LMFDB - Embedded newform 171.6.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,6,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, 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("171.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	171.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:	$27.4256331880$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1521 q^{2} +71.0645 q^{4} +99.7603 q^{5} -57.1289 q^{7} +396.585 q^{8} +O(q^{10})$	\(q+10.1521 q^{2} +71.0645 q^{4} +99.7603 q^{5} -57.1289 q^{7} +396.585 q^{8} +1012.77 q^{10} +438.977 q^{11} -477.590 q^{13} -579.977 q^{14} +1752.10 q^{16} -1671.39 q^{17} -361.000 q^{19} +7089.42 q^{20} +4456.52 q^{22} -459.514 q^{23} +6827.12 q^{25} -4848.53 q^{26} -4059.84 q^{28} +2696.78 q^{29} -2314.43 q^{31} +5096.67 q^{32} -16968.0 q^{34} -5699.20 q^{35} +3022.51 q^{37} -3664.90 q^{38} +39563.5 q^{40} -5483.05 q^{41} +5101.80 q^{43} +31195.7 q^{44} -4665.02 q^{46} -5803.63 q^{47} -13543.3 q^{49} +69309.4 q^{50} -33939.7 q^{52} +23207.5 q^{53} +43792.5 q^{55} -22656.5 q^{56} +27377.9 q^{58} +32849.5 q^{59} -46911.5 q^{61} -23496.2 q^{62} -4325.31 q^{64} -47644.5 q^{65} -3565.12 q^{67} -118776. q^{68} -57858.7 q^{70} -2346.44 q^{71} +57370.0 q^{73} +30684.7 q^{74} -25654.3 q^{76} -25078.3 q^{77} -29715.8 q^{79} +174790. q^{80} -55664.3 q^{82} -60526.0 q^{83} -166738. q^{85} +51793.8 q^{86} +174092. q^{88} -19134.9 q^{89} +27284.2 q^{91} -32655.1 q^{92} -58918.9 q^{94} -36013.5 q^{95} -43189.4 q^{97} -137492. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} + 49 q^{4} + 133 q^{5} + 72 q^{7} + 567 q^{8}+O(q^{10}) $ 2 * q + 7 * q^2 + 49 * q^4 + 133 * q^5 + 72 * q^7 + 567 * q^8	$ 2 q + 7 q^{2} + 49 q^{4} + 133 q^{5} + 72 q^{7} + 567 q^{8} + 908 q^{10} + 705 q^{11} - 1341 q^{13} - 987 q^{14} + 1921 q^{16} - 2784 q^{17} - 722 q^{19} + 6356 q^{20} + 3618 q^{22} + 2713 q^{23} + 4807 q^{25} - 2127 q^{26} - 6909 q^{28} + 7775 q^{29} + 7132 q^{31} - 889 q^{32} - 13461 q^{34} - 1407 q^{35} - 6248 q^{37} - 2527 q^{38} + 45228 q^{40} + 4174 q^{41} + 25357 q^{43} + 25326 q^{44} - 14665 q^{46} - 11727 q^{47} - 13676 q^{49} + 75677 q^{50} - 14889 q^{52} + 29133 q^{53} + 52635 q^{55} - 651 q^{56} + 11371 q^{58} + 64515 q^{59} - 40939 q^{61} - 53272 q^{62} + 9137 q^{64} - 76344 q^{65} + 19039 q^{67} - 94227 q^{68} - 71388 q^{70} + 70236 q^{71} + 67058 q^{73} + 59906 q^{74} - 17689 q^{76} + 9273 q^{77} - 32850 q^{79} + 180404 q^{80} - 86104 q^{82} - 71534 q^{83} - 203721 q^{85} - 12052 q^{86} + 219426 q^{88} + 87268 q^{89} - 84207 q^{91} - 102655 q^{92} - 40248 q^{94} - 48013 q^{95} - 62458 q^{97} - 137074 q^{98}+O(q^{100}) $ 2 * q + 7 * q^2 + 49 * q^4 + 133 * q^5 + 72 * q^7 + 567 * q^8 + 908 * q^10 + 705 * q^11 - 1341 * q^13 - 987 * q^14 + 1921 * q^16 - 2784 * q^17 - 722 * q^19 + 6356 * q^20 + 3618 * q^22 + 2713 * q^23 + 4807 * q^25 - 2127 * q^26 - 6909 * q^28 + 7775 * q^29 + 7132 * q^31 - 889 * q^32 - 13461 * q^34 - 1407 * q^35 - 6248 * q^37 - 2527 * q^38 + 45228 * q^40 + 4174 * q^41 + 25357 * q^43 + 25326 * q^44 - 14665 * q^46 - 11727 * q^47 - 13676 * q^49 + 75677 * q^50 - 14889 * q^52 + 29133 * q^53 + 52635 * q^55 - 651 * q^56 + 11371 * q^58 + 64515 * q^59 - 40939 * q^61 - 53272 * q^62 + 9137 * q^64 - 76344 * q^65 + 19039 * q^67 - 94227 * q^68 - 71388 * q^70 + 70236 * q^71 + 67058 * q^73 + 59906 * q^74 - 17689 * q^76 + 9273 * q^77 - 32850 * q^79 + 180404 * q^80 - 86104 * q^82 - 71534 * q^83 - 203721 * q^85 - 12052 * q^86 + 219426 * q^88 + 87268 * q^89 - 84207 * q^91 - 102655 * q^92 - 40248 * q^94 - 48013 * q^95 - 62458 * q^97 - 137074 * q^98

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 $

$2$	10.1521	1.79465	0.897324	−	0.441372i	$-0.145508\pi$
$2$	10.1521	1.79465	0.897324	+	0.441372i	$0.145508\pi$
$3$	0	0
$3$	0	0
$4$	71.0645	2.22076
$5$	99.7603	1.78457	0.892284	−	0.451475i	$-0.149102\pi$
$5$	99.7603	1.78457	0.892284	+	0.451475i	$0.149102\pi$
$6$	0	0
$7$	−57.1289	−0.440668	−0.220334	−	0.975425i	$-0.570715\pi$
$7$	−57.1289	−0.440668	−0.220334	+	0.975425i	$0.570715\pi$
$8$	396.585	2.19084
$9$	0	0
$10$	1012.77	3.20267
$11$	438.977	1.09386	0.546928	−	0.837180i	$-0.315797\pi$
$11$	438.977	1.09386	0.546928	+	0.837180i	$0.315797\pi$
$12$	0	0
$13$	−477.590	−0.783785	−0.391892	−	0.920011i	$-0.628179\pi$
$13$	−477.590	−0.783785	−0.391892	+	0.920011i	$0.628179\pi$
$14$	−579.977	−0.790844
$15$	0	0
$16$	1752.10	1.71103
$17$	−1671.39	−1.40267	−0.701334	−	0.712833i	$-0.747412\pi$
$17$	−1671.39	−1.40267	−0.701334	+	0.712833i	$0.747412\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	7089.42	3.96310
$21$	0	0
$22$	4456.52	1.96309
$23$	−459.514	−0.181125	−0.0905627	−	0.995891i	$-0.528867\pi$
$23$	−459.514	−0.181125	−0.0905627	+	0.995891i	$0.528867\pi$
$24$	0	0
$25$	6827.12	2.18468
$26$	−4848.53	−1.40662
$27$	0	0
$28$	−4059.84	−0.978619
$29$	2696.78	0.595457	0.297729	−	0.954651i	$-0.403771\pi$
$29$	2696.78	0.595457	0.297729	+	0.954651i	$0.403771\pi$
$30$	0	0
$31$	−2314.43	−0.432553	−0.216277	−	0.976332i	$-0.569391\pi$
$31$	−2314.43	−0.432553	−0.216277	+	0.976332i	$0.569391\pi$
$32$	5096.67	0.879856
$33$	0	0
$34$	−16968.0	−2.51730
$35$	−5699.20	−0.786401
$36$	0	0
$37$	3022.51	0.362964	0.181482	−	0.983394i	$-0.441911\pi$
$37$	3022.51	0.362964	0.181482	+	0.983394i	$0.441911\pi$
$38$	−3664.90	−0.411721
$39$	0	0
$40$	39563.5	3.90971
$41$	−5483.05	−0.509404	−0.254702	−	0.967020i	$-0.581977\pi$
$41$	−5483.05	−0.509404	−0.254702	+	0.967020i	$0.581977\pi$
$42$	0	0
$43$	5101.80	0.420777	0.210388	−	0.977618i	$-0.432527\pi$
$43$	5101.80	0.420777	0.210388	+	0.977618i	$0.432527\pi$
$44$	31195.7	2.42920
$45$	0	0
$46$	−4665.02	−0.325057
$47$	−5803.63	−0.383226	−0.191613	−	0.981471i	$-0.561372\pi$
$47$	−5803.63	−0.383226	−0.191613	+	0.981471i	$0.561372\pi$
$48$	0	0
$49$	−13543.3	−0.805812
$50$	69309.4	3.92073
$51$	0	0
$52$	−33939.7	−1.74060
$53$	23207.5	1.13485	0.567426	−	0.823424i	$-0.307939\pi$
$53$	23207.5	1.13485	0.567426	+	0.823424i	$0.307939\pi$
$54$	0	0
$55$	43792.5	1.95206
$56$	−22656.5	−0.965434
$57$	0	0
$58$	27377.9	1.06864
$59$	32849.5	1.22857	0.614284	−	0.789085i	$-0.289445\pi$
$59$	32849.5	1.22857	0.614284	+	0.789085i	$0.289445\pi$
$60$	0	0
$61$	−46911.5	−1.61419	−0.807095	−	0.590422i	$-0.798961\pi$
$61$	−46911.5	−1.61419	−0.807095	+	0.590422i	$0.798961\pi$
$62$	−23496.2	−0.776281
$63$	0	0
$64$	−4325.31	−0.131998
$65$	−47644.5	−1.39872
$66$	0	0
$67$	−3565.12	−0.0970257	−0.0485128	−	0.998823i	$-0.515448\pi$
$67$	−3565.12	−0.0970257	−0.0485128	+	0.998823i	$0.515448\pi$
$68$	−118776.	−3.11499
$69$	0	0
$70$	−57858.7	−1.41131
$71$	−2346.44	−0.0552413	−0.0276207	−	0.999618i	$-0.508793\pi$
$71$	−2346.44	−0.0552413	−0.0276207	+	0.999618i	$0.508793\pi$
$72$	0	0
$73$	57370.0	1.26002	0.630010	−	0.776587i	$-0.283051\pi$
$73$	57370.0	1.26002	0.630010	+	0.776587i	$0.283051\pi$
$74$	30684.7	0.651393
$75$	0	0
$76$	−25654.3	−0.509478
$77$	−25078.3	−0.482027
$78$	0	0
$79$	−29715.8	−0.535698	−0.267849	−	0.963461i	$-0.586313\pi$
$79$	−29715.8	−0.535698	−0.267849	+	0.963461i	$0.586313\pi$
$80$	174790.	3.05345
$81$	0	0
$82$	−55664.3	−0.914202
$83$	−60526.0	−0.964377	−0.482188	−	0.876068i	$-0.660158\pi$
$83$	−60526.0	−0.964377	−0.482188	+	0.876068i	$0.660158\pi$
$84$	0	0
$85$	−166738.	−2.50315
$86$	51793.8	0.755147
$87$	0	0
$88$	174092.	2.39647
$89$	−19134.9	−0.256066	−0.128033	−	0.991770i	$-0.540866\pi$
$89$	−19134.9	−0.256066	−0.128033	+	0.991770i	$0.540866\pi$
$90$	0	0
$91$	27284.2	0.345389
$92$	−32655.1	−0.402237
$93$	0	0
$94$	−58918.9	−0.687756
$95$	−36013.5	−0.409408
$96$	0	0
$97$	−43189.4	−0.466067	−0.233033	−	0.972469i	$-0.574865\pi$
$97$	−43189.4	−0.466067	−0.233033	+	0.972469i	$0.574865\pi$
$98$	−137492.	−1.44615
$99$	0	0
$100$	485166.	4.85166
$101$	55330.5	0.539711	0.269855	−	0.962901i	$-0.413024\pi$
$101$	55330.5	0.539711	0.269855	+	0.962901i	$0.413024\pi$
$102$	0	0
$103$	−205641.	−1.90993	−0.954964	−	0.296722i	$-0.904107\pi$
$103$	−205641.	−1.90993	−0.954964	+	0.296722i	$0.904107\pi$
$104$	−189405.	−1.71715
$105$	0	0
$106$	235604.	2.03666
$107$	−181889.	−1.53585	−0.767923	−	0.640543i	$-0.778709\pi$
$107$	−181889.	−1.53585	−0.767923	+	0.640543i	$0.778709\pi$
$108$	0	0
$109$	24288.8	0.195813	0.0979063	−	0.995196i	$-0.468785\pi$
$109$	24288.8	0.195813	0.0979063	+	0.995196i	$0.468785\pi$
$110$	444584.	3.50326
$111$	0	0
$112$	−100095.	−0.753996
$113$	229667.	1.69201	0.846006	−	0.533174i	$-0.179001\pi$
$113$	229667.	1.69201	0.846006	+	0.533174i	$0.179001\pi$
$114$	0	0
$115$	−45841.3	−0.323231
$116$	191645.	1.32237
$117$	0	0
$118$	333491.	2.20485
$119$	95484.6	0.618110
$120$	0	0
$121$	31649.7	0.196520
$122$	−476248.	−2.89690
$123$	0	0
$124$	−164474.	−0.960599
$125$	369325.	2.11414
$126$	0	0
$127$	8086.69	0.0444899	0.0222450	−	0.999753i	$-0.492919\pi$
$127$	8086.69	0.0444899	0.0222450	+	0.999753i	$0.492919\pi$
$128$	−207004.	−1.11675
$129$	0	0
$130$	−483691.	−2.51021
$131$	−118712.	−0.604390	−0.302195	−	0.953246i	$-0.597719\pi$
$131$	−118712.	−0.604390	−0.302195	+	0.953246i	$0.597719\pi$
$132$	0	0
$133$	20623.5	0.101096
$134$	−36193.3	−0.174127
$135$	0	0
$136$	−662847.	−3.07303
$137$	−65348.7	−0.297465	−0.148732	−	0.988877i	$-0.547519\pi$
$137$	−65348.7	−0.297465	−0.148732	+	0.988877i	$0.547519\pi$
$138$	0	0
$139$	−28437.6	−0.124841	−0.0624203	−	0.998050i	$-0.519882\pi$
$139$	−28437.6	−0.124841	−0.0624203	+	0.998050i	$0.519882\pi$
$140$	−405011.	−1.74641
$141$	0	0
$142$	−23821.3	−0.0991388
$143$	−209651.	−0.857347
$144$	0	0
$145$	269032.	1.06263
$146$	582424.	2.26130
$147$	0	0
$148$	214793.	0.806057
$149$	−115852.	−0.427502	−0.213751	−	0.976888i	$-0.568568\pi$
$149$	−115852.	−0.427502	−0.213751	+	0.976888i	$0.568568\pi$
$150$	0	0
$151$	35980.9	0.128419	0.0642096	−	0.997936i	$-0.479547\pi$
$151$	35980.9	0.128419	0.0642096	+	0.997936i	$0.479547\pi$
$152$	−143167.	−0.502614
$153$	0	0
$154$	−254596.	−0.865069
$155$	−230888.	−0.771920
$156$	0	0
$157$	566234.	1.83336	0.916678	−	0.399627i	$-0.130860\pi$
$157$	566234.	1.83336	0.916678	+	0.399627i	$0.130860\pi$
$158$	−301677.	−0.961390
$159$	0	0
$160$	508446.	1.57016
$161$	26251.6	0.0798161
$162$	0	0
$163$	−307841.	−0.907523	−0.453761	−	0.891123i	$-0.649918\pi$
$163$	−307841.	−0.907523	−0.453761	+	0.891123i	$0.649918\pi$
$164$	−389650.	−1.13127
$165$	0	0
$166$	−614464.	−1.73072
$167$	−126991.	−0.352357	−0.176179	−	0.984358i	$-0.556374\pi$
$167$	−126991.	−0.352357	−0.176179	+	0.984358i	$0.556374\pi$
$168$	0	0
$169$	−143201.	−0.385681
$170$	−1.69274e6	−4.49228
$171$	0	0
$172$	362556.	0.934446
$173$	582515.	1.47976	0.739881	−	0.672738i	$-0.234882\pi$
$173$	582515.	1.47976	0.739881	+	0.672738i	$0.234882\pi$
$174$	0	0
$175$	−390026.	−0.962718
$176$	769130.	1.87162
$177$	0	0
$178$	−194259.	−0.459548
$179$	233819.	0.545440	0.272720	−	0.962093i	$-0.412077\pi$
$179$	233819.	0.545440	0.272720	+	0.962093i	$0.412077\pi$
$180$	0	0
$181$	318180.	0.721899	0.360950	−	0.932585i	$-0.382453\pi$
$181$	318180.	0.721899	0.360950	+	0.932585i	$0.382453\pi$
$182$	276991.	0.619851
$183$	0	0
$184$	−182237.	−0.396818
$185$	301527.	0.647734
$186$	0	0
$187$	−733700.	−1.53432
$188$	−412432.	−0.851055
$189$	0	0
$190$	−365611.	−0.734743
$191$	182032.	0.361047	0.180523	−	0.983571i	$-0.442221\pi$
$191$	182032.	0.361047	0.180523	+	0.983571i	$0.442221\pi$
$192$	0	0
$193$	286026.	0.552729	0.276365	−	0.961053i	$-0.410870\pi$
$193$	286026.	0.552729	0.276365	+	0.961053i	$0.410870\pi$
$194$	−438462.	−0.836426
$195$	0	0
$196$	−962446.	−1.78952
$197$	465880.	0.855281	0.427640	−	0.903949i	$-0.359345\pi$
$197$	465880.	0.855281	0.427640	+	0.903949i	$0.359345\pi$
$198$	0	0
$199$	1.06447e6	1.90547	0.952736	−	0.303798i	$-0.0982548\pi$
$199$	1.06447e6	1.90547	0.952736	+	0.303798i	$0.0982548\pi$
$200$	2.70754e6	4.78629
$201$	0	0
$202$	561719.	0.968591
$203$	−154064.	−0.262399
$204$	0	0
$205$	−546991.	−0.909067
$206$	−2.08768e6	−3.42765
$207$	0	0
$208$	−836784.	−1.34108
$209$	−158471.	−0.250948
$210$	0	0
$211$	1.00586e6	1.55536	0.777679	−	0.628662i	$-0.216397\pi$
$211$	1.00586e6	1.55536	0.777679	+	0.628662i	$0.216397\pi$
$212$	1.64923e6	2.52024
$213$	0	0
$214$	−1.84655e6	−2.75630
$215$	508957.	0.750905
$216$	0	0
$217$	132221.	0.190612
$218$	246582.	0.351415
$219$	0	0
$220$	3.11209e6	4.33506
$221$	798238.	1.09939
$222$	0	0
$223$	−377403.	−0.508210	−0.254105	−	0.967177i	$-0.581781\pi$
$223$	−377403.	−0.508210	−0.254105	+	0.967177i	$0.581781\pi$
$224$	−291167.	−0.387724
$225$	0	0
$226$	2.33160e6	3.03657
$227$	1.28433e6	1.65429	0.827145	−	0.561989i	$-0.189964\pi$
$227$	1.28433e6	1.65429	0.827145	+	0.561989i	$0.189964\pi$
$228$	0	0
$229$	299103.	0.376905	0.188452	−	0.982082i	$-0.439653\pi$
$229$	299103.	0.376905	0.188452	+	0.982082i	$0.439653\pi$
$230$	−465384.	−0.580085
$231$	0	0
$232$	1.06950e6	1.30455
$233$	−564942.	−0.681732	−0.340866	−	0.940112i	$-0.610720\pi$
$233$	−564942.	−0.681732	−0.340866	+	0.940112i	$0.610720\pi$
$234$	0	0
$235$	−578972.	−0.683893
$236$	2.33443e6	2.72836
$237$	0	0
$238$	969366.	1.10929
$239$	1.03635e6	1.17358	0.586789	−	0.809740i	$-0.300392\pi$
$239$	1.03635e6	1.17358	0.586789	+	0.809740i	$0.300392\pi$
$240$	0	0
$241$	821274.	0.910847	0.455423	−	0.890275i	$-0.349488\pi$
$241$	821274.	0.910847	0.455423	+	0.890275i	$0.349488\pi$
$242$	321310.	0.352684
$243$	0	0
$244$	−3.33374e6	−3.58473
$245$	−1.35108e6	−1.43803
$246$	0	0
$247$	172410.	0.179813
$248$	−917868.	−0.947656
$249$	0	0
$250$	3.74941e6	3.79414
$251$	−1.23062e6	−1.23293	−0.616467	−	0.787380i	$-0.711437\pi$
$251$	−1.23062e6	−1.23293	−0.616467	+	0.787380i	$0.711437\pi$
$252$	0	0
$253$	−201716.	−0.198125
$254$	82096.7	0.0798438
$255$	0	0
$256$	−1.96311e6	−1.87217
$257$	1.05206e6	0.993592	0.496796	−	0.867867i	$-0.334510\pi$
$257$	1.05206e6	0.993592	0.496796	+	0.867867i	$0.334510\pi$
$258$	0	0
$259$	−172673.	−0.159946
$260$	−3.38583e6	−3.10622
$261$	0	0
$262$	−1.20517e6	−1.08467
$263$	−1.88157e6	−1.67738	−0.838688	−	0.544612i	$-0.816677\pi$
$263$	−1.88157e6	−1.67738	−0.838688	+	0.544612i	$0.816677\pi$
$264$	0	0
$265$	2.31519e6	2.02522
$266$	209372.	0.181432
$267$	0	0
$268$	−253353.	−0.215471
$269$	1.05389e6	0.888002	0.444001	−	0.896026i	$-0.353559\pi$
$269$	1.05389e6	0.888002	0.444001	+	0.896026i	$0.353559\pi$
$270$	0	0
$271$	688686.	0.569637	0.284818	−	0.958581i	$-0.408067\pi$
$271$	688686.	0.569637	0.284818	+	0.958581i	$0.408067\pi$
$272$	−2.92843e6	−2.40001
$273$	0	0
$274$	−663425.	−0.533845
$275$	2.99695e6	2.38972
$276$	0	0
$277$	−522720.	−0.409326	−0.204663	−	0.978832i	$-0.565610\pi$
$277$	−522720.	−0.409326	−0.204663	+	0.978832i	$0.565610\pi$
$278$	−288701.	−0.224045
$279$	0	0
$280$	−2.26022e6	−1.72288
$281$	2.00251e6	1.51290	0.756448	−	0.654053i	$-0.226933\pi$
$281$	2.00251e6	1.51290	0.756448	+	0.654053i	$0.226933\pi$
$282$	0	0
$283$	−2.25219e6	−1.67163	−0.835813	−	0.549014i	$-0.815003\pi$
$283$	−2.25219e6	−1.67163	−0.835813	+	0.549014i	$0.815003\pi$
$284$	−166749.	−0.122678
$285$	0	0
$286$	−2.12839e6	−1.53864
$287$	313241.	0.224478
$288$	0	0
$289$	1.37368e6	0.967476
$290$	2.73123e6	1.90705
$291$	0	0
$292$	4.07697e6	2.79821
$293$	−1.76627e6	−1.20195	−0.600976	−	0.799267i	$-0.705221\pi$
$293$	−1.76627e6	−1.20195	−0.600976	+	0.799267i	$0.705221\pi$
$294$	0	0
$295$	3.27708e6	2.19246
$296$	1.19868e6	0.795197
$297$	0	0
$298$	−1.17614e6	−0.767216
$299$	219459.	0.141963
$300$	0	0
$301$	−291460.	−0.185423
$302$	365281.	0.230467
$303$	0	0
$304$	−632507.	−0.392537
$305$	−4.67990e6	−2.88063
$306$	0	0
$307$	−1.71867e6	−1.04075	−0.520376	−	0.853937i	$-0.674208\pi$
$307$	−1.71867e6	−1.04075	−0.520376	+	0.853937i	$0.674208\pi$
$308$	−1.78218e6	−1.07047
$309$	0	0
$310$	−2.34399e6	−1.38533
$311$	−1.58149e6	−0.927182	−0.463591	−	0.886049i	$-0.653439\pi$
$311$	−1.58149e6	−0.927182	−0.463591	+	0.886049i	$0.653439\pi$
$312$	0	0
$313$	54617.0	0.0315114	0.0157557	−	0.999876i	$-0.494985\pi$
$313$	54617.0	0.0315114	0.0157557	+	0.999876i	$0.494985\pi$
$314$	5.74844e6	3.29023
$315$	0	0
$316$	−2.11174e6	−1.18966
$317$	1.03467e6	0.578303	0.289151	−	0.957283i	$-0.406627\pi$
$317$	1.03467e6	0.578303	0.289151	+	0.957283i	$0.406627\pi$
$318$	0	0
$319$	1.18382e6	0.651344
$320$	−431495.	−0.235559
$321$	0	0
$322$	266508.	0.143242
$323$	603371.	0.321794
$324$	0	0
$325$	−3.26057e6	−1.71232
$326$	−3.12522e6	−1.62868
$327$	0	0
$328$	−2.17450e6	−1.11603
$329$	331555.	0.168875
$330$	0	0
$331$	2.03987e6	1.02337	0.511685	−	0.859173i	$-0.329021\pi$
$331$	2.03987e6	1.02337	0.511685	+	0.859173i	$0.329021\pi$
$332$	−4.30125e6	−2.14165
$333$	0	0
$334$	−1.28923e6	−0.632358
$335$	−355657.	−0.173149
$336$	0	0
$337$	1.22774e6	0.588887	0.294444	−	0.955669i	$-0.404866\pi$
$337$	1.22774e6	0.588887	0.294444	+	0.955669i	$0.404866\pi$
$338$	−1.45378e6	−0.692162
$339$	0	0
$340$	−1.18492e7	−5.55892
$341$	−1.01598e6	−0.473150
$342$	0	0
$343$	1.73388e6	0.795763
$344$	2.02330e6	0.921857
$345$	0	0
$346$	5.91373e6	2.65565
$347$	−3.93267e6	−1.75333	−0.876665	−	0.481101i	$-0.840237\pi$
$347$	−3.93267e6	−1.75333	−0.876665	+	0.481101i	$0.840237\pi$
$348$	0	0
$349$	−3.46176e6	−1.52136	−0.760682	−	0.649125i	$-0.775135\pi$
$349$	−3.46176e6	−1.52136	−0.760682	+	0.649125i	$0.775135\pi$
$350$	−3.95957e6	−1.72774
$351$	0	0
$352$	2.23732e6	0.962436
$353$	3.91227e6	1.67106	0.835531	−	0.549443i	$-0.185160\pi$
$353$	3.91227e6	1.67106	0.835531	+	0.549443i	$0.185160\pi$
$354$	0	0
$355$	−234082.	−0.0985819
$356$	−1.35981e6	−0.568662
$357$	0	0
$358$	2.37375e6	0.978874
$359$	949856.	0.388975	0.194487	−	0.980905i	$-0.437696\pi$
$359$	949856.	0.388975	0.194487	+	0.980905i	$0.437696\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	3.23018e6	1.29556
$363$	0	0
$364$	1.93894e6	0.767027
$365$	5.72325e6	2.24859
$366$	0	0
$367$	3.72584e6	1.44397	0.721986	−	0.691908i	$-0.243230\pi$
$367$	3.72584e6	1.44397	0.721986	+	0.691908i	$0.243230\pi$
$368$	−805113.	−0.309911
$369$	0	0
$370$	3.06112e6	1.16245
$371$	−1.32582e6	−0.500093
$372$	0	0
$373$	−4.15376e6	−1.54586	−0.772929	−	0.634493i	$-0.781209\pi$
$373$	−4.15376e6	−1.54586	−0.772929	+	0.634493i	$0.781209\pi$
$374$	−7.44857e6	−2.75356
$375$	0	0
$376$	−2.30163e6	−0.839589
$377$	−1.28796e6	−0.466710
$378$	0	0
$379$	2.29264e6	0.819857	0.409928	−	0.912118i	$-0.365554\pi$
$379$	2.29264e6	0.819857	0.409928	+	0.912118i	$0.365554\pi$
$380$	−2.55928e6	−0.909198
$381$	0	0
$382$	1.84800e6	0.647952
$383$	−3.09365e6	−1.07764	−0.538821	−	0.842420i	$-0.681130\pi$
$383$	−3.09365e6	−1.07764	−0.538821	+	0.842420i	$0.681130\pi$
$384$	0	0
$385$	−2.50182e6	−0.860209
$386$	2.90376e6	0.991955
$387$	0	0
$388$	−3.06923e6	−1.03502
$389$	4.53732e6	1.52029	0.760144	−	0.649755i	$-0.225129\pi$
$389$	4.53732e6	1.52029	0.760144	+	0.649755i	$0.225129\pi$
$390$	0	0
$391$	768026.	0.254059
$392$	−5.37107e6	−1.76541
$393$	0	0
$394$	4.72965e6	1.53493
$395$	−2.96446e6	−0.955989
$396$	0	0
$397$	−404604.	−0.128841	−0.0644205	−	0.997923i	$-0.520520\pi$
$397$	−404604.	−0.128841	−0.0644205	+	0.997923i	$0.520520\pi$
$398$	1.08066e7	3.41965
$399$	0	0
$400$	1.19618e7	3.73806
$401$	5.08752e6	1.57996	0.789978	−	0.613135i	$-0.210092\pi$
$401$	5.08752e6	1.57996	0.789978	+	0.613135i	$0.210092\pi$
$402$	0	0
$403$	1.10535e6	0.339029
$404$	3.93203e6	1.19857
$405$	0	0
$406$	−1.56407e6	−0.470913
$407$	1.32681e6	0.397030
$408$	0	0
$409$	367772.	0.108710	0.0543551	−	0.998522i	$-0.482690\pi$
$409$	367772.	0.108710	0.0543551	+	0.998522i	$0.482690\pi$
$410$	−5.55309e6	−1.63146
$411$	0	0
$412$	−1.46138e7	−4.24150
$413$	−1.87666e6	−0.541390
$414$	0	0
$415$	−6.03809e6	−1.72099
$416$	−2.43412e6	−0.689618
$417$	0	0
$418$	−1.60880e6	−0.450363
$419$	4.12612e6	1.14817	0.574086	−	0.818795i	$-0.305357\pi$
$419$	4.12612e6	1.14817	0.574086	+	0.818795i	$0.305357\pi$
$420$	0	0
$421$	−2.05918e6	−0.566224	−0.283112	−	0.959087i	$-0.591367\pi$
$421$	−2.05918e6	−0.566224	−0.283112	+	0.959087i	$0.591367\pi$
$422$	1.02115e7	2.79132
$423$	0	0
$424$	9.20376e6	2.48628
$425$	−1.14108e7	−3.06438
$426$	0	0
$427$	2.68000e6	0.711321
$428$	−1.29259e7	−3.41075
$429$	0	0
$430$	5.16696e6	1.34761
$431$	3.23089e6	0.837777	0.418889	−	0.908038i	$-0.362420\pi$
$431$	3.23089e6	0.837777	0.418889	+	0.908038i	$0.362420\pi$
$432$	0	0
$433$	−4.46477e6	−1.14440	−0.572202	−	0.820113i	$-0.693911\pi$
$433$	−4.46477e6	−1.14440	−0.572202	+	0.820113i	$0.693911\pi$
$434$	1.34231e6	0.342082
$435$	0	0
$436$	1.72607e6	0.434854
$437$	165885.	0.0415530
$438$	0	0
$439$	3.96204e6	0.981201	0.490601	−	0.871385i	$-0.336777\pi$
$439$	3.96204e6	0.981201	0.490601	+	0.871385i	$0.336777\pi$
$440$	1.73674e7	4.27666
$441$	0	0
$442$	8.10376e6	1.97302
$443$	−5.87689e6	−1.42278	−0.711391	−	0.702797i	$-0.751934\pi$
$443$	−5.87689e6	−1.42278	−0.711391	+	0.702797i	$0.751934\pi$
$444$	0	0
$445$	−1.90890e6	−0.456966
$446$	−3.83142e6	−0.912059
$447$	0	0
$448$	247101.	0.0581673
$449$	2.62215e6	0.613820	0.306910	−	0.951739i	$-0.400705\pi$
$449$	2.62215e6	0.613820	0.306910	+	0.951739i	$0.400705\pi$
$450$	0	0
$451$	−2.40693e6	−0.557215
$452$	1.63212e7	3.75756
$453$	0	0
$454$	1.30386e7	2.96887
$455$	2.72188e6	0.616369
$456$	0	0
$457$	5.12578e6	1.14807	0.574037	−	0.818830i	$-0.305377\pi$
$457$	5.12578e6	1.14807	0.574037	+	0.818830i	$0.305377\pi$
$458$	3.03651e6	0.676411
$459$	0	0
$460$	−3.25769e6	−0.717819
$461$	−2.65571e6	−0.582008	−0.291004	−	0.956722i	$-0.593989\pi$
$461$	−2.65571e6	−0.582008	−0.291004	+	0.956722i	$0.593989\pi$
$462$	0	0
$463$	−2.94632e6	−0.638744	−0.319372	−	0.947629i	$-0.603472\pi$
$463$	−2.94632e6	−0.638744	−0.319372	+	0.947629i	$0.603472\pi$
$464$	4.72502e6	1.01885
$465$	0	0
$466$	−5.73533e6	−1.22347
$467$	−1.82563e6	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$467$	−1.82563e6	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$468$	0	0
$469$	203671.	0.0427561
$470$	−5.87776e6	−1.22735
$471$	0	0
$472$	1.30276e7	2.69160
$473$	2.23957e6	0.460269
$474$	0	0
$475$	−2.46459e6	−0.501200
$476$	6.78556e6	1.37268
$477$	0	0
$478$	1.05211e7	2.10616
$479$	−5.92850e6	−1.18061	−0.590304	−	0.807181i	$-0.700992\pi$
$479$	−5.92850e6	−1.18061	−0.590304	+	0.807181i	$0.700992\pi$
$480$	0	0
$481$	−1.44352e6	−0.284486
$482$	8.33763e6	1.63465
$483$	0	0
$484$	2.24917e6	0.436424
$485$	−4.30859e6	−0.831727
$486$	0	0
$487$	−3.61261e6	−0.690237	−0.345118	−	0.938559i	$-0.612161\pi$
$487$	−3.61261e6	−0.690237	−0.345118	+	0.938559i	$0.612161\pi$
$488$	−1.86044e7	−3.53644
$489$	0	0
$490$	−1.37163e7	−2.58075
$491$	4.32852e6	0.810281	0.405140	−	0.914255i	$-0.367223\pi$
$491$	4.32852e6	0.810281	0.405140	+	0.914255i	$0.367223\pi$
$492$	0	0
$493$	−4.50736e6	−0.835228
$494$	1.75032e6	0.322700
$495$	0	0
$496$	−4.05510e6	−0.740112
$497$	134050.	0.0243431
$498$	0	0
$499$	−4.93615e6	−0.887435	−0.443718	−	0.896167i	$-0.646341\pi$
$499$	−4.93615e6	−0.887435	−0.443718	+	0.896167i	$0.646341\pi$
$500$	2.62459e7	4.69501
$501$	0	0
$502$	−1.24933e7	−2.21268
$503$	−1.03023e6	−0.181557	−0.0907786	−	0.995871i	$-0.528936\pi$
$503$	−1.03023e6	−0.181557	−0.0907786	+	0.995871i	$0.528936\pi$
$504$	0	0
$505$	5.51979e6	0.963150
$506$	−2.04784e6	−0.355565
$507$	0	0
$508$	574677.	0.0988017
$509$	−4.99120e6	−0.853907	−0.426954	−	0.904274i	$-0.640413\pi$
$509$	−4.99120e6	−0.853907	−0.426954	+	0.904274i	$0.640413\pi$
$510$	0	0
$511$	−3.27749e6	−0.555250
$512$	−1.33055e7	−2.24314
$513$	0	0
$514$	1.06806e7	1.78315
$515$	−2.05148e7	−3.40839
$516$	0	0
$517$	−2.54766e6	−0.419194
$518$	−1.75299e6	−0.287048
$519$	0	0
$520$	−1.88951e7	−3.06437
$521$	−9.78050e6	−1.57858	−0.789291	−	0.614020i	$-0.789551\pi$
$521$	−9.78050e6	−1.57858	−0.789291	+	0.614020i	$0.789551\pi$
$522$	0	0
$523$	−6.73157e6	−1.07612	−0.538062	−	0.842905i	$-0.680843\pi$
$523$	−6.73157e6	−1.07612	−0.538062	+	0.842905i	$0.680843\pi$
$524$	−8.43622e6	−1.34221
$525$	0	0
$526$	−1.91018e7	−3.01030
$527$	3.86830e6	0.606728
$528$	0	0
$529$	−6.22519e6	−0.967194
$530$	2.35040e7	3.63456
$531$	0	0
$532$	1.46560e6	0.224511
$533$	2.61865e6	0.399264
$534$	0	0
$535$	−1.81453e7	−2.74082
$536$	−1.41387e6	−0.212568
$537$	0	0
$538$	1.06991e7	1.59365
$539$	−5.94519e6	−0.881442
$540$	0	0
$541$	−6.68215e6	−0.981574	−0.490787	−	0.871280i	$-0.663291\pi$
$541$	−6.68215e6	−0.981574	−0.490787	+	0.871280i	$0.663291\pi$
$542$	6.99159e6	1.02230
$543$	0	0
$544$	−8.51851e6	−1.23415
$545$	2.42306e6	0.349441
$546$	0	0
$547$	−235187.	−0.0336081	−0.0168041	−	0.999859i	$-0.505349\pi$
$547$	−235187.	−0.0336081	−0.0168041	+	0.999859i	$0.505349\pi$
$548$	−4.64397e6	−0.660600
$549$	0	0
$550$	3.04252e7	4.28872
$551$	−973538.	−0.136607
$552$	0	0
$553$	1.69763e6	0.236065
$554$	−5.30668e6	−0.734597
$555$	0	0
$556$	−2.02090e6	−0.277242
$557$	4.80203e6	0.655824	0.327912	−	0.944708i	$-0.393655\pi$
$557$	4.80203e6	0.655824	0.327912	+	0.944708i	$0.393655\pi$
$558$	0	0
$559$	−2.43657e6	−0.329799
$560$	−9.98555e6	−1.34556
$561$	0	0
$562$	2.03296e7	2.71512
$563$	8.37831e6	1.11400	0.557000	−	0.830512i	$-0.311952\pi$
$563$	8.37831e6	1.11400	0.557000	+	0.830512i	$0.311952\pi$
$564$	0	0
$565$	2.29117e7	3.01951
$566$	−2.28644e7	−2.99998
$567$	0	0
$568$	−930565.	−0.121025
$569$	8.38130e6	1.08525	0.542626	−	0.839974i	$-0.317430\pi$
$569$	8.38130e6	1.08525	0.542626	+	0.839974i	$0.317430\pi$
$570$	0	0
$571$	3.97400e6	0.510079	0.255040	−	0.966931i	$-0.417912\pi$
$571$	3.97400e6	0.510079	0.255040	+	0.966931i	$0.417912\pi$
$572$	−1.48987e7	−1.90397
$573$	0	0
$574$	3.18004e6	0.402859
$575$	−3.13716e6	−0.395701
$576$	0	0
$577$	1.40884e7	1.76166	0.880830	−	0.473432i	$-0.156985\pi$
$577$	1.40884e7	1.76166	0.880830	+	0.473432i	$0.156985\pi$
$578$	1.39457e7	1.73628
$579$	0	0
$580$	1.91186e7	2.35986
$581$	3.45779e6	0.424970
$582$	0	0
$583$	1.01876e7	1.24136
$584$	2.27521e7	2.76051
$585$	0	0
$586$	−1.79313e7	−2.15708
$587$	−3.14647e6	−0.376901	−0.188451	−	0.982083i	$-0.560347\pi$
$587$	−3.14647e6	−0.376901	−0.188451	+	0.982083i	$0.560347\pi$
$588$	0	0
$589$	835508.	0.0992345
$590$	3.32691e7	3.93470
$591$	0	0
$592$	5.29573e6	0.621043
$593$	4.29402e6	0.501450	0.250725	−	0.968058i	$-0.419331\pi$
$593$	4.29402e6	0.501450	0.250725	+	0.968058i	$0.419331\pi$
$594$	0	0
$595$	9.52557e6	1.10306
$596$	−8.23297e6	−0.949382
$597$	0	0
$598$	2.22797e6	0.254774
$599$	6.82650e6	0.777376	0.388688	−	0.921370i	$-0.372928\pi$
$599$	6.82650e6	0.777376	0.388688	+	0.921370i	$0.372928\pi$
$600$	0	0
$601$	1.82575e6	0.206184	0.103092	−	0.994672i	$-0.467126\pi$
$601$	1.82575e6	0.206184	0.103092	+	0.994672i	$0.467126\pi$
$602$	−2.95892e6	−0.332769
$603$	0	0
$604$	2.55697e6	0.285189
$605$	3.15738e6	0.350703
$606$	0	0
$607$	−2.44884e6	−0.269767	−0.134883	−	0.990861i	$-0.543066\pi$
$607$	−2.44884e6	−0.269767	−0.134883	+	0.990861i	$0.543066\pi$
$608$	−1.83990e6	−0.201853
$609$	0	0
$610$	−4.75107e7	−5.16972
$611$	2.77176e6	0.300367
$612$	0	0
$613$	−6.05994e6	−0.651354	−0.325677	−	0.945481i	$-0.605592\pi$
$613$	−6.05994e6	−0.651354	−0.325677	+	0.945481i	$0.605592\pi$
$614$	−1.74481e7	−1.86778
$615$	0	0
$616$	−9.94568e6	−1.05605
$617$	−9.33155e6	−0.986827	−0.493414	−	0.869795i	$-0.664251\pi$
$617$	−9.33155e6	−0.986827	−0.493414	+	0.869795i	$0.664251\pi$
$618$	0	0
$619$	−9.73252e6	−1.02094	−0.510468	−	0.859897i	$-0.670528\pi$
$619$	−9.73252e6	−1.02094	−0.510468	+	0.859897i	$0.670528\pi$
$620$	−1.64079e7	−1.71425
$621$	0	0
$622$	−1.60554e7	−1.66397
$623$	1.09316e6	0.112840
$624$	0	0
$625$	1.55092e7	1.58815
$626$	554475.	0.0565518
$627$	0	0
$628$	4.02391e7	4.07145
$629$	−5.05178e6	−0.509118
$630$	0	0
$631$	8.09784e6	0.809647	0.404824	−	0.914395i	$-0.367333\pi$
$631$	8.09784e6	0.809647	0.404824	+	0.914395i	$0.367333\pi$
$632$	−1.17849e7	−1.17363
$633$	0	0
$634$	1.05041e7	1.03785
$635$	806731.	0.0793953
$636$	0	0
$637$	6.46814e6	0.631583
$638$	1.20183e7	1.16893
$639$	0	0
$640$	−2.06508e7	−1.99291
$641$	1.54883e7	1.48887	0.744436	−	0.667694i	$-0.232718\pi$
$641$	1.54883e7	1.48887	0.744436	+	0.667694i	$0.232718\pi$
$642$	0	0
$643$	1.43966e7	1.37320	0.686600	−	0.727036i	$-0.259103\pi$
$643$	1.43966e7	1.37320	0.686600	+	0.727036i	$0.259103\pi$
$644$	1.86555e6	0.177253
$645$	0	0
$646$	6.12546e6	0.577507
$647$	6.47992e6	0.608568	0.304284	−	0.952581i	$-0.401583\pi$
$647$	6.47992e6	0.608568	0.304284	+	0.952581i	$0.401583\pi$
$648$	0	0
$649$	1.44202e7	1.34388
$650$	−3.31015e7	−3.07301
$651$	0	0
$652$	−2.18766e7	−2.01539
$653$	7.89619e6	0.724660	0.362330	−	0.932050i	$-0.381981\pi$
$653$	7.89619e6	0.724660	0.362330	+	0.932050i	$0.381981\pi$
$654$	0	0
$655$	−1.18428e7	−1.07857
$656$	−9.60683e6	−0.871607
$657$	0	0
$658$	3.36597e6	0.303072
$659$	1.08756e7	0.975530	0.487765	−	0.872975i	$-0.337812\pi$
$659$	1.08756e7	0.975530	0.487765	+	0.872975i	$0.337812\pi$
$660$	0	0
$661$	−8.09950e6	−0.721032	−0.360516	−	0.932753i	$-0.617399\pi$
$661$	−8.09950e6	−0.721032	−0.360516	+	0.932753i	$0.617399\pi$
$662$	2.07089e7	1.83659
$663$	0	0
$664$	−2.40037e7	−2.11280
$665$	2.05741e6	0.180413
$666$	0	0
$667$	−1.23921e6	−0.107852
$668$	−9.02458e6	−0.782503
$669$	0	0
$670$	−3.61066e6	−0.310741
$671$	−2.05930e7	−1.76569
$672$	0	0
$673$	6.23497e6	0.530636	0.265318	−	0.964161i	$-0.414523\pi$
$673$	6.23497e6	0.530636	0.265318	+	0.964161i	$0.414523\pi$
$674$	1.24641e7	1.05685
$675$	0	0
$676$	−1.01765e7	−0.856507
$677$	−1.16588e7	−0.977649	−0.488824	−	0.872382i	$-0.662574\pi$
$677$	−1.16588e7	−0.977649	−0.488824	+	0.872382i	$0.662574\pi$
$678$	0	0
$679$	2.46737e6	0.205380
$680$	−6.61259e7	−5.48402
$681$	0	0
$682$	−1.03143e7	−0.849139
$683$	1.22340e6	0.100350	0.0501748	−	0.998740i	$-0.484022\pi$
$683$	1.22340e6	0.100350	0.0501748	+	0.998740i	$0.484022\pi$
$684$	0	0
$685$	−6.51921e6	−0.530846
$686$	1.76025e7	1.42811
$687$	0	0
$688$	8.93884e6	0.719962
$689$	−1.10837e7	−0.889480
$690$	0	0
$691$	−5.82015e6	−0.463702	−0.231851	−	0.972751i	$-0.574478\pi$
$691$	−5.82015e6	−0.463702	−0.231851	+	0.972751i	$0.574478\pi$
$692$	4.13961e7	3.28620
$693$	0	0
$694$	−3.99247e7	−3.14661
$695$	−2.83695e6	−0.222787
$696$	0	0
$697$	9.16430e6	0.714525
$698$	−3.51440e7	−2.73031
$699$	0	0
$700$	−2.77170e7	−2.13797
$701$	−9.44191e6	−0.725713	−0.362856	−	0.931845i	$-0.618198\pi$
$701$	−9.44191e6	−0.725713	−0.362856	+	0.931845i	$0.618198\pi$
$702$	0	0
$703$	−1.09113e6	−0.0832696
$704$	−1.89871e6	−0.144387
$705$	0	0
$706$	3.97177e7	2.99897
$707$	−3.16097e6	−0.237833
$708$	0	0
$709$	7.88964e6	0.589443	0.294721	−	0.955583i	$-0.404773\pi$
$709$	7.88964e6	0.589443	0.294721	+	0.955583i	$0.404773\pi$
$710$	−2.37642e6	−0.176920
$711$	0	0
$712$	−7.58862e6	−0.561000
$713$	1.06351e6	0.0783464
$714$	0	0
$715$	−2.09149e7	−1.52999
$716$	1.66162e7	1.21129
$717$	0	0
$718$	9.64300e6	0.698073
$719$	−1.89500e7	−1.36706	−0.683530	−	0.729922i	$-0.739556\pi$
$719$	−1.89500e7	−1.36706	−0.683530	+	0.729922i	$0.739556\pi$
$720$	0	0
$721$	1.17481e7	0.841643
$722$	1.32303e6	0.0944552
$723$	0	0
$724$	2.26113e7	1.60317
$725$	1.84113e7	1.30088
$726$	0	0
$727$	6.46947e6	0.453976	0.226988	−	0.973898i	$-0.427112\pi$
$727$	6.46947e6	0.453976	0.226988	+	0.973898i	$0.427112\pi$
$728$	1.08205e7	0.756693
$729$	0	0
$730$	5.81028e7	4.03543
$731$	−8.52707e6	−0.590210
$732$	0	0
$733$	6.39936e6	0.439923	0.219962	−	0.975509i	$-0.429407\pi$
$733$	6.39936e6	0.439923	0.219962	+	0.975509i	$0.429407\pi$
$734$	3.78249e7	2.59142
$735$	0	0
$736$	−2.34199e6	−0.159364
$737$	−1.56500e6	−0.106132
$738$	0	0
$739$	−1.28565e7	−0.865987	−0.432993	−	0.901397i	$-0.642543\pi$
$739$	−1.28565e7	−0.865987	−0.432993	+	0.901397i	$0.642543\pi$
$740$	2.14278e7	1.43846
$741$	0	0
$742$	−1.34598e7	−0.897491
$743$	2.19966e7	1.46179	0.730893	−	0.682492i	$-0.239104\pi$
$743$	2.19966e7	1.46179	0.730893	+	0.682492i	$0.239104\pi$
$744$	0	0
$745$	−1.15574e7	−0.762906
$746$	−4.21693e7	−2.77427
$747$	0	0
$748$	−5.21400e7	−3.40735
$749$	1.03911e7	0.676797
$750$	0	0
$751$	4.73982e6	0.306663	0.153332	−	0.988175i	$-0.451000\pi$
$751$	4.73982e6	0.306663	0.153332	+	0.988175i	$0.451000\pi$
$752$	−1.01685e7	−0.655712
$753$	0	0
$754$	−1.30754e7	−0.837581
$755$	3.58947e6	0.229173
$756$	0	0
$757$	−8.15849e6	−0.517452	−0.258726	−	0.965951i	$-0.583303\pi$
$757$	−8.15849e6	−0.517452	−0.258726	+	0.965951i	$0.583303\pi$
$758$	2.32750e7	1.47135
$759$	0	0
$760$	−1.42824e7	−0.896949
$761$	2.40805e7	1.50731	0.753657	−	0.657268i	$-0.228288\pi$
$761$	2.40805e7	1.50731	0.753657	+	0.657268i	$0.228288\pi$
$762$	0	0
$763$	−1.38760e6	−0.0862883
$764$	1.29360e7	0.801800
$765$	0	0
$766$	−3.14069e7	−1.93399
$767$	−1.56886e7	−0.962933
$768$	0	0
$769$	−3.57395e6	−0.217938	−0.108969	−	0.994045i	$-0.534755\pi$
$769$	−3.57395e6	−0.217938	−0.108969	+	0.994045i	$0.534755\pi$
$770$	−2.53986e7	−1.54377
$771$	0	0
$772$	2.03263e7	1.22748
$773$	−1.38262e7	−0.832251	−0.416126	−	0.909307i	$-0.636612\pi$
$773$	−1.38262e7	−0.832251	−0.416126	+	0.909307i	$0.636612\pi$
$774$	0	0
$775$	−1.58009e7	−0.944990
$776$	−1.71283e7	−1.02108
$777$	0	0
$778$	4.60632e7	2.72838
$779$	1.97938e6	0.116865
$780$	0	0
$781$	−1.03003e6	−0.0604260
$782$	7.79705e6	0.455946
$783$	0	0
$784$	−2.37291e7	−1.37877
$785$	5.64877e7	3.27175
$786$	0	0
$787$	2.52043e7	1.45057	0.725283	−	0.688451i	$-0.241709\pi$
$787$	2.52043e7	1.45057	0.725283	+	0.688451i	$0.241709\pi$
$788$	3.31075e7	1.89938
$789$	0	0
$790$	−3.00954e7	−1.71567
$791$	−1.31207e7	−0.745615
$792$	0	0
$793$	2.24045e7	1.26518
$794$	−4.10757e6	−0.231224
$795$	0	0
$796$	7.56464e7	4.23161
$797$	−3.37562e7	−1.88238	−0.941190	−	0.337877i	$-0.890291\pi$
$797$	−3.37562e7	−1.88238	−0.941190	+	0.337877i	$0.890291\pi$
$798$	0	0
$799$	9.70011e6	0.537539
$800$	3.47956e7	1.92220
$801$	0	0
$802$	5.16488e7	2.83547
$803$	2.51841e7	1.37828
$804$	0	0
$805$	2.61887e6	0.142437
$806$	1.12216e7	0.608437
$807$	0	0
$808$	2.19433e7	1.18242
$809$	2.10318e7	1.12981	0.564906	−	0.825155i	$-0.308912\pi$
$809$	2.10318e7	1.12981	0.564906	+	0.825155i	$0.308912\pi$
$810$	0	0
$811$	−2.27692e7	−1.21561	−0.607807	−	0.794085i	$-0.707950\pi$
$811$	−2.27692e7	−1.21561	−0.607807	+	0.794085i	$0.707950\pi$
$812$	−1.09485e7	−0.582726
$813$	0	0
$814$	1.34699e7	0.712530
$815$	−3.07103e7	−1.61954
$816$	0	0
$817$	−1.84175e6	−0.0965328
$818$	3.73365e6	0.195097
$819$	0	0
$820$	−3.88716e7	−2.01882
$821$	2.30273e7	1.19230	0.596149	−	0.802874i	$-0.296697\pi$
$821$	2.30273e7	1.19230	0.596149	+	0.802874i	$0.296697\pi$
$822$	0	0
$823$	−1.70408e7	−0.876979	−0.438490	−	0.898736i	$-0.644486\pi$
$823$	−1.70408e7	−0.876979	−0.438490	+	0.898736i	$0.644486\pi$
$824$	−8.15542e7	−4.18435
$825$	0	0
$826$	−1.90520e7	−0.971605
$827$	−3.62225e6	−0.184168	−0.0920840	−	0.995751i	$-0.529353\pi$
$827$	−3.62225e6	−0.184168	−0.0920840	+	0.995751i	$0.529353\pi$
$828$	0	0
$829$	1.36980e7	0.692261	0.346131	−	0.938186i	$-0.387495\pi$
$829$	1.36980e7	0.692261	0.346131	+	0.938186i	$0.387495\pi$
$830$	−6.12991e7	−3.08858
$831$	0	0
$832$	2.06573e6	0.103458
$833$	2.26361e7	1.13029
$834$	0	0
$835$	−1.26687e7	−0.628805
$836$	−1.12616e7	−0.557296
$837$	0	0
$838$	4.18887e7	2.06057
$839$	8.15272e6	0.399850	0.199925	−	0.979811i	$-0.435930\pi$
$839$	8.15272e6	0.399850	0.199925	+	0.979811i	$0.435930\pi$
$840$	0	0
$841$	−1.32385e7	−0.645431
$842$	−2.09049e7	−1.01617
$843$	0	0
$844$	7.14807e7	3.45408
$845$	−1.42858e7	−0.688274
$846$	0	0
$847$	−1.80811e6	−0.0865999
$848$	4.06618e7	1.94177
$849$	0	0
$850$	−1.15843e8	−5.49948
$851$	−1.38889e6	−0.0657420
$852$	0	0
$853$	−3.76514e7	−1.77178	−0.885888	−	0.463899i	$-0.846450\pi$
$853$	−3.76514e7	−1.77178	−0.885888	+	0.463899i	$0.846450\pi$
$854$	2.72076e7	1.27657
$855$	0	0
$856$	−7.21345e7	−3.36480
$857$	−1.18679e7	−0.551980	−0.275990	−	0.961160i	$-0.589006\pi$
$857$	−1.18679e7	−0.551980	−0.275990	+	0.961160i	$0.589006\pi$
$858$	0	0
$859$	−8.63985e6	−0.399506	−0.199753	−	0.979846i	$-0.564014\pi$
$859$	−8.63985e6	−0.399506	−0.199753	+	0.979846i	$0.564014\pi$
$860$	3.61687e7	1.66758
$861$	0	0
$862$	3.28002e7	1.50352
$863$	1.39409e7	0.637181	0.318591	−	0.947892i	$-0.396790\pi$
$863$	1.39409e7	0.637181	0.318591	+	0.947892i	$0.396790\pi$
$864$	0	0
$865$	5.81119e7	2.64073
$866$	−4.53267e7	−2.05380
$867$	0	0
$868$	9.39620e6	0.423305
$869$	−1.30446e7	−0.585976
$870$	0	0
$871$	1.70266e6	0.0760473
$872$	9.63260e6	0.428995
$873$	0	0
$874$	1.68407e6	0.0745731
$875$	−2.10992e7	−0.931633
$876$	0	0
$877$	2.92561e7	1.28445	0.642224	−	0.766517i	$-0.278012\pi$
$877$	2.92561e7	1.28445	0.642224	+	0.766517i	$0.278012\pi$
$878$	4.02229e7	1.76091
$879$	0	0
$880$	7.67286e7	3.34003
$881$	1.44235e7	0.626081	0.313041	−	0.949740i	$-0.398652\pi$
$881$	1.44235e7	0.626081	0.313041	+	0.949740i	$0.398652\pi$
$882$	0	0
$883$	−4.94360e6	−0.213374	−0.106687	−	0.994293i	$-0.534024\pi$
$883$	−4.94360e6	−0.213374	−0.106687	+	0.994293i	$0.534024\pi$
$884$	5.67263e7	2.44149
$885$	0	0
$886$	−5.96626e7	−2.55339
$887$	1.32368e7	0.564903	0.282452	−	0.959282i	$-0.408852\pi$
$887$	1.32368e7	0.564903	0.282452	+	0.959282i	$0.408852\pi$
$888$	0	0
$889$	−461984.	−0.0196053
$890$	−1.93793e7	−0.820094
$891$	0	0
$892$	−2.68200e7	−1.12862
$893$	2.09511e6	0.0879181
$894$	0	0
$895$	2.33259e7	0.973375
$896$	1.18259e7	0.492114
$897$	0	0
$898$	2.66202e7	1.10159
$899$	−6.24150e6	−0.257567
$900$	0	0
$901$	−3.87888e7	−1.59182
$902$	−2.44353e7	−1.00001
$903$	0	0
$904$	9.10827e7	3.70693
$905$	3.17417e7	1.28828
$906$	0	0
$907$	7.51989e6	0.303524	0.151762	−	0.988417i	$-0.451505\pi$
$907$	7.51989e6	0.303524	0.151762	+	0.988417i	$0.451505\pi$
$908$	9.12701e7	3.67379
$909$	0	0
$910$	2.76327e7	1.10617
$911$	−4.53248e7	−1.80942	−0.904710	−	0.426027i	$-0.859913\pi$
$911$	−4.53248e7	−1.80942	−0.904710	+	0.426027i	$0.859913\pi$
$912$	0	0
$913$	−2.65695e7	−1.05489
$914$	5.20373e7	2.06039
$915$	0	0
$916$	2.12556e7	0.837016
$917$	6.78190e6	0.266335
$918$	0	0
$919$	1.47088e7	0.574497	0.287249	−	0.957856i	$-0.407259\pi$
$919$	1.47088e7	0.574497	0.287249	+	0.957856i	$0.407259\pi$
$920$	−1.81800e7	−0.708148
$921$	0	0
$922$	−2.69610e7	−1.04450
$923$	1.12064e6	0.0432973
$924$	0	0
$925$	2.06351e7	0.792960
$926$	−2.99112e7	−1.14632
$927$	0	0
$928$	1.37446e7	0.523917
$929$	1.35395e7	0.514711	0.257356	−	0.966317i	$-0.417149\pi$
$929$	1.35395e7	0.514711	0.257356	+	0.966317i	$0.417149\pi$
$930$	0	0
$931$	4.88913e6	0.184866
$932$	−4.01473e7	−1.51397
$933$	0	0
$934$	−1.85339e7	−0.695183
$935$	−7.31942e7	−2.73809
$936$	0	0
$937$	−1.21203e7	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$937$	−1.21203e7	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$938$	2.06769e6	0.0767321
$939$	0	0
$940$	−4.11444e7	−1.51877
$941$	−2.61927e7	−0.964288	−0.482144	−	0.876092i	$-0.660142\pi$
$941$	−2.61927e7	−0.964288	−0.482144	+	0.876092i	$0.660142\pi$
$942$	0	0
$943$	2.51954e6	0.0922661
$944$	5.75555e7	2.10212
$945$	0	0
$946$	2.27363e7	0.826021
$947$	3.47875e7	1.26051	0.630257	−	0.776386i	$-0.282949\pi$
$947$	3.47875e7	1.26051	0.630257	+	0.776386i	$0.282949\pi$
$948$	0	0
$949$	−2.73993e7	−0.987585
$950$	−2.50207e7	−0.899478
$951$	0	0
$952$	3.78678e7	1.35418
$953$	−1.16268e7	−0.414695	−0.207347	−	0.978267i	$-0.566483\pi$
$953$	−1.16268e7	−0.414695	−0.207347	+	0.978267i	$0.566483\pi$
$954$	0	0
$955$	1.81595e7	0.644312
$956$	7.36477e7	2.60624
$957$	0	0
$958$	−6.01865e7	−2.11878
$959$	3.73330e6	0.131083
$960$	0	0
$961$	−2.32726e7	−0.812898
$962$	−1.46547e7	−0.510552
$963$	0	0
$964$	5.83634e7	2.02278
$965$	2.85341e7	0.986383
$966$	0	0
$967$	−2.70316e7	−0.929622	−0.464811	−	0.885410i	$-0.653878\pi$
$967$	−2.70316e7	−0.929622	−0.464811	+	0.885410i	$0.653878\pi$
$968$	1.25518e7	0.430544
$969$	0	0
$970$	−4.37411e7	−1.49266
$971$	5.13472e6	0.174771	0.0873854	−	0.996175i	$-0.472149\pi$
$971$	5.13472e6	0.174771	0.0873854	+	0.996175i	$0.472149\pi$
$972$	0	0
$973$	1.62461e6	0.0550132
$974$	−3.66754e7	−1.23873
$975$	0	0
$976$	−8.21934e7	−2.76193
$977$	5.11922e6	0.171580	0.0857902	−	0.996313i	$-0.472659\pi$
$977$	5.11922e6	0.171580	0.0857902	+	0.996313i	$0.472659\pi$
$978$	0	0
$979$	−8.39978e6	−0.280099
$980$	−9.60140e7	−3.19352
$981$	0	0
$982$	4.39434e7	1.45417
$983$	−3.30087e7	−1.08954	−0.544771	−	0.838585i	$-0.683384\pi$
$983$	−3.30087e7	−1.08954	−0.544771	+	0.838585i	$0.683384\pi$
$984$	0	0
$985$	4.64764e7	1.52631
$986$	−4.57590e7	−1.49894
$987$	0	0
$988$	1.22522e7	0.399321
$989$	−2.34435e6	−0.0762134
$990$	0	0
$991$	−3.54462e7	−1.14653	−0.573265	−	0.819370i	$-0.694324\pi$
$991$	−3.54462e7	−1.14653	−0.573265	+	0.819370i	$0.694324\pi$
$992$	−1.17959e7	−0.380585
$993$	0	0
$994$	1.36088e6	0.0436873
$995$	1.06192e8	3.40044
$996$	0	0
$997$	4.51197e7	1.43757	0.718784	−	0.695233i	$-0.244699\pi$
$997$	4.51197e7	1.43757	0.718784	+	0.695233i	$0.244699\pi$
$998$	−5.01121e7	−1.59264
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.6.a.f.1.2		2
3.2	odd	2		19.6.a.c.1.1	✓	2
12.11	even	2		304.6.a.g.1.1		2
15.14	odd	2		475.6.a.d.1.2		2
21.20	even	2		931.6.a.c.1.1		2
57.56	even	2		361.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.6.a.c.1.1	✓	2	3.2	odd	2
171.6.a.f.1.2		2	1.1	even	1	trivial
304.6.a.g.1.1		2	12.11	even	2
361.6.a.d.1.2		2	57.56	even	2
475.6.a.d.1.2		2	15.14	odd	2
931.6.a.c.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+10.1521 q^{2} +71.0645 q^{4} +99.7603 q^{5} -57.1289 q^{7} +396.585 q^{8} +O(q^{10})\)	\(q+10.1521 q^{2} +71.0645 q^{4} +99.7603 q^{5} -57.1289 q^{7} +396.585 q^{8} +1012.77 q^{10} +438.977 q^{11} -477.590 q^{13} -579.977 q^{14} +1752.10 q^{16} -1671.39 q^{17} -361.000 q^{19} +7089.42 q^{20} +4456.52 q^{22} -459.514 q^{23} +6827.12 q^{25} -4848.53 q^{26} -4059.84 q^{28} +2696.78 q^{29} -2314.43 q^{31} +5096.67 q^{32} -16968.0 q^{34} -5699.20 q^{35} +3022.51 q^{37} -3664.90 q^{38} +39563.5 q^{40} -5483.05 q^{41} +5101.80 q^{43} +31195.7 q^{44} -4665.02 q^{46} -5803.63 q^{47} -13543.3 q^{49} +69309.4 q^{50} -33939.7 q^{52} +23207.5 q^{53} +43792.5 q^{55} -22656.5 q^{56} +27377.9 q^{58} +32849.5 q^{59} -46911.5 q^{61} -23496.2 q^{62} -4325.31 q^{64} -47644.5 q^{65} -3565.12 q^{67} -118776. q^{68} -57858.7 q^{70} -2346.44 q^{71} +57370.0 q^{73} +30684.7 q^{74} -25654.3 q^{76} -25078.3 q^{77} -29715.8 q^{79} +174790. q^{80} -55664.3 q^{82} -60526.0 q^{83} -166738. q^{85} +51793.8 q^{86} +174092. q^{88} -19134.9 q^{89} +27284.2 q^{91} -32655.1 q^{92} -58918.9 q^{94} -36013.5 q^{95} -43189.4 q^{97} -137492. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} + 49 q^{4} + 133 q^{5} + 72 q^{7} + 567 q^{8}+O(q^{10}) \) 2 * q + 7 * q^2 + 49 * q^4 + 133 * q^5 + 72 * q^7 + 567 * q^8	\( 2 q + 7 q^{2} + 49 q^{4} + 133 q^{5} + 72 q^{7} + 567 q^{8} + 908 q^{10} + 705 q^{11} - 1341 q^{13} - 987 q^{14} + 1921 q^{16} - 2784 q^{17} - 722 q^{19} + 6356 q^{20} + 3618 q^{22} + 2713 q^{23} + 4807 q^{25} - 2127 q^{26} - 6909 q^{28} + 7775 q^{29} + 7132 q^{31} - 889 q^{32} - 13461 q^{34} - 1407 q^{35} - 6248 q^{37} - 2527 q^{38} + 45228 q^{40} + 4174 q^{41} + 25357 q^{43} + 25326 q^{44} - 14665 q^{46} - 11727 q^{47} - 13676 q^{49} + 75677 q^{50} - 14889 q^{52} + 29133 q^{53} + 52635 q^{55} - 651 q^{56} + 11371 q^{58} + 64515 q^{59} - 40939 q^{61} - 53272 q^{62} + 9137 q^{64} - 76344 q^{65} + 19039 q^{67} - 94227 q^{68} - 71388 q^{70} + 70236 q^{71} + 67058 q^{73} + 59906 q^{74} - 17689 q^{76} + 9273 q^{77} - 32850 q^{79} + 180404 q^{80} - 86104 q^{82} - 71534 q^{83} - 203721 q^{85} - 12052 q^{86} + 219426 q^{88} + 87268 q^{89} - 84207 q^{91} - 102655 q^{92} - 40248 q^{94} - 48013 q^{95} - 62458 q^{97} - 137074 q^{98}+O(q^{100}) \) 2 * q + 7 * q^2 + 49 * q^4 + 133 * q^5 + 72 * q^7 + 567 * q^8 + 908 * q^10 + 705 * q^11 - 1341 * q^13 - 987 * q^14 + 1921 * q^16 - 2784 * q^17 - 722 * q^19 + 6356 * q^20 + 3618 * q^22 + 2713 * q^23 + 4807 * q^25 - 2127 * q^26 - 6909 * q^28 + 7775 * q^29 + 7132 * q^31 - 889 * q^32 - 13461 * q^34 - 1407 * q^35 - 6248 * q^37 - 2527 * q^38 + 45228 * q^40 + 4174 * q^41 + 25357 * q^43 + 25326 * q^44 - 14665 * q^46 - 11727 * q^47 - 13676 * q^49 + 75677 * q^50 - 14889 * q^52 + 29133 * q^53 + 52635 * q^55 - 651 * q^56 + 11371 * q^58 + 64515 * q^59 - 40939 * q^61 - 53272 * q^62 + 9137 * q^64 - 76344 * q^65 + 19039 * q^67 - 94227 * q^68 - 71388 * q^70 + 70236 * q^71 + 67058 * q^73 + 59906 * q^74 - 17689 * q^76 + 9273 * q^77 - 32850 * q^79 + 180404 * q^80 - 86104 * q^82 - 71534 * q^83 - 203721 * q^85 - 12052 * q^86 + 219426 * q^88 + 87268 * q^89 - 84207 * q^91 - 102655 * q^92 - 40248 * q^94 - 48013 * q^95 - 62458 * q^97 - 137074 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more