LMFDB - Embedded newform 128.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,6,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("128.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	128.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:	$20.5291289361$
Analytic rank:	$0$
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	$\chi$	$=$	128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{3} -94.0000 q^{5} -244.000 q^{7} -207.000 q^{9} +O(q^{10})$

$q-6.00000 q^{3} -94.0000 q^{5} -244.000 q^{7} -207.000 q^{9} +358.000 q^{11} +770.000 q^{13} +564.000 q^{15} +670.000 q^{17} -1030.00 q^{19} +1464.00 q^{21} -2828.00 q^{23} +5711.00 q^{25} +2700.00 q^{27} +762.000 q^{29} -4992.00 q^{31} -2148.00 q^{33} +22936.0 q^{35} +3562.00 q^{37} -4620.00 q^{39} +858.000 q^{41} -12786.0 q^{43} +19458.0 q^{45} -3560.00 q^{47} +42729.0 q^{49} -4020.00 q^{51} +9114.00 q^{53} -33652.0 q^{55} +6180.00 q^{57} +8246.00 q^{59} -4414.00 q^{61} +50508.0 q^{63} -72380.0 q^{65} +29986.0 q^{67} +16968.0 q^{69} -49572.0 q^{71} -24370.0 q^{73} -34266.0 q^{75} -87352.0 q^{77} +65176.0 q^{79} +34101.0 q^{81} +39378.0 q^{83} -62980.0 q^{85} -4572.00 q^{87} +11134.0 q^{89} -187880. q^{91} +29952.0 q^{93} +96820.0 q^{95} +478.000 q^{97} -74106.0 q^{99} +O(q^{100})$

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
$3$	−6.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−6.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	−94.0000	−1.68152	−0.840762	−	0.541406i	$-0.817892\pi$
$5$	−94.0000	−1.68152	−0.840762	+	0.541406i	$0.817892\pi$
$6$	0	0
$7$	−244.000	−1.88211	−0.941054	−	0.338255i	$-0.890163\pi$
$7$	−244.000	−1.88211	−0.941054	+	0.338255i	$0.890163\pi$
$8$	0	0
$9$	−207.000	−0.851852
$10$	0	0
$11$	358.000	0.892075	0.446037	−	0.895014i	$-0.352835\pi$
$11$	358.000	0.892075	0.446037	+	0.895014i	$0.352835\pi$
$12$	0	0
$13$	770.000	1.26367	0.631833	−	0.775104i	$-0.282303\pi$
$13$	770.000	1.26367	0.631833	+	0.775104i	$0.282303\pi$
$14$	0	0
$15$	564.000	0.647219
$16$	0	0
$17$	670.000	0.562280	0.281140	−	0.959667i	$-0.409288\pi$
$17$	670.000	0.562280	0.281140	+	0.959667i	$0.409288\pi$
$18$	0	0
$19$	−1030.00	−0.654566	−0.327283	−	0.944926i	$-0.606133\pi$
$19$	−1030.00	−0.654566	−0.327283	+	0.944926i	$0.606133\pi$
$20$	0	0
$21$	1464.00	0.724424
$22$	0	0
$23$	−2828.00	−1.11470	−0.557352	−	0.830276i	$-0.688183\pi$
$23$	−2828.00	−1.11470	−0.557352	+	0.830276i	$0.688183\pi$
$24$	0	0
$25$	5711.00	1.82752
$26$	0	0
$27$	2700.00	0.712778
$28$	0	0
$29$	762.000	0.168252	0.0841259	−	0.996455i	$-0.473190\pi$
$29$	762.000	0.168252	0.0841259	+	0.996455i	$0.473190\pi$
$30$	0	0
$31$	−4992.00	−0.932976	−0.466488	−	0.884528i	$-0.654481\pi$
$31$	−4992.00	−0.932976	−0.466488	+	0.884528i	$0.654481\pi$
$32$	0	0
$33$	−2148.00	−0.343360
$34$	0	0
$35$	22936.0	3.16481
$36$	0	0
$37$	3562.00	0.427750	0.213875	−	0.976861i	$-0.431392\pi$
$37$	3562.00	0.427750	0.213875	+	0.976861i	$0.431392\pi$
$38$	0	0
$39$	−4620.00	−0.486385
$40$	0	0
$41$	858.000	0.0797127	0.0398564	−	0.999205i	$-0.487310\pi$
$41$	858.000	0.0797127	0.0398564	+	0.999205i	$0.487310\pi$
$42$	0	0
$43$	−12786.0	−1.05454	−0.527271	−	0.849697i	$-0.676785\pi$
$43$	−12786.0	−1.05454	−0.527271	+	0.849697i	$0.676785\pi$
$44$	0	0
$45$	19458.0	1.43241
$46$	0	0
$47$	−3560.00	−0.235074	−0.117537	−	0.993068i	$-0.537500\pi$
$47$	−3560.00	−0.235074	−0.117537	+	0.993068i	$0.537500\pi$
$48$	0	0
$49$	42729.0	2.54233
$50$	0	0
$51$	−4020.00	−0.216422
$52$	0	0
$53$	9114.00	0.445676	0.222838	−	0.974855i	$-0.428468\pi$
$53$	9114.00	0.445676	0.222838	+	0.974855i	$0.428468\pi$
$54$	0	0
$55$	−33652.0	−1.50004
$56$	0	0
$57$	6180.00	0.251942
$58$	0	0
$59$	8246.00	0.308399	0.154200	−	0.988040i	$-0.450720\pi$
$59$	8246.00	0.308399	0.154200	+	0.988040i	$0.450720\pi$
$60$	0	0
$61$	−4414.00	−0.151883	−0.0759413	−	0.997112i	$-0.524196\pi$
$61$	−4414.00	−0.151883	−0.0759413	+	0.997112i	$0.524196\pi$
$62$	0	0
$63$	50508.0	1.60328
$64$	0	0
$65$	−72380.0	−2.12488
$66$	0	0
$67$	29986.0	0.816078	0.408039	−	0.912965i	$-0.366213\pi$
$67$	29986.0	0.816078	0.408039	+	0.912965i	$0.366213\pi$
$68$	0	0
$69$	16968.0	0.429050
$70$	0	0
$71$	−49572.0	−1.16705	−0.583526	−	0.812094i	$-0.698328\pi$
$71$	−49572.0	−1.16705	−0.583526	+	0.812094i	$0.698328\pi$
$72$	0	0
$73$	−24370.0	−0.535240	−0.267620	−	0.963525i	$-0.586237\pi$
$73$	−24370.0	−0.535240	−0.267620	+	0.963525i	$0.586237\pi$
$74$	0	0
$75$	−34266.0	−0.703413
$76$	0	0
$77$	−87352.0	−1.67898
$78$	0	0
$79$	65176.0	1.17495	0.587476	−	0.809242i	$-0.300122\pi$
$79$	65176.0	1.17495	0.587476	+	0.809242i	$0.300122\pi$
$80$	0	0
$81$	34101.0	0.577503
$82$	0	0
$83$	39378.0	0.627420	0.313710	−	0.949519i	$-0.398428\pi$
$83$	39378.0	0.627420	0.313710	+	0.949519i	$0.398428\pi$
$84$	0	0
$85$	−62980.0	−0.945487
$86$	0	0
$87$	−4572.00	−0.0647602
$88$	0	0
$89$	11134.0	0.148997	0.0744983	−	0.997221i	$-0.476264\pi$
$89$	11134.0	0.148997	0.0744983	+	0.997221i	$0.476264\pi$
$90$	0	0
$91$	−187880.	−2.37836
$92$	0	0
$93$	29952.0	0.359103
$94$	0	0
$95$	96820.0	1.10067
$96$	0	0
$97$	478.000	0.00515820	0.00257910	−	0.999997i	$-0.499179\pi$
$97$	478.000	0.00515820	0.00257910	+	0.999997i	$0.499179\pi$
$98$	0	0
$99$	−74106.0	−0.759916
$100$	0	0
$101$	−53670.0	−0.523514	−0.261757	−	0.965134i	$-0.584302\pi$
$101$	−53670.0	−0.523514	−0.261757	+	0.965134i	$0.584302\pi$
$102$	0	0
$103$	194444.	1.80593	0.902966	−	0.429712i	$-0.141385\pi$
$103$	194444.	1.80593	0.902966	+	0.429712i	$0.141385\pi$
$104$	0	0
$105$	−137616.	−1.21814
$106$	0	0
$107$	−65690.0	−0.554677	−0.277338	−	0.960772i	$-0.589452\pi$
$107$	−65690.0	−0.554677	−0.277338	+	0.960772i	$0.589452\pi$
$108$	0	0
$109$	110394.	0.889978	0.444989	−	0.895536i	$-0.353208\pi$
$109$	110394.	0.889978	0.444989	+	0.895536i	$0.353208\pi$
$110$	0	0
$111$	−21372.0	−0.164641
$112$	0	0
$113$	−196414.	−1.44703	−0.723513	−	0.690311i	$-0.757474\pi$
$113$	−196414.	−1.44703	−0.723513	+	0.690311i	$0.757474\pi$
$114$	0	0
$115$	265832.	1.87440
$116$	0	0
$117$	−159390.	−1.07646
$118$	0	0
$119$	−163480.	−1.05827
$120$	0	0
$121$	−32887.0	−0.204202
$122$	0	0
$123$	−5148.00	−0.0306814
$124$	0	0
$125$	−243084.	−1.39149
$126$	0	0
$127$	−107696.	−0.592503	−0.296251	−	0.955110i	$-0.595737\pi$
$127$	−107696.	−0.592503	−0.296251	+	0.955110i	$0.595737\pi$
$128$	0	0
$129$	76716.0	0.405893
$130$	0	0
$131$	−286254.	−1.45738	−0.728691	−	0.684843i	$-0.759871\pi$
$131$	−286254.	−1.45738	−0.728691	+	0.684843i	$0.759871\pi$
$132$	0	0
$133$	251320.	1.23196
$134$	0	0
$135$	−253800.	−1.19855
$136$	0	0
$137$	383722.	1.74669	0.873344	−	0.487104i	$-0.161947\pi$
$137$	383722.	1.74669	0.873344	+	0.487104i	$0.161947\pi$
$138$	0	0
$139$	200126.	0.878550	0.439275	−	0.898353i	$-0.355235\pi$
$139$	200126.	0.878550	0.439275	+	0.898353i	$0.355235\pi$
$140$	0	0
$141$	21360.0	0.0904802
$142$	0	0
$143$	275660.	1.12728
$144$	0	0
$145$	−71628.0	−0.282919
$146$	0	0
$147$	−256374.	−0.978545
$148$	0	0
$149$	215922.	0.796767	0.398383	−	0.917219i	$-0.369571\pi$
$149$	215922.	0.796767	0.398383	+	0.917219i	$0.369571\pi$
$150$	0	0
$151$	109300.	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$151$	109300.	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$152$	0	0
$153$	−138690.	−0.478979
$154$	0	0
$155$	469248.	1.56882
$156$	0	0
$157$	289746.	0.938141	0.469071	−	0.883161i	$-0.344589\pi$
$157$	289746.	0.938141	0.469071	+	0.883161i	$0.344589\pi$
$158$	0	0
$159$	−54684.0	−0.171541
$160$	0	0
$161$	690032.	2.09800
$162$	0	0
$163$	540410.	1.59314	0.796571	−	0.604545i	$-0.206645\pi$
$163$	540410.	1.59314	0.796571	+	0.604545i	$0.206645\pi$
$164$	0	0
$165$	201912.	0.577367
$166$	0	0
$167$	395836.	1.09831	0.549154	−	0.835721i	$-0.314950\pi$
$167$	395836.	1.09831	0.549154	+	0.835721i	$0.314950\pi$
$168$	0	0
$169$	221607.	0.596852
$170$	0	0
$171$	213210.	0.557593
$172$	0	0
$173$	551442.	1.40083	0.700414	−	0.713737i	$-0.252999\pi$
$173$	551442.	1.40083	0.700414	+	0.713737i	$0.252999\pi$
$174$	0	0
$175$	−1.39348e6	−3.43959
$176$	0	0
$177$	−49476.0	−0.118703
$178$	0	0
$179$	−469806.	−1.09594	−0.547969	−	0.836499i	$-0.684599\pi$
$179$	−469806.	−1.09594	−0.547969	+	0.836499i	$0.684599\pi$
$180$	0	0
$181$	−729502.	−1.65512	−0.827561	−	0.561376i	$-0.810272\pi$
$181$	−729502.	−1.65512	−0.827561	+	0.561376i	$0.810272\pi$
$182$	0	0
$183$	26484.0	0.0584596
$184$	0	0
$185$	−334828.	−0.719271
$186$	0	0
$187$	239860.	0.501596
$188$	0	0
$189$	−658800.	−1.34153
$190$	0	0
$191$	−532048.	−1.05528	−0.527640	−	0.849468i	$-0.676923\pi$
$191$	−532048.	−1.05528	−0.527640	+	0.849468i	$0.676923\pi$
$192$	0	0
$193$	195934.	0.378631	0.189316	−	0.981916i	$-0.439373\pi$
$193$	195934.	0.378631	0.189316	+	0.981916i	$0.439373\pi$
$194$	0	0
$195$	434280.	0.817868
$196$	0	0
$197$	683794.	1.25534	0.627668	−	0.778481i	$-0.284010\pi$
$197$	683794.	1.25534	0.627668	+	0.778481i	$0.284010\pi$
$198$	0	0
$199$	−186868.	−0.334505	−0.167252	−	0.985914i	$-0.553489\pi$
$199$	−186868.	−0.334505	−0.167252	+	0.985914i	$0.553489\pi$
$200$	0	0
$201$	−179916.	−0.314108
$202$	0	0
$203$	−185928.	−0.316668
$204$	0	0
$205$	−80652.0	−0.134039
$206$	0	0
$207$	585396.	0.949563
$208$	0	0
$209$	−368740.	−0.583922
$210$	0	0
$211$	−496510.	−0.767754	−0.383877	−	0.923384i	$-0.625411\pi$
$211$	−496510.	−0.767754	−0.383877	+	0.923384i	$0.625411\pi$
$212$	0	0
$213$	297432.	0.449199
$214$	0	0
$215$	1.20188e6	1.77324
$216$	0	0
$217$	1.21805e6	1.75596
$218$	0	0
$219$	146220.	0.206014
$220$	0	0
$221$	515900.	0.710534
$222$	0	0
$223$	26496.0	0.0356795	0.0178397	−	0.999841i	$-0.494321\pi$
$223$	26496.0	0.0356795	0.0178397	+	0.999841i	$0.494321\pi$
$224$	0	0
$225$	−1.18218e6	−1.55678
$226$	0	0
$227$	−1.06181e6	−1.36768	−0.683839	−	0.729633i	$-0.739691\pi$
$227$	−1.06181e6	−1.36768	−0.683839	+	0.729633i	$0.739691\pi$
$228$	0	0
$229$	1.13423e6	1.42926	0.714629	−	0.699503i	$-0.246595\pi$
$229$	1.13423e6	1.42926	0.714629	+	0.699503i	$0.246595\pi$
$230$	0	0
$231$	524112.	0.646240
$232$	0	0
$233$	−768210.	−0.927022	−0.463511	−	0.886091i	$-0.653411\pi$
$233$	−768210.	−0.927022	−0.463511	+	0.886091i	$0.653411\pi$
$234$	0	0
$235$	334640.	0.395283
$236$	0	0
$237$	−391056.	−0.452239
$238$	0	0
$239$	419784.	0.475369	0.237685	−	0.971342i	$-0.423612\pi$
$239$	419784.	0.475369	0.237685	+	0.971342i	$0.423612\pi$
$240$	0	0
$241$	880414.	0.976437	0.488219	−	0.872721i	$-0.337647\pi$
$241$	880414.	0.976437	0.488219	+	0.872721i	$0.337647\pi$
$242$	0	0
$243$	−860706.	−0.935059
$244$	0	0
$245$	−4.01653e6	−4.27499
$246$	0	0
$247$	−793100.	−0.827152
$248$	0	0
$249$	−236268.	−0.241494
$250$	0	0
$251$	16278.0	0.0163086	0.00815430	−	0.999967i	$-0.497404\pi$
$251$	16278.0	0.0163086	0.00815430	+	0.999967i	$0.497404\pi$
$252$	0	0
$253$	−1.01242e6	−0.994400
$254$	0	0
$255$	377880.	0.363918
$256$	0	0
$257$	−1.26257e6	−1.19241	−0.596203	−	0.802834i	$-0.703325\pi$
$257$	−1.26257e6	−1.19241	−0.596203	+	0.802834i	$0.703325\pi$
$258$	0	0
$259$	−869128.	−0.805071
$260$	0	0
$261$	−157734.	−0.143326
$262$	0	0
$263$	−402084.	−0.358449	−0.179225	−	0.983808i	$-0.557359\pi$
$263$	−402084.	−0.358449	−0.179225	+	0.983808i	$0.557359\pi$
$264$	0	0
$265$	−856716.	−0.749415
$266$	0	0
$267$	−66804.0	−0.0573488
$268$	0	0
$269$	115786.	0.0975608	0.0487804	−	0.998810i	$-0.484467\pi$
$269$	115786.	0.0975608	0.0487804	+	0.998810i	$0.484467\pi$
$270$	0	0
$271$	1.32375e6	1.09492	0.547461	−	0.836831i	$-0.315594\pi$
$271$	1.32375e6	1.09492	0.547461	+	0.836831i	$0.315594\pi$
$272$	0	0
$273$	1.12728e6	0.915430
$274$	0	0
$275$	2.04454e6	1.63028
$276$	0	0
$277$	971802.	0.760989	0.380494	−	0.924783i	$-0.375754\pi$
$277$	971802.	0.760989	0.380494	+	0.924783i	$0.375754\pi$
$278$	0	0
$279$	1.03334e6	0.794757
$280$	0	0
$281$	−2.00680e6	−1.51614	−0.758069	−	0.652174i	$-0.773857\pi$
$281$	−2.00680e6	−1.51614	−0.758069	+	0.652174i	$0.773857\pi$
$282$	0	0
$283$	−742770.	−0.551300	−0.275650	−	0.961258i	$-0.588893\pi$
$283$	−742770.	−0.551300	−0.275650	+	0.961258i	$0.588893\pi$
$284$	0	0
$285$	−580920.	−0.423647
$286$	0	0
$287$	−209352.	−0.150028
$288$	0	0
$289$	−970957.	−0.683841
$290$	0	0
$291$	−2868.00	−0.00198539
$292$	0	0
$293$	1.22639e6	0.834561	0.417281	−	0.908778i	$-0.362983\pi$
$293$	1.22639e6	0.834561	0.417281	+	0.908778i	$0.362983\pi$
$294$	0	0
$295$	−775124.	−0.518580
$296$	0	0
$297$	966600.	0.635851
$298$	0	0
$299$	−2.17756e6	−1.40861
$300$	0	0
$301$	3.11978e6	1.98476
$302$	0	0
$303$	322020.	0.201501
$304$	0	0
$305$	414916.	0.255394
$306$	0	0
$307$	2.82734e6	1.71211	0.856055	−	0.516884i	$-0.172908\pi$
$307$	2.82734e6	1.71211	0.856055	+	0.516884i	$0.172908\pi$
$308$	0	0
$309$	−1.16666e6	−0.695104
$310$	0	0
$311$	772052.	0.452632	0.226316	−	0.974054i	$-0.427332\pi$
$311$	772052.	0.452632	0.226316	+	0.974054i	$0.427332\pi$
$312$	0	0
$313$	3.15486e6	1.82020	0.910100	−	0.414389i	$-0.136005\pi$
$313$	3.15486e6	1.82020	0.910100	+	0.414389i	$0.136005\pi$
$314$	0	0
$315$	−4.74775e6	−2.69595
$316$	0	0
$317$	−3.50861e6	−1.96104	−0.980522	−	0.196408i	$-0.937072\pi$
$317$	−3.50861e6	−1.96104	−0.980522	+	0.196408i	$0.937072\pi$
$318$	0	0
$319$	272796.	0.150093
$320$	0	0
$321$	394140.	0.213495
$322$	0	0
$323$	−690100.	−0.368049
$324$	0	0
$325$	4.39747e6	2.30938
$326$	0	0
$327$	−662364.	−0.342553
$328$	0	0
$329$	868640.	0.442436
$330$	0	0
$331$	2.98540e6	1.49773	0.748863	−	0.662725i	$-0.230600\pi$
$331$	2.98540e6	1.49773	0.748863	+	0.662725i	$0.230600\pi$
$332$	0	0
$333$	−737334.	−0.364379
$334$	0	0
$335$	−2.81868e6	−1.37225
$336$	0	0
$337$	136114.	0.0652872	0.0326436	−	0.999467i	$-0.489607\pi$
$337$	136114.	0.0652872	0.0326436	+	0.999467i	$0.489607\pi$
$338$	0	0
$339$	1.17848e6	0.556961
$340$	0	0
$341$	−1.78714e6	−0.832284
$342$	0	0
$343$	−6.32497e6	−2.90284
$344$	0	0
$345$	−1.59499e6	−0.721458
$346$	0	0
$347$	4.01295e6	1.78912	0.894561	−	0.446945i	$-0.147488\pi$
$347$	4.01295e6	1.78912	0.894561	+	0.446945i	$0.147488\pi$
$348$	0	0
$349$	2.71835e6	1.19465	0.597326	−	0.801999i	$-0.296230\pi$
$349$	2.71835e6	1.19465	0.597326	+	0.801999i	$0.296230\pi$
$350$	0	0
$351$	2.07900e6	0.900714
$352$	0	0
$353$	992274.	0.423833	0.211917	−	0.977288i	$-0.432029\pi$
$353$	992274.	0.423833	0.211917	+	0.977288i	$0.432029\pi$
$354$	0	0
$355$	4.65977e6	1.96243
$356$	0	0
$357$	980880.	0.407329
$358$	0	0
$359$	−2.39372e6	−0.980249	−0.490125	−	0.871652i	$-0.663049\pi$
$359$	−2.39372e6	−0.980249	−0.490125	+	0.871652i	$0.663049\pi$
$360$	0	0
$361$	−1.41520e6	−0.571544
$362$	0	0
$363$	197322.	0.0785975
$364$	0	0
$365$	2.29078e6	0.900018
$366$	0	0
$367$	3.41788e6	1.32462	0.662311	−	0.749229i	$-0.269576\pi$
$367$	3.41788e6	1.32462	0.662311	+	0.749229i	$0.269576\pi$
$368$	0	0
$369$	−177606.	−0.0679034
$370$	0	0
$371$	−2.22382e6	−0.838811
$372$	0	0
$373$	−2.86357e6	−1.06570	−0.532852	−	0.846209i	$-0.678880\pi$
$373$	−2.86357e6	−1.06570	−0.532852	+	0.846209i	$0.678880\pi$
$374$	0	0
$375$	1.45850e6	0.535586
$376$	0	0
$377$	586740.	0.212614
$378$	0	0
$379$	2.60503e6	0.931568	0.465784	−	0.884898i	$-0.345772\pi$
$379$	2.60503e6	0.931568	0.465784	+	0.884898i	$0.345772\pi$
$380$	0	0
$381$	646176.	0.228054
$382$	0	0
$383$	1.76333e6	0.614237	0.307119	−	0.951671i	$-0.400635\pi$
$383$	1.76333e6	0.614237	0.307119	+	0.951671i	$0.400635\pi$
$384$	0	0
$385$	8.21109e6	2.82325
$386$	0	0
$387$	2.64670e6	0.898313
$388$	0	0
$389$	3.32939e6	1.11555	0.557776	−	0.829991i	$-0.311655\pi$
$389$	3.32939e6	1.11555	0.557776	+	0.829991i	$0.311655\pi$
$390$	0	0
$391$	−1.89476e6	−0.626776
$392$	0	0
$393$	1.71752e6	0.560946
$394$	0	0
$395$	−6.12654e6	−1.97571
$396$	0	0
$397$	2.26886e6	0.722489	0.361244	−	0.932471i	$-0.382352\pi$
$397$	2.26886e6	0.722489	0.361244	+	0.932471i	$0.382352\pi$
$398$	0	0
$399$	−1.50792e6	−0.474183
$400$	0	0
$401$	−5.33920e6	−1.65812	−0.829059	−	0.559161i	$-0.811123\pi$
$401$	−5.33920e6	−1.65812	−0.829059	+	0.559161i	$0.811123\pi$
$402$	0	0
$403$	−3.84384e6	−1.17897
$404$	0	0
$405$	−3.20549e6	−0.971085
$406$	0	0
$407$	1.27520e6	0.381585
$408$	0	0
$409$	3.65367e6	1.07999	0.539997	−	0.841667i	$-0.318425\pi$
$409$	3.65367e6	1.07999	0.539997	+	0.841667i	$0.318425\pi$
$410$	0	0
$411$	−2.30233e6	−0.672300
$412$	0	0
$413$	−2.01202e6	−0.580441
$414$	0	0
$415$	−3.70153e6	−1.05502
$416$	0	0
$417$	−1.20076e6	−0.338154
$418$	0	0
$419$	95090.0	0.0264606	0.0132303	−	0.999912i	$-0.495789\pi$
$419$	95090.0	0.0264606	0.0132303	+	0.999912i	$0.495789\pi$
$420$	0	0
$421$	1.16384e6	0.320029	0.160014	−	0.987115i	$-0.448846\pi$
$421$	1.16384e6	0.320029	0.160014	+	0.987115i	$0.448846\pi$
$422$	0	0
$423$	736920.	0.200249
$424$	0	0
$425$	3.82637e6	1.02758
$426$	0	0
$427$	1.07702e6	0.285859
$428$	0	0
$429$	−1.65396e6	−0.433892
$430$	0	0
$431$	1.59335e6	0.413160	0.206580	−	0.978430i	$-0.433767\pi$
$431$	1.59335e6	0.413160	0.206580	+	0.978430i	$0.433767\pi$
$432$	0	0
$433$	−4.29127e6	−1.09993	−0.549966	−	0.835187i	$-0.685359\pi$
$433$	−4.29127e6	−1.09993	−0.549966	+	0.835187i	$0.685359\pi$
$434$	0	0
$435$	429768.	0.108896
$436$	0	0
$437$	2.91284e6	0.729647
$438$	0	0
$439$	−829164.	−0.205343	−0.102671	−	0.994715i	$-0.532739\pi$
$439$	−829164.	−0.205343	−0.102671	+	0.994715i	$0.532739\pi$
$440$	0	0
$441$	−8.84490e6	−2.16569
$442$	0	0
$443$	3.67477e6	0.889652	0.444826	−	0.895617i	$-0.353265\pi$
$443$	3.67477e6	0.889652	0.444826	+	0.895617i	$0.353265\pi$
$444$	0	0
$445$	−1.04660e6	−0.250541
$446$	0	0
$447$	−1.29553e6	−0.306676
$448$	0	0
$449$	−4.55034e6	−1.06519	−0.532596	−	0.846370i	$-0.678783\pi$
$449$	−4.55034e6	−1.06519	−0.532596	+	0.846370i	$0.678783\pi$
$450$	0	0
$451$	307164.	0.0711097
$452$	0	0
$453$	−655800.	−0.150150
$454$	0	0
$455$	1.76607e7	3.99926
$456$	0	0
$457$	−1.53629e6	−0.344099	−0.172050	−	0.985088i	$-0.555039\pi$
$457$	−1.53629e6	−0.344099	−0.172050	+	0.985088i	$0.555039\pi$
$458$	0	0
$459$	1.80900e6	0.400781
$460$	0	0
$461$	−5.82393e6	−1.27633	−0.638166	−	0.769899i	$-0.720307\pi$
$461$	−5.82393e6	−1.27633	−0.638166	+	0.769899i	$0.720307\pi$
$462$	0	0
$463$	−6.25086e6	−1.35515	−0.677574	−	0.735454i	$-0.736969\pi$
$463$	−6.25086e6	−1.35515	−0.677574	+	0.735454i	$0.736969\pi$
$464$	0	0
$465$	−2.81549e6	−0.603839
$466$	0	0
$467$	−4.73020e6	−1.00366	−0.501831	−	0.864966i	$-0.667340\pi$
$467$	−4.73020e6	−1.00366	−0.501831	+	0.864966i	$0.667340\pi$
$468$	0	0
$469$	−7.31658e6	−1.53595
$470$	0	0
$471$	−1.73848e6	−0.361091
$472$	0	0
$473$	−4.57739e6	−0.940730
$474$	0	0
$475$	−5.88233e6	−1.19623
$476$	0	0
$477$	−1.88660e6	−0.379650
$478$	0	0
$479$	−2.07014e6	−0.412251	−0.206126	−	0.978526i	$-0.566086\pi$
$479$	−2.07014e6	−0.412251	−0.206126	+	0.978526i	$0.566086\pi$
$480$	0	0
$481$	2.74274e6	0.540533
$482$	0	0
$483$	−4.14019e6	−0.807519
$484$	0	0
$485$	−44932.0	−0.00867364
$486$	0	0
$487$	−4.44026e6	−0.848371	−0.424186	−	0.905575i	$-0.639440\pi$
$487$	−4.44026e6	−0.848371	−0.424186	+	0.905575i	$0.639440\pi$
$488$	0	0
$489$	−3.24246e6	−0.613200
$490$	0	0
$491$	−600130.	−0.112342	−0.0561709	−	0.998421i	$-0.517889\pi$
$491$	−600130.	−0.112342	−0.0561709	+	0.998421i	$0.517889\pi$
$492$	0	0
$493$	510540.	0.0946046
$494$	0	0
$495$	6.96596e6	1.27782
$496$	0	0
$497$	1.20956e7	2.19652
$498$	0	0
$499$	1.07866e6	0.193924	0.0969622	−	0.995288i	$-0.469087\pi$
$499$	1.07866e6	0.193924	0.0969622	+	0.995288i	$0.469087\pi$
$500$	0	0
$501$	−2.37502e6	−0.422739
$502$	0	0
$503$	7.93082e6	1.39765	0.698825	−	0.715293i	$-0.253707\pi$
$503$	7.93082e6	1.39765	0.698825	+	0.715293i	$0.253707\pi$
$504$	0	0
$505$	5.04498e6	0.880301
$506$	0	0
$507$	−1.32964e6	−0.229728
$508$	0	0
$509$	4.30213e6	0.736019	0.368010	−	0.929822i	$-0.380039\pi$
$509$	4.30213e6	0.736019	0.368010	+	0.929822i	$0.380039\pi$
$510$	0	0
$511$	5.94628e6	1.00738
$512$	0	0
$513$	−2.78100e6	−0.466560
$514$	0	0
$515$	−1.82777e7	−3.03672
$516$	0	0
$517$	−1.27448e6	−0.209704
$518$	0	0
$519$	−3.30865e6	−0.539179
$520$	0	0
$521$	1.80372e6	0.291122	0.145561	−	0.989349i	$-0.453501\pi$
$521$	1.80372e6	0.291122	0.145561	+	0.989349i	$0.453501\pi$
$522$	0	0
$523$	1.03090e7	1.64803	0.824013	−	0.566570i	$-0.191730\pi$
$523$	1.03090e7	1.64803	0.824013	+	0.566570i	$0.191730\pi$
$524$	0	0
$525$	8.36090e6	1.32390
$526$	0	0
$527$	−3.34464e6	−0.524593
$528$	0	0
$529$	1.56124e6	0.242566
$530$	0	0
$531$	−1.70692e6	−0.262710
$532$	0	0
$533$	660660.	0.100730
$534$	0	0
$535$	6.17486e6	0.932701
$536$	0	0
$537$	2.81884e6	0.421827
$538$	0	0
$539$	1.52970e7	2.26795
$540$	0	0
$541$	−1.24387e7	−1.82718	−0.913590	−	0.406637i	$-0.866702\pi$
$541$	−1.24387e7	−1.82718	−0.913590	+	0.406637i	$0.866702\pi$
$542$	0	0
$543$	4.37701e6	0.637057
$544$	0	0
$545$	−1.03770e7	−1.49652
$546$	0	0
$547$	4.25221e6	0.607640	0.303820	−	0.952729i	$-0.401738\pi$
$547$	4.25221e6	0.607640	0.303820	+	0.952729i	$0.401738\pi$
$548$	0	0
$549$	913698.	0.129381
$550$	0	0
$551$	−784860.	−0.110132
$552$	0	0
$553$	−1.59029e7	−2.21139
$554$	0	0
$555$	2.00897e6	0.276847
$556$	0	0
$557$	−3.85574e6	−0.526587	−0.263293	−	0.964716i	$-0.584809\pi$
$557$	−3.85574e6	−0.526587	−0.263293	+	0.964716i	$0.584809\pi$
$558$	0	0
$559$	−9.84522e6	−1.33259
$560$	0	0
$561$	−1.43916e6	−0.193064
$562$	0	0
$563$	4.77499e6	0.634894	0.317447	−	0.948276i	$-0.397174\pi$
$563$	4.77499e6	0.634894	0.317447	+	0.948276i	$0.397174\pi$
$564$	0	0
$565$	1.84629e7	2.43321
$566$	0	0
$567$	−8.32064e6	−1.08692
$568$	0	0
$569$	4.67014e6	0.604713	0.302356	−	0.953195i	$-0.402227\pi$
$569$	4.67014e6	0.604713	0.302356	+	0.953195i	$0.402227\pi$
$570$	0	0
$571$	−8.71957e6	−1.11919	−0.559596	−	0.828765i	$-0.689044\pi$
$571$	−8.71957e6	−1.11919	−0.559596	+	0.828765i	$0.689044\pi$
$572$	0	0
$573$	3.19229e6	0.406177
$574$	0	0
$575$	−1.61507e7	−2.03715
$576$	0	0
$577$	−243838.	−0.0304903	−0.0152452	−	0.999884i	$-0.504853\pi$
$577$	−243838.	−0.0304903	−0.0152452	+	0.999884i	$0.504853\pi$
$578$	0	0
$579$	−1.17560e6	−0.145735
$580$	0	0
$581$	−9.60823e6	−1.18087
$582$	0	0
$583$	3.26281e6	0.397576
$584$	0	0
$585$	1.49827e7	1.81009
$586$	0	0
$587$	5.70388e6	0.683243	0.341621	−	0.939838i	$-0.389024\pi$
$587$	5.70388e6	0.683243	0.341621	+	0.939838i	$0.389024\pi$
$588$	0	0
$589$	5.14176e6	0.610694
$590$	0	0
$591$	−4.10276e6	−0.483179
$592$	0	0
$593$	9.27343e6	1.08294	0.541469	−	0.840721i	$-0.317868\pi$
$593$	9.27343e6	1.08294	0.541469	+	0.840721i	$0.317868\pi$
$594$	0	0
$595$	1.53671e7	1.77951
$596$	0	0
$597$	1.12121e6	0.128751
$598$	0	0
$599$	−4.39408e6	−0.500381	−0.250190	−	0.968197i	$-0.580493\pi$
$599$	−4.39408e6	−0.500381	−0.250190	+	0.968197i	$0.580493\pi$
$600$	0	0
$601$	1.40871e7	1.59087	0.795435	−	0.606039i	$-0.207243\pi$
$601$	1.40871e7	1.59087	0.795435	+	0.606039i	$0.207243\pi$
$602$	0	0
$603$	−6.20710e6	−0.695177
$604$	0	0
$605$	3.09138e6	0.343371
$606$	0	0
$607$	−4.33403e6	−0.477442	−0.238721	−	0.971088i	$-0.576728\pi$
$607$	−4.33403e6	−0.477442	−0.238721	+	0.971088i	$0.576728\pi$
$608$	0	0
$609$	1.11557e6	0.121886
$610$	0	0
$611$	−2.74120e6	−0.297056
$612$	0	0
$613$	5.90471e6	0.634669	0.317334	−	0.948314i	$-0.397212\pi$
$613$	5.90471e6	0.634669	0.317334	+	0.948314i	$0.397212\pi$
$614$	0	0
$615$	483912.	0.0515916
$616$	0	0
$617$	−6.25210e6	−0.661170	−0.330585	−	0.943776i	$-0.607246\pi$
$617$	−6.25210e6	−0.661170	−0.330585	+	0.943776i	$0.607246\pi$
$618$	0	0
$619$	−1.30201e7	−1.36580	−0.682900	−	0.730512i	$-0.739281\pi$
$619$	−1.30201e7	−1.36580	−0.682900	+	0.730512i	$0.739281\pi$
$620$	0	0
$621$	−7.63560e6	−0.794537
$622$	0	0
$623$	−2.71670e6	−0.280428
$624$	0	0
$625$	5.00302e6	0.512309
$626$	0	0
$627$	2.21244e6	0.224752
$628$	0	0
$629$	2.38654e6	0.240515
$630$	0	0
$631$	1.05519e7	1.05502	0.527508	−	0.849550i	$-0.323127\pi$
$631$	1.05519e7	1.05502	0.527508	+	0.849550i	$0.323127\pi$
$632$	0	0
$633$	2.97906e6	0.295508
$634$	0	0
$635$	1.01234e7	0.996307
$636$	0	0
$637$	3.29013e7	3.21266
$638$	0	0
$639$	1.02614e7	0.994156
$640$	0	0
$641$	426366.	0.0409862	0.0204931	−	0.999790i	$-0.493476\pi$
$641$	426366.	0.0409862	0.0204931	+	0.999790i	$0.493476\pi$
$642$	0	0
$643$	1.93325e7	1.84400	0.922001	−	0.387186i	$-0.126553\pi$
$643$	1.93325e7	1.84400	0.922001	+	0.387186i	$0.126553\pi$
$644$	0	0
$645$	−7.21130e6	−0.682519
$646$	0	0
$647$	−1.88126e7	−1.76680	−0.883401	−	0.468619i	$-0.844752\pi$
$647$	−1.88126e7	−1.76680	−0.883401	+	0.468619i	$0.844752\pi$
$648$	0	0
$649$	2.95207e6	0.275115
$650$	0	0
$651$	−7.30829e6	−0.675870
$652$	0	0
$653$	1.17057e7	1.07427	0.537136	−	0.843495i	$-0.319506\pi$
$653$	1.17057e7	1.07427	0.537136	+	0.843495i	$0.319506\pi$
$654$	0	0
$655$	2.69079e7	2.45062
$656$	0	0
$657$	5.04459e6	0.455945
$658$	0	0
$659$	−8.70814e6	−0.781110	−0.390555	−	0.920580i	$-0.627717\pi$
$659$	−8.70814e6	−0.781110	−0.390555	+	0.920580i	$0.627717\pi$
$660$	0	0
$661$	753922.	0.0671155	0.0335577	−	0.999437i	$-0.489316\pi$
$661$	753922.	0.0671155	0.0335577	+	0.999437i	$0.489316\pi$
$662$	0	0
$663$	−3.09540e6	−0.273485
$664$	0	0
$665$	−2.36241e7	−2.07158
$666$	0	0
$667$	−2.15494e6	−0.187551
$668$	0	0
$669$	−158976.	−0.0137330
$670$	0	0
$671$	−1.58021e6	−0.135491
$672$	0	0
$673$	1.55952e6	0.132725	0.0663625	−	0.997796i	$-0.478861\pi$
$673$	1.55952e6	0.132725	0.0663625	+	0.997796i	$0.478861\pi$
$674$	0	0
$675$	1.54197e7	1.30262
$676$	0	0
$677$	−1.47183e7	−1.23420	−0.617099	−	0.786885i	$-0.711692\pi$
$677$	−1.47183e7	−1.23420	−0.617099	+	0.786885i	$0.711692\pi$
$678$	0	0
$679$	−116632.	−0.00970830
$680$	0	0
$681$	6.37088e6	0.526419
$682$	0	0
$683$	1.62332e7	1.33154	0.665769	−	0.746158i	$-0.268104\pi$
$683$	1.62332e7	1.33154	0.665769	+	0.746158i	$0.268104\pi$
$684$	0	0
$685$	−3.60699e7	−2.93710
$686$	0	0
$687$	−6.80536e6	−0.550122
$688$	0	0
$689$	7.01778e6	0.563186
$690$	0	0
$691$	−1.86468e7	−1.48563	−0.742814	−	0.669498i	$-0.766509\pi$
$691$	−1.86468e7	−1.48563	−0.742814	+	0.669498i	$0.766509\pi$
$692$	0	0
$693$	1.80819e7	1.43024
$694$	0	0
$695$	−1.88118e7	−1.47730
$696$	0	0
$697$	574860.	0.0448209
$698$	0	0
$699$	4.60926e6	0.356811
$700$	0	0
$701$	−1.46202e7	−1.12372	−0.561861	−	0.827232i	$-0.689914\pi$
$701$	−1.46202e7	−1.12372	−0.561861	+	0.827232i	$0.689914\pi$
$702$	0	0
$703$	−3.66886e6	−0.279990
$704$	0	0
$705$	−2.00784e6	−0.152145
$706$	0	0
$707$	1.30955e7	0.985310
$708$	0	0
$709$	−1.46011e7	−1.09087	−0.545433	−	0.838155i	$-0.683635\pi$
$709$	−1.46011e7	−1.09087	−0.545433	+	0.838155i	$0.683635\pi$
$710$	0	0
$711$	−1.34914e7	−1.00088
$712$	0	0
$713$	1.41174e7	1.03999
$714$	0	0
$715$	−2.59120e7	−1.89556
$716$	0	0
$717$	−2.51870e6	−0.182970
$718$	0	0
$719$	1.06325e7	0.767032	0.383516	−	0.923534i	$-0.374713\pi$
$719$	1.06325e7	0.767032	0.383516	+	0.923534i	$0.374713\pi$
$720$	0	0
$721$	−4.74443e7	−3.39896
$722$	0	0
$723$	−5.28248e6	−0.375831
$724$	0	0
$725$	4.35178e6	0.307484
$726$	0	0
$727$	−2.22678e6	−0.156258	−0.0781288	−	0.996943i	$-0.524895\pi$
$727$	−2.22678e6	−0.156258	−0.0781288	+	0.996943i	$0.524895\pi$
$728$	0	0
$729$	−3.12231e6	−0.217599
$730$	0	0
$731$	−8.56662e6	−0.592947
$732$	0	0
$733$	−1.35176e7	−0.929263	−0.464632	−	0.885504i	$-0.653813\pi$
$733$	−1.35176e7	−0.929263	−0.464632	+	0.885504i	$0.653813\pi$
$734$	0	0
$735$	2.40992e7	1.64545
$736$	0	0
$737$	1.07350e7	0.728002
$738$	0	0
$739$	−1.13008e7	−0.761198	−0.380599	−	0.924740i	$-0.624282\pi$
$739$	−1.13008e7	−0.761198	−0.380599	+	0.924740i	$0.624282\pi$
$740$	0	0
$741$	4.75860e6	0.318371
$742$	0	0
$743$	2.47261e7	1.64318	0.821588	−	0.570081i	$-0.193088\pi$
$743$	2.47261e7	1.64318	0.821588	+	0.570081i	$0.193088\pi$
$744$	0	0
$745$	−2.02967e7	−1.33978
$746$	0	0
$747$	−8.15125e6	−0.534469
$748$	0	0
$749$	1.60284e7	1.04396
$750$	0	0
$751$	2.53318e7	1.63895	0.819476	−	0.573113i	$-0.194264\pi$
$751$	2.53318e7	1.63895	0.819476	+	0.573113i	$0.194264\pi$
$752$	0	0
$753$	−97668.0	−0.00627718
$754$	0	0
$755$	−1.02742e7	−0.655965
$756$	0	0
$757$	1.18453e7	0.751288	0.375644	−	0.926764i	$-0.377422\pi$
$757$	1.18453e7	0.751288	0.375644	+	0.926764i	$0.377422\pi$
$758$	0	0
$759$	6.07454e6	0.382745
$760$	0	0
$761$	−1.00233e7	−0.627404	−0.313702	−	0.949522i	$-0.601569\pi$
$761$	−1.00233e7	−0.627404	−0.313702	+	0.949522i	$0.601569\pi$
$762$	0	0
$763$	−2.69361e7	−1.67504
$764$	0	0
$765$	1.30369e7	0.805414
$766$	0	0
$767$	6.34942e6	0.389714
$768$	0	0
$769$	2.28511e7	1.39345	0.696726	−	0.717338i	$-0.254639\pi$
$769$	2.28511e7	1.39345	0.696726	+	0.717338i	$0.254639\pi$
$770$	0	0
$771$	7.57544e6	0.458957
$772$	0	0
$773$	−2.86153e6	−0.172246	−0.0861230	−	0.996285i	$-0.527448\pi$
$773$	−2.86153e6	−0.172246	−0.0861230	+	0.996285i	$0.527448\pi$
$774$	0	0
$775$	−2.85093e7	−1.70503
$776$	0	0
$777$	5.21477e6	0.309872
$778$	0	0
$779$	−883740.	−0.0521772
$780$	0	0
$781$	−1.77468e7	−1.04110
$782$	0	0
$783$	2.05740e6	0.119926
$784$	0	0
$785$	−2.72361e7	−1.57751
$786$	0	0
$787$	2.49934e7	1.43843	0.719215	−	0.694788i	$-0.244502\pi$
$787$	2.49934e7	1.43843	0.719215	+	0.694788i	$0.244502\pi$
$788$	0	0
$789$	2.41250e6	0.137967
$790$	0	0
$791$	4.79250e7	2.72346
$792$	0	0
$793$	−3.39878e6	−0.191929
$794$	0	0
$795$	5.14030e6	0.288450
$796$	0	0
$797$	1.96103e7	1.09355	0.546775	−	0.837279i	$-0.315855\pi$
$797$	1.96103e7	1.09355	0.546775	+	0.837279i	$0.315855\pi$
$798$	0	0
$799$	−2.38520e6	−0.132178
$800$	0	0
$801$	−2.30474e6	−0.126923
$802$	0	0
$803$	−8.72446e6	−0.477474
$804$	0	0
$805$	−6.48630e7	−3.52783
$806$	0	0
$807$	−694716.	−0.0375512
$808$	0	0
$809$	−2.73783e6	−0.147074	−0.0735369	−	0.997292i	$-0.523429\pi$
$809$	−2.73783e6	−0.147074	−0.0735369	+	0.997292i	$0.523429\pi$
$810$	0	0
$811$	−2.34672e7	−1.25288	−0.626441	−	0.779469i	$-0.715489\pi$
$811$	−2.34672e7	−1.25288	−0.626441	+	0.779469i	$0.715489\pi$
$812$	0	0
$813$	−7.94251e6	−0.421436
$814$	0	0
$815$	−5.07985e7	−2.67890
$816$	0	0
$817$	1.31696e7	0.690266
$818$	0	0
$819$	3.88912e7	2.02601
$820$	0	0
$821$	2.54779e7	1.31919	0.659593	−	0.751623i	$-0.270729\pi$
$821$	2.54779e7	1.31919	0.659593	+	0.751623i	$0.270729\pi$
$822$	0	0
$823$	−2.40912e7	−1.23982	−0.619910	−	0.784673i	$-0.712831\pi$
$823$	−2.40912e7	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$824$	0	0
$825$	−1.22672e7	−0.627497
$826$	0	0
$827$	−1.43696e7	−0.730601	−0.365300	−	0.930890i	$-0.619034\pi$
$827$	−1.43696e7	−0.730601	−0.365300	+	0.930890i	$0.619034\pi$
$828$	0	0
$829$	−1.15102e7	−0.581698	−0.290849	−	0.956769i	$-0.593938\pi$
$829$	−1.15102e7	−0.581698	−0.290849	+	0.956769i	$0.593938\pi$
$830$	0	0
$831$	−5.83081e6	−0.292905
$832$	0	0
$833$	2.86284e7	1.42950
$834$	0	0
$835$	−3.72086e7	−1.84683
$836$	0	0
$837$	−1.34784e7	−0.665005
$838$	0	0
$839$	2.92310e7	1.43363	0.716817	−	0.697262i	$-0.245598\pi$
$839$	2.92310e7	1.43363	0.716817	+	0.697262i	$0.245598\pi$
$840$	0	0
$841$	−1.99305e7	−0.971691
$842$	0	0
$843$	1.20408e7	0.583562
$844$	0	0
$845$	−2.08311e7	−1.00362
$846$	0	0
$847$	8.02443e6	0.384331
$848$	0	0
$849$	4.45662e6	0.212196
$850$	0	0
$851$	−1.00733e7	−0.476814
$852$	0	0
$853$	−2.35284e6	−0.110718	−0.0553592	−	0.998467i	$-0.517630\pi$
$853$	−2.35284e6	−0.110718	−0.0553592	+	0.998467i	$0.517630\pi$
$854$	0	0
$855$	−2.00417e7	−0.937605
$856$	0	0
$857$	2.94700e7	1.37065	0.685327	−	0.728236i	$-0.259659\pi$
$857$	2.94700e7	1.37065	0.685327	+	0.728236i	$0.259659\pi$
$858$	0	0
$859$	3.77270e7	1.74449	0.872246	−	0.489067i	$-0.162663\pi$
$859$	3.77270e7	1.74449	0.872246	+	0.489067i	$0.162663\pi$
$860$	0	0
$861$	1.25611e6	0.0577458
$862$	0	0
$863$	−1.35022e7	−0.617133	−0.308567	−	0.951203i	$-0.599849\pi$
$863$	−1.35022e7	−0.617133	−0.308567	+	0.951203i	$0.599849\pi$
$864$	0	0
$865$	−5.18355e7	−2.35552
$866$	0	0
$867$	5.82574e6	0.263211
$868$	0	0
$869$	2.33330e7	1.04814
$870$	0	0
$871$	2.30892e7	1.03125
$872$	0	0
$873$	−98946.0	−0.00439403
$874$	0	0
$875$	5.93125e7	2.61894
$876$	0	0
$877$	4.35051e7	1.91003	0.955016	−	0.296553i	$-0.0958371\pi$
$877$	4.35051e7	1.91003	0.955016	+	0.296553i	$0.0958371\pi$
$878$	0	0
$879$	−7.35832e6	−0.321223
$880$	0	0
$881$	1.46741e6	0.0636959	0.0318480	−	0.999493i	$-0.489861\pi$
$881$	1.46741e6	0.0636959	0.0318480	+	0.999493i	$0.489861\pi$
$882$	0	0
$883$	−1.08478e7	−0.468211	−0.234106	−	0.972211i	$-0.575216\pi$
$883$	−1.08478e7	−0.468211	−0.234106	+	0.972211i	$0.575216\pi$
$884$	0	0
$885$	4.65074e6	0.199602
$886$	0	0
$887$	−1.11749e7	−0.476908	−0.238454	−	0.971154i	$-0.576641\pi$
$887$	−1.11749e7	−0.476908	−0.238454	+	0.971154i	$0.576641\pi$
$888$	0	0
$889$	2.62778e7	1.11515
$890$	0	0
$891$	1.22082e7	0.515176
$892$	0	0
$893$	3.66680e6	0.153872
$894$	0	0
$895$	4.41618e7	1.84285
$896$	0	0
$897$	1.30654e7	0.542176
$898$	0	0
$899$	−3.80390e6	−0.156975
$900$	0	0
$901$	6.10638e6	0.250595
$902$	0	0
$903$	−1.87187e7	−0.763935
$904$	0	0
$905$	6.85732e7	2.78313
$906$	0	0
$907$	2.65074e7	1.06992	0.534958	−	0.844879i	$-0.320327\pi$
$907$	2.65074e7	1.06992	0.534958	+	0.844879i	$0.320327\pi$
$908$	0	0
$909$	1.11097e7	0.445956
$910$	0	0
$911$	8.05639e6	0.321621	0.160811	−	0.986985i	$-0.448589\pi$
$911$	8.05639e6	0.321621	0.160811	+	0.986985i	$0.448589\pi$
$912$	0	0
$913$	1.40973e7	0.559706
$914$	0	0
$915$	−2.48950e6	−0.0983012
$916$	0	0
$917$	6.98460e7	2.74295
$918$	0	0
$919$	−3.32401e6	−0.129830	−0.0649148	−	0.997891i	$-0.520678\pi$
$919$	−3.32401e6	−0.129830	−0.0649148	+	0.997891i	$0.520678\pi$
$920$	0	0
$921$	−1.69640e7	−0.658992
$922$	0	0
$923$	−3.81704e7	−1.47477
$924$	0	0
$925$	2.03426e7	0.781721
$926$	0	0
$927$	−4.02499e7	−1.53839
$928$	0	0
$929$	−2.46446e7	−0.936877	−0.468439	−	0.883496i	$-0.655183\pi$
$929$	−2.46446e7	−0.936877	−0.468439	+	0.883496i	$0.655183\pi$
$930$	0	0
$931$	−4.40109e7	−1.66412
$932$	0	0
$933$	−4.63231e6	−0.174218
$934$	0	0
$935$	−2.25468e7	−0.843445
$936$	0	0
$937$	−1.93837e7	−0.721253	−0.360627	−	0.932710i	$-0.617437\pi$
$937$	−1.93837e7	−0.721253	−0.360627	+	0.932710i	$0.617437\pi$
$938$	0	0
$939$	−1.89291e7	−0.700595
$940$	0	0
$941$	−7.54660e6	−0.277829	−0.138914	−	0.990304i	$-0.544361\pi$
$941$	−7.54660e6	−0.277829	−0.138914	+	0.990304i	$0.544361\pi$
$942$	0	0
$943$	−2.42642e6	−0.0888561
$944$	0	0
$945$	6.19272e7	2.25581
$946$	0	0
$947$	1.02750e6	0.0372311	0.0186156	−	0.999827i	$-0.494074\pi$
$947$	1.02750e6	0.0372311	0.0186156	+	0.999827i	$0.494074\pi$
$948$	0	0
$949$	−1.87649e7	−0.676364
$950$	0	0
$951$	2.10517e7	0.754806
$952$	0	0
$953$	2.32289e7	0.828507	0.414254	−	0.910162i	$-0.364043\pi$
$953$	2.32289e7	0.828507	0.414254	+	0.910162i	$0.364043\pi$
$954$	0	0
$955$	5.00125e7	1.77448
$956$	0	0
$957$	−1.63678e6	−0.0577709
$958$	0	0
$959$	−9.36282e7	−3.28746
$960$	0	0
$961$	−3.70909e6	−0.129556
$962$	0	0
$963$	1.35978e7	0.472502
$964$	0	0
$965$	−1.84178e7	−0.636677
$966$	0	0
$967$	2.43535e7	0.837518	0.418759	−	0.908097i	$-0.362465\pi$
$967$	2.43535e7	0.837518	0.418759	+	0.908097i	$0.362465\pi$
$968$	0	0
$969$	4.14060e6	0.141662
$970$	0	0
$971$	2.69959e7	0.918860	0.459430	−	0.888214i	$-0.348054\pi$
$971$	2.69959e7	0.918860	0.459430	+	0.888214i	$0.348054\pi$
$972$	0	0
$973$	−4.88307e7	−1.65353
$974$	0	0
$975$	−2.63848e7	−0.888879
$976$	0	0
$977$	5.59558e6	0.187546	0.0937732	−	0.995594i	$-0.470107\pi$
$977$	5.59558e6	0.187546	0.0937732	+	0.995594i	$0.470107\pi$
$978$	0	0
$979$	3.98597e6	0.132916
$980$	0	0
$981$	−2.28516e7	−0.758129
$982$	0	0
$983$	−5.99648e6	−0.197930	−0.0989652	−	0.995091i	$-0.531553\pi$
$983$	−5.99648e6	−0.197930	−0.0989652	+	0.995091i	$0.531553\pi$
$984$	0	0
$985$	−6.42766e7	−2.11088
$986$	0	0
$987$	−5.21184e6	−0.170294
$988$	0	0
$989$	3.61588e7	1.17550
$990$	0	0
$991$	2.22784e7	0.720608	0.360304	−	0.932835i	$-0.382673\pi$
$991$	2.22784e7	0.720608	0.360304	+	0.932835i	$0.382673\pi$
$992$	0	0
$993$	−1.79124e7	−0.576475
$994$	0	0
$995$	1.75656e7	0.562477
$996$	0	0
$997$	3.05853e7	0.974483	0.487241	−	0.873267i	$-0.338003\pi$
$997$	3.05853e7	0.974483	0.487241	+	0.873267i	$0.338003\pi$
$998$	0	0
$999$	9.61740e6	0.304891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.6.a.a.1.1	✓	1
4.3	odd	2		128.6.a.c.1.1	yes	1
8.3	odd	2		128.6.a.b.1.1	yes	1
8.5	even	2		128.6.a.d.1.1	yes	1
16.3	odd	4		256.6.b.a.129.2		2
16.5	even	4		256.6.b.i.129.2		2
16.11	odd	4		256.6.b.a.129.1		2
16.13	even	4		256.6.b.i.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.6.a.a.1.1	✓	1	1.1	even	1	trivial
128.6.a.b.1.1	yes	1	8.3	odd	2
128.6.a.c.1.1	yes	1	4.3	odd	2
128.6.a.d.1.1	yes	1	8.5	even	2
256.6.b.a.129.1		2	16.11	odd	4
256.6.b.a.129.2		2	16.3	odd	4
256.6.b.i.129.1		2	16.13	even	4
256.6.b.i.129.2		2	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	128.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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