LMFDB - Embedded newform 48.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,6,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$7.69842335102$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} -66.0000 q^{5} -176.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} -66.0000 q^{5} -176.000 q^{7} +81.0000 q^{9} +60.0000 q^{11} -658.000 q^{13} -594.000 q^{15} -414.000 q^{17} -956.000 q^{19} -1584.00 q^{21} -600.000 q^{23} +1231.00 q^{25} +729.000 q^{27} +5574.00 q^{29} +3592.00 q^{31} +540.000 q^{33} +11616.0 q^{35} -8458.00 q^{37} -5922.00 q^{39} +19194.0 q^{41} -13316.0 q^{43} -5346.00 q^{45} +19680.0 q^{47} +14169.0 q^{49} -3726.00 q^{51} -31266.0 q^{53} -3960.00 q^{55} -8604.00 q^{57} -26340.0 q^{59} -31090.0 q^{61} -14256.0 q^{63} +43428.0 q^{65} +16804.0 q^{67} -5400.00 q^{69} -6120.00 q^{71} -25558.0 q^{73} +11079.0 q^{75} -10560.0 q^{77} -74408.0 q^{79} +6561.00 q^{81} +6468.00 q^{83} +27324.0 q^{85} +50166.0 q^{87} -32742.0 q^{89} +115808. q^{91} +32328.0 q^{93} +63096.0 q^{95} +166082. q^{97} +4860.00 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$5$	−66.0000	−1.18064	−0.590322	−	0.807168i	$-0.700999\pi$
$5$	−66.0000	−1.18064	−0.590322	+	0.807168i	$0.700999\pi$
$6$	0	0
$7$	−176.000	−1.35759	−0.678793	−	0.734329i	$-0.737497\pi$
$7$	−176.000	−1.35759	−0.678793	+	0.734329i	$0.737497\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	60.0000	0.149510	0.0747549	−	0.997202i	$-0.476183\pi$
$11$	60.0000	0.149510	0.0747549	+	0.997202i	$0.476183\pi$
$12$	0	0
$13$	−658.000	−1.07986	−0.539930	−	0.841710i	$-0.681549\pi$
$13$	−658.000	−1.07986	−0.539930	+	0.841710i	$0.681549\pi$
$14$	0	0
$15$	−594.000	−0.681645
$16$	0	0
$17$	−414.000	−0.347439	−0.173719	−	0.984795i	$-0.555579\pi$
$17$	−414.000	−0.347439	−0.173719	+	0.984795i	$0.555579\pi$
$18$	0	0
$19$	−956.000	−0.607539	−0.303769	−	0.952746i	$-0.598245\pi$
$19$	−956.000	−0.607539	−0.303769	+	0.952746i	$0.598245\pi$
$20$	0	0
$21$	−1584.00	−0.783803
$22$	0	0
$23$	−600.000	−0.236500	−0.118250	−	0.992984i	$-0.537728\pi$
$23$	−600.000	−0.236500	−0.118250	+	0.992984i	$0.537728\pi$
$24$	0	0
$25$	1231.00	0.393920
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	5574.00	1.23076	0.615378	−	0.788232i	$-0.289003\pi$
$29$	5574.00	1.23076	0.615378	+	0.788232i	$0.289003\pi$
$30$	0	0
$31$	3592.00	0.671324	0.335662	−	0.941983i	$-0.391040\pi$
$31$	3592.00	0.671324	0.335662	+	0.941983i	$0.391040\pi$
$32$	0	0
$33$	540.000	0.0863195
$34$	0	0
$35$	11616.0	1.60283
$36$	0	0
$37$	−8458.00	−1.01570	−0.507848	−	0.861447i	$-0.669559\pi$
$37$	−8458.00	−1.01570	−0.507848	+	0.861447i	$0.669559\pi$
$38$	0	0
$39$	−5922.00	−0.623458
$40$	0	0
$41$	19194.0	1.78322	0.891612	−	0.452800i	$-0.149575\pi$
$41$	19194.0	1.78322	0.891612	+	0.452800i	$0.149575\pi$
$42$	0	0
$43$	−13316.0	−1.09825	−0.549127	−	0.835739i	$-0.685040\pi$
$43$	−13316.0	−1.09825	−0.549127	+	0.835739i	$0.685040\pi$
$44$	0	0
$45$	−5346.00	−0.393548
$46$	0	0
$47$	19680.0	1.29951	0.649756	−	0.760143i	$-0.274871\pi$
$47$	19680.0	1.29951	0.649756	+	0.760143i	$0.274871\pi$
$48$	0	0
$49$	14169.0	0.843042
$50$	0	0
$51$	−3726.00	−0.200594
$52$	0	0
$53$	−31266.0	−1.52891	−0.764456	−	0.644676i	$-0.776992\pi$
$53$	−31266.0	−1.52891	−0.764456	+	0.644676i	$0.776992\pi$
$54$	0	0
$55$	−3960.00	−0.176518
$56$	0	0
$57$	−8604.00	−0.350763
$58$	0	0
$59$	−26340.0	−0.985112	−0.492556	−	0.870281i	$-0.663937\pi$
$59$	−26340.0	−0.985112	−0.492556	+	0.870281i	$0.663937\pi$
$60$	0	0
$61$	−31090.0	−1.06978	−0.534892	−	0.844920i	$-0.679648\pi$
$61$	−31090.0	−1.06978	−0.534892	+	0.844920i	$0.679648\pi$
$62$	0	0
$63$	−14256.0	−0.452529
$64$	0	0
$65$	43428.0	1.27493
$66$	0	0
$67$	16804.0	0.457326	0.228663	−	0.973506i	$-0.426565\pi$
$67$	16804.0	0.457326	0.228663	+	0.973506i	$0.426565\pi$
$68$	0	0
$69$	−5400.00	−0.136544
$70$	0	0
$71$	−6120.00	−0.144081	−0.0720403	−	0.997402i	$-0.522951\pi$
$71$	−6120.00	−0.144081	−0.0720403	+	0.997402i	$0.522951\pi$
$72$	0	0
$73$	−25558.0	−0.561332	−0.280666	−	0.959806i	$-0.590555\pi$
$73$	−25558.0	−0.561332	−0.280666	+	0.959806i	$0.590555\pi$
$74$	0	0
$75$	11079.0	0.227430
$76$	0	0
$77$	−10560.0	−0.202972
$78$	0	0
$79$	−74408.0	−1.34138	−0.670690	−	0.741738i	$-0.734002\pi$
$79$	−74408.0	−1.34138	−0.670690	+	0.741738i	$0.734002\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	6468.00	0.103056	0.0515282	−	0.998672i	$-0.483591\pi$
$83$	6468.00	0.103056	0.0515282	+	0.998672i	$0.483591\pi$
$84$	0	0
$85$	27324.0	0.410201
$86$	0	0
$87$	50166.0	0.710577
$88$	0	0
$89$	−32742.0	−0.438157	−0.219079	−	0.975707i	$-0.570305\pi$
$89$	−32742.0	−0.438157	−0.219079	+	0.975707i	$0.570305\pi$
$90$	0	0
$91$	115808.	1.46600
$92$	0	0
$93$	32328.0	0.387589
$94$	0	0
$95$	63096.0	0.717287
$96$	0	0
$97$	166082.	1.79223	0.896114	−	0.443824i	$-0.146378\pi$
$97$	166082.	1.79223	0.896114	+	0.443824i	$0.146378\pi$
$98$	0	0
$99$	4860.00	0.0498366
$100$	0	0
$101$	−22002.0	−0.214614	−0.107307	−	0.994226i	$-0.534223\pi$
$101$	−22002.0	−0.214614	−0.107307	+	0.994226i	$0.534223\pi$
$102$	0	0
$103$	79264.0	0.736178	0.368089	−	0.929791i	$-0.380012\pi$
$103$	79264.0	0.736178	0.368089	+	0.929791i	$0.380012\pi$
$104$	0	0
$105$	104544.	0.925392
$106$	0	0
$107$	−227988.	−1.92510	−0.962548	−	0.271110i	$-0.912609\pi$
$107$	−227988.	−1.92510	−0.962548	+	0.271110i	$0.912609\pi$
$108$	0	0
$109$	−8530.00	−0.0687674	−0.0343837	−	0.999409i	$-0.510947\pi$
$109$	−8530.00	−0.0687674	−0.0343837	+	0.999409i	$0.510947\pi$
$110$	0	0
$111$	−76122.0	−0.586412
$112$	0	0
$113$	−195438.	−1.43984	−0.719918	−	0.694059i	$-0.755821\pi$
$113$	−195438.	−1.43984	−0.719918	+	0.694059i	$0.755821\pi$
$114$	0	0
$115$	39600.0	0.279223
$116$	0	0
$117$	−53298.0	−0.359953
$118$	0	0
$119$	72864.0	0.471678
$120$	0	0
$121$	−157451.	−0.977647
$122$	0	0
$123$	172746.	1.02954
$124$	0	0
$125$	125004.	0.715565
$126$	0	0
$127$	−173000.	−0.951780	−0.475890	−	0.879505i	$-0.657874\pi$
$127$	−173000.	−0.951780	−0.475890	+	0.879505i	$0.657874\pi$
$128$	0	0
$129$	−119844.	−0.634077
$130$	0	0
$131$	−151260.	−0.770098	−0.385049	−	0.922896i	$-0.625815\pi$
$131$	−151260.	−0.770098	−0.385049	+	0.922896i	$0.625815\pi$
$132$	0	0
$133$	168256.	0.824786
$134$	0	0
$135$	−48114.0	−0.227215
$136$	0	0
$137$	−128454.	−0.584718	−0.292359	−	0.956309i	$-0.594440\pi$
$137$	−128454.	−0.584718	−0.292359	+	0.956309i	$0.594440\pi$
$138$	0	0
$139$	−154196.	−0.676918	−0.338459	−	0.940981i	$-0.609906\pi$
$139$	−154196.	−0.676918	−0.338459	+	0.940981i	$0.609906\pi$
$140$	0	0
$141$	177120.	0.750274
$142$	0	0
$143$	−39480.0	−0.161450
$144$	0	0
$145$	−367884.	−1.45308
$146$	0	0
$147$	127521.	0.486730
$148$	0	0
$149$	29454.0	0.108687	0.0543436	−	0.998522i	$-0.482693\pi$
$149$	29454.0	0.108687	0.0543436	+	0.998522i	$0.482693\pi$
$150$	0	0
$151$	203872.	0.727638	0.363819	−	0.931470i	$-0.381473\pi$
$151$	203872.	0.727638	0.363819	+	0.931470i	$0.381473\pi$
$152$	0	0
$153$	−33534.0	−0.115813
$154$	0	0
$155$	−237072.	−0.792594
$156$	0	0
$157$	136142.	0.440801	0.220401	−	0.975409i	$-0.429263\pi$
$157$	136142.	0.440801	0.220401	+	0.975409i	$0.429263\pi$
$158$	0	0
$159$	−281394.	−0.882718
$160$	0	0
$161$	105600.	0.321070
$162$	0	0
$163$	171124.	0.504478	0.252239	−	0.967665i	$-0.418833\pi$
$163$	171124.	0.504478	0.252239	+	0.967665i	$0.418833\pi$
$164$	0	0
$165$	−35640.0	−0.101913
$166$	0	0
$167$	676200.	1.87622	0.938110	−	0.346336i	$-0.112574\pi$
$167$	676200.	1.87622	0.938110	+	0.346336i	$0.112574\pi$
$168$	0	0
$169$	61671.0	0.166098
$170$	0	0
$171$	−77436.0	−0.202513
$172$	0	0
$173$	133158.	0.338261	0.169131	−	0.985594i	$-0.445904\pi$
$173$	133158.	0.338261	0.169131	+	0.985594i	$0.445904\pi$
$174$	0	0
$175$	−216656.	−0.534781
$176$	0	0
$177$	−237060.	−0.568755
$178$	0	0
$179$	693396.	1.61752	0.808758	−	0.588141i	$-0.200140\pi$
$179$	693396.	1.61752	0.808758	+	0.588141i	$0.200140\pi$
$180$	0	0
$181$	377174.	0.855747	0.427873	−	0.903839i	$-0.359263\pi$
$181$	377174.	0.855747	0.427873	+	0.903839i	$0.359263\pi$
$182$	0	0
$183$	−279810.	−0.617640
$184$	0	0
$185$	558228.	1.19917
$186$	0	0
$187$	−24840.0	−0.0519455
$188$	0	0
$189$	−128304.	−0.261268
$190$	0	0
$191$	265344.	0.526291	0.263145	−	0.964756i	$-0.415240\pi$
$191$	265344.	0.526291	0.263145	+	0.964756i	$0.415240\pi$
$192$	0	0
$193$	295298.	0.570647	0.285323	−	0.958431i	$-0.407899\pi$
$193$	295298.	0.570647	0.285323	+	0.958431i	$0.407899\pi$
$194$	0	0
$195$	390852.	0.736081
$196$	0	0
$197$	201294.	0.369543	0.184772	−	0.982781i	$-0.440845\pi$
$197$	201294.	0.369543	0.184772	+	0.982781i	$0.440845\pi$
$198$	0	0
$199$	−652448.	−1.16792	−0.583960	−	0.811782i	$-0.698498\pi$
$199$	−652448.	−1.16792	−0.583960	+	0.811782i	$0.698498\pi$
$200$	0	0
$201$	151236.	0.264037
$202$	0	0
$203$	−981024.	−1.67086
$204$	0	0
$205$	−1.26680e6	−2.10535
$206$	0	0
$207$	−48600.0	−0.0788334
$208$	0	0
$209$	−57360.0	−0.0908330
$210$	0	0
$211$	1.14706e6	1.77370	0.886850	−	0.462058i	$-0.152889\pi$
$211$	1.14706e6	1.77370	0.886850	+	0.462058i	$0.152889\pi$
$212$	0	0
$213$	−55080.0	−0.0831850
$214$	0	0
$215$	878856.	1.29665
$216$	0	0
$217$	−632192.	−0.911380
$218$	0	0
$219$	−230022.	−0.324085
$220$	0	0
$221$	272412.	0.375185
$222$	0	0
$223$	−701960.	−0.945258	−0.472629	−	0.881262i	$-0.656695\pi$
$223$	−701960.	−0.945258	−0.472629	+	0.881262i	$0.656695\pi$
$224$	0	0
$225$	99711.0	0.131307
$226$	0	0
$227$	−1.23611e6	−1.59218	−0.796089	−	0.605179i	$-0.793101\pi$
$227$	−1.23611e6	−1.59218	−0.796089	+	0.605179i	$0.793101\pi$
$228$	0	0
$229$	105830.	0.133358	0.0666792	−	0.997774i	$-0.478760\pi$
$229$	105830.	0.133358	0.0666792	+	0.997774i	$0.478760\pi$
$230$	0	0
$231$	−95040.0	−0.117186
$232$	0	0
$233$	−438678.	−0.529366	−0.264683	−	0.964335i	$-0.585267\pi$
$233$	−438678.	−0.529366	−0.264683	+	0.964335i	$0.585267\pi$
$234$	0	0
$235$	−1.29888e6	−1.53426
$236$	0	0
$237$	−669672.	−0.774446
$238$	0	0
$239$	−28464.0	−0.0322330	−0.0161165	−	0.999870i	$-0.505130\pi$
$239$	−28464.0	−0.0322330	−0.0161165	+	0.999870i	$0.505130\pi$
$240$	0	0
$241$	892562.	0.989910	0.494955	−	0.868919i	$-0.335185\pi$
$241$	892562.	0.989910	0.494955	+	0.868919i	$0.335185\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−935154.	−0.995332
$246$	0	0
$247$	629048.	0.656057
$248$	0	0
$249$	58212.0	0.0594996
$250$	0	0
$251$	110124.	0.110331	0.0551655	−	0.998477i	$-0.482431\pi$
$251$	110124.	0.110331	0.0551655	+	0.998477i	$0.482431\pi$
$252$	0	0
$253$	−36000.0	−0.0353591
$254$	0	0
$255$	245916.	0.236830
$256$	0	0
$257$	140802.	0.132977	0.0664884	−	0.997787i	$-0.478820\pi$
$257$	140802.	0.132977	0.0664884	+	0.997787i	$0.478820\pi$
$258$	0	0
$259$	1.48861e6	1.37889
$260$	0	0
$261$	451494.	0.410252
$262$	0	0
$263$	938760.	0.836884	0.418442	−	0.908244i	$-0.362576\pi$
$263$	938760.	0.836884	0.418442	+	0.908244i	$0.362576\pi$
$264$	0	0
$265$	2.06356e6	1.80510
$266$	0	0
$267$	−294678.	−0.252970
$268$	0	0
$269$	−1.11451e6	−0.939078	−0.469539	−	0.882912i	$-0.655580\pi$
$269$	−1.11451e6	−0.939078	−0.469539	+	0.882912i	$0.655580\pi$
$270$	0	0
$271$	−567704.	−0.469568	−0.234784	−	0.972048i	$-0.575438\pi$
$271$	−567704.	−0.469568	−0.234784	+	0.972048i	$0.575438\pi$
$272$	0	0
$273$	1.04227e6	0.846398
$274$	0	0
$275$	73860.0	0.0588949
$276$	0	0
$277$	−1.21326e6	−0.950066	−0.475033	−	0.879968i	$-0.657564\pi$
$277$	−1.21326e6	−0.950066	−0.475033	+	0.879968i	$0.657564\pi$
$278$	0	0
$279$	290952.	0.223775
$280$	0	0
$281$	687738.	0.519586	0.259793	−	0.965664i	$-0.416346\pi$
$281$	687738.	0.519586	0.259793	+	0.965664i	$0.416346\pi$
$282$	0	0
$283$	830908.	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$283$	830908.	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$284$	0	0
$285$	567864.	0.414126
$286$	0	0
$287$	−3.37814e6	−2.42088
$288$	0	0
$289$	−1.24846e6	−0.879286
$290$	0	0
$291$	1.49474e6	1.03474
$292$	0	0
$293$	−1.31263e6	−0.893248	−0.446624	−	0.894722i	$-0.647374\pi$
$293$	−1.31263e6	−0.893248	−0.446624	+	0.894722i	$0.647374\pi$
$294$	0	0
$295$	1.73844e6	1.16307
$296$	0	0
$297$	43740.0	0.0287732
$298$	0	0
$299$	394800.	0.255387
$300$	0	0
$301$	2.34362e6	1.49097
$302$	0	0
$303$	−198018.	−0.123908
$304$	0	0
$305$	2.05194e6	1.26303
$306$	0	0
$307$	−1.69022e6	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$307$	−1.69022e6	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$308$	0	0
$309$	713376.	0.425033
$310$	0	0
$311$	1.50204e6	0.880604	0.440302	−	0.897850i	$-0.354871\pi$
$311$	1.50204e6	0.880604	0.440302	+	0.897850i	$0.354871\pi$
$312$	0	0
$313$	810842.	0.467816	0.233908	−	0.972259i	$-0.424848\pi$
$313$	810842.	0.467816	0.233908	+	0.972259i	$0.424848\pi$
$314$	0	0
$315$	940896.	0.534275
$316$	0	0
$317$	903558.	0.505019	0.252510	−	0.967594i	$-0.418744\pi$
$317$	903558.	0.505019	0.252510	+	0.967594i	$0.418744\pi$
$318$	0	0
$319$	334440.	0.184010
$320$	0	0
$321$	−2.05189e6	−1.11146
$322$	0	0
$323$	395784.	0.211082
$324$	0	0
$325$	−809998.	−0.425379
$326$	0	0
$327$	−76770.0	−0.0397029
$328$	0	0
$329$	−3.46368e6	−1.76420
$330$	0	0
$331$	−1.12197e6	−0.562875	−0.281438	−	0.959580i	$-0.590811\pi$
$331$	−1.12197e6	−0.562875	−0.281438	+	0.959580i	$0.590811\pi$
$332$	0	0
$333$	−685098.	−0.338565
$334$	0	0
$335$	−1.10906e6	−0.539939
$336$	0	0
$337$	−2.75217e6	−1.32008	−0.660041	−	0.751229i	$-0.729461\pi$
$337$	−2.75217e6	−1.32008	−0.660041	+	0.751229i	$0.729461\pi$
$338$	0	0
$339$	−1.75894e6	−0.831289
$340$	0	0
$341$	215520.	0.100369
$342$	0	0
$343$	464288.	0.213085
$344$	0	0
$345$	356400.	0.161209
$346$	0	0
$347$	−1.91749e6	−0.854889	−0.427445	−	0.904042i	$-0.640586\pi$
$347$	−1.91749e6	−0.854889	−0.427445	+	0.904042i	$0.640586\pi$
$348$	0	0
$349$	1.83659e6	0.807140	0.403570	−	0.914949i	$-0.367769\pi$
$349$	1.83659e6	0.807140	0.403570	+	0.914949i	$0.367769\pi$
$350$	0	0
$351$	−479682.	−0.207819
$352$	0	0
$353$	−622014.	−0.265683	−0.132841	−	0.991137i	$-0.542410\pi$
$353$	−622014.	−0.265683	−0.132841	+	0.991137i	$0.542410\pi$
$354$	0	0
$355$	403920.	0.170108
$356$	0	0
$357$	655776.	0.272323
$358$	0	0
$359$	−3.74062e6	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$359$	−3.74062e6	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$360$	0	0
$361$	−1.56216e6	−0.630897
$362$	0	0
$363$	−1.41706e6	−0.564445
$364$	0	0
$365$	1.68683e6	0.662733
$366$	0	0
$367$	−16232.0	−0.00629081	−0.00314541	−	0.999995i	$-0.501001\pi$
$367$	−16232.0	−0.00629081	−0.00314541	+	0.999995i	$0.501001\pi$
$368$	0	0
$369$	1.55471e6	0.594408
$370$	0	0
$371$	5.50282e6	2.07563
$372$	0	0
$373$	293606.	0.109268	0.0546340	−	0.998506i	$-0.482601\pi$
$373$	293606.	0.109268	0.0546340	+	0.998506i	$0.482601\pi$
$374$	0	0
$375$	1.12504e6	0.413131
$376$	0	0
$377$	−3.66769e6	−1.32904
$378$	0	0
$379$	−3.18012e6	−1.13722	−0.568611	−	0.822607i	$-0.692519\pi$
$379$	−3.18012e6	−1.13722	−0.568611	+	0.822607i	$0.692519\pi$
$380$	0	0
$381$	−1.55700e6	−0.549511
$382$	0	0
$383$	2.97984e6	1.03800	0.518998	−	0.854775i	$-0.326305\pi$
$383$	2.97984e6	1.03800	0.518998	+	0.854775i	$0.326305\pi$
$384$	0	0
$385$	696960.	0.239638
$386$	0	0
$387$	−1.07860e6	−0.366085
$388$	0	0
$389$	3.45977e6	1.15924	0.579620	−	0.814887i	$-0.303201\pi$
$389$	3.45977e6	1.15924	0.579620	+	0.814887i	$0.303201\pi$
$390$	0	0
$391$	248400.	0.0821693
$392$	0	0
$393$	−1.36134e6	−0.444616
$394$	0	0
$395$	4.91093e6	1.58369
$396$	0	0
$397$	−3.90416e6	−1.24323	−0.621615	−	0.783323i	$-0.713523\pi$
$397$	−3.90416e6	−1.24323	−0.621615	+	0.783323i	$0.713523\pi$
$398$	0	0
$399$	1.51430e6	0.476191
$400$	0	0
$401$	5.44115e6	1.68978	0.844890	−	0.534940i	$-0.179666\pi$
$401$	5.44115e6	1.68978	0.844890	+	0.534940i	$0.179666\pi$
$402$	0	0
$403$	−2.36354e6	−0.724936
$404$	0	0
$405$	−433026.	−0.131183
$406$	0	0
$407$	−507480.	−0.151856
$408$	0	0
$409$	1.96995e6	0.582299	0.291150	−	0.956678i	$-0.405962\pi$
$409$	1.96995e6	0.582299	0.291150	+	0.956678i	$0.405962\pi$
$410$	0	0
$411$	−1.15609e6	−0.337587
$412$	0	0
$413$	4.63584e6	1.33738
$414$	0	0
$415$	−426888.	−0.121673
$416$	0	0
$417$	−1.38776e6	−0.390819
$418$	0	0
$419$	−139020.	−0.0386850	−0.0193425	−	0.999813i	$-0.506157\pi$
$419$	−139020.	−0.0386850	−0.0193425	+	0.999813i	$0.506157\pi$
$420$	0	0
$421$	4.32743e6	1.18994	0.594970	−	0.803748i	$-0.297164\pi$
$421$	4.32743e6	1.18994	0.594970	+	0.803748i	$0.297164\pi$
$422$	0	0
$423$	1.59408e6	0.433171
$424$	0	0
$425$	−509634.	−0.136863
$426$	0	0
$427$	5.47184e6	1.45232
$428$	0	0
$429$	−355320.	−0.0932130
$430$	0	0
$431$	2.79936e6	0.725881	0.362941	−	0.931812i	$-0.381773\pi$
$431$	2.79936e6	0.725881	0.362941	+	0.931812i	$0.381773\pi$
$432$	0	0
$433$	−5.90241e6	−1.51290	−0.756449	−	0.654052i	$-0.773068\pi$
$433$	−5.90241e6	−1.51290	−0.756449	+	0.654052i	$0.773068\pi$
$434$	0	0
$435$	−3.31096e6	−0.838939
$436$	0	0
$437$	573600.	0.143683
$438$	0	0
$439$	446512.	0.110579	0.0552894	−	0.998470i	$-0.482392\pi$
$439$	446512.	0.110579	0.0552894	+	0.998470i	$0.482392\pi$
$440$	0	0
$441$	1.14769e6	0.281014
$442$	0	0
$443$	−3.49525e6	−0.846193	−0.423096	−	0.906085i	$-0.639057\pi$
$443$	−3.49525e6	−0.846193	−0.423096	+	0.906085i	$0.639057\pi$
$444$	0	0
$445$	2.16097e6	0.517308
$446$	0	0
$447$	265086.	0.0627506
$448$	0	0
$449$	−1.20613e6	−0.282343	−0.141171	−	0.989985i	$-0.545087\pi$
$449$	−1.20613e6	−0.282343	−0.141171	+	0.989985i	$0.545087\pi$
$450$	0	0
$451$	1.15164e6	0.266609
$452$	0	0
$453$	1.83485e6	0.420102
$454$	0	0
$455$	−7.64333e6	−1.73083
$456$	0	0
$457$	233546.	0.0523097	0.0261548	−	0.999658i	$-0.491674\pi$
$457$	233546.	0.0523097	0.0261548	+	0.999658i	$0.491674\pi$
$458$	0	0
$459$	−301806.	−0.0668646
$460$	0	0
$461$	−1.74489e6	−0.382398	−0.191199	−	0.981551i	$-0.561238\pi$
$461$	−1.74489e6	−0.382398	−0.191199	+	0.981551i	$0.561238\pi$
$462$	0	0
$463$	2.91786e6	0.632576	0.316288	−	0.948663i	$-0.397563\pi$
$463$	2.91786e6	0.632576	0.316288	+	0.948663i	$0.397563\pi$
$464$	0	0
$465$	−2.13365e6	−0.457605
$466$	0	0
$467$	5.31076e6	1.12684	0.563422	−	0.826169i	$-0.309484\pi$
$467$	5.31076e6	1.12684	0.563422	+	0.826169i	$0.309484\pi$
$468$	0	0
$469$	−2.95750e6	−0.620859
$470$	0	0
$471$	1.22528e6	0.254497
$472$	0	0
$473$	−798960.	−0.164200
$474$	0	0
$475$	−1.17684e6	−0.239322
$476$	0	0
$477$	−2.53255e6	−0.509638
$478$	0	0
$479$	−2.34466e6	−0.466918	−0.233459	−	0.972367i	$-0.575004\pi$
$479$	−2.34466e6	−0.466918	−0.233459	+	0.972367i	$0.575004\pi$
$480$	0	0
$481$	5.56536e6	1.09681
$482$	0	0
$483$	950400.	0.185370
$484$	0	0
$485$	−1.09614e7	−2.11598
$486$	0	0
$487$	−9.81531e6	−1.87535	−0.937674	−	0.347517i	$-0.887025\pi$
$487$	−9.81531e6	−1.87535	−0.937674	+	0.347517i	$0.887025\pi$
$488$	0	0
$489$	1.54012e6	0.291260
$490$	0	0
$491$	5.94520e6	1.11292	0.556458	−	0.830876i	$-0.312160\pi$
$491$	5.94520e6	1.11292	0.556458	+	0.830876i	$0.312160\pi$
$492$	0	0
$493$	−2.30764e6	−0.427612
$494$	0	0
$495$	−320760.	−0.0588393
$496$	0	0
$497$	1.07712e6	0.195602
$498$	0	0
$499$	−6.47832e6	−1.16469	−0.582346	−	0.812941i	$-0.697865\pi$
$499$	−6.47832e6	−1.16469	−0.582346	+	0.812941i	$0.697865\pi$
$500$	0	0
$501$	6.08580e6	1.08324
$502$	0	0
$503$	−4.71794e6	−0.831444	−0.415722	−	0.909492i	$-0.636471\pi$
$503$	−4.71794e6	−0.831444	−0.415722	+	0.909492i	$0.636471\pi$
$504$	0	0
$505$	1.45213e6	0.253383
$506$	0	0
$507$	555039.	0.0958967
$508$	0	0
$509$	−1.90771e6	−0.326375	−0.163188	−	0.986595i	$-0.552178\pi$
$509$	−1.90771e6	−0.326375	−0.163188	+	0.986595i	$0.552178\pi$
$510$	0	0
$511$	4.49821e6	0.762057
$512$	0	0
$513$	−696924.	−0.116921
$514$	0	0
$515$	−5.23142e6	−0.869164
$516$	0	0
$517$	1.18080e6	0.194290
$518$	0	0
$519$	1.19842e6	0.195295
$520$	0	0
$521$	8.01974e6	1.29439	0.647196	−	0.762324i	$-0.275941\pi$
$521$	8.01974e6	1.29439	0.647196	+	0.762324i	$0.275941\pi$
$522$	0	0
$523$	−1.91162e6	−0.305596	−0.152798	−	0.988257i	$-0.548828\pi$
$523$	−1.91162e6	−0.305596	−0.152798	+	0.988257i	$0.548828\pi$
$524$	0	0
$525$	−1.94990e6	−0.308756
$526$	0	0
$527$	−1.48709e6	−0.233244
$528$	0	0
$529$	−6.07634e6	−0.944068
$530$	0	0
$531$	−2.13354e6	−0.328371
$532$	0	0
$533$	−1.26297e7	−1.92563
$534$	0	0
$535$	1.50472e7	2.27285
$536$	0	0
$537$	6.24056e6	0.933874
$538$	0	0
$539$	850140.	0.126043
$540$	0	0
$541$	−1.19900e7	−1.76128	−0.880639	−	0.473788i	$-0.842886\pi$
$541$	−1.19900e7	−1.76128	−0.880639	+	0.473788i	$0.842886\pi$
$542$	0	0
$543$	3.39457e6	0.494066
$544$	0	0
$545$	562980.	0.0811898
$546$	0	0
$547$	−4.45809e6	−0.637061	−0.318530	−	0.947913i	$-0.603189\pi$
$547$	−4.45809e6	−0.637061	−0.318530	+	0.947913i	$0.603189\pi$
$548$	0	0
$549$	−2.51829e6	−0.356595
$550$	0	0
$551$	−5.32874e6	−0.747732
$552$	0	0
$553$	1.30958e7	1.82104
$554$	0	0
$555$	5.02405e6	0.692344
$556$	0	0
$557$	9.02612e6	1.23272	0.616358	−	0.787466i	$-0.288607\pi$
$557$	9.02612e6	1.23272	0.616358	+	0.787466i	$0.288607\pi$
$558$	0	0
$559$	8.76193e6	1.18596
$560$	0	0
$561$	−223560.	−0.0299907
$562$	0	0
$563$	−6.84899e6	−0.910658	−0.455329	−	0.890323i	$-0.650478\pi$
$563$	−6.84899e6	−0.910658	−0.455329	+	0.890323i	$0.650478\pi$
$564$	0	0
$565$	1.28989e7	1.69993
$566$	0	0
$567$	−1.15474e6	−0.150843
$568$	0	0
$569$	−5.46322e6	−0.707405	−0.353703	−	0.935358i	$-0.615077\pi$
$569$	−5.46322e6	−0.707405	−0.353703	+	0.935358i	$0.615077\pi$
$570$	0	0
$571$	1.02324e7	1.31337	0.656684	−	0.754166i	$-0.271959\pi$
$571$	1.02324e7	1.31337	0.656684	+	0.754166i	$0.271959\pi$
$572$	0	0
$573$	2.38810e6	0.303854
$574$	0	0
$575$	−738600.	−0.0931622
$576$	0	0
$577$	1.59437e7	1.99365	0.996825	−	0.0796186i	$-0.0253702\pi$
$577$	1.59437e7	1.99365	0.996825	+	0.0796186i	$0.0253702\pi$
$578$	0	0
$579$	2.65768e6	0.329463
$580$	0	0
$581$	−1.13837e6	−0.139908
$582$	0	0
$583$	−1.87596e6	−0.228587
$584$	0	0
$585$	3.51767e6	0.424977
$586$	0	0
$587$	9.47713e6	1.13522	0.567612	−	0.823296i	$-0.307867\pi$
$587$	9.47713e6	1.13522	0.567612	+	0.823296i	$0.307867\pi$
$588$	0	0
$589$	−3.43395e6	−0.407855
$590$	0	0
$591$	1.81165e6	0.213356
$592$	0	0
$593$	2.45349e6	0.286515	0.143258	−	0.989685i	$-0.454242\pi$
$593$	2.45349e6	0.286515	0.143258	+	0.989685i	$0.454242\pi$
$594$	0	0
$595$	−4.80902e6	−0.556884
$596$	0	0
$597$	−5.87203e6	−0.674299
$598$	0	0
$599$	9.29978e6	1.05902	0.529512	−	0.848302i	$-0.322375\pi$
$599$	9.29978e6	1.05902	0.529512	+	0.848302i	$0.322375\pi$
$600$	0	0
$601$	−1.14617e7	−1.29438	−0.647192	−	0.762327i	$-0.724057\pi$
$601$	−1.14617e7	−1.29438	−0.647192	+	0.762327i	$0.724057\pi$
$602$	0	0
$603$	1.36112e6	0.152442
$604$	0	0
$605$	1.03918e7	1.15425
$606$	0	0
$607$	−1.12784e7	−1.24244	−0.621219	−	0.783637i	$-0.713362\pi$
$607$	−1.12784e7	−1.24244	−0.621219	+	0.783637i	$0.713362\pi$
$608$	0	0
$609$	−8.82922e6	−0.964670
$610$	0	0
$611$	−1.29494e7	−1.40329
$612$	0	0
$613$	93782.0	0.0100802	0.00504009	−	0.999987i	$-0.498396\pi$
$613$	93782.0	0.0100802	0.00504009	+	0.999987i	$0.498396\pi$
$614$	0	0
$615$	−1.14012e7	−1.21553
$616$	0	0
$617$	−1.49642e7	−1.58248	−0.791242	−	0.611504i	$-0.790565\pi$
$617$	−1.49642e7	−1.58248	−0.791242	+	0.611504i	$0.790565\pi$
$618$	0	0
$619$	5.06888e6	0.531723	0.265861	−	0.964011i	$-0.414344\pi$
$619$	5.06888e6	0.531723	0.265861	+	0.964011i	$0.414344\pi$
$620$	0	0
$621$	−437400.	−0.0455145
$622$	0	0
$623$	5.76259e6	0.594837
$624$	0	0
$625$	−1.20971e7	−1.23875
$626$	0	0
$627$	−516240.	−0.0524424
$628$	0	0
$629$	3.50161e6	0.352892
$630$	0	0
$631$	−1.55919e7	−1.55892	−0.779462	−	0.626450i	$-0.784507\pi$
$631$	−1.55919e7	−1.55892	−0.779462	+	0.626450i	$0.784507\pi$
$632$	0	0
$633$	1.03235e7	1.02405
$634$	0	0
$635$	1.14180e7	1.12371
$636$	0	0
$637$	−9.32320e6	−0.910367
$638$	0	0
$639$	−495720.	−0.0480269
$640$	0	0
$641$	1.09701e7	1.05455	0.527274	−	0.849695i	$-0.323214\pi$
$641$	1.09701e7	1.05455	0.527274	+	0.849695i	$0.323214\pi$
$642$	0	0
$643$	2.83704e6	0.270607	0.135303	−	0.990804i	$-0.456799\pi$
$643$	2.83704e6	0.270607	0.135303	+	0.990804i	$0.456799\pi$
$644$	0	0
$645$	7.90970e6	0.748619
$646$	0	0
$647$	6.05686e6	0.568835	0.284418	−	0.958700i	$-0.408200\pi$
$647$	6.05686e6	0.568835	0.284418	+	0.958700i	$0.408200\pi$
$648$	0	0
$649$	−1.58040e6	−0.147284
$650$	0	0
$651$	−5.68973e6	−0.526186
$652$	0	0
$653$	−1.08892e6	−0.0999341	−0.0499671	−	0.998751i	$-0.515912\pi$
$653$	−1.08892e6	−0.0999341	−0.0499671	+	0.998751i	$0.515912\pi$
$654$	0	0
$655$	9.98316e6	0.909211
$656$	0	0
$657$	−2.07020e6	−0.187111
$658$	0	0
$659$	−7.41803e6	−0.665388	−0.332694	−	0.943035i	$-0.607958\pi$
$659$	−7.41803e6	−0.665388	−0.332694	+	0.943035i	$0.607958\pi$
$660$	0	0
$661$	767654.	0.0683379	0.0341690	−	0.999416i	$-0.489122\pi$
$661$	767654.	0.0683379	0.0341690	+	0.999416i	$0.489122\pi$
$662$	0	0
$663$	2.45171e6	0.216613
$664$	0	0
$665$	−1.11049e7	−0.973779
$666$	0	0
$667$	−3.34440e6	−0.291074
$668$	0	0
$669$	−6.31764e6	−0.545745
$670$	0	0
$671$	−1.86540e6	−0.159943
$672$	0	0
$673$	1.42263e6	0.121075	0.0605373	−	0.998166i	$-0.480719\pi$
$673$	1.42263e6	0.121075	0.0605373	+	0.998166i	$0.480719\pi$
$674$	0	0
$675$	897399.	0.0758099
$676$	0	0
$677$	−6.16231e6	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$677$	−6.16231e6	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$678$	0	0
$679$	−2.92304e7	−2.43310
$680$	0	0
$681$	−1.11250e7	−0.919245
$682$	0	0
$683$	−1.50621e7	−1.23548	−0.617739	−	0.786383i	$-0.711951\pi$
$683$	−1.50621e7	−1.23548	−0.617739	+	0.786383i	$0.711951\pi$
$684$	0	0
$685$	8.47796e6	0.690343
$686$	0	0
$687$	952470.	0.0769945
$688$	0	0
$689$	2.05730e7	1.65101
$690$	0	0
$691$	5.87636e6	0.468180	0.234090	−	0.972215i	$-0.424789\pi$
$691$	5.87636e6	0.468180	0.234090	+	0.972215i	$0.424789\pi$
$692$	0	0
$693$	−855360.	−0.0676575
$694$	0	0
$695$	1.01769e7	0.799199
$696$	0	0
$697$	−7.94632e6	−0.619561
$698$	0	0
$699$	−3.94810e6	−0.305630
$700$	0	0
$701$	3.60077e6	0.276758	0.138379	−	0.990379i	$-0.455811\pi$
$701$	3.60077e6	0.276758	0.138379	+	0.990379i	$0.455811\pi$
$702$	0	0
$703$	8.08585e6	0.617074
$704$	0	0
$705$	−1.16899e7	−0.885806
$706$	0	0
$707$	3.87235e6	0.291358
$708$	0	0
$709$	9.22516e6	0.689221	0.344610	−	0.938746i	$-0.388011\pi$
$709$	9.22516e6	0.689221	0.344610	+	0.938746i	$0.388011\pi$
$710$	0	0
$711$	−6.02705e6	−0.447127
$712$	0	0
$713$	−2.15520e6	−0.158768
$714$	0	0
$715$	2.60568e6	0.190615
$716$	0	0
$717$	−256176.	−0.0186098
$718$	0	0
$719$	2.63923e7	1.90395	0.951975	−	0.306177i	$-0.0990500\pi$
$719$	2.63923e7	1.90395	0.951975	+	0.306177i	$0.0990500\pi$
$720$	0	0
$721$	−1.39505e7	−0.999426
$722$	0	0
$723$	8.03306e6	0.571525
$724$	0	0
$725$	6.86159e6	0.484819
$726$	0	0
$727$	9.79485e6	0.687324	0.343662	−	0.939093i	$-0.388333\pi$
$727$	9.79485e6	0.687324	0.343662	+	0.939093i	$0.388333\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	5.51282e6	0.381576
$732$	0	0
$733$	4.07584e6	0.280193	0.140096	−	0.990138i	$-0.455259\pi$
$733$	4.07584e6	0.280193	0.140096	+	0.990138i	$0.455259\pi$
$734$	0	0
$735$	−8.41639e6	−0.574655
$736$	0	0
$737$	1.00824e6	0.0683747
$738$	0	0
$739$	1.65709e7	1.11618	0.558089	−	0.829781i	$-0.311535\pi$
$739$	1.65709e7	1.11618	0.558089	+	0.829781i	$0.311535\pi$
$740$	0	0
$741$	5.66143e6	0.378775
$742$	0	0
$743$	−1.44141e7	−0.957892	−0.478946	−	0.877844i	$-0.658981\pi$
$743$	−1.44141e7	−0.957892	−0.478946	+	0.877844i	$0.658981\pi$
$744$	0	0
$745$	−1.94396e6	−0.128321
$746$	0	0
$747$	523908.	0.0343521
$748$	0	0
$749$	4.01259e7	2.61349
$750$	0	0
$751$	−1.67944e7	−1.08659	−0.543295	−	0.839542i	$-0.682823\pi$
$751$	−1.67944e7	−1.08659	−0.543295	+	0.839542i	$0.682823\pi$
$752$	0	0
$753$	991116.	0.0636997
$754$	0	0
$755$	−1.34556e7	−0.859081
$756$	0	0
$757$	1.32943e7	0.843188	0.421594	−	0.906785i	$-0.361471\pi$
$757$	1.32943e7	0.843188	0.421594	+	0.906785i	$0.361471\pi$
$758$	0	0
$759$	−324000.	−0.0204146
$760$	0	0
$761$	−2.14786e6	−0.134445	−0.0672225	−	0.997738i	$-0.521414\pi$
$761$	−2.14786e6	−0.134445	−0.0672225	+	0.997738i	$0.521414\pi$
$762$	0	0
$763$	1.50128e6	0.0933577
$764$	0	0
$765$	2.21324e6	0.136734
$766$	0	0
$767$	1.73317e7	1.06378
$768$	0	0
$769$	−1.31059e7	−0.799193	−0.399596	−	0.916691i	$-0.630850\pi$
$769$	−1.31059e7	−0.799193	−0.399596	+	0.916691i	$0.630850\pi$
$770$	0	0
$771$	1.26722e6	0.0767742
$772$	0	0
$773$	−2.37154e7	−1.42752	−0.713759	−	0.700392i	$-0.753009\pi$
$773$	−2.37154e7	−1.42752	−0.713759	+	0.700392i	$0.753009\pi$
$774$	0	0
$775$	4.42175e6	0.264448
$776$	0	0
$777$	1.33975e7	0.796105
$778$	0	0
$779$	−1.83495e7	−1.08338
$780$	0	0
$781$	−367200.	−0.0215415
$782$	0	0
$783$	4.06345e6	0.236859
$784$	0	0
$785$	−8.98537e6	−0.520430
$786$	0	0
$787$	8.40048e6	0.483468	0.241734	−	0.970343i	$-0.422284\pi$
$787$	8.40048e6	0.483468	0.241734	+	0.970343i	$0.422284\pi$
$788$	0	0
$789$	8.44884e6	0.483175
$790$	0	0
$791$	3.43971e7	1.95470
$792$	0	0
$793$	2.04572e7	1.15522
$794$	0	0
$795$	1.85720e7	1.04218
$796$	0	0
$797$	5.41023e6	0.301696	0.150848	−	0.988557i	$-0.451800\pi$
$797$	5.41023e6	0.301696	0.150848	+	0.988557i	$0.451800\pi$
$798$	0	0
$799$	−8.14752e6	−0.451501
$800$	0	0
$801$	−2.65210e6	−0.146052
$802$	0	0
$803$	−1.53348e6	−0.0839246
$804$	0	0
$805$	−6.96960e6	−0.379069
$806$	0	0
$807$	−1.00306e7	−0.542177
$808$	0	0
$809$	−2.60777e7	−1.40087	−0.700436	−	0.713715i	$-0.747011\pi$
$809$	−2.60777e7	−1.40087	−0.700436	+	0.713715i	$0.747011\pi$
$810$	0	0
$811$	−1.90021e7	−1.01449	−0.507247	−	0.861800i	$-0.669337\pi$
$811$	−1.90021e7	−1.01449	−0.507247	+	0.861800i	$0.669337\pi$
$812$	0	0
$813$	−5.10934e6	−0.271105
$814$	0	0
$815$	−1.12942e7	−0.595608
$816$	0	0
$817$	1.27301e7	0.667231
$818$	0	0
$819$	9.38045e6	0.488668
$820$	0	0
$821$	−3.10173e7	−1.60600	−0.803001	−	0.595978i	$-0.796764\pi$
$821$	−3.10173e7	−1.60600	−0.803001	+	0.595978i	$0.796764\pi$
$822$	0	0
$823$	1.56290e7	0.804323	0.402162	−	0.915569i	$-0.368259\pi$
$823$	1.56290e7	0.804323	0.402162	+	0.915569i	$0.368259\pi$
$824$	0	0
$825$	664740.	0.0340030
$826$	0	0
$827$	−1.58421e7	−0.805467	−0.402733	−	0.915317i	$-0.631940\pi$
$827$	−1.58421e7	−0.805467	−0.402733	+	0.915317i	$0.631940\pi$
$828$	0	0
$829$	2.06176e6	0.104196	0.0520980	−	0.998642i	$-0.483409\pi$
$829$	2.06176e6	0.104196	0.0520980	+	0.998642i	$0.483409\pi$
$830$	0	0
$831$	−1.09193e7	−0.548521
$832$	0	0
$833$	−5.86597e6	−0.292905
$834$	0	0
$835$	−4.46292e7	−2.21515
$836$	0	0
$837$	2.61857e6	0.129196
$838$	0	0
$839$	−3.03900e7	−1.49048	−0.745240	−	0.666796i	$-0.767665\pi$
$839$	−3.03900e7	−1.49048	−0.745240	+	0.666796i	$0.767665\pi$
$840$	0	0
$841$	1.05583e7	0.514760
$842$	0	0
$843$	6.18964e6	0.299983
$844$	0	0
$845$	−4.07029e6	−0.196103
$846$	0	0
$847$	2.77114e7	1.32724
$848$	0	0
$849$	7.47817e6	0.356062
$850$	0	0
$851$	5.07480e6	0.240212
$852$	0	0
$853$	−2.97738e7	−1.40108	−0.700538	−	0.713615i	$-0.747056\pi$
$853$	−2.97738e7	−1.40108	−0.700538	+	0.713615i	$0.747056\pi$
$854$	0	0
$855$	5.11078e6	0.239096
$856$	0	0
$857$	8.64100e6	0.401894	0.200947	−	0.979602i	$-0.435598\pi$
$857$	8.64100e6	0.401894	0.200947	+	0.979602i	$0.435598\pi$
$858$	0	0
$859$	3.35663e7	1.55210	0.776051	−	0.630670i	$-0.217220\pi$
$859$	3.35663e7	1.55210	0.776051	+	0.630670i	$0.217220\pi$
$860$	0	0
$861$	−3.04033e7	−1.39770
$862$	0	0
$863$	−3.90191e7	−1.78341	−0.891703	−	0.452621i	$-0.850489\pi$
$863$	−3.90191e7	−1.78341	−0.891703	+	0.452621i	$0.850489\pi$
$864$	0	0
$865$	−8.78843e6	−0.399366
$866$	0	0
$867$	−1.12361e7	−0.507656
$868$	0	0
$869$	−4.46448e6	−0.200549
$870$	0	0
$871$	−1.10570e7	−0.493848
$872$	0	0
$873$	1.34526e7	0.597409
$874$	0	0
$875$	−2.20007e7	−0.971441
$876$	0	0
$877$	−1.81382e7	−0.796333	−0.398166	−	0.917313i	$-0.630353\pi$
$877$	−1.81382e7	−0.796333	−0.398166	+	0.917313i	$0.630353\pi$
$878$	0	0
$879$	−1.18136e7	−0.515717
$880$	0	0
$881$	3.05312e7	1.32527	0.662634	−	0.748943i	$-0.269438\pi$
$881$	3.05312e7	1.32527	0.662634	+	0.748943i	$0.269438\pi$
$882$	0	0
$883$	4.35533e7	1.87983	0.939916	−	0.341405i	$-0.110903\pi$
$883$	4.35533e7	1.87983	0.939916	+	0.341405i	$0.110903\pi$
$884$	0	0
$885$	1.56460e7	0.671497
$886$	0	0
$887$	1.34152e7	0.572515	0.286257	−	0.958153i	$-0.407589\pi$
$887$	1.34152e7	0.572515	0.286257	+	0.958153i	$0.407589\pi$
$888$	0	0
$889$	3.04480e7	1.29212
$890$	0	0
$891$	393660.	0.0166122
$892$	0	0
$893$	−1.88141e7	−0.789504
$894$	0	0
$895$	−4.57641e7	−1.90971
$896$	0	0
$897$	3.55320e6	0.147448
$898$	0	0
$899$	2.00218e7	0.826236
$900$	0	0
$901$	1.29441e7	0.531203
$902$	0	0
$903$	2.10925e7	0.860815
$904$	0	0
$905$	−2.48935e7	−1.01033
$906$	0	0
$907$	−3.10816e6	−0.125454	−0.0627272	−	0.998031i	$-0.519980\pi$
$907$	−3.10816e6	−0.125454	−0.0627272	+	0.998031i	$0.519980\pi$
$908$	0	0
$909$	−1.78216e6	−0.0715381
$910$	0	0
$911$	−1.19035e6	−0.0475203	−0.0237602	−	0.999718i	$-0.507564\pi$
$911$	−1.19035e6	−0.0475203	−0.0237602	+	0.999718i	$0.507564\pi$
$912$	0	0
$913$	388080.	0.0154079
$914$	0	0
$915$	1.84675e7	0.729213
$916$	0	0
$917$	2.66218e7	1.04547
$918$	0	0
$919$	4.71996e7	1.84353	0.921764	−	0.387752i	$-0.126748\pi$
$919$	4.71996e7	1.84353	0.921764	+	0.387752i	$0.126748\pi$
$920$	0	0
$921$	−1.52120e7	−0.590931
$922$	0	0
$923$	4.02696e6	0.155587
$924$	0	0
$925$	−1.04118e7	−0.400103
$926$	0	0
$927$	6.42038e6	0.245393
$928$	0	0
$929$	1.33595e6	0.0507870	0.0253935	−	0.999678i	$-0.491916\pi$
$929$	1.33595e6	0.0507870	0.0253935	+	0.999678i	$0.491916\pi$
$930$	0	0
$931$	−1.35456e7	−0.512180
$932$	0	0
$933$	1.35184e7	0.508417
$934$	0	0
$935$	1.63944e6	0.0613291
$936$	0	0
$937$	1.47238e7	0.547861	0.273931	−	0.961749i	$-0.411676\pi$
$937$	1.47238e7	0.547861	0.273931	+	0.961749i	$0.411676\pi$
$938$	0	0
$939$	7.29758e6	0.270094
$940$	0	0
$941$	−2.69196e7	−0.991049	−0.495525	−	0.868594i	$-0.665024\pi$
$941$	−2.69196e7	−0.991049	−0.495525	+	0.868594i	$0.665024\pi$
$942$	0	0
$943$	−1.15164e7	−0.421733
$944$	0	0
$945$	8.46806e6	0.308464
$946$	0	0
$947$	3.73160e6	0.135214	0.0676068	−	0.997712i	$-0.478464\pi$
$947$	3.73160e6	0.135214	0.0676068	+	0.997712i	$0.478464\pi$
$948$	0	0
$949$	1.68172e7	0.606160
$950$	0	0
$951$	8.13202e6	0.291573
$952$	0	0
$953$	2.18735e7	0.780166	0.390083	−	0.920780i	$-0.372446\pi$
$953$	2.18735e7	0.780166	0.390083	+	0.920780i	$0.372446\pi$
$954$	0	0
$955$	−1.75127e7	−0.621362
$956$	0	0
$957$	3.00996e6	0.106238
$958$	0	0
$959$	2.26079e7	0.793805
$960$	0	0
$961$	−1.57267e7	−0.549324
$962$	0	0
$963$	−1.84670e7	−0.641699
$964$	0	0
$965$	−1.94897e7	−0.673730
$966$	0	0
$967$	−1.76025e7	−0.605352	−0.302676	−	0.953093i	$-0.597880\pi$
$967$	−1.76025e7	−0.605352	−0.302676	+	0.953093i	$0.597880\pi$
$968$	0	0
$969$	3.56206e6	0.121868
$970$	0	0
$971$	−1.67317e7	−0.569497	−0.284749	−	0.958602i	$-0.591910\pi$
$971$	−1.67317e7	−0.569497	−0.284749	+	0.958602i	$0.591910\pi$
$972$	0	0
$973$	2.71385e7	0.918975
$974$	0	0
$975$	−7.28998e6	−0.245592
$976$	0	0
$977$	5.55382e7	1.86147	0.930733	−	0.365699i	$-0.119170\pi$
$977$	5.55382e7	1.86147	0.930733	+	0.365699i	$0.119170\pi$
$978$	0	0
$979$	−1.96452e6	−0.0655088
$980$	0	0
$981$	−690930.	−0.0229225
$982$	0	0
$983$	3.86784e7	1.27669	0.638344	−	0.769751i	$-0.279620\pi$
$983$	3.86784e7	1.27669	0.638344	+	0.769751i	$0.279620\pi$
$984$	0	0
$985$	−1.32854e7	−0.436299
$986$	0	0
$987$	−3.11731e7	−1.01856
$988$	0	0
$989$	7.98960e6	0.259737
$990$	0	0
$991$	−9.58498e6	−0.310033	−0.155016	−	0.987912i	$-0.549543\pi$
$991$	−9.58498e6	−0.310033	−0.155016	+	0.987912i	$0.549543\pi$
$992$	0	0
$993$	−1.00977e7	−0.324976
$994$	0	0
$995$	4.30616e7	1.37890
$996$	0	0
$997$	−1.03650e7	−0.330242	−0.165121	−	0.986273i	$-0.552802\pi$
$997$	−1.03650e7	−0.330242	−0.165121	+	0.986273i	$0.552802\pi$
$998$	0	0
$999$	−6.16588e6	−0.195471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.a.c.1.1		1
3.2	odd	2		144.6.a.j.1.1		1
4.3	odd	2		6.6.a.a.1.1	✓	1
8.3	odd	2		192.6.a.o.1.1		1
8.5	even	2		192.6.a.g.1.1		1
12.11	even	2		18.6.a.b.1.1		1
16.3	odd	4		768.6.d.c.385.1		2
16.5	even	4		768.6.d.p.385.1		2
16.11	odd	4		768.6.d.c.385.2		2
16.13	even	4		768.6.d.p.385.2		2
20.3	even	4		150.6.c.b.49.1		2
20.7	even	4		150.6.c.b.49.2		2
20.19	odd	2		150.6.a.d.1.1		1
24.5	odd	2		576.6.a.i.1.1		1
24.11	even	2		576.6.a.j.1.1		1
28.3	even	6		294.6.e.a.79.1		2
28.11	odd	6		294.6.e.g.79.1		2
28.19	even	6		294.6.e.a.67.1		2
28.23	odd	6		294.6.e.g.67.1		2
28.27	even	2		294.6.a.m.1.1		1
36.7	odd	6		162.6.c.e.109.1		2
36.11	even	6		162.6.c.h.109.1		2
36.23	even	6		162.6.c.h.55.1		2
36.31	odd	6		162.6.c.e.55.1		2
44.43	even	2		726.6.a.a.1.1		1
52.51	odd	2		1014.6.a.c.1.1		1
60.23	odd	4		450.6.c.j.199.2		2
60.47	odd	4		450.6.c.j.199.1		2
60.59	even	2		450.6.a.m.1.1		1
84.83	odd	2		882.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.6.a.a.1.1	✓	1	4.3	odd	2
18.6.a.b.1.1		1	12.11	even	2
48.6.a.c.1.1		1	1.1	even	1	trivial
144.6.a.j.1.1		1	3.2	odd	2
150.6.a.d.1.1		1	20.19	odd	2
150.6.c.b.49.1		2	20.3	even	4
150.6.c.b.49.2		2	20.7	even	4
162.6.c.e.55.1		2	36.31	odd	6
162.6.c.e.109.1		2	36.7	odd	6
162.6.c.h.55.1		2	36.23	even	6
162.6.c.h.109.1		2	36.11	even	6
192.6.a.g.1.1		1	8.5	even	2
192.6.a.o.1.1		1	8.3	odd	2
294.6.a.m.1.1		1	28.27	even	2
294.6.e.a.67.1		2	28.19	even	6
294.6.e.a.79.1		2	28.3	even	6
294.6.e.g.67.1		2	28.23	odd	6
294.6.e.g.79.1		2	28.11	odd	6
450.6.a.m.1.1		1	60.59	even	2
450.6.c.j.199.1		2	60.47	odd	4
450.6.c.j.199.2		2	60.23	odd	4
576.6.a.i.1.1		1	24.5	odd	2
576.6.a.j.1.1		1	24.11	even	2
726.6.a.a.1.1		1	44.43	even	2
768.6.d.c.385.1		2	16.3	odd	4
768.6.d.c.385.2		2	16.11	odd	4
768.6.d.p.385.1		2	16.5	even	4
768.6.d.p.385.2		2	16.13	even	4
882.6.a.a.1.1		1	84.83	odd	2
1014.6.a.c.1.1		1	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more