LMFDB - Embedded newform 3047.1.d.b.3046.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3047,1,Mod(3046,3047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3047, 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("3047.3046");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3047 = 11 \cdot 277 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3047.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:	$1.52065109349$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			3046.3
Root			$0.803391$ of defining polynomial
Character	$\chi$	$=$	3047.3046

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09390 q^{2} -0.165159 q^{3} +0.196609 q^{4} +0.180666 q^{6} +0.878826 q^{8} -0.972723 q^{9} +O(q^{10})$	$q-1.09390 q^{2} -0.165159 q^{3} +0.196609 q^{4} +0.180666 q^{6} +0.878826 q^{8} -0.972723 q^{9} -1.00000 q^{11} -0.0324717 q^{12} -1.15795 q^{16} -1.89163 q^{17} +1.06406 q^{18} +1.09390 q^{22} +1.57828 q^{23} -0.145146 q^{24} +1.00000 q^{25} +0.325812 q^{27} +0.387855 q^{32} +0.165159 q^{33} +2.06925 q^{34} -0.191246 q^{36} +1.35456 q^{43} -0.196609 q^{44} -1.72648 q^{46} +1.09390 q^{47} +0.191246 q^{48} +1.00000 q^{49} -1.09390 q^{50} +0.312420 q^{51} -0.356405 q^{54} -1.35456 q^{59} -0.490971 q^{61} +0.733680 q^{64} -0.180666 q^{66} -1.75895 q^{67} -0.371913 q^{68} -0.260667 q^{69} +1.89163 q^{71} -0.854854 q^{72} +0.803391 q^{73} -0.165159 q^{75} +0.918912 q^{81} -1.48175 q^{86} -0.878826 q^{88} -1.75895 q^{89} +0.310304 q^{92} -1.19661 q^{94} -0.0640577 q^{96} -1.09390 q^{98} +0.972723 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} - q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{8} + 8 q^{9}+O(q^{10}) $ 9 * q + q^2 - q^3 + 8 * q^4 + 2 * q^6 + 2 * q^8 + 8 * q^9	$ 9 q + q^{2} - q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{8} + 8 q^{9} - 9 q^{11} - 3 q^{12} + 7 q^{16} + q^{17} + 3 q^{18} - q^{22} - q^{23} + 4 q^{24} + 9 q^{25} - 2 q^{27} + 3 q^{32} + q^{33} - 2 q^{34} + 5 q^{36} + q^{43} - 8 q^{44} + 2 q^{46} - q^{47} - 5 q^{48} + 9 q^{49} + q^{50} + 2 q^{51} + 4 q^{54} - q^{59} + q^{61} + 6 q^{64} - 2 q^{66} - q^{67} + 3 q^{68} - 2 q^{69} - q^{71} - 13 q^{72} + q^{73} - q^{75} + 7 q^{81} - 2 q^{86} - 2 q^{88} - q^{89} - 3 q^{92} - 17 q^{94} + 6 q^{96} + q^{98} - 8 q^{99}+O(q^{100}) $ 9 * q + q^2 - q^3 + 8 * q^4 + 2 * q^6 + 2 * q^8 + 8 * q^9 - 9 * q^11 - 3 * q^12 + 7 * q^16 + q^17 + 3 * q^18 - q^22 - q^23 + 4 * q^24 + 9 * q^25 - 2 * q^27 + 3 * q^32 + q^33 - 2 * q^34 + 5 * q^36 + q^43 - 8 * q^44 + 2 * q^46 - q^47 - 5 * q^48 + 9 * q^49 + q^50 + 2 * q^51 + 4 * q^54 - q^59 + q^61 + 6 * q^64 - 2 * q^66 - q^67 + 3 * q^68 - 2 * q^69 - q^71 - 13 * q^72 + q^73 - q^75 + 7 * q^81 - 2 * q^86 - 2 * q^88 - q^89 - 3 * q^92 - 17 * q^94 + 6 * q^96 + q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3047\mathbb{Z}\right)^\times$.

$n$	$1663$	$2498$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$2$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$3$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$3$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$4$	0.196609	0.196609
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.180666	0.180666
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.878826	0.878826
$9$	−0.972723	−0.972723
$10$	0	0
$11$	−1.00000	−1.00000
$11$	−1.00000	−1.00000
$12$	−0.0324717	−0.0324717
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−1.15795	−1.15795
$17$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$17$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$18$	1.06406	1.06406
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	1.09390	1.09390
$23$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$23$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$24$	−0.145146	−0.145146
$25$	1.00000	1.00000
$26$	0	0
$27$	0.325812	0.325812
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.387855	0.387855
$33$	0.165159	0.165159
$34$	2.06925	2.06925
$35$	0	0
$36$	−0.191246	−0.191246
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$43$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$44$	−0.196609	−0.196609
$45$	0	0
$46$	−1.72648	−1.72648
$47$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$47$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$48$	0.191246	0.191246
$49$	1.00000	1.00000
$50$	−1.09390	−1.09390
$51$	0.312420	0.312420
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−0.356405	−0.356405
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$59$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$60$	0	0
$61$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$61$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$62$	0	0
$63$	0	0
$64$	0.733680	0.733680
$65$	0	0
$66$	−0.180666	−0.180666
$67$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$67$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$68$	−0.371913	−0.371913
$69$	−0.260667	−0.260667
$70$	0	0
$71$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$71$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$72$	−0.854854	−0.854854
$73$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$73$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$74$	0	0
$75$	−0.165159	−0.165159
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0.918912	0.918912
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−1.48175	−1.48175
$87$	0	0
$88$	−0.878826	−0.878826
$89$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$89$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$90$	0	0
$91$	0	0
$92$	0.310304	0.310304
$93$	0	0
$94$	−1.19661	−1.19661
$95$	0	0
$96$	−0.0640577	−0.0640577
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.09390	−1.09390
$99$	0.972723	0.972723
$100$	0.196609	0.196609
$101$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$101$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$102$	−0.341755	−0.341755
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$107$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$108$	0.0640577	0.0640577
$109$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$109$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$113$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	1.48175	1.48175
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0.537071	0.537071
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$127$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$128$	−1.19043	−1.19043
$129$	−0.223718	−0.223718
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0.0324717	0.0324717
$133$	0	0
$134$	1.92411	1.92411
$135$	0	0
$136$	−1.66242	−1.66242
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.285142	0.285142
$139$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$139$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$140$	0	0
$141$	−0.180666	−0.180666
$142$	−2.06925	−2.06925
$143$	0	0
$144$	1.12637	1.12637
$145$	0	0
$146$	−0.878826	−0.878826
$147$	−0.165159	−0.165159
$148$	0	0
$149$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$149$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$150$	0.180666	0.180666
$151$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$151$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$152$	0	0
$153$	1.84004	1.84004
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$157$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.00519	−1.00519
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$167$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.266320	0.266320
$173$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$173$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$174$	0	0
$175$	0	0
$176$	1.15795	1.15795
$177$	0.223718	0.223718
$178$	1.92411	1.92411
$179$	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
$183$	0.0810881	0.0810881
$184$	1.38703	1.38703
$185$	0	0
$186$	0	0
$187$	1.89163	1.89163
$188$	0.215070	0.215070
$189$	0	0
$190$	0	0
$191$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$191$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$192$	−0.121174	−0.121174
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0.196609	0.196609
$197$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$197$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$198$	−1.06406	−1.06406
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.878826	0.878826
$201$	0.290505	0.290505
$202$	−1.48175	−1.48175
$203$	0	0
$204$	0.0614246	0.0614246
$205$	0	0
$206$	0	0
$207$	−1.53523	−1.53523
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−0.312420	−0.312420
$214$	−0.180666	−0.180666
$215$	0	0
$216$	0.286332	0.286332
$217$	0	0
$218$	−2.15795	−2.15795
$219$	−0.132687	−0.132687
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−0.972723	−0.972723
$226$	−0.537071	−0.537071
$227$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$227$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$228$	0	0
$229$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$229$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$233$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$234$	0	0
$235$	0	0
$236$	−0.266320	−0.266320
$237$	0	0
$238$	0	0
$239$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$239$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.09390	−1.09390
$243$	−0.477579	−0.477579
$244$	−0.0965294	−0.0965294
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	−1.57828	−1.57828
$254$	−1.92411	−1.92411
$255$	0	0
$256$	0.568522	0.568522
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.244724	0.244724
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$263$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$264$	0.145146	0.145146
$265$	0	0
$266$	0	0
$267$	0.290505	0.290505
$268$	−0.345825	−0.345825
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$271$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$272$	2.19043	2.19043
$273$	0	0
$274$	0	0
$275$	−1.00000	−1.00000
$276$	−0.0512495	−0.0512495
$277$	−1.00000	−1.00000
$277$	−1.00000	−1.00000
$278$	1.72648	1.72648
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.197630	0.197630
$283$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$283$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$284$	0.371913	0.371913
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−0.377276	−0.377276
$289$	2.57828	2.57828
$290$	0	0
$291$	0	0
$292$	0.157954	0.157954
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.180666	0.180666
$295$	0	0
$296$	0	0
$297$	−0.325812	−0.325812
$298$	−1.92411	−1.92411
$299$	0	0
$300$	−0.0324717	−0.0324717
$301$	0	0
$302$	−0.180666	−0.180666
$303$	−0.223718	−0.223718
$304$	0	0
$305$	0	0
$306$	−2.01281	−2.01281
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$311$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$312$	0	0
$313$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$313$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$314$	−2.06925	−2.06925
$315$	0	0
$316$	0	0
$317$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$317$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.0272774	−0.0272774
$322$	0	0
$323$	0	0
$324$	0.180666	0.180666
$325$	0	0
$326$	0	0
$327$	−0.325812	−0.325812
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	1.72648	1.72648
$335$	0	0
$336$	0	0
$337$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$337$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$338$	−1.09390	−1.09390
$339$	−0.0810881	−0.0810881
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	1.19043	1.19043
$345$	0	0
$346$	−0.878826	−0.878826
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$349$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$350$	0	0
$351$	0	0
$352$	−0.387855	−0.387855
$353$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$353$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$354$	−0.244724	−0.244724
$355$	0	0
$356$	−0.345825	−0.345825
$357$	0	0
$358$	0	0
$359$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$359$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	−0.165159	−0.165159
$364$	0	0
$365$	0	0
$366$	−0.0887020	−0.0887020
$367$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$367$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$368$	−1.82758	−1.82758
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$373$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$374$	−2.06925	−2.06925
$375$	0	0
$376$	0.961345	0.961345
$377$	0	0
$378$	0	0
$379$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$379$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$380$	0	0
$381$	−0.290505	−0.290505
$382$	0.878826	0.878826
$383$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$383$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$384$	0.196609	0.196609
$385$	0	0
$386$	0	0
$387$	−1.31761	−1.31761
$388$	0	0
$389$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$389$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$390$	0	0
$391$	−2.98553	−2.98553
$392$	0.878826	0.878826
$393$	0	0
$394$	1.72648	1.72648
$395$	0	0
$396$	0.191246	0.191246
$397$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$397$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$398$	0	0
$399$	0	0
$400$	−1.15795	−1.15795
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.317783	−0.317783
$403$	0	0
$404$	0.266320	0.266320
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0.274563	0.274563
$409$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$409$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.67938	1.67938
$415$	0	0
$416$	0	0
$417$	0.260667	0.260667
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$421$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$422$	0	0
$423$	−1.06406	−1.06406
$424$	0	0
$425$	−1.89163	−1.89163
$426$	0.341755	0.341755
$427$	0	0
$428$	0.0324717	0.0324717
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.377276	−0.377276
$433$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$433$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$434$	0	0
$435$	0	0
$436$	0.387855	0.387855
$437$	0	0
$438$	0.145146	0.145146
$439$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$439$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$440$	0	0
$441$	−0.972723	−0.972723
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.290505	−0.290505
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.06406	1.06406
$451$	0	0
$452$	0.0965294	0.0965294
$453$	−0.0272774	−0.0272774
$454$	−1.92411	−1.92411
$455$	0	0
$456$	0	0
$457$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$457$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$458$	−0.537071	−0.537071
$459$	−0.616318	−0.616318
$460$	0	0
$461$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$461$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$462$	0	0
$463$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$463$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$464$	0	0
$465$	0	0
$466$	0.537071	0.537071
$467$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$467$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.312420	−0.312420
$472$	−1.19043	−1.19043
$473$	−1.35456	−1.35456
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−0.878826	−0.878826
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0.196609	0.196609
$485$	0	0
$486$	0.522421	0.522421
$487$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$487$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$488$	−0.431478	−0.431478
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$499$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$500$	0	0
$501$	0.260667	0.260667
$502$	0	0
$503$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$503$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$504$	0	0
$505$	0	0
$506$	1.72648	1.72648
$507$	−0.165159	−0.165159
$508$	0.345825	0.345825
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0.568522	0.568522
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−0.0439850	−0.0439850
$517$	−1.09390	−1.09390
$518$	0	0
$519$	−0.132687	−0.132687
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$523$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$524$	0	0
$525$	0	0
$526$	0.537071	0.537071
$527$	0	0
$528$	−0.191246	−0.191246
$529$	1.49097	1.49097
$530$	0	0
$531$	1.31761	1.31761
$532$	0	0
$533$	0	0
$534$	−0.317783	−0.317783
$535$	0	0
$536$	−1.54581	−1.54581
$537$	0	0
$538$	0	0
$539$	−1.00000	−1.00000
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.15795	−2.15795
$543$	0	0
$544$	−0.733680	−0.733680
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0.477579	0.477579
$550$	1.09390	1.09390
$551$	0	0
$552$	−0.229081	−0.229081
$553$	0	0
$554$	1.09390	1.09390
$555$	0	0
$556$	−0.310304	−0.310304
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.312420	−0.312420
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−0.0355207	−0.0355207
$565$	0	0
$566$	2.06925	2.06925
$567$	0	0
$568$	1.66242	1.66242
$569$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$569$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$570$	0	0
$571$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$571$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$572$	0	0
$573$	0.132687	0.132687
$574$	0	0
$575$	1.57828	1.57828
$576$	−0.713668	−0.713668
$577$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$577$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$578$	−2.82037	−2.82037
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0.706041	0.706041
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.0324717	−0.0324717
$589$	0	0
$590$	0	0
$591$	0.260667	0.260667
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0.356405	0.356405
$595$	0	0
$596$	0.345825	0.345825
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−0.145146	−0.145146
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	1.71097	1.71097
$604$	0.0324717	0.0324717
$605$	0	0
$606$	0.244724	0.244724
$607$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$607$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0.361768	0.361768
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$617$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0.514223	0.514223
$622$	1.92411	1.92411
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	−1.19661	−1.19661
$627$	0	0
$628$	0.371913	0.371913
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0.878826	0.878826
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.84004	−1.84004
$640$	0	0
$641$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$641$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$642$	0.0298386	0.0298386
$643$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$643$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.807564	0.807564
$649$	1.35456	1.35456
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.356405	0.356405
$655$	0	0
$656$	0	0
$657$	−0.781476	−0.781476
$658$	0	0
$659$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$659$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−0.310304	−0.310304
$669$	0	0
$670$	0	0
$671$	0.490971	0.490971
$672$	0	0
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$674$	−1.92411	−1.92411
$675$	0.325812	0.325812
$676$	0.196609	0.196609
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0.0887020	0.0887020
$679$	0	0
$680$	0	0
$681$	−0.290505	−0.290505
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.0810881	−0.0810881
$688$	−1.56852	−1.56852
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0.157954	0.157954
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−1.92411	−1.92411
$699$	0.0810881	0.0810881
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−0.733680	−0.733680
$705$	0	0
$706$	0.878826	0.878826
$707$	0	0
$708$	0.0439850	0.0439850
$709$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$709$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$710$	0	0
$711$	0	0
$712$	−1.54581	−1.54581
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.132687	−0.132687
$718$	−1.48175	−1.48175
$719$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$719$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$720$	0	0
$721$	0	0
$722$	−1.09390	−1.09390
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.180666	0.180666
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.840036	−0.840036
$730$	0	0
$731$	−2.56234	−2.56234
$732$	0.0159427	0.0159427
$733$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$733$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$734$	2.15795	2.15795
$735$	0	0
$736$	0.612145	0.612145
$737$	1.75895	1.75895
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−0.180666	−0.180666
$747$	0	0
$748$	0.371913	0.371913
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1.26668	−1.26668
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$757$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$758$	−1.19661	−1.19661
$759$	0.260667	0.260667
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0.317783	0.317783
$763$	0	0
$764$	−0.157954	−0.157954
$765$	0	0
$766$	2.15795	2.15795
$767$	0	0
$768$	−0.0938963	−0.0938963
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.44133	1.44133
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−2.06925	−2.06925
$779$	0	0
$780$	0	0
$781$	−1.89163	−1.89163
$782$	3.26586	3.26586
$783$	0	0
$784$	−1.15795	−1.15795
$785$	0	0
$786$	0	0
$787$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$787$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$788$	−0.310304	−0.310304
$789$	0.0810881	0.0810881
$790$	0	0
$791$	0	0
$792$	0.854854	0.854854
$793$	0	0
$794$	−1.72648	−1.72648
$795$	0	0
$796$	0	0
$797$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$797$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$798$	0	0
$799$	−2.06925	−2.06925
$800$	0.387855	0.387855
$801$	1.71097	1.71097
$802$	0	0
$803$	−0.803391	−0.803391
$804$	0.0571160	0.0571160
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1.19043	1.19043
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$811$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$812$	0	0
$813$	−0.325812	−0.325812
$814$	0	0
$815$	0	0
$816$	−0.361768	−0.361768
$817$	0	0
$818$	1.19661	1.19661
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0.165159	0.165159
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−0.301840	−0.301840
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0.165159	0.165159
$832$	0	0
$833$	−1.89163	−1.89163
$834$	−0.285142	−0.285142
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	1.92411	1.92411
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1.16397	1.16397
$847$	0	0
$848$	0	0
$849$	0.312420	0.312420
$850$	2.06925	2.06925
$851$	0	0
$852$	−0.0614246	−0.0614246
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0.145146	0.145146
$857$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$857$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$858$	0	0
$859$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$859$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.126368	0.126368
$865$	0	0
$866$	1.48175	1.48175
$867$	−0.425826	−0.425826
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1.73368	1.73368
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−0.0260875	−0.0260875
$877$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$877$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$878$	−0.878826	−0.878826
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.06406	1.06406
$883$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$883$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$887$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.918912	−0.918912
$892$	0	0
$893$	0	0
$894$	0.317783	0.317783
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.191246	−0.191246
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0.431478	0.431478
$905$	0	0
$906$	0.0298386	0.0298386
$907$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$907$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$908$	0.345825	0.345825
$909$	−1.31761	−1.31761
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	1.72648	1.72648
$915$	0	0
$916$	0.0965294	0.0965294
$917$	0	0
$918$	0.674188	0.674188
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−0.180666	−0.180666
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.48175	1.48175
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.0965294	−0.0965294
$933$	0.290505	0.290505
$934$	2.15795	2.15795
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−0.180666	−0.180666
$940$	0	0
$941$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$941$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$942$	0.341755	0.341755
$943$	0	0
$944$	1.56852	1.56852
$945$	0	0
$946$	1.48175	1.48175
$947$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$947$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.132687	0.132687
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0.157954	0.157954
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	−0.160654	−0.160654
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.878826	0.878826
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.0938963	−0.0938963
$973$	0	0
$974$	−0.537071	−0.537071
$975$	0	0
$976$	0.568522	0.568522
$977$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$977$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$978$	0	0
$979$	1.75895	1.75895
$980$	0	0
$981$	−1.91891	−1.91891
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.13788	2.13788
$990$	0	0
$991$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$991$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$997$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$998$	2.15795	2.15795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3047.1.d.b.3046.3	yes	9
11.10	odd	2		3047.1.d.a.3046.7	✓	9
277.276	even	2		3047.1.d.a.3046.7	✓	9
3047.3046	odd	2	CM	3047.1.d.b.3046.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3047.1.d.a.3046.7	✓	9	11.10	odd	2
3047.1.d.a.3046.7	✓	9	277.276	even	2
3047.1.d.b.3046.3	yes	9	1.1	even	1	trivial
3047.1.d.b.3046.3	yes	9	3047.3046	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:	\( N \)	\(=\)	\( 3047 = 11 \cdot 277 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3047.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.09390 q^{2} -0.165159 q^{3} +0.196609 q^{4} +0.180666 q^{6} +0.878826 q^{8} -0.972723 q^{9} +O(q^{10})\)	\(q-1.09390 q^{2} -0.165159 q^{3} +0.196609 q^{4} +0.180666 q^{6} +0.878826 q^{8} -0.972723 q^{9} -1.00000 q^{11} -0.0324717 q^{12} -1.15795 q^{16} -1.89163 q^{17} +1.06406 q^{18} +1.09390 q^{22} +1.57828 q^{23} -0.145146 q^{24} +1.00000 q^{25} +0.325812 q^{27} +0.387855 q^{32} +0.165159 q^{33} +2.06925 q^{34} -0.191246 q^{36} +1.35456 q^{43} -0.196609 q^{44} -1.72648 q^{46} +1.09390 q^{47} +0.191246 q^{48} +1.00000 q^{49} -1.09390 q^{50} +0.312420 q^{51} -0.356405 q^{54} -1.35456 q^{59} -0.490971 q^{61} +0.733680 q^{64} -0.180666 q^{66} -1.75895 q^{67} -0.371913 q^{68} -0.260667 q^{69} +1.89163 q^{71} -0.854854 q^{72} +0.803391 q^{73} -0.165159 q^{75} +0.918912 q^{81} -1.48175 q^{86} -0.878826 q^{88} -1.75895 q^{89} +0.310304 q^{92} -1.19661 q^{94} -0.0640577 q^{96} -1.09390 q^{98} +0.972723 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) 9 * q + q^2 - q^3 + 8 * q^4 + 2 * q^6 + 2 * q^8 + 8 * q^9	\( 9 q + q^{2} - q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{8} + 8 q^{9} - 9 q^{11} - 3 q^{12} + 7 q^{16} + q^{17} + 3 q^{18} - q^{22} - q^{23} + 4 q^{24} + 9 q^{25} - 2 q^{27} + 3 q^{32} + q^{33} - 2 q^{34} + 5 q^{36} + q^{43} - 8 q^{44} + 2 q^{46} - q^{47} - 5 q^{48} + 9 q^{49} + q^{50} + 2 q^{51} + 4 q^{54} - q^{59} + q^{61} + 6 q^{64} - 2 q^{66} - q^{67} + 3 q^{68} - 2 q^{69} - q^{71} - 13 q^{72} + q^{73} - q^{75} + 7 q^{81} - 2 q^{86} - 2 q^{88} - q^{89} - 3 q^{92} - 17 q^{94} + 6 q^{96} + q^{98} - 8 q^{99}+O(q^{100}) \) 9 * q + q^2 - q^3 + 8 * q^4 + 2 * q^6 + 2 * q^8 + 8 * q^9 - 9 * q^11 - 3 * q^12 + 7 * q^16 + q^17 + 3 * q^18 - q^22 - q^23 + 4 * q^24 + 9 * q^25 - 2 * q^27 + 3 * q^32 + q^33 - 2 * q^34 + 5 * q^36 + q^43 - 8 * q^44 + 2 * q^46 - q^47 - 5 * q^48 + 9 * q^49 + q^50 + 2 * q^51 + 4 * q^54 - q^59 + q^61 + 6 * q^64 - 2 * q^66 - q^67 + 3 * q^68 - 2 * q^69 - q^71 - 13 * q^72 + q^73 - q^75 + 7 * q^81 - 2 * q^86 - 2 * q^88 - q^89 - 3 * q^92 - 17 * q^94 + 6 * q^96 + q^98 - 8 * q^99

Introduction

L-functions

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Varieties

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Database

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