LMFDB - Embedded newform 168.6.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [168,6,Mod(1,168)] 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("168.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	168.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,9,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.9444817286$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	168.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} -64.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -54.0000 q^{11} +738.000 q^{13} -576.000 q^{15} -848.000 q^{17} -1604.00 q^{19} +441.000 q^{21} -3670.00 q^{23} +971.000 q^{25} +729.000 q^{27} -4330.00 q^{29} -4760.00 q^{31} -486.000 q^{33} -3136.00 q^{35} -2094.00 q^{37} +6642.00 q^{39} -6116.00 q^{41} +7916.00 q^{43} -5184.00 q^{45} +6572.00 q^{47} +2401.00 q^{49} -7632.00 q^{51} -7894.00 q^{53} +3456.00 q^{55} -14436.0 q^{57} -41664.0 q^{59} -26570.0 q^{61} +3969.00 q^{63} -47232.0 q^{65} -41736.0 q^{67} -33030.0 q^{69} +83574.0 q^{71} -42314.0 q^{73} +8739.00 q^{75} -2646.00 q^{77} +508.000 q^{79} +6561.00 q^{81} -8364.00 q^{83} +54272.0 q^{85} -38970.0 q^{87} -49220.0 q^{89} +36162.0 q^{91} -42840.0 q^{93} +102656. q^{95} +159670. q^{97} -4374.00 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−64.0000	−1.14487	−0.572433	−	0.819951i	$-0.694000\pi$
$5$	−64.0000	−1.14487	−0.572433	+	0.819951i	$0.694000\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−54.0000	−0.134559	−0.0672794	−	0.997734i	$-0.521432\pi$
$11$	−54.0000	−0.134559	−0.0672794	+	0.997734i	$0.521432\pi$
$12$	0	0
$13$	738.000	1.21115	0.605575	−	0.795788i	$-0.292943\pi$
$13$	738.000	1.21115	0.605575	+	0.795788i	$0.292943\pi$
$14$	0	0
$15$	−576.000	−0.660989
$16$	0	0
$17$	−848.000	−0.711662	−0.355831	−	0.934550i	$-0.615802\pi$
$17$	−848.000	−0.711662	−0.355831	+	0.934550i	$0.615802\pi$
$18$	0	0
$19$	−1604.00	−1.01934	−0.509672	−	0.860369i	$-0.670233\pi$
$19$	−1604.00	−1.01934	−0.509672	+	0.860369i	$0.670233\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	−3670.00	−1.44659	−0.723297	−	0.690537i	$-0.757374\pi$
$23$	−3670.00	−1.44659	−0.723297	+	0.690537i	$0.757374\pi$
$24$	0	0
$25$	971.000	0.310720
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−4330.00	−0.956077	−0.478039	−	0.878339i	$-0.658652\pi$
$29$	−4330.00	−0.956077	−0.478039	+	0.878339i	$0.658652\pi$
$30$	0	0
$31$	−4760.00	−0.889616	−0.444808	−	0.895626i	$-0.646728\pi$
$31$	−4760.00	−0.889616	−0.444808	+	0.895626i	$0.646728\pi$
$32$	0	0
$33$	−486.000	−0.0776875
$34$	0	0
$35$	−3136.00	−0.432719
$36$	0	0
$37$	−2094.00	−0.251462	−0.125731	−	0.992064i	$-0.540128\pi$
$37$	−2094.00	−0.251462	−0.125731	+	0.992064i	$0.540128\pi$
$38$	0	0
$39$	6642.00	0.699258
$40$	0	0
$41$	−6116.00	−0.568209	−0.284104	−	0.958793i	$-0.591696\pi$
$41$	−6116.00	−0.568209	−0.284104	+	0.958793i	$0.591696\pi$
$42$	0	0
$43$	7916.00	0.652882	0.326441	−	0.945218i	$-0.394151\pi$
$43$	7916.00	0.652882	0.326441	+	0.945218i	$0.394151\pi$
$44$	0	0
$45$	−5184.00	−0.381622
$46$	0	0
$47$	6572.00	0.433963	0.216982	−	0.976176i	$-0.430379\pi$
$47$	6572.00	0.433963	0.216982	+	0.976176i	$0.430379\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−7632.00	−0.410878
$52$	0	0
$53$	−7894.00	−0.386018	−0.193009	−	0.981197i	$-0.561825\pi$
$53$	−7894.00	−0.386018	−0.193009	+	0.981197i	$0.561825\pi$
$54$	0	0
$55$	3456.00	0.154052
$56$	0	0
$57$	−14436.0	−0.588518
$58$	0	0
$59$	−41664.0	−1.55823	−0.779114	−	0.626882i	$-0.784331\pi$
$59$	−41664.0	−1.55823	−0.779114	+	0.626882i	$0.784331\pi$
$60$	0	0
$61$	−26570.0	−0.914254	−0.457127	−	0.889401i	$-0.651122\pi$
$61$	−26570.0	−0.914254	−0.457127	+	0.889401i	$0.651122\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−47232.0	−1.38661
$66$	0	0
$67$	−41736.0	−1.13586	−0.567929	−	0.823078i	$-0.692255\pi$
$67$	−41736.0	−1.13586	−0.567929	+	0.823078i	$0.692255\pi$
$68$	0	0
$69$	−33030.0	−0.835191
$70$	0	0
$71$	83574.0	1.96755	0.983774	−	0.179412i	$-0.0574196\pi$
$71$	83574.0	1.96755	0.983774	+	0.179412i	$0.0574196\pi$
$72$	0	0
$73$	−42314.0	−0.929345	−0.464672	−	0.885483i	$-0.653828\pi$
$73$	−42314.0	−0.929345	−0.464672	+	0.885483i	$0.653828\pi$
$74$	0	0
$75$	8739.00	0.179394
$76$	0	0
$77$	−2646.00	−0.0508584
$78$	0	0
$79$	508.000	0.00915790	0.00457895	−	0.999990i	$-0.498542\pi$
$79$	508.000	0.00915790	0.00457895	+	0.999990i	$0.498542\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−8364.00	−0.133266	−0.0666329	−	0.997778i	$-0.521226\pi$
$83$	−8364.00	−0.133266	−0.0666329	+	0.997778i	$0.521226\pi$
$84$	0	0
$85$	54272.0	0.814758
$86$	0	0
$87$	−38970.0	−0.551991
$88$	0	0
$89$	−49220.0	−0.658668	−0.329334	−	0.944213i	$-0.606824\pi$
$89$	−49220.0	−0.658668	−0.329334	+	0.944213i	$0.606824\pi$
$90$	0	0
$91$	36162.0	0.457772
$92$	0	0
$93$	−42840.0	−0.513620
$94$	0	0
$95$	102656.	1.16701
$96$	0	0
$97$	159670.	1.72303	0.861517	−	0.507728i	$-0.169515\pi$
$97$	159670.	1.72303	0.861517	+	0.507728i	$0.169515\pi$
$98$	0	0
$99$	−4374.00	−0.0448529
$100$	0	0
$101$	−67020.0	−0.653734	−0.326867	−	0.945070i	$-0.605993\pi$
$101$	−67020.0	−0.653734	−0.326867	+	0.945070i	$0.605993\pi$
$102$	0	0
$103$	165768.	1.53960	0.769800	−	0.638286i	$-0.220356\pi$
$103$	165768.	1.53960	0.769800	+	0.638286i	$0.220356\pi$
$104$	0	0
$105$	−28224.0	−0.249830
$106$	0	0
$107$	103146.	0.870949	0.435475	−	0.900201i	$-0.356581\pi$
$107$	103146.	0.870949	0.435475	+	0.900201i	$0.356581\pi$
$108$	0	0
$109$	60094.0	0.484468	0.242234	−	0.970218i	$-0.422120\pi$
$109$	60094.0	0.484468	0.242234	+	0.970218i	$0.422120\pi$
$110$	0	0
$111$	−18846.0	−0.145182
$112$	0	0
$113$	−126246.	−0.930083	−0.465041	−	0.885289i	$-0.653961\pi$
$113$	−126246.	−0.930083	−0.465041	+	0.885289i	$0.653961\pi$
$114$	0	0
$115$	234880.	1.65616
$116$	0	0
$117$	59778.0	0.403717
$118$	0	0
$119$	−41552.0	−0.268983
$120$	0	0
$121$	−158135.	−0.981894
$122$	0	0
$123$	−55044.0	−0.328055
$124$	0	0
$125$	137856.	0.789134
$126$	0	0
$127$	308636.	1.69800	0.848999	−	0.528394i	$-0.177206\pi$
$127$	308636.	1.69800	0.848999	+	0.528394i	$0.177206\pi$
$128$	0	0
$129$	71244.0	0.376942
$130$	0	0
$131$	−61012.0	−0.310625	−0.155313	−	0.987865i	$-0.549639\pi$
$131$	−61012.0	−0.310625	−0.155313	+	0.987865i	$0.549639\pi$
$132$	0	0
$133$	−78596.0	−0.385275
$134$	0	0
$135$	−46656.0	−0.220330
$136$	0	0
$137$	−317242.	−1.44407	−0.722037	−	0.691855i	$-0.756794\pi$
$137$	−317242.	−1.44407	−0.722037	+	0.691855i	$0.756794\pi$
$138$	0	0
$139$	−7236.00	−0.0317659	−0.0158830	−	0.999874i	$-0.505056\pi$
$139$	−7236.00	−0.0317659	−0.0158830	+	0.999874i	$0.505056\pi$
$140$	0	0
$141$	59148.0	0.250549
$142$	0	0
$143$	−39852.0	−0.162971
$144$	0	0
$145$	277120.	1.09458
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	126058.	0.465163	0.232581	−	0.972577i	$-0.425283\pi$
$149$	126058.	0.465163	0.232581	+	0.972577i	$0.425283\pi$
$150$	0	0
$151$	200296.	0.714875	0.357437	−	0.933937i	$-0.383651\pi$
$151$	200296.	0.714875	0.357437	+	0.933937i	$0.383651\pi$
$152$	0	0
$153$	−68688.0	−0.237221
$154$	0	0
$155$	304640.	1.01849
$156$	0	0
$157$	510894.	1.65418	0.827088	−	0.562073i	$-0.189996\pi$
$157$	510894.	1.65418	0.827088	+	0.562073i	$0.189996\pi$
$158$	0	0
$159$	−71046.0	−0.222868
$160$	0	0
$161$	−179830.	−0.546761
$162$	0	0
$163$	21184.0	0.0624509	0.0312255	−	0.999512i	$-0.490059\pi$
$163$	21184.0	0.0624509	0.0312255	+	0.999512i	$0.490059\pi$
$164$	0	0
$165$	31104.0	0.0889419
$166$	0	0
$167$	−267180.	−0.741332	−0.370666	−	0.928766i	$-0.620871\pi$
$167$	−267180.	−0.741332	−0.370666	+	0.928766i	$0.620871\pi$
$168$	0	0
$169$	173351.	0.466885
$170$	0	0
$171$	−129924.	−0.339781
$172$	0	0
$173$	−91948.0	−0.233575	−0.116788	−	0.993157i	$-0.537260\pi$
$173$	−91948.0	−0.233575	−0.116788	+	0.993157i	$0.537260\pi$
$174$	0	0
$175$	47579.0	0.117441
$176$	0	0
$177$	−374976.	−0.899643
$178$	0	0
$179$	−402826.	−0.939691	−0.469845	−	0.882749i	$-0.655690\pi$
$179$	−402826.	−0.939691	−0.469845	+	0.882749i	$0.655690\pi$
$180$	0	0
$181$	−796222.	−1.80650	−0.903250	−	0.429116i	$-0.858825\pi$
$181$	−796222.	−1.80650	−0.903250	+	0.429116i	$0.858825\pi$
$182$	0	0
$183$	−239130.	−0.527845
$184$	0	0
$185$	134016.	0.287890
$186$	0	0
$187$	45792.0	0.0957603
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	474934.	0.941998	0.470999	−	0.882134i	$-0.343894\pi$
$191$	474934.	0.941998	0.470999	+	0.882134i	$0.343894\pi$
$192$	0	0
$193$	716022.	1.38367	0.691836	−	0.722055i	$-0.256802\pi$
$193$	716022.	1.38367	0.691836	+	0.722055i	$0.256802\pi$
$194$	0	0
$195$	−425088.	−0.800557
$196$	0	0
$197$	−221814.	−0.407215	−0.203607	−	0.979053i	$-0.565267\pi$
$197$	−221814.	−0.407215	−0.203607	+	0.979053i	$0.565267\pi$
$198$	0	0
$199$	−333616.	−0.597192	−0.298596	−	0.954380i	$-0.596518\pi$
$199$	−333616.	−0.597192	−0.298596	+	0.954380i	$0.596518\pi$
$200$	0	0
$201$	−375624.	−0.655788
$202$	0	0
$203$	−212170.	−0.361363
$204$	0	0
$205$	391424.	0.650523
$206$	0	0
$207$	−297270.	−0.482198
$208$	0	0
$209$	86616.0	0.137162
$210$	0	0
$211$	−176404.	−0.272774	−0.136387	−	0.990656i	$-0.543549\pi$
$211$	−176404.	−0.272774	−0.136387	+	0.990656i	$0.543549\pi$
$212$	0	0
$213$	752166.	1.13596
$214$	0	0
$215$	−506624.	−0.747463
$216$	0	0
$217$	−233240.	−0.336243
$218$	0	0
$219$	−380826.	−0.536558
$220$	0	0
$221$	−625824.	−0.861929
$222$	0	0
$223$	−125016.	−0.168346	−0.0841731	−	0.996451i	$-0.526825\pi$
$223$	−125016.	−0.168346	−0.0841731	+	0.996451i	$0.526825\pi$
$224$	0	0
$225$	78651.0	0.103573
$226$	0	0
$227$	−272104.	−0.350486	−0.175243	−	0.984525i	$-0.556071\pi$
$227$	−272104.	−0.350486	−0.175243	+	0.984525i	$0.556071\pi$
$228$	0	0
$229$	−325822.	−0.410574	−0.205287	−	0.978702i	$-0.565813\pi$
$229$	−325822.	−0.410574	−0.205287	+	0.978702i	$0.565813\pi$
$230$	0	0
$231$	−23814.0	−0.0293631
$232$	0	0
$233$	−534682.	−0.645217	−0.322608	−	0.946533i	$-0.604560\pi$
$233$	−534682.	−0.645217	−0.322608	+	0.946533i	$0.604560\pi$
$234$	0	0
$235$	−420608.	−0.496830
$236$	0	0
$237$	4572.00	0.00528732
$238$	0	0
$239$	1.48512e6	1.68177	0.840887	−	0.541211i	$-0.182034\pi$
$239$	1.48512e6	1.68177	0.840887	+	0.541211i	$0.182034\pi$
$240$	0	0
$241$	1.17689e6	1.30524	0.652622	−	0.757684i	$-0.273669\pi$
$241$	1.17689e6	1.30524	0.652622	+	0.757684i	$0.273669\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−153664.	−0.163552
$246$	0	0
$247$	−1.18375e6	−1.23458
$248$	0	0
$249$	−75276.0	−0.0769411
$250$	0	0
$251$	14080.0	0.0141065	0.00705324	−	0.999975i	$-0.497755\pi$
$251$	14080.0	0.0141065	0.00705324	+	0.999975i	$0.497755\pi$
$252$	0	0
$253$	198180.	0.194652
$254$	0	0
$255$	488448.	0.470401
$256$	0	0
$257$	−1.86851e6	−1.76466	−0.882332	−	0.470627i	$-0.844028\pi$
$257$	−1.86851e6	−1.76466	−0.882332	+	0.470627i	$0.844028\pi$
$258$	0	0
$259$	−102606.	−0.0950437
$260$	0	0
$261$	−350730.	−0.318692
$262$	0	0
$263$	802890.	0.715759	0.357879	−	0.933768i	$-0.383500\pi$
$263$	802890.	0.715759	0.357879	+	0.933768i	$0.383500\pi$
$264$	0	0
$265$	505216.	0.441939
$266$	0	0
$267$	−442980.	−0.380282
$268$	0	0
$269$	−197448.	−0.166369	−0.0831844	−	0.996534i	$-0.526509\pi$
$269$	−197448.	−0.166369	−0.0831844	+	0.996534i	$0.526509\pi$
$270$	0	0
$271$	−2.01928e6	−1.67022	−0.835109	−	0.550084i	$-0.814596\pi$
$271$	−2.01928e6	−1.67022	−0.835109	+	0.550084i	$0.814596\pi$
$272$	0	0
$273$	325458.	0.264295
$274$	0	0
$275$	−52434.0	−0.0418101
$276$	0	0
$277$	1.57993e6	1.23720	0.618598	−	0.785708i	$-0.287701\pi$
$277$	1.57993e6	1.23720	0.618598	+	0.785708i	$0.287701\pi$
$278$	0	0
$279$	−385560.	−0.296539
$280$	0	0
$281$	1.44392e6	1.09088	0.545440	−	0.838150i	$-0.316363\pi$
$281$	1.44392e6	1.09088	0.545440	+	0.838150i	$0.316363\pi$
$282$	0	0
$283$	−1.68046e6	−1.24727	−0.623637	−	0.781714i	$-0.714346\pi$
$283$	−1.68046e6	−1.24727	−0.623637	+	0.781714i	$0.714346\pi$
$284$	0	0
$285$	923904.	0.673775
$286$	0	0
$287$	−299684.	−0.214763
$288$	0	0
$289$	−700753.	−0.493538
$290$	0	0
$291$	1.43703e6	0.994794
$292$	0	0
$293$	−2.31092e6	−1.57259	−0.786297	−	0.617849i	$-0.788004\pi$
$293$	−2.31092e6	−1.57259	−0.786297	+	0.617849i	$0.788004\pi$
$294$	0	0
$295$	2.66650e6	1.78396
$296$	0	0
$297$	−39366.0	−0.0258958
$298$	0	0
$299$	−2.70846e6	−1.75204
$300$	0	0
$301$	387884.	0.246766
$302$	0	0
$303$	−603180.	−0.377433
$304$	0	0
$305$	1.70048e6	1.04670
$306$	0	0
$307$	−793964.	−0.480789	−0.240395	−	0.970675i	$-0.577277\pi$
$307$	−793964.	−0.480789	−0.240395	+	0.970675i	$0.577277\pi$
$308$	0	0
$309$	1.49191e6	0.888888
$310$	0	0
$311$	1.18376e6	0.694007	0.347004	−	0.937864i	$-0.387199\pi$
$311$	1.18376e6	0.694007	0.347004	+	0.937864i	$0.387199\pi$
$312$	0	0
$313$	994970.	0.574049	0.287025	−	0.957923i	$-0.407334\pi$
$313$	994970.	0.574049	0.287025	+	0.957923i	$0.407334\pi$
$314$	0	0
$315$	−254016.	−0.144240
$316$	0	0
$317$	1.84619e6	1.03188	0.515940	−	0.856625i	$-0.327443\pi$
$317$	1.84619e6	1.03188	0.515940	+	0.856625i	$0.327443\pi$
$318$	0	0
$319$	233820.	0.128649
$320$	0	0
$321$	928314.	0.502843
$322$	0	0
$323$	1.36019e6	0.725427
$324$	0	0
$325$	716598.	0.376329
$326$	0	0
$327$	540846.	0.279708
$328$	0	0
$329$	322028.	0.164023
$330$	0	0
$331$	−1.55801e6	−0.781629	−0.390815	−	0.920469i	$-0.627807\pi$
$331$	−1.55801e6	−0.781629	−0.390815	+	0.920469i	$0.627807\pi$
$332$	0	0
$333$	−169614.	−0.0838207
$334$	0	0
$335$	2.67110e6	1.30041
$336$	0	0
$337$	3.28798e6	1.57708	0.788541	−	0.614982i	$-0.210837\pi$
$337$	3.28798e6	1.57708	0.788541	+	0.614982i	$0.210837\pi$
$338$	0	0
$339$	−1.13621e6	−0.536983
$340$	0	0
$341$	257040.	0.119706
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	2.11392e6	0.956183
$346$	0	0
$347$	3.66137e6	1.63238	0.816188	−	0.577786i	$-0.196083\pi$
$347$	3.66137e6	1.63238	0.816188	+	0.577786i	$0.196083\pi$
$348$	0	0
$349$	−3.76811e6	−1.65600	−0.827998	−	0.560730i	$-0.810520\pi$
$349$	−3.76811e6	−1.65600	−0.827998	+	0.560730i	$0.810520\pi$
$350$	0	0
$351$	538002.	0.233086
$352$	0	0
$353$	1.97794e6	0.844842	0.422421	−	0.906400i	$-0.361180\pi$
$353$	1.97794e6	0.844842	0.422421	+	0.906400i	$0.361180\pi$
$354$	0	0
$355$	−5.34874e6	−2.25258
$356$	0	0
$357$	−373968.	−0.155297
$358$	0	0
$359$	3.17410e6	1.29982	0.649912	−	0.760009i	$-0.274806\pi$
$359$	3.17410e6	1.29982	0.649912	+	0.760009i	$0.274806\pi$
$360$	0	0
$361$	96717.0	0.0390602
$362$	0	0
$363$	−1.42321e6	−0.566897
$364$	0	0
$365$	2.70810e6	1.06398
$366$	0	0
$367$	3.62163e6	1.40359	0.701793	−	0.712381i	$-0.252383\pi$
$367$	3.62163e6	1.40359	0.701793	+	0.712381i	$0.252383\pi$
$368$	0	0
$369$	−495396.	−0.189403
$370$	0	0
$371$	−386806.	−0.145901
$372$	0	0
$373$	−3.65737e6	−1.36112	−0.680561	−	0.732692i	$-0.738264\pi$
$373$	−3.65737e6	−1.36112	−0.680561	+	0.732692i	$0.738264\pi$
$374$	0	0
$375$	1.24070e6	0.455607
$376$	0	0
$377$	−3.19554e6	−1.15795
$378$	0	0
$379$	1.07802e6	0.385504	0.192752	−	0.981248i	$-0.438259\pi$
$379$	1.07802e6	0.385504	0.192752	+	0.981248i	$0.438259\pi$
$380$	0	0
$381$	2.77772e6	0.980340
$382$	0	0
$383$	3.86954e6	1.34792	0.673958	−	0.738770i	$-0.264593\pi$
$383$	3.86954e6	1.34792	0.673958	+	0.738770i	$0.264593\pi$
$384$	0	0
$385$	169344.	0.0582261
$386$	0	0
$387$	641196.	0.217627
$388$	0	0
$389$	−3.75845e6	−1.25932	−0.629658	−	0.776872i	$-0.716805\pi$
$389$	−3.75845e6	−1.25932	−0.629658	+	0.776872i	$0.716805\pi$
$390$	0	0
$391$	3.11216e6	1.02948
$392$	0	0
$393$	−549108.	−0.179340
$394$	0	0
$395$	−32512.0	−0.0104846
$396$	0	0
$397$	−1.47106e6	−0.468440	−0.234220	−	0.972184i	$-0.575254\pi$
$397$	−1.47106e6	−0.468440	−0.234220	+	0.972184i	$0.575254\pi$
$398$	0	0
$399$	−707364.	−0.222439
$400$	0	0
$401$	−5.30313e6	−1.64692	−0.823458	−	0.567378i	$-0.807958\pi$
$401$	−5.30313e6	−1.64692	−0.823458	+	0.567378i	$0.807958\pi$
$402$	0	0
$403$	−3.51288e6	−1.07746
$404$	0	0
$405$	−419904.	−0.127207
$406$	0	0
$407$	113076.	0.0338364
$408$	0	0
$409$	−6.46984e6	−1.91243	−0.956215	−	0.292666i	$-0.905458\pi$
$409$	−6.46984e6	−1.91243	−0.956215	+	0.292666i	$0.905458\pi$
$410$	0	0
$411$	−2.85518e6	−0.833736
$412$	0	0
$413$	−2.04154e6	−0.588955
$414$	0	0
$415$	535296.	0.152572
$416$	0	0
$417$	−65124.0	−0.0183401
$418$	0	0
$419$	554024.	0.154168	0.0770839	−	0.997025i	$-0.475439\pi$
$419$	554024.	0.154168	0.0770839	+	0.997025i	$0.475439\pi$
$420$	0	0
$421$	−3.37900e6	−0.929143	−0.464572	−	0.885536i	$-0.653792\pi$
$421$	−3.37900e6	−0.929143	−0.464572	+	0.885536i	$0.653792\pi$
$422$	0	0
$423$	532332.	0.144654
$424$	0	0
$425$	−823408.	−0.221128
$426$	0	0
$427$	−1.30193e6	−0.345556
$428$	0	0
$429$	−358668.	−0.0940913
$430$	0	0
$431$	2.90338e6	0.752854	0.376427	−	0.926446i	$-0.377152\pi$
$431$	2.90338e6	0.752854	0.376427	+	0.926446i	$0.377152\pi$
$432$	0	0
$433$	−5.05684e6	−1.29616	−0.648081	−	0.761571i	$-0.724428\pi$
$433$	−5.05684e6	−1.29616	−0.648081	+	0.761571i	$0.724428\pi$
$434$	0	0
$435$	2.49408e6	0.631957
$436$	0	0
$437$	5.88668e6	1.47457
$438$	0	0
$439$	−2.43257e6	−0.602426	−0.301213	−	0.953557i	$-0.597392\pi$
$439$	−2.43257e6	−0.602426	−0.301213	+	0.953557i	$0.597392\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−2.35832e6	−0.570944	−0.285472	−	0.958387i	$-0.592150\pi$
$443$	−2.35832e6	−0.570944	−0.285472	+	0.958387i	$0.592150\pi$
$444$	0	0
$445$	3.15008e6	0.754087
$446$	0	0
$447$	1.13452e6	0.268562
$448$	0	0
$449$	798466.	0.186913	0.0934567	−	0.995623i	$-0.470208\pi$
$449$	798466.	0.186913	0.0934567	+	0.995623i	$0.470208\pi$
$450$	0	0
$451$	330264.	0.0764575
$452$	0	0
$453$	1.80266e6	0.412733
$454$	0	0
$455$	−2.31437e6	−0.524088
$456$	0	0
$457$	−3.53337e6	−0.791404	−0.395702	−	0.918379i	$-0.629499\pi$
$457$	−3.53337e6	−0.791404	−0.395702	+	0.918379i	$0.629499\pi$
$458$	0	0
$459$	−618192.	−0.136959
$460$	0	0
$461$	−1.98709e6	−0.435477	−0.217739	−	0.976007i	$-0.569868\pi$
$461$	−1.98709e6	−0.435477	−0.217739	+	0.976007i	$0.569868\pi$
$462$	0	0
$463$	6.33175e6	1.37269	0.686343	−	0.727278i	$-0.259215\pi$
$463$	6.33175e6	1.37269	0.686343	+	0.727278i	$0.259215\pi$
$464$	0	0
$465$	2.74176e6	0.588027
$466$	0	0
$467$	274560.	0.0582566	0.0291283	−	0.999576i	$-0.490727\pi$
$467$	274560.	0.0582566	0.0291283	+	0.999576i	$0.490727\pi$
$468$	0	0
$469$	−2.04506e6	−0.429314
$470$	0	0
$471$	4.59805e6	0.955039
$472$	0	0
$473$	−427464.	−0.0878510
$474$	0	0
$475$	−1.55748e6	−0.316730
$476$	0	0
$477$	−639414.	−0.128673
$478$	0	0
$479$	933460.	0.185890	0.0929452	−	0.995671i	$-0.470372\pi$
$479$	933460.	0.185890	0.0929452	+	0.995671i	$0.470372\pi$
$480$	0	0
$481$	−1.54537e6	−0.304558
$482$	0	0
$483$	−1.61847e6	−0.315673
$484$	0	0
$485$	−1.02189e7	−1.97265
$486$	0	0
$487$	−6.05600e6	−1.15708	−0.578540	−	0.815654i	$-0.696377\pi$
$487$	−6.05600e6	−1.15708	−0.578540	+	0.815654i	$0.696377\pi$
$488$	0	0
$489$	190656.	0.0360561
$490$	0	0
$491$	1.65757e6	0.310290	0.155145	−	0.987892i	$-0.450416\pi$
$491$	1.65757e6	0.310290	0.155145	+	0.987892i	$0.450416\pi$
$492$	0	0
$493$	3.67184e6	0.680403
$494$	0	0
$495$	279936.	0.0513506
$496$	0	0
$497$	4.09513e6	0.743663
$498$	0	0
$499$	4.08804e6	0.734961	0.367480	−	0.930031i	$-0.380220\pi$
$499$	4.08804e6	0.734961	0.367480	+	0.930031i	$0.380220\pi$
$500$	0	0
$501$	−2.40462e6	−0.428008
$502$	0	0
$503$	−7.43036e6	−1.30945	−0.654726	−	0.755866i	$-0.727216\pi$
$503$	−7.43036e6	−1.30945	−0.654726	+	0.755866i	$0.727216\pi$
$504$	0	0
$505$	4.28928e6	0.748438
$506$	0	0
$507$	1.56016e6	0.269556
$508$	0	0
$509$	−3.51290e6	−0.600996	−0.300498	−	0.953782i	$-0.597153\pi$
$509$	−3.51290e6	−0.600996	−0.300498	+	0.953782i	$0.597153\pi$
$510$	0	0
$511$	−2.07339e6	−0.351259
$512$	0	0
$513$	−1.16932e6	−0.196173
$514$	0	0
$515$	−1.06092e7	−1.76264
$516$	0	0
$517$	−354888.	−0.0583936
$518$	0	0
$519$	−827532.	−0.134855
$520$	0	0
$521$	4.81406e6	0.776994	0.388497	−	0.921450i	$-0.372994\pi$
$521$	4.81406e6	0.776994	0.388497	+	0.921450i	$0.372994\pi$
$522$	0	0
$523$	−2.42660e6	−0.387921	−0.193960	−	0.981009i	$-0.562133\pi$
$523$	−2.42660e6	−0.387921	−0.193960	+	0.981009i	$0.562133\pi$
$524$	0	0
$525$	428211.	0.0678047
$526$	0	0
$527$	4.03648e6	0.633106
$528$	0	0
$529$	7.03256e6	1.09263
$530$	0	0
$531$	−3.37478e6	−0.519409
$532$	0	0
$533$	−4.51361e6	−0.688186
$534$	0	0
$535$	−6.60134e6	−0.997121
$536$	0	0
$537$	−3.62543e6	−0.542531
$538$	0	0
$539$	−129654.	−0.0192227
$540$	0	0
$541$	4.82543e6	0.708831	0.354415	−	0.935088i	$-0.384680\pi$
$541$	4.82543e6	0.708831	0.354415	+	0.935088i	$0.384680\pi$
$542$	0	0
$543$	−7.16600e6	−1.04298
$544$	0	0
$545$	−3.84602e6	−0.554651
$546$	0	0
$547$	1.34543e6	0.192262	0.0961310	−	0.995369i	$-0.469353\pi$
$547$	1.34543e6	0.192262	0.0961310	+	0.995369i	$0.469353\pi$
$548$	0	0
$549$	−2.15217e6	−0.304751
$550$	0	0
$551$	6.94532e6	0.974571
$552$	0	0
$553$	24892.0	0.00346136
$554$	0	0
$555$	1.20614e6	0.166214
$556$	0	0
$557$	−938438.	−0.128164	−0.0640822	−	0.997945i	$-0.520412\pi$
$557$	−938438.	−0.128164	−0.0640822	+	0.997945i	$0.520412\pi$
$558$	0	0
$559$	5.84201e6	0.790738
$560$	0	0
$561$	412128.	0.0552872
$562$	0	0
$563$	1.16124e7	1.54402	0.772008	−	0.635613i	$-0.219253\pi$
$563$	1.16124e7	1.54402	0.772008	+	0.635613i	$0.219253\pi$
$564$	0	0
$565$	8.07974e6	1.06482
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	−6.94876e6	−0.899760	−0.449880	−	0.893089i	$-0.648533\pi$
$569$	−6.94876e6	−0.899760	−0.449880	+	0.893089i	$0.648533\pi$
$570$	0	0
$571$	2.59412e6	0.332966	0.166483	−	0.986044i	$-0.446759\pi$
$571$	2.59412e6	0.332966	0.166483	+	0.986044i	$0.446759\pi$
$572$	0	0
$573$	4.27441e6	0.543863
$574$	0	0
$575$	−3.56357e6	−0.449485
$576$	0	0
$577$	−1.51612e7	−1.89581	−0.947906	−	0.318551i	$-0.896804\pi$
$577$	−1.51612e7	−1.89581	−0.947906	+	0.318551i	$0.896804\pi$
$578$	0	0
$579$	6.44420e6	0.798863
$580$	0	0
$581$	−409836.	−0.0503697
$582$	0	0
$583$	426276.	0.0519421
$584$	0	0
$585$	−3.82579e6	−0.462202
$586$	0	0
$587$	−1.29826e6	−0.155513	−0.0777567	−	0.996972i	$-0.524776\pi$
$587$	−1.29826e6	−0.155513	−0.0777567	+	0.996972i	$0.524776\pi$
$588$	0	0
$589$	7.63504e6	0.906824
$590$	0	0
$591$	−1.99633e6	−0.235105
$592$	0	0
$593$	1.15002e7	1.34298	0.671490	−	0.741014i	$-0.265655\pi$
$593$	1.15002e7	1.34298	0.671490	+	0.741014i	$0.265655\pi$
$594$	0	0
$595$	2.65933e6	0.307949
$596$	0	0
$597$	−3.00254e6	−0.344789
$598$	0	0
$599$	9.99854e6	1.13860	0.569298	−	0.822131i	$-0.307215\pi$
$599$	9.99854e6	1.13860	0.569298	+	0.822131i	$0.307215\pi$
$600$	0	0
$601$	8.05405e6	0.909553	0.454777	−	0.890606i	$-0.349719\pi$
$601$	8.05405e6	0.909553	0.454777	+	0.890606i	$0.349719\pi$
$602$	0	0
$603$	−3.38062e6	−0.378619
$604$	0	0
$605$	1.01206e7	1.12414
$606$	0	0
$607$	4.03667e6	0.444684	0.222342	−	0.974969i	$-0.428630\pi$
$607$	4.03667e6	0.444684	0.222342	+	0.974969i	$0.428630\pi$
$608$	0	0
$609$	−1.90953e6	−0.208633
$610$	0	0
$611$	4.85014e6	0.525595
$612$	0	0
$613$	1.60521e7	1.72536	0.862681	−	0.505748i	$-0.168784\pi$
$613$	1.60521e7	1.72536	0.862681	+	0.505748i	$0.168784\pi$
$614$	0	0
$615$	3.52282e6	0.375580
$616$	0	0
$617$	1.34770e7	1.42522	0.712609	−	0.701561i	$-0.247513\pi$
$617$	1.34770e7	1.42522	0.712609	+	0.701561i	$0.247513\pi$
$618$	0	0
$619$	−1.73797e7	−1.82312	−0.911559	−	0.411170i	$-0.865120\pi$
$619$	−1.73797e7	−1.82312	−0.911559	+	0.411170i	$0.865120\pi$
$620$	0	0
$621$	−2.67543e6	−0.278397
$622$	0	0
$623$	−2.41178e6	−0.248953
$624$	0	0
$625$	−1.18572e7	−1.21417
$626$	0	0
$627$	779544.	0.0791903
$628$	0	0
$629$	1.77571e6	0.178956
$630$	0	0
$631$	1.46908e7	1.46883	0.734416	−	0.678700i	$-0.237456\pi$
$631$	1.46908e7	1.46883	0.734416	+	0.678700i	$0.237456\pi$
$632$	0	0
$633$	−1.58764e6	−0.157486
$634$	0	0
$635$	−1.97527e7	−1.94398
$636$	0	0
$637$	1.77194e6	0.173021
$638$	0	0
$639$	6.76949e6	0.655849
$640$	0	0
$641$	1.60166e7	1.53966	0.769830	−	0.638249i	$-0.220341\pi$
$641$	1.60166e7	1.53966	0.769830	+	0.638249i	$0.220341\pi$
$642$	0	0
$643$	8.48624e6	0.809446	0.404723	−	0.914439i	$-0.367368\pi$
$643$	8.48624e6	0.809446	0.404723	+	0.914439i	$0.367368\pi$
$644$	0	0
$645$	−4.55962e6	−0.431548
$646$	0	0
$647$	−1.85487e7	−1.74202	−0.871008	−	0.491269i	$-0.836533\pi$
$647$	−1.85487e7	−1.74202	−0.871008	+	0.491269i	$0.836533\pi$
$648$	0	0
$649$	2.24986e6	0.209673
$650$	0	0
$651$	−2.09916e6	−0.194130
$652$	0	0
$653$	−3.01271e6	−0.276486	−0.138243	−	0.990398i	$-0.544146\pi$
$653$	−3.01271e6	−0.276486	−0.138243	+	0.990398i	$0.544146\pi$
$654$	0	0
$655$	3.90477e6	0.355625
$656$	0	0
$657$	−3.42743e6	−0.309782
$658$	0	0
$659$	−6.06060e6	−0.543628	−0.271814	−	0.962350i	$-0.587624\pi$
$659$	−6.06060e6	−0.543628	−0.271814	+	0.962350i	$0.587624\pi$
$660$	0	0
$661$	1.42899e7	1.27211	0.636057	−	0.771642i	$-0.280564\pi$
$661$	1.42899e7	1.27211	0.636057	+	0.771642i	$0.280564\pi$
$662$	0	0
$663$	−5.63242e6	−0.497635
$664$	0	0
$665$	5.03014e6	0.441089
$666$	0	0
$667$	1.58911e7	1.38305
$668$	0	0
$669$	−1.12514e6	−0.0971948
$670$	0	0
$671$	1.43478e6	0.123021
$672$	0	0
$673$	−5.29680e6	−0.450792	−0.225396	−	0.974267i	$-0.572368\pi$
$673$	−5.29680e6	−0.450792	−0.225396	+	0.974267i	$0.572368\pi$
$674$	0	0
$675$	707859.	0.0597981
$676$	0	0
$677$	1.18550e7	0.994096	0.497048	−	0.867723i	$-0.334417\pi$
$677$	1.18550e7	0.994096	0.497048	+	0.867723i	$0.334417\pi$
$678$	0	0
$679$	7.82383e6	0.651246
$680$	0	0
$681$	−2.44894e6	−0.202353
$682$	0	0
$683$	−1.65625e7	−1.35854	−0.679272	−	0.733886i	$-0.737704\pi$
$683$	−1.65625e7	−1.35854	−0.679272	+	0.733886i	$0.737704\pi$
$684$	0	0
$685$	2.03035e7	1.65327
$686$	0	0
$687$	−2.93240e6	−0.237045
$688$	0	0
$689$	−5.82577e6	−0.467526
$690$	0	0
$691$	−4.69748e6	−0.374257	−0.187128	−	0.982335i	$-0.559918\pi$
$691$	−4.69748e6	−0.374257	−0.187128	+	0.982335i	$0.559918\pi$
$692$	0	0
$693$	−214326.	−0.0169528
$694$	0	0
$695$	463104.	0.0363678
$696$	0	0
$697$	5.18637e6	0.404372
$698$	0	0
$699$	−4.81214e6	−0.372516
$700$	0	0
$701$	−2.10890e7	−1.62092	−0.810458	−	0.585797i	$-0.800781\pi$
$701$	−2.10890e7	−1.62092	−0.810458	+	0.585797i	$0.800781\pi$
$702$	0	0
$703$	3.35878e6	0.256326
$704$	0	0
$705$	−3.78547e6	−0.286845
$706$	0	0
$707$	−3.28398e6	−0.247088
$708$	0	0
$709$	−1.68683e7	−1.26025	−0.630123	−	0.776495i	$-0.716996\pi$
$709$	−1.68683e7	−1.26025	−0.630123	+	0.776495i	$0.716996\pi$
$710$	0	0
$711$	41148.0	0.00305263
$712$	0	0
$713$	1.74692e7	1.28691
$714$	0	0
$715$	2.55053e6	0.186580
$716$	0	0
$717$	1.33661e7	0.970972
$718$	0	0
$719$	−1.19606e7	−0.862844	−0.431422	−	0.902150i	$-0.641988\pi$
$719$	−1.19606e7	−0.862844	−0.431422	+	0.902150i	$0.641988\pi$
$720$	0	0
$721$	8.12263e6	0.581914
$722$	0	0
$723$	1.05920e7	0.753583
$724$	0	0
$725$	−4.20443e6	−0.297072
$726$	0	0
$727$	−2.20722e6	−0.154885	−0.0774427	−	0.996997i	$-0.524675\pi$
$727$	−2.20722e6	−0.154885	−0.0774427	+	0.996997i	$0.524675\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−6.71277e6	−0.464631
$732$	0	0
$733$	−384494.	−0.0264320	−0.0132160	−	0.999913i	$-0.504207\pi$
$733$	−384494.	−0.0264320	−0.0132160	+	0.999913i	$0.504207\pi$
$734$	0	0
$735$	−1.38298e6	−0.0944270
$736$	0	0
$737$	2.25374e6	0.152840
$738$	0	0
$739$	8.04242e6	0.541721	0.270860	−	0.962619i	$-0.412692\pi$
$739$	8.04242e6	0.541721	0.270860	+	0.962619i	$0.412692\pi$
$740$	0	0
$741$	−1.06538e7	−0.712784
$742$	0	0
$743$	−1.85291e6	−0.123135	−0.0615675	−	0.998103i	$-0.519610\pi$
$743$	−1.85291e6	−0.123135	−0.0615675	+	0.998103i	$0.519610\pi$
$744$	0	0
$745$	−8.06771e6	−0.532549
$746$	0	0
$747$	−677484.	−0.0444219
$748$	0	0
$749$	5.05415e6	0.329188
$750$	0	0
$751$	1.19326e7	0.772034	0.386017	−	0.922492i	$-0.373851\pi$
$751$	1.19326e7	0.772034	0.386017	+	0.922492i	$0.373851\pi$
$752$	0	0
$753$	126720.	0.00814437
$754$	0	0
$755$	−1.28189e7	−0.818436
$756$	0	0
$757$	−5.55886e6	−0.352570	−0.176285	−	0.984339i	$-0.556408\pi$
$757$	−5.55886e6	−0.352570	−0.176285	+	0.984339i	$0.556408\pi$
$758$	0	0
$759$	1.78362e6	0.112382
$760$	0	0
$761$	2.71599e7	1.70007	0.850033	−	0.526730i	$-0.176582\pi$
$761$	2.71599e7	1.70007	0.850033	+	0.526730i	$0.176582\pi$
$762$	0	0
$763$	2.94461e6	0.183112
$764$	0	0
$765$	4.39603e6	0.271586
$766$	0	0
$767$	−3.07480e7	−1.88725
$768$	0	0
$769$	−8.75668e6	−0.533978	−0.266989	−	0.963700i	$-0.586029\pi$
$769$	−8.75668e6	−0.533978	−0.266989	+	0.963700i	$0.586029\pi$
$770$	0	0
$771$	−1.68166e7	−1.01883
$772$	0	0
$773$	−3.96856e6	−0.238883	−0.119441	−	0.992841i	$-0.538110\pi$
$773$	−3.96856e6	−0.238883	−0.119441	+	0.992841i	$0.538110\pi$
$774$	0	0
$775$	−4.62196e6	−0.276422
$776$	0	0
$777$	−923454.	−0.0548735
$778$	0	0
$779$	9.81006e6	0.579200
$780$	0	0
$781$	−4.51300e6	−0.264751
$782$	0	0
$783$	−3.15657e6	−0.183997
$784$	0	0
$785$	−3.26972e7	−1.89381
$786$	0	0
$787$	−2.11112e7	−1.21500	−0.607501	−	0.794319i	$-0.707828\pi$
$787$	−2.11112e7	−1.21500	−0.607501	+	0.794319i	$0.707828\pi$
$788$	0	0
$789$	7.22601e6	0.413244
$790$	0	0
$791$	−6.18605e6	−0.351538
$792$	0	0
$793$	−1.96087e7	−1.10730
$794$	0	0
$795$	4.54694e6	0.255154
$796$	0	0
$797$	−2.95085e7	−1.64552	−0.822758	−	0.568392i	$-0.807566\pi$
$797$	−2.95085e7	−1.64552	−0.822758	+	0.568392i	$0.807566\pi$
$798$	0	0
$799$	−5.57306e6	−0.308835
$800$	0	0
$801$	−3.98682e6	−0.219556
$802$	0	0
$803$	2.28496e6	0.125052
$804$	0	0
$805$	1.15091e7	0.625968
$806$	0	0
$807$	−1.77703e6	−0.0960531
$808$	0	0
$809$	13002.0	0.000698456 0	0.000349228	−	1.00000i	$-0.499889\pi$
$809$	13002.0	0.000698456 0	0.000349228		1.00000i	$0.499889\pi$
$810$	0	0
$811$	2.61790e7	1.39766	0.698829	−	0.715289i	$-0.253705\pi$
$811$	2.61790e7	1.39766	0.698829	+	0.715289i	$0.253705\pi$
$812$	0	0
$813$	−1.81735e7	−0.964301
$814$	0	0
$815$	−1.35578e6	−0.0714980
$816$	0	0
$817$	−1.26973e7	−0.665511
$818$	0	0
$819$	2.92912e6	0.152591
$820$	0	0
$821$	3.21209e7	1.66314	0.831572	−	0.555417i	$-0.187441\pi$
$821$	3.21209e7	1.66314	0.831572	+	0.555417i	$0.187441\pi$
$822$	0	0
$823$	−2.49758e7	−1.28535	−0.642674	−	0.766140i	$-0.722175\pi$
$823$	−2.49758e7	−1.28535	−0.642674	+	0.766140i	$0.722175\pi$
$824$	0	0
$825$	−471906.	−0.0241391
$826$	0	0
$827$	−3.09229e7	−1.57223	−0.786116	−	0.618079i	$-0.787911\pi$
$827$	−3.09229e7	−1.57223	−0.786116	+	0.618079i	$0.787911\pi$
$828$	0	0
$829$	−1.69047e7	−0.854319	−0.427160	−	0.904176i	$-0.640486\pi$
$829$	−1.69047e7	−0.854319	−0.427160	+	0.904176i	$0.640486\pi$
$830$	0	0
$831$	1.42194e7	0.714295
$832$	0	0
$833$	−2.03605e6	−0.101666
$834$	0	0
$835$	1.70995e7	0.848726
$836$	0	0
$837$	−3.47004e6	−0.171207
$838$	0	0
$839$	2.77783e7	1.36239	0.681194	−	0.732103i	$-0.261461\pi$
$839$	2.77783e7	1.36239	0.681194	+	0.732103i	$0.261461\pi$
$840$	0	0
$841$	−1.76225e6	−0.0859166
$842$	0	0
$843$	1.29953e7	0.629820
$844$	0	0
$845$	−1.10945e7	−0.534521
$846$	0	0
$847$	−7.74862e6	−0.371121
$848$	0	0
$849$	−1.51241e7	−0.720114
$850$	0	0
$851$	7.68498e6	0.363763
$852$	0	0
$853$	−1.77504e7	−0.835289	−0.417645	−	0.908611i	$-0.637144\pi$
$853$	−1.77504e7	−0.835289	−0.417645	+	0.908611i	$0.637144\pi$
$854$	0	0
$855$	8.31514e6	0.389004
$856$	0	0
$857$	1.50040e7	0.697838	0.348919	−	0.937153i	$-0.386549\pi$
$857$	1.50040e7	0.697838	0.348919	+	0.937153i	$0.386549\pi$
$858$	0	0
$859$	910972.	0.0421233	0.0210616	−	0.999778i	$-0.493295\pi$
$859$	910972.	0.0421233	0.0210616	+	0.999778i	$0.493295\pi$
$860$	0	0
$861$	−2.69716e6	−0.123993
$862$	0	0
$863$	1.48837e7	0.680275	0.340138	−	0.940376i	$-0.389526\pi$
$863$	1.48837e7	0.680275	0.340138	+	0.940376i	$0.389526\pi$
$864$	0	0
$865$	5.88467e6	0.267413
$866$	0	0
$867$	−6.30678e6	−0.284944
$868$	0	0
$869$	−27432.0	−0.00123228
$870$	0	0
$871$	−3.08012e7	−1.37569
$872$	0	0
$873$	1.29333e7	0.574345
$874$	0	0
$875$	6.75494e6	0.298265
$876$	0	0
$877$	−2.39951e7	−1.05348	−0.526738	−	0.850028i	$-0.676585\pi$
$877$	−2.39951e7	−1.05348	−0.526738	+	0.850028i	$0.676585\pi$
$878$	0	0
$879$	−2.07983e7	−0.907938
$880$	0	0
$881$	7.85879e6	0.341127	0.170563	−	0.985347i	$-0.445441\pi$
$881$	7.85879e6	0.341127	0.170563	+	0.985347i	$0.445441\pi$
$882$	0	0
$883$	−1.74586e7	−0.753541	−0.376771	−	0.926307i	$-0.622966\pi$
$883$	−1.74586e7	−0.753541	−0.376771	+	0.926307i	$0.622966\pi$
$884$	0	0
$885$	2.39985e7	1.02997
$886$	0	0
$887$	−700092.	−0.0298776	−0.0149388	−	0.999888i	$-0.504755\pi$
$887$	−700092.	−0.0298776	−0.0149388	+	0.999888i	$0.504755\pi$
$888$	0	0
$889$	1.51232e7	0.641783
$890$	0	0
$891$	−354294.	−0.0149510
$892$	0	0
$893$	−1.05415e7	−0.442357
$894$	0	0
$895$	2.57809e7	1.07582
$896$	0	0
$897$	−2.43761e7	−1.01154
$898$	0	0
$899$	2.06108e7	0.850542
$900$	0	0
$901$	6.69411e6	0.274714
$902$	0	0
$903$	3.49096e6	0.142471
$904$	0	0
$905$	5.09582e7	2.06820
$906$	0	0
$907$	3.72979e7	1.50545	0.752724	−	0.658336i	$-0.228739\pi$
$907$	3.72979e7	1.50545	0.752724	+	0.658336i	$0.228739\pi$
$908$	0	0
$909$	−5.42862e6	−0.217911
$910$	0	0
$911$	−2.99873e7	−1.19713	−0.598564	−	0.801075i	$-0.704262\pi$
$911$	−2.99873e7	−1.19713	−0.598564	+	0.801075i	$0.704262\pi$
$912$	0	0
$913$	451656.	0.0179321
$914$	0	0
$915$	1.53043e7	0.604312
$916$	0	0
$917$	−2.98959e6	−0.117405
$918$	0	0
$919$	−2.78316e7	−1.08705	−0.543525	−	0.839393i	$-0.682911\pi$
$919$	−2.78316e7	−1.08705	−0.543525	+	0.839393i	$0.682911\pi$
$920$	0	0
$921$	−7.14568e6	−0.277584
$922$	0	0
$923$	6.16776e7	2.38300
$924$	0	0
$925$	−2.03327e6	−0.0781343
$926$	0	0
$927$	1.34272e7	0.513200
$928$	0	0
$929$	1.88191e7	0.715418	0.357709	−	0.933833i	$-0.383558\pi$
$929$	1.88191e7	0.715418	0.357709	+	0.933833i	$0.383558\pi$
$930$	0	0
$931$	−3.85120e6	−0.145620
$932$	0	0
$933$	1.06539e7	0.400685
$934$	0	0
$935$	−2.93069e6	−0.109633
$936$	0	0
$937$	5.39613e6	0.200786	0.100393	−	0.994948i	$-0.467990\pi$
$937$	5.39613e6	0.200786	0.100393	+	0.994948i	$0.467990\pi$
$938$	0	0
$939$	8.95473e6	0.331427
$940$	0	0
$941$	3.99942e7	1.47239	0.736194	−	0.676770i	$-0.236621\pi$
$941$	3.99942e7	1.47239	0.736194	+	0.676770i	$0.236621\pi$
$942$	0	0
$943$	2.24457e7	0.821967
$944$	0	0
$945$	−2.28614e6	−0.0832768
$946$	0	0
$947$	3.09314e7	1.12079	0.560395	−	0.828225i	$-0.310649\pi$
$947$	3.09314e7	1.12079	0.560395	+	0.828225i	$0.310649\pi$
$948$	0	0
$949$	−3.12277e7	−1.12558
$950$	0	0
$951$	1.66157e7	0.595756
$952$	0	0
$953$	−1.55848e7	−0.555865	−0.277933	−	0.960601i	$-0.589649\pi$
$953$	−1.55848e7	−0.555865	−0.277933	+	0.960601i	$0.589649\pi$
$954$	0	0
$955$	−3.03958e7	−1.07846
$956$	0	0
$957$	2.10438e6	0.0742753
$958$	0	0
$959$	−1.55449e7	−0.545808
$960$	0	0
$961$	−5.97155e6	−0.208583
$962$	0	0
$963$	8.35483e6	0.290316
$964$	0	0
$965$	−4.58254e7	−1.58412
$966$	0	0
$967$	−2.60131e7	−0.894593	−0.447297	−	0.894386i	$-0.647613\pi$
$967$	−2.60131e7	−0.894593	−0.447297	+	0.894386i	$0.647613\pi$
$968$	0	0
$969$	1.22417e7	0.418826
$970$	0	0
$971$	−8.84316e6	−0.300995	−0.150497	−	0.988610i	$-0.548088\pi$
$971$	−8.84316e6	−0.300995	−0.150497	+	0.988610i	$0.548088\pi$
$972$	0	0
$973$	−354564.	−0.0120064
$974$	0	0
$975$	6.44938e6	0.217273
$976$	0	0
$977$	−2.70010e7	−0.904988	−0.452494	−	0.891768i	$-0.649466\pi$
$977$	−2.70010e7	−0.904988	−0.452494	+	0.891768i	$0.649466\pi$
$978$	0	0
$979$	2.65788e6	0.0886296
$980$	0	0
$981$	4.86761e6	0.161489
$982$	0	0
$983$	−1.69892e7	−0.560775	−0.280387	−	0.959887i	$-0.590463\pi$
$983$	−1.69892e7	−0.560775	−0.280387	+	0.959887i	$0.590463\pi$
$984$	0	0
$985$	1.41961e7	0.466207
$986$	0	0
$987$	2.89825e6	0.0946985
$988$	0	0
$989$	−2.90517e7	−0.944455
$990$	0	0
$991$	3.77922e7	1.22241	0.611207	−	0.791470i	$-0.290684\pi$
$991$	3.77922e7	1.22241	0.611207	+	0.791470i	$0.290684\pi$
$992$	0	0
$993$	−1.40221e7	−0.451274
$994$	0	0
$995$	2.13514e7	0.683706
$996$	0	0
$997$	−5.16921e7	−1.64697	−0.823487	−	0.567336i	$-0.807974\pi$
$997$	−5.16921e7	−1.64697	−0.823487	+	0.567336i	$0.807974\pi$
$998$	0	0
$999$	−1.52653e6	−0.0483939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.6.a.d.1.1	✓	1
3.2	odd	2		504.6.a.h.1.1		1
4.3	odd	2		336.6.a.c.1.1		1
12.11	even	2		1008.6.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.d.1.1	✓	1	1.1	even	1	trivial
336.6.a.c.1.1		1	4.3	odd	2
504.6.a.h.1.1		1	3.2	odd	2
1008.6.a.z.1.1		1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	168.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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