LMFDB - Embedded newform 1151.1.b.a.1150.7

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			1150.7
Root			$1.33065$ 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)

$f(q)$	$=$	$q-1.08714 q^{2} +1.21245 q^{3} +0.181863 q^{4} -0.229367 q^{5} -1.31810 q^{6} +1.63586 q^{7} +0.889426 q^{8} +0.470037 q^{9} +O(q^{10})$	\(q-1.08714 q^{2} +1.21245 q^{3} +0.181863 q^{4} -0.229367 q^{5} -1.31810 q^{6} +1.63586 q^{7} +0.889426 q^{8} +0.470037 q^{9} +0.249353 q^{10} +0.380782 q^{11} +0.220500 q^{12} -1.77840 q^{14} -0.278096 q^{15} -1.14879 q^{16} -0.510994 q^{18} -0.0417133 q^{20} +1.98340 q^{21} -0.413962 q^{22} +1.07838 q^{24} -0.947391 q^{25} -0.642554 q^{27} +0.297502 q^{28} -1.54298 q^{29} +0.302328 q^{30} +0.359463 q^{32} +0.461680 q^{33} -0.375212 q^{35} +0.0854822 q^{36} +0.955440 q^{37} -0.204005 q^{40} -2.15622 q^{42} +1.97656 q^{43} +0.0692501 q^{44} -0.107811 q^{45} +1.79233 q^{47} -1.39285 q^{48} +1.67603 q^{49} +1.02994 q^{50} -1.99413 q^{53} +0.698543 q^{54} -0.0873388 q^{55} +1.45497 q^{56} +1.67743 q^{58} -1.94739 q^{59} -0.0505753 q^{60} +0.768914 q^{63} +0.758004 q^{64} -0.501908 q^{66} +0.0766055 q^{67} +0.407906 q^{70} +0.418063 q^{72} -1.03869 q^{74} -1.14866 q^{75} +0.622906 q^{77} +0.263494 q^{80} -1.24910 q^{81} +0.676034 q^{83} +0.360706 q^{84} -2.14879 q^{86} -1.87079 q^{87} +0.338678 q^{88} +0.117205 q^{90} -1.94851 q^{94} +0.435831 q^{96} -1.82208 q^{98} +0.178982 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$2$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$3$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$3$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$4$	0.181863	0.181863
$5$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$5$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$6$	−1.31810	−1.31810
$7$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$7$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$8$	0.889426	0.889426
$9$	0.470037	0.470037
$10$	0.249353	0.249353
$11$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$11$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$12$	0.220500	0.220500
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−1.77840	−1.77840
$15$	−0.278096	−0.278096
$16$	−1.14879	−1.14879
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.510994	−0.510994
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−0.0417133	−0.0417133
$21$	1.98340	1.98340
$22$	−0.413962	−0.413962
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.07838	1.07838
$25$	−0.947391	−0.947391
$26$	0	0
$27$	−0.642554	−0.642554
$28$	0.297502	0.297502
$29$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$29$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$30$	0.302328	0.302328
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.359463	0.359463
$33$	0.461680	0.461680
$34$	0	0
$35$	−0.375212	−0.375212
$36$	0.0854822	0.0854822
$37$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$37$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$38$	0	0
$39$	0	0
$40$	−0.204005	−0.204005
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−2.15622	−2.15622
$43$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$43$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$44$	0.0692501	0.0692501
$45$	−0.107811	−0.107811
$46$	0	0
$47$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$47$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$48$	−1.39285	−1.39285
$49$	1.67603	1.67603
$50$	1.02994	1.02994
$51$	0	0
$52$	0	0
$53$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$53$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$54$	0.698543	0.698543
$55$	−0.0873388	−0.0873388
$56$	1.45497	1.45497
$57$	0	0
$58$	1.67743	1.67743
$59$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$59$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$60$	−0.0505753	−0.0505753
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0.768914	0.768914
$64$	0.758004	0.758004
$65$	0	0
$66$	−0.501908	−0.501908
$67$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$67$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$68$	0	0
$69$	0	0
$70$	0.407906	0.407906
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.418063	0.418063
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.03869	−1.03869
$75$	−1.14866	−1.14866
$76$	0	0
$77$	0.622906	0.622906
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.263494	0.263494
$81$	−1.24910	−1.24910
$82$	0	0
$83$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$83$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$84$	0.360706	0.360706
$85$	0	0
$86$	−2.14879	−2.14879
$87$	−1.87079	−1.87079
$88$	0.338678	0.338678
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.117205	0.117205
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−1.94851	−1.94851
$95$	0	0
$96$	0.435831	0.435831
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.82208	−1.82208
$99$	0.178982	0.178982
$100$	−0.172295	−0.172295
$101$	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
$105$	−0.454926	−0.454926
$106$	2.16789	2.16789
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.116857	−0.116857
$109$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$109$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$110$	0.0949491	0.0949491
$111$	1.15842	1.15842
$112$	−1.87926	−1.87926
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−0.280610	−0.280610
$117$	0	0
$118$	2.11708	2.11708
$119$	0	0
$120$	−0.247346	−0.247346
$121$	−0.855005	−0.855005
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.446667	0.446667
$126$	−0.835914	−0.835914
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.18352	−1.18352
$129$	2.39648	2.39648
$130$	0	0
$131$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$131$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$132$	0.0839623	0.0839623
$133$	0	0
$134$	−0.0832805	−0.0832805
$135$	0.147381	0.147381
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$139$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$140$	−0.0682370	−0.0682370
$141$	2.17311	2.17311
$142$	0	0
$143$	0	0
$144$	−0.539973	−0.539973
$145$	0.353908	0.353908
$146$	0	0
$147$	2.03211	2.03211
$148$	0.173759	0.173759
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.24875	1.24875
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−0.677183	−0.677183
$155$	0	0
$156$	0	0
$157$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$157$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$158$	0	0
$159$	−2.41779	−2.41779
$160$	−0.0824488	−0.0824488
$161$	0	0
$162$	1.35794	1.35794
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−0.105894	−0.105894
$166$	−0.734940	−0.734940
$167$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$167$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$168$	1.76409	1.76409
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.359463	0.359463
$173$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$173$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$174$	2.03380	2.03380
$175$	−1.54980	−1.54980
$176$	−0.437438	−0.437438
$177$	−2.36112	−2.36112
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.0196068	−0.0196068
$181$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$181$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.219146	−0.219146
$186$	0	0
$187$	0	0
$188$	0.325958	0.325958
$189$	−1.05113	−1.05113
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0.919043	0.919043
$193$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$193$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$194$	0	0
$195$	0	0
$196$	0.304808	0.304808
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.194577	−0.194577
$199$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$199$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$200$	−0.842634	−0.842634
$201$	0.0928804	0.0928804
$202$	0	0
$203$	−2.52409	−2.52409
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0.494566	0.494566
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.362658	−0.362658
$213$	0	0
$214$	0	0
$215$	−0.453358	−0.453358
$216$	−0.571504	−0.571504
$217$	0	0
$218$	1.86894	1.86894
$219$	0	0
$220$	−0.0158837	−0.0158837
$221$	0	0
$222$	−1.25936	−1.25936
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.588030	0.588030
$225$	−0.445309	−0.445309
$226$	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
$231$	0.755243	0.755243
$232$	−1.37236	−1.37236
$233$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$233$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$234$	0	0
$235$	−0.411101	−0.411101
$236$	−0.354158	−0.354158
$237$	0	0
$238$	0	0
$239$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$239$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$240$	0.319474	0.319474
$241$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$241$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$242$	0.929506	0.929506
$243$	−0.871921	−0.871921
$244$	0	0
$245$	−0.384427	−0.384427
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.819658	0.819658
$250$	−0.485587	−0.485587
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0.139837	0.139837
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0.528637	0.528637
$257$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$257$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$258$	−2.60530	−2.60530
$259$	1.56296	1.56296
$260$	0	0
$261$	−0.725257	−0.725257
$262$	−2.07294	−2.07294
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0.410630	0.410630
$265$	0.457388	0.457388
$266$	0	0
$267$	0	0
$268$	0.0139317	0.0139317
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.160223	−0.160223
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.360750	−0.360750
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.576141	0.576141
$279$	0	0
$280$	−0.333723	−0.333723
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−2.36247	−2.36247
$283$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$283$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.168961	0.168961
$289$	1.00000	1.00000
$290$	−0.384746	−0.384746
$291$	0	0
$292$	0	0
$293$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$293$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$294$	−2.20918	−2.20918
$295$	0.446667	0.446667
$296$	0.849793	0.849793
$297$	−0.244673	−0.244673
$298$	0	0
$299$	0	0
$300$	−0.208899	−0.208899
$301$	3.23337	3.23337
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$307$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$308$	0.113283	0.113283
$309$	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			$\pi$
$314$	−0.413962	−0.413962
$315$	−0.176363	−0.176363
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	2.62846	2.62846
$319$	−0.587539	−0.587539
$320$	−0.173861	−0.173861
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.227165	−0.227165
$325$	0	0
$326$	0	0
$327$	−2.08437	−2.08437
$328$	0	0
$329$	2.93200	2.93200
$330$	0.115121	0.115121
$331$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$331$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$332$	0.122945	0.122945
$333$	0.449092	0.449092
$334$	0.889426	0.889426
$335$	−0.0175708	−0.0175708
$336$	−2.27851	−2.27851
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.08714	−1.08714
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.10590	1.10590
$344$	1.75800	1.75800
$345$	0	0
$346$	1.86894	1.86894
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−0.340226	−0.340226
$349$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$349$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$350$	1.68484	1.68484
$351$	0	0
$352$	0.136877	0.136877
$353$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$353$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$354$	2.56685	2.56685
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.0958898	−0.0958898
$361$	1.00000	1.00000
$362$	1.44660	1.44660
$363$	−1.03665	−1.03665
$364$	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
$370$	0.238242	0.238242
$371$	−3.26212	−3.26212
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.541562	0.541562
$376$	1.59415	1.59415
$377$	0	0
$378$	1.14272	1.14272
$379$	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			$\pi$
$384$	−1.43495	−1.43495
$385$	−0.142874	−0.142874
$386$	2.16789	2.16789
$387$	0.929057	0.929057
$388$	0	0
$389$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$389$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$390$	0	0
$391$	0	0
$392$	1.49071	1.49071
$393$	2.31189	2.31189
$394$	0	0
$395$	0	0
$396$	0.0325501	0.0325501
$397$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$397$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$398$	1.18186	1.18186
$399$	0	0
$400$	1.08835	1.08835
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.100974	−0.100974
$403$	0	0
$404$	0	0
$405$	0.286503	0.286503
$406$	2.74403	2.74403
$407$	0.363814	0.363814
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.18566	−3.18566
$414$	0	0
$415$	−0.155060	−0.155060
$416$	0	0
$417$	−0.642554	−0.642554
$418$	0	0
$419$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$419$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$420$	−0.0827340	−0.0827340
$421$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$421$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$422$	0	0
$423$	0.842462	0.842462
$424$	−1.77363	−1.77363
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0.492861	0.492861
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.738159	0.738159
$433$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$433$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$434$	0	0
$435$	0.429096	0.429096
$436$	−0.312647	−0.312647
$437$	0	0
$438$	0	0
$439$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$439$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$440$	−0.0776814	−0.0776814
$441$	0.787798	0.787798
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.210674	0.210674
$445$	0	0
$446$	0	0
$447$	0	0
$448$	1.23999	1.23999
$449$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$449$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$450$	0.484111	0.484111
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$457$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$461$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$462$	−0.821051	−0.821051
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	1.77256	1.77256
$465$	0	0
$466$	−2.07294	−2.07294
$467$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$467$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$468$	0	0
$469$	0.125316	0.125316
$470$	0.446923	0.446923
$471$	0.461680	0.461680
$472$	−1.73206	−1.73206
$473$	0.752639	0.752639
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.937316	−0.937316
$478$	2.01664	2.01664
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.0999652	−0.0999652
$481$	0	0
$482$	−2.14879	−2.14879
$483$	0	0
$484$	−0.155494	−0.155494
$485$	0	0
$486$	0.947896	0.947896
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0.417924	0.417924
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.0410525	−0.0410525
$496$	0	0
$497$	0	0
$498$	−0.891079	−0.891079
$499$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$499$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$500$	0.0812321	0.0812321
$501$	−0.991951	−0.991951
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0.683892	0.683892
$505$	0	0
$506$	0	0
$507$	1.21245	1.21245
$508$	0	0
$509$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$509$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$510$	0	0
$511$	0	0
$512$	0.608815	0.608815
$513$	0	0
$514$	1.86894	1.86894
$515$	0	0
$516$	0.435831	0.435831
$517$	0.682488	0.682488
$518$	−1.69915	−1.69915
$519$	−2.08437	−2.08437
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0.788452	0.788452
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0.346775	0.346775
$525$	−1.87905	−1.87905
$526$	0	0
$527$	0	0
$528$	−0.530372	−0.530372
$529$	1.00000	1.00000
$530$	−0.497242	−0.497242
$531$	−0.915346	−0.915346
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0.0681349	0.0681349
$537$	0	0
$538$	0	0
$539$	0.638204	0.638204
$540$	0.0268030	0.0268030
$541$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$541$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$542$	0	0
$543$	−1.61335	−1.61335
$544$	0	0
$545$	0.394314	0.394314
$546$	0	0
$547$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$547$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$548$	0	0
$549$	0	0
$550$	0.392184	0.392184
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.265704	−0.265704
$556$	−0.0963805	−0.0963805
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0.431039	0.431039
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.395208	0.395208
$565$	0	0
$566$	1.67743	1.67743
$567$	−2.04335	−2.04335
$568$	0	0
$569$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$569$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$570$	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
$576$	0.356290	0.356290
$577$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$577$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$578$	−1.08714	−1.08714
$579$	−2.41779	−2.41779
$580$	0.0643627	0.0643627
$581$	1.10590	1.10590
$582$	0	0
$583$	−0.759330	−0.759330
$584$	0	0
$585$	0	0
$586$	−1.77840	−1.77840
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.369565	0.369565
$589$	0	0
$590$	−0.485587	−0.485587
$591$	0	0
$592$	−1.09760	−1.09760
$593$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$593$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$594$	0.265993	0.265993
$595$	0	0
$596$	0	0
$597$	−1.31810	−1.31810
$598$	0	0
$599$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$599$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$600$	−1.02165	−1.02165
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−3.51511	−3.51511
$603$	0.0360074	0.0360074
$604$	0	0
$605$	0.196110	0.196110
$606$	0	0
$607$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$607$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$608$	0	0
$609$	−3.06034	−3.06034
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$613$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$614$	1.86894	1.86894
$615$	0	0
$616$	0.554029	0.554029
$617$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$617$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$618$	0	0
$619$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$619$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.844940	0.844940
$626$	0	0
$627$	0	0
$628$	0.0692501	0.0692501
$629$	0	0
$630$	0.191731	0.191731
$631$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$631$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.439705	−0.439705
$637$	0	0
$638$	0.638734	0.638734
$639$	0	0
$640$	0.271459	0.271459
$641$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$641$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$642$	0	0
$643$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$643$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$644$	0	0
$645$	−0.549674	−0.549674
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.11098	−1.11098
$649$	−0.741532	−0.741532
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	2.26599	2.26599
$655$	−0.437355	−0.437355
$656$	0	0
$657$	0	0
$658$	−3.18748	−3.18748
$659$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$659$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$660$	−0.0192582	−0.0192582
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.56661	−1.56661
$663$	0	0
$664$	0.601282	0.601282
$665$	0	0
$666$	−0.488224	−0.488224
$667$	0	0
$668$	−0.148789	−0.148789
$669$	0	0
$670$	0.0191018	0.0191018
$671$	0	0
$672$	0.712958	0.712958
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.608750	0.608750
$676$	0.181863	0.181863
$677$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$677$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$683$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$684$	0	0
$685$	0	0
$686$	−1.20226	−1.20226
$687$	0	0
$688$	−2.27065	−2.27065
$689$	0	0
$690$	0	0
$691$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$691$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$692$	−0.312647	−0.312647
$693$	0.292789	0.292789
$694$	0	0
$695$	0.121556	0.121556
$696$	−1.66392	−1.66392
$697$	0	0
$698$	2.16789	2.16789
$699$	2.31189	2.31189
$700$	−0.281850	−0.281850
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.288634	0.288634
$705$	−0.498440	−0.498440
$706$	−0.0832805	−0.0832805
$707$	0	0
$708$	−0.429399	−0.429399
$709$	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
$717$	−2.24910	−2.24910
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.123852	0.123852
$721$	0	0
$722$	−1.08714	−1.08714
$723$	2.39648	2.39648
$724$	−0.241996	−0.241996
$725$	1.46180	1.46180
$726$	1.12698	1.12698
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0.191941	0.191941
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$733$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$734$	0	0
$735$	−0.466098	−0.466098
$736$	0	0
$737$	0.0291700	0.0291700
$738$	0	0
$739$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$739$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$740$	−0.0398545	−0.0398545
$741$	0	0
$742$	3.54636	3.54636
$743$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$743$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.317761	0.317761
$748$	0	0
$749$	0	0
$750$	−0.588751	−0.588751
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−2.05901	−2.05901
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−0.191161	−0.191161
$757$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$757$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$761$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$762$	0	0
$763$	−2.81227	−2.81227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.640947	0.640947
$769$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$769$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$770$	0.155323	0.155323
$771$	−2.08437	−2.08437
$772$	−0.362658	−0.362658
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.01001	−1.01001
$775$	0	0
$776$	0	0
$777$	1.89502	1.89502
$778$	−1.31810	−1.31810
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.991447	0.991447
$784$	−1.92541	−1.92541
$785$	−0.0873388	−0.0873388
$786$	−2.51334	−2.51334
$787$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$787$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.159191	0.159191
$793$	0	0
$794$	1.18186	1.18186
$795$	0.554560	0.554560
$796$	−0.197709	−0.197709
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.340552	−0.340552
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0.0168915	0.0168915
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$809$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$810$	−0.311467	−0.311467
$811$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$811$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$812$	−0.459039	−0.459039
$813$	0	0
$814$	−0.395515	−0.395515
$815$	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
$823$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$823$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$824$	0	0
$825$	−0.437391	−0.437391
$826$	3.46324	3.46324
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$829$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$830$	0.168571	0.168571
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0.698543	0.698543
$835$	0.187654	0.187654
$836$	0	0
$837$	0	0
$838$	−1.56661	−1.56661
$839$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$839$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$840$	−0.404623	−0.404623
$841$	1.38078	1.38078
$842$	−2.14879	−2.14879
$843$	0	0
$844$	0	0
$845$	−0.229367	−0.229367
$846$	−0.915870	−0.915870
$847$	−1.39867	−1.39867
$848$	2.29084	2.29084
$849$	−1.87079	−1.87079
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$853$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$859$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$860$	−0.0824488	−0.0824488
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.230974	−0.230974
$865$	0.394314	0.394314
$866$	−2.07294	−2.07294
$867$	1.21245	1.21245
$868$	0	0
$869$	0	0
$870$	−0.466485	−0.466485
$871$	0	0
$872$	−1.52905	−1.52905
$873$	0	0
$874$	0	0
$875$	0.730684	0.730684
$876$	0	0
$877$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$877$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$878$	−0.413962	−0.413962
$879$	1.98340	1.98340
$880$	0.100334	0.100334
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.856443	−0.856443
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0.541562	0.541562
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	1.03033	1.03033
$889$	0	0
$890$	0	0
$891$	−0.475636	−0.475636
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.93606	−1.93606
$897$	0	0
$898$	−1.31810	−1.31810
$899$	0	0
$900$	−0.0809851	−0.0809851
$901$	0	0
$902$	0	0
$903$	3.92031	3.92031
$904$	0	0
$905$	0.305207	0.305207
$906$	0	0
$907$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$907$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0.257422	0.257422
$914$	1.67743	1.67743
$915$	0	0
$916$	0	0
$917$	3.11924	3.11924
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2.08437	−2.08437
$922$	1.44660	1.44660
$923$	0	0
$924$	0.137351	0.137351
$925$	−0.905175	−0.905175
$926$	0	0
$927$	0	0
$928$	−0.554643	−0.554643
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0.346775	0.346775
$933$	0	0
$934$	−1.56661	−1.56661
$935$	0	0
$936$	0	0
$937$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$937$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$938$	−0.136235	−0.136235
$939$	0	0
$940$	−0.0747640	−0.0747640
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.501908	−0.501908
$943$	0	0
$944$	2.23714	2.23714
$945$	0.241094	0.241094
$946$	−0.818221	−0.818221
$947$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$947$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$948$	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			$\pi$
$954$	1.01899	1.01899
$955$	0	0
$956$	−0.337356	−0.337356
$957$	−0.712362	−0.712362
$958$	0	0
$959$	0	0
$960$	−0.210798	−0.210798
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.359463	0.359463
$965$	0.457388	0.457388
$966$	0	0
$967$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$967$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$968$	−0.760463	−0.760463
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.158570	−0.158570
$973$	−0.866945	−0.866945
$974$	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
$980$	−0.0699129	−0.0699129
$981$	−0.808059	−0.808059
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.55491	3.55491
$988$	0	0
$989$	0	0
$990$	0.0446296	0.0446296
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	1.74719	1.74719
$994$	0	0
$995$	0.249353	0.249353
$996$	0.149065	0.149065
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	2.11708	2.11708
$999$	−0.613922	−0.613922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.7	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.7	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.7	✓	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)

\(f(q)\)	\(=\)	\(q-1.08714 q^{2} +1.21245 q^{3} +0.181863 q^{4} -0.229367 q^{5} -1.31810 q^{6} +1.63586 q^{7} +0.889426 q^{8} +0.470037 q^{9} +O(q^{10})\)	\(q-1.08714 q^{2} +1.21245 q^{3} +0.181863 q^{4} -0.229367 q^{5} -1.31810 q^{6} +1.63586 q^{7} +0.889426 q^{8} +0.470037 q^{9} +0.249353 q^{10} +0.380782 q^{11} +0.220500 q^{12} -1.77840 q^{14} -0.278096 q^{15} -1.14879 q^{16} -0.510994 q^{18} -0.0417133 q^{20} +1.98340 q^{21} -0.413962 q^{22} +1.07838 q^{24} -0.947391 q^{25} -0.642554 q^{27} +0.297502 q^{28} -1.54298 q^{29} +0.302328 q^{30} +0.359463 q^{32} +0.461680 q^{33} -0.375212 q^{35} +0.0854822 q^{36} +0.955440 q^{37} -0.204005 q^{40} -2.15622 q^{42} +1.97656 q^{43} +0.0692501 q^{44} -0.107811 q^{45} +1.79233 q^{47} -1.39285 q^{48} +1.67603 q^{49} +1.02994 q^{50} -1.99413 q^{53} +0.698543 q^{54} -0.0873388 q^{55} +1.45497 q^{56} +1.67743 q^{58} -1.94739 q^{59} -0.0505753 q^{60} +0.768914 q^{63} +0.758004 q^{64} -0.501908 q^{66} +0.0766055 q^{67} +0.407906 q^{70} +0.418063 q^{72} -1.03869 q^{74} -1.14866 q^{75} +0.622906 q^{77} +0.263494 q^{80} -1.24910 q^{81} +0.676034 q^{83} +0.360706 q^{84} -2.14879 q^{86} -1.87079 q^{87} +0.338678 q^{88} +0.117205 q^{90} -1.94851 q^{94} +0.435831 q^{96} -1.82208 q^{98} +0.178982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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