LMFDB - Embedded newform 546.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,6,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	546.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:	$87.5695656179$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +33.0000 q^{5} -36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +33.0000 q^{5} -36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +132.000 q^{10} -94.0000 q^{11} -144.000 q^{12} -169.000 q^{13} +196.000 q^{14} -297.000 q^{15} +256.000 q^{16} -1112.00 q^{17} +324.000 q^{18} -1909.00 q^{19} +528.000 q^{20} -441.000 q^{21} -376.000 q^{22} +961.000 q^{23} -576.000 q^{24} -2036.00 q^{25} -676.000 q^{26} -729.000 q^{27} +784.000 q^{28} -6337.00 q^{29} -1188.00 q^{30} +7015.00 q^{31} +1024.00 q^{32} +846.000 q^{33} -4448.00 q^{34} +1617.00 q^{35} +1296.00 q^{36} +928.000 q^{37} -7636.00 q^{38} +1521.00 q^{39} +2112.00 q^{40} +11442.0 q^{41} -1764.00 q^{42} -12711.0 q^{43} -1504.00 q^{44} +2673.00 q^{45} +3844.00 q^{46} -15107.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} -8144.00 q^{50} +10008.0 q^{51} -2704.00 q^{52} +18691.0 q^{53} -2916.00 q^{54} -3102.00 q^{55} +3136.00 q^{56} +17181.0 q^{57} -25348.0 q^{58} -12360.0 q^{59} -4752.00 q^{60} +14110.0 q^{61} +28060.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} -5577.00 q^{65} +3384.00 q^{66} -53746.0 q^{67} -17792.0 q^{68} -8649.00 q^{69} +6468.00 q^{70} -47748.0 q^{71} +5184.00 q^{72} -25301.0 q^{73} +3712.00 q^{74} +18324.0 q^{75} -30544.0 q^{76} -4606.00 q^{77} +6084.00 q^{78} -5447.00 q^{79} +8448.00 q^{80} +6561.00 q^{81} +45768.0 q^{82} +29393.0 q^{83} -7056.00 q^{84} -36696.0 q^{85} -50844.0 q^{86} +57033.0 q^{87} -6016.00 q^{88} -77621.0 q^{89} +10692.0 q^{90} -8281.00 q^{91} +15376.0 q^{92} -63135.0 q^{93} -60428.0 q^{94} -62997.0 q^{95} -9216.00 q^{96} +73607.0 q^{97} +9604.00 q^{98} -7614.00 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	33.0000	0.590322	0.295161	−	0.955448i	$-0.404627\pi$
$5$	33.0000	0.590322	0.295161	+	0.955448i	$0.404627\pi$
$6$	−36.0000	−0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	132.000	0.417421
$11$	−94.0000	−0.234232	−0.117116	−	0.993118i	$-0.537365\pi$
$11$	−94.0000	−0.234232	−0.117116	+	0.993118i	$0.537365\pi$
$12$	−144.000	−0.288675
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	196.000	0.267261
$15$	−297.000	−0.340823
$16$	256.000	0.250000
$17$	−1112.00	−0.933217	−0.466608	−	0.884464i	$-0.654524\pi$
$17$	−1112.00	−0.933217	−0.466608	+	0.884464i	$0.654524\pi$
$18$	324.000	0.235702
$19$	−1909.00	−1.21317	−0.606585	−	0.795018i	$-0.707461\pi$
$19$	−1909.00	−1.21317	−0.606585	+	0.795018i	$0.707461\pi$
$20$	528.000	0.295161
$21$	−441.000	−0.218218
$22$	−376.000	−0.165627
$23$	961.000	0.378795	0.189397	−	0.981901i	$-0.439347\pi$
$23$	961.000	0.378795	0.189397	+	0.981901i	$0.439347\pi$
$24$	−576.000	−0.204124
$25$	−2036.00	−0.651520
$26$	−676.000	−0.196116
$27$	−729.000	−0.192450
$28$	784.000	0.188982
$29$	−6337.00	−1.39923	−0.699614	−	0.714521i	$-0.746645\pi$
$29$	−6337.00	−1.39923	−0.699614	+	0.714521i	$0.746645\pi$
$30$	−1188.00	−0.240998
$31$	7015.00	1.31106	0.655531	−	0.755168i	$-0.272445\pi$
$31$	7015.00	1.31106	0.655531	+	0.755168i	$0.272445\pi$
$32$	1024.00	0.176777
$33$	846.000	0.135234
$34$	−4448.00	−0.659884
$35$	1617.00	0.223121
$36$	1296.00	0.166667
$37$	928.000	0.111441	0.0557203	−	0.998446i	$-0.482254\pi$
$37$	928.000	0.111441	0.0557203	+	0.998446i	$0.482254\pi$
$38$	−7636.00	−0.857841
$39$	1521.00	0.160128
$40$	2112.00	0.208710
$41$	11442.0	1.06302	0.531511	−	0.847051i	$-0.321624\pi$
$41$	11442.0	1.06302	0.531511	+	0.847051i	$0.321624\pi$
$42$	−1764.00	−0.154303
$43$	−12711.0	−1.04836	−0.524178	−	0.851609i	$-0.675627\pi$
$43$	−12711.0	−1.04836	−0.524178	+	0.851609i	$0.675627\pi$
$44$	−1504.00	−0.117116
$45$	2673.00	0.196774
$46$	3844.00	0.267848
$47$	−15107.0	−0.997547	−0.498774	−	0.866732i	$-0.666216\pi$
$47$	−15107.0	−0.997547	−0.498774	+	0.866732i	$0.666216\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	−8144.00	−0.460694
$51$	10008.0	0.538793
$52$	−2704.00	−0.138675
$53$	18691.0	0.913993	0.456996	−	0.889468i	$-0.348925\pi$
$53$	18691.0	0.913993	0.456996	+	0.889468i	$0.348925\pi$
$54$	−2916.00	−0.136083
$55$	−3102.00	−0.138272
$56$	3136.00	0.133631
$57$	17181.0	0.700424
$58$	−25348.0	−0.989404
$59$	−12360.0	−0.462262	−0.231131	−	0.972923i	$-0.574243\pi$
$59$	−12360.0	−0.462262	−0.231131	+	0.972923i	$0.574243\pi$
$60$	−4752.00	−0.170411
$61$	14110.0	0.485515	0.242757	−	0.970087i	$-0.421948\pi$
$61$	14110.0	0.485515	0.242757	+	0.970087i	$0.421948\pi$
$62$	28060.0	0.927061
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	−5577.00	−0.163726
$66$	3384.00	0.0956248
$67$	−53746.0	−1.46271	−0.731357	−	0.681995i	$-0.761112\pi$
$67$	−53746.0	−1.46271	−0.731357	+	0.681995i	$0.761112\pi$
$68$	−17792.0	−0.466608
$69$	−8649.00	−0.218697
$70$	6468.00	0.157770
$71$	−47748.0	−1.12411	−0.562056	−	0.827099i	$-0.689989\pi$
$71$	−47748.0	−1.12411	−0.562056	+	0.827099i	$0.689989\pi$
$72$	5184.00	0.117851
$73$	−25301.0	−0.555687	−0.277844	−	0.960626i	$-0.589620\pi$
$73$	−25301.0	−0.555687	−0.277844	+	0.960626i	$0.589620\pi$
$74$	3712.00	0.0788004
$75$	18324.0	0.376155
$76$	−30544.0	−0.606585
$77$	−4606.00	−0.0885314
$78$	6084.00	0.113228
$79$	−5447.00	−0.0981951	−0.0490975	−	0.998794i	$-0.515635\pi$
$79$	−5447.00	−0.0981951	−0.0490975	+	0.998794i	$0.515635\pi$
$80$	8448.00	0.147580
$81$	6561.00	0.111111
$82$	45768.0	0.751670
$83$	29393.0	0.468326	0.234163	−	0.972197i	$-0.424765\pi$
$83$	29393.0	0.468326	0.234163	+	0.972197i	$0.424765\pi$
$84$	−7056.00	−0.109109
$85$	−36696.0	−0.550898
$86$	−50844.0	−0.741299
$87$	57033.0	0.807845
$88$	−6016.00	−0.0828135
$89$	−77621.0	−1.03873	−0.519367	−	0.854551i	$-0.673832\pi$
$89$	−77621.0	−1.03873	−0.519367	+	0.854551i	$0.673832\pi$
$90$	10692.0	0.139140
$91$	−8281.00	−0.104828
$92$	15376.0	0.189397
$93$	−63135.0	−0.756942
$94$	−60428.0	−0.705373
$95$	−62997.0	−0.716161
$96$	−9216.00	−0.102062
$97$	73607.0	0.794310	0.397155	−	0.917752i	$-0.369998\pi$
$97$	73607.0	0.794310	0.397155	+	0.917752i	$0.369998\pi$
$98$	9604.00	0.101015
$99$	−7614.00	−0.0780773
$100$	−32576.0	−0.325760
$101$	−44706.0	−0.436076	−0.218038	−	0.975940i	$-0.569966\pi$
$101$	−44706.0	−0.436076	−0.218038	+	0.975940i	$0.569966\pi$
$102$	40032.0	0.380984
$103$	13572.0	0.126052	0.0630262	−	0.998012i	$-0.479925\pi$
$103$	13572.0	0.126052	0.0630262	+	0.998012i	$0.479925\pi$
$104$	−10816.0	−0.0980581
$105$	−14553.0	−0.128819
$106$	74764.0	0.646291
$107$	71900.0	0.607113	0.303556	−	0.952813i	$-0.401826\pi$
$107$	71900.0	0.607113	0.303556	+	0.952813i	$0.401826\pi$
$108$	−11664.0	−0.0962250
$109$	−115806.	−0.933609	−0.466804	−	0.884361i	$-0.654595\pi$
$109$	−115806.	−0.933609	−0.466804	+	0.884361i	$0.654595\pi$
$110$	−12408.0	−0.0977733
$111$	−8352.00	−0.0643403
$112$	12544.0	0.0944911
$113$	−229915.	−1.69384	−0.846918	−	0.531724i	$-0.821544\pi$
$113$	−229915.	−1.69384	−0.846918	+	0.531724i	$0.821544\pi$
$114$	68724.0	0.495275
$115$	31713.0	0.223611
$116$	−101392.	−0.699614
$117$	−13689.0	−0.0924500
$118$	−49440.0	−0.326869
$119$	−54488.0	−0.352723
$120$	−19008.0	−0.120499
$121$	−152215.	−0.945135
$122$	56440.0	0.343311
$123$	−102978.	−0.613736
$124$	112240.	0.655531
$125$	−170313.	−0.974929
$126$	15876.0	0.0890871
$127$	134652.	0.740804	0.370402	−	0.928871i	$-0.379220\pi$
$127$	134652.	0.740804	0.370402	+	0.928871i	$0.379220\pi$
$128$	16384.0	0.0883883
$129$	114399.	0.605268
$130$	−22308.0	−0.115772
$131$	−6608.00	−0.0336428	−0.0168214	−	0.999859i	$-0.505355\pi$
$131$	−6608.00	−0.0336428	−0.0168214	+	0.999859i	$0.505355\pi$
$132$	13536.0	0.0676169
$133$	−93541.0	−0.458535
$134$	−214984.	−1.03429
$135$	−24057.0	−0.113608
$136$	−71168.0	−0.329942
$137$	385392.	1.75429	0.877145	−	0.480226i	$-0.159445\pi$
$137$	385392.	1.75429	0.877145	+	0.480226i	$0.159445\pi$
$138$	−34596.0	−0.154642
$139$	−248546.	−1.09111	−0.545557	−	0.838074i	$-0.683682\pi$
$139$	−248546.	−1.09111	−0.545557	+	0.838074i	$0.683682\pi$
$140$	25872.0	0.111560
$141$	135963.	0.575934
$142$	−190992.	−0.794867
$143$	15886.0	0.0649643
$144$	20736.0	0.0833333
$145$	−209121.	−0.825995
$146$	−101204.	−0.392930
$147$	−21609.0	−0.0824786
$148$	14848.0	0.0557203
$149$	−284302.	−1.04909	−0.524547	−	0.851382i	$-0.675765\pi$
$149$	−284302.	−1.04909	−0.524547	+	0.851382i	$0.675765\pi$
$150$	73296.0	0.265982
$151$	313644.	1.11942	0.559712	−	0.828687i	$-0.310912\pi$
$151$	313644.	1.11942	0.559712	+	0.828687i	$0.310912\pi$
$152$	−122176.	−0.428921
$153$	−90072.0	−0.311072
$154$	−18424.0	−0.0626011
$155$	231495.	0.773949
$156$	24336.0	0.0800641
$157$	−406868.	−1.31736	−0.658680	−	0.752423i	$-0.728885\pi$
$157$	−406868.	−1.31736	−0.658680	+	0.752423i	$0.728885\pi$
$158$	−21788.0	−0.0694344
$159$	−168219.	−0.527694
$160$	33792.0	0.104355
$161$	47089.0	0.143171
$162$	26244.0	0.0785674
$163$	−480068.	−1.41525	−0.707626	−	0.706587i	$-0.750234\pi$
$163$	−480068.	−1.41525	−0.707626	+	0.706587i	$0.750234\pi$
$164$	183072.	0.531511
$165$	27918.0	0.0798315
$166$	117572.	0.331157
$167$	−326393.	−0.905628	−0.452814	−	0.891605i	$-0.649580\pi$
$167$	−326393.	−0.905628	−0.452814	+	0.891605i	$0.649580\pi$
$168$	−28224.0	−0.0771517
$169$	28561.0	0.0769231
$170$	−146784.	−0.389544
$171$	−154629.	−0.404390
$172$	−203376.	−0.524178
$173$	−178548.	−0.453565	−0.226783	−	0.973945i	$-0.572821\pi$
$173$	−178548.	−0.453565	−0.226783	+	0.973945i	$0.572821\pi$
$174$	228132.	0.571233
$175$	−99764.0	−0.246251
$176$	−24064.0	−0.0585580
$177$	111240.	0.266887
$178$	−310484.	−0.734496
$179$	−336439.	−0.784827	−0.392413	−	0.919789i	$-0.628360\pi$
$179$	−336439.	−0.784827	−0.392413	+	0.919789i	$0.628360\pi$
$180$	42768.0	0.0983870
$181$	−308606.	−0.700177	−0.350089	−	0.936717i	$-0.613849\pi$
$181$	−308606.	−0.700177	−0.350089	+	0.936717i	$0.613849\pi$
$182$	−33124.0	−0.0741249
$183$	−126990.	−0.280312
$184$	61504.0	0.133924
$185$	30624.0	0.0657859
$186$	−252540.	−0.535239
$187$	104528.	0.218589
$188$	−241712.	−0.498774
$189$	−35721.0	−0.0727393
$190$	−251988.	−0.506403
$191$	−299672.	−0.594378	−0.297189	−	0.954819i	$-0.596049\pi$
$191$	−299672.	−0.594378	−0.297189	+	0.954819i	$0.596049\pi$
$192$	−36864.0	−0.0721688
$193$	151822.	0.293387	0.146694	−	0.989182i	$-0.453137\pi$
$193$	151822.	0.293387	0.146694	+	0.989182i	$0.453137\pi$
$194$	294428.	0.561662
$195$	50193.0	0.0945272
$196$	38416.0	0.0714286
$197$	−150606.	−0.276488	−0.138244	−	0.990398i	$-0.544146\pi$
$197$	−150606.	−0.276488	−0.138244	+	0.990398i	$0.544146\pi$
$198$	−30456.0	−0.0552090
$199$	652312.	1.16768	0.583838	−	0.811870i	$-0.301550\pi$
$199$	652312.	1.16768	0.583838	+	0.811870i	$0.301550\pi$
$200$	−130304.	−0.230347
$201$	483714.	0.844498
$202$	−178824.	−0.308352
$203$	−310513.	−0.528859
$204$	160128.	0.269396
$205$	377586.	0.627525
$206$	54288.0	0.0891324
$207$	77841.0	0.126265
$208$	−43264.0	−0.0693375
$209$	179446.	0.284163
$210$	−58212.0	−0.0910887
$211$	790909.	1.22298	0.611491	−	0.791251i	$-0.290570\pi$
$211$	790909.	1.22298	0.611491	+	0.791251i	$0.290570\pi$
$212$	299056.	0.456996
$213$	429732.	0.649006
$214$	287600.	0.429294
$215$	−419463.	−0.618867
$216$	−46656.0	−0.0680414
$217$	343735.	0.495535
$218$	−463224.	−0.660161
$219$	227709.	0.320826
$220$	−49632.0	−0.0691361
$221$	187928.	0.258828
$222$	−33408.0	−0.0454955
$223$	652701.	0.878926	0.439463	−	0.898261i	$-0.355169\pi$
$223$	652701.	0.878926	0.439463	+	0.898261i	$0.355169\pi$
$224$	50176.0	0.0668153
$225$	−164916.	−0.217173
$226$	−919660.	−1.19772
$227$	489132.	0.630030	0.315015	−	0.949087i	$-0.397990\pi$
$227$	489132.	0.630030	0.315015	+	0.949087i	$0.397990\pi$
$228$	274896.	0.350212
$229$	−147026.	−0.185270	−0.0926351	−	0.995700i	$-0.529529\pi$
$229$	−147026.	−0.185270	−0.0926351	+	0.995700i	$0.529529\pi$
$230$	126852.	0.158117
$231$	41454.0	0.0511136
$232$	−405568.	−0.494702
$233$	168831.	0.203733	0.101867	−	0.994798i	$-0.467518\pi$
$233$	168831.	0.203733	0.101867	+	0.994798i	$0.467518\pi$
$234$	−54756.0	−0.0653720
$235$	−498531.	−0.588874
$236$	−197760.	−0.231131
$237$	49023.0	0.0566929
$238$	−217952.	−0.249413
$239$	682836.	0.773253	0.386627	−	0.922236i	$-0.373640\pi$
$239$	682836.	0.773253	0.386627	+	0.922236i	$0.373640\pi$
$240$	−76032.0	−0.0852056
$241$	196647.	0.218095	0.109047	−	0.994037i	$-0.465220\pi$
$241$	196647.	0.218095	0.109047	+	0.994037i	$0.465220\pi$
$242$	−608860.	−0.668312
$243$	−59049.0	−0.0641500
$244$	225760.	0.242757
$245$	79233.0	0.0843317
$246$	−411912.	−0.433977
$247$	322621.	0.336473
$248$	448960.	0.463531
$249$	−264537.	−0.270388
$250$	−681252.	−0.689379
$251$	1.14257e6	1.14471	0.572357	−	0.820005i	$-0.306029\pi$
$251$	1.14257e6	1.14471	0.572357	+	0.820005i	$0.306029\pi$
$252$	63504.0	0.0629941
$253$	−90334.0	−0.0887258
$254$	538608.	0.523828
$255$	330264.	0.318061
$256$	65536.0	0.0625000
$257$	488762.	0.461599	0.230799	−	0.973001i	$-0.425866\pi$
$257$	488762.	0.461599	0.230799	+	0.973001i	$0.425866\pi$
$258$	457596.	0.427989
$259$	45472.0	0.0421206
$260$	−89232.0	−0.0818629
$261$	−513297.	−0.466410
$262$	−26432.0	−0.0237890
$263$	−122445.	−0.109157	−0.0545785	−	0.998509i	$-0.517382\pi$
$263$	−122445.	−0.109157	−0.0545785	+	0.998509i	$0.517382\pi$
$264$	54144.0	0.0478124
$265$	616803.	0.539550
$266$	−374164.	−0.324234
$267$	698589.	0.599713
$268$	−859936.	−0.731357
$269$	1.40926e6	1.18744	0.593720	−	0.804672i	$-0.297659\pi$
$269$	1.40926e6	1.18744	0.593720	+	0.804672i	$0.297659\pi$
$270$	−96228.0	−0.0803326
$271$	−249784.	−0.206605	−0.103303	−	0.994650i	$-0.532941\pi$
$271$	−249784.	−0.206605	−0.103303	+	0.994650i	$0.532941\pi$
$272$	−284672.	−0.233304
$273$	74529.0	0.0605228
$274$	1.54157e6	1.24047
$275$	191384.	0.152607
$276$	−138384.	−0.109349
$277$	−2.25376e6	−1.76485	−0.882427	−	0.470449i	$-0.844092\pi$
$277$	−2.25376e6	−1.76485	−0.882427	+	0.470449i	$0.844092\pi$
$278$	−994184.	−0.771533
$279$	568215.	0.437021
$280$	103488.	0.0788851
$281$	−968738.	−0.731881	−0.365941	−	0.930638i	$-0.619253\pi$
$281$	−968738.	−0.731881	−0.365941	+	0.930638i	$0.619253\pi$
$282$	543852.	0.407247
$283$	899352.	0.667519	0.333759	−	0.942658i	$-0.391683\pi$
$283$	899352.	0.667519	0.333759	+	0.942658i	$0.391683\pi$
$284$	−763968.	−0.562056
$285$	566973.	0.413476
$286$	63544.0	0.0459367
$287$	560658.	0.401785
$288$	82944.0	0.0589256
$289$	−183313.	−0.129107
$290$	−836484.	−0.584067
$291$	−662463.	−0.458595
$292$	−404816.	−0.277844
$293$	88569.0	0.0602716	0.0301358	−	0.999546i	$-0.490406\pi$
$293$	88569.0	0.0602716	0.0301358	+	0.999546i	$0.490406\pi$
$294$	−86436.0	−0.0583212
$295$	−407880.	−0.272884
$296$	59392.0	0.0394002
$297$	68526.0	0.0450780
$298$	−1.13721e6	−0.741821
$299$	−162409.	−0.105059
$300$	293184.	0.188078
$301$	−622839.	−0.396241
$302$	1.25458e6	0.791552
$303$	402354.	0.251769
$304$	−488704.	−0.303293
$305$	465630.	0.286610
$306$	−360288.	−0.219961
$307$	−66007.0	−0.0399709	−0.0199855	−	0.999800i	$-0.506362\pi$
$307$	−66007.0	−0.0399709	−0.0199855	+	0.999800i	$0.506362\pi$
$308$	−73696.0	−0.0442657
$309$	−122148.	−0.0727763
$310$	925980.	0.547265
$311$	1.91235e6	1.12116	0.560580	−	0.828101i	$-0.310578\pi$
$311$	1.91235e6	1.12116	0.560580	+	0.828101i	$0.310578\pi$
$312$	97344.0	0.0566139
$313$	1.74251e6	1.00535	0.502673	−	0.864477i	$-0.332350\pi$
$313$	1.74251e6	1.00535	0.502673	+	0.864477i	$0.332350\pi$
$314$	−1.62747e6	−0.931514
$315$	130977.	0.0743736
$316$	−87152.0	−0.0490975
$317$	−3.33277e6	−1.86276	−0.931380	−	0.364048i	$-0.881394\pi$
$317$	−3.33277e6	−1.86276	−0.931380	+	0.364048i	$0.881394\pi$
$318$	−672876.	−0.373136
$319$	595678.	0.327744
$320$	135168.	0.0737902
$321$	−647100.	−0.350517
$322$	188356.	0.101237
$323$	2.12281e6	1.13215
$324$	104976.	0.0555556
$325$	344084.	0.180699
$326$	−1.92027e6	−1.00073
$327$	1.04225e6	0.539019
$328$	732288.	0.375835
$329$	−740243.	−0.377038
$330$	111672.	0.0564494
$331$	1.01402e6	0.508716	0.254358	−	0.967110i	$-0.418136\pi$
$331$	1.01402e6	0.508716	0.254358	+	0.967110i	$0.418136\pi$
$332$	470288.	0.234163
$333$	75168.0	0.0371469
$334$	−1.30557e6	−0.640375
$335$	−1.77362e6	−0.863472
$336$	−112896.	−0.0545545
$337$	−3.11781e6	−1.49546	−0.747729	−	0.664004i	$-0.768856\pi$
$337$	−3.11781e6	−1.49546	−0.747729	+	0.664004i	$0.768856\pi$
$338$	114244.	0.0543928
$339$	2.06924e6	0.977936
$340$	−587136.	−0.275449
$341$	−659410.	−0.307093
$342$	−618516.	−0.285947
$343$	117649.	0.0539949
$344$	−813504.	−0.370650
$345$	−285417.	−0.129102
$346$	−714192.	−0.320719
$347$	1.94994e6	0.869354	0.434677	−	0.900586i	$-0.356863\pi$
$347$	1.94994e6	0.869354	0.434677	+	0.900586i	$0.356863\pi$
$348$	912528.	0.403923
$349$	710591.	0.312289	0.156144	−	0.987734i	$-0.450094\pi$
$349$	710591.	0.312289	0.156144	+	0.987734i	$0.450094\pi$
$350$	−399056.	−0.174126
$351$	123201.	0.0533761
$352$	−96256.0	−0.0414068
$353$	3.71809e6	1.58812	0.794059	−	0.607841i	$-0.207964\pi$
$353$	3.71809e6	1.58812	0.794059	+	0.607841i	$0.207964\pi$
$354$	444960.	0.188718
$355$	−1.57568e6	−0.663588
$356$	−1.24194e6	−0.519367
$357$	490392.	0.203645
$358$	−1.34576e6	−0.554956
$359$	−3.17622e6	−1.30069	−0.650345	−	0.759639i	$-0.725376\pi$
$359$	−3.17622e6	−1.30069	−0.650345	+	0.759639i	$0.725376\pi$
$360$	171072.	0.0695701
$361$	1.16818e6	0.471783
$362$	−1.23442e6	−0.495100
$363$	1.36994e6	0.545674
$364$	−132496.	−0.0524142
$365$	−834933.	−0.328034
$366$	−507960.	−0.198211
$367$	223286.	0.0865359	0.0432680	−	0.999064i	$-0.486223\pi$
$367$	223286.	0.0865359	0.0432680	+	0.999064i	$0.486223\pi$
$368$	246016.	0.0946987
$369$	926802.	0.354341
$370$	122496.	0.0465176
$371$	915859.	0.345457
$372$	−1.01016e6	−0.378471
$373$	2.56341e6	0.953996	0.476998	−	0.878904i	$-0.341725\pi$
$373$	2.56341e6	0.953996	0.476998	+	0.878904i	$0.341725\pi$
$374$	418112.	0.154566
$375$	1.53282e6	0.562875
$376$	−966848.	−0.352686
$377$	1.07095e6	0.388076
$378$	−142884.	−0.0514344
$379$	−1.24118e6	−0.443851	−0.221926	−	0.975064i	$-0.571234\pi$
$379$	−1.24118e6	−0.443851	−0.221926	+	0.975064i	$0.571234\pi$
$380$	−1.00795e6	−0.358081
$381$	−1.21187e6	−0.427704
$382$	−1.19869e6	−0.420289
$383$	−3.30661e6	−1.15182	−0.575912	−	0.817512i	$-0.695353\pi$
$383$	−3.30661e6	−1.15182	−0.575912	+	0.817512i	$0.695353\pi$
$384$	−147456.	−0.0510310
$385$	−151998.	−0.0522620
$386$	607288.	0.207456
$387$	−1.02959e6	−0.349452
$388$	1.17771e6	0.397155
$389$	−143426.	−0.0480567	−0.0240283	−	0.999711i	$-0.507649\pi$
$389$	−143426.	−0.0480567	−0.0240283	+	0.999711i	$0.507649\pi$
$390$	200772.	0.0668408
$391$	−1.06863e6	−0.353497
$392$	153664.	0.0505076
$393$	59472.0	0.0194237
$394$	−602424.	−0.195507
$395$	−179751.	−0.0579667
$396$	−121824.	−0.0390387
$397$	3.59272e6	1.14406	0.572028	−	0.820234i	$-0.306157\pi$
$397$	3.59272e6	1.14406	0.572028	+	0.820234i	$0.306157\pi$
$398$	2.60925e6	0.825672
$399$	841869.	0.264736
$400$	−521216.	−0.162880
$401$	844840.	0.262370	0.131185	−	0.991358i	$-0.458122\pi$
$401$	844840.	0.262370	0.131185	+	0.991358i	$0.458122\pi$
$402$	1.93486e6	0.597150
$403$	−1.18554e6	−0.363623
$404$	−715296.	−0.218038
$405$	216513.	0.0655913
$406$	−1.24205e6	−0.373960
$407$	−87232.0	−0.0261030
$408$	640512.	0.190492
$409$	−3.22387e6	−0.952948	−0.476474	−	0.879189i	$-0.658085\pi$
$409$	−3.22387e6	−0.952948	−0.476474	+	0.879189i	$0.658085\pi$
$410$	1.51034e6	0.443727
$411$	−3.46853e6	−1.01284
$412$	217152.	0.0630262
$413$	−605640.	−0.174719
$414$	311364.	0.0892827
$415$	969969.	0.276463
$416$	−173056.	−0.0490290
$417$	2.23691e6	0.629954
$418$	717784.	0.200934
$419$	6.27207e6	1.74532	0.872662	−	0.488326i	$-0.162392\pi$
$419$	6.27207e6	1.74532	0.872662	+	0.488326i	$0.162392\pi$
$420$	−232848.	−0.0644094
$421$	4.62438e6	1.27159	0.635797	−	0.771856i	$-0.280671\pi$
$421$	4.62438e6	1.27159	0.635797	+	0.771856i	$0.280671\pi$
$422$	3.16364e6	0.864779
$423$	−1.22367e6	−0.332516
$424$	1.19622e6	0.323145
$425$	2.26403e6	0.608009
$426$	1.71893e6	0.458917
$427$	691390.	0.183507
$428$	1.15040e6	0.303556
$429$	−142974.	−0.0375071
$430$	−1.67785e6	−0.437605
$431$	2.43777e6	0.632120	0.316060	−	0.948739i	$-0.397640\pi$
$431$	2.43777e6	0.632120	0.316060	+	0.948739i	$0.397640\pi$
$432$	−186624.	−0.0481125
$433$	−2.13286e6	−0.546692	−0.273346	−	0.961916i	$-0.588130\pi$
$433$	−2.13286e6	−0.546692	−0.273346	+	0.961916i	$0.588130\pi$
$434$	1.37494e6	0.350396
$435$	1.88209e6	0.476889
$436$	−1.85290e6	−0.466804
$437$	−1.83455e6	−0.459543
$438$	910836.	0.226858
$439$	−2.68932e6	−0.666011	−0.333006	−	0.942925i	$-0.608063\pi$
$439$	−2.68932e6	−0.666011	−0.333006	+	0.942925i	$0.608063\pi$
$440$	−198528.	−0.0488866
$441$	194481.	0.0476190
$442$	751712.	0.183019
$443$	3.59786e6	0.871034	0.435517	−	0.900181i	$-0.356566\pi$
$443$	3.59786e6	0.871034	0.435517	+	0.900181i	$0.356566\pi$
$444$	−133632.	−0.0321701
$445$	−2.56149e6	−0.613187
$446$	2.61080e6	0.621494
$447$	2.55872e6	0.605694
$448$	200704.	0.0472456
$449$	−388068.	−0.0908431	−0.0454216	−	0.998968i	$-0.514463\pi$
$449$	−388068.	−0.0908431	−0.0454216	+	0.998968i	$0.514463\pi$
$450$	−659664.	−0.153565
$451$	−1.07555e6	−0.248994
$452$	−3.67864e6	−0.846918
$453$	−2.82280e6	−0.646300
$454$	1.95653e6	0.445499
$455$	−273273.	−0.0618826
$456$	1.09958e6	0.247637
$457$	−2.58041e6	−0.577961	−0.288981	−	0.957335i	$-0.593316\pi$
$457$	−2.58041e6	−0.577961	−0.288981	+	0.957335i	$0.593316\pi$
$458$	−588104.	−0.131006
$459$	810648.	0.179598
$460$	507408.	0.111805
$461$	2.65634e6	0.582145	0.291072	−	0.956701i	$-0.405988\pi$
$461$	2.65634e6	0.582145	0.291072	+	0.956701i	$0.405988\pi$
$462$	165816.	0.0361428
$463$	4.42007e6	0.958246	0.479123	−	0.877748i	$-0.340955\pi$
$463$	4.42007e6	0.958246	0.479123	+	0.877748i	$0.340955\pi$
$464$	−1.62227e6	−0.349807
$465$	−2.08345e6	−0.446840
$466$	675324.	0.144061
$467$	7.87055e6	1.66999	0.834994	−	0.550260i	$-0.185471\pi$
$467$	7.87055e6	1.66999	0.834994	+	0.550260i	$0.185471\pi$
$468$	−219024.	−0.0462250
$469$	−2.63355e6	−0.552854
$470$	−1.99412e6	−0.416397
$471$	3.66181e6	0.760578
$472$	−791040.	−0.163434
$473$	1.19483e6	0.245558
$474$	196092.	0.0400880
$475$	3.88672e6	0.790405
$476$	−871808.	−0.176361
$477$	1.51397e6	0.304664
$478$	2.73134e6	0.546772
$479$	9.49657e6	1.89116	0.945579	−	0.325392i	$-0.105496\pi$
$479$	9.49657e6	1.89116	0.945579	+	0.325392i	$0.105496\pi$
$480$	−304128.	−0.0602495
$481$	−156832.	−0.0309081
$482$	786588.	0.154216
$483$	−423801.	−0.0826598
$484$	−2.43544e6	−0.472568
$485$	2.42903e6	0.468898
$486$	−236196.	−0.0453609
$487$	−9.61252e6	−1.83660	−0.918300	−	0.395884i	$-0.870438\pi$
$487$	−9.61252e6	−1.83660	−0.918300	+	0.395884i	$0.870438\pi$
$488$	903040.	0.171655
$489$	4.32061e6	0.817096
$490$	316932.	0.0596315
$491$	4.52985e6	0.847970	0.423985	−	0.905669i	$-0.360631\pi$
$491$	4.52985e6	0.847970	0.423985	+	0.905669i	$0.360631\pi$
$492$	−1.64765e6	−0.306868
$493$	7.04674e6	1.30578
$494$	1.29048e6	0.237922
$495$	−251262.	−0.0460908
$496$	1.79584e6	0.327766
$497$	−2.33965e6	−0.424874
$498$	−1.05815e6	−0.191193
$499$	−9.86186e6	−1.77299	−0.886497	−	0.462734i	$-0.846869\pi$
$499$	−9.86186e6	−1.77299	−0.886497	+	0.462734i	$0.846869\pi$
$500$	−2.72501e6	−0.487464
$501$	2.93754e6	0.522864
$502$	4.57026e6	0.809435
$503$	7.85853e6	1.38491	0.692455	−	0.721461i	$-0.256529\pi$
$503$	7.85853e6	1.38491	0.692455	+	0.721461i	$0.256529\pi$
$504$	254016.	0.0445435
$505$	−1.47530e6	−0.257425
$506$	−361336.	−0.0627386
$507$	−257049.	−0.0444116
$508$	2.15443e6	0.370402
$509$	2.14808e6	0.367499	0.183750	−	0.982973i	$-0.441176\pi$
$509$	2.14808e6	0.367499	0.183750	+	0.982973i	$0.441176\pi$
$510$	1.32106e6	0.224903
$511$	−1.23975e6	−0.210030
$512$	262144.	0.0441942
$513$	1.39166e6	0.233475
$514$	1.95505e6	0.326400
$515$	447876.	0.0744114
$516$	1.83038e6	0.302634
$517$	1.42006e6	0.233657
$518$	181888.	0.0297838
$519$	1.60693e6	0.261866
$520$	−356928.	−0.0578858
$521$	−2.79345e6	−0.450865	−0.225432	−	0.974259i	$-0.572379\pi$
$521$	−2.79345e6	−0.450865	−0.225432	+	0.974259i	$0.572379\pi$
$522$	−2.05319e6	−0.329801
$523$	−355354.	−0.0568077	−0.0284038	−	0.999597i	$-0.509042\pi$
$523$	−355354.	−0.0568077	−0.0284038	+	0.999597i	$0.509042\pi$
$524$	−105728.	−0.0168214
$525$	897876.	0.142173
$526$	−489780.	−0.0771857
$527$	−7.80068e6	−1.22351
$528$	216576.	0.0338085
$529$	−5.51282e6	−0.856515
$530$	2.46721e6	0.381520
$531$	−1.00116e6	−0.154087
$532$	−1.49666e6	−0.229268
$533$	−1.93370e6	−0.294829
$534$	2.79436e6	0.424061
$535$	2.37270e6	0.358392
$536$	−3.43974e6	−0.517147
$537$	3.02795e6	0.453120
$538$	5.63706e6	0.839647
$539$	−225694.	−0.0334617
$540$	−384912.	−0.0568038
$541$	−7.45645e6	−1.09532	−0.547658	−	0.836703i	$-0.684480\pi$
$541$	−7.45645e6	−1.09532	−0.547658	+	0.836703i	$0.684480\pi$
$542$	−999136.	−0.146092
$543$	2.77745e6	0.404247
$544$	−1.13869e6	−0.164971
$545$	−3.82160e6	−0.551130
$546$	298116.	0.0427960
$547$	1.10886e6	0.158455	0.0792276	−	0.996857i	$-0.474755\pi$
$547$	1.10886e6	0.158455	0.0792276	+	0.996857i	$0.474755\pi$
$548$	6.16627e6	0.877145
$549$	1.14291e6	0.161838
$550$	765536.	0.107909
$551$	1.20973e7	1.69750
$552$	−553536.	−0.0773211
$553$	−266903.	−0.0371142
$554$	−9.01505e6	−1.24794
$555$	−275616.	−0.0379815
$556$	−3.97674e6	−0.545557
$557$	536532.	0.0732753	0.0366377	−	0.999329i	$-0.488335\pi$
$557$	536532.	0.0732753	0.0366377	+	0.999329i	$0.488335\pi$
$558$	2.27286e6	0.309020
$559$	2.14816e6	0.290762
$560$	413952.	0.0557802
$561$	−940752.	−0.126203
$562$	−3.87495e6	−0.517518
$563$	−88124.0	−0.0117172	−0.00585859	−	0.999983i	$-0.501865\pi$
$563$	−88124.0	−0.0117172	−0.00585859	+	0.999983i	$0.501865\pi$
$564$	2.17541e6	0.287967
$565$	−7.58720e6	−0.999908
$566$	3.59741e6	0.472007
$567$	321489.	0.0419961
$568$	−3.05587e6	−0.397433
$569$	8.43743e6	1.09252	0.546260	−	0.837616i	$-0.316051\pi$
$569$	8.43743e6	1.09252	0.546260	+	0.837616i	$0.316051\pi$
$570$	2.26789e6	0.292372
$571$	290617.	0.0373019	0.0186509	−	0.999826i	$-0.494063\pi$
$571$	290617.	0.0373019	0.0186509	+	0.999826i	$0.494063\pi$
$572$	254176.	0.0324821
$573$	2.69705e6	0.343164
$574$	2.24263e6	0.284105
$575$	−1.95660e6	−0.246792
$576$	331776.	0.0416667
$577$	−1.49644e7	−1.87120	−0.935598	−	0.353067i	$-0.885139\pi$
$577$	−1.49644e7	−1.87120	−0.935598	+	0.353067i	$0.885139\pi$
$578$	−733252.	−0.0912922
$579$	−1.36640e6	−0.169387
$580$	−3.34594e6	−0.412998
$581$	1.44026e6	0.177011
$582$	−2.64985e6	−0.324276
$583$	−1.75695e6	−0.214086
$584$	−1.61926e6	−0.196465
$585$	−451737.	−0.0545753
$586$	354276.	0.0426185
$587$	1.03540e7	1.24027	0.620133	−	0.784497i	$-0.287079\pi$
$587$	1.03540e7	1.24027	0.620133	+	0.784497i	$0.287079\pi$
$588$	−345744.	−0.0412393
$589$	−1.33916e7	−1.59054
$590$	−1.63152e6	−0.192958
$591$	1.35545e6	0.159631
$592$	237568.	0.0278602
$593$	−353147.	−0.0412400	−0.0206200	−	0.999787i	$-0.506564\pi$
$593$	−353147.	−0.0412400	−0.0206200	+	0.999787i	$0.506564\pi$
$594$	274104.	0.0318749
$595$	−1.79810e6	−0.208220
$596$	−4.54883e6	−0.524547
$597$	−5.87081e6	−0.674159
$598$	−649636.	−0.0742877
$599$	7.97451e6	0.908107	0.454054	−	0.890974i	$-0.349977\pi$
$599$	7.97451e6	0.908107	0.454054	+	0.890974i	$0.349977\pi$
$600$	1.17274e6	0.132991
$601$	8.15272e6	0.920696	0.460348	−	0.887739i	$-0.347725\pi$
$601$	8.15272e6	0.920696	0.460348	+	0.887739i	$0.347725\pi$
$602$	−2.49136e6	−0.280185
$603$	−4.35343e6	−0.487571
$604$	5.01830e6	0.559712
$605$	−5.02310e6	−0.557934
$606$	1.60942e6	0.178027
$607$	2.04774e6	0.225581	0.112791	−	0.993619i	$-0.464021\pi$
$607$	2.04774e6	0.225581	0.112791	+	0.993619i	$0.464021\pi$
$608$	−1.95482e6	−0.214460
$609$	2.79462e6	0.305337
$610$	1.86252e6	0.202664
$611$	2.55308e6	0.276670
$612$	−1.44115e6	−0.155536
$613$	8.15418e6	0.876454	0.438227	−	0.898864i	$-0.355607\pi$
$613$	8.15418e6	0.876454	0.438227	+	0.898864i	$0.355607\pi$
$614$	−264028.	−0.0282637
$615$	−3.39827e6	−0.362302
$616$	−294784.	−0.0313006
$617$	1.28930e7	1.36345	0.681726	−	0.731608i	$-0.261230\pi$
$617$	1.28930e7	1.36345	0.681726	+	0.731608i	$0.261230\pi$
$618$	−488592.	−0.0514606
$619$	−5.76347e6	−0.604585	−0.302292	−	0.953215i	$-0.597752\pi$
$619$	−5.76347e6	−0.604585	−0.302292	+	0.953215i	$0.597752\pi$
$620$	3.70392e6	0.386975
$621$	−700569.	−0.0728991
$622$	7.64942e6	0.792779
$623$	−3.80343e6	−0.392604
$624$	389376.	0.0400320
$625$	742171.	0.0759983
$626$	6.97006e6	0.710887
$627$	−1.61501e6	−0.164062
$628$	−6.50989e6	−0.658680
$629$	−1.03194e6	−0.103998
$630$	523908.	0.0525901
$631$	−1.97615e6	−0.197582	−0.0987908	−	0.995108i	$-0.531497\pi$
$631$	−1.97615e6	−0.197582	−0.0987908	+	0.995108i	$0.531497\pi$
$632$	−348608.	−0.0347172
$633$	−7.11818e6	−0.706089
$634$	−1.33311e7	−1.31717
$635$	4.44352e6	0.437313
$636$	−2.69150e6	−0.263847
$637$	−405769.	−0.0396214
$638$	2.38271e6	0.231750
$639$	−3.86759e6	−0.374704
$640$	540672.	0.0521776
$641$	−1.04631e7	−1.00581	−0.502906	−	0.864341i	$-0.667736\pi$
$641$	−1.04631e7	−1.00581	−0.502906	+	0.864341i	$0.667736\pi$
$642$	−2.58840e6	−0.247853
$643$	−1.33585e7	−1.27418	−0.637090	−	0.770789i	$-0.719862\pi$
$643$	−1.33585e7	−1.27418	−0.637090	+	0.770789i	$0.719862\pi$
$644$	753424.	0.0715855
$645$	3.77517e6	0.357303
$646$	8.49123e6	0.800552
$647$	3.41848e6	0.321050	0.160525	−	0.987032i	$-0.448681\pi$
$647$	3.41848e6	0.321050	0.160525	+	0.987032i	$0.448681\pi$
$648$	419904.	0.0392837
$649$	1.16184e6	0.108277
$650$	1.37634e6	0.127774
$651$	−3.09362e6	−0.286097
$652$	−7.68109e6	−0.707626
$653$	−3.41811e6	−0.313692	−0.156846	−	0.987623i	$-0.550133\pi$
$653$	−3.41811e6	−0.313692	−0.156846	+	0.987623i	$0.550133\pi$
$654$	4.16902e6	0.381144
$655$	−218064.	−0.0198601
$656$	2.92915e6	0.265756
$657$	−2.04938e6	−0.185229
$658$	−2.96097e6	−0.266606
$659$	1.31180e7	1.17667	0.588333	−	0.808619i	$-0.299784\pi$
$659$	1.31180e7	1.17667	0.588333	+	0.808619i	$0.299784\pi$
$660$	446688.	0.0399158
$661$	743707.	0.0662061	0.0331031	−	0.999452i	$-0.489461\pi$
$661$	743707.	0.0662061	0.0331031	+	0.999452i	$0.489461\pi$
$662$	4.05607e6	0.359717
$663$	−1.69135e6	−0.149434
$664$	1.88115e6	0.165578
$665$	−3.08685e6	−0.270684
$666$	300672.	0.0262668
$667$	−6.08986e6	−0.530020
$668$	−5.22229e6	−0.452814
$669$	−5.87431e6	−0.507448
$670$	−7.09447e6	−0.610567
$671$	−1.32634e6	−0.113723
$672$	−451584.	−0.0385758
$673$	169353.	0.0144130	0.00720651	−	0.999974i	$-0.497706\pi$
$673$	169353.	0.0144130	0.00720651	+	0.999974i	$0.497706\pi$
$674$	−1.24712e7	−1.05745
$675$	1.48424e6	0.125385
$676$	456976.	0.0384615
$677$	−7.32685e6	−0.614392	−0.307196	−	0.951646i	$-0.599391\pi$
$677$	−7.32685e6	−0.614392	−0.307196	+	0.951646i	$0.599391\pi$
$678$	8.27694e6	0.691505
$679$	3.60674e6	0.300221
$680$	−2.34854e6	−0.194772
$681$	−4.40219e6	−0.363748
$682$	−2.63764e6	−0.217147
$683$	−2.27710e7	−1.86780	−0.933902	−	0.357530i	$-0.883619\pi$
$683$	−2.27710e7	−1.86780	−0.933902	+	0.357530i	$0.883619\pi$
$684$	−2.47406e6	−0.202195
$685$	1.27179e7	1.03560
$686$	470596.	0.0381802
$687$	1.32323e6	0.106966
$688$	−3.25402e6	−0.262089
$689$	−3.15878e6	−0.253496
$690$	−1.14167e6	−0.0912887
$691$	−9.31443e6	−0.742098	−0.371049	−	0.928613i	$-0.621002\pi$
$691$	−9.31443e6	−0.742098	−0.371049	+	0.928613i	$0.621002\pi$
$692$	−2.85677e6	−0.226783
$693$	−373086.	−0.0295105
$694$	7.79974e6	0.614726
$695$	−8.20202e6	−0.644108
$696$	3.65011e6	0.285616
$697$	−1.27235e7	−0.992030
$698$	2.84236e6	0.220821
$699$	−1.51948e6	−0.117626
$700$	−1.59622e6	−0.123126
$701$	−2.74209e6	−0.210759	−0.105380	−	0.994432i	$-0.533606\pi$
$701$	−2.74209e6	−0.210759	−0.105380	+	0.994432i	$0.533606\pi$
$702$	492804.	0.0377426
$703$	−1.77155e6	−0.135197
$704$	−385024.	−0.0292790
$705$	4.48678e6	0.339987
$706$	1.48723e7	1.12297
$707$	−2.19059e6	−0.164821
$708$	1.77984e6	0.133444
$709$	−2.58345e7	−1.93012	−0.965062	−	0.262023i	$-0.915610\pi$
$709$	−2.58345e7	−1.93012	−0.965062	+	0.262023i	$0.915610\pi$
$710$	−6.30274e6	−0.469227
$711$	−441207.	−0.0327317
$712$	−4.96774e6	−0.367248
$713$	6.74142e6	0.496623
$714$	1.96157e6	0.143998
$715$	524238.	0.0383498
$716$	−5.38302e6	−0.392413
$717$	−6.14552e6	−0.446438
$718$	−1.27049e7	−0.919727
$719$	2.18226e7	1.57429	0.787145	−	0.616768i	$-0.211558\pi$
$719$	2.18226e7	1.57429	0.787145	+	0.616768i	$0.211558\pi$
$720$	684288.	0.0491935
$721$	665028.	0.0476433
$722$	4.67273e6	0.333601
$723$	−1.76982e6	−0.125917
$724$	−4.93770e6	−0.350089
$725$	1.29021e7	0.911625
$726$	5.47974e6	0.385850
$727$	−2.51304e7	−1.76345	−0.881724	−	0.471766i	$-0.843617\pi$
$727$	−2.51304e7	−1.76345	−0.881724	+	0.471766i	$0.843617\pi$
$728$	−529984.	−0.0370625
$729$	531441.	0.0370370
$730$	−3.33973e6	−0.231955
$731$	1.41346e7	0.978343
$732$	−2.03184e6	−0.140156
$733$	−1.89702e7	−1.30410	−0.652052	−	0.758174i	$-0.726092\pi$
$733$	−1.89702e7	−1.30410	−0.652052	+	0.758174i	$0.726092\pi$
$734$	893144.	0.0611901
$735$	−713097.	−0.0486889
$736$	984064.	0.0669621
$737$	5.05212e6	0.342614
$738$	3.70721e6	0.250557
$739$	5.69725e6	0.383755	0.191878	−	0.981419i	$-0.438542\pi$
$739$	5.69725e6	0.383755	0.191878	+	0.981419i	$0.438542\pi$
$740$	489984.	0.0328929
$741$	−2.90359e6	−0.194263
$742$	3.66344e6	0.244275
$743$	−535500.	−0.0355867	−0.0177933	−	0.999842i	$-0.505664\pi$
$743$	−535500.	−0.0355867	−0.0177933	+	0.999842i	$0.505664\pi$
$744$	−4.04064e6	−0.267620
$745$	−9.38197e6	−0.619303
$746$	1.02537e7	0.674577
$747$	2.38083e6	0.156109
$748$	1.67245e6	0.109295
$749$	3.52310e6	0.229467
$750$	6.13127e6	0.398013
$751$	−2.50574e7	−1.62120	−0.810599	−	0.585601i	$-0.800858\pi$
$751$	−2.50574e7	−1.62120	−0.810599	+	0.585601i	$0.800858\pi$
$752$	−3.86739e6	−0.249387
$753$	−1.02831e7	−0.660901
$754$	4.28381e6	0.274411
$755$	1.03503e7	0.660821
$756$	−571536.	−0.0363696
$757$	−2.25991e7	−1.43335	−0.716674	−	0.697408i	$-0.754337\pi$
$757$	−2.25991e7	−1.43335	−0.716674	+	0.697408i	$0.754337\pi$
$758$	−4.96473e6	−0.313850
$759$	813006.	0.0512259
$760$	−4.03181e6	−0.253201
$761$	1.98800e6	0.124439	0.0622193	−	0.998063i	$-0.480182\pi$
$761$	1.98800e6	0.124439	0.0622193	+	0.998063i	$0.480182\pi$
$762$	−4.84747e6	−0.302432
$763$	−5.67449e6	−0.352871
$764$	−4.79475e6	−0.297189
$765$	−2.97238e6	−0.183633
$766$	−1.32264e7	−0.814463
$767$	2.08884e6	0.128208
$768$	−589824.	−0.0360844
$769$	4.40979e6	0.268907	0.134453	−	0.990920i	$-0.457072\pi$
$769$	4.40979e6	0.268907	0.134453	+	0.990920i	$0.457072\pi$
$770$	−607992.	−0.0369548
$771$	−4.39886e6	−0.266504
$772$	2.42915e6	0.146694
$773$	1.29494e7	0.779473	0.389737	−	0.920926i	$-0.372566\pi$
$773$	1.29494e7	0.779473	0.389737	+	0.920926i	$0.372566\pi$
$774$	−4.11836e6	−0.247100
$775$	−1.42825e7	−0.854184
$776$	4.71085e6	0.280831
$777$	−409248.	−0.0243183
$778$	−573704.	−0.0339812
$779$	−2.18428e7	−1.28963
$780$	803088.	0.0472636
$781$	4.48831e6	0.263303
$782$	−4.27453e6	−0.249960
$783$	4.61967e6	0.269282
$784$	614656.	0.0357143
$785$	−1.34266e7	−0.777666
$786$	237888.	0.0137346
$787$	−2.39811e7	−1.38017	−0.690085	−	0.723729i	$-0.742427\pi$
$787$	−2.39811e7	−1.38017	−0.690085	+	0.723729i	$0.742427\pi$
$788$	−2.40970e6	−0.138244
$789$	1.10200e6	0.0630218
$790$	−719004.	−0.0409886
$791$	−1.12658e7	−0.640210
$792$	−487296.	−0.0276045
$793$	−2.38459e6	−0.134658
$794$	1.43709e7	0.808969
$795$	−5.55123e6	−0.311509
$796$	1.04370e7	0.583838
$797$	−1.20817e7	−0.673723	−0.336862	−	0.941554i	$-0.609365\pi$
$797$	−1.20817e7	−0.673723	−0.336862	+	0.941554i	$0.609365\pi$
$798$	3.36748e6	0.187196
$799$	1.67990e7	0.930928
$800$	−2.08486e6	−0.115174
$801$	−6.28730e6	−0.346245
$802$	3.37936e6	0.185523
$803$	2.37829e6	0.130160
$804$	7.73942e6	0.422249
$805$	1.55394e6	0.0845169
$806$	−4.74214e6	−0.257121
$807$	−1.26834e7	−0.685569
$808$	−2.86118e6	−0.154176
$809$	−4.19257e6	−0.225221	−0.112611	−	0.993639i	$-0.535921\pi$
$809$	−4.19257e6	−0.225221	−0.112611	+	0.993639i	$0.535921\pi$
$810$	866052.	0.0463801
$811$	7.72928e6	0.412655	0.206327	−	0.978483i	$-0.433849\pi$
$811$	7.72928e6	0.412655	0.206327	+	0.978483i	$0.433849\pi$
$812$	−4.96821e6	−0.264429
$813$	2.24806e6	0.119284
$814$	−348928.	−0.0184576
$815$	−1.58422e7	−0.835454
$816$	2.56205e6	0.134698
$817$	2.42653e7	1.27183
$818$	−1.28955e7	−0.673836
$819$	−670761.	−0.0349428
$820$	6.04138e6	0.313763
$821$	−3.58460e7	−1.85602	−0.928012	−	0.372551i	$-0.878483\pi$
$821$	−3.58460e7	−1.85602	−0.928012	+	0.372551i	$0.878483\pi$
$822$	−1.38741e7	−0.716186
$823$	2.70595e7	1.39258	0.696289	−	0.717761i	$-0.254833\pi$
$823$	2.70595e7	1.39258	0.696289	+	0.717761i	$0.254833\pi$
$824$	868608.	0.0445662
$825$	−1.72246e6	−0.0881076
$826$	−2.42256e6	−0.123545
$827$	−5.34505e6	−0.271762	−0.135881	−	0.990725i	$-0.543386\pi$
$827$	−5.34505e6	−0.271762	−0.135881	+	0.990725i	$0.543386\pi$
$828$	1.24546e6	0.0631324
$829$	−2.75307e7	−1.39133	−0.695665	−	0.718366i	$-0.744890\pi$
$829$	−2.75307e7	−1.39133	−0.695665	+	0.718366i	$0.744890\pi$
$830$	3.87988e6	0.195489
$831$	2.02839e7	1.01894
$832$	−692224.	−0.0346688
$833$	−2.66991e6	−0.133317
$834$	8.94766e6	0.445445
$835$	−1.07710e7	−0.534612
$836$	2.87114e6	0.142082
$837$	−5.11394e6	−0.252314
$838$	2.50883e7	1.23413
$839$	−2.80851e6	−0.137744	−0.0688718	−	0.997626i	$-0.521940\pi$
$839$	−2.80851e6	−0.137744	−0.0688718	+	0.997626i	$0.521940\pi$
$840$	−931392.	−0.0455443
$841$	1.96464e7	0.957841
$842$	1.84975e7	0.899153
$843$	8.71864e6	0.422552
$844$	1.26545e7	0.611491
$845$	942513.	0.0454094
$846$	−4.89467e6	−0.235124
$847$	−7.45854e6	−0.357228
$848$	4.78490e6	0.228498
$849$	−8.09417e6	−0.385392
$850$	9.05613e6	0.429928
$851$	891808.	0.0422131
$852$	6.87571e6	0.324503
$853$	1.30735e7	0.615204	0.307602	−	0.951515i	$-0.400474\pi$
$853$	1.30735e7	0.615204	0.307602	+	0.951515i	$0.400474\pi$
$854$	2.76556e6	0.129759
$855$	−5.10276e6	−0.238720
$856$	4.60160e6	0.214647
$857$	1.73465e7	0.806787	0.403394	−	0.915027i	$-0.367831\pi$
$857$	1.73465e7	0.806787	0.403394	+	0.915027i	$0.367831\pi$
$858$	−571896.	−0.0265215
$859$	3.22830e7	1.49276	0.746381	−	0.665519i	$-0.231790\pi$
$859$	3.22830e7	1.49276	0.746381	+	0.665519i	$0.231790\pi$
$860$	−6.71141e6	−0.309434
$861$	−5.04592e6	−0.231970
$862$	9.75108e6	0.446976
$863$	1.21181e7	0.553869	0.276934	−	0.960889i	$-0.410682\pi$
$863$	1.21181e7	0.553869	0.276934	+	0.960889i	$0.410682\pi$
$864$	−746496.	−0.0340207
$865$	−5.89208e6	−0.267749
$866$	−8.53144e6	−0.386570
$867$	1.64982e6	0.0745398
$868$	5.49976e6	0.247768
$869$	512018.	0.0230004
$870$	7.52836e6	0.337211
$871$	9.08307e6	0.405684
$872$	−7.41158e6	−0.330080
$873$	5.96217e6	0.264770
$874$	−7.33820e6	−0.324946
$875$	−8.34534e6	−0.368488
$876$	3.64334e6	0.160413
$877$	3.95515e7	1.73646	0.868228	−	0.496165i	$-0.165259\pi$
$877$	3.95515e7	1.73646	0.868228	+	0.496165i	$0.165259\pi$
$878$	−1.07573e7	−0.470941
$879$	−797121.	−0.0347978
$880$	−794112.	−0.0345681
$881$	−4.28140e7	−1.85843	−0.929215	−	0.369538i	$-0.879516\pi$
$881$	−4.28140e7	−1.85843	−0.929215	+	0.369538i	$0.879516\pi$
$882$	777924.	0.0336718
$883$	3.02925e7	1.30747	0.653737	−	0.756721i	$-0.273200\pi$
$883$	3.02925e7	1.30747	0.653737	+	0.756721i	$0.273200\pi$
$884$	3.00685e6	0.129414
$885$	3.67092e6	0.157549
$886$	1.43914e7	0.615914
$887$	−1.71988e7	−0.733988	−0.366994	−	0.930223i	$-0.619613\pi$
$887$	−1.71988e7	−0.733988	−0.366994	+	0.930223i	$0.619613\pi$
$888$	−534528.	−0.0227477
$889$	6.59795e6	0.279998
$890$	−1.02460e7	−0.433589
$891$	−616734.	−0.0260258
$892$	1.04432e7	0.439463
$893$	2.88393e7	1.21020
$894$	1.02349e7	0.428291
$895$	−1.11025e7	−0.463300
$896$	802816.	0.0334077
$897$	1.46168e6	0.0606557
$898$	−1.55227e6	−0.0642358
$899$	−4.44541e7	−1.83448
$900$	−2.63866e6	−0.108587
$901$	−2.07844e7	−0.852953
$902$	−4.30219e6	−0.176065
$903$	5.60555e6	0.228770
$904$	−1.47146e7	−0.598861
$905$	−1.01840e7	−0.413330
$906$	−1.12912e7	−0.457003
$907$	−8.87007e6	−0.358021	−0.179011	−	0.983847i	$-0.557290\pi$
$907$	−8.87007e6	−0.358021	−0.179011	+	0.983847i	$0.557290\pi$
$908$	7.82611e6	0.315015
$909$	−3.62119e6	−0.145359
$910$	−1.09309e6	−0.0437576
$911$	−3.30818e7	−1.32066	−0.660332	−	0.750973i	$-0.729585\pi$
$911$	−3.30818e7	−1.32066	−0.660332	+	0.750973i	$0.729585\pi$
$912$	4.39834e6	0.175106
$913$	−2.76294e6	−0.109697
$914$	−1.03216e7	−0.408680
$915$	−4.19067e6	−0.165474
$916$	−2.35242e6	−0.0926351
$917$	−323792.	−0.0127158
$918$	3.24259e6	0.126995
$919$	1.92856e6	0.0753259	0.0376630	−	0.999290i	$-0.488009\pi$
$919$	1.92856e6	0.0753259	0.0376630	+	0.999290i	$0.488009\pi$
$920$	2.02963e6	0.0790583
$921$	594063.	0.0230772
$922$	1.06254e7	0.411639
$923$	8.06941e6	0.311772
$924$	663264.	0.0255568
$925$	−1.88941e6	−0.0726058
$926$	1.76803e7	0.677582
$927$	1.09933e6	0.0420174
$928$	−6.48909e6	−0.247351
$929$	3.07955e7	1.17071	0.585353	−	0.810778i	$-0.300956\pi$
$929$	3.07955e7	1.17071	0.585353	+	0.810778i	$0.300956\pi$
$930$	−8.33382e6	−0.315963
$931$	−4.58351e6	−0.173310
$932$	2.70130e6	0.101867
$933$	−1.72112e7	−0.647302
$934$	3.14822e7	1.18086
$935$	3.44942e6	0.129038
$936$	−876096.	−0.0326860
$937$	1.41435e7	0.526269	0.263134	−	0.964759i	$-0.415244\pi$
$937$	1.41435e7	0.526269	0.263134	+	0.964759i	$0.415244\pi$
$938$	−1.05342e7	−0.390927
$939$	−1.56826e7	−0.580437
$940$	−7.97650e6	−0.294437
$941$	2.65039e7	0.975742	0.487871	−	0.872916i	$-0.337774\pi$
$941$	2.65039e7	0.975742	0.487871	+	0.872916i	$0.337774\pi$
$942$	1.46472e7	0.537810
$943$	1.09958e7	0.402667
$944$	−3.16416e6	−0.115566
$945$	−1.17879e6	−0.0429396
$946$	4.77934e6	0.173636
$947$	3.15186e7	1.14207	0.571034	−	0.820926i	$-0.306542\pi$
$947$	3.15186e7	1.14207	0.571034	+	0.820926i	$0.306542\pi$
$948$	784368.	0.0283465
$949$	4.27587e6	0.154120
$950$	1.55469e7	0.558901
$951$	2.99949e7	1.07547
$952$	−3.48723e6	−0.124706
$953$	2.89866e7	1.03387	0.516934	−	0.856025i	$-0.327073\pi$
$953$	2.89866e7	1.03387	0.516934	+	0.856025i	$0.327073\pi$
$954$	6.05588e6	0.215430
$955$	−9.88918e6	−0.350874
$956$	1.09254e7	0.386627
$957$	−5.36110e6	−0.189223
$958$	3.79863e7	1.33725
$959$	1.88842e7	0.663059
$960$	−1.21651e6	−0.0426028
$961$	2.05811e7	0.718885
$962$	−627328.	−0.0218553
$963$	5.82390e6	0.202371
$964$	3.14635e6	0.109047
$965$	5.01013e6	0.173193
$966$	−1.69520e6	−0.0584493
$967$	4.68788e7	1.61217	0.806084	−	0.591801i	$-0.201583\pi$
$967$	4.68788e7	1.61217	0.806084	+	0.591801i	$0.201583\pi$
$968$	−9.74176e6	−0.334156
$969$	−1.91053e7	−0.653648
$970$	9.71612e6	0.331561
$971$	2.56806e7	0.874093	0.437047	−	0.899439i	$-0.356024\pi$
$971$	2.56806e7	0.874093	0.437047	+	0.899439i	$0.356024\pi$
$972$	−944784.	−0.0320750
$973$	−1.21788e7	−0.412402
$974$	−3.84501e7	−1.29867
$975$	−3.09676e6	−0.104327
$976$	3.61216e6	0.121379
$977$	2.39913e7	0.804113	0.402056	−	0.915615i	$-0.368296\pi$
$977$	2.39913e7	0.804113	0.402056	+	0.915615i	$0.368296\pi$
$978$	1.72824e7	0.577774
$979$	7.29637e6	0.243305
$980$	1.26773e6	0.0421659
$981$	−9.38029e6	−0.311203
$982$	1.81194e7	0.599605
$983$	−4.77551e6	−0.157629	−0.0788145	−	0.996889i	$-0.525113\pi$
$983$	−4.77551e6	−0.157629	−0.0788145	+	0.996889i	$0.525113\pi$
$984$	−6.59059e6	−0.216988
$985$	−4.97000e6	−0.163217
$986$	2.81870e7	0.923328
$987$	6.66219e6	0.217683
$988$	5.16194e6	0.168237
$989$	−1.22153e7	−0.397111
$990$	−1.00505e6	−0.0325911
$991$	−9.82040e6	−0.317647	−0.158824	−	0.987307i	$-0.550770\pi$
$991$	−9.82040e6	−0.317647	−0.158824	+	0.987307i	$0.550770\pi$
$992$	7.18336e6	0.231765
$993$	−9.12616e6	−0.293708
$994$	−9.35861e6	−0.300431
$995$	2.15263e7	0.689305
$996$	−4.23259e6	−0.135194
$997$	−4.09143e7	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$997$	−4.09143e7	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$998$	−3.94474e7	−1.25370
$999$	−676512.	−0.0214468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.6.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.6.a.d.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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