LMFDB - Embedded newform 1151.1.b.a.1150.1

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			1150.1
Root			$-1.90679$ 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)

$f(q)$	$=$	$q-1.99413 q^{2} +0.380782 q^{3} +2.97656 q^{4} +1.63586 q^{5} -0.759330 q^{6} +1.21245 q^{7} -3.94152 q^{8} -0.855005 q^{9} +O(q^{10})$	\(q-1.99413 q^{2} +0.380782 q^{3} +2.97656 q^{4} +1.63586 q^{5} -0.759330 q^{6} +1.21245 q^{7} -3.94152 q^{8} -0.855005 q^{9} -3.26212 q^{10} +0.955440 q^{11} +1.13342 q^{12} -2.41779 q^{14} +0.622906 q^{15} +4.88335 q^{16} +1.70499 q^{18} +4.86923 q^{20} +0.461680 q^{21} -1.90527 q^{22} -1.50086 q^{24} +1.67603 q^{25} -0.706353 q^{27} +3.60893 q^{28} -1.71914 q^{29} -1.24216 q^{30} -5.79653 q^{32} +0.363814 q^{33} +1.98340 q^{35} -2.54497 q^{36} +0.0766055 q^{37} -6.44777 q^{40} -0.920650 q^{42} -1.94739 q^{43} +2.84392 q^{44} -1.39867 q^{45} -1.54298 q^{47} +1.85949 q^{48} +0.470037 q^{49} -3.34223 q^{50} -0.229367 q^{53} +1.40856 q^{54} +1.56296 q^{55} -4.77890 q^{56} +3.42819 q^{58} +0.676034 q^{59} +1.85412 q^{60} -1.03665 q^{63} +6.67568 q^{64} -0.725494 q^{66} -1.33065 q^{67} -3.95516 q^{70} +3.37002 q^{72} -0.152761 q^{74} +0.638204 q^{75} +1.15842 q^{77} +7.98848 q^{80} +0.586038 q^{81} -0.529963 q^{83} +1.37422 q^{84} +3.88335 q^{86} -0.654618 q^{87} -3.76589 q^{88} +2.78913 q^{90} +3.07690 q^{94} -2.20721 q^{96} -0.937316 q^{98} -0.816906 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$2$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$3$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$3$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$4$	2.97656	2.97656
$5$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$5$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$6$	−0.759330	−0.759330
$7$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$7$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$8$	−3.94152	−3.94152
$9$	−0.855005	−0.855005
$10$	−3.26212	−3.26212
$11$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$11$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$12$	1.13342	1.13342
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.41779	−2.41779
$15$	0.622906	0.622906
$16$	4.88335	4.88335
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.70499	1.70499
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	4.86923	4.86923
$21$	0.461680	0.461680
$22$	−1.90527	−1.90527
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.50086	−1.50086
$25$	1.67603	1.67603
$26$	0	0
$27$	−0.706353	−0.706353
$28$	3.60893	3.60893
$29$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$29$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$30$	−1.24216	−1.24216
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−5.79653	−5.79653
$33$	0.363814	0.363814
$34$	0	0
$35$	1.98340	1.98340
$36$	−2.54497	−2.54497
$37$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$37$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$38$	0	0
$39$	0	0
$40$	−6.44777	−6.44777
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.920650	−0.920650
$43$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$43$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$44$	2.84392	2.84392
$45$	−1.39867	−1.39867
$46$	0	0
$47$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$47$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$48$	1.85949	1.85949
$49$	0.470037	0.470037
$50$	−3.34223	−3.34223
$51$	0	0
$52$	0	0
$53$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$53$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$54$	1.40856	1.40856
$55$	1.56296	1.56296
$56$	−4.77890	−4.77890
$57$	0	0
$58$	3.42819	3.42819
$59$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$59$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$60$	1.85412	1.85412
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−1.03665	−1.03665
$64$	6.67568	6.67568
$65$	0	0
$66$	−0.725494	−0.725494
$67$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$67$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$68$	0	0
$69$	0	0
$70$	−3.95516	−3.95516
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	3.37002	3.37002
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.152761	−0.152761
$75$	0.638204	0.638204
$76$	0	0
$77$	1.15842	1.15842
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	7.98848	7.98848
$81$	0.586038	0.586038
$82$	0	0
$83$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$83$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$84$	1.37422	1.37422
$85$	0	0
$86$	3.88335	3.88335
$87$	−0.654618	−0.654618
$88$	−3.76589	−3.76589
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	2.78913	2.78913
$91$	0	0
$92$	0	0
$93$	0	0
$94$	3.07690	3.07690
$95$	0	0
$96$	−2.20721	−2.20721
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.937316	−0.937316
$99$	−0.816906	−0.816906
$100$	4.98882	4.98882
$101$	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
$105$	0.755243	0.755243
$106$	0.457388	0.457388
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−2.10250	−2.10250
$109$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$109$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$110$	−3.11676	−3.11676
$111$	0.0291700	0.0291700
$112$	5.92083	5.92083
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−5.11712	−5.11712
$117$	0	0
$118$	−1.34810	−1.34810
$119$	0	0
$120$	−2.45520	−2.45520
$121$	−0.0871351	−0.0871351
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.10590	1.10590
$126$	2.06722	2.06722
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−7.51567	−7.51567
$129$	−0.741532	−0.741532
$130$	0	0
$131$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$131$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$132$	1.08292	1.08292
$133$	0	0
$134$	2.65349	2.65349
$135$	−1.15549	−1.15549
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$139$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$140$	5.90371	5.90371
$141$	−0.587539	−0.587539
$142$	0	0
$143$	0	0
$144$	−4.17529	−4.17529
$145$	−2.81227	−2.81227
$146$	0	0
$147$	0.178982	0.178982
$148$	0.228021	0.228021
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.27266	−1.27266
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−2.31005	−2.31005
$155$	0	0
$156$	0	0
$157$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$157$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$158$	0	0
$159$	−0.0873388	−0.0873388
$160$	−9.48230	−9.48230
$161$	0	0
$162$	−1.16864	−1.16864
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0.595149	0.595149
$166$	1.05682	1.05682
$167$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$167$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$168$	−1.81972	−1.81972
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	−5.79653	−5.79653
$173$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$173$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$174$	1.30539	1.30539
$175$	2.03211	2.03211
$176$	4.66575	4.66575
$177$	0.257422	0.257422
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−4.16322	−4.16322
$181$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$181$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.125316	0.125316
$186$	0	0
$187$	0	0
$188$	−4.59277	−4.59277
$189$	−0.856418	−0.856418
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2.54198	2.54198
$193$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$193$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$194$	0	0
$195$	0	0
$196$	1.39909	1.39909
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	1.62902	1.62902
$199$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$199$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$200$	−6.60612	−6.60612
$201$	−0.506688	−0.506688
$202$	0	0
$203$	−2.08437	−2.08437
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−1.50605	−1.50605
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.682724	−0.682724
$213$	0	0
$214$	0	0
$215$	−3.18566	−3.18566
$216$	2.78411	2.78411
$217$	0	0
$218$	−2.87363	−2.87363
$219$	0	0
$220$	4.65226	4.65226
$221$	0	0
$222$	−0.0581688	−0.0581688
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−7.02800	−7.02800
$225$	−1.43302	−1.43302
$226$	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
$231$	0.441107	0.441107
$232$	6.77603	6.77603
$233$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$233$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$234$	0	0
$235$	−2.52409	−2.52409
$236$	2.01226	2.01226
$237$	0	0
$238$	0	0
$239$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$239$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$240$	3.04187	3.04187
$241$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$241$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$242$	0.173759	0.173759
$243$	0.929506	0.929506
$244$	0	0
$245$	0.768914	0.768914
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.201800	−0.201800
$250$	−2.20530	−2.20530
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−3.08566	−3.08566
$253$	0	0
$254$	0	0
$255$	0	0
$256$	8.31154	8.31154
$257$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$257$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$258$	1.47871	1.47871
$259$	0.0928804	0.0928804
$260$	0	0
$261$	1.46987	1.46987
$262$	−3.57414	−3.57414
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.43398	−1.43398
$265$	−0.375212	−0.375212
$266$	0	0
$267$	0	0
$268$	−3.96076	−3.96076
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	2.30421	2.30421
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.60135	1.60135
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	3.69912	3.69912
$279$	0	0
$280$	−7.81761	−7.81761
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.17163	1.17163
$283$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$283$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	4.95606	4.95606
$289$	1.00000	1.00000
$290$	5.60803	5.60803
$291$	0	0
$292$	0	0
$293$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$293$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$294$	−0.356913	−0.356913
$295$	1.10590	1.10590
$296$	−0.301942	−0.301942
$297$	−0.674878	−0.674878
$298$	0	0
$299$	0	0
$300$	1.89965	1.89965
$301$	−2.36112	−2.36112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$307$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$308$	3.44812	3.44812
$309$	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			$\pi$
$314$	−1.90527	−1.90527
$315$	−1.69582	−1.69582
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.174165	0.174165
$319$	−1.64253	−1.64253
$320$	10.9205	10.9205
$321$	0	0
$322$	0	0
$323$	0	0
$324$	1.74438	1.74438
$325$	0	0
$326$	0	0
$327$	0.548724	0.548724
$328$	0	0
$329$	−1.87079	−1.87079
$330$	−1.18681	−1.18681
$331$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$331$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$332$	−1.57747	−1.57747
$333$	−0.0654981	−0.0654981
$334$	−3.94152	−3.94152
$335$	−2.17676	−2.17676
$336$	2.25455	2.25455
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.99413	−1.99413
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−0.642554	−0.642554
$344$	7.67568	7.67568
$345$	0	0
$346$	−2.87363	−2.87363
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.94851	−1.94851
$349$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$349$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$350$	−4.05229	−4.05229
$351$	0	0
$352$	−5.53823	−5.53823
$353$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$353$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$354$	−0.513333	−0.513333
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	5.51288	5.51288
$361$	1.00000	1.00000
$362$	−3.80240	−3.80240
$363$	−0.0331795	−0.0331795
$364$	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
$370$	−0.249896	−0.249896
$371$	−0.278096	−0.278096
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.421105	0.421105
$376$	6.08168	6.08168
$377$	0	0
$378$	1.70781	1.70781
$379$	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			$\pi$
$384$	−2.86183	−2.86183
$385$	1.89502	1.89502
$386$	0.457388	0.457388
$387$	1.66503	1.66503
$388$	0	0
$389$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$389$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$390$	0	0
$391$	0	0
$392$	−1.85266	−1.85266
$393$	0.682488	0.682488
$394$	0	0
$395$	0	0
$396$	−2.43157	−2.43157
$397$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$397$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$398$	3.97656	3.97656
$399$	0	0
$400$	8.18467	8.18467
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	1.01040	1.01040
$403$	0	0
$404$	0	0
$405$	0.958676	0.958676
$406$	4.15651	4.15651
$407$	0.0731919	0.0731919
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.819658	0.819658
$414$	0	0
$415$	−0.866945	−0.866945
$416$	0	0
$417$	−0.706353	−0.706353
$418$	0	0
$419$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$419$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$420$	2.24803	2.24803
$421$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$421$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$422$	0	0
$423$	1.31925	1.31925
$424$	0.904055	0.904055
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	6.35262	6.35262
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−3.44937	−3.44937
$433$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$433$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$434$	0	0
$435$	−1.07086	−1.07086
$436$	4.28935	4.28935
$437$	0	0
$438$	0	0
$439$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$439$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$440$	−6.16046	−6.16046
$441$	−0.401884	−0.401884
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.0868263	0.0868263
$445$	0	0
$446$	0	0
$447$	0	0
$448$	8.09394	8.09394
$449$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$449$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$450$	2.85762	2.85762
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$457$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$461$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$462$	−0.879626	−0.879626
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−8.39516	−8.39516
$465$	0	0
$466$	−3.57414	−3.57414
$467$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$467$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$468$	0	0
$469$	−1.61335	−1.61335
$470$	5.03338	5.03338
$471$	0.363814	0.363814
$472$	−2.66460	−2.66460
$473$	−1.86061	−1.86061
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.196110	0.196110
$478$	2.16789	2.16789
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−3.61069	−3.61069
$481$	0	0
$482$	3.88335	3.88335
$483$	0	0
$484$	−0.259363	−0.259363
$485$	0	0
$486$	−1.85356	−1.85356
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−1.53332	−1.53332
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−1.33634	−1.33634
$496$	0	0
$497$	0	0
$498$	0.402417	0.402417
$499$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$499$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$500$	3.29177	3.29177
$501$	0.752639	0.752639
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	4.08598	4.08598
$505$	0	0
$506$	0	0
$507$	0.380782	0.380782
$508$	0	0
$509$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$509$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$510$	0	0
$511$	0	0
$512$	−9.05864	−9.05864
$513$	0	0
$514$	−2.87363	−2.87363
$515$	0	0
$516$	−2.20721	−2.20721
$517$	−1.47422	−1.47422
$518$	−0.185216	−0.185216
$519$	0.548724	0.548724
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−2.93112	−2.93112
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	5.33498	5.33498
$525$	0.773791	0.773791
$526$	0	0
$527$	0	0
$528$	1.77663	1.77663
$529$	1.00000	1.00000
$530$	0.748222	0.748222
$531$	−0.578012	−0.578012
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	5.24479	5.24479
$537$	0	0
$538$	0	0
$539$	0.449092	0.449092
$540$	−3.43940	−3.43940
$541$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$541$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$542$	0	0
$543$	0.726073	0.726073
$544$	0	0
$545$	2.35734	2.35734
$546$	0	0
$547$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$547$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$548$	0	0
$549$	0	0
$550$	−3.19330	−3.19330
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.0477180	0.0477180
$556$	−5.52153	−5.52153
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	9.68564	9.68564
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−1.74884	−1.74884
$565$	0	0
$566$	3.42819	3.42819
$567$	0.710543	0.710543
$568$	0	0
$569$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$569$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$570$	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
$576$	−5.70774	−5.70774
$577$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$577$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$578$	−1.99413	−1.99413
$579$	−0.0873388	−0.0873388
$580$	−8.37089	−8.37089
$581$	−0.642554	−0.642554
$582$	0	0
$583$	−0.219146	−0.219146
$584$	0	0
$585$	0	0
$586$	−2.41779	−2.41779
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.532750	0.532750
$589$	0	0
$590$	−2.20530	−2.20530
$591$	0	0
$592$	0.374092	0.374092
$593$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$593$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$594$	1.34579	1.34579
$595$	0	0
$596$	0	0
$597$	−0.759330	−0.759330
$598$	0	0
$599$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$599$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$600$	−2.51549	−2.51549
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	4.70838	4.70838
$603$	1.13771	1.13771
$604$	0	0
$605$	−0.142541	−0.142541
$606$	0	0
$607$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$607$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$608$	0	0
$609$	−0.793692	−0.793692
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$613$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$614$	−2.87363	−2.87363
$615$	0	0
$616$	−4.56595	−4.56595
$617$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$617$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$618$	0	0
$619$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$619$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.133055	0.133055
$626$	0	0
$627$	0	0
$628$	2.84392	2.84392
$629$	0	0
$630$	3.38168	3.38168
$631$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$631$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.259969	−0.259969
$637$	0	0
$638$	3.27543	3.27543
$639$	0	0
$640$	−12.2946	−12.2946
$641$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$641$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$642$	0	0
$643$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$643$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$644$	0	0
$645$	−1.21304	−1.21304
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.30988	−2.30988
$649$	0.645909	0.645909
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−1.09423	−1.09423
$655$	2.93200	2.93200
$656$	0	0
$657$	0	0
$658$	3.73059	3.73059
$659$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$659$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$660$	1.77150	1.77150
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.63147	1.63147
$663$	0	0
$664$	2.08886	2.08886
$665$	0	0
$666$	0.130612	0.130612
$667$	0	0
$668$	5.88335	5.88335
$669$	0	0
$670$	4.34074	4.34074
$671$	0	0
$672$	−2.67614	−2.67614
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−1.18387	−1.18387
$676$	2.97656	2.97656
$677$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$677$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$683$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$684$	0	0
$685$	0	0
$686$	1.28134	1.28134
$687$	0	0
$688$	−9.50980	−9.50980
$689$	0	0
$690$	0	0
$691$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$691$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$692$	4.28935	4.28935
$693$	−0.990458	−0.990458
$694$	0	0
$695$	−3.03453	−3.03453
$696$	2.58019	2.58019
$697$	0	0
$698$	0.457388	0.457388
$699$	0.682488	0.682488
$700$	6.04869	6.04869
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	6.37821	6.37821
$705$	−0.961130	−0.961130
$706$	2.65349	2.65349
$707$	0	0
$708$	0.766231	0.766231
$709$	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
$717$	−0.413962	−0.413962
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−6.83019	−6.83019
$721$	0	0
$722$	−1.99413	−1.99413
$723$	−0.741532	−0.741532
$724$	5.67568	5.67568
$725$	−2.88134	−2.88134
$726$	0.0661643	0.0661643
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.232099	−0.232099
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$733$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$734$	0	0
$735$	0.292789	0.292789
$736$	0	0
$737$	−1.27136	−1.27136
$738$	0	0
$739$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$739$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$740$	0.373010	0.373010
$741$	0	0
$742$	0.554560	0.554560
$743$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$743$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.453121	0.453121
$748$	0	0
$749$	0	0
$750$	−0.839740	−0.839740
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−7.53491	−7.53491
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−2.54918	−2.54918
$757$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$757$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$761$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$762$	0	0
$763$	1.74719	1.74719
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	3.16489	3.16489
$769$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$769$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$770$	−3.77891	−3.77891
$771$	0.548724	0.548724
$772$	−0.682724	−0.682724
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−3.32029	−3.32029
$775$	0	0
$776$	0	0
$777$	0.0353672	0.0353672
$778$	−0.759330	−0.759330
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.21432	1.21432
$784$	2.29536	2.29536
$785$	1.56296	1.56296
$786$	−1.36097	−1.36097
$787$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$787$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	3.21985	3.21985
$793$	0	0
$794$	3.97656	3.97656
$795$	−0.142874	−0.142874
$796$	−5.93565	−5.93565
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−9.71518	−9.71518
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1.50819	−1.50819
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$809$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$810$	−1.91173	−1.91173
$811$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$811$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$812$	−6.20426	−6.20426
$813$	0	0
$814$	−0.145954	−0.145954
$815$	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
$823$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$823$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$824$	0	0
$825$	0.609765	0.609765
$826$	−1.63451	−1.63451
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$829$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$830$	1.72880	1.72880
$831$	0	0
$832$	0	0
$833$	0	0
$834$	1.40856	1.40856
$835$	3.23337	3.23337
$836$	0	0
$837$	0	0
$838$	1.63147	1.63147
$839$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$839$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$840$	−2.97681	−2.97681
$841$	1.95544	1.95544
$842$	3.88335	3.88335
$843$	0	0
$844$	0	0
$845$	1.63586	1.63586
$846$	−2.63077	−2.63077
$847$	−0.105647	−0.105647
$848$	−1.12008	−1.12008
$849$	−0.654618	−0.654618
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$853$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$859$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$860$	−9.48230	−9.48230
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	4.09439	4.09439
$865$	2.35734	2.35734
$866$	−3.57414	−3.57414
$867$	0.380782	0.380782
$868$	0	0
$869$	0	0
$870$	2.13544	2.13544
$871$	0	0
$872$	−5.67990	−5.67990
$873$	0	0
$874$	0	0
$875$	1.34084	1.34084
$876$	0	0
$877$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$877$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$878$	−1.90527	−1.90527
$879$	0.461680	0.461680
$880$	7.63251	7.63251
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.801409	0.801409
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0.421105	0.421105
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.114974	−0.114974
$889$	0	0
$890$	0	0
$891$	0.559924	0.559924
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−9.11238	−9.11238
$897$	0	0
$898$	−0.759330	−0.759330
$899$	0	0
$900$	−4.26546	−4.26546
$901$	0	0
$902$	0	0
$903$	−0.899071	−0.899071
$904$	0	0
$905$	3.11924	3.11924
$906$	0	0
$907$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$907$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.506348	−0.506348
$914$	3.42819	3.42819
$915$	0	0
$916$	0	0
$917$	2.17311	2.17311
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0.548724	0.548724
$922$	−3.80240	−3.80240
$923$	0	0
$924$	1.31298	1.31298
$925$	0.128393	0.128393
$926$	0	0
$927$	0	0
$928$	9.96504	9.96504
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	5.33498	5.33498
$933$	0	0
$934$	1.63147	1.63147
$935$	0	0
$936$	0	0
$937$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$937$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$938$	3.21723	3.21723
$939$	0	0
$940$	−7.51312	−7.51312
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.725494	−0.725494
$943$	0	0
$944$	3.30131	3.30131
$945$	−1.40098	−1.40098
$946$	3.71031	3.71031
$947$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$947$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$948$	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			$\pi$
$954$	−0.391069	−0.391069
$955$	0	0
$956$	−3.23592	−3.23592
$957$	−0.625448	−0.625448
$958$	0	0
$959$	0	0
$960$	4.15832	4.15832
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	−5.79653	−5.79653
$965$	−0.375212	−0.375212
$966$	0	0
$967$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$967$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$968$	0.343445	0.343445
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.76673	2.76673
$973$	−2.24910	−2.24910
$974$	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
$980$	2.28872	2.28872
$981$	−1.23210	−1.23210
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.712362	−0.712362
$988$	0	0
$989$	0	0
$990$	2.66484	2.66484
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−0.311532	−0.311532
$994$	0	0
$995$	−3.26212	−3.26212
$996$	−0.600671	−0.600671
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.34810	−1.34810
$999$	−0.0541105	−0.0541105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.1	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.1	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.1	✓	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)

\(f(q)\)	\(=\)	\(q-1.99413 q^{2} +0.380782 q^{3} +2.97656 q^{4} +1.63586 q^{5} -0.759330 q^{6} +1.21245 q^{7} -3.94152 q^{8} -0.855005 q^{9} +O(q^{10})\)	\(q-1.99413 q^{2} +0.380782 q^{3} +2.97656 q^{4} +1.63586 q^{5} -0.759330 q^{6} +1.21245 q^{7} -3.94152 q^{8} -0.855005 q^{9} -3.26212 q^{10} +0.955440 q^{11} +1.13342 q^{12} -2.41779 q^{14} +0.622906 q^{15} +4.88335 q^{16} +1.70499 q^{18} +4.86923 q^{20} +0.461680 q^{21} -1.90527 q^{22} -1.50086 q^{24} +1.67603 q^{25} -0.706353 q^{27} +3.60893 q^{28} -1.71914 q^{29} -1.24216 q^{30} -5.79653 q^{32} +0.363814 q^{33} +1.98340 q^{35} -2.54497 q^{36} +0.0766055 q^{37} -6.44777 q^{40} -0.920650 q^{42} -1.94739 q^{43} +2.84392 q^{44} -1.39867 q^{45} -1.54298 q^{47} +1.85949 q^{48} +0.470037 q^{49} -3.34223 q^{50} -0.229367 q^{53} +1.40856 q^{54} +1.56296 q^{55} -4.77890 q^{56} +3.42819 q^{58} +0.676034 q^{59} +1.85412 q^{60} -1.03665 q^{63} +6.67568 q^{64} -0.725494 q^{66} -1.33065 q^{67} -3.95516 q^{70} +3.37002 q^{72} -0.152761 q^{74} +0.638204 q^{75} +1.15842 q^{77} +7.98848 q^{80} +0.586038 q^{81} -0.529963 q^{83} +1.37422 q^{84} +3.88335 q^{86} -0.654618 q^{87} -3.76589 q^{88} +2.78913 q^{90} +3.07690 q^{94} -2.20721 q^{96} -0.937316 q^{98} -0.816906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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