LMFDB - Embedded newform 1151.1.b.a.1150.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1151,1,Mod(1150,1151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1151.1150");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1151 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1151.b (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:	$0.574423829541$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{82})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 19 x^{18} + 18 x^{17} + 153 x^{16} - 136 x^{15} - 680 x^{14} + 560 x^{13} + 1820 x^{12} + \cdots + 1 $ x^20 - x^19 - 19x^18 + 18x^17 + 153x^16 - 136x^15 - 680x^14 + 560x^13 + 1820x^12 - 1365x^11 - 3003x^10 + 2002x^9 + 3003x^8 - 1716x^7 - 1716x^6 + 792x^5 + 495x^4 - 165x^3 - 55x^2 + 10x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{41}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{41} - \cdots)$

Embedding invariants

Embedding label			1150.10
Root			$-1.79233$ of defining polynomial
Character	$\chi$	$=$	1151.1150

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.229367 q^{2} +0.955440 q^{3} -0.947391 q^{4} +1.21245 q^{5} -0.219146 q^{6} +0.380782 q^{7} +0.446667 q^{8} -0.0871351 q^{9} +O(q^{10})$	\(q-0.229367 q^{2} +0.955440 q^{3} -0.947391 q^{4} +1.21245 q^{5} -0.219146 q^{6} +0.380782 q^{7} +0.446667 q^{8} -0.0871351 q^{9} -0.278096 q^{10} +0.0766055 q^{11} -0.905175 q^{12} -0.0873388 q^{14} +1.15842 q^{15} +0.844940 q^{16} +0.0199859 q^{18} -1.14866 q^{20} +0.363814 q^{21} -0.0175708 q^{22} +0.426763 q^{24} +0.470037 q^{25} -1.03869 q^{27} -0.360750 q^{28} +1.44104 q^{29} -0.265704 q^{30} -0.640468 q^{32} +0.0731919 q^{33} +0.461680 q^{35} +0.0825510 q^{36} -1.33065 q^{37} +0.541562 q^{40} -0.0834470 q^{42} +0.676034 q^{43} -0.0725753 q^{44} -0.105647 q^{45} -1.71914 q^{47} +0.807289 q^{48} -0.855005 q^{49} -0.107811 q^{50} +1.63586 q^{53} +0.238242 q^{54} +0.0928804 q^{55} +0.170083 q^{56} -0.330528 q^{58} -0.529963 q^{59} -1.09748 q^{60} -0.0331795 q^{63} -0.698038 q^{64} -0.0167878 q^{66} +1.90679 q^{67} -0.105894 q^{70} -0.0389204 q^{72} +0.305207 q^{74} +0.449092 q^{75} +0.0291700 q^{77} +1.02445 q^{80} -0.905272 q^{81} -1.85500 q^{83} -0.344674 q^{84} -0.155060 q^{86} +1.37683 q^{87} +0.0342171 q^{88} +0.0242319 q^{90} +0.394314 q^{94} -0.611929 q^{96} +0.196110 q^{98} -0.00667503 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{14} - 2 q^{15} + 18 q^{16} - 3 q^{18} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 19 q^{25} - 2 q^{27} - 3 q^{28} - q^{29} - 4 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 16 q^{36} - q^{37} - 4 q^{40} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - q^{47} - 5 q^{48} + 19 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} - 4 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 3 q^{63} + 17 q^{64} - 4 q^{66} - q^{67} - 4 q^{70} - 6 q^{72} - 2 q^{74} - 3 q^{75} - 2 q^{77} - 5 q^{80} + 18 q^{81} - q^{83} - 6 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - 6 q^{90} - 2 q^{94} - 6 q^{96} - 3 q^{98} - 3 q^{99}+O(q^{100}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^14 - 2 * q^15 + 18 * q^16 - 3 * q^18 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 19 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - 4 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 16 * q^36 - q^37 - 4 * q^40 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - q^47 - 5 * q^48 + 19 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 - 4 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 3 * q^63 + 17 * q^64 - 4 * q^66 - q^67 - 4 * q^70 - 6 * q^72 - 2 * q^74 - 3 * q^75 - 2 * q^77 - 5 * q^80 + 18 * q^81 - q^83 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - 6 * q^90 - 2 * q^94 - 6 * q^96 - 3 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1151\mathbb{Z}\right)^\times$.

$n$	$17$
$\chi(n)$	$-1$

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 $

$2$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$2$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$3$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$3$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$4$	−0.947391	−0.947391
$5$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$5$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$6$	−0.219146	−0.219146
$7$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$7$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$8$	0.446667	0.446667
$9$	−0.0871351	−0.0871351
$10$	−0.278096	−0.278096
$11$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$11$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$12$	−0.905175	−0.905175
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.0873388	−0.0873388
$15$	1.15842	1.15842
$16$	0.844940	0.844940
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.0199859	0.0199859
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.14866	−1.14866
$21$	0.363814	0.363814
$22$	−0.0175708	−0.0175708
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.426763	0.426763
$25$	0.470037	0.470037
$26$	0	0
$27$	−1.03869	−1.03869
$28$	−0.360750	−0.360750
$29$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$29$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$30$	−0.265704	−0.265704
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−0.640468	−0.640468
$33$	0.0731919	0.0731919
$34$	0	0
$35$	0.461680	0.461680
$36$	0.0825510	0.0825510
$37$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$37$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$38$	0	0
$39$	0	0
$40$	0.541562	0.541562
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.0834470	−0.0834470
$43$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$43$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$44$	−0.0725753	−0.0725753
$45$	−0.105647	−0.105647
$46$	0	0
$47$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$47$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$48$	0.807289	0.807289
$49$	−0.855005	−0.855005
$50$	−0.107811	−0.107811
$51$	0	0
$52$	0	0
$53$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$53$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$54$	0.238242	0.238242
$55$	0.0928804	0.0928804
$56$	0.170083	0.170083
$57$	0	0
$58$	−0.330528	−0.330528
$59$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$59$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$60$	−1.09748	−1.09748
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.0331795	−0.0331795
$64$	−0.698038	−0.698038
$65$	0	0
$66$	−0.0167878	−0.0167878
$67$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$67$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$68$	0	0
$69$	0	0
$70$	−0.105894	−0.105894
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.0389204	−0.0389204
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.305207	0.305207
$75$	0.449092	0.449092
$76$	0	0
$77$	0.0291700	0.0291700
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.02445	1.02445
$81$	−0.905272	−0.905272
$82$	0	0
$83$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$83$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$84$	−0.344674	−0.344674
$85$	0	0
$86$	−0.155060	−0.155060
$87$	1.37683	1.37683
$88$	0.0342171	0.0342171
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.0242319	0.0242319
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.394314	0.394314
$95$	0	0
$96$	−0.611929	−0.611929
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.196110	0.196110
$99$	−0.00667503	−0.00667503
$100$	−0.445309	−0.445309
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0.441107	0.441107
$106$	−0.375212	−0.375212
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.984047	0.984047
$109$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$109$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$110$	−0.0213037	−0.0213037
$111$	−1.27136	−1.27136
$112$	0.321738	0.321738
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.36523	−1.36523
$117$	0	0
$118$	0.121556	0.121556
$119$	0	0
$120$	0.517429	0.517429
$121$	−0.994132	−0.994132
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.642554	−0.642554
$126$	0.00761028	0.00761028
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.800575	0.800575
$129$	0.645909	0.645909
$130$	0	0
$131$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$131$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$132$	−0.0693413	−0.0693413
$133$	0	0
$134$	−0.437355	−0.437355
$135$	−1.25936	−1.25936
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$139$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$140$	−0.437391	−0.437391
$141$	−1.64253	−1.64253
$142$	0	0
$143$	0	0
$144$	−0.0736240	−0.0736240
$145$	1.74719	1.74719
$146$	0	0
$147$	−0.816906	−0.816906
$148$	1.26065	1.26065
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.103007	−0.103007
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−0.00669063	−0.00669063
$155$	0	0
$156$	0	0
$157$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$157$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$158$	0	0
$159$	1.56296	1.56296
$160$	−0.776536	−0.776536
$161$	0	0
$162$	0.207639	0.207639
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0.0887416	0.0887416
$166$	0.425477	0.425477
$167$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$167$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$168$	0.162504	0.162504
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−0.640468	−0.640468
$173$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$173$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$174$	−0.315799	−0.315799
$175$	0.178982	0.178982
$176$	0.0647270	0.0647270
$177$	−0.506348	−0.506348
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0.100089	0.100089
$181$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$181$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.61335	−1.61335
$186$	0	0
$187$	0	0
$188$	1.62870	1.62870
$189$	−0.395515	−0.395515
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.666933	−0.666933
$193$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$193$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$194$	0	0
$195$	0	0
$196$	0.810024	0.810024
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.00153103	0.00153103
$199$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$199$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$200$	0.209950	0.209950
$201$	1.82183	1.82183
$202$	0	0
$203$	0.548724	0.548724
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.101175	−0.101175
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−1.54980	−1.54980
$213$	0	0
$214$	0	0
$215$	0.819658	0.819658
$216$	−0.463949	−0.463949
$217$	0	0
$218$	0.187654	0.187654
$219$	0	0
$220$	−0.0879940	−0.0879940
$221$	0	0
$222$	0.291607	0.291607
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.243879	−0.243879
$225$	−0.0409567	−0.0409567
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0.0278702	0.0278702
$232$	0.643666	0.643666
$233$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$233$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$234$	0	0
$235$	−2.08437	−2.08437
$236$	0.502082	0.502082
$237$	0	0
$238$	0	0
$239$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$239$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$240$	0.978799	0.978799
$241$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$241$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$242$	0.228021	0.228021
$243$	0.173759	0.173759
$244$	0	0
$245$	−1.03665	−1.03665
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.77235	−1.77235
$250$	0.147381	0.147381
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0.0314340	0.0314340
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0.514413	0.514413
$257$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$257$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$258$	−0.148150	−0.148150
$259$	−0.506688	−0.506688
$260$	0	0
$261$	−0.125565	−0.125565
$262$	0.353908	0.353908
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0.0326924	0.0326924
$265$	1.98340	1.98340
$266$	0	0
$267$	0	0
$268$	−1.80648	−1.80648
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.288856	0.288856
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.0360074	0.0360074
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.249353	0.249353
$279$	0	0
$280$	0.206217	0.206217
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.376743	0.376743
$283$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$283$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.0558073	0.0558073
$289$	1.00000	1.00000
$290$	−0.400748	−0.400748
$291$	0	0
$292$	0	0
$293$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$293$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$294$	0.187371	0.187371
$295$	−0.642554	−0.642554
$296$	−0.594358	−0.594358
$297$	−0.0795695	−0.0795695
$298$	0	0
$299$	0	0
$300$	−0.425466	−0.425466
$301$	0.257422	0.257422
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$307$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$308$	−0.0276354	−0.0276354
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−0.0175708	−0.0175708
$315$	−0.0402285	−0.0402285
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.358492	−0.358492
$319$	0.110392	0.110392
$320$	−0.846337	−0.846337
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.857647	0.857647
$325$	0	0
$326$	0	0
$327$	−0.781681	−0.781681
$328$	0	0
$329$	−0.654618	−0.654618
$330$	−0.0203544	−0.0203544
$331$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$331$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$332$	1.75741	1.75741
$333$	0.115946	0.115946
$334$	0.446667	0.446667
$335$	2.31189	2.31189
$336$	0.307401	0.307401
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.229367	−0.229367
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−0.706353	−0.706353
$344$	0.301962	0.301962
$345$	0	0
$346$	0.187654	0.187654
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.30440	−1.30440
$349$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$349$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$350$	−0.0410525	−0.0410525
$351$	0	0
$352$	−0.0490634	−0.0490634
$353$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$353$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$354$	0.116139	0.116139
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.0471890	−0.0471890
$361$	1.00000	1.00000
$362$	−0.411101	−0.411101
$363$	−0.949833	−0.949833
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0.370049	0.370049
$371$	0.622906	0.622906
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.613922	−0.613922
$376$	−0.767883	−0.767883
$377$	0	0
$378$	0.0907181	0.0907181
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.764901	0.764901
$385$	0.0353672	0.0353672
$386$	−0.375212	−0.375212
$387$	−0.0589063	−0.0589063
$388$	0	0
$389$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$389$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$390$	0	0
$391$	0	0
$392$	−0.381902	−0.381902
$393$	−1.47422	−1.47422
$394$	0	0
$395$	0	0
$396$	0.00632386	0.00632386
$397$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$397$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$398$	0.0526092	0.0526092
$399$	0	0
$400$	0.397153	0.397153
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.417866	−0.417866
$403$	0	0
$404$	0	0
$405$	−1.09760	−1.09760
$406$	−0.125859	−0.125859
$407$	−0.101935	−0.101935
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.201800	−0.201800
$414$	0	0
$415$	−2.24910	−2.24910
$416$	0	0
$417$	−1.03869	−1.03869
$418$	0	0
$419$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$419$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$420$	−0.417901	−0.417901
$421$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$421$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$422$	0	0
$423$	0.149797	0.149797
$424$	0.730684	0.730684
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.188002	−0.188002
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.877633	−0.877633
$433$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$433$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$434$	0	0
$435$	1.66934	1.66934
$436$	0.775096	0.775096
$437$	0	0
$438$	0	0
$439$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$439$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$440$	0.0414866	0.0414866
$441$	0.0745009	0.0745009
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	1.20447	1.20447
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−0.265800	−0.265800
$449$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$449$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$450$	0.00939411	0.00939411
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$457$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$461$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$462$	−0.00639249	−0.00639249
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	1.21760	1.21760
$465$	0	0
$466$	0.353908	0.353908
$467$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$467$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$468$	0	0
$469$	0.726073	0.726073
$470$	0.478086	0.478086
$471$	0.0731919	0.0731919
$472$	−0.236717	−0.236717
$473$	0.0517879	0.0517879
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.142541	−0.142541
$478$	0.457388	0.457388
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.741933	−0.741933
$481$	0	0
$482$	−0.155060	−0.155060
$483$	0	0
$484$	0.941831	0.941831
$485$	0	0
$486$	−0.0398545	−0.0398545
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0.237773	0.237773
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.00809314	−0.00809314
$496$	0	0
$497$	0	0
$498$	0.406517	0.406517
$499$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$499$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$500$	0.608750	0.608750
$501$	−1.86061	−1.86061
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.0148202	−0.0148202
$505$	0	0
$506$	0	0
$507$	0.955440	0.955440
$508$	0	0
$509$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$509$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$510$	0	0
$511$	0	0
$512$	−0.918564	−0.918564
$513$	0	0
$514$	0.187654	0.187654
$515$	0	0
$516$	−0.611929	−0.611929
$517$	−0.131695	−0.131695
$518$	0.116218	0.116218
$519$	−0.781681	−0.781681
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0.0288006	0.0288006
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.46180	1.46180
$525$	0.171006	0.171006
$526$	0	0
$527$	0	0
$528$	0.0618428	0.0618428
$529$	1.00000	1.00000
$530$	−0.454926	−0.454926
$531$	0.0461784	0.0461784
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0.851701	0.851701
$537$	0	0
$538$	0	0
$539$	−0.0654981	−0.0654981
$540$	1.19311	1.19311
$541$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$541$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$542$	0	0
$543$	1.71246	1.71246
$544$	0	0
$545$	−0.991951	−0.991951
$546$	0	0
$547$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$547$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$548$	0	0
$549$	0	0
$550$	−0.00825890	−0.00825890
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.54146	−1.54146
$556$	1.02994	1.02994
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0.390092	0.390092
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.55612	1.55612
$565$	0	0
$566$	−0.330528	−0.330528
$567$	−0.344712	−0.344712
$568$	0	0
$569$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$569$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0.0608236	0.0608236
$577$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$577$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$578$	−0.229367	−0.229367
$579$	1.56296	1.56296
$580$	−1.65528	−1.65528
$581$	−0.706353	−0.706353
$582$	0	0
$583$	0.125316	0.125316
$584$	0	0
$585$	0	0
$586$	−0.0873388	−0.0873388
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.773929	0.773929
$589$	0	0
$590$	0.147381	0.147381
$591$	0	0
$592$	−1.12432	−1.12432
$593$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$593$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$594$	0.0182506	0.0182506
$595$	0	0
$596$	0	0
$597$	−0.219146	−0.219146
$598$	0	0
$599$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$599$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$600$	0.200595	0.200595
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.0590440	−0.0590440
$603$	−0.166149	−0.166149
$604$	0	0
$605$	−1.20534	−1.20534
$606$	0	0
$607$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$607$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$608$	0	0
$609$	0.524272	0.524272
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$613$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$614$	0.187654	0.187654
$615$	0	0
$616$	0.0130293	0.0130293
$617$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$617$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$618$	0	0
$619$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$619$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.24910	−1.24910
$626$	0	0
$627$	0	0
$628$	−0.0725753	−0.0725753
$629$	0	0
$630$	0.00922709	0.00922709
$631$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$631$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−1.48074	−1.48074
$637$	0	0
$638$	−0.0253202	−0.0253202
$639$	0	0
$640$	0.970658	0.970658
$641$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$641$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$642$	0	0
$643$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$643$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$644$	0	0
$645$	0.783133	0.783133
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.404355	−0.404355
$649$	−0.0405981	−0.0405981
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.179292	0.179292
$655$	−1.87079	−1.87079
$656$	0	0
$657$	0	0
$658$	0.150148	0.150148
$659$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$659$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$660$	−0.0840730	−0.0840730
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.453358	−0.453358
$663$	0	0
$664$	−0.828569	−0.828569
$665$	0	0
$666$	−0.0265943	−0.0265943
$667$	0	0
$668$	1.84494	1.84494
$669$	0	0
$670$	−0.530271	−0.530271
$671$	0	0
$672$	−0.233012	−0.233012
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−0.488224	−0.488224
$676$	−0.947391	−0.947391
$677$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$677$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$683$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$684$	0	0
$685$	0	0
$686$	0.162014	0.162014
$687$	0	0
$688$	0.571208	0.571208
$689$	0	0
$690$	0	0
$691$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$691$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$692$	0.775096	0.775096
$693$	−0.00254173	−0.00254173
$694$	0	0
$695$	−1.31810	−1.31810
$696$	0.614984	0.614984
$697$	0	0
$698$	−0.375212	−0.375212
$699$	−1.47422	−1.47422
$700$	−0.169566	−0.169566
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−0.0534735	−0.0534735
$705$	−1.99149	−1.99149
$706$	−0.437355	−0.437355
$707$	0	0
$708$	0.479709	0.479709
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.90527	−1.90527
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.0892654	−0.0892654
$721$	0	0
$722$	−0.229367	−0.229367
$723$	0.645909	0.645909
$724$	−1.69804	−1.69804
$725$	0.677344	0.677344
$726$	0.217860	0.217860
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.07129	1.07129
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$733$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$734$	0	0
$735$	−0.990458	−0.990458
$736$	0	0
$737$	0.146071	0.146071
$738$	0	0
$739$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$739$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$740$	1.52847	1.52847
$741$	0	0
$742$	−0.142874	−0.142874
$743$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$743$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.161636	0.161636
$748$	0	0
$749$	0	0
$750$	0.140813	0.140813
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1.45257	−1.45257
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0.374708	0.374708
$757$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$757$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$761$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$762$	0	0
$763$	−0.311532	−0.311532
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.491490	0.491490
$769$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$769$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$770$	−0.00811206	−0.00811206
$771$	−0.781681	−0.781681
$772$	−1.54980	−1.54980
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0.0135111	0.0135111
$775$	0	0
$776$	0	0
$777$	−0.484110	−0.484110
$778$	−0.219146	−0.219146
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.49680	−1.49680
$784$	−0.722428	−0.722428
$785$	0.0928804	0.0928804
$786$	0.338138	0.338138
$787$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$787$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−0.00298151	−0.00298151
$793$	0	0
$794$	0.0526092	0.0526092
$795$	1.89502	1.89502
$796$	0.217300	0.217300
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.301044	−0.301044
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1.72598	−1.72598
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$809$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$810$	0.251753	0.251753
$811$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$811$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$812$	−0.519856	−0.519856
$813$	0	0
$814$	0.0233805	0.0233805
$815$	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	0
$823$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$823$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$824$	0	0
$825$	0.0344029	0.0344029
$826$	0.0462863	0.0462863
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$829$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$830$	0.515869	0.515869
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0.238242	0.238242
$835$	−2.36112	−2.36112
$836$	0	0
$837$	0	0
$838$	−0.453358	−0.453358
$839$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$839$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$840$	0.197028	0.197028
$841$	1.07661	1.07661
$842$	−0.155060	−0.155060
$843$	0	0
$844$	0	0
$845$	1.21245	1.21245
$846$	−0.0343586	−0.0343586
$847$	−0.378548	−0.378548
$848$	1.38220	1.38220
$849$	1.37683	1.37683
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$853$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$859$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$860$	−0.776536	−0.776536
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.665249	0.665249
$865$	−0.991951	−0.991951
$866$	0.353908	0.353908
$867$	0.955440	0.955440
$868$	0	0
$869$	0	0
$870$	−0.382891	−0.382891
$871$	0	0
$872$	−0.365435	−0.365435
$873$	0	0
$874$	0	0
$875$	−0.244673	−0.244673
$876$	0	0
$877$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$877$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$878$	−0.0175708	−0.0175708
$879$	0.363814	0.363814
$880$	0.0784784	0.0784784
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.0170880	−0.0170880
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.613922	−0.613922
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.567873	−0.567873
$889$	0	0
$890$	0	0
$891$	−0.0693488	−0.0693488
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.304845	0.304845
$897$	0	0
$898$	−0.219146	−0.219146
$899$	0	0
$900$	0.0388020	0.0388020
$901$	0	0
$902$	0	0
$903$	0.245951	0.245951
$904$	0	0
$905$	2.17311	2.17311
$906$	0	0
$907$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$907$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.142104	−0.142104
$914$	−0.330528	−0.330528
$915$	0	0
$916$	0	0
$917$	−0.587539	−0.587539
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−0.781681	−0.781681
$922$	−0.411101	−0.411101
$923$	0	0
$924$	−0.0264039	−0.0264039
$925$	−0.625455	−0.625455
$926$	0	0
$927$	0	0
$928$	−0.922942	−0.922942
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	1.46180	1.46180
$933$	0	0
$934$	−0.453358	−0.453358
$935$	0	0
$936$	0	0
$937$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$937$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$938$	−0.166537	−0.166537
$939$	0	0
$940$	1.97471	1.97471
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.0167878	−0.0167878
$943$	0	0
$944$	−0.447787	−0.447787
$945$	−0.479543	−0.479543
$946$	−0.0118784	−0.0118784
$947$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$947$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.0326941	0.0326941
$955$	0	0
$956$	1.88922	1.88922
$957$	0.105473	0.105473
$958$	0	0
$959$	0	0
$960$	−0.808624	−0.808624
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−0.640468	−0.640468
$965$	1.98340	1.98340
$966$	0	0
$967$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$967$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$968$	−0.444046	−0.444046
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.164618	−0.164618
$973$	−0.413962	−0.413962
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0.982114	0.982114
$981$	0.0712885	0.0712885
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.625448	−0.625448
$988$	0	0
$989$	0	0
$990$	0.00185630	0.00185630
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	1.88848	1.88848
$994$	0	0
$995$	−0.278096	−0.278096
$996$	1.67910	1.67910
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.121556	0.121556
$999$	1.38214	1.38214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.10	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.10	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.10	✓	20	1151.1150	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:	\( N \)	\(=\)	\( 1151 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.229367 q^{2} +0.955440 q^{3} -0.947391 q^{4} +1.21245 q^{5} -0.219146 q^{6} +0.380782 q^{7} +0.446667 q^{8} -0.0871351 q^{9} +O(q^{10})\)	\(q-0.229367 q^{2} +0.955440 q^{3} -0.947391 q^{4} +1.21245 q^{5} -0.219146 q^{6} +0.380782 q^{7} +0.446667 q^{8} -0.0871351 q^{9} -0.278096 q^{10} +0.0766055 q^{11} -0.905175 q^{12} -0.0873388 q^{14} +1.15842 q^{15} +0.844940 q^{16} +0.0199859 q^{18} -1.14866 q^{20} +0.363814 q^{21} -0.0175708 q^{22} +0.426763 q^{24} +0.470037 q^{25} -1.03869 q^{27} -0.360750 q^{28} +1.44104 q^{29} -0.265704 q^{30} -0.640468 q^{32} +0.0731919 q^{33} +0.461680 q^{35} +0.0825510 q^{36} -1.33065 q^{37} +0.541562 q^{40} -0.0834470 q^{42} +0.676034 q^{43} -0.0725753 q^{44} -0.105647 q^{45} -1.71914 q^{47} +0.807289 q^{48} -0.855005 q^{49} -0.107811 q^{50} +1.63586 q^{53} +0.238242 q^{54} +0.0928804 q^{55} +0.170083 q^{56} -0.330528 q^{58} -0.529963 q^{59} -1.09748 q^{60} -0.0331795 q^{63} -0.698038 q^{64} -0.0167878 q^{66} +1.90679 q^{67} -0.105894 q^{70} -0.0389204 q^{72} +0.305207 q^{74} +0.449092 q^{75} +0.0291700 q^{77} +1.02445 q^{80} -0.905272 q^{81} -1.85500 q^{83} -0.344674 q^{84} -0.155060 q^{86} +1.37683 q^{87} +0.0342171 q^{88} +0.0242319 q^{90} +0.394314 q^{94} -0.611929 q^{96} +0.196110 q^{98} -0.00667503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{14} - 2 q^{15} + 18 q^{16} - 3 q^{18} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 19 q^{25} - 2 q^{27} - 3 q^{28} - q^{29} - 4 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 16 q^{36} - q^{37} - 4 q^{40} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - q^{47} - 5 q^{48} + 19 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} - 4 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 3 q^{63} + 17 q^{64} - 4 q^{66} - q^{67} - 4 q^{70} - 6 q^{72} - 2 q^{74} - 3 q^{75} - 2 q^{77} - 5 q^{80} + 18 q^{81} - q^{83} - 6 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - 6 q^{90} - 2 q^{94} - 6 q^{96} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^14 - 2 * q^15 + 18 * q^16 - 3 * q^18 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 19 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - 4 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 16 * q^36 - q^37 - 4 * q^40 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - q^47 - 5 * q^48 + 19 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 - 4 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 3 * q^63 + 17 * q^64 - 4 * q^66 - q^67 - 4 * q^70 - 6 * q^72 - 2 * q^74 - 3 * q^75 - 2 * q^77 - 5 * q^80 + 18 * q^81 - q^83 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - 6 * q^90 - 2 * q^94 - 6 * q^96 - 3 * q^98 - 3 * q^99

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