LMFDB - Embedded newform 1151.1.b.a.1150.16

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			1150.16
Root			$1.99413$ 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)

$f(q)$	$=$	$q+1.44104 q^{2} +0.676034 q^{3} +1.07661 q^{4} +1.97656 q^{5} +0.974194 q^{6} -1.94739 q^{7} +0.110392 q^{8} -0.542978 q^{9} +O(q^{10})$	\(q+1.44104 q^{2} +0.676034 q^{3} +1.07661 q^{4} +1.97656 q^{5} +0.974194 q^{6} -1.94739 q^{7} +0.110392 q^{8} -0.542978 q^{9} +2.84831 q^{10} -0.529963 q^{11} +0.727822 q^{12} -2.80627 q^{14} +1.33622 q^{15} -0.917526 q^{16} -0.782455 q^{18} +2.12798 q^{20} -1.31650 q^{21} -0.763700 q^{22} +0.0746286 q^{24} +2.90679 q^{25} -1.04311 q^{27} -2.09657 q^{28} +1.21245 q^{29} +1.92555 q^{30} -1.43259 q^{32} -0.358273 q^{33} -3.84914 q^{35} -0.584573 q^{36} -1.85500 q^{37} +0.218196 q^{40} -1.89714 q^{42} -1.33065 q^{43} -0.570561 q^{44} -1.07323 q^{45} +1.63586 q^{47} -0.620279 q^{48} +2.79233 q^{49} +4.18881 q^{50} -0.818137 q^{53} -1.50316 q^{54} -1.04750 q^{55} -0.214976 q^{56} +1.74719 q^{58} +1.90679 q^{59} +1.43858 q^{60} +1.05739 q^{63} -1.14689 q^{64} -0.516287 q^{66} -1.08714 q^{67} -5.54677 q^{70} -0.0599404 q^{72} -2.67314 q^{74} +1.96509 q^{75} +1.03205 q^{77} -1.81355 q^{80} -0.162196 q^{81} +1.79233 q^{83} -1.41735 q^{84} -1.91753 q^{86} +0.819658 q^{87} -0.0585036 q^{88} -1.54657 q^{90} +2.35734 q^{94} -0.968477 q^{96} +4.02387 q^{98} +0.287758 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$2$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$3$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$3$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$4$	1.07661	1.07661
$5$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$5$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$6$	0.974194	0.974194
$7$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$7$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$8$	0.110392	0.110392
$9$	−0.542978	−0.542978
$10$	2.84831	2.84831
$11$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$11$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$12$	0.727822	0.727822
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.80627	−2.80627
$15$	1.33622	1.33622
$16$	−0.917526	−0.917526
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.782455	−0.782455
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	2.12798	2.12798
$21$	−1.31650	−1.31650
$22$	−0.763700	−0.763700
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.0746286	0.0746286
$25$	2.90679	2.90679
$26$	0	0
$27$	−1.04311	−1.04311
$28$	−2.09657	−2.09657
$29$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$29$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$30$	1.92555	1.92555
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.43259	−1.43259
$33$	−0.358273	−0.358273
$34$	0	0
$35$	−3.84914	−3.84914
$36$	−0.584573	−0.584573
$37$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$37$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$38$	0	0
$39$	0	0
$40$	0.218196	0.218196
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.89714	−1.89714
$43$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$43$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$44$	−0.570561	−0.570561
$45$	−1.07323	−1.07323
$46$	0	0
$47$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$47$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$48$	−0.620279	−0.620279
$49$	2.79233	2.79233
$50$	4.18881	4.18881
$51$	0	0
$52$	0	0
$53$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$53$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$54$	−1.50316	−1.50316
$55$	−1.04750	−1.04750
$56$	−0.214976	−0.214976
$57$	0	0
$58$	1.74719	1.74719
$59$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$59$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$60$	1.43858	1.43858
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	1.05739	1.05739
$64$	−1.14689	−1.14689
$65$	0	0
$66$	−0.516287	−0.516287
$67$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$67$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$68$	0	0
$69$	0	0
$70$	−5.54677	−5.54677
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.0599404	−0.0599404
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−2.67314	−2.67314
$75$	1.96509	1.96509
$76$	0	0
$77$	1.03205	1.03205
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.81355	−1.81355
$81$	−0.162196	−0.162196
$82$	0	0
$83$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$83$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$84$	−1.41735	−1.41735
$85$	0	0
$86$	−1.91753	−1.91753
$87$	0.819658	0.819658
$88$	−0.0585036	−0.0585036
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1.54657	−1.54657
$91$	0	0
$92$	0	0
$93$	0	0
$94$	2.35734	2.35734
$95$	0	0
$96$	−0.968477	−0.968477
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	4.02387	4.02387
$99$	0.287758	0.287758
$100$	3.12947	3.12947
$101$	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
$105$	−2.60215	−2.60215
$106$	−1.17897	−1.17897
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.12301	−1.12301
$109$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$109$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$110$	−1.50950	−1.50950
$111$	−1.25405	−1.25405
$112$	1.78678	1.78678
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	1.30533	1.30533
$117$	0	0
$118$	2.74777	2.74777
$119$	0	0
$120$	0.147508	0.147508
$121$	−0.719139	−0.719139
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.76889	3.76889
$126$	1.52375	1.52375
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.220136	−0.220136
$129$	−0.899565	−0.899565
$130$	0	0
$131$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$131$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$132$	−0.385719	−0.385719
$133$	0	0
$134$	−1.56661	−1.56661
$135$	−2.06176	−2.06176
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$139$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$140$	−4.14400	−4.14400
$141$	1.10590	1.10590
$142$	0	0
$143$	0	0
$144$	0.498197	0.498197
$145$	2.39648	2.39648
$146$	0	0
$147$	1.88771	1.88771
$148$	−1.99711	−1.99711
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	2.83178	2.83178
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	1.48722	1.48722
$155$	0	0
$156$	0	0
$157$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$157$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$158$	0	0
$159$	−0.553088	−0.553088
$160$	−2.83159	−2.83159
$161$	0	0
$162$	−0.233732	−0.233732
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−0.708148	−0.708148
$166$	2.58283	2.58283
$167$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$167$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$168$	−0.145331	−0.145331
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−1.43259	−1.43259
$173$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$173$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$174$	1.18116	1.18116
$175$	−5.66066	−5.66066
$176$	0.486255	0.486255
$177$	1.28906	1.28906
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.15545	−1.15545
$181$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$181$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.66653	−3.66653
$186$	0	0
$187$	0	0
$188$	1.76117	1.76117
$189$	2.03133	2.03133
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.775338	−0.775338
$193$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$193$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$194$	0	0
$195$	0	0
$196$	3.00624	3.00624
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.414672	0.414672
$199$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$199$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$200$	0.320886	0.320886
$201$	−0.734940	−0.734940
$202$	0	0
$203$	−2.36112	−2.36112
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−3.74981	−3.74981
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.880811	−0.880811
$213$	0	0
$214$	0	0
$215$	−2.63011	−2.63011
$216$	−0.115150	−0.115150
$217$	0	0
$218$	0.548724	0.548724
$219$	0	0
$220$	−1.12775	−1.12775
$221$	0	0
$222$	−1.80713	−1.80713
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	2.78981	2.78981
$225$	−1.57833	−1.57833
$226$	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
$231$	0.697697	0.697697
$232$	0.133845	0.133845
$233$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$233$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$234$	0	0
$235$	3.23337	3.23337
$236$	2.05286	2.05286
$237$	0	0
$238$	0	0
$239$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$239$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$240$	−1.22602	−1.22602
$241$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$241$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$242$	−1.03631	−1.03631
$243$	0.933455	0.933455
$244$	0	0
$245$	5.51921	5.51921
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.21168	1.21168
$250$	5.43114	5.43114
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.13839	1.13839
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0.829668	0.829668
$257$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$257$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$258$	−1.29631	−1.29631
$259$	3.61242	3.61242
$260$	0	0
$261$	−0.658335	−0.658335
$262$	−0.330528	−0.330528
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.0395504	−0.0395504
$265$	−1.61710	−1.61710
$266$	0	0
$267$	0	0
$268$	−1.17042	−1.17042
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−2.97109	−2.97109
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.54049	−1.54049
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−2.22350	−2.22350
$279$	0	0
$280$	−0.424913	−0.424913
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.59364	1.59364
$283$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$283$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.777864	0.777864
$289$	1.00000	1.00000
$290$	3.45344	3.45344
$291$	0	0
$292$	0	0
$293$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$293$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$294$	2.72027	2.72027
$295$	3.76889	3.76889
$296$	−0.204777	−0.204777
$297$	0.552807	0.552807
$298$	0	0
$299$	0	0
$300$	2.11563	2.11563
$301$	2.59130	2.59130
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$307$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$308$	1.11111	1.11111
$309$	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			$\pi$
$314$	−0.763700	−0.763700
$315$	2.09000	2.09000
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.797024	−0.797024
$319$	−0.642554	−0.642554
$320$	−2.26690	−2.26690
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.174621	−0.174621
$325$	0	0
$326$	0	0
$327$	0.257422	0.257422
$328$	0	0
$329$	−3.18566	−3.18566
$330$	−1.02047	−1.02047
$331$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$331$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$332$	1.92963	1.92963
$333$	1.00723	1.00723
$334$	0.110392	0.110392
$335$	−2.14879	−2.14879
$336$	1.20792	1.20792
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.44104	1.44104
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3.49037	−3.49037
$344$	−0.146893	−0.146893
$345$	0	0
$346$	0.548724	0.548724
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.882448	0.882448
$349$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$349$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$350$	−8.15726	−8.15726
$351$	0	0
$352$	0.759218	0.759218
$353$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$353$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$354$	1.85759	1.85759
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.118476	−0.118476
$361$	1.00000	1.00000
$362$	−2.87363	−2.87363
$363$	−0.486162	−0.486162
$364$	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
$370$	−5.28363	−5.28363
$371$	1.59323	1.59323
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	2.54790	2.54790
$376$	0.180585	0.180585
$377$	0	0
$378$	2.92724	2.92724
$379$	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			$\pi$
$384$	−0.148819	−0.148819
$385$	2.03990	2.03990
$386$	−1.17897	−1.17897
$387$	0.722515	0.722515
$388$	0	0
$389$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$389$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$390$	0	0
$391$	0	0
$392$	0.308250	0.308250
$393$	−0.155060	−0.155060
$394$	0	0
$395$	0	0
$396$	0.309802	0.309802
$397$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$397$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$398$	2.07661	2.07661
$399$	0	0
$400$	−2.66706	−2.66706
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−1.05908	−1.05908
$403$	0	0
$404$	0	0
$405$	−0.320591	−0.320591
$406$	−3.40247	−3.40247
$407$	0.983084	0.983084
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.71327	−3.71327
$414$	0	0
$415$	3.54265	3.54265
$416$	0	0
$417$	−1.04311	−1.04311
$418$	0	0
$419$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$419$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$420$	−2.80148	−2.80148
$421$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$421$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$422$	0	0
$423$	−0.888236	−0.888236
$424$	−0.0903156	−0.0903156
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−3.79011	−3.79011
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.957077	0.957077
$433$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$433$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$434$	0	0
$435$	1.62010	1.62010
$436$	0.409952	0.409952
$437$	0	0
$438$	0	0
$439$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$439$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$440$	−0.115636	−0.115636
$441$	−1.51618	−1.51618
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−1.35011	−1.35011
$445$	0	0
$446$	0	0
$447$	0	0
$448$	2.23345	2.23345
$449$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$449$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$450$	−2.27444	−2.27444
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$457$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$461$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$462$	1.00541	1.00541
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.11246	−1.11246
$465$	0	0
$466$	−0.330528	−0.330528
$467$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$467$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$468$	0	0
$469$	2.11708	2.11708
$470$	4.65943	4.65943
$471$	−0.358273	−0.358273
$472$	0.210494	0.210494
$473$	0.705196	0.705196
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.444231	0.444231
$478$	−2.47735	−2.47735
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.91425	−1.91425
$481$	0	0
$482$	−1.91753	−1.91753
$483$	0	0
$484$	−0.774229	−0.774229
$485$	0	0
$486$	1.34515	1.34515
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	7.95342	7.95342
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.568772	0.568772
$496$	0	0
$497$	0	0
$498$	1.74608	1.74608
$499$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$499$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$500$	4.05761	4.05761
$501$	0.0517879	0.0517879
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0.116727	0.116727
$505$	0	0
$506$	0	0
$507$	0.676034	0.676034
$508$	0	0
$509$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$509$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$510$	0	0
$511$	0	0
$512$	1.41572	1.41572
$513$	0	0
$514$	0.548724	0.548724
$515$	0	0
$516$	−0.968477	−0.968477
$517$	−0.866945	−0.866945
$518$	5.20565	5.20565
$519$	0.257422	0.257422
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−0.948689	−0.948689
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.246938	−0.246938
$525$	−3.82680	−3.82680
$526$	0	0
$527$	0	0
$528$	0.328725	0.328725
$529$	1.00000	1.00000
$530$	−2.33031	−2.33031
$531$	−1.03535	−1.03535
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−0.120011	−0.120011
$537$	0	0
$538$	0	0
$539$	−1.47983	−1.47983
$540$	−2.21970	−2.21970
$541$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$541$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$542$	0	0
$543$	−1.34810	−1.34810
$544$	0	0
$545$	0.752639	0.752639
$546$	0	0
$547$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$547$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$548$	0	0
$549$	0	0
$550$	−2.21992	−2.21992
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.47870	−2.47870
$556$	−1.66118	−1.66118
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	3.53168	3.53168
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.19061	1.19061
$565$	0	0
$566$	1.74719	1.74719
$567$	0.315859	0.315859
$568$	0	0
$569$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$569$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$570$	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
$576$	0.622738	0.622738
$577$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$577$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$578$	1.44104	1.44104
$579$	−0.553088	−0.553088
$580$	2.58007	2.58007
$581$	−3.49037	−3.49037
$582$	0	0
$583$	0.433582	0.433582
$584$	0	0
$585$	0	0
$586$	−2.80627	−2.80627
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	2.03232	2.03232
$589$	0	0
$590$	5.43114	5.43114
$591$	0	0
$592$	1.70202	1.70202
$593$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$593$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$594$	0.796619	0.796619
$595$	0	0
$596$	0	0
$597$	0.974194	0.974194
$598$	0	0
$599$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$599$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$600$	0.216930	0.216930
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	3.73417	3.73417
$603$	0.590291	0.590291
$604$	0	0
$605$	−1.42142	−1.42142
$606$	0	0
$607$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$607$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$608$	0	0
$609$	−1.59619	−1.59619
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$613$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$614$	0.548724	0.548724
$615$	0	0
$616$	0.113929	0.113929
$617$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$617$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$618$	0	0
$619$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$619$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	4.54265	4.54265
$626$	0	0
$627$	0	0
$628$	−0.570561	−0.570561
$629$	0	0
$630$	3.01178	3.01178
$631$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$631$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.595458	−0.595458
$637$	0	0
$638$	−0.925948	−0.925948
$639$	0	0
$640$	−0.435112	−0.435112
$641$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$641$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$642$	0	0
$643$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$643$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$644$	0	0
$645$	−1.77805	−1.77805
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.0179051	−0.0179051
$649$	−1.01053	−1.01053
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.370956	0.370956
$655$	−0.453358	−0.453358
$656$	0	0
$657$	0	0
$658$	−4.59067	−4.59067
$659$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$659$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$660$	−0.762396	−0.762396
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.37683	1.37683
$663$	0	0
$664$	0.197859	0.197859
$665$	0	0
$666$	1.45146	1.45146
$667$	0	0
$668$	0.0824739	0.0824739
$669$	0	0
$670$	−3.09650	−3.09650
$671$	0	0
$672$	1.88600	1.88600
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.03209	−3.03209
$676$	1.07661	1.07661
$677$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$677$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$683$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$684$	0	0
$685$	0	0
$686$	−5.02977	−5.02977
$687$	0	0
$688$	1.22091	1.22091
$689$	0	0
$690$	0	0
$691$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$691$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$692$	0.409952	0.409952
$693$	−0.560378	−0.560378
$694$	0	0
$695$	−3.04979	−3.04979
$696$	0.0904835	0.0904835
$697$	0	0
$698$	−1.17897	−1.17897
$699$	−0.155060	−0.155060
$700$	−6.09430	−6.09430
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0.607811	0.607811
$705$	2.18587	2.18587
$706$	−1.56661	−1.56661
$707$	0	0
$708$	1.38781	1.38781
$709$	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
$717$	−1.16220	−1.16220
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.984716	0.984716
$721$	0	0
$722$	1.44104	1.44104
$723$	−0.899565	−0.899565
$724$	−2.14689	−2.14689
$725$	3.52434	3.52434
$726$	−0.700581	−0.700581
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0.793243	0.793243
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$733$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$734$	0	0
$735$	3.73117	3.73117
$736$	0	0
$737$	0.576141	0.576141
$738$	0	0
$739$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$739$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$740$	−3.94741	−3.94741
$741$	0	0
$742$	2.29592	2.29592
$743$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$743$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.973197	−0.973197
$748$	0	0
$749$	0	0
$750$	3.67163	3.67163
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1.50094	−1.50094
$753$	0	0
$754$	0	0
$755$	0	0
$756$	2.18695	2.18695
$757$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$757$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$761$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$762$	0	0
$763$	−0.741532	−0.741532
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.560883	0.560883
$769$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$769$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$770$	2.93958	2.93958
$771$	0.257422	0.257422
$772$	−0.880811	−0.880811
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.04118	1.04118
$775$	0	0
$776$	0	0
$777$	2.44212	2.44212
$778$	0.974194	0.974194
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.26471	−1.26471
$784$	−2.56204	−2.56204
$785$	−1.04750	−1.04750
$786$	−0.223448	−0.223448
$787$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$787$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.0317662	0.0317662
$793$	0	0
$794$	2.07661	2.07661
$795$	−1.09321	−1.09321
$796$	1.55143	1.55143
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−4.16423	−4.16423
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−0.791240	−0.791240
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$809$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$810$	−0.461985	−0.461985
$811$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$811$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$812$	−2.54199	−2.54199
$813$	0	0
$814$	1.41667	1.41667
$815$	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
$823$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$823$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$824$	0	0
$825$	−1.04143	−1.04143
$826$	−5.35098	−5.35098
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$829$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$830$	5.10511	5.10511
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−1.50316	−1.50316
$835$	0.151415	0.151415
$836$	0	0
$837$	0	0
$838$	1.37683	1.37683
$839$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$839$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$840$	−0.287256	−0.287256
$841$	0.470037	0.470037
$842$	−1.91753	−1.91753
$843$	0	0
$844$	0	0
$845$	1.97656	1.97656
$846$	−1.27999	−1.27999
$847$	1.40045	1.40045
$848$	0.750662	0.750662
$849$	0.819658	0.819658
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$853$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$859$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$860$	−2.83159	−2.83159
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.49434	1.49434
$865$	0.752639	0.752639
$866$	−0.330528	−0.330528
$867$	0.676034	0.676034
$868$	0	0
$869$	0	0
$870$	2.33464	2.33464
$871$	0	0
$872$	0.0420352	0.0420352
$873$	0	0
$874$	0	0
$875$	−7.33951	−7.33951
$876$	0	0
$877$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$877$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$878$	−0.763700	−0.763700
$879$	−1.31650	−1.31650
$880$	0.961112	0.961112
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−2.18487	−2.18487
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	2.54790	2.54790
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.138436	−0.138436
$889$	0	0
$890$	0	0
$891$	0.0859580	0.0859580
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.428690	0.428690
$897$	0	0
$898$	0.974194	0.974194
$899$	0	0
$900$	−1.69923	−1.69923
$901$	0	0
$902$	0	0
$903$	1.75181	1.75181
$904$	0	0
$905$	−3.94152	−3.94152
$906$	0	0
$907$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$907$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.949869	−0.949869
$914$	1.74719	1.74719
$915$	0	0
$916$	0	0
$917$	0.446667	0.446667
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0.257422	0.257422
$922$	−2.87363	−2.87363
$923$	0	0
$924$	0.751145	0.751145
$925$	−5.39211	−5.39211
$926$	0	0
$927$	0	0
$928$	−1.73694	−1.73694
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−0.246938	−0.246938
$933$	0	0
$934$	1.37683	1.37683
$935$	0	0
$936$	0	0
$937$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$937$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$938$	3.05080	3.05080
$939$	0	0
$940$	3.48107	3.48107
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.516287	−0.516287
$943$	0	0
$944$	−1.74953	−1.74953
$945$	4.01506	4.01506
$946$	1.01622	1.01622
$947$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$947$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$948$	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			$\pi$
$954$	0.640156	0.640156
$955$	0	0
$956$	−1.85083	−1.85083
$957$	−0.434388	−0.434388
$958$	0	0
$959$	0	0
$960$	−1.53250	−1.53250
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−1.43259	−1.43259
$965$	−1.61710	−1.61710
$966$	0	0
$967$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$967$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$968$	−0.0793871	−0.0793871
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.00496	1.00496
$973$	3.00478	3.00478
$974$	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
$980$	5.94201	5.94201
$981$	−0.206757	−0.206757
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.15361	−2.15361
$988$	0	0
$989$	0	0
$990$	0.819625	0.819625
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0.645909	0.645909
$994$	0	0
$995$	2.84831	2.84831
$996$	1.30450	1.30450
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	2.74777	2.74777
$999$	1.93497	1.93497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.16	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.16	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.44104 q^{2} +0.676034 q^{3} +1.07661 q^{4} +1.97656 q^{5} +0.974194 q^{6} -1.94739 q^{7} +0.110392 q^{8} -0.542978 q^{9} +O(q^{10})\)	\(q+1.44104 q^{2} +0.676034 q^{3} +1.07661 q^{4} +1.97656 q^{5} +0.974194 q^{6} -1.94739 q^{7} +0.110392 q^{8} -0.542978 q^{9} +2.84831 q^{10} -0.529963 q^{11} +0.727822 q^{12} -2.80627 q^{14} +1.33622 q^{15} -0.917526 q^{16} -0.782455 q^{18} +2.12798 q^{20} -1.31650 q^{21} -0.763700 q^{22} +0.0746286 q^{24} +2.90679 q^{25} -1.04311 q^{27} -2.09657 q^{28} +1.21245 q^{29} +1.92555 q^{30} -1.43259 q^{32} -0.358273 q^{33} -3.84914 q^{35} -0.584573 q^{36} -1.85500 q^{37} +0.218196 q^{40} -1.89714 q^{42} -1.33065 q^{43} -0.570561 q^{44} -1.07323 q^{45} +1.63586 q^{47} -0.620279 q^{48} +2.79233 q^{49} +4.18881 q^{50} -0.818137 q^{53} -1.50316 q^{54} -1.04750 q^{55} -0.214976 q^{56} +1.74719 q^{58} +1.90679 q^{59} +1.43858 q^{60} +1.05739 q^{63} -1.14689 q^{64} -0.516287 q^{66} -1.08714 q^{67} -5.54677 q^{70} -0.0599404 q^{72} -2.67314 q^{74} +1.96509 q^{75} +1.03205 q^{77} -1.81355 q^{80} -0.162196 q^{81} +1.79233 q^{83} -1.41735 q^{84} -1.91753 q^{86} +0.819658 q^{87} -0.0585036 q^{88} -1.54657 q^{90} +2.35734 q^{94} -0.968477 q^{96} +4.02387 q^{98} +0.287758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more