LMFDB - Embedded newform 6480.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +0.514137 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.514137 q^{7} -3.32088 q^{11} -1.32088 q^{13} +3.32088 q^{17} +1.32088 q^{19} +4.12763 q^{23} +1.00000 q^{25} +1.38650 q^{29} -8.73566 q^{31} -0.514137 q^{35} +0.292611 q^{37} +11.3492 q^{41} -10.3492 q^{43} -4.86330 q^{47} -6.73566 q^{49} +5.02827 q^{53} +3.32088 q^{55} -5.02827 q^{59} +7.34916 q^{61} +1.32088 q^{65} -9.44852 q^{67} +8.99093 q^{71} +6.05655 q^{73} -1.70739 q^{77} +8.05655 q^{79} +1.54241 q^{83} -3.32088 q^{85} +3.00000 q^{89} -0.679116 q^{91} -1.32088 q^{95} -12.2553 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - 5 * q^7	$ 3 q - 3 q^{5} - 5 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 3 q^{23} + 3 q^{25} + 7 q^{29} - 8 q^{31} + 5 q^{35} + 6 q^{37} + 13 q^{41} - 10 q^{43} + 13 q^{47} - 2 q^{49} + 2 q^{53} + 2 q^{55} - 2 q^{59} + q^{61} - 4 q^{65} - 11 q^{67} - 10 q^{71} - 8 q^{73} - 2 q^{79} - 15 q^{83} - 2 q^{85} + 9 q^{89} - 10 q^{91} + 4 q^{95} - 18 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 5 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 4 * q^19 + 3 * q^23 + 3 * q^25 + 7 * q^29 - 8 * q^31 + 5 * q^35 + 6 * q^37 + 13 * q^41 - 10 * q^43 + 13 * q^47 - 2 * q^49 + 2 * q^53 + 2 * q^55 - 2 * q^59 + q^61 - 4 * q^65 - 11 * q^67 - 10 * q^71 - 8 * q^73 - 2 * q^79 - 15 * q^83 - 2 * q^85 + 9 * q^89 - 10 * q^91 + 4 * q^95 - 18 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.514137	0.194325	0.0971627	−	0.995269i	$-0.469023\pi$
$7$	0.514137	0.194325	0.0971627	+	0.995269i	$0.469023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.32088	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$11$	−3.32088	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$12$	0	0
$13$	−1.32088	−0.366347	−0.183174	−	0.983081i	$-0.558637\pi$
$13$	−1.32088	−0.366347	−0.183174	+	0.983081i	$0.558637\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.32088	0.805433	0.402716	−	0.915325i	$-0.368066\pi$
$17$	3.32088	0.805433	0.402716	+	0.915325i	$0.368066\pi$
$18$	0	0
$19$	1.32088	0.303032	0.151516	−	0.988455i	$-0.451585\pi$
$19$	1.32088	0.303032	0.151516	+	0.988455i	$0.451585\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.12763	0.860671	0.430335	−	0.902669i	$-0.358395\pi$
$23$	4.12763	0.860671	0.430335	+	0.902669i	$0.358395\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.38650	0.257467	0.128734	−	0.991679i	$-0.458909\pi$
$29$	1.38650	0.257467	0.128734	+	0.991679i	$0.458909\pi$
$30$	0	0
$31$	−8.73566	−1.56897	−0.784486	−	0.620147i	$-0.787073\pi$
$31$	−8.73566	−1.56897	−0.784486	+	0.620147i	$0.787073\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.514137	−0.0869050
$36$	0	0
$37$	0.292611	0.0481049	0.0240524	−	0.999711i	$-0.492343\pi$
$37$	0.292611	0.0481049	0.0240524	+	0.999711i	$0.492343\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.3492	1.77244	0.886220	−	0.463264i	$-0.153322\pi$
$41$	11.3492	1.77244	0.886220	+	0.463264i	$0.153322\pi$
$42$	0	0
$43$	−10.3492	−1.57823	−0.789116	−	0.614244i	$-0.789461\pi$
$43$	−10.3492	−1.57823	−0.789116	+	0.614244i	$0.789461\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.86330	−0.709385	−0.354692	−	0.934983i	$-0.615414\pi$
$47$	−4.86330	−0.709385	−0.354692	+	0.934983i	$0.615414\pi$
$48$	0	0
$49$	−6.73566	−0.962238
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.02827	0.690687	0.345343	−	0.938476i	$-0.387762\pi$
$53$	5.02827	0.690687	0.345343	+	0.938476i	$0.387762\pi$
$54$	0	0
$55$	3.32088	0.447788
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.02827	−0.654625	−0.327313	−	0.944916i	$-0.606143\pi$
$59$	−5.02827	−0.654625	−0.327313	+	0.944916i	$0.606143\pi$
$60$	0	0
$61$	7.34916	0.940963	0.470482	−	0.882410i	$-0.344080\pi$
$61$	7.34916	0.940963	0.470482	+	0.882410i	$0.344080\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.32088	0.163836
$66$	0	0
$67$	−9.44852	−1.15432	−0.577160	−	0.816631i	$-0.695839\pi$
$67$	−9.44852	−1.15432	−0.577160	+	0.816631i	$0.695839\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.99093	1.06703	0.533513	−	0.845792i	$-0.320871\pi$
$71$	8.99093	1.06703	0.533513	+	0.845792i	$0.320871\pi$
$72$	0	0
$73$	6.05655	0.708865	0.354433	−	0.935082i	$-0.384674\pi$
$73$	6.05655	0.708865	0.354433	+	0.935082i	$0.384674\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.70739	−0.194575
$78$	0	0
$79$	8.05655	0.906432	0.453216	−	0.891401i	$-0.350277\pi$
$79$	8.05655	0.906432	0.453216	+	0.891401i	$0.350277\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.54241	0.169302	0.0846508	−	0.996411i	$-0.473023\pi$
$83$	1.54241	0.169302	0.0846508	+	0.996411i	$0.473023\pi$
$84$	0	0
$85$	−3.32088	−0.360200
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−0.679116	−0.0711906
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.32088	−0.135520
$96$	0	0
$97$	−12.2553	−1.24433	−0.622167	−	0.782885i	$-0.713747\pi$
$97$	−12.2553	−1.24433	−0.622167	+	0.782885i	$0.713747\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.6700	−1.16121	−0.580606	−	0.814184i	$-0.697184\pi$
$101$	−11.6700	−1.16121	−0.580606	+	0.814184i	$0.697184\pi$
$102$	0	0
$103$	−0.292611	−0.0288318	−0.0144159	−	0.999896i	$-0.504589\pi$
$103$	−0.292611	−0.0288318	−0.0144159	+	0.999896i	$0.504589\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.87237	0.181009	0.0905043	−	0.995896i	$-0.471152\pi$
$107$	1.87237	0.181009	0.0905043	+	0.995896i	$0.471152\pi$
$108$	0	0
$109$	5.54787	0.531390	0.265695	−	0.964057i	$-0.414399\pi$
$109$	5.54787	0.531390	0.265695	+	0.964057i	$0.414399\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.80128	−0.733883	−0.366942	−	0.930244i	$-0.619595\pi$
$113$	−7.80128	−0.733883	−0.366942	+	0.930244i	$0.619595\pi$
$114$	0	0
$115$	−4.12763	−0.384904
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.70739	0.156516
$120$	0	0
$121$	0.0282739	0.00257035
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.8916	−1.58762	−0.793810	−	0.608166i	$-0.791906\pi$
$127$	−17.8916	−1.58762	−0.793810	+	0.608166i	$0.791906\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0.679116	0.0588868
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.67004	−0.484424	−0.242212	−	0.970223i	$-0.577873\pi$
$137$	−5.67004	−0.484424	−0.242212	+	0.970223i	$0.577873\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.38650	0.366818
$144$	0	0
$145$	−1.38650	−0.115143
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.6610	−1.44684	−0.723422	−	0.690407i	$-0.757432\pi$
$149$	−17.6610	−1.44684	−0.723422	+	0.690407i	$0.757432\pi$
$150$	0	0
$151$	−1.26434	−0.102890	−0.0514451	−	0.998676i	$-0.516383\pi$
$151$	−1.26434	−0.102890	−0.0514451	+	0.998676i	$0.516383\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.73566	0.701665
$156$	0	0
$157$	−15.6700	−1.25061	−0.625303	−	0.780382i	$-0.715025\pi$
$157$	−15.6700	−1.25061	−0.625303	+	0.780382i	$0.715025\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.12217	0.167250
$162$	0	0
$163$	−15.7074	−1.23030	−0.615149	−	0.788411i	$-0.710904\pi$
$163$	−15.7074	−1.23030	−0.615149	+	0.788411i	$0.710904\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.16498	0.477060	0.238530	−	0.971135i	$-0.423334\pi$
$167$	6.16498	0.477060	0.238530	+	0.971135i	$0.423334\pi$
$168$	0	0
$169$	−11.2553	−0.865790
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.58522	−0.652722	−0.326361	−	0.945245i	$-0.605823\pi$
$173$	−8.58522	−0.652722	−0.326361	+	0.945245i	$0.605823\pi$
$174$	0	0
$175$	0.514137	0.0388651
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.06562	−0.0796482	−0.0398241	−	0.999207i	$-0.512680\pi$
$179$	−1.06562	−0.0796482	−0.0398241	+	0.999207i	$0.512680\pi$
$180$	0	0
$181$	−12.6700	−0.941757	−0.470878	−	0.882198i	$-0.656063\pi$
$181$	−12.6700	−0.941757	−0.470878	+	0.882198i	$0.656063\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.292611	−0.0215132
$186$	0	0
$187$	−11.0283	−0.806467
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9344	−1.22533	−0.612664	−	0.790343i	$-0.709902\pi$
$191$	−16.9344	−1.22533	−0.612664	+	0.790343i	$0.709902\pi$
$192$	0	0
$193$	26.7175	1.92317	0.961585	−	0.274509i	$-0.0885153\pi$
$193$	26.7175	1.92317	0.961585	+	0.274509i	$0.0885153\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.2553	1.01565	0.507823	−	0.861462i	$-0.330450\pi$
$197$	14.2553	1.01565	0.507823	+	0.861462i	$0.330450\pi$
$198$	0	0
$199$	24.6610	1.74817	0.874085	−	0.485773i	$-0.161462\pi$
$199$	24.6610	1.74817	0.874085	+	0.485773i	$0.161462\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.712853	0.0500325
$204$	0	0
$205$	−11.3492	−0.792660
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.38650	−0.303421
$210$	0	0
$211$	5.37743	0.370198	0.185099	−	0.982720i	$-0.440739\pi$
$211$	5.37743	0.370198	0.185099	+	0.982720i	$0.440739\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.3492	0.705807
$216$	0	0
$217$	−4.49133	−0.304891
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.38650	−0.295068
$222$	0	0
$223$	−8.66458	−0.580223	−0.290112	−	0.956993i	$-0.593692\pi$
$223$	−8.66458	−0.580223	−0.290112	+	0.956993i	$0.593692\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.32088	0.220415	0.110207	−	0.993909i	$-0.464848\pi$
$227$	3.32088	0.220415	0.110207	+	0.993909i	$0.464848\pi$
$228$	0	0
$229$	−25.3118	−1.67265	−0.836326	−	0.548233i	$-0.815301\pi$
$229$	−25.3118	−1.67265	−0.836326	+	0.548233i	$0.815301\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.6327	−1.81028	−0.905139	−	0.425116i	$-0.860233\pi$
$233$	−27.6327	−1.81028	−0.905139	+	0.425116i	$0.860233\pi$
$234$	0	0
$235$	4.86330	0.317246
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.19872	−0.271592	−0.135796	−	0.990737i	$-0.543359\pi$
$239$	−4.19872	−0.271592	−0.135796	+	0.990737i	$0.543359\pi$
$240$	0	0
$241$	3.60442	0.232181	0.116091	−	0.993239i	$-0.462964\pi$
$241$	3.60442	0.232181	0.116091	+	0.993239i	$0.462964\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.73566	0.430326
$246$	0	0
$247$	−1.74474	−0.111015
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.87783	−0.434125	−0.217062	−	0.976158i	$-0.569648\pi$
$251$	−6.87783	−0.434125	−0.217062	+	0.976158i	$0.569648\pi$
$252$	0	0
$253$	−13.7074	−0.861776
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	0.150442	0.00934801
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.23606	0.384532	0.192266	−	0.981343i	$-0.438416\pi$
$263$	6.23606	0.384532	0.192266	+	0.981343i	$0.438416\pi$
$264$	0	0
$265$	−5.02827	−0.308884
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.92345	−0.605044	−0.302522	−	0.953142i	$-0.597828\pi$
$269$	−9.92345	−0.605044	−0.302522	+	0.953142i	$0.597828\pi$
$270$	0	0
$271$	−6.60442	−0.401190	−0.200595	−	0.979674i	$-0.564288\pi$
$271$	−6.60442	−0.401190	−0.200595	+	0.979674i	$0.564288\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.32088	−0.200257
$276$	0	0
$277$	22.6610	1.36157	0.680783	−	0.732485i	$-0.261640\pi$
$277$	22.6610	1.36157	0.680783	+	0.732485i	$0.261640\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.5479	0.927508	0.463754	−	0.885964i	$-0.346502\pi$
$281$	15.5479	0.927508	0.463754	+	0.885964i	$0.346502\pi$
$282$	0	0
$283$	−0.645378	−0.0383637	−0.0191819	−	0.999816i	$-0.506106\pi$
$283$	−0.645378	−0.0383637	−0.0191819	+	0.999816i	$0.506106\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.83502	0.344430
$288$	0	0
$289$	−5.97173	−0.351278
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.37743	0.0804704	0.0402352	−	0.999190i	$-0.487189\pi$
$293$	1.37743	0.0804704	0.0402352	+	0.999190i	$0.487189\pi$
$294$	0	0
$295$	5.02827	0.292757
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.45213	−0.315305
$300$	0	0
$301$	−5.32088	−0.306691
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.34916	−0.420812
$306$	0	0
$307$	7.98546	0.455754	0.227877	−	0.973690i	$-0.426822\pi$
$307$	7.98546	0.455754	0.227877	+	0.973690i	$0.426822\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.63270	0.546220	0.273110	−	0.961983i	$-0.411948\pi$
$311$	9.63270	0.546220	0.273110	+	0.961983i	$0.411948\pi$
$312$	0	0
$313$	−24.5369	−1.38691	−0.693455	−	0.720500i	$-0.743912\pi$
$313$	−24.5369	−1.38691	−0.693455	+	0.720500i	$0.743912\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.3492	1.14292	0.571461	−	0.820629i	$-0.306377\pi$
$317$	20.3492	1.14292	0.571461	+	0.820629i	$0.306377\pi$
$318$	0	0
$319$	−4.60442	−0.257798
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.38650	0.244072
$324$	0	0
$325$	−1.32088	−0.0732695
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.50040	−0.137851
$330$	0	0
$331$	−16.4431	−0.903792	−0.451896	−	0.892071i	$-0.649252\pi$
$331$	−16.4431	−0.903792	−0.451896	+	0.892071i	$0.649252\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.44852	0.516228
$336$	0	0
$337$	4.89703	0.266758	0.133379	−	0.991065i	$-0.457417\pi$
$337$	4.89703	0.266758	0.133379	+	0.991065i	$0.457417\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	29.0101	1.57099
$342$	0	0
$343$	−7.06201	−0.381313
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.2745	1.19576	0.597878	−	0.801587i	$-0.296011\pi$
$347$	22.2745	1.19576	0.597878	+	0.801587i	$0.296011\pi$
$348$	0	0
$349$	−2.94345	−0.157559	−0.0787797	−	0.996892i	$-0.525102\pi$
$349$	−2.94345	−0.157559	−0.0787797	+	0.996892i	$0.525102\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.8296	−1.00220	−0.501098	−	0.865390i	$-0.667070\pi$
$353$	−18.8296	−1.00220	−0.501098	+	0.865390i	$0.667070\pi$
$354$	0	0
$355$	−8.99093	−0.477189
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.8770	−1.68241	−0.841203	−	0.540720i	$-0.818152\pi$
$359$	−31.8770	−1.68241	−0.841203	+	0.540720i	$0.818152\pi$
$360$	0	0
$361$	−17.2553	−0.908172
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.05655	−0.317014
$366$	0	0
$367$	18.3492	0.957818	0.478909	−	0.877864i	$-0.341032\pi$
$367$	18.3492	0.957818	0.478909	+	0.877864i	$0.341032\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.58522	0.134218
$372$	0	0
$373$	−2.19872	−0.113845	−0.0569226	−	0.998379i	$-0.518129\pi$
$373$	−2.19872	−0.113845	−0.0569226	+	0.998379i	$0.518129\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.83141	−0.0943226
$378$	0	0
$379$	−15.4713	−0.794709	−0.397354	−	0.917665i	$-0.630072\pi$
$379$	−15.4713	−0.794709	−0.397354	+	0.917665i	$0.630072\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.70739	0.393829	0.196915	−	0.980421i	$-0.436908\pi$
$383$	7.70739	0.393829	0.196915	+	0.980421i	$0.436908\pi$
$384$	0	0
$385$	1.70739	0.0870166
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.6327	1.24893	0.624464	−	0.781054i	$-0.285318\pi$
$389$	24.6327	1.24893	0.624464	+	0.781054i	$0.285318\pi$
$390$	0	0
$391$	13.7074	0.693212
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.05655	−0.405369
$396$	0	0
$397$	−6.77301	−0.339928	−0.169964	−	0.985450i	$-0.554365\pi$
$397$	−6.77301	−0.339928	−0.169964	+	0.985450i	$0.554365\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.4996	0.923826	0.461913	−	0.886925i	$-0.347163\pi$
$401$	18.4996	0.923826	0.461913	+	0.886925i	$0.347163\pi$
$402$	0	0
$403$	11.5388	0.574789
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.971726	−0.0481667
$408$	0	0
$409$	−13.4148	−0.663318	−0.331659	−	0.943399i	$-0.607608\pi$
$409$	−13.4148	−0.663318	−0.331659	+	0.943399i	$0.607608\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.58522	−0.127210
$414$	0	0
$415$	−1.54241	−0.0757140
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.1150	−1.61777	−0.808886	−	0.587966i	$-0.799929\pi$
$419$	−33.1150	−1.61777	−0.808886	+	0.587966i	$0.799929\pi$
$420$	0	0
$421$	−14.6983	−0.716352	−0.358176	−	0.933654i	$-0.616601\pi$
$421$	−14.6983	−0.716352	−0.358176	+	0.933654i	$0.616601\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.32088	0.161087
$426$	0	0
$427$	3.77847	0.182853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.7549	−1.57775	−0.788873	−	0.614556i	$-0.789335\pi$
$431$	−32.7549	−1.57775	−0.788873	+	0.614556i	$0.789335\pi$
$432$	0	0
$433$	−11.8314	−0.568581	−0.284291	−	0.958738i	$-0.591758\pi$
$433$	−11.8314	−0.568581	−0.284291	+	0.958738i	$0.591758\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.45213	0.260811
$438$	0	0
$439$	−8.31181	−0.396701	−0.198351	−	0.980131i	$-0.563558\pi$
$439$	−8.31181	−0.396701	−0.198351	+	0.980131i	$0.563558\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.1751	−1.38615	−0.693076	−	0.720865i	$-0.743745\pi$
$443$	−29.1751	−1.38615	−0.693076	+	0.720865i	$0.743745\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.9717	−0.895331	−0.447666	−	0.894201i	$-0.647744\pi$
$449$	−18.9717	−0.895331	−0.447666	+	0.894201i	$0.647744\pi$
$450$	0	0
$451$	−37.6892	−1.77472
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.679116	0.0318374
$456$	0	0
$457$	−23.2353	−1.08690	−0.543450	−	0.839442i	$-0.682882\pi$
$457$	−23.2353	−1.08690	−0.543450	+	0.839442i	$0.682882\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.42571	−0.206126	−0.103063	−	0.994675i	$-0.532864\pi$
$461$	−4.42571	−0.206126	−0.103063	+	0.994675i	$0.532864\pi$
$462$	0	0
$463$	19.5087	0.906645	0.453322	−	0.891347i	$-0.350239\pi$
$463$	19.5087	0.906645	0.453322	+	0.891347i	$0.350239\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.5935	−1.13805	−0.569026	−	0.822320i	$-0.692679\pi$
$467$	−24.5935	−1.13805	−0.569026	+	0.822320i	$0.692679\pi$
$468$	0	0
$469$	−4.85783	−0.224314
$470$	0	0
$471$	0	0
$472$	0	0
$473$	34.3684	1.58026
$474$	0	0
$475$	1.32088	0.0606063
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.7549	1.49661	0.748304	−	0.663356i	$-0.230868\pi$
$479$	32.7549	1.49661	0.748304	+	0.663356i	$0.230868\pi$
$480$	0	0
$481$	−0.386505	−0.0176231
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.2553	0.556483
$486$	0	0
$487$	6.03735	0.273578	0.136789	−	0.990600i	$-0.456322\pi$
$487$	6.03735	0.273578	0.136789	+	0.990600i	$0.456322\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.4431	0.651806	0.325903	−	0.945403i	$-0.394332\pi$
$491$	14.4431	0.651806	0.325903	+	0.945403i	$0.394332\pi$
$492$	0	0
$493$	4.60442	0.207373
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.62257	0.207351
$498$	0	0
$499$	−20.9717	−0.938823	−0.469412	−	0.882979i	$-0.655534\pi$
$499$	−20.9717	−0.938823	−0.469412	+	0.882979i	$0.655534\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.31728	−0.237086	−0.118543	−	0.992949i	$-0.537822\pi$
$503$	−5.31728	−0.237086	−0.118543	+	0.992949i	$0.537822\pi$
$504$	0	0
$505$	11.6700	0.519310
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.2270	−0.807897	−0.403949	−	0.914782i	$-0.632363\pi$
$509$	−18.2270	−0.807897	−0.403949	+	0.914782i	$0.632363\pi$
$510$	0	0
$511$	3.11389	0.137751
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.292611	0.0128940
$516$	0	0
$517$	16.1504	0.710296
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−40.1232	−1.75783	−0.878915	−	0.476978i	$-0.841732\pi$
$521$	−40.1232	−1.75783	−0.878915	+	0.476978i	$0.841732\pi$
$522$	0	0
$523$	−18.9873	−0.830257	−0.415129	−	0.909763i	$-0.636263\pi$
$523$	−18.9873	−0.830257	−0.415129	+	0.909763i	$0.636263\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.0101	−1.26370
$528$	0	0
$529$	−5.96265	−0.259246
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14.9909	−0.649329
$534$	0	0
$535$	−1.87237	−0.0809495
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.3684	0.963473
$540$	0	0
$541$	16.5279	0.710589	0.355294	−	0.934754i	$-0.384381\pi$
$541$	16.5279	0.710589	0.355294	+	0.934754i	$0.384381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.54787	−0.237645
$546$	0	0
$547$	−17.6737	−0.755671	−0.377835	−	0.925873i	$-0.623331\pi$
$547$	−17.6737	−0.755671	−0.377835	+	0.925873i	$0.623331\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.83141	0.0780208
$552$	0	0
$553$	4.14217	0.176143
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.3401	−0.734723	−0.367362	−	0.930078i	$-0.619739\pi$
$557$	−17.3401	−0.734723	−0.367362	+	0.930078i	$0.619739\pi$
$558$	0	0
$559$	13.6700	0.578181
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.9945	0.547654	0.273827	−	0.961779i	$-0.411710\pi$
$563$	12.9945	0.547654	0.273827	+	0.961779i	$0.411710\pi$
$564$	0	0
$565$	7.80128	0.328202
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.6802	0.699269	0.349635	−	0.936886i	$-0.386306\pi$
$569$	16.6802	0.699269	0.349635	+	0.936886i	$0.386306\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.12763	0.172134
$576$	0	0
$577$	−23.5953	−0.982287	−0.491144	−	0.871079i	$-0.663421\pi$
$577$	−23.5953	−0.982287	−0.491144	+	0.871079i	$0.663421\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.793010	0.0328996
$582$	0	0
$583$	−16.6983	−0.691574
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.1276	1.16095	0.580476	−	0.814277i	$-0.302867\pi$
$587$	28.1276	1.16095	0.580476	+	0.814277i	$0.302867\pi$
$588$	0	0
$589$	−11.5388	−0.475448
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.17872	0.376925	0.188462	−	0.982080i	$-0.439650\pi$
$593$	9.17872	0.376925	0.188462	+	0.982080i	$0.439650\pi$
$594$	0	0
$595$	−1.70739	−0.0699961
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.4713	−1.28588	−0.642942	−	0.765915i	$-0.722286\pi$
$599$	−31.4713	−1.28588	−0.642942	+	0.765915i	$0.722286\pi$
$600$	0	0
$601$	−29.2654	−1.19376	−0.596880	−	0.802330i	$-0.703593\pi$
$601$	−29.2654	−1.19376	−0.596880	+	0.802330i	$0.703593\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.0282739	−0.00114950
$606$	0	0
$607$	44.2034	1.79416	0.897080	−	0.441868i	$-0.145684\pi$
$607$	44.2034	1.79416	0.897080	+	0.441868i	$0.145684\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.42385	0.259881
$612$	0	0
$613$	−35.1715	−1.42056	−0.710282	−	0.703918i	$-0.751432\pi$
$613$	−35.1715	−1.42056	−0.710282	+	0.703918i	$0.751432\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.42571	0.298948	0.149474	−	0.988766i	$-0.452242\pi$
$617$	7.42571	0.298948	0.149474	+	0.988766i	$0.452242\pi$
$618$	0	0
$619$	−8.54787	−0.343568	−0.171784	−	0.985135i	$-0.554953\pi$
$619$	−8.54787	−0.343568	−0.171784	+	0.985135i	$0.554953\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.54241	0.0617954
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.971726	0.0387453
$630$	0	0
$631$	2.36836	0.0942829	0.0471415	−	0.998888i	$-0.484989\pi$
$631$	2.36836	0.0942829	0.0471415	+	0.998888i	$0.484989\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.8916	0.710005
$636$	0	0
$637$	8.89703	0.352513
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.133096	0.00525698	0.00262849	−	0.999997i	$-0.499163\pi$
$641$	0.133096	0.00525698	0.00262849	+	0.999997i	$0.499163\pi$
$642$	0	0
$643$	22.6464	0.893088	0.446544	−	0.894762i	$-0.352655\pi$
$643$	22.6464	0.893088	0.446544	+	0.894762i	$0.352655\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.3912	1.82383	0.911913	−	0.410385i	$-0.134606\pi$
$647$	46.3912	1.82383	0.911913	+	0.410385i	$0.134606\pi$
$648$	0	0
$649$	16.6983	0.655466
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.4057	−1.42467	−0.712333	−	0.701842i	$-0.752361\pi$
$653$	−36.4057	−1.42467	−0.712333	+	0.701842i	$0.752361\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19.1414	0.745642	0.372821	−	0.927903i	$-0.378391\pi$
$659$	19.1414	0.745642	0.372821	+	0.927903i	$0.378391\pi$
$660$	0	0
$661$	39.9072	1.55221	0.776104	−	0.630605i	$-0.217193\pi$
$661$	39.9072	1.55221	0.776104	+	0.630605i	$0.217193\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.679116	−0.0263350
$666$	0	0
$667$	5.72298	0.221595
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.4057	−0.942172
$672$	0	0
$673$	23.6508	0.911673	0.455836	−	0.890064i	$-0.349340\pi$
$673$	23.6508	0.911673	0.455836	+	0.890064i	$0.349340\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.8031	0.568931	0.284465	−	0.958686i	$-0.408184\pi$
$677$	14.8031	0.568931	0.284465	+	0.958686i	$0.408184\pi$
$678$	0	0
$679$	−6.30088	−0.241806
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.95252	−0.189503	−0.0947515	−	0.995501i	$-0.530206\pi$
$683$	−4.95252	−0.189503	−0.0947515	+	0.995501i	$0.530206\pi$
$684$	0	0
$685$	5.67004	0.216641
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.64177	−0.253031
$690$	0	0
$691$	19.2088	0.730739	0.365369	−	0.930863i	$-0.380943\pi$
$691$	19.2088	0.730739	0.365369	+	0.930863i	$0.380943\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	37.6892	1.42758
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.3492	1.10850	0.554251	−	0.832349i	$-0.313005\pi$
$701$	29.3492	1.10850	0.554251	+	0.832349i	$0.313005\pi$
$702$	0	0
$703$	0.386505	0.0145773
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	38.7266	1.45441	0.727204	−	0.686422i	$-0.240820\pi$
$709$	38.7266	1.45441	0.727204	+	0.686422i	$0.240820\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.0576	−1.35037
$714$	0	0
$715$	−4.38650	−0.164046
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.0848	−0.562569	−0.281284	−	0.959624i	$-0.590760\pi$
$719$	−15.0848	−0.562569	−0.281284	+	0.959624i	$0.590760\pi$
$720$	0	0
$721$	−0.150442	−0.00560275
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.38650	0.0514935
$726$	0	0
$727$	−12.3455	−0.457871	−0.228936	−	0.973442i	$-0.573525\pi$
$727$	−12.3455	−0.457871	−0.228936	+	0.973442i	$0.573525\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−34.3684	−1.27116
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.3774	1.15580
$738$	0	0
$739$	−29.7266	−1.09351	−0.546755	−	0.837293i	$-0.684137\pi$
$739$	−29.7266	−1.09351	−0.546755	+	0.837293i	$0.684137\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.3648	1.77433	0.887165	−	0.461452i	$-0.152671\pi$
$743$	48.3648	1.77433	0.887165	+	0.461452i	$0.152671\pi$
$744$	0	0
$745$	17.6610	0.647048
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.962653	0.0351746
$750$	0	0
$751$	31.8205	1.16115	0.580573	−	0.814208i	$-0.302829\pi$
$751$	31.8205	1.16115	0.580573	+	0.814208i	$0.302829\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.26434	0.0460139
$756$	0	0
$757$	4.94531	0.179740	0.0898701	−	0.995953i	$-0.471355\pi$
$757$	4.94531	0.179740	0.0898701	+	0.995953i	$0.471355\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.4249	−1.28415	−0.642076	−	0.766641i	$-0.721927\pi$
$761$	−35.4249	−1.28415	−0.642076	+	0.766641i	$0.721927\pi$
$762$	0	0
$763$	2.85237	0.103263
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.64177	0.239820
$768$	0	0
$769$	49.4249	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$769$	49.4249	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.6599	−0.455345	−0.227673	−	0.973738i	$-0.573112\pi$
$773$	−12.6599	−0.455345	−0.227673	+	0.973738i	$0.573112\pi$
$774$	0	0
$775$	−8.73566	−0.313794
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.9909	0.537106
$780$	0	0
$781$	−29.8578	−1.06840
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.6700	0.559288
$786$	0	0
$787$	−30.9344	−1.10269	−0.551346	−	0.834277i	$-0.685885\pi$
$787$	−30.9344	−1.10269	−0.551346	+	0.834277i	$0.685885\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.01093	−0.142612
$792$	0	0
$793$	−9.70739	−0.344720
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.5935	1.08368	0.541839	−	0.840483i	$-0.317728\pi$
$797$	30.5935	1.08368	0.541839	+	0.840483i	$0.317728\pi$
$798$	0	0
$799$	−16.1504	−0.571362
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.1131	−0.709776
$804$	0	0
$805$	−2.12217	−0.0747966
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.89703	0.101854	0.0509271	−	0.998702i	$-0.483782\pi$
$809$	2.89703	0.101854	0.0509271	+	0.998702i	$0.483782\pi$
$810$	0	0
$811$	14.8861	0.522722	0.261361	−	0.965241i	$-0.415829\pi$
$811$	14.8861	0.522722	0.261361	+	0.965241i	$0.415829\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.7074	0.550206
$816$	0	0
$817$	−13.6700	−0.478254
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.95173	0.312417	0.156209	−	0.987724i	$-0.450073\pi$
$821$	8.95173	0.312417	0.156209	+	0.987724i	$0.450073\pi$
$822$	0	0
$823$	−2.99454	−0.104383	−0.0521915	−	0.998637i	$-0.516621\pi$
$823$	−2.99454	−0.104383	−0.0521915	+	0.998637i	$0.516621\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.9663	1.11158	0.555788	−	0.831324i	$-0.312417\pi$
$827$	31.9663	1.11158	0.555788	+	0.831324i	$0.312417\pi$
$828$	0	0
$829$	22.7458	0.789994	0.394997	−	0.918682i	$-0.370746\pi$
$829$	22.7458	0.789994	0.394997	+	0.918682i	$0.370746\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.3684	−0.775018
$834$	0	0
$835$	−6.16498	−0.213348
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.2643	−0.803174	−0.401587	−	0.915821i	$-0.631541\pi$
$839$	−23.2643	−0.803174	−0.401587	+	0.915821i	$0.631541\pi$
$840$	0	0
$841$	−27.0776	−0.933710
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.2553	0.387193
$846$	0	0
$847$	0.0145366	0.000499485 0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.20779	0.0414025
$852$	0	0
$853$	10.9909	0.376322	0.188161	−	0.982138i	$-0.439747\pi$
$853$	10.9909	0.376322	0.188161	+	0.982138i	$0.439747\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.1504	0.551689	0.275844	−	0.961202i	$-0.411043\pi$
$857$	16.1504	0.551689	0.275844	+	0.961202i	$0.411043\pi$
$858$	0	0
$859$	−28.5188	−0.973049	−0.486524	−	0.873667i	$-0.661736\pi$
$859$	−28.5188	−0.973049	−0.486524	+	0.873667i	$0.661736\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.2890	0.418322	0.209161	−	0.977881i	$-0.432927\pi$
$863$	12.2890	0.418322	0.209161	+	0.977881i	$0.432927\pi$
$864$	0	0
$865$	8.58522	0.291906
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−26.7549	−0.907597
$870$	0	0
$871$	12.4804	0.422882
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.514137	−0.0173810
$876$	0	0
$877$	−39.7002	−1.34058	−0.670290	−	0.742099i	$-0.733830\pi$
$877$	−39.7002	−1.34058	−0.670290	+	0.742099i	$0.733830\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.1040	−1.08161	−0.540806	−	0.841147i	$-0.681881\pi$
$881$	−32.1040	−1.08161	−0.540806	+	0.841147i	$0.681881\pi$
$882$	0	0
$883$	−13.5051	−0.454482	−0.227241	−	0.973839i	$-0.572970\pi$
$883$	−13.5051	−0.454482	−0.227241	+	0.973839i	$0.572970\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.1222	1.17929	0.589643	−	0.807664i	$-0.299268\pi$
$887$	35.1222	1.17929	0.589643	+	0.807664i	$0.299268\pi$
$888$	0	0
$889$	−9.19872	−0.308515
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.42385	−0.214966
$894$	0	0
$895$	1.06562	0.0356198
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.1120	−0.403959
$900$	0	0
$901$	16.6983	0.556302
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.6700	0.421166
$906$	0	0
$907$	−15.1186	−0.502004	−0.251002	−	0.967987i	$-0.580760\pi$
$907$	−15.1186	−0.502004	−0.251002	+	0.967987i	$0.580760\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.5561	1.74126	0.870631	−	0.491936i	$-0.163711\pi$
$911$	52.5561	1.74126	0.870631	+	0.491936i	$0.163711\pi$
$912$	0	0
$913$	−5.12217	−0.169519
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.08482	−0.101870
$918$	0	0
$919$	54.5489	1.79940	0.899702	−	0.436505i	$-0.143784\pi$
$919$	54.5489	1.79940	0.899702	+	0.436505i	$0.143784\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.8760	−0.390903
$924$	0	0
$925$	0.292611	0.00962098
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.3793	−0.668623	−0.334311	−	0.942463i	$-0.608504\pi$
$929$	−20.3793	−0.668623	−0.334311	+	0.942463i	$0.608504\pi$
$930$	0	0
$931$	−8.89703	−0.291588
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.0283	0.360663
$936$	0	0
$937$	49.1979	1.60723	0.803613	−	0.595152i	$-0.202908\pi$
$937$	49.1979	1.60723	0.803613	+	0.595152i	$0.202908\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23.2371	0.757508	0.378754	−	0.925497i	$-0.376353\pi$
$941$	23.2371	0.757508	0.378754	+	0.925497i	$0.376353\pi$
$942$	0	0
$943$	46.8452	1.52549
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.1642	1.20767	0.603837	−	0.797108i	$-0.293638\pi$
$947$	37.1642	1.20767	0.603837	+	0.797108i	$0.293638\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.5761	0.763706	0.381853	−	0.924223i	$-0.375286\pi$
$953$	23.5761	0.763706	0.381853	+	0.924223i	$0.375286\pi$
$954$	0	0
$955$	16.9344	0.547984
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.91518	−0.0941360
$960$	0	0
$961$	45.3118	1.46167
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.7175	−0.860067
$966$	0	0
$967$	−8.38290	−0.269576	−0.134788	−	0.990874i	$-0.543035\pi$
$967$	−8.38290	−0.269576	−0.134788	+	0.990874i	$0.543035\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.2078	0.423858	0.211929	−	0.977285i	$-0.432025\pi$
$971$	13.2078	0.423858	0.211929	+	0.977285i	$0.432025\pi$
$972$	0	0
$973$	−4.11310	−0.131860
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.3310	0.458490	0.229245	−	0.973369i	$-0.426374\pi$
$977$	14.3310	0.458490	0.229245	+	0.973369i	$0.426374\pi$
$978$	0	0
$979$	−9.96265	−0.318408
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.3082	−1.03047	−0.515236	−	0.857048i	$-0.672296\pi$
$983$	−32.3082	−1.03047	−0.515236	+	0.857048i	$0.672296\pi$
$984$	0	0
$985$	−14.2553	−0.454210
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.7175	−1.35834
$990$	0	0
$991$	39.6700	1.26016	0.630080	−	0.776530i	$-0.283022\pi$
$991$	39.6700	1.26016	0.630080	+	0.776530i	$0.283022\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.6610	−0.781805
$996$	0	0
$997$	38.6874	1.22524	0.612621	−	0.790377i	$-0.290115\pi$
$997$	38.6874	1.22524	0.612621	+	0.790377i	$0.290115\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bs.1.3		3
3.2	odd	2		6480.2.a.bv.1.3		3
4.3	odd	2		405.2.a.i.1.1		3
9.2	odd	6		720.2.q.i.481.1		6
9.4	even	3		2160.2.q.k.721.1		6
9.5	odd	6		720.2.q.i.241.1		6
9.7	even	3		2160.2.q.k.1441.1		6
12.11	even	2		405.2.a.j.1.3		3
20.3	even	4		2025.2.b.m.649.6		6
20.7	even	4		2025.2.b.m.649.1		6
20.19	odd	2		2025.2.a.o.1.3		3
36.7	odd	6		135.2.e.b.91.3		6
36.11	even	6		45.2.e.b.31.1	yes	6
36.23	even	6		45.2.e.b.16.1	✓	6
36.31	odd	6		135.2.e.b.46.3		6
60.23	odd	4		2025.2.b.l.649.1		6
60.47	odd	4		2025.2.b.l.649.6		6
60.59	even	2		2025.2.a.n.1.1		3
180.7	even	12		675.2.k.b.199.1		12
180.23	odd	12		225.2.k.b.124.6		12
180.43	even	12		675.2.k.b.199.6		12
180.47	odd	12		225.2.k.b.49.6		12
180.59	even	6		225.2.e.b.151.3		6
180.67	even	12		675.2.k.b.424.6		12
180.79	odd	6		675.2.e.b.226.1		6
180.83	odd	12		225.2.k.b.49.1		12
180.103	even	12		675.2.k.b.424.1		12
180.119	even	6		225.2.e.b.76.3		6
180.139	odd	6		675.2.e.b.451.1		6
180.167	odd	12		225.2.k.b.124.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.1	✓	6	36.23	even	6
45.2.e.b.31.1	yes	6	36.11	even	6
135.2.e.b.46.3		6	36.31	odd	6
135.2.e.b.91.3		6	36.7	odd	6
225.2.e.b.76.3		6	180.119	even	6
225.2.e.b.151.3		6	180.59	even	6
225.2.k.b.49.1		12	180.83	odd	12
225.2.k.b.49.6		12	180.47	odd	12
225.2.k.b.124.1		12	180.167	odd	12
225.2.k.b.124.6		12	180.23	odd	12
405.2.a.i.1.1		3	4.3	odd	2
405.2.a.j.1.3		3	12.11	even	2
675.2.e.b.226.1		6	180.79	odd	6
675.2.e.b.451.1		6	180.139	odd	6
675.2.k.b.199.1		12	180.7	even	12
675.2.k.b.199.6		12	180.43	even	12
675.2.k.b.424.1		12	180.103	even	12
675.2.k.b.424.6		12	180.67	even	12
720.2.q.i.241.1		6	9.5	odd	6
720.2.q.i.481.1		6	9.2	odd	6
2025.2.a.n.1.1		3	60.59	even	2
2025.2.a.o.1.3		3	20.19	odd	2
2025.2.b.l.649.1		6	60.23	odd	4
2025.2.b.l.649.6		6	60.47	odd	4
2025.2.b.m.649.1		6	20.7	even	4
2025.2.b.m.649.6		6	20.3	even	4
2160.2.q.k.721.1		6	9.4	even	3
2160.2.q.k.1441.1		6	9.7	even	3
6480.2.a.bs.1.3		3	1.1	even	1	trivial
6480.2.a.bv.1.3		3	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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.514137 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.514137 q^{7} -3.32088 q^{11} -1.32088 q^{13} +3.32088 q^{17} +1.32088 q^{19} +4.12763 q^{23} +1.00000 q^{25} +1.38650 q^{29} -8.73566 q^{31} -0.514137 q^{35} +0.292611 q^{37} +11.3492 q^{41} -10.3492 q^{43} -4.86330 q^{47} -6.73566 q^{49} +5.02827 q^{53} +3.32088 q^{55} -5.02827 q^{59} +7.34916 q^{61} +1.32088 q^{65} -9.44852 q^{67} +8.99093 q^{71} +6.05655 q^{73} -1.70739 q^{77} +8.05655 q^{79} +1.54241 q^{83} -3.32088 q^{85} +3.00000 q^{89} -0.679116 q^{91} -1.32088 q^{95} -12.2553 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - 5 * q^7	\( 3 q - 3 q^{5} - 5 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 3 q^{23} + 3 q^{25} + 7 q^{29} - 8 q^{31} + 5 q^{35} + 6 q^{37} + 13 q^{41} - 10 q^{43} + 13 q^{47} - 2 q^{49} + 2 q^{53} + 2 q^{55} - 2 q^{59} + q^{61} - 4 q^{65} - 11 q^{67} - 10 q^{71} - 8 q^{73} - 2 q^{79} - 15 q^{83} - 2 q^{85} + 9 q^{89} - 10 q^{91} + 4 q^{95} - 18 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 5 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 4 * q^19 + 3 * q^23 + 3 * q^25 + 7 * q^29 - 8 * q^31 + 5 * q^35 + 6 * q^37 + 13 * q^41 - 10 * q^43 + 13 * q^47 - 2 * q^49 + 2 * q^53 + 2 * q^55 - 2 * q^59 + q^61 - 4 * q^65 - 11 * q^67 - 10 * q^71 - 8 * q^73 - 2 * q^79 - 15 * q^83 - 2 * q^85 + 9 * q^89 - 10 * q^91 + 4 * q^95 - 18 * q^97

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