LMFDB - Embedded newform 2175.1.b.c.2174.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,1,Mod(2174,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.2174"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2175.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,-6,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.08546640248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.895152515625.1

Embedding invariants

Embedding label			2174.1
Root			$1.53209i$ of defining polynomial
Character	$\chi$	$=$	2175.2174
Dual form			2175.1.b.c.2174.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87939i q^{2} -1.00000i q^{3} -2.53209 q^{4} -1.87939 q^{6} -1.53209i q^{7} +2.87939i q^{8} -1.00000 q^{9} -0.347296 q^{11} +2.53209i q^{12} -1.87939i q^{13} -2.87939 q^{14} +2.87939 q^{16} +0.347296i q^{17} +1.87939i q^{18} -1.53209 q^{21} +0.652704i q^{22} +2.87939 q^{24} -3.53209 q^{26} +1.00000i q^{27} +3.87939i q^{28} +1.00000 q^{29} -2.53209i q^{32} +0.347296i q^{33} +0.652704 q^{34} +2.53209 q^{36} -1.87939 q^{39} +1.00000 q^{41} +2.87939i q^{42} +0.879385 q^{44} +1.53209i q^{47} -2.87939i q^{48} -1.34730 q^{49} +0.347296 q^{51} +4.75877i q^{52} +1.87939 q^{54} +4.41147 q^{56} -1.87939i q^{58} +1.53209i q^{63} -1.87939 q^{64} +0.652704 q^{66} -0.347296i q^{67} -0.879385i q^{68} -2.87939i q^{72} +0.532089i q^{77} +3.53209i q^{78} +1.00000 q^{81} -1.87939i q^{82} +3.87939 q^{84} -1.00000i q^{87} -1.00000i q^{88} -1.87939 q^{89} -2.87939 q^{91} +2.87939 q^{94} -2.53209 q^{96} +2.53209i q^{98} +0.347296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{9} - 6 q^{14} + 6 q^{16} + 6 q^{24} - 12 q^{26} + 6 q^{29} + 6 q^{34} + 6 q^{36} + 6 q^{41} - 6 q^{44} - 6 q^{49} + 6 q^{56} + 6 q^{66} + 6 q^{81} + 12 q^{84} - 6 q^{91} + 6 q^{94}+ \cdots - 6 q^{96}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^9 - 6 * q^14 + 6 * q^16 + 6 * q^24 - 12 * q^26 + 6 * q^29 + 6 * q^34 + 6 * q^36 + 6 * q^41 - 6 * q^44 - 6 * q^49 + 6 * q^56 + 6 * q^66 + 6 * q^81 + 12 * q^84 - 6 * q^91 + 6 * q^94 - 6 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$2$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$3$	− 1.00000i	− 1.00000i
$3$	− 1.00000i	− 1.00000i
$4$	−2.53209	−2.53209
$5$	0	0
$5$	0	0
$6$	−1.87939	−1.87939
$7$	− 1.53209i	− 1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$7$	− 1.53209i	− 1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$8$	2.87939i	2.87939i
$9$	−1.00000	−1.00000
$10$	0	0
$11$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$11$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$12$	2.53209i	2.53209i
$13$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$13$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$14$	−2.87939	−2.87939
$15$	0	0
$16$	2.87939	2.87939
$17$	0.347296i	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$17$	0.347296i	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$18$	1.87939i	1.87939i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−1.53209	−1.53209
$22$	0.652704i	0.652704i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.87939	2.87939
$25$	0	0
$26$	−3.53209	−3.53209
$27$	1.00000i	1.00000i
$28$	3.87939i	3.87939i
$29$	1.00000	1.00000
$29$	1.00000	1.00000
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 2.53209i	− 2.53209i
$33$	0.347296i	0.347296i
$34$	0.652704	0.652704
$35$	0	0
$36$	2.53209	2.53209
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−1.87939	−1.87939
$40$	0	0
$41$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$41$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$42$	2.87939i	2.87939i
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0.879385	0.879385
$45$	0	0
$46$	0	0
$47$	1.53209i	1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$47$	1.53209i	1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$48$	− 2.87939i	− 2.87939i
$49$	−1.34730	−1.34730
$50$	0	0
$51$	0.347296	0.347296
$52$	4.75877i	4.75877i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	1.87939	1.87939
$55$	0	0
$56$	4.41147	4.41147
$57$	0	0
$58$	− 1.87939i	− 1.87939i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	1.53209i	1.53209i
$64$	−1.87939	−1.87939
$65$	0	0
$66$	0.652704	0.652704
$67$	− 0.347296i	− 0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$67$	− 0.347296i	− 0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$68$	− 0.879385i	− 0.879385i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 2.87939i	− 2.87939i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.532089i	0.532089i
$78$	3.53209i	3.53209i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	− 1.87939i	− 1.87939i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	3.87939	3.87939
$85$	0	0
$86$	0	0
$87$	− 1.00000i	− 1.00000i
$88$	− 1.00000i	− 1.00000i
$89$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$89$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$90$	0	0
$91$	−2.87939	−2.87939
$92$	0	0
$93$	0	0
$94$	2.87939	2.87939
$95$	0	0
$96$	−2.53209	−2.53209
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	2.53209i	2.53209i
$99$	0.347296	0.347296
$100$	0	0
$101$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$101$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$102$	− 0.652704i	− 0.652704i
$103$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$103$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$104$	5.41147	5.41147
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 2.53209i	− 2.53209i
$109$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$109$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$110$	0	0
$111$	0	0
$112$	− 4.41147i	− 4.41147i
$113$	− 1.53209i	− 1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$113$	− 1.53209i	− 1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$114$	0	0
$115$	0	0
$116$	−2.53209	−2.53209
$117$	1.87939i	1.87939i
$118$	0	0
$119$	0.532089	0.532089
$120$	0	0
$121$	−0.879385	−0.879385
$122$	0	0
$123$	− 1.00000i	− 1.00000i
$124$	0	0
$125$	0	0
$126$	2.87939	2.87939
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000i	1.00000i
$129$	0	0
$130$	0	0
$131$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$131$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$132$	− 0.879385i	− 0.879385i
$133$	0	0
$134$	−0.652704	−0.652704
$135$	0	0
$136$	−1.00000	−1.00000
$137$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$138$	0	0
$139$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$139$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$140$	0	0
$141$	1.53209	1.53209
$142$	0	0
$143$	0.652704i	0.652704i
$144$	−2.87939	−2.87939
$145$	0	0
$146$	0	0
$147$	1.34730i	1.34730i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$152$	0	0
$153$	− 0.347296i	− 0.347296i
$154$	1.00000	1.00000
$155$	0	0
$156$	4.75877	4.75877
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	− 1.87939i	− 1.87939i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−2.53209	−2.53209
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	− 4.41147i	− 4.41147i
$169$	−2.53209	−2.53209
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−1.87939	−1.87939
$175$	0	0
$176$	−1.00000	−1.00000
$177$	0	0
$178$	3.53209i	3.53209i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$181$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$182$	5.41147i	5.41147i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.120615i	− 0.120615i
$188$	− 3.87939i	− 3.87939i
$189$	1.53209	1.53209
$190$	0	0
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$191$	−2.00000	−2.00000	−1.00000			$\pi$
$192$	1.87939i	1.87939i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	3.41147	3.41147
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	− 0.652704i	− 0.652704i
$199$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$199$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$200$	0	0
$201$	−0.347296	−0.347296
$202$	− 3.53209i	− 3.53209i
$203$	− 1.53209i	− 1.53209i
$204$	−0.879385	−0.879385
$205$	0	0
$206$	−1.87939	−1.87939
$207$	0	0
$208$	− 5.41147i	− 5.41147i
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−2.87939	−2.87939
$217$	0	0
$218$	− 3.53209i	− 3.53209i
$219$	0	0
$220$	0	0
$221$	0.652704	0.652704
$222$	0	0
$223$	0.347296i	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$223$	0.347296i	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$224$	−3.87939	−3.87939
$225$	0	0
$226$	−2.87939	−2.87939
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0.532089	0.532089
$232$	2.87939i	2.87939i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	3.53209	3.53209
$235$	0	0
$236$	0	0
$237$	0	0
$238$	− 1.00000i	− 1.00000i
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$241$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$242$	1.65270i	1.65270i
$243$	− 1.00000i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	−1.87939	−1.87939
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$251$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$252$	− 3.87939i	− 3.87939i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.00000	−1.00000
$262$	2.87939i	2.87939i
$263$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$263$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$264$	−1.00000	−1.00000
$265$	0	0
$266$	0	0
$267$	1.87939i	1.87939i
$268$	0.879385i	0.879385i
$269$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$269$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.00000i	1.00000i
$273$	2.87939i	2.87939i
$274$	−1.87939	−1.87939
$275$	0	0
$276$	0	0
$277$	− 0.347296i	− 0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$277$	− 0.347296i	− 0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$278$	2.87939i	2.87939i
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	− 2.87939i	− 2.87939i
$283$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$284$	0	0
$285$	0	0
$286$	1.22668	1.22668
$287$	− 1.53209i	− 1.53209i
$288$	2.53209i	2.53209i
$289$	0.879385	0.879385
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.87939i	1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$293$	1.87939i	1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$294$	2.53209	2.53209
$295$	0	0
$296$	0	0
$297$	− 0.347296i	− 0.347296i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	1.87939i	1.87939i
$303$	− 1.87939i	− 1.87939i
$304$	0	0
$305$	0	0
$306$	−0.652704	−0.652704
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	− 1.34730i	− 1.34730i
$309$	−1.00000	−1.00000
$310$	0	0
$311$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$311$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$312$	− 5.41147i	− 5.41147i
$313$	1.53209i	1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$313$	1.53209i	1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$317$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$318$	0	0
$319$	−0.347296	−0.347296
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−2.53209	−2.53209
$325$	0	0
$326$	0	0
$327$	− 1.87939i	− 1.87939i
$328$	2.87939i	2.87939i
$329$	2.34730	2.34730
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−4.41147	−4.41147
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	4.75877i	4.75877i
$339$	−1.53209	−1.53209
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.532089i	0.532089i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	2.53209i	2.53209i
$349$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$350$	0	0
$351$	1.87939	1.87939
$352$	0.879385i	0.879385i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	4.75877	4.75877
$357$	− 0.532089i	− 0.532089i
$358$	0	0
$359$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$359$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	− 0.652704i	− 0.652704i
$363$	0.879385i	0.879385i
$364$	7.29086	7.29086
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−1.00000	−1.00000
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$373$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$374$	−0.226682	−0.226682
$375$	0	0
$376$	−4.41147	−4.41147
$377$	− 1.87939i	− 1.87939i
$378$	− 2.87939i	− 2.87939i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	3.75877i	3.75877i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$389$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$390$	0	0
$391$	0	0
$392$	− 3.87939i	− 3.87939i
$393$	1.53209i	1.53209i
$394$	0	0
$395$	0	0
$396$	−0.879385	−0.879385
$397$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$397$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$398$	− 3.53209i	− 3.53209i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0.652704i	0.652704i
$403$	0	0
$404$	−4.75877	−4.75877
$405$	0	0
$406$	−2.87939	−2.87939
$407$	0	0
$408$	1.00000i	1.00000i
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−1.00000	−1.00000
$412$	2.53209i	2.53209i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−4.75877	−4.75877
$417$	1.53209i	1.53209i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 1.53209i	− 1.53209i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.652704	0.652704
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	2.87939i	2.87939i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−4.75877	−4.75877
$437$	0	0
$438$	0	0
$439$	−0.347296	−0.347296	−0.173648	−	0.984808i	$-0.555556\pi$
$439$	−0.347296	−0.347296	−0.173648	+	0.984808i	$0.555556\pi$
$440$	0	0
$441$	1.34730	1.34730
$442$	− 1.22668i	− 1.22668i
$443$	1.87939i	1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$443$	1.87939i	1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$444$	0	0
$445$	0	0
$446$	0.652704	0.652704
$447$	0	0
$448$	2.87939i	2.87939i
$449$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$449$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$450$	0	0
$451$	−0.347296	−0.347296
$452$	3.87939i	3.87939i
$453$	1.00000i	1.00000i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.53209i	− 1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$457$	− 1.53209i	− 1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$458$	0	0
$459$	−0.347296	−0.347296
$460$	0	0
$461$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$461$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$462$	− 1.00000i	− 1.00000i
$463$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$463$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$464$	2.87939	2.87939
$465$	0	0
$466$	0	0
$467$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$467$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$468$	− 4.75877i	− 4.75877i
$469$	−0.532089	−0.532089
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−1.34730	−1.34730
$477$	0	0
$478$	0	0
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	0	0
$481$	0	0
$482$	− 2.87939i	− 2.87939i
$483$	0	0
$484$	2.22668	2.22668
$485$	0	0
$486$	−1.87939	−1.87939
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$487$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$491$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$492$	2.53209i	2.53209i
$493$	0.347296i	0.347296i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$499$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$500$	0	0
$501$	0	0
$502$	− 3.53209i	− 3.53209i
$503$	1.87939i	1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$503$	1.87939i	1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$504$	−4.41147	−4.41147
$505$	0	0
$506$	0	0
$507$	2.53209i	2.53209i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.532089i	− 0.532089i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.87939i	1.87939i
$523$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$523$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$524$	3.87939	3.87939
$525$	0	0
$526$	1.87939	1.87939
$527$	0	0
$528$	1.00000i	1.00000i
$529$	−1.00000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.87939i	− 1.87939i
$534$	3.53209	3.53209
$535$	0	0
$536$	1.00000	1.00000
$537$	0	0
$538$	− 0.652704i	− 0.652704i
$539$	0.467911	0.467911
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	− 0.347296i	− 0.347296i
$544$	0.879385	0.879385
$545$	0	0
$546$	5.41147	5.41147
$547$	− 0.347296i	− 0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$547$	− 0.347296i	− 0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$548$	2.53209i	2.53209i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−0.652704	−0.652704
$555$	0	0
$556$	3.87939	3.87939
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.120615	−0.120615
$562$	0	0
$563$	− 0.347296i	− 0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$563$	− 0.347296i	− 0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$564$	−3.87939	−3.87939
$565$	0	0
$566$	−1.87939	−1.87939
$567$	− 1.53209i	− 1.53209i
$568$	0	0
$569$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$569$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$570$	0	0
$571$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$572$	− 1.65270i	− 1.65270i
$573$	2.00000i	2.00000i
$574$	−2.87939	−2.87939
$575$	0	0
$576$	1.87939	1.87939
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 1.65270i	− 1.65270i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	3.53209	3.53209
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	− 3.41147i	− 3.41147i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	−0.652704	−0.652704
$595$	0	0
$596$	0	0
$597$	− 1.87939i	− 1.87939i
$598$	0	0
$599$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$599$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0.347296i	0.347296i
$604$	2.53209	2.53209
$605$	0	0
$606$	−3.53209	−3.53209
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−1.53209	−1.53209
$610$	0	0
$611$	2.87939	2.87939
$612$	0.879385i	0.879385i
$613$	1.53209i	1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$613$	1.53209i	1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$614$	0	0
$615$	0	0
$616$	−1.53209	−1.53209
$617$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$617$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$618$	1.87939i	1.87939i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	2.87939i	2.87939i
$623$	2.87939i	2.87939i
$624$	−5.41147	−5.41147
$625$	0	0
$626$	2.87939	2.87939
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$631$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$632$	0	0
$633$	0	0
$634$	−3.53209	−3.53209
$635$	0	0
$636$	0	0
$637$	2.53209i	2.53209i
$638$	0.652704i	0.652704i
$639$	0	0
$640$	0	0
$641$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$641$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$642$	0	0
$643$	0.347296i	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$643$	0.347296i	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	2.87939i	2.87939i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 0.347296i	− 0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$653$	− 0.347296i	− 0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$654$	−3.53209	−3.53209
$655$	0	0
$656$	2.87939	2.87939
$657$	0	0
$658$	− 4.41147i	− 4.41147i
$659$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$659$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$660$	0	0
$661$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$662$	0	0
$663$	− 0.652704i	− 0.652704i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0.347296	0.347296
$670$	0	0
$671$	0	0
$672$	3.87939i	3.87939i
$673$	− 1.87939i	− 1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$673$	− 1.87939i	− 1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$674$	0	0
$675$	0	0
$676$	6.41147	6.41147
$677$	0.347296i	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$677$	0.347296i	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$678$	2.87939i	2.87939i
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	1.00000
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$692$	0	0
$693$	− 0.532089i	− 0.532089i
$694$	0	0
$695$	0	0
$696$	2.87939	2.87939
$697$	0.347296i	0.347296i
$698$	− 1.87939i	− 1.87939i
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	− 3.53209i	− 3.53209i
$703$	0	0
$704$	0.652704	0.652704
$705$	0	0
$706$	0	0
$707$	− 2.87939i	− 2.87939i
$708$	0	0
$709$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	0	0
$711$	0	0
$712$	− 5.41147i	− 5.41147i
$713$	0	0
$714$	−1.00000	−1.00000
$715$	0	0
$716$	0	0
$717$	0	0
$718$	1.87939i	1.87939i
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1.53209	−1.53209
$722$	− 1.87939i	− 1.87939i
$723$	− 1.53209i	− 1.53209i
$724$	−0.879385	−0.879385
$725$	0	0
$726$	1.65270	1.65270
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	− 8.29086i	− 8.29086i
$729$	−1.00000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.120615i	0.120615i
$738$	1.87939i	1.87939i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1.53209i	− 1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$743$	− 1.53209i	− 1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$744$	0	0
$745$	0	0
$746$	3.75877	3.75877
$747$	0	0
$748$	0.305407i	0.305407i
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	4.41147i	4.41147i
$753$	− 1.87939i	− 1.87939i
$754$	−3.53209	−3.53209
$755$	0	0
$756$	−3.87939	−3.87939
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 2.87939i	− 2.87939i
$764$	5.06418	5.06418
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$773$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	3.53209i	3.53209i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.00000i	1.00000i
$784$	−3.87939	−3.87939
$785$	0	0
$786$	2.87939	2.87939
$787$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$787$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$788$	0	0
$789$	1.00000	1.00000
$790$	0	0
$791$	−2.34730	−2.34730
$792$	1.00000i	1.00000i
$793$	0	0
$794$	1.87939	1.87939
$795$	0	0
$796$	−4.75877	−4.75877
$797$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$797$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$798$	0	0
$799$	−0.532089	−0.532089
$800$	0	0
$801$	1.87939	1.87939
$802$	0	0
$803$	0	0
$804$	0.879385	0.879385
$805$	0	0
$806$	0	0
$807$	− 0.347296i	− 0.347296i
$808$	5.41147i	5.41147i
$809$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$809$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$810$	0	0
$811$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$811$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$812$	3.87939i	3.87939i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	1.00000	1.00000
$817$	0	0
$818$	0	0
$819$	2.87939	2.87939
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	1.87939i	1.87939i
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	2.87939	2.87939
$825$	0	0
$826$	0	0
$827$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−0.347296	−0.347296
$832$	3.53209i	3.53209i
$833$	− 0.467911i	− 0.467911i
$834$	2.87939	2.87939
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$839$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	−2.87939	−2.87939
$847$	1.34730i	1.34730i
$848$	0	0
$849$	−1.00000	−1.00000
$850$	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
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	− 1.22668i	− 1.22668i
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	−1.53209	−1.53209
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	2.53209	2.53209
$865$	0	0
$866$	0	0
$867$	− 0.879385i	− 0.879385i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.652704	−0.652704
$872$	5.41147i	5.41147i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$877$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$878$	0.652704i	0.652704i
$879$	1.87939	1.87939
$880$	0	0
$881$	−1.53209	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$881$	−1.53209	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$882$	− 2.53209i	− 2.53209i
$883$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$883$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$884$	−1.65270	−1.65270
$885$	0	0
$886$	3.53209	3.53209
$887$	1.53209i	1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$887$	1.53209i	1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.347296	−0.347296
$892$	− 0.879385i	− 0.879385i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.53209	1.53209
$897$	0	0
$898$	− 2.87939i	− 2.87939i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0.652704i	0.652704i
$903$	0	0
$904$	4.41147	4.41147
$905$	0	0
$906$	1.87939	1.87939
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−1.87939	−1.87939
$910$	0	0
$911$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$911$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$912$	0	0
$913$	0	0
$914$	−2.87939	−2.87939
$915$	0	0
$916$	0	0
$917$	2.34730i	2.34730i
$918$	0.652704i	0.652704i
$919$	1.87939	1.87939	0.939693	−	0.342020i	$-0.111111\pi$
$919$	1.87939	1.87939	0.939693	+	0.342020i	$0.111111\pi$
$920$	0	0
$921$	0	0
$922$	− 1.87939i	− 1.87939i
$923$	0	0
$924$	−1.34730	−1.34730
$925$	0	0
$926$	−3.53209	−3.53209
$927$	1.00000i	1.00000i
$928$	− 2.53209i	− 2.53209i
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.53209i	1.53209i
$934$	−1.87939	−1.87939
$935$	0	0
$936$	−5.41147	−5.41147
$937$	1.87939i	1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$937$	1.87939i	1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$938$	1.00000i	1.00000i
$939$	1.53209	1.53209
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.347296i	0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$947$	0.347296i	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1.87939	−1.87939
$952$	1.53209i	1.53209i
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.347296i	0.347296i
$958$	1.87939i	1.87939i
$959$	−1.53209	−1.53209
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−3.87939	−3.87939
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 2.53209i	− 2.53209i
$969$	0	0
$970$	0	0
$971$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$971$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$972$	2.53209i	2.53209i
$973$	2.34730i	2.34730i
$974$	−3.75877	−3.75877
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0.652704	0.652704
$980$	0	0
$981$	−1.87939	−1.87939
$982$	− 1.87939i	− 1.87939i
$983$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$983$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$	2.87939	2.87939
$985$	0	0
$986$	0.652704	0.652704
$987$	− 2.34730i	− 2.34730i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$991$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 3.53209i	− 3.53209i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.1.b.c.2174.1		6
3.2	odd	2		2175.1.b.d.2174.6		6
5.2	odd	4		2175.1.h.d.1826.3	yes	3
5.3	odd	4		2175.1.h.f.1826.1	yes	3
5.4	even	2	inner	2175.1.b.c.2174.6		6
15.2	even	4		2175.1.h.e.1826.1	yes	3
15.8	even	4		2175.1.h.c.1826.3	✓	3
15.14	odd	2		2175.1.b.d.2174.1		6
29.28	even	2		2175.1.b.d.2174.6		6
87.86	odd	2	CM	2175.1.b.c.2174.1		6
145.28	odd	4		2175.1.h.c.1826.3	✓	3
145.57	odd	4		2175.1.h.e.1826.1	yes	3
145.144	even	2		2175.1.b.d.2174.1		6
435.173	even	4		2175.1.h.f.1826.1	yes	3
435.347	even	4		2175.1.h.d.1826.3	yes	3
435.434	odd	2	inner	2175.1.b.c.2174.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.1.b.c.2174.1		6	1.1	even	1	trivial
2175.1.b.c.2174.1		6	87.86	odd	2	CM
2175.1.b.c.2174.6		6	5.4	even	2	inner
2175.1.b.c.2174.6		6	435.434	odd	2	inner
2175.1.b.d.2174.1		6	15.14	odd	2
2175.1.b.d.2174.1		6	145.144	even	2
2175.1.b.d.2174.6		6	3.2	odd	2
2175.1.b.d.2174.6		6	29.28	even	2
2175.1.h.c.1826.3	✓	3	15.8	even	4
2175.1.h.c.1826.3	✓	3	145.28	odd	4
2175.1.h.d.1826.3	yes	3	5.2	odd	4
2175.1.h.d.1826.3	yes	3	435.347	even	4
2175.1.h.e.1826.1	yes	3	15.2	even	4
2175.1.h.e.1826.1	yes	3	145.57	odd	4
2175.1.h.f.1826.1	yes	3	5.3	odd	4
2175.1.h.f.1826.1	yes	3	435.173	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.87939i q^{2} -1.00000i q^{3} -2.53209 q^{4} -1.87939 q^{6} -1.53209i q^{7} +2.87939i q^{8} -1.00000 q^{9} -0.347296 q^{11} +2.53209i q^{12} -1.87939i q^{13} -2.87939 q^{14} +2.87939 q^{16} +0.347296i q^{17} +1.87939i q^{18} -1.53209 q^{21} +0.652704i q^{22} +2.87939 q^{24} -3.53209 q^{26} +1.00000i q^{27} +3.87939i q^{28} +1.00000 q^{29} -2.53209i q^{32} +0.347296i q^{33} +0.652704 q^{34} +2.53209 q^{36} -1.87939 q^{39} +1.00000 q^{41} +2.87939i q^{42} +0.879385 q^{44} +1.53209i q^{47} -2.87939i q^{48} -1.34730 q^{49} +0.347296 q^{51} +4.75877i q^{52} +1.87939 q^{54} +4.41147 q^{56} -1.87939i q^{58} +1.53209i q^{63} -1.87939 q^{64} +0.652704 q^{66} -0.347296i q^{67} -0.879385i q^{68} -2.87939i q^{72} +0.532089i q^{77} +3.53209i q^{78} +1.00000 q^{81} -1.87939i q^{82} +3.87939 q^{84} -1.00000i q^{87} -1.00000i q^{88} -1.87939 q^{89} -2.87939 q^{91} +2.87939 q^{94} -2.53209 q^{96} +2.53209i q^{98} +0.347296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{9} - 6 q^{14} + 6 q^{16} + 6 q^{24} - 12 q^{26} + 6 q^{29} + 6 q^{34} + 6 q^{36} + 6 q^{41} - 6 q^{44} - 6 q^{49} + 6 q^{56} + 6 q^{66} + 6 q^{81} + 12 q^{84} - 6 q^{91} + 6 q^{94}+ \cdots - 6 q^{96}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^9 - 6 * q^14 + 6 * q^16 + 6 * q^24 - 12 * q^26 + 6 * q^29 + 6 * q^34 + 6 * q^36 + 6 * q^41 - 6 * q^44 - 6 * q^49 + 6 * q^56 + 6 * q^66 + 6 * q^81 + 12 * q^84 - 6 * q^91 + 6 * q^94 - 6 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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