LMFDB - Embedded newform 1151.1.b.a.1150.13

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			1150.13
Root			$-0.380782$ 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)

$f(q)$	$=$	$q+0.676034 q^{2} -1.99413 q^{3} -0.542978 q^{4} -1.85500 q^{5} -1.34810 q^{6} -1.08714 q^{7} -1.04311 q^{8} +2.97656 q^{9} +O(q^{10})$	\(q+0.676034 q^{2} -1.99413 q^{3} -0.542978 q^{4} -1.85500 q^{5} -1.34810 q^{6} -1.08714 q^{7} -1.04311 q^{8} +2.97656 q^{9} -1.25405 q^{10} -0.229367 q^{11} +1.08277 q^{12} -0.734940 q^{14} +3.69912 q^{15} -0.162196 q^{16} +2.01226 q^{18} +1.00723 q^{20} +2.16789 q^{21} -0.155060 q^{22} +2.08009 q^{24} +2.44104 q^{25} -3.94152 q^{27} +0.590291 q^{28} -1.33065 q^{29} +2.50073 q^{30} +0.933455 q^{32} +0.457388 q^{33} +2.01664 q^{35} -1.61621 q^{36} +1.63586 q^{37} +1.93497 q^{40} +1.46557 q^{42} -1.71914 q^{43} +0.124541 q^{44} -5.52153 q^{45} +0.0766055 q^{47} +0.323440 q^{48} +0.181863 q^{49} +1.65023 q^{50} -0.529963 q^{53} -2.66460 q^{54} +0.425477 q^{55} +1.13400 q^{56} -0.899565 q^{58} +1.44104 q^{59} -2.00854 q^{60} -3.23592 q^{63} +0.793243 q^{64} +0.309210 q^{66} +1.21245 q^{67} +1.36332 q^{70} -3.10487 q^{72} +1.10590 q^{74} -4.86776 q^{75} +0.249353 q^{77} +0.300875 q^{80} +4.88335 q^{81} -0.818137 q^{83} -1.17712 q^{84} -1.16220 q^{86} +2.65349 q^{87} +0.239254 q^{88} -3.73274 q^{90} +0.0517879 q^{94} -1.86143 q^{96} +0.122945 q^{98} -0.682724 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$2$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$3$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$3$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$4$	−0.542978	−0.542978
$5$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$5$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$6$	−1.34810	−1.34810
$7$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$7$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$8$	−1.04311	−1.04311
$9$	2.97656	2.97656
$10$	−1.25405	−1.25405
$11$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$11$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$12$	1.08277	1.08277
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.734940	−0.734940
$15$	3.69912	3.69912
$16$	−0.162196	−0.162196
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.01226	2.01226
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.00723	1.00723
$21$	2.16789	2.16789
$22$	−0.155060	−0.155060
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.08009	2.08009
$25$	2.44104	2.44104
$26$	0	0
$27$	−3.94152	−3.94152
$28$	0.590291	0.590291
$29$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$29$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$30$	2.50073	2.50073
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.933455	0.933455
$33$	0.457388	0.457388
$34$	0	0
$35$	2.01664	2.01664
$36$	−1.61621	−1.61621
$37$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$37$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$38$	0	0
$39$	0	0
$40$	1.93497	1.93497
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	1.46557	1.46557
$43$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$43$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$44$	0.124541	0.124541
$45$	−5.52153	−5.52153
$46$	0	0
$47$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$47$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$48$	0.323440	0.323440
$49$	0.181863	0.181863
$50$	1.65023	1.65023
$51$	0	0
$52$	0	0
$53$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$53$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$54$	−2.66460	−2.66460
$55$	0.425477	0.425477
$56$	1.13400	1.13400
$57$	0	0
$58$	−0.899565	−0.899565
$59$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$59$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$60$	−2.00854	−2.00854
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−3.23592	−3.23592
$64$	0.793243	0.793243
$65$	0	0
$66$	0.309210	0.309210
$67$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$67$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$68$	0	0
$69$	0	0
$70$	1.36332	1.36332
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.10487	−3.10487
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.10590	1.10590
$75$	−4.86776	−4.86776
$76$	0	0
$77$	0.249353	0.249353
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.300875	0.300875
$81$	4.88335	4.88335
$82$	0	0
$83$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$83$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$84$	−1.17712	−1.17712
$85$	0	0
$86$	−1.16220	−1.16220
$87$	2.65349	2.65349
$88$	0.239254	0.239254
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.73274	−3.73274
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.0517879	0.0517879
$95$	0	0
$96$	−1.86143	−1.86143
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.122945	0.122945
$99$	−0.682724	−0.682724
$100$	−1.32543	−1.32543
$101$	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
$105$	−4.02145	−4.02145
$106$	−0.358273	−0.358273
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.14016	2.14016
$109$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$109$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$110$	0.287637	0.287637
$111$	−3.26212	−3.26212
$112$	0.176329	0.176329
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0.722515	0.722515
$117$	0	0
$118$	0.974194	0.974194
$119$	0	0
$120$	−3.85858	−3.85858
$121$	−0.947391	−0.947391
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.67314	−2.67314
$126$	−2.18759	−2.18759
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.397196	−0.397196
$129$	3.42819	3.42819
$130$	0	0
$131$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$131$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$132$	−0.248352	−0.248352
$133$	0	0
$134$	0.819658	0.819658
$135$	7.31154	7.31154
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$139$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$140$	−1.09499	−1.09499
$141$	−0.152761	−0.152761
$142$	0	0
$143$	0	0
$144$	−0.482787	−0.482787
$145$	2.46836	2.46836
$146$	0	0
$147$	−0.362658	−0.362658
$148$	−0.888236	−0.888236
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−3.29077	−3.29077
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0.168571	0.168571
$155$	0	0
$156$	0	0
$157$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$157$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$158$	0	0
$159$	1.05682	1.05682
$160$	−1.73156	−1.73156
$161$	0	0
$162$	3.30131	3.30131
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−0.848456	−0.848456
$166$	−0.553088	−0.553088
$167$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$167$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$168$	−2.26134	−2.26134
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.933455	0.933455
$173$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$173$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$174$	1.79385	1.79385
$175$	−2.65374	−2.65374
$176$	0.0372024	0.0372024
$177$	−2.87363	−2.87363
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	2.99807	2.99807
$181$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$181$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.03453	−3.03453
$186$	0	0
$187$	0	0
$188$	−0.0415951	−0.0415951
$189$	4.28497	4.28497
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.58183	−1.58183
$193$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$193$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$194$	0	0
$195$	0	0
$196$	−0.0987475	−0.0987475
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.461545	−0.461545
$199$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$199$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$200$	−2.54627	−2.54627
$201$	−2.41779	−2.41779
$202$	0	0
$203$	1.44660	1.44660
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−2.71863	−2.71863
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0.287758	0.287758
$213$	0	0
$214$	0	0
$215$	3.18901	3.18901
$216$	4.11142	4.11142
$217$	0	0
$218$	1.28906	1.28906
$219$	0	0
$220$	−0.231025	−0.231025
$221$	0	0
$222$	−2.20530	−2.20530
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.01479	−1.01479
$225$	7.26591	7.26591
$226$	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
$231$	−0.497242	−0.497242
$232$	1.38801	1.38801
$233$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$233$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$234$	0	0
$235$	−0.142104	−0.142104
$236$	−0.782455	−0.782455
$237$	0	0
$238$	0	0
$239$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$239$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$240$	−0.599984	−0.599984
$241$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$241$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$242$	−0.640468	−0.640468
$243$	−5.79653	−5.79653
$244$	0	0
$245$	−0.337356	−0.337356
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.63147	1.63147
$250$	−1.80713	−1.80713
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.75704	1.75704
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.06176	−1.06176
$257$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$257$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$258$	2.31757	2.31757
$259$	−1.77840	−1.77840
$260$	0	0
$261$	−3.96076	−3.96076
$262$	0.645909	0.645909
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.477104	−0.477104
$265$	0.983084	0.983084
$266$	0	0
$267$	0	0
$268$	−0.658335	−0.658335
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	4.94285	4.94285
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.559894	−0.559894
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.33622	1.33622
$279$	0	0
$280$	−2.10357	−2.10357
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−0.103272	−0.103272
$283$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$283$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	2.77849	2.77849
$289$	1.00000	1.00000
$290$	1.66870	1.66870
$291$	0	0
$292$	0	0
$293$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$293$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$294$	−0.245169	−0.245169
$295$	−2.67314	−2.67314
$296$	−1.70637	−1.70637
$297$	0.904055	0.904055
$298$	0	0
$299$	0	0
$300$	2.64309	2.64309
$301$	1.86894	1.86894
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$307$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$308$	−0.135393	−0.135393
$309$	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			$\pi$
$314$	−0.155060	−0.155060
$315$	6.00265	6.00265
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.714443	0.714443
$319$	0.305207	0.305207
$320$	−1.47147	−1.47147
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−2.65156	−2.65156
$325$	0	0
$326$	0	0
$327$	−3.80240	−3.80240
$328$	0	0
$329$	−0.0832805	−0.0832805
$330$	−0.573585	−0.573585
$331$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$331$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$332$	0.444231	0.444231
$333$	4.86923	4.86923
$334$	−1.04311	−1.04311
$335$	−2.24910	−2.24910
$336$	−0.351623	−0.351623
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.676034	0.676034
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.889426	0.889426
$344$	1.79324	1.79324
$345$	0	0
$346$	1.28906	1.28906
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.44079	−1.44079
$349$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$349$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$350$	−1.79402	−1.79402
$351$	0	0
$352$	−0.214104	−0.214104
$353$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$353$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$354$	−1.94267	−1.94267
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	5.75954	5.75954
$361$	1.00000	1.00000
$362$	0.257422	0.257422
$363$	1.88922	1.88922
$364$	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
$370$	−2.05144	−2.05144
$371$	0.576141	0.576141
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	5.33060	5.33060
$376$	−0.0799076	−0.0799076
$377$	0	0
$378$	2.89678	2.89678
$379$	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			$\pi$
$384$	0.792061	0.792061
$385$	−0.462551	−0.462551
$386$	−0.358273	−0.358273
$387$	−5.11712	−5.11712
$388$	0	0
$389$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$389$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$390$	0	0
$391$	0	0
$392$	−0.189702	−0.189702
$393$	−1.90527	−1.90527
$394$	0	0
$395$	0	0
$396$	0.370705	0.370705
$397$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$397$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$398$	0.457022	0.457022
$399$	0	0
$400$	−0.395928	−0.395928
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−1.63451	−1.63451
$403$	0	0
$404$	0	0
$405$	−9.05864	−9.05864
$406$	0.977949	0.977949
$407$	−0.375212	−0.375212
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.56661	−1.56661
$414$	0	0
$415$	1.51765	1.51765
$416$	0	0
$417$	−3.94152	−3.94152
$418$	0	0
$419$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$419$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$420$	2.18356	2.18356
$421$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$421$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$422$	0	0
$423$	0.228021	0.228021
$424$	0.552807	0.552807
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	2.15588	2.15588
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.639300	0.639300
$433$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$433$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$434$	0	0
$435$	−4.92224	−4.92224
$436$	−1.03535	−1.03535
$437$	0	0
$438$	0	0
$439$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$439$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$440$	−0.443817	−0.443817
$441$	0.541325	0.541325
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	1.77126	1.77126
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−0.862363	−0.862363
$449$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$449$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$450$	4.91200	4.91200
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$457$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$461$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$462$	−0.336153	−0.336153
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0.215827	0.215827
$465$	0	0
$466$	0.645909	0.645909
$467$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$467$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$468$	0	0
$469$	−1.31810	−1.31810
$470$	−0.0960668	−0.0960668
$471$	0.457388	0.457388
$472$	−1.50316	−1.50316
$473$	0.394314	0.394314
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.57747	−1.57747
$478$	−1.31650	−1.31650
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	3.45297	3.45297
$481$	0	0
$482$	−1.16220	−1.16220
$483$	0	0
$484$	0.514413	0.514413
$485$	0	0
$486$	−3.91865	−3.91865
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−0.228064	−0.228064
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.26646	1.26646
$496$	0	0
$497$	0	0
$498$	1.10293	1.10293
$499$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$499$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$500$	1.45146	1.45146
$501$	3.07690	3.07690
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	3.37541	3.37541
$505$	0	0
$506$	0	0
$507$	−1.99413	−1.99413
$508$	0	0
$509$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$509$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$510$	0	0
$511$	0	0
$512$	−0.320591	−0.320591
$513$	0	0
$514$	1.28906	1.28906
$515$	0	0
$516$	−1.86143	−1.86143
$517$	−0.0175708	−0.0175708
$518$	−1.20226	−1.20226
$519$	−3.80240	−3.80240
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.67761	−2.67761
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.518783	−0.518783
$525$	5.29191	5.29191
$526$	0	0
$527$	0	0
$528$	−0.0741865	−0.0741865
$529$	1.00000	1.00000
$530$	0.664598	0.664598
$531$	4.28935	4.28935
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−1.26471	−1.26471
$537$	0	0
$538$	0	0
$539$	−0.0417133	−0.0417133
$540$	−3.97001	−3.97001
$541$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$541$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$542$	0	0
$543$	−0.759330	−0.759330
$544$	0	0
$545$	−3.53711	−3.53711
$546$	0	0
$547$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$547$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$548$	0	0
$549$	0	0
$550$	−0.378508	−0.378508
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.05124	6.05124
$556$	−1.07323	−1.07323
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−0.327091	−0.327091
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.0829461	0.0829461
$565$	0	0
$566$	−0.899565	−0.899565
$567$	−5.30887	−5.30887
$568$	0	0
$569$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$569$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$570$	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
$576$	2.36114	2.36114
$577$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$577$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$578$	0.676034	0.676034
$579$	1.05682	1.05682
$580$	−1.34027	−1.34027
$581$	0.889426	0.889426
$582$	0	0
$583$	0.121556	0.121556
$584$	0	0
$585$	0	0
$586$	−0.734940	−0.734940
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.196916	0.196916
$589$	0	0
$590$	−1.80713	−1.80713
$591$	0	0
$592$	−0.265330	−0.265330
$593$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$593$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$594$	0.611171	0.611171
$595$	0	0
$596$	0	0
$597$	−1.34810	−1.34810
$598$	0	0
$599$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$599$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$600$	5.07759	5.07759
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.26346	1.26346
$603$	3.60893	3.60893
$604$	0	0
$605$	1.75741	1.75741
$606$	0	0
$607$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$607$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$608$	0	0
$609$	−2.88471	−2.88471
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$613$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$614$	1.28906	1.28906
$615$	0	0
$616$	−0.260101	−0.260101
$617$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$617$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$618$	0	0
$619$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$619$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.51765	2.51765
$626$	0	0
$627$	0	0
$628$	0.124541	0.124541
$629$	0	0
$630$	4.05800	4.05800
$631$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$631$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.573828	−0.573828
$637$	0	0
$638$	0.206330	0.206330
$639$	0	0
$640$	0.736801	0.736801
$641$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$641$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$642$	0	0
$643$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$643$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$644$	0	0
$645$	−6.35931	−6.35931
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−5.09385	−5.09385
$649$	−0.330528	−0.330528
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−2.57055	−2.57055
$655$	−1.77235	−1.77235
$656$	0	0
$657$	0	0
$658$	−0.0563004	−0.0563004
$659$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$659$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$660$	0.460693	0.460693
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.21168	1.21168
$663$	0	0
$664$	0.853403	0.853403
$665$	0	0
$666$	3.29177	3.29177
$667$	0	0
$668$	0.837804	0.837804
$669$	0	0
$670$	−1.52047	−1.52047
$671$	0	0
$672$	2.02363	2.02363
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−9.62143	−9.62143
$676$	−0.542978	−0.542978
$677$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$677$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$683$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$684$	0	0
$685$	0	0
$686$	0.601282	0.601282
$687$	0	0
$688$	0.278838	0.278838
$689$	0	0
$690$	0	0
$691$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$691$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$692$	−1.03535	−1.03535
$693$	0.742214	0.742214
$694$	0	0
$695$	−3.66653	−3.66653
$696$	−2.76787	−2.76787
$697$	0	0
$698$	−0.358273	−0.358273
$699$	−1.90527	−1.90527
$700$	1.44093	1.44093
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−0.181944	−0.181944
$705$	0.283373	0.283373
$706$	0.819658	0.819658
$707$	0	0
$708$	1.56032	1.56032
$709$	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
$717$	3.88335	3.88335
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.895572	0.895572
$721$	0	0
$722$	0.676034	0.676034
$723$	3.42819	3.42819
$724$	−0.206757	−0.206757
$725$	−3.24818	−3.24818
$726$	1.27718	1.27718
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	6.67568	6.67568
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$733$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$734$	0	0
$735$	0.672733	0.672733
$736$	0	0
$737$	−0.278096	−0.278096
$738$	0	0
$739$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$739$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$740$	1.64768	1.64768
$741$	0	0
$742$	0.389491	0.389491
$743$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$743$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.43524	−2.43524
$748$	0	0
$749$	0	0
$750$	3.60366	3.60366
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−0.0124251	−0.0124251
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−2.32664	−2.32664
$757$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$757$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$761$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$762$	0	0
$763$	−2.07294	−2.07294
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	2.11729	2.11729
$769$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$769$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$770$	−0.312700	−0.312700
$771$	−3.80240	−3.80240
$772$	0.287758	0.287758
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−3.45935	−3.45935
$775$	0	0
$776$	0	0
$777$	3.54636	3.54636
$778$	−1.34810	−1.34810
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.24479	5.24479
$784$	−0.0294974	−0.0294974
$785$	0.425477	0.425477
$786$	−1.28803	−1.28803
$787$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$787$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.712154	0.712154
$793$	0	0
$794$	0.457022	0.457022
$795$	−1.96040	−1.96040
$796$	−0.367072	−0.367072
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.27860	2.27860
$801$	0	0
$802$	0	0
$803$	0	0
$804$	1.31281	1.31281
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$809$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$810$	−6.12395	−6.12395
$811$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$811$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$812$	−0.785471	−0.785471
$813$	0	0
$814$	−0.253656	−0.253656
$815$	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
$823$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$823$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$824$	0	0
$825$	1.11650	1.11650
$826$	−1.05908	−1.05908
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$829$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$830$	1.02598	1.02598
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−2.66460	−2.66460
$835$	2.86223	2.86223
$836$	0	0
$837$	0	0
$838$	1.21168	1.21168
$839$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$839$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$840$	4.19479	4.19479
$841$	0.770633	0.770633
$842$	−1.16220	−1.16220
$843$	0	0
$844$	0	0
$845$	−1.85500	−1.85500
$846$	0.154150	0.154150
$847$	1.02994	1.02994
$848$	0.0859580	0.0859580
$849$	2.65349	2.65349
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$853$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$859$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$860$	−1.73156	−1.73156
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−3.67924	−3.67924
$865$	−3.53711	−3.53711
$866$	0.645909	0.645909
$867$	−1.99413	−1.99413
$868$	0	0
$869$	0	0
$870$	−3.32760	−3.32760
$871$	0	0
$872$	−1.98899	−1.98899
$873$	0	0
$874$	0	0
$875$	2.90607	2.90607
$876$	0	0
$877$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$877$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$878$	−0.155060	−0.155060
$879$	2.16789	2.16789
$880$	−0.0690107	−0.0690107
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.365954	0.365954
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	5.33060	5.33060
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	3.40273	3.40273
$889$	0	0
$890$	0	0
$891$	−1.12008	−1.12008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.431806	0.431806
$897$	0	0
$898$	−1.34810	−1.34810
$899$	0	0
$900$	−3.94523	−3.94523
$901$	0	0
$902$	0	0
$903$	−3.72691	−3.72691
$904$	0	0
$905$	−0.706353	−0.706353
$906$	0	0
$907$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$907$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0.187654	0.187654
$914$	−0.899565	−0.899565
$915$	0	0
$916$	0	0
$917$	−1.03869	−1.03869
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−3.80240	−3.80240
$922$	0.257422	0.257422
$923$	0	0
$924$	0.269992	0.269992
$925$	3.99320	3.99320
$926$	0	0
$927$	0	0
$928$	−1.24210	−1.24210
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.518783	−0.518783
$933$	0	0
$934$	1.21168	1.21168
$935$	0	0
$936$	0	0
$937$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$937$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$938$	−0.891079	−0.891079
$939$	0	0
$940$	0.0771591	0.0771591
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0.309210	0.309210
$943$	0	0
$944$	−0.233732	−0.233732
$945$	−7.94864	−7.94864
$946$	0.266569	0.266569
$947$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$947$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$948$	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			$\pi$
$954$	−1.06642	−1.06642
$955$	0	0
$956$	1.05739	1.05739
$957$	−0.608624	−0.608624
$958$	0	0
$959$	0	0
$960$	2.93431	2.93431
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.933455	0.933455
$965$	0.983084	0.983084
$966$	0	0
$967$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$967$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$968$	0.988229	0.988229
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	3.14739	3.14739
$973$	−2.14879	−2.14879
$974$	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
$980$	0.183177	0.183177
$981$	5.67568	5.67568
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.166072	0.166072
$988$	0	0
$989$	0	0
$990$	0.856168	0.856168
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−3.57414	−3.57414
$994$	0	0
$995$	−1.25405	−1.25405
$996$	−0.885855	−0.885855
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.974194	0.974194
$999$	−6.44777	−6.44777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.13	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.13	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.13	✓	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)

\(f(q)\)	\(=\)	\(q+0.676034 q^{2} -1.99413 q^{3} -0.542978 q^{4} -1.85500 q^{5} -1.34810 q^{6} -1.08714 q^{7} -1.04311 q^{8} +2.97656 q^{9} +O(q^{10})\)	\(q+0.676034 q^{2} -1.99413 q^{3} -0.542978 q^{4} -1.85500 q^{5} -1.34810 q^{6} -1.08714 q^{7} -1.04311 q^{8} +2.97656 q^{9} -1.25405 q^{10} -0.229367 q^{11} +1.08277 q^{12} -0.734940 q^{14} +3.69912 q^{15} -0.162196 q^{16} +2.01226 q^{18} +1.00723 q^{20} +2.16789 q^{21} -0.155060 q^{22} +2.08009 q^{24} +2.44104 q^{25} -3.94152 q^{27} +0.590291 q^{28} -1.33065 q^{29} +2.50073 q^{30} +0.933455 q^{32} +0.457388 q^{33} +2.01664 q^{35} -1.61621 q^{36} +1.63586 q^{37} +1.93497 q^{40} +1.46557 q^{42} -1.71914 q^{43} +0.124541 q^{44} -5.52153 q^{45} +0.0766055 q^{47} +0.323440 q^{48} +0.181863 q^{49} +1.65023 q^{50} -0.529963 q^{53} -2.66460 q^{54} +0.425477 q^{55} +1.13400 q^{56} -0.899565 q^{58} +1.44104 q^{59} -2.00854 q^{60} -3.23592 q^{63} +0.793243 q^{64} +0.309210 q^{66} +1.21245 q^{67} +1.36332 q^{70} -3.10487 q^{72} +1.10590 q^{74} -4.86776 q^{75} +0.249353 q^{77} +0.300875 q^{80} +4.88335 q^{81} -0.818137 q^{83} -1.17712 q^{84} -1.16220 q^{86} +2.65349 q^{87} +0.239254 q^{88} -3.73274 q^{90} +0.0517879 q^{94} -1.86143 q^{96} +0.122945 q^{98} -0.682724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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