LMFDB - Embedded newform 1151.1.b.a.1150.6

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			1150.6
Root			$1.94739$ 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)

$f(q)$	$=$	$q-1.33065 q^{2} -1.71914 q^{3} +0.770633 q^{4} +1.79233 q^{5} +2.28758 q^{6} -1.54298 q^{7} +0.305207 q^{8} +1.95544 q^{9} +O(q^{10})$	\(q-1.33065 q^{2} -1.71914 q^{3} +0.770633 q^{4} +1.79233 q^{5} +2.28758 q^{6} -1.54298 q^{7} +0.305207 q^{8} +1.95544 q^{9} -2.38497 q^{10} +1.44104 q^{11} -1.32483 q^{12} +2.05317 q^{14} -3.08127 q^{15} -1.17676 q^{16} -2.60201 q^{18} +1.38123 q^{20} +2.65259 q^{21} -1.91753 q^{22} -0.524694 q^{24} +2.21245 q^{25} -1.64253 q^{27} -1.18907 q^{28} -1.85500 q^{29} +4.10009 q^{30} +1.26065 q^{32} -2.47735 q^{33} -2.76553 q^{35} +1.50693 q^{36} -0.818137 q^{37} +0.547033 q^{40} -3.52968 q^{42} +1.63586 q^{43} +1.11052 q^{44} +3.50480 q^{45} -0.529963 q^{47} +2.02301 q^{48} +1.38078 q^{49} -2.94400 q^{50} +1.90679 q^{53} +2.18564 q^{54} +2.58283 q^{55} -0.470928 q^{56} +2.46836 q^{58} +1.21245 q^{59} -2.37453 q^{60} -3.01720 q^{63} -0.500724 q^{64} +3.29649 q^{66} +1.97656 q^{67} +3.67995 q^{70} +0.596814 q^{72} +1.08866 q^{74} -3.80351 q^{75} -2.22350 q^{77} -2.10914 q^{80} +0.868305 q^{81} +0.380782 q^{83} +2.04418 q^{84} -2.17676 q^{86} +3.18901 q^{87} +0.439817 q^{88} -4.66366 q^{90} +0.705196 q^{94} -2.16723 q^{96} -1.83734 q^{98} +2.81787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$2$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$3$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$3$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$4$	0.770633	0.770633
$5$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$5$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$6$	2.28758	2.28758
$7$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$7$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$8$	0.305207	0.305207
$9$	1.95544	1.95544
$10$	−2.38497	−2.38497
$11$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$11$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$12$	−1.32483	−1.32483
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	2.05317	2.05317
$15$	−3.08127	−3.08127
$16$	−1.17676	−1.17676
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−2.60201	−2.60201
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.38123	1.38123
$21$	2.65259	2.65259
$22$	−1.91753	−1.91753
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.524694	−0.524694
$25$	2.21245	2.21245
$26$	0	0
$27$	−1.64253	−1.64253
$28$	−1.18907	−1.18907
$29$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$29$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$30$	4.10009	4.10009
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.26065	1.26065
$33$	−2.47735	−2.47735
$34$	0	0
$35$	−2.76553	−2.76553
$36$	1.50693	1.50693
$37$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$37$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$38$	0	0
$39$	0	0
$40$	0.547033	0.547033
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−3.52968	−3.52968
$43$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$43$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$44$	1.11052	1.11052
$45$	3.50480	3.50480
$46$	0	0
$47$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$47$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$48$	2.02301	2.02301
$49$	1.38078	1.38078
$50$	−2.94400	−2.94400
$51$	0	0
$52$	0	0
$53$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$53$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$54$	2.18564	2.18564
$55$	2.58283	2.58283
$56$	−0.470928	−0.470928
$57$	0	0
$58$	2.46836	2.46836
$59$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$59$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$60$	−2.37453	−2.37453
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−3.01720	−3.01720
$64$	−0.500724	−0.500724
$65$	0	0
$66$	3.29649	3.29649
$67$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$67$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$68$	0	0
$69$	0	0
$70$	3.67995	3.67995
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.596814	0.596814
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.08866	1.08866
$75$	−3.80351	−3.80351
$76$	0	0
$77$	−2.22350	−2.22350
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−2.10914	−2.10914
$81$	0.868305	0.868305
$82$	0	0
$83$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$83$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$84$	2.04418	2.04418
$85$	0	0
$86$	−2.17676	−2.17676
$87$	3.18901	3.18901
$88$	0.439817	0.439817
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−4.66366	−4.66366
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.705196	0.705196
$95$	0	0
$96$	−2.16723	−2.16723
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.83734	−1.83734
$99$	2.81787	2.81787
$100$	1.70499	1.70499
$101$	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
$105$	4.75433	4.75433
$106$	−2.53728	−2.53728
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.26579	−1.26579
$109$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$109$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$110$	−3.43684	−3.43684
$111$	1.40649	1.40649
$112$	1.81571	1.81571
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.42953	−1.42953
$117$	0	0
$118$	−1.61335	−1.61335
$119$	0	0
$120$	−0.940425	−0.940425
$121$	1.07661	1.07661
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.17311	2.17311
$126$	4.01484	4.01484
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.594358	−0.594358
$129$	−2.81227	−2.81227
$130$	0	0
$131$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$131$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$132$	−1.90913	−1.90913
$133$	0	0
$134$	−2.63011	−2.63011
$135$	−2.94396	−2.94396
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$139$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$140$	−2.13121	−2.13121
$141$	0.911080	0.911080
$142$	0	0
$143$	0	0
$144$	−2.30108	−2.30108
$145$	−3.32478	−3.32478
$146$	0	0
$147$	−2.37376	−2.37376
$148$	−0.630484	−0.630484
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	5.06115	5.06115
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	2.95870	2.95870
$155$	0	0
$156$	0	0
$157$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$157$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$158$	0	0
$159$	−3.27804	−3.27804
$160$	2.25950	2.25950
$161$	0	0
$162$	−1.15541	−1.15541
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−4.44024	−4.44024
$166$	−0.506688	−0.506688
$167$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$167$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$168$	0.809591	0.809591
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	1.26065	1.26065
$173$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$173$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$174$	−4.24346	−4.24346
$175$	−3.41376	−3.41376
$176$	−1.69576	−1.69576
$177$	−2.08437	−2.08437
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	2.70091	2.70091
$181$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$181$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.46637	−1.46637
$186$	0	0
$187$	0	0
$188$	−0.408407	−0.408407
$189$	2.53439	2.53439
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0.860814	0.860814
$193$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$193$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$194$	0	0
$195$	0	0
$196$	1.06408	1.06408
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.74961	−3.74961
$199$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$199$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$200$	0.675256	0.675256
$201$	−3.39798	−3.39798
$202$	0	0
$203$	2.86223	2.86223
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−6.32635	−6.32635
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.46944	1.46944
$213$	0	0
$214$	0	0
$215$	2.93200	2.93200
$216$	−0.501313	−0.501313
$217$	0	0
$218$	1.44660	1.44660
$219$	0	0
$220$	1.99041	1.99041
$221$	0	0
$222$	−1.87155	−1.87155
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.94515	−1.94515
$225$	4.32631	4.32631
$226$	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
$231$	3.82250	3.82250
$232$	−0.566161	−0.566161
$233$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$233$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$234$	0	0
$235$	−0.949869	−0.949869
$236$	0.934355	0.934355
$237$	0	0
$238$	0	0
$239$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$239$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$240$	3.62590	3.62590
$241$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$241$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$242$	−1.43259	−1.43259
$243$	0.149797	0.149797
$244$	0	0
$245$	2.47482	2.47482
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.654618	−0.654618
$250$	−2.89166	−2.89166
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−2.32516	−2.32516
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.29161	1.29161
$257$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$257$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$258$	3.74215	3.74215
$259$	1.26237	1.26237
$260$	0	0
$261$	−3.62735	−3.62735
$262$	−0.899565	−0.899565
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.756107	−0.756107
$265$	3.41760	3.41760
$266$	0	0
$267$	0	0
$268$	1.52320	1.52320
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	3.91739	3.91739
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.18824	3.18824
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.27136	−1.27136
$279$	0	0
$280$	−0.844059	−0.844059
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−1.21233	−1.21233
$283$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$283$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	2.46512	2.46512
$289$	1.00000	1.00000
$290$	4.42413	4.42413
$291$	0	0
$292$	0	0
$293$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$293$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$294$	3.15864	3.15864
$295$	2.17311	2.17311
$296$	−0.249701	−0.249701
$297$	−2.36696	−2.36696
$298$	0	0
$299$	0	0
$300$	−2.93111	−2.93111
$301$	−2.52409	−2.52409
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$307$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$308$	−1.71350	−1.71350
$309$	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			$\pi$
$314$	−1.91753	−1.91753
$315$	−5.40782	−5.40782
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	4.36193	4.36193
$319$	−2.67314	−2.67314
$320$	−0.897463	−0.897463
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.669144	0.669144
$325$	0	0
$326$	0	0
$327$	1.86894	1.86894
$328$	0	0
$329$	0.817721	0.817721
$330$	5.90841	5.90841
$331$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$331$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$332$	0.293443	0.293443
$333$	−1.59982	−1.59982
$334$	0.305207	0.305207
$335$	3.54265	3.54265
$336$	−3.12146	−3.12146
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.33065	−1.33065
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−0.587539	−0.587539
$344$	0.499276	0.499276
$345$	0	0
$346$	1.44660	1.44660
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	2.45756	2.45756
$349$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$349$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$350$	4.54253	4.54253
$351$	0	0
$352$	1.81665	1.81665
$353$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$353$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$354$	2.77357	2.77357
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.06969	1.06969
$361$	1.00000	1.00000
$362$	2.59130	2.59130
$363$	−1.85083	−1.85083
$364$	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
$370$	1.95123	1.95123
$371$	−2.94214	−2.94214
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−3.73588	−3.73588
$376$	−0.161749	−0.161749
$377$	0	0
$378$	−3.37239	−3.37239
$379$	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			$\pi$
$384$	1.02178	1.02178
$385$	−3.98525	−3.98525
$386$	−2.53728	−2.53728
$387$	3.19882	3.19882
$388$	0	0
$389$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$389$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$390$	0	0
$391$	0	0
$392$	0.421425	0.421425
$393$	−1.16220	−1.16220
$394$	0	0
$395$	0	0
$396$	2.17155	2.17155
$397$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$397$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$398$	1.77063	1.77063
$399$	0	0
$400$	−2.60352	−2.60352
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	4.52153	4.52153
$403$	0	0
$404$	0	0
$405$	1.55629	1.55629
$406$	−3.80863	−3.80863
$407$	−1.17897	−1.17897
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.87079	−1.87079
$414$	0	0
$415$	0.682488	0.682488
$416$	0	0
$417$	−1.64253	−1.64253
$418$	0	0
$419$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$419$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$420$	3.66384	3.66384
$421$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$421$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$422$	0	0
$423$	−1.03631	−1.03631
$424$	0.581967	0.581967
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−3.90147	−3.90147
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.93286	1.93286
$433$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$433$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$434$	0	0
$435$	5.71576	5.71576
$436$	−0.837782	−0.837782
$437$	0	0
$438$	0	0
$439$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$439$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$440$	0.788298	0.788298
$441$	2.70004	2.70004
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	1.08389	1.08389
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0.772606	0.772606
$449$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$449$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$450$	−5.75682	−5.75682
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$457$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$461$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$462$	−5.08642	−5.08642
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	2.18289	2.18289
$465$	0	0
$466$	−0.899565	−0.899565
$467$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$467$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$468$	0	0
$469$	−3.04979	−3.04979
$470$	1.26394	1.26394
$471$	−2.47735	−2.47735
$472$	0.370049	0.370049
$473$	2.35734	2.35734
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.72862	3.72862
$478$	−0.101935	−0.101935
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−3.88439	−3.88439
$481$	0	0
$482$	−2.17676	−2.17676
$483$	0	0
$484$	0.829668	0.829668
$485$	0	0
$486$	−0.199328	−0.199328
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−3.29312	−3.29312
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	5.05056	5.05056
$496$	0	0
$497$	0	0
$498$	0.871068	0.871068
$499$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$499$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$500$	1.67467	1.67467
$501$	0.394314	0.394314
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.920872	−0.920872
$505$	0	0
$506$	0	0
$507$	−1.71914	−1.71914
$508$	0	0
$509$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$509$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$510$	0	0
$511$	0	0
$512$	−1.12432	−1.12432
$513$	0	0
$514$	1.44660	1.44660
$515$	0	0
$516$	−2.16723	−2.16723
$517$	−0.763700	−0.763700
$518$	−1.67977	−1.67977
$519$	1.86894	1.86894
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	4.82674	4.82674
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0.520974	0.520974
$525$	5.86874	5.86874
$526$	0	0
$527$	0	0
$528$	2.91525	2.91525
$529$	1.00000	1.00000
$530$	−4.54764	−4.54764
$531$	2.37087	2.37087
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0.603261	0.603261
$537$	0	0
$538$	0	0
$539$	1.98977	1.98977
$540$	−2.26872	−2.26872
$541$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$541$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$542$	0	0
$543$	3.34784	3.34784
$544$	0	0
$545$	−1.94851	−1.94851
$546$	0	0
$547$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$547$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$548$	0	0
$549$	0	0
$550$	−4.24243	−4.24243
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.52090	2.52090
$556$	0.736293	0.736293
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	3.25436	3.25436
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0.702109	0.702109
$565$	0	0
$566$	2.46836	2.46836
$567$	−1.33978	−1.33978
$568$	0	0
$569$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$569$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−0.979135	−0.979135
$577$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$577$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$578$	−1.33065	−1.33065
$579$	−3.27804	−3.27804
$580$	−2.56219	−2.56219
$581$	−0.587539	−0.587539
$582$	0	0
$583$	2.74777	2.74777
$584$	0	0
$585$	0	0
$586$	2.05317	2.05317
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−1.82930	−1.82930
$589$	0	0
$590$	−2.89166	−2.89166
$591$	0	0
$592$	0.962749	0.962749
$593$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$593$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$594$	3.14960	3.14960
$595$	0	0
$596$	0	0
$597$	2.28758	2.28758
$598$	0	0
$599$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$599$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$600$	−1.16086	−1.16086
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	3.35869	3.35869
$603$	3.86505	3.86505
$604$	0	0
$605$	1.92963	1.92963
$606$	0	0
$607$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$607$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$608$	0	0
$609$	−4.92058	−4.92058
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$613$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$614$	1.44660	1.44660
$615$	0	0
$616$	−0.678628	−0.678628
$617$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$617$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$618$	0	0
$619$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$619$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.68249	1.68249
$626$	0	0
$627$	0	0
$628$	1.11052	1.11052
$629$	0	0
$630$	7.19593	7.19593
$631$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$631$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−2.52617	−2.52617
$637$	0	0
$638$	3.55702	3.55702
$639$	0	0
$640$	−1.06529	−1.06529
$641$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$641$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$642$	0	0
$643$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$643$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$644$	0	0
$645$	−5.04052	−5.04052
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.265013	0.265013
$649$	1.74719	1.74719
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−2.48690	−2.48690
$655$	1.21168	1.21168
$656$	0	0
$657$	0	0
$658$	−1.08810	−1.08810
$659$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$659$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$660$	−3.42179	−3.42179
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	2.65349	2.65349
$663$	0	0
$664$	0.116218	0.116218
$665$	0	0
$666$	2.12880	2.12880
$667$	0	0
$668$	−0.176758	−0.176758
$669$	0	0
$670$	−4.71403	−4.71403
$671$	0	0
$672$	3.34399	3.34399
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.63403	−3.63403
$676$	0.770633	0.770633
$677$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$677$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$683$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$684$	0	0
$685$	0	0
$686$	0.781809	0.781809
$687$	0	0
$688$	−1.92501	−1.92501
$689$	0	0
$690$	0	0
$691$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$691$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$692$	−0.837782	−0.837782
$693$	−4.34792	−4.34792
$694$	0	0
$695$	1.71246	1.71246
$696$	0.973310	0.973310
$697$	0	0
$698$	−2.53728	−2.53728
$699$	−1.16220	−1.16220
$700$	−2.63076	−2.63076
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−0.721565	−0.721565
$705$	1.63296	1.63296
$706$	−2.63011	−2.63011
$707$	0	0
$708$	−1.60629	−1.60629
$709$	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
$717$	−0.131695	−0.131695
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−4.12429	−4.12429
$721$	0	0
$722$	−1.33065	−1.33065
$723$	−2.81227	−2.81227
$724$	−1.50072	−1.50072
$725$	−4.10411	−4.10411
$726$	2.46282	2.46282
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.12583	−1.12583
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$733$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$734$	0	0
$735$	−4.25456	−4.25456
$736$	0	0
$737$	2.84831	2.84831
$738$	0	0
$739$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$739$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$740$	−1.13004	−1.13004
$741$	0	0
$742$	3.91496	3.91496
$743$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$743$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.744597	0.744597
$748$	0	0
$749$	0	0
$750$	4.97116	4.97116
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0.623638	0.623638
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.95309	1.95309
$757$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$757$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$761$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$762$	0	0
$763$	1.67743	1.67743
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−2.22045	−2.22045
$769$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$769$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$770$	5.30297	5.30297
$771$	1.86894	1.86894
$772$	1.46944	1.46944
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−4.25652	−4.25652
$775$	0	0
$776$	0	0
$777$	−2.17019	−2.17019
$778$	2.28758	2.28758
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.04691	3.04691
$784$	−1.62485	−1.62485
$785$	2.58283	2.58283
$786$	1.54648	1.54648
$787$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$787$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.860035	0.860035
$793$	0	0
$794$	1.77063	1.77063
$795$	−5.87534	−5.87534
$796$	−1.02544	−1.02544
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.78912	2.78912
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−2.61860	−2.61860
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$809$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$810$	−2.07088	−2.07088
$811$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$811$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$812$	2.20573	2.20573
$813$	0	0
$814$	1.56880	1.56880
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$823$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$824$	0	0
$825$	−5.48102	−5.48102
$826$	2.48936	2.48936
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$829$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$830$	−0.908153	−0.908153
$831$	0	0
$832$	0	0
$833$	0	0
$834$	2.18564	2.18564
$835$	−0.411101	−0.411101
$836$	0	0
$837$	0	0
$838$	2.65349	2.65349
$839$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$839$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$840$	1.45106	1.45106
$841$	2.44104	2.44104
$842$	−2.17676	−2.17676
$843$	0	0
$844$	0	0
$845$	1.79233	1.79233
$846$	1.37897	1.37897
$847$	−1.66118	−1.66118
$848$	−2.24383	−2.24383
$849$	3.18901	3.18901
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$853$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$859$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$860$	2.25950	2.25950
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−2.07066	−2.07066
$865$	−1.94851	−1.94851
$866$	−0.899565	−0.899565
$867$	−1.71914	−1.71914
$868$	0	0
$869$	0	0
$870$	−7.60569	−7.60569
$871$	0	0
$872$	−0.331802	−0.331802
$873$	0	0
$874$	0	0
$875$	−3.35307	−3.35307
$876$	0	0
$877$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$877$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$878$	−1.91753	−1.91753
$879$	2.65259	2.65259
$880$	−3.03936	−3.03936
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−3.59281	−3.59281
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−3.73588	−3.73588
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0.429272	0.429272
$889$	0	0
$890$	0	0
$891$	1.25126	1.25126
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.917081	0.917081
$897$	0	0
$898$	2.28758	2.28758
$899$	0	0
$900$	3.33400	3.33400
$901$	0	0
$902$	0	0
$903$	4.33927	4.33927
$904$	0	0
$905$	−3.49037	−3.49037
$906$	0	0
$907$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$907$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0.548724	0.548724
$914$	2.46836	2.46836
$915$	0	0
$916$	0	0
$917$	−1.04311	−1.04311
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.86894	1.86894
$922$	2.59130	2.59130
$923$	0	0
$924$	2.94575	2.94575
$925$	−1.81009	−1.81009
$926$	0	0
$927$	0	0
$928$	−2.33851	−2.33851
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0.520974	0.520974
$933$	0	0
$934$	2.65349	2.65349
$935$	0	0
$936$	0	0
$937$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$937$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$938$	4.05821	4.05821
$939$	0	0
$940$	−0.732001	−0.732001
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	3.29649	3.29649
$943$	0	0
$944$	−1.42676	−1.42676
$945$	4.54247	4.54247
$946$	−3.13680	−3.13680
$947$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$947$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$948$	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			$\pi$
$954$	−4.96149	−4.96149
$955$	0	0
$956$	0.0590347	0.0590347
$957$	4.59550	4.59550
$958$	0	0
$959$	0	0
$960$	1.54286	1.54286
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	1.26065	1.26065
$965$	3.41760	3.41760
$966$	0	0
$967$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$967$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$968$	0.328588	0.328588
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.115439	0.115439
$973$	−1.47422	−1.47422
$974$	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
$980$	1.90718	1.90718
$981$	−2.12583	−2.12583
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.40578	−1.40578
$988$	0	0
$989$	0	0
$990$	−6.72054	−6.72054
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	3.42819	3.42819
$994$	0	0
$995$	−2.38497	−2.38497
$996$	−0.504470	−0.504470
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.61335	−1.61335
$999$	1.34382	1.34382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.6	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.6	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.33065 q^{2} -1.71914 q^{3} +0.770633 q^{4} +1.79233 q^{5} +2.28758 q^{6} -1.54298 q^{7} +0.305207 q^{8} +1.95544 q^{9} +O(q^{10})\)	\(q-1.33065 q^{2} -1.71914 q^{3} +0.770633 q^{4} +1.79233 q^{5} +2.28758 q^{6} -1.54298 q^{7} +0.305207 q^{8} +1.95544 q^{9} -2.38497 q^{10} +1.44104 q^{11} -1.32483 q^{12} +2.05317 q^{14} -3.08127 q^{15} -1.17676 q^{16} -2.60201 q^{18} +1.38123 q^{20} +2.65259 q^{21} -1.91753 q^{22} -0.524694 q^{24} +2.21245 q^{25} -1.64253 q^{27} -1.18907 q^{28} -1.85500 q^{29} +4.10009 q^{30} +1.26065 q^{32} -2.47735 q^{33} -2.76553 q^{35} +1.50693 q^{36} -0.818137 q^{37} +0.547033 q^{40} -3.52968 q^{42} +1.63586 q^{43} +1.11052 q^{44} +3.50480 q^{45} -0.529963 q^{47} +2.02301 q^{48} +1.38078 q^{49} -2.94400 q^{50} +1.90679 q^{53} +2.18564 q^{54} +2.58283 q^{55} -0.470928 q^{56} +2.46836 q^{58} +1.21245 q^{59} -2.37453 q^{60} -3.01720 q^{63} -0.500724 q^{64} +3.29649 q^{66} +1.97656 q^{67} +3.67995 q^{70} +0.596814 q^{72} +1.08866 q^{74} -3.80351 q^{75} -2.22350 q^{77} -2.10914 q^{80} +0.868305 q^{81} +0.380782 q^{83} +2.04418 q^{84} -2.17676 q^{86} +3.18901 q^{87} +0.439817 q^{88} -4.66366 q^{90} +0.705196 q^{94} -2.16723 q^{96} -1.83734 q^{98} +2.81787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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