LMFDB - Embedded newform 447.1.d.b.446.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [447,1,Mod(446,447)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(447, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("447.446");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 447 = 3 \cdot 149 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	447.d (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:	$0.223082060647$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.89314623.1
Artin image:	$D_{14}$
Artin field:	Galois closure of 14.2.1188588180363187221.1

Embedding invariants

Embedding label			446.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	447.446

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{2} -1.00000 q^{3} -0.801938 q^{4} -0.445042 q^{6} +1.24698 q^{7} -0.801938 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+0.445042 q^{2} -1.00000 q^{3} -0.801938 q^{4} -0.445042 q^{6} +1.24698 q^{7} -0.801938 q^{8} +1.00000 q^{9} +1.80194 q^{11} +0.801938 q^{12} +0.554958 q^{14} +0.445042 q^{16} +0.445042 q^{18} -0.445042 q^{19} -1.24698 q^{21} +0.801938 q^{22} -1.24698 q^{23} +0.801938 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{33} -0.801938 q^{36} -0.445042 q^{37} -0.198062 q^{38} +1.80194 q^{41} -0.554958 q^{42} -1.44504 q^{44} -0.554958 q^{46} -0.445042 q^{48} +0.554958 q^{49} +0.445042 q^{50} -0.445042 q^{54} -1.00000 q^{56} +0.445042 q^{57} +0.445042 q^{59} -1.80194 q^{61} -0.801938 q^{62} +1.24698 q^{63} -0.801938 q^{66} -1.80194 q^{67} +1.24698 q^{69} +0.445042 q^{71} -0.801938 q^{72} +1.24698 q^{73} -0.198062 q^{74} -1.00000 q^{75} +0.356896 q^{76} +2.24698 q^{77} +1.00000 q^{81} +0.801938 q^{82} -1.24698 q^{83} +1.00000 q^{84} -1.44504 q^{88} -1.24698 q^{89} +1.00000 q^{92} +1.80194 q^{93} -1.00000 q^{96} +0.246980 q^{98} +1.80194 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 - 3 * q^3 + 2 * q^4 - q^6 - q^7 + 2 * q^8 + 3 * q^9	$ 3 q + q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - q^{7} + 2 q^{8} + 3 q^{9} + q^{11} - 2 q^{12} + 2 q^{14} + q^{16} + q^{18} - q^{19} + q^{21} - 2 q^{22} + q^{23} - 2 q^{24} + 3 q^{25} - 3 q^{27} - 3 q^{28} - q^{31} + 3 q^{32} - q^{33} + 2 q^{36} - q^{37} - 5 q^{38} + q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{46} - q^{48} + 2 q^{49} + q^{50} - q^{54} - 3 q^{56} + q^{57} + q^{59} - q^{61} + 2 q^{62} - q^{63} + 2 q^{66} - q^{67} - q^{69} + q^{71} + 2 q^{72} - q^{73} - 5 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{77} + 3 q^{81} - 2 q^{82} + q^{83} + 3 q^{84} - 4 q^{88} + q^{89} + 3 q^{92} + q^{93} - 3 q^{96} - 4 q^{98} + q^{99}+O(q^{100}) $ 3 * q + q^2 - 3 * q^3 + 2 * q^4 - q^6 - q^7 + 2 * q^8 + 3 * q^9 + q^11 - 2 * q^12 + 2 * q^14 + q^16 + q^18 - q^19 + q^21 - 2 * q^22 + q^23 - 2 * q^24 + 3 * q^25 - 3 * q^27 - 3 * q^28 - q^31 + 3 * q^32 - q^33 + 2 * q^36 - q^37 - 5 * q^38 + q^41 - 2 * q^42 - 4 * q^44 - 2 * q^46 - q^48 + 2 * q^49 + q^50 - q^54 - 3 * q^56 + q^57 + q^59 - q^61 + 2 * q^62 - q^63 + 2 * q^66 - q^67 - q^69 + q^71 + 2 * q^72 - q^73 - 5 * q^74 - 3 * q^75 - 3 * q^76 + 2 * q^77 + 3 * q^81 - 2 * q^82 + q^83 + 3 * q^84 - 4 * q^88 + q^89 + 3 * q^92 + q^93 - 3 * q^96 - 4 * q^98 + q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/447\mathbb{Z}\right)^\times$.

$n$	$151$	$299$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$2$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	−0.801938	−0.801938
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−0.445042	−0.445042
$7$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$7$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$8$	−0.801938	−0.801938
$9$	1.00000	1.00000
$10$	0	0
$11$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$11$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$12$	0.801938	0.801938
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.554958	0.554958
$15$	0	0
$16$	0.445042	0.445042
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.445042	0.445042
$19$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$19$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$20$	0	0
$21$	−1.24698	−1.24698
$22$	0.801938	0.801938
$23$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$23$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$24$	0.801938	0.801938
$25$	1.00000	1.00000
$26$	0	0
$27$	−1.00000	−1.00000
$28$	−1.00000	−1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$31$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$32$	1.00000	1.00000
$33$	−1.80194	−1.80194
$34$	0	0
$35$	0	0
$36$	−0.801938	−0.801938
$37$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$37$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$38$	−0.198062	−0.198062
$39$	0	0
$40$	0	0
$41$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$41$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$42$	−0.554958	−0.554958
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−1.44504	−1.44504
$45$	0	0
$46$	−0.554958	−0.554958
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.445042	−0.445042
$49$	0.554958	0.554958
$50$	0.445042	0.445042
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−0.445042	−0.445042
$55$	0	0
$56$	−1.00000	−1.00000
$57$	0.445042	0.445042
$58$	0	0
$59$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$59$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$60$	0	0
$61$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$62$	−0.801938	−0.801938
$63$	1.24698	1.24698
$64$	0	0
$65$	0	0
$66$	−0.801938	−0.801938
$67$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$67$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$68$	0	0
$69$	1.24698	1.24698
$70$	0	0
$71$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$71$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$72$	−0.801938	−0.801938
$73$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$73$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$74$	−0.198062	−0.198062
$75$	−1.00000	−1.00000
$76$	0.356896	0.356896
$77$	2.24698	2.24698
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	0.801938	0.801938
$83$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$83$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$84$	1.00000	1.00000
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−1.44504	−1.44504
$89$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$89$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	1.00000
$93$	1.80194	1.80194
$94$	0	0
$95$	0	0
$96$	−1.00000	−1.00000
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.246980	0.246980
$99$	1.80194	1.80194
$100$	−0.801938	−0.801938
$101$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$101$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$102$	0	0
$103$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$103$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.801938	0.801938
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0.445042	0.445042
$112$	0.554958	0.554958
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.198062	0.198062
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0.198062	0.198062
$119$	0	0
$120$	0	0
$121$	2.24698	2.24698
$122$	−0.801938	−0.801938
$123$	−1.80194	−1.80194
$124$	1.44504	1.44504
$125$	0	0
$126$	0.554958	0.554958
$127$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$127$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	0	0
$131$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$131$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$132$	1.44504	1.44504
$133$	−0.554958	−0.554958
$134$	−0.801938	−0.801938
$135$	0	0
$136$	0	0
$137$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$137$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$138$	0.554958	0.554958
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0.198062	0.198062
$143$	0	0
$144$	0.445042	0.445042
$145$	0	0
$146$	0.554958	0.554958
$147$	−0.554958	−0.554958
$148$	0.356896	0.356896
$149$	−1.00000	−1.00000
$149$	−1.00000	−1.00000
$150$	−0.445042	−0.445042
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.356896	0.356896
$153$	0	0
$154$	1.00000	1.00000
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.55496	−1.55496
$162$	0.445042	0.445042
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−1.44504	−1.44504
$165$	0	0
$166$	−0.554958	−0.554958
$167$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$167$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$168$	1.00000	1.00000
$169$	1.00000	1.00000
$170$	0	0
$171$	−0.445042	−0.445042
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.24698	1.24698
$176$	0.801938	0.801938
$177$	−0.445042	−0.445042
$178$	−0.554958	−0.554958
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	1.80194	1.80194
$184$	1.00000	1.00000
$185$	0	0
$186$	0.801938	0.801938
$187$	0	0
$188$	0	0
$189$	−1.24698	−1.24698
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−0.445042	−0.445042
$197$	−2.00000	−2.00000	−1.00000			$\pi$
$197$	−2.00000	−2.00000	−1.00000			$\pi$
$198$	0.801938	0.801938
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.801938	−0.801938
$201$	1.80194	1.80194
$202$	0.198062	0.198062
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−0.198062	−0.198062
$207$	−1.24698	−1.24698
$208$	0	0
$209$	−0.801938	−0.801938
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−0.445042	−0.445042
$214$	0	0
$215$	0	0
$216$	0.801938	0.801938
$217$	−2.24698	−2.24698
$218$	0	0
$219$	−1.24698	−1.24698
$220$	0	0
$221$	0	0
$222$	0.198062	0.198062
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.24698	1.24698
$225$	1.00000	1.00000
$226$	0	0
$227$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$227$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$228$	−0.356896	−0.356896
$229$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$229$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$230$	0	0
$231$	−2.24698	−2.24698
$232$	0	0
$233$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$233$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$234$	0	0
$235$	0	0
$236$	−0.356896	−0.356896
$237$	0	0
$238$	0	0
$239$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$239$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000	1.00000
$243$	−1.00000	−1.00000
$244$	1.44504	1.44504
$245$	0	0
$246$	−0.801938	−0.801938
$247$	0	0
$248$	1.44504	1.44504
$249$	1.24698	1.24698
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.00000	−1.00000
$253$	−2.24698	−2.24698
$254$	−0.198062	−0.198062
$255$	0	0
$256$	−0.445042	−0.445042
$257$	−2.00000	−2.00000	−1.00000			$\pi$
$257$	−2.00000	−2.00000	−1.00000			$\pi$
$258$	0	0
$259$	−0.554958	−0.554958
$260$	0	0
$261$	0	0
$262$	−0.554958	−0.554958
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.44504	1.44504
$265$	0	0
$266$	−0.246980	−0.246980
$267$	1.24698	1.24698
$268$	1.44504	1.44504
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−0.554958	−0.554958
$275$	1.80194	1.80194
$276$	−1.00000	−1.00000
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.80194	−1.80194
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−0.356896	−0.356896
$285$	0	0
$286$	0	0
$287$	2.24698	2.24698
$288$	1.00000	1.00000
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	−1.00000	−1.00000
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.246980	−0.246980
$295$	0	0
$296$	0.356896	0.356896
$297$	−1.80194	−1.80194
$298$	−0.445042	−0.445042
$299$	0	0
$300$	0.801938	0.801938
$301$	0	0
$302$	0	0
$303$	−0.445042	−0.445042
$304$	−0.198062	−0.198062
$305$	0	0
$306$	0	0
$307$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$307$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$308$	−1.80194	−1.80194
$309$	0.445042	0.445042
$310$	0	0
$311$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$311$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−0.692021	−0.692021
$323$	0	0
$324$	−0.801938	−0.801938
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−1.44504	−1.44504
$329$	0	0
$330$	0	0
$331$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$331$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$332$	1.00000	1.00000
$333$	−0.445042	−0.445042
$334$	0.801938	0.801938
$335$	0	0
$336$	−0.554958	−0.554958
$337$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$338$	0.445042	0.445042
$339$	0	0
$340$	0	0
$341$	−3.24698	−3.24698
$342$	−0.198062	−0.198062
$343$	−0.554958	−0.554958
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0.554958	0.554958
$351$	0	0
$352$	1.80194	1.80194
$353$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$353$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$354$	−0.198062	−0.198062
$355$	0	0
$356$	1.00000	1.00000
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−0.801938	−0.801938
$362$	0	0
$363$	−2.24698	−2.24698
$364$	0	0
$365$	0	0
$366$	0.801938	0.801938
$367$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$367$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$368$	−0.554958	−0.554958
$369$	1.80194	1.80194
$370$	0	0
$371$	0	0
$372$	−1.44504	−1.44504
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	−0.554958	−0.554958
$379$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$380$	0	0
$381$	0.445042	0.445042
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$389$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$390$	0	0
$391$	0	0
$392$	−0.445042	−0.445042
$393$	1.24698	1.24698
$394$	−0.890084	−0.890084
$395$	0	0
$396$	−1.44504	−1.44504
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0.554958	0.554958
$400$	0.445042	0.445042
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0.801938	0.801938
$403$	0	0
$404$	−0.356896	−0.356896
$405$	0	0
$406$	0	0
$407$	−0.801938	−0.801938
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	1.24698	1.24698
$412$	0.356896	0.356896
$413$	0.554958	0.554958
$414$	−0.554958	−0.554958
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−0.356896	−0.356896
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$421$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	−0.198062	−0.198062
$427$	−2.24698	−2.24698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.445042	−0.445042
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−1.00000	−1.00000
$435$	0	0
$436$	0	0
$437$	0.554958	0.554958
$438$	−0.554958	−0.554958
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0.554958	0.554958
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−0.356896	−0.356896
$445$	0	0
$446$	0	0
$447$	1.00000	1.00000
$448$	0	0
$449$	−2.00000	−2.00000	−1.00000			$\pi$
$449$	−2.00000	−2.00000	−1.00000			$\pi$
$450$	0.445042	0.445042
$451$	3.24698	3.24698
$452$	0	0
$453$	0	0
$454$	0.198062	0.198062
$455$	0	0
$456$	−0.356896	−0.356896
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.554958	0.554958
$459$	0	0
$460$	0	0
$461$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$461$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$462$	−1.00000	−1.00000
$463$	2.00000	2.00000	1.00000			$0$
$463$	2.00000	2.00000	1.00000			$0$
$464$	0	0
$465$	0	0
$466$	0.801938	0.801938
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−2.24698	−2.24698
$470$	0	0
$471$	0	0
$472$	−0.356896	−0.356896
$473$	0	0
$474$	0	0
$475$	−0.445042	−0.445042
$476$	0	0
$477$	0	0
$478$	0.801938	0.801938
$479$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$479$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.55496	1.55496
$484$	−1.80194	−1.80194
$485$	0	0
$486$	−0.445042	−0.445042
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.44504	1.44504
$489$	0	0
$490$	0	0
$491$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$491$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$492$	1.44504	1.44504
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−0.801938	−0.801938
$497$	0.554958	0.554958
$498$	0.554958	0.554958
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1.80194	−1.80194
$502$	0	0
$503$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$503$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$504$	−1.00000	−1.00000
$505$	0	0
$506$	−1.00000	−1.00000
$507$	−1.00000	−1.00000
$508$	0.356896	0.356896
$509$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$509$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$510$	0	0
$511$	1.55496	1.55496
$512$	0.801938	0.801938
$513$	0.445042	0.445042
$514$	−0.890084	−0.890084
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−0.246980	−0.246980
$519$	0	0
$520$	0	0
$521$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$521$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$522$	0	0
$523$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$523$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$524$	1.00000	1.00000
$525$	−1.24698	−1.24698
$526$	0	0
$527$	0	0
$528$	−0.801938	−0.801938
$529$	0.554958	0.554958
$530$	0	0
$531$	0.445042	0.445042
$532$	0.445042	0.445042
$533$	0	0
$534$	0.554958	0.554958
$535$	0	0
$536$	1.44504	1.44504
$537$	0	0
$538$	0	0
$539$	1.00000	1.00000
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.00000	2.00000	1.00000			$0$
$547$	2.00000	2.00000	1.00000			$0$
$548$	1.00000	1.00000
$549$	−1.80194	−1.80194
$550$	0.801938	0.801938
$551$	0	0
$552$	−1.00000	−1.00000
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−0.801938	−0.801938
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.24698	1.24698
$568$	−0.356896	−0.356896
$569$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$569$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$570$	0	0
$571$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$571$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$572$	0	0
$573$	0	0
$574$	1.00000	1.00000
$575$	−1.24698	−1.24698
$576$	0	0
$577$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$577$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$578$	0.445042	0.445042
$579$	0	0
$580$	0	0
$581$	−1.55496	−1.55496
$582$	0	0
$583$	0	0
$584$	−1.00000	−1.00000
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.445042	0.445042
$589$	0.801938	0.801938
$590$	0	0
$591$	2.00000	2.00000
$592$	−0.198062	−0.198062
$593$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$593$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$594$	−0.801938	−0.801938
$595$	0	0
$596$	0.801938	0.801938
$597$	0	0
$598$	0	0
$599$	−2.00000	−2.00000	−1.00000			$\pi$
$599$	−2.00000	−2.00000	−1.00000			$\pi$
$600$	0.801938	0.801938
$601$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$601$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$602$	0	0
$603$	−1.80194	−1.80194
$604$	0	0
$605$	0	0
$606$	−0.198062	−0.198062
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−0.445042	−0.445042
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$613$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$614$	0.554958	0.554958
$615$	0	0
$616$	−1.80194	−1.80194
$617$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$617$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$618$	0.198062	0.198062
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	1.24698	1.24698
$622$	−0.554958	−0.554958
$623$	−1.55496	−1.55496
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0.801938	0.801938
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$631$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.445042	0.445042
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	2.00000	2.00000	1.00000			$0$
$643$	2.00000	2.00000	1.00000			$0$
$644$	1.24698	1.24698
$645$	0	0
$646$	0	0
$647$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$647$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$648$	−0.801938	−0.801938
$649$	0.801938	0.801938
$650$	0	0
$651$	2.24698	2.24698
$652$	0	0
$653$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$653$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$654$	0	0
$655$	0	0
$656$	0.801938	0.801938
$657$	1.24698	1.24698
$658$	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			$\pi$
$662$	0.554958	0.554958
$663$	0	0
$664$	1.00000	1.00000
$665$	0	0
$666$	−0.198062	−0.198062
$667$	0	0
$668$	−1.44504	−1.44504
$669$	0	0
$670$	0	0
$671$	−3.24698	−3.24698
$672$	−1.24698	−1.24698
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−0.801938	−0.801938
$675$	−1.00000	−1.00000
$676$	−0.801938	−0.801938
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.445042	−0.445042
$682$	−1.44504	−1.44504
$683$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$683$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$684$	0.356896	0.356896
$685$	0	0
$686$	−0.246980	−0.246980
$687$	−1.24698	−1.24698
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$691$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$692$	0	0
$693$	2.24698	2.24698
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.80194	−1.80194
$700$	−1.00000	−1.00000
$701$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$701$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$702$	0	0
$703$	0.198062	0.198062
$704$	0	0
$705$	0	0
$706$	−0.554958	−0.554958
$707$	0.554958	0.554958
$708$	0.356896	0.356896
$709$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$709$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$710$	0	0
$711$	0	0
$712$	1.00000	1.00000
$713$	2.24698	2.24698
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.80194	−1.80194
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−0.554958	−0.554958
$722$	−0.356896	−0.356896
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−1.00000	−1.00000
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	−1.44504	−1.44504
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−0.801938	−0.801938
$735$	0	0
$736$	−1.24698	−1.24698
$737$	−3.24698	−3.24698
$738$	0.801938	0.801938
$739$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$739$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$743$	−2.00000	−2.00000	−1.00000			$\pi$
$744$	−1.44504	−1.44504
$745$	0	0
$746$	0	0
$747$	−1.24698	−1.24698
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$751$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.00000	1.00000
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−0.198062	−0.198062
$759$	2.24698	2.24698
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0.198062	0.198062
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.445042	0.445042
$769$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$769$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$770$	0	0
$771$	2.00000	2.00000
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−1.80194	−1.80194
$776$	0	0
$777$	0.554958	0.554958
$778$	0.801938	0.801938
$779$	−0.801938	−0.801938
$780$	0	0
$781$	0.801938	0.801938
$782$	0	0
$783$	0	0
$784$	0.246980	0.246980
$785$	0	0
$786$	0.554958	0.554958
$787$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$787$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$788$	1.60388	1.60388
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.44504	−1.44504
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$797$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$798$	0.246980	0.246980
$799$	0	0
$800$	1.00000	1.00000
$801$	−1.24698	−1.24698
$802$	0	0
$803$	2.24698	2.24698
$804$	−1.44504	−1.44504
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−0.356896	−0.356896
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	−0.356896	−0.356896
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0.554958	0.554958
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0.356896	0.356896
$825$	−1.80194	−1.80194
$826$	0.246980	0.246980
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.00000	1.00000
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0.643104	0.643104
$837$	1.80194	1.80194
$838$	0	0
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$839$	−2.00000	−2.00000	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.554958	0.554958
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.80194	2.80194
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.554958	0.554958
$852$	0.356896	0.356896
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	−1.00000	−1.00000
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$859$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$860$	0	0
$861$	−2.24698	−2.24698
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.00000	−1.00000
$865$	0	0
$866$	0	0
$867$	−1.00000	−1.00000
$868$	1.80194	1.80194
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0.246980	0.246980
$875$	0	0
$876$	1.00000	1.00000
$877$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$877$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$881$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$882$	0.246980	0.246980
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.356896	−0.356896
$889$	−0.554958	−0.554958
$890$	0	0
$891$	1.80194	1.80194
$892$	0	0
$893$	0	0
$894$	0.445042	0.445042
$895$	0	0
$896$	−1.24698	−1.24698
$897$	0	0
$898$	−0.890084	−0.890084
$899$	0	0
$900$	−0.801938	−0.801938
$901$	0	0
$902$	1.44504	1.44504
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.356896	−0.356896
$909$	0.445042	0.445042
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.198062	0.198062
$913$	−2.24698	−2.24698
$914$	0	0
$915$	0	0
$916$	−1.00000	−1.00000
$917$	−1.55496	−1.55496
$918$	0	0
$919$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$919$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$920$	0	0
$921$	−1.24698	−1.24698
$922$	0.801938	0.801938
$923$	0	0
$924$	1.80194	1.80194
$925$	−0.445042	−0.445042
$926$	0.890084	0.890084
$927$	−0.445042	−0.445042
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.246980	−0.246980
$932$	−1.44504	−1.44504
$933$	1.24698	1.24698
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−1.00000	−1.00000
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−2.24698	−2.24698
$944$	0.198062	0.198062
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−0.198062	−0.198062
$951$	0	0
$952$	0	0
$953$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$953$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$954$	0	0
$955$	0	0
$956$	−1.44504	−1.44504
$957$	0	0
$958$	0.801938	0.801938
$959$	−1.55496	−1.55496
$960$	0	0
$961$	2.24698	2.24698
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0.692021	0.692021
$967$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$967$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$968$	−1.80194	−1.80194
$969$	0	0
$970$	0	0
$971$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$971$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$972$	0.801938	0.801938
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−0.801938	−0.801938
$977$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$977$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$978$	0	0
$979$	−2.24698	−2.24698
$980$	0	0
$981$	0	0
$982$	0.198062	0.198062
$983$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$983$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$984$	1.44504	1.44504
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−1.80194	−1.80194
$993$	−1.24698	−1.24698
$994$	0.246980	0.246980
$995$	0	0
$996$	−1.00000	−1.00000
$997$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$997$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$998$	0	0
$999$	0.445042	0.445042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	447.1.d.b.446.2	yes	3
3.2	odd	2		447.1.d.a.446.2	✓	3
149.148	even	2		447.1.d.a.446.2	✓	3
447.446	odd	2	CM	447.1.d.b.446.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
447.1.d.a.446.2	✓	3	3.2	odd	2
447.1.d.a.446.2	✓	3	149.148	even	2
447.1.d.b.446.2	yes	3	1.1	even	1	trivial
447.1.d.b.446.2	yes	3	447.446	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 447 = 3 \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	447.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} -1.00000 q^{3} -0.801938 q^{4} -0.445042 q^{6} +1.24698 q^{7} -0.801938 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.445042 q^{2} -1.00000 q^{3} -0.801938 q^{4} -0.445042 q^{6} +1.24698 q^{7} -0.801938 q^{8} +1.00000 q^{9} +1.80194 q^{11} +0.801938 q^{12} +0.554958 q^{14} +0.445042 q^{16} +0.445042 q^{18} -0.445042 q^{19} -1.24698 q^{21} +0.801938 q^{22} -1.24698 q^{23} +0.801938 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -1.80194 q^{31} +1.00000 q^{32} -1.80194 q^{33} -0.801938 q^{36} -0.445042 q^{37} -0.198062 q^{38} +1.80194 q^{41} -0.554958 q^{42} -1.44504 q^{44} -0.554958 q^{46} -0.445042 q^{48} +0.554958 q^{49} +0.445042 q^{50} -0.445042 q^{54} -1.00000 q^{56} +0.445042 q^{57} +0.445042 q^{59} -1.80194 q^{61} -0.801938 q^{62} +1.24698 q^{63} -0.801938 q^{66} -1.80194 q^{67} +1.24698 q^{69} +0.445042 q^{71} -0.801938 q^{72} +1.24698 q^{73} -0.198062 q^{74} -1.00000 q^{75} +0.356896 q^{76} +2.24698 q^{77} +1.00000 q^{81} +0.801938 q^{82} -1.24698 q^{83} +1.00000 q^{84} -1.44504 q^{88} -1.24698 q^{89} +1.00000 q^{92} +1.80194 q^{93} -1.00000 q^{96} +0.246980 q^{98} +1.80194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 - 3 * q^3 + 2 * q^4 - q^6 - q^7 + 2 * q^8 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - q^{7} + 2 q^{8} + 3 q^{9} + q^{11} - 2 q^{12} + 2 q^{14} + q^{16} + q^{18} - q^{19} + q^{21} - 2 q^{22} + q^{23} - 2 q^{24} + 3 q^{25} - 3 q^{27} - 3 q^{28} - q^{31} + 3 q^{32} - q^{33} + 2 q^{36} - q^{37} - 5 q^{38} + q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{46} - q^{48} + 2 q^{49} + q^{50} - q^{54} - 3 q^{56} + q^{57} + q^{59} - q^{61} + 2 q^{62} - q^{63} + 2 q^{66} - q^{67} - q^{69} + q^{71} + 2 q^{72} - q^{73} - 5 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{77} + 3 q^{81} - 2 q^{82} + q^{83} + 3 q^{84} - 4 q^{88} + q^{89} + 3 q^{92} + q^{93} - 3 q^{96} - 4 q^{98} + q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 + 2 * q^4 - q^6 - q^7 + 2 * q^8 + 3 * q^9 + q^11 - 2 * q^12 + 2 * q^14 + q^16 + q^18 - q^19 + q^21 - 2 * q^22 + q^23 - 2 * q^24 + 3 * q^25 - 3 * q^27 - 3 * q^28 - q^31 + 3 * q^32 - q^33 + 2 * q^36 - q^37 - 5 * q^38 + q^41 - 2 * q^42 - 4 * q^44 - 2 * q^46 - q^48 + 2 * q^49 + q^50 - q^54 - 3 * q^56 + q^57 + q^59 - q^61 + 2 * q^62 - q^63 + 2 * q^66 - q^67 - q^69 + q^71 + 2 * q^72 - q^73 - 5 * q^74 - 3 * q^75 - 3 * q^76 + 2 * q^77 + 3 * q^81 - 2 * q^82 + q^83 + 3 * q^84 - 4 * q^88 + q^89 + 3 * q^92 + q^93 - 3 * q^96 - 4 * q^98 + q^99

Introduction

L-functions

Modular forms

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