LMFDB - Embedded newform 1151.1.b.a.1150.14

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			1150.14
Root			$0.818137$ 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)

$f(q)$	$=$	$q+0.955440 q^{2} +1.79233 q^{3} -0.0871351 q^{4} -1.33065 q^{5} +1.71246 q^{6} +1.90679 q^{7} -1.03869 q^{8} +2.21245 q^{9} +O(q^{10})$	\(q+0.955440 q^{2} +1.79233 q^{3} -0.0871351 q^{4} -1.33065 q^{5} +1.71246 q^{6} +1.90679 q^{7} -1.03869 q^{8} +2.21245 q^{9} -1.27136 q^{10} -1.54298 q^{11} -0.156175 q^{12} +1.82183 q^{14} -2.38497 q^{15} -0.905272 q^{16} +2.11386 q^{18} +0.115946 q^{20} +3.41760 q^{21} -1.47422 q^{22} -1.86168 q^{24} +0.770633 q^{25} +2.17311 q^{27} -0.166149 q^{28} +0.676034 q^{29} -2.27869 q^{30} +0.173759 q^{32} -2.76553 q^{33} -2.53728 q^{35} -0.192782 q^{36} -1.71914 q^{37} +1.38214 q^{40} +3.26531 q^{42} -1.99413 q^{43} +0.134448 q^{44} -2.94400 q^{45} -1.94739 q^{47} -1.62255 q^{48} +2.63586 q^{49} +0.736293 q^{50} +0.0766055 q^{53} +2.07628 q^{54} +2.05317 q^{55} -1.98057 q^{56} +0.645909 q^{58} -0.229367 q^{59} +0.207814 q^{60} +4.21869 q^{63} +1.07129 q^{64} -2.64230 q^{66} +1.44104 q^{67} -2.42421 q^{70} -2.29805 q^{72} -1.64253 q^{74} +1.38123 q^{75} -2.94214 q^{77} +1.20460 q^{80} +1.68249 q^{81} +1.63586 q^{83} -0.297793 q^{84} -1.90527 q^{86} +1.21168 q^{87} +1.60268 q^{88} -2.81282 q^{90} -1.86061 q^{94} +0.311433 q^{96} +2.51840 q^{98} -3.41376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$2$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$3$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$3$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$4$	−0.0871351	−0.0871351
$5$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$5$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$6$	1.71246	1.71246
$7$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$7$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$8$	−1.03869	−1.03869
$9$	2.21245	2.21245
$10$	−1.27136	−1.27136
$11$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$11$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$12$	−0.156175	−0.156175
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.82183	1.82183
$15$	−2.38497	−2.38497
$16$	−0.905272	−0.905272
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.11386	2.11386
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.115946	0.115946
$21$	3.41760	3.41760
$22$	−1.47422	−1.47422
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.86168	−1.86168
$25$	0.770633	0.770633
$26$	0	0
$27$	2.17311	2.17311
$28$	−0.166149	−0.166149
$29$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$29$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$30$	−2.27869	−2.27869
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.173759	0.173759
$33$	−2.76553	−2.76553
$34$	0	0
$35$	−2.53728	−2.53728
$36$	−0.192782	−0.192782
$37$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$37$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$38$	0	0
$39$	0	0
$40$	1.38214	1.38214
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	3.26531	3.26531
$43$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$43$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$44$	0.134448	0.134448
$45$	−2.94400	−2.94400
$46$	0	0
$47$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$47$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$48$	−1.62255	−1.62255
$49$	2.63586	2.63586
$50$	0.736293	0.736293
$51$	0	0
$52$	0	0
$53$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$53$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$54$	2.07628	2.07628
$55$	2.05317	2.05317
$56$	−1.98057	−1.98057
$57$	0	0
$58$	0.645909	0.645909
$59$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$59$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$60$	0.207814	0.207814
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	4.21869	4.21869
$64$	1.07129	1.07129
$65$	0	0
$66$	−2.64230	−2.64230
$67$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$67$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$68$	0	0
$69$	0	0
$70$	−2.42421	−2.42421
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.29805	−2.29805
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.64253	−1.64253
$75$	1.38123	1.38123
$76$	0	0
$77$	−2.94214	−2.94214
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.20460	1.20460
$81$	1.68249	1.68249
$82$	0	0
$83$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$83$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$84$	−0.297793	−0.297793
$85$	0	0
$86$	−1.90527	−1.90527
$87$	1.21168	1.21168
$88$	1.60268	1.60268
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−2.81282	−2.81282
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−1.86061	−1.86061
$95$	0	0
$96$	0.311433	0.311433
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	2.51840	2.51840
$99$	−3.41376	−3.41376
$100$	−0.0671492	−0.0671492
$101$	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
$105$	−4.54764	−4.54764
$106$	0.0731919	0.0731919
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.189354	−0.189354
$109$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$109$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$110$	1.96168	1.96168
$111$	−3.08127	−3.08127
$112$	−1.72617	−1.72617
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−0.0589063	−0.0589063
$117$	0	0
$118$	−0.219146	−0.219146
$119$	0	0
$120$	2.47725	2.47725
$121$	1.38078	1.38078
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.305207	0.305207
$126$	4.03070	4.03070
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.849793	0.849793
$129$	−3.57414	−3.57414
$130$	0	0
$131$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$131$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$132$	0.240975	0.240975
$133$	0	0
$134$	1.37683	1.37683
$135$	−2.89166	−2.89166
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$139$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$140$	0.221086	0.221086
$141$	−3.49037	−3.49037
$142$	0	0
$143$	0	0
$144$	−2.00287	−2.00287
$145$	−0.899565	−0.899565
$146$	0	0
$147$	4.72433	4.72433
$148$	0.149797	0.149797
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.31968	1.31968
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−2.81104	−2.81104
$155$	0	0
$156$	0	0
$157$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$157$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$158$	0	0
$159$	0.137302	0.137302
$160$	−0.231212	−0.231212
$161$	0	0
$162$	1.60752	1.60752
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	3.67995	3.67995
$166$	1.56296	1.56296
$167$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$167$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$168$	−3.54984	−3.54984
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.173759	0.173759
$173$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$173$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$174$	1.15768	1.15768
$175$	1.46944	1.46944
$176$	1.39682	1.39682
$177$	−0.411101	−0.411101
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0.256526	0.256526
$181$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$181$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.28758	2.28758
$186$	0	0
$187$	0	0
$188$	0.169686	0.169686
$189$	4.14368	4.14368
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.92010	1.92010
$193$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$193$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$194$	0	0
$195$	0	0
$196$	−0.229676	−0.229676
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.26165	−3.26165
$199$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$199$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$200$	−0.800450	−0.800450
$201$	2.58283	2.58283
$202$	0	0
$203$	1.28906	1.28906
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−4.34500	−4.34500
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.00667503	−0.00667503
$213$	0	0
$214$	0	0
$215$	2.65349	2.65349
$216$	−2.25720	−2.25720
$217$	0	0
$218$	−0.506348	−0.506348
$219$	0	0
$220$	−0.178903	−0.178903
$221$	0	0
$222$	−2.94396	−2.94396
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.331322	0.331322
$225$	1.70499	1.70499
$226$	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
$231$	−5.27329	−5.27329
$232$	−0.702191	−0.702191
$233$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$233$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$234$	0	0
$235$	2.59130	2.59130
$236$	0.0199859	0.0199859
$237$	0	0
$238$	0	0
$239$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$239$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$240$	2.15905	2.15905
$241$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$241$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$242$	1.31925	1.31925
$243$	0.842462	0.842462
$244$	0	0
$245$	−3.50741	−3.50741
$246$	0	0
$247$	0	0
$248$	0	0
$249$	2.93200	2.93200
$250$	0.291607	0.291607
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.367596	−0.367596
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.259363	−0.259363
$257$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$257$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$258$	−3.41488	−3.41488
$259$	−3.27804	−3.27804
$260$	0	0
$261$	1.49569	1.49569
$262$	1.88848	1.88848
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.87253	2.87253
$265$	−0.101935	−0.101935
$266$	0	0
$267$	0	0
$268$	−0.125565	−0.125565
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−2.76280	−2.76280
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.18907	−1.18907
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.15842	1.15842
$279$	0	0
$280$	2.63545	2.63545
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−3.33484	−3.33484
$283$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$283$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.384433	0.384433
$289$	1.00000	1.00000
$290$	−0.859480	−0.859480
$291$	0	0
$292$	0	0
$293$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$293$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$294$	4.51381	4.51381
$295$	0.305207	0.305207
$296$	1.78566	1.78566
$297$	−3.35307	−3.35307
$298$	0	0
$299$	0	0
$300$	−0.120354	−0.120354
$301$	−3.80240	−3.80240
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$307$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$308$	0.256364	0.256364
$309$	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			$\pi$
$314$	−1.47422	−1.47422
$315$	−5.61360	−5.61360
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.131184	0.131184
$319$	−1.04311	−1.04311
$320$	−1.42551	−1.42551
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.146604	−0.146604
$325$	0	0
$326$	0	0
$327$	−0.949869	−0.949869
$328$	0	0
$329$	−3.71327	−3.71327
$330$	3.51597	3.51597
$331$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$331$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$332$	−0.142541	−0.142541
$333$	−3.80351	−3.80351
$334$	−1.03869	−1.03869
$335$	−1.91753	−1.91753
$336$	−3.09386	−3.09386
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.955440	0.955440
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3.11924	3.11924
$344$	2.07129	2.07129
$345$	0	0
$346$	−0.506348	−0.506348
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−0.105580	−0.105580
$349$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$349$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$350$	1.40396	1.40396
$351$	0	0
$352$	−0.268106	−0.268106
$353$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$353$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$354$	−0.392783	−0.392783
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	3.05791	3.05791
$361$	1.00000	1.00000
$362$	−0.781681	−0.781681
$363$	2.47482	2.47482
$364$	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
$370$	2.18564	2.18564
$371$	0.146071	0.146071
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.547033	0.547033
$376$	2.02274	2.02274
$377$	0	0
$378$	3.95903	3.95903
$379$	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			$\pi$
$384$	1.52311	1.52311
$385$	3.91496	3.91496
$386$	0.0731919	0.0731919
$387$	−4.41192	−4.41192
$388$	0	0
$389$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$389$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$390$	0	0
$391$	0	0
$392$	−2.73785	−2.73785
$393$	3.54265	3.54265
$394$	0	0
$395$	0	0
$396$	0.297459	0.297459
$397$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$397$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$398$	0.912865	0.912865
$399$	0	0
$400$	−0.697633	−0.697633
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	2.46773	2.46773
$403$	0	0
$404$	0	0
$405$	−2.23880	−2.23880
$406$	1.23162	1.23162
$407$	2.65259	2.65259
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.437355	−0.437355
$414$	0	0
$415$	−2.17676	−2.17676
$416$	0	0
$417$	2.17311	2.17311
$418$	0	0
$419$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$419$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$420$	0.396259	0.396259
$421$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$421$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$422$	0	0
$423$	−4.30851	−4.30851
$424$	−0.0795695	−0.0795695
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	2.53525	2.53525
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.96726	−1.96726
$433$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$433$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$434$	0	0
$435$	−1.61232	−1.61232
$436$	0.0461784	0.0461784
$437$	0	0
$438$	0	0
$439$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$439$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$440$	−2.13261	−2.13261
$441$	5.83171	5.83171
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.268486	0.268486
$445$	0	0
$446$	0	0
$447$	0	0
$448$	2.04273	2.04273
$449$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$449$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$450$	1.62901	1.62901
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$457$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$461$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$462$	−5.03831	−5.03831
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−0.611995	−0.611995
$465$	0	0
$466$	1.88848	1.88848
$467$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$467$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$468$	0	0
$469$	2.74777	2.74777
$470$	2.47583	2.47583
$471$	−2.76553	−2.76553
$472$	0.238242	0.238242
$473$	3.07690	3.07690
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.169486	0.169486
$478$	0.363814	0.363814
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.414409	−0.414409
$481$	0	0
$482$	−1.90527	−1.90527
$483$	0	0
$484$	−0.120315	−0.120315
$485$	0	0
$486$	0.804922	0.804922
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−3.35112	−3.35112
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.54253	4.54253
$496$	0	0
$497$	0	0
$498$	2.80135	2.80135
$499$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$499$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$500$	−0.0265943	−0.0265943
$501$	−1.94851	−1.94851
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−4.38191	−4.38191
$505$	0	0
$506$	0	0
$507$	1.79233	1.79233
$508$	0	0
$509$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$509$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$510$	0	0
$511$	0	0
$512$	−1.09760	−1.09760
$513$	0	0
$514$	−0.506348	−0.506348
$515$	0	0
$516$	0.311433	0.311433
$517$	3.00478	3.00478
$518$	−3.13197	−3.13197
$519$	−0.949869	−0.949869
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.42904	1.42904
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.172228	−0.172228
$525$	2.63372	2.63372
$526$	0	0
$527$	0	0
$528$	2.50356	2.50356
$529$	1.00000	1.00000
$530$	−0.0973929	−0.0973929
$531$	−0.507463	−0.507463
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.49680	−1.49680
$537$	0	0
$538$	0	0
$539$	−4.06707	−4.06707
$540$	0.251965	0.251965
$541$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$541$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$542$	0	0
$543$	−1.46637	−1.46637
$544$	0	0
$545$	0.705196	0.705196
$546$	0	0
$547$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$547$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$548$	0	0
$549$	0	0
$550$	−1.13608	−1.13608
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.10009	4.10009
$556$	−0.105647	−0.105647
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	2.29693	2.29693
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.304134	0.304134
$565$	0	0
$566$	0.645909	0.645909
$567$	3.20816	3.20816
$568$	0	0
$569$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$569$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$570$	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
$576$	2.37017	2.37017
$577$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$577$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$578$	0.955440	0.955440
$579$	0.137302	0.137302
$580$	0.0783837	0.0783837
$581$	3.11924	3.11924
$582$	0	0
$583$	−0.118201	−0.118201
$584$	0	0
$585$	0	0
$586$	1.82183	1.82183
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−0.411655	−0.411655
$589$	0	0
$590$	0.291607	0.291607
$591$	0	0
$592$	1.55629	1.55629
$593$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$593$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$594$	−3.20365	−3.20365
$595$	0	0
$596$	0	0
$597$	1.71246	1.71246
$598$	0	0
$599$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$599$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$600$	−1.43467	−1.43467
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−3.63296	−3.63296
$603$	3.18824	3.18824
$604$	0	0
$605$	−1.83734	−1.83734
$606$	0	0
$607$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$607$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$608$	0	0
$609$	2.31042	2.31042
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$613$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$614$	−0.506348	−0.506348
$615$	0	0
$616$	3.05598	3.05598
$617$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$617$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$618$	0	0
$619$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$619$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.17676	−1.17676
$626$	0	0
$627$	0	0
$628$	0.134448	0.134448
$629$	0	0
$630$	−5.36346	−5.36346
$631$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$631$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.0119639	−0.0119639
$637$	0	0
$638$	−0.996624	−0.996624
$639$	0	0
$640$	−1.13078	−1.13078
$641$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$641$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$642$	0	0
$643$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$643$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$644$	0	0
$645$	4.75594	4.75594
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.74759	−1.74759
$649$	0.353908	0.353908
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−0.907543	−0.907543
$655$	−2.63011	−2.63011
$656$	0	0
$657$	0	0
$658$	−3.54781	−3.54781
$659$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$659$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$660$	−0.320653	−0.320653
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.77235	−1.77235
$663$	0	0
$664$	−1.69915	−1.69915
$665$	0	0
$666$	−3.63403	−3.63403
$667$	0	0
$668$	0.0947276	0.0947276
$669$	0	0
$670$	−1.83208	−1.83208
$671$	0	0
$672$	0.593839	0.593839
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	1.67467	1.67467
$676$	−0.0871351	−0.0871351
$677$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$677$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$683$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$684$	0	0
$685$	0	0
$686$	2.98025	2.98025
$687$	0	0
$688$	1.80523	1.80523
$689$	0	0
$690$	0	0
$691$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$691$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$692$	0.0461784	0.0461784
$693$	−6.50934	−6.50934
$694$	0	0
$695$	−1.61335	−1.61335
$696$	−1.25856	−1.25856
$697$	0	0
$698$	0.0731919	0.0731919
$699$	3.54265	3.54265
$700$	−0.128040	−0.128040
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.65297	−1.65297
$705$	4.64446	4.64446
$706$	1.37683	1.37683
$707$	0	0
$708$	0.0358214	0.0358214
$709$	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
$717$	0.682488	0.682488
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	2.66512	2.66512
$721$	0	0
$722$	0.955440	0.955440
$723$	−3.57414	−3.57414
$724$	0.0712885	0.0712885
$725$	0.520974	0.520974
$726$	2.36454	2.36454
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.172517	−0.172517
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$733$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$734$	0	0
$735$	−6.28644	−6.28644
$736$	0	0
$737$	−2.22350	−2.22350
$738$	0	0
$739$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$739$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$740$	−0.199328	−0.199328
$741$	0	0
$742$	0.139562	0.139562
$743$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$743$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.61926	3.61926
$748$	0	0
$749$	0	0
$750$	0.522657	0.522657
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.76292	1.76292
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−0.361060	−0.361060
$757$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$757$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$761$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$762$	0	0
$763$	−1.01053	−1.01053
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−0.464864	−0.464864
$769$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$769$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$770$	3.74051	3.74051
$771$	−0.949869	−0.949869
$772$	−0.00667503	−0.00667503
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−4.21532	−4.21532
$775$	0	0
$776$	0	0
$777$	−5.87534	−5.87534
$778$	1.71246	1.71246
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.46910	1.46910
$784$	−2.38617	−2.38617
$785$	2.05317	2.05317
$786$	3.38479	3.38479
$787$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$787$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	3.54585	3.54585
$793$	0	0
$794$	0.912865	0.912865
$795$	−0.182702	−0.182702
$796$	−0.0832523	−0.0832523
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.133904	0.133904
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−0.225055	−0.225055
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$809$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$810$	−2.13904	−2.13904
$811$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$811$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$812$	−0.112322	−0.112322
$813$	0	0
$814$	2.53439	2.53439
$815$	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
$823$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$823$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$824$	0	0
$825$	−2.13121	−2.13121
$826$	−0.417866	−0.417866
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$829$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$830$	−2.07976	−2.07976
$831$	0	0
$832$	0	0
$833$	0	0
$834$	2.07628	2.07628
$835$	1.44660	1.44660
$836$	0	0
$837$	0	0
$838$	−1.77235	−1.77235
$839$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$839$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$840$	4.72360	4.72360
$841$	−0.542978	−0.542978
$842$	−1.90527	−1.90527
$843$	0	0
$844$	0	0
$845$	−1.33065	−1.33065
$846$	−4.11652	−4.11652
$847$	2.63287	2.63287
$848$	−0.0693488	−0.0693488
$849$	1.21168	1.21168
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$853$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$859$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$860$	−0.231212	−0.231212
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.377598	0.377598
$865$	0.705196	0.705196
$866$	1.88848	1.88848
$867$	1.79233	1.79233
$868$	0	0
$869$	0	0
$870$	−1.54047	−1.54047
$871$	0	0
$872$	0.550468	0.550468
$873$	0	0
$874$	0	0
$875$	0.581967	0.581967
$876$	0	0
$877$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$877$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$878$	−1.47422	−1.47422
$879$	3.41760	3.41760
$880$	−1.85867	−1.85867
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	5.57184	5.57184
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0.547033	0.547033
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	3.20049	3.20049
$889$	0	0
$890$	0	0
$891$	−2.59604	−2.59604
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.62038	1.62038
$897$	0	0
$898$	1.71246	1.71246
$899$	0	0
$900$	−0.148564	−0.148564
$901$	0	0
$902$	0	0
$903$	−6.81515	−6.81515
$904$	0	0
$905$	1.08866	1.08866
$906$	0	0
$907$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$907$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.52409	−2.52409
$914$	0.645909	0.645909
$915$	0	0
$916$	0	0
$917$	3.76889	3.76889
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−0.949869	−0.949869
$922$	−0.781681	−0.781681
$923$	0	0
$924$	0.459489	0.459489
$925$	−1.32483	−1.32483
$926$	0	0
$927$	0	0
$928$	0.117467	0.117467
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.172228	−0.172228
$933$	0	0
$934$	−1.77235	−1.77235
$935$	0	0
$936$	0	0
$937$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$937$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$938$	2.62533	2.62533
$939$	0	0
$940$	−0.225793	−0.225793
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−2.64230	−2.64230
$943$	0	0
$944$	0.207639	0.207639
$945$	−5.51379	−5.51379
$946$	2.93979	2.93979
$947$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$947$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$948$	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			$\pi$
$954$	0.161933	0.161933
$955$	0	0
$956$	−0.0331795	−0.0331795
$957$	−1.86959	−1.86959
$958$	0	0
$959$	0	0
$960$	−2.55499	−2.55499
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.173759	0.173759
$965$	−0.101935	−0.101935
$966$	0	0
$967$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$967$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$968$	−1.43421	−1.43421
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.0734080	−0.0734080
$973$	2.31189	2.31189
$974$	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
$980$	0.305618	0.305618
$981$	−1.17252	−1.17252
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.65541	−6.65541
$988$	0	0
$989$	0	0
$990$	4.34011	4.34011
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−3.32478	−3.32478
$994$	0	0
$995$	−1.27136	−1.27136
$996$	−0.255480	−0.255480
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−0.219146	−0.219146
$999$	−3.73588	−3.73588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.14	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.14	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.14	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.14	✓	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)

\(f(q)\)	\(=\)	\(q+0.955440 q^{2} +1.79233 q^{3} -0.0871351 q^{4} -1.33065 q^{5} +1.71246 q^{6} +1.90679 q^{7} -1.03869 q^{8} +2.21245 q^{9} +O(q^{10})\)	\(q+0.955440 q^{2} +1.79233 q^{3} -0.0871351 q^{4} -1.33065 q^{5} +1.71246 q^{6} +1.90679 q^{7} -1.03869 q^{8} +2.21245 q^{9} -1.27136 q^{10} -1.54298 q^{11} -0.156175 q^{12} +1.82183 q^{14} -2.38497 q^{15} -0.905272 q^{16} +2.11386 q^{18} +0.115946 q^{20} +3.41760 q^{21} -1.47422 q^{22} -1.86168 q^{24} +0.770633 q^{25} +2.17311 q^{27} -0.166149 q^{28} +0.676034 q^{29} -2.27869 q^{30} +0.173759 q^{32} -2.76553 q^{33} -2.53728 q^{35} -0.192782 q^{36} -1.71914 q^{37} +1.38214 q^{40} +3.26531 q^{42} -1.99413 q^{43} +0.134448 q^{44} -2.94400 q^{45} -1.94739 q^{47} -1.62255 q^{48} +2.63586 q^{49} +0.736293 q^{50} +0.0766055 q^{53} +2.07628 q^{54} +2.05317 q^{55} -1.98057 q^{56} +0.645909 q^{58} -0.229367 q^{59} +0.207814 q^{60} +4.21869 q^{63} +1.07129 q^{64} -2.64230 q^{66} +1.44104 q^{67} -2.42421 q^{70} -2.29805 q^{72} -1.64253 q^{74} +1.38123 q^{75} -2.94214 q^{77} +1.20460 q^{80} +1.68249 q^{81} +1.63586 q^{83} -0.297793 q^{84} -1.90527 q^{86} +1.21168 q^{87} +1.60268 q^{88} -2.81282 q^{90} -1.86061 q^{94} +0.311433 q^{96} +2.51840 q^{98} -3.41376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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