LMFDB - Embedded newform 1151.1.b.a.1150.19

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			1150.19
Root			$-0.676034$ 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)

$f(q)$	$=$	$q+1.90679 q^{2} +1.44104 q^{3} +2.63586 q^{4} -1.54298 q^{5} +2.74777 q^{6} -1.71914 q^{7} +3.11924 q^{8} +1.07661 q^{9} +O(q^{10})$	\(q+1.90679 q^{2} +1.44104 q^{3} +2.63586 q^{4} -1.54298 q^{5} +2.74777 q^{6} -1.71914 q^{7} +3.11924 q^{8} +1.07661 q^{9} -2.94214 q^{10} -0.818137 q^{11} +3.79839 q^{12} -3.27804 q^{14} -2.22350 q^{15} +3.31189 q^{16} +2.05286 q^{18} -4.06707 q^{20} -2.47735 q^{21} -1.56002 q^{22} +4.49496 q^{24} +1.38078 q^{25} +0.110392 q^{27} -4.53141 q^{28} -1.08714 q^{29} -4.23975 q^{30} +3.19585 q^{32} -1.17897 q^{33} +2.65259 q^{35} +2.83778 q^{36} +1.97656 q^{37} -4.81293 q^{40} -4.72380 q^{42} +1.21245 q^{43} -2.15649 q^{44} -1.66118 q^{45} -1.85500 q^{47} +4.77258 q^{48} +1.95544 q^{49} +2.63287 q^{50} +1.79233 q^{53} +0.210494 q^{54} +1.26237 q^{55} -5.36241 q^{56} -2.07294 q^{58} +0.380782 q^{59} -5.86083 q^{60} -1.85083 q^{63} +2.78193 q^{64} -2.24805 q^{66} -1.94739 q^{67} +5.05795 q^{70} +3.35819 q^{72} +3.76889 q^{74} +1.98977 q^{75} +1.40649 q^{77} -5.11018 q^{80} -0.917526 q^{81} +0.955440 q^{83} -6.52995 q^{84} +2.31189 q^{86} -1.56661 q^{87} -2.55197 q^{88} -3.16752 q^{90} -3.53711 q^{94} +4.60536 q^{96} +3.72862 q^{98} -0.880811 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$2$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$3$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$3$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$4$	2.63586	2.63586
$5$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$5$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$6$	2.74777	2.74777
$7$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$7$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$8$	3.11924	3.11924
$9$	1.07661	1.07661
$10$	−2.94214	−2.94214
$11$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$11$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$12$	3.79839	3.79839
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−3.27804	−3.27804
$15$	−2.22350	−2.22350
$16$	3.31189	3.31189
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.05286	2.05286
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−4.06707	−4.06707
$21$	−2.47735	−2.47735
$22$	−1.56002	−1.56002
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	4.49496	4.49496
$25$	1.38078	1.38078
$26$	0	0
$27$	0.110392	0.110392
$28$	−4.53141	−4.53141
$29$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$29$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$30$	−4.23975	−4.23975
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	3.19585	3.19585
$33$	−1.17897	−1.17897
$34$	0	0
$35$	2.65259	2.65259
$36$	2.83778	2.83778
$37$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$37$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$38$	0	0
$39$	0	0
$40$	−4.81293	−4.81293
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−4.72380	−4.72380
$43$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$43$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$44$	−2.15649	−2.15649
$45$	−1.66118	−1.66118
$46$	0	0
$47$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$47$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$48$	4.77258	4.77258
$49$	1.95544	1.95544
$50$	2.63287	2.63287
$51$	0	0
$52$	0	0
$53$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$53$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$54$	0.210494	0.210494
$55$	1.26237	1.26237
$56$	−5.36241	−5.36241
$57$	0	0
$58$	−2.07294	−2.07294
$59$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$59$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$60$	−5.86083	−5.86083
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−1.85083	−1.85083
$64$	2.78193	2.78193
$65$	0	0
$66$	−2.24805	−2.24805
$67$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$67$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$68$	0	0
$69$	0	0
$70$	5.05795	5.05795
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	3.35819	3.35819
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	3.76889	3.76889
$75$	1.98977	1.98977
$76$	0	0
$77$	1.40649	1.40649
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−5.11018	−5.11018
$81$	−0.917526	−0.917526
$82$	0	0
$83$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$83$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$84$	−6.52995	−6.52995
$85$	0	0
$86$	2.31189	2.31189
$87$	−1.56661	−1.56661
$88$	−2.55197	−2.55197
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.16752	−3.16752
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−3.53711	−3.53711
$95$	0	0
$96$	4.60536	4.60536
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	3.72862	3.72862
$99$	−0.880811	−0.880811
$100$	3.63955	3.63955
$101$	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
$105$	3.82250	3.82250
$106$	3.41760	3.41760
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.290977	0.290977
$109$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$109$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$110$	2.40707	2.40707
$111$	2.84831	2.84831
$112$	−5.69360	−5.69360
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−2.86553	−2.86553
$117$	0	0
$118$	0.726073	0.726073
$119$	0	0
$120$	−6.93563	−6.93563
$121$	−0.330651	−0.330651
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.587539	−0.587539
$126$	−3.52916	−3.52916
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	2.10871	2.10871
$129$	1.74719	1.74719
$130$	0	0
$131$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$131$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$132$	−3.10760	−3.10760
$133$	0	0
$134$	−3.71327	−3.71327
$135$	−0.170332	−0.170332
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$139$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$140$	6.99186	6.99186
$141$	−2.67314	−2.67314
$142$	0	0
$143$	0	0
$144$	3.56560	3.56560
$145$	1.67743	1.67743
$146$	0	0
$147$	2.81787	2.81787
$148$	5.20994	5.20994
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	3.79407	3.79407
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	2.68189	2.68189
$155$	0	0
$156$	0	0
$157$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$157$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$158$	0	0
$159$	2.58283	2.58283
$160$	−4.93113	−4.93113
$161$	0	0
$162$	−1.74953	−1.74953
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	1.81913	1.81913
$166$	1.82183	1.82183
$167$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$167$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$168$	−7.72747	−7.72747
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	3.19585	3.19585
$173$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$173$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$174$	−2.98720	−2.98720
$175$	−2.37376	−2.37376
$176$	−2.70958	−2.70958
$177$	0.548724	0.548724
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−4.37863	−4.37863
$181$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$181$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.04979	−3.04979
$186$	0	0
$187$	0	0
$188$	−4.88953	−4.88953
$189$	−0.189779	−0.189779
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	4.00888	4.00888
$193$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$193$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$194$	0	0
$195$	0	0
$196$	5.15426	5.15426
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.67952	−1.67952
$199$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$199$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$200$	4.30700	4.30700
$201$	−2.80627	−2.80627
$202$	0	0
$203$	1.86894	1.86894
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	7.28872	7.28872
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	4.72433	4.72433
$213$	0	0
$214$	0	0
$215$	−1.87079	−1.87079
$216$	0.344339	0.344339
$217$	0	0
$218$	−3.80240	−3.80240
$219$	0	0
$220$	3.32742	3.32742
$221$	0	0
$222$	5.43114	5.43114
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−5.49411	−5.49411
$225$	1.48656	1.48656
$226$	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
$231$	2.02682	2.02682
$232$	−3.39104	−3.39104
$233$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$233$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$234$	0	0
$235$	2.86223	2.86223
$236$	1.00369	1.00369
$237$	0	0
$238$	0	0
$239$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$239$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$240$	−7.36399	−7.36399
$241$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$241$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$242$	−0.630484	−0.630484
$243$	−1.43259	−1.43259
$244$	0	0
$245$	−3.01720	−3.01720
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.37683	1.37683
$250$	−1.12031	−1.12031
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−4.87854	−4.87854
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.23895	1.23895
$257$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$257$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$258$	3.33154	3.33154
$259$	−3.39798	−3.39798
$260$	0	0
$261$	−1.17042	−1.17042
$262$	−1.01053	−1.01053
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−3.67750	−3.67750
$265$	−2.76553	−2.76553
$266$	0	0
$267$	0	0
$268$	−5.13305	−5.13305
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.324788	−0.324788
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.12967	−1.12967
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.146071	0.146071
$279$	0	0
$280$	8.27409	8.27409
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−5.09713	−5.09713
$283$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$283$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	3.44067	3.44067
$289$	1.00000	1.00000
$290$	3.19850	3.19850
$291$	0	0
$292$	0	0
$293$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$293$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$294$	5.37310	5.37310
$295$	−0.587539	−0.587539
$296$	6.16537	6.16537
$297$	−0.0903156	−0.0903156
$298$	0	0
$299$	0	0
$300$	5.24474	5.24474
$301$	−2.08437	−2.08437
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$307$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$308$	3.70731	3.70731
$309$	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			$\pi$
$314$	−1.56002	−1.56002
$315$	2.85580	2.85580
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	4.92492	4.92492
$319$	0.889426	0.889426
$320$	−4.29246	−4.29246
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−2.41847	−2.41847
$325$	0	0
$326$	0	0
$327$	−2.87363	−2.87363
$328$	0	0
$329$	3.18901	3.18901
$330$	3.46870	3.46870
$331$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$331$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$332$	2.51840	2.51840
$333$	2.12798	2.12798
$334$	3.11924	3.11924
$335$	3.00478	3.00478
$336$	−8.20473	−8.20473
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.90679	1.90679
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.64253	−1.64253
$344$	3.78193	3.78193
$345$	0	0
$346$	−3.80240	−3.80240
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−4.12936	−4.12936
$349$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$349$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$350$	−4.52626	−4.52626
$351$	0	0
$352$	−2.61464	−2.61464
$353$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$353$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$354$	1.04630	1.04630
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−5.18162	−5.18162
$361$	1.00000	1.00000
$362$	1.28906	1.28906
$363$	−0.476483	−0.476483
$364$	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
$370$	−5.81532	−5.81532
$371$	−3.08127	−3.08127
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.846669	−0.846669
$376$	−5.78621	−5.78621
$377$	0	0
$378$	−0.361869	−0.361869
$379$	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			$\pi$
$384$	3.03875	3.03875
$385$	−2.17019	−2.17019
$386$	3.41760	3.41760
$387$	1.30533	1.30533
$388$	0	0
$389$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$389$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$390$	0	0
$391$	0	0
$392$	6.09949	6.09949
$393$	−0.763700	−0.763700
$394$	0	0
$395$	0	0
$396$	−2.32169	−2.32169
$397$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$397$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$398$	3.63586	3.63586
$399$	0	0
$400$	4.57300	4.57300
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−5.35098	−5.35098
$403$	0	0
$404$	0	0
$405$	1.41572	1.41572
$406$	3.56367	3.56367
$407$	−1.61710	−1.61710
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.654618	−0.654618
$414$	0	0
$415$	−1.47422	−1.47422
$416$	0	0
$417$	0.110392	0.110392
$418$	0	0
$419$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$419$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$420$	10.0756	10.0756
$421$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$421$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$422$	0	0
$423$	−1.99711	−1.99711
$424$	5.59072	5.59072
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−3.56720	−3.56720
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.365606	0.365606
$433$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$433$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$434$	0	0
$435$	2.41724	2.41724
$436$	−5.25625	−5.25625
$437$	0	0
$438$	0	0
$439$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$439$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$440$	3.93763	3.93763
$441$	2.10524	2.10524
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	7.50774	7.50774
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−4.78252	−4.78252
$449$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$449$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$450$	2.83456	2.83456
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$457$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$461$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$462$	3.86472	3.86472
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−3.60047	−3.60047
$465$	0	0
$466$	−1.01053	−1.01053
$467$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$467$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$468$	0	0
$469$	3.34784	3.34784
$470$	5.45768	5.45768
$471$	−1.17897	−1.17897
$472$	1.18775	1.18775
$473$	−0.991951	−0.991951
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.92963	1.92963
$478$	−2.53728	−2.53728
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−7.10597	−7.10597
$481$	0	0
$482$	2.31189	2.31189
$483$	0	0
$484$	−0.871550	−0.871550
$485$	0	0
$486$	−2.73165	−2.73165
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−5.75318	−5.75318
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.35907	1.35907
$496$	0	0
$497$	0	0
$498$	2.62533	2.62533
$499$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$499$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$500$	−1.54867	−1.54867
$501$	2.35734	2.35734
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−5.77320	−5.77320
$505$	0	0
$506$	0	0
$507$	1.44104	1.44104
$508$	0	0
$509$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$509$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$510$	0	0
$511$	0	0
$512$	0.253709	0.253709
$513$	0	0
$514$	−3.80240	−3.80240
$515$	0	0
$516$	4.60536	4.60536
$517$	1.51765	1.51765
$518$	−6.47925	−6.47925
$519$	−2.87363	−2.87363
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.23174	−2.23174
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.39691	−1.39691
$525$	−3.42069	−3.42069
$526$	0	0
$527$	0	0
$528$	−3.90463	−3.90463
$529$	1.00000	1.00000
$530$	−5.27329	−5.27329
$531$	0.409952	0.409952
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−6.07439	−6.07439
$537$	0	0
$538$	0	0
$539$	−1.59982	−1.59982
$540$	−0.448971	−0.448971
$541$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$541$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$542$	0	0
$543$	0.974194	0.974194
$544$	0	0
$545$	3.07690	3.07690
$546$	0	0
$547$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$547$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$548$	0	0
$549$	0	0
$550$	−2.15405	−2.15405
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.39488	−4.39488
$556$	0.201921	0.201921
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	8.78511	8.78511
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−7.04603	−7.04603
$565$	0	0
$566$	−2.07294	−2.07294
$567$	1.57736	1.57736
$568$	0	0
$569$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$569$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$570$	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
$576$	2.99504	2.99504
$577$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$577$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$578$	1.90679	1.90679
$579$	2.58283	2.58283
$580$	4.42146	4.42146
$581$	−1.64253	−1.64253
$582$	0	0
$583$	−1.46637	−1.46637
$584$	0	0
$585$	0	0
$586$	−3.27804	−3.27804
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	7.42752	7.42752
$589$	0	0
$590$	−1.12031	−1.12031
$591$	0	0
$592$	6.54616	6.54616
$593$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$593$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$594$	−0.172213	−0.172213
$595$	0	0
$596$	0	0
$597$	2.74777	2.74777
$598$	0	0
$599$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$599$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$600$	6.20657	6.20657
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−3.97447	−3.97447
$603$	−2.09657	−2.09657
$604$	0	0
$605$	0.510188	0.510188
$606$	0	0
$607$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$607$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$608$	0	0
$609$	2.69322	2.69322
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$613$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$614$	−3.80240	−3.80240
$615$	0	0
$616$	4.38719	4.38719
$617$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$617$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$618$	0	0
$619$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$619$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−0.474223	−0.474223
$626$	0	0
$627$	0	0
$628$	−2.15649	−2.15649
$629$	0	0
$630$	5.44541	5.44541
$631$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$631$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	6.80797	6.80797
$637$	0	0
$638$	1.69595	1.69595
$639$	0	0
$640$	−3.25370	−3.25370
$641$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$641$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$642$	0	0
$643$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$643$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$644$	0	0
$645$	−2.69588	−2.69588
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.86199	−2.86199
$649$	−0.311532	−0.311532
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−5.47942	−5.47942
$655$	0.817721	0.817721
$656$	0	0
$657$	0	0
$658$	6.08078	6.08078
$659$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$659$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$660$	4.79496	4.79496
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.437355	−0.437355
$663$	0	0
$664$	2.98025	2.98025
$665$	0	0
$666$	4.05761	4.05761
$667$	0	0
$668$	4.31189	4.31189
$669$	0	0
$670$	5.72950	5.72950
$671$	0	0
$672$	−7.91725	−7.91725
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.152427	0.152427
$676$	2.63586	2.63586
$677$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$677$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$683$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$684$	0	0
$685$	0	0
$686$	−3.13197	−3.13197
$687$	0	0
$688$	4.01551	4.01551
$689$	0	0
$690$	0	0
$691$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$691$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$692$	−5.25625	−5.25625
$693$	1.51424	1.51424
$694$	0	0
$695$	−0.118201	−0.118201
$696$	−4.88663	−4.88663
$697$	0	0
$698$	3.41760	3.41760
$699$	−0.763700	−0.763700
$700$	−6.25689	−6.25689
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−2.27600	−2.27600
$705$	4.12460	4.12460
$706$	−3.71327	−3.71327
$707$	0	0
$708$	1.44636	1.44636
$709$	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
$717$	−1.91753	−1.91753
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−5.50165	−5.50165
$721$	0	0
$722$	1.90679	1.90679
$723$	1.74719	1.74719
$724$	1.78193	1.78193
$725$	−1.50110	−1.50110
$726$	−0.908554	−0.908554
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.14689	−1.14689
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$733$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$734$	0	0
$735$	−4.34792	−4.34792
$736$	0	0
$737$	1.59323	1.59323
$738$	0	0
$739$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$739$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$740$	−8.03882	−8.03882
$741$	0	0
$742$	−5.87534	−5.87534
$743$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$743$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.02863	1.02863
$748$	0	0
$749$	0	0
$750$	−1.61442	−1.61442
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−6.14358	−6.14358
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−0.500230	−0.500230
$757$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$757$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$761$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$762$	0	0
$763$	3.42819	3.42819
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.78538	1.78538
$769$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$769$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$770$	−4.13810	−4.13810
$771$	−2.87363	−2.87363
$772$	4.72433	4.72433
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.48900	2.48900
$775$	0	0
$776$	0	0
$777$	−4.89664	−4.89664
$778$	2.74777	2.74777
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.120011	−0.120011
$784$	6.47621	6.47621
$785$	1.26237	1.26237
$786$	−1.45622	−1.45622
$787$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$787$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−2.74746	−2.74746
$793$	0	0
$794$	3.63586	3.63586
$795$	−3.98525	−3.98525
$796$	5.02604	5.02604
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	4.41277	4.41277
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−7.39694	−7.39694
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$809$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$810$	2.69949	2.69949
$811$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$811$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$812$	4.92625	4.92625
$813$	0	0
$814$	−3.08347	−3.08347
$815$	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
$823$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$823$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$824$	0	0
$825$	−1.62790	−1.62790
$826$	−1.24822	−1.24822
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$829$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$830$	−2.81104	−2.81104
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0.210494	0.210494
$835$	−2.52409	−2.52409
$836$	0	0
$837$	0	0
$838$	−0.437355	−0.437355
$839$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$839$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$840$	11.9233	11.9233
$841$	0.181863	0.181863
$842$	2.31189	2.31189
$843$	0	0
$844$	0	0
$845$	−1.54298	−1.54298
$846$	−3.80807	−3.80807
$847$	0.568436	0.568436
$848$	5.93601	5.93601
$849$	−1.56661	−1.56661
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$853$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$859$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$860$	−4.93113	−4.93113
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.352795	0.352795
$865$	3.07690	3.07690
$866$	−1.01053	−1.01053
$867$	1.44104	1.44104
$868$	0	0
$869$	0	0
$870$	4.60918	4.60918
$871$	0	0
$872$	−6.22018	−6.22018
$873$	0	0
$874$	0	0
$875$	1.01006	1.01006
$876$	0	0
$877$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$877$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$878$	−1.56002	−1.56002
$879$	−2.47735	−2.47735
$880$	4.18083	4.18083
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	4.01425	4.01425
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.846669	−0.846669
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	8.88457	8.88457
$889$	0	0
$890$	0	0
$891$	0.750662	0.750662
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−3.62517	−3.62517
$897$	0	0
$898$	2.74777	2.74777
$899$	0	0
$900$	3.91836	3.91836
$901$	0	0
$902$	0	0
$903$	−3.00367	−3.00367
$904$	0	0
$905$	−1.04311	−1.04311
$906$	0	0
$907$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$907$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.781681	−0.781681
$914$	−2.07294	−2.07294
$915$	0	0
$916$	0	0
$917$	0.911080	0.911080
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−2.87363	−2.87363
$922$	1.28906	1.28906
$923$	0	0
$924$	5.34240	5.34240
$925$	2.72920	2.72920
$926$	0	0
$927$	0	0
$928$	−3.47432	−3.47432
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−1.39691	−1.39691
$933$	0	0
$934$	−0.437355	−0.437355
$935$	0	0
$936$	0	0
$937$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$937$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$938$	6.38363	6.38363
$939$	0	0
$940$	7.54444	7.54444
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−2.24805	−2.24805
$943$	0	0
$944$	1.26111	1.26111
$945$	0.292825	0.292825
$946$	−1.89145	−1.89145
$947$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$947$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$948$	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			$\pi$
$954$	3.67941	3.67941
$955$	0	0
$956$	−3.50741	−3.50741
$957$	1.28170	1.28170
$958$	0	0
$959$	0	0
$960$	−6.18562	−6.18562
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	3.19585	3.19585
$965$	−2.76553	−2.76553
$966$	0	0
$967$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$967$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$968$	−1.03138	−1.03138
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−3.77610	−3.77610
$973$	−0.131695	−0.131695
$974$	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
$980$	−7.95292	−7.95292
$981$	−2.14689	−2.14689
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.59550	4.59550
$988$	0	0
$989$	0	0
$990$	2.59147	2.59147
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−0.330528	−0.330528
$994$	0	0
$995$	−2.94214	−2.94214
$996$	3.62913	3.62913
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.726073	0.726073
$999$	0.218196	0.218196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.19	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.19	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.19	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.19	✓	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)

\(f(q)\)	\(=\)	\(q+1.90679 q^{2} +1.44104 q^{3} +2.63586 q^{4} -1.54298 q^{5} +2.74777 q^{6} -1.71914 q^{7} +3.11924 q^{8} +1.07661 q^{9} +O(q^{10})\)	\(q+1.90679 q^{2} +1.44104 q^{3} +2.63586 q^{4} -1.54298 q^{5} +2.74777 q^{6} -1.71914 q^{7} +3.11924 q^{8} +1.07661 q^{9} -2.94214 q^{10} -0.818137 q^{11} +3.79839 q^{12} -3.27804 q^{14} -2.22350 q^{15} +3.31189 q^{16} +2.05286 q^{18} -4.06707 q^{20} -2.47735 q^{21} -1.56002 q^{22} +4.49496 q^{24} +1.38078 q^{25} +0.110392 q^{27} -4.53141 q^{28} -1.08714 q^{29} -4.23975 q^{30} +3.19585 q^{32} -1.17897 q^{33} +2.65259 q^{35} +2.83778 q^{36} +1.97656 q^{37} -4.81293 q^{40} -4.72380 q^{42} +1.21245 q^{43} -2.15649 q^{44} -1.66118 q^{45} -1.85500 q^{47} +4.77258 q^{48} +1.95544 q^{49} +2.63287 q^{50} +1.79233 q^{53} +0.210494 q^{54} +1.26237 q^{55} -5.36241 q^{56} -2.07294 q^{58} +0.380782 q^{59} -5.86083 q^{60} -1.85083 q^{63} +2.78193 q^{64} -2.24805 q^{66} -1.94739 q^{67} +5.05795 q^{70} +3.35819 q^{72} +3.76889 q^{74} +1.98977 q^{75} +1.40649 q^{77} -5.11018 q^{80} -0.917526 q^{81} +0.955440 q^{83} -6.52995 q^{84} +2.31189 q^{86} -1.56661 q^{87} -2.55197 q^{88} -3.16752 q^{90} -3.53711 q^{94} +4.60536 q^{96} +3.72862 q^{98} -0.880811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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