LMFDB - Embedded newform 1151.1.b.a.1150.8

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			1150.8
Root			$0.229367$ 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)

$f(q)$	$=$	$q-0.818137 q^{2} -0.529963 q^{3} -0.330651 q^{4} -1.94739 q^{5} +0.433582 q^{6} +0.676034 q^{7} +1.08866 q^{8} -0.719139 q^{9} +O(q^{10})$	\(q-0.818137 q^{2} -0.529963 q^{3} -0.330651 q^{4} -1.94739 q^{5} +0.433582 q^{6} +0.676034 q^{7} +1.08866 q^{8} -0.719139 q^{9} +1.59323 q^{10} -1.85500 q^{11} +0.175233 q^{12} -0.553088 q^{14} +1.03205 q^{15} -0.560018 q^{16} +0.588355 q^{18} +0.643908 q^{20} -0.358273 q^{21} +1.51765 q^{22} -0.576947 q^{24} +2.79233 q^{25} +0.911080 q^{27} -0.223532 q^{28} +0.380782 q^{29} -0.844355 q^{30} -0.630484 q^{32} +0.983084 q^{33} -1.31650 q^{35} +0.237784 q^{36} -1.08714 q^{37} -2.12004 q^{40} +0.293116 q^{42} +1.90679 q^{43} +0.613360 q^{44} +1.40045 q^{45} +1.21245 q^{47} +0.296789 q^{48} -0.542978 q^{49} -2.28451 q^{50} +1.97656 q^{53} -0.745389 q^{54} +3.61242 q^{55} +0.735968 q^{56} -0.311532 q^{58} +1.79233 q^{59} -0.341247 q^{60} -0.486162 q^{63} +1.07584 q^{64} -0.804298 q^{66} -1.99413 q^{67} +1.07708 q^{70} -0.782895 q^{72} +0.889426 q^{74} -1.47983 q^{75} -1.25405 q^{77} +1.09057 q^{80} +0.236300 q^{81} -1.54298 q^{83} +0.118463 q^{84} -1.56002 q^{86} -0.201800 q^{87} -2.01946 q^{88} -1.14576 q^{90} -0.991951 q^{94} +0.334133 q^{96} +0.444231 q^{98} +1.33401 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$2$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$3$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$3$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$4$	−0.330651	−0.330651
$5$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$5$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$6$	0.433582	0.433582
$7$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$7$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$8$	1.08866	1.08866
$9$	−0.719139	−0.719139
$10$	1.59323	1.59323
$11$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$11$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$12$	0.175233	0.175233
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.553088	−0.553088
$15$	1.03205	1.03205
$16$	−0.560018	−0.560018
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.588355	0.588355
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.643908	0.643908
$21$	−0.358273	−0.358273
$22$	1.51765	1.51765
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.576947	−0.576947
$25$	2.79233	2.79233
$26$	0	0
$27$	0.911080	0.911080
$28$	−0.223532	−0.223532
$29$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$29$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$30$	−0.844355	−0.844355
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−0.630484	−0.630484
$33$	0.983084	0.983084
$34$	0	0
$35$	−1.31650	−1.31650
$36$	0.237784	0.237784
$37$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$37$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$38$	0	0
$39$	0	0
$40$	−2.12004	−2.12004
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0.293116	0.293116
$43$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$43$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$44$	0.613360	0.613360
$45$	1.40045	1.40045
$46$	0	0
$47$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$47$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$48$	0.296789	0.296789
$49$	−0.542978	−0.542978
$50$	−2.28451	−2.28451
$51$	0	0
$52$	0	0
$53$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$53$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$54$	−0.745389	−0.745389
$55$	3.61242	3.61242
$56$	0.735968	0.735968
$57$	0	0
$58$	−0.311532	−0.311532
$59$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$59$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$60$	−0.341247	−0.341247
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.486162	−0.486162
$64$	1.07584	1.07584
$65$	0	0
$66$	−0.804298	−0.804298
$67$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$67$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$68$	0	0
$69$	0	0
$70$	1.07708	1.07708
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.782895	−0.782895
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.889426	0.889426
$75$	−1.47983	−1.47983
$76$	0	0
$77$	−1.25405	−1.25405
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.09057	1.09057
$81$	0.236300	0.236300
$82$	0	0
$83$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$83$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$84$	0.118463	0.118463
$85$	0	0
$86$	−1.56002	−1.56002
$87$	−0.201800	−0.201800
$88$	−2.01946	−2.01946
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1.14576	−1.14576
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−0.991951	−0.991951
$95$	0	0
$96$	0.334133	0.334133
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.444231	0.444231
$99$	1.33401	1.33401
$100$	−0.923288	−0.923288
$101$	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
$105$	0.697697	0.697697
$106$	−1.61710	−1.61710
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.301250	−0.301250
$109$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$109$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$110$	−2.95546	−2.95546
$111$	0.576141	0.576141
$112$	−0.378591	−0.378591
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−0.125906	−0.125906
$117$	0	0
$118$	−1.46637	−1.46637
$119$	0	0
$120$	1.12354	1.12354
$121$	2.44104	2.44104
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.49037	−3.49037
$126$	0.397748	0.397748
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.249701	−0.249701
$129$	−1.01053	−1.01053
$130$	0	0
$131$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$131$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$132$	−0.325058	−0.325058
$133$	0	0
$134$	1.63147	1.63147
$135$	−1.77423	−1.77423
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$139$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$140$	0.435303	0.435303
$141$	−0.642554	−0.642554
$142$	0	0
$143$	0	0
$144$	0.402731	0.402731
$145$	−0.741532	−0.741532
$146$	0	0
$147$	0.287758	0.287758
$148$	0.359463	0.359463
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.21071	1.21071
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	1.02598	1.02598
$155$	0	0
$156$	0	0
$157$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$157$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$158$	0	0
$159$	−1.04750	−1.04750
$160$	1.22780	1.22780
$161$	0	0
$162$	−0.193326	−0.193326
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−1.91445	−1.91445
$166$	1.26237	1.26237
$167$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$167$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$168$	−0.390036	−0.390036
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−0.630484	−0.630484
$173$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$173$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$174$	0.165101	0.165101
$175$	1.88771	1.88771
$176$	1.03884	1.03884
$177$	−0.949869	−0.949869
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.463059	−0.463059
$181$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$181$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.11708	2.11708
$186$	0	0
$187$	0	0
$188$	−0.400899	−0.400899
$189$	0.615921	0.615921
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.570156	−0.570156
$193$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$193$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$194$	0	0
$195$	0	0
$196$	0.179537	0.179537
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.09140	−1.09140
$199$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$199$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$200$	3.03989	3.03989
$201$	1.05682	1.05682
$202$	0	0
$203$	0.257422	0.257422
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.570812	−0.570812
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.653553	−0.653553
$213$	0	0
$214$	0	0
$215$	−3.71327	−3.71327
$216$	0.991852	0.991852
$217$	0	0
$218$	−0.781681	−0.781681
$219$	0	0
$220$	−1.19445	−1.19445
$221$	0	0
$222$	−0.471363	−0.471363
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−0.426228	−0.426228
$225$	−2.00807	−2.00807
$226$	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
$231$	0.664598	0.664598
$232$	0.414541	0.414541
$233$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$233$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$234$	0	0
$235$	−2.36112	−2.36112
$236$	−0.592637	−0.592637
$237$	0	0
$238$	0	0
$239$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$239$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$240$	−0.577964	−0.577964
$241$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$241$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$242$	−1.99711	−1.99711
$243$	−1.03631	−1.03631
$244$	0	0
$245$	1.05739	1.05739
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.817721	0.817721
$250$	2.85560	2.85560
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0.160750	0.160750
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.871550	−0.871550
$257$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$257$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$258$	0.826752	0.826752
$259$	−0.734940	−0.734940
$260$	0	0
$261$	−0.273835	−0.273835
$262$	−1.33836	−1.33836
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.07024	1.07024
$265$	−3.84914	−3.84914
$266$	0	0
$267$	0	0
$268$	0.659362	0.659362
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.45156	1.45156
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.17979	−5.17979
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.40649	1.40649
$279$	0	0
$280$	−1.43322	−1.43322
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.525697	0.525697
$283$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$283$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.453406	0.453406
$289$	1.00000	1.00000
$290$	0.606675	0.606675
$291$	0	0
$292$	0	0
$293$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$293$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$294$	−0.235426	−0.235426
$295$	−3.49037	−3.49037
$296$	−1.18352	−1.18352
$297$	−1.69006	−1.69006
$298$	0	0
$299$	0	0
$300$	0.489309	0.489309
$301$	1.28906	1.28906
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$307$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$308$	0.414652	0.414652
$309$	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			$\pi$
$314$	1.51765	1.51765
$315$	0.946748	0.946748
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.857002	0.857002
$319$	−0.706353	−0.706353
$320$	−2.09508	−2.09508
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.0781331	−0.0781331
$325$	0	0
$326$	0	0
$327$	−0.506348	−0.506348
$328$	0	0
$329$	0.819658	0.819658
$330$	1.56628	1.56628
$331$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$331$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$332$	0.510188	0.510188
$333$	0.781801	0.781801
$334$	1.08866	1.08866
$335$	3.88335	3.88335
$336$	0.200639	0.200639
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.818137	−0.818137
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.04311	−1.04311
$344$	2.07584	2.07584
$345$	0	0
$346$	−0.781681	−0.781681
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.0667256	0.0667256
$349$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$349$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$350$	−1.54441	−1.54441
$351$	0	0
$352$	1.16955	1.16955
$353$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$353$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$354$	0.777123	0.777123
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.52460	1.52460
$361$	1.00000	1.00000
$362$	0.187654	0.187654
$363$	−1.29366	−1.29366
$364$	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
$370$	−1.73206	−1.73206
$371$	1.33622	1.33622
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	1.84977	1.84977
$376$	1.31994	1.31994
$377$	0	0
$378$	−0.503908	−0.503908
$379$	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			$\pi$
$384$	0.132333	0.132333
$385$	2.44212	2.44212
$386$	−1.61710	−1.61710
$387$	−1.37125	−1.37125
$388$	0	0
$389$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$389$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$390$	0	0
$391$	0	0
$392$	−0.591116	−0.591116
$393$	−0.866945	−0.866945
$394$	0	0
$395$	0	0
$396$	−0.441091	−0.441091
$397$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$397$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$398$	0.669349	0.669349
$399$	0	0
$400$	−1.56376	−1.56376
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−0.864621	−0.864621
$403$	0	0
$404$	0	0
$405$	−0.460169	−0.460169
$406$	−0.210606	−0.210606
$407$	2.01664	2.01664
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.21168	1.21168
$414$	0	0
$415$	3.00478	3.00478
$416$	0	0
$417$	0.911080	0.911080
$418$	0	0
$419$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$419$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$420$	−0.230695	−0.230695
$421$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$421$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$422$	0	0
$423$	−0.871921	−0.871921
$424$	2.15179	2.15179
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	3.03797	3.03797
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.510222	−0.510222
$433$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$433$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$434$	0	0
$435$	0.392984	0.392984
$436$	−0.315917	−0.315917
$437$	0	0
$438$	0	0
$439$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$439$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$440$	3.93268	3.93268
$441$	0.390477	0.390477
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−0.190502	−0.190502
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0.727304	0.727304
$449$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$449$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$450$	1.64288	1.64288
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$457$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$461$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$462$	−0.543732	−0.543732
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−0.213245	−0.213245
$465$	0	0
$466$	−1.33836	−1.33836
$467$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$467$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$468$	0	0
$469$	−1.34810	−1.34810
$470$	1.93172	1.93172
$471$	0.983084	0.983084
$472$	1.95123	1.95123
$473$	−3.53711	−3.53711
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.42142	−1.42142
$478$	−1.17897	−1.17897
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.650688	−0.650688
$481$	0	0
$482$	−1.56002	−1.56002
$483$	0	0
$484$	−0.807134	−0.807134
$485$	0	0
$486$	0.847844	0.847844
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−0.865091	−0.865091
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.59783	−2.59783
$496$	0	0
$497$	0	0
$498$	−0.669008	−0.669008
$499$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$499$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$500$	1.15410	1.15410
$501$	0.705196	0.705196
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.529263	−0.529263
$505$	0	0
$506$	0	0
$507$	−0.529963	−0.529963
$508$	0	0
$509$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$509$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$510$	0	0
$511$	0	0
$512$	0.962749	0.962749
$513$	0	0
$514$	−0.781681	−0.781681
$515$	0	0
$516$	0.334133	0.334133
$517$	−2.24910	−2.24910
$518$	0.601282	0.601282
$519$	−0.506348	−0.506348
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0.224035	0.224035
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.540899	−0.540899
$525$	−1.00042	−1.00042
$526$	0	0
$527$	0	0
$528$	−0.550545	−0.550545
$529$	1.00000	1.00000
$530$	3.14912	3.14912
$531$	−1.28894	−1.28894
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−2.17092	−2.17092
$537$	0	0
$538$	0	0
$539$	1.00723	1.00723
$540$	0.586651	0.586651
$541$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$541$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$542$	0	0
$543$	0.121556	0.121556
$544$	0	0
$545$	−1.86061	−1.86061
$546$	0	0
$547$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$547$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$548$	0	0
$549$	0	0
$550$	4.23778	4.23778
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.12197	−1.12197
$556$	0.568436	0.568436
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0.737265	0.737265
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.212461	0.212461
$565$	0	0
$566$	−0.311532	−0.311532
$567$	0.159747	0.159747
$568$	0	0
$569$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$569$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$570$	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
$576$	−0.773679	−0.773679
$577$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$577$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$578$	−0.818137	−0.818137
$579$	−1.04750	−1.04750
$580$	0.245189	0.245189
$581$	−1.04311	−1.04311
$582$	0	0
$583$	−3.66653	−3.66653
$584$	0	0
$585$	0	0
$586$	−0.553088	−0.553088
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.0951477	−0.0951477
$589$	0	0
$590$	2.85560	2.85560
$591$	0	0
$592$	0.608815	0.608815
$593$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$593$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$594$	1.38270	1.38270
$595$	0	0
$596$	0	0
$597$	0.433582	0.433582
$598$	0	0
$599$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$599$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$600$	−1.61103	−1.61103
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.05462	−1.05462
$603$	1.43406	1.43406
$604$	0	0
$605$	−4.75367	−4.75367
$606$	0	0
$607$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$607$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$608$	0	0
$609$	−0.136424	−0.136424
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$613$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$614$	−0.781681	−0.781681
$615$	0	0
$616$	−1.36522	−1.36522
$617$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$617$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$618$	0	0
$619$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$619$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.00478	4.00478
$626$	0	0
$627$	0	0
$628$	0.613360	0.613360
$629$	0	0
$630$	−0.774570	−0.774570
$631$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$631$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0.346359	0.346359
$637$	0	0
$638$	0.577894	0.577894
$639$	0	0
$640$	0.486266	0.486266
$641$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$641$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$642$	0	0
$643$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$643$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$644$	0	0
$645$	1.96790	1.96790
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.257250	0.257250
$649$	−3.32478	−3.32478
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.414262	0.414262
$655$	−3.18566	−3.18566
$656$	0	0
$657$	0	0
$658$	−0.670593	−0.670593
$659$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$659$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$660$	0.633015	0.633015
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.0626738	−0.0626738
$663$	0	0
$664$	−1.67977	−1.67977
$665$	0	0
$666$	−0.639621	−0.639621
$667$	0	0
$668$	0.439982	0.439982
$669$	0	0
$670$	−3.17712	−3.17712
$671$	0	0
$672$	0.225885	0.225885
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	2.54404	2.54404
$676$	−0.330651	−0.330651
$677$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$677$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$683$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$684$	0	0
$685$	0	0
$686$	0.853403	0.853403
$687$	0	0
$688$	−1.06784	−1.06784
$689$	0	0
$690$	0	0
$691$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$691$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$692$	−0.315917	−0.315917
$693$	0.901834	0.901834
$694$	0	0
$695$	3.34784	3.34784
$696$	−0.219691	−0.219691
$697$	0	0
$698$	−1.61710	−1.61710
$699$	−0.866945	−0.866945
$700$	−0.624174	−0.624174
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.99569	−1.99569
$705$	1.25130	1.25130
$706$	1.63147	1.63147
$707$	0	0
$708$	0.314076	0.314076
$709$	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
$717$	−0.763700	−0.763700
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.784275	−0.784275
$721$	0	0
$722$	−0.818137	−0.818137
$723$	−1.01053	−1.01053
$724$	0.0758405	0.0758405
$725$	1.06327	1.06327
$726$	1.05839	1.05839
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0.312906	0.312906
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$733$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$734$	0	0
$735$	−0.560378	−0.560378
$736$	0	0
$737$	3.69912	3.69912
$738$	0	0
$739$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$739$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$740$	−0.700014	−0.700014
$741$	0	0
$742$	−1.09321	−1.09321
$743$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$743$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.10962	1.10962
$748$	0	0
$749$	0	0
$750$	−1.51336	−1.51336
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−0.678995	−0.678995
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−0.203655	−0.203655
$757$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$757$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$761$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$762$	0	0
$763$	0.645909	0.645909
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.461889	0.461889
$769$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$769$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$770$	−1.99799	−1.99799
$771$	−0.506348	−0.506348
$772$	−0.653553	−0.653553
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.12187	1.12187
$775$	0	0
$776$	0	0
$777$	0.389491	0.389491
$778$	0.433582	0.433582
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.346923	0.346923
$784$	0.304078	0.304078
$785$	3.61242	3.61242
$786$	0.709280	0.709280
$787$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$787$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.45227	1.45227
$793$	0	0
$794$	0.669349	0.669349
$795$	2.03990	2.03990
$796$	0.270518	0.270518
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.76052	−1.76052
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−0.349438	−0.349438
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$809$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$810$	0.376482	0.376482
$811$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$811$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$812$	−0.0851168	−0.0851168
$813$	0	0
$814$	−1.64989	−1.64989
$815$	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
$823$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$823$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$824$	0	0
$825$	2.74510	2.74510
$826$	−0.991318	−0.991318
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$829$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$830$	−2.45832	−2.45832
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−0.745389	−0.745389
$835$	2.59130	2.59130
$836$	0	0
$837$	0	0
$838$	−0.0626738	−0.0626738
$839$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$839$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$840$	0.759552	0.759552
$841$	−0.855005	−0.855005
$842$	−1.56002	−1.56002
$843$	0	0
$844$	0	0
$845$	−1.94739	−1.94739
$846$	0.713351	0.713351
$847$	1.65023	1.65023
$848$	−1.10691	−1.10691
$849$	−0.201800	−0.201800
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$853$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$859$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$860$	1.22780	1.22780
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.574421	−0.574421
$865$	−1.86061	−1.86061
$866$	−1.33836	−1.33836
$867$	−0.529963	−0.529963
$868$	0	0
$869$	0	0
$870$	−0.321515	−0.321515
$871$	0	0
$872$	1.04014	1.04014
$873$	0	0
$874$	0	0
$875$	−2.35961	−2.35961
$876$	0	0
$877$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$877$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$878$	1.51765	1.51765
$879$	−0.358273	−0.358273
$880$	−2.02302	−2.02302
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.319464	−0.319464
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	1.84977	1.84977
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0.627219	0.627219
$889$	0	0
$890$	0	0
$891$	−0.438338	−0.438338
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−0.168807	−0.168807
$897$	0	0
$898$	0.433582	0.433582
$899$	0	0
$900$	0.663973	0.663973
$901$	0	0
$902$	0	0
$903$	−0.683152	−0.683152
$904$	0	0
$905$	0.446667	0.446667
$906$	0	0
$907$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$907$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	2.86223	2.86223
$914$	−0.311532	−0.311532
$915$	0	0
$916$	0	0
$917$	1.10590	1.10590
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−0.506348	−0.506348
$922$	0.187654	0.187654
$923$	0	0
$924$	−0.219750	−0.219750
$925$	−3.03564	−3.03564
$926$	0	0
$927$	0	0
$928$	−0.240077	−0.240077
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.540899	−0.540899
$933$	0	0
$934$	−0.0626738	−0.0626738
$935$	0	0
$936$	0	0
$937$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$937$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$938$	1.10293	1.10293
$939$	0	0
$940$	0.780706	0.780706
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.804298	−0.804298
$943$	0	0
$944$	−1.00374	−1.00374
$945$	−1.19944	−1.19944
$946$	2.89384	2.89384
$947$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$947$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$948$	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			$\pi$
$954$	1.16292	1.16292
$955$	0	0
$956$	−0.476483	−0.476483
$957$	0.374341	0.374341
$958$	0	0
$959$	0	0
$960$	1.11032	1.11032
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−0.630484	−0.630484
$965$	−3.84914	−3.84914
$966$	0	0
$967$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$967$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$968$	2.65746	2.65746
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.342658	0.342658
$973$	−1.16220	−1.16220
$974$	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
$980$	−0.349628	−0.349628
$981$	−0.687094	−0.687094
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.434388	−0.434388
$988$	0	0
$989$	0	0
$990$	2.12538	2.12538
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−0.0405981	−0.0405981
$994$	0	0
$995$	1.59323	1.59323
$996$	−0.270381	−0.270381
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.46637	−1.46637
$999$	−0.990467	−0.990467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.8	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.8	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.8	✓	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)

\(f(q)\)	\(=\)	\(q-0.818137 q^{2} -0.529963 q^{3} -0.330651 q^{4} -1.94739 q^{5} +0.433582 q^{6} +0.676034 q^{7} +1.08866 q^{8} -0.719139 q^{9} +O(q^{10})\)	\(q-0.818137 q^{2} -0.529963 q^{3} -0.330651 q^{4} -1.94739 q^{5} +0.433582 q^{6} +0.676034 q^{7} +1.08866 q^{8} -0.719139 q^{9} +1.59323 q^{10} -1.85500 q^{11} +0.175233 q^{12} -0.553088 q^{14} +1.03205 q^{15} -0.560018 q^{16} +0.588355 q^{18} +0.643908 q^{20} -0.358273 q^{21} +1.51765 q^{22} -0.576947 q^{24} +2.79233 q^{25} +0.911080 q^{27} -0.223532 q^{28} +0.380782 q^{29} -0.844355 q^{30} -0.630484 q^{32} +0.983084 q^{33} -1.31650 q^{35} +0.237784 q^{36} -1.08714 q^{37} -2.12004 q^{40} +0.293116 q^{42} +1.90679 q^{43} +0.613360 q^{44} +1.40045 q^{45} +1.21245 q^{47} +0.296789 q^{48} -0.542978 q^{49} -2.28451 q^{50} +1.97656 q^{53} -0.745389 q^{54} +3.61242 q^{55} +0.735968 q^{56} -0.311532 q^{58} +1.79233 q^{59} -0.341247 q^{60} -0.486162 q^{63} +1.07584 q^{64} -0.804298 q^{66} -1.99413 q^{67} +1.07708 q^{70} -0.782895 q^{72} +0.889426 q^{74} -1.47983 q^{75} -1.25405 q^{77} +1.09057 q^{80} +0.236300 q^{81} -1.54298 q^{83} +0.118463 q^{84} -1.56002 q^{86} -0.201800 q^{87} -2.01946 q^{88} -1.14576 q^{90} -0.991951 q^{94} +0.334133 q^{96} +0.444231 q^{98} +1.33401 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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