LMFDB - Embedded newform 324.6.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,6,Mod(1,324)] 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("324.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0] 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:	$51.9643576194$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{91})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 47x^{2} + 484 $ x^4 - 47*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.63572$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+83.2171 q^{5} +143.136 q^{7} -431.674 q^{11} -1036.95 q^{13} -1659.30 q^{17} -2024.77 q^{19} -2345.43 q^{23} +3800.09 q^{25} -3722.73 q^{29} -441.182 q^{31} +11911.4 q^{35} +7655.95 q^{37} -1349.74 q^{41} -19394.5 q^{43} +14675.8 q^{47} +3680.99 q^{49} +20190.2 q^{53} -35922.7 q^{55} +16468.6 q^{59} -52354.7 q^{61} -86292.3 q^{65} -31879.9 q^{67} +47866.9 q^{71} +5456.25 q^{73} -61788.2 q^{77} +91569.1 q^{79} -14016.9 q^{83} -138082. q^{85} +104964. q^{89} -148426. q^{91} -168496. q^{95} -90500.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 176 q^{7} - 1372 q^{13} - 2944 q^{19} + 3304 q^{25} - 4144 q^{31} + 1676 q^{37} - 38320 q^{43} - 20172 q^{49} - 80640 q^{55} - 120196 q^{61} - 107296 q^{67} - 92380 q^{73} + 5024 q^{79} - 329868 q^{85}+ \cdots - 331864 q^{97}+O(q^{100}) $ 4 * q + 176 * q^7 - 1372 * q^13 - 2944 * q^19 + 3304 * q^25 - 4144 * q^31 + 1676 * q^37 - 38320 * q^43 - 20172 * q^49 - 80640 * q^55 - 120196 * q^61 - 107296 * q^67 - 92380 * q^73 + 5024 * q^79 - 329868 * q^85 - 335552 * q^91 - 331864 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	83.2171	1.48863	0.744316	−	0.667827i	$-0.232775\pi$
$5$	83.2171	1.48863	0.744316	+	0.667827i	$0.232775\pi$
$6$	0	0
$7$	143.136	1.10409	0.552045	−	0.833814i	$-0.313848\pi$
$7$	143.136	1.10409	0.552045	+	0.833814i	$0.313848\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−431.674	−1.07566	−0.537829	−	0.843054i	$-0.680755\pi$
$11$	−431.674	−1.07566	−0.537829	+	0.843054i	$0.680755\pi$
$12$	0	0
$13$	−1036.95	−1.70177	−0.850885	−	0.525351i	$-0.823934\pi$
$13$	−1036.95	−1.70177	−0.850885	+	0.525351i	$0.823934\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1659.30	−1.39253	−0.696263	−	0.717786i	$-0.745155\pi$
$17$	−1659.30	−1.39253	−0.696263	+	0.717786i	$0.745155\pi$
$18$	0	0
$19$	−2024.77	−1.28674	−0.643372	−	0.765554i	$-0.722465\pi$
$19$	−2024.77	−1.28674	−0.643372	+	0.765554i	$0.722465\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2345.43	−0.924492	−0.462246	−	0.886752i	$-0.652956\pi$
$23$	−2345.43	−0.924492	−0.462246	+	0.886752i	$0.652956\pi$
$24$	0	0
$25$	3800.09	1.21603
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3722.73	−0.821989	−0.410995	−	0.911638i	$-0.634819\pi$
$29$	−3722.73	−0.821989	−0.410995	+	0.911638i	$0.634819\pi$
$30$	0	0
$31$	−441.182	−0.0824544	−0.0412272	−	0.999150i	$-0.513127\pi$
$31$	−441.182	−0.0824544	−0.0412272	+	0.999150i	$0.513127\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11911.4	1.64359
$36$	0	0
$37$	7655.95	0.919379	0.459690	−	0.888080i	$-0.347961\pi$
$37$	7655.95	0.919379	0.459690	+	0.888080i	$0.347961\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1349.74	−0.125398	−0.0626990	−	0.998032i	$-0.519971\pi$
$41$	−1349.74	−0.125398	−0.0626990	+	0.998032i	$0.519971\pi$
$42$	0	0
$43$	−19394.5	−1.59958	−0.799792	−	0.600277i	$-0.795057\pi$
$43$	−19394.5	−1.59958	−0.799792	+	0.600277i	$0.795057\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	14675.8	0.969075	0.484537	−	0.874770i	$-0.338988\pi$
$47$	14675.8	0.969075	0.484537	+	0.874770i	$0.338988\pi$
$48$	0	0
$49$	3680.99	0.219015
$50$	0	0
$51$	0	0
$52$	0	0
$53$	20190.2	0.987304	0.493652	−	0.869659i	$-0.335662\pi$
$53$	20190.2	0.987304	0.493652	+	0.869659i	$0.335662\pi$
$54$	0	0
$55$	−35922.7	−1.60126
$56$	0	0
$57$	0	0
$58$	0	0
$59$	16468.6	0.615925	0.307962	−	0.951399i	$-0.400353\pi$
$59$	16468.6	0.615925	0.307962	+	0.951399i	$0.400353\pi$
$60$	0	0
$61$	−52354.7	−1.80149	−0.900743	−	0.434353i	$-0.856977\pi$
$61$	−52354.7	−1.80149	−0.900743	+	0.434353i	$0.856977\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−86292.3	−2.53331
$66$	0	0
$67$	−31879.9	−0.867622	−0.433811	−	0.901004i	$-0.642831\pi$
$67$	−31879.9	−0.867622	−0.433811	+	0.901004i	$0.642831\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	47866.9	1.12691	0.563455	−	0.826147i	$-0.309472\pi$
$71$	47866.9	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$72$	0	0
$73$	5456.25	0.119836	0.0599179	−	0.998203i	$-0.480916\pi$
$73$	5456.25	0.119836	0.0599179	+	0.998203i	$0.480916\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−61788.2	−1.18762
$78$	0	0
$79$	91569.1	1.65075	0.825375	−	0.564585i	$-0.190964\pi$
$79$	91569.1	1.65075	0.825375	+	0.564585i	$0.190964\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14016.9	−0.223335	−0.111667	−	0.993746i	$-0.535619\pi$
$83$	−14016.9	−0.223335	−0.111667	+	0.993746i	$0.535619\pi$
$84$	0	0
$85$	−138082.	−2.07296
$86$	0	0
$87$	0	0
$88$	0	0
$89$	104964.	1.40465	0.702323	−	0.711859i	$-0.252146\pi$
$89$	104964.	1.40465	0.702323	+	0.711859i	$0.252146\pi$
$90$	0	0
$91$	−148426.	−1.87891
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−168496.	−1.91549
$96$	0	0
$97$	−90500.4	−0.976610	−0.488305	−	0.872673i	$-0.662385\pi$
$97$	−90500.4	−0.976610	−0.488305	+	0.872673i	$0.662385\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−109179.	−1.06496	−0.532481	−	0.846442i	$-0.678740\pi$
$101$	−109179.	−1.06496	−0.532481	+	0.846442i	$0.678740\pi$
$102$	0	0
$103$	17554.9	0.163045	0.0815223	−	0.996672i	$-0.474022\pi$
$103$	17554.9	0.163045	0.0815223	+	0.996672i	$0.474022\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	208161.	1.75768	0.878838	−	0.477120i	$-0.158319\pi$
$107$	208161.	1.75768	0.878838	+	0.477120i	$0.158319\pi$
$108$	0	0
$109$	10653.4	0.0858862	0.0429431	−	0.999078i	$-0.486327\pi$
$109$	10653.4	0.0858862	0.0429431	+	0.999078i	$0.486327\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	52969.1	0.390235	0.195117	−	0.980780i	$-0.437491\pi$
$113$	52969.1	0.390235	0.195117	+	0.980780i	$0.437491\pi$
$114$	0	0
$115$	−195180.	−1.37623
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−237507.	−1.53748
$120$	0	0
$121$	25291.5	0.157040
$122$	0	0
$123$	0	0
$124$	0	0
$125$	56178.9	0.321587
$126$	0	0
$127$	−24494.8	−0.134761	−0.0673805	−	0.997727i	$-0.521464\pi$
$127$	−24494.8	−0.134761	−0.0673805	+	0.997727i	$0.521464\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	345250.	1.75774	0.878872	−	0.477058i	$-0.158297\pi$
$131$	345250.	1.75774	0.878872	+	0.477058i	$0.158297\pi$
$132$	0	0
$133$	−289818.	−1.42068
$134$	0	0
$135$	0	0
$136$	0	0
$137$	285552.	1.29982	0.649911	−	0.760010i	$-0.274806\pi$
$137$	285552.	1.29982	0.649911	+	0.760010i	$0.274806\pi$
$138$	0	0
$139$	−172780.	−0.758502	−0.379251	−	0.925294i	$-0.623818\pi$
$139$	−172780.	−0.758502	−0.379251	+	0.925294i	$0.623818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	447626.	1.83052
$144$	0	0
$145$	−309795.	−1.22364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−277228.	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$149$	−277228.	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$150$	0	0
$151$	−101615.	−0.362674	−0.181337	−	0.983421i	$-0.558043\pi$
$151$	−101615.	−0.362674	−0.181337	+	0.983421i	$0.558043\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−36713.9	−0.122744
$156$	0	0
$157$	−175821.	−0.569275	−0.284638	−	0.958635i	$-0.591873\pi$
$157$	−175821.	−0.569275	−0.284638	+	0.958635i	$0.591873\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−335716.	−1.02072
$162$	0	0
$163$	533498.	1.57277	0.786383	−	0.617740i	$-0.211951\pi$
$163$	533498.	1.57277	0.786383	+	0.617740i	$0.211951\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−637208.	−1.76803	−0.884016	−	0.467457i	$-0.845170\pi$
$167$	−637208.	−1.76803	−0.884016	+	0.467457i	$0.845170\pi$
$168$	0	0
$169$	703980.	1.89602
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−507234.	−1.28853	−0.644263	−	0.764804i	$-0.722836\pi$
$173$	−507234.	−1.28853	−0.644263	+	0.764804i	$0.722836\pi$
$174$	0	0
$175$	543930.	1.34260
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−576195.	−1.34412	−0.672058	−	0.740498i	$-0.734590\pi$
$179$	−576195.	−1.34412	−0.672058	+	0.740498i	$0.734590\pi$
$180$	0	0
$181$	−474013.	−1.07546	−0.537730	−	0.843117i	$-0.680718\pi$
$181$	−474013.	−1.07546	−0.537730	+	0.843117i	$0.680718\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	637106.	1.36862
$186$	0	0
$187$	716278.	1.49788
$188$	0	0
$189$	0	0
$190$	0	0
$191$	239287.	0.474608	0.237304	−	0.971435i	$-0.423736\pi$
$191$	239287.	0.474608	0.237304	+	0.971435i	$0.423736\pi$
$192$	0	0
$193$	−139272.	−0.269134	−0.134567	−	0.990904i	$-0.542964\pi$
$193$	−139272.	−0.269134	−0.134567	+	0.990904i	$0.542964\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	350045.	0.642626	0.321313	−	0.946973i	$-0.395876\pi$
$197$	350045.	0.642626	0.321313	+	0.946973i	$0.395876\pi$
$198$	0	0
$199$	494321.	0.884863	0.442432	−	0.896802i	$-0.354116\pi$
$199$	494321.	0.884863	0.442432	+	0.896802i	$0.354116\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−532857.	−0.907550
$204$	0	0
$205$	−112322.	−0.186672
$206$	0	0
$207$	0	0
$208$	0	0
$209$	874041.	1.38410
$210$	0	0
$211$	−504307.	−0.779809	−0.389905	−	0.920855i	$-0.627492\pi$
$211$	−504307.	−0.779809	−0.389905	+	0.920855i	$0.627492\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.61395e6	−2.38119
$216$	0	0
$217$	−63149.2	−0.0910371
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.72062e6	2.36976
$222$	0	0
$223$	−607535.	−0.818105	−0.409052	−	0.912511i	$-0.634141\pi$
$223$	−607535.	−0.818105	−0.409052	+	0.912511i	$0.634141\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	924635.	1.19098	0.595491	−	0.803362i	$-0.296957\pi$
$227$	924635.	1.19098	0.595491	+	0.803362i	$0.296957\pi$
$228$	0	0
$229$	804772.	1.01411	0.507054	−	0.861914i	$-0.330735\pi$
$229$	804772.	1.01411	0.507054	+	0.861914i	$0.330735\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.16946e6	−1.41122	−0.705611	−	0.708599i	$-0.749327\pi$
$233$	−1.16946e6	−1.41122	−0.705611	+	0.708599i	$0.749327\pi$
$234$	0	0
$235$	1.22128e6	1.44260
$236$	0	0
$237$	0	0
$238$	0	0
$239$	311300.	0.352520	0.176260	−	0.984344i	$-0.443600\pi$
$239$	311300.	0.352520	0.176260	+	0.984344i	$0.443600\pi$
$240$	0	0
$241$	−384343.	−0.426262	−0.213131	−	0.977024i	$-0.568366\pi$
$241$	−384343.	−0.426262	−0.213131	+	0.977024i	$0.568366\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	306322.	0.326034
$246$	0	0
$247$	2.09959e6	2.18974
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.01398e6	1.01589	0.507944	−	0.861390i	$-0.330406\pi$
$251$	1.01398e6	1.01589	0.507944	+	0.861390i	$0.330406\pi$
$252$	0	0
$253$	1.01246e6	0.994437
$254$	0	0
$255$	0	0
$256$	0	0
$257$	814731.	0.769452	0.384726	−	0.923031i	$-0.374296\pi$
$257$	814731.	0.769452	0.384726	+	0.923031i	$0.374296\pi$
$258$	0	0
$259$	1.09584e6	1.01508
$260$	0	0
$261$	0	0
$262$	0	0
$263$	611094.	0.544777	0.272388	−	0.962187i	$-0.412186\pi$
$263$	611094.	0.544777	0.272388	+	0.962187i	$0.412186\pi$
$264$	0	0
$265$	1.68017e6	1.46973
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.21595e6	−1.02456	−0.512279	−	0.858819i	$-0.671198\pi$
$269$	−1.21595e6	−1.02456	−0.512279	+	0.858819i	$0.671198\pi$
$270$	0	0
$271$	−1.06241e6	−0.878759	−0.439379	−	0.898302i	$-0.644802\pi$
$271$	−1.06241e6	−0.878759	−0.439379	+	0.898302i	$0.644802\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.64040e6	−1.30803
$276$	0	0
$277$	1.05276e6	0.824386	0.412193	−	0.911097i	$-0.364763\pi$
$277$	1.05276e6	0.824386	0.412193	+	0.911097i	$0.364763\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−52689.9	−0.0398072	−0.0199036	−	0.999802i	$-0.506336\pi$
$281$	−52689.9	−0.0398072	−0.0199036	+	0.999802i	$0.506336\pi$
$282$	0	0
$283$	−1.59276e6	−1.18218	−0.591091	−	0.806605i	$-0.701302\pi$
$283$	−1.59276e6	−1.18218	−0.591091	+	0.806605i	$0.701302\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−193197.	−0.138451
$288$	0	0
$289$	1.33343e6	0.939131
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−365026.	−0.248402	−0.124201	−	0.992257i	$-0.539637\pi$
$293$	−365026.	−0.248402	−0.124201	+	0.992257i	$0.539637\pi$
$294$	0	0
$295$	1.37047e6	0.916886
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.43210e6	1.57327
$300$	0	0
$301$	−2.77606e6	−1.76609
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.35680e6	−2.68175
$306$	0	0
$307$	1.58303e6	0.958613	0.479307	−	0.877648i	$-0.340888\pi$
$307$	1.58303e6	0.958613	0.479307	+	0.877648i	$0.340888\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−603966.	−0.354088	−0.177044	−	0.984203i	$-0.556654\pi$
$311$	−603966.	−0.354088	−0.177044	+	0.984203i	$0.556654\pi$
$312$	0	0
$313$	−212876.	−0.122819	−0.0614097	−	0.998113i	$-0.519560\pi$
$313$	−212876.	−0.122819	−0.0614097	+	0.998113i	$0.519560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.05080e6	−1.14624	−0.573120	−	0.819471i	$-0.694267\pi$
$317$	−2.05080e6	−1.14624	−0.573120	+	0.819471i	$0.694267\pi$
$318$	0	0
$319$	1.60700e6	0.884179
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.35971e6	1.79182
$324$	0	0
$325$	−3.94052e6	−2.06940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.10064e6	1.06995
$330$	0	0
$331$	2.88851e6	1.44912	0.724560	−	0.689211i	$-0.242043\pi$
$331$	2.88851e6	1.44912	0.724560	+	0.689211i	$0.242043\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.65296e6	−1.29157
$336$	0	0
$337$	−2.43591e6	−1.16838	−0.584192	−	0.811615i	$-0.698589\pi$
$337$	−2.43591e6	−1.16838	−0.584192	+	0.811615i	$0.698589\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	190447.	0.0886928
$342$	0	0
$343$	−1.87881e6	−0.862278
$344$	0	0
$345$	0	0
$346$	0	0
$347$	377025.	0.168092	0.0840459	−	0.996462i	$-0.473216\pi$
$347$	377025.	0.168092	0.0840459	+	0.996462i	$0.473216\pi$
$348$	0	0
$349$	−4.41483e6	−1.94022	−0.970109	−	0.242671i	$-0.921977\pi$
$349$	−4.41483e6	−1.94022	−0.970109	+	0.242671i	$0.921977\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.16286e6	−1.35096	−0.675481	−	0.737378i	$-0.736064\pi$
$353$	−3.16286e6	−1.35096	−0.675481	+	0.737378i	$0.736064\pi$
$354$	0	0
$355$	3.98334e6	1.67756
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−140929.	−0.0577116	−0.0288558	−	0.999584i	$-0.509186\pi$
$359$	−140929.	−0.0577116	−0.0288558	+	0.999584i	$0.509186\pi$
$360$	0	0
$361$	1.62360e6	0.655709
$362$	0	0
$363$	0	0
$364$	0	0
$365$	454053.	0.178392
$366$	0	0
$367$	−2.21624e6	−0.858919	−0.429460	−	0.903086i	$-0.641296\pi$
$367$	−2.21624e6	−0.858919	−0.429460	+	0.903086i	$0.641296\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.88995e6	1.09007
$372$	0	0
$373$	−3.86169e6	−1.43716	−0.718581	−	0.695444i	$-0.755208\pi$
$373$	−3.86169e6	−1.43716	−0.718581	+	0.695444i	$0.755208\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.86030e6	1.39884
$378$	0	0
$379$	1.70193e6	0.608616	0.304308	−	0.952574i	$-0.401575\pi$
$379$	1.70193e6	0.608616	0.304308	+	0.952574i	$0.401575\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.20471e6	−1.11633	−0.558164	−	0.829731i	$-0.688494\pi$
$383$	−3.20471e6	−1.11633	−0.558164	+	0.829731i	$0.688494\pi$
$384$	0	0
$385$	−5.14184e6	−1.76794
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.80401e6	1.60964	0.804821	−	0.593517i	$-0.202261\pi$
$389$	4.80401e6	1.60964	0.804821	+	0.593517i	$0.202261\pi$
$390$	0	0
$391$	3.89178e6	1.28738
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.62012e6	2.45736
$396$	0	0
$397$	−1.31595e6	−0.419047	−0.209524	−	0.977804i	$-0.567191\pi$
$397$	−1.31595e6	−0.419047	−0.209524	+	0.977804i	$0.567191\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.47348e6	−1.07871	−0.539354	−	0.842079i	$-0.681331\pi$
$401$	−3.47348e6	−1.07871	−0.539354	+	0.842079i	$0.681331\pi$
$402$	0	0
$403$	457486.	0.140319
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.30487e6	−0.988938
$408$	0	0
$409$	−3.66596e6	−1.08363	−0.541813	−	0.840499i	$-0.682262\pi$
$409$	−3.66596e6	−1.08363	−0.541813	+	0.840499i	$0.682262\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.35726e6	0.680036
$414$	0	0
$415$	−1.16644e6	−0.332464
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.32948e6	0.369953	0.184977	−	0.982743i	$-0.440779\pi$
$419$	1.32948e6	0.369953	0.184977	+	0.982743i	$0.440779\pi$
$420$	0	0
$421$	−2.40202e6	−0.660498	−0.330249	−	0.943894i	$-0.607133\pi$
$421$	−2.40202e6	−0.660498	−0.330249	+	0.943894i	$0.607133\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.30550e6	−1.69335
$426$	0	0
$427$	−7.49385e6	−1.98900
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.55904e6	1.44147	0.720737	−	0.693208i	$-0.243803\pi$
$431$	5.55904e6	1.44147	0.720737	+	0.693208i	$0.243803\pi$
$432$	0	0
$433$	−4.82056e6	−1.23560	−0.617800	−	0.786335i	$-0.711976\pi$
$433$	−4.82056e6	−1.23560	−0.617800	+	0.786335i	$0.711976\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.74896e6	1.18958
$438$	0	0
$439$	5.68814e6	1.40867	0.704334	−	0.709868i	$-0.251246\pi$
$439$	5.68814e6	1.40867	0.704334	+	0.709868i	$0.251246\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.06512e6	−0.499960	−0.249980	−	0.968251i	$-0.580424\pi$
$443$	−2.06512e6	−0.499960	−0.249980	+	0.968251i	$0.580424\pi$
$444$	0	0
$445$	8.73483e6	2.09100
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.12412e6	−0.263146	−0.131573	−	0.991306i	$-0.542003\pi$
$449$	−1.12412e6	−0.263146	−0.131573	+	0.991306i	$0.542003\pi$
$450$	0	0
$451$	582648.	0.134885
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.23516e7	−2.79701
$456$	0	0
$457$	−83388.1	−0.0186773	−0.00933863	−	0.999956i	$-0.502973\pi$
$457$	−83388.1	−0.0186773	−0.00933863	+	0.999956i	$0.502973\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.17734e6	−1.79209	−0.896045	−	0.443963i	$-0.853572\pi$
$461$	−8.17734e6	−1.79209	−0.896045	+	0.443963i	$0.853572\pi$
$462$	0	0
$463$	−6.44616e6	−1.39749	−0.698745	−	0.715371i	$-0.746258\pi$
$463$	−6.44616e6	−1.39749	−0.698745	+	0.715371i	$0.746258\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.62654e6	−0.981666	−0.490833	−	0.871254i	$-0.663308\pi$
$467$	−4.62654e6	−0.981666	−0.490833	+	0.871254i	$0.663308\pi$
$468$	0	0
$469$	−4.56318e6	−0.957933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.37210e6	1.72061
$474$	0	0
$475$	−7.69431e6	−1.56472
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.78505e6	−0.554618	−0.277309	−	0.960781i	$-0.589443\pi$
$479$	−2.78505e6	−0.554618	−0.277309	+	0.960781i	$0.589443\pi$
$480$	0	0
$481$	−7.93886e6	−1.56457
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.53118e6	−1.45381
$486$	0	0
$487$	−437677.	−0.0836241	−0.0418120	−	0.999125i	$-0.513313\pi$
$487$	−437677.	−0.0836241	−0.0418120	+	0.999125i	$0.513313\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.94924e6	−1.86246	−0.931228	−	0.364437i	$-0.881261\pi$
$491$	−9.94924e6	−1.86246	−0.931228	+	0.364437i	$0.881261\pi$
$492$	0	0
$493$	6.17713e6	1.14464
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.85149e6	1.24421
$498$	0	0
$499$	−2.59137e6	−0.465885	−0.232942	−	0.972491i	$-0.574835\pi$
$499$	−2.59137e6	−0.465885	−0.232942	+	0.972491i	$0.574835\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.37988e6	1.47679	0.738393	−	0.674370i	$-0.235585\pi$
$503$	8.37988e6	1.47679	0.738393	+	0.674370i	$0.235585\pi$
$504$	0	0
$505$	−9.08552e6	−1.58534
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.72852e6	1.15113	0.575566	−	0.817755i	$-0.304782\pi$
$509$	6.72852e6	1.15113	0.575566	+	0.817755i	$0.304782\pi$
$510$	0	0
$511$	780987.	0.132310
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.46087e6	0.242714
$516$	0	0
$517$	−6.33517e6	−1.04239
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.01162e6	1.45448	0.727241	−	0.686382i	$-0.240802\pi$
$521$	9.01162e6	1.45448	0.727241	+	0.686382i	$0.240802\pi$
$522$	0	0
$523$	5.74204e6	0.917936	0.458968	−	0.888453i	$-0.348219\pi$
$523$	5.74204e6	0.917936	0.458968	+	0.888453i	$0.348219\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	732055.	0.114820
$528$	0	0
$529$	−935293.	−0.145314
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.39962e6	0.213399
$534$	0	0
$535$	1.73225e7	2.61654
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.58899e6	−0.235586
$540$	0	0
$541$	8.40455e6	1.23459	0.617293	−	0.786733i	$-0.288229\pi$
$541$	8.40455e6	1.23459	0.617293	+	0.786733i	$0.288229\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	886548.	0.127853
$546$	0	0
$547$	−3.27309e6	−0.467724	−0.233862	−	0.972270i	$-0.575136\pi$
$547$	−3.27309e6	−0.467724	−0.233862	+	0.972270i	$0.575136\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.53767e6	1.05769
$552$	0	0
$553$	1.31069e7	1.82258
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.71812e6	−0.371219	−0.185610	−	0.982624i	$-0.559426\pi$
$557$	−2.71812e6	−0.371219	−0.185610	+	0.982624i	$0.559426\pi$
$558$	0	0
$559$	2.01112e7	2.72213
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.84509e6	0.511252	0.255626	−	0.966776i	$-0.417718\pi$
$563$	3.84509e6	0.511252	0.255626	+	0.966776i	$0.417718\pi$
$564$	0	0
$565$	4.40793e6	0.580917
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.68747e6	−0.347986	−0.173993	−	0.984747i	$-0.555667\pi$
$569$	−2.68747e6	−0.347986	−0.173993	+	0.984747i	$0.555667\pi$
$570$	0	0
$571$	2.24220e6	0.287796	0.143898	−	0.989593i	$-0.454036\pi$
$571$	2.24220e6	0.287796	0.143898	+	0.989593i	$0.454036\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.91285e6	−1.12421
$576$	0	0
$577$	−484542.	−0.0605887	−0.0302944	−	0.999541i	$-0.509644\pi$
$577$	−484542.	−0.0605887	−0.0302944	+	0.999541i	$0.509644\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.00633e6	−0.246582
$582$	0	0
$583$	−8.71559e6	−1.06200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.37503e6	0.883422	0.441711	−	0.897157i	$-0.354372\pi$
$587$	7.37503e6	0.883422	0.441711	+	0.897157i	$0.354372\pi$
$588$	0	0
$589$	893294.	0.106098
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−43672.8	−0.00510005	−0.00255003	−	0.999997i	$-0.500812\pi$
$593$	−43672.8	−0.00510005	−0.00255003	+	0.999997i	$0.500812\pi$
$594$	0	0
$595$	−1.97646e7	−2.28874
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.11973e6	0.469140	0.234570	−	0.972099i	$-0.424632\pi$
$599$	4.11973e6	0.469140	0.234570	+	0.972099i	$0.424632\pi$
$600$	0	0
$601$	−1.78283e6	−0.201337	−0.100669	−	0.994920i	$-0.532098\pi$
$601$	−1.78283e6	−0.201337	−0.100669	+	0.994920i	$0.532098\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.10468e6	0.233775
$606$	0	0
$607$	3.36543e6	0.370739	0.185370	−	0.982669i	$-0.440652\pi$
$607$	3.36543e6	0.370739	0.185370	+	0.982669i	$0.440652\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.52181e7	−1.64914
$612$	0	0
$613$	2.13132e6	0.229085	0.114543	−	0.993418i	$-0.463460\pi$
$613$	2.13132e6	0.229085	0.114543	+	0.993418i	$0.463460\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.43801e6	0.469326	0.234663	−	0.972077i	$-0.424601\pi$
$617$	4.43801e6	0.469326	0.234663	+	0.972077i	$0.424601\pi$
$618$	0	0
$619$	1.54196e7	1.61751	0.808753	−	0.588148i	$-0.200143\pi$
$619$	1.54196e7	1.61751	0.808753	+	0.588148i	$0.200143\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.50242e7	1.55086
$624$	0	0
$625$	−7.20023e6	−0.737304
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.27035e7	−1.28026
$630$	0	0
$631$	1.30923e7	1.30901	0.654505	−	0.756058i	$-0.272877\pi$
$631$	1.30923e7	1.30901	0.654505	+	0.756058i	$0.272877\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.03839e6	−0.200610
$636$	0	0
$637$	−3.81702e6	−0.372714
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.35657e6	0.514922	0.257461	−	0.966289i	$-0.417114\pi$
$641$	5.35657e6	0.514922	0.257461	+	0.966289i	$0.417114\pi$
$642$	0	0
$643$	−2.26379e6	−0.215928	−0.107964	−	0.994155i	$-0.534433\pi$
$643$	−2.26379e6	−0.215928	−0.107964	+	0.994155i	$0.534433\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.79069e6	−0.449922	−0.224961	−	0.974368i	$-0.572226\pi$
$647$	−4.79069e6	−0.449922	−0.224961	+	0.974368i	$0.572226\pi$
$648$	0	0
$649$	−7.10908e6	−0.662524
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.44356e6	0.774895	0.387447	−	0.921892i	$-0.373357\pi$
$653$	8.44356e6	0.774895	0.387447	+	0.921892i	$0.373357\pi$
$654$	0	0
$655$	2.87307e7	2.61664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.73666e7	−1.55777	−0.778883	−	0.627170i	$-0.784213\pi$
$659$	−1.73666e7	−1.55777	−0.778883	+	0.627170i	$0.784213\pi$
$660$	0	0
$661$	3.06846e6	0.273160	0.136580	−	0.990629i	$-0.456389\pi$
$661$	3.06846e6	0.273160	0.136580	+	0.990629i	$0.456389\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.41178e7	−2.11487
$666$	0	0
$667$	8.73140e6	0.759923
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.26001e7	1.93778
$672$	0	0
$673$	1.32622e7	1.12870	0.564348	−	0.825537i	$-0.309128\pi$
$673$	1.32622e7	1.12870	0.564348	+	0.825537i	$0.309128\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.99829e6	0.670696	0.335348	−	0.942094i	$-0.391146\pi$
$677$	7.99829e6	0.670696	0.335348	+	0.942094i	$0.391146\pi$
$678$	0	0
$679$	−1.29539e7	−1.07827
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−119801.	−0.00982673	−0.00491336	−	0.999988i	$-0.501564\pi$
$683$	−119801.	−0.00982673	−0.00491336	+	0.999988i	$0.501564\pi$
$684$	0	0
$685$	2.37628e7	1.93496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.09363e7	−1.68017
$690$	0	0
$691$	−511463.	−0.0407492	−0.0203746	−	0.999792i	$-0.506486\pi$
$691$	−511463.	−0.0407492	−0.0203746	+	0.999792i	$0.506486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.43783e7	−1.12913
$696$	0	0
$697$	2.23963e6	0.174620
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.23020e7	−0.945543	−0.472772	−	0.881185i	$-0.656746\pi$
$701$	−1.23020e7	−0.945543	−0.472772	+	0.881185i	$0.656746\pi$
$702$	0	0
$703$	−1.55015e7	−1.18301
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.56274e7	−1.17581
$708$	0	0
$709$	−1.06449e6	−0.0795290	−0.0397645	−	0.999209i	$-0.512661\pi$
$709$	−1.06449e6	−0.0795290	−0.0397645	+	0.999209i	$0.512661\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.03476e6	0.0762285
$714$	0	0
$715$	3.72501e7	2.72498
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.11931e7	1.52887	0.764437	−	0.644699i	$-0.223017\pi$
$719$	2.11931e7	1.52887	0.764437	+	0.644699i	$0.223017\pi$
$720$	0	0
$721$	2.51275e6	0.180016
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.41467e7	−0.999562
$726$	0	0
$727$	524302.	0.0367913	0.0183957	−	0.999831i	$-0.494144\pi$
$727$	524302.	0.0367913	0.0183957	+	0.999831i	$0.494144\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.21813e7	2.22746
$732$	0	0
$733$	1.31522e7	0.904149	0.452074	−	0.891980i	$-0.350684\pi$
$733$	1.31522e7	0.904149	0.452074	+	0.891980i	$0.350684\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.37617e7	0.933265
$738$	0	0
$739$	−1.22510e7	−0.825204	−0.412602	−	0.910912i	$-0.635380\pi$
$739$	−1.22510e7	−0.825204	−0.412602	+	0.910912i	$0.635380\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.36833e7	0.909321	0.454661	−	0.890665i	$-0.349761\pi$
$743$	1.36833e7	0.909321	0.454661	+	0.890665i	$0.349761\pi$
$744$	0	0
$745$	−2.30701e7	−1.52286
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.97953e7	1.94063
$750$	0	0
$751$	−2.36812e7	−1.53216	−0.766080	−	0.642745i	$-0.777795\pi$
$751$	−2.36812e7	−1.53216	−0.766080	+	0.642745i	$0.777795\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.45613e6	−0.539889
$756$	0	0
$757$	3.77298e6	0.239301	0.119651	−	0.992816i	$-0.461823\pi$
$757$	3.77298e6	0.239301	0.119651	+	0.992816i	$0.461823\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.87112e7	1.17122	0.585612	−	0.810592i	$-0.300854\pi$
$761$	1.87112e7	1.17122	0.585612	+	0.810592i	$0.300854\pi$
$762$	0	0
$763$	1.52489e6	0.0948261
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.70772e7	−1.04816
$768$	0	0
$769$	8.60825e6	0.524927	0.262464	−	0.964942i	$-0.415465\pi$
$769$	8.60825e6	0.524927	0.262464	+	0.964942i	$0.415465\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.47263e6	−0.389612	−0.194806	−	0.980842i	$-0.562408\pi$
$773$	−6.47263e6	−0.389612	−0.194806	+	0.980842i	$0.562408\pi$
$774$	0	0
$775$	−1.67653e6	−0.100267
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.73292e6	0.161355
$780$	0	0
$781$	−2.06629e7	−1.21217
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.46313e7	−0.847442
$786$	0	0
$787$	8.79244e6	0.506026	0.253013	−	0.967463i	$-0.418579\pi$
$787$	8.79244e6	0.506026	0.253013	+	0.967463i	$0.418579\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.58179e6	0.430855
$792$	0	0
$793$	5.42894e7	3.06572
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.46426e7	−0.816530	−0.408265	−	0.912864i	$-0.633866\pi$
$797$	−1.46426e7	−0.816530	−0.408265	+	0.912864i	$0.633866\pi$
$798$	0	0
$799$	−2.43516e7	−1.34946
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.35532e6	−0.128902
$804$	0	0
$805$	−2.79373e7	−1.51948
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.23770e7	1.73926	0.869631	−	0.493702i	$-0.164357\pi$
$809$	3.23770e7	1.73926	0.869631	+	0.493702i	$0.164357\pi$
$810$	0	0
$811$	−3.44288e7	−1.83810	−0.919051	−	0.394139i	$-0.871043\pi$
$811$	−3.44288e7	−1.83810	−0.919051	+	0.394139i	$0.871043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.43962e7	2.34127
$816$	0	0
$817$	3.92694e7	2.05826
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.73653e7	−1.93469	−0.967343	−	0.253469i	$-0.918428\pi$
$821$	−3.73653e7	−1.93469	−0.967343	+	0.253469i	$0.918428\pi$
$822$	0	0
$823$	−5.45022e6	−0.280488	−0.140244	−	0.990117i	$-0.544789\pi$
$823$	−5.45022e6	−0.280488	−0.140244	+	0.990117i	$0.544789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.99406e6	−0.203072	−0.101536	−	0.994832i	$-0.532376\pi$
$827$	−3.99406e6	−0.203072	−0.101536	+	0.994832i	$0.532376\pi$
$828$	0	0
$829$	−3.12054e7	−1.57704	−0.788520	−	0.615009i	$-0.789152\pi$
$829$	−3.12054e7	−1.57704	−0.788520	+	0.615009i	$0.789152\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.10788e6	−0.304985
$834$	0	0
$835$	−5.30266e7	−2.63195
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.91123e6	−0.486096	−0.243048	−	0.970014i	$-0.578147\pi$
$839$	−9.91123e6	−0.486096	−0.243048	+	0.970014i	$0.578147\pi$
$840$	0	0
$841$	−6.65245e6	−0.324334
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.85832e7	2.82248
$846$	0	0
$847$	3.62013e6	0.173386
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.79565e7	−0.849959
$852$	0	0
$853$	1.67315e7	0.787340	0.393670	−	0.919252i	$-0.371205\pi$
$853$	1.67315e7	0.787340	0.393670	+	0.919252i	$0.371205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.03493e6	−0.0481347	−0.0240674	−	0.999710i	$-0.507662\pi$
$857$	−1.03493e6	−0.0481347	−0.0240674	+	0.999710i	$0.507662\pi$
$858$	0	0
$859$	−2.33995e7	−1.08199	−0.540995	−	0.841026i	$-0.681952\pi$
$859$	−2.33995e7	−1.08199	−0.540995	+	0.841026i	$0.681952\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.55276e7	−1.62382	−0.811912	−	0.583780i	$-0.801573\pi$
$863$	−3.55276e7	−1.62382	−0.811912	+	0.583780i	$0.801573\pi$
$864$	0	0
$865$	−4.22106e7	−1.91814
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.95280e7	−1.77564
$870$	0	0
$871$	3.30580e7	1.47649
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.04124e6	0.355061
$876$	0	0
$877$	1.64714e7	0.723155	0.361577	−	0.932342i	$-0.382238\pi$
$877$	1.64714e7	0.723155	0.361577	+	0.932342i	$0.382238\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.34334e7	−1.01717	−0.508587	−	0.861011i	$-0.669832\pi$
$881$	−2.34334e7	−1.01717	−0.508587	+	0.861011i	$0.669832\pi$
$882$	0	0
$883$	−2.95106e6	−0.127373	−0.0636863	−	0.997970i	$-0.520286\pi$
$883$	−2.95106e6	−0.127373	−0.0636863	+	0.997970i	$0.520286\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.45216e6	−0.190004	−0.0950019	−	0.995477i	$-0.530286\pi$
$887$	−4.45216e6	−0.190004	−0.0950019	+	0.995477i	$0.530286\pi$
$888$	0	0
$889$	−3.50609e6	−0.148788
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.97152e7	−1.24695
$894$	0	0
$895$	−4.79493e7	−2.00090
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.64240e6	0.0677766
$900$	0	0
$901$	−3.35017e7	−1.37485
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.94460e7	−1.60097
$906$	0	0
$907$	−2.68053e7	−1.08194	−0.540969	−	0.841042i	$-0.681942\pi$
$907$	−2.68053e7	−1.08194	−0.540969	+	0.841042i	$0.681942\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.05552e7	−0.820589	−0.410294	−	0.911953i	$-0.634574\pi$
$911$	−2.05552e7	−0.820589	−0.410294	+	0.911953i	$0.634574\pi$
$912$	0	0
$913$	6.05073e6	0.240232
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.94178e7	1.94071
$918$	0	0
$919$	1.44716e7	0.565233	0.282616	−	0.959233i	$-0.408798\pi$
$919$	1.44716e7	0.565233	0.282616	+	0.959233i	$0.408798\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.96358e7	−1.91774
$924$	0	0
$925$	2.90933e7	1.11799
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.43904e6	0.168752	0.0843761	−	0.996434i	$-0.473110\pi$
$929$	4.43904e6	0.168752	0.0843761	+	0.996434i	$0.473110\pi$
$930$	0	0
$931$	−7.45317e6	−0.281817
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.96066e7	2.22980
$936$	0	0
$937$	3.40690e7	1.26768	0.633842	−	0.773463i	$-0.281477\pi$
$937$	3.40690e7	1.26768	0.633842	+	0.773463i	$0.281477\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.69962e7	0.625718	0.312859	−	0.949800i	$-0.398713\pi$
$941$	1.69962e7	0.625718	0.312859	+	0.949800i	$0.398713\pi$
$942$	0	0
$943$	3.16572e6	0.115929
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.68415e7	−0.610247	−0.305124	−	0.952313i	$-0.598698\pi$
$947$	−1.68415e7	−0.610247	−0.305124	+	0.952313i	$0.598698\pi$
$948$	0	0
$949$	−5.65788e6	−0.203933
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.76148e7	0.628268	0.314134	−	0.949379i	$-0.398286\pi$
$953$	1.76148e7	0.628268	0.314134	+	0.949379i	$0.398286\pi$
$954$	0	0
$955$	1.99127e7	0.706517
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.08729e7	1.43512
$960$	0	0
$961$	−2.84345e7	−0.993201
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.15898e7	−0.400642
$966$	0	0
$967$	−9.77590e6	−0.336194	−0.168097	−	0.985770i	$-0.553762\pi$
$967$	−9.77590e6	−0.336194	−0.168097	+	0.985770i	$0.553762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.47619e6	−0.118319	−0.0591595	−	0.998249i	$-0.518842\pi$
$971$	−3.47619e6	−0.118319	−0.0591595	+	0.998249i	$0.518842\pi$
$972$	0	0
$973$	−2.47311e7	−0.837455
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.45057e7	0.486185	0.243093	−	0.970003i	$-0.421838\pi$
$977$	1.45057e7	0.486185	0.243093	+	0.970003i	$0.421838\pi$
$978$	0	0
$979$	−4.53104e7	−1.51092
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.00490e7	−0.991849	−0.495925	−	0.868366i	$-0.665171\pi$
$983$	−3.00490e7	−0.991849	−0.495925	+	0.868366i	$0.665171\pi$
$984$	0	0
$985$	2.91297e7	0.956635
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.54885e7	1.47880
$990$	0	0
$991$	−1.91325e7	−0.618855	−0.309427	−	0.950923i	$-0.600137\pi$
$991$	−1.91325e7	−0.618855	−0.309427	+	0.950923i	$0.600137\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.11359e7	1.31724
$996$	0	0
$997$	2.30499e7	0.734398	0.367199	−	0.930142i	$-0.380317\pi$
$997$	2.30499e7	0.734398	0.367199	+	0.930142i	$0.380317\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.a.c.1.4	yes	4
3.2	odd	2	inner	324.6.a.c.1.1	✓	4
9.2	odd	6		324.6.e.j.109.4		8
9.4	even	3		324.6.e.j.217.1		8
9.5	odd	6		324.6.e.j.217.4		8
9.7	even	3		324.6.e.j.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.6.a.c.1.1	✓	4	3.2	odd	2	inner
324.6.a.c.1.4	yes	4	1.1	even	1	trivial
324.6.e.j.109.1		8	9.7	even	3
324.6.e.j.109.4		8	9.2	odd	6
324.6.e.j.217.1		8	9.4	even	3
324.6.e.j.217.4		8	9.5	odd	6

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+83.2171 q^{5} +143.136 q^{7} -431.674 q^{11} -1036.95 q^{13} -1659.30 q^{17} -2024.77 q^{19} -2345.43 q^{23} +3800.09 q^{25} -3722.73 q^{29} -441.182 q^{31} +11911.4 q^{35} +7655.95 q^{37} -1349.74 q^{41} -19394.5 q^{43} +14675.8 q^{47} +3680.99 q^{49} +20190.2 q^{53} -35922.7 q^{55} +16468.6 q^{59} -52354.7 q^{61} -86292.3 q^{65} -31879.9 q^{67} +47866.9 q^{71} +5456.25 q^{73} -61788.2 q^{77} +91569.1 q^{79} -14016.9 q^{83} -138082. q^{85} +104964. q^{89} -148426. q^{91} -168496. q^{95} -90500.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 176 q^{7} - 1372 q^{13} - 2944 q^{19} + 3304 q^{25} - 4144 q^{31} + 1676 q^{37} - 38320 q^{43} - 20172 q^{49} - 80640 q^{55} - 120196 q^{61} - 107296 q^{67} - 92380 q^{73} + 5024 q^{79} - 329868 q^{85}+ \cdots - 331864 q^{97}+O(q^{100}) \) 4 * q + 176 * q^7 - 1372 * q^13 - 2944 * q^19 + 3304 * q^25 - 4144 * q^31 + 1676 * q^37 - 38320 * q^43 - 20172 * q^49 - 80640 * q^55 - 120196 * q^61 - 107296 * q^67 - 92380 * q^73 + 5024 * q^79 - 329868 * q^85 - 335552 * q^91 - 331864 * q^97

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