LMFDB - Embedded newform 6579.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6579,2,Mod(1,6579)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6579, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6579.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6579 = 3^{2} \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6579.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:	$52.5335794898$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 731)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6579.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} -1.56155 q^{5} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} -1.56155 q^{5} -3.00000 q^{8} -1.56155 q^{10} +2.00000 q^{11} -0.438447 q^{13} -1.00000 q^{16} +1.00000 q^{17} +7.12311 q^{19} +1.56155 q^{20} +2.00000 q^{22} -2.00000 q^{23} -2.56155 q^{25} -0.438447 q^{26} -3.12311 q^{29} -6.00000 q^{31} +5.00000 q^{32} +1.00000 q^{34} -5.56155 q^{37} +7.12311 q^{38} +4.68466 q^{40} +5.12311 q^{41} -1.00000 q^{43} -2.00000 q^{44} -2.00000 q^{46} +9.56155 q^{47} -7.00000 q^{49} -2.56155 q^{50} +0.438447 q^{52} +10.6847 q^{53} -3.12311 q^{55} -3.12311 q^{58} +11.8078 q^{59} +0.684658 q^{61} -6.00000 q^{62} +7.00000 q^{64} +0.684658 q^{65} +5.56155 q^{67} -1.00000 q^{68} -5.56155 q^{71} -11.8078 q^{73} -5.56155 q^{74} -7.12311 q^{76} -9.12311 q^{79} +1.56155 q^{80} +5.12311 q^{82} -15.8078 q^{83} -1.56155 q^{85} -1.00000 q^{86} -6.00000 q^{88} +0.246211 q^{89} +2.00000 q^{92} +9.56155 q^{94} -11.1231 q^{95} +16.2462 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 + q^5 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} + q^{5} - 6 q^{8} + q^{10} + 4 q^{11} - 5 q^{13} - 2 q^{16} + 2 q^{17} + 6 q^{19} - q^{20} + 4 q^{22} - 4 q^{23} - q^{25} - 5 q^{26} + 2 q^{29} - 12 q^{31} + 10 q^{32} + 2 q^{34} - 7 q^{37} + 6 q^{38} - 3 q^{40} + 2 q^{41} - 2 q^{43} - 4 q^{44} - 4 q^{46} + 15 q^{47} - 14 q^{49} - q^{50} + 5 q^{52} + 9 q^{53} + 2 q^{55} + 2 q^{58} + 3 q^{59} - 11 q^{61} - 12 q^{62} + 14 q^{64} - 11 q^{65} + 7 q^{67} - 2 q^{68} - 7 q^{71} - 3 q^{73} - 7 q^{74} - 6 q^{76} - 10 q^{79} - q^{80} + 2 q^{82} - 11 q^{83} + q^{85} - 2 q^{86} - 12 q^{88} - 16 q^{89} + 4 q^{92} + 15 q^{94} - 14 q^{95} + 16 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + q^5 - 6 * q^8 + q^10 + 4 * q^11 - 5 * q^13 - 2 * q^16 + 2 * q^17 + 6 * q^19 - q^20 + 4 * q^22 - 4 * q^23 - q^25 - 5 * q^26 + 2 * q^29 - 12 * q^31 + 10 * q^32 + 2 * q^34 - 7 * q^37 + 6 * q^38 - 3 * q^40 + 2 * q^41 - 2 * q^43 - 4 * q^44 - 4 * q^46 + 15 * q^47 - 14 * q^49 - q^50 + 5 * q^52 + 9 * q^53 + 2 * q^55 + 2 * q^58 + 3 * q^59 - 11 * q^61 - 12 * q^62 + 14 * q^64 - 11 * q^65 + 7 * q^67 - 2 * q^68 - 7 * q^71 - 3 * q^73 - 7 * q^74 - 6 * q^76 - 10 * q^79 - q^80 + 2 * q^82 - 11 * q^83 + q^85 - 2 * q^86 - 12 * q^88 - 16 * q^89 + 4 * q^92 + 15 * q^94 - 14 * q^95 + 16 * q^97 - 14 * 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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.56155	−0.493806
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−0.438447	−0.121603	−0.0608017	−	0.998150i	$-0.519366\pi$
$13$	−0.438447	−0.121603	−0.0608017	+	0.998150i	$0.519366\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	1.56155	0.349174
$21$	0	0
$22$	2.00000	0.426401
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	−0.438447	−0.0859866
$27$	0	0
$28$	0	0
$29$	−3.12311	−0.579946	−0.289973	−	0.957035i	$-0.593646\pi$
$29$	−3.12311	−0.579946	−0.289973	+	0.957035i	$0.593646\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−5.56155	−0.914314	−0.457157	−	0.889386i	$-0.651132\pi$
$37$	−5.56155	−0.914314	−0.457157	+	0.889386i	$0.651132\pi$
$38$	7.12311	1.15552
$39$	0	0
$40$	4.68466	0.740710
$41$	5.12311	0.800095	0.400047	−	0.916494i	$-0.368994\pi$
$41$	5.12311	0.800095	0.400047	+	0.916494i	$0.368994\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−2.00000	−0.294884
$47$	9.56155	1.39470	0.697348	−	0.716733i	$-0.254363\pi$
$47$	9.56155	1.39470	0.697348	+	0.716733i	$0.254363\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−2.56155	−0.362258
$51$	0	0
$52$	0.438447	0.0608017
$53$	10.6847	1.46765	0.733825	−	0.679338i	$-0.237733\pi$
$53$	10.6847	1.46765	0.733825	+	0.679338i	$0.237733\pi$
$54$	0	0
$55$	−3.12311	−0.421119
$56$	0	0
$57$	0	0
$58$	−3.12311	−0.410084
$59$	11.8078	1.53724	0.768620	−	0.639706i	$-0.220944\pi$
$59$	11.8078	1.53724	0.768620	+	0.639706i	$0.220944\pi$
$60$	0	0
$61$	0.684658	0.0876615	0.0438308	−	0.999039i	$-0.486044\pi$
$61$	0.684658	0.0876615	0.0438308	+	0.999039i	$0.486044\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	7.00000	0.875000
$65$	0.684658	0.0849214
$66$	0	0
$67$	5.56155	0.679452	0.339726	−	0.940524i	$-0.389666\pi$
$67$	5.56155	0.679452	0.339726	+	0.940524i	$0.389666\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	−5.56155	−0.660035	−0.330017	−	0.943975i	$-0.607055\pi$
$71$	−5.56155	−0.660035	−0.330017	+	0.943975i	$0.607055\pi$
$72$	0	0
$73$	−11.8078	−1.38199	−0.690997	−	0.722858i	$-0.742828\pi$
$73$	−11.8078	−1.38199	−0.690997	+	0.722858i	$0.742828\pi$
$74$	−5.56155	−0.646517
$75$	0	0
$76$	−7.12311	−0.817076
$77$	0	0
$78$	0	0
$79$	−9.12311	−1.02643	−0.513215	−	0.858260i	$-0.671546\pi$
$79$	−9.12311	−1.02643	−0.513215	+	0.858260i	$0.671546\pi$
$80$	1.56155	0.174587
$81$	0	0
$82$	5.12311	0.565752
$83$	−15.8078	−1.73513	−0.867564	−	0.497326i	$-0.834315\pi$
$83$	−15.8078	−1.73513	−0.867564	+	0.497326i	$0.834315\pi$
$84$	0	0
$85$	−1.56155	−0.169374
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−6.00000	−0.639602
$89$	0.246211	0.0260983	0.0130492	−	0.999915i	$-0.495846\pi$
$89$	0.246211	0.0260983	0.0130492	+	0.999915i	$0.495846\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	0	0
$94$	9.56155	0.986199
$95$	−11.1231	−1.14121
$96$	0	0
$97$	16.2462	1.64955	0.824776	−	0.565459i	$-0.191301\pi$
$97$	16.2462	1.64955	0.824776	+	0.565459i	$0.191301\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	2.56155	0.256155
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−17.5616	−1.73039	−0.865196	−	0.501435i	$-0.832806\pi$
$103$	−17.5616	−1.73039	−0.865196	+	0.501435i	$0.832806\pi$
$104$	1.31534	0.128980
$105$	0	0
$106$	10.6847	1.03779
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−4.24621	−0.406713	−0.203357	−	0.979105i	$-0.565185\pi$
$109$	−4.24621	−0.406713	−0.203357	+	0.979105i	$0.565185\pi$
$110$	−3.12311	−0.297776
$111$	0	0
$112$	0	0
$113$	11.8078	1.11078	0.555391	−	0.831590i	$-0.312569\pi$
$113$	11.8078	1.11078	0.555391	+	0.831590i	$0.312569\pi$
$114$	0	0
$115$	3.12311	0.291231
$116$	3.12311	0.289973
$117$	0	0
$118$	11.8078	1.08699
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0.684658	0.0619861
$123$	0	0
$124$	6.00000	0.538816
$125$	11.8078	1.05612
$126$	0	0
$127$	22.0540	1.95697	0.978487	−	0.206309i	$-0.0661452\pi$
$127$	22.0540	1.95697	0.978487	+	0.206309i	$0.0661452\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0.684658	0.0600485
$131$	−11.8078	−1.03165	−0.515825	−	0.856694i	$-0.672514\pi$
$131$	−11.8078	−1.03165	−0.515825	+	0.856694i	$0.672514\pi$
$132$	0	0
$133$	0	0
$134$	5.56155	0.480445
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−12.2462	−1.04626	−0.523132	−	0.852252i	$-0.675237\pi$
$137$	−12.2462	−1.04626	−0.523132	+	0.852252i	$0.675237\pi$
$138$	0	0
$139$	−19.3693	−1.64288	−0.821442	−	0.570292i	$-0.806830\pi$
$139$	−19.3693	−1.64288	−0.821442	+	0.570292i	$0.806830\pi$
$140$	0	0
$141$	0	0
$142$	−5.56155	−0.466715
$143$	−0.876894	−0.0733296
$144$	0	0
$145$	4.87689	0.405004
$146$	−11.8078	−0.977218
$147$	0	0
$148$	5.56155	0.457157
$149$	8.24621	0.675556	0.337778	−	0.941226i	$-0.390325\pi$
$149$	8.24621	0.675556	0.337778	+	0.941226i	$0.390325\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−21.3693	−1.73328
$153$	0	0
$154$	0	0
$155$	9.36932	0.752562
$156$	0	0
$157$	−22.4924	−1.79509	−0.897545	−	0.440922i	$-0.854651\pi$
$157$	−22.4924	−1.79509	−0.897545	+	0.440922i	$0.854651\pi$
$158$	−9.12311	−0.725795
$159$	0	0
$160$	−7.80776	−0.617258
$161$	0	0
$162$	0	0
$163$	−21.5616	−1.68883	−0.844416	−	0.535689i	$-0.820052\pi$
$163$	−21.5616	−1.68883	−0.844416	+	0.535689i	$0.820052\pi$
$164$	−5.12311	−0.400047
$165$	0	0
$166$	−15.8078	−1.22692
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	−1.56155	−0.119766
$171$	0	0
$172$	1.00000	0.0762493
$173$	5.12311	0.389503	0.194751	−	0.980853i	$-0.437610\pi$
$173$	5.12311	0.389503	0.194751	+	0.980853i	$0.437610\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	0.246211	0.0184543
$179$	−8.87689	−0.663490	−0.331745	−	0.943369i	$-0.607637\pi$
$179$	−8.87689	−0.663490	−0.331745	+	0.943369i	$0.607637\pi$
$180$	0	0
$181$	−19.3693	−1.43971	−0.719855	−	0.694124i	$-0.755792\pi$
$181$	−19.3693	−1.43971	−0.719855	+	0.694124i	$0.755792\pi$
$182$	0	0
$183$	0	0
$184$	6.00000	0.442326
$185$	8.68466	0.638509
$186$	0	0
$187$	2.00000	0.146254
$188$	−9.56155	−0.697348
$189$	0	0
$190$	−11.1231	−0.806955
$191$	−11.1231	−0.804840	−0.402420	−	0.915455i	$-0.631831\pi$
$191$	−11.1231	−0.804840	−0.402420	+	0.915455i	$0.631831\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	16.2462	1.16641
$195$	0	0
$196$	7.00000	0.500000
$197$	−17.6155	−1.25505	−0.627527	−	0.778595i	$-0.715933\pi$
$197$	−17.6155	−1.25505	−0.627527	+	0.778595i	$0.715933\pi$
$198$	0	0
$199$	−2.93087	−0.207764	−0.103882	−	0.994590i	$-0.533126\pi$
$199$	−2.93087	−0.207764	−0.103882	+	0.994590i	$0.533126\pi$
$200$	7.68466	0.543387
$201$	0	0
$202$	2.00000	0.140720
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−17.5616	−1.22357
$207$	0	0
$208$	0.438447	0.0304008
$209$	14.2462	0.985431
$210$	0	0
$211$	−4.19224	−0.288605	−0.144303	−	0.989534i	$-0.546094\pi$
$211$	−4.19224	−0.288605	−0.144303	+	0.989534i	$0.546094\pi$
$212$	−10.6847	−0.733825
$213$	0	0
$214$	−2.00000	−0.136717
$215$	1.56155	0.106497
$216$	0	0
$217$	0	0
$218$	−4.24621	−0.287590
$219$	0	0
$220$	3.12311	0.210560
$221$	−0.438447	−0.0294931
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	11.8078	0.785441
$227$	−20.4924	−1.36013	−0.680065	−	0.733152i	$-0.738048\pi$
$227$	−20.4924	−1.36013	−0.680065	+	0.733152i	$0.738048\pi$
$228$	0	0
$229$	5.80776	0.383788	0.191894	−	0.981416i	$-0.438537\pi$
$229$	5.80776	0.383788	0.191894	+	0.981416i	$0.438537\pi$
$230$	3.12311	0.205931
$231$	0	0
$232$	9.36932	0.615126
$233$	−19.1231	−1.25280	−0.626398	−	0.779503i	$-0.715472\pi$
$233$	−19.1231	−1.25280	−0.626398	+	0.779503i	$0.715472\pi$
$234$	0	0
$235$	−14.9309	−0.973983
$236$	−11.8078	−0.768620
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−5.36932	−0.345868	−0.172934	−	0.984933i	$-0.555325\pi$
$241$	−5.36932	−0.345868	−0.172934	+	0.984933i	$0.555325\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−0.684658	−0.0438308
$245$	10.9309	0.698348
$246$	0	0
$247$	−3.12311	−0.198718
$248$	18.0000	1.14300
$249$	0	0
$250$	11.8078	0.746789
$251$	6.43845	0.406391	0.203196	−	0.979138i	$-0.434867\pi$
$251$	6.43845	0.406391	0.203196	+	0.979138i	$0.434867\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	22.0540	1.38379
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0	0
$259$	0	0
$260$	−0.684658	−0.0424607
$261$	0	0
$262$	−11.8078	−0.729486
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	−16.6847	−1.02493
$266$	0	0
$267$	0	0
$268$	−5.56155	−0.339726
$269$	−8.24621	−0.502780	−0.251390	−	0.967886i	$-0.580888\pi$
$269$	−8.24621	−0.502780	−0.251390	+	0.967886i	$0.580888\pi$
$270$	0	0
$271$	−18.9309	−1.14997	−0.574984	−	0.818164i	$-0.694992\pi$
$271$	−18.9309	−1.14997	−0.574984	+	0.818164i	$0.694992\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−12.2462	−0.739821
$275$	−5.12311	−0.308935
$276$	0	0
$277$	30.9309	1.85846	0.929228	−	0.369507i	$-0.120473\pi$
$277$	30.9309	1.85846	0.929228	+	0.369507i	$0.120473\pi$
$278$	−19.3693	−1.16169
$279$	0	0
$280$	0	0
$281$	20.9309	1.24863	0.624316	−	0.781172i	$-0.285378\pi$
$281$	20.9309	1.24863	0.624316	+	0.781172i	$0.285378\pi$
$282$	0	0
$283$	12.2462	0.727962	0.363981	−	0.931406i	$-0.381417\pi$
$283$	12.2462	0.727962	0.363981	+	0.931406i	$0.381417\pi$
$284$	5.56155	0.330017
$285$	0	0
$286$	−0.876894	−0.0518519
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	4.87689	0.286381
$291$	0	0
$292$	11.8078	0.690997
$293$	0.246211	0.0143838	0.00719191	−	0.999974i	$-0.497711\pi$
$293$	0.246211	0.0143838	0.00719191	+	0.999974i	$0.497711\pi$
$294$	0	0
$295$	−18.4384	−1.07353
$296$	16.6847	0.969776
$297$	0	0
$298$	8.24621	0.477690
$299$	0.876894	0.0507121
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	−7.12311	−0.408538
$305$	−1.06913	−0.0612182
$306$	0	0
$307$	−14.2462	−0.813074	−0.406537	−	0.913634i	$-0.633264\pi$
$307$	−14.2462	−0.813074	−0.406537	+	0.913634i	$0.633264\pi$
$308$	0	0
$309$	0	0
$310$	9.36932	0.532141
$311$	−15.3693	−0.871514	−0.435757	−	0.900064i	$-0.643519\pi$
$311$	−15.3693	−0.871514	−0.435757	+	0.900064i	$0.643519\pi$
$312$	0	0
$313$	15.8078	0.893508	0.446754	−	0.894657i	$-0.352580\pi$
$313$	15.8078	0.893508	0.446754	+	0.894657i	$0.352580\pi$
$314$	−22.4924	−1.26932
$315$	0	0
$316$	9.12311	0.513215
$317$	19.3693	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$317$	19.3693	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$318$	0	0
$319$	−6.24621	−0.349721
$320$	−10.9309	−0.611054
$321$	0	0
$322$	0	0
$323$	7.12311	0.396340
$324$	0	0
$325$	1.12311	0.0622987
$326$	−21.5616	−1.19418
$327$	0	0
$328$	−15.3693	−0.848629
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	15.8078	0.867564
$333$	0	0
$334$	−22.0000	−1.20379
$335$	−8.68466	−0.474494
$336$	0	0
$337$	−3.75379	−0.204482	−0.102241	−	0.994760i	$-0.532601\pi$
$337$	−3.75379	−0.204482	−0.102241	+	0.994760i	$0.532601\pi$
$338$	−12.8078	−0.696651
$339$	0	0
$340$	1.56155	0.0846871
$341$	−12.0000	−0.649836
$342$	0	0
$343$	0	0
$344$	3.00000	0.161749
$345$	0	0
$346$	5.12311	0.275420
$347$	18.4384	0.989828	0.494914	−	0.868942i	$-0.335200\pi$
$347$	18.4384	0.989828	0.494914	+	0.868942i	$0.335200\pi$
$348$	0	0
$349$	−10.8769	−0.582227	−0.291113	−	0.956689i	$-0.594026\pi$
$349$	−10.8769	−0.582227	−0.291113	+	0.956689i	$0.594026\pi$
$350$	0	0
$351$	0	0
$352$	10.0000	0.533002
$353$	19.1771	1.02069	0.510347	−	0.859969i	$-0.329517\pi$
$353$	19.1771	1.02069	0.510347	+	0.859969i	$0.329517\pi$
$354$	0	0
$355$	8.68466	0.460934
$356$	−0.246211	−0.0130492
$357$	0	0
$358$	−8.87689	−0.469158
$359$	3.80776	0.200966	0.100483	−	0.994939i	$-0.467961\pi$
$359$	3.80776	0.200966	0.100483	+	0.994939i	$0.467961\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	−19.3693	−1.01803
$363$	0	0
$364$	0	0
$365$	18.4384	0.965112
$366$	0	0
$367$	18.4924	0.965297	0.482648	−	0.875814i	$-0.339675\pi$
$367$	18.4924	0.965297	0.482648	+	0.875814i	$0.339675\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	8.68466	0.451494
$371$	0	0
$372$	0	0
$373$	−12.6307	−0.653992	−0.326996	−	0.945026i	$-0.606036\pi$
$373$	−12.6307	−0.653992	−0.326996	+	0.945026i	$0.606036\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−28.6847	−1.47930
$377$	1.36932	0.0705234
$378$	0	0
$379$	−9.12311	−0.468622	−0.234311	−	0.972162i	$-0.575283\pi$
$379$	−9.12311	−0.468622	−0.234311	+	0.972162i	$0.575283\pi$
$380$	11.1231	0.570603
$381$	0	0
$382$	−11.1231	−0.569108
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	−16.2462	−0.824776
$389$	−2.49242	−0.126371	−0.0631854	−	0.998002i	$-0.520126\pi$
$389$	−2.49242	−0.126371	−0.0631854	+	0.998002i	$0.520126\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	21.0000	1.06066
$393$	0	0
$394$	−17.6155	−0.887457
$395$	14.2462	0.716805
$396$	0	0
$397$	6.87689	0.345141	0.172571	−	0.984997i	$-0.444793\pi$
$397$	6.87689	0.345141	0.172571	+	0.984997i	$0.444793\pi$
$398$	−2.93087	−0.146911
$399$	0	0
$400$	2.56155	0.128078
$401$	3.75379	0.187455	0.0937276	−	0.995598i	$-0.470122\pi$
$401$	3.75379	0.187455	0.0937276	+	0.995598i	$0.470122\pi$
$402$	0	0
$403$	2.63068	0.131044
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	−11.1231	−0.551352
$408$	0	0
$409$	−35.8617	−1.77325	−0.886624	−	0.462490i	$-0.846956\pi$
$409$	−35.8617	−1.77325	−0.886624	+	0.462490i	$0.846956\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	17.5616	0.865196
$413$	0	0
$414$	0	0
$415$	24.6847	1.21172
$416$	−2.19224	−0.107483
$417$	0	0
$418$	14.2462	0.696805
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	12.7386	0.620843	0.310422	−	0.950599i	$-0.399530\pi$
$421$	12.7386	0.620843	0.310422	+	0.950599i	$0.399530\pi$
$422$	−4.19224	−0.204075
$423$	0	0
$424$	−32.0540	−1.55668
$425$	−2.56155	−0.124254
$426$	0	0
$427$	0	0
$428$	2.00000	0.0966736
$429$	0	0
$430$	1.56155	0.0753048
$431$	13.1231	0.632118	0.316059	−	0.948740i	$-0.397640\pi$
$431$	13.1231	0.632118	0.316059	+	0.948740i	$0.397640\pi$
$432$	0	0
$433$	11.3693	0.546375	0.273187	−	0.961961i	$-0.411922\pi$
$433$	11.3693	0.546375	0.273187	+	0.961961i	$0.411922\pi$
$434$	0	0
$435$	0	0
$436$	4.24621	0.203357
$437$	−14.2462	−0.681489
$438$	0	0
$439$	15.3693	0.733537	0.366769	−	0.930312i	$-0.380464\pi$
$439$	15.3693	0.733537	0.366769	+	0.930312i	$0.380464\pi$
$440$	9.36932	0.446665
$441$	0	0
$442$	−0.438447	−0.0208548
$443$	31.4233	1.49297	0.746483	−	0.665405i	$-0.231741\pi$
$443$	31.4233	1.49297	0.746483	+	0.665405i	$0.231741\pi$
$444$	0	0
$445$	−0.384472	−0.0182257
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	30.4384	1.43648	0.718240	−	0.695796i	$-0.244948\pi$
$449$	30.4384	1.43648	0.718240	+	0.695796i	$0.244948\pi$
$450$	0	0
$451$	10.2462	0.482475
$452$	−11.8078	−0.555391
$453$	0	0
$454$	−20.4924	−0.961757
$455$	0	0
$456$	0	0
$457$	−4.63068	−0.216614	−0.108307	−	0.994117i	$-0.534543\pi$
$457$	−4.63068	−0.216614	−0.108307	+	0.994117i	$0.534543\pi$
$458$	5.80776	0.271379
$459$	0	0
$460$	−3.12311	−0.145616
$461$	17.8078	0.829390	0.414695	−	0.909960i	$-0.363888\pi$
$461$	17.8078	0.829390	0.414695	+	0.909960i	$0.363888\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	3.12311	0.144987
$465$	0	0
$466$	−19.1231	−0.885861
$467$	32.4924	1.50357	0.751785	−	0.659408i	$-0.229193\pi$
$467$	32.4924	1.50357	0.751785	+	0.659408i	$0.229193\pi$
$468$	0	0
$469$	0	0
$470$	−14.9309	−0.688710
$471$	0	0
$472$	−35.4233	−1.63049
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−18.2462	−0.837194
$476$	0	0
$477$	0	0
$478$	8.00000	0.365911
$479$	−31.3693	−1.43330	−0.716650	−	0.697433i	$-0.754326\pi$
$479$	−31.3693	−1.43330	−0.716650	+	0.697433i	$0.754326\pi$
$480$	0	0
$481$	2.43845	0.111184
$482$	−5.36932	−0.244566
$483$	0	0
$484$	7.00000	0.318182
$485$	−25.3693	−1.15196
$486$	0	0
$487$	−26.4924	−1.20049	−0.600243	−	0.799818i	$-0.704930\pi$
$487$	−26.4924	−1.20049	−0.600243	+	0.799818i	$0.704930\pi$
$488$	−2.05398	−0.0929791
$489$	0	0
$490$	10.9309	0.493806
$491$	5.36932	0.242314	0.121157	−	0.992633i	$-0.461340\pi$
$491$	5.36932	0.242314	0.121157	+	0.992633i	$0.461340\pi$
$492$	0	0
$493$	−3.12311	−0.140658
$494$	−3.12311	−0.140515
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	0	0
$499$	4.68466	0.209714	0.104857	−	0.994487i	$-0.466561\pi$
$499$	4.68466	0.209714	0.104857	+	0.994487i	$0.466561\pi$
$500$	−11.8078	−0.528059
$501$	0	0
$502$	6.43845	0.287362
$503$	−39.4233	−1.75780	−0.878899	−	0.477008i	$-0.841721\pi$
$503$	−39.4233	−1.75780	−0.878899	+	0.477008i	$0.841721\pi$
$504$	0	0
$505$	−3.12311	−0.138976
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−22.0540	−0.978487
$509$	−4.05398	−0.179689	−0.0898446	−	0.995956i	$-0.528637\pi$
$509$	−4.05398	−0.179689	−0.0898446	+	0.995956i	$0.528637\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−26.0000	−1.14681
$515$	27.4233	1.20841
$516$	0	0
$517$	19.1231	0.841033
$518$	0	0
$519$	0	0
$520$	−2.05398	−0.0900728
$521$	2.63068	0.115252	0.0576262	−	0.998338i	$-0.481647\pi$
$521$	2.63068	0.115252	0.0576262	+	0.998338i	$0.481647\pi$
$522$	0	0
$523$	39.1231	1.71073	0.855367	−	0.518023i	$-0.173332\pi$
$523$	39.1231	1.71073	0.855367	+	0.518023i	$0.173332\pi$
$524$	11.8078	0.515825
$525$	0	0
$526$	−12.4924	−0.544696
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−16.6847	−0.724735
$531$	0	0
$532$	0	0
$533$	−2.24621	−0.0972942
$534$	0	0
$535$	3.12311	0.135024
$536$	−16.6847	−0.720667
$537$	0	0
$538$	−8.24621	−0.355519
$539$	−14.0000	−0.603023
$540$	0	0
$541$	21.6155	0.929324	0.464662	−	0.885488i	$-0.346176\pi$
$541$	21.6155	0.929324	0.464662	+	0.885488i	$0.346176\pi$
$542$	−18.9309	−0.813150
$543$	0	0
$544$	5.00000	0.214373
$545$	6.63068	0.284027
$546$	0	0
$547$	34.4924	1.47479	0.737395	−	0.675462i	$-0.236056\pi$
$547$	34.4924	1.47479	0.737395	+	0.675462i	$0.236056\pi$
$548$	12.2462	0.523132
$549$	0	0
$550$	−5.12311	−0.218450
$551$	−22.2462	−0.947720
$552$	0	0
$553$	0	0
$554$	30.9309	1.31413
$555$	0	0
$556$	19.3693	0.821442
$557$	−19.1771	−0.812559	−0.406279	−	0.913749i	$-0.633174\pi$
$557$	−19.1771	−0.812559	−0.406279	+	0.913749i	$0.633174\pi$
$558$	0	0
$559$	0.438447	0.0185443
$560$	0	0
$561$	0	0
$562$	20.9309	0.882915
$563$	1.75379	0.0739134	0.0369567	−	0.999317i	$-0.488234\pi$
$563$	1.75379	0.0739134	0.0369567	+	0.999317i	$0.488234\pi$
$564$	0	0
$565$	−18.4384	−0.775711
$566$	12.2462	0.514747
$567$	0	0
$568$	16.6847	0.700073
$569$	−3.56155	−0.149308	−0.0746540	−	0.997209i	$-0.523785\pi$
$569$	−3.56155	−0.149308	−0.0746540	+	0.997209i	$0.523785\pi$
$570$	0	0
$571$	−30.2462	−1.26576	−0.632882	−	0.774248i	$-0.718128\pi$
$571$	−30.2462	−1.26576	−0.632882	+	0.774248i	$0.718128\pi$
$572$	0.876894	0.0366648
$573$	0	0
$574$	0	0
$575$	5.12311	0.213648
$576$	0	0
$577$	−4.73863	−0.197272	−0.0986360	−	0.995124i	$-0.531448\pi$
$577$	−4.73863	−0.197272	−0.0986360	+	0.995124i	$0.531448\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−4.87689	−0.202502
$581$	0	0
$582$	0	0
$583$	21.3693	0.885027
$584$	35.4233	1.46583
$585$	0	0
$586$	0.246211	0.0101709
$587$	2.63068	0.108580	0.0542900	−	0.998525i	$-0.482710\pi$
$587$	2.63068	0.108580	0.0542900	+	0.998525i	$0.482710\pi$
$588$	0	0
$589$	−42.7386	−1.76101
$590$	−18.4384	−0.759099
$591$	0	0
$592$	5.56155	0.228578
$593$	−12.2462	−0.502892	−0.251446	−	0.967871i	$-0.580906\pi$
$593$	−12.2462	−0.502892	−0.251446	+	0.967871i	$0.580906\pi$
$594$	0	0
$595$	0	0
$596$	−8.24621	−0.337778
$597$	0	0
$598$	0.876894	0.0358589
$599$	−24.6847	−1.00859	−0.504294	−	0.863532i	$-0.668247\pi$
$599$	−24.6847	−1.00859	−0.504294	+	0.863532i	$0.668247\pi$
$600$	0	0
$601$	2.43845	0.0994663	0.0497332	−	0.998763i	$-0.484163\pi$
$601$	2.43845	0.0994663	0.0497332	+	0.998763i	$0.484163\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	10.9309	0.444403
$606$	0	0
$607$	19.3153	0.783986	0.391993	−	0.919968i	$-0.371786\pi$
$607$	19.3153	0.783986	0.391993	+	0.919968i	$0.371786\pi$
$608$	35.6155	1.44440
$609$	0	0
$610$	−1.06913	−0.0432878
$611$	−4.19224	−0.169600
$612$	0	0
$613$	−1.50758	−0.0608905	−0.0304452	−	0.999536i	$-0.509693\pi$
$613$	−1.50758	−0.0608905	−0.0304452	+	0.999536i	$0.509693\pi$
$614$	−14.2462	−0.574930
$615$	0	0
$616$	0	0
$617$	−40.2462	−1.62025	−0.810126	−	0.586256i	$-0.800601\pi$
$617$	−40.2462	−1.62025	−0.810126	+	0.586256i	$0.800601\pi$
$618$	0	0
$619$	48.7386	1.95897	0.979486	−	0.201514i	$-0.0645863\pi$
$619$	48.7386	1.95897	0.979486	+	0.201514i	$0.0645863\pi$
$620$	−9.36932	−0.376281
$621$	0	0
$622$	−15.3693	−0.616253
$623$	0	0
$624$	0	0
$625$	−5.63068	−0.225227
$626$	15.8078	0.631805
$627$	0	0
$628$	22.4924	0.897545
$629$	−5.56155	−0.221754
$630$	0	0
$631$	−0.384472	−0.0153056	−0.00765279	−	0.999971i	$-0.502436\pi$
$631$	−0.384472	−0.0153056	−0.00765279	+	0.999971i	$0.502436\pi$
$632$	27.3693	1.08869
$633$	0	0
$634$	19.3693	0.769254
$635$	−34.4384	−1.36665
$636$	0	0
$637$	3.06913	0.121603
$638$	−6.24621	−0.247290
$639$	0	0
$640$	4.68466	0.185177
$641$	15.4233	0.609183	0.304592	−	0.952483i	$-0.401480\pi$
$641$	15.4233	0.609183	0.304592	+	0.952483i	$0.401480\pi$
$642$	0	0
$643$	44.2462	1.74490	0.872450	−	0.488703i	$-0.162530\pi$
$643$	44.2462	1.74490	0.872450	+	0.488703i	$0.162530\pi$
$644$	0	0
$645$	0	0
$646$	7.12311	0.280255
$647$	36.1080	1.41955	0.709775	−	0.704428i	$-0.248797\pi$
$647$	36.1080	1.41955	0.709775	+	0.704428i	$0.248797\pi$
$648$	0	0
$649$	23.6155	0.926991
$650$	1.12311	0.0440518
$651$	0	0
$652$	21.5616	0.844416
$653$	11.1231	0.435281	0.217640	−	0.976029i	$-0.430164\pi$
$653$	11.1231	0.435281	0.217640	+	0.976029i	$0.430164\pi$
$654$	0	0
$655$	18.4384	0.720450
$656$	−5.12311	−0.200024
$657$	0	0
$658$	0	0
$659$	46.2462	1.80150	0.900748	−	0.434341i	$-0.143019\pi$
$659$	46.2462	1.80150	0.900748	+	0.434341i	$0.143019\pi$
$660$	0	0
$661$	−40.4384	−1.57287	−0.786437	−	0.617671i	$-0.788076\pi$
$661$	−40.4384	−1.57287	−0.786437	+	0.617671i	$0.788076\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	47.4233	1.84038
$665$	0	0
$666$	0	0
$667$	6.24621	0.241854
$668$	22.0000	0.851206
$669$	0	0
$670$	−8.68466	−0.335518
$671$	1.36932	0.0528619
$672$	0	0
$673$	28.1922	1.08673	0.543365	−	0.839496i	$-0.317150\pi$
$673$	28.1922	1.08673	0.543365	+	0.839496i	$0.317150\pi$
$674$	−3.75379	−0.144591
$675$	0	0
$676$	12.8078	0.492606
$677$	0.384472	0.0147765	0.00738823	−	0.999973i	$-0.497648\pi$
$677$	0.384472	0.0147765	0.00738823	+	0.999973i	$0.497648\pi$
$678$	0	0
$679$	0	0
$680$	4.68466	0.179648
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−43.3693	−1.65948	−0.829740	−	0.558150i	$-0.811512\pi$
$683$	−43.3693	−1.65948	−0.829740	+	0.558150i	$0.811512\pi$
$684$	0	0
$685$	19.1231	0.730656
$686$	0	0
$687$	0	0
$688$	1.00000	0.0381246
$689$	−4.68466	−0.178471
$690$	0	0
$691$	−0.492423	−0.0187326	−0.00936632	−	0.999956i	$-0.502981\pi$
$691$	−0.492423	−0.0187326	−0.00936632	+	0.999956i	$0.502981\pi$
$692$	−5.12311	−0.194751
$693$	0	0
$694$	18.4384	0.699914
$695$	30.2462	1.14730
$696$	0	0
$697$	5.12311	0.194051
$698$	−10.8769	−0.411697
$699$	0	0
$700$	0	0
$701$	−28.0540	−1.05958	−0.529792	−	0.848128i	$-0.677730\pi$
$701$	−28.0540	−1.05958	−0.529792	+	0.848128i	$0.677730\pi$
$702$	0	0
$703$	−39.6155	−1.49413
$704$	14.0000	0.527645
$705$	0	0
$706$	19.1771	0.721739
$707$	0	0
$708$	0	0
$709$	5.50758	0.206841	0.103421	−	0.994638i	$-0.467021\pi$
$709$	5.50758	0.206841	0.103421	+	0.994638i	$0.467021\pi$
$710$	8.68466	0.325929
$711$	0	0
$712$	−0.738634	−0.0276815
$713$	12.0000	0.449404
$714$	0	0
$715$	1.36932	0.0512095
$716$	8.87689	0.331745
$717$	0	0
$718$	3.80776	0.142104
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	0	0
$722$	31.7386	1.18119
$723$	0	0
$724$	19.3693	0.719855
$725$	8.00000	0.297113
$726$	0	0
$727$	−50.7386	−1.88179	−0.940896	−	0.338696i	$-0.890014\pi$
$727$	−50.7386	−1.88179	−0.940896	+	0.338696i	$0.890014\pi$
$728$	0	0
$729$	0	0
$730$	18.4384	0.682438
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	8.24621	0.304581	0.152290	−	0.988336i	$-0.451335\pi$
$733$	8.24621	0.304581	0.152290	+	0.988336i	$0.451335\pi$
$734$	18.4924	0.682568
$735$	0	0
$736$	−10.0000	−0.368605
$737$	11.1231	0.409725
$738$	0	0
$739$	−40.1080	−1.47539	−0.737697	−	0.675131i	$-0.764087\pi$
$739$	−40.1080	−1.47539	−0.737697	+	0.675131i	$0.764087\pi$
$740$	−8.68466	−0.319254
$741$	0	0
$742$	0	0
$743$	−34.5464	−1.26738	−0.633692	−	0.773585i	$-0.718461\pi$
$743$	−34.5464	−1.26738	−0.633692	+	0.773585i	$0.718461\pi$
$744$	0	0
$745$	−12.8769	−0.471773
$746$	−12.6307	−0.462442
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	0	0
$751$	−30.0540	−1.09669	−0.548343	−	0.836254i	$-0.684741\pi$
$751$	−30.0540	−1.09669	−0.548343	+	0.836254i	$0.684741\pi$
$752$	−9.56155	−0.348674
$753$	0	0
$754$	1.36932	0.0498676
$755$	−12.4924	−0.454646
$756$	0	0
$757$	26.4924	0.962883	0.481442	−	0.876478i	$-0.340113\pi$
$757$	26.4924	0.962883	0.481442	+	0.876478i	$0.340113\pi$
$758$	−9.12311	−0.331366
$759$	0	0
$760$	33.3693	1.21043
$761$	−54.6004	−1.97926	−0.989631	−	0.143633i	$-0.954121\pi$
$761$	−54.6004	−1.97926	−0.989631	+	0.143633i	$0.954121\pi$
$762$	0	0
$763$	0	0
$764$	11.1231	0.402420
$765$	0	0
$766$	0	0
$767$	−5.17708	−0.186934
$768$	0	0
$769$	24.5464	0.885166	0.442583	−	0.896728i	$-0.354062\pi$
$769$	24.5464	0.885166	0.442583	+	0.896728i	$0.354062\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	22.4924	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$773$	22.4924	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$774$	0	0
$775$	15.3693	0.552082
$776$	−48.7386	−1.74961
$777$	0	0
$778$	−2.49242	−0.0893577
$779$	36.4924	1.30748
$780$	0	0
$781$	−11.1231	−0.398016
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	7.00000	0.250000
$785$	35.1231	1.25360
$786$	0	0
$787$	−34.9848	−1.24708	−0.623538	−	0.781793i	$-0.714305\pi$
$787$	−34.9848	−1.24708	−0.623538	+	0.781793i	$0.714305\pi$
$788$	17.6155	0.627527
$789$	0	0
$790$	14.2462	0.506857
$791$	0	0
$792$	0	0
$793$	−0.300187	−0.0106599
$794$	6.87689	0.244052
$795$	0	0
$796$	2.93087	0.103882
$797$	−12.9309	−0.458035	−0.229017	−	0.973422i	$-0.573551\pi$
$797$	−12.9309	−0.458035	−0.229017	+	0.973422i	$0.573551\pi$
$798$	0	0
$799$	9.56155	0.338263
$800$	−12.8078	−0.452823
$801$	0	0
$802$	3.75379	0.132551
$803$	−23.6155	−0.833374
$804$	0	0
$805$	0	0
$806$	2.63068	0.0926619
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−39.3693	−1.38415	−0.692076	−	0.721825i	$-0.743304\pi$
$809$	−39.3693	−1.38415	−0.692076	+	0.721825i	$0.743304\pi$
$810$	0	0
$811$	−7.50758	−0.263627	−0.131813	−	0.991275i	$-0.542080\pi$
$811$	−7.50758	−0.263627	−0.131813	+	0.991275i	$0.542080\pi$
$812$	0	0
$813$	0	0
$814$	−11.1231	−0.389865
$815$	33.6695	1.17939
$816$	0	0
$817$	−7.12311	−0.249206
$818$	−35.8617	−1.25388
$819$	0	0
$820$	8.00000	0.279372
$821$	16.2462	0.566997	0.283498	−	0.958973i	$-0.408505\pi$
$821$	16.2462	0.566997	0.283498	+	0.958973i	$0.408505\pi$
$822$	0	0
$823$	9.50758	0.331413	0.165707	−	0.986175i	$-0.447010\pi$
$823$	9.50758	0.331413	0.165707	+	0.986175i	$0.447010\pi$
$824$	52.6847	1.83536
$825$	0	0
$826$	0	0
$827$	28.7386	0.999340	0.499670	−	0.866216i	$-0.333455\pi$
$827$	28.7386	0.999340	0.499670	+	0.866216i	$0.333455\pi$
$828$	0	0
$829$	41.1231	1.42826	0.714132	−	0.700011i	$-0.246822\pi$
$829$	41.1231	1.42826	0.714132	+	0.700011i	$0.246822\pi$
$830$	24.6847	0.856817
$831$	0	0
$832$	−3.06913	−0.106403
$833$	−7.00000	−0.242536
$834$	0	0
$835$	34.3542	1.18887
$836$	−14.2462	−0.492716
$837$	0	0
$838$	20.0000	0.690889
$839$	−19.8078	−0.683840	−0.341920	−	0.939729i	$-0.611077\pi$
$839$	−19.8078	−0.683840	−0.341920	+	0.939729i	$0.611077\pi$
$840$	0	0
$841$	−19.2462	−0.663662
$842$	12.7386	0.439002
$843$	0	0
$844$	4.19224	0.144303
$845$	20.0000	0.688021
$846$	0	0
$847$	0	0
$848$	−10.6847	−0.366913
$849$	0	0
$850$	−2.56155	−0.0878605
$851$	11.1231	0.381295
$852$	0	0
$853$	−54.9848	−1.88265	−0.941323	−	0.337508i	$-0.890416\pi$
$853$	−54.9848	−1.88265	−0.941323	+	0.337508i	$0.890416\pi$
$854$	0	0
$855$	0	0
$856$	6.00000	0.205076
$857$	−37.1231	−1.26810	−0.634051	−	0.773292i	$-0.718609\pi$
$857$	−37.1231	−1.26810	−0.634051	+	0.773292i	$0.718609\pi$
$858$	0	0
$859$	2.63068	0.0897577	0.0448789	−	0.998992i	$-0.485710\pi$
$859$	2.63068	0.0897577	0.0448789	+	0.998992i	$0.485710\pi$
$860$	−1.56155	−0.0532485
$861$	0	0
$862$	13.1231	0.446975
$863$	12.4924	0.425247	0.212624	−	0.977134i	$-0.431799\pi$
$863$	12.4924	0.425247	0.212624	+	0.977134i	$0.431799\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	11.3693	0.386345
$867$	0	0
$868$	0	0
$869$	−18.2462	−0.618960
$870$	0	0
$871$	−2.43845	−0.0826236
$872$	12.7386	0.431385
$873$	0	0
$874$	−14.2462	−0.481885
$875$	0	0
$876$	0	0
$877$	−21.1231	−0.713277	−0.356638	−	0.934243i	$-0.616077\pi$
$877$	−21.1231	−0.713277	−0.356638	+	0.934243i	$0.616077\pi$
$878$	15.3693	0.518689
$879$	0	0
$880$	3.12311	0.105280
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−51.9157	−1.74710	−0.873551	−	0.486732i	$-0.838189\pi$
$883$	−51.9157	−1.74710	−0.873551	+	0.486732i	$0.838189\pi$
$884$	0.438447	0.0147466
$885$	0	0
$886$	31.4233	1.05569
$887$	−38.2462	−1.28418	−0.642091	−	0.766628i	$-0.721933\pi$
$887$	−38.2462	−1.28418	−0.642091	+	0.766628i	$0.721933\pi$
$888$	0	0
$889$	0	0
$890$	−0.384472	−0.0128875
$891$	0	0
$892$	16.0000	0.535720
$893$	68.1080	2.27915
$894$	0	0
$895$	13.8617	0.463347
$896$	0	0
$897$	0	0
$898$	30.4384	1.01574
$899$	18.7386	0.624968
$900$	0	0
$901$	10.6847	0.355958
$902$	10.2462	0.341162
$903$	0	0
$904$	−35.4233	−1.17816
$905$	30.2462	1.00542
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	20.4924	0.680065
$909$	0	0
$910$	0	0
$911$	−31.8078	−1.05384	−0.526919	−	0.849915i	$-0.676653\pi$
$911$	−31.8078	−1.05384	−0.526919	+	0.849915i	$0.676653\pi$
$912$	0	0
$913$	−31.6155	−1.04632
$914$	−4.63068	−0.153169
$915$	0	0
$916$	−5.80776	−0.191894
$917$	0	0
$918$	0	0
$919$	−11.8078	−0.389502	−0.194751	−	0.980853i	$-0.562390\pi$
$919$	−11.8078	−0.389502	−0.194751	+	0.980853i	$0.562390\pi$
$920$	−9.36932	−0.308897
$921$	0	0
$922$	17.8078	0.586467
$923$	2.43845	0.0802625
$924$	0	0
$925$	14.2462	0.468413
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−15.6155	−0.512605
$929$	25.8617	0.848496	0.424248	−	0.905546i	$-0.360538\pi$
$929$	25.8617	0.848496	0.424248	+	0.905546i	$0.360538\pi$
$930$	0	0
$931$	−49.8617	−1.63415
$932$	19.1231	0.626398
$933$	0	0
$934$	32.4924	1.06318
$935$	−3.12311	−0.102136
$936$	0	0
$937$	34.1080	1.11426	0.557129	−	0.830426i	$-0.311903\pi$
$937$	34.1080	1.11426	0.557129	+	0.830426i	$0.311903\pi$
$938$	0	0
$939$	0	0
$940$	14.9309	0.486991
$941$	−28.6307	−0.933334	−0.466667	−	0.884433i	$-0.654545\pi$
$941$	−28.6307	−0.933334	−0.466667	+	0.884433i	$0.654545\pi$
$942$	0	0
$943$	−10.2462	−0.333663
$944$	−11.8078	−0.384310
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	−0.246211	−0.00800079	−0.00400040	−	0.999992i	$-0.501273\pi$
$947$	−0.246211	−0.00800079	−0.00400040	+	0.999992i	$0.501273\pi$
$948$	0	0
$949$	5.17708	0.168055
$950$	−18.2462	−0.591985
$951$	0	0
$952$	0	0
$953$	29.1231	0.943390	0.471695	−	0.881762i	$-0.343642\pi$
$953$	29.1231	0.943390	0.471695	+	0.881762i	$0.343642\pi$
$954$	0	0
$955$	17.3693	0.562058
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−31.3693	−1.01350
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	2.43845	0.0786187
$963$	0	0
$964$	5.36932	0.172934
$965$	−3.12311	−0.100536
$966$	0	0
$967$	−24.9848	−0.803458	−0.401729	−	0.915758i	$-0.631591\pi$
$967$	−24.9848	−0.803458	−0.401729	+	0.915758i	$0.631591\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−25.3693	−0.814560
$971$	−17.1771	−0.551239	−0.275619	−	0.961267i	$-0.588883\pi$
$971$	−17.1771	−0.551239	−0.275619	+	0.961267i	$0.588883\pi$
$972$	0	0
$973$	0	0
$974$	−26.4924	−0.848872
$975$	0	0
$976$	−0.684658	−0.0219154
$977$	−23.7538	−0.759951	−0.379976	−	0.924997i	$-0.624068\pi$
$977$	−23.7538	−0.759951	−0.379976	+	0.924997i	$0.624068\pi$
$978$	0	0
$979$	0.492423	0.0157379
$980$	−10.9309	−0.349174
$981$	0	0
$982$	5.36932	0.171342
$983$	42.5464	1.35702	0.678510	−	0.734591i	$-0.262626\pi$
$983$	42.5464	1.35702	0.678510	+	0.734591i	$0.262626\pi$
$984$	0	0
$985$	27.5076	0.876464
$986$	−3.12311	−0.0994599
$987$	0	0
$988$	3.12311	0.0993592
$989$	2.00000	0.0635963
$990$	0	0
$991$	−55.9157	−1.77622	−0.888111	−	0.459630i	$-0.847982\pi$
$991$	−55.9157	−1.77622	−0.888111	+	0.459630i	$0.847982\pi$
$992$	−30.0000	−0.952501
$993$	0	0
$994$	0	0
$995$	4.57671	0.145091
$996$	0	0
$997$	−0.876894	−0.0277715	−0.0138858	−	0.999904i	$-0.504420\pi$
$997$	−0.876894	−0.0277715	−0.0138858	+	0.999904i	$0.504420\pi$
$998$	4.68466	0.148290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6579.2.a.f.1.1		2
3.2	odd	2		731.2.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.a.b.1.1	✓	2	3.2	odd	2
6579.2.a.f.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6579 = 3^{2} \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6579.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} -1.56155 q^{5} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} -1.56155 q^{5} -3.00000 q^{8} -1.56155 q^{10} +2.00000 q^{11} -0.438447 q^{13} -1.00000 q^{16} +1.00000 q^{17} +7.12311 q^{19} +1.56155 q^{20} +2.00000 q^{22} -2.00000 q^{23} -2.56155 q^{25} -0.438447 q^{26} -3.12311 q^{29} -6.00000 q^{31} +5.00000 q^{32} +1.00000 q^{34} -5.56155 q^{37} +7.12311 q^{38} +4.68466 q^{40} +5.12311 q^{41} -1.00000 q^{43} -2.00000 q^{44} -2.00000 q^{46} +9.56155 q^{47} -7.00000 q^{49} -2.56155 q^{50} +0.438447 q^{52} +10.6847 q^{53} -3.12311 q^{55} -3.12311 q^{58} +11.8078 q^{59} +0.684658 q^{61} -6.00000 q^{62} +7.00000 q^{64} +0.684658 q^{65} +5.56155 q^{67} -1.00000 q^{68} -5.56155 q^{71} -11.8078 q^{73} -5.56155 q^{74} -7.12311 q^{76} -9.12311 q^{79} +1.56155 q^{80} +5.12311 q^{82} -15.8078 q^{83} -1.56155 q^{85} -1.00000 q^{86} -6.00000 q^{88} +0.246211 q^{89} +2.00000 q^{92} +9.56155 q^{94} -11.1231 q^{95} +16.2462 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 + q^5 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} + q^{5} - 6 q^{8} + q^{10} + 4 q^{11} - 5 q^{13} - 2 q^{16} + 2 q^{17} + 6 q^{19} - q^{20} + 4 q^{22} - 4 q^{23} - q^{25} - 5 q^{26} + 2 q^{29} - 12 q^{31} + 10 q^{32} + 2 q^{34} - 7 q^{37} + 6 q^{38} - 3 q^{40} + 2 q^{41} - 2 q^{43} - 4 q^{44} - 4 q^{46} + 15 q^{47} - 14 q^{49} - q^{50} + 5 q^{52} + 9 q^{53} + 2 q^{55} + 2 q^{58} + 3 q^{59} - 11 q^{61} - 12 q^{62} + 14 q^{64} - 11 q^{65} + 7 q^{67} - 2 q^{68} - 7 q^{71} - 3 q^{73} - 7 q^{74} - 6 q^{76} - 10 q^{79} - q^{80} + 2 q^{82} - 11 q^{83} + q^{85} - 2 q^{86} - 12 q^{88} - 16 q^{89} + 4 q^{92} + 15 q^{94} - 14 q^{95} + 16 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + q^5 - 6 * q^8 + q^10 + 4 * q^11 - 5 * q^13 - 2 * q^16 + 2 * q^17 + 6 * q^19 - q^20 + 4 * q^22 - 4 * q^23 - q^25 - 5 * q^26 + 2 * q^29 - 12 * q^31 + 10 * q^32 + 2 * q^34 - 7 * q^37 + 6 * q^38 - 3 * q^40 + 2 * q^41 - 2 * q^43 - 4 * q^44 - 4 * q^46 + 15 * q^47 - 14 * q^49 - q^50 + 5 * q^52 + 9 * q^53 + 2 * q^55 + 2 * q^58 + 3 * q^59 - 11 * q^61 - 12 * q^62 + 14 * q^64 - 11 * q^65 + 7 * q^67 - 2 * q^68 - 7 * q^71 - 3 * q^73 - 7 * q^74 - 6 * q^76 - 10 * q^79 - q^80 + 2 * q^82 - 11 * q^83 + q^85 - 2 * q^86 - 12 * q^88 - 16 * q^89 + 4 * q^92 + 15 * q^94 - 14 * q^95 + 16 * q^97 - 14 * q^98

Introduction

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