LMFDB - Embedded newform 1151.1.b.a.1150.11

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			1150.11
Root			$-1.97656$ 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)

$f(q)$	$=$	$q+0.0766055 q^{2} -1.54298 q^{3} -0.994132 q^{4} +1.90679 q^{5} -0.118201 q^{6} +1.79233 q^{7} -0.152761 q^{8} +1.38078 q^{9} +O(q^{10})$	\(q+0.0766055 q^{2} -1.54298 q^{3} -0.994132 q^{4} +1.90679 q^{5} -0.118201 q^{6} +1.79233 q^{7} -0.152761 q^{8} +1.38078 q^{9} +0.146071 q^{10} -1.71914 q^{11} +1.53392 q^{12} +0.137302 q^{14} -2.94214 q^{15} +0.982429 q^{16} +0.105775 q^{18} -1.89560 q^{20} -2.76553 q^{21} -0.131695 q^{22} +0.235708 q^{24} +2.63586 q^{25} -0.587539 q^{27} -1.78181 q^{28} -0.529963 q^{29} -0.225384 q^{30} +0.228021 q^{32} +2.65259 q^{33} +3.41760 q^{35} -1.37268 q^{36} +1.44104 q^{37} -0.291284 q^{40} -0.211855 q^{42} -0.229367 q^{43} +1.70905 q^{44} +2.63287 q^{45} +0.676034 q^{47} -1.51587 q^{48} +2.21245 q^{49} +0.201921 q^{50} -1.33065 q^{53} -0.0450087 q^{54} -3.27804 q^{55} -0.273799 q^{56} -0.0405981 q^{58} +1.63586 q^{59} +2.92487 q^{60} +2.47482 q^{63} -0.964962 q^{64} +0.203203 q^{66} -0.818137 q^{67} +0.261807 q^{70} -0.210930 q^{72} +0.110392 q^{74} -4.06707 q^{75} -3.08127 q^{77} +1.87329 q^{80} -0.474223 q^{81} +1.21245 q^{83} +2.74930 q^{84} -0.0175708 q^{86} +0.817721 q^{87} +0.262618 q^{88} +0.201692 q^{90} +0.0517879 q^{94} -0.351831 q^{96} +0.169486 q^{98} -2.37376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$2$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$3$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$3$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$4$	−0.994132	−0.994132
$5$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$5$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$6$	−0.118201	−0.118201
$7$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$7$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$8$	−0.152761	−0.152761
$9$	1.38078	1.38078
$10$	0.146071	0.146071
$11$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$11$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$12$	1.53392	1.53392
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.137302	0.137302
$15$	−2.94214	−2.94214
$16$	0.982429	0.982429
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.105775	0.105775
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.89560	−1.89560
$21$	−2.76553	−2.76553
$22$	−0.131695	−0.131695
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.235708	0.235708
$25$	2.63586	2.63586
$26$	0	0
$27$	−0.587539	−0.587539
$28$	−1.78181	−1.78181
$29$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$29$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$30$	−0.225384	−0.225384
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.228021	0.228021
$33$	2.65259	2.65259
$34$	0	0
$35$	3.41760	3.41760
$36$	−1.37268	−1.37268
$37$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$37$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$38$	0	0
$39$	0	0
$40$	−0.291284	−0.291284
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.211855	−0.211855
$43$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$43$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$44$	1.70905	1.70905
$45$	2.63287	2.63287
$46$	0	0
$47$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$47$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$48$	−1.51587	−1.51587
$49$	2.21245	2.21245
$50$	0.201921	0.201921
$51$	0	0
$52$	0	0
$53$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$53$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$54$	−0.0450087	−0.0450087
$55$	−3.27804	−3.27804
$56$	−0.273799	−0.273799
$57$	0	0
$58$	−0.0405981	−0.0405981
$59$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$59$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$60$	2.92487	2.92487
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	2.47482	2.47482
$64$	−0.964962	−0.964962
$65$	0	0
$66$	0.203203	0.203203
$67$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$67$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$68$	0	0
$69$	0	0
$70$	0.261807	0.261807
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.210930	−0.210930
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.110392	0.110392
$75$	−4.06707	−4.06707
$76$	0	0
$77$	−3.08127	−3.08127
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.87329	1.87329
$81$	−0.474223	−0.474223
$82$	0	0
$83$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$83$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$84$	2.74930	2.74930
$85$	0	0
$86$	−0.0175708	−0.0175708
$87$	0.817721	0.817721
$88$	0.262618	0.262618
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.201692	0.201692
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0.0517879	0.0517879
$95$	0	0
$96$	−0.351831	−0.351831
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.169486	0.169486
$99$	−2.37376	−2.37376
$100$	−2.62039	−2.62039
$101$	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
$105$	−5.27329	−5.27329
$106$	−0.101935	−0.101935
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.584091	0.584091
$109$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$109$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$110$	−0.251116	−0.251116
$111$	−2.22350	−2.22350
$112$	1.76084	1.76084
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0.526853	0.526853
$117$	0	0
$118$	0.125316	0.125316
$119$	0	0
$120$	0.449445	0.449445
$121$	1.95544	1.95544
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.11924	3.11924
$126$	0.189585	0.189585
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.301942	−0.301942
$129$	0.353908	0.353908
$130$	0	0
$131$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$131$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$132$	−2.63703	−2.63703
$133$	0	0
$134$	−0.0626738	−0.0626738
$135$	−1.12031	−1.12031
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$139$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$140$	−3.39755	−3.39755
$141$	−1.04311	−1.04311
$142$	0	0
$143$	0	0
$144$	1.35652	1.35652
$145$	−1.01053	−1.01053
$146$	0	0
$147$	−3.41376	−3.41376
$148$	−1.43259	−1.43259
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.311560	−0.311560
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−0.236042	−0.236042
$155$	0	0
$156$	0	0
$157$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$157$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$158$	0	0
$159$	2.05317	2.05317
$160$	0.434788	0.434788
$161$	0	0
$162$	−0.0363281	−0.0363281
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	5.05795	5.05795
$166$	0.0928804	0.0928804
$167$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$167$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$168$	0.422466	0.422466
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.228021	0.228021
$173$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$173$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$174$	0.0626419	0.0626419
$175$	4.72433	4.72433
$176$	−1.68893	−1.68893
$177$	−2.52409	−2.52409
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−2.61741	−2.61741
$181$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$181$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.74777	2.74777
$186$	0	0
$187$	0	0
$188$	−0.672067	−0.672067
$189$	−1.05306	−1.05306
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.48891	1.48891
$193$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$193$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$194$	0	0
$195$	0	0
$196$	−2.19947	−2.19947
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−0.181843	−0.181843
$199$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$199$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$200$	−0.402657	−0.402657
$201$	1.26237	1.26237
$202$	0	0
$203$	−0.949869	−0.949869
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−0.403963	−0.403963
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.32284	1.32284
$213$	0	0
$214$	0	0
$215$	−0.437355	−0.437355
$216$	0.0897532	0.0897532
$217$	0	0
$218$	−0.142104	−0.142104
$219$	0	0
$220$	3.25881	3.25881
$221$	0	0
$222$	−0.170332	−0.170332
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.408689	0.408689
$225$	3.63955	3.63955
$226$	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
$231$	4.75433	4.75433
$232$	0.0809579	0.0809579
$233$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$233$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$234$	0	0
$235$	1.28906	1.28906
$236$	−1.62626	−1.62626
$237$	0	0
$238$	0	0
$239$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$239$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$240$	−2.89044	−2.89044
$241$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$241$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$242$	0.149797	0.149797
$243$	1.31925	1.31925
$244$	0	0
$245$	4.21869	4.21869
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.87079	−1.87079
$250$	0.238951	0.238951
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−2.46030	−2.46030
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0.941831	0.941831
$257$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$257$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$258$	0.0271113	0.0271113
$259$	2.58283	2.58283
$260$	0	0
$261$	−0.731763	−0.731763
$262$	−0.149181	−0.149181
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.405214	−0.405214
$265$	−2.53728	−2.53728
$266$	0	0
$267$	0	0
$268$	0.813336	0.813336
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.0858222	−0.0858222
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.53141	−4.53141
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.0291700	0.0291700
$279$	0	0
$280$	−0.522078	−0.522078
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−0.0799076	−0.0799076
$283$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$283$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0.314847	0.314847
$289$	1.00000	1.00000
$290$	−0.0774121	−0.0774121
$291$	0	0
$292$	0	0
$293$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$293$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$294$	−0.261513	−0.261513
$295$	3.11924	3.11924
$296$	−0.220136	−0.220136
$297$	1.01006	1.01006
$298$	0	0
$299$	0	0
$300$	4.04321	4.04321
$301$	−0.411101	−0.411101
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$307$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$308$	3.06318	3.06318
$309$	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			$\pi$
$314$	−0.131695	−0.131695
$315$	4.71897	4.71897
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.157284	0.157284
$319$	0.911080	0.911080
$320$	−1.83998	−1.83998
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.471440	0.471440
$325$	0	0
$326$	0	0
$327$	2.86223	2.86223
$328$	0	0
$329$	1.21168	1.21168
$330$	0.387466	0.387466
$331$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$331$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$332$	−1.20534	−1.20534
$333$	1.98977	1.98977
$334$	−0.152761	−0.152761
$335$	−1.56002	−1.56002
$336$	−2.71694	−2.71694
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.0766055	0.0766055
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.17311	2.17311
$344$	0.0350384	0.0350384
$345$	0	0
$346$	−0.142104	−0.142104
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−0.812923	−0.812923
$349$	−1.33065	−1.33065	−0.665326	−	0.746553i	$-0.731707\pi$
$349$	−1.33065	−1.33065	−0.665326	+	0.746553i	$0.731707\pi$
$350$	0.361910	0.361910
$351$	0	0
$352$	−0.392000	−0.392000
$353$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$353$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$354$	−0.193359	−0.193359
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−0.402200	−0.402200
$361$	1.00000	1.00000
$362$	0.151415	0.151415
$363$	−3.01720	−3.01720
$364$	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
$370$	0.210494	0.210494
$371$	−2.38497	−2.38497
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−4.81293	−4.81293
$376$	−0.103272	−0.103272
$377$	0	0
$378$	−0.0806705	−0.0806705
$379$	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			$\pi$
$384$	0.465890	0.465890
$385$	−5.87534	−5.87534
$386$	−0.101935	−0.101935
$387$	−0.316706	−0.316706
$388$	0	0
$389$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$389$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$390$	0	0
$391$	0	0
$392$	−0.337977	−0.337977
$393$	3.00478	3.00478
$394$	0	0
$395$	0	0
$396$	2.35983	2.35983
$397$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$397$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$398$	0.00586840	0.00586840
$399$	0	0
$400$	2.58954	2.58954
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0.0967043	0.0967043
$403$	0	0
$404$	0	0
$405$	−0.904244	−0.904244
$406$	−0.0727652	−0.0727652
$407$	−2.47735	−2.47735
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.93200	2.93200
$414$	0	0
$415$	2.31189	2.31189
$416$	0	0
$417$	−0.587539	−0.587539
$418$	0	0
$419$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$419$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$420$	5.24234	5.24234
$421$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$421$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$422$	0	0
$423$	0.933455	0.933455
$424$	0.203272	0.203272
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.0335038	−0.0335038
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.577215	−0.577215
$433$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$433$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$434$	0	0
$435$	1.55923	1.55923
$436$	1.84412	1.84412
$437$	0	0
$438$	0	0
$439$	−1.71914	−1.71914	−0.859570	−	0.511019i	$-0.829268\pi$
$439$	−1.71914	−1.71914	−0.859570	+	0.511019i	$0.829268\pi$
$440$	0.500758	0.500758
$441$	3.05491	3.05491
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	2.21045	2.21045
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−1.72953	−1.72953
$449$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$449$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$450$	0.278809	0.278809
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$457$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$461$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$462$	0.364208	0.364208
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−0.520651	−0.520651
$465$	0	0
$466$	−0.149181	−0.149181
$467$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$467$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$468$	0	0
$469$	−1.46637	−1.46637
$470$	0.0987488	0.0987488
$471$	2.65259	2.65259
$472$	−0.249896	−0.249896
$473$	0.394314	0.394314
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.83734	−1.83734
$478$	0.0731919	0.0731919
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.670869	−0.670869
$481$	0	0
$482$	−0.0175708	−0.0175708
$483$	0	0
$484$	−1.94396	−1.94396
$485$	0	0
$486$	0.101062	0.101062
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0.323174	0.323174
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.52626	−4.52626
$496$	0	0
$497$	0	0
$498$	−0.143312	−0.143312
$499$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$499$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$500$	−3.10094	−3.10094
$501$	3.07690	3.07690
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.378057	−0.378057
$505$	0	0
$506$	0	0
$507$	−1.54298	−1.54298
$508$	0	0
$509$	−0.229367	−0.229367	−0.114683	−	0.993402i	$-0.536585\pi$
$509$	−0.229367	−0.229367	−0.114683	+	0.993402i	$0.536585\pi$
$510$	0	0
$511$	0	0
$512$	0.374092	0.374092
$513$	0	0
$514$	−0.142104	−0.142104
$515$	0	0
$516$	−0.351831	−0.351831
$517$	−1.16220	−1.16220
$518$	0.197859	0.197859
$519$	2.86223	2.86223
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−0.0560571	−0.0560571
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.93596	1.93596
$525$	−7.28954	−7.28954
$526$	0	0
$527$	0	0
$528$	2.60599	2.60599
$529$	1.00000	1.00000
$530$	−0.194369	−0.194369
$531$	2.25876	2.25876
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0.124980	0.124980
$537$	0	0
$538$	0	0
$539$	−3.80351	−3.80351
$540$	1.11374	1.11374
$541$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$541$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$542$	0	0
$543$	−3.04979	−3.04979
$544$	0	0
$545$	−3.53711	−3.53711
$546$	0	0
$547$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$547$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$548$	0	0
$549$	0	0
$550$	−0.347131	−0.347131
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.23975	−4.23975
$556$	−0.378548	−0.378548
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	3.35755	3.35755
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.03698	1.03698
$565$	0	0
$566$	−0.0405981	−0.0405981
$567$	−0.849964	−0.849964
$568$	0	0
$569$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$569$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$570$	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
$576$	−1.33240	−1.33240
$577$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$577$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$578$	0.0766055	0.0766055
$579$	2.05317	2.05317
$580$	1.00460	1.00460
$581$	2.17311	2.17311
$582$	0	0
$583$	2.28758	2.28758
$584$	0	0
$585$	0	0
$586$	0.137302	0.137302
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	3.39373	3.39373
$589$	0	0
$590$	0.238951	0.238951
$591$	0	0
$592$	1.41572	1.41572
$593$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$593$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$594$	0.0773762	0.0773762
$595$	0	0
$596$	0	0
$597$	−0.118201	−0.118201
$598$	0	0
$599$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$599$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$600$	0.621292	0.621292
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.0314926	−0.0314926
$603$	−1.12967	−1.12967
$604$	0	0
$605$	3.72862	3.72862
$606$	0	0
$607$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$607$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$608$	0	0
$609$	1.46563	1.46563
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$613$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$614$	−0.142104	−0.142104
$615$	0	0
$616$	0.470699	0.470699
$617$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$617$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$618$	0	0
$619$	1.44104	1.44104	0.720522	−	0.693433i	$-0.243902\pi$
$619$	1.44104	1.44104	0.720522	+	0.693433i	$0.243902\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.31189	3.31189
$626$	0	0
$627$	0	0
$628$	1.70905	1.70905
$629$	0	0
$630$	0.361499	0.361499
$631$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$631$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−2.04112	−2.04112
$637$	0	0
$638$	0.0697937	0.0697937
$639$	0	0
$640$	−0.575741	−0.575741
$641$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$641$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$642$	0	0
$643$	−1.08714	−1.08714	−0.543568	−	0.839365i	$-0.682927\pi$
$643$	−1.08714	−1.08714	−0.543568	+	0.839365i	$0.682927\pi$
$644$	0	0
$645$	0.674829	0.674829
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.0724429	0.0724429
$649$	−2.81227	−2.81227
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.219263	0.219263
$655$	−3.71327	−3.71327
$656$	0	0
$657$	0	0
$658$	0.0928210	0.0928210
$659$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$659$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$660$	−5.02827	−5.02827
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.0832805	−0.0832805
$663$	0	0
$664$	−0.185216	−0.185216
$665$	0	0
$666$	0.152427	0.152427
$667$	0	0
$668$	1.98243	1.98243
$669$	0	0
$670$	−0.119506	−0.119506
$671$	0	0
$672$	−0.630598	−0.630598
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−1.54867	−1.54867
$676$	−0.994132	−0.994132
$677$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$677$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$683$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$684$	0	0
$685$	0	0
$686$	0.166472	0.166472
$687$	0	0
$688$	−0.225337	−0.225337
$689$	0	0
$690$	0	0
$691$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$691$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$692$	1.84412	1.84412
$693$	−4.25456	−4.25456
$694$	0	0
$695$	0.726073	0.726073
$696$	−0.124916	−0.124916
$697$	0	0
$698$	−0.101935	−0.101935
$699$	3.00478	3.00478
$700$	−4.69661	−4.69661
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.65890	1.65890
$705$	−1.98899	−1.98899
$706$	−0.0626738	−0.0626738
$707$	0	0
$708$	2.50928	2.50928
$709$	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
$717$	−1.47422	−1.47422
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	2.58660	2.58660
$721$	0	0
$722$	0.0766055	0.0766055
$723$	0.353908	0.353908
$724$	−1.96496	−1.96496
$725$	−1.39691	−1.39691
$726$	−0.231134	−0.231134
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.56136	−1.56136
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$733$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$734$	0	0
$735$	−6.50934	−6.50934
$736$	0	0
$737$	1.40649	1.40649
$738$	0	0
$739$	0.0766055	0.0766055	0.0383027	−	0.999266i	$-0.487805\pi$
$739$	0.0766055	0.0766055	0.0383027	+	0.999266i	$0.487805\pi$
$740$	−2.73165	−2.73165
$741$	0	0
$742$	−0.182702	−0.182702
$743$	1.63586	1.63586	0.817929	−	0.575319i	$-0.195122\pi$
$743$	1.63586	1.63586	0.817929	+	0.575319i	$0.195122\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.67413	1.67413
$748$	0	0
$749$	0	0
$750$	−0.368696	−0.368696
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0.664155	0.664155
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.04688	1.04688
$757$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$757$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$761$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$762$	0	0
$763$	−3.32478	−3.32478
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.45323	−1.45323
$769$	−1.54298	−1.54298	−0.771489	−	0.636242i	$-0.780488\pi$
$769$	−1.54298	−1.54298	−0.771489	+	0.636242i	$0.780488\pi$
$770$	−0.450083	−0.450083
$771$	2.86223	2.86223
$772$	1.32284	1.32284
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−0.0242614	−0.0242614
$775$	0	0
$776$	0	0
$777$	−3.98525	−3.98525
$778$	−0.118201	−0.118201
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.311374	0.311374
$784$	2.17358	2.17358
$785$	−3.27804	−3.27804
$786$	0.230183	0.230183
$787$	−1.94739	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$787$	−1.94739	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0.362618	0.362618
$793$	0	0
$794$	0.00586840	0.00586840
$795$	3.91496	3.91496
$796$	−0.0761559	−0.0761559
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.601031	0.601031
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1.25496	−1.25496
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$809$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$810$	−0.0692701	−0.0692701
$811$	0.955440	0.955440	0.477720	−	0.878512i	$-0.341463\pi$
$811$	0.955440	0.955440	0.477720	+	0.878512i	$0.341463\pi$
$812$	0.944295	0.944295
$813$	0	0
$814$	−0.189779	−0.189779
$815$	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
$823$	−1.99413	−1.99413	−0.997066	−	0.0765493i	$-0.975610\pi$
$823$	−1.99413	−1.99413	−0.997066	+	0.0765493i	$0.975610\pi$
$824$	0	0
$825$	6.99186	6.99186
$826$	0.224607	0.224607
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−0.529963	−0.529963	−0.264982	−	0.964253i	$-0.585366\pi$
$829$	−0.529963	−0.529963	−0.264982	+	0.964253i	$0.585366\pi$
$830$	0.177104	0.177104
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−0.0450087	−0.0450087
$835$	−3.80240	−3.80240
$836$	0	0
$837$	0	0
$838$	−0.0832805	−0.0832805
$839$	0.380782	0.380782	0.190391	−	0.981708i	$-0.439024\pi$
$839$	0.380782	0.380782	0.190391	+	0.981708i	$0.439024\pi$
$840$	0.805555	0.805555
$841$	−0.719139	−0.719139
$842$	−0.0175708	−0.0175708
$843$	0	0
$844$	0	0
$845$	1.90679	1.90679
$846$	0.0715078	0.0715078
$847$	3.50480	3.50480
$848$	−1.30727	−1.30727
$849$	0.817721	0.817721
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.21245	1.21245	0.606225	−	0.795293i	$-0.292683\pi$
$853$	1.21245	1.21245	0.606225	+	0.795293i	$0.292683\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.85500	−1.85500	−0.927502	−	0.373817i	$-0.878049\pi$
$859$	−1.85500	−1.85500	−0.927502	+	0.373817i	$0.878049\pi$
$860$	0.434788	0.434788
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−0.133971	−0.133971
$865$	−3.53711	−3.53711
$866$	−0.149181	−0.149181
$867$	−1.54298	−1.54298
$868$	0	0
$869$	0	0
$870$	0.119445	0.119445
$871$	0	0
$872$	0.283373	0.283373
$873$	0	0
$874$	0	0
$875$	5.59072	5.59072
$876$	0	0
$877$	−0.818137	−0.818137	−0.409069	−	0.912504i	$-0.634146\pi$
$877$	−0.818137	−0.818137	−0.409069	+	0.912504i	$0.634146\pi$
$878$	−0.131695	−0.131695
$879$	−2.76553	−2.76553
$880$	−3.22044	−3.22044
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.234023	0.234023
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−4.81293	−4.81293
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0.339665	0.339665
$889$	0	0
$890$	0	0
$891$	0.815255	0.815255
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−0.541180	−0.541180
$897$	0	0
$898$	−0.118201	−0.118201
$899$	0	0
$900$	−3.61819	−3.61819
$901$	0	0
$902$	0	0
$903$	0.634320	0.634320
$904$	0	0
$905$	3.76889	3.76889
$906$	0	0
$907$	0.676034	0.676034	0.338017	−	0.941140i	$-0.390244\pi$
$907$	0.676034	0.676034	0.338017	+	0.941140i	$0.390244\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.08437	−2.08437
$914$	−0.0405981	−0.0405981
$915$	0	0
$916$	0	0
$917$	−3.49037	−3.49037
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	2.86223	2.86223
$922$	0.151415	0.151415
$923$	0	0
$924$	−4.72643	−4.72643
$925$	3.79839	3.79839
$926$	0	0
$927$	0	0
$928$	−0.120843	−0.120843
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	1.93596	1.93596
$933$	0	0
$934$	−0.0832805	−0.0832805
$935$	0	0
$936$	0	0
$937$	1.90679	1.90679	0.953396	−	0.301721i	$-0.0975610\pi$
$937$	1.90679	1.90679	0.953396	+	0.301721i	$0.0975610\pi$
$938$	−0.112332	−0.112332
$939$	0	0
$940$	−1.28149	−1.28149
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0.203203	0.203203
$943$	0	0
$944$	1.60712	1.60712
$945$	−2.00797	−2.00797
$946$	0.0302066	0.0302066
$947$	1.79233	1.79233	0.896166	−	0.443720i	$-0.146341\pi$
$947$	1.79233	1.79233	0.896166	+	0.443720i	$0.146341\pi$
$948$	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			$\pi$
$954$	−0.140750	−0.140750
$955$	0	0
$956$	−0.949833	−0.949833
$957$	−1.40578	−1.40578
$958$	0	0
$959$	0	0
$960$	2.83905	2.83905
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0.228021	0.228021
$965$	−2.53728	−2.53728
$966$	0	0
$967$	1.97656	1.97656	0.988280	−	0.152649i	$-0.0487805\pi$
$967$	1.97656	1.97656	0.988280	+	0.152649i	$0.0487805\pi$
$968$	−0.298716	−0.298716
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.31151	−1.31151
$973$	0.682488	0.682488
$974$	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
$980$	−4.19393	−4.19393
$981$	−2.56136	−2.56136
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.86959	−1.86959
$988$	0	0
$989$	0	0
$990$	−0.346736	−0.346736
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	1.67743	1.67743
$994$	0	0
$995$	0.146071	0.146071
$996$	1.85981	1.85981
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.125316	0.125316
$999$	−0.846669	−0.846669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1151.1.b.a.1150.11	✓	20
1151.1150	odd	2	CM	1151.1.b.a.1150.11	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1151.1.b.a.1150.11	✓	20	1.1	even	1	trivial
1151.1.b.a.1150.11	✓	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)

\(f(q)\)	\(=\)	\(q+0.0766055 q^{2} -1.54298 q^{3} -0.994132 q^{4} +1.90679 q^{5} -0.118201 q^{6} +1.79233 q^{7} -0.152761 q^{8} +1.38078 q^{9} +O(q^{10})\)	\(q+0.0766055 q^{2} -1.54298 q^{3} -0.994132 q^{4} +1.90679 q^{5} -0.118201 q^{6} +1.79233 q^{7} -0.152761 q^{8} +1.38078 q^{9} +0.146071 q^{10} -1.71914 q^{11} +1.53392 q^{12} +0.137302 q^{14} -2.94214 q^{15} +0.982429 q^{16} +0.105775 q^{18} -1.89560 q^{20} -2.76553 q^{21} -0.131695 q^{22} +0.235708 q^{24} +2.63586 q^{25} -0.587539 q^{27} -1.78181 q^{28} -0.529963 q^{29} -0.225384 q^{30} +0.228021 q^{32} +2.65259 q^{33} +3.41760 q^{35} -1.37268 q^{36} +1.44104 q^{37} -0.291284 q^{40} -0.211855 q^{42} -0.229367 q^{43} +1.70905 q^{44} +2.63287 q^{45} +0.676034 q^{47} -1.51587 q^{48} +2.21245 q^{49} +0.201921 q^{50} -1.33065 q^{53} -0.0450087 q^{54} -3.27804 q^{55} -0.273799 q^{56} -0.0405981 q^{58} +1.63586 q^{59} +2.92487 q^{60} +2.47482 q^{63} -0.964962 q^{64} +0.203203 q^{66} -0.818137 q^{67} +0.261807 q^{70} -0.210930 q^{72} +0.110392 q^{74} -4.06707 q^{75} -3.08127 q^{77} +1.87329 q^{80} -0.474223 q^{81} +1.21245 q^{83} +2.74930 q^{84} -0.0175708 q^{86} +0.817721 q^{87} +0.262618 q^{88} +0.201692 q^{90} +0.0517879 q^{94} -0.351831 q^{96} +0.169486 q^{98} -2.37376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - q^{3} + 19 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 19 q^{9}+O(q^{10}) \) 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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