LMFDB - Embedded newform 171.6.a.k.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,6,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,42,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$27.4256331880$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 117x^{4} + 2916x^{2} - 1216 $ x^6 - 117x^4 + 2916x^2 - 1216
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$9.01831$ 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)

$f(q)$	$=$	\(q+9.01831 q^{2} +49.3298 q^{4} -69.9349 q^{5} -24.9809 q^{7} +156.286 q^{8} -630.694 q^{10} +15.2536 q^{11} -878.013 q^{13} -225.286 q^{14} -169.122 q^{16} +39.2385 q^{17} -361.000 q^{19} -3449.88 q^{20} +137.562 q^{22} -1235.03 q^{23} +1765.89 q^{25} -7918.19 q^{26} -1232.31 q^{28} -3698.42 q^{29} -3303.45 q^{31} -6526.34 q^{32} +353.865 q^{34} +1747.04 q^{35} +845.412 q^{37} -3255.61 q^{38} -10929.8 q^{40} +18819.0 q^{41} +7092.14 q^{43} +752.457 q^{44} -11137.9 q^{46} +8219.10 q^{47} -16183.0 q^{49} +15925.3 q^{50} -43312.3 q^{52} -4978.22 q^{53} -1066.76 q^{55} -3904.16 q^{56} -33353.5 q^{58} +36585.9 q^{59} +18413.6 q^{61} -29791.5 q^{62} -53444.6 q^{64} +61403.7 q^{65} -56841.1 q^{67} +1935.63 q^{68} +15755.3 q^{70} -31417.8 q^{71} +62822.4 q^{73} +7624.19 q^{74} -17808.1 q^{76} -381.049 q^{77} -44851.7 q^{79} +11827.5 q^{80} +169715. q^{82} +41714.2 q^{83} -2744.14 q^{85} +63959.0 q^{86} +2383.92 q^{88} +54448.5 q^{89} +21933.6 q^{91} -60923.9 q^{92} +74122.4 q^{94} +25246.5 q^{95} -43487.7 q^{97} -145943. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{4} - 10 q^{7} - 788 q^{10} - 1256 q^{13} - 606 q^{16} - 2166 q^{19} - 2524 q^{22} - 3944 q^{25} - 9632 q^{28} - 18136 q^{31} - 14072 q^{34} - 23764 q^{37} - 34284 q^{40} - 24606 q^{43} - 45640 q^{46}+ \cdots - 16492 q^{97}+O(q^{100}) $ 6 * q + 42 * q^4 - 10 * q^7 - 788 * q^10 - 1256 * q^13 - 606 * q^16 - 2166 * q^19 - 2524 * q^22 - 3944 * q^25 - 9632 * q^28 - 18136 * q^31 - 14072 * q^34 - 23764 * q^37 - 34284 * q^40 - 24606 * q^43 - 45640 * q^46 - 63256 * q^49 - 89128 * q^52 - 63402 * q^55 - 78572 * q^58 - 42334 * q^61 - 109662 * q^64 - 129696 * q^67 - 15096 * q^70 - 16998 * q^73 - 15162 * q^76 - 129004 * q^79 + 211396 * q^82 - 11754 * q^85 + 55044 * q^88 + 23932 * q^91 + 235368 * q^94 - 16492 * q^97

Coefficient data

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

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

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

$2$	9.01831	1.59423	0.797113	−	0.603830i	$-0.206359\pi$
$2$	9.01831	1.59423	0.797113	+	0.603830i	$0.206359\pi$
$3$	0	0
$3$	0	0
$4$	49.3298	1.54156
$5$	−69.9349	−1.25103	−0.625516	−	0.780211i	$-0.715112\pi$
$5$	−69.9349	−1.25103	−0.625516	+	0.780211i	$0.715112\pi$
$6$	0	0
$7$	−24.9809	−0.192692	−0.0963460	−	0.995348i	$-0.530716\pi$
$7$	−24.9809	−0.192692	−0.0963460	+	0.995348i	$0.530716\pi$
$8$	156.286	0.863365
$9$	0	0
$10$	−630.694	−1.99443
$11$	15.2536	0.0380093	0.0190047	−	0.999819i	$-0.493950\pi$
$11$	15.2536	0.0380093	0.0190047	+	0.999819i	$0.493950\pi$
$12$	0	0
$13$	−878.013	−1.44093	−0.720465	−	0.693492i	$-0.756071\pi$
$13$	−878.013	−1.44093	−0.720465	+	0.693492i	$0.756071\pi$
$14$	−225.286	−0.307195
$15$	0	0
$16$	−169.122	−0.165158
$17$	39.2385	0.0329299	0.0164649	−	0.999864i	$-0.494759\pi$
$17$	39.2385	0.0329299	0.0164649	+	0.999864i	$0.494759\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	−3449.88	−1.92854
$21$	0	0
$22$	137.562	0.0605955
$23$	−1235.03	−0.486809	−0.243404	−	0.969925i	$-0.578264\pi$
$23$	−1235.03	−0.486809	−0.243404	+	0.969925i	$0.578264\pi$
$24$	0	0
$25$	1765.89	0.565083
$26$	−7918.19	−2.29717
$27$	0	0
$28$	−1232.31	−0.297046
$29$	−3698.42	−0.816622	−0.408311	−	0.912843i	$-0.633882\pi$
$29$	−3698.42	−0.816622	−0.408311	+	0.912843i	$0.633882\pi$
$30$	0	0
$31$	−3303.45	−0.617396	−0.308698	−	0.951160i	$-0.599893\pi$
$31$	−3303.45	−0.617396	−0.308698	+	0.951160i	$0.599893\pi$
$32$	−6526.34	−1.12666
$33$	0	0
$34$	353.865	0.0524977
$35$	1747.04	0.241064
$36$	0	0
$37$	845.412	0.101523	0.0507615	−	0.998711i	$-0.483835\pi$
$37$	845.412	0.101523	0.0507615	+	0.998711i	$0.483835\pi$
$38$	−3255.61	−0.365741
$39$	0	0
$40$	−10929.8	−1.08010
$41$	18819.0	1.74838	0.874192	−	0.485581i	$-0.161392\pi$
$41$	18819.0	1.74838	0.874192	+	0.485581i	$0.161392\pi$
$42$	0	0
$43$	7092.14	0.584933	0.292466	−	0.956276i	$-0.405524\pi$
$43$	7092.14	0.584933	0.292466	+	0.956276i	$0.405524\pi$
$44$	752.457	0.0585936
$45$	0	0
$46$	−11137.9	−0.776084
$47$	8219.10	0.542725	0.271362	−	0.962477i	$-0.412526\pi$
$47$	8219.10	0.542725	0.271362	+	0.962477i	$0.412526\pi$
$48$	0	0
$49$	−16183.0	−0.962870
$50$	15925.3	0.900871
$51$	0	0
$52$	−43312.3	−2.22128
$53$	−4978.22	−0.243436	−0.121718	−	0.992565i	$-0.538840\pi$
$53$	−4978.22	−0.243436	−0.121718	+	0.992565i	$0.538840\pi$
$54$	0	0
$55$	−1066.76	−0.0475509
$56$	−3904.16	−0.166363
$57$	0	0
$58$	−33353.5	−1.30188
$59$	36585.9	1.36831	0.684153	−	0.729338i	$-0.260172\pi$
$59$	36585.9	1.36831	0.684153	+	0.729338i	$0.260172\pi$
$60$	0	0
$61$	18413.6	0.633599	0.316799	−	0.948493i	$-0.397392\pi$
$61$	18413.6	0.633599	0.316799	+	0.948493i	$0.397392\pi$
$62$	−29791.5	−0.984269
$63$	0	0
$64$	−53444.6	−1.63100
$65$	61403.7	1.80265
$66$	0	0
$67$	−56841.1	−1.54695	−0.773474	−	0.633828i	$-0.781483\pi$
$67$	−56841.1	−1.54695	−0.773474	+	0.633828i	$0.781483\pi$
$68$	1935.63	0.0507633
$69$	0	0
$70$	15755.3	0.384310
$71$	−31417.8	−0.739657	−0.369829	−	0.929100i	$-0.620584\pi$
$71$	−31417.8	−0.739657	−0.369829	+	0.929100i	$0.620584\pi$
$72$	0	0
$73$	62822.4	1.37977	0.689886	−	0.723918i	$-0.257661\pi$
$73$	62822.4	1.37977	0.689886	+	0.723918i	$0.257661\pi$
$74$	7624.19	0.161851
$75$	0	0
$76$	−17808.1	−0.353658
$77$	−381.049	−0.00732409
$78$	0	0
$79$	−44851.7	−0.808558	−0.404279	−	0.914636i	$-0.632477\pi$
$79$	−44851.7	−0.808558	−0.404279	+	0.914636i	$0.632477\pi$
$80$	11827.5	0.206618
$81$	0	0
$82$	169715.	2.78732
$83$	41714.2	0.664643	0.332322	−	0.943166i	$-0.392168\pi$
$83$	41714.2	0.664643	0.332322	+	0.943166i	$0.392168\pi$
$84$	0	0
$85$	−2744.14	−0.0411964
$86$	63959.0	0.932515
$87$	0	0
$88$	2383.92	0.0328159
$89$	54448.5	0.728636	0.364318	−	0.931275i	$-0.381302\pi$
$89$	54448.5	0.728636	0.364318	+	0.931275i	$0.381302\pi$
$90$	0	0
$91$	21933.6	0.277655
$92$	−60923.9	−0.750444
$93$	0	0
$94$	74122.4	0.865226
$95$	25246.5	0.287007
$96$	0	0
$97$	−43487.7	−0.469285	−0.234642	−	0.972082i	$-0.575392\pi$
$97$	−43487.7	−0.469285	−0.234642	+	0.972082i	$0.575392\pi$
$98$	−145943.	−1.53503
$99$	0	0
$100$	87110.8	0.871108
$101$	−10927.3	−0.106588	−0.0532940	−	0.998579i	$-0.516972\pi$
$101$	−10927.3	−0.106588	−0.0532940	+	0.998579i	$0.516972\pi$
$102$	0	0
$103$	−76020.3	−0.706052	−0.353026	−	0.935614i	$-0.614847\pi$
$103$	−76020.3	−0.706052	−0.353026	+	0.935614i	$0.614847\pi$
$104$	−137221.	−1.24405
$105$	0	0
$106$	−44895.1	−0.388092
$107$	−123618.	−1.04381	−0.521906	−	0.853003i	$-0.674779\pi$
$107$	−123618.	−1.04381	−0.521906	+	0.853003i	$0.674779\pi$
$108$	0	0
$109$	−145642.	−1.17414	−0.587070	−	0.809536i	$-0.699719\pi$
$109$	−145642.	−1.17414	−0.587070	+	0.809536i	$0.699719\pi$
$110$	−9620.35	−0.0758070
$111$	0	0
$112$	4224.82	0.0318246
$113$	−4514.38	−0.0332584	−0.0166292	−	0.999862i	$-0.505293\pi$
$113$	−4514.38	−0.0332584	−0.0166292	+	0.999862i	$0.505293\pi$
$114$	0	0
$115$	86371.8	0.609014
$116$	−182442.	−1.25887
$117$	0	0
$118$	329943.	2.18139
$119$	−980.214	−0.00634532
$120$	0	0
$121$	−160818.	−0.998555
$122$	166060.	1.01010
$123$	0	0
$124$	−162959.	−0.951751
$125$	95049.5	0.544095
$126$	0	0
$127$	71367.3	0.392636	0.196318	−	0.980540i	$-0.437102\pi$
$127$	71367.3	0.392636	0.196318	+	0.980540i	$0.437102\pi$
$128$	−273137.	−1.47352
$129$	0	0
$130$	553758.	2.87383
$131$	−244939.	−1.24704	−0.623519	−	0.781808i	$-0.714298\pi$
$131$	−244939.	−1.24704	−0.623519	+	0.781808i	$0.714298\pi$
$132$	0	0
$133$	9018.12	0.0442066
$134$	−512610.	−2.46618
$135$	0	0
$136$	6132.42	0.0284305
$137$	357466.	1.62717	0.813585	−	0.581446i	$-0.197513\pi$
$137$	357466.	1.62717	0.813585	+	0.581446i	$0.197513\pi$
$138$	0	0
$139$	116186.	0.510056	0.255028	−	0.966934i	$-0.417915\pi$
$139$	116186.	0.510056	0.255028	+	0.966934i	$0.417915\pi$
$140$	86181.1	0.371614
$141$	0	0
$142$	−283336.	−1.17918
$143$	−13392.9	−0.0547688
$144$	0	0
$145$	258648.	1.02162
$146$	566552.	2.19967
$147$	0	0
$148$	41704.1	0.156503
$149$	237480.	0.876318	0.438159	−	0.898897i	$-0.355631\pi$
$149$	237480.	0.876318	0.438159	+	0.898897i	$0.355631\pi$
$150$	0	0
$151$	67124.3	0.239573	0.119786	−	0.992800i	$-0.461779\pi$
$151$	67124.3	0.239573	0.119786	+	0.992800i	$0.461779\pi$
$152$	−56419.2	−0.198070
$153$	0	0
$154$	−3436.42	−0.0116763
$155$	231026.	0.772382
$156$	0	0
$157$	−381859.	−1.23639	−0.618193	−	0.786026i	$-0.712135\pi$
$157$	−381859.	−1.23639	−0.618193	+	0.786026i	$0.712135\pi$
$158$	−404486.	−1.28902
$159$	0	0
$160$	456419.	1.40949
$161$	30852.2	0.0938042
$162$	0	0
$163$	−30881.4	−0.0910391	−0.0455195	−	0.998963i	$-0.514494\pi$
$163$	−30881.4	−0.0910391	−0.0455195	+	0.998963i	$0.514494\pi$
$164$	928338.	2.69523
$165$	0	0
$166$	376191.	1.05959
$167$	−44367.3	−0.123104	−0.0615519	−	0.998104i	$-0.519605\pi$
$167$	−44367.3	−0.123104	−0.0615519	+	0.998104i	$0.519605\pi$
$168$	0	0
$169$	399614.	1.07628
$170$	−24747.5	−0.0656763
$171$	0	0
$172$	349854.	0.901707
$173$	498738.	1.26694	0.633471	−	0.773766i	$-0.281629\pi$
$173$	498738.	1.26694	0.633471	+	0.773766i	$0.281629\pi$
$174$	0	0
$175$	−44113.5	−0.108887
$176$	−2579.72	−0.00627755
$177$	0	0
$178$	491033.	1.16161
$179$	632626.	1.47576	0.737878	−	0.674934i	$-0.235828\pi$
$179$	632626.	1.47576	0.737878	+	0.674934i	$0.235828\pi$
$180$	0	0
$181$	−810854.	−1.83970	−0.919849	−	0.392274i	$-0.871689\pi$
$181$	−810854.	−1.83970	−0.919849	+	0.392274i	$0.871689\pi$
$182$	197804.	0.442646
$183$	0	0
$184$	−193018.	−0.420294
$185$	−59123.8	−0.127009
$186$	0	0
$187$	598.528	0.00125164
$188$	405447.	0.836641
$189$	0	0
$190$	227681.	0.457554
$191$	−299878.	−0.594786	−0.297393	−	0.954755i	$-0.596117\pi$
$191$	−299878.	−0.594786	−0.297393	+	0.954755i	$0.596117\pi$
$192$	0	0
$193$	−616954.	−1.19223	−0.596114	−	0.802900i	$-0.703289\pi$
$193$	−616954.	−1.19223	−0.596114	+	0.802900i	$0.703289\pi$
$194$	−392185.	−0.748146
$195$	0	0
$196$	−798302.	−1.48432
$197$	−687623.	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$197$	−687623.	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$198$	0	0
$199$	−800129.	−1.43228	−0.716139	−	0.697957i	$-0.754092\pi$
$199$	−800129.	−1.43228	−0.716139	+	0.697957i	$0.754092\pi$
$200$	275983.	0.487873
$201$	0	0
$202$	−98545.6	−0.169926
$203$	92389.9	0.157356
$204$	0	0
$205$	−1.31610e6	−2.18729
$206$	−685574.	−1.12561
$207$	0	0
$208$	148491.	0.237981
$209$	−5506.55	−0.00871994
$210$	0	0
$211$	−1.26274e6	−1.95258	−0.976291	−	0.216460i	$-0.930549\pi$
$211$	−1.26274e6	−1.95258	−0.976291	+	0.216460i	$0.930549\pi$
$212$	−245575.	−0.375270
$213$	0	0
$214$	−1.11482e6	−1.66407
$215$	−495988.	−0.731770
$216$	0	0
$217$	82523.3	0.118967
$218$	−1.31344e6	−1.87184
$219$	0	0
$220$	−52623.0	−0.0733025
$221$	−34451.9	−0.0474496
$222$	0	0
$223$	1.38236e6	1.86148	0.930741	−	0.365679i	$-0.119163\pi$
$223$	1.38236e6	1.86148	0.930741	+	0.365679i	$0.119163\pi$
$224$	163034.	0.217099
$225$	0	0
$226$	−40712.0	−0.0530214
$227$	−1.45586e6	−1.87523	−0.937615	−	0.347674i	$-0.886972\pi$
$227$	−1.45586e6	−1.87523	−0.937615	+	0.347674i	$0.886972\pi$
$228$	0	0
$229$	1.05680e6	1.33169	0.665845	−	0.746090i	$-0.268071\pi$
$229$	1.05680e6	1.33169	0.665845	+	0.746090i	$0.268071\pi$
$230$	778927.	0.970906
$231$	0	0
$232$	−578010.	−0.705043
$233$	125063.	0.150917	0.0754586	−	0.997149i	$-0.475958\pi$
$233$	125063.	0.150917	0.0754586	+	0.997149i	$0.475958\pi$
$234$	0	0
$235$	−574802.	−0.678966
$236$	1.80477e6	2.10932
$237$	0	0
$238$	−8839.87	−0.0101159
$239$	−1.01600e6	−1.15054	−0.575268	−	0.817965i	$-0.695102\pi$
$239$	−1.01600e6	−1.15054	−0.575268	+	0.817965i	$0.695102\pi$
$240$	0	0
$241$	735229.	0.815417	0.407709	−	0.913112i	$-0.366328\pi$
$241$	735229.	0.815417	0.407709	+	0.913112i	$0.366328\pi$
$242$	−1.45031e6	−1.59192
$243$	0	0
$244$	908340.	0.976729
$245$	1.13175e6	1.20458
$246$	0	0
$247$	316963.	0.330572
$248$	−516282.	−0.533038
$249$	0	0
$250$	857185.	0.867411
$251$	−1.38044e6	−1.38303	−0.691517	−	0.722360i	$-0.743057\pi$
$251$	−1.38044e6	−1.38303	−0.691517	+	0.722360i	$0.743057\pi$
$252$	0	0
$253$	−18838.7	−0.0185033
$254$	643612.	0.625950
$255$	0	0
$256$	−753005.	−0.718122
$257$	−753987.	−0.712084	−0.356042	−	0.934470i	$-0.615874\pi$
$257$	−753987.	−0.712084	−0.356042	+	0.934470i	$0.615874\pi$
$258$	0	0
$259$	−21119.2	−0.0195627
$260$	3.02904e6	2.77889
$261$	0	0
$262$	−2.20894e6	−1.98806
$263$	−329348.	−0.293607	−0.146803	−	0.989166i	$-0.546898\pi$
$263$	−329348.	−0.293607	−0.146803	+	0.989166i	$0.546898\pi$
$264$	0	0
$265$	348151.	0.304546
$266$	81328.1	0.0704753
$267$	0	0
$268$	−2.80396e6	−2.38471
$269$	−2.00722e6	−1.69128	−0.845638	−	0.533756i	$-0.820780\pi$
$269$	−2.00722e6	−1.69128	−0.845638	+	0.533756i	$0.820780\pi$
$270$	0	0
$271$	−969156.	−0.801623	−0.400812	−	0.916160i	$-0.631272\pi$
$271$	−969156.	−0.801623	−0.400812	+	0.916160i	$0.631272\pi$
$272$	−6636.09	−0.00543864
$273$	0	0
$274$	3.22374e6	2.59408
$275$	26936.1	0.0214785
$276$	0	0
$277$	78953.9	0.0618264	0.0309132	−	0.999522i	$-0.490158\pi$
$277$	78953.9	0.0618264	0.0309132	+	0.999522i	$0.490158\pi$
$278$	1.04780e6	0.813145
$279$	0	0
$280$	273037.	0.208126
$281$	581164.	0.439069	0.219535	−	0.975605i	$-0.429546\pi$
$281$	581164.	0.439069	0.219535	+	0.975605i	$0.429546\pi$
$282$	0	0
$283$	1.88724e6	1.40075	0.700375	−	0.713775i	$-0.253016\pi$
$283$	1.88724e6	1.40075	0.700375	+	0.713775i	$0.253016\pi$
$284$	−1.54984e6	−1.14022
$285$	0	0
$286$	−120781.	−0.0873138
$287$	−470116.	−0.336899
$288$	0	0
$289$	−1.41832e6	−0.998916
$290$	2.33257e6	1.62870
$291$	0	0
$292$	3.09902e6	2.12700
$293$	−2.13183e6	−1.45072	−0.725360	−	0.688370i	$-0.758326\pi$
$293$	−2.13183e6	−1.45072	−0.725360	+	0.688370i	$0.758326\pi$
$294$	0	0
$295$	−2.55863e6	−1.71180
$296$	132126.	0.0876514
$297$	0	0
$298$	2.14167e6	1.39705
$299$	1.08437e6	0.701457
$300$	0	0
$301$	−177168.	−0.112712
$302$	605347.	0.381933
$303$	0	0
$304$	61053.0	0.0378899
$305$	−1.28775e6	−0.792653
$306$	0	0
$307$	−1.32016e6	−0.799428	−0.399714	−	0.916640i	$-0.630891\pi$
$307$	−1.32016e6	−0.799428	−0.399714	+	0.916640i	$0.630891\pi$
$308$	−18797.1	−0.0112905
$309$	0	0
$310$	2.08347e6	1.23135
$311$	1.81117e6	1.06184	0.530918	−	0.847423i	$-0.321847\pi$
$311$	1.81117e6	1.06184	0.530918	+	0.847423i	$0.321847\pi$
$312$	0	0
$313$	297903.	0.171876	0.0859378	−	0.996301i	$-0.472611\pi$
$313$	297903.	0.171876	0.0859378	+	0.996301i	$0.472611\pi$
$314$	−3.44372e6	−1.97108
$315$	0	0
$316$	−2.21253e6	−1.24644
$317$	2.85867e6	1.59777	0.798887	−	0.601481i	$-0.205422\pi$
$317$	2.85867e6	1.59777	0.798887	+	0.601481i	$0.205422\pi$
$318$	0	0
$319$	−56414.2	−0.0310393
$320$	3.73764e6	2.04043
$321$	0	0
$322$	278235.	0.149545
$323$	−14165.1	−0.00755463
$324$	0	0
$325$	−1.55047e6	−0.814245
$326$	−278498.	−0.145137
$327$	0	0
$328$	2.94114e6	1.50949
$329$	−205321.	−0.104579
$330$	0	0
$331$	−1.63518e6	−0.820342	−0.410171	−	0.912009i	$-0.634531\pi$
$331$	−1.63518e6	−0.820342	−0.410171	+	0.912009i	$0.634531\pi$
$332$	2.05775e6	1.02459
$333$	0	0
$334$	−400118.	−0.196255
$335$	3.97518e6	1.93528
$336$	0	0
$337$	741151.	0.355494	0.177747	−	0.984076i	$-0.443119\pi$
$337$	741151.	0.355494	0.177747	+	0.984076i	$0.443119\pi$
$338$	3.60385e6	1.71583
$339$	0	0
$340$	−135368.	−0.0635066
$341$	−50389.5	−0.0234668
$342$	0	0
$343$	824120.	0.378229
$344$	1.10840e6	0.505010
$345$	0	0
$346$	4.49777e6	2.01979
$347$	−404168.	−0.180193	−0.0900966	−	0.995933i	$-0.528718\pi$
$347$	−404168.	−0.180193	−0.0900966	+	0.995933i	$0.528718\pi$
$348$	0	0
$349$	−2.94995e6	−1.29644	−0.648219	−	0.761454i	$-0.724486\pi$
$349$	−2.94995e6	−1.29644	−0.648219	+	0.761454i	$0.724486\pi$
$350$	−397829.	−0.173591
$351$	0	0
$352$	−99550.1	−0.0428238
$353$	244773.	0.104551	0.0522754	−	0.998633i	$-0.483353\pi$
$353$	244773.	0.104551	0.0522754	+	0.998633i	$0.483353\pi$
$354$	0	0
$355$	2.19720e6	0.925335
$356$	2.68594e6	1.12323
$357$	0	0
$358$	5.70522e6	2.35269
$359$	−1.10852e6	−0.453948	−0.226974	−	0.973901i	$-0.572883\pi$
$359$	−1.10852e6	−0.453948	−0.226974	+	0.973901i	$0.572883\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−7.31253e6	−2.93289
$363$	0	0
$364$	1.08198e6	0.428022
$365$	−4.39348e6	−1.72614
$366$	0	0
$367$	2.98676e6	1.15754	0.578768	−	0.815492i	$-0.303534\pi$
$367$	2.98676e6	1.15754	0.578768	+	0.815492i	$0.303534\pi$
$368$	208871.	0.0804005
$369$	0	0
$370$	−533197.	−0.202480
$371$	124361.	0.0469081
$372$	0	0
$373$	3.57172e6	1.32925	0.664623	−	0.747179i	$-0.268592\pi$
$373$	3.57172e6	1.32925	0.664623	+	0.747179i	$0.268592\pi$
$374$	5397.71	0.00199540
$375$	0	0
$376$	1.28453e6	0.468570
$377$	3.24726e6	1.17669
$378$	0	0
$379$	5.41131e6	1.93510	0.967551	−	0.252675i	$-0.0813102\pi$
$379$	5.41131e6	1.93510	0.967551	+	0.252675i	$0.0813102\pi$
$380$	1.24541e6	0.442437
$381$	0	0
$382$	−2.70439e6	−0.948223
$383$	54664.9	0.0190419	0.00952097	−	0.999955i	$-0.496969\pi$
$383$	54664.9	0.0190419	0.00952097	+	0.999955i	$0.496969\pi$
$384$	0	0
$385$	26648.6	0.00916268
$386$	−5.56388e6	−1.90068
$387$	0	0
$388$	−2.14524e6	−0.723430
$389$	−2.24796e6	−0.753208	−0.376604	−	0.926374i	$-0.622908\pi$
$389$	−2.24796e6	−0.753208	−0.376604	+	0.926374i	$0.622908\pi$
$390$	0	0
$391$	−48460.8	−0.0160306
$392$	−2.52917e6	−0.831308
$393$	0	0
$394$	−6.20120e6	−2.01250
$395$	3.13670e6	1.01153
$396$	0	0
$397$	1.76879e6	0.563249	0.281624	−	0.959525i	$-0.409127\pi$
$397$	1.76879e6	0.563249	0.281624	+	0.959525i	$0.409127\pi$
$398$	−7.21581e6	−2.28338
$399$	0	0
$400$	−298650.	−0.0933281
$401$	4.31762e6	1.34086	0.670431	−	0.741972i	$-0.266109\pi$
$401$	4.31762e6	1.34086	0.670431	+	0.741972i	$0.266109\pi$
$402$	0	0
$403$	2.90047e6	0.889624
$404$	−539041.	−0.164312
$405$	0	0
$406$	833201.	0.250862
$407$	12895.6	0.00385882
$408$	0	0
$409$	−1.56828e6	−0.463569	−0.231784	−	0.972767i	$-0.574456\pi$
$409$	−1.56828e6	−0.463569	−0.231784	+	0.972767i	$0.574456\pi$
$410$	−1.18690e7	−3.48703
$411$	0	0
$412$	−3.75007e6	−1.08842
$413$	−913949.	−0.263662
$414$	0	0
$415$	−2.91728e6	−0.831491
$416$	5.73021e6	1.62344
$417$	0	0
$418$	−49659.7	−0.0139016
$419$	−92273.8	−0.0256769	−0.0128385	−	0.999918i	$-0.504087\pi$
$419$	−92273.8	−0.0256769	−0.0128385	+	0.999918i	$0.504087\pi$
$420$	0	0
$421$	−6.80057e6	−1.86999	−0.934997	−	0.354655i	$-0.884598\pi$
$421$	−6.80057e6	−1.86999	−0.934997	+	0.354655i	$0.884598\pi$
$422$	−1.13878e7	−3.11286
$423$	0	0
$424$	−778025.	−0.210174
$425$	69290.7	0.0186081
$426$	0	0
$427$	−459989.	−0.122089
$428$	−6.09805e6	−1.60910
$429$	0	0
$430$	−4.47297e6	−1.16661
$431$	4.66801e6	1.21043	0.605214	−	0.796063i	$-0.293088\pi$
$431$	4.66801e6	1.21043	0.605214	+	0.796063i	$0.293088\pi$
$432$	0	0
$433$	3.00139e6	0.769312	0.384656	−	0.923060i	$-0.374320\pi$
$433$	3.00139e6	0.769312	0.384656	+	0.923060i	$0.374320\pi$
$434$	744220.	0.189661
$435$	0	0
$436$	−7.18449e6	−1.81000
$437$	445847.	0.111682
$438$	0	0
$439$	6.91030e6	1.71134	0.855669	−	0.517524i	$-0.173146\pi$
$439$	6.91030e6	1.71134	0.855669	+	0.517524i	$0.173146\pi$
$440$	−166719.	−0.0410538
$441$	0	0
$442$	−310698.	−0.0756455
$443$	6.81040e6	1.64878	0.824391	−	0.566021i	$-0.191518\pi$
$443$	6.81040e6	1.64878	0.824391	+	0.566021i	$0.191518\pi$
$444$	0	0
$445$	−3.80785e6	−0.911548
$446$	1.24665e7	2.96762
$447$	0	0
$448$	1.33510e6	0.314281
$449$	−1.64802e6	−0.385787	−0.192893	−	0.981220i	$-0.561787\pi$
$449$	−1.64802e6	−0.385787	−0.192893	+	0.981220i	$0.561787\pi$
$450$	0	0
$451$	287057.	0.0664549
$452$	−222693.	−0.0512698
$453$	0	0
$454$	−1.31294e7	−2.98954
$455$	−1.53392e6	−0.347356
$456$	0	0
$457$	−3.88919e6	−0.871101	−0.435551	−	0.900164i	$-0.643446\pi$
$457$	−3.88919e6	−0.871101	−0.435551	+	0.900164i	$0.643446\pi$
$458$	9.53052e6	2.12301
$459$	0	0
$460$	4.26071e6	0.938830
$461$	−2.50854e6	−0.549753	−0.274877	−	0.961479i	$-0.588637\pi$
$461$	−2.50854e6	−0.549753	−0.274877	+	0.961479i	$0.588637\pi$
$462$	0	0
$463$	−212544.	−0.0460782	−0.0230391	−	0.999735i	$-0.507334\pi$
$463$	−212544.	−0.0460782	−0.0230391	+	0.999735i	$0.507334\pi$
$464$	625484.	0.134872
$465$	0	0
$466$	1.12786e6	0.240596
$467$	−8.46883e6	−1.79693	−0.898465	−	0.439046i	$-0.855317\pi$
$467$	−8.46883e6	−1.79693	−0.898465	+	0.439046i	$0.855317\pi$
$468$	0	0
$469$	1.41994e6	0.298084
$470$	−5.18374e6	−1.08243
$471$	0	0
$472$	5.71785e6	1.18135
$473$	108181.	0.0222329
$474$	0	0
$475$	−637485.	−0.129639
$476$	−48353.8	−0.00978168
$477$	0	0
$478$	−9.16263e6	−1.83422
$479$	−2.49078e6	−0.496016	−0.248008	−	0.968758i	$-0.579776\pi$
$479$	−2.49078e6	−0.496016	−0.248008	+	0.968758i	$0.579776\pi$
$480$	0	0
$481$	−742283.	−0.146287
$482$	6.63052e6	1.29996
$483$	0	0
$484$	−7.93314e6	−1.53933
$485$	3.04130e6	0.587091
$486$	0	0
$487$	4.86094e6	0.928748	0.464374	−	0.885639i	$-0.346279\pi$
$487$	4.86094e6	0.928748	0.464374	+	0.885639i	$0.346279\pi$
$488$	2.87778e6	0.547027
$489$	0	0
$490$	1.02065e7	1.92038
$491$	8.88281e6	1.66283	0.831413	−	0.555655i	$-0.187533\pi$
$491$	8.88281e6	1.66283	0.831413	+	0.555655i	$0.187533\pi$
$492$	0	0
$493$	−145120.	−0.0268913
$494$	2.85847e6	0.527006
$495$	0	0
$496$	558686.	0.101968
$497$	784847.	0.142526
$498$	0	0
$499$	1.33020e6	0.239148	0.119574	−	0.992825i	$-0.461847\pi$
$499$	1.33020e6	0.239148	0.119574	+	0.992825i	$0.461847\pi$
$500$	4.68878e6	0.838754
$501$	0	0
$502$	−1.24492e7	−2.20487
$503$	4.82565e6	0.850425	0.425212	−	0.905094i	$-0.360199\pi$
$503$	4.82565e6	0.850425	0.425212	+	0.905094i	$0.360199\pi$
$504$	0	0
$505$	764198.	0.133345
$506$	−169893.	−0.0294984
$507$	0	0
$508$	3.52054e6	0.605271
$509$	3.34728e6	0.572662	0.286331	−	0.958131i	$-0.407564\pi$
$509$	3.34728e6	0.572662	0.286331	+	0.958131i	$0.407564\pi$
$510$	0	0
$511$	−1.56936e6	−0.265871
$512$	1.94955e6	0.328670
$513$	0	0
$514$	−6.79969e6	−1.13522
$515$	5.31647e6	0.883294
$516$	0	0
$517$	125371.	0.0206286
$518$	−190459.	−0.0311873
$519$	0	0
$520$	9.59653e6	1.55635
$521$	7.67213e6	1.23829	0.619144	−	0.785277i	$-0.287480\pi$
$521$	7.67213e6	1.23829	0.619144	+	0.785277i	$0.287480\pi$
$522$	0	0
$523$	8.45117e6	1.35102	0.675511	−	0.737349i	$-0.263923\pi$
$523$	8.45117e6	1.35102	0.675511	+	0.737349i	$0.263923\pi$
$524$	−1.20828e7	−1.92238
$525$	0	0
$526$	−2.97016e6	−0.468076
$527$	−129622.	−0.0203308
$528$	0	0
$529$	−4.91104e6	−0.763017
$530$	3.13974e6	0.485516
$531$	0	0
$532$	444862.	0.0681469
$533$	−1.65233e7	−2.51930
$534$	0	0
$535$	8.64520e6	1.30584
$536$	−8.88346e6	−1.33558
$537$	0	0
$538$	−1.81017e7	−2.69628
$539$	−246848.	−0.0365981
$540$	0	0
$541$	9.89761e6	1.45391	0.726954	−	0.686686i	$-0.240935\pi$
$541$	9.89761e6	1.45391	0.726954	+	0.686686i	$0.240935\pi$
$542$	−8.74014e6	−1.27797
$543$	0	0
$544$	−256084.	−0.0371009
$545$	1.01854e7	1.46889
$546$	0	0
$547$	−86566.6	−0.0123703	−0.00618517	−	0.999981i	$-0.501969\pi$
$547$	−86566.6	−0.0123703	−0.00618517	+	0.999981i	$0.501969\pi$
$548$	1.76337e7	2.50838
$549$	0	0
$550$	242918.	0.0342415
$551$	1.33513e6	0.187346
$552$	0	0
$553$	1.12044e6	0.155803
$554$	712030.	0.0985653
$555$	0	0
$556$	5.73145e6	0.786281
$557$	−5.91849e6	−0.808300	−0.404150	−	0.914693i	$-0.632433\pi$
$557$	−5.91849e6	−0.808300	−0.404150	+	0.914693i	$0.632433\pi$
$558$	0	0
$559$	−6.22699e6	−0.842847
$560$	−295462.	−0.0398137
$561$	0	0
$562$	5.24112e6	0.699976
$563$	−801516.	−0.106571	−0.0532857	−	0.998579i	$-0.516969\pi$
$563$	−801516.	−0.106571	−0.0532857	+	0.998579i	$0.516969\pi$
$564$	0	0
$565$	315712.	0.0416074
$566$	1.70197e7	2.23311
$567$	0	0
$568$	−4.91016e6	−0.638594
$569$	−1.10482e7	−1.43058	−0.715289	−	0.698829i	$-0.753705\pi$
$569$	−1.10482e7	−1.43058	−0.715289	+	0.698829i	$0.753705\pi$
$570$	0	0
$571$	2.87709e6	0.369286	0.184643	−	0.982806i	$-0.440887\pi$
$571$	2.87709e6	0.369286	0.184643	+	0.982806i	$0.440887\pi$
$572$	−660667.	−0.0844292
$573$	0	0
$574$	−4.23965e6	−0.537094
$575$	−2.18092e6	−0.275088
$576$	0	0
$577$	3.25493e6	0.407007	0.203503	−	0.979074i	$-0.434767\pi$
$577$	3.25493e6	0.407007	0.203503	+	0.979074i	$0.434767\pi$
$578$	−1.27908e7	−1.59250
$579$	0	0
$580$	1.27591e7	1.57489
$581$	−1.04206e6	−0.128071
$582$	0	0
$583$	−75935.8	−0.00925284
$584$	9.81825e6	1.19125
$585$	0	0
$586$	−1.92255e7	−2.31277
$587$	−9.44870e6	−1.13182	−0.565910	−	0.824467i	$-0.691475\pi$
$587$	−9.44870e6	−1.13182	−0.565910	+	0.824467i	$0.691475\pi$
$588$	0	0
$589$	1.19255e6	0.141640
$590$	−2.30745e7	−2.72899
$591$	0	0
$592$	−142978.	−0.0167673
$593$	−1.20293e6	−0.140477	−0.0702383	−	0.997530i	$-0.522376\pi$
$593$	−1.20293e6	−0.140477	−0.0702383	+	0.997530i	$0.522376\pi$
$594$	0	0
$595$	68551.2	0.00793821
$596$	1.17149e7	1.35089
$597$	0	0
$598$	9.77922e6	1.11828
$599$	−9.23082e6	−1.05117	−0.525585	−	0.850741i	$-0.676154\pi$
$599$	−9.23082e6	−1.05117	−0.525585	+	0.850741i	$0.676154\pi$
$600$	0	0
$601$	−9.23923e6	−1.04340	−0.521699	−	0.853130i	$-0.674701\pi$
$601$	−9.23923e6	−1.04340	−0.521699	+	0.853130i	$0.674701\pi$
$602$	−1.59776e6	−0.179688
$603$	0	0
$604$	3.31123e6	0.369315
$605$	1.12468e7	1.24923
$606$	0	0
$607$	−384978.	−0.0424096	−0.0212048	−	0.999775i	$-0.506750\pi$
$607$	−384978.	−0.0424096	−0.0212048	+	0.999775i	$0.506750\pi$
$608$	2.35601e6	0.258475
$609$	0	0
$610$	−1.16133e7	−1.26367
$611$	−7.21648e6	−0.782028
$612$	0	0
$613$	−3.47518e6	−0.373530	−0.186765	−	0.982405i	$-0.559800\pi$
$613$	−3.47518e6	−0.373530	−0.186765	+	0.982405i	$0.559800\pi$
$614$	−1.19056e7	−1.27447
$615$	0	0
$616$	−59552.5	−0.00632337
$617$	3.15989e6	0.334164	0.167082	−	0.985943i	$-0.446566\pi$
$617$	3.15989e6	0.334164	0.167082	+	0.985943i	$0.446566\pi$
$618$	0	0
$619$	1.82717e7	1.91670	0.958348	−	0.285604i	$-0.0921943\pi$
$619$	1.82717e7	1.91670	0.958348	+	0.285604i	$0.0921943\pi$
$620$	1.13965e7	1.19067
$621$	0	0
$622$	1.63337e7	1.69281
$623$	−1.36017e6	−0.140402
$624$	0	0
$625$	−1.21657e7	−1.24576
$626$	2.68658e6	0.274009
$627$	0	0
$628$	−1.88370e7	−1.90596
$629$	33172.7	0.00334314
$630$	0	0
$631$	−1.44229e6	−0.144204	−0.0721021	−	0.997397i	$-0.522971\pi$
$631$	−1.44229e6	−0.144204	−0.0721021	+	0.997397i	$0.522971\pi$
$632$	−7.00968e6	−0.698080
$633$	0	0
$634$	2.57803e7	2.54721
$635$	−4.99106e6	−0.491200
$636$	0	0
$637$	1.42088e7	1.38743
$638$	−508760.	−0.0494836
$639$	0	0
$640$	1.91018e7	1.84342
$641$	7.73894e6	0.743937	0.371969	−	0.928245i	$-0.378683\pi$
$641$	7.73894e6	0.743937	0.371969	+	0.928245i	$0.378683\pi$
$642$	0	0
$643$	−1.43398e7	−1.36778	−0.683888	−	0.729587i	$-0.739712\pi$
$643$	−1.43398e7	−1.36778	−0.683888	+	0.729587i	$0.739712\pi$
$644$	1.52194e6	0.144604
$645$	0	0
$646$	−127745.	−0.0120438
$647$	−4.01534e6	−0.377105	−0.188552	−	0.982063i	$-0.560380\pi$
$647$	−4.01534e6	−0.377105	−0.188552	+	0.982063i	$0.560380\pi$
$648$	0	0
$649$	558066.	0.0520084
$650$	−1.39826e7	−1.29809
$651$	0	0
$652$	−1.52337e6	−0.140342
$653$	−2.59212e6	−0.237888	−0.118944	−	0.992901i	$-0.537951\pi$
$653$	−2.59212e6	−0.237888	−0.118944	+	0.992901i	$0.537951\pi$
$654$	0	0
$655$	1.71298e7	1.56009
$656$	−3.18271e6	−0.288760
$657$	0	0
$658$	−1.85165e6	−0.166722
$659$	1.02865e7	0.922686	0.461343	−	0.887222i	$-0.347368\pi$
$659$	1.02865e7	0.922686	0.461343	+	0.887222i	$0.347368\pi$
$660$	0	0
$661$	1.39262e7	1.23974	0.619868	−	0.784706i	$-0.287186\pi$
$661$	1.39262e7	1.23974	0.619868	+	0.784706i	$0.287186\pi$
$662$	−1.47465e7	−1.30781
$663$	0	0
$664$	6.51934e6	0.573830
$665$	−630681.	−0.0553039
$666$	0	0
$667$	4.56767e6	0.397539
$668$	−2.18863e6	−0.189772
$669$	0	0
$670$	3.58493e7	3.08528
$671$	280874.	0.0240827
$672$	0	0
$673$	−582222.	−0.0495508	−0.0247754	−	0.999693i	$-0.507887\pi$
$673$	−582222.	−0.0495508	−0.0247754	+	0.999693i	$0.507887\pi$
$674$	6.68393e6	0.566738
$675$	0	0
$676$	1.97129e7	1.65914
$677$	−1.14185e7	−0.957500	−0.478750	−	0.877951i	$-0.658910\pi$
$677$	−1.14185e7	−0.957500	−0.478750	+	0.877951i	$0.658910\pi$
$678$	0	0
$679$	1.08636e6	0.0904274
$680$	−428870.	−0.0355675
$681$	0	0
$682$	−454428.	−0.0374114
$683$	−1.84937e7	−1.51695	−0.758477	−	0.651700i	$-0.774056\pi$
$683$	−1.84937e7	−1.51695	−0.758477	+	0.651700i	$0.774056\pi$
$684$	0	0
$685$	−2.49993e7	−2.03564
$686$	7.43216e6	0.602983
$687$	0	0
$688$	−1.19944e6	−0.0966064
$689$	4.37095e6	0.350774
$690$	0	0
$691$	1.36336e7	1.08621	0.543107	−	0.839663i	$-0.317248\pi$
$691$	1.36336e7	1.08621	0.543107	+	0.839663i	$0.317248\pi$
$692$	2.46027e7	1.95306
$693$	0	0
$694$	−3.64491e6	−0.287269
$695$	−8.12548e6	−0.638097
$696$	0	0
$697$	738429.	0.0575741
$698$	−2.66036e7	−2.06681
$699$	0	0
$700$	−2.17611e6	−0.167856
$701$	−1.97549e6	−0.151838	−0.0759190	−	0.997114i	$-0.524189\pi$
$701$	−1.97549e6	−0.151838	−0.0759190	+	0.997114i	$0.524189\pi$
$702$	0	0
$703$	−305194.	−0.0232910
$704$	−815222.	−0.0619932
$705$	0	0
$706$	2.20744e6	0.166678
$707$	272974.	0.0205387
$708$	0	0
$709$	−1.56906e7	−1.17226	−0.586128	−	0.810218i	$-0.699349\pi$
$709$	−1.56906e7	−1.17226	−0.586128	+	0.810218i	$0.699349\pi$
$710$	1.98150e7	1.47519
$711$	0	0
$712$	8.50953e6	0.629079
$713$	4.07987e6	0.300554
$714$	0	0
$715$	936628.	0.0685176
$716$	3.12074e7	2.27496
$717$	0	0
$718$	−9.99695e6	−0.723696
$719$	−1.50922e6	−0.108875	−0.0544377	−	0.998517i	$-0.517337\pi$
$719$	−1.50922e6	−0.108875	−0.0544377	+	0.998517i	$0.517337\pi$
$720$	0	0
$721$	1.89906e6	0.136050
$722$	1.17527e6	0.0839066
$723$	0	0
$724$	−3.99993e7	−2.83600
$725$	−6.53098e6	−0.461460
$726$	0	0
$727$	2.42082e6	0.169874	0.0849369	−	0.996386i	$-0.472931\pi$
$727$	2.42082e6	0.169874	0.0849369	+	0.996386i	$0.472931\pi$
$728$	3.42791e6	0.239718
$729$	0	0
$730$	−3.96217e7	−2.75186
$731$	278285.	0.0192618
$732$	0	0
$733$	−2.40820e7	−1.65552	−0.827758	−	0.561086i	$-0.810384\pi$
$733$	−2.40820e7	−1.65552	−0.827758	+	0.561086i	$0.810384\pi$
$734$	2.69355e7	1.84537
$735$	0	0
$736$	8.06024e6	0.548470
$737$	−867031.	−0.0587985
$738$	0	0
$739$	−9.91699e6	−0.667988	−0.333994	−	0.942575i	$-0.608397\pi$
$739$	−9.91699e6	−0.667988	−0.333994	+	0.942575i	$0.608397\pi$
$740$	−2.91657e6	−0.195791
$741$	0	0
$742$	1.12152e6	0.0747822
$743$	2.39963e7	1.59468	0.797338	−	0.603533i	$-0.206241\pi$
$743$	2.39963e7	1.59468	0.797338	+	0.603533i	$0.206241\pi$
$744$	0	0
$745$	−1.66081e7	−1.09630
$746$	3.22109e7	2.11912
$747$	0	0
$748$	29525.3	0.00192948
$749$	3.08809e6	0.201134
$750$	0	0
$751$	1.75635e7	1.13634	0.568172	−	0.822909i	$-0.307651\pi$
$751$	1.75635e7	1.13634	0.568172	+	0.822909i	$0.307651\pi$
$752$	−1.39003e6	−0.0896354
$753$	0	0
$754$	2.92848e7	1.87592
$755$	−4.69433e6	−0.299713
$756$	0	0
$757$	3.44163e6	0.218285	0.109143	−	0.994026i	$-0.465189\pi$
$757$	3.44163e6	0.218285	0.109143	+	0.994026i	$0.465189\pi$
$758$	4.88008e7	3.08499
$759$	0	0
$760$	3.94567e6	0.247791
$761$	2.71265e7	1.69798	0.848990	−	0.528409i	$-0.177211\pi$
$761$	2.71265e7	1.69798	0.848990	+	0.528409i	$0.177211\pi$
$762$	0	0
$763$	3.63827e6	0.226247
$764$	−1.47929e7	−0.916896
$765$	0	0
$766$	492985.	0.0303572
$767$	−3.21229e7	−1.97163
$768$	0	0
$769$	−9.55287e6	−0.582530	−0.291265	−	0.956642i	$-0.594076\pi$
$769$	−9.55287e6	−0.582530	−0.291265	+	0.956642i	$0.594076\pi$
$770$	240325.	0.0146074
$771$	0	0
$772$	−3.04342e7	−1.83789
$773$	3.86711e6	0.232776	0.116388	−	0.993204i	$-0.462868\pi$
$773$	3.86711e6	0.232776	0.116388	+	0.993204i	$0.462868\pi$
$774$	0	0
$775$	−5.83352e6	−0.348880
$776$	−6.79650e6	−0.405164
$777$	0	0
$778$	−2.02728e7	−1.20078
$779$	−6.79366e6	−0.401107
$780$	0	0
$781$	−479235.	−0.0281139
$782$	−437034.	−0.0255563
$783$	0	0
$784$	2.73689e6	0.159026
$785$	2.67053e7	1.54676
$786$	0	0
$787$	1.47900e7	0.851197	0.425598	−	0.904912i	$-0.360064\pi$
$787$	1.47900e7	0.851197	0.425598	+	0.904912i	$0.360064\pi$
$788$	−3.39204e7	−1.94601
$789$	0	0
$790$	2.82877e7	1.61261
$791$	112773.	0.00640863
$792$	0	0
$793$	−1.61674e7	−0.912971
$794$	1.59515e7	0.897946
$795$	0	0
$796$	−3.94702e7	−2.20794
$797$	−1.59707e7	−0.890589	−0.445294	−	0.895384i	$-0.646901\pi$
$797$	−1.59707e7	−0.890589	−0.445294	+	0.895384i	$0.646901\pi$
$798$	0	0
$799$	322505.	0.0178719
$800$	−1.15248e7	−0.636659
$801$	0	0
$802$	3.89376e7	2.13764
$803$	958267.	0.0524442
$804$	0	0
$805$	−2.15765e6	−0.117352
$806$	2.61574e7	1.41826
$807$	0	0
$808$	−1.70778e6	−0.0920244
$809$	711077.	0.0381984	0.0190992	−	0.999818i	$-0.493920\pi$
$809$	711077.	0.0381984	0.0190992	+	0.999818i	$0.493920\pi$
$810$	0	0
$811$	−1.71319e7	−0.914648	−0.457324	−	0.889300i	$-0.651192\pi$
$811$	−1.71319e7	−0.914648	−0.457324	+	0.889300i	$0.651192\pi$
$812$	4.55758e6	0.242574
$813$	0	0
$814$	116296.	0.00615184
$815$	2.15969e6	0.113893
$816$	0	0
$817$	−2.56026e6	−0.134193
$818$	−1.41432e7	−0.739033
$819$	0	0
$820$	−6.49232e7	−3.37183
$821$	−9.14797e6	−0.473660	−0.236830	−	0.971551i	$-0.576108\pi$
$821$	−9.14797e6	−0.473660	−0.236830	+	0.971551i	$0.576108\pi$
$822$	0	0
$823$	1.37403e7	0.707128	0.353564	−	0.935410i	$-0.384970\pi$
$823$	1.37403e7	0.707128	0.353564	+	0.935410i	$0.384970\pi$
$824$	−1.18809e7	−0.609581
$825$	0	0
$826$	−8.24227e6	−0.420336
$827$	−5.10837e6	−0.259728	−0.129864	−	0.991532i	$-0.541454\pi$
$827$	−5.10837e6	−0.259728	−0.129864	+	0.991532i	$0.541454\pi$
$828$	0	0
$829$	−1.36897e7	−0.691844	−0.345922	−	0.938263i	$-0.612434\pi$
$829$	−1.36897e7	−0.691844	−0.345922	+	0.938263i	$0.612434\pi$
$830$	−2.63089e7	−1.32558
$831$	0	0
$832$	4.69251e7	2.35016
$833$	−634995.	−0.0317072
$834$	0	0
$835$	3.10282e6	0.154007
$836$	−271637.	−0.0134423
$837$	0	0
$838$	−832153.	−0.0409348
$839$	−7.59127e6	−0.372314	−0.186157	−	0.982520i	$-0.559603\pi$
$839$	−7.59127e6	−0.372314	−0.186157	+	0.982520i	$0.559603\pi$
$840$	0	0
$841$	−6.83285e6	−0.333128
$842$	−6.13296e7	−2.98119
$843$	0	0
$844$	−6.22910e7	−3.01002
$845$	−2.79470e7	−1.34646
$846$	0	0
$847$	4.01739e6	0.192414
$848$	841927.	0.0402054
$849$	0	0
$850$	624885.	0.0296656
$851$	−1.04411e6	−0.0494223
$852$	0	0
$853$	1.52482e6	0.0717542	0.0358771	−	0.999356i	$-0.488578\pi$
$853$	1.52482e6	0.0717542	0.0358771	+	0.999356i	$0.488578\pi$
$854$	−4.14832e6	−0.194638
$855$	0	0
$856$	−1.93197e7	−0.901190
$857$	−7.90135e6	−0.367493	−0.183747	−	0.982974i	$-0.558823\pi$
$857$	−7.90135e6	−0.367493	−0.183747	+	0.982974i	$0.558823\pi$
$858$	0	0
$859$	4.18519e7	1.93523	0.967615	−	0.252432i	$-0.0812302\pi$
$859$	4.18519e7	1.93523	0.967615	+	0.252432i	$0.0812302\pi$
$860$	−2.44670e7	−1.12807
$861$	0	0
$862$	4.20976e7	1.92970
$863$	−3.23045e7	−1.47651	−0.738254	−	0.674523i	$-0.764350\pi$
$863$	−3.23045e7	−1.47651	−0.738254	+	0.674523i	$0.764350\pi$
$864$	0	0
$865$	−3.48792e7	−1.58499
$866$	2.70674e7	1.22646
$867$	0	0
$868$	4.07086e6	0.183395
$869$	−684149.	−0.0307327
$870$	0	0
$871$	4.99073e7	2.22904
$872$	−2.27617e7	−1.01371
$873$	0	0
$874$	4.02078e6	0.178046
$875$	−2.37442e6	−0.104843
$876$	0	0
$877$	−8.40521e6	−0.369020	−0.184510	−	0.982831i	$-0.559070\pi$
$877$	−8.40521e6	−0.369020	−0.184510	+	0.982831i	$0.559070\pi$
$878$	6.23192e7	2.72826
$879$	0	0
$880$	180412.	0.00785343
$881$	−2.14767e7	−0.932238	−0.466119	−	0.884722i	$-0.654348\pi$
$881$	−2.14767e7	−0.932238	−0.466119	+	0.884722i	$0.654348\pi$
$882$	0	0
$883$	−4.00787e7	−1.72986	−0.864931	−	0.501890i	$-0.832638\pi$
$883$	−4.00787e7	−1.72986	−0.864931	+	0.501890i	$0.832638\pi$
$884$	−1.69951e6	−0.0731463
$885$	0	0
$886$	6.14183e7	2.62853
$887$	2.81155e7	1.19988	0.599938	−	0.800046i	$-0.295192\pi$
$887$	2.81155e7	1.19988	0.599938	+	0.800046i	$0.295192\pi$
$888$	0	0
$889$	−1.78282e6	−0.0756577
$890$	−3.43403e7	−1.45321
$891$	0	0
$892$	6.81916e7	2.86958
$893$	−2.96709e6	−0.124510
$894$	0	0
$895$	−4.42426e7	−1.84622
$896$	6.82322e6	0.283935
$897$	0	0
$898$	−1.48624e7	−0.615031
$899$	1.22175e7	0.504179
$900$	0	0
$901$	−195338.	−0.00801632
$902$	2.58877e6	0.105944
$903$	0	0
$904$	−705533.	−0.0287142
$905$	5.67070e7	2.30152
$906$	0	0
$907$	−1.84010e7	−0.742717	−0.371358	−	0.928490i	$-0.621108\pi$
$907$	−1.84010e7	−0.742717	−0.371358	+	0.928490i	$0.621108\pi$
$908$	−7.18173e7	−2.89078
$909$	0	0
$910$	−1.38334e7	−0.553764
$911$	4.13921e7	1.65242	0.826211	−	0.563361i	$-0.190492\pi$
$911$	4.13921e7	1.65242	0.826211	+	0.563361i	$0.190492\pi$
$912$	0	0
$913$	636291.	0.0252627
$914$	−3.50739e7	−1.38873
$915$	0	0
$916$	5.21316e7	2.05288
$917$	6.11881e6	0.240294
$918$	0	0
$919$	7.77910e6	0.303837	0.151918	−	0.988393i	$-0.451455\pi$
$919$	7.77910e6	0.303837	0.151918	+	0.988393i	$0.451455\pi$
$920$	1.34987e7	0.525801
$921$	0	0
$922$	−2.26227e7	−0.876431
$923$	2.75853e7	1.06579
$924$	0	0
$925$	1.49290e6	0.0573689
$926$	−1.91678e6	−0.0734591
$927$	0	0
$928$	2.41371e7	0.920059
$929$	−3.18869e7	−1.21220	−0.606098	−	0.795390i	$-0.707266\pi$
$929$	−3.18869e7	−1.21220	−0.606098	+	0.795390i	$0.707266\pi$
$930$	0	0
$931$	5.84205e6	0.220897
$932$	6.16934e6	0.232648
$933$	0	0
$934$	−7.63745e7	−2.86471
$935$	−41858.0	−0.00156585
$936$	0	0
$937$	−1.23641e7	−0.460061	−0.230030	−	0.973183i	$-0.573883\pi$
$937$	−1.23641e7	−0.460061	−0.230030	+	0.973183i	$0.573883\pi$
$938$	1.28055e7	0.475214
$939$	0	0
$940$	−2.83549e7	−1.04667
$941$	7.79985e6	0.287152	0.143576	−	0.989639i	$-0.454140\pi$
$941$	7.79985e6	0.287152	0.143576	+	0.989639i	$0.454140\pi$
$942$	0	0
$943$	−2.32421e7	−0.851129
$944$	−6.18747e6	−0.225987
$945$	0	0
$946$	975605.	0.0354443
$947$	3.53833e7	1.28211	0.641053	−	0.767497i	$-0.278498\pi$
$947$	3.53833e7	1.28211	0.641053	+	0.767497i	$0.278498\pi$
$948$	0	0
$949$	−5.51589e7	−1.98815
$950$	−5.74903e6	−0.206674
$951$	0	0
$952$	−153194.	−0.00547833
$953$	−1.37685e6	−0.0491084	−0.0245542	−	0.999699i	$-0.507817\pi$
$953$	−1.37685e6	−0.0491084	−0.0245542	+	0.999699i	$0.507817\pi$
$954$	0	0
$955$	2.09719e7	0.744097
$956$	−5.01193e7	−1.77362
$957$	0	0
$958$	−2.24626e7	−0.790762
$959$	−8.92983e6	−0.313543
$960$	0	0
$961$	−1.77164e7	−0.618822
$962$	−6.69414e6	−0.233215
$963$	0	0
$964$	3.62687e7	1.25701
$965$	4.31466e7	1.49152
$966$	0	0
$967$	−1.49803e7	−0.515175	−0.257588	−	0.966255i	$-0.582928\pi$
$967$	−1.49803e7	−0.515175	−0.257588	+	0.966255i	$0.582928\pi$
$968$	−2.51336e7	−0.862118
$969$	0	0
$970$	2.74274e7	0.935956
$971$	−1.07440e7	−0.365695	−0.182847	−	0.983141i	$-0.558531\pi$
$971$	−1.07440e7	−0.365695	−0.182847	+	0.983141i	$0.558531\pi$
$972$	0	0
$973$	−2.90244e6	−0.0982837
$974$	4.38374e7	1.48063
$975$	0	0
$976$	−3.11414e6	−0.104644
$977$	−1.81622e7	−0.608741	−0.304371	−	0.952554i	$-0.598446\pi$
$977$	−1.81622e7	−0.608741	−0.304371	+	0.952554i	$0.598446\pi$
$978$	0	0
$979$	830535.	0.0276950
$980$	5.58292e7	1.85693
$981$	0	0
$982$	8.01079e7	2.65092
$983$	2.71045e7	0.894660	0.447330	−	0.894369i	$-0.352375\pi$
$983$	2.71045e7	0.894660	0.447330	+	0.894369i	$0.352375\pi$
$984$	0	0
$985$	4.80889e7	1.57926
$986$	−1.30874e6	−0.0428708
$987$	0	0
$988$	1.56357e7	0.509596
$989$	−8.75901e6	−0.284750
$990$	0	0
$991$	6.68184e6	0.216128	0.108064	−	0.994144i	$-0.465535\pi$
$991$	6.68184e6	0.216128	0.108064	+	0.994144i	$0.465535\pi$
$992$	2.15594e7	0.695598
$993$	0	0
$994$	7.07799e6	0.227219
$995$	5.59569e7	1.79183
$996$	0	0
$997$	−2.45982e7	−0.783729	−0.391864	−	0.920023i	$-0.628170\pi$
$997$	−2.45982e7	−0.783729	−0.391864	+	0.920023i	$0.628170\pi$
$998$	1.19962e7	0.381256
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.6.a.k.1.6	yes	6
3.2	odd	2	inner	171.6.a.k.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.6.a.k.1.1	✓	6	3.2	odd	2	inner
171.6.a.k.1.6	yes	6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+9.01831 q^{2} +49.3298 q^{4} -69.9349 q^{5} -24.9809 q^{7} +156.286 q^{8} -630.694 q^{10} +15.2536 q^{11} -878.013 q^{13} -225.286 q^{14} -169.122 q^{16} +39.2385 q^{17} -361.000 q^{19} -3449.88 q^{20} +137.562 q^{22} -1235.03 q^{23} +1765.89 q^{25} -7918.19 q^{26} -1232.31 q^{28} -3698.42 q^{29} -3303.45 q^{31} -6526.34 q^{32} +353.865 q^{34} +1747.04 q^{35} +845.412 q^{37} -3255.61 q^{38} -10929.8 q^{40} +18819.0 q^{41} +7092.14 q^{43} +752.457 q^{44} -11137.9 q^{46} +8219.10 q^{47} -16183.0 q^{49} +15925.3 q^{50} -43312.3 q^{52} -4978.22 q^{53} -1066.76 q^{55} -3904.16 q^{56} -33353.5 q^{58} +36585.9 q^{59} +18413.6 q^{61} -29791.5 q^{62} -53444.6 q^{64} +61403.7 q^{65} -56841.1 q^{67} +1935.63 q^{68} +15755.3 q^{70} -31417.8 q^{71} +62822.4 q^{73} +7624.19 q^{74} -17808.1 q^{76} -381.049 q^{77} -44851.7 q^{79} +11827.5 q^{80} +169715. q^{82} +41714.2 q^{83} -2744.14 q^{85} +63959.0 q^{86} +2383.92 q^{88} +54448.5 q^{89} +21933.6 q^{91} -60923.9 q^{92} +74122.4 q^{94} +25246.5 q^{95} -43487.7 q^{97} -145943. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{4} - 10 q^{7} - 788 q^{10} - 1256 q^{13} - 606 q^{16} - 2166 q^{19} - 2524 q^{22} - 3944 q^{25} - 9632 q^{28} - 18136 q^{31} - 14072 q^{34} - 23764 q^{37} - 34284 q^{40} - 24606 q^{43} - 45640 q^{46}+ \cdots - 16492 q^{97}+O(q^{100}) \) 6 * q + 42 * q^4 - 10 * q^7 - 788 * q^10 - 1256 * q^13 - 606 * q^16 - 2166 * q^19 - 2524 * q^22 - 3944 * q^25 - 9632 * q^28 - 18136 * q^31 - 14072 * q^34 - 23764 * q^37 - 34284 * q^40 - 24606 * q^43 - 45640 * q^46 - 63256 * q^49 - 89128 * q^52 - 63402 * q^55 - 78572 * q^58 - 42334 * q^61 - 109662 * q^64 - 129696 * q^67 - 15096 * q^70 - 16998 * q^73 - 15162 * q^76 - 129004 * q^79 + 211396 * q^82 - 11754 * q^85 + 55044 * q^88 + 23932 * q^91 + 235368 * q^94 - 16492 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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