LMFDB - Embedded newform 6384.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.46410 q^{11} +3.46410 q^{13} +3.46410 q^{15} +0.535898 q^{17} -1.00000 q^{19} +1.00000 q^{21} -1.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.92820 q^{31} -1.46410 q^{33} +3.46410 q^{35} +10.0000 q^{37} +3.46410 q^{39} -2.00000 q^{41} -6.92820 q^{43} +3.46410 q^{45} +2.92820 q^{47} +1.00000 q^{49} +0.535898 q^{51} +2.00000 q^{53} -5.07180 q^{55} -1.00000 q^{57} -4.00000 q^{59} +8.92820 q^{61} +1.00000 q^{63} +12.0000 q^{65} -2.53590 q^{67} -1.46410 q^{69} +10.9282 q^{71} +10.0000 q^{73} +7.00000 q^{75} -1.46410 q^{77} -8.39230 q^{79} +1.00000 q^{81} -8.00000 q^{83} +1.85641 q^{85} -6.00000 q^{87} -2.00000 q^{89} +3.46410 q^{91} +6.92820 q^{93} -3.46410 q^{95} +11.4641 q^{97} -1.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{17} - 2 q^{19} + 2 q^{21} + 4 q^{23} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{33} + 20 q^{37} - 4 q^{41} - 8 q^{47} + 2 q^{49} + 8 q^{51} + 4 q^{53} - 24 q^{55} - 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 24 q^{65} - 12 q^{67} + 4 q^{69} + 8 q^{71} + 20 q^{73} + 14 q^{75} + 4 q^{77} + 4 q^{79} + 2 q^{81} - 16 q^{83} - 24 q^{85} - 12 q^{87} - 4 q^{89} + 16 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^17 - 2 * q^19 + 2 * q^21 + 4 * q^23 + 14 * q^25 + 2 * q^27 - 12 * q^29 + 4 * q^33 + 20 * q^37 - 4 * q^41 - 8 * q^47 + 2 * q^49 + 8 * q^51 + 4 * q^53 - 24 * q^55 - 2 * q^57 - 8 * q^59 + 4 * q^61 + 2 * q^63 + 24 * q^65 - 12 * q^67 + 4 * q^69 + 8 * q^71 + 20 * q^73 + 14 * q^75 + 4 * q^77 + 4 * q^79 + 2 * q^81 - 16 * q^83 - 24 * q^85 - 12 * q^87 - 4 * q^89 + 16 * q^97 + 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.46410	−0.305286	−0.152643	−	0.988281i	$-0.548779\pi$
$23$	−1.46410	−0.305286	−0.152643	+	0.988281i	$0.548779\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	3.46410	0.585540
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	3.46410	0.516398
$46$	0	0
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.535898	0.0750408
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−5.07180	−0.683881
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	−2.53590	−0.309809	−0.154905	−	0.987929i	$-0.549507\pi$
$67$	−2.53590	−0.309809	−0.154905	+	0.987929i	$0.549507\pi$
$68$	0	0
$69$	−1.46410	−0.176257
$70$	0	0
$71$	10.9282	1.29694	0.648470	−	0.761241i	$-0.275409\pi$
$71$	10.9282	1.29694	0.648470	+	0.761241i	$0.275409\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	−1.46410	−0.166850
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	1.85641	0.201356
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	0	0
$93$	6.92820	0.718421
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	11.4641	1.16400	0.582002	−	0.813188i	$-0.302270\pi$
$97$	11.4641	1.16400	0.582002	+	0.813188i	$0.302270\pi$
$98$	0	0
$99$	−1.46410	−0.147148
$100$	0	0
$101$	11.4641	1.14072	0.570360	−	0.821395i	$-0.306804\pi$
$101$	11.4641	1.14072	0.570360	+	0.821395i	$0.306804\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	0	0
$105$	3.46410	0.338062
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−15.8564	−1.49165	−0.745823	−	0.666145i	$-0.767943\pi$
$113$	−15.8564	−1.49165	−0.745823	+	0.666145i	$0.767943\pi$
$114$	0	0
$115$	−5.07180	−0.472947
$116$	0	0
$117$	3.46410	0.320256
$118$	0	0
$119$	0.535898	0.0491257
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	10.5359	0.934910	0.467455	−	0.884017i	$-0.345171\pi$
$127$	10.5359	0.934910	0.467455	+	0.884017i	$0.345171\pi$
$128$	0	0
$129$	−6.92820	−0.609994
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	17.8564	1.51456	0.757280	−	0.653090i	$-0.226528\pi$
$139$	17.8564	1.51456	0.757280	+	0.653090i	$0.226528\pi$
$140$	0	0
$141$	2.92820	0.246599
$142$	0	0
$143$	−5.07180	−0.424125
$144$	0	0
$145$	−20.7846	−1.72607
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−16.9282	−1.38681	−0.693406	−	0.720547i	$-0.743891\pi$
$149$	−16.9282	−1.38681	−0.693406	+	0.720547i	$0.743891\pi$
$150$	0	0
$151$	5.46410	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$151$	5.46410	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$152$	0	0
$153$	0.535898	0.0433248
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	−7.85641	−0.627009	−0.313505	−	0.949587i	$-0.601503\pi$
$157$	−7.85641	−0.627009	−0.313505	+	0.949587i	$0.601503\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−1.46410	−0.115387
$162$	0	0
$163$	−14.9282	−1.16927	−0.584634	−	0.811297i	$-0.698762\pi$
$163$	−14.9282	−1.16927	−0.584634	+	0.811297i	$0.698762\pi$
$164$	0	0
$165$	−5.07180	−0.394839
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	12.9282	0.982913	0.491457	−	0.870902i	$-0.336465\pi$
$173$	12.9282	0.982913	0.491457	+	0.870902i	$0.336465\pi$
$174$	0	0
$175$	7.00000	0.529150
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−17.8564	−1.33465	−0.667325	−	0.744766i	$-0.732561\pi$
$179$	−17.8564	−1.33465	−0.667325	+	0.744766i	$0.732561\pi$
$180$	0	0
$181$	−15.4641	−1.14944	−0.574719	−	0.818351i	$-0.694889\pi$
$181$	−15.4641	−1.14944	−0.574719	+	0.818351i	$0.694889\pi$
$182$	0	0
$183$	8.92820	0.659992
$184$	0	0
$185$	34.6410	2.54686
$186$	0	0
$187$	−0.784610	−0.0573763
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−1.46410	−0.105939	−0.0529693	−	0.998596i	$-0.516869\pi$
$191$	−1.46410	−0.105939	−0.0529693	+	0.998596i	$0.516869\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	−8.92820	−0.636108	−0.318054	−	0.948073i	$-0.603029\pi$
$197$	−8.92820	−0.636108	−0.318054	+	0.948073i	$0.603029\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	−2.53590	−0.178868
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−6.92820	−0.483887
$206$	0	0
$207$	−1.46410	−0.101762
$208$	0	0
$209$	1.46410	0.101274
$210$	0	0
$211$	5.46410	0.376164	0.188082	−	0.982153i	$-0.439773\pi$
$211$	5.46410	0.376164	0.188082	+	0.982153i	$0.439773\pi$
$212$	0	0
$213$	10.9282	0.748788
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	6.92820	0.470317
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	1.85641	0.124875
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−1.07180	−0.0711377	−0.0355688	−	0.999367i	$-0.511324\pi$
$227$	−1.07180	−0.0711377	−0.0355688	+	0.999367i	$0.511324\pi$
$228$	0	0
$229$	16.9282	1.11865	0.559324	−	0.828949i	$-0.311061\pi$
$229$	16.9282	1.11865	0.559324	+	0.828949i	$0.311061\pi$
$230$	0	0
$231$	−1.46410	−0.0963308
$232$	0	0
$233$	−14.7846	−0.968572	−0.484286	−	0.874910i	$-0.660921\pi$
$233$	−14.7846	−0.968572	−0.484286	+	0.874910i	$0.660921\pi$
$234$	0	0
$235$	10.1436	0.661695
$236$	0	0
$237$	−8.39230	−0.545139
$238$	0	0
$239$	18.2487	1.18041	0.590206	−	0.807253i	$-0.299047\pi$
$239$	18.2487	1.18041	0.590206	+	0.807253i	$0.299047\pi$
$240$	0	0
$241$	−13.3205	−0.858049	−0.429025	−	0.903293i	$-0.641143\pi$
$241$	−13.3205	−0.858049	−0.429025	+	0.903293i	$0.641143\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.46410	0.221313
$246$	0	0
$247$	−3.46410	−0.220416
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−18.9282	−1.19474	−0.597369	−	0.801967i	$-0.703787\pi$
$251$	−18.9282	−1.19474	−0.597369	+	0.801967i	$0.703787\pi$
$252$	0	0
$253$	2.14359	0.134767
$254$	0	0
$255$	1.85641	0.116253
$256$	0	0
$257$	11.0718	0.690640	0.345320	−	0.938485i	$-0.387770\pi$
$257$	11.0718	0.690640	0.345320	+	0.938485i	$0.387770\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	23.3205	1.43800	0.719002	−	0.695008i	$-0.244599\pi$
$263$	23.3205	1.43800	0.719002	+	0.695008i	$0.244599\pi$
$264$	0	0
$265$	6.92820	0.425596
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−3.07180	−0.187291	−0.0936454	−	0.995606i	$-0.529852\pi$
$269$	−3.07180	−0.187291	−0.0936454	+	0.995606i	$0.529852\pi$
$270$	0	0
$271$	2.14359	0.130214	0.0651070	−	0.997878i	$-0.479261\pi$
$271$	2.14359	0.130214	0.0651070	+	0.997878i	$0.479261\pi$
$272$	0	0
$273$	3.46410	0.209657
$274$	0	0
$275$	−10.2487	−0.618021
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	6.92820	0.414781
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−3.46410	−0.205196
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	11.4641	0.672038
$292$	0	0
$293$	20.9282	1.22264	0.611319	−	0.791384i	$-0.290639\pi$
$293$	20.9282	1.22264	0.611319	+	0.791384i	$0.290639\pi$
$294$	0	0
$295$	−13.8564	−0.806751
$296$	0	0
$297$	−1.46410	−0.0849558
$298$	0	0
$299$	−5.07180	−0.293310
$300$	0	0
$301$	−6.92820	−0.399335
$302$	0	0
$303$	11.4641	0.658595
$304$	0	0
$305$	30.9282	1.77094
$306$	0	0
$307$	20.7846	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$307$	20.7846	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$308$	0	0
$309$	−6.92820	−0.394132
$310$	0	0
$311$	−29.8564	−1.69300	−0.846501	−	0.532388i	$-0.821295\pi$
$311$	−29.8564	−1.69300	−0.846501	+	0.532388i	$0.821295\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	3.46410	0.195180
$316$	0	0
$317$	−19.8564	−1.11525	−0.557623	−	0.830094i	$-0.688287\pi$
$317$	−19.8564	−1.11525	−0.557623	+	0.830094i	$0.688287\pi$
$318$	0	0
$319$	8.78461	0.491844
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	−0.535898	−0.0298182
$324$	0	0
$325$	24.2487	1.34508
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	2.92820	0.161437
$330$	0	0
$331$	−21.4641	−1.17977	−0.589887	−	0.807486i	$-0.700828\pi$
$331$	−21.4641	−1.17977	−0.589887	+	0.807486i	$0.700828\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	−8.78461	−0.479954
$336$	0	0
$337$	−30.7846	−1.67694	−0.838472	−	0.544944i	$-0.816551\pi$
$337$	−30.7846	−1.67694	−0.838472	+	0.544944i	$0.816551\pi$
$338$	0	0
$339$	−15.8564	−0.861202
$340$	0	0
$341$	−10.1436	−0.549306
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.07180	−0.273056
$346$	0	0
$347$	7.32051	0.392985	0.196493	−	0.980505i	$-0.437045\pi$
$347$	7.32051	0.392985	0.196493	+	0.980505i	$0.437045\pi$
$348$	0	0
$349$	8.14359	0.435917	0.217958	−	0.975958i	$-0.430060\pi$
$349$	8.14359	0.435917	0.217958	+	0.975958i	$0.430060\pi$
$350$	0	0
$351$	3.46410	0.184900
$352$	0	0
$353$	−29.3205	−1.56057	−0.780287	−	0.625422i	$-0.784927\pi$
$353$	−29.3205	−1.56057	−0.780287	+	0.625422i	$0.784927\pi$
$354$	0	0
$355$	37.8564	2.00921
$356$	0	0
$357$	0.535898	0.0283628
$358$	0	0
$359$	−0.679492	−0.0358622	−0.0179311	−	0.999839i	$-0.505708\pi$
$359$	−0.679492	−0.0358622	−0.0179311	+	0.999839i	$0.505708\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.85641	−0.464841
$364$	0	0
$365$	34.6410	1.81319
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−30.7846	−1.59397	−0.796983	−	0.604001i	$-0.793572\pi$
$373$	−30.7846	−1.59397	−0.796983	+	0.604001i	$0.793572\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−20.7846	−1.07046
$378$	0	0
$379$	−5.46410	−0.280672	−0.140336	−	0.990104i	$-0.544818\pi$
$379$	−5.46410	−0.280672	−0.140336	+	0.990104i	$0.544818\pi$
$380$	0	0
$381$	10.5359	0.539770
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−5.07180	−0.258483
$386$	0	0
$387$	−6.92820	−0.352180
$388$	0	0
$389$	−25.7128	−1.30369	−0.651846	−	0.758352i	$-0.726005\pi$
$389$	−25.7128	−1.30369	−0.651846	+	0.758352i	$0.726005\pi$
$390$	0	0
$391$	−0.784610	−0.0396794
$392$	0	0
$393$	18.9282	0.954802
$394$	0	0
$395$	−29.0718	−1.46276
$396$	0	0
$397$	−7.85641	−0.394302	−0.197151	−	0.980373i	$-0.563169\pi$
$397$	−7.85641	−0.394302	−0.197151	+	0.980373i	$0.563169\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−7.85641	−0.392330	−0.196165	−	0.980571i	$-0.562849\pi$
$401$	−7.85641	−0.392330	−0.196165	+	0.980571i	$0.562849\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	−14.6410	−0.725728
$408$	0	0
$409$	31.1769	1.54160	0.770800	−	0.637078i	$-0.219857\pi$
$409$	31.1769	1.54160	0.770800	+	0.637078i	$0.219857\pi$
$410$	0	0
$411$	12.9282	0.637701
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−27.7128	−1.36037
$416$	0	0
$417$	17.8564	0.874432
$418$	0	0
$419$	−5.07180	−0.247773	−0.123887	−	0.992296i	$-0.539536\pi$
$419$	−5.07180	−0.247773	−0.123887	+	0.992296i	$0.539536\pi$
$420$	0	0
$421$	−22.7846	−1.11045	−0.555227	−	0.831699i	$-0.687369\pi$
$421$	−22.7846	−1.11045	−0.555227	+	0.831699i	$0.687369\pi$
$422$	0	0
$423$	2.92820	0.142374
$424$	0	0
$425$	3.75129	0.181964
$426$	0	0
$427$	8.92820	0.432066
$428$	0	0
$429$	−5.07180	−0.244869
$430$	0	0
$431$	21.0718	1.01499	0.507496	−	0.861654i	$-0.330571\pi$
$431$	21.0718	1.01499	0.507496	+	0.861654i	$0.330571\pi$
$432$	0	0
$433$	41.3205	1.98574	0.992868	−	0.119215i	$-0.0380378\pi$
$433$	41.3205	1.98574	0.992868	+	0.119215i	$0.0380378\pi$
$434$	0	0
$435$	−20.7846	−0.996546
$436$	0	0
$437$	1.46410	0.0700375
$438$	0	0
$439$	−36.7846	−1.75563	−0.877817	−	0.478996i	$-0.841001\pi$
$439$	−36.7846	−1.75563	−0.877817	+	0.478996i	$0.841001\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.3205	−1.10799	−0.553995	−	0.832520i	$-0.686897\pi$
$443$	−23.3205	−1.10799	−0.553995	+	0.832520i	$0.686897\pi$
$444$	0	0
$445$	−6.92820	−0.328428
$446$	0	0
$447$	−16.9282	−0.800677
$448$	0	0
$449$	19.8564	0.937082	0.468541	−	0.883442i	$-0.344780\pi$
$449$	19.8564	0.937082	0.468541	+	0.883442i	$0.344780\pi$
$450$	0	0
$451$	2.92820	0.137884
$452$	0	0
$453$	5.46410	0.256726
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	39.8564	1.86440	0.932202	−	0.361938i	$-0.117885\pi$
$457$	39.8564	1.86440	0.932202	+	0.361938i	$0.117885\pi$
$458$	0	0
$459$	0.535898	0.0250136
$460$	0	0
$461$	−38.1051	−1.77473	−0.887366	−	0.461065i	$-0.847467\pi$
$461$	−38.1051	−1.77473	−0.887366	+	0.461065i	$0.847467\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	19.7128	0.912200	0.456100	−	0.889928i	$-0.349246\pi$
$467$	19.7128	0.912200	0.456100	+	0.889928i	$0.349246\pi$
$468$	0	0
$469$	−2.53590	−0.117097
$470$	0	0
$471$	−7.85641	−0.362004
$472$	0	0
$473$	10.1436	0.466403
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	27.7128	1.26623	0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128	1.26623	0.633115	+	0.774057i	$0.281776\pi$
$480$	0	0
$481$	34.6410	1.57949
$482$	0	0
$483$	−1.46410	−0.0666189
$484$	0	0
$485$	39.7128	1.80327
$486$	0	0
$487$	4.67949	0.212048	0.106024	−	0.994364i	$-0.466188\pi$
$487$	4.67949	0.212048	0.106024	+	0.994364i	$0.466188\pi$
$488$	0	0
$489$	−14.9282	−0.675077
$490$	0	0
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	0	0
$493$	−3.21539	−0.144814
$494$	0	0
$495$	−5.07180	−0.227960
$496$	0	0
$497$	10.9282	0.490197
$498$	0	0
$499$	9.07180	0.406109	0.203055	−	0.979167i	$-0.434913\pi$
$499$	9.07180	0.406109	0.203055	+	0.979167i	$0.434913\pi$
$500$	0	0
$501$	5.07180	0.226591
$502$	0	0
$503$	−5.85641	−0.261124	−0.130562	−	0.991440i	$-0.541678\pi$
$503$	−5.85641	−0.261124	−0.130562	+	0.991440i	$0.541678\pi$
$504$	0	0
$505$	39.7128	1.76720
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	24.6410	1.09219	0.546097	−	0.837722i	$-0.316113\pi$
$509$	24.6410	1.09219	0.546097	+	0.837722i	$0.316113\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−24.0000	−1.05757
$516$	0	0
$517$	−4.28719	−0.188550
$518$	0	0
$519$	12.9282	0.567485
$520$	0	0
$521$	−29.7128	−1.30174	−0.650871	−	0.759188i	$-0.725596\pi$
$521$	−29.7128	−1.30174	−0.650871	+	0.759188i	$0.725596\pi$
$522$	0	0
$523$	42.6410	1.86456	0.932281	−	0.361736i	$-0.117816\pi$
$523$	42.6410	1.86456	0.932281	+	0.361736i	$0.117816\pi$
$524$	0	0
$525$	7.00000	0.305505
$526$	0	0
$527$	3.71281	0.161733
$528$	0	0
$529$	−20.8564	−0.906800
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−6.92820	−0.300094
$534$	0	0
$535$	13.8564	0.599065
$536$	0	0
$537$	−17.8564	−0.770561
$538$	0	0
$539$	−1.46410	−0.0630633
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	−15.4641	−0.663628
$544$	0	0
$545$	6.92820	0.296772
$546$	0	0
$547$	−38.2487	−1.63540	−0.817698	−	0.575647i	$-0.804750\pi$
$547$	−38.2487	−1.63540	−0.817698	+	0.575647i	$0.804750\pi$
$548$	0	0
$549$	8.92820	0.381046
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−8.39230	−0.356877
$554$	0	0
$555$	34.6410	1.47043
$556$	0	0
$557$	−8.92820	−0.378300	−0.189150	−	0.981948i	$-0.560573\pi$
$557$	−8.92820	−0.378300	−0.189150	+	0.981948i	$0.560573\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−0.784610	−0.0331262
$562$	0	0
$563$	1.85641	0.0782382	0.0391191	−	0.999235i	$-0.487545\pi$
$563$	1.85641	0.0782382	0.0391191	+	0.999235i	$0.487545\pi$
$564$	0	0
$565$	−54.9282	−2.31085
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	9.71281	0.407182	0.203591	−	0.979056i	$-0.434739\pi$
$569$	9.71281	0.407182	0.203591	+	0.979056i	$0.434739\pi$
$570$	0	0
$571$	45.5692	1.90701	0.953506	−	0.301373i	$-0.0974450\pi$
$571$	45.5692	1.90701	0.953506	+	0.301373i	$0.0974450\pi$
$572$	0	0
$573$	−1.46410	−0.0611637
$574$	0	0
$575$	−10.2487	−0.427401
$576$	0	0
$577$	23.8564	0.993155	0.496578	−	0.867992i	$-0.334590\pi$
$577$	23.8564	0.993155	0.496578	+	0.867992i	$0.334590\pi$
$578$	0	0
$579$	−22.0000	−0.914289
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−2.92820	−0.121274
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	0	0
$591$	−8.92820	−0.367257
$592$	0	0
$593$	19.4641	0.799295	0.399647	−	0.916669i	$-0.369133\pi$
$593$	19.4641	0.799295	0.399647	+	0.916669i	$0.369133\pi$
$594$	0	0
$595$	1.85641	0.0761052
$596$	0	0
$597$	−16.7846	−0.686948
$598$	0	0
$599$	−26.9282	−1.10026	−0.550128	−	0.835080i	$-0.685421\pi$
$599$	−26.9282	−1.10026	−0.550128	+	0.835080i	$0.685421\pi$
$600$	0	0
$601$	41.3205	1.68550	0.842749	−	0.538306i	$-0.180936\pi$
$601$	41.3205	1.68550	0.842749	+	0.538306i	$0.180936\pi$
$602$	0	0
$603$	−2.53590	−0.103270
$604$	0	0
$605$	−30.6795	−1.24730
$606$	0	0
$607$	−9.85641	−0.400059	−0.200030	−	0.979790i	$-0.564104\pi$
$607$	−9.85641	−0.400059	−0.200030	+	0.979790i	$0.564104\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	10.1436	0.410366
$612$	0	0
$613$	3.85641	0.155759	0.0778794	−	0.996963i	$-0.475185\pi$
$613$	3.85641	0.155759	0.0778794	+	0.996963i	$0.475185\pi$
$614$	0	0
$615$	−6.92820	−0.279372
$616$	0	0
$617$	−0.928203	−0.0373681	−0.0186840	−	0.999825i	$-0.505948\pi$
$617$	−0.928203	−0.0373681	−0.0186840	+	0.999825i	$0.505948\pi$
$618$	0	0
$619$	42.6410	1.71389	0.856944	−	0.515410i	$-0.172360\pi$
$619$	42.6410	1.71389	0.856944	+	0.515410i	$0.172360\pi$
$620$	0	0
$621$	−1.46410	−0.0587524
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	1.46410	0.0584706
$628$	0	0
$629$	5.35898	0.213677
$630$	0	0
$631$	24.7846	0.986660	0.493330	−	0.869842i	$-0.335780\pi$
$631$	24.7846	0.986660	0.493330	+	0.869842i	$0.335780\pi$
$632$	0	0
$633$	5.46410	0.217179
$634$	0	0
$635$	36.4974	1.44836
$636$	0	0
$637$	3.46410	0.137253
$638$	0	0
$639$	10.9282	0.432313
$640$	0	0
$641$	11.8564	0.468300	0.234150	−	0.972200i	$-0.424769\pi$
$641$	11.8564	0.468300	0.234150	+	0.972200i	$0.424769\pi$
$642$	0	0
$643$	1.07180	0.0422675	0.0211338	−	0.999777i	$-0.493272\pi$
$643$	1.07180	0.0422675	0.0211338	+	0.999777i	$0.493272\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	5.85641	0.229884
$650$	0	0
$651$	6.92820	0.271538
$652$	0	0
$653$	21.7128	0.849688	0.424844	−	0.905267i	$-0.360329\pi$
$653$	21.7128	0.849688	0.424844	+	0.905267i	$0.360329\pi$
$654$	0	0
$655$	65.5692	2.56200
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−41.8564	−1.63049	−0.815247	−	0.579113i	$-0.803399\pi$
$659$	−41.8564	−1.63049	−0.815247	+	0.579113i	$0.803399\pi$
$660$	0	0
$661$	10.6795	0.415384	0.207692	−	0.978194i	$-0.433405\pi$
$661$	10.6795	0.415384	0.207692	+	0.978194i	$0.433405\pi$
$662$	0	0
$663$	1.85641	0.0720969
$664$	0	0
$665$	−3.46410	−0.134332
$666$	0	0
$667$	8.78461	0.340141
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	−13.0718	−0.504631
$672$	0	0
$673$	−3.07180	−0.118409	−0.0592045	−	0.998246i	$-0.518856\pi$
$673$	−3.07180	−0.118409	−0.0592045	+	0.998246i	$0.518856\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	0	0
$677$	7.85641	0.301946	0.150973	−	0.988538i	$-0.451759\pi$
$677$	7.85641	0.301946	0.150973	+	0.988538i	$0.451759\pi$
$678$	0	0
$679$	11.4641	0.439952
$680$	0	0
$681$	−1.07180	−0.0410713
$682$	0	0
$683$	−17.0718	−0.653234	−0.326617	−	0.945157i	$-0.605909\pi$
$683$	−17.0718	−0.653234	−0.326617	+	0.945157i	$0.605909\pi$
$684$	0	0
$685$	44.7846	1.71113
$686$	0	0
$687$	16.9282	0.645851
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−1.46410	−0.0556166
$694$	0	0
$695$	61.8564	2.34635
$696$	0	0
$697$	−1.07180	−0.0405972
$698$	0	0
$699$	−14.7846	−0.559205
$700$	0	0
$701$	−41.7128	−1.57547	−0.787736	−	0.616013i	$-0.788747\pi$
$701$	−41.7128	−1.57547	−0.787736	+	0.616013i	$0.788747\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	10.1436	0.382030
$706$	0	0
$707$	11.4641	0.431152
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−8.39230	−0.314736
$712$	0	0
$713$	−10.1436	−0.379881
$714$	0	0
$715$	−17.5692	−0.657052
$716$	0	0
$717$	18.2487	0.681511
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−6.92820	−0.258020
$722$	0	0
$723$	−13.3205	−0.495395
$724$	0	0
$725$	−42.0000	−1.55984
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.71281	−0.137323
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	3.46410	0.127775
$736$	0	0
$737$	3.71281	0.136763
$738$	0	0
$739$	−17.8564	−0.656859	−0.328429	−	0.944529i	$-0.606519\pi$
$739$	−17.8564	−0.656859	−0.328429	+	0.944529i	$0.606519\pi$
$740$	0	0
$741$	−3.46410	−0.127257
$742$	0	0
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	−58.6410	−2.14844
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	0.392305	0.0143154	0.00715770	−	0.999974i	$-0.497722\pi$
$751$	0.392305	0.0143154	0.00715770	+	0.999974i	$0.497722\pi$
$752$	0	0
$753$	−18.9282	−0.689782
$754$	0	0
$755$	18.9282	0.688868
$756$	0	0
$757$	−23.8564	−0.867076	−0.433538	−	0.901135i	$-0.642735\pi$
$757$	−23.8564	−0.867076	−0.433538	+	0.901135i	$0.642735\pi$
$758$	0	0
$759$	2.14359	0.0778075
$760$	0	0
$761$	11.4641	0.415573	0.207787	−	0.978174i	$-0.433374\pi$
$761$	11.4641	0.415573	0.207787	+	0.978174i	$0.433374\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	1.85641	0.0671185
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	7.85641	0.283309	0.141655	−	0.989916i	$-0.454758\pi$
$769$	7.85641	0.283309	0.141655	+	0.989916i	$0.454758\pi$
$770$	0	0
$771$	11.0718	0.398741
$772$	0	0
$773$	−38.7846	−1.39499	−0.697493	−	0.716592i	$-0.745701\pi$
$773$	−38.7846	−1.39499	−0.697493	+	0.716592i	$0.745701\pi$
$774$	0	0
$775$	48.4974	1.74208
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−27.2154	−0.971359
$786$	0	0
$787$	52.7846	1.88157	0.940784	−	0.339006i	$-0.110091\pi$
$787$	52.7846	1.88157	0.940784	+	0.339006i	$0.110091\pi$
$788$	0	0
$789$	23.3205	0.830232
$790$	0	0
$791$	−15.8564	−0.563789
$792$	0	0
$793$	30.9282	1.09829
$794$	0	0
$795$	6.92820	0.245718
$796$	0	0
$797$	24.6410	0.872830	0.436415	−	0.899746i	$-0.356248\pi$
$797$	24.6410	0.872830	0.436415	+	0.899746i	$0.356248\pi$
$798$	0	0
$799$	1.56922	0.0555150
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−14.6410	−0.516670
$804$	0	0
$805$	−5.07180	−0.178757
$806$	0	0
$807$	−3.07180	−0.108132
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−46.9282	−1.64787	−0.823936	−	0.566683i	$-0.808227\pi$
$811$	−46.9282	−1.64787	−0.823936	+	0.566683i	$0.808227\pi$
$812$	0	0
$813$	2.14359	0.0751791
$814$	0	0
$815$	−51.7128	−1.81142
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	3.46410	0.121046
$820$	0	0
$821$	−35.0718	−1.22401	−0.612007	−	0.790852i	$-0.709638\pi$
$821$	−35.0718	−1.22401	−0.612007	+	0.790852i	$0.709638\pi$
$822$	0	0
$823$	−5.85641	−0.204141	−0.102071	−	0.994777i	$-0.532547\pi$
$823$	−5.85641	−0.204141	−0.102071	+	0.994777i	$0.532547\pi$
$824$	0	0
$825$	−10.2487	−0.356814
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−26.3923	−0.916643	−0.458321	−	0.888787i	$-0.651549\pi$
$829$	−26.3923	−0.916643	−0.458321	+	0.888787i	$0.651549\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	0.535898	0.0185678
$834$	0	0
$835$	17.5692	0.608008
$836$	0	0
$837$	6.92820	0.239474
$838$	0	0
$839$	18.9282	0.653474	0.326737	−	0.945115i	$-0.394051\pi$
$839$	18.9282	0.653474	0.326737	+	0.945115i	$0.394051\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−26.0000	−0.895488
$844$	0	0
$845$	−3.46410	−0.119169
$846$	0	0
$847$	−8.85641	−0.304310
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−14.6410	−0.501888
$852$	0	0
$853$	48.9282	1.67527	0.837635	−	0.546231i	$-0.183938\pi$
$853$	48.9282	1.67527	0.837635	+	0.546231i	$0.183938\pi$
$854$	0	0
$855$	−3.46410	−0.118470
$856$	0	0
$857$	−35.5692	−1.21502	−0.607511	−	0.794311i	$-0.707832\pi$
$857$	−35.5692	−1.21502	−0.607511	+	0.794311i	$0.707832\pi$
$858$	0	0
$859$	−12.7846	−0.436205	−0.218103	−	0.975926i	$-0.569987\pi$
$859$	−12.7846	−0.436205	−0.218103	+	0.975926i	$0.569987\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	−45.8564	−1.56097	−0.780485	−	0.625174i	$-0.785028\pi$
$863$	−45.8564	−1.56097	−0.780485	+	0.625174i	$0.785028\pi$
$864$	0	0
$865$	44.7846	1.52272
$866$	0	0
$867$	−16.7128	−0.567597
$868$	0	0
$869$	12.2872	0.416814
$870$	0	0
$871$	−8.78461	−0.297655
$872$	0	0
$873$	11.4641	0.388001
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	20.9282	0.705891
$880$	0	0
$881$	−19.1769	−0.646087	−0.323043	−	0.946384i	$-0.604706\pi$
$881$	−19.1769	−0.646087	−0.323043	+	0.946384i	$0.604706\pi$
$882$	0	0
$883$	48.4974	1.63207	0.816034	−	0.578004i	$-0.196168\pi$
$883$	48.4974	1.63207	0.816034	+	0.578004i	$0.196168\pi$
$884$	0	0
$885$	−13.8564	−0.465778
$886$	0	0
$887$	−27.7128	−0.930505	−0.465253	−	0.885178i	$-0.654037\pi$
$887$	−27.7128	−0.930505	−0.465253	+	0.885178i	$0.654037\pi$
$888$	0	0
$889$	10.5359	0.353363
$890$	0	0
$891$	−1.46410	−0.0490492
$892$	0	0
$893$	−2.92820	−0.0979886
$894$	0	0
$895$	−61.8564	−2.06763
$896$	0	0
$897$	−5.07180	−0.169342
$898$	0	0
$899$	−41.5692	−1.38641
$900$	0	0
$901$	1.07180	0.0357067
$902$	0	0
$903$	−6.92820	−0.230556
$904$	0	0
$905$	−53.5692	−1.78070
$906$	0	0
$907$	34.5359	1.14675	0.573373	−	0.819295i	$-0.305635\pi$
$907$	34.5359	1.14675	0.573373	+	0.819295i	$0.305635\pi$
$908$	0	0
$909$	11.4641	0.380240
$910$	0	0
$911$	46.6410	1.54529	0.772643	−	0.634841i	$-0.218934\pi$
$911$	46.6410	1.54529	0.772643	+	0.634841i	$0.218934\pi$
$912$	0	0
$913$	11.7128	0.387638
$914$	0	0
$915$	30.9282	1.02245
$916$	0	0
$917$	18.9282	0.625064
$918$	0	0
$919$	−33.5692	−1.10735	−0.553673	−	0.832734i	$-0.686774\pi$
$919$	−33.5692	−1.10735	−0.553673	+	0.832734i	$0.686774\pi$
$920$	0	0
$921$	20.7846	0.684876
$922$	0	0
$923$	37.8564	1.24606
$924$	0	0
$925$	70.0000	2.30159
$926$	0	0
$927$	−6.92820	−0.227552
$928$	0	0
$929$	46.3923	1.52208	0.761041	−	0.648704i	$-0.224689\pi$
$929$	46.3923	1.52208	0.761041	+	0.648704i	$0.224689\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−29.8564	−0.977455
$934$	0	0
$935$	−2.71797	−0.0888870
$936$	0	0
$937$	7.07180	0.231026	0.115513	−	0.993306i	$-0.463149\pi$
$937$	7.07180	0.231026	0.115513	+	0.993306i	$0.463149\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−46.7846	−1.52513	−0.762567	−	0.646909i	$-0.776061\pi$
$941$	−46.7846	−1.52513	−0.762567	+	0.646909i	$0.776061\pi$
$942$	0	0
$943$	2.92820	0.0953554
$944$	0	0
$945$	3.46410	0.112687
$946$	0	0
$947$	−48.1051	−1.56321	−0.781603	−	0.623776i	$-0.785598\pi$
$947$	−48.1051	−1.56321	−0.781603	+	0.623776i	$0.785598\pi$
$948$	0	0
$949$	34.6410	1.12449
$950$	0	0
$951$	−19.8564	−0.643888
$952$	0	0
$953$	−4.14359	−0.134224	−0.0671121	−	0.997745i	$-0.521379\pi$
$953$	−4.14359	−0.134224	−0.0671121	+	0.997745i	$0.521379\pi$
$954$	0	0
$955$	−5.07180	−0.164119
$956$	0	0
$957$	8.78461	0.283966
$958$	0	0
$959$	12.9282	0.417473
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−76.2102	−2.45329
$966$	0	0
$967$	5.07180	0.163098	0.0815490	−	0.996669i	$-0.474013\pi$
$967$	5.07180	0.163098	0.0815490	+	0.996669i	$0.474013\pi$
$968$	0	0
$969$	−0.535898	−0.0172155
$970$	0	0
$971$	−58.6410	−1.88188	−0.940940	−	0.338574i	$-0.890056\pi$
$971$	−58.6410	−1.88188	−0.940940	+	0.338574i	$0.890056\pi$
$972$	0	0
$973$	17.8564	0.572450
$974$	0	0
$975$	24.2487	0.776580
$976$	0	0
$977$	19.8564	0.635263	0.317631	−	0.948214i	$-0.397113\pi$
$977$	19.8564	0.635263	0.317631	+	0.948214i	$0.397113\pi$
$978$	0	0
$979$	2.92820	0.0935858
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−8.78461	−0.280186	−0.140093	−	0.990138i	$-0.544740\pi$
$983$	−8.78461	−0.280186	−0.140093	+	0.990138i	$0.544740\pi$
$984$	0	0
$985$	−30.9282	−0.985454
$986$	0	0
$987$	2.92820	0.0932057
$988$	0	0
$989$	10.1436	0.322548
$990$	0	0
$991$	−56.3923	−1.79136	−0.895680	−	0.444699i	$-0.853311\pi$
$991$	−56.3923	−1.79136	−0.895680	+	0.444699i	$0.853311\pi$
$992$	0	0
$993$	−21.4641	−0.681143
$994$	0	0
$995$	−58.1436	−1.84328
$996$	0	0
$997$	−20.1436	−0.637954	−0.318977	−	0.947762i	$-0.603339\pi$
$997$	−20.1436	−0.637954	−0.318977	+	0.947762i	$0.603339\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.br.1.2		2
4.3	odd	2		798.2.a.k.1.2	✓	2
12.11	even	2		2394.2.a.x.1.1		2
28.27	even	2		5586.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.k.1.2	✓	2	4.3	odd	2
2394.2.a.x.1.1		2	12.11	even	2
5586.2.a.bd.1.1		2	28.27	even	2
6384.2.a.br.1.2		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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.46410 q^{11} +3.46410 q^{13} +3.46410 q^{15} +0.535898 q^{17} -1.00000 q^{19} +1.00000 q^{21} -1.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.92820 q^{31} -1.46410 q^{33} +3.46410 q^{35} +10.0000 q^{37} +3.46410 q^{39} -2.00000 q^{41} -6.92820 q^{43} +3.46410 q^{45} +2.92820 q^{47} +1.00000 q^{49} +0.535898 q^{51} +2.00000 q^{53} -5.07180 q^{55} -1.00000 q^{57} -4.00000 q^{59} +8.92820 q^{61} +1.00000 q^{63} +12.0000 q^{65} -2.53590 q^{67} -1.46410 q^{69} +10.9282 q^{71} +10.0000 q^{73} +7.00000 q^{75} -1.46410 q^{77} -8.39230 q^{79} +1.00000 q^{81} -8.00000 q^{83} +1.85641 q^{85} -6.00000 q^{87} -2.00000 q^{89} +3.46410 q^{91} +6.92820 q^{93} -3.46410 q^{95} +11.4641 q^{97} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{17} - 2 q^{19} + 2 q^{21} + 4 q^{23} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{33} + 20 q^{37} - 4 q^{41} - 8 q^{47} + 2 q^{49} + 8 q^{51} + 4 q^{53} - 24 q^{55} - 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 24 q^{65} - 12 q^{67} + 4 q^{69} + 8 q^{71} + 20 q^{73} + 14 q^{75} + 4 q^{77} + 4 q^{79} + 2 q^{81} - 16 q^{83} - 24 q^{85} - 12 q^{87} - 4 q^{89} + 16 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^17 - 2 * q^19 + 2 * q^21 + 4 * q^23 + 14 * q^25 + 2 * q^27 - 12 * q^29 + 4 * q^33 + 20 * q^37 - 4 * q^41 - 8 * q^47 + 2 * q^49 + 8 * q^51 + 4 * q^53 - 24 * q^55 - 2 * q^57 - 8 * q^59 + 4 * q^61 + 2 * q^63 + 24 * q^65 - 12 * q^67 + 4 * q^69 + 8 * q^71 + 20 * q^73 + 14 * q^75 + 4 * q^77 + 4 * q^79 + 2 * q^81 - 16 * q^83 - 24 * q^85 - 12 * q^87 - 4 * q^89 + 16 * q^97 + 4 * q^99

Introduction

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