LMFDB - Embedded newform 3575.1.c.d.3574.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3575,1,Mod(3574,3575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3575, 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("3575.3574");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3575 = 5^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3575.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.78415742016$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.20449.1
Artin image:	$C_4\times D_5$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			3574.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	3575.3574
Dual form			3575.1.c.d.3574.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} -0.618034i q^{3} +0.618034 q^{4} +0.381966 q^{6} -1.61803i q^{7} +1.00000i q^{8} +0.618034 q^{9} +O(q^{10})$	$q+0.618034i q^{2} -0.618034i q^{3} +0.618034 q^{4} +0.381966 q^{6} -1.61803i q^{7} +1.00000i q^{8} +0.618034 q^{9} +1.00000 q^{11} -0.381966i q^{12} -1.00000i q^{13} +1.00000 q^{14} +0.381966i q^{18} -0.618034 q^{19} -1.00000 q^{21} +0.618034i q^{22} +1.61803i q^{23} +0.618034 q^{24} +0.618034 q^{26} -1.00000i q^{27} -1.00000i q^{28} +1.00000i q^{32} -0.618034i q^{33} +0.381966 q^{36} -0.381966i q^{38} -0.618034 q^{39} -1.61803 q^{41} -0.618034i q^{42} +0.618034 q^{44} -1.00000 q^{46} -1.61803 q^{49} -0.618034i q^{52} -0.618034i q^{53} +0.618034 q^{54} +1.61803 q^{56} +0.381966i q^{57} -1.00000i q^{63} -0.618034 q^{64} +0.381966 q^{66} +1.00000 q^{69} +0.618034i q^{72} -0.618034i q^{73} -0.381966 q^{76} -1.61803i q^{77} -0.381966i q^{78} -1.00000i q^{82} +1.61803i q^{83} -0.618034 q^{84} +1.00000i q^{88} -1.61803 q^{91} +1.00000i q^{92} +0.618034 q^{96} -1.00000i q^{98} +0.618034 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9	$ 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{19} - 4 q^{21} - 2 q^{24} - 2 q^{26} + 6 q^{36} + 2 q^{39} - 2 q^{41} - 2 q^{44} - 4 q^{46} - 2 q^{49} - 2 q^{54} + 2 q^{56} + 2 q^{64} + 6 q^{66} + 4 q^{69} - 6 q^{76} + 2 q^{84} - 2 q^{91} - 2 q^{96} - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^19 - 4 * q^21 - 2 * q^24 - 2 * q^26 + 6 * q^36 + 2 * q^39 - 2 * q^41 - 2 * q^44 - 4 * q^46 - 2 * q^49 - 2 * q^54 + 2 * q^56 + 2 * q^64 + 6 * q^66 + 4 * q^69 - 6 * q^76 + 2 * q^84 - 2 * q^91 - 2 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3575\mathbb{Z}\right)^\times$.

$n$	$651$	$1002$	$1926$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$2$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$3$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$3$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$4$	0.618034	0.618034
$5$	0	0
$5$	0	0
$6$	0.381966	0.381966
$7$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$7$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$8$	1.00000i	1.00000i
$9$	0.618034	0.618034
$10$	0	0
$11$	1.00000	1.00000
$11$	1.00000	1.00000
$12$	− 0.381966i	− 0.381966i
$13$	− 1.00000i	− 1.00000i
$13$	− 1.00000i	− 1.00000i
$14$	1.00000	1.00000
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0.381966i	0.381966i
$19$	−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	0
$21$	−1.00000	−1.00000
$22$	0.618034i	0.618034i
$23$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$23$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$24$	0.618034	0.618034
$25$	0	0
$26$	0.618034	0.618034
$27$	− 1.00000i	− 1.00000i
$28$	− 1.00000i	− 1.00000i
$29$	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			$\pi$
$32$	1.00000i	1.00000i
$33$	− 0.618034i	− 0.618034i
$34$	0	0
$35$	0	0
$36$	0.381966	0.381966
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	− 0.381966i	− 0.381966i
$39$	−0.618034	−0.618034
$40$	0	0
$41$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$41$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$42$	− 0.618034i	− 0.618034i
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0.618034	0.618034
$45$	0	0
$46$	−1.00000	−1.00000
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−1.61803	−1.61803
$50$	0	0
$51$	0	0
$52$	− 0.618034i	− 0.618034i
$53$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$53$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$54$	0.618034	0.618034
$55$	0	0
$56$	1.61803	1.61803
$57$	0.381966i	0.381966i
$58$	0	0
$59$	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
$63$	− 1.00000i	− 1.00000i
$64$	−0.618034	−0.618034
$65$	0	0
$66$	0.381966	0.381966
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	1.00000	1.00000
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.618034i	0.618034i
$73$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$73$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$74$	0	0
$75$	0	0
$76$	−0.381966	−0.381966
$77$	− 1.61803i	− 1.61803i
$78$	− 0.381966i	− 0.381966i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	− 1.00000i	− 1.00000i
$83$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$83$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$84$	−0.618034	−0.618034
$85$	0	0
$86$	0	0
$87$	0	0
$88$	1.00000i	1.00000i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−1.61803	−1.61803
$92$	1.00000i	1.00000i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0.618034	0.618034
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 1.00000i	− 1.00000i
$99$	0.618034	0.618034
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$103$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	0.381966	0.381966
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 0.618034i	− 0.618034i
$109$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$109$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$113$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$114$	−0.236068	−0.236068
$115$	0	0
$116$	0	0
$117$	− 0.618034i	− 0.618034i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	1.00000i	1.00000i
$124$	0	0
$125$	0	0
$126$	0.618034	0.618034
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0.618034i	0.618034i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	− 0.381966i	− 0.381966i
$133$	1.00000i	1.00000i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0.618034i	0.618034i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.00000i	− 1.00000i
$144$	0	0
$145$	0	0
$146$	0.381966	0.381966
$147$	1.00000i	1.00000i
$148$	0	0
$149$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$149$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$150$	0	0
$151$	2.00000	2.00000	1.00000			$0$
$151$	2.00000	2.00000	1.00000			$0$
$152$	− 0.618034i	− 0.618034i
$153$	0	0
$154$	1.00000	1.00000
$155$	0	0
$156$	−0.381966	−0.381966
$157$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$157$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$158$	0	0
$159$	−0.381966	−0.381966
$160$	0	0
$161$	2.61803	2.61803
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−1.00000	−1.00000
$165$	0	0
$166$	−1.00000	−1.00000
$167$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$167$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$168$	− 1.00000i	− 1.00000i
$169$	−1.00000	−1.00000
$170$	0	0
$171$	−0.381966	−0.381966
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$180$	0	0
$181$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$181$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$182$	− 1.00000i	− 1.00000i
$183$	0	0
$184$	−1.61803	−1.61803
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.61803	−1.61803
$190$	0	0
$191$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$191$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$192$	0.381966i	0.381966i
$193$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$193$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$194$	0	0
$195$	0	0
$196$	−1.00000	−1.00000
$197$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$197$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$198$	0.381966i	0.381966i
$199$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$199$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.00000	−1.00000
$207$	1.00000i	1.00000i
$208$	0	0
$209$	−0.618034	−0.618034
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 0.381966i	− 0.381966i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	1.00000	1.00000
$217$	0	0
$218$	− 0.381966i	− 0.381966i
$219$	−0.381966	−0.381966
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.61803	1.61803
$225$	0	0
$226$	0.381966	0.381966
$227$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$227$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$228$	0.236068i	0.236068i
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−1.00000	−1.00000
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0.381966	0.381966
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$239$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$240$	0	0
$241$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$242$	0.618034i	0.618034i
$243$	− 1.00000i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	−0.618034	−0.618034
$247$	0.618034i	0.618034i
$248$	0	0
$249$	1.00000	1.00000
$250$	0	0
$251$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$251$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$252$	− 0.618034i	− 0.618034i
$253$	1.61803i	1.61803i
$254$	0	0
$255$	0	0
$256$	−1.00000	−1.00000
$257$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$257$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0.618034	0.618034
$265$	0	0
$266$	−0.618034	−0.618034
$267$	0	0
$268$	0	0
$269$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$269$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$270$	0	0
$271$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$271$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$272$	0	0
$273$	1.00000i	1.00000i
$274$	0	0
$275$	0	0
$276$	0.618034	0.618034
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	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
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0.618034	0.618034
$287$	2.61803i	2.61803i
$288$	0.618034i	0.618034i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	0	0
$292$	− 0.381966i	− 0.381966i
$293$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$293$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$294$	−0.618034	−0.618034
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 1.00000i
$298$	1.00000i	1.00000i
$299$	1.61803	1.61803
$300$	0	0
$301$	0	0
$302$	1.23607i	1.23607i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$307$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$308$	− 1.00000i	− 1.00000i
$309$	1.00000	1.00000
$310$	0	0
$311$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$311$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$312$	− 0.618034i	− 0.618034i
$313$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$313$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$314$	1.00000	1.00000
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	− 0.236068i	− 0.236068i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	1.61803i	1.61803i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.381966i	0.381966i
$328$	− 1.61803i	− 1.61803i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.00000i	1.00000i
$333$	0	0
$334$	−0.381966	−0.381966
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	− 0.618034i	− 0.618034i
$339$	−0.381966	−0.381966
$340$	0	0
$341$	0	0
$342$	− 0.236068i	− 0.236068i
$343$	1.00000i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$349$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$350$	0	0
$351$	−1.00000	−1.00000
$352$	1.00000i	1.00000i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	− 1.23607i	− 1.23607i
$359$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$359$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$360$	0	0
$361$	−0.618034	−0.618034
$362$	0.381966i	0.381966i
$363$	− 0.618034i	− 0.618034i
$364$	−1.00000	−1.00000
$365$	0	0
$366$	0	0
$367$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$367$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$368$	0	0
$369$	−1.00000	−1.00000
$370$	0	0
$371$	−1.00000	−1.00000
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	− 1.00000i	− 1.00000i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0.381966i	0.381966i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0.381966	0.381966
$385$	0	0
$386$	−1.00000	−1.00000
$387$	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
$392$	− 1.61803i	− 1.61803i
$393$	0	0
$394$	−0.381966	−0.381966
$395$	0	0
$396$	0.381966	0.381966
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	− 0.381966i	− 0.381966i
$399$	0.618034	0.618034
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.00000	−2.00000	−1.00000			$\pi$
$409$	−2.00000	−2.00000	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	1.00000i	1.00000i
$413$	0	0
$414$	−0.618034	−0.618034
$415$	0	0
$416$	1.00000	1.00000
$417$	0	0
$418$	− 0.381966i	− 0.381966i
$419$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$419$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0.618034	0.618034
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.618034	−0.618034
$430$	0	0
$431$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$432$	0	0
$433$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$433$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$434$	0	0
$435$	0	0
$436$	−0.381966	−0.381966
$437$	− 1.00000i	− 1.00000i
$438$	− 0.236068i	− 0.236068i
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.00000	−1.00000
$442$	0	0
$443$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$443$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.00000i	− 1.00000i
$448$	1.00000i	1.00000i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−1.61803	−1.61803
$452$	− 0.381966i	− 0.381966i
$453$	− 1.23607i	− 1.23607i
$454$	−0.381966	−0.381966
$455$	0	0
$456$	−0.381966	−0.381966
$457$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$457$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$461$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$462$	− 0.618034i	− 0.618034i
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$467$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$468$	− 0.381966i	− 0.381966i
$469$	0	0
$470$	0	0
$471$	−1.00000	−1.00000
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.381966i	− 0.381966i
$478$	− 0.381966i	− 0.381966i
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$479$	−2.00000	−2.00000	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	− 1.00000i	− 1.00000i
$483$	− 1.61803i	− 1.61803i
$484$	0.618034	0.618034
$485$	0	0
$486$	0.618034	0.618034
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.618034i	0.618034i
$493$	0	0
$494$	−0.381966	−0.381966
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0.618034i	0.618034i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0.381966	0.381966
$502$	− 1.00000i	− 1.00000i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	1.00000	1.00000
$505$	0	0
$506$	−1.00000	−1.00000
$507$	0.618034i	0.618034i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1.00000	−1.00000
$512$	0	0
$513$	0.618034i	0.618034i
$514$	1.00000	1.00000
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$521$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.61803	−1.61803
$530$	0	0
$531$	0	0
$532$	0.618034i	0.618034i
$533$	1.61803i	1.61803i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.23607i	1.23607i
$538$	1.00000i	1.00000i
$539$	−1.61803	−1.61803
$540$	0	0
$541$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$542$	0.381966i	0.381966i
$543$	− 0.381966i	− 0.381966i
$544$	0	0
$545$	0	0
$546$	−0.618034	−0.618034
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	1.00000i	1.00000i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$557$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0.381966i	0.381966i
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	− 0.618034i	− 0.618034i
$573$	− 0.381966i	− 0.381966i
$574$	−1.61803	−1.61803
$575$	0	0
$576$	−0.381966	−0.381966
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 0.618034i	− 0.618034i
$579$	1.00000	1.00000
$580$	0	0
$581$	2.61803	2.61803
$582$	0	0
$583$	− 0.618034i	− 0.618034i
$584$	0.618034	0.618034
$585$	0	0
$586$	1.23607	1.23607
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0.618034i	0.618034i
$589$	0	0
$590$	0	0
$591$	0.381966	0.381966
$592$	0	0
$593$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$593$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$594$	0.618034	0.618034
$595$	0	0
$596$	1.00000	1.00000
$597$	0.381966i	0.381966i
$598$	1.00000i	1.00000i
$599$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$599$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	1.23607	1.23607
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	− 0.618034i	− 0.618034i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$613$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$614$	−1.23607	−1.23607
$615$	0	0
$616$	1.61803	1.61803
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0.618034i	0.618034i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	1.61803	1.61803
$622$	0.381966i	0.381966i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−1.00000	−1.00000
$627$	0.381966i	0.381966i
$628$	− 1.00000i	− 1.00000i
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.236068	−0.236068
$637$	1.61803i	1.61803i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$641$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	1.61803	1.61803
$645$	0	0
$646$	0	0
$647$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$647$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$653$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$654$	−0.236068	−0.236068
$655$	0	0
$656$	0	0
$657$	− 0.381966i	− 0.381966i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−1.61803	−1.61803
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0.381966i	0.381966i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	− 1.00000i	− 1.00000i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	−0.618034	−0.618034
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	− 0.236068i	− 0.236068i
$679$	0	0
$680$	0	0
$681$	0.381966	0.381966
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−0.236068	−0.236068
$685$	0	0
$686$	−0.618034	−0.618034
$687$	0	0
$688$	0	0
$689$	−0.618034	−0.618034
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	− 1.00000i	− 1.00000i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.00000i	1.00000i
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	− 0.618034i	− 0.618034i
$703$	0	0
$704$	−0.618034	−0.618034
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.23607	−1.23607
$717$	0.381966i	0.381966i
$718$	1.00000i	1.00000i
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$719$	−2.00000	−2.00000	−1.00000			$\pi$
$720$	0	0
$721$	2.61803	2.61803
$722$	− 0.381966i	− 0.381966i
$723$	1.00000i	1.00000i
$724$	0.381966	0.381966
$725$	0	0
$726$	0.381966	0.381966
$727$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$727$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$728$	− 1.61803i	− 1.61803i
$729$	−0.618034	−0.618034
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$733$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$734$	−0.381966	−0.381966
$735$	0	0
$736$	−1.61803	−1.61803
$737$	0	0
$738$	− 0.618034i	− 0.618034i
$739$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$739$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$740$	0	0
$741$	0.381966	0.381966
$742$	− 0.618034i	− 0.618034i
$743$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$743$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.00000i	1.00000i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$751$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$752$	0	0
$753$	1.00000i	1.00000i
$754$	0	0
$755$	0	0
$756$	−1.00000	−1.00000
$757$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$757$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$758$	0	0
$759$	1.00000	1.00000
$760$	0	0
$761$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$761$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$762$	0	0
$763$	1.00000i	1.00000i
$764$	0.381966	0.381966
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.618034i	0.618034i
$769$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$769$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$770$	0	0
$771$	−1.00000	−1.00000
$772$	1.00000i	1.00000i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 0.381966i	− 0.381966i
$779$	1.00000	1.00000
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$787$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$788$	0.381966i	0.381966i
$789$	0	0
$790$	0	0
$791$	−1.00000	−1.00000
$792$	0.618034i	0.618034i
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−0.381966	−0.381966
$797$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$797$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$798$	0.381966i	0.381966i
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 0.618034i	− 0.618034i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.00000i	− 1.00000i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$812$	0	0
$813$	− 0.381966i	− 0.381966i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 1.23607i	− 1.23607i
$819$	−1.00000	−1.00000
$820$	0	0
$821$	2.00000	2.00000	1.00000			$0$
$821$	2.00000	2.00000	1.00000			$0$
$822$	0	0
$823$	1.61803i	1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$823$	1.61803i	1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$824$	−1.61803	−1.61803
$825$	0	0
$826$	0	0
$827$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$827$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$828$	0.618034i	0.618034i
$829$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$829$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$830$	0	0
$831$	0	0
$832$	0.618034i	0.618034i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−0.381966	−0.381966
$837$	0	0
$838$	1.00000i	1.00000i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	− 0.381966i	− 0.381966i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.61803i	− 1.61803i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$853$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	− 0.381966i	− 0.381966i
$859$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$859$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$860$	0	0
$861$	1.61803	1.61803
$862$	− 1.00000i	− 1.00000i
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	1.00000	1.00000
$865$	0	0
$866$	0.381966	0.381966
$867$	0.618034i	0.618034i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 0.618034i	− 0.618034i
$873$	0	0
$874$	0.618034	0.618034
$875$	0	0
$876$	−0.236068	−0.236068
$877$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$877$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$878$	0	0
$879$	−1.23607	−1.23607
$880$	0	0
$881$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$881$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$882$	− 0.618034i	− 0.618034i
$883$	− 0.618034i	− 0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$883$	− 0.618034i	− 0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$884$	0	0
$885$	0	0
$886$	−1.00000	−1.00000
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0.618034	0.618034
$895$	0	0
$896$	1.00000	1.00000
$897$	− 1.00000i	− 1.00000i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 1.00000i	− 1.00000i
$903$	0	0
$904$	0.618034	0.618034
$905$	0	0
$906$	0.763932	0.763932
$907$	− 1.61803i	− 1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$907$	− 1.61803i	− 1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$908$	0.381966i	0.381966i
$909$	0	0
$910$	0	0
$911$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$911$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$912$	0	0
$913$	1.61803i	1.61803i
$914$	−0.381966	−0.381966
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.23607	1.23607
$922$	0.381966i	0.381966i
$923$	0	0
$924$	−0.618034	−0.618034
$925$	0	0
$926$	0	0
$927$	1.00000i	1.00000i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.00000	1.00000
$932$	0	0
$933$	− 0.381966i	− 0.381966i
$934$	−1.23607	−1.23607
$935$	0	0
$936$	0.618034	0.618034
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	1.00000	1.00000
$940$	0	0
$941$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$941$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$942$	− 0.618034i	− 0.618034i
$943$	− 2.61803i	− 2.61803i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−0.618034	−0.618034
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0.236068	0.236068
$955$	0	0
$956$	−0.381966	−0.381966
$957$	0	0
$958$	− 1.23607i	− 1.23607i
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.00000	−1.00000
$965$	0	0
$966$	1.00000	1.00000
$967$	0.618034i	0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$967$	0.618034i	0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$968$	1.00000i	1.00000i
$969$	0	0
$970$	0	0
$971$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$971$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$972$	− 0.618034i	− 0.618034i
$973$	0	0
$974$	0	0
$975$	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	0
$980$	0	0
$981$	−0.381966	−0.381966
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−1.00000	−1.00000
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0.381966i	0.381966i
$989$	0	0
$990$	0	0
$991$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$991$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0.618034	0.618034
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3575.1.c.d.3574.3		4
5.2	odd	4		3575.1.h.f.2001.1		2
5.3	odd	4		143.1.d.a.142.2	✓	2
5.4	even	2	inner	3575.1.c.d.3574.2		4
11.10	odd	2		3575.1.c.c.3574.2		4
13.12	even	2		3575.1.c.c.3574.2		4
15.8	even	4		1287.1.g.b.1000.1		2
20.3	even	4		2288.1.m.b.2001.1		2
55.3	odd	20		1573.1.l.d.233.1		4
55.8	even	20		1573.1.l.a.233.1		4
55.13	even	20		1573.1.l.c.766.1		4
55.18	even	20		1573.1.l.a.1546.1		4
55.28	even	20		1573.1.l.c.844.1		4
55.32	even	4		3575.1.h.e.2001.2		2
55.38	odd	20		1573.1.l.b.844.1		4
55.43	even	4		143.1.d.b.142.1	yes	2
55.48	odd	20		1573.1.l.d.1546.1		4
55.53	odd	20		1573.1.l.b.766.1		4
55.54	odd	2		3575.1.c.c.3574.3		4
65.3	odd	12		1859.1.i.b.1330.1		4
65.8	even	4		1859.1.c.c.846.3		4
65.12	odd	4		3575.1.h.e.2001.2		2
65.18	even	4		1859.1.c.c.846.2		4
65.23	odd	12		1859.1.i.a.1330.2		4
65.28	even	12		1859.1.k.c.1374.3		8
65.33	even	12		1859.1.k.c.1836.3		8
65.38	odd	4		143.1.d.b.142.1	yes	2
65.43	odd	12		1859.1.i.a.868.2		4
65.48	odd	12		1859.1.i.b.868.1		4
65.58	even	12		1859.1.k.c.1836.2		8
65.63	even	12		1859.1.k.c.1374.2		8
65.64	even	2		3575.1.c.c.3574.3		4
143.142	odd	2	CM	3575.1.c.d.3574.3		4
165.98	odd	4		1287.1.g.a.1000.2		2
195.38	even	4		1287.1.g.a.1000.2		2
220.43	odd	4		2288.1.m.a.2001.1		2
260.103	even	4		2288.1.m.a.2001.1		2
715.38	odd	20		1573.1.l.c.844.1		4
715.43	even	12		1859.1.i.b.868.1		4
715.98	odd	12		1859.1.k.c.1836.2		8
715.103	odd	20		1573.1.l.a.1546.1		4
715.142	even	4		3575.1.h.f.2001.1		2
715.153	even	12		1859.1.i.b.1330.1		4
715.168	odd	20		1573.1.l.a.233.1		4
715.233	even	20		1573.1.l.b.766.1		4
715.263	even	12		1859.1.i.a.1330.2		4
715.318	odd	12		1859.1.k.c.1836.3		8
715.373	even	12		1859.1.i.a.868.2		4
715.428	even	4		143.1.d.a.142.2	✓	2
715.483	odd	12		1859.1.k.c.1374.2		8
715.493	odd	20		1573.1.l.c.766.1		4
715.538	odd	4		1859.1.c.c.846.3		4
715.558	even	20		1573.1.l.d.233.1		4
715.593	odd	4		1859.1.c.c.846.2		4
715.623	even	20		1573.1.l.d.1546.1		4
715.648	odd	12		1859.1.k.c.1374.3		8
715.688	even	20		1573.1.l.b.844.1		4
715.714	odd	2	inner	3575.1.c.d.3574.2		4
2145.428	odd	4		1287.1.g.b.1000.1		2
2860.1143	odd	4		2288.1.m.b.2001.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.1.d.a.142.2	✓	2	5.3	odd	4
143.1.d.a.142.2	✓	2	715.428	even	4
143.1.d.b.142.1	yes	2	55.43	even	4
143.1.d.b.142.1	yes	2	65.38	odd	4
1287.1.g.a.1000.2		2	165.98	odd	4
1287.1.g.a.1000.2		2	195.38	even	4
1287.1.g.b.1000.1		2	15.8	even	4
1287.1.g.b.1000.1		2	2145.428	odd	4
1573.1.l.a.233.1		4	55.8	even	20
1573.1.l.a.233.1		4	715.168	odd	20
1573.1.l.a.1546.1		4	55.18	even	20
1573.1.l.a.1546.1		4	715.103	odd	20
1573.1.l.b.766.1		4	55.53	odd	20
1573.1.l.b.766.1		4	715.233	even	20
1573.1.l.b.844.1		4	55.38	odd	20
1573.1.l.b.844.1		4	715.688	even	20
1573.1.l.c.766.1		4	55.13	even	20
1573.1.l.c.766.1		4	715.493	odd	20
1573.1.l.c.844.1		4	55.28	even	20
1573.1.l.c.844.1		4	715.38	odd	20
1573.1.l.d.233.1		4	55.3	odd	20
1573.1.l.d.233.1		4	715.558	even	20
1573.1.l.d.1546.1		4	55.48	odd	20
1573.1.l.d.1546.1		4	715.623	even	20
1859.1.c.c.846.2		4	65.18	even	4
1859.1.c.c.846.2		4	715.593	odd	4
1859.1.c.c.846.3		4	65.8	even	4
1859.1.c.c.846.3		4	715.538	odd	4
1859.1.i.a.868.2		4	65.43	odd	12
1859.1.i.a.868.2		4	715.373	even	12
1859.1.i.a.1330.2		4	65.23	odd	12
1859.1.i.a.1330.2		4	715.263	even	12
1859.1.i.b.868.1		4	65.48	odd	12
1859.1.i.b.868.1		4	715.43	even	12
1859.1.i.b.1330.1		4	65.3	odd	12
1859.1.i.b.1330.1		4	715.153	even	12
1859.1.k.c.1374.2		8	65.63	even	12
1859.1.k.c.1374.2		8	715.483	odd	12
1859.1.k.c.1374.3		8	65.28	even	12
1859.1.k.c.1374.3		8	715.648	odd	12
1859.1.k.c.1836.2		8	65.58	even	12
1859.1.k.c.1836.2		8	715.98	odd	12
1859.1.k.c.1836.3		8	65.33	even	12
1859.1.k.c.1836.3		8	715.318	odd	12
2288.1.m.a.2001.1		2	220.43	odd	4
2288.1.m.a.2001.1		2	260.103	even	4
2288.1.m.b.2001.1		2	20.3	even	4
2288.1.m.b.2001.1		2	2860.1143	odd	4
3575.1.c.c.3574.2		4	11.10	odd	2
3575.1.c.c.3574.2		4	13.12	even	2
3575.1.c.c.3574.3		4	55.54	odd	2
3575.1.c.c.3574.3		4	65.64	even	2
3575.1.c.d.3574.2		4	5.4	even	2	inner
3575.1.c.d.3574.2		4	715.714	odd	2	inner
3575.1.c.d.3574.3		4	1.1	even	1	trivial
3575.1.c.d.3574.3		4	143.142	odd	2	CM
3575.1.h.e.2001.2		2	55.32	even	4
3575.1.h.e.2001.2		2	65.12	odd	4
3575.1.h.f.2001.1		2	5.2	odd	4
3575.1.h.f.2001.1		2	715.142	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3575 = 5^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3575.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} -0.618034i q^{3} +0.618034 q^{4} +0.381966 q^{6} -1.61803i q^{7} +1.00000i q^{8} +0.618034 q^{9} +O(q^{10})\)	\(q+0.618034i q^{2} -0.618034i q^{3} +0.618034 q^{4} +0.381966 q^{6} -1.61803i q^{7} +1.00000i q^{8} +0.618034 q^{9} +1.00000 q^{11} -0.381966i q^{12} -1.00000i q^{13} +1.00000 q^{14} +0.381966i q^{18} -0.618034 q^{19} -1.00000 q^{21} +0.618034i q^{22} +1.61803i q^{23} +0.618034 q^{24} +0.618034 q^{26} -1.00000i q^{27} -1.00000i q^{28} +1.00000i q^{32} -0.618034i q^{33} +0.381966 q^{36} -0.381966i q^{38} -0.618034 q^{39} -1.61803 q^{41} -0.618034i q^{42} +0.618034 q^{44} -1.00000 q^{46} -1.61803 q^{49} -0.618034i q^{52} -0.618034i q^{53} +0.618034 q^{54} +1.61803 q^{56} +0.381966i q^{57} -1.00000i q^{63} -0.618034 q^{64} +0.381966 q^{66} +1.00000 q^{69} +0.618034i q^{72} -0.618034i q^{73} -0.381966 q^{76} -1.61803i q^{77} -0.381966i q^{78} -1.00000i q^{82} +1.61803i q^{83} -0.618034 q^{84} +1.00000i q^{88} -1.61803 q^{91} +1.00000i q^{92} +0.618034 q^{96} -1.00000i q^{98} +0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9	\( 4 q - 2 q^{4} + 6 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{19} - 4 q^{21} - 2 q^{24} - 2 q^{26} + 6 q^{36} + 2 q^{39} - 2 q^{41} - 2 q^{44} - 4 q^{46} - 2 q^{49} - 2 q^{54} + 2 q^{56} + 2 q^{64} + 6 q^{66} + 4 q^{69} - 6 q^{76} + 2 q^{84} - 2 q^{91} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^4 + 6 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^19 - 4 * q^21 - 2 * q^24 - 2 * q^26 + 6 * q^36 + 2 * q^39 - 2 * q^41 - 2 * q^44 - 4 * q^46 - 2 * q^49 - 2 * q^54 + 2 * q^56 + 2 * q^64 + 6 * q^66 + 4 * q^69 - 6 * q^76 + 2 * q^84 - 2 * q^91 - 2 * q^96 - 2 * q^99

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