LMFDB - Embedded newform 7440.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
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:	no (minimal twist has level 3720)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} -5.46410 q^{11} +4.73205 q^{13} -1.00000 q^{15} +1.46410 q^{19} +1.26795 q^{21} -3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.73205 q^{29} +1.00000 q^{31} +5.46410 q^{33} -1.26795 q^{35} -7.66025 q^{37} -4.73205 q^{39} +3.46410 q^{41} +1.00000 q^{45} +7.46410 q^{47} -5.39230 q^{49} -5.46410 q^{55} -1.46410 q^{57} +14.1962 q^{59} -0.535898 q^{61} -1.26795 q^{63} +4.73205 q^{65} -9.66025 q^{67} +3.46410 q^{69} +3.66025 q^{71} +4.73205 q^{73} -1.00000 q^{75} +6.92820 q^{77} -16.9282 q^{79} +1.00000 q^{81} -3.46410 q^{83} -2.73205 q^{87} -6.73205 q^{89} -6.00000 q^{91} -1.00000 q^{93} +1.46410 q^{95} -16.9282 q^{97} -5.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{19} + 6 q^{21} + 2 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + 4 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^19 + 6 * q^21 + 2 * q^25 - 2 * q^27 + 2 * q^29 + 2 * q^31 + 4 * q^33 - 6 * q^35 + 2 * q^37 - 6 * q^39 + 2 * q^45 + 8 * q^47 + 10 * q^49 - 4 * q^55 + 4 * q^57 + 18 * q^59 - 8 * q^61 - 6 * q^63 + 6 * q^65 - 2 * q^67 - 10 * q^71 + 6 * q^73 - 2 * q^75 - 20 * q^79 + 2 * q^81 - 2 * q^87 - 10 * q^89 - 12 * q^91 - 2 * q^93 - 4 * q^95 - 20 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	4.73205	1.31243	0.656217	−	0.754572i	$-0.272155\pi$
$13$	4.73205	1.31243	0.656217	+	0.754572i	$0.272155\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	1.26795	0.276689
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.73205	0.507329	0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205	0.507329	0.253665	+	0.967292i	$0.418364\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	5.46410	0.951178
$34$	0	0
$35$	−1.26795	−0.214323
$36$	0	0
$37$	−7.66025	−1.25934	−0.629669	−	0.776864i	$-0.716809\pi$
$37$	−7.66025	−1.25934	−0.629669	+	0.776864i	$0.716809\pi$
$38$	0	0
$39$	−4.73205	−0.757735
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	7.46410	1.08875	0.544376	−	0.838842i	$-0.316767\pi$
$47$	7.46410	1.08875	0.544376	+	0.838842i	$0.316767\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	14.1962	1.84818	0.924091	−	0.382173i	$-0.124824\pi$
$59$	14.1962	1.84818	0.924091	+	0.382173i	$0.124824\pi$
$60$	0	0
$61$	−0.535898	−0.0686148	−0.0343074	−	0.999411i	$-0.510923\pi$
$61$	−0.535898	−0.0686148	−0.0343074	+	0.999411i	$0.510923\pi$
$62$	0	0
$63$	−1.26795	−0.159747
$64$	0	0
$65$	4.73205	0.586939
$66$	0	0
$67$	−9.66025	−1.18019	−0.590094	−	0.807335i	$-0.700909\pi$
$67$	−9.66025	−1.18019	−0.590094	+	0.807335i	$0.700909\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	0	0
$71$	3.66025	0.434392	0.217196	−	0.976128i	$-0.430309\pi$
$71$	3.66025	0.434392	0.217196	+	0.976128i	$0.430309\pi$
$72$	0	0
$73$	4.73205	0.553845	0.276922	−	0.960892i	$-0.410686\pi$
$73$	4.73205	0.553845	0.276922	+	0.960892i	$0.410686\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	6.92820	0.789542
$78$	0	0
$79$	−16.9282	−1.90457	−0.952286	−	0.305208i	$-0.901274\pi$
$79$	−16.9282	−1.90457	−0.952286	+	0.305208i	$0.901274\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.73205	−0.292907
$88$	0	0
$89$	−6.73205	−0.713596	−0.356798	−	0.934182i	$-0.616132\pi$
$89$	−6.73205	−0.713596	−0.356798	+	0.934182i	$0.616132\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	1.46410	0.150214
$96$	0	0
$97$	−16.9282	−1.71880	−0.859399	−	0.511305i	$-0.829162\pi$
$97$	−16.9282	−1.71880	−0.859399	+	0.511305i	$0.829162\pi$
$98$	0	0
$99$	−5.46410	−0.549163
$100$	0	0
$101$	−15.4641	−1.53874	−0.769368	−	0.638806i	$-0.779429\pi$
$101$	−15.4641	−1.53874	−0.769368	+	0.638806i	$0.779429\pi$
$102$	0	0
$103$	17.6603	1.74012	0.870058	−	0.492949i	$-0.164081\pi$
$103$	17.6603	1.74012	0.870058	+	0.492949i	$0.164081\pi$
$104$	0	0
$105$	1.26795	0.123739
$106$	0	0
$107$	14.3923	1.39136	0.695678	−	0.718353i	$-0.255104\pi$
$107$	14.3923	1.39136	0.695678	+	0.718353i	$0.255104\pi$
$108$	0	0
$109$	−4.39230	−0.420707	−0.210353	−	0.977625i	$-0.567461\pi$
$109$	−4.39230	−0.420707	−0.210353	+	0.977625i	$0.567461\pi$
$110$	0	0
$111$	7.66025	0.727079
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	−3.46410	−0.323029
$116$	0	0
$117$	4.73205	0.437478
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	−3.46410	−0.312348
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.39230	−0.744697	−0.372348	−	0.928093i	$-0.621447\pi$
$127$	−8.39230	−0.744697	−0.372348	+	0.928093i	$0.621447\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.26795	0.285522	0.142761	−	0.989757i	$-0.454402\pi$
$131$	3.26795	0.285522	0.142761	+	0.989757i	$0.454402\pi$
$132$	0	0
$133$	−1.85641	−0.160971
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	5.46410	0.466830	0.233415	−	0.972377i	$-0.425010\pi$
$137$	5.46410	0.466830	0.233415	+	0.972377i	$0.425010\pi$
$138$	0	0
$139$	−17.8564	−1.51456	−0.757280	−	0.653090i	$-0.773472\pi$
$139$	−17.8564	−1.51456	−0.757280	+	0.653090i	$0.773472\pi$
$140$	0	0
$141$	−7.46410	−0.628591
$142$	0	0
$143$	−25.8564	−2.16222
$144$	0	0
$145$	2.73205	0.226884
$146$	0	0
$147$	5.39230	0.444750
$148$	0	0
$149$	11.4641	0.939176	0.469588	−	0.882886i	$-0.344403\pi$
$149$	11.4641	0.939176	0.469588	+	0.882886i	$0.344403\pi$
$150$	0	0
$151$	−3.07180	−0.249979	−0.124990	−	0.992158i	$-0.539890\pi$
$151$	−3.07180	−0.249979	−0.124990	+	0.992158i	$0.539890\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	0.928203	0.0740787	0.0370393	−	0.999314i	$-0.488207\pi$
$157$	0.928203	0.0740787	0.0370393	+	0.999314i	$0.488207\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.39230	0.346162
$162$	0	0
$163$	−2.73205	−0.213991	−0.106995	−	0.994260i	$-0.534123\pi$
$163$	−2.73205	−0.213991	−0.106995	+	0.994260i	$0.534123\pi$
$164$	0	0
$165$	5.46410	0.425380
$166$	0	0
$167$	−24.7846	−1.91789	−0.958945	−	0.283591i	$-0.908474\pi$
$167$	−24.7846	−1.91789	−0.958945	+	0.283591i	$0.908474\pi$
$168$	0	0
$169$	9.39230	0.722485
$170$	0	0
$171$	1.46410	0.111963
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	−1.26795	−0.0958479
$176$	0	0
$177$	−14.1962	−1.06705
$178$	0	0
$179$	−5.07180	−0.379084	−0.189542	−	0.981873i	$-0.560700\pi$
$179$	−5.07180	−0.379084	−0.189542	+	0.981873i	$0.560700\pi$
$180$	0	0
$181$	−14.3923	−1.06977	−0.534886	−	0.844924i	$-0.679645\pi$
$181$	−14.3923	−1.06977	−0.534886	+	0.844924i	$0.679645\pi$
$182$	0	0
$183$	0.535898	0.0396147
$184$	0	0
$185$	−7.66025	−0.563193
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.26795	0.0922297
$190$	0	0
$191$	−7.26795	−0.525890	−0.262945	−	0.964811i	$-0.584694\pi$
$191$	−7.26795	−0.525890	−0.262945	+	0.964811i	$0.584694\pi$
$192$	0	0
$193$	4.92820	0.354740	0.177370	−	0.984144i	$-0.443241\pi$
$193$	4.92820	0.354740	0.177370	+	0.984144i	$0.443241\pi$
$194$	0	0
$195$	−4.73205	−0.338869
$196$	0	0
$197$	−13.4641	−0.959278	−0.479639	−	0.877466i	$-0.659232\pi$
$197$	−13.4641	−0.959278	−0.479639	+	0.877466i	$0.659232\pi$
$198$	0	0
$199$	12.9282	0.916456	0.458228	−	0.888835i	$-0.348484\pi$
$199$	12.9282	0.916456	0.458228	+	0.888835i	$0.348484\pi$
$200$	0	0
$201$	9.66025	0.681382
$202$	0	0
$203$	−3.46410	−0.243132
$204$	0	0
$205$	3.46410	0.241943
$206$	0	0
$207$	−3.46410	−0.240772
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−7.60770	−0.523735	−0.261868	−	0.965104i	$-0.584338\pi$
$211$	−7.60770	−0.523735	−0.261868	+	0.965104i	$0.584338\pi$
$212$	0	0
$213$	−3.66025	−0.250796
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.26795	−0.0860740
$218$	0	0
$219$	−4.73205	−0.319762
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.2487	−1.75774	−0.878872	−	0.477058i	$-0.841703\pi$
$223$	−26.2487	−1.75774	−0.878872	+	0.477058i	$0.841703\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.07180	−0.469372	−0.234686	−	0.972071i	$-0.575406\pi$
$227$	−7.07180	−0.469372	−0.234686	+	0.972071i	$0.575406\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−6.92820	−0.455842
$232$	0	0
$233$	9.85641	0.645715	0.322857	−	0.946448i	$-0.395357\pi$
$233$	9.85641	0.645715	0.322857	+	0.946448i	$0.395357\pi$
$234$	0	0
$235$	7.46410	0.486904
$236$	0	0
$237$	16.9282	1.09960
$238$	0	0
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	4.92820	0.317453	0.158727	−	0.987323i	$-0.449261\pi$
$241$	4.92820	0.317453	0.158727	+	0.987323i	$0.449261\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.39230	−0.344502
$246$	0	0
$247$	6.92820	0.440831
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	14.9282	0.942260	0.471130	−	0.882064i	$-0.343846\pi$
$251$	14.9282	0.942260	0.471130	+	0.882064i	$0.343846\pi$
$252$	0	0
$253$	18.9282	1.19001
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.7846	−1.04700	−0.523498	−	0.852027i	$-0.675373\pi$
$257$	−16.7846	−1.04700	−0.523498	+	0.852027i	$0.675373\pi$
$258$	0	0
$259$	9.71281	0.603525
$260$	0	0
$261$	2.73205	0.169110
$262$	0	0
$263$	20.7846	1.28163	0.640817	−	0.767694i	$-0.278596\pi$
$263$	20.7846	1.28163	0.640817	+	0.767694i	$0.278596\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.73205	0.411995
$268$	0	0
$269$	−28.9808	−1.76699	−0.883494	−	0.468442i	$-0.844815\pi$
$269$	−28.9808	−1.76699	−0.883494	+	0.468442i	$0.844815\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−5.46410	−0.329498
$276$	0	0
$277$	10.5885	0.636199	0.318099	−	0.948057i	$-0.396955\pi$
$277$	10.5885	0.636199	0.318099	+	0.948057i	$0.396955\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−12.5359	−0.747829	−0.373915	−	0.927463i	$-0.621985\pi$
$281$	−12.5359	−0.747829	−0.373915	+	0.927463i	$0.621985\pi$
$282$	0	0
$283$	−14.3397	−0.852409	−0.426205	−	0.904627i	$-0.640150\pi$
$283$	−14.3397	−0.852409	−0.426205	+	0.904627i	$0.640150\pi$
$284$	0	0
$285$	−1.46410	−0.0867259
$286$	0	0
$287$	−4.39230	−0.259270
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	16.9282	0.992349
$292$	0	0
$293$	−9.85641	−0.575817	−0.287909	−	0.957658i	$-0.592960\pi$
$293$	−9.85641	−0.575817	−0.287909	+	0.957658i	$0.592960\pi$
$294$	0	0
$295$	14.1962	0.826532
$296$	0	0
$297$	5.46410	0.317059
$298$	0	0
$299$	−16.3923	−0.947991
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.4641	0.888389
$304$	0	0
$305$	−0.535898	−0.0306855
$306$	0	0
$307$	−18.7321	−1.06909	−0.534547	−	0.845139i	$-0.679518\pi$
$307$	−18.7321	−1.06909	−0.534547	+	0.845139i	$0.679518\pi$
$308$	0	0
$309$	−17.6603	−1.00466
$310$	0	0
$311$	−12.7321	−0.721968	−0.360984	−	0.932572i	$-0.617559\pi$
$311$	−12.7321	−0.721968	−0.360984	+	0.932572i	$0.617559\pi$
$312$	0	0
$313$	−7.66025	−0.432983	−0.216492	−	0.976284i	$-0.569461\pi$
$313$	−7.66025	−0.432983	−0.216492	+	0.976284i	$0.569461\pi$
$314$	0	0
$315$	−1.26795	−0.0714408
$316$	0	0
$317$	2.92820	0.164464	0.0822321	−	0.996613i	$-0.473795\pi$
$317$	2.92820	0.164464	0.0822321	+	0.996613i	$0.473795\pi$
$318$	0	0
$319$	−14.9282	−0.835819
$320$	0	0
$321$	−14.3923	−0.803300
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.73205	0.262487
$326$	0	0
$327$	4.39230	0.242895
$328$	0	0
$329$	−9.46410	−0.521773
$330$	0	0
$331$	−22.7846	−1.25236	−0.626178	−	0.779680i	$-0.715382\pi$
$331$	−22.7846	−1.25236	−0.626178	+	0.779680i	$0.715382\pi$
$332$	0	0
$333$	−7.66025	−0.419779
$334$	0	0
$335$	−9.66025	−0.527796
$336$	0	0
$337$	15.2679	0.831698	0.415849	−	0.909434i	$-0.363484\pi$
$337$	15.2679	0.831698	0.415849	+	0.909434i	$0.363484\pi$
$338$	0	0
$339$	−12.9282	−0.702164
$340$	0	0
$341$	−5.46410	−0.295898
$342$	0	0
$343$	15.7128	0.848412
$344$	0	0
$345$	3.46410	0.186501
$346$	0	0
$347$	17.0718	0.916462	0.458231	−	0.888833i	$-0.348483\pi$
$347$	17.0718	0.916462	0.458231	+	0.888833i	$0.348483\pi$
$348$	0	0
$349$	0.392305	0.0209996	0.0104998	−	0.999945i	$-0.496658\pi$
$349$	0.392305	0.0209996	0.0104998	+	0.999945i	$0.496658\pi$
$350$	0	0
$351$	−4.73205	−0.252578
$352$	0	0
$353$	−5.85641	−0.311705	−0.155853	−	0.987780i	$-0.549812\pi$
$353$	−5.85641	−0.311705	−0.155853	+	0.987780i	$0.549812\pi$
$354$	0	0
$355$	3.66025	0.194266
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.5885	1.61440	0.807199	−	0.590280i	$-0.200983\pi$
$359$	30.5885	1.61440	0.807199	+	0.590280i	$0.200983\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	−18.8564	−0.989705
$364$	0	0
$365$	4.73205	0.247687
$366$	0	0
$367$	1.07180	0.0559473	0.0279737	−	0.999609i	$-0.491095\pi$
$367$	1.07180	0.0559473	0.0279737	+	0.999609i	$0.491095\pi$
$368$	0	0
$369$	3.46410	0.180334
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.2487	−1.04844	−0.524219	−	0.851583i	$-0.675643\pi$
$373$	−20.2487	−1.04844	−0.524219	+	0.851583i	$0.675643\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	12.9282	0.665836
$378$	0	0
$379$	−11.6077	−0.596247	−0.298124	−	0.954527i	$-0.596361\pi$
$379$	−11.6077	−0.596247	−0.298124	+	0.954527i	$0.596361\pi$
$380$	0	0
$381$	8.39230	0.429951
$382$	0	0
$383$	3.46410	0.177007	0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410	0.177007	0.0885037	+	0.996076i	$0.471792\pi$
$384$	0	0
$385$	6.92820	0.353094
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.9808	0.860959	0.430479	−	0.902600i	$-0.358345\pi$
$389$	16.9808	0.860959	0.430479	+	0.902600i	$0.358345\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.26795	−0.164846
$394$	0	0
$395$	−16.9282	−0.851750
$396$	0	0
$397$	−28.9282	−1.45186	−0.725932	−	0.687766i	$-0.758592\pi$
$397$	−28.9282	−1.45186	−0.725932	+	0.687766i	$0.758592\pi$
$398$	0	0
$399$	1.85641	0.0929366
$400$	0	0
$401$	4.58846	0.229137	0.114568	−	0.993415i	$-0.463452\pi$
$401$	4.58846	0.229137	0.114568	+	0.993415i	$0.463452\pi$
$402$	0	0
$403$	4.73205	0.235720
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	41.8564	2.07475
$408$	0	0
$409$	6.39230	0.316079	0.158040	−	0.987433i	$-0.449483\pi$
$409$	6.39230	0.316079	0.158040	+	0.987433i	$0.449483\pi$
$410$	0	0
$411$	−5.46410	−0.269524
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	−3.46410	−0.170046
$416$	0	0
$417$	17.8564	0.874432
$418$	0	0
$419$	−6.58846	−0.321867	−0.160934	−	0.986965i	$-0.551450\pi$
$419$	−6.58846	−0.321867	−0.160934	+	0.986965i	$0.551450\pi$
$420$	0	0
$421$	14.2487	0.694440	0.347220	−	0.937784i	$-0.387126\pi$
$421$	14.2487	0.694440	0.347220	+	0.937784i	$0.387126\pi$
$422$	0	0
$423$	7.46410	0.362917
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.679492	0.0328829
$428$	0	0
$429$	25.8564	1.24836
$430$	0	0
$431$	−0.732051	−0.0352616	−0.0176308	−	0.999845i	$-0.505612\pi$
$431$	−0.732051	−0.0352616	−0.0176308	+	0.999845i	$0.505612\pi$
$432$	0	0
$433$	35.6603	1.71372	0.856861	−	0.515547i	$-0.172411\pi$
$433$	35.6603	1.71372	0.856861	+	0.515547i	$0.172411\pi$
$434$	0	0
$435$	−2.73205	−0.130992
$436$	0	0
$437$	−5.07180	−0.242617
$438$	0	0
$439$	−24.7846	−1.18290	−0.591452	−	0.806340i	$-0.701445\pi$
$439$	−24.7846	−1.18290	−0.591452	+	0.806340i	$0.701445\pi$
$440$	0	0
$441$	−5.39230	−0.256776
$442$	0	0
$443$	−38.0000	−1.80543	−0.902717	−	0.430234i	$-0.858431\pi$
$443$	−38.0000	−1.80543	−0.902717	+	0.430234i	$0.858431\pi$
$444$	0	0
$445$	−6.73205	−0.319130
$446$	0	0
$447$	−11.4641	−0.542233
$448$	0	0
$449$	−6.33975	−0.299191	−0.149596	−	0.988747i	$-0.547797\pi$
$449$	−6.33975	−0.299191	−0.149596	+	0.988747i	$0.547797\pi$
$450$	0	0
$451$	−18.9282	−0.891294
$452$	0	0
$453$	3.07180	0.144326
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	21.5167	1.00651	0.503253	−	0.864139i	$-0.332136\pi$
$457$	21.5167	1.00651	0.503253	+	0.864139i	$0.332136\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−35.5167	−1.65418	−0.827088	−	0.562073i	$-0.810004\pi$
$461$	−35.5167	−1.65418	−0.827088	+	0.562073i	$0.810004\pi$
$462$	0	0
$463$	12.7846	0.594151	0.297076	−	0.954854i	$-0.403989\pi$
$463$	12.7846	0.594151	0.297076	+	0.954854i	$0.403989\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	35.1769	1.62779	0.813897	−	0.581010i	$-0.197342\pi$
$467$	35.1769	1.62779	0.813897	+	0.581010i	$0.197342\pi$
$468$	0	0
$469$	12.2487	0.565593
$470$	0	0
$471$	−0.928203	−0.0427693
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.46410	0.0671776
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.80385	0.265185	0.132592	−	0.991171i	$-0.457670\pi$
$479$	5.80385	0.265185	0.132592	+	0.991171i	$0.457670\pi$
$480$	0	0
$481$	−36.2487	−1.65280
$482$	0	0
$483$	−4.39230	−0.199857
$484$	0	0
$485$	−16.9282	−0.768670
$486$	0	0
$487$	24.3923	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$487$	24.3923	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$488$	0	0
$489$	2.73205	0.123548
$490$	0	0
$491$	−30.2487	−1.36511	−0.682553	−	0.730836i	$-0.739130\pi$
$491$	−30.2487	−1.36511	−0.682553	+	0.730836i	$0.739130\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−5.46410	−0.245593
$496$	0	0
$497$	−4.64102	−0.208178
$498$	0	0
$499$	−17.8564	−0.799363	−0.399681	−	0.916654i	$-0.630879\pi$
$499$	−17.8564	−0.799363	−0.399681	+	0.916654i	$0.630879\pi$
$500$	0	0
$501$	24.7846	1.10729
$502$	0	0
$503$	−39.1769	−1.74681	−0.873406	−	0.486993i	$-0.838094\pi$
$503$	−39.1769	−1.74681	−0.873406	+	0.486993i	$0.838094\pi$
$504$	0	0
$505$	−15.4641	−0.688143
$506$	0	0
$507$	−9.39230	−0.417127
$508$	0	0
$509$	16.5885	0.735270	0.367635	−	0.929970i	$-0.380168\pi$
$509$	16.5885	0.735270	0.367635	+	0.929970i	$0.380168\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−1.46410	−0.0646417
$514$	0	0
$515$	17.6603	0.778204
$516$	0	0
$517$	−40.7846	−1.79371
$518$	0	0
$519$	10.0000	0.438951
$520$	0	0
$521$	−8.53590	−0.373964	−0.186982	−	0.982363i	$-0.559871\pi$
$521$	−8.53590	−0.373964	−0.186982	+	0.982363i	$0.559871\pi$
$522$	0	0
$523$	−38.2487	−1.67250	−0.836250	−	0.548349i	$-0.815257\pi$
$523$	−38.2487	−1.67250	−0.836250	+	0.548349i	$0.815257\pi$
$524$	0	0
$525$	1.26795	0.0553378
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	14.1962	0.616061
$532$	0	0
$533$	16.3923	0.710030
$534$	0	0
$535$	14.3923	0.622234
$536$	0	0
$537$	5.07180	0.218864
$538$	0	0
$539$	29.4641	1.26911
$540$	0	0
$541$	−16.3923	−0.704760	−0.352380	−	0.935857i	$-0.614628\pi$
$541$	−16.3923	−0.704760	−0.352380	+	0.935857i	$0.614628\pi$
$542$	0	0
$543$	14.3923	0.617633
$544$	0	0
$545$	−4.39230	−0.188146
$546$	0	0
$547$	−3.80385	−0.162641	−0.0813204	−	0.996688i	$-0.525914\pi$
$547$	−3.80385	−0.162641	−0.0813204	+	0.996688i	$0.525914\pi$
$548$	0	0
$549$	−0.535898	−0.0228716
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	21.4641	0.912746
$554$	0	0
$555$	7.66025	0.325160
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.24871	0.347642	0.173821	−	0.984777i	$-0.444389\pi$
$563$	8.24871	0.347642	0.173821	+	0.984777i	$0.444389\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	0	0
$567$	−1.26795	−0.0532489
$568$	0	0
$569$	−19.5167	−0.818181	−0.409090	−	0.912494i	$-0.634154\pi$
$569$	−19.5167	−0.818181	−0.409090	+	0.912494i	$0.634154\pi$
$570$	0	0
$571$	−17.0718	−0.714432	−0.357216	−	0.934022i	$-0.616274\pi$
$571$	−17.0718	−0.714432	−0.357216	+	0.934022i	$0.616274\pi$
$572$	0	0
$573$	7.26795	0.303623
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	18.3923	0.765682	0.382841	−	0.923814i	$-0.374946\pi$
$577$	18.3923	0.765682	0.382841	+	0.923814i	$0.374946\pi$
$578$	0	0
$579$	−4.92820	−0.204809
$580$	0	0
$581$	4.39230	0.182224
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.73205	0.195646
$586$	0	0
$587$	2.14359	0.0884756	0.0442378	−	0.999021i	$-0.485914\pi$
$587$	2.14359	0.0884756	0.0442378	+	0.999021i	$0.485914\pi$
$588$	0	0
$589$	1.46410	0.0603273
$590$	0	0
$591$	13.4641	0.553839
$592$	0	0
$593$	−23.0718	−0.947445	−0.473723	−	0.880674i	$-0.657090\pi$
$593$	−23.0718	−0.947445	−0.473723	+	0.880674i	$0.657090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.9282	−0.529116
$598$	0	0
$599$	11.2679	0.460396	0.230198	−	0.973144i	$-0.426063\pi$
$599$	11.2679	0.460396	0.230198	+	0.973144i	$0.426063\pi$
$600$	0	0
$601$	−38.3923	−1.56605	−0.783027	−	0.621987i	$-0.786326\pi$
$601$	−38.3923	−1.56605	−0.783027	+	0.621987i	$0.786326\pi$
$602$	0	0
$603$	−9.66025	−0.393396
$604$	0	0
$605$	18.8564	0.766622
$606$	0	0
$607$	25.6603	1.04152	0.520759	−	0.853704i	$-0.325649\pi$
$607$	25.6603	1.04152	0.520759	+	0.853704i	$0.325649\pi$
$608$	0	0
$609$	3.46410	0.140372
$610$	0	0
$611$	35.3205	1.42891
$612$	0	0
$613$	30.1962	1.21961	0.609806	−	0.792551i	$-0.291248\pi$
$613$	30.1962	1.21961	0.609806	+	0.792551i	$0.291248\pi$
$614$	0	0
$615$	−3.46410	−0.139686
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	35.8564	1.44119	0.720595	−	0.693356i	$-0.243869\pi$
$619$	35.8564	1.44119	0.720595	+	0.693356i	$0.243869\pi$
$620$	0	0
$621$	3.46410	0.139010
$622$	0	0
$623$	8.53590	0.341984
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−49.5692	−1.97332	−0.986660	−	0.162796i	$-0.947949\pi$
$631$	−49.5692	−1.97332	−0.986660	+	0.162796i	$0.947949\pi$
$632$	0	0
$633$	7.60770	0.302379
$634$	0	0
$635$	−8.39230	−0.333038
$636$	0	0
$637$	−25.5167	−1.01101
$638$	0	0
$639$	3.66025	0.144797
$640$	0	0
$641$	−41.6603	−1.64548	−0.822741	−	0.568417i	$-0.807556\pi$
$641$	−41.6603	−1.64548	−0.822741	+	0.568417i	$0.807556\pi$
$642$	0	0
$643$	−44.3923	−1.75066	−0.875331	−	0.483525i	$-0.839356\pi$
$643$	−44.3923	−1.75066	−0.875331	+	0.483525i	$0.839356\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.0718	0.671162	0.335581	−	0.942011i	$-0.391067\pi$
$647$	17.0718	0.671162	0.335581	+	0.942011i	$0.391067\pi$
$648$	0	0
$649$	−77.5692	−3.04486
$650$	0	0
$651$	1.26795	0.0496948
$652$	0	0
$653$	−16.0000	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$653$	−16.0000	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$654$	0	0
$655$	3.26795	0.127689
$656$	0	0
$657$	4.73205	0.184615
$658$	0	0
$659$	22.1962	0.864639	0.432320	−	0.901720i	$-0.357695\pi$
$659$	22.1962	0.864639	0.432320	+	0.901720i	$0.357695\pi$
$660$	0	0
$661$	−38.7846	−1.50855	−0.754273	−	0.656561i	$-0.772011\pi$
$661$	−38.7846	−1.50855	−0.754273	+	0.656561i	$0.772011\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.85641	−0.0719884
$666$	0	0
$667$	−9.46410	−0.366451
$668$	0	0
$669$	26.2487	1.01483
$670$	0	0
$671$	2.92820	0.113042
$672$	0	0
$673$	−15.2679	−0.588536	−0.294268	−	0.955723i	$-0.595076\pi$
$673$	−15.2679	−0.588536	−0.294268	+	0.955723i	$0.595076\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.14359	−0.0823850	−0.0411925	−	0.999151i	$-0.513116\pi$
$677$	−2.14359	−0.0823850	−0.0411925	+	0.999151i	$0.513116\pi$
$678$	0	0
$679$	21.4641	0.823717
$680$	0	0
$681$	7.07180	0.270992
$682$	0	0
$683$	−3.07180	−0.117539	−0.0587695	−	0.998272i	$-0.518718\pi$
$683$	−3.07180	−0.117539	−0.0587695	+	0.998272i	$0.518718\pi$
$684$	0	0
$685$	5.46410	0.208773
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	6.92820	0.263181
$694$	0	0
$695$	−17.8564	−0.677332
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9.85641	−0.372804
$700$	0	0
$701$	−50.7846	−1.91811	−0.959054	−	0.283223i	$-0.908596\pi$
$701$	−50.7846	−1.91811	−0.959054	+	0.283223i	$0.908596\pi$
$702$	0	0
$703$	−11.2154	−0.422996
$704$	0	0
$705$	−7.46410	−0.281114
$706$	0	0
$707$	19.6077	0.737423
$708$	0	0
$709$	−23.8564	−0.895946	−0.447973	−	0.894047i	$-0.647854\pi$
$709$	−23.8564	−0.895946	−0.447973	+	0.894047i	$0.647854\pi$
$710$	0	0
$711$	−16.9282	−0.634857
$712$	0	0
$713$	−3.46410	−0.129732
$714$	0	0
$715$	−25.8564	−0.966975
$716$	0	0
$717$	−6.92820	−0.258738
$718$	0	0
$719$	−13.1769	−0.491416	−0.245708	−	0.969344i	$-0.579020\pi$
$719$	−13.1769	−0.491416	−0.245708	+	0.969344i	$0.579020\pi$
$720$	0	0
$721$	−22.3923	−0.833933
$722$	0	0
$723$	−4.92820	−0.183282
$724$	0	0
$725$	2.73205	0.101466
$726$	0	0
$727$	30.0526	1.11459	0.557294	−	0.830315i	$-0.311840\pi$
$727$	30.0526	1.11459	0.557294	+	0.830315i	$0.311840\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	47.8564	1.76762	0.883808	−	0.467849i	$-0.154971\pi$
$733$	47.8564	1.76762	0.883808	+	0.467849i	$0.154971\pi$
$734$	0	0
$735$	5.39230	0.198898
$736$	0	0
$737$	52.7846	1.94435
$738$	0	0
$739$	−20.9282	−0.769856	−0.384928	−	0.922947i	$-0.625774\pi$
$739$	−20.9282	−0.769856	−0.384928	+	0.922947i	$0.625774\pi$
$740$	0	0
$741$	−6.92820	−0.254514
$742$	0	0
$743$	22.9282	0.841154	0.420577	−	0.907257i	$-0.361828\pi$
$743$	22.9282	0.841154	0.420577	+	0.907257i	$0.361828\pi$
$744$	0	0
$745$	11.4641	0.420012
$746$	0	0
$747$	−3.46410	−0.126745
$748$	0	0
$749$	−18.2487	−0.666793
$750$	0	0
$751$	29.8564	1.08948	0.544738	−	0.838606i	$-0.316629\pi$
$751$	29.8564	1.08948	0.544738	+	0.838606i	$0.316629\pi$
$752$	0	0
$753$	−14.9282	−0.544014
$754$	0	0
$755$	−3.07180	−0.111794
$756$	0	0
$757$	24.4449	0.888464	0.444232	−	0.895912i	$-0.353477\pi$
$757$	24.4449	0.888464	0.444232	+	0.895912i	$0.353477\pi$
$758$	0	0
$759$	−18.9282	−0.687050
$760$	0	0
$761$	35.1244	1.27326	0.636628	−	0.771171i	$-0.280329\pi$
$761$	35.1244	1.27326	0.636628	+	0.771171i	$0.280329\pi$
$762$	0	0
$763$	5.56922	0.201619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	67.1769	2.42562
$768$	0	0
$769$	−35.3205	−1.27369	−0.636845	−	0.770992i	$-0.719761\pi$
$769$	−35.3205	−1.27369	−0.636845	+	0.770992i	$0.719761\pi$
$770$	0	0
$771$	16.7846	0.604483
$772$	0	0
$773$	−17.0718	−0.614030	−0.307015	−	0.951705i	$-0.599330\pi$
$773$	−17.0718	−0.614030	−0.307015	+	0.951705i	$0.599330\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−9.71281	−0.348445
$778$	0	0
$779$	5.07180	0.181716
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	−2.73205	−0.0976355
$784$	0	0
$785$	0.928203	0.0331290
$786$	0	0
$787$	19.3205	0.688702	0.344351	−	0.938841i	$-0.388099\pi$
$787$	19.3205	0.688702	0.344351	+	0.938841i	$0.388099\pi$
$788$	0	0
$789$	−20.7846	−0.739952
$790$	0	0
$791$	−16.3923	−0.582843
$792$	0	0
$793$	−2.53590	−0.0900524
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.3205	0.542680	0.271340	−	0.962484i	$-0.412533\pi$
$797$	15.3205	0.542680	0.271340	+	0.962484i	$0.412533\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.73205	−0.237865
$802$	0	0
$803$	−25.8564	−0.912453
$804$	0	0
$805$	4.39230	0.154808
$806$	0	0
$807$	28.9808	1.02017
$808$	0	0
$809$	−16.9808	−0.597012	−0.298506	−	0.954408i	$-0.596488\pi$
$809$	−16.9808	−0.597012	−0.298506	+	0.954408i	$0.596488\pi$
$810$	0	0
$811$	29.1769	1.02454	0.512270	−	0.858824i	$-0.328805\pi$
$811$	29.1769	1.02454	0.512270	+	0.858824i	$0.328805\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.73205	−0.0956996
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−38.7321	−1.35176	−0.675879	−	0.737013i	$-0.736236\pi$
$821$	−38.7321	−1.35176	−0.675879	+	0.737013i	$0.736236\pi$
$822$	0	0
$823$	−48.4974	−1.69051	−0.845257	−	0.534360i	$-0.820553\pi$
$823$	−48.4974	−1.69051	−0.845257	+	0.534360i	$0.820553\pi$
$824$	0	0
$825$	5.46410	0.190236
$826$	0	0
$827$	−41.3205	−1.43685	−0.718427	−	0.695602i	$-0.755138\pi$
$827$	−41.3205	−1.43685	−0.718427	+	0.695602i	$0.755138\pi$
$828$	0	0
$829$	−44.2487	−1.53682	−0.768411	−	0.639957i	$-0.778952\pi$
$829$	−44.2487	−1.53682	−0.768411	+	0.639957i	$0.778952\pi$
$830$	0	0
$831$	−10.5885	−0.367310
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.7846	−0.857707
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−32.0526	−1.10658	−0.553289	−	0.832990i	$-0.686627\pi$
$839$	−32.0526	−1.10658	−0.553289	+	0.832990i	$0.686627\pi$
$840$	0	0
$841$	−21.5359	−0.742617
$842$	0	0
$843$	12.5359	0.431759
$844$	0	0
$845$	9.39230	0.323105
$846$	0	0
$847$	−23.9090	−0.821522
$848$	0	0
$849$	14.3397	0.492139
$850$	0	0
$851$	26.5359	0.909639
$852$	0	0
$853$	−9.32051	−0.319128	−0.159564	−	0.987188i	$-0.551009\pi$
$853$	−9.32051	−0.319128	−0.159564	+	0.987188i	$0.551009\pi$
$854$	0	0
$855$	1.46410	0.0500712
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	42.6410	1.45489	0.727446	−	0.686165i	$-0.240707\pi$
$859$	42.6410	1.45489	0.727446	+	0.686165i	$0.240707\pi$
$860$	0	0
$861$	4.39230	0.149689
$862$	0	0
$863$	34.9282	1.18897	0.594485	−	0.804107i	$-0.297356\pi$
$863$	34.9282	1.18897	0.594485	+	0.804107i	$0.297356\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	92.4974	3.13776
$870$	0	0
$871$	−45.7128	−1.54892
$872$	0	0
$873$	−16.9282	−0.572933
$874$	0	0
$875$	−1.26795	−0.0428645
$876$	0	0
$877$	23.1769	0.782629	0.391314	−	0.920257i	$-0.372021\pi$
$877$	23.1769	0.782629	0.391314	+	0.920257i	$0.372021\pi$
$878$	0	0
$879$	9.85641	0.332448
$880$	0	0
$881$	9.26795	0.312245	0.156123	−	0.987738i	$-0.450101\pi$
$881$	9.26795	0.312245	0.156123	+	0.987738i	$0.450101\pi$
$882$	0	0
$883$	−42.5359	−1.43145	−0.715723	−	0.698384i	$-0.753903\pi$
$883$	−42.5359	−1.43145	−0.715723	+	0.698384i	$0.753903\pi$
$884$	0	0
$885$	−14.1962	−0.477198
$886$	0	0
$887$	50.7846	1.70518	0.852590	−	0.522580i	$-0.175030\pi$
$887$	50.7846	1.70518	0.852590	+	0.522580i	$0.175030\pi$
$888$	0	0
$889$	10.6410	0.356888
$890$	0	0
$891$	−5.46410	−0.183054
$892$	0	0
$893$	10.9282	0.365698
$894$	0	0
$895$	−5.07180	−0.169531
$896$	0	0
$897$	16.3923	0.547323
$898$	0	0
$899$	2.73205	0.0911190
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.3923	−0.478416
$906$	0	0
$907$	21.2679	0.706191	0.353095	−	0.935587i	$-0.385129\pi$
$907$	21.2679	0.706191	0.353095	+	0.935587i	$0.385129\pi$
$908$	0	0
$909$	−15.4641	−0.512912
$910$	0	0
$911$	28.3923	0.940679	0.470340	−	0.882486i	$-0.344131\pi$
$911$	28.3923	0.940679	0.470340	+	0.882486i	$0.344131\pi$
$912$	0	0
$913$	18.9282	0.626432
$914$	0	0
$915$	0.535898	0.0177163
$916$	0	0
$917$	−4.14359	−0.136834
$918$	0	0
$919$	−17.4641	−0.576088	−0.288044	−	0.957617i	$-0.593005\pi$
$919$	−17.4641	−0.576088	−0.288044	+	0.957617i	$0.593005\pi$
$920$	0	0
$921$	18.7321	0.617242
$922$	0	0
$923$	17.3205	0.570111
$924$	0	0
$925$	−7.66025	−0.251868
$926$	0	0
$927$	17.6603	0.580039
$928$	0	0
$929$	−28.9808	−0.950828	−0.475414	−	0.879762i	$-0.657702\pi$
$929$	−28.9808	−0.950828	−0.475414	+	0.879762i	$0.657702\pi$
$930$	0	0
$931$	−7.89488	−0.258744
$932$	0	0
$933$	12.7321	0.416829
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.0718	−0.753723	−0.376861	−	0.926270i	$-0.622997\pi$
$937$	−23.0718	−0.753723	−0.376861	+	0.926270i	$0.622997\pi$
$938$	0	0
$939$	7.66025	0.249983
$940$	0	0
$941$	−51.5167	−1.67940	−0.839698	−	0.543054i	$-0.817268\pi$
$941$	−51.5167	−1.67940	−0.839698	+	0.543054i	$0.817268\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	0	0
$945$	1.26795	0.0412464
$946$	0	0
$947$	27.4641	0.892463	0.446232	−	0.894917i	$-0.352766\pi$
$947$	27.4641	0.892463	0.446232	+	0.894917i	$0.352766\pi$
$948$	0	0
$949$	22.3923	0.726885
$950$	0	0
$951$	−2.92820	−0.0949535
$952$	0	0
$953$	34.2487	1.10942	0.554712	−	0.832042i	$-0.312828\pi$
$953$	34.2487	1.10942	0.554712	+	0.832042i	$0.312828\pi$
$954$	0	0
$955$	−7.26795	−0.235185
$956$	0	0
$957$	14.9282	0.482560
$958$	0	0
$959$	−6.92820	−0.223723
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	14.3923	0.463786
$964$	0	0
$965$	4.92820	0.158644
$966$	0	0
$967$	23.3205	0.749937	0.374968	−	0.927038i	$-0.377654\pi$
$967$	23.3205	0.749937	0.374968	+	0.927038i	$0.377654\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.41154	0.173665	0.0868323	−	0.996223i	$-0.472326\pi$
$971$	5.41154	0.173665	0.0868323	+	0.996223i	$0.472326\pi$
$972$	0	0
$973$	22.6410	0.725838
$974$	0	0
$975$	−4.73205	−0.151547
$976$	0	0
$977$	−17.7128	−0.566683	−0.283342	−	0.959019i	$-0.591443\pi$
$977$	−17.7128	−0.566683	−0.283342	+	0.959019i	$0.591443\pi$
$978$	0	0
$979$	36.7846	1.17564
$980$	0	0
$981$	−4.39230	−0.140236
$982$	0	0
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	−13.4641	−0.429002
$986$	0	0
$987$	9.46410	0.301246
$988$	0	0
$989$	0	0
$990$	0	0
$991$	13.0718	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$991$	13.0718	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$992$	0	0
$993$	22.7846	0.723048
$994$	0	0
$995$	12.9282	0.409852
$996$	0	0
$997$	29.3205	0.928590	0.464295	−	0.885681i	$-0.346308\pi$
$997$	29.3205	0.928590	0.464295	+	0.885681i	$0.346308\pi$
$998$	0	0
$999$	7.66025	0.242360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bf.1.2		2
4.3	odd	2		3720.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.k.1.1	✓	2	4.3	odd	2
7440.2.a.bf.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{9} -5.46410 q^{11} +4.73205 q^{13} -1.00000 q^{15} +1.46410 q^{19} +1.26795 q^{21} -3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.73205 q^{29} +1.00000 q^{31} +5.46410 q^{33} -1.26795 q^{35} -7.66025 q^{37} -4.73205 q^{39} +3.46410 q^{41} +1.00000 q^{45} +7.46410 q^{47} -5.39230 q^{49} -5.46410 q^{55} -1.46410 q^{57} +14.1962 q^{59} -0.535898 q^{61} -1.26795 q^{63} +4.73205 q^{65} -9.66025 q^{67} +3.46410 q^{69} +3.66025 q^{71} +4.73205 q^{73} -1.00000 q^{75} +6.92820 q^{77} -16.9282 q^{79} +1.00000 q^{81} -3.46410 q^{83} -2.73205 q^{87} -6.73205 q^{89} -6.00000 q^{91} -1.00000 q^{93} +1.46410 q^{95} -16.9282 q^{97} -5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{19} + 6 q^{21} + 2 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + 4 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 2 * q^9 - 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^19 + 6 * q^21 + 2 * q^25 - 2 * q^27 + 2 * q^29 + 2 * q^31 + 4 * q^33 - 6 * q^35 + 2 * q^37 - 6 * q^39 + 2 * q^45 + 8 * q^47 + 10 * q^49 - 4 * q^55 + 4 * q^57 + 18 * q^59 - 8 * q^61 - 6 * q^63 + 6 * q^65 - 2 * q^67 - 10 * q^71 + 6 * q^73 - 2 * q^75 - 20 * q^79 + 2 * q^81 - 2 * q^87 - 10 * q^89 - 12 * q^91 - 2 * q^93 - 4 * q^95 - 20 * q^97 - 4 * q^99

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