LMFDB - Embedded newform 3240.2.a.g.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -2.73205 q^{7} +0.267949 q^{11} +1.46410 q^{13} +6.19615 q^{17} +1.00000 q^{19} -7.46410 q^{23} +1.00000 q^{25} +0.267949 q^{29} -2.46410 q^{31} +2.73205 q^{35} -7.26795 q^{37} +7.73205 q^{41} +6.19615 q^{43} -5.26795 q^{47} +0.464102 q^{49} +4.73205 q^{53} -0.267949 q^{55} -1.19615 q^{59} -14.9282 q^{61} -1.46410 q^{65} -0.535898 q^{67} +12.1244 q^{71} -1.26795 q^{73} -0.732051 q^{77} -8.53590 q^{79} -11.1244 q^{83} -6.19615 q^{85} +5.19615 q^{89} -4.00000 q^{91} -1.00000 q^{95} -10.1962 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 4 q^{13} + 2 q^{17} + 2 q^{19} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 2 q^{31} + 2 q^{35} - 18 q^{37} + 12 q^{41} + 2 q^{43} - 14 q^{47} - 6 q^{49} + 6 q^{53} - 4 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 4 * q^13 + 2 * q^17 + 2 * q^19 - 8 * q^23 + 2 * q^25 + 4 * q^29 + 2 * q^31 + 2 * q^35 - 18 * q^37 + 12 * q^41 + 2 * q^43 - 14 * q^47 - 6 * q^49 + 6 * q^53 - 4 * q^55 + 8 * q^59 - 16 * q^61 + 4 * q^65 - 8 * q^67 - 6 * q^73 + 2 * q^77 - 24 * q^79 + 2 * q^83 - 2 * q^85 - 8 * q^91 - 2 * q^95 - 10 * q^97

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
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.267949	0.0807897	0.0403949	−	0.999184i	$-0.487138\pi$
$11$	0.267949	0.0807897	0.0403949	+	0.999184i	$0.487138\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.19615	1.50279	0.751394	−	0.659854i	$-0.229382\pi$
$17$	6.19615	1.50279	0.751394	+	0.659854i	$0.229382\pi$
$18$	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
$23$	−7.46410	−1.55637	−0.778186	−	0.628033i	$-0.783860\pi$
$23$	−7.46410	−1.55637	−0.778186	+	0.628033i	$0.783860\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.267949	0.0497569	0.0248785	−	0.999690i	$-0.492080\pi$
$29$	0.267949	0.0497569	0.0248785	+	0.999690i	$0.492080\pi$
$30$	0	0
$31$	−2.46410	−0.442566	−0.221283	−	0.975210i	$-0.571024\pi$
$31$	−2.46410	−0.442566	−0.221283	+	0.975210i	$0.571024\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−7.26795	−1.19484	−0.597422	−	0.801927i	$-0.703808\pi$
$37$	−7.26795	−1.19484	−0.597422	+	0.801927i	$0.703808\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.73205	1.20754	0.603772	−	0.797157i	$-0.293664\pi$
$41$	7.73205	1.20754	0.603772	+	0.797157i	$0.293664\pi$
$42$	0	0
$43$	6.19615	0.944904	0.472452	−	0.881356i	$-0.343369\pi$
$43$	6.19615	0.944904	0.472452	+	0.881356i	$0.343369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.26795	−0.768409	−0.384205	−	0.923248i	$-0.625524\pi$
$47$	−5.26795	−0.768409	−0.384205	+	0.923248i	$0.625524\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.73205	0.649997	0.324999	−	0.945715i	$-0.394636\pi$
$53$	4.73205	0.649997	0.324999	+	0.945715i	$0.394636\pi$
$54$	0	0
$55$	−0.267949	−0.0361303
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.19615	−0.155726	−0.0778629	−	0.996964i	$-0.524810\pi$
$59$	−1.19615	−0.155726	−0.0778629	+	0.996964i	$0.524810\pi$
$60$	0	0
$61$	−14.9282	−1.91136	−0.955680	−	0.294407i	$-0.904878\pi$
$61$	−14.9282	−1.91136	−0.955680	+	0.294407i	$0.904878\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.46410	−0.181599
$66$	0	0
$67$	−0.535898	−0.0654704	−0.0327352	−	0.999464i	$-0.510422\pi$
$67$	−0.535898	−0.0654704	−0.0327352	+	0.999464i	$0.510422\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.1244	1.43890	0.719448	−	0.694546i	$-0.244395\pi$
$71$	12.1244	1.43890	0.719448	+	0.694546i	$0.244395\pi$
$72$	0	0
$73$	−1.26795	−0.148402	−0.0742011	−	0.997243i	$-0.523641\pi$
$73$	−1.26795	−0.148402	−0.0742011	+	0.997243i	$0.523641\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.732051	−0.0834249
$78$	0	0
$79$	−8.53590	−0.960364	−0.480182	−	0.877169i	$-0.659429\pi$
$79$	−8.53590	−0.960364	−0.480182	+	0.877169i	$0.659429\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.1244	−1.22106	−0.610528	−	0.791994i	$-0.709043\pi$
$83$	−11.1244	−1.22106	−0.610528	+	0.791994i	$0.709043\pi$
$84$	0	0
$85$	−6.19615	−0.672067
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−10.1962	−1.03526	−0.517631	−	0.855604i	$-0.673186\pi$
$97$	−10.1962	−1.03526	−0.517631	+	0.855604i	$0.673186\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.66025	−0.264705	−0.132353	−	0.991203i	$-0.542253\pi$
$101$	−2.66025	−0.264705	−0.132353	+	0.991203i	$0.542253\pi$
$102$	0	0
$103$	−18.3923	−1.81225	−0.906124	−	0.423013i	$-0.860973\pi$
$103$	−18.3923	−1.81225	−0.906124	+	0.423013i	$0.860973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.53590	−0.825196	−0.412598	−	0.910913i	$-0.635379\pi$
$107$	−8.53590	−0.825196	−0.412598	+	0.910913i	$0.635379\pi$
$108$	0	0
$109$	−7.92820	−0.759384	−0.379692	−	0.925113i	$-0.623970\pi$
$109$	−7.92820	−0.759384	−0.379692	+	0.925113i	$0.623970\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.19615	−0.206597	−0.103298	−	0.994650i	$-0.532940\pi$
$113$	−2.19615	−0.206597	−0.103298	+	0.994650i	$0.532940\pi$
$114$	0	0
$115$	7.46410	0.696031
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.9282	−1.55181
$120$	0	0
$121$	−10.9282	−0.993473
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.66025	−0.502266	−0.251133	−	0.967953i	$-0.580803\pi$
$127$	−5.66025	−0.502266	−0.251133	+	0.967953i	$0.580803\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.26795	−0.372892	−0.186446	−	0.982465i	$-0.559697\pi$
$131$	−4.26795	−0.372892	−0.186446	+	0.982465i	$0.559697\pi$
$132$	0	0
$133$	−2.73205	−0.236899
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.46410	−0.125087	−0.0625433	−	0.998042i	$-0.519921\pi$
$137$	−1.46410	−0.125087	−0.0625433	+	0.998042i	$0.519921\pi$
$138$	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
$143$	0.392305	0.0328062
$144$	0	0
$145$	−0.267949	−0.0222520
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	12.8564	1.04624	0.523120	−	0.852259i	$-0.324768\pi$
$151$	12.8564	1.04624	0.523120	+	0.852259i	$0.324768\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.46410	0.197921
$156$	0	0
$157$	20.0526	1.60037	0.800184	−	0.599754i	$-0.204735\pi$
$157$	20.0526	1.60037	0.800184	+	0.599754i	$0.204735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.3923	1.60714
$162$	0	0
$163$	−23.1244	−1.81124	−0.905620	−	0.424091i	$-0.860594\pi$
$163$	−23.1244	−1.81124	−0.905620	+	0.424091i	$0.860594\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.26795	−0.562411	−0.281205	−	0.959648i	$-0.590734\pi$
$167$	−7.26795	−0.562411	−0.281205	+	0.959648i	$0.590734\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.4641	−0.871600	−0.435800	−	0.900044i	$-0.643534\pi$
$173$	−11.4641	−0.871600	−0.435800	+	0.900044i	$0.643534\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.0526	1.27457	0.637284	−	0.770629i	$-0.280058\pi$
$179$	17.0526	1.27457	0.637284	+	0.770629i	$0.280058\pi$
$180$	0	0
$181$	16.3205	1.21309	0.606547	−	0.795048i	$-0.292554\pi$
$181$	16.3205	1.21309	0.606547	+	0.795048i	$0.292554\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.26795	0.534350
$186$	0	0
$187$	1.66025	0.121410
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7321	0.848901	0.424451	−	0.905451i	$-0.360467\pi$
$191$	11.7321	0.848901	0.424451	+	0.905451i	$0.360467\pi$
$192$	0	0
$193$	−5.80385	−0.417770	−0.208885	−	0.977940i	$-0.566983\pi$
$193$	−5.80385	−0.417770	−0.208885	+	0.977940i	$0.566983\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.8564	1.55720	0.778602	−	0.627518i	$-0.215929\pi$
$197$	21.8564	1.55720	0.778602	+	0.627518i	$0.215929\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.732051	−0.0513799
$204$	0	0
$205$	−7.73205	−0.540030
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.267949	0.0185344
$210$	0	0
$211$	−12.3205	−0.848179	−0.424089	−	0.905620i	$-0.639406\pi$
$211$	−12.3205	−0.848179	−0.424089	+	0.905620i	$0.639406\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.19615	−0.422574
$216$	0	0
$217$	6.73205	0.457001
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.07180	0.610235
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.6603	−0.773918	−0.386959	−	0.922097i	$-0.626475\pi$
$227$	−11.6603	−0.773918	−0.386959	+	0.922097i	$0.626475\pi$
$228$	0	0
$229$	17.8564	1.17998	0.589992	−	0.807409i	$-0.299131\pi$
$229$	17.8564	1.17998	0.589992	+	0.807409i	$0.299131\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.26795	−0.0830661	−0.0415331	−	0.999137i	$-0.513224\pi$
$233$	−1.26795	−0.0830661	−0.0415331	+	0.999137i	$0.513224\pi$
$234$	0	0
$235$	5.26795	0.343643
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.4641	−1.00029	−0.500145	−	0.865942i	$-0.666720\pi$
$239$	−15.4641	−1.00029	−0.500145	+	0.865942i	$0.666720\pi$
$240$	0	0
$241$	−3.53590	−0.227767	−0.113884	−	0.993494i	$-0.536329\pi$
$241$	−3.53590	−0.227767	−0.113884	+	0.993494i	$0.536329\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.464102	−0.0296504
$246$	0	0
$247$	1.46410	0.0931586
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.5359	1.29621	0.648107	−	0.761549i	$-0.275561\pi$
$251$	20.5359	1.29621	0.648107	+	0.761549i	$0.275561\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.3205	−1.82896	−0.914482	−	0.404628i	$-0.867401\pi$
$257$	−29.3205	−1.82896	−0.914482	+	0.404628i	$0.867401\pi$
$258$	0	0
$259$	19.8564	1.23382
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.53590	−0.526346	−0.263173	−	0.964749i	$-0.584769\pi$
$263$	−8.53590	−0.526346	−0.263173	+	0.964749i	$0.584769\pi$
$264$	0	0
$265$	−4.73205	−0.290688
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.267949	−0.0163372	−0.00816858	−	0.999967i	$-0.502600\pi$
$269$	−0.267949	−0.0163372	−0.00816858	+	0.999967i	$0.502600\pi$
$270$	0	0
$271$	27.7128	1.68343	0.841717	−	0.539919i	$-0.181545\pi$
$271$	27.7128	1.68343	0.841717	+	0.539919i	$0.181545\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.267949	0.0161579
$276$	0	0
$277$	−15.6603	−0.940933	−0.470467	−	0.882418i	$-0.655914\pi$
$277$	−15.6603	−0.940933	−0.470467	+	0.882418i	$0.655914\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.3923	1.09719	0.548596	−	0.836087i	$-0.315162\pi$
$281$	18.3923	1.09719	0.548596	+	0.836087i	$0.315162\pi$
$282$	0	0
$283$	11.4641	0.681470	0.340735	−	0.940159i	$-0.389324\pi$
$283$	11.4641	0.681470	0.340735	+	0.940159i	$0.389324\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−21.1244	−1.24693
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.73205	0.393291	0.196645	−	0.980475i	$-0.436995\pi$
$293$	6.73205	0.393291	0.196645	+	0.980475i	$0.436995\pi$
$294$	0	0
$295$	1.19615	0.0696427
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.9282	−0.631994
$300$	0	0
$301$	−16.9282	−0.975725
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.9282	0.854786
$306$	0	0
$307$	−28.4449	−1.62343	−0.811717	−	0.584051i	$-0.801467\pi$
$307$	−28.4449	−1.62343	−0.811717	+	0.584051i	$0.801467\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.0526	−0.853552	−0.426776	−	0.904357i	$-0.640351\pi$
$311$	−15.0526	−0.853552	−0.426776	+	0.904357i	$0.640351\pi$
$312$	0	0
$313$	−12.7846	−0.722629	−0.361314	−	0.932444i	$-0.617672\pi$
$313$	−12.7846	−0.722629	−0.361314	+	0.932444i	$0.617672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.5167	−1.09616	−0.548082	−	0.836424i	$-0.684642\pi$
$317$	−19.5167	−1.09616	−0.548082	+	0.836424i	$0.684642\pi$
$318$	0	0
$319$	0.0717968	0.00401985
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.19615	0.344763
$324$	0	0
$325$	1.46410	0.0812137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.3923	0.793473
$330$	0	0
$331$	11.7846	0.647741	0.323870	−	0.946101i	$-0.395016\pi$
$331$	11.7846	0.647741	0.323870	+	0.946101i	$0.395016\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.535898	0.0292793
$336$	0	0
$337$	3.32051	0.180880	0.0904398	−	0.995902i	$-0.471173\pi$
$337$	3.32051	0.180880	0.0904398	+	0.995902i	$0.471173\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.660254	−0.0357548
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.5885	0.568418	0.284209	−	0.958762i	$-0.408269\pi$
$347$	10.5885	0.568418	0.284209	+	0.958762i	$0.408269\pi$
$348$	0	0
$349$	−17.7846	−0.951988	−0.475994	−	0.879448i	$-0.657912\pi$
$349$	−17.7846	−0.951988	−0.475994	+	0.879448i	$0.657912\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.7321	1.42280	0.711402	−	0.702786i	$-0.248061\pi$
$353$	26.7321	1.42280	0.711402	+	0.702786i	$0.248061\pi$
$354$	0	0
$355$	−12.1244	−0.643494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.1962	−1.54091	−0.770457	−	0.637492i	$-0.779972\pi$
$359$	−29.1962	−1.54091	−0.770457	+	0.637492i	$0.779972\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.26795	0.0663675
$366$	0	0
$367$	2.67949	0.139868	0.0699342	−	0.997552i	$-0.477721\pi$
$367$	2.67949	0.139868	0.0699342	+	0.997552i	$0.477721\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.9282	−0.671199
$372$	0	0
$373$	26.7321	1.38413	0.692067	−	0.721834i	$-0.256700\pi$
$373$	26.7321	1.38413	0.692067	+	0.721834i	$0.256700\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.392305	0.0202047
$378$	0	0
$379$	−20.2487	−1.04011	−0.520053	−	0.854134i	$-0.674088\pi$
$379$	−20.2487	−1.04011	−0.520053	+	0.854134i	$0.674088\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.3205	0.578451	0.289225	−	0.957261i	$-0.406602\pi$
$383$	11.3205	0.578451	0.289225	+	0.957261i	$0.406602\pi$
$384$	0	0
$385$	0.732051	0.0373088
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.2487	1.83788	0.918941	−	0.394394i	$-0.129046\pi$
$389$	36.2487	1.83788	0.918941	+	0.394394i	$0.129046\pi$
$390$	0	0
$391$	−46.2487	−2.33890
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.53590	0.429488
$396$	0	0
$397$	−20.2487	−1.01625	−0.508127	−	0.861282i	$-0.669662\pi$
$397$	−20.2487	−1.01625	−0.508127	+	0.861282i	$0.669662\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.8564	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$401$	31.8564	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$402$	0	0
$403$	−3.60770	−0.179712
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.94744	−0.0965311
$408$	0	0
$409$	36.7846	1.81888	0.909441	−	0.415833i	$-0.136510\pi$
$409$	36.7846	1.81888	0.909441	+	0.415833i	$0.136510\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.26795	0.160805
$414$	0	0
$415$	11.1244	0.546073
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.3205	0.553043	0.276522	−	0.961008i	$-0.410818\pi$
$419$	11.3205	0.553043	0.276522	+	0.961008i	$0.410818\pi$
$420$	0	0
$421$	26.0718	1.27066	0.635331	−	0.772240i	$-0.280864\pi$
$421$	26.0718	1.27066	0.635331	+	0.772240i	$0.280864\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.19615	0.300558
$426$	0	0
$427$	40.7846	1.97371
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.2679	−0.879936	−0.439968	−	0.898013i	$-0.645010\pi$
$431$	−18.2679	−0.879936	−0.439968	+	0.898013i	$0.645010\pi$
$432$	0	0
$433$	11.4641	0.550930	0.275465	−	0.961311i	$-0.411168\pi$
$433$	11.4641	0.550930	0.275465	+	0.961311i	$0.411168\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.46410	−0.357056
$438$	0	0
$439$	−6.85641	−0.327238	−0.163619	−	0.986524i	$-0.552317\pi$
$439$	−6.85641	−0.327238	−0.163619	+	0.986524i	$0.552317\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.1244	0.718580	0.359290	−	0.933226i	$-0.383019\pi$
$443$	15.1244	0.718580	0.359290	+	0.933226i	$0.383019\pi$
$444$	0	0
$445$	−5.19615	−0.246321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.4449	−0.823274	−0.411637	−	0.911348i	$-0.635043\pi$
$449$	−17.4449	−0.823274	−0.411637	+	0.911348i	$0.635043\pi$
$450$	0	0
$451$	2.07180	0.0975571
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−26.5885	−1.24376	−0.621878	−	0.783114i	$-0.713630\pi$
$457$	−26.5885	−1.24376	−0.621878	+	0.783114i	$0.713630\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.6603	0.682796	0.341398	−	0.939919i	$-0.389100\pi$
$461$	14.6603	0.682796	0.341398	+	0.939919i	$0.389100\pi$
$462$	0	0
$463$	−1.60770	−0.0747159	−0.0373580	−	0.999302i	$-0.511894\pi$
$463$	−1.60770	−0.0747159	−0.0373580	+	0.999302i	$0.511894\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.5167	−1.55097	−0.775483	−	0.631368i	$-0.782494\pi$
$467$	−33.5167	−1.55097	−0.775483	+	0.631368i	$0.782494\pi$
$468$	0	0
$469$	1.46410	0.0676059
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.66025	0.0763386
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.6603	−0.852609	−0.426304	−	0.904580i	$-0.640185\pi$
$479$	−18.6603	−0.852609	−0.426304	+	0.904580i	$0.640185\pi$
$480$	0	0
$481$	−10.6410	−0.485189
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.1962	0.462983
$486$	0	0
$487$	13.4641	0.610117	0.305058	−	0.952334i	$-0.401324\pi$
$487$	13.4641	0.610117	0.305058	+	0.952334i	$0.401324\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.1962	1.31760	0.658802	−	0.752316i	$-0.271064\pi$
$491$	29.1962	1.31760	0.658802	+	0.752316i	$0.271064\pi$
$492$	0	0
$493$	1.66025	0.0747741
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−33.1244	−1.48583
$498$	0	0
$499$	−18.0718	−0.809005	−0.404502	−	0.914537i	$-0.632555\pi$
$499$	−18.0718	−0.809005	−0.404502	+	0.914537i	$0.632555\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.39230	−0.195843	−0.0979216	−	0.995194i	$-0.531219\pi$
$503$	−4.39230	−0.195843	−0.0979216	+	0.995194i	$0.531219\pi$
$504$	0	0
$505$	2.66025	0.118380
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.143594	0.00636467	0.00318234	−	0.999995i	$-0.498987\pi$
$509$	0.143594	0.00636467	0.00318234	+	0.999995i	$0.498987\pi$
$510$	0	0
$511$	3.46410	0.153243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.3923	0.810462
$516$	0	0
$517$	−1.41154	−0.0620796
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.2487	−0.711869	−0.355934	−	0.934511i	$-0.615837\pi$
$521$	−16.2487	−0.711869	−0.355934	+	0.934511i	$0.615837\pi$
$522$	0	0
$523$	−0.392305	−0.0171543	−0.00857715	−	0.999963i	$-0.502730\pi$
$523$	−0.392305	−0.0171543	−0.00857715	+	0.999963i	$0.502730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.2679	−0.665082
$528$	0	0
$529$	32.7128	1.42230
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.3205	0.490346
$534$	0	0
$535$	8.53590	0.369039
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.124356	0.00535638
$540$	0	0
$541$	24.3205	1.04562	0.522810	−	0.852449i	$-0.324884\pi$
$541$	24.3205	1.04562	0.522810	+	0.852449i	$0.324884\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.92820	0.339607
$546$	0	0
$547$	40.7846	1.74382	0.871912	−	0.489663i	$-0.162880\pi$
$547$	40.7846	1.74382	0.871912	+	0.489663i	$0.162880\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.267949	0.0114150
$552$	0	0
$553$	23.3205	0.991689
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.5359	−0.954877	−0.477438	−	0.878665i	$-0.658435\pi$
$557$	−22.5359	−0.954877	−0.477438	+	0.878665i	$0.658435\pi$
$558$	0	0
$559$	9.07180	0.383696
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.9090	−1.68196	−0.840981	−	0.541064i	$-0.818022\pi$
$563$	−39.9090	−1.68196	−0.840981	+	0.541064i	$0.818022\pi$
$564$	0	0
$565$	2.19615	0.0923928
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.2679	−1.77196	−0.885982	−	0.463719i	$-0.846515\pi$
$569$	−42.2679	−1.77196	−0.885982	+	0.463719i	$0.846515\pi$
$570$	0	0
$571$	−30.1769	−1.26286	−0.631432	−	0.775431i	$-0.717533\pi$
$571$	−30.1769	−1.26286	−0.631432	+	0.775431i	$0.717533\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.46410	−0.311275
$576$	0	0
$577$	−31.1244	−1.29572	−0.647862	−	0.761758i	$-0.724337\pi$
$577$	−31.1244	−1.29572	−0.647862	+	0.761758i	$0.724337\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30.3923	1.26089
$582$	0	0
$583$	1.26795	0.0525131
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.4449	1.09150	0.545748	−	0.837949i	$-0.316246\pi$
$587$	26.4449	1.09150	0.545748	+	0.837949i	$0.316246\pi$
$588$	0	0
$589$	−2.46410	−0.101532
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.71281	−0.398857	−0.199429	−	0.979912i	$-0.563909\pi$
$593$	−9.71281	−0.398857	−0.199429	+	0.979912i	$0.563909\pi$
$594$	0	0
$595$	16.9282	0.693989
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−46.5167	−1.90062	−0.950310	−	0.311306i	$-0.899233\pi$
$599$	−46.5167	−1.90062	−0.950310	+	0.311306i	$0.899233\pi$
$600$	0	0
$601$	30.3205	1.23680	0.618400	−	0.785864i	$-0.287781\pi$
$601$	30.3205	1.23680	0.618400	+	0.785864i	$0.287781\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.9282	0.444295
$606$	0	0
$607$	21.8038	0.884991	0.442495	−	0.896771i	$-0.354093\pi$
$607$	21.8038	0.884991	0.442495	+	0.896771i	$0.354093\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.71281	−0.312027
$612$	0	0
$613$	26.5359	1.07177	0.535887	−	0.844289i	$-0.319977\pi$
$613$	26.5359	1.07177	0.535887	+	0.844289i	$0.319977\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.78461	0.353655	0.176828	−	0.984242i	$-0.443417\pi$
$617$	8.78461	0.353655	0.176828	+	0.984242i	$0.443417\pi$
$618$	0	0
$619$	−15.0718	−0.605787	−0.302893	−	0.953024i	$-0.597953\pi$
$619$	−15.0718	−0.605787	−0.302893	+	0.953024i	$0.597953\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.1962	−0.568757
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.0333	−1.79560
$630$	0	0
$631$	38.1769	1.51980	0.759899	−	0.650041i	$-0.225248\pi$
$631$	38.1769	1.51980	0.759899	+	0.650041i	$0.225248\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.66025	0.224620
$636$	0	0
$637$	0.679492	0.0269225
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.2679	1.59049	0.795244	−	0.606289i	$-0.207343\pi$
$641$	40.2679	1.59049	0.795244	+	0.606289i	$0.207343\pi$
$642$	0	0
$643$	35.6603	1.40630	0.703152	−	0.711040i	$-0.251776\pi$
$643$	35.6603	1.40630	0.703152	+	0.711040i	$0.251776\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−50.1051	−1.96984	−0.984918	−	0.173023i	$-0.944646\pi$
$647$	−50.1051	−1.96984	−0.984918	+	0.173023i	$0.944646\pi$
$648$	0	0
$649$	−0.320508	−0.0125810
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.2487	0.401063	0.200532	−	0.979687i	$-0.435733\pi$
$653$	10.2487	0.401063	0.200532	+	0.979687i	$0.435733\pi$
$654$	0	0
$655$	4.26795	0.166763
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.5359	1.03369	0.516846	−	0.856078i	$-0.327106\pi$
$659$	26.5359	1.03369	0.516846	+	0.856078i	$0.327106\pi$
$660$	0	0
$661$	−29.1051	−1.13206	−0.566029	−	0.824385i	$-0.691521\pi$
$661$	−29.1051	−1.13206	−0.566029	+	0.824385i	$0.691521\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.73205	0.105944
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−24.2487	−0.934719	−0.467360	−	0.884067i	$-0.654795\pi$
$673$	−24.2487	−0.934719	−0.467360	+	0.884067i	$0.654795\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.7128	0.680759	0.340379	−	0.940288i	$-0.389445\pi$
$677$	17.7128	0.680759	0.340379	+	0.940288i	$0.389445\pi$
$678$	0	0
$679$	27.8564	1.06903
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.3205	0.433167	0.216584	−	0.976264i	$-0.430509\pi$
$683$	11.3205	0.433167	0.216584	+	0.976264i	$0.430509\pi$
$684$	0	0
$685$	1.46410	0.0559404
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	46.7846	1.77977	0.889885	−	0.456185i	$-0.150784\pi$
$691$	46.7846	1.77977	0.889885	+	0.456185i	$0.150784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.00000	0.341389
$696$	0	0
$697$	47.9090	1.81468
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32.2679	1.21874	0.609372	−	0.792885i	$-0.291422\pi$
$701$	32.2679	1.21874	0.609372	+	0.792885i	$0.291422\pi$
$702$	0	0
$703$	−7.26795	−0.274116
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.26795	0.273339
$708$	0	0
$709$	10.5359	0.395684	0.197842	−	0.980234i	$-0.436607\pi$
$709$	10.5359	0.395684	0.197842	+	0.980234i	$0.436607\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.3923	0.688797
$714$	0	0
$715$	−0.392305	−0.0146714
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.9808	1.93856	0.969278	−	0.245969i	$-0.0791062\pi$
$719$	51.9808	1.93856	0.969278	+	0.245969i	$0.0791062\pi$
$720$	0	0
$721$	50.2487	1.87136
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.267949	0.00995138
$726$	0	0
$727$	10.5359	0.390755	0.195377	−	0.980728i	$-0.437407\pi$
$727$	10.5359	0.390755	0.195377	+	0.980728i	$0.437407\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.3923	1.41999
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.143594	−0.00528934
$738$	0	0
$739$	−22.8564	−0.840787	−0.420393	−	0.907342i	$-0.638108\pi$
$739$	−22.8564	−0.840787	−0.420393	+	0.907342i	$0.638108\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.4449	1.19029	0.595143	−	0.803620i	$-0.297095\pi$
$743$	32.4449	1.19029	0.595143	+	0.803620i	$0.297095\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	23.3205	0.852113
$750$	0	0
$751$	−3.71281	−0.135482	−0.0677412	−	0.997703i	$-0.521579\pi$
$751$	−3.71281	−0.135482	−0.0677412	+	0.997703i	$0.521579\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.8564	−0.467893
$756$	0	0
$757$	−6.53590	−0.237551	−0.118776	−	0.992921i	$-0.537897\pi$
$757$	−6.53590	−0.237551	−0.118776	+	0.992921i	$0.537897\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.0526	−1.77815	−0.889077	−	0.457758i	$-0.848653\pi$
$761$	−49.0526	−1.77815	−0.889077	+	0.457758i	$0.848653\pi$
$762$	0	0
$763$	21.6603	0.784154
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.75129	−0.0632354
$768$	0	0
$769$	10.4641	0.377345	0.188673	−	0.982040i	$-0.439582\pi$
$769$	10.4641	0.377345	0.188673	+	0.982040i	$0.439582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.1962	−1.15802	−0.579008	−	0.815322i	$-0.696560\pi$
$773$	−32.1962	−1.15802	−0.579008	+	0.815322i	$0.696560\pi$
$774$	0	0
$775$	−2.46410	−0.0885131
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.73205	0.277029
$780$	0	0
$781$	3.24871	0.116248
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20.0526	−0.715707
$786$	0	0
$787$	41.3731	1.47479	0.737395	−	0.675461i	$-0.236056\pi$
$787$	41.3731	1.47479	0.737395	+	0.675461i	$0.236056\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−21.8564	−0.776144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.4641	−0.760297	−0.380149	−	0.924925i	$-0.624127\pi$
$797$	−21.4641	−0.760297	−0.380149	+	0.924925i	$0.624127\pi$
$798$	0	0
$799$	−32.6410	−1.15476
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.339746	−0.0119894
$804$	0	0
$805$	−20.3923	−0.718734
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−49.9808	−1.75723	−0.878615	−	0.477531i	$-0.841532\pi$
$809$	−49.9808	−1.75723	−0.878615	+	0.477531i	$0.841532\pi$
$810$	0	0
$811$	−27.7846	−0.975650	−0.487825	−	0.872942i	$-0.662210\pi$
$811$	−27.7846	−0.975650	−0.487825	+	0.872942i	$0.662210\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.1244	0.810011
$816$	0	0
$817$	6.19615	0.216776
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.58846	−0.264839	−0.132419	−	0.991194i	$-0.542275\pi$
$821$	−7.58846	−0.264839	−0.132419	+	0.991194i	$0.542275\pi$
$822$	0	0
$823$	−39.8564	−1.38931	−0.694653	−	0.719345i	$-0.744442\pi$
$823$	−39.8564	−1.38931	−0.694653	+	0.719345i	$0.744442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.1051	−1.11640	−0.558202	−	0.829705i	$-0.688509\pi$
$827$	−32.1051	−1.11640	−0.558202	+	0.829705i	$0.688509\pi$
$828$	0	0
$829$	4.21539	0.146407	0.0732033	−	0.997317i	$-0.476678\pi$
$829$	4.21539	0.146407	0.0732033	+	0.997317i	$0.476678\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.87564	0.0996352
$834$	0	0
$835$	7.26795	0.251518
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.8038	−0.718229	−0.359114	−	0.933294i	$-0.616921\pi$
$839$	−20.8038	−0.718229	−0.359114	+	0.933294i	$0.616921\pi$
$840$	0	0
$841$	−28.9282	−0.997524
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.8564	0.373472
$846$	0	0
$847$	29.8564	1.02588
$848$	0	0
$849$	0	0
$850$	0	0
$851$	54.2487	1.85962
$852$	0	0
$853$	32.5885	1.11581	0.557904	−	0.829906i	$-0.311606\pi$
$853$	32.5885	1.11581	0.557904	+	0.829906i	$0.311606\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.07180	−0.173249	−0.0866246	−	0.996241i	$-0.527608\pi$
$857$	−5.07180	−0.173249	−0.0866246	+	0.996241i	$0.527608\pi$
$858$	0	0
$859$	56.9615	1.94350	0.971751	−	0.236008i	$-0.0758392\pi$
$859$	56.9615	1.94350	0.971751	+	0.236008i	$0.0758392\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.5167	−0.868597	−0.434299	−	0.900769i	$-0.643004\pi$
$863$	−25.5167	−0.868597	−0.434299	+	0.900769i	$0.643004\pi$
$864$	0	0
$865$	11.4641	0.389791
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.28719	−0.0775875
$870$	0	0
$871$	−0.784610	−0.0265855
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.73205	0.0923602
$876$	0	0
$877$	−1.75129	−0.0591368	−0.0295684	−	0.999563i	$-0.509413\pi$
$877$	−1.75129	−0.0591368	−0.0295684	+	0.999563i	$0.509413\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.33975	−0.112519	−0.0562595	−	0.998416i	$-0.517917\pi$
$881$	−3.33975	−0.112519	−0.0562595	+	0.998416i	$0.517917\pi$
$882$	0	0
$883$	4.87564	0.164078	0.0820392	−	0.996629i	$-0.473857\pi$
$883$	4.87564	0.164078	0.0820392	+	0.996629i	$0.473857\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.8038	−1.06787	−0.533934	−	0.845526i	$-0.679287\pi$
$887$	−31.8038	−1.06787	−0.533934	+	0.845526i	$0.679287\pi$
$888$	0	0
$889$	15.4641	0.518649
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.26795	−0.176285
$894$	0	0
$895$	−17.0526	−0.570004
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.660254	−0.0220207
$900$	0	0
$901$	29.3205	0.976808
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.3205	−0.542512
$906$	0	0
$907$	4.14359	0.137586	0.0687929	−	0.997631i	$-0.478085\pi$
$907$	4.14359	0.137586	0.0687929	+	0.997631i	$0.478085\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.6603	0.353190	0.176595	−	0.984284i	$-0.443492\pi$
$911$	10.6603	0.353190	0.176595	+	0.984284i	$0.443492\pi$
$912$	0	0
$913$	−2.98076	−0.0986488
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.6603	0.385056
$918$	0	0
$919$	38.7128	1.27702	0.638509	−	0.769614i	$-0.279552\pi$
$919$	38.7128	1.27702	0.638509	+	0.769614i	$0.279552\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17.7513	0.584291
$924$	0	0
$925$	−7.26795	−0.238969
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.9808	0.721165	0.360583	−	0.932727i	$-0.382578\pi$
$929$	21.9808	0.721165	0.360583	+	0.932727i	$0.382578\pi$
$930$	0	0
$931$	0.464102	0.0152103
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.66025	−0.0542961
$936$	0	0
$937$	−36.6410	−1.19701	−0.598505	−	0.801119i	$-0.704238\pi$
$937$	−36.6410	−1.19701	−0.598505	+	0.801119i	$0.704238\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.535898	−0.0174698	−0.00873489	−	0.999962i	$-0.502780\pi$
$941$	−0.535898	−0.0174698	−0.00873489	+	0.999962i	$0.502780\pi$
$942$	0	0
$943$	−57.7128	−1.87939
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.9282	1.07002	0.535011	−	0.844845i	$-0.320307\pi$
$947$	32.9282	1.07002	0.535011	+	0.844845i	$0.320307\pi$
$948$	0	0
$949$	−1.85641	−0.0602615
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−51.0333	−1.65313	−0.826566	−	0.562840i	$-0.809709\pi$
$953$	−51.0333	−1.65313	−0.826566	+	0.562840i	$0.809709\pi$
$954$	0	0
$955$	−11.7321	−0.379640
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−24.9282	−0.804136
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.80385	0.186832
$966$	0	0
$967$	52.9808	1.70375	0.851873	−	0.523748i	$-0.175467\pi$
$967$	52.9808	1.70375	0.851873	+	0.523748i	$0.175467\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.8038	0.924359	0.462180	−	0.886786i	$-0.347067\pi$
$971$	28.8038	0.924359	0.462180	+	0.886786i	$0.347067\pi$
$972$	0	0
$973$	24.5885	0.788270
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.392305	−0.0125509	−0.00627547	−	0.999980i	$-0.501998\pi$
$977$	−0.392305	−0.0125509	−0.00627547	+	0.999980i	$0.501998\pi$
$978$	0	0
$979$	1.39230	0.0444983
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.6603	−0.371904	−0.185952	−	0.982559i	$-0.559537\pi$
$983$	−11.6603	−0.371904	−0.185952	+	0.982559i	$0.559537\pi$
$984$	0	0
$985$	−21.8564	−0.696403
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−46.2487	−1.47062
$990$	0	0
$991$	43.1051	1.36928	0.684640	−	0.728882i	$-0.259960\pi$
$991$	43.1051	1.36928	0.684640	+	0.728882i	$0.259960\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.0000	0.697447
$996$	0	0
$997$	−46.7321	−1.48002	−0.740009	−	0.672596i	$-0.765179\pi$
$997$	−46.7321	−1.48002	−0.740009	+	0.672596i	$0.765179\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.a.g.1.1	✓	2
3.2	odd	2		3240.2.a.l.1.1	yes	2
4.3	odd	2		6480.2.a.bg.1.2		2
9.2	odd	6		3240.2.q.bb.1081.2		4
9.4	even	3		3240.2.q.bf.2161.2		4
9.5	odd	6		3240.2.q.bb.2161.2		4
9.7	even	3		3240.2.q.bf.1081.2		4
12.11	even	2		6480.2.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.g.1.1	✓	2	1.1	even	1	trivial
3240.2.a.l.1.1	yes	2	3.2	odd	2
3240.2.q.bb.1081.2		4	9.2	odd	6
3240.2.q.bb.2161.2		4	9.5	odd	6
3240.2.q.bf.1081.2		4	9.7	even	3
3240.2.q.bf.2161.2		4	9.4	even	3
6480.2.a.bg.1.2		2	4.3	odd	2
6480.2.a.bq.1.2		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.73205 q^{7} +0.267949 q^{11} +1.46410 q^{13} +6.19615 q^{17} +1.00000 q^{19} -7.46410 q^{23} +1.00000 q^{25} +0.267949 q^{29} -2.46410 q^{31} +2.73205 q^{35} -7.26795 q^{37} +7.73205 q^{41} +6.19615 q^{43} -5.26795 q^{47} +0.464102 q^{49} +4.73205 q^{53} -0.267949 q^{55} -1.19615 q^{59} -14.9282 q^{61} -1.46410 q^{65} -0.535898 q^{67} +12.1244 q^{71} -1.26795 q^{73} -0.732051 q^{77} -8.53590 q^{79} -11.1244 q^{83} -6.19615 q^{85} +5.19615 q^{89} -4.00000 q^{91} -1.00000 q^{95} -10.1962 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 4 q^{13} + 2 q^{17} + 2 q^{19} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 2 q^{31} + 2 q^{35} - 18 q^{37} + 12 q^{41} + 2 q^{43} - 14 q^{47} - 6 q^{49} + 6 q^{53} - 4 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 4 * q^13 + 2 * q^17 + 2 * q^19 - 8 * q^23 + 2 * q^25 + 4 * q^29 + 2 * q^31 + 2 * q^35 - 18 * q^37 + 12 * q^41 + 2 * q^43 - 14 * q^47 - 6 * q^49 + 6 * q^53 - 4 * q^55 + 8 * q^59 - 16 * q^61 + 4 * q^65 - 8 * q^67 - 6 * q^73 + 2 * q^77 - 24 * q^79 + 2 * q^83 - 2 * q^85 - 8 * q^91 - 2 * q^95 - 10 * q^97

Introduction

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