LMFDB - Embedded newform 1151.1.b.a.1150.9

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			1150.9
Root			$-0.955440$ of defining polynomial
Character	$\chi$	$=$	1151.1150

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.529963 q^{2} -0.229367 q^{3} -0.719139 q^{4} -1.08714 q^{5} +0.121556 q^{6} -1.99413 q^{7} +0.911080 q^{8} -0.947391 q^{9} +O(q^{10})$	\(q-0.529963 q^{2} -0.229367 q^{3} -0.719139 q^{4} -1.08714 q^{5} +0.121556 q^{6} -1.99413 q^{7} +0.911080 q^{8} -0.947391 q^{9} +0.576141 q^{10} +1.63586 q^{11} +0.164947 q^{12} +1.05682 q^{14} +0.249353 q^{15} +0.236300 q^{16} +0.502082 q^{18} +0.781801 q^{20} +0.457388 q^{21} -0.866945 q^{22} -0.208972 q^{24} +0.181863 q^{25} +0.446667 q^{27} +1.43406 q^{28} +1.90679 q^{29} -0.132148 q^{30} -1.03631 q^{32} -0.375212 q^{33} +2.16789 q^{35} +0.681306 q^{36} +1.21245 q^{37} -0.990467 q^{40} -0.242399 q^{42} +1.44104 q^{43} -1.17641 q^{44} +1.02994 q^{45} -1.33065 q^{47} -0.0541995 q^{48} +2.97656 q^{49} -0.0963805 q^{50} -1.85500 q^{53} -0.236717 q^{54} -1.77840 q^{55} -1.81681 q^{56} -1.01053 q^{58} -0.818137 q^{59} -0.179319 q^{60} +1.88922 q^{63} +0.312906 q^{64} +0.198848 q^{66} +0.380782 q^{67} -1.14890 q^{70} -0.863149 q^{72} -0.642554 q^{74} -0.0417133 q^{75} -3.26212 q^{77} -0.256890 q^{80} +0.844940 q^{81} +1.97656 q^{83} -0.328925 q^{84} -0.763700 q^{86} -0.437355 q^{87} +1.49040 q^{88} -0.545831 q^{90} +0.705196 q^{94} +0.237695 q^{96} -1.57747 q^{98} -1.54980 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9	$ 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - 2 q^{14} - 2 q^{15} + 18 q^{16} - 3 q^{18} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 19 q^{25} - 2 q^{27} - 3 q^{28} - q^{29} - 4 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 16 q^{36} - q^{37} - 4 q^{40} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - q^{47} - 5 q^{48} + 19 q^{49} - 3 q^{50} - q^{53} - 4 q^{54} - 2 q^{55} - 4 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 3 q^{63} + 17 q^{64} - 4 q^{66} - q^{67} - 4 q^{70} - 6 q^{72} - 2 q^{74} - 3 q^{75} - 2 q^{77} - 5 q^{80} + 18 q^{81} - q^{83} - 6 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - 6 q^{90} - 2 q^{94} - 6 q^{96} - 3 q^{98} - 3 q^{99}+O(q^{100}) $ 20 * q - q^2 - q^3 + 19 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 19 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - 2 * q^14 - 2 * q^15 + 18 * q^16 - 3 * q^18 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 19 * q^25 - 2 * q^27 - 3 * q^28 - q^29 - 4 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 16 * q^36 - q^37 - 4 * q^40 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - q^47 - 5 * q^48 + 19 * q^49 - 3 * q^50 - q^53 - 4 * q^54 - 2 * q^55 - 4 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 3 * q^63 + 17 * q^64 - 4 * q^66 - q^67 - 4 * q^70 - 6 * q^72 - 2 * q^74 - 3 * q^75 - 2 * q^77 - 5 * q^80 + 18 * q^81 - q^83 - 6 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - 6 * q^90 - 2 * q^94 - 6 * q^96 - 3 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$2$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$3$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$3$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$4$	−0.719139	−0.719139
$5$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$5$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$6$	0.121556	0.121556
$7$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$7$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$8$	0.911080	0.911080
$9$	−0.947391	−0.947391
$10$	0.576141	0.576141
$11$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$11$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$12$	0.164947	0.164947
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.05682	1.05682
$15$	0.249353	0.249353
$16$	0.236300	0.236300
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.502082	0.502082
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.781801	0.781801
$21$	0.457388	0.457388
$22$	−0.866945	−0.866945
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−0.208972	−0.208972
$25$	0.181863	0.181863
$26$	0	0
$27$	0.446667	0.446667
$28$	1.43406	1.43406
$29$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$29$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$30$	−0.132148	−0.132148
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.03631	−1.03631
$33$	−0.375212	−0.375212
$34$	0	0
$35$	2.16789	2.16789
$36$	0.681306	0.681306
$37$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$37$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$38$	0	0
$39$	0	0
$40$	−0.990467	−0.990467
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.242399	−0.242399
$43$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$43$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$44$	−1.17641	−1.17641
$45$	1.02994	1.02994
$46$	0	0
$47$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$47$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$48$	−0.0541995	−0.0541995
$49$	2.97656	2.97656
$50$	−0.0963805	−0.0963805
$51$	0	0
$52$	0	0
$53$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$53$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$54$	−0.236717	−0.236717
$55$	−1.77840	−1.77840
$56$	−1.81681	−1.81681
$57$	0	0
$58$	−1.01053	−1.01053
$59$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$59$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$60$	−0.179319	−0.179319
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	1.88922	1.88922
$64$	0.312906	0.312906
$65$	0	0
$66$	0.198848	0.198848
$67$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$67$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$68$	0	0
$69$	0	0
$70$	−1.14890	−1.14890
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.863149	−0.863149
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.642554	−0.642554
$75$	−0.0417133	−0.0417133
$76$	0	0
$77$	−3.26212	−3.26212
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.256890	−0.256890
$81$	0.844940	0.844940
$82$	0	0
$83$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$83$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$84$	−0.328925	−0.328925
$85$	0	0
$86$	−0.763700	−0.763700
$87$	−0.437355	−0.437355
$88$	1.49040	1.49040
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−0.545831	−0.545831
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.705196	0.705196
$95$	0	0
$96$	0.237695	0.237695
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.57747	−1.57747
$99$	−1.54980	−1.54980
$100$	−0.130785	−0.130785
$101$	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
$105$	−0.497242	−0.497242
$106$	0.983084	0.983084
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.321216	−0.321216
$109$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$109$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$110$	0.942486	0.942486
$111$	−0.278096	−0.278096
$112$	−0.471214	−0.471214
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.37125	−1.37125
$117$	0	0
$118$	0.433582	0.433582
$119$	0	0
$120$	0.227180	0.227180
$121$	1.67603	1.67603
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.889426	0.889426
$126$	−1.00122	−1.00122
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.870482	0.870482
$129$	−0.330528	−0.330528
$130$	0	0
$131$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$131$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$132$	0.269829	0.269829
$133$	0	0
$134$	−0.201800	−0.201800
$135$	−0.485587	−0.485587
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$139$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$140$	−1.55902	−1.55902
$141$	0.305207	0.305207
$142$	0	0
$143$	0	0
$144$	−0.223869	−0.223869
$145$	−2.07294	−2.07294
$146$	0	0
$147$	−0.682724	−0.682724
$148$	−0.871921	−0.871921
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.0221065	0.0221065
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	1.72880	1.72880
$155$	0	0
$156$	0	0
$157$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$157$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$158$	0	0
$159$	0.425477	0.425477
$160$	1.12661	1.12661
$161$	0	0
$162$	−0.447787	−0.447787
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0.407906	0.407906
$166$	−1.04750	−1.04750
$167$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$167$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$168$	0.416717	0.416717
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−1.03631	−1.03631
$173$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$173$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$174$	0.231782	0.231782
$175$	−0.362658	−0.362658
$176$	0.386554	0.386554
$177$	0.187654	0.187654
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.740672	−0.740672
$181$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$181$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.31810	−1.31810
$186$	0	0
$187$	0	0
$188$	0.956924	0.956924
$189$	−0.890713	−0.890713
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.0717702	−0.0717702
$193$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$193$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$194$	0	0
$195$	0	0
$196$	−2.14056	−2.14056
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.821335	0.821335
$199$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$199$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$200$	0.165692	0.165692
$201$	−0.0873388	−0.0873388
$202$	0	0
$203$	−3.80240	−3.80240
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0.263520	0.263520
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.33401	1.33401
$213$	0	0
$214$	0	0
$215$	−1.56661	−1.56661
$216$	0.406949	0.406949
$217$	0	0
$218$	−0.949869	−0.949869
$219$	0	0
$220$	1.27892	1.27892
$221$	0	0
$222$	0.147381	0.147381
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	2.06654	2.06654
$225$	−0.172295	−0.172295
$226$	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
$231$	0.748222	0.748222
$232$	1.73724	1.73724
$233$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$233$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$234$	0	0
$235$	1.44660	1.44660
$236$	0.588355	0.588355
$237$	0	0
$238$	0	0
$239$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$239$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$240$	0.0589222	0.0589222
$241$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$241$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$242$	−0.888236	−0.888236
$243$	−0.640468	−0.640468
$244$	0	0
$245$	−3.23592	−3.23592
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.453358	−0.453358
$250$	−0.471363	−0.471363
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.35861	−1.35861
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.774229	−0.774229
$257$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$257$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$258$	0.175167	0.175167
$259$	−2.41779	−2.41779
$260$	0	0
$261$	−1.80648	−1.80648
$262$	−0.0405981	−0.0405981
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.341848	−0.341848
$265$	2.01664	2.01664
$266$	0	0
$267$	0	0
$268$	−0.273835	−0.273835
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.257343	0.257343
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.297502	0.297502
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.03205	1.03205
$279$	0	0
$280$	1.97512	1.97512
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−0.161749	−0.161749
$283$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$283$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.981791	0.981791
$289$	1.00000	1.00000
$290$	1.09858	1.09858
$291$	0	0
$292$	0	0
$293$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$293$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$294$	0.361819	0.361819
$295$	0.889426	0.889426
$296$	1.10464	1.10464
$297$	0.730684	0.730684
$298$	0	0
$299$	0	0
$300$	0.0299977	0.0299977
$301$	−2.87363	−2.87363
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$307$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$308$	2.34592	2.34592
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−0.866945	−0.866945
$315$	−2.05384	−2.05384
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.225487	−0.225487
$319$	3.11924	3.11924
$320$	−0.340171	−0.340171
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.607630	−0.607630
$325$	0	0
$326$	0	0
$327$	−0.411101	−0.411101
$328$	0	0
$329$	2.65349	2.65349
$330$	−0.216175	−0.216175
$331$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$331$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$332$	−1.42142	−1.42142
$333$	−1.14866	−1.14866
$334$	0.911080	0.911080
$335$	−0.413962	−0.413962
$336$	0.108081	0.108081
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.529963	−0.529963
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.94152	−3.94152
$344$	1.31291	1.31291
$345$	0	0
$346$	−0.949869	−0.949869
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.314519	0.314519
$349$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$349$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$350$	0.192195	0.192195
$351$	0	0
$352$	−1.69526	−1.69526
$353$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$353$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$354$	−0.0994494	−0.0994494
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.938360	0.938360
$361$	1.00000	1.00000
$362$	−0.506348	−0.506348
$363$	−0.384427	−0.384427
$364$	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
$370$	0.698543	0.698543
$371$	3.69912	3.69912
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.204005	−0.204005
$376$	−1.21233	−1.21233
$377$	0	0
$378$	0.472045	0.472045
$379$	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			$\pi$
$384$	−0.199660	−0.199660
$385$	3.54636	3.54636
$386$	0.983084	0.983084
$387$	−1.36523	−1.36523
$388$	0	0
$389$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$389$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$390$	0	0
$391$	0	0
$392$	2.71189	2.71189
$393$	−0.0175708	−0.0175708
$394$	0	0
$395$	0	0
$396$	1.11452	1.11452
$397$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$397$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$398$	0.280861	0.280861
$399$	0	0
$400$	0.0429742	0.0429742
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0.0462863	0.0462863
$403$	0	0
$404$	0	0
$405$	−0.918564	−0.918564
$406$	2.01513	2.01513
$407$	1.98340	1.98340
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.63147	1.63147
$414$	0	0
$415$	−2.14879	−2.14879
$416$	0	0
$417$	0.446667	0.446667
$418$	0	0
$419$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$419$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$420$	0.357586	0.357586
$421$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$421$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$422$	0	0
$423$	1.26065	1.26065
$424$	−1.69006	−1.69006
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0.830245	0.830245
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.105548	0.105548
$433$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$433$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$434$	0	0
$435$	0.475464	0.475464
$436$	−1.28894	−1.28894
$437$	0	0
$438$	0	0
$439$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$439$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$440$	−1.62026	−1.62026
$441$	−2.81997	−2.81997
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.199990	0.199990
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−0.623976	−0.623976
$449$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$449$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$450$	0.0913100	0.0913100
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$457$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$461$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$462$	−0.396530	−0.396530
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0.450576	0.450576
$465$	0	0
$466$	−0.0405981	−0.0405981
$467$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$467$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$468$	0	0
$469$	−0.759330	−0.759330
$470$	−0.766643	−0.766643
$471$	−0.375212	−0.375212
$472$	−0.745389	−0.745389
$473$	2.35734	2.35734
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.75741	1.75741
$478$	−0.358273	−0.358273
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.258407	−0.258407
$481$	0	0
$482$	−0.763700	−0.763700
$483$	0	0
$484$	−1.20530	−1.20530
$485$	0	0
$486$	0.339424	0.339424
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	1.71492	1.71492
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.68484	1.68484
$496$	0	0
$497$	0	0
$498$	0.240263	0.240263
$499$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$499$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$500$	−0.639621	−0.639621
$501$	0.394314	0.394314
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.72123	1.72123
$505$	0	0
$506$	0	0
$507$	−0.229367	−0.229367
$508$	0	0
$509$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$509$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$510$	0	0
$511$	0	0
$512$	−0.460169	−0.460169
$513$	0	0
$514$	−0.949869	−0.949869
$515$	0	0
$516$	0.237695	0.237695
$517$	−2.17676	−2.17676
$518$	1.28134	1.28134
$519$	−0.411101	−0.411101
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0.957367	0.957367
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.0550900	−0.0550900
$525$	0.0831818	0.0831818
$526$	0	0
$527$	0	0
$528$	−0.0886627	−0.0886627
$529$	1.00000	1.00000
$530$	−1.06875	−1.06875
$531$	0.775096	0.775096
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0.346923	0.346923
$537$	0	0
$538$	0	0
$539$	4.86923	4.86923
$540$	0.349205	0.349205
$541$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$541$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$542$	0	0
$543$	−0.219146	−0.219146
$544$	0	0
$545$	−1.94851	−1.94851
$546$	0	0
$547$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$547$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$548$	0	0
$549$	0	0
$550$	−0.157665	−0.157665
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.302328	0.302328
$556$	1.40045	1.40045
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0.512273	0.512273
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−0.219487	−0.219487
$565$	0	0
$566$	−1.01053	−1.01053
$567$	−1.68492	−1.68492
$568$	0	0
$569$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$569$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$570$	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
$576$	−0.296444	−0.296444
$577$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$577$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$578$	−0.529963	−0.529963
$579$	0.425477	0.425477
$580$	1.49073	1.49073
$581$	−3.94152	−3.94152
$582$	0	0
$583$	−3.03453	−3.03453
$584$	0	0
$585$	0	0
$586$	1.05682	1.05682
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.490974	0.490974
$589$	0	0
$590$	−0.471363	−0.471363
$591$	0	0
$592$	0.286503	0.286503
$593$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$593$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$594$	−0.387235	−0.387235
$595$	0	0
$596$	0	0
$597$	0.121556	0.121556
$598$	0	0
$599$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$599$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$600$	−0.0380041	−0.0380041
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	1.52292	1.52292
$603$	−0.360750	−0.360750
$604$	0	0
$605$	−1.82208	−1.82208
$606$	0	0
$607$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$607$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$608$	0	0
$609$	0.872144	0.872144
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$613$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$614$	−0.949869	−0.949869
$615$	0	0
$616$	−2.97205	−2.97205
$617$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$617$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$618$	0	0
$619$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$619$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.14879	−1.14879
$626$	0	0
$627$	0	0
$628$	−1.17641	−1.17641
$629$	0	0
$630$	1.08846	1.08846
$631$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$631$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.305977	−0.305977
$637$	0	0
$638$	−1.65308	−1.65308
$639$	0	0
$640$	−0.946332	−0.946332
$641$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$641$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$642$	0	0
$643$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$643$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$644$	0	0
$645$	0.359328	0.359328
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.769808	0.769808
$649$	−1.33836	−1.33836
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.217869	0.217869
$655$	−0.0832805	−0.0832805
$656$	0	0
$657$	0	0
$658$	−1.40625	−1.40625
$659$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$659$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$660$	−0.293341	−0.293341
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0.817721	0.817721
$663$	0	0
$664$	1.80081	1.80081
$665$	0	0
$666$	0.608750	0.608750
$667$	0	0
$668$	1.23630	1.23630
$669$	0	0
$670$	0.219384	0.219384
$671$	0	0
$672$	−0.473996	−0.473996
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.0812321	0.0812321
$676$	−0.719139	−0.719139
$677$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$677$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$683$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$684$	0	0
$685$	0	0
$686$	2.08886	2.08886
$687$	0	0
$688$	0.340519	0.340519
$689$	0	0
$690$	0	0
$691$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$691$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$692$	−1.28894	−1.28894
$693$	3.09050	3.09050
$694$	0	0
$695$	2.11708	2.11708
$696$	−0.398466	−0.398466
$697$	0	0
$698$	0.983084	0.983084
$699$	−0.0175708	−0.0175708
$700$	0.260802	0.260802
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.511870	0.511870
$705$	−0.331802	−0.331802
$706$	−0.201800	−0.201800
$707$	0	0
$708$	−0.134949	−0.134949
$709$	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
$717$	−0.155060	−0.155060
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.243376	0.243376
$721$	0	0
$722$	−0.529963	−0.529963
$723$	−0.330528	−0.330528
$724$	−0.687094	−0.687094
$725$	0.346775	0.346775
$726$	0.203732	0.203732
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.698038	−0.698038
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$733$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$734$	0	0
$735$	0.742214	0.742214
$736$	0	0
$737$	0.622906	0.622906
$738$	0	0
$739$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$739$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$740$	0.947896	0.947896
$741$	0	0
$742$	−1.96040	−1.96040
$743$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$743$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.87258	−1.87258
$748$	0	0
$749$	0	0
$750$	0.108115	0.108115
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−0.314433	−0.314433
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0.640546	0.640546
$757$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$757$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$761$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$762$	0	0
$763$	−3.57414	−3.57414
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.177583	0.177583
$769$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$769$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$770$	−1.87944	−1.87944
$771$	−0.411101	−0.411101
$772$	1.33401	1.33401
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0.723522	0.723522
$775$	0	0
$776$	0	0
$777$	0.554560	0.554560
$778$	0.121556	0.121556
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.851701	0.851701
$784$	0.703363	0.703363
$785$	−1.77840	−1.77840
$786$	0.00931185	0.00931185
$787$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$787$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.41199	−1.41199
$793$	0	0
$794$	0.280861	0.280861
$795$	−0.462551	−0.462551
$796$	0.381117	0.381117
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.188466	−0.188466
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0.0628088	0.0628088
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$809$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$810$	0.486805	0.486805
$811$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$811$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$812$	2.73445	2.73445
$813$	0	0
$814$	−1.05113	−1.05113
$815$	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
$823$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$823$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$824$	0	0
$825$	−0.0682370	−0.0682370
$826$	−0.864621	−0.864621
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$829$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$830$	1.13878	1.13878
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−0.236717	−0.236717
$835$	1.86894	1.86894
$836$	0	0
$837$	0	0
$838$	0.817721	0.817721
$839$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$839$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$840$	−0.453028	−0.453028
$841$	2.63586	2.63586
$842$	−0.763700	−0.763700
$843$	0	0
$844$	0	0
$845$	−1.08714	−1.08714
$846$	−0.668096	−0.668096
$847$	−3.34223	−3.34223
$848$	−0.438338	−0.438338
$849$	−0.437355	−0.437355
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$853$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$859$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$860$	1.12661	1.12661
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.462886	−0.462886
$865$	−1.94851	−1.94851
$866$	−0.0405981	−0.0405981
$867$	−0.229367	−0.229367
$868$	0	0
$869$	0	0
$870$	−0.251978	−0.251978
$871$	0	0
$872$	1.63296	1.63296
$873$	0	0
$874$	0	0
$875$	−1.77363	−1.77363
$876$	0	0
$877$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$877$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$878$	−0.866945	−0.866945
$879$	0.457388	0.457388
$880$	−0.420237	−0.420237
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.49448	1.49448
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.204005	−0.204005
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.253368	−0.253368
$889$	0	0
$890$	0	0
$891$	1.38220	1.38220
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.73586	−1.73586
$897$	0	0
$898$	0.121556	0.121556
$899$	0	0
$900$	0.123904	0.123904
$901$	0	0
$902$	0	0
$903$	0.659115	0.659115
$904$	0	0
$905$	−1.03869	−1.03869
$906$	0	0
$907$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$907$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	3.23337	3.23337
$914$	−1.01053	−1.01053
$915$	0	0
$916$	0	0
$917$	−0.152761	−0.152761
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−0.411101	−0.411101
$922$	−0.506348	−0.506348
$923$	0	0
$924$	−0.538076	−0.538076
$925$	0.220500	0.220500
$926$	0	0
$927$	0	0
$928$	−1.97603	−1.97603
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.0550900	−0.0550900
$933$	0	0
$934$	0.817721	0.817721
$935$	0	0
$936$	0	0
$937$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$937$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$938$	0.402417	0.402417
$939$	0	0
$940$	−1.04031	−1.04031
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0.198848	0.198848
$943$	0	0
$944$	−0.193326	−0.193326
$945$	0.968325	0.968325
$946$	−1.24930	−1.24930
$947$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$947$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−0.931365	−0.931365
$955$	0	0
$956$	−0.486162	−0.486162
$957$	−0.715451	−0.715451
$958$	0	0
$959$	0	0
$960$	0.0780239	0.0780239
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.03631	−1.03631
$965$	2.01664	2.01664
$966$	0	0
$967$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$967$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$968$	1.52700	1.52700
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.460586	0.460586
$973$	3.88335	3.88335
$974$	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
$980$	2.32708	2.32708
$981$	−1.69804	−1.69804
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.608624	−0.608624
$988$	0	0
$989$	0	0
$990$	−0.892903	−0.892903
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0.353908	0.353908
$994$	0	0
$995$	0.576141	0.576141
$996$	0.326027	0.326027
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.433582	0.433582
$999$	0.541562	0.541562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.9	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.9	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.9	✓	20	1151.1150	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.529963 q^{2} -0.229367 q^{3} -0.719139 q^{4} -1.08714 q^{5} +0.121556 q^{6} -1.99413 q^{7} +0.911080 q^{8} -0.947391 q^{9} +O(q^{10})\)	\(q-0.529963 q^{2} -0.229367 q^{3} -0.719139 q^{4} -1.08714 q^{5} +0.121556 q^{6} -1.99413 q^{7} +0.911080 q^{8} -0.947391 q^{9} +0.576141 q^{10} +1.63586 q^{11} +0.164947 q^{12} +1.05682 q^{14} +0.249353 q^{15} +0.236300 q^{16} +0.502082 q^{18} +0.781801 q^{20} +0.457388 q^{21} -0.866945 q^{22} -0.208972 q^{24} +0.181863 q^{25} +0.446667 q^{27} +1.43406 q^{28} +1.90679 q^{29} -0.132148 q^{30} -1.03631 q^{32} -0.375212 q^{33} +2.16789 q^{35} +0.681306 q^{36} +1.21245 q^{37} -0.990467 q^{40} -0.242399 q^{42} +1.44104 q^{43} -1.17641 q^{44} +1.02994 q^{45} -1.33065 q^{47} -0.0541995 q^{48} +2.97656 q^{49} -0.0963805 q^{50} -1.85500 q^{53} -0.236717 q^{54} -1.77840 q^{55} -1.81681 q^{56} -1.01053 q^{58} -0.818137 q^{59} -0.179319 q^{60} +1.88922 q^{63} +0.312906 q^{64} +0.198848 q^{66} +0.380782 q^{67} -1.14890 q^{70} -0.863149 q^{72} -0.642554 q^{74} -0.0417133 q^{75} -3.26212 q^{77} -0.256890 q^{80} +0.844940 q^{81} +1.97656 q^{83} -0.328925 q^{84} -0.763700 q^{86} -0.437355 q^{87} +1.49040 q^{88} -0.545831 q^{90} +0.705196 q^{94} +0.237695 q^{96} -1.57747 q^{98} -1.54980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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