1.00000
$6$	2.23607	2.23607
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.726543	0.726543
$9$	2.61803	2.61803
$10$	1.17557	1.17557
$11$	1.00000	1.00000
$11$	1.00000	1.00000
$12$	−0.726543	−0.726543
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	1.90211	1.90211
$16$	−1.23607	−1.23607
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.07768	−3.07768
$19$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$19$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$20$	−0.381966	−0.381966
$21$	0	0
$22$	−1.17557	−1.17557
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.38197	−1.38197
$25$	1.00000	1.00000
$26$	0	0
$27$	−3.07768	−3.07768
$28$	0	0
$29$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$29$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$30$	−2.23607	−2.23607
$31$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$31$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$32$	0.726543	0.726543
$33$	−1.90211	−1.90211
$34$	0	0
$35$	0	0
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.726543	0.726543
$39$	0	0
$40$	−0.726543	−0.726543
$41$	−1.00000	−1.00000
$41$	−1.00000	−1.00000
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0.381966	0.381966
$45$	−2.61803	−2.61803
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.35114	2.35114
$49$	1.00000	1.00000
$50$	−1.17557	−1.17557
$51$	0	0
$52$	0	0
$53$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$53$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$54$	3.61803	3.61803
$55$	−1.00000	−1.00000
$56$	0	0
$57$	1.17557	1.17557
$58$	1.90211	1.90211
$59$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$59$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$60$	0.726543	0.726543
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.726543	−0.726543
$63$	0	0
$64$	0.381966	0.381966
$65$	0	0
$66$	2.23607	2.23607
$67$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$67$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.90211	1.90211
$73$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$73$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$74$	0	0
$75$	−1.90211	−1.90211
$76$	−0.236068	−0.236068
$77$	0	0
$78$	0	0
$79$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$79$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$80$	1.23607	1.23607
$81$	3.23607	3.23607
$82$	1.17557	1.17557
$83$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$83$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.07768	3.07768
$88$	0.726543	0.726543
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	3.07768	3.07768
$91$	0	0
$92$	0	0
$93$	−1.17557	−1.17557
$94$	0	0
$95$	0.618034	0.618034
$96$	−1.38197	−1.38197
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−1.17557	−1.17557
$99$	2.61803	2.61803
$100$	0.381966	0.381966
$101$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$101$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−1.38197	−1.38197
$107$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$107$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$108$	−1.17557	−1.17557
$109$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$109$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$110$	1.17557	1.17557
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−1.38197	−1.38197
$115$	0	0
$116$	−0.618034	−0.618034
$117$	0	0
$118$	1.90211	1.90211
$119$	0	0
$120$	1.38197	1.38197
$121$	1.00000	1.00000
$122$	0	0
$123$	1.90211	1.90211
$124$	0.236068	0.236068
$125$	−1.00000	−1.00000
$126$	0	0
$127$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$127$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$128$	−1.17557	−1.17557
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−0.726543	−0.726543
$133$	0	0
$134$	−1.38197	−1.38197
$135$	3.07768	3.07768
$136$	0	0
$137$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$137$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−3.23607	−3.23607
$145$	1.61803	1.61803
$146$	−2.23607	−2.23607
$147$	−1.90211	−1.90211
$148$	0	0
$149$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$149$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$150$	2.23607	2.23607
$151$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$151$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$152$	−0.449028	−0.449028
$153$	0	0
$154$	0	0
$155$	−0.618034	−0.618034
$156$	0	0
$157$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$157$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$158$	1.90211	1.90211
$159$	−2.23607	−2.23607
$160$	−0.726543	−0.726543
$161$	0	0
$162$	−3.80423	−3.80423
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−0.381966	−0.381966
$165$	1.90211	1.90211
$166$	2.23607	2.23607
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	−1.61803	−1.61803
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−3.61803	−3.61803
$175$	0	0
$176$	−1.23607	−1.23607
$177$	3.07768	3.07768
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.00000	−1.00000
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	1.38197	1.38197
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−0.726543	−0.726543
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.726543	−0.726543
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0.381966	0.381966
$197$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$197$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$198$	−3.07768	−3.07768
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.726543	0.726543
$201$	−2.23607	−2.23607
$202$	−1.90211	−1.90211
$203$	0	0
$204$	0	0
$205$	1.00000	1.00000
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.618034	−0.618034
$210$	0	0
$211$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$211$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$212$	0.449028	0.449028
$213$	0	0
$214$	−1.38197	−1.38197
$215$	0	0
$216$	−2.23607	−2.23607
$217$	0	0
$218$	−0.726543	−0.726543
$219$	−3.61803	−3.61803
$220$	−0.381966	−0.381966
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.61803	2.61803
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0.449028	0.449028
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−1.17557	−1.17557
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	−0.618034	−0.618034
$237$	3.07768	3.07768
$238$	0	0
$239$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$239$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$240$	−2.35114	−2.35114
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.17557	−1.17557
$243$	−3.07768	−3.07768
$244$	0	0
$245$	−1.00000	−1.00000
$246$	−2.23607	−2.23607
$247$	0	0
$248$	0.449028	0.449028
$249$	3.61803	3.61803
$250$	1.17557	1.17557
$251$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$251$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$252$	0	0
$253$	0	0
$254$	2.23607	2.23607
$255$	0	0
$256$	1.00000	1.00000
$257$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$257$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.23607	−4.23607
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.38197	−1.38197
$265$	−1.17557	−1.17557
$266$	0	0
$267$	0	0
$268$	0.449028	0.449028
$269$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$269$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$270$	−3.61803	−3.61803
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	1.38197	1.38197
$275$	1.00000	1.00000
$276$	0	0
$277$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$277$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$278$	0	0
$279$	1.61803	1.61803
$280$	0	0
$281$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$281$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$282$	0	0
$283$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$283$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$284$	0	0
$285$	−1.17557	−1.17557
$286$	0	0
$287$	0	0
$288$	1.90211	1.90211
$289$	1.00000	1.00000
$290$	−1.90211	−1.90211
$291$	0	0
$292$	0.726543	0.726543
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.23607	2.23607
$295$	1.61803	1.61803
$296$	0	0
$297$	−3.07768	−3.07768
$298$	0.726543	0.726543
$299$	0	0
$300$	−0.726543	−0.726543
$301$	0	0
$302$	−1.90211	−1.90211
$303$	−3.07768	−3.07768
$304$	0.763932	0.763932
$305$	0	0
$306$	0	0
$307$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$307$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$308$	0	0
$309$	0	0
$310$	0.726543	0.726543
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$313$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$314$	−2.23607	−2.23607
$315$	0	0
$316$	−0.618034	−0.618034
$317$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$317$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$318$	2.62866	2.62866
$319$	−1.61803	−1.61803
$320$	−0.381966	−0.381966
$321$	−2.23607	−2.23607
$322$	0	0
$323$	0	0
$324$	1.23607	1.23607
$325$	0	0
$326$	0	0
$327$	−1.17557	−1.17557
$328$	−0.726543	−0.726543
$329$	0	0
$330$	−2.23607	−2.23607
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−0.726543	−0.726543
$333$	0	0
$334$	0	0
$335$	−1.17557	−1.17557
$336$	0	0
$337$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$337$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$338$	−1.17557	−1.17557
$339$	0	0
$340$	0	0
$341$	0.618034	0.618034
$342$	1.90211	1.90211
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	1.17557	1.17557
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0.726543	0.726543
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−3.61803	−3.61803
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.90211	−1.90211
$361$	−0.618034	−0.618034
$362$	0	0
$363$	−1.90211	−1.90211
$364$	0	0
$365$	−1.90211	−1.90211
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−2.61803	−2.61803
$370$	0	0
$371$	0	0
$372$	−0.449028	−0.449028
$373$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$373$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$374$	0	0
$375$	1.90211	1.90211
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$379$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$380$	0.236068	0.236068
$381$	3.61803	3.61803
$382$	0	0
$383$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$383$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$384$	2.23607	2.23607
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$389$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$390$	0	0
$391$	0	0
$392$	0.726543	0.726543
$393$	0	0
$394$	−1.38197	−1.38197
$395$	1.61803	1.61803
$396$	1.00000	1.00000
$397$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$397$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$398$	0	0
$399$	0	0
$400$	−1.23607	−1.23607
$401$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$401$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$402$	2.62866	2.62866
$403$	0	0
$404$	0.618034	0.618034
$405$	−3.23607	−3.23607
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−1.17557	−1.17557
$411$	2.23607	2.23607
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.90211	1.90211
$416$	0	0
$417$	0	0
$418$	0.726543	0.726543
$419$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$419$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−0.726543	−0.726543
$423$	0	0
$424$	0.854102	0.854102
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.449028	0.449028
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	3.80423	3.80423
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	−3.07768	−3.07768
$436$	0.236068	0.236068
$437$	0	0
$438$	4.25325	4.25325
$439$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$439$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$440$	−0.726543	−0.726543
$441$	2.61803	2.61803
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.17557	1.17557
$448$	0	0
$449$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$449$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$450$	−3.07768	−3.07768
$451$	−1.00000	−1.00000
$452$	0	0
$453$	−3.07768	−3.07768
$454$	0	0
$455$	0	0
$456$	0.854102	0.854102
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$463$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$464$	2.00000	2.00000
$465$	1.17557	1.17557
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.61803	−3.61803
$472$	−1.17557	−1.17557
$473$	0	0
$474$	−3.61803	−3.61803
$475$	−0.618034	−0.618034
$476$	0	0
$477$	3.07768	3.07768
$478$	−1.90211	−1.90211
$479$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$479$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$480$	1.38197	1.38197
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0.381966	0.381966
$485$	0	0
$486$	3.61803	3.61803
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	1.17557	1.17557
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.726543	0.726543
$493$	0	0
$494$	0	0
$495$	−2.61803	−2.61803
$496$	−0.763932	−0.763932
$497$	0	0
$498$	−4.25325	−4.25325
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.381966	−0.381966
$501$	0	0
$502$	−1.90211	−1.90211
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1.61803	−1.61803
$506$	0	0
$507$	−1.90211	−1.90211
$508$	−0.726543	−0.726543
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.90211	1.90211
$514$	−2.23607	−2.23607
$515$	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			$\pi$
$522$	4.97980	4.97980
$523$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$523$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.35114	2.35114
$529$	1.00000	1.00000
$530$	1.38197	1.38197
$531$	−4.23607	−4.23607
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.17557	−1.17557
$536$	0.854102	0.854102
$537$	0	0
$538$	−0.726543	−0.726543
$539$	1.00000	1.00000
$540$	1.17557	1.17557
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.618034	−0.618034
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−0.449028	−0.449028
$549$	0	0
$550$	−1.17557	−1.17557
$551$	1.00000	1.00000
$552$	0	0
$553$	0	0
$554$	2.23607	2.23607
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−1.90211	−1.90211
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−0.726543	−0.726543
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	1.38197	1.38197
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	1.38197	1.38197
$571$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$571$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	1.00000
$577$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$577$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$578$	−1.17557	−1.17557
$579$	0	0
$580$	0.618034	0.618034
$581$	0	0
$582$	0	0
$583$	1.17557	1.17557
$584$	1.38197	1.38197
$585$	0	0
$586$	0	0
$587$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$587$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$588$	−0.726543	−0.726543
$589$	−0.381966	−0.381966
$590$	−1.90211	−1.90211
$591$	−2.23607	−2.23607
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	3.61803	3.61803
$595$	0	0
$596$	−0.236068	−0.236068
$597$	0	0
$598$	0	0
$599$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$599$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$600$	−1.38197	−1.38197
$601$	2.00000	2.00000	1.00000			$0$
$601$	2.00000	2.00000	1.00000			$0$
$602$	0	0
$603$	3.07768	3.07768
$604$	0.618034	0.618034
$605$	−1.00000	−1.00000
$606$	3.61803	3.61803
$607$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$607$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$608$	−0.449028	−0.449028
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$613$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$614$	−2.23607	−2.23607
$615$	−1.90211	−1.90211
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$619$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$620$	−0.236068	−0.236068
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	−1.38197	−1.38197
$627$	1.17557	1.17557
$628$	0.726543	0.726543
$629$	0	0
$630$	0	0
$631$	−2.00000	−2.00000	−1.00000			$\pi$
$631$	−2.00000	−2.00000	−1.00000			$\pi$
$632$	−1.17557	−1.17557
$633$	−1.17557	−1.17557
$634$	−2.23607	−2.23607
$635$	1.90211	1.90211
$636$	−0.854102	−0.854102
$637$	0	0
$638$	1.90211	1.90211
$639$	0	0
$640$	1.17557	1.17557
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	2.62866	2.62866
$643$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$643$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	2.35114	2.35114
$649$	−1.61803	−1.61803
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$653$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$654$	1.38197	1.38197
$655$	0	0
$656$	1.23607	1.23607
$657$	4.97980	4.97980
$658$	0	0
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$660$	0.726543	0.726543
$661$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$661$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$662$	0	0
$663$	0	0
$664$	−1.38197	−1.38197
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	1.38197	1.38197
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	1.38197	1.38197
$675$	−3.07768	−3.07768
$676$	0.381966	0.381966
$677$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$677$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−0.726543	−0.726543
$683$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$683$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$684$	−0.618034	−0.618034
$685$	1.17557	1.17557
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	2.23607	2.23607
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.381966	0.381966
$705$	0	0
$706$	0	0
$707$	0	0
$708$	1.17557	1.17557
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−4.23607	−4.23607
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.07768	−3.07768
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	3.23607	3.23607
$721$	0	0
$722$	0.726543	0.726543
$723$	0	0
$724$	0	0
$725$	−1.61803	−1.61803
$726$	2.23607	2.23607
$727$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$727$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$728$	0	0
$729$	2.61803	2.61803
$730$	2.23607	2.23607
$731$	0	0
$732$	0	0
$733$	1.17557	1.17557	0.587785	−	0.809017i	$-0.300000\pi$
$733$	1.17557	1.17557	0.587785	+	0.809017i	$0.300000\pi$
$734$	0	0
$735$	1.90211	1.90211
$736$	0	0
$737$	1.17557	1.17557
$738$	3.07768	3.07768
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$743$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$744$	−0.854102	−0.854102
$745$	0.618034	0.618034
$746$	1.38197	1.38197
$747$	−4.97980	−4.97980
$748$	0	0
$749$	0	0
$750$	−2.23607	−2.23607
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−3.07768	−3.07768
$754$	0	0
$755$	−1.61803	−1.61803
$756$	0	0
$757$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$757$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$758$	−1.90211	−1.90211
$759$	0	0
$760$	0.449028	0.449028
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−4.25325	−4.25325
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−1.38197	−1.38197
$767$	0	0
$768$	−1.90211	−1.90211
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−3.61803	−3.61803
$772$	0	0
$773$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$773$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$774$	0	0
$775$	0.618034	0.618034
$776$	0	0
$777$	0	0
$778$	0.726543	0.726543
$779$	0.618034	0.618034
$780$	0	0
$781$	0	0
$782$	0	0
$783$	4.97980	4.97980
$784$	−1.23607	−1.23607
$785$	−1.90211	−1.90211
$786$	0	0
$787$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$787$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$788$	0.449028	0.449028
$789$	0	0
$790$	−1.90211	−1.90211
$791$	0	0
$792$	1.90211	1.90211
$793$	0	0
$794$	1.38197	1.38197
$795$	2.23607	2.23607
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.726543	0.726543
$801$	0	0
$802$	1.90211	1.90211
$803$	1.90211	1.90211
$804$	−0.854102	−0.854102
$805$	0	0
$806$	0	0
$807$	−1.17557	−1.17557
$808$	1.17557	1.17557
$809$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$809$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$810$	3.80423	3.80423
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0.381966	0.381966
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−2.62866	−2.62866
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−1.90211	−1.90211
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$829$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$830$	−2.23607	−2.23607
$831$	3.61803	3.61803
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.236068	−0.236068
$837$	−1.90211	−1.90211
$838$	−0.726543	−0.726543
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.61803	1.61803
$842$	0	0
$843$	−1.17557	−1.17557
$844$	0.236068	0.236068
$845$	−1.00000	−1.00000
$846$	0	0
$847$	0	0
$848$	−1.45309	−1.45309
$849$	2.23607	2.23607
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	1.61803	1.61803
$856$	0.854102	0.854102
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$859$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−2.23607	−2.23607
$865$	0	0
$866$	0	0
$867$	−1.90211	−1.90211
$868$	0	0
$869$	−1.61803	−1.61803
$870$	3.61803	3.61803
$871$	0	0
$872$	0.449028	0.449028
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−1.38197	−1.38197
$877$	−1.17557	−1.17557	−0.587785	−	0.809017i	$-0.700000\pi$
$877$	−1.17557	−1.17557	−0.587785	+	0.809017i	$0.700000\pi$
$878$	−0.726543	−0.726543
$879$	0	0
$880$	1.23607	1.23607
$881$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$881$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$882$	−3.07768	−3.07768
$883$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$883$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$884$	0	0
$885$	−3.07768	−3.07768
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.23607	3.23607
$892$	0	0
$893$	0	0
$894$	−1.38197	−1.38197
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−0.726543	−0.726543
$899$	−1.00000	−1.00000
$900$	1.00000	1.00000
$901$	0	0
$902$	1.17557	1.17557
$903$	0	0
$904$	0	0
$905$	0	0
$906$	3.61803	3.61803
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	4.23607	4.23607
$910$	0	0
$911$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$911$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$912$	−1.45309	−1.45309
$913$	−1.90211	−1.90211
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.00000	2.00000	1.00000			$0$
$919$	2.00000	2.00000	1.00000			$0$
$920$	0	0
$921$	−3.61803	−3.61803
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.38197	1.38197
$927$	0	0
$928$	−1.17557	−1.17557
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	−1.38197	−1.38197
$931$	−0.618034	−0.618034
$932$	0	0
$933$	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
$939$	−2.23607	−2.23607
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	4.25325	4.25325
$943$	0	0
$944$	2.00000	2.00000
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	1.17557	1.17557
$949$	0	0
$950$	0.726543	0.726543
$951$	−3.61803	−3.61803
$952$	0	0
$953$	1.90211	1.90211	0.951057	−	0.309017i	$-0.100000\pi$
$953$	1.90211	1.90211	0.951057	+	0.309017i	$0.100000\pi$
$954$	−3.61803	−3.61803
$955$	0	0
$956$	0.618034	0.618034
$957$	3.07768	3.07768
$958$	0.726543	0.726543
$959$	0	0
$960$	0.726543	0.726543
$961$	−0.618034	−0.618034
$962$	0	0
$963$	3.07768	3.07768
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.726543	0.726543
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.17557	−1.17557
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.90211	−1.90211	−0.951057	−	0.309017i	$-0.900000\pi$
$977$	−1.90211	−1.90211	−0.951057	+	0.309017i	$0.900000\pi$
$978$	0	0
$979$	0	0
$980$	−0.381966	−0.381966
$981$	1.61803	1.61803
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.38197	1.38197
$985$	−1.17557	−1.17557
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	3.07768	3.07768
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0.449028	0.449028
$993$	0	0
$994$	0	0
$995$	0	0
$996$	1.38197	1.38197
$997$	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	2255.1.h.j.2254.2	yes	4
5.4	even	2	inner	2255.1.h.j.2254.3	yes	4
11.10	odd	2		2255.1.h.i.2254.3	yes	4
41.40	even	2		2255.1.h.i.2254.2	✓	4
55.54	odd	2		2255.1.h.i.2254.2	✓	4
205.204	even	2		2255.1.h.i.2254.3	yes	4
451.450	odd	2	inner	2255.1.h.j.2254.3	yes	4
2255.2254	odd	2	CM	2255.1.h.j.2254.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2255.1.h.i.2254.2	✓	4	41.40	even	2
2255.1.h.i.2254.2	✓	4	55.54	odd	2
2255.1.h.i.2254.3	yes	4	11.10	odd	2
2255.1.h.i.2254.3	yes	4	205.204	even	2
2255.1.h.j.2254.2	yes	4	1.1	even	1	trivial
2255.1.h.j.2254.2	yes	4	2255.2254	odd	2	CM
2255.1.h.j.2254.3	yes	4	5.4	even	2	inner
2255.1.h.j.2254.3	yes	4	451.450	odd	2	inner

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:	\( N \)	\(=\)	\( 2255 = 5 \cdot 11 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2255.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.17557 q^{2} -1.90211 q^{3} +0.381966 q^{4} -1.00000 q^{5} +2.23607 q^{6} +0.726543 q^{8} +2.61803 q^{9} +O(q^{10})\)	\(q-1.17557 q^{2} -1.90211 q^{3} +0.381966 q^{4} -1.00000 q^{5} +2.23607 q^{6} +0.726543 q^{8} +2.61803 q^{9} +1.17557 q^{10} +1.00000 q^{11} -0.726543 q^{12} +1.90211 q^{15} -1.23607 q^{16} -3.07768 q^{18} -0.618034 q^{19} -0.381966 q^{20} -1.17557 q^{22} -1.38197 q^{24} +1.00000 q^{25} -3.07768 q^{27} -1.61803 q^{29} -2.23607 q^{30} +0.618034 q^{31} +0.726543 q^{32} -1.90211 q^{33} +1.00000 q^{36} +0.726543 q^{38} -0.726543 q^{40} -1.00000 q^{41} +0.381966 q^{44} -2.61803 q^{45} +2.35114 q^{48} +1.00000 q^{49} -1.17557 q^{50} +1.17557 q^{53} +3.61803 q^{54} -1.00000 q^{55} +1.17557 q^{57} +1.90211 q^{58} -1.61803 q^{59} +0.726543 q^{60} -0.726543 q^{62} +0.381966 q^{64} +2.23607 q^{66} +1.17557 q^{67} +1.90211 q^{72} +1.90211 q^{73} -1.90211 q^{75} -0.236068 q^{76} -1.61803 q^{79} +1.23607 q^{80} +3.23607 q^{81} +1.17557 q^{82} -1.90211 q^{83} +3.07768 q^{87} +0.726543 q^{88} +3.07768 q^{90} -1.17557 q^{93} +0.618034 q^{95} -1.38197 q^{96} -1.17557 q^{98} +2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} - 4 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^4 - 4 * q^5 + 6 * q^9	\( 4 q + 6 q^{4} - 4 q^{5} + 6 q^{9} + 4 q^{11} + 4 q^{16} + 2 q^{19} - 6 q^{20} - 10 q^{24} + 4 q^{25} - 2 q^{29} - 2 q^{31} + 4 q^{36} - 4 q^{41} + 6 q^{44} - 6 q^{45} + 4 q^{49} + 10 q^{54} - 4 q^{55} - 2 q^{59} + 6 q^{64} + 8 q^{76} - 2 q^{79} - 4 q^{80} + 4 q^{81} - 2 q^{95} - 10 q^{96} + 6 q^{99}+O(q^{100}) \) 4 * q + 6 * q^4 - 4 * q^5 + 6 * q^9 + 4 * q^11 + 4 * q^16 + 2 * q^19 - 6 * q^20 - 10 * q^24 + 4 * q^25 - 2 * q^29 - 2 * q^31 + 4 * q^36 - 4 * q^41 + 6 * q^44 - 6 * q^45 + 4 * q^49 + 10 * q^54 - 4 * q^55 - 2 * q^59 + 6 * q^64 + 8 * q^76 - 2 * q^79 - 4 * q^80 + 4 * q^81 - 2 * q^95 - 10 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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