LMFDB - Embedded newform 3240.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, 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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	3240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.27492 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +4.27492 q^{7} -1.27492 q^{11} +6.27492 q^{13} +2.00000 q^{17} +1.00000 q^{19} +0.274917 q^{23} +1.00000 q^{25} -1.27492 q^{29} -1.27492 q^{31} +4.27492 q^{35} +4.54983 q^{37} -7.54983 q^{41} +4.00000 q^{43} +6.27492 q^{47} +11.2749 q^{49} +8.27492 q^{53} -1.27492 q^{55} -13.0000 q^{59} -6.54983 q^{61} +6.27492 q^{65} -14.5498 q^{67} -0.725083 q^{71} -15.0997 q^{73} -5.45017 q^{77} -4.54983 q^{79} +12.5498 q^{83} +2.00000 q^{85} -9.82475 q^{89} +26.8248 q^{91} +1.00000 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + q^7	$ 2 q + 2 q^{5} + q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 5 q^{29} + 5 q^{31} + q^{35} - 6 q^{37} + 8 q^{43} + 5 q^{47} + 15 q^{49} + 9 q^{53} + 5 q^{55} - 26 q^{59} + 2 q^{61} + 5 q^{65} - 14 q^{67} - 9 q^{71} - 26 q^{77} + 6 q^{79} + 10 q^{83} + 4 q^{85} + 3 q^{89} + 31 q^{91} + 2 q^{95} + 32 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + q^7 + 5 * q^11 + 5 * q^13 + 4 * q^17 + 2 * q^19 - 7 * q^23 + 2 * q^25 + 5 * q^29 + 5 * q^31 + q^35 - 6 * q^37 + 8 * q^43 + 5 * q^47 + 15 * q^49 + 9 * q^53 + 5 * q^55 - 26 * q^59 + 2 * q^61 + 5 * q^65 - 14 * q^67 - 9 * q^71 - 26 * q^77 + 6 * q^79 + 10 * q^83 + 4 * q^85 + 3 * q^89 + 31 * q^91 + 2 * q^95 + 32 * 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
$7$	4.27492	1.61577	0.807883	−	0.589342i	$-0.200613\pi$
$7$	4.27492	1.61577	0.807883	+	0.589342i	$0.200613\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.27492	−0.384402	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	−1.27492	−0.384402	−0.192201	+	0.981356i	$0.561563\pi$
$12$	0	0
$13$	6.27492	1.74035	0.870174	−	0.492744i	$-0.164006\pi$
$13$	6.27492	1.74035	0.870174	+	0.492744i	$0.164006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.274917	0.0573242	0.0286621	−	0.999589i	$-0.490875\pi$
$23$	0.274917	0.0573242	0.0286621	+	0.999589i	$0.490875\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.27492	−0.236746	−0.118373	−	0.992969i	$-0.537768\pi$
$29$	−1.27492	−0.236746	−0.118373	+	0.992969i	$0.537768\pi$
$30$	0	0
$31$	−1.27492	−0.228982	−0.114491	−	0.993424i	$-0.536524\pi$
$31$	−1.27492	−0.228982	−0.114491	+	0.993424i	$0.536524\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.27492	0.722593
$36$	0	0
$37$	4.54983	0.747988	0.373994	−	0.927431i	$-0.377988\pi$
$37$	4.54983	0.747988	0.373994	+	0.927431i	$0.377988\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.54983	−1.17909	−0.589543	−	0.807737i	$-0.700692\pi$
$41$	−7.54983	−1.17909	−0.589543	+	0.807737i	$0.700692\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.27492	0.915291	0.457645	−	0.889135i	$-0.348693\pi$
$47$	6.27492	0.915291	0.457645	+	0.889135i	$0.348693\pi$
$48$	0	0
$49$	11.2749	1.61070
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.27492	1.13665	0.568324	−	0.822805i	$-0.307592\pi$
$53$	8.27492	1.13665	0.568324	+	0.822805i	$0.307592\pi$
$54$	0	0
$55$	−1.27492	−0.171910
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.0000	−1.69246	−0.846228	−	0.532821i	$-0.821132\pi$
$59$	−13.0000	−1.69246	−0.846228	+	0.532821i	$0.821132\pi$
$60$	0	0
$61$	−6.54983	−0.838620	−0.419310	−	0.907843i	$-0.637728\pi$
$61$	−6.54983	−0.838620	−0.419310	+	0.907843i	$0.637728\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.27492	0.778308
$66$	0	0
$67$	−14.5498	−1.77755	−0.888773	−	0.458348i	$-0.848441\pi$
$67$	−14.5498	−1.77755	−0.888773	+	0.458348i	$0.848441\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.725083	−0.0860515	−0.0430257	−	0.999074i	$-0.513700\pi$
$71$	−0.725083	−0.0860515	−0.0430257	+	0.999074i	$0.513700\pi$
$72$	0	0
$73$	−15.0997	−1.76728	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	−15.0997	−1.76728	−0.883641	+	0.468165i	$0.844915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.45017	−0.621104
$78$	0	0
$79$	−4.54983	−0.511896	−0.255948	−	0.966691i	$-0.582388\pi$
$79$	−4.54983	−0.511896	−0.255948	+	0.966691i	$0.582388\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5498	1.37752	0.688762	−	0.724988i	$-0.258155\pi$
$83$	12.5498	1.37752	0.688762	+	0.724988i	$0.258155\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.82475	−1.04142	−0.520711	−	0.853733i	$-0.674333\pi$
$89$	−9.82475	−1.04142	−0.520711	+	0.853733i	$0.674333\pi$
$90$	0	0
$91$	26.8248	2.81200
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.2749	1.71892	0.859459	−	0.511204i	$-0.170800\pi$
$101$	17.2749	1.71892	0.859459	+	0.511204i	$0.170800\pi$
$102$	0	0
$103$	16.8248	1.65779	0.828896	−	0.559403i	$-0.188969\pi$
$103$	16.8248	1.65779	0.828896	+	0.559403i	$0.188969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.0997	−1.45974	−0.729870	−	0.683586i	$-0.760419\pi$
$107$	−15.0997	−1.45974	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	1.27492	0.122115	0.0610575	−	0.998134i	$-0.480553\pi$
$109$	1.27492	0.122115	0.0610575	+	0.998134i	$0.480553\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0.274917	0.0256362
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.54983	0.783762
$120$	0	0
$121$	−9.37459	−0.852235
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.72508	0.330548	0.165274	−	0.986248i	$-0.447149\pi$
$127$	3.72508	0.330548	0.165274	+	0.986248i	$0.447149\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.5498	−1.18385	−0.591927	−	0.805991i	$-0.701633\pi$
$131$	−13.5498	−1.18385	−0.591927	+	0.805991i	$0.701633\pi$
$132$	0	0
$133$	4.27492	0.370682
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.549834	−0.0469755	−0.0234878	−	0.999724i	$-0.507477\pi$
$137$	−0.549834	−0.0469755	−0.0234878	+	0.999724i	$0.507477\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−1.27492	−0.105876
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−4.72508	−0.384522	−0.192261	−	0.981344i	$-0.561582\pi$
$151$	−4.72508	−0.384522	−0.192261	+	0.981344i	$0.561582\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.27492	−0.102404
$156$	0	0
$157$	18.2749	1.45850	0.729249	−	0.684249i	$-0.239870\pi$
$157$	18.2749	1.45850	0.729249	+	0.684249i	$0.239870\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.17525	0.0926225
$162$	0	0
$163$	5.45017	0.426890	0.213445	−	0.976955i	$-0.431532\pi$
$163$	5.45017	0.426890	0.213445	+	0.976955i	$0.431532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.5498	−1.74496	−0.872479	−	0.488651i	$-0.837489\pi$
$167$	−22.5498	−1.74496	−0.872479	+	0.488651i	$0.837489\pi$
$168$	0	0
$169$	26.3746	2.02881
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.2749	1.23736	0.618680	−	0.785643i	$-0.287668\pi$
$173$	16.2749	1.23736	0.618680	+	0.785643i	$0.287668\pi$
$174$	0	0
$175$	4.27492	0.323153
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.54983	0.265327	0.132664	−	0.991161i	$-0.457647\pi$
$179$	3.54983	0.265327	0.132664	+	0.991161i	$0.457647\pi$
$180$	0	0
$181$	8.72508	0.648530	0.324265	−	0.945966i	$-0.394883\pi$
$181$	8.72508	0.648530	0.324265	+	0.945966i	$0.394883\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.54983	0.334510
$186$	0	0
$187$	−2.54983	−0.186462
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.3746	−1.32954	−0.664769	−	0.747049i	$-0.731470\pi$
$191$	−18.3746	−1.32954	−0.664769	+	0.747049i	$0.731470\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.3746	1.09539	0.547697	−	0.836677i	$-0.315505\pi$
$197$	15.3746	1.09539	0.547697	+	0.836677i	$0.315505\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.45017	−0.382527
$204$	0	0
$205$	−7.54983	−0.527303
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.27492	−0.0881879
$210$	0	0
$211$	14.0997	0.970661	0.485331	−	0.874331i	$-0.338699\pi$
$211$	14.0997	0.970661	0.485331	+	0.874331i	$0.338699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−5.45017	−0.369981
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.5498	0.844193
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.5498	−1.49669	−0.748343	−	0.663312i	$-0.769150\pi$
$227$	−22.5498	−1.49669	−0.748343	+	0.663312i	$0.769150\pi$
$228$	0	0
$229$	−6.54983	−0.432825	−0.216413	−	0.976302i	$-0.569436\pi$
$229$	−6.54983	−0.432825	−0.216413	+	0.976302i	$0.569436\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0997	1.38229	0.691143	−	0.722718i	$-0.257108\pi$
$233$	21.0997	1.38229	0.691143	+	0.722718i	$0.257108\pi$
$234$	0	0
$235$	6.27492	0.409330
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.5498	−1.84674	−0.923368	−	0.383917i	$-0.874575\pi$
$239$	−28.5498	−1.84674	−0.923368	+	0.383917i	$0.874575\pi$
$240$	0	0
$241$	−16.7251	−1.07736	−0.538679	−	0.842511i	$-0.681076\pi$
$241$	−16.7251	−1.07736	−0.538679	+	0.842511i	$0.681076\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.2749	0.720328
$246$	0	0
$247$	6.27492	0.399263
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.2749	1.65846	0.829229	−	0.558909i	$-0.188780\pi$
$251$	26.2749	1.65846	0.829229	+	0.558909i	$0.188780\pi$
$252$	0	0
$253$	−0.350497	−0.0220355
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.0997	−1.69043	−0.845215	−	0.534426i	$-0.820528\pi$
$257$	−27.0997	−1.69043	−0.845215	+	0.534426i	$0.820528\pi$
$258$	0	0
$259$	19.4502	1.20857
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.27492	−0.510253	−0.255127	−	0.966908i	$-0.582117\pi$
$263$	−8.27492	−0.510253	−0.255127	+	0.966908i	$0.582117\pi$
$264$	0	0
$265$	8.27492	0.508324
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.9244	1.64161	0.820805	−	0.571208i	$-0.193525\pi$
$269$	26.9244	1.64161	0.820805	+	0.571208i	$0.193525\pi$
$270$	0	0
$271$	28.5498	1.73428	0.867139	−	0.498065i	$-0.165956\pi$
$271$	28.5498	1.73428	0.867139	+	0.498065i	$0.165956\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.27492	−0.0768804
$276$	0	0
$277$	17.7251	1.06500	0.532499	−	0.846431i	$-0.321253\pi$
$277$	17.7251	1.06500	0.532499	+	0.846431i	$0.321253\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.8248	1.71954	0.859770	−	0.510681i	$-0.170607\pi$
$281$	28.8248	1.71954	0.859770	+	0.510681i	$0.170607\pi$
$282$	0	0
$283$	−26.5498	−1.57822	−0.789112	−	0.614249i	$-0.789459\pi$
$283$	−26.5498	−1.57822	−0.789112	+	0.614249i	$0.789459\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.2749	−1.90513
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.2749	1.06763	0.533816	−	0.845601i	$-0.320757\pi$
$293$	18.2749	1.06763	0.533816	+	0.845601i	$0.320757\pi$
$294$	0	0
$295$	−13.0000	−0.756889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.72508	0.0997641
$300$	0	0
$301$	17.0997	0.985609
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.54983	−0.375042
$306$	0	0
$307$	−29.0997	−1.66081	−0.830403	−	0.557163i	$-0.811890\pi$
$307$	−29.0997	−1.66081	−0.830403	+	0.557163i	$0.811890\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.9244	1.52674	0.763372	−	0.645959i	$-0.223542\pi$
$311$	26.9244	1.52674	0.763372	+	0.645959i	$0.223542\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.27492	−0.127772	−0.0638860	−	0.997957i	$-0.520349\pi$
$317$	−2.27492	−0.127772	−0.0638860	+	0.997957i	$0.520349\pi$
$318$	0	0
$319$	1.62541	0.0910057
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	6.27492	0.348070
$326$	0	0
$327$	0	0
$328$	0	0
$329$	26.8248	1.47890
$330$	0	0
$331$	3.82475	0.210227	0.105114	−	0.994460i	$-0.466479\pi$
$331$	3.82475	0.210227	0.105114	+	0.994460i	$0.466479\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−14.5498	−0.794942
$336$	0	0
$337$	20.5498	1.11942	0.559710	−	0.828688i	$-0.310912\pi$
$337$	20.5498	1.11942	0.559710	+	0.828688i	$0.310912\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.62541	0.0880211
$342$	0	0
$343$	18.2749	0.986753
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	0	0
$349$	−9.82475	−0.525907	−0.262953	−	0.964809i	$-0.584697\pi$
$349$	−9.82475	−0.525907	−0.262953	+	0.964809i	$0.584697\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.54983	−0.455062	−0.227531	−	0.973771i	$-0.573065\pi$
$353$	−8.54983	−0.455062	−0.227531	+	0.973771i	$0.573065\pi$
$354$	0	0
$355$	−0.725083	−0.0384834
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.8248	−1.15187	−0.575933	−	0.817497i	$-0.695361\pi$
$359$	−21.8248	−1.15187	−0.575933	+	0.817497i	$0.695361\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.0997	−0.790353
$366$	0	0
$367$	−2.54983	−0.133100	−0.0665501	−	0.997783i	$-0.521199\pi$
$367$	−2.54983	−0.133100	−0.0665501	+	0.997783i	$0.521199\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.3746	1.83656
$372$	0	0
$373$	0.900331	0.0466174	0.0233087	−	0.999728i	$-0.492580\pi$
$373$	0.900331	0.0466174	0.0233087	+	0.999728i	$0.492580\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	1.72508	0.0886116	0.0443058	−	0.999018i	$-0.485892\pi$
$379$	1.72508	0.0886116	0.0443058	+	0.999018i	$0.485892\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−35.3746	−1.80756	−0.903778	−	0.428001i	$-0.859218\pi$
$383$	−35.3746	−1.80756	−0.903778	+	0.428001i	$0.859218\pi$
$384$	0	0
$385$	−5.45017	−0.277766
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.45017	−0.0735263	−0.0367632	−	0.999324i	$-0.511705\pi$
$389$	−1.45017	−0.0735263	−0.0367632	+	0.999324i	$0.511705\pi$
$390$	0	0
$391$	0.549834	0.0278063
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.54983	−0.228927
$396$	0	0
$397$	−23.0997	−1.15934	−0.579670	−	0.814852i	$-0.696818\pi$
$397$	−23.0997	−1.15934	−0.579670	+	0.814852i	$0.696818\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.7251	0.785273	0.392637	−	0.919694i	$-0.371563\pi$
$401$	15.7251	0.785273	0.392637	+	0.919694i	$0.371563\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.80066	−0.287528
$408$	0	0
$409$	−7.17525	−0.354793	−0.177397	−	0.984139i	$-0.556768\pi$
$409$	−7.17525	−0.354793	−0.177397	+	0.984139i	$0.556768\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−55.5739	−2.73461
$414$	0	0
$415$	12.5498	0.616047
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.0997	1.03079	0.515393	−	0.856954i	$-0.327646\pi$
$419$	21.0997	1.03079	0.515393	+	0.856954i	$0.327646\pi$
$420$	0	0
$421$	−24.7251	−1.20503	−0.602513	−	0.798109i	$-0.705834\pi$
$421$	−24.7251	−1.20503	−0.602513	+	0.798109i	$0.705834\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−28.0000	−1.35501
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.27492	0.0614106	0.0307053	−	0.999528i	$-0.490225\pi$
$431$	1.27492	0.0614106	0.0307053	+	0.999528i	$0.490225\pi$
$432$	0	0
$433$	−2.54983	−0.122537	−0.0612686	−	0.998121i	$-0.519515\pi$
$433$	−2.54983	−0.122537	−0.0612686	+	0.998121i	$0.519515\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.274917	0.0131511
$438$	0	0
$439$	12.3746	0.590607	0.295303	−	0.955404i	$-0.404579\pi$
$439$	12.3746	0.590607	0.295303	+	0.955404i	$0.404579\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.54983	−0.216169	−0.108085	−	0.994142i	$-0.534472\pi$
$443$	−4.54983	−0.216169	−0.108085	+	0.994142i	$0.534472\pi$
$444$	0	0
$445$	−9.82475	−0.465738
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.5498	−0.639456	−0.319728	−	0.947509i	$-0.603592\pi$
$449$	−13.5498	−0.639456	−0.319728	+	0.947509i	$0.603592\pi$
$450$	0	0
$451$	9.62541	0.453243
$452$	0	0
$453$	0	0
$454$	0	0
$455$	26.8248	1.25756
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.3746	0.669491	0.334746	−	0.942309i	$-0.391350\pi$
$461$	14.3746	0.669491	0.334746	+	0.942309i	$0.391350\pi$
$462$	0	0
$463$	−26.4743	−1.23036	−0.615181	−	0.788386i	$-0.710917\pi$
$463$	−26.4743	−1.23036	−0.615181	+	0.788386i	$0.710917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.1993	0.934714	0.467357	−	0.884069i	$-0.345206\pi$
$467$	20.1993	0.934714	0.467357	+	0.884069i	$0.345206\pi$
$468$	0	0
$469$	−62.1993	−2.87210
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.09967	−0.234483
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.27492	0.149635	0.0748174	−	0.997197i	$-0.476163\pi$
$479$	3.27492	0.149635	0.0748174	+	0.997197i	$0.476163\pi$
$480$	0	0
$481$	28.5498	1.30176
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	−29.7251	−1.34697	−0.673486	−	0.739200i	$-0.735204\pi$
$487$	−29.7251	−1.34697	−0.673486	+	0.739200i	$0.735204\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	−2.54983	−0.114839
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.09967	−0.139039
$498$	0	0
$499$	−14.4502	−0.646878	−0.323439	−	0.946249i	$-0.604839\pi$
$499$	−14.4502	−0.646878	−0.323439	+	0.946249i	$0.604839\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	17.2749	0.768724
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.1993	−1.42721	−0.713605	−	0.700548i	$-0.752939\pi$
$509$	−32.1993	−1.42721	−0.713605	+	0.700548i	$0.752939\pi$
$510$	0	0
$511$	−64.5498	−2.85552
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.8248	0.741387
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.9244	1.13577	0.567885	−	0.823108i	$-0.307762\pi$
$521$	25.9244	1.13577	0.567885	+	0.823108i	$0.307762\pi$
$522$	0	0
$523$	5.64950	0.247036	0.123518	−	0.992342i	$-0.460582\pi$
$523$	5.64950	0.247036	0.123518	+	0.992342i	$0.460582\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.54983	−0.111073
$528$	0	0
$529$	−22.9244	−0.996714
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−47.3746	−2.05202
$534$	0	0
$535$	−15.0997	−0.652816
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.3746	−0.619157
$540$	0	0
$541$	−8.92442	−0.383691	−0.191845	−	0.981425i	$-0.561447\pi$
$541$	−8.92442	−0.383691	−0.191845	+	0.981425i	$0.561447\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.27492	0.0546115
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.27492	−0.0543133
$552$	0	0
$553$	−19.4502	−0.827105
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.9244	−0.844225	−0.422112	−	0.906544i	$-0.638711\pi$
$557$	−19.9244	−0.844225	−0.422112	+	0.906544i	$0.638711\pi$
$558$	0	0
$559$	25.0997	1.06160
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.90033	0.290814	0.145407	−	0.989372i	$-0.453551\pi$
$563$	6.90033	0.290814	0.145407	+	0.989372i	$0.453551\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.5498	−1.74186	−0.870930	−	0.491407i	$-0.836483\pi$
$569$	−41.5498	−1.74186	−0.870930	+	0.491407i	$0.836483\pi$
$570$	0	0
$571$	−23.8248	−0.997035	−0.498517	−	0.866880i	$-0.666122\pi$
$571$	−23.8248	−0.997035	−0.498517	+	0.866880i	$0.666122\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.274917	0.0114648
$576$	0	0
$577$	−26.5498	−1.10528	−0.552642	−	0.833419i	$-0.686380\pi$
$577$	−26.5498	−1.10528	−0.552642	+	0.833419i	$0.686380\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	53.6495	2.22576
$582$	0	0
$583$	−10.5498	−0.436929
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.4502	−0.802794	−0.401397	−	0.915904i	$-0.631475\pi$
$587$	−19.4502	−0.802794	−0.401397	+	0.915904i	$0.631475\pi$
$588$	0	0
$589$	−1.27492	−0.0525320
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.1993	0.500967	0.250483	−	0.968121i	$-0.419410\pi$
$593$	12.1993	0.500967	0.250483	+	0.968121i	$0.419410\pi$
$594$	0	0
$595$	8.54983	0.350509
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.2749	−0.950987	−0.475494	−	0.879719i	$-0.657731\pi$
$599$	−23.2749	−0.950987	−0.475494	+	0.879719i	$0.657731\pi$
$600$	0	0
$601$	−32.0997	−1.30937	−0.654686	−	0.755901i	$-0.727199\pi$
$601$	−32.0997	−1.30937	−0.654686	+	0.755901i	$0.727199\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.37459	−0.381131
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.3746	1.59293
$612$	0	0
$613$	37.0241	1.49539	0.747694	−	0.664043i	$-0.231161\pi$
$613$	37.0241	1.49539	0.747694	+	0.664043i	$0.231161\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.64950	0.0664065	0.0332033	−	0.999449i	$-0.489429\pi$
$617$	1.64950	0.0664065	0.0332033	+	0.999449i	$0.489429\pi$
$618$	0	0
$619$	19.3746	0.778730	0.389365	−	0.921083i	$-0.372694\pi$
$619$	19.3746	0.778730	0.389365	+	0.921083i	$0.372694\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−42.0000	−1.68269
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.09967	0.362828
$630$	0	0
$631$	−43.4743	−1.73068	−0.865341	−	0.501183i	$-0.832898\pi$
$631$	−43.4743	−1.73068	−0.865341	+	0.501183i	$0.832898\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.72508	0.147825
$636$	0	0
$637$	70.7492	2.80318
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.9244	−0.747470	−0.373735	−	0.927536i	$-0.621923\pi$
$641$	−18.9244	−0.747470	−0.373735	+	0.927536i	$0.621923\pi$
$642$	0	0
$643$	−2.90033	−0.114378	−0.0571889	−	0.998363i	$-0.518214\pi$
$643$	−2.90033	−0.114378	−0.0571889	+	0.998363i	$0.518214\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.4502	−0.528781	−0.264390	−	0.964416i	$-0.585171\pi$
$647$	−13.4502	−0.528781	−0.264390	+	0.964416i	$0.585171\pi$
$648$	0	0
$649$	16.5739	0.650583
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.5498	0.960709	0.480355	−	0.877074i	$-0.340508\pi$
$653$	24.5498	0.960709	0.480355	+	0.877074i	$0.340508\pi$
$654$	0	0
$655$	−13.5498	−0.529436
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.72508	−0.378835	−0.189418	−	0.981897i	$-0.560660\pi$
$659$	−9.72508	−0.378835	−0.189418	+	0.981897i	$0.560660\pi$
$660$	0	0
$661$	12.9244	0.502702	0.251351	−	0.967896i	$-0.419125\pi$
$661$	12.9244	0.502702	0.251351	+	0.967896i	$0.419125\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.27492	0.165774
$666$	0	0
$667$	−0.350497	−0.0135713
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.35050	0.322367
$672$	0	0
$673$	−39.0997	−1.50718	−0.753591	−	0.657344i	$-0.771680\pi$
$673$	−39.0997	−1.50718	−0.753591	+	0.657344i	$0.771680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.9244	0.996356	0.498178	−	0.867075i	$-0.334003\pi$
$677$	25.9244	0.996356	0.498178	+	0.867075i	$0.334003\pi$
$678$	0	0
$679$	68.3987	2.62490
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.5498	−0.633262	−0.316631	−	0.948549i	$-0.602552\pi$
$683$	−16.5498	−0.633262	−0.316631	+	0.948549i	$0.602552\pi$
$684$	0	0
$685$	−0.549834	−0.0210081
$686$	0	0
$687$	0	0
$688$	0	0
$689$	51.9244	1.97816
$690$	0	0
$691$	14.8248	0.563960	0.281980	−	0.959420i	$-0.409009\pi$
$691$	14.8248	0.563960	0.281980	+	0.959420i	$0.409009\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.00000	−0.341389
$696$	0	0
$697$	−15.0997	−0.571941
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.37459	0.165226	0.0826129	−	0.996582i	$-0.473673\pi$
$701$	4.37459	0.165226	0.0826129	+	0.996582i	$0.473673\pi$
$702$	0	0
$703$	4.54983	0.171600
$704$	0	0
$705$	0	0
$706$	0	0
$707$	73.8488	2.77737
$708$	0	0
$709$	−31.0997	−1.16797	−0.583986	−	0.811764i	$-0.698508\pi$
$709$	−31.0997	−1.16797	−0.583986	+	0.811764i	$0.698508\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.350497	−0.0131262
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.1752	−0.826997	−0.413499	−	0.910505i	$-0.635693\pi$
$719$	−22.1752	−0.826997	−0.413499	+	0.910505i	$0.635693\pi$
$720$	0	0
$721$	71.9244	2.67861
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.27492	−0.0473492
$726$	0	0
$727$	7.37459	0.273508	0.136754	−	0.990605i	$-0.456333\pi$
$727$	7.37459	0.273508	0.136754	+	0.990605i	$0.456333\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.5498	0.683292
$738$	0	0
$739$	−36.9244	−1.35829	−0.679143	−	0.734006i	$-0.737649\pi$
$739$	−36.9244	−1.35829	−0.679143	+	0.734006i	$0.737649\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.4502	−0.493439	−0.246719	−	0.969087i	$-0.579353\pi$
$743$	−13.4502	−0.493439	−0.246719	+	0.969087i	$0.579353\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−64.5498	−2.35860
$750$	0	0
$751$	19.4502	0.709747	0.354873	−	0.934914i	$-0.384524\pi$
$751$	19.4502	0.709747	0.354873	+	0.934914i	$0.384524\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.72508	−0.171963
$756$	0	0
$757$	−3.37459	−0.122651	−0.0613257	−	0.998118i	$-0.519533\pi$
$757$	−3.37459	−0.122651	−0.0613257	+	0.998118i	$0.519533\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.4502	1.24882	0.624409	−	0.781098i	$-0.285340\pi$
$761$	34.4502	1.24882	0.624409	+	0.781098i	$0.285340\pi$
$762$	0	0
$763$	5.45017	0.197309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−81.5739	−2.94546
$768$	0	0
$769$	−8.72508	−0.314635	−0.157317	−	0.987548i	$-0.550285\pi$
$769$	−8.72508	−0.314635	−0.157317	+	0.987548i	$0.550285\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.6495	−0.490939	−0.245469	−	0.969404i	$-0.578942\pi$
$773$	−13.6495	−0.490939	−0.245469	+	0.969404i	$0.578942\pi$
$774$	0	0
$775$	−1.27492	−0.0457964
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.54983	−0.270501
$780$	0	0
$781$	0.924421	0.0330784
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.2749	0.652260
$786$	0	0
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−25.6495	−0.911991
$792$	0	0
$793$	−41.0997	−1.45949
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40.5498	−1.43635	−0.718174	−	0.695863i	$-0.755022\pi$
$797$	−40.5498	−1.43635	−0.718174	+	0.695863i	$0.755022\pi$
$798$	0	0
$799$	12.5498	0.443981
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.2508	0.679347
$804$	0	0
$805$	1.17525	0.0414221
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28.6495	−1.00726	−0.503631	−	0.863919i	$-0.668003\pi$
$809$	−28.6495	−1.00726	−0.503631	+	0.863919i	$0.668003\pi$
$810$	0	0
$811$	35.8248	1.25798	0.628989	−	0.777415i	$-0.283469\pi$
$811$	35.8248	1.25798	0.628989	+	0.777415i	$0.283469\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.45017	0.190911
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.8248	−0.622088	−0.311044	−	0.950395i	$-0.600679\pi$
$821$	−17.8248	−0.622088	−0.311044	+	0.950395i	$0.600679\pi$
$822$	0	0
$823$	38.5498	1.34376	0.671881	−	0.740659i	$-0.265486\pi$
$823$	38.5498	1.34376	0.671881	+	0.740659i	$0.265486\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.09967	−0.316426	−0.158213	−	0.987405i	$-0.550573\pi$
$827$	−9.09967	−0.316426	−0.158213	+	0.987405i	$0.550573\pi$
$828$	0	0
$829$	−35.8248	−1.24425	−0.622123	−	0.782920i	$-0.713729\pi$
$829$	−35.8248	−1.24425	−0.622123	+	0.782920i	$0.713729\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22.5498	0.781305
$834$	0	0
$835$	−22.5498	−0.780369
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15.4743	−0.534231	−0.267115	−	0.963665i	$-0.586070\pi$
$839$	−15.4743	−0.534231	−0.267115	+	0.963665i	$0.586070\pi$
$840$	0	0
$841$	−27.3746	−0.943951
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26.3746	0.907313
$846$	0	0
$847$	−40.0756	−1.37701
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.25083	0.0428778
$852$	0	0
$853$	41.2990	1.41405	0.707026	−	0.707188i	$-0.250037\pi$
$853$	41.2990	1.41405	0.707026	+	0.707188i	$0.250037\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.7492	0.777097	0.388548	−	0.921428i	$-0.372977\pi$
$857$	22.7492	0.777097	0.388548	+	0.921428i	$0.372977\pi$
$858$	0	0
$859$	−15.4743	−0.527975	−0.263987	−	0.964526i	$-0.585038\pi$
$859$	−15.4743	−0.527975	−0.263987	+	0.964526i	$0.585038\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.7251	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$863$	15.7251	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$864$	0	0
$865$	16.2749	0.553364
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.80066	0.196774
$870$	0	0
$871$	−91.2990	−3.09355
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.27492	0.144519
$876$	0	0
$877$	−5.72508	−0.193322	−0.0966612	−	0.995317i	$-0.530816\pi$
$877$	−5.72508	−0.193322	−0.0966612	+	0.995317i	$0.530816\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	56.3746	1.89931	0.949654	−	0.313301i	$-0.101435\pi$
$881$	56.3746	1.89931	0.949654	+	0.313301i	$0.101435\pi$
$882$	0	0
$883$	−28.1993	−0.948983	−0.474492	−	0.880260i	$-0.657368\pi$
$883$	−28.1993	−0.948983	−0.474492	+	0.880260i	$0.657368\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.8248	0.766380	0.383190	−	0.923670i	$-0.374825\pi$
$887$	22.8248	0.766380	0.383190	+	0.923670i	$0.374825\pi$
$888$	0	0
$889$	15.9244	0.534088
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.27492	0.209982
$894$	0	0
$895$	3.54983	0.118658
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.62541	0.0542106
$900$	0	0
$901$	16.5498	0.551355
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.72508	0.290032
$906$	0	0
$907$	20.9003	0.693984	0.346992	−	0.937868i	$-0.387203\pi$
$907$	20.9003	0.693984	0.346992	+	0.937868i	$0.387203\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	56.0241	1.85616	0.928080	−	0.372380i	$-0.121458\pi$
$911$	56.0241	1.85616	0.928080	+	0.372380i	$0.121458\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−57.9244	−1.91283
$918$	0	0
$919$	−40.9244	−1.34997	−0.674986	−	0.737831i	$-0.735850\pi$
$919$	−40.9244	−1.34997	−0.674986	+	0.737831i	$0.735850\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.54983	−0.149760
$924$	0	0
$925$	4.54983	0.149598
$926$	0	0
$927$	0	0
$928$	0	0
$929$	53.1238	1.74293	0.871467	−	0.490454i	$-0.163169\pi$
$929$	53.1238	1.74293	0.871467	+	0.490454i	$0.163169\pi$
$930$	0	0
$931$	11.2749	0.369520
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.54983	−0.0833885
$936$	0	0
$937$	14.5498	0.475322	0.237661	−	0.971348i	$-0.423619\pi$
$937$	14.5498	0.475322	0.237661	+	0.971348i	$0.423619\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−23.0997	−0.753028	−0.376514	−	0.926411i	$-0.622877\pi$
$941$	−23.0997	−0.753028	−0.376514	+	0.926411i	$0.622877\pi$
$942$	0	0
$943$	−2.07558	−0.0675902
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.5498	−1.38268	−0.691342	−	0.722528i	$-0.742980\pi$
$947$	−42.5498	−1.38268	−0.691342	+	0.722528i	$0.742980\pi$
$948$	0	0
$949$	−94.7492	−3.07569
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.90033	−0.0939509	−0.0469755	−	0.998896i	$-0.514958\pi$
$953$	−2.90033	−0.0939509	−0.0469755	+	0.998896i	$0.514958\pi$
$954$	0	0
$955$	−18.3746	−0.594588
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.35050	−0.0759015
$960$	0	0
$961$	−29.3746	−0.947567
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	55.6495	1.78957	0.894784	−	0.446500i	$-0.147330\pi$
$967$	55.6495	1.78957	0.894784	+	0.446500i	$0.147330\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.64950	−0.277576	−0.138788	−	0.990322i	$-0.544321\pi$
$971$	−8.64950	−0.277576	−0.138788	+	0.990322i	$0.544321\pi$
$972$	0	0
$973$	−38.4743	−1.23343
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.6495	−1.33249	−0.666243	−	0.745735i	$-0.732099\pi$
$977$	−41.6495	−1.33249	−0.666243	+	0.745735i	$0.732099\pi$
$978$	0	0
$979$	12.5257	0.400325
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.7492	0.916956	0.458478	−	0.888706i	$-0.348395\pi$
$983$	28.7492	0.916956	0.458478	+	0.888706i	$0.348395\pi$
$984$	0	0
$985$	15.3746	0.489875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.09967	0.0349674
$990$	0	0
$991$	−9.27492	−0.294627	−0.147314	−	0.989090i	$-0.547063\pi$
$991$	−9.27492	−0.294627	−0.147314	+	0.989090i	$0.547063\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−29.3746	−0.930302	−0.465151	−	0.885231i	$-0.654000\pi$
$997$	−29.3746	−0.930302	−0.465151	+	0.885231i	$0.654000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.a.n.1.2	yes	2
3.2	odd	2		3240.2.a.h.1.2	✓	2
4.3	odd	2		6480.2.a.bm.1.1		2
9.2	odd	6		3240.2.q.be.1081.1		4
9.4	even	3		3240.2.q.y.2161.1		4
9.5	odd	6		3240.2.q.be.2161.1		4
9.7	even	3		3240.2.q.y.1081.1		4
12.11	even	2		6480.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.h.1.2	✓	2	3.2	odd	2
3240.2.a.n.1.2	yes	2	1.1	even	1	trivial
3240.2.q.y.1081.1		4	9.7	even	3
3240.2.q.y.2161.1		4	9.4	even	3
3240.2.q.be.1081.1		4	9.2	odd	6
3240.2.q.be.2161.1		4	9.5	odd	6
6480.2.a.bf.1.1		2	12.11	even	2
6480.2.a.bm.1.1		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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.27492 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +4.27492 q^{7} -1.27492 q^{11} +6.27492 q^{13} +2.00000 q^{17} +1.00000 q^{19} +0.274917 q^{23} +1.00000 q^{25} -1.27492 q^{29} -1.27492 q^{31} +4.27492 q^{35} +4.54983 q^{37} -7.54983 q^{41} +4.00000 q^{43} +6.27492 q^{47} +11.2749 q^{49} +8.27492 q^{53} -1.27492 q^{55} -13.0000 q^{59} -6.54983 q^{61} +6.27492 q^{65} -14.5498 q^{67} -0.725083 q^{71} -15.0997 q^{73} -5.45017 q^{77} -4.54983 q^{79} +12.5498 q^{83} +2.00000 q^{85} -9.82475 q^{89} +26.8248 q^{91} +1.00000 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + q^7	\( 2 q + 2 q^{5} + q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 5 q^{29} + 5 q^{31} + q^{35} - 6 q^{37} + 8 q^{43} + 5 q^{47} + 15 q^{49} + 9 q^{53} + 5 q^{55} - 26 q^{59} + 2 q^{61} + 5 q^{65} - 14 q^{67} - 9 q^{71} - 26 q^{77} + 6 q^{79} + 10 q^{83} + 4 q^{85} + 3 q^{89} + 31 q^{91} + 2 q^{95} + 32 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + q^7 + 5 * q^11 + 5 * q^13 + 4 * q^17 + 2 * q^19 - 7 * q^23 + 2 * q^25 + 5 * q^29 + 5 * q^31 + q^35 - 6 * q^37 + 8 * q^43 + 5 * q^47 + 15 * q^49 + 9 * q^53 + 5 * q^55 - 26 * q^59 + 2 * q^61 + 5 * q^65 - 14 * q^67 - 9 * q^71 - 26 * q^77 + 6 * q^79 + 10 * q^83 + 4 * q^85 + 3 * q^89 + 31 * q^91 + 2 * q^95 + 32 * q^97

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