LMFDB - Embedded newform 567.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(1,567)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("567.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	567.a (trivial)

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:	$4.52751779461$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66908 q^{2} +5.12398 q^{4} +1.45490 q^{5} +1.00000 q^{7} -8.33816 q^{8} +O(q^{10})$	\(q-2.66908 q^{2} +5.12398 q^{4} +1.45490 q^{5} +1.00000 q^{7} -8.33816 q^{8} -3.88325 q^{10} +1.54510 q^{11} +5.88325 q^{13} -2.66908 q^{14} +12.0072 q^{16} +6.79306 q^{17} -6.24797 q^{19} +7.45490 q^{20} -4.12398 q^{22} -2.90981 q^{23} -2.88325 q^{25} -15.7029 q^{26} +5.12398 q^{28} -3.88325 q^{29} +2.00000 q^{31} -15.3719 q^{32} -18.1312 q^{34} +1.45490 q^{35} +5.00000 q^{37} +16.6763 q^{38} -12.1312 q^{40} +2.24797 q^{41} -7.13122 q^{43} +7.91705 q^{44} +7.76651 q^{46} +5.33816 q^{47} +1.00000 q^{49} +7.69563 q^{50} +30.1457 q^{52} +9.79306 q^{53} +2.24797 q^{55} -8.33816 q^{56} +10.3647 q^{58} +4.67632 q^{59} -2.36471 q^{61} -5.33816 q^{62} +17.0145 q^{64} +8.55957 q^{65} +3.36471 q^{67} +34.8075 q^{68} -3.88325 q^{70} -1.36471 q^{71} -1.88325 q^{73} -13.3454 q^{74} -32.0145 q^{76} +1.54510 q^{77} +3.36471 q^{79} +17.4694 q^{80} -6.00000 q^{82} -2.24797 q^{83} +9.88325 q^{85} +19.0338 q^{86} -12.8833 q^{88} -0.793062 q^{89} +5.88325 q^{91} -14.9098 q^{92} -14.2480 q^{94} -9.09019 q^{95} +10.2480 q^{97} -2.66908 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 + 3 * q^5 + 3 * q^7 - 9 * q^8	$ 3 q + 6 q^{4} + 3 q^{5} + 3 q^{7} - 9 q^{8} + 3 q^{10} + 6 q^{11} + 3 q^{13} + 12 q^{16} + 3 q^{17} + 21 q^{20} - 3 q^{22} - 6 q^{23} + 6 q^{25} - 27 q^{26} + 6 q^{28} + 3 q^{29} + 6 q^{31} - 18 q^{32} - 21 q^{34} + 3 q^{35} + 15 q^{37} + 18 q^{38} - 3 q^{40} - 12 q^{41} + 12 q^{43} - 3 q^{44} - 6 q^{46} + 3 q^{49} + 27 q^{50} + 9 q^{52} + 12 q^{53} - 12 q^{55} - 9 q^{56} + 27 q^{58} - 18 q^{59} - 3 q^{61} + 3 q^{64} - 21 q^{65} + 6 q^{67} + 39 q^{68} + 3 q^{70} + 9 q^{73} - 48 q^{76} + 6 q^{77} + 6 q^{79} + 3 q^{80} - 18 q^{82} + 12 q^{83} + 15 q^{85} + 45 q^{86} - 24 q^{88} + 15 q^{89} + 3 q^{91} - 42 q^{92} - 24 q^{94} - 30 q^{95} + 12 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^5 + 3 * q^7 - 9 * q^8 + 3 * q^10 + 6 * q^11 + 3 * q^13 + 12 * q^16 + 3 * q^17 + 21 * q^20 - 3 * q^22 - 6 * q^23 + 6 * q^25 - 27 * q^26 + 6 * q^28 + 3 * q^29 + 6 * q^31 - 18 * q^32 - 21 * q^34 + 3 * q^35 + 15 * q^37 + 18 * q^38 - 3 * q^40 - 12 * q^41 + 12 * q^43 - 3 * q^44 - 6 * q^46 + 3 * q^49 + 27 * q^50 + 9 * q^52 + 12 * q^53 - 12 * q^55 - 9 * q^56 + 27 * q^58 - 18 * q^59 - 3 * q^61 + 3 * q^64 - 21 * q^65 + 6 * q^67 + 39 * q^68 + 3 * q^70 + 9 * q^73 - 48 * q^76 + 6 * q^77 + 6 * q^79 + 3 * q^80 - 18 * q^82 + 12 * q^83 + 15 * q^85 + 45 * q^86 - 24 * q^88 + 15 * q^89 + 3 * q^91 - 42 * q^92 - 24 * q^94 - 30 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.66908	−1.88732	−0.943662	−	0.330911i	$-0.892644\pi$
$2$	−2.66908	−1.88732	−0.943662	+	0.330911i	$0.892644\pi$
$3$	0	0
$3$	0	0
$4$	5.12398	2.56199
$5$	1.45490	0.650653	0.325326	−	0.945602i	$-0.394526\pi$
$5$	1.45490	0.650653	0.325326	+	0.945602i	$0.394526\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−8.33816	−2.94798
$9$	0	0
$10$	−3.88325	−1.22799
$11$	1.54510	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$11$	1.54510	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$12$	0	0
$13$	5.88325	1.63172	0.815861	−	0.578249i	$-0.196264\pi$
$13$	5.88325	1.63172	0.815861	+	0.578249i	$0.196264\pi$
$14$	−2.66908	−0.713341
$15$	0	0
$16$	12.0072	3.00181
$17$	6.79306	1.64756	0.823780	−	0.566910i	$-0.191861\pi$
$17$	6.79306	1.64756	0.823780	+	0.566910i	$0.191861\pi$
$18$	0	0
$19$	−6.24797	−1.43338	−0.716691	−	0.697391i	$-0.754344\pi$
$19$	−6.24797	−1.43338	−0.716691	+	0.697391i	$0.754344\pi$
$20$	7.45490	1.66697
$21$	0	0
$22$	−4.12398	−0.879236
$23$	−2.90981	−0.606737	−0.303368	−	0.952873i	$-0.598111\pi$
$23$	−2.90981	−0.606737	−0.303368	+	0.952873i	$0.598111\pi$
$24$	0	0
$25$	−2.88325	−0.576651
$26$	−15.7029	−3.07959
$27$	0	0
$28$	5.12398	0.968342
$29$	−3.88325	−0.721102	−0.360551	−	0.932739i	$-0.617411\pi$
$29$	−3.88325	−0.721102	−0.360551	+	0.932739i	$0.617411\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−15.3719	−2.71740
$33$	0	0
$34$	−18.1312	−3.10948
$35$	1.45490	0.245924
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	16.6763	2.70526
$39$	0	0
$40$	−12.1312	−1.91811
$41$	2.24797	0.351073	0.175537	−	0.984473i	$-0.443834\pi$
$41$	2.24797	0.351073	0.175537	+	0.984473i	$0.443834\pi$
$42$	0	0
$43$	−7.13122	−1.08750	−0.543750	−	0.839247i	$-0.682996\pi$
$43$	−7.13122	−1.08750	−0.543750	+	0.839247i	$0.682996\pi$
$44$	7.91705	1.19354
$45$	0	0
$46$	7.76651	1.14511
$47$	5.33816	0.778650	0.389325	−	0.921100i	$-0.372708\pi$
$47$	5.33816	0.778650	0.389325	+	0.921100i	$0.372708\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.69563	1.08833
$51$	0	0
$52$	30.1457	4.18046
$53$	9.79306	1.34518	0.672590	−	0.740015i	$-0.265182\pi$
$53$	9.79306	1.34518	0.672590	+	0.740015i	$0.265182\pi$
$54$	0	0
$55$	2.24797	0.303116
$56$	−8.33816	−1.11423
$57$	0	0
$58$	10.3647	1.36095
$59$	4.67632	0.608805	0.304402	−	0.952544i	$-0.401543\pi$
$59$	4.67632	0.608805	0.304402	+	0.952544i	$0.401543\pi$
$60$	0	0
$61$	−2.36471	−0.302770	−0.151385	−	0.988475i	$-0.548373\pi$
$61$	−2.36471	−0.302770	−0.151385	+	0.988475i	$0.548373\pi$
$62$	−5.33816	−0.677947
$63$	0	0
$64$	17.0145	2.12681
$65$	8.55957	1.06168
$66$	0	0
$67$	3.36471	0.411065	0.205533	−	0.978650i	$-0.434107\pi$
$67$	3.36471	0.411065	0.205533	+	0.978650i	$0.434107\pi$
$68$	34.8075	4.22103
$69$	0	0
$70$	−3.88325	−0.464138
$71$	−1.36471	−0.161962	−0.0809808	−	0.996716i	$-0.525805\pi$
$71$	−1.36471	−0.161962	−0.0809808	+	0.996716i	$0.525805\pi$
$72$	0	0
$73$	−1.88325	−0.220418	−0.110209	−	0.993908i	$-0.535152\pi$
$73$	−1.88325	−0.220418	−0.110209	+	0.993908i	$0.535152\pi$
$74$	−13.3454	−1.55137
$75$	0	0
$76$	−32.0145	−3.67231
$77$	1.54510	0.176080
$78$	0	0
$79$	3.36471	0.378560	0.189280	−	0.981923i	$-0.439385\pi$
$79$	3.36471	0.378560	0.189280	+	0.981923i	$0.439385\pi$
$80$	17.4694	1.95314
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−2.24797	−0.246746	−0.123373	−	0.992360i	$-0.539371\pi$
$83$	−2.24797	−0.246746	−0.123373	+	0.992360i	$0.539371\pi$
$84$	0	0
$85$	9.88325	1.07199
$86$	19.0338	2.05247
$87$	0	0
$88$	−12.8833	−1.37336
$89$	−0.793062	−0.0840644	−0.0420322	−	0.999116i	$-0.513383\pi$
$89$	−0.793062	−0.0840644	−0.0420322	+	0.999116i	$0.513383\pi$
$90$	0	0
$91$	5.88325	0.616733
$92$	−14.9098	−1.55445
$93$	0	0
$94$	−14.2480	−1.46957
$95$	−9.09019	−0.932634
$96$	0	0
$97$	10.2480	1.04052	0.520262	−	0.854007i	$-0.325834\pi$
$97$	10.2480	1.04052	0.520262	+	0.854007i	$0.325834\pi$
$98$	−2.66908	−0.269618
$99$	0	0
$100$	−14.7737	−1.47737
$101$	−9.09019	−0.904508	−0.452254	−	0.891889i	$-0.649380\pi$
$101$	−9.09019	−0.904508	−0.452254	+	0.891889i	$0.649380\pi$
$102$	0	0
$103$	18.0145	1.77502	0.887509	−	0.460789i	$-0.152434\pi$
$103$	18.0145	1.77502	0.887509	+	0.460789i	$0.152434\pi$
$104$	−49.0555	−4.81029
$105$	0	0
$106$	−26.1385	−2.53879
$107$	18.2214	1.76153	0.880765	−	0.473553i	$-0.157029\pi$
$107$	18.2214	1.76153	0.880765	+	0.473553i	$0.157029\pi$
$108$	0	0
$109$	16.3792	1.56884	0.784421	−	0.620229i	$-0.212960\pi$
$109$	16.3792	1.56884	0.784421	+	0.620229i	$0.212960\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	12.0072	1.13458
$113$	−7.49593	−0.705158	−0.352579	−	0.935782i	$-0.614695\pi$
$113$	−7.49593	−0.705158	−0.352579	+	0.935782i	$0.614695\pi$
$114$	0	0
$115$	−4.23349	−0.394775
$116$	−19.8977	−1.84746
$117$	0	0
$118$	−12.4815	−1.14901
$119$	6.79306	0.622719
$120$	0	0
$121$	−8.61268	−0.782971
$122$	6.31160	0.571426
$123$	0	0
$124$	10.2480	0.920295
$125$	−11.4694	−1.02585
$126$	0	0
$127$	−7.13122	−0.632793	−0.316397	−	0.948627i	$-0.602473\pi$
$127$	−7.13122	−0.632793	−0.316397	+	0.948627i	$0.602473\pi$
$128$	−14.6691	−1.29658
$129$	0	0
$130$	−22.8462	−2.00374
$131$	−18.9243	−1.65342	−0.826711	−	0.562627i	$-0.809791\pi$
$131$	−18.9243	−1.65342	−0.826711	+	0.562627i	$0.809791\pi$
$132$	0	0
$133$	−6.24797	−0.541767
$134$	−8.98068	−0.775813
$135$	0	0
$136$	−56.6416	−4.85698
$137$	5.90981	0.504909	0.252454	−	0.967609i	$-0.418762\pi$
$137$	5.90981	0.504909	0.252454	+	0.967609i	$0.418762\pi$
$138$	0	0
$139$	−1.75203	−0.148606	−0.0743028	−	0.997236i	$-0.523673\pi$
$139$	−1.75203	−0.148606	−0.0743028	+	0.997236i	$0.523673\pi$
$140$	7.45490	0.630054
$141$	0	0
$142$	3.64252	0.305674
$143$	9.09019	0.760160
$144$	0	0
$145$	−5.64976	−0.469187
$146$	5.02655	0.416001
$147$	0	0
$148$	25.6199	2.10594
$149$	6.09019	0.498928	0.249464	−	0.968384i	$-0.419746\pi$
$149$	6.09019	0.498928	0.249464	+	0.968384i	$0.419746\pi$
$150$	0	0
$151$	0.635288	0.0516990	0.0258495	−	0.999666i	$-0.491771\pi$
$151$	0.635288	0.0516990	0.0258495	+	0.999666i	$0.491771\pi$
$152$	52.0965	4.22559
$153$	0	0
$154$	−4.12398	−0.332320
$155$	2.90981	0.233721
$156$	0	0
$157$	−5.15383	−0.411320	−0.205660	−	0.978623i	$-0.565934\pi$
$157$	−5.15383	−0.411320	−0.205660	+	0.978623i	$0.565934\pi$
$158$	−8.98068	−0.714465
$159$	0	0
$160$	−22.3647	−1.76809
$161$	−2.90981	−0.229325
$162$	0	0
$163$	−5.36471	−0.420197	−0.210098	−	0.977680i	$-0.567378\pi$
$163$	−5.36471	−0.420197	−0.210098	+	0.977680i	$0.567378\pi$
$164$	11.5185	0.899447
$165$	0	0
$166$	6.00000	0.465690
$167$	−9.75203	−0.754635	−0.377318	−	0.926084i	$-0.623153\pi$
$167$	−9.75203	−0.754635	−0.377318	+	0.926084i	$0.623153\pi$
$168$	0	0
$169$	21.6127	1.66251
$170$	−26.3792	−2.02319
$171$	0	0
$172$	−36.5403	−2.78617
$173$	−24.3116	−1.84838	−0.924189	−	0.381937i	$-0.875257\pi$
$173$	−24.3116	−1.84838	−0.924189	+	0.381937i	$0.875257\pi$
$174$	0	0
$175$	−2.88325	−0.217954
$176$	18.5523	1.39843
$177$	0	0
$178$	2.11675	0.158657
$179$	−13.5861	−1.01547	−0.507737	−	0.861512i	$-0.669518\pi$
$179$	−13.5861	−1.01547	−0.507737	+	0.861512i	$0.669518\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−15.7029	−1.16397
$183$	0	0
$184$	24.2624	1.78865
$185$	7.27452	0.534833
$186$	0	0
$187$	10.4959	0.767539
$188$	27.3526	1.99490
$189$	0	0
$190$	24.2624	1.76018
$191$	−15.3116	−1.10791	−0.553954	−	0.832547i	$-0.686882\pi$
$191$	−15.3116	−1.10791	−0.553954	+	0.832547i	$0.686882\pi$
$192$	0	0
$193$	−12.3792	−0.891073	−0.445537	−	0.895264i	$-0.646987\pi$
$193$	−12.3792	−0.891073	−0.445537	+	0.895264i	$0.646987\pi$
$194$	−27.3526	−1.96380
$195$	0	0
$196$	5.12398	0.365999
$197$	5.72942	0.408205	0.204102	−	0.978950i	$-0.434572\pi$
$197$	5.72942	0.408205	0.204102	+	0.978950i	$0.434572\pi$
$198$	0	0
$199$	12.0145	0.851684	0.425842	−	0.904798i	$-0.359978\pi$
$199$	12.0145	0.851684	0.425842	+	0.904798i	$0.359978\pi$
$200$	24.0410	1.69996
$201$	0	0
$202$	24.2624	1.70710
$203$	−3.88325	−0.272551
$204$	0	0
$205$	3.27058	0.228427
$206$	−48.0821	−3.35004
$207$	0	0
$208$	70.6416	4.89812
$209$	−9.65371	−0.667761
$210$	0	0
$211$	−13.1312	−0.903990	−0.451995	−	0.892020i	$-0.649288\pi$
$211$	−13.1312	−0.903990	−0.451995	+	0.892020i	$0.649288\pi$
$212$	50.1795	3.44634
$213$	0	0
$214$	−48.6344	−3.32458
$215$	−10.3752	−0.707586
$216$	0	0
$217$	2.00000	0.135769
$218$	−43.7173	−2.96091
$219$	0	0
$220$	11.5185	0.776580
$221$	39.9653	2.68836
$222$	0	0
$223$	−1.27058	−0.0850840	−0.0425420	−	0.999095i	$-0.513546\pi$
$223$	−1.27058	−0.0850840	−0.0425420	+	0.999095i	$0.513546\pi$
$224$	−15.3719	−1.02708
$225$	0	0
$226$	20.0072	1.33086
$227$	16.1949	1.07489	0.537445	−	0.843299i	$-0.319389\pi$
$227$	16.1949	1.07489	0.537445	+	0.843299i	$0.319389\pi$
$228$	0	0
$229$	12.3647	0.817083	0.408542	−	0.912740i	$-0.366037\pi$
$229$	12.3647	0.817083	0.408542	+	0.912740i	$0.366037\pi$
$230$	11.2995	0.745069
$231$	0	0
$232$	32.3792	2.12580
$233$	−18.7931	−1.23117	−0.615587	−	0.788069i	$-0.711081\pi$
$233$	−18.7931	−1.23117	−0.615587	+	0.788069i	$0.711081\pi$
$234$	0	0
$235$	7.76651	0.506631
$236$	23.9614	1.55975
$237$	0	0
$238$	−18.1312	−1.17527
$239$	−21.1312	−1.36687	−0.683433	−	0.730014i	$-0.739514\pi$
$239$	−21.1312	−1.36687	−0.683433	+	0.730014i	$0.739514\pi$
$240$	0	0
$241$	6.36471	0.409987	0.204994	−	0.978763i	$-0.434283\pi$
$241$	6.36471	0.409987	0.204994	+	0.978763i	$0.434283\pi$
$242$	22.9879	1.47772
$243$	0	0
$244$	−12.1167	−0.775695
$245$	1.45490	0.0929504
$246$	0	0
$247$	−36.7584	−2.33888
$248$	−16.6763	−1.05895
$249$	0	0
$250$	30.6127	1.93612
$251$	16.0145	1.01082	0.505412	−	0.862878i	$-0.331340\pi$
$251$	16.0145	1.01082	0.505412	+	0.862878i	$0.331340\pi$
$252$	0	0
$253$	−4.49593	−0.282657
$254$	19.0338	1.19429
$255$	0	0
$256$	5.12398	0.320249
$257$	−17.2890	−1.07846	−0.539229	−	0.842159i	$-0.681284\pi$
$257$	−17.2890	−1.07846	−0.539229	+	0.842159i	$0.681284\pi$
$258$	0	0
$259$	5.00000	0.310685
$260$	43.8591	2.72003
$261$	0	0
$262$	50.5104	3.12054
$263$	13.5451	0.835226	0.417613	−	0.908625i	$-0.362867\pi$
$263$	13.5451	0.835226	0.417613	+	0.908625i	$0.362867\pi$
$264$	0	0
$265$	14.2480	0.875246
$266$	16.6763	1.02249
$267$	0	0
$268$	17.2407	1.05315
$269$	12.7931	0.780007	0.390003	−	0.920813i	$-0.372474\pi$
$269$	12.7931	0.780007	0.390003	+	0.920813i	$0.372474\pi$
$270$	0	0
$271$	−32.0145	−1.94474	−0.972370	−	0.233443i	$-0.925001\pi$
$271$	−32.0145	−1.94474	−0.972370	+	0.233443i	$0.925001\pi$
$272$	81.5659	4.94566
$273$	0	0
$274$	−15.7737	−0.952927
$275$	−4.45490	−0.268641
$276$	0	0
$277$	13.3792	0.803877	0.401939	−	0.915667i	$-0.368336\pi$
$277$	13.3792	0.803877	0.401939	+	0.915667i	$0.368336\pi$
$278$	4.67632	0.280467
$279$	0	0
$280$	−12.1312	−0.724979
$281$	10.7665	0.642276	0.321138	−	0.947032i	$-0.395935\pi$
$281$	10.7665	0.642276	0.321138	+	0.947032i	$0.395935\pi$
$282$	0	0
$283$	−28.7439	−1.70865	−0.854324	−	0.519741i	$-0.826028\pi$
$283$	−28.7439	−1.70865	−0.854324	+	0.519741i	$0.826028\pi$
$284$	−6.99276	−0.414944
$285$	0	0
$286$	−24.2624	−1.43467
$287$	2.24797	0.132693
$288$	0	0
$289$	29.1457	1.71445
$290$	15.0797	0.885508
$291$	0	0
$292$	−9.64976	−0.564710
$293$	−6.13122	−0.358190	−0.179095	−	0.983832i	$-0.557317\pi$
$293$	−6.13122	−0.358190	−0.179095	+	0.983832i	$0.557317\pi$
$294$	0	0
$295$	6.80359	0.396120
$296$	−41.6908	−2.42323
$297$	0	0
$298$	−16.2552	−0.941639
$299$	−17.1191	−0.990025
$300$	0	0
$301$	−7.13122	−0.411037
$302$	−1.69563	−0.0975727
$303$	0	0
$304$	−75.0208	−4.30274
$305$	−3.44043	−0.196998
$306$	0	0
$307$	−5.76651	−0.329112	−0.164556	−	0.986368i	$-0.552619\pi$
$307$	−5.76651	−0.329112	−0.164556	+	0.986368i	$0.552619\pi$
$308$	7.91705	0.451116
$309$	0	0
$310$	−7.76651	−0.441108
$311$	−5.33816	−0.302699	−0.151350	−	0.988480i	$-0.548362\pi$
$311$	−5.33816	−0.302699	−0.151350	+	0.988480i	$0.548362\pi$
$312$	0	0
$313$	7.38732	0.417556	0.208778	−	0.977963i	$-0.433051\pi$
$313$	7.38732	0.417556	0.208778	+	0.977963i	$0.433051\pi$
$314$	13.7560	0.776295
$315$	0	0
$316$	17.2407	0.969867
$317$	−11.0266	−0.619313	−0.309656	−	0.950848i	$-0.600214\pi$
$317$	−11.0266	−0.619313	−0.309656	+	0.950848i	$0.600214\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	24.7544	1.38381
$321$	0	0
$322$	7.76651	0.432811
$323$	−42.4428	−2.36158
$324$	0	0
$325$	−16.9629	−0.940933
$326$	14.3188	0.793047
$327$	0	0
$328$	−18.7439	−1.03496
$329$	5.33816	0.294302
$330$	0	0
$331$	−31.5330	−1.73321	−0.866606	−	0.498994i	$-0.833703\pi$
$331$	−31.5330	−1.73321	−0.866606	+	0.498994i	$0.833703\pi$
$332$	−11.5185	−0.632162
$333$	0	0
$334$	26.0289	1.42424
$335$	4.89533	0.267461
$336$	0	0
$337$	34.6498	1.88749	0.943746	−	0.330670i	$-0.107275\pi$
$337$	34.6498	1.88749	0.943746	+	0.330670i	$0.107275\pi$
$338$	−57.6859	−3.13770
$339$	0	0
$340$	50.6416	2.74643
$341$	3.09019	0.167343
$342$	0	0
$343$	1.00000	0.0539949
$344$	59.4612	3.20594
$345$	0	0
$346$	64.8896	3.48849
$347$	−2.68840	−0.144321	−0.0721603	−	0.997393i	$-0.522989\pi$
$347$	−2.68840	−0.144321	−0.0721603	+	0.997393i	$0.522989\pi$
$348$	0	0
$349$	−4.48146	−0.239887	−0.119943	−	0.992781i	$-0.538271\pi$
$349$	−4.48146	−0.239887	−0.119943	+	0.992781i	$0.538271\pi$
$350$	7.69563	0.411349
$351$	0	0
$352$	−23.7511	−1.26594
$353$	−31.1047	−1.65553	−0.827767	−	0.561072i	$-0.810389\pi$
$353$	−31.1047	−1.65553	−0.827767	+	0.561072i	$0.810389\pi$
$354$	0	0
$355$	−1.98553	−0.105381
$356$	−4.06364	−0.215372
$357$	0	0
$358$	36.2624	1.91653
$359$	18.2214	0.961689	0.480845	−	0.876806i	$-0.340330\pi$
$359$	18.2214	0.961689	0.480845	+	0.876806i	$0.340330\pi$
$360$	0	0
$361$	20.0371	1.05458
$362$	−5.33816	−0.280567
$363$	0	0
$364$	30.1457	1.58006
$365$	−2.73995	−0.143416
$366$	0	0
$367$	−28.2624	−1.47529	−0.737644	−	0.675190i	$-0.764062\pi$
$367$	−28.2624	−1.47529	−0.737644	+	0.675190i	$0.764062\pi$
$368$	−34.9388	−1.82131
$369$	0	0
$370$	−19.4163	−1.00940
$371$	9.79306	0.508430
$372$	0	0
$373$	−2.15383	−0.111521	−0.0557605	−	0.998444i	$-0.517758\pi$
$373$	−2.15383	−0.111521	−0.0557605	+	0.998444i	$0.517758\pi$
$374$	−28.0145	−1.44859
$375$	0	0
$376$	−44.5104	−2.29545
$377$	−22.8462	−1.17664
$378$	0	0
$379$	−7.67237	−0.394103	−0.197052	−	0.980393i	$-0.563137\pi$
$379$	−7.67237	−0.394103	−0.197052	+	0.980393i	$0.563137\pi$
$380$	−46.5780	−2.38940
$381$	0	0
$382$	40.8679	2.09098
$383$	17.8196	0.910540	0.455270	−	0.890353i	$-0.349543\pi$
$383$	17.8196	0.910540	0.455270	+	0.890353i	$0.349543\pi$
$384$	0	0
$385$	2.24797	0.114567
$386$	33.0410	1.68174
$387$	0	0
$388$	52.5104	2.66581
$389$	0.180384	0.00914581	0.00457291	−	0.999990i	$-0.498544\pi$
$389$	0.180384	0.00914581	0.00457291	+	0.999990i	$0.498544\pi$
$390$	0	0
$391$	−19.7665	−0.999635
$392$	−8.33816	−0.421141
$393$	0	0
$394$	−15.2923	−0.770414
$395$	4.89533	0.246311
$396$	0	0
$397$	−22.3937	−1.12391	−0.561953	−	0.827169i	$-0.689950\pi$
$397$	−22.3937	−1.12391	−0.561953	+	0.827169i	$0.689950\pi$
$398$	−32.0676	−1.60740
$399$	0	0
$400$	−34.6199	−1.73100
$401$	12.0902	0.603755	0.301878	−	0.953347i	$-0.402387\pi$
$401$	12.0902	0.603755	0.301878	+	0.953347i	$0.402387\pi$
$402$	0	0
$403$	11.7665	0.586132
$404$	−46.5780	−2.31734
$405$	0	0
$406$	10.3647	0.514392
$407$	7.72548	0.382938
$408$	0	0
$409$	−20.3647	−1.00697	−0.503485	−	0.864004i	$-0.667949\pi$
$409$	−20.3647	−1.00697	−0.503485	+	0.864004i	$0.667949\pi$
$410$	−8.72942	−0.431116
$411$	0	0
$412$	92.3059	4.54758
$413$	4.67632	0.230106
$414$	0	0
$415$	−3.27058	−0.160546
$416$	−90.4371	−4.43404
$417$	0	0
$418$	25.7665	1.26028
$419$	10.9774	0.536281	0.268140	−	0.963380i	$-0.413591\pi$
$419$	10.9774	0.536281	0.268140	+	0.963380i	$0.413591\pi$
$420$	0	0
$421$	7.72942	0.376709	0.188355	−	0.982101i	$-0.439685\pi$
$421$	7.72942	0.376709	0.188355	+	0.982101i	$0.439685\pi$
$422$	35.0483	1.70612
$423$	0	0
$424$	−81.6561	−3.96557
$425$	−19.5861	−0.950067
$426$	0	0
$427$	−2.36471	−0.114436
$428$	93.3662	4.51303
$429$	0	0
$430$	27.6923	1.33544
$431$	27.1722	1.30884	0.654421	−	0.756131i	$-0.272913\pi$
$431$	27.1722	1.30884	0.654421	+	0.756131i	$0.272913\pi$
$432$	0	0
$433$	4.59820	0.220976	0.110488	−	0.993877i	$-0.464759\pi$
$433$	4.59820	0.220976	0.110488	+	0.993877i	$0.464759\pi$
$434$	−5.33816	−0.256240
$435$	0	0
$436$	83.9267	4.01936
$437$	18.1804	0.869686
$438$	0	0
$439$	−24.2480	−1.15729	−0.578646	−	0.815579i	$-0.696419\pi$
$439$	−24.2480	−1.15729	−0.578646	+	0.815579i	$0.696419\pi$
$440$	−18.7439	−0.893580
$441$	0	0
$442$	−106.671	−5.07380
$443$	30.0821	1.42924	0.714621	−	0.699512i	$-0.246599\pi$
$443$	30.0821	1.42924	0.714621	+	0.699512i	$0.246599\pi$
$444$	0	0
$445$	−1.15383	−0.0546968
$446$	3.39127	0.160581
$447$	0	0
$448$	17.0145	0.803858
$449$	−29.3792	−1.38649	−0.693245	−	0.720702i	$-0.743820\pi$
$449$	−29.3792	−1.38649	−0.693245	+	0.720702i	$0.743820\pi$
$450$	0	0
$451$	3.47332	0.163552
$452$	−38.4090	−1.80661
$453$	0	0
$454$	−43.2254	−2.02867
$455$	8.55957	0.401279
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−33.0024	−1.54210
$459$	0	0
$460$	−21.6923	−1.01141
$461$	8.51041	0.396369	0.198185	−	0.980165i	$-0.436495\pi$
$461$	8.51041	0.396369	0.198185	+	0.980165i	$0.436495\pi$
$462$	0	0
$463$	23.3937	1.08720	0.543598	−	0.839346i	$-0.317062\pi$
$463$	23.3937	1.08720	0.543598	+	0.839346i	$0.317062\pi$
$464$	−46.6272	−2.16461
$465$	0	0
$466$	50.1602	2.32362
$467$	−25.8872	−1.19792	−0.598958	−	0.800780i	$-0.704419\pi$
$467$	−25.8872	−1.19792	−0.598958	+	0.800780i	$0.704419\pi$
$468$	0	0
$469$	3.36471	0.155368
$470$	−20.7294	−0.956177
$471$	0	0
$472$	−38.9919	−1.79475
$473$	−11.0184	−0.506627
$474$	0	0
$475$	18.0145	0.826561
$476$	34.8075	1.59540
$477$	0	0
$478$	56.4009	2.57972
$479$	−11.8196	−0.540052	−0.270026	−	0.962853i	$-0.587032\pi$
$479$	−11.8196	−0.540052	−0.270026	+	0.962853i	$0.587032\pi$
$480$	0	0
$481$	29.4163	1.34127
$482$	−16.9879	−0.773779
$483$	0	0
$484$	−44.1312	−2.00596
$485$	14.9098	0.677020
$486$	0	0
$487$	31.8606	1.44374	0.721872	−	0.692027i	$-0.243282\pi$
$487$	31.8606	1.44374	0.721872	+	0.692027i	$0.243282\pi$
$488$	19.7173	0.892562
$489$	0	0
$490$	−3.88325	−0.175428
$491$	15.5330	0.700995	0.350498	−	0.936564i	$-0.386012\pi$
$491$	15.5330	0.700995	0.350498	+	0.936564i	$0.386012\pi$
$492$	0	0
$493$	−26.3792	−1.18806
$494$	98.1110	4.41422
$495$	0	0
$496$	24.0145	1.07828
$497$	−1.36471	−0.0612157
$498$	0	0
$499$	24.7584	1.10834	0.554169	−	0.832405i	$-0.313036\pi$
$499$	24.7584	1.10834	0.554169	+	0.832405i	$0.313036\pi$
$500$	−58.7689	−2.62823
$501$	0	0
$502$	−42.7439	−1.90775
$503$	−35.2995	−1.57393	−0.786964	−	0.616999i	$-0.788348\pi$
$503$	−35.2995	−1.57393	−0.786964	+	0.616999i	$0.788348\pi$
$504$	0	0
$505$	−13.2254	−0.588521
$506$	12.0000	0.533465
$507$	0	0
$508$	−36.5403	−1.62121
$509$	−0.180384	−0.00799536	−0.00399768	−	0.999992i	$-0.501273\pi$
$509$	−0.180384	−0.00799536	−0.00399768	+	0.999992i	$0.501273\pi$
$510$	0	0
$511$	−1.88325	−0.0833103
$512$	15.6618	0.692162
$513$	0	0
$514$	46.1457	2.03540
$515$	26.2093	1.15492
$516$	0	0
$517$	8.24797	0.362745
$518$	−13.3454	−0.586363
$519$	0	0
$520$	−71.3711	−3.12983
$521$	−10.8567	−0.475641	−0.237820	−	0.971309i	$-0.576433\pi$
$521$	−10.8567	−0.475641	−0.237820	+	0.971309i	$0.576433\pi$
$522$	0	0
$523$	−40.7439	−1.78161	−0.890803	−	0.454389i	$-0.849857\pi$
$523$	−40.7439	−1.78161	−0.890803	+	0.454389i	$0.849857\pi$
$524$	−96.9677	−4.23605
$525$	0	0
$526$	−36.1529	−1.57634
$527$	13.5861	0.591821
$528$	0	0
$529$	−14.5330	−0.631870
$530$	−38.0289	−1.65187
$531$	0	0
$532$	−32.0145	−1.38800
$533$	13.2254	0.572854
$534$	0	0
$535$	26.5104	1.14614
$536$	−28.0555	−1.21181
$537$	0	0
$538$	−34.1457	−1.47213
$539$	1.54510	0.0665520
$540$	0	0
$541$	−17.2335	−0.740926	−0.370463	−	0.928847i	$-0.620801\pi$
$541$	−17.2335	−0.740926	−0.370463	+	0.928847i	$0.620801\pi$
$542$	85.4492	3.67036
$543$	0	0
$544$	−104.423	−4.47708
$545$	23.8301	1.02077
$546$	0	0
$547$	−39.8606	−1.70432	−0.852159	−	0.523283i	$-0.824707\pi$
$547$	−39.8606	−1.70432	−0.852159	+	0.523283i	$0.824707\pi$
$548$	30.2818	1.29357
$549$	0	0
$550$	11.8905	0.507012
$551$	24.2624	1.03361
$552$	0	0
$553$	3.36471	0.143082
$554$	−35.7101	−1.51718
$555$	0	0
$556$	−8.97739	−0.380726
$557$	−28.5861	−1.21123	−0.605616	−	0.795757i	$-0.707073\pi$
$557$	−28.5861	−1.21123	−0.605616	+	0.795757i	$0.707073\pi$
$558$	0	0
$559$	−41.9548	−1.77450
$560$	17.4694	0.738216
$561$	0	0
$562$	−28.7367	−1.21218
$563$	−24.0821	−1.01494	−0.507469	−	0.861670i	$-0.669419\pi$
$563$	−24.0821	−1.01494	−0.507469	+	0.861670i	$0.669419\pi$
$564$	0	0
$565$	−10.9059	−0.458813
$566$	76.7197	3.22477
$567$	0	0
$568$	11.3792	0.477460
$569$	1.05311	0.0441486	0.0220743	−	0.999756i	$-0.492973\pi$
$569$	1.05311	0.0441486	0.0220743	+	0.999756i	$0.492973\pi$
$570$	0	0
$571$	−16.2624	−0.680562	−0.340281	−	0.940324i	$-0.610522\pi$
$571$	−16.2624	−0.680562	−0.340281	+	0.940324i	$0.610522\pi$
$572$	46.5780	1.94752
$573$	0	0
$574$	−6.00000	−0.250435
$575$	8.38972	0.349875
$576$	0	0
$577$	38.3937	1.59835	0.799175	−	0.601099i	$-0.205270\pi$
$577$	38.3937	1.59835	0.799175	+	0.601099i	$0.205270\pi$
$578$	−77.7922	−3.23573
$579$	0	0
$580$	−28.9493	−1.20205
$581$	−2.24797	−0.0932614
$582$	0	0
$583$	15.1312	0.626671
$584$	15.7029	0.649789
$585$	0	0
$586$	16.3647	0.676020
$587$	27.9548	1.15382	0.576909	−	0.816809i	$-0.304259\pi$
$587$	27.9548	1.15382	0.576909	+	0.816809i	$0.304259\pi$
$588$	0	0
$589$	−12.4959	−0.514886
$590$	−18.1593	−0.747608
$591$	0	0
$592$	60.0362	2.46747
$593$	−3.52249	−0.144651	−0.0723256	−	0.997381i	$-0.523042\pi$
$593$	−3.52249	−0.144651	−0.0723256	+	0.997381i	$0.523042\pi$
$594$	0	0
$595$	9.88325	0.405174
$596$	31.2060	1.27825
$597$	0	0
$598$	45.6923	1.86850
$599$	−5.95897	−0.243477	−0.121738	−	0.992562i	$-0.538847\pi$
$599$	−5.95897	−0.243477	−0.121738	+	0.992562i	$0.538847\pi$
$600$	0	0
$601$	−0.860645	−0.0351064	−0.0175532	−	0.999846i	$-0.505588\pi$
$601$	−0.860645	−0.0351064	−0.0175532	+	0.999846i	$0.505588\pi$
$602$	19.0338	0.775759
$603$	0	0
$604$	3.25520	0.132452
$605$	−12.5306	−0.509442
$606$	0	0
$607$	−32.9774	−1.33851	−0.669256	−	0.743032i	$-0.733387\pi$
$607$	−32.9774	−1.33851	−0.669256	+	0.743032i	$0.733387\pi$
$608$	96.0434	3.89508
$609$	0	0
$610$	9.18278	0.371800
$611$	31.4057	1.27054
$612$	0	0
$613$	−27.9203	−1.12769	−0.563846	−	0.825880i	$-0.690679\pi$
$613$	−27.9203	−1.12769	−0.563846	+	0.825880i	$0.690679\pi$
$614$	15.3913	0.621141
$615$	0	0
$616$	−12.8833	−0.519081
$617$	41.3711	1.66554	0.832768	−	0.553622i	$-0.186755\pi$
$617$	41.3711	1.66554	0.832768	+	0.553622i	$0.186755\pi$
$618$	0	0
$619$	−11.2850	−0.453584	−0.226792	−	0.973943i	$-0.572824\pi$
$619$	−11.2850	−0.453584	−0.226792	+	0.973943i	$0.572824\pi$
$620$	14.9098	0.598792
$621$	0	0
$622$	14.2480	0.571291
$623$	−0.793062	−0.0317734
$624$	0	0
$625$	−2.27058	−0.0908230
$626$	−19.7173	−0.788064
$627$	0	0
$628$	−26.4081	−1.05380
$629$	33.9653	1.35429
$630$	0	0
$631$	32.2624	1.28435	0.642174	−	0.766559i	$-0.278033\pi$
$631$	32.2624	1.28435	0.642174	+	0.766559i	$0.278033\pi$
$632$	−28.0555	−1.11599
$633$	0	0
$634$	29.4307	1.16884
$635$	−10.3752	−0.411729
$636$	0	0
$637$	5.88325	0.233103
$638$	16.0145	0.634019
$639$	0	0
$640$	−21.3421	−0.843621
$641$	37.9388	1.49849	0.749245	−	0.662292i	$-0.230416\pi$
$641$	37.9388	1.49849	0.749245	+	0.662292i	$0.230416\pi$
$642$	0	0
$643$	38.8036	1.53026	0.765132	−	0.643873i	$-0.222674\pi$
$643$	38.8036	1.53026	0.765132	+	0.643873i	$0.222674\pi$
$644$	−14.9098	−0.587529
$645$	0	0
$646$	113.283	4.45707
$647$	43.7279	1.71912	0.859560	−	0.511035i	$-0.170738\pi$
$647$	43.7279	1.71912	0.859560	+	0.511035i	$0.170738\pi$
$648$	0	0
$649$	7.22536	0.283620
$650$	45.2754	1.77585
$651$	0	0
$652$	−27.4887	−1.07654
$653$	35.1988	1.37744	0.688718	−	0.725030i	$-0.258174\pi$
$653$	35.1988	1.37744	0.688718	+	0.725030i	$0.258174\pi$
$654$	0	0
$655$	−27.5330	−1.07580
$656$	26.9919	1.05386
$657$	0	0
$658$	−14.2480	−0.555444
$659$	4.37285	0.170342	0.0851710	−	0.996366i	$-0.472856\pi$
$659$	4.37285	0.170342	0.0851710	+	0.996366i	$0.472856\pi$
$660$	0	0
$661$	−22.6127	−0.879531	−0.439766	−	0.898113i	$-0.644938\pi$
$661$	−22.6127	−0.879531	−0.439766	+	0.898113i	$0.644938\pi$
$662$	84.1641	3.27113
$663$	0	0
$664$	18.7439	0.727404
$665$	−9.09019	−0.352503
$666$	0	0
$667$	11.2995	0.437519
$668$	−49.9693	−1.93337
$669$	0	0
$670$	−13.0660	−0.504785
$671$	−3.65371	−0.141050
$672$	0	0
$673$	25.7294	0.991796	0.495898	−	0.868381i	$-0.334839\pi$
$673$	25.7294	0.991796	0.495898	+	0.868381i	$0.334839\pi$
$674$	−92.4830	−3.56231
$675$	0	0
$676$	110.743	4.25935
$677$	−34.1352	−1.31192	−0.655960	−	0.754795i	$-0.727736\pi$
$677$	−34.1352	−1.31192	−0.655960	+	0.754795i	$0.727736\pi$
$678$	0	0
$679$	10.2480	0.393281
$680$	−82.4081	−3.16021
$681$	0	0
$682$	−8.24797	−0.315831
$683$	13.8075	0.528331	0.264165	−	0.964477i	$-0.414904\pi$
$683$	13.8075	0.528331	0.264165	+	0.964477i	$0.414904\pi$
$684$	0	0
$685$	8.59820	0.328520
$686$	−2.66908	−0.101906
$687$	0	0
$688$	−85.6263	−3.26447
$689$	57.6151	2.19496
$690$	0	0
$691$	−1.27058	−0.0483350	−0.0241675	−	0.999708i	$-0.507693\pi$
$691$	−1.27058	−0.0483350	−0.0241675	+	0.999708i	$0.507693\pi$
$692$	−124.572	−4.73553
$693$	0	0
$694$	7.17554	0.272380
$695$	−2.54904	−0.0966906
$696$	0	0
$697$	15.2706	0.578414
$698$	11.9614	0.452744
$699$	0	0
$700$	−14.7737	−0.558395
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	−31.2398	−1.17823
$704$	26.2890	0.990804
$705$	0	0
$706$	83.0208	3.12453
$707$	−9.09019	−0.341872
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	5.29952	0.198888
$711$	0	0
$712$	6.61268	0.247821
$713$	−5.81962	−0.217946
$714$	0	0
$715$	13.2254	0.494600
$716$	−69.6151	−2.60164
$717$	0	0
$718$	−48.6344	−1.81502
$719$	−36.4428	−1.35909	−0.679544	−	0.733635i	$-0.737822\pi$
$719$	−36.4428	−1.35909	−0.679544	+	0.733635i	$0.737822\pi$
$720$	0	0
$721$	18.0145	0.670894
$722$	−53.4806	−1.99034
$723$	0	0
$724$	10.2480	0.380863
$725$	11.1964	0.415824
$726$	0	0
$727$	−15.7810	−0.585284	−0.292642	−	0.956222i	$-0.594534\pi$
$727$	−15.7810	−0.585284	−0.292642	+	0.956222i	$0.594534\pi$
$728$	−49.0555	−1.81812
$729$	0	0
$730$	7.31315	0.270672
$731$	−48.4428	−1.79172
$732$	0	0
$733$	34.7728	1.28436	0.642182	−	0.766552i	$-0.278029\pi$
$733$	34.7728	1.28436	0.642182	+	0.766552i	$0.278029\pi$
$734$	75.4347	2.78435
$735$	0	0
$736$	44.7294	1.64875
$737$	5.19880	0.191500
$738$	0	0
$739$	0.635288	0.0233694	0.0116847	−	0.999932i	$-0.496281\pi$
$739$	0.635288	0.0233694	0.0116847	+	0.999932i	$0.496281\pi$
$740$	37.2745	1.37024
$741$	0	0
$742$	−26.1385	−0.959573
$743$	−49.8896	−1.83027	−0.915136	−	0.403146i	$-0.867917\pi$
$743$	−49.8896	−1.83027	−0.915136	+	0.403146i	$0.867917\pi$
$744$	0	0
$745$	8.86064	0.324629
$746$	5.74874	0.210476
$747$	0	0
$748$	53.7810	1.96643
$749$	18.2214	0.665796
$750$	0	0
$751$	−18.1683	−0.662971	−0.331485	−	0.943460i	$-0.607550\pi$
$751$	−18.1683	−0.662971	−0.331485	+	0.943460i	$0.607550\pi$
$752$	64.0965	2.33736
$753$	0	0
$754$	60.9782	2.22070
$755$	0.924283	0.0336381
$756$	0	0
$757$	−14.4959	−0.526864	−0.263432	−	0.964678i	$-0.584854\pi$
$757$	−14.4959	−0.526864	−0.263432	+	0.964678i	$0.584854\pi$
$758$	20.4782	0.743800
$759$	0	0
$760$	75.7955	2.74939
$761$	−22.0039	−0.797642	−0.398821	−	0.917029i	$-0.630581\pi$
$761$	−22.0039	−0.797642	−0.398821	+	0.917029i	$0.630581\pi$
$762$	0	0
$763$	16.3792	0.592966
$764$	−78.4564	−2.83845
$765$	0	0
$766$	−47.5620	−1.71848
$767$	27.5120	0.993399
$768$	0	0
$769$	27.6353	0.996554	0.498277	−	0.867018i	$-0.333966\pi$
$769$	27.6353	0.996554	0.498277	+	0.867018i	$0.333966\pi$
$770$	−6.00000	−0.216225
$771$	0	0
$772$	−63.4307	−2.28292
$773$	22.0636	0.793574	0.396787	−	0.917911i	$-0.370125\pi$
$773$	22.0636	0.793574	0.396787	+	0.917911i	$0.370125\pi$
$774$	0	0
$775$	−5.76651	−0.207139
$776$	−85.4492	−3.06745
$777$	0	0
$778$	−0.481458	−0.0172611
$779$	−14.0452	−0.503222
$780$	0	0
$781$	−2.10861	−0.0754520
$782$	52.7584	1.88664
$783$	0	0
$784$	12.0072	0.428830
$785$	−7.49833	−0.267627
$786$	0	0
$787$	−26.0145	−0.927316	−0.463658	−	0.886014i	$-0.653463\pi$
$787$	−26.0145	−0.927316	−0.463658	+	0.886014i	$0.653463\pi$
$788$	29.3575	1.04582
$789$	0	0
$790$	−13.0660	−0.464869
$791$	−7.49593	−0.266525
$792$	0	0
$793$	−13.9122	−0.494037
$794$	59.7705	2.12117
$795$	0	0
$796$	61.5620	2.18201
$797$	−30.8751	−1.09365	−0.546826	−	0.837246i	$-0.684164\pi$
$797$	−30.8751	−1.09365	−0.546826	+	0.837246i	$0.684164\pi$
$798$	0	0
$799$	36.2624	1.28287
$800$	44.3212	1.56699
$801$	0	0
$802$	−32.2697	−1.13948
$803$	−2.90981	−0.102685
$804$	0	0
$805$	−4.23349	−0.149211
$806$	−31.4057	−1.10622
$807$	0	0
$808$	75.7955	2.66647
$809$	−43.8567	−1.54192	−0.770960	−	0.636884i	$-0.780223\pi$
$809$	−43.8567	−1.54192	−0.770960	+	0.636884i	$0.780223\pi$
$810$	0	0
$811$	11.2706	0.395763	0.197882	−	0.980226i	$-0.436594\pi$
$811$	11.2706	0.395763	0.197882	+	0.980226i	$0.436594\pi$
$812$	−19.8977	−0.698273
$813$	0	0
$814$	−20.6199	−0.722728
$815$	−7.80514	−0.273402
$816$	0	0
$817$	44.5556	1.55880
$818$	54.3550	1.90048
$819$	0	0
$820$	16.7584	0.585228
$821$	−34.4057	−1.20077	−0.600384	−	0.799712i	$-0.704986\pi$
$821$	−34.4057	−1.20077	−0.600384	+	0.799712i	$0.704986\pi$
$822$	0	0
$823$	2.54115	0.0885790	0.0442895	−	0.999019i	$-0.485898\pi$
$823$	2.54115	0.0885790	0.0442895	+	0.999019i	$0.485898\pi$
$824$	−150.208	−5.23273
$825$	0	0
$826$	−12.4815	−0.434285
$827$	−7.90586	−0.274914	−0.137457	−	0.990508i	$-0.543893\pi$
$827$	−7.90586	−0.274914	−0.137457	+	0.990508i	$0.543893\pi$
$828$	0	0
$829$	−40.2624	−1.39837	−0.699186	−	0.714940i	$-0.746454\pi$
$829$	−40.2624	−1.39837	−0.699186	+	0.714940i	$0.746454\pi$
$830$	8.72942	0.303003
$831$	0	0
$832$	100.100	3.47036
$833$	6.79306	0.235366
$834$	0	0
$835$	−14.1883	−0.491005
$836$	−49.4654	−1.71080
$837$	0	0
$838$	−29.2995	−1.01214
$839$	21.0516	0.726780	0.363390	−	0.931637i	$-0.381619\pi$
$839$	21.0516	0.726780	0.363390	+	0.931637i	$0.381619\pi$
$840$	0	0
$841$	−13.9203	−0.480012
$842$	−20.6304	−0.710972
$843$	0	0
$844$	−67.2842	−2.31602
$845$	31.4444	1.08172
$846$	0	0
$847$	−8.61268	−0.295935
$848$	117.588	4.03798
$849$	0	0
$850$	52.2769	1.79308
$851$	−14.5490	−0.498735
$852$	0	0
$853$	−25.4733	−0.872190	−0.436095	−	0.899901i	$-0.643639\pi$
$853$	−25.4733	−0.872190	−0.436095	+	0.899901i	$0.643639\pi$
$854$	6.31160	0.215979
$855$	0	0
$856$	−151.933	−5.19296
$857$	35.2890	1.20545	0.602725	−	0.797949i	$-0.294082\pi$
$857$	35.2890	1.20545	0.602725	+	0.797949i	$0.294082\pi$
$858$	0	0
$859$	−24.0289	−0.819857	−0.409929	−	0.912118i	$-0.634446\pi$
$859$	−24.0289	−0.819857	−0.409929	+	0.912118i	$0.634446\pi$
$860$	−53.1626	−1.81283
$861$	0	0
$862$	−72.5249	−2.47021
$863$	−39.8365	−1.35605	−0.678025	−	0.735039i	$-0.737164\pi$
$863$	−39.8365	−1.35605	−0.678025	+	0.735039i	$0.737164\pi$
$864$	0	0
$865$	−35.3711	−1.20265
$866$	−12.2730	−0.417053
$867$	0	0
$868$	10.2480	0.347839
$869$	5.19880	0.176357
$870$	0	0
$871$	19.7955	0.670743
$872$	−136.572	−4.62492
$873$	0	0
$874$	−48.5249	−1.64138
$875$	−11.4694	−0.387736
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	64.7197	2.18419
$879$	0	0
$880$	26.9919	0.909896
$881$	−1.28505	−0.0432944	−0.0216472	−	0.999766i	$-0.506891\pi$
$881$	−1.28505	−0.0432944	−0.0216472	+	0.999766i	$0.506891\pi$
$882$	0	0
$883$	−25.9348	−0.872776	−0.436388	−	0.899759i	$-0.643742\pi$
$883$	−25.9348	−0.872776	−0.436388	+	0.899759i	$0.643742\pi$
$884$	204.782	6.88755
$885$	0	0
$886$	−80.2914	−2.69744
$887$	41.1191	1.38065	0.690323	−	0.723501i	$-0.257469\pi$
$887$	41.1191	1.38065	0.690323	+	0.723501i	$0.257469\pi$
$888$	0	0
$889$	−7.13122	−0.239173
$890$	3.07966	0.103231
$891$	0	0
$892$	−6.51041	−0.217985
$893$	−33.3526	−1.11610
$894$	0	0
$895$	−19.7665	−0.660721
$896$	−14.6691	−0.490060
$897$	0	0
$898$	78.4154	2.61675
$899$	−7.76651	−0.259028
$900$	0	0
$901$	66.5249	2.21627
$902$	−9.27058	−0.308676
$903$	0	0
$904$	62.5023	2.07879
$905$	2.90981	0.0967253
$906$	0	0
$907$	33.3647	1.10786	0.553929	−	0.832564i	$-0.313128\pi$
$907$	33.3647	1.10786	0.553929	+	0.832564i	$0.313128\pi$
$908$	82.9822	2.75386
$909$	0	0
$910$	−22.8462	−0.757343
$911$	−20.9919	−0.695492	−0.347746	−	0.937589i	$-0.613053\pi$
$911$	−20.9919	−0.695492	−0.347746	+	0.937589i	$0.613053\pi$
$912$	0	0
$913$	−3.47332	−0.114950
$914$	2.66908	0.0882853
$915$	0	0
$916$	63.3566	2.09336
$917$	−18.9243	−0.624935
$918$	0	0
$919$	24.6353	0.812643	0.406322	−	0.913730i	$-0.366811\pi$
$919$	24.6353	0.812643	0.406322	+	0.913730i	$0.366811\pi$
$920$	35.2995	1.16379
$921$	0	0
$922$	−22.7150	−0.748077
$923$	−8.02895	−0.264276
$924$	0	0
$925$	−14.4163	−0.474004
$926$	−62.4395	−2.05189
$927$	0	0
$928$	59.6932	1.95952
$929$	46.9879	1.54162	0.770812	−	0.637063i	$-0.219851\pi$
$929$	46.9879	1.54162	0.770812	+	0.637063i	$0.219851\pi$
$930$	0	0
$931$	−6.24797	−0.204769
$932$	−96.2953	−3.15426
$933$	0	0
$934$	69.0950	2.26086
$935$	15.2706	0.499401
$936$	0	0
$937$	58.6416	1.91574	0.957869	−	0.287205i	$-0.0927260\pi$
$937$	58.6416	1.91574	0.957869	+	0.287205i	$0.0927260\pi$
$938$	−8.98068	−0.293230
$939$	0	0
$940$	39.7955	1.29798
$941$	−23.7318	−0.773635	−0.386818	−	0.922156i	$-0.626426\pi$
$941$	−23.7318	−0.773635	−0.386818	+	0.922156i	$0.626426\pi$
$942$	0	0
$943$	−6.54115	−0.213009
$944$	56.1496	1.82752
$945$	0	0
$946$	29.4090	0.956170
$947$	−16.1352	−0.524322	−0.262161	−	0.965024i	$-0.584435\pi$
$947$	−16.1352	−0.524322	−0.262161	+	0.965024i	$0.584435\pi$
$948$	0	0
$949$	−11.0797	−0.359661
$950$	−48.0821	−1.55999
$951$	0	0
$952$	−56.6416	−1.83577
$953$	20.3792	0.660147	0.330073	−	0.943955i	$-0.392927\pi$
$953$	20.3792	0.660147	0.330073	+	0.943955i	$0.392927\pi$
$954$	0	0
$955$	−22.2769	−0.720864
$956$	−108.276	−3.50190
$957$	0	0
$958$	31.5475	1.01925
$959$	5.90981	0.190838
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−78.5144	−2.53140
$963$	0	0
$964$	32.6127	1.05038
$965$	−18.0105	−0.579779
$966$	0	0
$967$	24.2335	0.779297	0.389648	−	0.920964i	$-0.372597\pi$
$967$	24.2335	0.779297	0.389648	+	0.920964i	$0.372597\pi$
$968$	71.8139	2.30819
$969$	0	0
$970$	−39.7955	−1.27776
$971$	48.3445	1.55145	0.775724	−	0.631072i	$-0.217385\pi$
$971$	48.3445	1.55145	0.775724	+	0.631072i	$0.217385\pi$
$972$	0	0
$973$	−1.75203	−0.0561676
$974$	−85.0386	−2.72481
$975$	0	0
$976$	−28.3937	−0.908859
$977$	−27.3526	−0.875088	−0.437544	−	0.899197i	$-0.644152\pi$
$977$	−27.3526	−0.875088	−0.437544	+	0.899197i	$0.644152\pi$
$978$	0	0
$979$	−1.22536	−0.0391626
$980$	7.45490	0.238138
$981$	0	0
$982$	−41.4588	−1.32301
$983$	−17.0757	−0.544631	−0.272315	−	0.962208i	$-0.587789\pi$
$983$	−17.0757	−0.544631	−0.272315	+	0.962208i	$0.587789\pi$
$984$	0	0
$985$	8.33576	0.265600
$986$	70.4081	2.24225
$987$	0	0
$988$	−188.349	−5.99219
$989$	20.7505	0.659827
$990$	0	0
$991$	38.1231	1.21102	0.605510	−	0.795838i	$-0.292969\pi$
$991$	38.1231	1.21102	0.605510	+	0.795838i	$0.292969\pi$
$992$	−30.7439	−0.976120
$993$	0	0
$994$	3.64252	0.115534
$995$	17.4799	0.554150
$996$	0	0
$997$	7.64976	0.242270	0.121135	−	0.992636i	$-0.461347\pi$
$997$	7.64976	0.242270	0.121135	+	0.992636i	$0.461347\pi$
$998$	−66.0821	−2.09179
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.a.f.1.1	yes	3
3.2	odd	2		567.2.a.e.1.3	✓	3
4.3	odd	2		9072.2.a.cb.1.2		3
7.6	odd	2		3969.2.a.n.1.1		3
9.2	odd	6		567.2.f.m.190.1		6
9.4	even	3		567.2.f.l.379.3		6
9.5	odd	6		567.2.f.m.379.1		6
9.7	even	3		567.2.f.l.190.3		6
12.11	even	2		9072.2.a.bu.1.2		3
21.20	even	2		3969.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.3	✓	3	3.2	odd	2
567.2.a.f.1.1	yes	3	1.1	even	1	trivial
567.2.f.l.190.3		6	9.7	even	3
567.2.f.l.379.3		6	9.4	even	3
567.2.f.m.190.1		6	9.2	odd	6
567.2.f.m.379.1		6	9.5	odd	6
3969.2.a.n.1.1		3	7.6	odd	2
3969.2.a.o.1.3		3	21.20	even	2
9072.2.a.bu.1.2		3	12.11	even	2
9072.2.a.cb.1.2		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66908 q^{2} +5.12398 q^{4} +1.45490 q^{5} +1.00000 q^{7} -8.33816 q^{8} +O(q^{10})\)	\(q-2.66908 q^{2} +5.12398 q^{4} +1.45490 q^{5} +1.00000 q^{7} -8.33816 q^{8} -3.88325 q^{10} +1.54510 q^{11} +5.88325 q^{13} -2.66908 q^{14} +12.0072 q^{16} +6.79306 q^{17} -6.24797 q^{19} +7.45490 q^{20} -4.12398 q^{22} -2.90981 q^{23} -2.88325 q^{25} -15.7029 q^{26} +5.12398 q^{28} -3.88325 q^{29} +2.00000 q^{31} -15.3719 q^{32} -18.1312 q^{34} +1.45490 q^{35} +5.00000 q^{37} +16.6763 q^{38} -12.1312 q^{40} +2.24797 q^{41} -7.13122 q^{43} +7.91705 q^{44} +7.76651 q^{46} +5.33816 q^{47} +1.00000 q^{49} +7.69563 q^{50} +30.1457 q^{52} +9.79306 q^{53} +2.24797 q^{55} -8.33816 q^{56} +10.3647 q^{58} +4.67632 q^{59} -2.36471 q^{61} -5.33816 q^{62} +17.0145 q^{64} +8.55957 q^{65} +3.36471 q^{67} +34.8075 q^{68} -3.88325 q^{70} -1.36471 q^{71} -1.88325 q^{73} -13.3454 q^{74} -32.0145 q^{76} +1.54510 q^{77} +3.36471 q^{79} +17.4694 q^{80} -6.00000 q^{82} -2.24797 q^{83} +9.88325 q^{85} +19.0338 q^{86} -12.8833 q^{88} -0.793062 q^{89} +5.88325 q^{91} -14.9098 q^{92} -14.2480 q^{94} -9.09019 q^{95} +10.2480 q^{97} -2.66908 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 + 3 * q^5 + 3 * q^7 - 9 * q^8	\( 3 q + 6 q^{4} + 3 q^{5} + 3 q^{7} - 9 q^{8} + 3 q^{10} + 6 q^{11} + 3 q^{13} + 12 q^{16} + 3 q^{17} + 21 q^{20} - 3 q^{22} - 6 q^{23} + 6 q^{25} - 27 q^{26} + 6 q^{28} + 3 q^{29} + 6 q^{31} - 18 q^{32} - 21 q^{34} + 3 q^{35} + 15 q^{37} + 18 q^{38} - 3 q^{40} - 12 q^{41} + 12 q^{43} - 3 q^{44} - 6 q^{46} + 3 q^{49} + 27 q^{50} + 9 q^{52} + 12 q^{53} - 12 q^{55} - 9 q^{56} + 27 q^{58} - 18 q^{59} - 3 q^{61} + 3 q^{64} - 21 q^{65} + 6 q^{67} + 39 q^{68} + 3 q^{70} + 9 q^{73} - 48 q^{76} + 6 q^{77} + 6 q^{79} + 3 q^{80} - 18 q^{82} + 12 q^{83} + 15 q^{85} + 45 q^{86} - 24 q^{88} + 15 q^{89} + 3 q^{91} - 42 q^{92} - 24 q^{94} - 30 q^{95} + 12 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^5 + 3 * q^7 - 9 * q^8 + 3 * q^10 + 6 * q^11 + 3 * q^13 + 12 * q^16 + 3 * q^17 + 21 * q^20 - 3 * q^22 - 6 * q^23 + 6 * q^25 - 27 * q^26 + 6 * q^28 + 3 * q^29 + 6 * q^31 - 18 * q^32 - 21 * q^34 + 3 * q^35 + 15 * q^37 + 18 * q^38 - 3 * q^40 - 12 * q^41 + 12 * q^43 - 3 * q^44 - 6 * q^46 + 3 * q^49 + 27 * q^50 + 9 * q^52 + 12 * q^53 - 12 * q^55 - 9 * q^56 + 27 * q^58 - 18 * q^59 - 3 * q^61 + 3 * q^64 - 21 * q^65 + 6 * q^67 + 39 * q^68 + 3 * q^70 + 9 * q^73 - 48 * q^76 + 6 * q^77 + 6 * q^79 + 3 * q^80 - 18 * q^82 + 12 * q^83 + 15 * q^85 + 45 * q^86 - 24 * q^88 + 15 * q^89 + 3 * q^91 - 42 * q^92 - 24 * q^94 - 30 * q^95 + 12 * q^97

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