LMFDB - Embedded newform 7440.2.a.bj.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,2,0,-2,0,0,0,2,0,0,0,-4,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.44949 q^{7} +1.00000 q^{9} -4.44949 q^{13} -1.00000 q^{15} +2.00000 q^{17} +4.89898 q^{19} -2.44949 q^{21} +2.89898 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.449490 q^{29} -1.00000 q^{31} +2.44949 q^{35} -3.55051 q^{37} -4.44949 q^{39} +7.79796 q^{41} +3.10102 q^{43} -1.00000 q^{45} -8.89898 q^{47} -1.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} +4.89898 q^{57} +7.34847 q^{59} -6.89898 q^{61} -2.44949 q^{63} +4.44949 q^{65} +6.44949 q^{67} +2.89898 q^{69} -1.55051 q^{71} -12.4495 q^{73} +1.00000 q^{75} +2.89898 q^{79} +1.00000 q^{81} +1.10102 q^{83} -2.00000 q^{85} +0.449490 q^{87} -4.44949 q^{89} +10.8990 q^{91} -1.00000 q^{93} -4.89898 q^{95} -5.10102 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{31} - 12 q^{37} - 4 q^{39} - 4 q^{41} + 16 q^{43} - 2 q^{45} - 8 q^{47} - 2 q^{49} + 4 q^{51}+ \cdots - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 - 2 * q^15 + 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 2 * q^31 - 12 * q^37 - 4 * q^39 - 4 * q^41 + 16 * q^43 - 2 * q^45 - 8 * q^47 - 2 * q^49 + 4 * q^51 - 12 * q^53 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 4 * q^69 - 8 * q^71 - 20 * q^73 + 2 * q^75 - 4 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 - 4 * q^87 - 4 * q^89 + 12 * q^91 - 2 * q^93 - 20 * 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
$3$	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
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	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
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	2.89898	0.604479	0.302240	−	0.953232i	$-0.402266\pi$
$23$	2.89898	0.604479	0.302240	+	0.953232i	$0.402266\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.449490	0.0834681	0.0417341	−	0.999129i	$-0.486712\pi$
$29$	0.449490	0.0834681	0.0417341	+	0.999129i	$0.486712\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.44949	0.414039
$36$	0	0
$37$	−3.55051	−0.583700	−0.291850	−	0.956464i	$-0.594271\pi$
$37$	−3.55051	−0.583700	−0.291850	+	0.956464i	$0.594271\pi$
$38$	0	0
$39$	−4.44949	−0.712489
$40$	0	0
$41$	7.79796	1.21784	0.608918	−	0.793233i	$-0.291604\pi$
$41$	7.79796	1.21784	0.608918	+	0.793233i	$0.291604\pi$
$42$	0	0
$43$	3.10102	0.472901	0.236451	−	0.971644i	$-0.424016\pi$
$43$	3.10102	0.472901	0.236451	+	0.971644i	$0.424016\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.89898	−1.29805	−0.649025	−	0.760767i	$-0.724823\pi$
$47$	−8.89898	−1.29805	−0.649025	+	0.760767i	$0.724823\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.89898	0.648886
$58$	0	0
$59$	7.34847	0.956689	0.478345	−	0.878172i	$-0.341237\pi$
$59$	7.34847	0.956689	0.478345	+	0.878172i	$0.341237\pi$
$60$	0	0
$61$	−6.89898	−0.883324	−0.441662	−	0.897182i	$-0.645611\pi$
$61$	−6.89898	−0.883324	−0.441662	+	0.897182i	$0.645611\pi$
$62$	0	0
$63$	−2.44949	−0.308607
$64$	0	0
$65$	4.44949	0.551891
$66$	0	0
$67$	6.44949	0.787931	0.393965	−	0.919125i	$-0.371103\pi$
$67$	6.44949	0.787931	0.393965	+	0.919125i	$0.371103\pi$
$68$	0	0
$69$	2.89898	0.348996
$70$	0	0
$71$	−1.55051	−0.184012	−0.0920059	−	0.995758i	$-0.529328\pi$
$71$	−1.55051	−0.184012	−0.0920059	+	0.995758i	$0.529328\pi$
$72$	0	0
$73$	−12.4495	−1.45710	−0.728551	−	0.684991i	$-0.759806\pi$
$73$	−12.4495	−1.45710	−0.728551	+	0.684991i	$0.759806\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.89898	0.326161	0.163080	−	0.986613i	$-0.447857\pi$
$79$	2.89898	0.326161	0.163080	+	0.986613i	$0.447857\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.10102	0.120853	0.0604264	−	0.998173i	$-0.480754\pi$
$83$	1.10102	0.120853	0.0604264	+	0.998173i	$0.480754\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0.449490	0.0481904
$88$	0	0
$89$	−4.44949	−0.471645	−0.235822	−	0.971796i	$-0.575778\pi$
$89$	−4.44949	−0.471645	−0.235822	+	0.971796i	$0.575778\pi$
$90$	0	0
$91$	10.8990	1.14252
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−4.89898	−0.502625
$96$	0	0
$97$	−5.10102	−0.517930	−0.258965	−	0.965887i	$-0.583381\pi$
$97$	−5.10102	−0.517930	−0.258965	+	0.965887i	$0.583381\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	5.55051	0.546908	0.273454	−	0.961885i	$-0.411834\pi$
$103$	5.55051	0.546908	0.273454	+	0.961885i	$0.411834\pi$
$104$	0	0
$105$	2.44949	0.239046
$106$	0	0
$107$	4.89898	0.473602	0.236801	−	0.971558i	$-0.423901\pi$
$107$	4.89898	0.473602	0.236801	+	0.971558i	$0.423901\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−3.55051	−0.337000
$112$	0	0
$113$	−15.7980	−1.48615	−0.743073	−	0.669210i	$-0.766633\pi$
$113$	−15.7980	−1.48615	−0.743073	+	0.669210i	$0.766633\pi$
$114$	0	0
$115$	−2.89898	−0.270331
$116$	0	0
$117$	−4.44949	−0.411355
$118$	0	0
$119$	−4.89898	−0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	7.79796	0.703118
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.89898	0.789657	0.394828	−	0.918755i	$-0.370804\pi$
$127$	8.89898	0.789657	0.394828	+	0.918755i	$0.370804\pi$
$128$	0	0
$129$	3.10102	0.273030
$130$	0	0
$131$	−12.2474	−1.07006	−0.535032	−	0.844832i	$-0.679701\pi$
$131$	−12.2474	−1.07006	−0.535032	+	0.844832i	$0.679701\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.8990	−1.27291	−0.636453	−	0.771316i	$-0.719599\pi$
$137$	−14.8990	−1.27291	−0.636453	+	0.771316i	$0.719599\pi$
$138$	0	0
$139$	−13.7980	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$139$	−13.7980	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$140$	0	0
$141$	−8.89898	−0.749429
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.449490	−0.0373281
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	0.202041	0.0165518	0.00827592	−	0.999966i	$-0.497366\pi$
$149$	0.202041	0.0165518	0.00827592	+	0.999966i	$0.497366\pi$
$150$	0	0
$151$	10.8990	0.886946	0.443473	−	0.896288i	$-0.353746\pi$
$151$	10.8990	0.886946	0.443473	+	0.896288i	$0.353746\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−6.89898	−0.550599	−0.275299	−	0.961359i	$-0.588777\pi$
$157$	−6.89898	−0.550599	−0.275299	+	0.961359i	$0.588777\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−7.10102	−0.559639
$162$	0	0
$163$	−13.1464	−1.02971	−0.514854	−	0.857278i	$-0.672154\pi$
$163$	−13.1464	−1.02971	−0.514854	+	0.857278i	$0.672154\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	4.89898	0.374634
$172$	0	0
$173$	−0.202041	−0.0153609	−0.00768045	−	0.999971i	$-0.502445\pi$
$173$	−0.202041	−0.0153609	−0.00768045	+	0.999971i	$0.502445\pi$
$174$	0	0
$175$	−2.44949	−0.185164
$176$	0	0
$177$	7.34847	0.552345
$178$	0	0
$179$	15.5959	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$179$	15.5959	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$180$	0	0
$181$	−2.89898	−0.215479	−0.107740	−	0.994179i	$-0.534361\pi$
$181$	−2.89898	−0.215479	−0.107740	+	0.994179i	$0.534361\pi$
$182$	0	0
$183$	−6.89898	−0.509987
$184$	0	0
$185$	3.55051	0.261039
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.44949	−0.178174
$190$	0	0
$191$	−1.55051	−0.112191	−0.0560955	−	0.998425i	$-0.517865\pi$
$191$	−1.55051	−0.112191	−0.0560955	+	0.998425i	$0.517865\pi$
$192$	0	0
$193$	−8.69694	−0.626019	−0.313010	−	0.949750i	$-0.601337\pi$
$193$	−8.69694	−0.626019	−0.313010	+	0.949750i	$0.601337\pi$
$194$	0	0
$195$	4.44949	0.318635
$196$	0	0
$197$	−20.6969	−1.47460	−0.737298	−	0.675568i	$-0.763899\pi$
$197$	−20.6969	−1.47460	−0.737298	+	0.675568i	$0.763899\pi$
$198$	0	0
$199$	−24.6969	−1.75072	−0.875360	−	0.483472i	$-0.839375\pi$
$199$	−24.6969	−1.75072	−0.875360	+	0.483472i	$0.839375\pi$
$200$	0	0
$201$	6.44949	0.454912
$202$	0	0
$203$	−1.10102	−0.0772765
$204$	0	0
$205$	−7.79796	−0.544633
$206$	0	0
$207$	2.89898	0.201493
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.4949	−1.41093	−0.705463	−	0.708746i	$-0.749261\pi$
$211$	−20.4949	−1.41093	−0.705463	+	0.708746i	$0.749261\pi$
$212$	0	0
$213$	−1.55051	−0.106239
$214$	0	0
$215$	−3.10102	−0.211488
$216$	0	0
$217$	2.44949	0.166282
$218$	0	0
$219$	−12.4495	−0.841259
$220$	0	0
$221$	−8.89898	−0.598610
$222$	0	0
$223$	9.79796	0.656120	0.328060	−	0.944657i	$-0.393605\pi$
$223$	9.79796	0.656120	0.328060	+	0.944657i	$0.393605\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−21.7980	−1.44678	−0.723391	−	0.690439i	$-0.757417\pi$
$227$	−21.7980	−1.44678	−0.723391	+	0.690439i	$0.757417\pi$
$228$	0	0
$229$	19.7980	1.30829	0.654143	−	0.756371i	$-0.273029\pi$
$229$	19.7980	1.30829	0.654143	+	0.756371i	$0.273029\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.20204	−0.144261	−0.0721303	−	0.997395i	$-0.522980\pi$
$233$	−2.20204	−0.144261	−0.0721303	+	0.997395i	$0.522980\pi$
$234$	0	0
$235$	8.89898	0.580505
$236$	0	0
$237$	2.89898	0.188309
$238$	0	0
$239$	−16.8990	−1.09310	−0.546552	−	0.837425i	$-0.684060\pi$
$239$	−16.8990	−1.09310	−0.546552	+	0.837425i	$0.684060\pi$
$240$	0	0
$241$	3.79796	0.244648	0.122324	−	0.992490i	$-0.460965\pi$
$241$	3.79796	0.244648	0.122324	+	0.992490i	$0.460965\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−21.7980	−1.38697
$248$	0	0
$249$	1.10102	0.0697743
$250$	0	0
$251$	9.79796	0.618442	0.309221	−	0.950990i	$-0.399932\pi$
$251$	9.79796	0.618442	0.309221	+	0.950990i	$0.399932\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	16.0000	0.998053	0.499026	−	0.866587i	$-0.333691\pi$
$257$	16.0000	0.998053	0.499026	+	0.866587i	$0.333691\pi$
$258$	0	0
$259$	8.69694	0.540401
$260$	0	0
$261$	0.449490	0.0278227
$262$	0	0
$263$	−29.3939	−1.81250	−0.906252	−	0.422738i	$-0.861069\pi$
$263$	−29.3939	−1.81250	−0.906252	+	0.422738i	$0.861069\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−4.44949	−0.272304
$268$	0	0
$269$	12.0454	0.734421	0.367211	−	0.930138i	$-0.380313\pi$
$269$	12.0454	0.734421	0.367211	+	0.930138i	$0.380313\pi$
$270$	0	0
$271$	17.7980	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$271$	17.7980	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$272$	0	0
$273$	10.8990	0.659636
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.1464	0.669724	0.334862	−	0.942267i	$-0.391310\pi$
$277$	11.1464	0.669724	0.334862	+	0.942267i	$0.391310\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	14.8990	0.888799	0.444399	−	0.895829i	$-0.353417\pi$
$281$	14.8990	0.888799	0.444399	+	0.895829i	$0.353417\pi$
$282$	0	0
$283$	9.55051	0.567719	0.283859	−	0.958866i	$-0.408385\pi$
$283$	9.55051	0.567719	0.283859	+	0.958866i	$0.408385\pi$
$284$	0	0
$285$	−4.89898	−0.290191
$286$	0	0
$287$	−19.1010	−1.12750
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−5.10102	−0.299027
$292$	0	0
$293$	−21.7980	−1.27345	−0.636725	−	0.771091i	$-0.719711\pi$
$293$	−21.7980	−1.27345	−0.636725	+	0.771091i	$0.719711\pi$
$294$	0	0
$295$	−7.34847	−0.427844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.8990	−0.745967
$300$	0	0
$301$	−7.59592	−0.437821
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	6.89898	0.395034
$306$	0	0
$307$	22.4495	1.28126	0.640630	−	0.767850i	$-0.278673\pi$
$307$	22.4495	1.28126	0.640630	+	0.767850i	$0.278673\pi$
$308$	0	0
$309$	5.55051	0.315757
$310$	0	0
$311$	−4.65153	−0.263764	−0.131882	−	0.991265i	$-0.542102\pi$
$311$	−4.65153	−0.263764	−0.131882	+	0.991265i	$0.542102\pi$
$312$	0	0
$313$	−13.3485	−0.754500	−0.377250	−	0.926111i	$-0.623130\pi$
$313$	−13.3485	−0.754500	−0.377250	+	0.926111i	$0.623130\pi$
$314$	0	0
$315$	2.44949	0.138013
$316$	0	0
$317$	−15.5959	−0.875954	−0.437977	−	0.898986i	$-0.644305\pi$
$317$	−15.5959	−0.875954	−0.437977	+	0.898986i	$0.644305\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.89898	0.273434
$322$	0	0
$323$	9.79796	0.545173
$324$	0	0
$325$	−4.44949	−0.246813
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	21.7980	1.20176
$330$	0	0
$331$	1.10102	0.0605176	0.0302588	−	0.999542i	$-0.490367\pi$
$331$	1.10102	0.0605176	0.0302588	+	0.999542i	$0.490367\pi$
$332$	0	0
$333$	−3.55051	−0.194567
$334$	0	0
$335$	−6.44949	−0.352373
$336$	0	0
$337$	−19.1464	−1.04297	−0.521486	−	0.853260i	$-0.674622\pi$
$337$	−19.1464	−1.04297	−0.521486	+	0.853260i	$0.674622\pi$
$338$	0	0
$339$	−15.7980	−0.858027
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	−2.89898	−0.156076
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	14.2020	0.760218	0.380109	−	0.924942i	$-0.375887\pi$
$349$	14.2020	0.760218	0.380109	+	0.924942i	$0.375887\pi$
$350$	0	0
$351$	−4.44949	−0.237496
$352$	0	0
$353$	−15.7980	−0.840841	−0.420420	−	0.907329i	$-0.638117\pi$
$353$	−15.7980	−0.840841	−0.420420	+	0.907329i	$0.638117\pi$
$354$	0	0
$355$	1.55051	0.0822925
$356$	0	0
$357$	−4.89898	−0.259281
$358$	0	0
$359$	−9.55051	−0.504057	−0.252028	−	0.967720i	$-0.581098\pi$
$359$	−9.55051	−0.504057	−0.252028	+	0.967720i	$0.581098\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	12.4495	0.651636
$366$	0	0
$367$	11.5959	0.605302	0.302651	−	0.953101i	$-0.402128\pi$
$367$	11.5959	0.605302	0.302651	+	0.953101i	$0.402128\pi$
$368$	0	0
$369$	7.79796	0.405946
$370$	0	0
$371$	14.6969	0.763027
$372$	0	0
$373$	−3.30306	−0.171026	−0.0855130	−	0.996337i	$-0.527253\pi$
$373$	−3.30306	−0.171026	−0.0855130	+	0.996337i	$0.527253\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−16.8990	−0.868042	−0.434021	−	0.900903i	$-0.642906\pi$
$379$	−16.8990	−0.868042	−0.434021	+	0.900903i	$0.642906\pi$
$380$	0	0
$381$	8.89898	0.455909
$382$	0	0
$383$	−10.8990	−0.556912	−0.278456	−	0.960449i	$-0.589823\pi$
$383$	−10.8990	−0.556912	−0.278456	+	0.960449i	$0.589823\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.10102	0.157634
$388$	0	0
$389$	−6.65153	−0.337246	−0.168623	−	0.985681i	$-0.553932\pi$
$389$	−6.65153	−0.337246	−0.168623	+	0.985681i	$0.553932\pi$
$390$	0	0
$391$	5.79796	0.293215
$392$	0	0
$393$	−12.2474	−0.617802
$394$	0	0
$395$	−2.89898	−0.145863
$396$	0	0
$397$	11.7980	0.592123	0.296061	−	0.955169i	$-0.404327\pi$
$397$	11.7980	0.592123	0.296061	+	0.955169i	$0.404327\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	29.8434	1.49031	0.745153	−	0.666893i	$-0.232376\pi$
$401$	29.8434	1.49031	0.745153	+	0.666893i	$0.232376\pi$
$402$	0	0
$403$	4.44949	0.221645
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−0.696938	−0.0344614	−0.0172307	−	0.999852i	$-0.505485\pi$
$409$	−0.696938	−0.0344614	−0.0172307	+	0.999852i	$0.505485\pi$
$410$	0	0
$411$	−14.8990	−0.734912
$412$	0	0
$413$	−18.0000	−0.885722
$414$	0	0
$415$	−1.10102	−0.0540470
$416$	0	0
$417$	−13.7980	−0.675689
$418$	0	0
$419$	−26.4495	−1.29214	−0.646071	−	0.763277i	$-0.723589\pi$
$419$	−26.4495	−1.29214	−0.646071	+	0.763277i	$0.723589\pi$
$420$	0	0
$421$	5.79796	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$421$	5.79796	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$422$	0	0
$423$	−8.89898	−0.432683
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	16.8990	0.817799
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.5505	0.845378	0.422689	−	0.906275i	$-0.361086\pi$
$431$	17.5505	0.845378	0.422689	+	0.906275i	$0.361086\pi$
$432$	0	0
$433$	31.1464	1.49680	0.748401	−	0.663247i	$-0.230822\pi$
$433$	31.1464	1.49680	0.748401	+	0.663247i	$0.230822\pi$
$434$	0	0
$435$	−0.449490	−0.0215514
$436$	0	0
$437$	14.2020	0.679376
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	4.44949	0.210926
$446$	0	0
$447$	0.202041	0.00955621
$448$	0	0
$449$	−3.14643	−0.148489	−0.0742446	−	0.997240i	$-0.523655\pi$
$449$	−3.14643	−0.148489	−0.0742446	+	0.997240i	$0.523655\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.8990	0.512079
$454$	0	0
$455$	−10.8990	−0.510952
$456$	0	0
$457$	5.34847	0.250191	0.125095	−	0.992145i	$-0.460076\pi$
$457$	5.34847	0.250191	0.125095	+	0.992145i	$0.460076\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	3.55051	0.165364	0.0826819	−	0.996576i	$-0.473651\pi$
$461$	3.55051	0.165364	0.0826819	+	0.996576i	$0.473651\pi$
$462$	0	0
$463$	−1.30306	−0.0605584	−0.0302792	−	0.999541i	$-0.509640\pi$
$463$	−1.30306	−0.0605584	−0.0302792	+	0.999541i	$0.509640\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−19.1010	−0.883890	−0.441945	−	0.897042i	$-0.645711\pi$
$467$	−19.1010	−0.883890	−0.441945	+	0.897042i	$0.645711\pi$
$468$	0	0
$469$	−15.7980	−0.729482
$470$	0	0
$471$	−6.89898	−0.317888
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.89898	0.224781
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−19.3485	−0.884054	−0.442027	−	0.897002i	$-0.645740\pi$
$479$	−19.3485	−0.884054	−0.442027	+	0.897002i	$0.645740\pi$
$480$	0	0
$481$	15.7980	0.720325
$482$	0	0
$483$	−7.10102	−0.323108
$484$	0	0
$485$	5.10102	0.231625
$486$	0	0
$487$	32.8990	1.49080	0.745398	−	0.666620i	$-0.232260\pi$
$487$	32.8990	1.49080	0.745398	+	0.666620i	$0.232260\pi$
$488$	0	0
$489$	−13.1464	−0.594502
$490$	0	0
$491$	11.1010	0.500982	0.250491	−	0.968119i	$-0.419408\pi$
$491$	11.1010	0.500982	0.250491	+	0.968119i	$0.419408\pi$
$492$	0	0
$493$	0.898979	0.0404880
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.79796	0.170362
$498$	0	0
$499$	−13.7980	−0.617681	−0.308841	−	0.951114i	$-0.599941\pi$
$499$	−13.7980	−0.617681	−0.308841	+	0.951114i	$0.599941\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−27.1010	−1.20837	−0.604187	−	0.796842i	$-0.706502\pi$
$503$	−27.1010	−1.20837	−0.604187	+	0.796842i	$0.706502\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	6.79796	0.301908
$508$	0	0
$509$	6.24745	0.276913	0.138457	−	0.990368i	$-0.455786\pi$
$509$	6.24745	0.276913	0.138457	+	0.990368i	$0.455786\pi$
$510$	0	0
$511$	30.4949	1.34901
$512$	0	0
$513$	4.89898	0.216295
$514$	0	0
$515$	−5.55051	−0.244585
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.202041	−0.00886862
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	11.1010	0.485414	0.242707	−	0.970100i	$-0.421965\pi$
$523$	11.1010	0.485414	0.242707	+	0.970100i	$0.421965\pi$
$524$	0	0
$525$	−2.44949	−0.106904
$526$	0	0
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	−14.5959	−0.634605
$530$	0	0
$531$	7.34847	0.318896
$532$	0	0
$533$	−34.6969	−1.50289
$534$	0	0
$535$	−4.89898	−0.211801
$536$	0	0
$537$	15.5959	0.673014
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.404082	−0.0173728	−0.00868642	−	0.999962i	$-0.502765\pi$
$541$	−0.404082	−0.0173728	−0.00868642	+	0.999962i	$0.502765\pi$
$542$	0	0
$543$	−2.89898	−0.124407
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−30.9444	−1.32309	−0.661543	−	0.749907i	$-0.730098\pi$
$547$	−30.9444	−1.32309	−0.661543	+	0.749907i	$0.730098\pi$
$548$	0	0
$549$	−6.89898	−0.294441
$550$	0	0
$551$	2.20204	0.0938101
$552$	0	0
$553$	−7.10102	−0.301966
$554$	0	0
$555$	3.55051	0.150711
$556$	0	0
$557$	21.5959	0.915048	0.457524	−	0.889197i	$-0.348736\pi$
$557$	21.5959	0.915048	0.457524	+	0.889197i	$0.348736\pi$
$558$	0	0
$559$	−13.7980	−0.583591
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.898979	−0.0378875	−0.0189437	−	0.999821i	$-0.506030\pi$
$563$	−0.898979	−0.0378875	−0.0189437	+	0.999821i	$0.506030\pi$
$564$	0	0
$565$	15.7980	0.664625
$566$	0	0
$567$	−2.44949	−0.102869
$568$	0	0
$569$	−40.4495	−1.69573	−0.847865	−	0.530212i	$-0.822113\pi$
$569$	−40.4495	−1.69573	−0.847865	+	0.530212i	$0.822113\pi$
$570$	0	0
$571$	10.2020	0.426942	0.213471	−	0.976949i	$-0.431523\pi$
$571$	10.2020	0.426942	0.213471	+	0.976949i	$0.431523\pi$
$572$	0	0
$573$	−1.55051	−0.0647735
$574$	0	0
$575$	2.89898	0.120896
$576$	0	0
$577$	2.40408	0.100083	0.0500416	−	0.998747i	$-0.484065\pi$
$577$	2.40408	0.100083	0.0500416	+	0.998747i	$0.484065\pi$
$578$	0	0
$579$	−8.69694	−0.361432
$580$	0	0
$581$	−2.69694	−0.111888
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.44949	0.183964
$586$	0	0
$587$	−15.5959	−0.643712	−0.321856	−	0.946789i	$-0.604307\pi$
$587$	−15.5959	−0.643712	−0.321856	+	0.946789i	$0.604307\pi$
$588$	0	0
$589$	−4.89898	−0.201859
$590$	0	0
$591$	−20.6969	−0.851358
$592$	0	0
$593$	−33.5959	−1.37962	−0.689809	−	0.723991i	$-0.742306\pi$
$593$	−33.5959	−1.37962	−0.689809	+	0.723991i	$0.742306\pi$
$594$	0	0
$595$	4.89898	0.200839
$596$	0	0
$597$	−24.6969	−1.01078
$598$	0	0
$599$	14.9444	0.610611	0.305306	−	0.952254i	$-0.401241\pi$
$599$	14.9444	0.610611	0.305306	+	0.952254i	$0.401241\pi$
$600$	0	0
$601$	3.30306	0.134735	0.0673673	−	0.997728i	$-0.478540\pi$
$601$	3.30306	0.134735	0.0673673	+	0.997728i	$0.478540\pi$
$602$	0	0
$603$	6.44949	0.262644
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	42.9444	1.74306	0.871529	−	0.490343i	$-0.163129\pi$
$607$	42.9444	1.74306	0.871529	+	0.490343i	$0.163129\pi$
$608$	0	0
$609$	−1.10102	−0.0446156
$610$	0	0
$611$	39.5959	1.60188
$612$	0	0
$613$	36.0454	1.45586	0.727930	−	0.685651i	$-0.240482\pi$
$613$	36.0454	1.45586	0.727930	+	0.685651i	$0.240482\pi$
$614$	0	0
$615$	−7.79796	−0.314444
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−46.4949	−1.86879	−0.934394	−	0.356242i	$-0.884058\pi$
$619$	−46.4949	−1.86879	−0.934394	+	0.356242i	$0.884058\pi$
$620$	0	0
$621$	2.89898	0.116332
$622$	0	0
$623$	10.8990	0.436658
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.10102	−0.283136
$630$	0	0
$631$	−3.59592	−0.143151	−0.0715756	−	0.997435i	$-0.522803\pi$
$631$	−3.59592	−0.143151	−0.0715756	+	0.997435i	$0.522803\pi$
$632$	0	0
$633$	−20.4949	−0.814599
$634$	0	0
$635$	−8.89898	−0.353145
$636$	0	0
$637$	4.44949	0.176295
$638$	0	0
$639$	−1.55051	−0.0613372
$640$	0	0
$641$	−44.0454	−1.73969	−0.869845	−	0.493326i	$-0.835781\pi$
$641$	−44.0454	−1.73969	−0.869845	+	0.493326i	$0.835781\pi$
$642$	0	0
$643$	20.4949	0.808240	0.404120	−	0.914706i	$-0.367578\pi$
$643$	20.4949	0.808240	0.404120	+	0.914706i	$0.367578\pi$
$644$	0	0
$645$	−3.10102	−0.122103
$646$	0	0
$647$	−19.5959	−0.770395	−0.385198	−	0.922834i	$-0.625867\pi$
$647$	−19.5959	−0.770395	−0.385198	+	0.922834i	$0.625867\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.44949	0.0960031
$652$	0	0
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	12.2474	0.478547
$656$	0	0
$657$	−12.4495	−0.485701
$658$	0	0
$659$	−0.651531	−0.0253800	−0.0126900	−	0.999919i	$-0.504039\pi$
$659$	−0.651531	−0.0253800	−0.0126900	+	0.999919i	$0.504039\pi$
$660$	0	0
$661$	39.3939	1.53224	0.766122	−	0.642695i	$-0.222184\pi$
$661$	39.3939	1.53224	0.766122	+	0.642695i	$0.222184\pi$
$662$	0	0
$663$	−8.89898	−0.345608
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	1.30306	0.0504547
$668$	0	0
$669$	9.79796	0.378811
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.9444	1.11572	0.557862	−	0.829934i	$-0.311622\pi$
$673$	28.9444	1.11572	0.557862	+	0.829934i	$0.311622\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−8.20204	−0.315230	−0.157615	−	0.987501i	$-0.550381\pi$
$677$	−8.20204	−0.315230	−0.157615	+	0.987501i	$0.550381\pi$
$678$	0	0
$679$	12.4949	0.479510
$680$	0	0
$681$	−21.7980	−0.835300
$682$	0	0
$683$	−3.59592	−0.137594	−0.0687970	−	0.997631i	$-0.521916\pi$
$683$	−3.59592	−0.137594	−0.0687970	+	0.997631i	$0.521916\pi$
$684$	0	0
$685$	14.8990	0.569261
$686$	0	0
$687$	19.7980	0.755339
$688$	0	0
$689$	26.6969	1.01707
$690$	0	0
$691$	45.3939	1.72686	0.863432	−	0.504465i	$-0.168310\pi$
$691$	45.3939	1.72686	0.863432	+	0.504465i	$0.168310\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.7980	0.523386
$696$	0	0
$697$	15.5959	0.590738
$698$	0	0
$699$	−2.20204	−0.0832888
$700$	0	0
$701$	44.6969	1.68818	0.844090	−	0.536201i	$-0.180141\pi$
$701$	44.6969	1.68818	0.844090	+	0.536201i	$0.180141\pi$
$702$	0	0
$703$	−17.3939	−0.656022
$704$	0	0
$705$	8.89898	0.335155
$706$	0	0
$707$	−4.89898	−0.184245
$708$	0	0
$709$	−7.79796	−0.292858	−0.146429	−	0.989221i	$-0.546778\pi$
$709$	−7.79796	−0.292858	−0.146429	+	0.989221i	$0.546778\pi$
$710$	0	0
$711$	2.89898	0.108720
$712$	0	0
$713$	−2.89898	−0.108568
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.8990	−0.631104
$718$	0	0
$719$	−17.7980	−0.663752	−0.331876	−	0.943323i	$-0.607682\pi$
$719$	−17.7980	−0.663752	−0.331876	+	0.943323i	$0.607682\pi$
$720$	0	0
$721$	−13.5959	−0.506338
$722$	0	0
$723$	3.79796	0.141248
$724$	0	0
$725$	0.449490	0.0166936
$726$	0	0
$727$	−17.1464	−0.635926	−0.317963	−	0.948103i	$-0.602999\pi$
$727$	−17.1464	−0.635926	−0.317963	+	0.948103i	$0.602999\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.20204	0.229391
$732$	0	0
$733$	10.4041	0.384284	0.192142	−	0.981367i	$-0.438457\pi$
$733$	10.4041	0.384284	0.192142	+	0.981367i	$0.438457\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	0	0
$738$	0	0
$739$	40.6969	1.49706	0.748531	−	0.663100i	$-0.230760\pi$
$739$	40.6969	1.49706	0.748531	+	0.663100i	$0.230760\pi$
$740$	0	0
$741$	−21.7980	−0.800768
$742$	0	0
$743$	1.79796	0.0659607	0.0329804	−	0.999456i	$-0.489500\pi$
$743$	1.79796	0.0659607	0.0329804	+	0.999456i	$0.489500\pi$
$744$	0	0
$745$	−0.202041	−0.00740221
$746$	0	0
$747$	1.10102	0.0402842
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	5.79796	0.211571	0.105785	−	0.994389i	$-0.466264\pi$
$751$	5.79796	0.211571	0.105785	+	0.994389i	$0.466264\pi$
$752$	0	0
$753$	9.79796	0.357057
$754$	0	0
$755$	−10.8990	−0.396654
$756$	0	0
$757$	−12.4495	−0.452484	−0.226242	−	0.974071i	$-0.572644\pi$
$757$	−12.4495	−0.452484	−0.226242	+	0.974071i	$0.572644\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.8434	−1.37182	−0.685910	−	0.727686i	$-0.740596\pi$
$761$	−37.8434	−1.37182	−0.685910	+	0.727686i	$0.740596\pi$
$762$	0	0
$763$	9.79796	0.354710
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	−32.6969	−1.18062
$768$	0	0
$769$	33.7980	1.21879	0.609393	−	0.792868i	$-0.291413\pi$
$769$	33.7980	1.21879	0.609393	+	0.792868i	$0.291413\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	0	0
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	8.69694	0.312001
$778$	0	0
$779$	38.2020	1.36873
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.449490	0.0160635
$784$	0	0
$785$	6.89898	0.246235
$786$	0	0
$787$	47.1918	1.68221	0.841104	−	0.540874i	$-0.181906\pi$
$787$	47.1918	1.68221	0.841104	+	0.540874i	$0.181906\pi$
$788$	0	0
$789$	−29.3939	−1.04645
$790$	0	0
$791$	38.6969	1.37590
$792$	0	0
$793$	30.6969	1.09008
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	26.4949	0.938497	0.469249	−	0.883066i	$-0.344525\pi$
$797$	26.4949	0.938497	0.469249	+	0.883066i	$0.344525\pi$
$798$	0	0
$799$	−17.7980	−0.629647
$800$	0	0
$801$	−4.44949	−0.157215
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7.10102	0.250278
$806$	0	0
$807$	12.0454	0.424018
$808$	0	0
$809$	−17.3485	−0.609940	−0.304970	−	0.952362i	$-0.598646\pi$
$809$	−17.3485	−0.609940	−0.304970	+	0.952362i	$0.598646\pi$
$810$	0	0
$811$	−46.2929	−1.62556	−0.812781	−	0.582569i	$-0.802047\pi$
$811$	−46.2929	−1.62556	−0.812781	+	0.582569i	$0.802047\pi$
$812$	0	0
$813$	17.7980	0.624202
$814$	0	0
$815$	13.1464	0.460499
$816$	0	0
$817$	15.1918	0.531495
$818$	0	0
$819$	10.8990	0.380841
$820$	0	0
$821$	−55.6413	−1.94190	−0.970948	−	0.239291i	$-0.923085\pi$
$821$	−55.6413	−1.94190	−0.970948	+	0.239291i	$0.923085\pi$
$822$	0	0
$823$	−24.4949	−0.853838	−0.426919	−	0.904290i	$-0.640401\pi$
$823$	−24.4949	−0.853838	−0.426919	+	0.904290i	$0.640401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.0908	1.46364	0.731821	−	0.681497i	$-0.238671\pi$
$827$	42.0908	1.46364	0.731821	+	0.681497i	$0.238671\pi$
$828$	0	0
$829$	2.89898	0.100686	0.0503429	−	0.998732i	$-0.483969\pi$
$829$	2.89898	0.100686	0.0503429	+	0.998732i	$0.483969\pi$
$830$	0	0
$831$	11.1464	0.386665
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	14.4495	0.498852	0.249426	−	0.968394i	$-0.419758\pi$
$839$	14.4495	0.498852	0.249426	+	0.968394i	$0.419758\pi$
$840$	0	0
$841$	−28.7980	−0.993033
$842$	0	0
$843$	14.8990	0.513148
$844$	0	0
$845$	−6.79796	−0.233857
$846$	0	0
$847$	26.9444	0.925820
$848$	0	0
$849$	9.55051	0.327773
$850$	0	0
$851$	−10.2929	−0.352835
$852$	0	0
$853$	−19.3939	−0.664034	−0.332017	−	0.943273i	$-0.607729\pi$
$853$	−19.3939	−0.664034	−0.332017	+	0.943273i	$0.607729\pi$
$854$	0	0
$855$	−4.89898	−0.167542
$856$	0	0
$857$	33.1918	1.13381	0.566906	−	0.823783i	$-0.308140\pi$
$857$	33.1918	1.13381	0.566906	+	0.823783i	$0.308140\pi$
$858$	0	0
$859$	47.5959	1.62395	0.811976	−	0.583691i	$-0.198392\pi$
$859$	47.5959	1.62395	0.811976	+	0.583691i	$0.198392\pi$
$860$	0	0
$861$	−19.1010	−0.650961
$862$	0	0
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	0	0
$865$	0.202041	0.00686960
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−28.6969	−0.972359
$872$	0	0
$873$	−5.10102	−0.172643
$874$	0	0
$875$	2.44949	0.0828079
$876$	0	0
$877$	−16.2020	−0.547104	−0.273552	−	0.961857i	$-0.588199\pi$
$877$	−16.2020	−0.547104	−0.273552	+	0.961857i	$0.588199\pi$
$878$	0	0
$879$	−21.7980	−0.735227
$880$	0	0
$881$	1.75255	0.0590450	0.0295225	−	0.999564i	$-0.490601\pi$
$881$	1.75255	0.0590450	0.0295225	+	0.999564i	$0.490601\pi$
$882$	0	0
$883$	21.3939	0.719961	0.359981	−	0.932960i	$-0.382783\pi$
$883$	21.3939	0.719961	0.359981	+	0.932960i	$0.382783\pi$
$884$	0	0
$885$	−7.34847	−0.247016
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	−21.7980	−0.731080
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−43.5959	−1.45888
$894$	0	0
$895$	−15.5959	−0.521314
$896$	0	0
$897$	−12.8990	−0.430684
$898$	0	0
$899$	−0.449490	−0.0149913
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−7.59592	−0.252776
$904$	0	0
$905$	2.89898	0.0963653
$906$	0	0
$907$	−37.1464	−1.23343	−0.616713	−	0.787188i	$-0.711536\pi$
$907$	−37.1464	−1.23343	−0.616713	+	0.787188i	$0.711536\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	6.20204	0.205483	0.102741	−	0.994708i	$-0.467239\pi$
$911$	6.20204	0.205483	0.102741	+	0.994708i	$0.467239\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	6.89898	0.228073
$916$	0	0
$917$	30.0000	0.990687
$918$	0	0
$919$	−25.3031	−0.834671	−0.417335	−	0.908752i	$-0.637036\pi$
$919$	−25.3031	−0.834671	−0.417335	+	0.908752i	$0.637036\pi$
$920$	0	0
$921$	22.4495	0.739736
$922$	0	0
$923$	6.89898	0.227083
$924$	0	0
$925$	−3.55051	−0.116740
$926$	0	0
$927$	5.55051	0.182303
$928$	0	0
$929$	−40.9444	−1.34334	−0.671671	−	0.740850i	$-0.734423\pi$
$929$	−40.9444	−1.34334	−0.671671	+	0.740850i	$0.734423\pi$
$930$	0	0
$931$	−4.89898	−0.160558
$932$	0	0
$933$	−4.65153	−0.152284
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.4949	−0.734876	−0.367438	−	0.930048i	$-0.619765\pi$
$937$	−22.4949	−0.734876	−0.367438	+	0.930048i	$0.619765\pi$
$938$	0	0
$939$	−13.3485	−0.435611
$940$	0	0
$941$	−2.24745	−0.0732647	−0.0366324	−	0.999329i	$-0.511663\pi$
$941$	−2.24745	−0.0732647	−0.0366324	+	0.999329i	$0.511663\pi$
$942$	0	0
$943$	22.6061	0.736157
$944$	0	0
$945$	2.44949	0.0796819
$946$	0	0
$947$	6.49490	0.211056	0.105528	−	0.994416i	$-0.466347\pi$
$947$	6.49490	0.211056	0.105528	+	0.994416i	$0.466347\pi$
$948$	0	0
$949$	55.3939	1.79816
$950$	0	0
$951$	−15.5959	−0.505732
$952$	0	0
$953$	40.6969	1.31830	0.659152	−	0.752010i	$-0.270916\pi$
$953$	40.6969	1.31830	0.659152	+	0.752010i	$0.270916\pi$
$954$	0	0
$955$	1.55051	0.0501733
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.4949	1.17848
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.89898	0.157867
$964$	0	0
$965$	8.69694	0.279964
$966$	0	0
$967$	−29.3939	−0.945243	−0.472622	−	0.881265i	$-0.656692\pi$
$967$	−29.3939	−0.945243	−0.472622	+	0.881265i	$0.656692\pi$
$968$	0	0
$969$	9.79796	0.314756
$970$	0	0
$971$	21.5505	0.691589	0.345794	−	0.938310i	$-0.387609\pi$
$971$	21.5505	0.691589	0.345794	+	0.938310i	$0.387609\pi$
$972$	0	0
$973$	33.7980	1.08351
$974$	0	0
$975$	−4.44949	−0.142498
$976$	0	0
$977$	−25.1918	−0.805958	−0.402979	−	0.915209i	$-0.632025\pi$
$977$	−25.1918	−0.805958	−0.402979	+	0.915209i	$0.632025\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−6.20204	−0.197814	−0.0989072	−	0.995097i	$-0.531535\pi$
$983$	−6.20204	−0.197814	−0.0989072	+	0.995097i	$0.531535\pi$
$984$	0	0
$985$	20.6969	0.659459
$986$	0	0
$987$	21.7980	0.693837
$988$	0	0
$989$	8.98979	0.285859
$990$	0	0
$991$	−37.3939	−1.18786	−0.593928	−	0.804518i	$-0.702424\pi$
$991$	−37.3939	−1.18786	−0.593928	+	0.804518i	$0.702424\pi$
$992$	0	0
$993$	1.10102	0.0349398
$994$	0	0
$995$	24.6969	0.782946
$996$	0	0
$997$	−28.2929	−0.896044	−0.448022	−	0.894023i	$-0.647871\pi$
$997$	−28.2929	−0.896044	−0.448022	+	0.894023i	$0.647871\pi$
$998$	0	0
$999$	−3.55051	−0.112333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bj.1.1		2
4.3	odd	2		1860.2.a.d.1.2	✓	2
12.11	even	2		5580.2.a.g.1.2		2
20.3	even	4		9300.2.g.l.3349.1		4
20.7	even	4		9300.2.g.l.3349.4		4
20.19	odd	2		9300.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.d.1.2	✓	2	4.3	odd	2
5580.2.a.g.1.2		2	12.11	even	2
7440.2.a.bj.1.1		2	1.1	even	1	trivial
9300.2.a.p.1.1		2	20.19	odd	2
9300.2.g.l.3349.1		4	20.3	even	4
9300.2.g.l.3349.4		4	20.7	even	4

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.44949 q^{7} +1.00000 q^{9} -4.44949 q^{13} -1.00000 q^{15} +2.00000 q^{17} +4.89898 q^{19} -2.44949 q^{21} +2.89898 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.449490 q^{29} -1.00000 q^{31} +2.44949 q^{35} -3.55051 q^{37} -4.44949 q^{39} +7.79796 q^{41} +3.10102 q^{43} -1.00000 q^{45} -8.89898 q^{47} -1.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} +4.89898 q^{57} +7.34847 q^{59} -6.89898 q^{61} -2.44949 q^{63} +4.44949 q^{65} +6.44949 q^{67} +2.89898 q^{69} -1.55051 q^{71} -12.4495 q^{73} +1.00000 q^{75} +2.89898 q^{79} +1.00000 q^{81} +1.10102 q^{83} -2.00000 q^{85} +0.449490 q^{87} -4.44949 q^{89} +10.8990 q^{91} -1.00000 q^{93} -4.89898 q^{95} -5.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{31} - 12 q^{37} - 4 q^{39} - 4 q^{41} + 16 q^{43} - 2 q^{45} - 8 q^{47} - 2 q^{49} + 4 q^{51}+ \cdots - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 - 2 * q^15 + 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 2 * q^31 - 12 * q^37 - 4 * q^39 - 4 * q^41 + 16 * q^43 - 2 * q^45 - 8 * q^47 - 2 * q^49 + 4 * q^51 - 12 * q^53 - 4 * q^61 + 4 * q^65 + 8 * q^67 - 4 * q^69 - 8 * q^71 - 20 * q^73 + 2 * q^75 - 4 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 - 4 * q^87 - 4 * q^89 + 12 * q^91 - 2 * q^93 - 20 * q^97

Introduction

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Database

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