LMFDB - Embedded newform 8640.2.a.db.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8640,2,Mod(1,8640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8640, 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("8640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8640 = 2^{6} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8640.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:	$68.9907473464$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.77200$ of defining polynomial
Character	$\chi$	$=$	8640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.77200 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.77200 q^{7} -3.77200 q^{11} -3.00000 q^{13} +1.77200 q^{17} -2.77200 q^{19} -7.77200 q^{23} +1.00000 q^{25} -3.77200 q^{29} -2.22800 q^{31} -4.77200 q^{35} +0.772002 q^{37} -9.54400 q^{41} -1.77200 q^{43} -5.77200 q^{47} +15.7720 q^{49} +9.54400 q^{53} -3.77200 q^{55} +12.0000 q^{59} -0.772002 q^{61} -3.00000 q^{65} +14.7720 q^{67} -13.5440 q^{71} +4.77200 q^{73} +18.0000 q^{77} +5.00000 q^{79} +6.00000 q^{83} +1.77200 q^{85} +8.00000 q^{89} +14.3160 q^{91} -2.77200 q^{95} -18.3160 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - q^7	$ 2 q + 2 q^{5} - q^{7} + q^{11} - 6 q^{13} - 5 q^{17} + 3 q^{19} - 7 q^{23} + 2 q^{25} + q^{29} - 13 q^{31} - q^{35} - 7 q^{37} - 2 q^{41} + 5 q^{43} - 3 q^{47} + 23 q^{49} + 2 q^{53} + q^{55} + 24 q^{59} + 7 q^{61} - 6 q^{65} + 21 q^{67} - 10 q^{71} + q^{73} + 36 q^{77} + 10 q^{79} + 12 q^{83} - 5 q^{85} + 16 q^{89} + 3 q^{91} + 3 q^{95} - 11 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 + q^11 - 6 * q^13 - 5 * q^17 + 3 * q^19 - 7 * q^23 + 2 * q^25 + q^29 - 13 * q^31 - q^35 - 7 * q^37 - 2 * q^41 + 5 * q^43 - 3 * q^47 + 23 * q^49 + 2 * q^53 + q^55 + 24 * q^59 + 7 * q^61 - 6 * q^65 + 21 * q^67 - 10 * q^71 + q^73 + 36 * q^77 + 10 * q^79 + 12 * q^83 - 5 * q^85 + 16 * q^89 + 3 * q^91 + 3 * q^95 - 11 * 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
$7$	−4.77200	−1.80365	−0.901824	−	0.432104i	$-0.857771\pi$
$7$	−4.77200	−1.80365	−0.901824	+	0.432104i	$0.857771\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.77200	−1.13730	−0.568651	−	0.822579i	$-0.692534\pi$
$11$	−3.77200	−1.13730	−0.568651	+	0.822579i	$0.692534\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.77200	0.429774	0.214887	−	0.976639i	$-0.431062\pi$
$17$	1.77200	0.429774	0.214887	+	0.976639i	$0.431062\pi$
$18$	0	0
$19$	−2.77200	−0.635941	−0.317970	−	0.948101i	$-0.603001\pi$
$19$	−2.77200	−0.635941	−0.317970	+	0.948101i	$0.603001\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.77200	−1.62057	−0.810287	−	0.586033i	$-0.800689\pi$
$23$	−7.77200	−1.62057	−0.810287	+	0.586033i	$0.800689\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.77200	−0.700443	−0.350222	−	0.936667i	$-0.613894\pi$
$29$	−3.77200	−0.700443	−0.350222	+	0.936667i	$0.613894\pi$
$30$	0	0
$31$	−2.22800	−0.400160	−0.200080	−	0.979780i	$-0.564120\pi$
$31$	−2.22800	−0.400160	−0.200080	+	0.979780i	$0.564120\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.77200	−0.806616
$36$	0	0
$37$	0.772002	0.126916	0.0634582	−	0.997984i	$-0.479787\pi$
$37$	0.772002	0.126916	0.0634582	+	0.997984i	$0.479787\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.54400	−1.49052	−0.745261	−	0.666772i	$-0.767675\pi$
$41$	−9.54400	−1.49052	−0.745261	+	0.666772i	$0.767675\pi$
$42$	0	0
$43$	−1.77200	−0.270228	−0.135114	−	0.990830i	$-0.543140\pi$
$43$	−1.77200	−0.270228	−0.135114	+	0.990830i	$0.543140\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.77200	−0.841933	−0.420967	−	0.907076i	$-0.638309\pi$
$47$	−5.77200	−0.841933	−0.420967	+	0.907076i	$0.638309\pi$
$48$	0	0
$49$	15.7720	2.25314
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.54400	1.31097	0.655485	−	0.755208i	$-0.272464\pi$
$53$	9.54400	1.31097	0.655485	+	0.755208i	$0.272464\pi$
$54$	0	0
$55$	−3.77200	−0.508617
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−0.772002	−0.0988447	−0.0494224	−	0.998778i	$-0.515738\pi$
$61$	−0.772002	−0.0988447	−0.0494224	+	0.998778i	$0.515738\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	14.7720	1.80469	0.902344	−	0.431017i	$-0.141845\pi$
$67$	14.7720	1.80469	0.902344	+	0.431017i	$0.141845\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.5440	−1.60738	−0.803689	−	0.595050i	$-0.797132\pi$
$71$	−13.5440	−1.60738	−0.803689	+	0.595050i	$0.797132\pi$
$72$	0	0
$73$	4.77200	0.558521	0.279260	−	0.960215i	$-0.409911\pi$
$73$	4.77200	0.558521	0.279260	+	0.960215i	$0.409911\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0000	2.05129
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	1.77200	0.192201
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	14.3160	1.50073
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.77200	−0.284401
$96$	0	0
$97$	−18.3160	−1.85971	−0.929854	−	0.367928i	$-0.880067\pi$
$97$	−18.3160	−1.85971	−0.929854	+	0.367928i	$0.880067\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2280	1.01772	0.508862	−	0.860848i	$-0.330066\pi$
$101$	10.2280	1.01772	0.508862	+	0.860848i	$0.330066\pi$
$102$	0	0
$103$	−16.7720	−1.65259	−0.826297	−	0.563234i	$-0.809557\pi$
$103$	−16.7720	−1.65259	−0.826297	+	0.563234i	$0.809557\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	0	0
$109$	−9.54400	−0.914150	−0.457075	−	0.889428i	$-0.651103\pi$
$109$	−9.54400	−0.914150	−0.457075	+	0.889428i	$0.651103\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.3160	1.06452	0.532260	−	0.846581i	$-0.321343\pi$
$113$	11.3160	1.06452	0.532260	+	0.846581i	$0.321343\pi$
$114$	0	0
$115$	−7.77200	−0.724743
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.45600	−0.775160
$120$	0	0
$121$	3.22800	0.293454
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.5440	1.02436	0.512182	−	0.858877i	$-0.328837\pi$
$127$	11.5440	1.02436	0.512182	+	0.858877i	$0.328837\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.22800	0.544143	0.272071	−	0.962277i	$-0.412291\pi$
$131$	6.22800	0.544143	0.272071	+	0.962277i	$0.412291\pi$
$132$	0	0
$133$	13.2280	1.14701
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.0880	−1.80167	−0.900835	−	0.434162i	$-0.857045\pi$
$137$	−21.0880	−1.80167	−0.900835	+	0.434162i	$0.857045\pi$
$138$	0	0
$139$	−12.7720	−1.08331	−0.541654	−	0.840602i	$-0.682202\pi$
$139$	−12.7720	−1.08331	−0.541654	+	0.840602i	$0.682202\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.3160	0.946292
$144$	0	0
$145$	−3.77200	−0.313248
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.77200	0.800554	0.400277	−	0.916394i	$-0.368914\pi$
$149$	9.77200	0.800554	0.400277	+	0.916394i	$0.368914\pi$
$150$	0	0
$151$	24.0880	1.96025	0.980127	−	0.198370i	$-0.0635649\pi$
$151$	24.0880	1.96025	0.980127	+	0.198370i	$0.0635649\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.22800	−0.178957
$156$	0	0
$157$	−7.77200	−0.620273	−0.310137	−	0.950692i	$-0.600375\pi$
$157$	−7.77200	−0.620273	−0.310137	+	0.950692i	$0.600375\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	37.0880	2.92294
$162$	0	0
$163$	−8.54400	−0.669218	−0.334609	−	0.942357i	$-0.608604\pi$
$163$	−8.54400	−0.669218	−0.334609	+	0.942357i	$0.608604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.0880	1.78660	0.893302	−	0.449457i	$-0.148383\pi$
$167$	23.0880	1.78660	0.893302	+	0.449457i	$0.148383\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.54400	−0.573560	−0.286780	−	0.957996i	$-0.592585\pi$
$173$	−7.54400	−0.573560	−0.286780	+	0.957996i	$0.592585\pi$
$174$	0	0
$175$	−4.77200	−0.360729
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.54400	0.713352	0.356676	−	0.934228i	$-0.383910\pi$
$179$	9.54400	0.713352	0.356676	+	0.934228i	$0.383910\pi$
$180$	0	0
$181$	24.3160	1.80739	0.903697	−	0.428172i	$-0.140842\pi$
$181$	24.3160	1.80739	0.903697	+	0.428172i	$0.140842\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.772002	0.0567587
$186$	0	0
$187$	−6.68399	−0.488782
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.54400	−0.256435	−0.128218	−	0.991746i	$-0.540926\pi$
$191$	−3.54400	−0.256435	−0.128218	+	0.991746i	$0.540926\pi$
$192$	0	0
$193$	−12.7720	−0.919349	−0.459674	−	0.888088i	$-0.652034\pi$
$193$	−12.7720	−0.919349	−0.459674	+	0.888088i	$0.652034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	−9.54400	−0.666582
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.4560	0.723256
$210$	0	0
$211$	14.3160	0.985554	0.492777	−	0.870156i	$-0.335982\pi$
$211$	14.3160	0.985554	0.492777	+	0.870156i	$0.335982\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.77200	−0.120850
$216$	0	0
$217$	10.6320	0.721748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.31601	−0.357593
$222$	0	0
$223$	−15.0880	−1.01037	−0.505184	−	0.863012i	$-0.668575\pi$
$223$	−15.0880	−1.01037	−0.505184	+	0.863012i	$0.668575\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.54400	0.102479	0.0512396	−	0.998686i	$-0.483683\pi$
$227$	1.54400	0.102479	0.0512396	+	0.998686i	$0.483683\pi$
$228$	0	0
$229$	−10.4560	−0.690952	−0.345476	−	0.938428i	$-0.612282\pi$
$229$	−10.4560	−0.690952	−0.345476	+	0.938428i	$0.612282\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.4560	0.816019	0.408010	−	0.912978i	$-0.366223\pi$
$233$	12.4560	0.816019	0.408010	+	0.912978i	$0.366223\pi$
$234$	0	0
$235$	−5.77200	−0.376524
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	8.08801	0.520994	0.260497	−	0.965475i	$-0.416114\pi$
$241$	8.08801	0.520994	0.260497	+	0.965475i	$0.416114\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.7720	1.00764
$246$	0	0
$247$	8.31601	0.529135
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.227998	0.0143911	0.00719556	−	0.999974i	$-0.497710\pi$
$251$	0.227998	0.0143911	0.00719556	+	0.999974i	$0.497710\pi$
$252$	0	0
$253$	29.3160	1.84308
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.22800	−0.138979	−0.0694894	−	0.997583i	$-0.522137\pi$
$257$	−2.22800	−0.138979	−0.0694894	+	0.997583i	$0.522137\pi$
$258$	0	0
$259$	−3.68399	−0.228912
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	9.54400	0.586283
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.31601	0.324123	0.162061	−	0.986781i	$-0.448186\pi$
$269$	5.31601	0.324123	0.162061	+	0.986781i	$0.448186\pi$
$270$	0	0
$271$	10.7720	0.654353	0.327176	−	0.944963i	$-0.393903\pi$
$271$	10.7720	0.654353	0.327176	+	0.944963i	$0.393903\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.77200	−0.227460
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.5440	−0.688658	−0.344329	−	0.938849i	$-0.611893\pi$
$281$	−11.5440	−0.688658	−0.344329	+	0.938849i	$0.611893\pi$
$282$	0	0
$283$	−11.0880	−0.659114	−0.329557	−	0.944136i	$-0.606899\pi$
$283$	−11.0880	−0.659114	−0.329557	+	0.944136i	$0.606899\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	45.5440	2.68838
$288$	0	0
$289$	−13.8600	−0.815295
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.0880	1.69934	0.849670	−	0.527315i	$-0.176801\pi$
$293$	29.0880	1.69934	0.849670	+	0.527315i	$0.176801\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	23.3160	1.34840
$300$	0	0
$301$	8.45600	0.487396
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.772002	−0.0442047
$306$	0	0
$307$	−5.77200	−0.329426	−0.164713	−	0.986342i	$-0.552670\pi$
$307$	−5.77200	−0.329426	−0.164713	+	0.986342i	$0.552670\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−28.3160	−1.60052	−0.800258	−	0.599656i	$-0.795304\pi$
$313$	−28.3160	−1.60052	−0.800258	+	0.599656i	$0.795304\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.0880	−0.622765	−0.311382	−	0.950285i	$-0.600792\pi$
$317$	−11.0880	−0.622765	−0.311382	+	0.950285i	$0.600792\pi$
$318$	0	0
$319$	14.2280	0.796615
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.91199	−0.273311
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.5440	1.51855
$330$	0	0
$331$	3.22800	0.177427	0.0887134	−	0.996057i	$-0.471724\pi$
$331$	3.22800	0.177427	0.0887134	+	0.996057i	$0.471724\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.7720	0.807081
$336$	0	0
$337$	−26.7720	−1.45836	−0.729182	−	0.684320i	$-0.760099\pi$
$337$	−26.7720	−1.45836	−0.729182	+	0.684320i	$0.760099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.40401	0.455103
$342$	0	0
$343$	−41.8600	−2.26023
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.6320	1.53705	0.768523	−	0.639822i	$-0.220992\pi$
$347$	28.6320	1.53705	0.768523	+	0.639822i	$0.220992\pi$
$348$	0	0
$349$	25.4040	1.35985	0.679923	−	0.733284i	$-0.262013\pi$
$349$	25.4040	1.35985	0.679923	+	0.733284i	$0.262013\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.2280	0.757280	0.378640	−	0.925544i	$-0.376392\pi$
$353$	14.2280	0.757280	0.378640	+	0.925544i	$0.376392\pi$
$354$	0	0
$355$	−13.5440	−0.718841
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.45600	0.129623	0.0648113	−	0.997898i	$-0.479355\pi$
$359$	2.45600	0.129623	0.0648113	+	0.997898i	$0.479355\pi$
$360$	0	0
$361$	−11.3160	−0.595579
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.77200	0.249778
$366$	0	0
$367$	10.3160	0.538491	0.269246	−	0.963072i	$-0.413226\pi$
$367$	10.3160	0.538491	0.269246	+	0.963072i	$0.413226\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−45.5440	−2.36453
$372$	0	0
$373$	−20.0880	−1.04012	−0.520059	−	0.854130i	$-0.674090\pi$
$373$	−20.0880	−1.04012	−0.520059	+	0.854130i	$0.674090\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.3160	0.582804
$378$	0	0
$379$	0.316006	0.0162321	0.00811606	−	0.999967i	$-0.497417\pi$
$379$	0.316006	0.0162321	0.00811606	+	0.999967i	$0.497417\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.7720	0.703716	0.351858	−	0.936053i	$-0.385550\pi$
$383$	13.7720	0.703716	0.351858	+	0.936053i	$0.385550\pi$
$384$	0	0
$385$	18.0000	0.917365
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.3160	−1.48638	−0.743190	−	0.669080i	$-0.766688\pi$
$389$	−29.3160	−1.48638	−0.743190	+	0.669080i	$0.766688\pi$
$390$	0	0
$391$	−13.7720	−0.696480
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.00000	0.251577
$396$	0	0
$397$	−31.3160	−1.57171	−0.785853	−	0.618414i	$-0.787776\pi$
$397$	−31.3160	−1.57171	−0.785853	+	0.618414i	$0.787776\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.54400	−0.0771039	−0.0385519	−	0.999257i	$-0.512275\pi$
$401$	−1.54400	−0.0771039	−0.0385519	+	0.999257i	$0.512275\pi$
$402$	0	0
$403$	6.68399	0.332953
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.91199	−0.144342
$408$	0	0
$409$	20.5440	1.01584	0.507918	−	0.861406i	$-0.330415\pi$
$409$	20.5440	1.01584	0.507918	+	0.861406i	$0.330415\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−57.2640	−2.81778
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.77200	−0.0865680	−0.0432840	−	0.999063i	$-0.513782\pi$
$419$	−1.77200	−0.0865680	−0.0432840	+	0.999063i	$0.513782\pi$
$420$	0	0
$421$	5.22800	0.254797	0.127399	−	0.991852i	$-0.459337\pi$
$421$	5.22800	0.254797	0.127399	+	0.991852i	$0.459337\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.77200	0.0859547
$426$	0	0
$427$	3.68399	0.178281
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.6320	1.18648	0.593241	−	0.805025i	$-0.297848\pi$
$431$	24.6320	1.18648	0.593241	+	0.805025i	$0.297848\pi$
$432$	0	0
$433$	1.08801	0.0522863	0.0261432	−	0.999658i	$-0.491677\pi$
$433$	1.08801	0.0522863	0.0261432	+	0.999658i	$0.491677\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.5440	1.03059
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.0880	−1.28699	−0.643495	−	0.765450i	$-0.722516\pi$
$443$	−27.0880	−1.28699	−0.643495	+	0.765450i	$0.722516\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.5440	1.29988	0.649941	−	0.759985i	$-0.274794\pi$
$449$	27.5440	1.29988	0.649941	+	0.759985i	$0.274794\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.3160	0.671145
$456$	0	0
$457$	17.5440	0.820674	0.410337	−	0.911934i	$-0.365411\pi$
$457$	17.5440	0.820674	0.410337	+	0.911934i	$0.365411\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.5440	1.65545	0.827725	−	0.561134i	$-0.189635\pi$
$461$	35.5440	1.65545	0.827725	+	0.561134i	$0.189635\pi$
$462$	0	0
$463$	−7.86001	−0.365286	−0.182643	−	0.983179i	$-0.558465\pi$
$463$	−7.86001	−0.365286	−0.182643	+	0.983179i	$0.558465\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	0	0
$469$	−70.4920	−3.25502
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.68399	0.307330
$474$	0	0
$475$	−2.77200	−0.127188
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.4560	−0.477747	−0.238873	−	0.971051i	$-0.576778\pi$
$479$	−10.4560	−0.477747	−0.238873	+	0.971051i	$0.576778\pi$
$480$	0	0
$481$	−2.31601	−0.105601
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.3160	−0.831687
$486$	0	0
$487$	6.77200	0.306869	0.153434	−	0.988159i	$-0.450967\pi$
$487$	6.77200	0.306869	0.153434	+	0.988159i	$0.450967\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	33.5440	1.51382	0.756910	−	0.653519i	$-0.226708\pi$
$491$	33.5440	1.51382	0.756910	+	0.653519i	$0.226708\pi$
$492$	0	0
$493$	−6.68399	−0.301032
$494$	0	0
$495$	0	0
$496$	0	0
$497$	64.6320	2.89914
$498$	0	0
$499$	−0.911993	−0.0408264	−0.0204132	−	0.999792i	$-0.506498\pi$
$499$	−0.911993	−0.0408264	−0.0204132	+	0.999792i	$0.506498\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.8600	−0.840926	−0.420463	−	0.907310i	$-0.638132\pi$
$503$	−18.8600	−0.840926	−0.420463	+	0.907310i	$0.638132\pi$
$504$	0	0
$505$	10.2280	0.455140
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.22800	0.364700	0.182350	−	0.983234i	$-0.441630\pi$
$509$	8.22800	0.364700	0.182350	+	0.983234i	$0.441630\pi$
$510$	0	0
$511$	−22.7720	−1.00737
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.7720	−0.739063
$516$	0	0
$517$	21.7720	0.957532
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.4560	1.15906	0.579529	−	0.814952i	$-0.303236\pi$
$521$	26.4560	1.15906	0.579529	+	0.814952i	$0.303236\pi$
$522$	0	0
$523$	1.45600	0.0636663	0.0318331	−	0.999493i	$-0.489865\pi$
$523$	1.45600	0.0636663	0.0318331	+	0.999493i	$0.489865\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.94802	−0.171978
$528$	0	0
$529$	37.4040	1.62626
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.6320	1.24019
$534$	0	0
$535$	−14.0000	−0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−59.4920	−2.56250
$540$	0	0
$541$	16.7720	0.721085	0.360542	−	0.932743i	$-0.382592\pi$
$541$	16.7720	0.721085	0.360542	+	0.932743i	$0.382592\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.54400	−0.408820
$546$	0	0
$547$	−0.544004	−0.0232599	−0.0116300	−	0.999932i	$-0.503702\pi$
$547$	−0.544004	−0.0232599	−0.0116300	+	0.999932i	$0.503702\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.4560	0.445440
$552$	0	0
$553$	−23.8600	−1.01463
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.1760	1.61757	0.808785	−	0.588105i	$-0.200126\pi$
$557$	38.1760	1.61757	0.808785	+	0.588105i	$0.200126\pi$
$558$	0	0
$559$	5.31601	0.224843
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.54400	−0.317942	−0.158971	−	0.987283i	$-0.550818\pi$
$563$	−7.54400	−0.317942	−0.158971	+	0.987283i	$0.550818\pi$
$564$	0	0
$565$	11.3160	0.476068
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.0880	0.800211	0.400105	−	0.916469i	$-0.368973\pi$
$569$	19.0880	0.800211	0.400105	+	0.916469i	$0.368973\pi$
$570$	0	0
$571$	−19.2280	−0.804667	−0.402333	−	0.915493i	$-0.631801\pi$
$571$	−19.2280	−0.804667	−0.402333	+	0.915493i	$0.631801\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.77200	−0.324115
$576$	0	0
$577$	−19.4040	−0.807800	−0.403900	−	0.914803i	$-0.632346\pi$
$577$	−19.4040	−0.807800	−0.403900	+	0.914803i	$0.632346\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−28.6320	−1.18786
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.91199	0.120191	0.0600954	−	0.998193i	$-0.480860\pi$
$587$	2.91199	0.120191	0.0600954	+	0.998193i	$0.480860\pi$
$588$	0	0
$589$	6.17601	0.254478
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.8600	−0.610227	−0.305114	−	0.952316i	$-0.598694\pi$
$593$	−14.8600	−0.610227	−0.305114	+	0.952316i	$0.598694\pi$
$594$	0	0
$595$	−8.45600	−0.346662
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.08801	−0.207890	−0.103945	−	0.994583i	$-0.533147\pi$
$599$	−5.08801	−0.207890	−0.103945	+	0.994583i	$0.533147\pi$
$600$	0	0
$601$	20.6840	0.843718	0.421859	−	0.906662i	$-0.361378\pi$
$601$	20.6840	0.843718	0.421859	+	0.906662i	$0.361378\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.22800	0.131237
$606$	0	0
$607$	−39.8600	−1.61787	−0.808934	−	0.587900i	$-0.799955\pi$
$607$	−39.8600	−1.61787	−0.808934	+	0.587900i	$0.799955\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.3160	0.700531
$612$	0	0
$613$	2.54400	0.102751	0.0513757	−	0.998679i	$-0.483639\pi$
$613$	2.54400	0.102751	0.0513757	+	0.998679i	$0.483639\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.31601	−0.294531	−0.147266	−	0.989097i	$-0.547047\pi$
$617$	−7.31601	−0.294531	−0.147266	+	0.989097i	$0.547047\pi$
$618$	0	0
$619$	5.68399	0.228459	0.114230	−	0.993454i	$-0.463560\pi$
$619$	5.68399	0.228459	0.114230	+	0.993454i	$0.463560\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38.1760	−1.52949
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.36799	0.0545453
$630$	0	0
$631$	2.77200	0.110352	0.0551758	−	0.998477i	$-0.482428\pi$
$631$	2.77200	0.110352	0.0551758	+	0.998477i	$0.482428\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.5440	0.458110
$636$	0	0
$637$	−47.3160	−1.87473
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.6320	0.419939	0.209970	−	0.977708i	$-0.432663\pi$
$641$	10.6320	0.419939	0.209970	+	0.977708i	$0.432663\pi$
$642$	0	0
$643$	−9.77200	−0.385370	−0.192685	−	0.981261i	$-0.561720\pi$
$643$	−9.77200	−0.385370	−0.192685	+	0.981261i	$0.561720\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.6320	−1.12564	−0.562820	−	0.826579i	$-0.690284\pi$
$647$	−28.6320	−1.12564	−0.562820	+	0.826579i	$0.690284\pi$
$648$	0	0
$649$	−45.2640	−1.77677
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.0880	1.06004	0.530018	−	0.847987i	$-0.322185\pi$
$653$	27.0880	1.06004	0.530018	+	0.847987i	$0.322185\pi$
$654$	0	0
$655$	6.22800	0.243348
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	7.22800	0.281137	0.140568	−	0.990071i	$-0.455107\pi$
$661$	7.22800	0.281137	0.140568	+	0.990071i	$0.455107\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.2280	0.512960
$666$	0	0
$667$	29.3160	1.13512
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.91199	0.112416
$672$	0	0
$673$	13.4040	0.516687	0.258343	−	0.966053i	$-0.416823\pi$
$673$	13.4040	0.516687	0.258343	+	0.966053i	$0.416823\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.91199	−0.342516	−0.171258	−	0.985226i	$-0.554783\pi$
$677$	−8.91199	−0.342516	−0.171258	+	0.985226i	$0.554783\pi$
$678$	0	0
$679$	87.4040	3.35426
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.4560	0.400088	0.200044	−	0.979787i	$-0.435892\pi$
$683$	10.4560	0.400088	0.200044	+	0.979787i	$0.435892\pi$
$684$	0	0
$685$	−21.0880	−0.805731
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−28.6320	−1.09079
$690$	0	0
$691$	38.6320	1.46963	0.734815	−	0.678267i	$-0.237269\pi$
$691$	38.6320	1.46963	0.734815	+	0.678267i	$0.237269\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.7720	−0.484470
$696$	0	0
$697$	−16.9120	−0.640587
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.7720	−0.746778	−0.373389	−	0.927675i	$-0.621804\pi$
$701$	−19.7720	−0.746778	−0.373389	+	0.927675i	$0.621804\pi$
$702$	0	0
$703$	−2.13999	−0.0807113
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−48.8080	−1.83561
$708$	0	0
$709$	−49.8600	−1.87253	−0.936266	−	0.351292i	$-0.885742\pi$
$709$	−49.8600	−1.87253	−0.936266	+	0.351292i	$0.885742\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.3160	0.648490
$714$	0	0
$715$	11.3160	0.423195
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	80.0360	2.98070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.77200	−0.140089
$726$	0	0
$727$	0.911993	0.0338239	0.0169120	−	0.999857i	$-0.494616\pi$
$727$	0.911993	0.0338239	0.0169120	+	0.999857i	$0.494616\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.13999	−0.116137
$732$	0	0
$733$	6.91199	0.255300	0.127650	−	0.991819i	$-0.459257\pi$
$733$	6.91199	0.255300	0.127650	+	0.991819i	$0.459257\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−55.7200	−2.05247
$738$	0	0
$739$	−35.0880	−1.29073	−0.645367	−	0.763873i	$-0.723295\pi$
$739$	−35.0880	−1.29073	−0.645367	+	0.763873i	$0.723295\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.4040	1.62903	0.814513	−	0.580146i	$-0.197004\pi$
$743$	44.4040	1.62903	0.814513	+	0.580146i	$0.197004\pi$
$744$	0	0
$745$	9.77200	0.358018
$746$	0	0
$747$	0	0
$748$	0	0
$749$	66.8080	2.44111
$750$	0	0
$751$	39.1760	1.42955	0.714777	−	0.699353i	$-0.246528\pi$
$751$	39.1760	1.42955	0.714777	+	0.699353i	$0.246528\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.0880	0.876652
$756$	0	0
$757$	46.7200	1.69807	0.849034	−	0.528338i	$-0.177185\pi$
$757$	46.7200	1.69807	0.849034	+	0.528338i	$0.177185\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0880	1.05444	0.527220	−	0.849729i	$-0.323234\pi$
$761$	29.0880	1.05444	0.527220	+	0.849729i	$0.323234\pi$
$762$	0	0
$763$	45.5440	1.64880
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.0000	−1.29988
$768$	0	0
$769$	48.2640	1.74045	0.870223	−	0.492659i	$-0.163975\pi$
$769$	48.2640	1.74045	0.870223	+	0.492659i	$0.163975\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.9120	0.392477	0.196239	−	0.980556i	$-0.437127\pi$
$773$	10.9120	0.392477	0.196239	+	0.980556i	$0.437127\pi$
$774$	0	0
$775$	−2.22800	−0.0800321
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.4560	0.947884
$780$	0	0
$781$	51.0880	1.82807
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.77200	−0.277395
$786$	0	0
$787$	39.1760	1.39647	0.698237	−	0.715867i	$-0.253968\pi$
$787$	39.1760	1.39647	0.698237	+	0.715867i	$0.253968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−54.0000	−1.92002
$792$	0	0
$793$	2.31601	0.0822438
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.1760	−1.63564	−0.817819	−	0.575475i	$-0.804817\pi$
$797$	−46.1760	−1.63564	−0.817819	+	0.575475i	$0.804817\pi$
$798$	0	0
$799$	−10.2280	−0.361841
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	37.0880	1.30718
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	−31.5440	−1.10766	−0.553830	−	0.832630i	$-0.686834\pi$
$811$	−31.5440	−1.10766	−0.553830	+	0.832630i	$0.686834\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.54400	−0.299283
$816$	0	0
$817$	4.91199	0.171849
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.5440	0.821691	0.410846	−	0.911705i	$-0.365233\pi$
$821$	23.5440	0.821691	0.410846	+	0.911705i	$0.365233\pi$
$822$	0	0
$823$	39.4040	1.37354	0.686769	−	0.726876i	$-0.259028\pi$
$823$	39.4040	1.37354	0.686769	+	0.726876i	$0.259028\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.54400	−0.123237	−0.0616185	−	0.998100i	$-0.519626\pi$
$827$	−3.54400	−0.123237	−0.0616185	+	0.998100i	$0.519626\pi$
$828$	0	0
$829$	−38.7720	−1.34661	−0.673304	−	0.739366i	$-0.735125\pi$
$829$	−38.7720	−1.34661	−0.673304	+	0.739366i	$0.735125\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27.9480	0.968341
$834$	0	0
$835$	23.0880	0.798993
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.6320	−1.88611	−0.943053	−	0.332642i	$-0.892060\pi$
$839$	−54.6320	−1.88611	−0.943053	+	0.332642i	$0.892060\pi$
$840$	0	0
$841$	−14.7720	−0.509379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	−15.4040	−0.529288
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	15.4560	0.529203	0.264602	−	0.964358i	$-0.414760\pi$
$853$	15.4560	0.529203	0.264602	+	0.964358i	$0.414760\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.0880	1.40354	0.701770	−	0.712404i	$-0.252394\pi$
$857$	41.0880	1.40354	0.701770	+	0.712404i	$0.252394\pi$
$858$	0	0
$859$	−24.7720	−0.845210	−0.422605	−	0.906314i	$-0.638884\pi$
$859$	−24.7720	−0.845210	−0.422605	+	0.906314i	$0.638884\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.3160	0.861767	0.430883	−	0.902408i	$-0.358202\pi$
$863$	25.3160	0.861767	0.430883	+	0.902408i	$0.358202\pi$
$864$	0	0
$865$	−7.54400	−0.256504
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.8600	−0.639782
$870$	0	0
$871$	−44.3160	−1.50159
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.77200	−0.161323
$876$	0	0
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.91199	−0.232871	−0.116435	−	0.993198i	$-0.537147\pi$
$881$	−6.91199	−0.232871	−0.116435	+	0.993198i	$0.537147\pi$
$882$	0	0
$883$	12.3160	0.414467	0.207233	−	0.978292i	$-0.433554\pi$
$883$	12.3160	0.414467	0.207233	+	0.978292i	$0.433554\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.68399	−0.224427	−0.112213	−	0.993684i	$-0.535794\pi$
$887$	−6.68399	−0.224427	−0.112213	+	0.993684i	$0.535794\pi$
$888$	0	0
$889$	−55.0880	−1.84759
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	9.54400	0.319021
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.40401	0.280290
$900$	0	0
$901$	16.9120	0.563420
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.3160	0.808291
$906$	0	0
$907$	−35.0000	−1.16216	−0.581078	−	0.813848i	$-0.697369\pi$
$907$	−35.0000	−1.16216	−0.581078	+	0.813848i	$0.697369\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38.1760	1.26483	0.632414	−	0.774631i	$-0.282064\pi$
$911$	38.1760	1.26483	0.632414	+	0.774631i	$0.282064\pi$
$912$	0	0
$913$	−22.6320	−0.749010
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29.7200	−0.981441
$918$	0	0
$919$	−4.86001	−0.160317	−0.0801585	−	0.996782i	$-0.525543\pi$
$919$	−4.86001	−0.160317	−0.0801585	+	0.996782i	$0.525543\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.6320	1.33742
$924$	0	0
$925$	0.772002	0.0253833
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.6320	−0.545679	−0.272839	−	0.962060i	$-0.587963\pi$
$929$	−16.6320	−0.545679	−0.272839	+	0.962060i	$0.587963\pi$
$930$	0	0
$931$	−43.7200	−1.43287
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.68399	−0.218590
$936$	0	0
$937$	−16.7720	−0.547917	−0.273959	−	0.961741i	$-0.588333\pi$
$937$	−16.7720	−0.547917	−0.273959	+	0.961741i	$0.588333\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.4040	0.730350	0.365175	−	0.930939i	$-0.381009\pi$
$941$	22.4040	0.730350	0.365175	+	0.930939i	$0.381009\pi$
$942$	0	0
$943$	74.1760	2.41550
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.45600	0.274783	0.137391	−	0.990517i	$-0.456128\pi$
$947$	8.45600	0.274783	0.137391	+	0.990517i	$0.456128\pi$
$948$	0	0
$949$	−14.3160	−0.464717
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.8600	−1.58273	−0.791365	−	0.611343i	$-0.790629\pi$
$953$	−48.8600	−1.58273	−0.791365	+	0.611343i	$0.790629\pi$
$954$	0	0
$955$	−3.54400	−0.114681
$956$	0	0
$957$	0	0
$958$	0	0
$959$	100.632	3.24958
$960$	0	0
$961$	−26.0360	−0.839872
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.7720	−0.411145
$966$	0	0
$967$	4.77200	0.153457	0.0767286	−	0.997052i	$-0.475552\pi$
$967$	4.77200	0.153457	0.0767286	+	0.997052i	$0.475552\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.7720	1.21216	0.606081	−	0.795403i	$-0.292741\pi$
$971$	37.7720	1.21216	0.606081	+	0.795403i	$0.292741\pi$
$972$	0	0
$973$	60.9480	1.95390
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.227998	0.00729431	0.00364715	−	0.999993i	$-0.498839\pi$
$977$	0.227998	0.00729431	0.00364715	+	0.999993i	$0.498839\pi$
$978$	0	0
$979$	−30.1760	−0.964430
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.4040	0.459417	0.229708	−	0.973260i	$-0.426223\pi$
$983$	14.4040	0.459417	0.229708	+	0.973260i	$0.426223\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.7720	0.437924
$990$	0	0
$991$	−8.08801	−0.256924	−0.128462	−	0.991714i	$-0.541004\pi$
$991$	−8.08801	−0.256924	−0.128462	+	0.991714i	$0.541004\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.0000	0.475532
$996$	0	0
$997$	−12.6840	−0.401706	−0.200853	−	0.979621i	$-0.564371\pi$
$997$	−12.6840	−0.401706	−0.200853	+	0.979621i	$0.564371\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8640.2.a.db.1.1		2
3.2	odd	2		8640.2.a.cn.1.1		2
4.3	odd	2		8640.2.a.de.1.2		2
8.3	odd	2		1080.2.a.m.1.2	✓	2
8.5	even	2		2160.2.a.z.1.1		2
12.11	even	2		8640.2.a.cq.1.2		2
24.5	odd	2		2160.2.a.bb.1.1		2
24.11	even	2		1080.2.a.n.1.2	yes	2
40.3	even	4		5400.2.f.be.649.1		4
40.19	odd	2		5400.2.a.cb.1.1		2
40.27	even	4		5400.2.f.be.649.4		4
72.11	even	6		3240.2.q.z.1081.1		4
72.43	odd	6		3240.2.q.bc.1081.1		4
72.59	even	6		3240.2.q.z.2161.1		4
72.67	odd	6		3240.2.q.bc.2161.1		4
120.59	even	2		5400.2.a.ca.1.1		2
120.83	odd	4		5400.2.f.bd.649.1		4
120.107	odd	4		5400.2.f.bd.649.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.m.1.2	✓	2	8.3	odd	2
1080.2.a.n.1.2	yes	2	24.11	even	2
2160.2.a.z.1.1		2	8.5	even	2
2160.2.a.bb.1.1		2	24.5	odd	2
3240.2.q.z.1081.1		4	72.11	even	6
3240.2.q.z.2161.1		4	72.59	even	6
3240.2.q.bc.1081.1		4	72.43	odd	6
3240.2.q.bc.2161.1		4	72.67	odd	6
5400.2.a.ca.1.1		2	120.59	even	2
5400.2.a.cb.1.1		2	40.19	odd	2
5400.2.f.bd.649.1		4	120.83	odd	4
5400.2.f.bd.649.4		4	120.107	odd	4
5400.2.f.be.649.1		4	40.3	even	4
5400.2.f.be.649.4		4	40.27	even	4
8640.2.a.cn.1.1		2	3.2	odd	2
8640.2.a.cq.1.2		2	12.11	even	2
8640.2.a.db.1.1		2	1.1	even	1	trivial
8640.2.a.de.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.77200 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.77200 q^{7} -3.77200 q^{11} -3.00000 q^{13} +1.77200 q^{17} -2.77200 q^{19} -7.77200 q^{23} +1.00000 q^{25} -3.77200 q^{29} -2.22800 q^{31} -4.77200 q^{35} +0.772002 q^{37} -9.54400 q^{41} -1.77200 q^{43} -5.77200 q^{47} +15.7720 q^{49} +9.54400 q^{53} -3.77200 q^{55} +12.0000 q^{59} -0.772002 q^{61} -3.00000 q^{65} +14.7720 q^{67} -13.5440 q^{71} +4.77200 q^{73} +18.0000 q^{77} +5.00000 q^{79} +6.00000 q^{83} +1.77200 q^{85} +8.00000 q^{89} +14.3160 q^{91} -2.77200 q^{95} -18.3160 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - q^7	\( 2 q + 2 q^{5} - q^{7} + q^{11} - 6 q^{13} - 5 q^{17} + 3 q^{19} - 7 q^{23} + 2 q^{25} + q^{29} - 13 q^{31} - q^{35} - 7 q^{37} - 2 q^{41} + 5 q^{43} - 3 q^{47} + 23 q^{49} + 2 q^{53} + q^{55} + 24 q^{59} + 7 q^{61} - 6 q^{65} + 21 q^{67} - 10 q^{71} + q^{73} + 36 q^{77} + 10 q^{79} + 12 q^{83} - 5 q^{85} + 16 q^{89} + 3 q^{91} + 3 q^{95} - 11 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 + q^11 - 6 * q^13 - 5 * q^17 + 3 * q^19 - 7 * q^23 + 2 * q^25 + q^29 - 13 * q^31 - q^35 - 7 * q^37 - 2 * q^41 + 5 * q^43 - 3 * q^47 + 23 * q^49 + 2 * q^53 + q^55 + 24 * q^59 + 7 * q^61 - 6 * q^65 + 21 * q^67 - 10 * q^71 + q^73 + 36 * q^77 + 10 * q^79 + 12 * q^83 - 5 * q^85 + 16 * q^89 + 3 * q^91 + 3 * q^95 - 11 * q^97

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