LMFDB - Embedded newform 1151.1.b.a.1150.12

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			1150.12
Root			$-1.44104$ 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)

$f(q)$	$=$	$q+0.380782 q^{2} +1.90679 q^{3} -0.855005 q^{4} +0.0766055 q^{5} +0.726073 q^{6} -1.33065 q^{7} -0.706353 q^{8} +2.63586 q^{9} +O(q^{10})$	\(q+0.380782 q^{2} +1.90679 q^{3} -0.855005 q^{4} +0.0766055 q^{5} +0.726073 q^{6} -1.33065 q^{7} -0.706353 q^{8} +2.63586 q^{9} +0.0291700 q^{10} +1.79233 q^{11} -1.63032 q^{12} -0.506688 q^{14} +0.146071 q^{15} +0.586038 q^{16} +1.00369 q^{18} -0.0654981 q^{20} -2.53728 q^{21} +0.682488 q^{22} -1.34687 q^{24} -0.994132 q^{25} +3.11924 q^{27} +1.13771 q^{28} -1.94739 q^{29} +0.0556211 q^{30} +0.929506 q^{32} +3.41760 q^{33} -0.101935 q^{35} -2.25367 q^{36} -1.54298 q^{37} -0.0541105 q^{40} -0.966150 q^{42} -1.08714 q^{43} -1.53245 q^{44} +0.201921 q^{45} +1.97656 q^{47} +1.11745 q^{48} +0.770633 q^{49} -0.378548 q^{50} +0.955440 q^{53} +1.18775 q^{54} +0.137302 q^{55} +0.939909 q^{56} -0.741532 q^{58} -1.99413 q^{59} -0.124891 q^{60} -3.50741 q^{63} -0.232099 q^{64} +1.30136 q^{66} -1.71914 q^{67} -0.0388151 q^{70} -1.86185 q^{72} -0.587539 q^{74} -1.89560 q^{75} -2.38497 q^{77} +0.0448937 q^{80} +3.31189 q^{81} -0.229367 q^{83} +2.16938 q^{84} -0.413962 q^{86} -3.71327 q^{87} -1.26602 q^{88} +0.0768880 q^{90} +0.752639 q^{94} +1.77238 q^{96} +0.293443 q^{98} +4.72433 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$2$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$3$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$3$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$4$	−0.855005	−0.855005
$5$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$5$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$6$	0.726073	0.726073
$7$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$7$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$8$	−0.706353	−0.706353
$9$	2.63586	2.63586
$10$	0.0291700	0.0291700
$11$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$11$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$12$	−1.63032	−1.63032
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.506688	−0.506688
$15$	0.146071	0.146071
$16$	0.586038	0.586038
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00369	1.00369
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−0.0654981	−0.0654981
$21$	−2.53728	−2.53728
$22$	0.682488	0.682488
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.34687	−1.34687
$25$	−0.994132	−0.994132
$26$	0	0
$27$	3.11924	3.11924
$28$	1.13771	1.13771
$29$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$29$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$30$	0.0556211	0.0556211
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.929506	0.929506
$33$	3.41760	3.41760
$34$	0	0
$35$	−0.101935	−0.101935
$36$	−2.25367	−2.25367
$37$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$37$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$38$	0	0
$39$	0	0
$40$	−0.0541105	−0.0541105
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.966150	−0.966150
$43$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$43$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$44$	−1.53245	−1.53245
$45$	0.201921	0.201921
$46$	0	0
$47$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$47$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$48$	1.11745	1.11745
$49$	0.770633	0.770633
$50$	−0.378548	−0.378548
$51$	0	0
$52$	0	0
$53$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$53$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$54$	1.18775	1.18775
$55$	0.137302	0.137302
$56$	0.939909	0.939909
$57$	0	0
$58$	−0.741532	−0.741532
$59$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$59$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$60$	−0.124891	−0.124891
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−3.50741	−3.50741
$64$	−0.232099	−0.232099
$65$	0	0
$66$	1.30136	1.30136
$67$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$67$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$68$	0	0
$69$	0	0
$70$	−0.0388151	−0.0388151
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.86185	−1.86185
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.587539	−0.587539
$75$	−1.89560	−1.89560
$76$	0	0
$77$	−2.38497	−2.38497
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.0448937	0.0448937
$81$	3.31189	3.31189
$82$	0	0
$83$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$83$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$84$	2.16938	2.16938
$85$	0	0
$86$	−0.413962	−0.413962
$87$	−3.71327	−3.71327
$88$	−1.26602	−1.26602
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.0768880	0.0768880
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.752639	0.752639
$95$	0	0
$96$	1.77238	1.77238
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.293443	0.293443
$99$	4.72433	4.72433
$100$	0.849987	0.849987
$101$	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
$105$	−0.194369	−0.194369
$106$	0.363814	0.363814
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−2.66697	−2.66697
$109$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$109$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$110$	0.0522823	0.0522823
$111$	−2.94214	−2.94214
$112$	−0.779813	−0.779813
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	1.66503	1.66503
$117$	0	0
$118$	−0.759330	−0.759330
$119$	0	0
$120$	−0.103177	−0.103177
$121$	2.21245	2.21245
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.152761	−0.152761
$126$	−1.33556	−1.33556
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.01789	−1.01789
$129$	−2.07294	−2.07294
$130$	0	0
$131$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$131$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$132$	−2.92207	−2.92207
$133$	0	0
$134$	−0.654618	−0.654618
$135$	0.238951	0.238951
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$139$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$140$	0.0871551	0.0871551
$141$	3.76889	3.76889
$142$	0	0
$143$	0	0
$144$	1.54471	1.54471
$145$	−0.149181	−0.149181
$146$	0	0
$147$	1.46944	1.46944
$148$	1.31925	1.31925
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.721812	−0.721812
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−0.908153	−0.908153
$155$	0	0
$156$	0	0
$157$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$157$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$158$	0	0
$159$	1.82183	1.82183
$160$	0.0712052	0.0712052
$161$	0	0
$162$	1.26111	1.26111
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0.261807	0.261807
$166$	−0.0873388	−0.0873388
$167$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$167$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$168$	1.79221	1.79221
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.929506	0.929506
$173$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$173$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$174$	−1.41395	−1.41395
$175$	1.32284	1.32284
$176$	1.05037	1.05037
$177$	−3.80240	−3.80240
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.172644	−0.172644
$181$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$181$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.118201	−0.118201
$186$	0	0
$187$	0	0
$188$	−1.68997	−1.68997
$189$	−4.15063	−4.15063
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.442565	−0.442565
$193$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$193$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$194$	0	0
$195$	0	0
$196$	−0.658895	−0.658895
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	1.79894	1.79894
$199$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$199$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$200$	0.702208	0.702208
$201$	−3.27804	−3.27804
$202$	0	0
$203$	2.59130	2.59130
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.0740124	−0.0740124
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.816906	−0.816906
$213$	0	0
$214$	0	0
$215$	−0.0832805	−0.0832805
$216$	−2.20329	−2.20329
$217$	0	0
$218$	0.257422	0.257422
$219$	0	0
$220$	−0.117394	−0.117394
$221$	0	0
$222$	−1.12031	−1.12031
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−1.23685	−1.23685
$225$	−2.62039	−2.62039
$226$	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
$231$	−4.54764	−4.54764
$232$	1.37555	1.37555
$233$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$233$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$234$	0	0
$235$	0.151415	0.151415
$236$	1.70499	1.70499
$237$	0	0
$238$	0	0
$239$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$239$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$240$	0.0856031	0.0856031
$241$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$241$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$242$	0.842462	0.842462
$243$	3.19585	3.19585
$244$	0	0
$245$	0.0590347	0.0590347
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.437355	−0.437355
$250$	−0.0581688	−0.0581688
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	2.99885	2.99885
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.155494	−0.155494
$257$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$257$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$258$	−0.789339	−0.789339
$259$	2.05317	2.05317
$260$	0	0
$261$	−5.13305	−5.13305
$262$	−0.311532	−0.311532
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−2.41403	−2.41403
$265$	0.0731919	0.0731919
$266$	0	0
$267$	0	0
$268$	1.46987	1.46987
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.0909883	0.0909883
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.78181	−1.78181
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.622906	0.622906
$279$	0	0
$280$	0.0720022	0.0720022
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.43513	1.43513
$283$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$283$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	2.45005	2.45005
$289$	1.00000	1.00000
$290$	−0.0568054	−0.0568054
$291$	0	0
$292$	0	0
$293$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$293$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$294$	0.559536	0.559536
$295$	−0.152761	−0.152761
$296$	1.08989	1.08989
$297$	5.59072	5.59072
$298$	0	0
$299$	0	0
$300$	1.62075	1.62075
$301$	1.44660	1.44660
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$307$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$308$	2.03916	2.03916
$309$	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			$\pi$
$314$	0.682488	0.682488
$315$	−0.268687	−0.268687
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.693719	0.693719
$319$	−3.49037	−3.49037
$320$	−0.0177801	−0.0177801
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−2.83168	−2.83168
$325$	0	0
$326$	0	0
$327$	1.28906	1.28906
$328$	0	0
$329$	−2.63011	−2.63011
$330$	0.0996915	0.0996915
$331$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$331$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$332$	0.196110	0.196110
$333$	−4.06707	−4.06707
$334$	−0.706353	−0.706353
$335$	−0.131695	−0.131695
$336$	−1.48694	−1.48694
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.380782	0.380782
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.305207	0.305207
$344$	0.767901	0.767901
$345$	0	0
$346$	0.257422	0.257422
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	3.17486	3.17486
$349$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$349$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$350$	0.503715	0.503715
$351$	0	0
$352$	1.66598	1.66598
$353$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$353$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$354$	−1.44788	−1.44788
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.142628	−0.142628
$361$	1.00000	1.00000
$362$	0.548724	0.548724
$363$	4.21869	4.21869
$364$	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
$370$	−0.0450087	−0.0450087
$371$	−1.27136	−1.27136
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.291284	−0.291284
$376$	−1.39615	−1.39615
$377$	0	0
$378$	−1.58048	−1.58048
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.94090	−1.94090
$385$	−0.182702	−0.182702
$386$	0.363814	0.363814
$387$	−2.86553	−2.86553
$388$	0	0
$389$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$389$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$390$	0	0
$391$	0	0
$392$	−0.544339	−0.544339
$393$	−1.56002	−1.56002
$394$	0	0
$395$	0	0
$396$	−4.03933	−4.03933
$397$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$397$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$398$	0.144995	0.144995
$399$	0	0
$400$	−0.582599	−0.582599
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−1.24822	−1.24822
$403$	0	0
$404$	0	0
$405$	0.253709	0.253709
$406$	0.986720	0.986720
$407$	−2.76553	−2.76553
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.65349	2.65349
$414$	0	0
$415$	−0.0175708	−0.0175708
$416$	0	0
$417$	3.11924	3.11924
$418$	0	0
$419$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$419$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$420$	0.166187	0.166187
$421$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$421$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$422$	0	0
$423$	5.20994	5.20994
$424$	−0.674878	−0.674878
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.0317117	−0.0317117
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.82800	1.82800
$433$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$433$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$434$	0	0
$435$	−0.284457	−0.284457
$436$	−0.578012	−0.578012
$437$	0	0
$438$	0	0
$439$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$439$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$440$	−0.0969839	−0.0969839
$441$	2.03128	2.03128
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	2.51554	2.51554
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0.308843	0.308843
$449$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$449$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$450$	−0.997798	−0.997798
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$457$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$461$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$462$	−1.73166	−1.73166
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−1.14125	−1.14125
$465$	0	0
$466$	−0.311532	−0.311532
$467$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$467$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$468$	0	0
$469$	2.28758	2.28758
$470$	0.0576563	0.0576563
$471$	3.41760	3.41760
$472$	1.40856	1.40856
$473$	−1.94851	−1.94851
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.51840	2.51840
$478$	0.461680	0.461680
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0.135774	0.135774
$481$	0	0
$482$	−0.413962	−0.413962
$483$	0	0
$484$	−1.89166	−1.89166
$485$	0	0
$486$	1.21692	1.21692
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0.0224794	0.0224794
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.361910	0.361910
$496$	0	0
$497$	0	0
$498$	−0.166537	−0.166537
$499$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$499$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$500$	0.130612	0.130612
$501$	−3.53711	−3.53711
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	2.47747	2.47747
$505$	0	0
$506$	0	0
$507$	1.90679	1.90679
$508$	0	0
$509$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$509$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$510$	0	0
$511$	0	0
$512$	0.958676	0.958676
$513$	0	0
$514$	0.257422	0.257422
$515$	0	0
$516$	1.77238	1.77238
$517$	3.54265	3.54265
$518$	0.781809	0.781809
$519$	1.28906	1.28906
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.95457	−1.95457
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0.699511	0.699511
$525$	2.52239	2.52239
$526$	0	0
$527$	0	0
$528$	2.00285	2.00285
$529$	1.00000	1.00000
$530$	0.0278702	0.0278702
$531$	−5.25625	−5.25625
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1.21432	1.21432
$537$	0	0
$538$	0	0
$539$	1.38123	1.38123
$540$	−0.204304	−0.204304
$541$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$541$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$542$	0	0
$543$	2.74777	2.74777
$544$	0	0
$545$	0.0517879	0.0517879
$546$	0	0
$547$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$547$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$548$	0	0
$549$	0	0
$550$	−0.678483	−0.678483
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.225384	−0.225384
$556$	−1.39867	−1.39867
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−0.0597379	−0.0597379
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−3.22242	−3.22242
$565$	0	0
$566$	−0.741532	−0.741532
$567$	−4.40697	−4.40697
$568$	0	0
$569$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$569$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$570$	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
$576$	−0.611780	−0.611780
$577$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$577$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$578$	0.380782	0.380782
$579$	1.82183	1.82183
$580$	0.127550	0.127550
$581$	0.305207	0.305207
$582$	0	0
$583$	1.71246	1.71246
$584$	0	0
$585$	0	0
$586$	−0.506688	−0.506688
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−1.25638	−1.25638
$589$	0	0
$590$	−0.0581688	−0.0581688
$591$	0	0
$592$	−0.904244	−0.904244
$593$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$593$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$594$	2.12885	2.12885
$595$	0	0
$596$	0	0
$597$	0.726073	0.726073
$598$	0	0
$599$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$599$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$600$	1.33896	1.33896
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0.550839	0.550839
$603$	−4.53141	−4.53141
$604$	0	0
$605$	0.169486	0.169486
$606$	0	0
$607$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$607$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$608$	0	0
$609$	4.94107	4.94107
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$613$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$614$	0.257422	0.257422
$615$	0	0
$616$	1.68463	1.68463
$617$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$617$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$618$	0	0
$619$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$619$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.982429	0.982429
$626$	0	0
$627$	0	0
$628$	−1.53245	−1.53245
$629$	0	0
$630$	−0.102311	−0.102311
$631$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$631$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−1.55767	−1.55767
$637$	0	0
$638$	−1.32907	−1.32907
$639$	0	0
$640$	−0.0779756	−0.0779756
$641$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$641$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$642$	0	0
$643$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$643$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$644$	0	0
$645$	−0.158799	−0.158799
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.33936	−2.33936
$649$	−3.57414	−3.57414
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.490850	0.490850
$655$	−0.0626738	−0.0626738
$656$	0	0
$657$	0	0
$658$	−1.00150	−1.00150
$659$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$659$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$660$	−0.223846	−0.223846
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.201800	−0.201800
$663$	0	0
$664$	0.162014	0.162014
$665$	0	0
$666$	−1.54867	−1.54867
$667$	0	0
$668$	1.58604	1.58604
$669$	0	0
$670$	−0.0501473	−0.0501473
$671$	0	0
$672$	−2.35841	−2.35841
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.10094	−3.10094
$676$	−0.855005	−0.855005
$677$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$677$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$683$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$684$	0	0
$685$	0	0
$686$	0.116218	0.116218
$687$	0	0
$688$	−0.637103	−0.637103
$689$	0	0
$690$	0	0
$691$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$691$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$692$	−0.578012	−0.578012
$693$	−6.28644	−6.28644
$694$	0	0
$695$	0.125316	0.125316
$696$	2.62288	2.62288
$697$	0	0
$698$	0.363814	0.363814
$699$	−1.56002	−1.56002
$700$	−1.13104	−1.13104
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−0.415998	−0.415998
$705$	0.288718	0.288718
$706$	−0.654618	−0.654618
$707$	0	0
$708$	3.25107	3.25107
$709$	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
$717$	2.31189	2.31189
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0.118334	0.118334
$721$	0	0
$722$	0.380782	0.380782
$723$	−2.07294	−2.07294
$724$	−1.23210	−1.23210
$725$	1.93596	1.93596
$726$	1.60640	1.60640
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.78193	2.78193
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$733$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$734$	0	0
$735$	0.112567	0.112567
$736$	0	0
$737$	−3.08127	−3.08127
$738$	0	0
$739$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$739$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$740$	0.101062	0.101062
$741$	0	0
$742$	−0.484110	−0.484110
$743$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$743$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.604579	−0.604579
$748$	0	0
$749$	0	0
$750$	−0.110916	−0.110916
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.15834	1.15834
$753$	0	0
$754$	0	0
$755$	0	0
$756$	3.54881	3.54881
$757$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$757$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$761$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$762$	0	0
$763$	−0.899565	−0.899565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−0.296494	−0.296494
$769$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$769$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$770$	−0.0695695	−0.0695695
$771$	1.28906	1.28906
$772$	−0.816906	−0.816906
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.09114	−1.09114
$775$	0	0
$776$	0	0
$777$	3.91496	3.91496
$778$	0.726073	0.726073
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.07439	−6.07439
$784$	0.451621	0.451621
$785$	0.137302	0.137302
$786$	−0.594027	−0.594027
$787$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$787$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.33705	−3.33705
$793$	0	0
$794$	0.144995	0.144995
$795$	0.139562	0.139562
$796$	−0.325571	−0.325571
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−0.924051	−0.924051
$801$	0	0
$802$	0	0
$803$	0	0
$804$	2.80274	2.80274
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$809$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$810$	0.0966079	0.0966079
$811$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$811$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$812$	−2.21557	−2.21557
$813$	0	0
$814$	−1.05306	−1.05306
$815$	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
$823$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$823$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$824$	0	0
$825$	−3.39755	−3.39755
$826$	1.01040	1.01040
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$829$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$830$	−0.00669063	−0.00669063
$831$	0	0
$832$	0	0
$833$	0	0
$834$	1.18775	1.18775
$835$	−0.142104	−0.142104
$836$	0	0
$837$	0	0
$838$	−0.201800	−0.201800
$839$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$839$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$840$	0.137293	0.137293
$841$	2.79233	2.79233
$842$	−0.413962	−0.413962
$843$	0	0
$844$	0	0
$845$	0.0766055	0.0766055
$846$	1.98385	1.98385
$847$	−2.94400	−2.94400
$848$	0.559924	0.559924
$849$	−3.71327	−3.71327
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$853$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$859$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$860$	0.0712052	0.0712052
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	2.89936	2.89936
$865$	0.0517879	0.0517879
$866$	−0.311532	−0.311532
$867$	1.90679	1.90679
$868$	0	0
$869$	0	0
$870$	−0.108316	−0.108316
$871$	0	0
$872$	−0.477518	−0.477518
$873$	0	0
$874$	0	0
$875$	0.203272	0.203272
$876$	0	0
$877$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$877$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$878$	0.682488	0.682488
$879$	−2.53728	−2.53728
$880$	0.0804644	0.0804644
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.773475	0.773475
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.291284	−0.291284
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	2.07819	2.07819
$889$	0	0
$890$	0	0
$891$	5.93601	5.93601
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.35445	1.35445
$897$	0	0
$898$	0.726073	0.726073
$899$	0	0
$900$	2.24045	2.24045
$901$	0	0
$902$	0	0
$903$	2.75836	2.75836
$904$	0	0
$905$	0.110392	0.110392
$906$	0	0
$907$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$907$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−0.411101	−0.411101
$914$	−0.741532	−0.741532
$915$	0	0
$916$	0	0
$917$	1.08866	1.08866
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.28906	1.28906
$922$	0.548724	0.548724
$923$	0	0
$924$	3.88825	3.88825
$925$	1.53392	1.53392
$926$	0	0
$927$	0	0
$928$	−1.81011	−1.81011
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0.699511	0.699511
$933$	0	0
$934$	−0.201800	−0.201800
$935$	0	0
$936$	0	0
$937$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$937$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$938$	0.871068	0.871068
$939$	0	0
$940$	−0.129461	−0.129461
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	1.30136	1.30136
$943$	0	0
$944$	−1.16864	−1.16864
$945$	−0.317961	−0.317961
$946$	−0.741956	−0.741956
$947$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$947$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$948$	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			$\pi$
$954$	0.958963	0.958963
$955$	0	0
$956$	−1.03665	−1.03665
$957$	−6.65541	−6.65541
$958$	0	0
$959$	0	0
$960$	−0.0339029	−0.0339029
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.929506	0.929506
$965$	0.0731919	0.0731919
$966$	0	0
$967$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$967$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$968$	−1.56277	−1.56277
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.73247	−2.73247
$973$	−2.17676	−2.17676
$974$	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
$980$	−0.0504750	−0.0504750
$981$	1.78193	1.78193
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.01508	−5.01508
$988$	0	0
$989$	0	0
$990$	0.137809	0.137809
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−1.01053	−1.01053
$994$	0	0
$995$	0.0291700	0.0291700
$996$	0.373941	0.373941
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−0.759330	−0.759330
$999$	−4.81293	−4.81293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.12	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.12	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.12	✓	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)

\(f(q)\)	\(=\)	\(q+0.380782 q^{2} +1.90679 q^{3} -0.855005 q^{4} +0.0766055 q^{5} +0.726073 q^{6} -1.33065 q^{7} -0.706353 q^{8} +2.63586 q^{9} +O(q^{10})\)	\(q+0.380782 q^{2} +1.90679 q^{3} -0.855005 q^{4} +0.0766055 q^{5} +0.726073 q^{6} -1.33065 q^{7} -0.706353 q^{8} +2.63586 q^{9} +0.0291700 q^{10} +1.79233 q^{11} -1.63032 q^{12} -0.506688 q^{14} +0.146071 q^{15} +0.586038 q^{16} +1.00369 q^{18} -0.0654981 q^{20} -2.53728 q^{21} +0.682488 q^{22} -1.34687 q^{24} -0.994132 q^{25} +3.11924 q^{27} +1.13771 q^{28} -1.94739 q^{29} +0.0556211 q^{30} +0.929506 q^{32} +3.41760 q^{33} -0.101935 q^{35} -2.25367 q^{36} -1.54298 q^{37} -0.0541105 q^{40} -0.966150 q^{42} -1.08714 q^{43} -1.53245 q^{44} +0.201921 q^{45} +1.97656 q^{47} +1.11745 q^{48} +0.770633 q^{49} -0.378548 q^{50} +0.955440 q^{53} +1.18775 q^{54} +0.137302 q^{55} +0.939909 q^{56} -0.741532 q^{58} -1.99413 q^{59} -0.124891 q^{60} -3.50741 q^{63} -0.232099 q^{64} +1.30136 q^{66} -1.71914 q^{67} -0.0388151 q^{70} -1.86185 q^{72} -0.587539 q^{74} -1.89560 q^{75} -2.38497 q^{77} +0.0448937 q^{80} +3.31189 q^{81} -0.229367 q^{83} +2.16938 q^{84} -0.413962 q^{86} -3.71327 q^{87} -1.26602 q^{88} +0.0768880 q^{90} +0.752639 q^{94} +1.77238 q^{96} +0.293443 q^{98} +4.72433 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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