LMFDB - Embedded newform 3047.1.d.b.3046.7

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			3046.7
Root			$0.165159$ 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)

$f(q)$	$=$	$q+1.35456 q^{2} +1.89163 q^{3} +0.834841 q^{4} +2.56234 q^{6} -0.223718 q^{8} +2.57828 q^{9} +O(q^{10})$	$q+1.35456 q^{2} +1.89163 q^{3} +0.834841 q^{4} +2.56234 q^{6} -0.223718 q^{8} +2.57828 q^{9} -1.00000 q^{11} +1.57921 q^{12} -1.13788 q^{16} -0.490971 q^{17} +3.49244 q^{18} -1.35456 q^{22} -1.75895 q^{23} -0.423192 q^{24} +1.00000 q^{25} +2.98553 q^{27} -1.31761 q^{32} -1.89163 q^{33} -0.665051 q^{34} +2.15246 q^{36} +1.97272 q^{43} -0.834841 q^{44} -2.38261 q^{46} -1.35456 q^{47} -2.15246 q^{48} +1.00000 q^{49} +1.35456 q^{50} -0.928738 q^{51} +4.04409 q^{54} -1.97272 q^{59} -1.09390 q^{61} -0.646910 q^{64} -2.56234 q^{66} -0.803391 q^{67} -0.409883 q^{68} -3.32729 q^{69} +0.490971 q^{71} -0.576808 q^{72} +0.165159 q^{73} +1.89163 q^{75} +3.06925 q^{81} +2.67218 q^{86} +0.223718 q^{88} -0.803391 q^{89} -1.46844 q^{92} -1.83484 q^{94} -2.49244 q^{96} +1.35456 q^{98} -2.57828 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$2$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$3$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$3$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$4$	0.834841	0.834841
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	2.56234	2.56234
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.223718	−0.223718
$9$	2.57828	2.57828
$10$	0	0
$11$	−1.00000	−1.00000
$11$	−1.00000	−1.00000
$12$	1.57921	1.57921
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−1.13788	−1.13788
$17$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$17$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$18$	3.49244	3.49244
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	−1.35456	−1.35456
$23$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$23$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$24$	−0.423192	−0.423192
$25$	1.00000	1.00000
$26$	0	0
$27$	2.98553	2.98553
$28$	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			$\pi$
$32$	−1.31761	−1.31761
$33$	−1.89163	−1.89163
$34$	−0.665051	−0.665051
$35$	0	0
$36$	2.15246	2.15246
$37$	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
$43$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$43$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$44$	−0.834841	−0.834841
$45$	0	0
$46$	−2.38261	−2.38261
$47$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$47$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$48$	−2.15246	−2.15246
$49$	1.00000	1.00000
$50$	1.35456	1.35456
$51$	−0.928738	−0.928738
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	4.04409	4.04409
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$59$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$60$	0	0
$61$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$61$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$62$	0	0
$63$	0	0
$64$	−0.646910	−0.646910
$65$	0	0
$66$	−2.56234	−2.56234
$67$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$67$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$68$	−0.409883	−0.409883
$69$	−3.32729	−3.32729
$70$	0	0
$71$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$71$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$72$	−0.576808	−0.576808
$73$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$73$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$74$	0	0
$75$	1.89163	1.89163
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	3.06925	3.06925
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	2.67218	2.67218
$87$	0	0
$88$	0.223718	0.223718
$89$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$89$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$90$	0	0
$91$	0	0
$92$	−1.46844	−1.46844
$93$	0	0
$94$	−1.83484	−1.83484
$95$	0	0
$96$	−2.49244	−2.49244
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.35456	1.35456
$99$	−2.57828	−2.57828
$100$	0.834841	0.834841
$101$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$101$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$102$	−1.25803	−1.25803
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$107$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$108$	2.49244	2.49244
$109$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$109$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$113$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2.67218	−2.67218
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	−1.48175	−1.48175
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$127$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$128$	0.441333	0.441333
$129$	3.73167	3.73167
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−1.57921	−1.57921
$133$	0	0
$134$	−1.08824	−1.08824
$135$	0	0
$136$	0.109839	0.109839
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−4.50702	−4.50702
$139$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$139$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$140$	0	0
$141$	−2.56234	−2.56234
$142$	0.665051	0.665051
$143$	0	0
$144$	−2.93378	−2.93378
$145$	0	0
$146$	0.223718	0.223718
$147$	1.89163	1.89163
$148$	0	0
$149$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$149$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$150$	2.56234	2.56234
$151$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$151$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$152$	0	0
$153$	−1.26586	−1.26586
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$157$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	4.15750	4.15750
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$167$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	1.64691	1.64691
$173$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$173$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$174$	0	0
$175$	0	0
$176$	1.13788	1.13788
$177$	−3.73167	−3.73167
$178$	−1.08824	−1.08824
$179$	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
$183$	−2.06925	−2.06925
$184$	0.393508	0.393508
$185$	0	0
$186$	0	0
$187$	0.490971	0.490971
$188$	−1.13085	−1.13085
$189$	0	0
$190$	0	0
$191$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$191$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$192$	−1.22372	−1.22372
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0.834841	0.834841
$197$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$197$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$198$	−3.49244	−3.49244
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.223718	−0.223718
$201$	−1.51972	−1.51972
$202$	2.67218	2.67218
$203$	0	0
$204$	−0.775349	−0.775349
$205$	0	0
$206$	0	0
$207$	−4.53506	−4.53506
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0.928738	0.928738
$214$	−2.56234	−2.56234
$215$	0	0
$216$	−0.667917	−0.667917
$217$	0	0
$218$	−2.13788	−2.13788
$219$	0.312420	0.312420
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.57828	2.57828
$226$	1.48175	1.48175
$227$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$227$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$228$	0	0
$229$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$229$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$233$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$234$	0	0
$235$	0	0
$236$	−1.64691	−1.64691
$237$	0	0
$238$	0	0
$239$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$239$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.35456	1.35456
$243$	2.82037	2.82037
$244$	−0.913230	−0.913230
$245$	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
$253$	1.75895	1.75895
$254$	1.08824	1.08824
$255$	0	0
$256$	1.24472	1.24472
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	5.05478	5.05478
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$263$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$264$	0.423192	0.423192
$265$	0	0
$266$	0	0
$267$	−1.51972	−1.51972
$268$	−0.670704	−0.670704
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$271$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$272$	0.558667	0.558667
$273$	0	0
$274$	0	0
$275$	−1.00000	−1.00000
$276$	−2.77776	−2.77776
$277$	−1.00000	−1.00000
$277$	−1.00000	−1.00000
$278$	2.38261	2.38261
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−3.47085	−3.47085
$283$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$283$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$284$	0.409883	0.409883
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−3.39718	−3.39718
$289$	−0.758948	−0.758948
$290$	0	0
$291$	0	0
$292$	0.137881	0.137881
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.56234	2.56234
$295$	0	0
$296$	0	0
$297$	−2.98553	−2.98553
$298$	1.08824	1.08824
$299$	0	0
$300$	1.57921	1.57921
$301$	0	0
$302$	−2.56234	−2.56234
$303$	3.73167	3.73167
$304$	0	0
$305$	0	0
$306$	−1.71469	−1.71469
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$311$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$312$	0	0
$313$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$313$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$314$	0.665051	0.665051
$315$	0	0
$316$	0	0
$317$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$317$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.57828	−3.57828
$322$	0	0
$323$	0	0
$324$	2.56234	2.56234
$325$	0	0
$326$	0	0
$327$	−2.98553	−2.98553
$328$	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
$334$	2.38261	2.38261
$335$	0	0
$336$	0	0
$337$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$337$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$338$	1.35456	1.35456
$339$	2.06925	2.06925
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−0.441333	−0.441333
$345$	0	0
$346$	0.223718	0.223718
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$349$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$350$	0	0
$351$	0	0
$352$	1.31761	1.31761
$353$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$353$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$354$	−5.05478	−5.05478
$355$	0	0
$356$	−0.670704	−0.670704
$357$	0	0
$358$	0	0
$359$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$359$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	1.89163	1.89163
$364$	0	0
$365$	0	0
$366$	−2.80293	−2.80293
$367$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$367$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$368$	2.00147	2.00147
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$373$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$374$	0.665051	0.665051
$375$	0	0
$376$	0.303040	0.303040
$377$	0	0
$378$	0	0
$379$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$379$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$380$	0	0
$381$	1.51972	1.51972
$382$	−0.223718	−0.223718
$383$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$383$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$384$	0.834841	0.834841
$385$	0	0
$386$	0	0
$387$	5.08623	5.08623
$388$	0	0
$389$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$389$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$390$	0	0
$391$	0.863592	0.863592
$392$	−0.223718	−0.223718
$393$	0	0
$394$	2.38261	2.38261
$395$	0	0
$396$	−2.15246	−2.15246
$397$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$397$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$398$	0	0
$399$	0	0
$400$	−1.13788	−1.13788
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−2.05856	−2.05856
$403$	0	0
$404$	1.64691	1.64691
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0.207775	0.207775
$409$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$409$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−6.14303	−6.14303
$415$	0	0
$416$	0	0
$417$	3.32729	3.32729
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$421$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$422$	0	0
$423$	−3.49244	−3.49244
$424$	0	0
$425$	−0.490971	−0.490971
$426$	1.25803	1.25803
$427$	0	0
$428$	−1.57921	−1.57921
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−3.39718	−3.39718
$433$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$433$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$434$	0	0
$435$	0	0
$436$	−1.31761	−1.31761
$437$	0	0
$438$	0.423192	0.423192
$439$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$439$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$440$	0	0
$441$	2.57828	2.57828
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.51972	1.51972
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	3.49244	3.49244
$451$	0	0
$452$	0.913230	0.913230
$453$	−3.57828	−3.57828
$454$	1.08824	1.08824
$455$	0	0
$456$	0	0
$457$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$457$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$458$	1.48175	1.48175
$459$	−1.46581	−1.46581
$460$	0	0
$461$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$461$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$462$	0	0
$463$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$463$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$464$	0	0
$465$	0	0
$466$	−1.48175	−1.48175
$467$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$467$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.928738	0.928738
$472$	0.441333	0.441333
$473$	−1.97272	−1.97272
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0.223718	0.223718
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0.834841	0.834841
$485$	0	0
$486$	3.82037	3.82037
$487$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$487$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$488$	0.244724	0.244724
$489$	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
$499$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$499$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$500$	0	0
$501$	3.32729	3.32729
$502$	0	0
$503$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$503$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$504$	0	0
$505$	0	0
$506$	2.38261	2.38261
$507$	1.89163	1.89163
$508$	0.670704	0.670704
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.24472	1.24472
$513$	0	0
$514$	0	0
$515$	0	0
$516$	3.11535	3.11535
$517$	1.35456	1.35456
$518$	0	0
$519$	0.312420	0.312420
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$523$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$524$	0	0
$525$	0	0
$526$	−1.48175	−1.48175
$527$	0	0
$528$	2.15246	2.15246
$529$	2.09390	2.09390
$530$	0	0
$531$	−5.08623	−5.08623
$532$	0	0
$533$	0	0
$534$	−2.05856	−2.05856
$535$	0	0
$536$	0.179733	0.179733
$537$	0	0
$538$	0	0
$539$	−1.00000	−1.00000
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.13788	−2.13788
$543$	0	0
$544$	0.646910	0.646910
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−2.82037	−2.82037
$550$	−1.35456	−1.35456
$551$	0	0
$552$	0.744373	0.744373
$553$	0	0
$554$	−1.35456	−1.35456
$555$	0	0
$556$	1.46844	1.46844
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.928738	0.928738
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−2.13915	−2.13915
$565$	0	0
$566$	−0.665051	−0.665051
$567$	0	0
$568$	−0.109839	−0.109839
$569$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$569$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$570$	0	0
$571$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$571$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$572$	0	0
$573$	−0.312420	−0.312420
$574$	0	0
$575$	−1.75895	−1.75895
$576$	−1.66792	−1.66792
$577$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$577$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$578$	−1.02804	−1.02804
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−0.0369490	−0.0369490
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	1.57921	1.57921
$589$	0	0
$590$	0	0
$591$	3.32729	3.32729
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−4.04409	−4.04409
$595$	0	0
$596$	0.670704	0.670704
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−0.423192	−0.423192
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−2.07137	−2.07137
$604$	−1.57921	−1.57921
$605$	0	0
$606$	5.05478	5.05478
$607$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$607$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−1.05679	−1.05679
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$617$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−5.25139	−5.25139
$622$	−1.08824	−1.08824
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	−1.83484	−1.83484
$627$	0	0
$628$	0.409883	0.409883
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−0.223718	−0.223718
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.26586	1.26586
$640$	0	0
$641$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$641$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$642$	−4.84701	−4.84701
$643$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$643$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.686647	−0.686647
$649$	1.97272	1.97272
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−4.04409	−4.04409
$655$	0	0
$656$	0	0
$657$	0.425826	0.425826
$658$	0	0
$659$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$659$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$660$	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
$668$	1.46844	1.46844
$669$	0	0
$670$	0	0
$671$	1.09390	1.09390
$672$	0	0
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$673$	−2.00000	−2.00000	−1.00000			$\pi$
$674$	1.08824	1.08824
$675$	2.98553	2.98553
$676$	0.834841	0.834841
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	2.80293	2.80293
$679$	0	0
$680$	0	0
$681$	1.51972	1.51972
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.06925	2.06925
$688$	−2.24472	−2.24472
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0.137881	0.137881
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.08824	1.08824
$699$	−2.06925	−2.06925
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.646910	0.646910
$705$	0	0
$706$	−0.223718	−0.223718
$707$	0	0
$708$	−3.11535	−3.11535
$709$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$709$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$710$	0	0
$711$	0	0
$712$	0.179733	0.179733
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.312420	0.312420
$718$	2.67218	2.67218
$719$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$719$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$720$	0	0
$721$	0	0
$722$	1.35456	1.35456
$723$	0	0
$724$	0	0
$725$	0	0
$726$	2.56234	2.56234
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.26586	2.26586
$730$	0	0
$731$	−0.968550	−0.968550
$732$	−1.72750	−1.72750
$733$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$733$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$734$	2.13788	2.13788
$735$	0	0
$736$	2.31761	2.31761
$737$	0.803391	0.803391
$738$	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
$746$	−2.56234	−2.56234
$747$	0	0
$748$	0.409883	0.409883
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.54133	1.54133
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$757$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$758$	−1.83484	−1.83484
$759$	3.32729	3.32729
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	2.05856	2.05856
$763$	0	0
$764$	−0.137881	−0.137881
$765$	0	0
$766$	2.13788	2.13788
$767$	0	0
$768$	2.35456	2.35456
$769$	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			$\pi$
$774$	6.88962	6.88962
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0.665051	0.665051
$779$	0	0
$780$	0	0
$781$	−0.490971	−0.490971
$782$	1.16979	1.16979
$783$	0	0
$784$	−1.13788	−1.13788
$785$	0	0
$786$	0	0
$787$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$787$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$788$	1.46844	1.46844
$789$	−2.06925	−2.06925
$790$	0	0
$791$	0	0
$792$	0.576808	0.576808
$793$	0	0
$794$	−2.38261	−2.38261
$795$	0	0
$796$	0	0
$797$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$797$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$798$	0	0
$799$	0.665051	0.665051
$800$	−1.31761	−1.31761
$801$	−2.07137	−2.07137
$802$	0	0
$803$	−0.165159	−0.165159
$804$	−1.26873	−1.26873
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−0.441333	−0.441333
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$811$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$812$	0	0
$813$	−2.98553	−2.98553
$814$	0	0
$815$	0	0
$816$	1.05679	1.05679
$817$	0	0
$818$	1.83484	1.83484
$819$	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
$825$	−1.89163	−1.89163
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−3.78606	−3.78606
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	−1.89163	−1.89163
$832$	0	0
$833$	−0.490971	−0.490971
$834$	4.50702	4.50702
$835$	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
$842$	−1.08824	−1.08824
$843$	0	0
$844$	0	0
$845$	0	0
$846$	−4.73074	−4.73074
$847$	0	0
$848$	0	0
$849$	−0.928738	−0.928738
$850$	−0.665051	−0.665051
$851$	0	0
$852$	0.775349	0.775349
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0.423192	0.423192
$857$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$857$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$858$	0	0
$859$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$859$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−3.93378	−3.93378
$865$	0	0
$866$	−2.67218	−2.67218
$867$	−1.43565	−1.43565
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0.353090	0.353090
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0.260821	0.260821
$877$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$877$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$878$	0.223718	0.223718
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3.49244	3.49244
$883$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$883$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$887$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.06925	−3.06925
$892$	0	0
$893$	0	0
$894$	2.05856	2.05856
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	2.15246	2.15246
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−0.244724	−0.244724
$905$	0	0
$906$	−4.84701	−4.84701
$907$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$907$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$908$	0.670704	0.670704
$909$	5.08623	5.08623
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	2.38261	2.38261
$915$	0	0
$916$	0.913230	0.913230
$917$	0	0
$918$	−1.98553	−1.98553
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−2.56234	−2.56234
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−2.67218	−2.67218
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.913230	−0.913230
$933$	−1.51972	−1.51972
$934$	2.13788	2.13788
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−2.56234	−2.56234
$940$	0	0
$941$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$941$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$942$	1.25803	1.25803
$943$	0	0
$944$	2.24472	2.24472
$945$	0	0
$946$	−2.67218	−2.67218
$947$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$947$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−0.312420	−0.312420
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0.137881	0.137881
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	−4.87717	−4.87717
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.223718	−0.223718
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.35456	2.35456
$973$	0	0
$974$	1.48175	1.48175
$975$	0	0
$976$	1.24472	1.24472
$977$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$977$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$978$	0	0
$979$	0.803391	0.803391
$980$	0	0
$981$	−4.06925	−4.06925
$982$	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
$989$	−3.46992	−3.46992
$990$	0	0
$991$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$991$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$997$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$998$	2.13788	2.13788
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3047.1.d.a.3046.3	✓	9	11.10	odd	2
3047.1.d.a.3046.3	✓	9	277.276	even	2
3047.1.d.b.3046.7	yes	9	1.1	even	1	trivial
3047.1.d.b.3046.7	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)

\(f(q)\)	\(=\)	\(q+1.35456 q^{2} +1.89163 q^{3} +0.834841 q^{4} +2.56234 q^{6} -0.223718 q^{8} +2.57828 q^{9} +O(q^{10})\)	\(q+1.35456 q^{2} +1.89163 q^{3} +0.834841 q^{4} +2.56234 q^{6} -0.223718 q^{8} +2.57828 q^{9} -1.00000 q^{11} +1.57921 q^{12} -1.13788 q^{16} -0.490971 q^{17} +3.49244 q^{18} -1.35456 q^{22} -1.75895 q^{23} -0.423192 q^{24} +1.00000 q^{25} +2.98553 q^{27} -1.31761 q^{32} -1.89163 q^{33} -0.665051 q^{34} +2.15246 q^{36} +1.97272 q^{43} -0.834841 q^{44} -2.38261 q^{46} -1.35456 q^{47} -2.15246 q^{48} +1.00000 q^{49} +1.35456 q^{50} -0.928738 q^{51} +4.04409 q^{54} -1.97272 q^{59} -1.09390 q^{61} -0.646910 q^{64} -2.56234 q^{66} -0.803391 q^{67} -0.409883 q^{68} -3.32729 q^{69} +0.490971 q^{71} -0.576808 q^{72} +0.165159 q^{73} +1.89163 q^{75} +3.06925 q^{81} +2.67218 q^{86} +0.223718 q^{88} -0.803391 q^{89} -1.46844 q^{92} -1.83484 q^{94} -2.49244 q^{96} +1.35456 q^{98} -2.57828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) 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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