LMFDB - Embedded newform 3240.2.a.n.1.1

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			1.1
Root			$-3.27492$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.27492 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.27492 q^{7} +6.27492 q^{11} -1.27492 q^{13} +2.00000 q^{17} +1.00000 q^{19} -7.27492 q^{23} +1.00000 q^{25} +6.27492 q^{29} +6.27492 q^{31} -3.27492 q^{35} -10.5498 q^{37} +7.54983 q^{41} +4.00000 q^{43} -1.27492 q^{47} +3.72508 q^{49} +0.725083 q^{53} +6.27492 q^{55} -13.0000 q^{59} +8.54983 q^{61} -1.27492 q^{65} +0.549834 q^{67} -8.27492 q^{71} +15.0997 q^{73} -20.5498 q^{77} +10.5498 q^{79} -2.54983 q^{83} +2.00000 q^{85} +12.8248 q^{89} +4.17525 q^{91} +1.00000 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.27492	−1.23780	−0.618901	−	0.785469i	$-0.712422\pi$
$7$	−3.27492	−1.23780	−0.618901	+	0.785469i	$0.712422\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.27492	1.89196	0.945979	−	0.324227i	$-0.105104\pi$
$11$	6.27492	1.89196	0.945979	+	0.324227i	$0.105104\pi$
$12$	0	0
$13$	−1.27492	−0.353598	−0.176799	−	0.984247i	$-0.556574\pi$
$13$	−1.27492	−0.353598	−0.176799	+	0.984247i	$0.556574\pi$
$14$	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
$23$	−7.27492	−1.51693	−0.758463	−	0.651717i	$-0.774049\pi$
$23$	−7.27492	−1.51693	−0.758463	+	0.651717i	$0.774049\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.27492	1.16522	0.582611	−	0.812751i	$-0.302031\pi$
$29$	6.27492	1.16522	0.582611	+	0.812751i	$0.302031\pi$
$30$	0	0
$31$	6.27492	1.12701	0.563504	−	0.826113i	$-0.309453\pi$
$31$	6.27492	1.12701	0.563504	+	0.826113i	$0.309453\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.27492	−0.553562
$36$	0	0
$37$	−10.5498	−1.73438	−0.867191	−	0.497976i	$-0.834077\pi$
$37$	−10.5498	−1.73438	−0.867191	+	0.497976i	$0.834077\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.54983	1.17909	0.589543	−	0.807737i	$-0.299308\pi$
$41$	7.54983	1.17909	0.589543	+	0.807737i	$0.299308\pi$
$42$	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
$47$	−1.27492	−0.185966	−0.0929829	−	0.995668i	$-0.529640\pi$
$47$	−1.27492	−0.185966	−0.0929829	+	0.995668i	$0.529640\pi$
$48$	0	0
$49$	3.72508	0.532155
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.725083	0.0995978	0.0497989	−	0.998759i	$-0.484142\pi$
$53$	0.725083	0.0995978	0.0497989	+	0.998759i	$0.484142\pi$
$54$	0	0
$55$	6.27492	0.846110
$56$	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
$61$	8.54983	1.09469	0.547347	−	0.836906i	$-0.315638\pi$
$61$	8.54983	1.09469	0.547347	+	0.836906i	$0.315638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.27492	−0.158134
$66$	0	0
$67$	0.549834	0.0671730	0.0335865	−	0.999436i	$-0.489307\pi$
$67$	0.549834	0.0671730	0.0335865	+	0.999436i	$0.489307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.27492	−0.982052	−0.491026	−	0.871145i	$-0.663378\pi$
$71$	−8.27492	−0.982052	−0.491026	+	0.871145i	$0.663378\pi$
$72$	0	0
$73$	15.0997	1.76728	0.883641	−	0.468165i	$-0.155085\pi$
$73$	15.0997	1.76728	0.883641	+	0.468165i	$0.155085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.5498	−2.34187
$78$	0	0
$79$	10.5498	1.18695	0.593475	−	0.804853i	$-0.297756\pi$
$79$	10.5498	1.18695	0.593475	+	0.804853i	$0.297756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.54983	−0.279881	−0.139940	−	0.990160i	$-0.544691\pi$
$83$	−2.54983	−0.279881	−0.139940	+	0.990160i	$0.544691\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.8248	1.35942	0.679710	−	0.733481i	$-0.262105\pi$
$89$	12.8248	1.35942	0.679710	+	0.733481i	$0.262105\pi$
$90$	0	0
$91$	4.17525	0.437685
$92$	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
$101$	9.72508	0.967682	0.483841	−	0.875156i	$-0.339241\pi$
$101$	9.72508	0.967682	0.483841	+	0.875156i	$0.339241\pi$
$102$	0	0
$103$	−5.82475	−0.573930	−0.286965	−	0.957941i	$-0.592646\pi$
$103$	−5.82475	−0.573930	−0.286965	+	0.957941i	$0.592646\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0997	1.45974	0.729870	−	0.683586i	$-0.239581\pi$
$107$	15.0997	1.45974	0.729870	+	0.683586i	$0.239581\pi$
$108$	0	0
$109$	−6.27492	−0.601028	−0.300514	−	0.953777i	$-0.597158\pi$
$109$	−6.27492	−0.601028	−0.300514	+	0.953777i	$0.597158\pi$
$110$	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
$115$	−7.27492	−0.678390
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.54983	−0.600422
$120$	0	0
$121$	28.3746	2.57951
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.2749	1.00049	0.500244	−	0.865885i	$-0.333244\pi$
$127$	11.2749	1.00049	0.500244	+	0.865885i	$0.333244\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.54983	0.135410	0.0677048	−	0.997705i	$-0.478432\pi$
$131$	1.54983	0.135410	0.0677048	+	0.997705i	$0.478432\pi$
$132$	0	0
$133$	−3.27492	−0.283971
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.5498	1.24308	0.621538	−	0.783384i	$-0.286508\pi$
$137$	14.5498	1.24308	0.621538	+	0.783384i	$0.286508\pi$
$138$	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
$145$	6.27492	0.521104
$146$	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
$151$	−12.2749	−0.998919	−0.499459	−	0.866337i	$-0.666468\pi$
$151$	−12.2749	−0.998919	−0.499459	+	0.866337i	$0.666468\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.27492	0.504013
$156$	0	0
$157$	10.7251	0.855955	0.427977	−	0.903789i	$-0.359226\pi$
$157$	10.7251	0.855955	0.427977	+	0.903789i	$0.359226\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.8248	1.87765
$162$	0	0
$163$	20.5498	1.60959	0.804794	−	0.593555i	$-0.202276\pi$
$163$	20.5498	1.60959	0.804794	+	0.593555i	$0.202276\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.45017	−0.576511	−0.288256	−	0.957554i	$-0.593075\pi$
$167$	−7.45017	−0.576511	−0.288256	+	0.957554i	$0.593075\pi$
$168$	0	0
$169$	−11.3746	−0.874968
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.72508	0.663356	0.331678	−	0.943393i	$-0.392385\pi$
$173$	8.72508	0.663356	0.331678	+	0.943393i	$0.392385\pi$
$174$	0	0
$175$	−3.27492	−0.247560
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5498	−0.863275	−0.431638	−	0.902047i	$-0.642064\pi$
$179$	−11.5498	−0.863275	−0.431638	+	0.902047i	$0.642064\pi$
$180$	0	0
$181$	16.2749	1.20971	0.604853	−	0.796337i	$-0.293232\pi$
$181$	16.2749	1.20971	0.604853	+	0.796337i	$0.293232\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.5498	−0.775639
$186$	0	0
$187$	12.5498	0.917735
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.3746	1.40190	0.700948	−	0.713212i	$-0.252761\pi$
$191$	19.3746	1.40190	0.700948	+	0.713212i	$0.252761\pi$
$192$	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
$197$	−22.3746	−1.59412	−0.797062	−	0.603898i	$-0.793613\pi$
$197$	−22.3746	−1.59412	−0.797062	+	0.603898i	$0.793613\pi$
$198$	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
$203$	−20.5498	−1.44232
$204$	0	0
$205$	7.54983	0.527303
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.27492	0.434045
$210$	0	0
$211$	−16.0997	−1.10835	−0.554173	−	0.832401i	$-0.686966\pi$
$211$	−16.0997	−1.10835	−0.554173	+	0.832401i	$0.686966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−20.5498	−1.39501
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.54983	−0.171520
$222$	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
$227$	−7.45017	−0.494485	−0.247242	−	0.968954i	$-0.579524\pi$
$227$	−7.45017	−0.494485	−0.247242	+	0.968954i	$0.579524\pi$
$228$	0	0
$229$	8.54983	0.564989	0.282494	−	0.959269i	$-0.408838\pi$
$229$	8.54983	0.564989	0.282494	+	0.959269i	$0.408838\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.09967	−0.596139	−0.298070	−	0.954544i	$-0.596343\pi$
$233$	−9.09967	−0.596139	−0.298070	+	0.954544i	$0.596343\pi$
$234$	0	0
$235$	−1.27492	−0.0831664
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.4502	−0.870019	−0.435009	−	0.900426i	$-0.643255\pi$
$239$	−13.4502	−0.870019	−0.435009	+	0.900426i	$0.643255\pi$
$240$	0	0
$241$	−24.2749	−1.56368	−0.781842	−	0.623476i	$-0.785720\pi$
$241$	−24.2749	−1.56368	−0.781842	+	0.623476i	$0.785720\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.72508	0.237987
$246$	0	0
$247$	−1.27492	−0.0811210
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.7251	1.18192	0.590958	−	0.806702i	$-0.298750\pi$
$251$	18.7251	1.18192	0.590958	+	0.806702i	$0.298750\pi$
$252$	0	0
$253$	−45.6495	−2.86996
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.09967	0.193352	0.0966760	−	0.995316i	$-0.469179\pi$
$257$	3.09967	0.193352	0.0966760	+	0.995316i	$0.469179\pi$
$258$	0	0
$259$	34.5498	2.14682
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.725083	−0.0447105	−0.0223553	−	0.999750i	$-0.507116\pi$
$263$	−0.725083	−0.0447105	−0.0223553	+	0.999750i	$0.507116\pi$
$264$	0	0
$265$	0.725083	0.0445415
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.9244	−1.58064	−0.790320	−	0.612694i	$-0.790086\pi$
$269$	−25.9244	−1.58064	−0.790320	+	0.612694i	$0.790086\pi$
$270$	0	0
$271$	13.4502	0.817039	0.408520	−	0.912750i	$-0.366045\pi$
$271$	13.4502	0.817039	0.408520	+	0.912750i	$0.366045\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.27492	0.378392
$276$	0	0
$277$	25.2749	1.51862	0.759311	−	0.650728i	$-0.225536\pi$
$277$	25.2749	1.51862	0.759311	+	0.650728i	$0.225536\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.17525	0.368384	0.184192	−	0.982890i	$-0.441033\pi$
$281$	6.17525	0.368384	0.184192	+	0.982890i	$0.441033\pi$
$282$	0	0
$283$	−11.4502	−0.680642	−0.340321	−	0.940309i	$-0.610536\pi$
$283$	−11.4502	−0.680642	−0.340321	+	0.940309i	$0.610536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.7251	−1.45948
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.7251	0.626566	0.313283	−	0.949660i	$-0.398571\pi$
$293$	10.7251	0.626566	0.313283	+	0.949660i	$0.398571\pi$
$294$	0	0
$295$	−13.0000	−0.756889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.27492	0.536382
$300$	0	0
$301$	−13.0997	−0.755052
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.54983	0.489562
$306$	0	0
$307$	1.09967	0.0627614	0.0313807	−	0.999508i	$-0.490010\pi$
$307$	1.09967	0.0627614	0.0313807	+	0.999508i	$0.490010\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.9244	−1.47004	−0.735020	−	0.678046i	$-0.762827\pi$
$311$	−25.9244	−1.47004	−0.735020	+	0.678046i	$0.762827\pi$
$312$	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
$317$	5.27492	0.296269	0.148134	−	0.988967i	$-0.452673\pi$
$317$	5.27492	0.296269	0.148134	+	0.988967i	$0.452673\pi$
$318$	0	0
$319$	39.3746	2.20455
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−1.27492	−0.0707197
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.17525	0.230189
$330$	0	0
$331$	−18.8248	−1.03470	−0.517351	−	0.855773i	$-0.673082\pi$
$331$	−18.8248	−1.03470	−0.517351	+	0.855773i	$0.673082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.549834	0.0300407
$336$	0	0
$337$	5.45017	0.296889	0.148445	−	0.988921i	$-0.452573\pi$
$337$	5.45017	0.296889	0.148445	+	0.988921i	$0.452573\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	39.3746	2.13225
$342$	0	0
$343$	10.7251	0.579100
$344$	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
$349$	12.8248	0.686493	0.343247	−	0.939245i	$-0.388473\pi$
$349$	12.8248	0.686493	0.343247	+	0.939245i	$0.388473\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.54983	0.348613	0.174306	−	0.984691i	$-0.444232\pi$
$353$	6.54983	0.348613	0.174306	+	0.984691i	$0.444232\pi$
$354$	0	0
$355$	−8.27492	−0.439187
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.824752	0.0435287	0.0217644	−	0.999763i	$-0.493072\pi$
$359$	0.824752	0.0435287	0.0217644	+	0.999763i	$0.493072\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.0997	0.790353
$366$	0	0
$367$	12.5498	0.655096	0.327548	−	0.944835i	$-0.393778\pi$
$367$	12.5498	0.655096	0.327548	+	0.944835i	$0.393778\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.37459	−0.123282
$372$	0	0
$373$	31.0997	1.61028	0.805140	−	0.593085i	$-0.202090\pi$
$373$	31.0997	1.61028	0.805140	+	0.593085i	$0.202090\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	9.27492	0.476420	0.238210	−	0.971214i	$-0.423439\pi$
$379$	9.27492	0.476420	0.238210	+	0.971214i	$0.423439\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.37459	0.121336	0.0606678	−	0.998158i	$-0.480677\pi$
$383$	2.37459	0.121336	0.0606678	+	0.998158i	$0.480677\pi$
$384$	0	0
$385$	−20.5498	−1.04732
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.5498	−0.839110	−0.419555	−	0.907730i	$-0.637814\pi$
$389$	−16.5498	−0.839110	−0.419555	+	0.907730i	$0.637814\pi$
$390$	0	0
$391$	−14.5498	−0.735817
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.5498	0.530820
$396$	0	0
$397$	7.09967	0.356322	0.178161	−	0.984001i	$-0.442985\pi$
$397$	7.09967	0.356322	0.178161	+	0.984001i	$0.442985\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.2749	1.16229	0.581147	−	0.813799i	$-0.302604\pi$
$401$	23.2749	1.16229	0.581147	+	0.813799i	$0.302604\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−66.1993	−3.28138
$408$	0	0
$409$	−29.8248	−1.47474	−0.737370	−	0.675490i	$-0.763932\pi$
$409$	−29.8248	−1.47474	−0.737370	+	0.675490i	$0.763932\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	42.5739	2.09493
$414$	0	0
$415$	−2.54983	−0.125166
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.09967	−0.444548	−0.222274	−	0.974984i	$-0.571348\pi$
$419$	−9.09967	−0.444548	−0.222274	+	0.974984i	$0.571348\pi$
$420$	0	0
$421$	−32.2749	−1.57298	−0.786492	−	0.617601i	$-0.788105\pi$
$421$	−32.2749	−1.57298	−0.786492	+	0.617601i	$0.788105\pi$
$422$	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
$431$	−6.27492	−0.302252	−0.151126	−	0.988514i	$-0.548290\pi$
$431$	−6.27492	−0.302252	−0.151126	+	0.988514i	$0.548290\pi$
$432$	0	0
$433$	12.5498	0.603107	0.301553	−	0.953449i	$-0.402495\pi$
$433$	12.5498	0.603107	0.301553	+	0.953449i	$0.402495\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.27492	−0.348006
$438$	0	0
$439$	−25.3746	−1.21106	−0.605531	−	0.795821i	$-0.707039\pi$
$439$	−25.3746	−1.21106	−0.605531	+	0.795821i	$0.707039\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.5498	0.501238	0.250619	−	0.968086i	$-0.419366\pi$
$443$	10.5498	0.501238	0.250619	+	0.968086i	$0.419366\pi$
$444$	0	0
$445$	12.8248	0.607952
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.54983	0.0731412	0.0365706	−	0.999331i	$-0.488357\pi$
$449$	1.54983	0.0731412	0.0365706	+	0.999331i	$0.488357\pi$
$450$	0	0
$451$	47.3746	2.23078
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.17525	0.195739
$456$	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
$461$	−23.3746	−1.08866	−0.544332	−	0.838870i	$-0.683217\pi$
$461$	−23.3746	−1.08866	−0.544332	+	0.838870i	$0.683217\pi$
$462$	0	0
$463$	41.4743	1.92747	0.963736	−	0.266857i	$-0.0859853\pi$
$463$	41.4743	1.92747	0.963736	+	0.266857i	$0.0859853\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−40.1993	−1.86020	−0.930102	−	0.367302i	$-0.880282\pi$
$467$	−40.1993	−1.86020	−0.930102	+	0.367302i	$0.880282\pi$
$468$	0	0
$469$	−1.80066	−0.0831469
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.0997	1.15408
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.27492	−0.195326	−0.0976630	−	0.995220i	$-0.531137\pi$
$479$	−4.27492	−0.195326	−0.0976630	+	0.995220i	$0.531137\pi$
$480$	0	0
$481$	13.4502	0.613275
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	−37.2749	−1.68909	−0.844544	−	0.535486i	$-0.820128\pi$
$487$	−37.2749	−1.68909	−0.844544	+	0.535486i	$0.820128\pi$
$488$	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
$493$	12.5498	0.565216
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.0997	1.21559
$498$	0	0
$499$	−29.5498	−1.32283	−0.661416	−	0.750019i	$-0.730044\pi$
$499$	−29.5498	−1.32283	−0.661416	+	0.750019i	$0.730044\pi$
$500$	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
$505$	9.72508	0.432761
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.1993	1.24991	0.624957	−	0.780659i	$-0.285117\pi$
$509$	28.1993	1.24991	0.624957	+	0.780659i	$0.285117\pi$
$510$	0	0
$511$	−49.4502	−2.18755
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.82475	−0.256669
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.9244	−1.17958	−0.589790	−	0.807557i	$-0.700790\pi$
$521$	−26.9244	−1.17958	−0.589790	+	0.807557i	$0.700790\pi$
$522$	0	0
$523$	−39.6495	−1.73375	−0.866876	−	0.498524i	$-0.833876\pi$
$523$	−39.6495	−1.73375	−0.866876	+	0.498524i	$0.833876\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.5498	0.546679
$528$	0	0
$529$	29.9244	1.30106
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.62541	−0.416923
$534$	0	0
$535$	15.0997	0.652816
$536$	0	0
$537$	0	0
$538$	0	0
$539$	23.3746	1.00681
$540$	0	0
$541$	43.9244	1.88846	0.944229	−	0.329289i	$-0.106809\pi$
$541$	43.9244	1.88846	0.944229	+	0.329289i	$0.106809\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.27492	−0.268788
$546$	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
$551$	6.27492	0.267320
$552$	0	0
$553$	−34.5498	−1.46921
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.9244	1.39505	0.697526	−	0.716559i	$-0.254284\pi$
$557$	32.9244	1.39505	0.697526	+	0.716559i	$0.254284\pi$
$558$	0	0
$559$	−5.09967	−0.215693
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.0997	1.56356	0.781782	−	0.623551i	$-0.214311\pi$
$563$	37.0997	1.56356	0.781782	+	0.623551i	$0.214311\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.4502	−1.10885	−0.554424	−	0.832234i	$-0.687062\pi$
$569$	−26.4502	−1.10885	−0.554424	+	0.832234i	$0.687062\pi$
$570$	0	0
$571$	−1.17525	−0.0491826	−0.0245913	−	0.999698i	$-0.507828\pi$
$571$	−1.17525	−0.0491826	−0.0245913	+	0.999698i	$0.507828\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.27492	−0.303385
$576$	0	0
$577$	−11.4502	−0.476677	−0.238338	−	0.971182i	$-0.576603\pi$
$577$	−11.4502	−0.476677	−0.238338	+	0.971182i	$0.576603\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.35050	0.346437
$582$	0	0
$583$	4.54983	0.188435
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.5498	−1.42602	−0.713012	−	0.701152i	$-0.752670\pi$
$587$	−34.5498	−1.42602	−0.713012	+	0.701152i	$0.752670\pi$
$588$	0	0
$589$	6.27492	0.258553
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−48.1993	−1.97931	−0.989655	−	0.143469i	$-0.954174\pi$
$593$	−48.1993	−1.97931	−0.989655	+	0.143469i	$0.954174\pi$
$594$	0	0
$595$	−6.54983	−0.268517
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−15.7251	−0.642509	−0.321255	−	0.946993i	$-0.604105\pi$
$599$	−15.7251	−0.642509	−0.321255	+	0.946993i	$0.604105\pi$
$600$	0	0
$601$	−1.90033	−0.0775161	−0.0387581	−	0.999249i	$-0.512340\pi$
$601$	−1.90033	−0.0775161	−0.0387581	+	0.999249i	$0.512340\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28.3746	1.15359
$606$	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
$611$	1.62541	0.0657572
$612$	0	0
$613$	−46.0241	−1.85890	−0.929448	−	0.368954i	$-0.879716\pi$
$613$	−46.0241	−1.85890	−0.929448	+	0.368954i	$0.879716\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.6495	−1.75726	−0.878631	−	0.477501i	$-0.841543\pi$
$617$	−43.6495	−1.75726	−0.878631	+	0.477501i	$0.841543\pi$
$618$	0	0
$619$	−18.3746	−0.738537	−0.369268	−	0.929323i	$-0.620392\pi$
$619$	−18.3746	−0.738537	−0.369268	+	0.929323i	$0.620392\pi$
$620$	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
$629$	−21.0997	−0.841299
$630$	0	0
$631$	24.4743	0.974305	0.487152	−	0.873317i	$-0.338036\pi$
$631$	24.4743	0.974305	0.487152	+	0.873317i	$0.338036\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.2749	0.447431
$636$	0	0
$637$	−4.74917	−0.188169
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.9244	1.33993	0.669967	−	0.742391i	$-0.266308\pi$
$641$	33.9244	1.33993	0.669967	+	0.742391i	$0.266308\pi$
$642$	0	0
$643$	−33.0997	−1.30532	−0.652662	−	0.757649i	$-0.726348\pi$
$643$	−33.0997	−1.30532	−0.652662	+	0.757649i	$0.726348\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.5498	−1.12241	−0.561205	−	0.827677i	$-0.689662\pi$
$647$	−28.5498	−1.12241	−0.561205	+	0.827677i	$0.689662\pi$
$648$	0	0
$649$	−81.5739	−3.20206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.45017	0.369814	0.184907	−	0.982756i	$-0.440802\pi$
$653$	9.45017	0.369814	0.184907	+	0.982756i	$0.440802\pi$
$654$	0	0
$655$	1.54983	0.0605570
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.2749	−0.672935	−0.336468	−	0.941695i	$-0.609232\pi$
$659$	−17.2749	−0.672935	−0.336468	+	0.941695i	$0.609232\pi$
$660$	0	0
$661$	−39.9244	−1.55288	−0.776440	−	0.630191i	$-0.782977\pi$
$661$	−39.9244	−1.55288	−0.776440	+	0.630191i	$0.782977\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.27492	−0.126996
$666$	0	0
$667$	−45.6495	−1.76756
$668$	0	0
$669$	0	0
$670$	0	0
$671$	53.6495	2.07112
$672$	0	0
$673$	−8.90033	−0.343083	−0.171541	−	0.985177i	$-0.554875\pi$
$673$	−8.90033	−0.343083	−0.171541	+	0.985177i	$0.554875\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.9244	−1.03479	−0.517395	−	0.855747i	$-0.673098\pi$
$677$	−26.9244	−1.03479	−0.517395	+	0.855747i	$0.673098\pi$
$678$	0	0
$679$	−52.3987	−2.01088
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.45017	−0.0554890	−0.0277445	−	0.999615i	$-0.508832\pi$
$683$	−1.45017	−0.0554890	−0.0277445	+	0.999615i	$0.508832\pi$
$684$	0	0
$685$	14.5498	0.555921
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.924421	−0.0352176
$690$	0	0
$691$	−7.82475	−0.297668	−0.148834	−	0.988862i	$-0.547552\pi$
$691$	−7.82475	−0.297668	−0.148834	+	0.988862i	$0.547552\pi$
$692$	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
$701$	−33.3746	−1.26054	−0.630270	−	0.776376i	$-0.717056\pi$
$701$	−33.3746	−1.26054	−0.630270	+	0.776376i	$0.717056\pi$
$702$	0	0
$703$	−10.5498	−0.397895
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31.8488	−1.19780
$708$	0	0
$709$	−0.900331	−0.0338126	−0.0169063	−	0.999857i	$-0.505382\pi$
$709$	−0.900331	−0.0338126	−0.0169063	+	0.999857i	$0.505382\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−45.6495	−1.70959
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.8248	−1.67168	−0.835841	−	0.548972i	$-0.815019\pi$
$719$	−44.8248	−1.67168	−0.835841	+	0.548972i	$0.815019\pi$
$720$	0	0
$721$	19.0756	0.710412
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.27492	0.233045
$726$	0	0
$727$	−30.3746	−1.12653	−0.563266	−	0.826276i	$-0.690455\pi$
$727$	−30.3746	−1.12653	−0.563266	+	0.826276i	$0.690455\pi$
$728$	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
$737$	3.45017	0.127088
$738$	0	0
$739$	15.9244	0.585789	0.292895	−	0.956145i	$-0.405381\pi$
$739$	15.9244	0.585789	0.292895	+	0.956145i	$0.405381\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.5498	−1.04739	−0.523696	−	0.851905i	$-0.675447\pi$
$743$	−28.5498	−1.04739	−0.523696	+	0.851905i	$0.675447\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−49.4502	−1.80687
$750$	0	0
$751$	34.5498	1.26074	0.630371	−	0.776294i	$-0.282903\pi$
$751$	34.5498	1.26074	0.630371	+	0.776294i	$0.282903\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.2749	−0.446730
$756$	0	0
$757$	34.3746	1.24937	0.624683	−	0.780879i	$-0.285228\pi$
$757$	34.3746	1.24937	0.624683	+	0.780879i	$0.285228\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.5498	1.79618	0.898090	−	0.439812i	$-0.144955\pi$
$761$	49.5498	1.79618	0.898090	+	0.439812i	$0.144955\pi$
$762$	0	0
$763$	20.5498	0.743954
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.5739	0.598450
$768$	0	0
$769$	−16.2749	−0.586889	−0.293444	−	0.955976i	$-0.594802\pi$
$769$	−16.2749	−0.586889	−0.293444	+	0.955976i	$0.594802\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.6495	1.13835	0.569177	−	0.822215i	$-0.307262\pi$
$773$	31.6495	1.13835	0.569177	+	0.822215i	$0.307262\pi$
$774$	0	0
$775$	6.27492	0.225402
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.54983	0.270501
$780$	0	0
$781$	−51.9244	−1.85800
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.7251	0.382795
$786$	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
$791$	19.6495	0.698656
$792$	0	0
$793$	−10.9003	−0.387082
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.4502	−0.901491	−0.450746	−	0.892652i	$-0.648842\pi$
$797$	−25.4502	−0.901491	−0.450746	+	0.892652i	$0.648842\pi$
$798$	0	0
$799$	−2.54983	−0.0902067
$800$	0	0
$801$	0	0
$802$	0	0
$803$	94.7492	3.34363
$804$	0	0
$805$	23.8248	0.839712
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.6495	0.585365	0.292683	−	0.956210i	$-0.405452\pi$
$809$	16.6495	0.585365	0.292683	+	0.956210i	$0.405452\pi$
$810$	0	0
$811$	13.1752	0.462646	0.231323	−	0.972877i	$-0.425695\pi$
$811$	13.1752	0.462646	0.231323	+	0.972877i	$0.425695\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.5498	0.719829
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.82475	0.168385	0.0841925	−	0.996450i	$-0.473169\pi$
$821$	4.82475	0.168385	0.0841925	+	0.996450i	$0.473169\pi$
$822$	0	0
$823$	23.4502	0.817421	0.408711	−	0.912664i	$-0.365979\pi$
$823$	23.4502	0.817421	0.408711	+	0.912664i	$0.365979\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.0997	0.733707	0.366854	−	0.930279i	$-0.380435\pi$
$827$	21.0997	0.733707	0.366854	+	0.930279i	$0.380435\pi$
$828$	0	0
$829$	−13.1752	−0.457595	−0.228798	−	0.973474i	$-0.573479\pi$
$829$	−13.1752	−0.457595	−0.228798	+	0.973474i	$0.573479\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.45017	0.258133
$834$	0	0
$835$	−7.45017	−0.257824
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.4743	1.81161	0.905806	−	0.423692i	$-0.139266\pi$
$839$	52.4743	1.81161	0.905806	+	0.423692i	$0.139266\pi$
$840$	0	0
$841$	10.3746	0.357744
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.3746	−0.391298
$846$	0	0
$847$	−92.9244	−3.19292
$848$	0	0
$849$	0	0
$850$	0	0
$851$	76.7492	2.63093
$852$	0	0
$853$	−49.2990	−1.68797	−0.843983	−	0.536370i	$-0.819795\pi$
$853$	−49.2990	−1.68797	−0.843983	+	0.536370i	$0.819795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.7492	−1.80188	−0.900939	−	0.433946i	$-0.857121\pi$
$857$	−52.7492	−1.80188	−0.900939	+	0.433946i	$0.857121\pi$
$858$	0	0
$859$	52.4743	1.79040	0.895199	−	0.445666i	$-0.147033\pi$
$859$	52.4743	1.79040	0.895199	+	0.445666i	$0.147033\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.2749	0.792287	0.396144	−	0.918189i	$-0.370348\pi$
$863$	23.2749	0.792287	0.396144	+	0.918189i	$0.370348\pi$
$864$	0	0
$865$	8.72508	0.296662
$866$	0	0
$867$	0	0
$868$	0	0
$869$	66.1993	2.24566
$870$	0	0
$871$	−0.700993	−0.0237523
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.27492	−0.110712
$876$	0	0
$877$	−13.2749	−0.448262	−0.224131	−	0.974559i	$-0.571954\pi$
$877$	−13.2749	−0.448262	−0.224131	+	0.974559i	$0.571954\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.6254	0.627506	0.313753	−	0.949505i	$-0.398414\pi$
$881$	18.6254	0.627506	0.313753	+	0.949505i	$0.398414\pi$
$882$	0	0
$883$	32.1993	1.08359	0.541797	−	0.840509i	$-0.317744\pi$
$883$	32.1993	1.08359	0.541797	+	0.840509i	$0.317744\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.175248	0.00588426	0.00294213	−	0.999996i	$-0.499063\pi$
$887$	0.175248	0.00588426	0.00294213	+	0.999996i	$0.499063\pi$
$888$	0	0
$889$	−36.9244	−1.23841
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.27492	−0.0426635
$894$	0	0
$895$	−11.5498	−0.386068
$896$	0	0
$897$	0	0
$898$	0	0
$899$	39.3746	1.31322
$900$	0	0
$901$	1.45017	0.0483120
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.2749	0.540997
$906$	0	0
$907$	51.0997	1.69674	0.848368	−	0.529406i	$-0.177585\pi$
$907$	51.0997	1.69674	0.848368	+	0.529406i	$0.177585\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.0241	−0.895348	−0.447674	−	0.894197i	$-0.647747\pi$
$911$	−27.0241	−0.895348	−0.447674	+	0.894197i	$0.647747\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.07558	−0.167610
$918$	0	0
$919$	11.9244	0.393350	0.196675	−	0.980469i	$-0.436986\pi$
$919$	11.9244	0.393350	0.196675	+	0.980469i	$0.436986\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.5498	0.347252
$924$	0	0
$925$	−10.5498	−0.346876
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−60.1238	−1.97260	−0.986298	−	0.164972i	$-0.947247\pi$
$929$	−60.1238	−1.97260	−0.986298	+	0.164972i	$0.947247\pi$
$930$	0	0
$931$	3.72508	0.122085
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.5498	0.410423
$936$	0	0
$937$	−0.549834	−0.0179623	−0.00898115	−	0.999960i	$-0.502859\pi$
$937$	−0.549834	−0.0179623	−0.00898115	+	0.999960i	$0.502859\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.09967	0.231443	0.115721	−	0.993282i	$-0.463082\pi$
$941$	7.09967	0.231443	0.115721	+	0.993282i	$0.463082\pi$
$942$	0	0
$943$	−54.9244	−1.78859
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.4502	−0.892011	−0.446005	−	0.895030i	$-0.647154\pi$
$947$	−27.4502	−0.892011	−0.446005	+	0.895030i	$0.647154\pi$
$948$	0	0
$949$	−19.2508	−0.624908
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33.0997	−1.07220	−0.536102	−	0.844153i	$-0.680104\pi$
$953$	−33.0997	−1.07220	−0.536102	+	0.844153i	$0.680104\pi$
$954$	0	0
$955$	19.3746	0.626947
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−47.6495	−1.53868
$960$	0	0
$961$	8.37459	0.270148
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	10.3505	0.332850	0.166425	−	0.986054i	$-0.446778\pi$
$967$	10.3505	0.332850	0.166425	+	0.986054i	$0.446778\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.6495	1.17614	0.588069	−	0.808811i	$-0.299888\pi$
$971$	36.6495	1.17614	0.588069	+	0.808811i	$0.299888\pi$
$972$	0	0
$973$	29.4743	0.944901
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.64950	0.116758	0.0583790	−	0.998294i	$-0.481407\pi$
$977$	3.64950	0.116758	0.0583790	+	0.998294i	$0.481407\pi$
$978$	0	0
$979$	80.4743	2.57197
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.7492	−1.49107	−0.745534	−	0.666468i	$-0.767805\pi$
$983$	−46.7492	−1.49107	−0.745534	+	0.666468i	$0.767805\pi$
$984$	0	0
$985$	−22.3746	−0.712914
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−29.0997	−0.925316
$990$	0	0
$991$	−1.72508	−0.0547991	−0.0273995	−	0.999625i	$-0.508723\pi$
$991$	−1.72508	−0.0547991	−0.0273995	+	0.999625i	$0.508723\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	8.37459	0.265226	0.132613	−	0.991168i	$-0.457663\pi$
$997$	8.37459	0.265226	0.132613	+	0.991168i	$0.457663\pi$
$998$	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.1.1	yes	2
3.2	odd	2		3240.2.a.h.1.1	✓	2
4.3	odd	2		6480.2.a.bm.1.2		2
9.2	odd	6		3240.2.q.be.1081.2		4
9.4	even	3		3240.2.q.y.2161.2		4
9.5	odd	6		3240.2.q.be.2161.2		4
9.7	even	3		3240.2.q.y.1081.2		4
12.11	even	2		6480.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.h.1.1	✓	2	3.2	odd	2
3240.2.a.n.1.1	yes	2	1.1	even	1	trivial
3240.2.q.y.1081.2		4	9.7	even	3
3240.2.q.y.2161.2		4	9.4	even	3
3240.2.q.be.1081.2		4	9.2	odd	6
3240.2.q.be.2161.2		4	9.5	odd	6
6480.2.a.bf.1.2		2	12.11	even	2
6480.2.a.bm.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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.27492 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.27492 q^{7} +6.27492 q^{11} -1.27492 q^{13} +2.00000 q^{17} +1.00000 q^{19} -7.27492 q^{23} +1.00000 q^{25} +6.27492 q^{29} +6.27492 q^{31} -3.27492 q^{35} -10.5498 q^{37} +7.54983 q^{41} +4.00000 q^{43} -1.27492 q^{47} +3.72508 q^{49} +0.725083 q^{53} +6.27492 q^{55} -13.0000 q^{59} +8.54983 q^{61} -1.27492 q^{65} +0.549834 q^{67} -8.27492 q^{71} +15.0997 q^{73} -20.5498 q^{77} +10.5498 q^{79} -2.54983 q^{83} +2.00000 q^{85} +12.8248 q^{89} +4.17525 q^{91} +1.00000 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

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Database

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