LMFDB - Embedded newform 1151.1.b.a.1150.3

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			1150.3
Root			$-0.0766055$ of defining polynomial
Character	$\chi$	$=$	1151.1150

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85500 q^{2} +1.63586 q^{3} +2.44104 q^{4} -1.99413 q^{5} -3.03453 q^{6} -0.229367 q^{7} -2.67314 q^{8} +1.67603 q^{9} +O(q^{10})$	\(q-1.85500 q^{2} +1.63586 q^{3} +2.44104 q^{4} -1.99413 q^{5} -3.03453 q^{6} -0.229367 q^{7} -2.67314 q^{8} +1.67603 q^{9} +3.69912 q^{10} +1.21245 q^{11} +3.99320 q^{12} +0.425477 q^{14} -3.26212 q^{15} +2.51765 q^{16} -3.10905 q^{18} -4.86776 q^{20} -0.375212 q^{21} -2.24910 q^{22} -4.37288 q^{24} +2.97656 q^{25} +1.10590 q^{27} -0.559894 q^{28} +1.79233 q^{29} +6.05124 q^{30} -1.99711 q^{32} +1.98340 q^{33} +0.457388 q^{35} +4.09127 q^{36} +0.380782 q^{37} +5.33060 q^{40} +0.696020 q^{42} -0.818137 q^{43} +2.95964 q^{44} -3.34223 q^{45} +1.90679 q^{47} +4.11852 q^{48} -0.947391 q^{49} -5.52153 q^{50} -1.08714 q^{53} -2.05144 q^{54} -2.41779 q^{55} +0.613130 q^{56} -3.32478 q^{58} +1.97656 q^{59} -7.96297 q^{60} -0.384427 q^{63} +1.18700 q^{64} -3.67921 q^{66} +0.955440 q^{67} -0.848456 q^{70} -4.48028 q^{72} -0.706353 q^{74} +4.86923 q^{75} -0.278096 q^{77} -5.02052 q^{80} +0.133055 q^{81} -1.94739 q^{83} -0.915908 q^{84} +1.51765 q^{86} +2.93200 q^{87} -3.24105 q^{88} +6.19986 q^{90} -3.53711 q^{94} -3.26699 q^{96} +1.75741 q^{98} +2.03211 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{14} - 2 q^{15} + 18 q^{16} - 3 q^{18} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 19 q^{25} - 2 q^{27} - 3 q^{28} - q^{29} - 4 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 16 q^{36} - q^{37} - 4 q^{40} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - q^{47} - 5 q^{48} + 19 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} - 4 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 3 q^{63} + 17 q^{64} - 4 q^{66} - q^{67} - 4 q^{70} - 6 q^{72} - 2 q^{74} - 3 q^{75} - 2 q^{77} - 5 q^{80} + 18 q^{81} - q^{83} - 6 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - 6 q^{90} - 2 q^{94} - 6 q^{96} - 3 q^{98} - 3 q^{99}+O(q^{100}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^14 - 2 * q^15 + 18 * q^16 - 3 * q^18 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 19 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - 4 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 16 * q^36 - q^37 - 4 * q^40 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - q^47 - 5 * q^48 + 19 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 - 4 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 3 * q^63 + 17 * q^64 - 4 * q^66 - q^67 - 4 * q^70 - 6 * q^72 - 2 * q^74 - 3 * q^75 - 2 * q^77 - 5 * q^80 + 18 * q^81 - q^83 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - 6 * q^90 - 2 * q^94 - 6 * q^96 - 3 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$2$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$3$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$3$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$4$	2.44104	2.44104
$5$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$5$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$6$	−3.03453	−3.03453
$7$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$7$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$8$	−2.67314	−2.67314
$9$	1.67603	1.67603
$10$	3.69912	3.69912
$11$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$11$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$12$	3.99320	3.99320
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.425477	0.425477
$15$	−3.26212	−3.26212
$16$	2.51765	2.51765
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.10905	−3.10905
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−4.86776	−4.86776
$21$	−0.375212	−0.375212
$22$	−2.24910	−2.24910
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−4.37288	−4.37288
$25$	2.97656	2.97656
$26$	0	0
$27$	1.10590	1.10590
$28$	−0.559894	−0.559894
$29$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$29$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$30$	6.05124	6.05124
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.99711	−1.99711
$33$	1.98340	1.98340
$34$	0	0
$35$	0.457388	0.457388
$36$	4.09127	4.09127
$37$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$37$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$38$	0	0
$39$	0	0
$40$	5.33060	5.33060
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0.696020	0.696020
$43$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$43$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$44$	2.95964	2.95964
$45$	−3.34223	−3.34223
$46$	0	0
$47$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$47$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$48$	4.11852	4.11852
$49$	−0.947391	−0.947391
$50$	−5.52153	−5.52153
$51$	0	0
$52$	0	0
$53$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$53$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$54$	−2.05144	−2.05144
$55$	−2.41779	−2.41779
$56$	0.613130	0.613130
$57$	0	0
$58$	−3.32478	−3.32478
$59$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$59$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$60$	−7.96297	−7.96297
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−0.384427	−0.384427
$64$	1.18700	1.18700
$65$	0	0
$66$	−3.67921	−3.67921
$67$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$67$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$68$	0	0
$69$	0	0
$70$	−0.848456	−0.848456
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−4.48028	−4.48028
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.706353	−0.706353
$75$	4.86923	4.86923
$76$	0	0
$77$	−0.278096	−0.278096
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−5.02052	−5.02052
$81$	0.133055	0.133055
$82$	0	0
$83$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$83$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$84$	−0.915908	−0.915908
$85$	0	0
$86$	1.51765	1.51765
$87$	2.93200	2.93200
$88$	−3.24105	−3.24105
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	6.19986	6.19986
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−3.53711	−3.53711
$95$	0	0
$96$	−3.26699	−3.26699
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.75741	1.75741
$99$	2.03211	2.03211
$100$	7.26591	7.26591
$101$	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
$105$	0.748222	0.748222
$106$	2.01664	2.01664
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.69954	2.69954
$109$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$109$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$110$	4.48501	4.48501
$111$	0.622906	0.622906
$112$	−0.577465	−0.577465
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	4.37516	4.37516
$117$	0	0
$118$	−3.66653	−3.66653
$119$	0	0
$120$	8.72010	8.72010
$121$	0.470037	0.470037
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.94152	−3.94152
$126$	0.713113	0.713113
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.204777	−0.204777
$129$	−1.33836	−1.33836
$130$	0	0
$131$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$131$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$132$	4.84156	4.84156
$133$	0	0
$134$	−1.77235	−1.77235
$135$	−2.20530	−2.20530
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$139$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$140$	1.11650	1.11650
$141$	3.11924	3.11924
$142$	0	0
$143$	0	0
$144$	4.21966	4.21966
$145$	−3.57414	−3.57414
$146$	0	0
$147$	−1.54980	−1.54980
$148$	0.929506	0.929506
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−9.03245	−9.03245
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0.515869	0.515869
$155$	0	0
$156$	0	0
$157$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$157$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$158$	0	0
$159$	−1.77840	−1.77840
$160$	3.98250	3.98250
$161$	0	0
$162$	−0.246818	−0.246818
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−3.95516	−3.95516
$166$	3.61242	3.61242
$167$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$167$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$168$	1.00299	1.00299
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−1.99711	−1.99711
$173$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$173$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$174$	−5.43888	−5.43888
$175$	−0.682724	−0.682724
$176$	3.05253	3.05253
$177$	3.23337	3.23337
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−8.15853	−8.15853
$181$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$181$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.759330	−0.759330
$186$	0	0
$187$	0	0
$188$	4.65456	4.65456
$189$	−0.253656	−0.253656
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.94176	1.94176
$193$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$193$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$194$	0	0
$195$	0	0
$196$	−2.31262	−2.31262
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.76957	−3.76957
$199$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$199$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$200$	−7.95677	−7.95677
$201$	1.56296	1.56296
$202$	0	0
$203$	−0.411101	−0.411101
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−1.38795	−1.38795
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−2.65374	−2.65374
$213$	0	0
$214$	0	0
$215$	1.63147	1.63147
$216$	−2.95622	−2.95622
$217$	0	0
$218$	2.86223	2.86223
$219$	0	0
$220$	−5.90192	−5.90192
$221$	0	0
$222$	−1.15549	−1.15549
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.458070	0.458070
$225$	4.98882	4.98882
$226$	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
$231$	−0.454926	−0.454926
$232$	−4.79116	−4.79116
$233$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$233$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$234$	0	0
$235$	−3.80240	−3.80240
$236$	4.82487	4.82487
$237$	0	0
$238$	0	0
$239$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$239$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$240$	−8.21287	−8.21287
$241$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$241$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$242$	−0.871921	−0.871921
$243$	−0.888236	−0.888236
$244$	0	0
$245$	1.88922	1.88922
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.18566	−3.18566
$250$	7.31154	7.31154
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−0.938402	−0.938402
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.807134	−0.807134
$257$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$257$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$258$	2.48266	2.48266
$259$	−0.0873388	−0.0873388
$260$	0	0
$261$	3.00401	3.00401
$262$	2.46836	2.46836
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−5.30191	−5.30191
$265$	2.16789	2.16789
$266$	0	0
$267$	0	0
$268$	2.33227	2.33227
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	4.09085	4.09085
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.60893	3.60893
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.25405	−1.25405
$279$	0	0
$280$	−1.22266	−1.22266
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−5.78621	−5.78621
$283$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$283$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−3.34722	−3.34722
$289$	1.00000	1.00000
$290$	6.63005	6.63005
$291$	0	0
$292$	0	0
$293$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$293$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$294$	2.87488	2.87488
$295$	−3.94152	−3.94152
$296$	−1.01789	−1.01789
$297$	1.34084	1.34084
$298$	0	0
$299$	0	0
$300$	11.8860	11.8860
$301$	0.187654	0.187654
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$307$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$308$	−0.678844	−0.678844
$309$	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			$\pi$
$314$	−2.24910	−2.24910
$315$	0.766597	0.766597
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	3.29894	3.29894
$319$	2.17311	2.17311
$320$	−2.36703	−2.36703
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.324794	0.324794
$325$	0	0
$326$	0	0
$327$	−2.52409	−2.52409
$328$	0	0
$329$	−0.437355	−0.437355
$330$	7.33684	7.33684
$331$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$331$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$332$	−4.75367	−4.75367
$333$	0.638204	0.638204
$334$	−2.67314	−2.67314
$335$	−1.90527	−1.90527
$336$	−0.944651	−0.944651
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.85500	−1.85500
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.446667	0.446667
$344$	2.18700	2.18700
$345$	0	0
$346$	2.86223	2.86223
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	7.15714	7.15714
$349$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$349$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$350$	1.26646	1.26646
$351$	0	0
$352$	−2.42140	−2.42140
$353$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$353$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$354$	−5.99793	−5.99793
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	8.93426	8.93426
$361$	1.00000	1.00000
$362$	−0.142104	−0.142104
$363$	0.768914	0.768914
$364$	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
$370$	1.40856	1.40856
$371$	0.249353	0.249353
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−6.44777	−6.44777
$376$	−5.09713	−5.09713
$377$	0	0
$378$	0.470533	0.470533
$379$	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			$\pi$
$384$	−0.334987	−0.334987
$385$	0.554560	0.554560
$386$	2.01664	2.01664
$387$	−1.37123	−1.37123
$388$	0	0
$389$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$389$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$390$	0	0
$391$	0	0
$392$	2.53251	2.53251
$393$	−2.17676	−2.17676
$394$	0	0
$395$	0	0
$396$	4.96046	4.96046
$397$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$397$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$398$	3.44104	3.44104
$399$	0	0
$400$	7.49393	7.49393
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−2.89931	−2.89931
$403$	0	0
$404$	0	0
$405$	−0.265330	−0.265330
$406$	0.762595	0.762595
$407$	0.461680	0.461680
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.453358	−0.453358
$414$	0	0
$415$	3.88335	3.88335
$416$	0	0
$417$	1.10590	1.10590
$418$	0	0
$419$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$419$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$420$	1.82644	1.82644
$421$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$421$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$422$	0	0
$423$	3.19585	3.19585
$424$	2.90607	2.90607
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−3.02639	−3.02639
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	2.78426	2.78426
$433$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$433$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$434$	0	0
$435$	−5.84679	−5.84679
$436$	−3.76648	−3.76648
$437$	0	0
$438$	0	0
$439$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$439$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$440$	6.46309	6.46309
$441$	−1.58786	−1.58786
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	1.52054	1.52054
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−0.272258	−0.272258
$449$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$449$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$450$	−9.25428	−9.25428
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$457$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$461$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$462$	0.843890	0.843890
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	4.51246	4.51246
$465$	0	0
$466$	2.46836	2.46836
$467$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$467$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$468$	0	0
$469$	−0.219146	−0.219146
$470$	7.05346	7.05346
$471$	1.98340	1.98340
$472$	−5.28363	−5.28363
$473$	−0.991951	−0.991951
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.82208	−1.82208
$478$	0.983084	0.983084
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	6.51480	6.51480
$481$	0	0
$482$	1.51765	1.51765
$483$	0	0
$484$	1.14738	1.14738
$485$	0	0
$486$	1.64768	1.64768
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−3.50452	−3.50452
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.05229	−4.05229
$496$	0	0
$497$	0	0
$498$	5.90941	5.90941
$499$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$499$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$500$	−9.62143	−9.62143
$501$	2.35734	2.35734
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.02763	1.02763
$505$	0	0
$506$	0	0
$507$	1.63586	1.63586
$508$	0	0
$509$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$509$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$510$	0	0
$511$	0	0
$512$	1.70202	1.70202
$513$	0	0
$514$	2.86223	2.86223
$515$	0	0
$516$	−3.26699	−3.26699
$517$	2.31189	2.31189
$518$	0.162014	0.162014
$519$	−2.52409	−2.52409
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−5.57245	−5.57245
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−3.24818	−3.24818
$525$	−1.11684	−1.11684
$526$	0	0
$527$	0	0
$528$	4.99350	4.99350
$529$	1.00000	1.00000
$530$	−4.02145	−4.02145
$531$	3.31278	3.31278
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−2.55403	−2.55403
$537$	0	0
$538$	0	0
$539$	−1.14866	−1.14866
$540$	−5.38324	−5.38324
$541$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$541$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$542$	0	0
$543$	0.125316	0.125316
$544$	0	0
$545$	3.07690	3.07690
$546$	0	0
$547$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$547$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$548$	0	0
$549$	0	0
$550$	−6.69459	−6.69459
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.24216	−1.24216
$556$	1.65023	1.65023
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	1.15154	1.15154
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	7.61421	7.61421
$565$	0	0
$566$	−3.32478	−3.32478
$567$	−0.0305185	−0.0305185
$568$	0	0
$569$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$569$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$570$	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
$576$	1.98945	1.98945
$577$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$577$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$578$	−1.85500	−1.85500
$579$	−1.77840	−1.77840
$580$	−8.72464	−8.72464
$581$	0.446667	0.446667
$582$	0	0
$583$	−1.31810	−1.31810
$584$	0	0
$585$	0	0
$586$	0.425477	0.425477
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−3.78312	−3.78312
$589$	0	0
$590$	7.31154	7.31154
$591$	0	0
$592$	0.958676	0.958676
$593$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$593$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$594$	−2.48727	−2.48727
$595$	0	0
$596$	0	0
$597$	−3.03453	−3.03453
$598$	0	0
$599$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$599$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$600$	−13.0162	−13.0162
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.348098	−0.348098
$603$	1.60135	1.60135
$604$	0	0
$605$	−0.937316	−0.937316
$606$	0	0
$607$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$607$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$608$	0	0
$609$	−0.672504	−0.672504
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$613$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$614$	2.86223	2.86223
$615$	0	0
$616$	0.743390	0.743390
$617$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$617$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$618$	0	0
$619$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$619$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.88335	4.88335
$626$	0	0
$627$	0	0
$628$	2.95964	2.95964
$629$	0	0
$630$	−1.42204	−1.42204
$631$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$631$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−4.34115	−4.34115
$637$	0	0
$638$	−4.03114	−4.03114
$639$	0	0
$640$	0.408353	0.408353
$641$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$641$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$642$	0	0
$643$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$643$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$644$	0	0
$645$	2.66886	2.66886
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.355676	−0.355676
$649$	2.39648	2.39648
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	4.68221	4.68221
$655$	2.65349	2.65349
$656$	0	0
$657$	0	0
$658$	0.811296	0.811296
$659$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$659$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$660$	−9.65471	−9.65471
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	3.18901	3.18901
$663$	0	0
$664$	5.20565	5.20565
$665$	0	0
$666$	−1.18387	−1.18387
$667$	0	0
$668$	3.51765	3.51765
$669$	0	0
$670$	3.53429	3.53429
$671$	0	0
$672$	0.749339	0.749339
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	3.29177	3.29177
$676$	2.44104	2.44104
$677$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$677$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$683$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$684$	0	0
$685$	0	0
$686$	−0.828569	−0.828569
$687$	0	0
$688$	−2.05978	−2.05978
$689$	0	0
$690$	0	0
$691$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$691$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$692$	−3.76648	−3.76648
$693$	−0.466098	−0.466098
$694$	0	0
$695$	−1.34810	−1.34810
$696$	−7.83765	−7.83765
$697$	0	0
$698$	2.01664	2.01664
$699$	−2.17676	−2.17676
$700$	−1.66656	−1.66656
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.43918	1.43918
$705$	−6.22018	−6.22018
$706$	−1.77235	−1.77235
$707$	0	0
$708$	7.89281	7.89281
$709$	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
$717$	−0.866945	−0.866945
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−8.41457	−8.41457
$721$	0	0
$722$	−1.85500	−1.85500
$723$	−1.33836	−1.33836
$724$	0.186997	0.186997
$725$	5.33498	5.33498
$726$	−1.42634	−1.42634
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.58608	−1.58608
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$733$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$734$	0	0
$735$	3.09050	3.09050
$736$	0	0
$737$	1.15842	1.15842
$738$	0	0
$739$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$739$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$740$	−1.85356	−1.85356
$741$	0	0
$742$	−0.462551	−0.462551
$743$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$743$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.26389	−3.26389
$748$	0	0
$749$	0	0
$750$	11.9607	11.9607
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	4.80063	4.80063
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−0.619185	−0.619185
$757$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$757$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$761$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$762$	0	0
$763$	0.353908	0.353908
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.32036	−1.32036
$769$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$769$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$770$	−1.02871	−1.02871
$771$	−2.52409	−2.52409
$772$	−2.65374	−2.65374
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.54363	2.54363
$775$	0	0
$776$	0	0
$777$	−0.142874	−0.142874
$778$	−3.03453	−3.03453
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.98213	1.98213
$784$	−2.38520	−2.38520
$785$	−2.41779	−2.41779
$786$	4.03790	4.03790
$787$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$787$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−5.43211	−5.43211
$793$	0	0
$794$	3.44104	3.44104
$795$	3.54636	3.54636
$796$	−4.52815	−4.52815
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−5.94451	−5.94451
$801$	0	0
$802$	0	0
$803$	0	0
$804$	3.81526	3.81526
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$809$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$810$	0.492188	0.492188
$811$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$811$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$812$	−1.00352	−1.00352
$813$	0	0
$814$	−0.856418	−0.856418
$815$	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
$823$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$823$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$824$	0	0
$825$	5.90371	5.90371
$826$	0.840980	0.840980
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$829$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$830$	−7.20364	−7.20364
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−2.05144	−2.05144
$835$	−2.87363	−2.87363
$836$	0	0
$837$	0	0
$838$	3.18901	3.18901
$839$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$839$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$840$	−2.00010	−2.00010
$841$	2.21245	2.21245
$842$	1.51765	1.51765
$843$	0	0
$844$	0	0
$845$	−1.99413	−1.99413
$846$	−5.92832	−5.92832
$847$	−0.107811	−0.107811
$848$	−2.73702	−2.73702
$849$	2.93200	2.93200
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$853$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$859$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$860$	3.98250	3.98250
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−2.20859	−2.20859
$865$	3.07690	3.07690
$866$	2.46836	2.46836
$867$	1.63586	1.63586
$868$	0	0
$869$	0	0
$870$	10.8458	10.8458
$871$	0	0
$872$	4.12460	4.12460
$873$	0	0
$874$	0	0
$875$	0.904055	0.904055
$876$	0	0
$877$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$877$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$878$	−2.24910	−2.24910
$879$	−0.375212	−0.375212
$880$	−6.08714	−6.08714
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.94549	2.94549
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−6.44777	−6.44777
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−1.66512	−1.66512
$889$	0	0
$890$	0	0
$891$	0.161323	0.161323
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.0469691	0.0469691
$897$	0	0
$898$	−3.03453	−3.03453
$899$	0	0
$900$	12.1779	12.1779
$901$	0	0
$902$	0	0
$903$	0.306975	0.306975
$904$	0	0
$905$	−0.152761	−0.152761
$906$	0	0
$907$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$907$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.36112	−2.36112
$914$	−3.32478	−3.32478
$915$	0	0
$916$	0	0
$917$	0.305207	0.305207
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2.52409	−2.52409
$922$	−0.142104	−0.142104
$923$	0	0
$924$	−1.11049	−1.11049
$925$	1.13342	1.13342
$926$	0	0
$927$	0	0
$928$	−3.57948	−3.57948
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−3.24818	−3.24818
$933$	0	0
$934$	3.18901	3.18901
$935$	0	0
$936$	0	0
$937$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$937$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$938$	0.406517	0.406517
$939$	0	0
$940$	−9.28181	−9.28181
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−3.67921	−3.67921
$943$	0	0
$944$	4.97629	4.97629
$945$	0.505823	0.505823
$946$	1.84007	1.84007
$947$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$947$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$948$	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			$\pi$
$954$	3.37996	3.37996
$955$	0	0
$956$	−1.29366	−1.29366
$957$	3.55491	3.55491
$958$	0	0
$959$	0	0
$960$	−3.87212	−3.87212
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.99711	−1.99711
$965$	2.16789	2.16789
$966$	0	0
$967$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$967$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$968$	−1.25648	−1.25648
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.16822	−2.16822
$973$	−0.155060	−0.155060
$974$	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
$980$	4.61167	4.61167
$981$	−2.58608	−2.58608
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.715451	−0.715451
$988$	0	0
$989$	0	0
$990$	7.51702	7.51702
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−2.81227	−2.81227
$994$	0	0
$995$	3.69912	3.69912
$996$	−7.77632	−7.77632
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−3.66653	−3.66653
$999$	0.421105	0.421105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.3	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.3	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.3	✓	20	1151.1150	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1151 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.85500 q^{2} +1.63586 q^{3} +2.44104 q^{4} -1.99413 q^{5} -3.03453 q^{6} -0.229367 q^{7} -2.67314 q^{8} +1.67603 q^{9} +O(q^{10})\)	\(q-1.85500 q^{2} +1.63586 q^{3} +2.44104 q^{4} -1.99413 q^{5} -3.03453 q^{6} -0.229367 q^{7} -2.67314 q^{8} +1.67603 q^{9} +3.69912 q^{10} +1.21245 q^{11} +3.99320 q^{12} +0.425477 q^{14} -3.26212 q^{15} +2.51765 q^{16} -3.10905 q^{18} -4.86776 q^{20} -0.375212 q^{21} -2.24910 q^{22} -4.37288 q^{24} +2.97656 q^{25} +1.10590 q^{27} -0.559894 q^{28} +1.79233 q^{29} +6.05124 q^{30} -1.99711 q^{32} +1.98340 q^{33} +0.457388 q^{35} +4.09127 q^{36} +0.380782 q^{37} +5.33060 q^{40} +0.696020 q^{42} -0.818137 q^{43} +2.95964 q^{44} -3.34223 q^{45} +1.90679 q^{47} +4.11852 q^{48} -0.947391 q^{49} -5.52153 q^{50} -1.08714 q^{53} -2.05144 q^{54} -2.41779 q^{55} +0.613130 q^{56} -3.32478 q^{58} +1.97656 q^{59} -7.96297 q^{60} -0.384427 q^{63} +1.18700 q^{64} -3.67921 q^{66} +0.955440 q^{67} -0.848456 q^{70} -4.48028 q^{72} -0.706353 q^{74} +4.86923 q^{75} -0.278096 q^{77} -5.02052 q^{80} +0.133055 q^{81} -1.94739 q^{83} -0.915908 q^{84} +1.51765 q^{86} +2.93200 q^{87} -3.24105 q^{88} +6.19986 q^{90} -3.53711 q^{94} -3.26699 q^{96} +1.75741 q^{98} +2.03211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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