LMFDB - Embedded newform 6534.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, 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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 594)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.23607 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.23607 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.38197 q^{13} -4.23607 q^{14} +1.00000 q^{16} -4.61803 q^{17} -3.61803 q^{19} -1.00000 q^{20} +5.47214 q^{23} -4.00000 q^{25} -2.38197 q^{26} +4.23607 q^{28} -7.09017 q^{29} -1.47214 q^{31} -1.00000 q^{32} +4.61803 q^{34} -4.23607 q^{35} +2.61803 q^{37} +3.61803 q^{38} +1.00000 q^{40} +2.47214 q^{41} +9.09017 q^{43} -5.47214 q^{46} -6.23607 q^{47} +10.9443 q^{49} +4.00000 q^{50} +2.38197 q^{52} -2.85410 q^{53} -4.23607 q^{56} +7.09017 q^{58} -14.8541 q^{59} -10.7984 q^{61} +1.47214 q^{62} +1.00000 q^{64} -2.38197 q^{65} -14.7082 q^{67} -4.61803 q^{68} +4.23607 q^{70} -5.38197 q^{71} -4.23607 q^{73} -2.61803 q^{74} -3.61803 q^{76} +4.47214 q^{79} -1.00000 q^{80} -2.47214 q^{82} -14.6525 q^{83} +4.61803 q^{85} -9.09017 q^{86} -0.291796 q^{89} +10.0902 q^{91} +5.47214 q^{92} +6.23607 q^{94} +3.61803 q^{95} -12.6180 q^{97} -10.9443 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 7 q^{13} - 4 q^{14} + 2 q^{16} - 7 q^{17} - 5 q^{19} - 2 q^{20} + 2 q^{23} - 8 q^{25} - 7 q^{26} + 4 q^{28} - 3 q^{29} + 6 q^{31} - 2 q^{32} + 7 q^{34} - 4 q^{35} + 3 q^{37} + 5 q^{38} + 2 q^{40} - 4 q^{41} + 7 q^{43} - 2 q^{46} - 8 q^{47} + 4 q^{49} + 8 q^{50} + 7 q^{52} + q^{53} - 4 q^{56} + 3 q^{58} - 23 q^{59} + 3 q^{61} - 6 q^{62} + 2 q^{64} - 7 q^{65} - 16 q^{67} - 7 q^{68} + 4 q^{70} - 13 q^{71} - 4 q^{73} - 3 q^{74} - 5 q^{76} - 2 q^{80} + 4 q^{82} + 2 q^{83} + 7 q^{85} - 7 q^{86} - 14 q^{89} + 9 q^{91} + 2 q^{92} + 8 q^{94} + 5 q^{95} - 23 q^{97} - 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 7 * q^13 - 4 * q^14 + 2 * q^16 - 7 * q^17 - 5 * q^19 - 2 * q^20 + 2 * q^23 - 8 * q^25 - 7 * q^26 + 4 * q^28 - 3 * q^29 + 6 * q^31 - 2 * q^32 + 7 * q^34 - 4 * q^35 + 3 * q^37 + 5 * q^38 + 2 * q^40 - 4 * q^41 + 7 * q^43 - 2 * q^46 - 8 * q^47 + 4 * q^49 + 8 * q^50 + 7 * q^52 + q^53 - 4 * q^56 + 3 * q^58 - 23 * q^59 + 3 * q^61 - 6 * q^62 + 2 * q^64 - 7 * q^65 - 16 * q^67 - 7 * q^68 + 4 * q^70 - 13 * q^71 - 4 * q^73 - 3 * q^74 - 5 * q^76 - 2 * q^80 + 4 * q^82 + 2 * q^83 + 7 * q^85 - 7 * q^86 - 14 * q^89 + 9 * q^91 + 2 * q^92 + 8 * q^94 + 5 * q^95 - 23 * q^97 - 4 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.38197	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$13$	2.38197	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$14$	−4.23607	−1.13214
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.61803	−1.12004	−0.560019	−	0.828480i	$-0.689206\pi$
$17$	−4.61803	−1.12004	−0.560019	+	0.828480i	$0.689206\pi$
$18$	0	0
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	5.47214	1.14102	0.570510	−	0.821291i	$-0.306746\pi$
$23$	5.47214	1.14102	0.570510	+	0.821291i	$0.306746\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−2.38197	−0.467142
$27$	0	0
$28$	4.23607	0.800542
$29$	−7.09017	−1.31661	−0.658306	−	0.752751i	$-0.728727\pi$
$29$	−7.09017	−1.31661	−0.658306	+	0.752751i	$0.728727\pi$
$30$	0	0
$31$	−1.47214	−0.264403	−0.132202	−	0.991223i	$-0.542205\pi$
$31$	−1.47214	−0.264403	−0.132202	+	0.991223i	$0.542205\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.61803	0.791986
$35$	−4.23607	−0.716026
$36$	0	0
$37$	2.61803	0.430402	0.215201	−	0.976570i	$-0.430959\pi$
$37$	2.61803	0.430402	0.215201	+	0.976570i	$0.430959\pi$
$38$	3.61803	0.586923
$39$	0	0
$40$	1.00000	0.158114
$41$	2.47214	0.386083	0.193041	−	0.981191i	$-0.438165\pi$
$41$	2.47214	0.386083	0.193041	+	0.981191i	$0.438165\pi$
$42$	0	0
$43$	9.09017	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$43$	9.09017	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$44$	0	0
$45$	0	0
$46$	−5.47214	−0.806822
$47$	−6.23607	−0.909624	−0.454812	−	0.890587i	$-0.650294\pi$
$47$	−6.23607	−0.909624	−0.454812	+	0.890587i	$0.650294\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	4.00000	0.565685
$51$	0	0
$52$	2.38197	0.330319
$53$	−2.85410	−0.392041	−0.196021	−	0.980600i	$-0.562802\pi$
$53$	−2.85410	−0.392041	−0.196021	+	0.980600i	$0.562802\pi$
$54$	0	0
$55$	0	0
$56$	−4.23607	−0.566068
$57$	0	0
$58$	7.09017	0.930985
$59$	−14.8541	−1.93384	−0.966920	−	0.255081i	$-0.917898\pi$
$59$	−14.8541	−1.93384	−0.966920	+	0.255081i	$0.917898\pi$
$60$	0	0
$61$	−10.7984	−1.38259	−0.691295	−	0.722573i	$-0.742960\pi$
$61$	−10.7984	−1.38259	−0.691295	+	0.722573i	$0.742960\pi$
$62$	1.47214	0.186961
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.38197	−0.295447
$66$	0	0
$67$	−14.7082	−1.79689	−0.898447	−	0.439083i	$-0.855303\pi$
$67$	−14.7082	−1.79689	−0.898447	+	0.439083i	$0.855303\pi$
$68$	−4.61803	−0.560019
$69$	0	0
$70$	4.23607	0.506307
$71$	−5.38197	−0.638722	−0.319361	−	0.947633i	$-0.603468\pi$
$71$	−5.38197	−0.638722	−0.319361	+	0.947633i	$0.603468\pi$
$72$	0	0
$73$	−4.23607	−0.495794	−0.247897	−	0.968786i	$-0.579739\pi$
$73$	−4.23607	−0.495794	−0.247897	+	0.968786i	$0.579739\pi$
$74$	−2.61803	−0.304340
$75$	0	0
$76$	−3.61803	−0.415017
$77$	0	0
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.47214	−0.273002
$83$	−14.6525	−1.60832	−0.804159	−	0.594414i	$-0.797384\pi$
$83$	−14.6525	−1.60832	−0.804159	+	0.594414i	$0.797384\pi$
$84$	0	0
$85$	4.61803	0.500896
$86$	−9.09017	−0.980218
$87$	0	0
$88$	0	0
$89$	−0.291796	−0.0309303	−0.0154652	−	0.999880i	$-0.504923\pi$
$89$	−0.291796	−0.0309303	−0.0154652	+	0.999880i	$0.504923\pi$
$90$	0	0
$91$	10.0902	1.05774
$92$	5.47214	0.570510
$93$	0	0
$94$	6.23607	0.643201
$95$	3.61803	0.371202
$96$	0	0
$97$	−12.6180	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$97$	−12.6180	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$98$	−10.9443	−1.10554
$99$	0	0
$100$	−4.00000	−0.400000
$101$	19.4721	1.93755	0.968775	−	0.247942i	$-0.0797541\pi$
$101$	19.4721	1.93755	0.968775	+	0.247942i	$0.0797541\pi$
$102$	0	0
$103$	−9.18034	−0.904566	−0.452283	−	0.891875i	$-0.649390\pi$
$103$	−9.18034	−0.904566	−0.452283	+	0.891875i	$0.649390\pi$
$104$	−2.38197	−0.233571
$105$	0	0
$106$	2.85410	0.277215
$107$	3.23607	0.312842	0.156421	−	0.987690i	$-0.450004\pi$
$107$	3.23607	0.312842	0.156421	+	0.987690i	$0.450004\pi$
$108$	0	0
$109$	6.56231	0.628555	0.314277	−	0.949331i	$-0.398238\pi$
$109$	6.56231	0.628555	0.314277	+	0.949331i	$0.398238\pi$
$110$	0	0
$111$	0	0
$112$	4.23607	0.400271
$113$	−7.76393	−0.730369	−0.365185	−	0.930935i	$-0.618994\pi$
$113$	−7.76393	−0.730369	−0.365185	+	0.930935i	$0.618994\pi$
$114$	0	0
$115$	−5.47214	−0.510279
$116$	−7.09017	−0.658306
$117$	0	0
$118$	14.8541	1.36743
$119$	−19.5623	−1.79327
$120$	0	0
$121$	0	0
$122$	10.7984	0.977639
$123$	0	0
$124$	−1.47214	−0.132202
$125$	9.00000	0.804984
$126$	0	0
$127$	−11.5623	−1.02599	−0.512994	−	0.858392i	$-0.671464\pi$
$127$	−11.5623	−1.02599	−0.512994	+	0.858392i	$0.671464\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.38197	0.208912
$131$	9.70820	0.848210	0.424105	−	0.905613i	$-0.360589\pi$
$131$	9.70820	0.848210	0.424105	+	0.905613i	$0.360589\pi$
$132$	0	0
$133$	−15.3262	−1.32895
$134$	14.7082	1.27060
$135$	0	0
$136$	4.61803	0.395993
$137$	−7.14590	−0.610515	−0.305258	−	0.952270i	$-0.598743\pi$
$137$	−7.14590	−0.610515	−0.305258	+	0.952270i	$0.598743\pi$
$138$	0	0
$139$	3.47214	0.294503	0.147251	−	0.989099i	$-0.452957\pi$
$139$	3.47214	0.294503	0.147251	+	0.989099i	$0.452957\pi$
$140$	−4.23607	−0.358013
$141$	0	0
$142$	5.38197	0.451645
$143$	0	0
$144$	0	0
$145$	7.09017	0.588807
$146$	4.23607	0.350579
$147$	0	0
$148$	2.61803	0.215201
$149$	19.2361	1.57588	0.787940	−	0.615752i	$-0.211148\pi$
$149$	19.2361	1.57588	0.787940	+	0.615752i	$0.211148\pi$
$150$	0	0
$151$	7.52786	0.612609	0.306304	−	0.951934i	$-0.400907\pi$
$151$	7.52786	0.612609	0.306304	+	0.951934i	$0.400907\pi$
$152$	3.61803	0.293461
$153$	0	0
$154$	0	0
$155$	1.47214	0.118245
$156$	0	0
$157$	8.23607	0.657310	0.328655	−	0.944450i	$-0.393405\pi$
$157$	8.23607	0.657310	0.328655	+	0.944450i	$0.393405\pi$
$158$	−4.47214	−0.355784
$159$	0	0
$160$	1.00000	0.0790569
$161$	23.1803	1.82687
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	2.47214	0.193041
$165$	0	0
$166$	14.6525	1.13725
$167$	12.9443	1.00166	0.500829	−	0.865546i	$-0.333029\pi$
$167$	12.9443	1.00166	0.500829	+	0.865546i	$0.333029\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	−4.61803	−0.354187
$171$	0	0
$172$	9.09017	0.693119
$173$	2.56231	0.194809	0.0974043	−	0.995245i	$-0.468946\pi$
$173$	2.56231	0.194809	0.0974043	+	0.995245i	$0.468946\pi$
$174$	0	0
$175$	−16.9443	−1.28087
$176$	0	0
$177$	0	0
$178$	0.291796	0.0218710
$179$	−25.3607	−1.89555	−0.947773	−	0.318945i	$-0.896671\pi$
$179$	−25.3607	−1.89555	−0.947773	+	0.318945i	$0.896671\pi$
$180$	0	0
$181$	8.61803	0.640573	0.320287	−	0.947321i	$-0.396221\pi$
$181$	8.61803	0.640573	0.320287	+	0.947321i	$0.396221\pi$
$182$	−10.0902	−0.747933
$183$	0	0
$184$	−5.47214	−0.403411
$185$	−2.61803	−0.192482
$186$	0	0
$187$	0	0
$188$	−6.23607	−0.454812
$189$	0	0
$190$	−3.61803	−0.262480
$191$	19.7426	1.42853	0.714264	−	0.699877i	$-0.246762\pi$
$191$	19.7426	1.42853	0.714264	+	0.699877i	$0.246762\pi$
$192$	0	0
$193$	−3.76393	−0.270934	−0.135467	−	0.990782i	$-0.543253\pi$
$193$	−3.76393	−0.270934	−0.135467	+	0.990782i	$0.543253\pi$
$194$	12.6180	0.905922
$195$	0	0
$196$	10.9443	0.781734
$197$	−19.1803	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$197$	−19.1803	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−19.4721	−1.37005
$203$	−30.0344	−2.10800
$204$	0	0
$205$	−2.47214	−0.172661
$206$	9.18034	0.639625
$207$	0	0
$208$	2.38197	0.165160
$209$	0	0
$210$	0	0
$211$	−13.7639	−0.947548	−0.473774	−	0.880646i	$-0.657109\pi$
$211$	−13.7639	−0.947548	−0.473774	+	0.880646i	$0.657109\pi$
$212$	−2.85410	−0.196021
$213$	0	0
$214$	−3.23607	−0.221213
$215$	−9.09017	−0.619944
$216$	0	0
$217$	−6.23607	−0.423332
$218$	−6.56231	−0.444455
$219$	0	0
$220$	0	0
$221$	−11.0000	−0.739940
$222$	0	0
$223$	13.7426	0.920276	0.460138	−	0.887848i	$-0.347800\pi$
$223$	13.7426	0.920276	0.460138	+	0.887848i	$0.347800\pi$
$224$	−4.23607	−0.283034
$225$	0	0
$226$	7.76393	0.516449
$227$	−0.437694	−0.0290508	−0.0145254	−	0.999895i	$-0.504624\pi$
$227$	−0.437694	−0.0290508	−0.0145254	+	0.999895i	$0.504624\pi$
$228$	0	0
$229$	−15.9443	−1.05363	−0.526814	−	0.849981i	$-0.676613\pi$
$229$	−15.9443	−1.05363	−0.526814	+	0.849981i	$0.676613\pi$
$230$	5.47214	0.360822
$231$	0	0
$232$	7.09017	0.465492
$233$	21.7639	1.42580	0.712901	−	0.701264i	$-0.247381\pi$
$233$	21.7639	1.42580	0.712901	+	0.701264i	$0.247381\pi$
$234$	0	0
$235$	6.23607	0.406796
$236$	−14.8541	−0.966920
$237$	0	0
$238$	19.5623	1.26804
$239$	−12.7639	−0.825630	−0.412815	−	0.910815i	$-0.635454\pi$
$239$	−12.7639	−0.825630	−0.412815	+	0.910815i	$0.635454\pi$
$240$	0	0
$241$	20.4164	1.31514	0.657568	−	0.753395i	$-0.271585\pi$
$241$	20.4164	1.31514	0.657568	+	0.753395i	$0.271585\pi$
$242$	0	0
$243$	0	0
$244$	−10.7984	−0.691295
$245$	−10.9443	−0.699204
$246$	0	0
$247$	−8.61803	−0.548352
$248$	1.47214	0.0934807
$249$	0	0
$250$	−9.00000	−0.569210
$251$	4.65248	0.293662	0.146831	−	0.989162i	$-0.453093\pi$
$251$	4.65248	0.293662	0.146831	+	0.989162i	$0.453093\pi$
$252$	0	0
$253$	0	0
$254$	11.5623	0.725484
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.965558	−0.0602299	−0.0301149	−	0.999546i	$-0.509587\pi$
$257$	−0.965558	−0.0602299	−0.0301149	+	0.999546i	$0.509587\pi$
$258$	0	0
$259$	11.0902	0.689110
$260$	−2.38197	−0.147723
$261$	0	0
$262$	−9.70820	−0.599775
$263$	−17.4164	−1.07394	−0.536971	−	0.843601i	$-0.680431\pi$
$263$	−17.4164	−1.07394	−0.536971	+	0.843601i	$0.680431\pi$
$264$	0	0
$265$	2.85410	0.175326
$266$	15.3262	0.939712
$267$	0	0
$268$	−14.7082	−0.898447
$269$	5.18034	0.315851	0.157925	−	0.987451i	$-0.449519\pi$
$269$	5.18034	0.315851	0.157925	+	0.987451i	$0.449519\pi$
$270$	0	0
$271$	28.7082	1.74390	0.871950	−	0.489596i	$-0.162856\pi$
$271$	28.7082	1.74390	0.871950	+	0.489596i	$0.162856\pi$
$272$	−4.61803	−0.280009
$273$	0	0
$274$	7.14590	0.431699
$275$	0	0
$276$	0	0
$277$	2.29180	0.137701	0.0688503	−	0.997627i	$-0.478067\pi$
$277$	2.29180	0.137701	0.0688503	+	0.997627i	$0.478067\pi$
$278$	−3.47214	−0.208245
$279$	0	0
$280$	4.23607	0.253153
$281$	−8.56231	−0.510784	−0.255392	−	0.966838i	$-0.582205\pi$
$281$	−8.56231	−0.510784	−0.255392	+	0.966838i	$0.582205\pi$
$282$	0	0
$283$	−6.38197	−0.379369	−0.189684	−	0.981845i	$-0.560746\pi$
$283$	−6.38197	−0.379369	−0.189684	+	0.981845i	$0.560746\pi$
$284$	−5.38197	−0.319361
$285$	0	0
$286$	0	0
$287$	10.4721	0.618151
$288$	0	0
$289$	4.32624	0.254485
$290$	−7.09017	−0.416349
$291$	0	0
$292$	−4.23607	−0.247897
$293$	−5.14590	−0.300627	−0.150313	−	0.988638i	$-0.548028\pi$
$293$	−5.14590	−0.300627	−0.150313	+	0.988638i	$0.548028\pi$
$294$	0	0
$295$	14.8541	0.864839
$296$	−2.61803	−0.152170
$297$	0	0
$298$	−19.2361	−1.11432
$299$	13.0344	0.753801
$300$	0	0
$301$	38.5066	2.21948
$302$	−7.52786	−0.433180
$303$	0	0
$304$	−3.61803	−0.207508
$305$	10.7984	0.618313
$306$	0	0
$307$	−24.8885	−1.42046	−0.710232	−	0.703968i	$-0.751410\pi$
$307$	−24.8885	−1.42046	−0.710232	+	0.703968i	$0.751410\pi$
$308$	0	0
$309$	0	0
$310$	−1.47214	−0.0836117
$311$	−17.6180	−0.999027	−0.499514	−	0.866306i	$-0.666488\pi$
$311$	−17.6180	−0.999027	−0.499514	+	0.866306i	$0.666488\pi$
$312$	0	0
$313$	25.8328	1.46016	0.730079	−	0.683363i	$-0.239483\pi$
$313$	25.8328	1.46016	0.730079	+	0.683363i	$0.239483\pi$
$314$	−8.23607	−0.464788
$315$	0	0
$316$	4.47214	0.251577
$317$	−9.88854	−0.555396	−0.277698	−	0.960668i	$-0.589571\pi$
$317$	−9.88854	−0.555396	−0.277698	+	0.960668i	$0.589571\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−23.1803	−1.29179
$323$	16.7082	0.929669
$324$	0	0
$325$	−9.52786	−0.528511
$326$	−6.00000	−0.332309
$327$	0	0
$328$	−2.47214	−0.136501
$329$	−26.4164	−1.45638
$330$	0	0
$331$	−29.0689	−1.59777	−0.798885	−	0.601484i	$-0.794577\pi$
$331$	−29.0689	−1.59777	−0.798885	+	0.601484i	$0.794577\pi$
$332$	−14.6525	−0.804159
$333$	0	0
$334$	−12.9443	−0.708279
$335$	14.7082	0.803595
$336$	0	0
$337$	−13.6738	−0.744857	−0.372429	−	0.928061i	$-0.621475\pi$
$337$	−13.6738	−0.744857	−0.372429	+	0.928061i	$0.621475\pi$
$338$	7.32624	0.398495
$339$	0	0
$340$	4.61803	0.250448
$341$	0	0
$342$	0	0
$343$	16.7082	0.902158
$344$	−9.09017	−0.490109
$345$	0	0
$346$	−2.56231	−0.137750
$347$	−24.1246	−1.29508	−0.647539	−	0.762033i	$-0.724202\pi$
$347$	−24.1246	−1.29508	−0.647539	+	0.762033i	$0.724202\pi$
$348$	0	0
$349$	7.65248	0.409628	0.204814	−	0.978801i	$-0.434341\pi$
$349$	7.65248	0.409628	0.204814	+	0.978801i	$0.434341\pi$
$350$	16.9443	0.905709
$351$	0	0
$352$	0	0
$353$	2.94427	0.156708	0.0783539	−	0.996926i	$-0.475034\pi$
$353$	2.94427	0.156708	0.0783539	+	0.996926i	$0.475034\pi$
$354$	0	0
$355$	5.38197	0.285645
$356$	−0.291796	−0.0154652
$357$	0	0
$358$	25.3607	1.34035
$359$	34.3050	1.81055	0.905273	−	0.424830i	$-0.139666\pi$
$359$	34.3050	1.81055	0.905273	+	0.424830i	$0.139666\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−8.61803	−0.452954
$363$	0	0
$364$	10.0902	0.528869
$365$	4.23607	0.221726
$366$	0	0
$367$	36.9443	1.92848	0.964238	−	0.265039i	$-0.0853849\pi$
$367$	36.9443	1.92848	0.964238	+	0.265039i	$0.0853849\pi$
$368$	5.47214	0.285255
$369$	0	0
$370$	2.61803	0.136105
$371$	−12.0902	−0.627690
$372$	0	0
$373$	−21.4721	−1.11179	−0.555893	−	0.831254i	$-0.687623\pi$
$373$	−21.4721	−1.11179	−0.555893	+	0.831254i	$0.687623\pi$
$374$	0	0
$375$	0	0
$376$	6.23607	0.321601
$377$	−16.8885	−0.869804
$378$	0	0
$379$	−33.7082	−1.73147	−0.865737	−	0.500499i	$-0.833150\pi$
$379$	−33.7082	−1.73147	−0.865737	+	0.500499i	$0.833150\pi$
$380$	3.61803	0.185601
$381$	0	0
$382$	−19.7426	−1.01012
$383$	−32.5967	−1.66562	−0.832808	−	0.553562i	$-0.813268\pi$
$383$	−32.5967	−1.66562	−0.832808	+	0.553562i	$0.813268\pi$
$384$	0	0
$385$	0	0
$386$	3.76393	0.191579
$387$	0	0
$388$	−12.6180	−0.640584
$389$	−28.7426	−1.45731	−0.728655	−	0.684881i	$-0.759854\pi$
$389$	−28.7426	−1.45731	−0.728655	+	0.684881i	$0.759854\pi$
$390$	0	0
$391$	−25.2705	−1.27798
$392$	−10.9443	−0.552769
$393$	0	0
$394$	19.1803	0.966292
$395$	−4.47214	−0.225018
$396$	0	0
$397$	16.3820	0.822187	0.411094	−	0.911593i	$-0.365147\pi$
$397$	16.3820	0.822187	0.411094	+	0.911593i	$0.365147\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	−3.50658	−0.174675
$404$	19.4721	0.968775
$405$	0	0
$406$	30.0344	1.49058
$407$	0	0
$408$	0	0
$409$	−32.7639	−1.62007	−0.810036	−	0.586380i	$-0.800553\pi$
$409$	−32.7639	−1.62007	−0.810036	+	0.586380i	$0.800553\pi$
$410$	2.47214	0.122090
$411$	0	0
$412$	−9.18034	−0.452283
$413$	−62.9230	−3.09624
$414$	0	0
$415$	14.6525	0.719262
$416$	−2.38197	−0.116785
$417$	0	0
$418$	0	0
$419$	16.8328	0.822337	0.411168	−	0.911559i	$-0.365121\pi$
$419$	16.8328	0.822337	0.411168	+	0.911559i	$0.365121\pi$
$420$	0	0
$421$	−16.8541	−0.821419	−0.410709	−	0.911766i	$-0.634719\pi$
$421$	−16.8541	−0.821419	−0.410709	+	0.911766i	$0.634719\pi$
$422$	13.7639	0.670018
$423$	0	0
$424$	2.85410	0.138607
$425$	18.4721	0.896030
$426$	0	0
$427$	−45.7426	−2.21364
$428$	3.23607	0.156421
$429$	0	0
$430$	9.09017	0.438367
$431$	−13.2148	−0.636534	−0.318267	−	0.948001i	$-0.603101\pi$
$431$	−13.2148	−0.636534	−0.318267	+	0.948001i	$0.603101\pi$
$432$	0	0
$433$	6.38197	0.306698	0.153349	−	0.988172i	$-0.450994\pi$
$433$	6.38197	0.306698	0.153349	+	0.988172i	$0.450994\pi$
$434$	6.23607	0.299341
$435$	0	0
$436$	6.56231	0.314277
$437$	−19.7984	−0.947085
$438$	0	0
$439$	−21.3262	−1.01785	−0.508923	−	0.860812i	$-0.669956\pi$
$439$	−21.3262	−1.01785	−0.508923	+	0.860812i	$0.669956\pi$
$440$	0	0
$441$	0	0
$442$	11.0000	0.523217
$443$	−2.79837	−0.132955	−0.0664774	−	0.997788i	$-0.521176\pi$
$443$	−2.79837	−0.132955	−0.0664774	+	0.997788i	$0.521176\pi$
$444$	0	0
$445$	0.291796	0.0138325
$446$	−13.7426	−0.650733
$447$	0	0
$448$	4.23607	0.200135
$449$	40.0344	1.88934	0.944671	−	0.328019i	$-0.106381\pi$
$449$	40.0344	1.88934	0.944671	+	0.328019i	$0.106381\pi$
$450$	0	0
$451$	0	0
$452$	−7.76393	−0.365185
$453$	0	0
$454$	0.437694	0.0205420
$455$	−10.0902	−0.473034
$456$	0	0
$457$	−24.9787	−1.16845	−0.584227	−	0.811590i	$-0.698602\pi$
$457$	−24.9787	−1.16845	−0.584227	+	0.811590i	$0.698602\pi$
$458$	15.9443	0.745027
$459$	0	0
$460$	−5.47214	−0.255140
$461$	28.9787	1.34967	0.674837	−	0.737967i	$-0.264214\pi$
$461$	28.9787	1.34967	0.674837	+	0.737967i	$0.264214\pi$
$462$	0	0
$463$	−13.0902	−0.608352	−0.304176	−	0.952616i	$-0.598381\pi$
$463$	−13.0902	−0.608352	−0.304176	+	0.952616i	$0.598381\pi$
$464$	−7.09017	−0.329153
$465$	0	0
$466$	−21.7639	−1.00819
$467$	12.5623	0.581314	0.290657	−	0.956827i	$-0.406126\pi$
$467$	12.5623	0.581314	0.290657	+	0.956827i	$0.406126\pi$
$468$	0	0
$469$	−62.3050	−2.87698
$470$	−6.23607	−0.287648
$471$	0	0
$472$	14.8541	0.683715
$473$	0	0
$474$	0	0
$475$	14.4721	0.664027
$476$	−19.5623	−0.896637
$477$	0	0
$478$	12.7639	0.583809
$479$	30.0344	1.37231	0.686154	−	0.727456i	$-0.259297\pi$
$479$	30.0344	1.37231	0.686154	+	0.727456i	$0.259297\pi$
$480$	0	0
$481$	6.23607	0.284340
$482$	−20.4164	−0.929942
$483$	0	0
$484$	0	0
$485$	12.6180	0.572955
$486$	0	0
$487$	−20.8328	−0.944025	−0.472012	−	0.881592i	$-0.656472\pi$
$487$	−20.8328	−0.944025	−0.472012	+	0.881592i	$0.656472\pi$
$488$	10.7984	0.488819
$489$	0	0
$490$	10.9443	0.494412
$491$	−39.2492	−1.77129	−0.885646	−	0.464360i	$-0.846284\pi$
$491$	−39.2492	−1.77129	−0.885646	+	0.464360i	$0.846284\pi$
$492$	0	0
$493$	32.7426	1.47465
$494$	8.61803	0.387744
$495$	0	0
$496$	−1.47214	−0.0661009
$497$	−22.7984	−1.02265
$498$	0	0
$499$	−23.2918	−1.04268	−0.521342	−	0.853348i	$-0.674568\pi$
$499$	−23.2918	−1.04268	−0.521342	+	0.853348i	$0.674568\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	−4.65248	−0.207650
$503$	35.0689	1.56364	0.781822	−	0.623502i	$-0.214290\pi$
$503$	35.0689	1.56364	0.781822	+	0.623502i	$0.214290\pi$
$504$	0	0
$505$	−19.4721	−0.866499
$506$	0	0
$507$	0	0
$508$	−11.5623	−0.512994
$509$	−10.6738	−0.473106	−0.236553	−	0.971619i	$-0.576018\pi$
$509$	−10.6738	−0.473106	−0.236553	+	0.971619i	$0.576018\pi$
$510$	0	0
$511$	−17.9443	−0.793808
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0.965558	0.0425889
$515$	9.18034	0.404534
$516$	0	0
$517$	0	0
$518$	−11.0902	−0.487274
$519$	0	0
$520$	2.38197	0.104456
$521$	−23.8328	−1.04413	−0.522067	−	0.852904i	$-0.674839\pi$
$521$	−23.8328	−1.04413	−0.522067	+	0.852904i	$0.674839\pi$
$522$	0	0
$523$	25.6525	1.12170	0.560852	−	0.827916i	$-0.310474\pi$
$523$	25.6525	1.12170	0.560852	+	0.827916i	$0.310474\pi$
$524$	9.70820	0.424105
$525$	0	0
$526$	17.4164	0.759391
$527$	6.79837	0.296142
$528$	0	0
$529$	6.94427	0.301925
$530$	−2.85410	−0.123974
$531$	0	0
$532$	−15.3262	−0.664477
$533$	5.88854	0.255061
$534$	0	0
$535$	−3.23607	−0.139907
$536$	14.7082	0.635298
$537$	0	0
$538$	−5.18034	−0.223340
$539$	0	0
$540$	0	0
$541$	−9.81966	−0.422180	−0.211090	−	0.977467i	$-0.567701\pi$
$541$	−9.81966	−0.422180	−0.211090	+	0.977467i	$0.567701\pi$
$542$	−28.7082	−1.23312
$543$	0	0
$544$	4.61803	0.197997
$545$	−6.56231	−0.281098
$546$	0	0
$547$	32.7984	1.40236	0.701179	−	0.712986i	$-0.252658\pi$
$547$	32.7984	1.40236	0.701179	+	0.712986i	$0.252658\pi$
$548$	−7.14590	−0.305258
$549$	0	0
$550$	0	0
$551$	25.6525	1.09283
$552$	0	0
$553$	18.9443	0.805592
$554$	−2.29180	−0.0973691
$555$	0	0
$556$	3.47214	0.147251
$557$	−1.49342	−0.0632783	−0.0316392	−	0.999499i	$-0.510073\pi$
$557$	−1.49342	−0.0632783	−0.0316392	+	0.999499i	$0.510073\pi$
$558$	0	0
$559$	21.6525	0.915802
$560$	−4.23607	−0.179007
$561$	0	0
$562$	8.56231	0.361179
$563$	12.4164	0.523289	0.261645	−	0.965164i	$-0.415735\pi$
$563$	12.4164	0.523289	0.261645	+	0.965164i	$0.415735\pi$
$564$	0	0
$565$	7.76393	0.326631
$566$	6.38197	0.268254
$567$	0	0
$568$	5.38197	0.225822
$569$	32.7082	1.37120	0.685600	−	0.727979i	$-0.259540\pi$
$569$	32.7082	1.37120	0.685600	+	0.727979i	$0.259540\pi$
$570$	0	0
$571$	6.65248	0.278397	0.139199	−	0.990264i	$-0.455547\pi$
$571$	6.65248	0.278397	0.139199	+	0.990264i	$0.455547\pi$
$572$	0	0
$573$	0	0
$574$	−10.4721	−0.437099
$575$	−21.8885	−0.912815
$576$	0	0
$577$	11.1803	0.465444	0.232722	−	0.972543i	$-0.425237\pi$
$577$	11.1803	0.465444	0.232722	+	0.972543i	$0.425237\pi$
$578$	−4.32624	−0.179948
$579$	0	0
$580$	7.09017	0.294403
$581$	−62.0689	−2.57505
$582$	0	0
$583$	0	0
$584$	4.23607	0.175290
$585$	0	0
$586$	5.14590	0.212575
$587$	−26.0902	−1.07686	−0.538428	−	0.842671i	$-0.680982\pi$
$587$	−26.0902	−1.07686	−0.538428	+	0.842671i	$0.680982\pi$
$588$	0	0
$589$	5.32624	0.219464
$590$	−14.8541	−0.611534
$591$	0	0
$592$	2.61803	0.107601
$593$	−26.2148	−1.07651	−0.538256	−	0.842781i	$-0.680917\pi$
$593$	−26.2148	−1.07651	−0.538256	+	0.842781i	$0.680917\pi$
$594$	0	0
$595$	19.5623	0.801976
$596$	19.2361	0.787940
$597$	0	0
$598$	−13.0344	−0.533018
$599$	−21.6180	−0.883289	−0.441644	−	0.897190i	$-0.645605\pi$
$599$	−21.6180	−0.883289	−0.441644	+	0.897190i	$0.645605\pi$
$600$	0	0
$601$	−22.3820	−0.912979	−0.456490	−	0.889729i	$-0.650893\pi$
$601$	−22.3820	−0.912979	−0.456490	+	0.889729i	$0.650893\pi$
$602$	−38.5066	−1.56941
$603$	0	0
$604$	7.52786	0.306304
$605$	0	0
$606$	0	0
$607$	28.3050	1.14886	0.574431	−	0.818553i	$-0.305223\pi$
$607$	28.3050	1.14886	0.574431	+	0.818553i	$0.305223\pi$
$608$	3.61803	0.146731
$609$	0	0
$610$	−10.7984	−0.437213
$611$	−14.8541	−0.600933
$612$	0	0
$613$	−29.2918	−1.18308	−0.591542	−	0.806274i	$-0.701481\pi$
$613$	−29.2918	−1.18308	−0.591542	+	0.806274i	$0.701481\pi$
$614$	24.8885	1.00442
$615$	0	0
$616$	0	0
$617$	−27.8541	−1.12136	−0.560682	−	0.828031i	$-0.689461\pi$
$617$	−27.8541	−1.12136	−0.560682	+	0.828031i	$0.689461\pi$
$618$	0	0
$619$	25.4721	1.02381	0.511906	−	0.859042i	$-0.328940\pi$
$619$	25.4721	1.02381	0.511906	+	0.859042i	$0.328940\pi$
$620$	1.47214	0.0591224
$621$	0	0
$622$	17.6180	0.706419
$623$	−1.23607	−0.0495220
$624$	0	0
$625$	11.0000	0.440000
$626$	−25.8328	−1.03249
$627$	0	0
$628$	8.23607	0.328655
$629$	−12.0902	−0.482067
$630$	0	0
$631$	12.6869	0.505058	0.252529	−	0.967589i	$-0.418738\pi$
$631$	12.6869	0.505058	0.252529	+	0.967589i	$0.418738\pi$
$632$	−4.47214	−0.177892
$633$	0	0
$634$	9.88854	0.392724
$635$	11.5623	0.458836
$636$	0	0
$637$	26.0689	1.03289
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	3.27051	0.129177	0.0645887	−	0.997912i	$-0.479426\pi$
$641$	3.27051	0.129177	0.0645887	+	0.997912i	$0.479426\pi$
$642$	0	0
$643$	−11.9656	−0.471876	−0.235938	−	0.971768i	$-0.575816\pi$
$643$	−11.9656	−0.471876	−0.235938	+	0.971768i	$0.575816\pi$
$644$	23.1803	0.913433
$645$	0	0
$646$	−16.7082	−0.657375
$647$	26.0000	1.02217	0.511083	−	0.859532i	$-0.329245\pi$
$647$	26.0000	1.02217	0.511083	+	0.859532i	$0.329245\pi$
$648$	0	0
$649$	0	0
$650$	9.52786	0.373714
$651$	0	0
$652$	6.00000	0.234978
$653$	19.8328	0.776118	0.388059	−	0.921635i	$-0.373146\pi$
$653$	19.8328	0.776118	0.388059	+	0.921635i	$0.373146\pi$
$654$	0	0
$655$	−9.70820	−0.379331
$656$	2.47214	0.0965207
$657$	0	0
$658$	26.4164	1.02982
$659$	−22.4164	−0.873219	−0.436610	−	0.899651i	$-0.643821\pi$
$659$	−22.4164	−0.873219	−0.436610	+	0.899651i	$0.643821\pi$
$660$	0	0
$661$	28.8885	1.12363	0.561817	−	0.827261i	$-0.310102\pi$
$661$	28.8885	1.12363	0.561817	+	0.827261i	$0.310102\pi$
$662$	29.0689	1.12979
$663$	0	0
$664$	14.6525	0.568626
$665$	15.3262	0.594326
$666$	0	0
$667$	−38.7984	−1.50228
$668$	12.9443	0.500829
$669$	0	0
$670$	−14.7082	−0.568227
$671$	0	0
$672$	0	0
$673$	40.6525	1.56704	0.783519	−	0.621368i	$-0.213423\pi$
$673$	40.6525	1.56704	0.783519	+	0.621368i	$0.213423\pi$
$674$	13.6738	0.526694
$675$	0	0
$676$	−7.32624	−0.281778
$677$	11.3262	0.435303	0.217651	−	0.976027i	$-0.430160\pi$
$677$	11.3262	0.435303	0.217651	+	0.976027i	$0.430160\pi$
$678$	0	0
$679$	−53.4508	−2.05126
$680$	−4.61803	−0.177094
$681$	0	0
$682$	0	0
$683$	−14.6525	−0.560661	−0.280331	−	0.959903i	$-0.590444\pi$
$683$	−14.6525	−0.560661	−0.280331	+	0.959903i	$0.590444\pi$
$684$	0	0
$685$	7.14590	0.273031
$686$	−16.7082	−0.637922
$687$	0	0
$688$	9.09017	0.346559
$689$	−6.79837	−0.258997
$690$	0	0
$691$	−49.9787	−1.90128	−0.950640	−	0.310296i	$-0.899572\pi$
$691$	−49.9787	−1.90128	−0.950640	+	0.310296i	$0.899572\pi$
$692$	2.56231	0.0974043
$693$	0	0
$694$	24.1246	0.915758
$695$	−3.47214	−0.131706
$696$	0	0
$697$	−11.4164	−0.432427
$698$	−7.65248	−0.289650
$699$	0	0
$700$	−16.9443	−0.640433
$701$	−0.944272	−0.0356647	−0.0178323	−	0.999841i	$-0.505677\pi$
$701$	−0.944272	−0.0356647	−0.0178323	+	0.999841i	$0.505677\pi$
$702$	0	0
$703$	−9.47214	−0.357248
$704$	0	0
$705$	0	0
$706$	−2.94427	−0.110809
$707$	82.4853	3.10218
$708$	0	0
$709$	12.8197	0.481452	0.240726	−	0.970593i	$-0.422614\pi$
$709$	12.8197	0.481452	0.240726	+	0.970593i	$0.422614\pi$
$710$	−5.38197	−0.201982
$711$	0	0
$712$	0.291796	0.0109355
$713$	−8.05573	−0.301689
$714$	0	0
$715$	0	0
$716$	−25.3607	−0.947773
$717$	0	0
$718$	−34.3050	−1.28025
$719$	20.0902	0.749237	0.374618	−	0.927179i	$-0.377774\pi$
$719$	20.0902	0.749237	0.374618	+	0.927179i	$0.377774\pi$
$720$	0	0
$721$	−38.8885	−1.44829
$722$	5.90983	0.219941
$723$	0	0
$724$	8.61803	0.320287
$725$	28.3607	1.05329
$726$	0	0
$727$	4.72949	0.175407	0.0877035	−	0.996147i	$-0.472047\pi$
$727$	4.72949	0.175407	0.0877035	+	0.996147i	$0.472047\pi$
$728$	−10.0902	−0.373967
$729$	0	0
$730$	−4.23607	−0.156784
$731$	−41.9787	−1.55264
$732$	0	0
$733$	−13.2705	−0.490157	−0.245079	−	0.969503i	$-0.578814\pi$
$733$	−13.2705	−0.490157	−0.245079	+	0.969503i	$0.578814\pi$
$734$	−36.9443	−1.36364
$735$	0	0
$736$	−5.47214	−0.201706
$737$	0	0
$738$	0	0
$739$	9.65248	0.355072	0.177536	−	0.984114i	$-0.443187\pi$
$739$	9.65248	0.355072	0.177536	+	0.984114i	$0.443187\pi$
$740$	−2.61803	−0.0962408
$741$	0	0
$742$	12.0902	0.443844
$743$	8.59675	0.315384	0.157692	−	0.987488i	$-0.449595\pi$
$743$	8.59675	0.315384	0.157692	+	0.987488i	$0.449595\pi$
$744$	0	0
$745$	−19.2361	−0.704755
$746$	21.4721	0.786151
$747$	0	0
$748$	0	0
$749$	13.7082	0.500887
$750$	0	0
$751$	0.0901699	0.00329035	0.00164517	−	0.999999i	$-0.499476\pi$
$751$	0.0901699	0.00329035	0.00164517	+	0.999999i	$0.499476\pi$
$752$	−6.23607	−0.227406
$753$	0	0
$754$	16.8885	0.615044
$755$	−7.52786	−0.273967
$756$	0	0
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	33.7082	1.22434
$759$	0	0
$760$	−3.61803	−0.131240
$761$	−5.87539	−0.212983	−0.106491	−	0.994314i	$-0.533962\pi$
$761$	−5.87539	−0.212983	−0.106491	+	0.994314i	$0.533962\pi$
$762$	0	0
$763$	27.7984	1.00637
$764$	19.7426	0.714264
$765$	0	0
$766$	32.5967	1.17777
$767$	−35.3820	−1.27757
$768$	0	0
$769$	−51.3951	−1.85336	−0.926678	−	0.375857i	$-0.877348\pi$
$769$	−51.3951	−1.85336	−0.926678	+	0.375857i	$0.877348\pi$
$770$	0	0
$771$	0	0
$772$	−3.76393	−0.135467
$773$	18.9656	0.682144	0.341072	−	0.940037i	$-0.389210\pi$
$773$	18.9656	0.682144	0.341072	+	0.940037i	$0.389210\pi$
$774$	0	0
$775$	5.88854	0.211523
$776$	12.6180	0.452961
$777$	0	0
$778$	28.7426	1.03047
$779$	−8.94427	−0.320462
$780$	0	0
$781$	0	0
$782$	25.2705	0.903672
$783$	0	0
$784$	10.9443	0.390867
$785$	−8.23607	−0.293958
$786$	0	0
$787$	−2.63932	−0.0940816	−0.0470408	−	0.998893i	$-0.514979\pi$
$787$	−2.63932	−0.0940816	−0.0470408	+	0.998893i	$0.514979\pi$
$788$	−19.1803	−0.683271
$789$	0	0
$790$	4.47214	0.159111
$791$	−32.8885	−1.16938
$792$	0	0
$793$	−25.7214	−0.913392
$794$	−16.3820	−0.581374
$795$	0	0
$796$	3.00000	0.106332
$797$	−13.5967	−0.481622	−0.240811	−	0.970572i	$-0.577413\pi$
$797$	−13.5967	−0.481622	−0.240811	+	0.970572i	$0.577413\pi$
$798$	0	0
$799$	28.7984	1.01881
$800$	4.00000	0.141421
$801$	0	0
$802$	5.00000	0.176556
$803$	0	0
$804$	0	0
$805$	−23.1803	−0.817000
$806$	3.50658	0.123514
$807$	0	0
$808$	−19.4721	−0.685027
$809$	30.8328	1.08402	0.542012	−	0.840371i	$-0.317663\pi$
$809$	30.8328	1.08402	0.542012	+	0.840371i	$0.317663\pi$
$810$	0	0
$811$	−29.3607	−1.03099	−0.515496	−	0.856892i	$-0.672392\pi$
$811$	−29.3607	−1.03099	−0.515496	+	0.856892i	$0.672392\pi$
$812$	−30.0344	−1.05400
$813$	0	0
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	−32.8885	−1.15062
$818$	32.7639	1.14556
$819$	0	0
$820$	−2.47214	−0.0863307
$821$	−12.3050	−0.429446	−0.214723	−	0.976675i	$-0.568885\pi$
$821$	−12.3050	−0.429446	−0.214723	+	0.976675i	$0.568885\pi$
$822$	0	0
$823$	26.6738	0.929789	0.464894	−	0.885366i	$-0.346092\pi$
$823$	26.6738	0.929789	0.464894	+	0.885366i	$0.346092\pi$
$824$	9.18034	0.319812
$825$	0	0
$826$	62.9230	2.18937
$827$	47.8328	1.66331	0.831655	−	0.555293i	$-0.187394\pi$
$827$	47.8328	1.66331	0.831655	+	0.555293i	$0.187394\pi$
$828$	0	0
$829$	2.67376	0.0928636	0.0464318	−	0.998921i	$-0.485215\pi$
$829$	2.67376	0.0928636	0.0464318	+	0.998921i	$0.485215\pi$
$830$	−14.6525	−0.508595
$831$	0	0
$832$	2.38197	0.0825798
$833$	−50.5410	−1.75114
$834$	0	0
$835$	−12.9443	−0.447955
$836$	0	0
$837$	0	0
$838$	−16.8328	−0.581480
$839$	−21.1459	−0.730037	−0.365019	−	0.931000i	$-0.618937\pi$
$839$	−21.1459	−0.730037	−0.365019	+	0.931000i	$0.618937\pi$
$840$	0	0
$841$	21.2705	0.733466
$842$	16.8541	0.580831
$843$	0	0
$844$	−13.7639	−0.473774
$845$	7.32624	0.252030
$846$	0	0
$847$	0	0
$848$	−2.85410	−0.0980103
$849$	0	0
$850$	−18.4721	−0.633589
$851$	14.3262	0.491097
$852$	0	0
$853$	−34.7426	−1.18957	−0.594783	−	0.803886i	$-0.702762\pi$
$853$	−34.7426	−1.18957	−0.594783	+	0.803886i	$0.702762\pi$
$854$	45.7426	1.56528
$855$	0	0
$856$	−3.23607	−0.110607
$857$	50.7426	1.73334	0.866668	−	0.498886i	$-0.166257\pi$
$857$	50.7426	1.73334	0.866668	+	0.498886i	$0.166257\pi$
$858$	0	0
$859$	−39.0689	−1.33301	−0.666507	−	0.745499i	$-0.732211\pi$
$859$	−39.0689	−1.33301	−0.666507	+	0.745499i	$0.732211\pi$
$860$	−9.09017	−0.309972
$861$	0	0
$862$	13.2148	0.450097
$863$	30.8885	1.05146	0.525729	−	0.850652i	$-0.323793\pi$
$863$	30.8885	1.05146	0.525729	+	0.850652i	$0.323793\pi$
$864$	0	0
$865$	−2.56231	−0.0871210
$866$	−6.38197	−0.216868
$867$	0	0
$868$	−6.23607	−0.211666
$869$	0	0
$870$	0	0
$871$	−35.0344	−1.18710
$872$	−6.56231	−0.222228
$873$	0	0
$874$	19.7984	0.669690
$875$	38.1246	1.28885
$876$	0	0
$877$	−9.63932	−0.325497	−0.162748	−	0.986668i	$-0.552036\pi$
$877$	−9.63932	−0.325497	−0.162748	+	0.986668i	$0.552036\pi$
$878$	21.3262	0.719726
$879$	0	0
$880$	0	0
$881$	3.29180	0.110903	0.0554517	−	0.998461i	$-0.482340\pi$
$881$	3.29180	0.110903	0.0554517	+	0.998461i	$0.482340\pi$
$882$	0	0
$883$	−45.6525	−1.53633	−0.768164	−	0.640253i	$-0.778829\pi$
$883$	−45.6525	−1.53633	−0.768164	+	0.640253i	$0.778829\pi$
$884$	−11.0000	−0.369970
$885$	0	0
$886$	2.79837	0.0940132
$887$	−17.9787	−0.603666	−0.301833	−	0.953361i	$-0.597599\pi$
$887$	−17.9787	−0.603666	−0.301833	+	0.953361i	$0.597599\pi$
$888$	0	0
$889$	−48.9787	−1.64269
$890$	−0.291796	−0.00978103
$891$	0	0
$892$	13.7426	0.460138
$893$	22.5623	0.755019
$894$	0	0
$895$	25.3607	0.847714
$896$	−4.23607	−0.141517
$897$	0	0
$898$	−40.0344	−1.33597
$899$	10.4377	0.348117
$900$	0	0
$901$	13.1803	0.439101
$902$	0	0
$903$	0	0
$904$	7.76393	0.258225
$905$	−8.61803	−0.286473
$906$	0	0
$907$	−29.9443	−0.994283	−0.497142	−	0.867669i	$-0.665617\pi$
$907$	−29.9443	−0.994283	−0.497142	+	0.867669i	$0.665617\pi$
$908$	−0.437694	−0.0145254
$909$	0	0
$910$	10.0902	0.334486
$911$	−11.4164	−0.378242	−0.189121	−	0.981954i	$-0.560564\pi$
$911$	−11.4164	−0.378242	−0.189121	+	0.981954i	$0.560564\pi$
$912$	0	0
$913$	0	0
$914$	24.9787	0.826222
$915$	0	0
$916$	−15.9443	−0.526814
$917$	41.1246	1.35805
$918$	0	0
$919$	5.97871	0.197220	0.0986098	−	0.995126i	$-0.468560\pi$
$919$	5.97871	0.197220	0.0986098	+	0.995126i	$0.468560\pi$
$920$	5.47214	0.180411
$921$	0	0
$922$	−28.9787	−0.954363
$923$	−12.8197	−0.421964
$924$	0	0
$925$	−10.4721	−0.344322
$926$	13.0902	0.430170
$927$	0	0
$928$	7.09017	0.232746
$929$	−53.6656	−1.76071	−0.880356	−	0.474313i	$-0.842696\pi$
$929$	−53.6656	−1.76071	−0.880356	+	0.474313i	$0.842696\pi$
$930$	0	0
$931$	−39.5967	−1.29773
$932$	21.7639	0.712901
$933$	0	0
$934$	−12.5623	−0.411051
$935$	0	0
$936$	0	0
$937$	−24.9098	−0.813769	−0.406884	−	0.913480i	$-0.633385\pi$
$937$	−24.9098	−0.813769	−0.406884	+	0.913480i	$0.633385\pi$
$938$	62.3050	2.03433
$939$	0	0
$940$	6.23607	0.203398
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	13.5279	0.440528
$944$	−14.8541	−0.483460
$945$	0	0
$946$	0	0
$947$	−53.9230	−1.75226	−0.876131	−	0.482073i	$-0.839884\pi$
$947$	−53.9230	−1.75226	−0.876131	+	0.482073i	$0.839884\pi$
$948$	0	0
$949$	−10.0902	−0.327541
$950$	−14.4721	−0.469538
$951$	0	0
$952$	19.5623	0.634018
$953$	29.8673	0.967495	0.483748	−	0.875208i	$-0.339275\pi$
$953$	29.8673	0.967495	0.483748	+	0.875208i	$0.339275\pi$
$954$	0	0
$955$	−19.7426	−0.638857
$956$	−12.7639	−0.412815
$957$	0	0
$958$	−30.0344	−0.970369
$959$	−30.2705	−0.977486
$960$	0	0
$961$	−28.8328	−0.930091
$962$	−6.23607	−0.201059
$963$	0	0
$964$	20.4164	0.657568
$965$	3.76393	0.121165
$966$	0	0
$967$	1.14590	0.0368496	0.0184248	−	0.999830i	$-0.494135\pi$
$967$	1.14590	0.0368496	0.0184248	+	0.999830i	$0.494135\pi$
$968$	0	0
$969$	0	0
$970$	−12.6180	−0.405141
$971$	22.2361	0.713589	0.356795	−	0.934183i	$-0.383870\pi$
$971$	22.2361	0.713589	0.356795	+	0.934183i	$0.383870\pi$
$972$	0	0
$973$	14.7082	0.471523
$974$	20.8328	0.667526
$975$	0	0
$976$	−10.7984	−0.345648
$977$	−44.8885	−1.43611	−0.718056	−	0.695985i	$-0.754968\pi$
$977$	−44.8885	−1.43611	−0.718056	+	0.695985i	$0.754968\pi$
$978$	0	0
$979$	0	0
$980$	−10.9443	−0.349602
$981$	0	0
$982$	39.2492	1.25249
$983$	22.0000	0.701691	0.350846	−	0.936433i	$-0.385894\pi$
$983$	22.0000	0.701691	0.350846	+	0.936433i	$0.385894\pi$
$984$	0	0
$985$	19.1803	0.611136
$986$	−32.7426	−1.04274
$987$	0	0
$988$	−8.61803	−0.274176
$989$	49.7426	1.58172
$990$	0	0
$991$	−30.4164	−0.966209	−0.483105	−	0.875563i	$-0.660491\pi$
$991$	−30.4164	−0.966209	−0.483105	+	0.875563i	$0.660491\pi$
$992$	1.47214	0.0467404
$993$	0	0
$994$	22.7984	0.723121
$995$	−3.00000	−0.0951064
$996$	0	0
$997$	43.1246	1.36577	0.682885	−	0.730526i	$-0.260725\pi$
$997$	43.1246	1.36577	0.682885	+	0.730526i	$0.260725\pi$
$998$	23.2918	0.737289
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.bg.1.2		2
3.2	odd	2		6534.2.a.cn.1.2		2
11.5	even	5		594.2.f.h.487.1	yes	4
11.9	even	5		594.2.f.h.433.1	yes	4
11.10	odd	2		6534.2.a.bz.1.1		2
33.5	odd	10		594.2.f.c.487.1	yes	4
33.20	odd	10		594.2.f.c.433.1	✓	4
33.32	even	2		6534.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
594.2.f.c.433.1	✓	4	33.20	odd	10
594.2.f.c.487.1	yes	4	33.5	odd	10
594.2.f.h.433.1	yes	4	11.9	even	5
594.2.f.h.487.1	yes	4	11.5	even	5
6534.2.a.bg.1.2		2	1.1	even	1	trivial
6534.2.a.bt.1.1		2	33.32	even	2
6534.2.a.bz.1.1		2	11.10	odd	2
6534.2.a.cn.1.2		2	3.2	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.23607 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.23607 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.38197 q^{13} -4.23607 q^{14} +1.00000 q^{16} -4.61803 q^{17} -3.61803 q^{19} -1.00000 q^{20} +5.47214 q^{23} -4.00000 q^{25} -2.38197 q^{26} +4.23607 q^{28} -7.09017 q^{29} -1.47214 q^{31} -1.00000 q^{32} +4.61803 q^{34} -4.23607 q^{35} +2.61803 q^{37} +3.61803 q^{38} +1.00000 q^{40} +2.47214 q^{41} +9.09017 q^{43} -5.47214 q^{46} -6.23607 q^{47} +10.9443 q^{49} +4.00000 q^{50} +2.38197 q^{52} -2.85410 q^{53} -4.23607 q^{56} +7.09017 q^{58} -14.8541 q^{59} -10.7984 q^{61} +1.47214 q^{62} +1.00000 q^{64} -2.38197 q^{65} -14.7082 q^{67} -4.61803 q^{68} +4.23607 q^{70} -5.38197 q^{71} -4.23607 q^{73} -2.61803 q^{74} -3.61803 q^{76} +4.47214 q^{79} -1.00000 q^{80} -2.47214 q^{82} -14.6525 q^{83} +4.61803 q^{85} -9.09017 q^{86} -0.291796 q^{89} +10.0902 q^{91} +5.47214 q^{92} +6.23607 q^{94} +3.61803 q^{95} -12.6180 q^{97} -10.9443 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 7 q^{13} - 4 q^{14} + 2 q^{16} - 7 q^{17} - 5 q^{19} - 2 q^{20} + 2 q^{23} - 8 q^{25} - 7 q^{26} + 4 q^{28} - 3 q^{29} + 6 q^{31} - 2 q^{32} + 7 q^{34} - 4 q^{35} + 3 q^{37} + 5 q^{38} + 2 q^{40} - 4 q^{41} + 7 q^{43} - 2 q^{46} - 8 q^{47} + 4 q^{49} + 8 q^{50} + 7 q^{52} + q^{53} - 4 q^{56} + 3 q^{58} - 23 q^{59} + 3 q^{61} - 6 q^{62} + 2 q^{64} - 7 q^{65} - 16 q^{67} - 7 q^{68} + 4 q^{70} - 13 q^{71} - 4 q^{73} - 3 q^{74} - 5 q^{76} - 2 q^{80} + 4 q^{82} + 2 q^{83} + 7 q^{85} - 7 q^{86} - 14 q^{89} + 9 q^{91} + 2 q^{92} + 8 q^{94} + 5 q^{95} - 23 q^{97} - 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 7 * q^13 - 4 * q^14 + 2 * q^16 - 7 * q^17 - 5 * q^19 - 2 * q^20 + 2 * q^23 - 8 * q^25 - 7 * q^26 + 4 * q^28 - 3 * q^29 + 6 * q^31 - 2 * q^32 + 7 * q^34 - 4 * q^35 + 3 * q^37 + 5 * q^38 + 2 * q^40 - 4 * q^41 + 7 * q^43 - 2 * q^46 - 8 * q^47 + 4 * q^49 + 8 * q^50 + 7 * q^52 + q^53 - 4 * q^56 + 3 * q^58 - 23 * q^59 + 3 * q^61 - 6 * q^62 + 2 * q^64 - 7 * q^65 - 16 * q^67 - 7 * q^68 + 4 * q^70 - 13 * q^71 - 4 * q^73 - 3 * q^74 - 5 * q^76 - 2 * q^80 + 4 * q^82 + 2 * q^83 + 7 * q^85 - 7 * q^86 - 14 * q^89 + 9 * q^91 + 2 * q^92 + 8 * q^94 + 5 * q^95 - 23 * q^97 - 4 * q^98

Introduction

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