LMFDB - Embedded newform 171.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,4,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, 4, names="a")

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

chi := DirichletCharacter("171.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
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:	$10.0893266110$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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			1.1
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)

$f(q)$

$=$

$q+3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +11.0000 q^{7} -21.0000 q^{8} +O(q^{10})$

\(q+3.00000 q^{2} +1.00000 q^{4} +12.0000 q^{5} +11.0000 q^{7} -21.0000 q^{8} +36.0000 q^{10} +54.0000 q^{11} +11.0000 q^{13} +33.0000 q^{14} -71.0000 q^{16} +93.0000 q^{17} +19.0000 q^{19} +12.0000 q^{20} +162.000 q^{22} -183.000 q^{23} +19.0000 q^{25} +33.0000 q^{26} +11.0000 q^{28} +249.000 q^{29} +56.0000 q^{31} -45.0000 q^{32} +279.000 q^{34} +132.000 q^{35} -250.000 q^{37} +57.0000 q^{38} -252.000 q^{40} -240.000 q^{41} -196.000 q^{43} +54.0000 q^{44} -549.000 q^{46} +168.000 q^{47} -222.000 q^{49} +57.0000 q^{50} +11.0000 q^{52} -435.000 q^{53} +648.000 q^{55} -231.000 q^{56} +747.000 q^{58} -195.000 q^{59} -358.000 q^{61} +168.000 q^{62} +433.000 q^{64} +132.000 q^{65} -961.000 q^{67} +93.0000 q^{68} +396.000 q^{70} +246.000 q^{71} +353.000 q^{73} -750.000 q^{74} +19.0000 q^{76} +594.000 q^{77} -34.0000 q^{79} -852.000 q^{80} -720.000 q^{82} -234.000 q^{83} +1116.00 q^{85} -588.000 q^{86} -1134.00 q^{88} +168.000 q^{89} +121.000 q^{91} -183.000 q^{92} +504.000 q^{94} +228.000 q^{95} +758.000 q^{97} -666.000 q^{98} +O(q^{100})\)

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	−21.0000	−0.928078
$9$	0	0
$10$	36.0000	1.13842
$11$	54.0000	1.48015	0.740073	−	0.672526i	$-0.234791\pi$
$11$	54.0000	1.48015	0.740073	+	0.672526i	$0.234791\pi$
$12$	0	0
$13$	11.0000	0.234681	0.117340	−	0.993092i	$-0.462563\pi$
$13$	11.0000	0.234681	0.117340	+	0.993092i	$0.462563\pi$
$14$	33.0000	0.629973
$15$	0	0
$16$	−71.0000	−1.10938
$17$	93.0000	1.32681	0.663406	−	0.748259i	$-0.269110\pi$
$17$	93.0000	1.32681	0.663406	+	0.748259i	$0.269110\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	12.0000	0.134164
$21$	0	0
$22$	162.000	1.56993
$23$	−183.000	−1.65905	−0.829525	−	0.558470i	$-0.811389\pi$
$23$	−183.000	−1.65905	−0.829525	+	0.558470i	$0.811389\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	33.0000	0.248917
$27$	0	0
$28$	11.0000	0.0742430
$29$	249.000	1.59442	0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000	1.59442	0.797209	+	0.603703i	$0.206309\pi$
$30$	0	0
$31$	56.0000	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$31$	56.0000	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	279.000	1.40730
$35$	132.000	0.637488
$36$	0	0
$37$	−250.000	−1.11080	−0.555402	−	0.831582i	$-0.687436\pi$
$37$	−250.000	−1.11080	−0.555402	+	0.831582i	$0.687436\pi$
$38$	57.0000	0.243332
$39$	0	0
$40$	−252.000	−0.996117
$41$	−240.000	−0.914188	−0.457094	−	0.889418i	$-0.651110\pi$
$41$	−240.000	−0.914188	−0.457094	+	0.889418i	$0.651110\pi$
$42$	0	0
$43$	−196.000	−0.695110	−0.347555	−	0.937660i	$-0.612988\pi$
$43$	−196.000	−0.695110	−0.347555	+	0.937660i	$0.612988\pi$
$44$	54.0000	0.185018
$45$	0	0
$46$	−549.000	−1.75969
$47$	168.000	0.521390	0.260695	−	0.965421i	$-0.416048\pi$
$47$	168.000	0.521390	0.260695	+	0.965421i	$0.416048\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	57.0000	0.161220
$51$	0	0
$52$	11.0000	0.0293351
$53$	−435.000	−1.12739	−0.563697	−	0.825982i	$-0.690621\pi$
$53$	−435.000	−1.12739	−0.563697	+	0.825982i	$0.690621\pi$
$54$	0	0
$55$	648.000	1.58866
$56$	−231.000	−0.551226
$57$	0	0
$58$	747.000	1.69114
$59$	−195.000	−0.430285	−0.215143	−	0.976583i	$-0.569022\pi$
$59$	−195.000	−0.430285	−0.215143	+	0.976583i	$0.569022\pi$
$60$	0	0
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	168.000	0.344129
$63$	0	0
$64$	433.000	0.845703
$65$	132.000	0.251886
$66$	0	0
$67$	−961.000	−1.75231	−0.876155	−	0.482029i	$-0.839900\pi$
$67$	−961.000	−1.75231	−0.876155	+	0.482029i	$0.839900\pi$
$68$	93.0000	0.165852
$69$	0	0
$70$	396.000	0.676158
$71$	246.000	0.411195	0.205597	−	0.978637i	$-0.434086\pi$
$71$	246.000	0.411195	0.205597	+	0.978637i	$0.434086\pi$
$72$	0	0
$73$	353.000	0.565966	0.282983	−	0.959125i	$-0.408676\pi$
$73$	353.000	0.565966	0.282983	+	0.959125i	$0.408676\pi$
$74$	−750.000	−1.17819
$75$	0	0
$76$	19.0000	0.0286770
$77$	594.000	0.879124
$78$	0	0
$79$	−34.0000	−0.0484215	−0.0242108	−	0.999707i	$-0.507707\pi$
$79$	−34.0000	−0.0484215	−0.0242108	+	0.999707i	$0.507707\pi$
$80$	−852.000	−1.19071
$81$	0	0
$82$	−720.000	−0.969643
$83$	−234.000	−0.309456	−0.154728	−	0.987957i	$-0.549450\pi$
$83$	−234.000	−0.309456	−0.154728	+	0.987957i	$0.549450\pi$
$84$	0	0
$85$	1116.00	1.42408
$86$	−588.000	−0.737275
$87$	0	0
$88$	−1134.00	−1.37369
$89$	168.000	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$89$	168.000	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$90$	0	0
$91$	121.000	0.139387
$92$	−183.000	−0.207381
$93$	0	0
$94$	504.000	0.553017
$95$	228.000	0.246235
$96$	0	0
$97$	758.000	0.793435	0.396718	−	0.917941i	$-0.370149\pi$
$97$	758.000	0.793435	0.396718	+	0.917941i	$0.370149\pi$
$98$	−666.000	−0.686491
$99$	0	0
$100$	19.0000	0.0190000
$101$	726.000	0.715245	0.357622	−	0.933866i	$-0.383588\pi$
$101$	726.000	0.715245	0.357622	+	0.933866i	$0.383588\pi$
$102$	0	0
$103$	2.00000	0.00191326	0.000956630	−	1.00000i	$-0.499695\pi$
$103$	2.00000	0.00191326	0.000956630		1.00000i	$0.499695\pi$
$104$	−231.000	−0.217802
$105$	0	0
$106$	−1305.00	−1.19578
$107$	−1413.00	−1.27663	−0.638317	−	0.769773i	$-0.720369\pi$
$107$	−1413.00	−1.27663	−0.638317	+	0.769773i	$0.720369\pi$
$108$	0	0
$109$	389.000	0.341830	0.170915	−	0.985286i	$-0.445328\pi$
$109$	389.000	0.341830	0.170915	+	0.985286i	$0.445328\pi$
$110$	1944.00	1.68503
$111$	0	0
$112$	−781.000	−0.658907
$113$	−342.000	−0.284714	−0.142357	−	0.989815i	$-0.545468\pi$
$113$	−342.000	−0.284714	−0.142357	+	0.989815i	$0.545468\pi$
$114$	0	0
$115$	−2196.00	−1.78068
$116$	249.000	0.199302
$117$	0	0
$118$	−585.000	−0.456387
$119$	1023.00	0.788053
$120$	0	0
$121$	1585.00	1.19083
$122$	−1074.00	−0.797011
$123$	0	0
$124$	56.0000	0.0405560
$125$	−1272.00	−0.910169
$126$	0	0
$127$	−1150.00	−0.803512	−0.401756	−	0.915747i	$-0.631600\pi$
$127$	−1150.00	−0.803512	−0.401756	+	0.915747i	$0.631600\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	396.000	0.267165
$131$	1452.00	0.968411	0.484205	−	0.874954i	$-0.339109\pi$
$131$	1452.00	0.968411	0.484205	+	0.874954i	$0.339109\pi$
$132$	0	0
$133$	209.000	0.136260
$134$	−2883.00	−1.85861
$135$	0	0
$136$	−1953.00	−1.23139
$137$	1689.00	1.05329	0.526646	−	0.850085i	$-0.323449\pi$
$137$	1689.00	1.05329	0.526646	+	0.850085i	$0.323449\pi$
$138$	0	0
$139$	2144.00	1.30829	0.654143	−	0.756371i	$-0.273030\pi$
$139$	2144.00	1.30829	0.654143	+	0.756371i	$0.273030\pi$
$140$	132.000	0.0796860
$141$	0	0
$142$	738.000	0.436138
$143$	594.000	0.347362
$144$	0	0
$145$	2988.00	1.71131
$146$	1059.00	0.600298
$147$	0	0
$148$	−250.000	−0.138850
$149$	3000.00	1.64946	0.824730	−	0.565527i	$-0.191327\pi$
$149$	3000.00	1.64946	0.824730	+	0.565527i	$0.191327\pi$
$150$	0	0
$151$	−1006.00	−0.542166	−0.271083	−	0.962556i	$-0.587382\pi$
$151$	−1006.00	−0.542166	−0.271083	+	0.962556i	$0.587382\pi$
$152$	−399.000	−0.212916
$153$	0	0
$154$	1782.00	0.932452
$155$	672.000	0.348234
$156$	0	0
$157$	2846.00	1.44672	0.723362	−	0.690469i	$-0.242596\pi$
$157$	2846.00	1.44672	0.723362	+	0.690469i	$0.242596\pi$
$158$	−102.000	−0.0513588
$159$	0	0
$160$	−540.000	−0.266817
$161$	−2013.00	−0.985383
$162$	0	0
$163$	−1600.00	−0.768845	−0.384422	−	0.923157i	$-0.625599\pi$
$163$	−1600.00	−0.768845	−0.384422	+	0.923157i	$0.625599\pi$
$164$	−240.000	−0.114273
$165$	0	0
$166$	−702.000	−0.328228
$167$	2004.00	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	2004.00	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−2076.00	−0.944925
$170$	3348.00	1.51047
$171$	0	0
$172$	−196.000	−0.0868887
$173$	462.000	0.203036	0.101518	−	0.994834i	$-0.467630\pi$
$173$	462.000	0.203036	0.101518	+	0.994834i	$0.467630\pi$
$174$	0	0
$175$	209.000	0.0902795
$176$	−3834.00	−1.64204
$177$	0	0
$178$	504.000	0.212227
$179$	−720.000	−0.300644	−0.150322	−	0.988637i	$-0.548031\pi$
$179$	−720.000	−0.300644	−0.150322	+	0.988637i	$0.548031\pi$
$180$	0	0
$181$	−2338.00	−0.960122	−0.480061	−	0.877235i	$-0.659386\pi$
$181$	−2338.00	−0.960122	−0.480061	+	0.877235i	$0.659386\pi$
$182$	363.000	0.147843
$183$	0	0
$184$	3843.00	1.53973
$185$	−3000.00	−1.19224
$186$	0	0
$187$	5022.00	1.96388
$188$	168.000	0.0651737
$189$	0	0
$190$	684.000	0.261171
$191$	−2871.00	−1.08763	−0.543817	−	0.839204i	$-0.683022\pi$
$191$	−2871.00	−1.08763	−0.543817	+	0.839204i	$0.683022\pi$
$192$	0	0
$193$	1658.00	0.618370	0.309185	−	0.951002i	$-0.399944\pi$
$193$	1658.00	0.618370	0.309185	+	0.951002i	$0.399944\pi$
$194$	2274.00	0.841565
$195$	0	0
$196$	−222.000	−0.0809038
$197$	4176.00	1.51029	0.755146	−	0.655556i	$-0.227566\pi$
$197$	4176.00	1.51029	0.755146	+	0.655556i	$0.227566\pi$
$198$	0	0
$199$	−241.000	−0.0858494	−0.0429247	−	0.999078i	$-0.513668\pi$
$199$	−241.000	−0.0858494	−0.0429247	+	0.999078i	$0.513668\pi$
$200$	−399.000	−0.141068
$201$	0	0
$202$	2178.00	0.758631
$203$	2739.00	0.946996
$204$	0	0
$205$	−2880.00	−0.981209
$206$	6.00000	0.00202932
$207$	0	0
$208$	−781.000	−0.260349
$209$	1026.00	0.339569
$210$	0	0
$211$	−745.000	−0.243071	−0.121535	−	0.992587i	$-0.538782\pi$
$211$	−745.000	−0.243071	−0.121535	+	0.992587i	$0.538782\pi$
$212$	−435.000	−0.140924
$213$	0	0
$214$	−4239.00	−1.35408
$215$	−2352.00	−0.746070
$216$	0	0
$217$	616.000	0.192704
$218$	1167.00	0.362565
$219$	0	0
$220$	648.000	0.198583
$221$	1023.00	0.311377
$222$	0	0
$223$	−1978.00	−0.593976	−0.296988	−	0.954881i	$-0.595982\pi$
$223$	−1978.00	−0.593976	−0.296988	+	0.954881i	$0.595982\pi$
$224$	−495.000	−0.147650
$225$	0	0
$226$	−1026.00	−0.301985
$227$	−5355.00	−1.56574	−0.782872	−	0.622183i	$-0.786246\pi$
$227$	−5355.00	−1.56574	−0.782872	+	0.622183i	$0.786246\pi$
$228$	0	0
$229$	−6370.00	−1.83817	−0.919086	−	0.394057i	$-0.871071\pi$
$229$	−6370.00	−1.83817	−0.919086	+	0.394057i	$0.871071\pi$
$230$	−6588.00	−1.88870
$231$	0	0
$232$	−5229.00	−1.47974
$233$	2838.00	0.797955	0.398978	−	0.916961i	$-0.369365\pi$
$233$	2838.00	0.797955	0.398978	+	0.916961i	$0.369365\pi$
$234$	0	0
$235$	2016.00	0.559614
$236$	−195.000	−0.0537857
$237$	0	0
$238$	3069.00	0.835856
$239$	369.000	0.0998687	0.0499344	−	0.998753i	$-0.484099\pi$
$239$	369.000	0.0998687	0.0499344	+	0.998753i	$0.484099\pi$
$240$	0	0
$241$	6608.00	1.76622	0.883109	−	0.469167i	$-0.155446\pi$
$241$	6608.00	1.76622	0.883109	+	0.469167i	$0.155446\pi$
$242$	4755.00	1.26307
$243$	0	0
$244$	−358.000	−0.0939287
$245$	−2664.00	−0.694680
$246$	0	0
$247$	209.000	0.0538395
$248$	−1176.00	−0.301113
$249$	0	0
$250$	−3816.00	−0.965380
$251$	−4674.00	−1.17538	−0.587690	−	0.809086i	$-0.699962\pi$
$251$	−4674.00	−1.17538	−0.587690	+	0.809086i	$0.699962\pi$
$252$	0	0
$253$	−9882.00	−2.45564
$254$	−3450.00	−0.852253
$255$	0	0
$256$	1513.00	0.369385
$257$	−4512.00	−1.09514	−0.547570	−	0.836760i	$-0.684447\pi$
$257$	−4512.00	−1.09514	−0.547570	+	0.836760i	$0.684447\pi$
$258$	0	0
$259$	−2750.00	−0.659756
$260$	132.000	0.0314857
$261$	0	0
$262$	4356.00	1.02715
$263$	−3768.00	−0.883440	−0.441720	−	0.897153i	$-0.645632\pi$
$263$	−3768.00	−0.883440	−0.441720	+	0.897153i	$0.645632\pi$
$264$	0	0
$265$	−5220.00	−1.21005
$266$	627.000	0.144526
$267$	0	0
$268$	−961.000	−0.219039
$269$	−4758.00	−1.07844	−0.539220	−	0.842165i	$-0.681281\pi$
$269$	−4758.00	−1.07844	−0.539220	+	0.842165i	$0.681281\pi$
$270$	0	0
$271$	−2041.00	−0.457498	−0.228749	−	0.973485i	$-0.573463\pi$
$271$	−2041.00	−0.457498	−0.228749	+	0.973485i	$0.573463\pi$
$272$	−6603.00	−1.47193
$273$	0	0
$274$	5067.00	1.11718
$275$	1026.00	0.224982
$276$	0	0
$277$	1964.00	0.426012	0.213006	−	0.977051i	$-0.431675\pi$
$277$	1964.00	0.426012	0.213006	+	0.977051i	$0.431675\pi$
$278$	6432.00	1.38765
$279$	0	0
$280$	−2772.00	−0.591638
$281$	5496.00	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$281$	5496.00	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$282$	0	0
$283$	3098.00	0.650731	0.325366	−	0.945588i	$-0.394513\pi$
$283$	3098.00	0.650731	0.325366	+	0.945588i	$0.394513\pi$
$284$	246.000	0.0513993
$285$	0	0
$286$	1782.00	0.368433
$287$	−2640.00	−0.542977
$288$	0	0
$289$	3736.00	0.760432
$290$	8964.00	1.81512
$291$	0	0
$292$	353.000	0.0707458
$293$	−117.000	−0.0233284	−0.0116642	−	0.999932i	$-0.503713\pi$
$293$	−117.000	−0.0233284	−0.0116642	+	0.999932i	$0.503713\pi$
$294$	0	0
$295$	−2340.00	−0.461831
$296$	5250.00	1.03091
$297$	0	0
$298$	9000.00	1.74952
$299$	−2013.00	−0.389347
$300$	0	0
$301$	−2156.00	−0.412856
$302$	−3018.00	−0.575054
$303$	0	0
$304$	−1349.00	−0.254508
$305$	−4296.00	−0.806519
$306$	0	0
$307$	−1420.00	−0.263986	−0.131993	−	0.991251i	$-0.542138\pi$
$307$	−1420.00	−0.263986	−0.131993	+	0.991251i	$0.542138\pi$
$308$	594.000	0.109891
$309$	0	0
$310$	2016.00	0.369358
$311$	6561.00	1.19627	0.598135	−	0.801395i	$-0.295909\pi$
$311$	6561.00	1.19627	0.598135	+	0.801395i	$0.295909\pi$
$312$	0	0
$313$	−1483.00	−0.267809	−0.133904	−	0.990994i	$-0.542751\pi$
$313$	−1483.00	−0.267809	−0.133904	+	0.990994i	$0.542751\pi$
$314$	8538.00	1.53448
$315$	0	0
$316$	−34.0000	−0.00605269
$317$	1239.00	0.219524	0.109762	−	0.993958i	$-0.464991\pi$
$317$	1239.00	0.219524	0.109762	+	0.993958i	$0.464991\pi$
$318$	0	0
$319$	13446.0	2.35997
$320$	5196.00	0.907704
$321$	0	0
$322$	−6039.00	−1.04516
$323$	1767.00	0.304392
$324$	0	0
$325$	209.000	0.0356715
$326$	−4800.00	−0.815483
$327$	0	0
$328$	5040.00	0.848437
$329$	1848.00	0.309676
$330$	0	0
$331$	−8899.00	−1.47774	−0.738872	−	0.673846i	$-0.764641\pi$
$331$	−8899.00	−1.47774	−0.738872	+	0.673846i	$0.764641\pi$
$332$	−234.000	−0.0386820
$333$	0	0
$334$	6012.00	0.984916
$335$	−11532.0	−1.88078
$336$	0	0
$337$	5816.00	0.940112	0.470056	−	0.882637i	$-0.344234\pi$
$337$	5816.00	0.940112	0.470056	+	0.882637i	$0.344234\pi$
$338$	−6228.00	−1.00224
$339$	0	0
$340$	1116.00	0.178011
$341$	3024.00	0.480231
$342$	0	0
$343$	−6215.00	−0.978363
$344$	4116.00	0.645116
$345$	0	0
$346$	1386.00	0.215352
$347$	1578.00	0.244125	0.122063	−	0.992522i	$-0.461049\pi$
$347$	1578.00	0.244125	0.122063	+	0.992522i	$0.461049\pi$
$348$	0	0
$349$	1658.00	0.254300	0.127150	−	0.991883i	$-0.459417\pi$
$349$	1658.00	0.254300	0.127150	+	0.991883i	$0.459417\pi$
$350$	627.000	0.0957559
$351$	0	0
$352$	−2430.00	−0.367953
$353$	11367.0	1.71389	0.856947	−	0.515405i	$-0.172359\pi$
$353$	11367.0	1.71389	0.856947	+	0.515405i	$0.172359\pi$
$354$	0	0
$355$	2952.00	0.441341
$356$	168.000	0.0250112
$357$	0	0
$358$	−2160.00	−0.318881
$359$	−2553.00	−0.375326	−0.187663	−	0.982233i	$-0.560091\pi$
$359$	−2553.00	−0.375326	−0.187663	+	0.982233i	$0.560091\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−7014.00	−1.01836
$363$	0	0
$364$	121.000	0.0174234
$365$	4236.00	0.607459
$366$	0	0
$367$	−196.000	−0.0278777	−0.0139389	−	0.999903i	$-0.504437\pi$
$367$	−196.000	−0.0278777	−0.0139389	+	0.999903i	$0.504437\pi$
$368$	12993.0	1.84051
$369$	0	0
$370$	−9000.00	−1.26456
$371$	−4785.00	−0.669609
$372$	0	0
$373$	9353.00	1.29834	0.649169	−	0.760644i	$-0.275117\pi$
$373$	9353.00	1.29834	0.649169	+	0.760644i	$0.275117\pi$
$374$	15066.0	2.08301
$375$	0	0
$376$	−3528.00	−0.483890
$377$	2739.00	0.374180
$378$	0	0
$379$	3827.00	0.518680	0.259340	−	0.965786i	$-0.416495\pi$
$379$	3827.00	0.518680	0.259340	+	0.965786i	$0.416495\pi$
$380$	228.000	0.0307794
$381$	0	0
$382$	−8613.00	−1.15361
$383$	−5694.00	−0.759660	−0.379830	−	0.925056i	$-0.624018\pi$
$383$	−5694.00	−0.759660	−0.379830	+	0.925056i	$0.624018\pi$
$384$	0	0
$385$	7128.00	0.943575
$386$	4974.00	0.655881
$387$	0	0
$388$	758.000	0.0991794
$389$	−1290.00	−0.168138	−0.0840689	−	0.996460i	$-0.526792\pi$
$389$	−1290.00	−0.168138	−0.0840689	+	0.996460i	$0.526792\pi$
$390$	0	0
$391$	−17019.0	−2.20125
$392$	4662.00	0.600680
$393$	0	0
$394$	12528.0	1.60191
$395$	−408.000	−0.0519714
$396$	0	0
$397$	6536.00	0.826278	0.413139	−	0.910668i	$-0.364432\pi$
$397$	6536.00	0.826278	0.413139	+	0.910668i	$0.364432\pi$
$398$	−723.000	−0.0910571
$399$	0	0
$400$	−1349.00	−0.168625
$401$	−2328.00	−0.289912	−0.144956	−	0.989438i	$-0.546304\pi$
$401$	−2328.00	−0.289912	−0.144956	+	0.989438i	$0.546304\pi$
$402$	0	0
$403$	616.000	0.0761418
$404$	726.000	0.0894056
$405$	0	0
$406$	8217.00	1.00444
$407$	−13500.0	−1.64415
$408$	0	0
$409$	−6676.00	−0.807107	−0.403554	−	0.914956i	$-0.632225\pi$
$409$	−6676.00	−0.807107	−0.403554	+	0.914956i	$0.632225\pi$
$410$	−8640.00	−1.04073
$411$	0	0
$412$	2.00000	0.000239158 0
$413$	−2145.00	−0.255565
$414$	0	0
$415$	−2808.00	−0.332143
$416$	−495.000	−0.0583398
$417$	0	0
$418$	3078.00	0.360167
$419$	8136.00	0.948615	0.474307	−	0.880359i	$-0.342699\pi$
$419$	8136.00	0.948615	0.474307	+	0.880359i	$0.342699\pi$
$420$	0	0
$421$	−8665.00	−1.00310	−0.501551	−	0.865128i	$-0.667237\pi$
$421$	−8665.00	−1.00310	−0.501551	+	0.865128i	$0.667237\pi$
$422$	−2235.00	−0.257815
$423$	0	0
$424$	9135.00	1.04631
$425$	1767.00	0.201676
$426$	0	0
$427$	−3938.00	−0.446307
$428$	−1413.00	−0.159579
$429$	0	0
$430$	−7056.00	−0.791327
$431$	−750.000	−0.0838196	−0.0419098	−	0.999121i	$-0.513344\pi$
$431$	−750.000	−0.0838196	−0.0419098	+	0.999121i	$0.513344\pi$
$432$	0	0
$433$	−4858.00	−0.539170	−0.269585	−	0.962977i	$-0.586887\pi$
$433$	−4858.00	−0.539170	−0.269585	+	0.962977i	$0.586887\pi$
$434$	1848.00	0.204394
$435$	0	0
$436$	389.000	0.0427287
$437$	−3477.00	−0.380612
$438$	0	0
$439$	6500.00	0.706670	0.353335	−	0.935497i	$-0.385048\pi$
$439$	6500.00	0.706670	0.353335	+	0.935497i	$0.385048\pi$
$440$	−13608.0	−1.47440
$441$	0	0
$442$	3069.00	0.330266
$443$	−3486.00	−0.373871	−0.186936	−	0.982372i	$-0.559856\pi$
$443$	−3486.00	−0.373871	−0.186936	+	0.982372i	$0.559856\pi$
$444$	0	0
$445$	2016.00	0.214759
$446$	−5934.00	−0.630007
$447$	0	0
$448$	4763.00	0.502300
$449$	15030.0	1.57975	0.789877	−	0.613265i	$-0.210144\pi$
$449$	15030.0	1.57975	0.789877	+	0.613265i	$0.210144\pi$
$450$	0	0
$451$	−12960.0	−1.35313
$452$	−342.000	−0.0355892
$453$	0	0
$454$	−16065.0	−1.66072
$455$	1452.00	0.149606
$456$	0	0
$457$	−2959.00	−0.302880	−0.151440	−	0.988466i	$-0.548391\pi$
$457$	−2959.00	−0.302880	−0.151440	+	0.988466i	$0.548391\pi$
$458$	−19110.0	−1.94968
$459$	0	0
$460$	−2196.00	−0.222585
$461$	156.000	0.0157606	0.00788031	−	0.999969i	$-0.497492\pi$
$461$	156.000	0.0157606	0.00788031	+	0.999969i	$0.497492\pi$
$462$	0	0
$463$	4484.00	0.450085	0.225042	−	0.974349i	$-0.427748\pi$
$463$	4484.00	0.450085	0.225042	+	0.974349i	$0.427748\pi$
$464$	−17679.0	−1.76881
$465$	0	0
$466$	8514.00	0.846359
$467$	−8766.00	−0.868613	−0.434306	−	0.900765i	$-0.643006\pi$
$467$	−8766.00	−0.868613	−0.434306	+	0.900765i	$0.643006\pi$
$468$	0	0
$469$	−10571.0	−1.04077
$470$	6048.00	0.593561
$471$	0	0
$472$	4095.00	0.399338
$473$	−10584.0	−1.02886
$474$	0	0
$475$	361.000	0.0348712
$476$	1023.00	0.0985066
$477$	0	0
$478$	1107.00	0.105927
$479$	18996.0	1.81200	0.906001	−	0.423275i	$-0.139119\pi$
$479$	18996.0	1.81200	0.906001	+	0.423275i	$0.139119\pi$
$480$	0	0
$481$	−2750.00	−0.260684
$482$	19824.0	1.87336
$483$	0	0
$484$	1585.00	0.148854
$485$	9096.00	0.851604
$486$	0	0
$487$	−7450.00	−0.693207	−0.346603	−	0.938012i	$-0.612665\pi$
$487$	−7450.00	−0.693207	−0.346603	+	0.938012i	$0.612665\pi$
$488$	7518.00	0.697385
$489$	0	0
$490$	−7992.00	−0.736820
$491$	−6180.00	−0.568023	−0.284012	−	0.958821i	$-0.591665\pi$
$491$	−6180.00	−0.568023	−0.284012	+	0.958821i	$0.591665\pi$
$492$	0	0
$493$	23157.0	2.11549
$494$	627.000	0.0571054
$495$	0	0
$496$	−3976.00	−0.359935
$497$	2706.00	0.244227
$498$	0	0
$499$	2576.00	0.231097	0.115549	−	0.993302i	$-0.463137\pi$
$499$	2576.00	0.231097	0.115549	+	0.993302i	$0.463137\pi$
$500$	−1272.00	−0.113771
$501$	0	0
$502$	−14022.0	−1.24668
$503$	10545.0	0.934748	0.467374	−	0.884060i	$-0.345200\pi$
$503$	10545.0	0.934748	0.467374	+	0.884060i	$0.345200\pi$
$504$	0	0
$505$	8712.00	0.767681
$506$	−29646.0	−2.60460
$507$	0	0
$508$	−1150.00	−0.100439
$509$	14694.0	1.27957	0.639784	−	0.768555i	$-0.279024\pi$
$509$	14694.0	1.27957	0.639784	+	0.768555i	$0.279024\pi$
$510$	0	0
$511$	3883.00	0.336152
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−13536.0	−1.16157
$515$	24.0000	0.00205353
$516$	0	0
$517$	9072.00	0.771733
$518$	−8250.00	−0.699776
$519$	0	0
$520$	−2772.00	−0.233770
$521$	−10332.0	−0.868816	−0.434408	−	0.900716i	$-0.643042\pi$
$521$	−10332.0	−0.868816	−0.434408	+	0.900716i	$0.643042\pi$
$522$	0	0
$523$	10937.0	0.914420	0.457210	−	0.889359i	$-0.348849\pi$
$523$	10937.0	0.914420	0.457210	+	0.889359i	$0.348849\pi$
$524$	1452.00	0.121051
$525$	0	0
$526$	−11304.0	−0.937030
$527$	5208.00	0.430482
$528$	0	0
$529$	21322.0	1.75245
$530$	−15660.0	−1.28345
$531$	0	0
$532$	209.000	0.0170325
$533$	−2640.00	−0.214542
$534$	0	0
$535$	−16956.0	−1.37023
$536$	20181.0	1.62628
$537$	0	0
$538$	−14274.0	−1.14386
$539$	−11988.0	−0.957996
$540$	0	0
$541$	18578.0	1.47640	0.738198	−	0.674584i	$-0.235677\pi$
$541$	18578.0	1.47640	0.738198	+	0.674584i	$0.235677\pi$
$542$	−6123.00	−0.485250
$543$	0	0
$544$	−4185.00	−0.329835
$545$	4668.00	0.366890
$546$	0	0
$547$	21404.0	1.67307	0.836535	−	0.547914i	$-0.184578\pi$
$547$	21404.0	1.67307	0.836535	+	0.547914i	$0.184578\pi$
$548$	1689.00	0.131662
$549$	0	0
$550$	3078.00	0.238630
$551$	4731.00	0.365785
$552$	0	0
$553$	−374.000	−0.0287597
$554$	5892.00	0.451854
$555$	0	0
$556$	2144.00	0.163536
$557$	3948.00	0.300327	0.150163	−	0.988661i	$-0.452020\pi$
$557$	3948.00	0.300327	0.150163	+	0.988661i	$0.452020\pi$
$558$	0	0
$559$	−2156.00	−0.163129
$560$	−9372.00	−0.707213
$561$	0	0
$562$	16488.0	1.23755
$563$	−5724.00	−0.428486	−0.214243	−	0.976780i	$-0.568729\pi$
$563$	−5724.00	−0.428486	−0.214243	+	0.976780i	$0.568729\pi$
$564$	0	0
$565$	−4104.00	−0.305587
$566$	9294.00	0.690205
$567$	0	0
$568$	−5166.00	−0.381621
$569$	20592.0	1.51716	0.758578	−	0.651582i	$-0.225895\pi$
$569$	20592.0	1.51716	0.758578	+	0.651582i	$0.225895\pi$
$570$	0	0
$571$	20684.0	1.51593	0.757967	−	0.652293i	$-0.226193\pi$
$571$	20684.0	1.51593	0.757967	+	0.652293i	$0.226193\pi$
$572$	594.000	0.0434203
$573$	0	0
$574$	−7920.00	−0.575914
$575$	−3477.00	−0.252176
$576$	0	0
$577$	−19573.0	−1.41219	−0.706096	−	0.708116i	$-0.749545\pi$
$577$	−19573.0	−1.41219	−0.706096	+	0.708116i	$0.749545\pi$
$578$	11208.0	0.806559
$579$	0	0
$580$	2988.00	0.213914
$581$	−2574.00	−0.183800
$582$	0	0
$583$	−23490.0	−1.66871
$584$	−7413.00	−0.525260
$585$	0	0
$586$	−351.000	−0.0247435
$587$	−13524.0	−0.950929	−0.475464	−	0.879735i	$-0.657720\pi$
$587$	−13524.0	−0.950929	−0.475464	+	0.879735i	$0.657720\pi$
$588$	0	0
$589$	1064.00	0.0744335
$590$	−7020.00	−0.489845
$591$	0	0
$592$	17750.0	1.23230
$593$	−8994.00	−0.622832	−0.311416	−	0.950274i	$-0.600803\pi$
$593$	−8994.00	−0.622832	−0.311416	+	0.950274i	$0.600803\pi$
$594$	0	0
$595$	12276.0	0.845827
$596$	3000.00	0.206183
$597$	0	0
$598$	−6039.00	−0.412965
$599$	−10128.0	−0.690850	−0.345425	−	0.938446i	$-0.612265\pi$
$599$	−10128.0	−0.690850	−0.345425	+	0.938446i	$0.612265\pi$
$600$	0	0
$601$	−22696.0	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$601$	−22696.0	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$602$	−6468.00	−0.437900
$603$	0	0
$604$	−1006.00	−0.0677708
$605$	19020.0	1.27814
$606$	0	0
$607$	−5182.00	−0.346509	−0.173254	−	0.984877i	$-0.555428\pi$
$607$	−5182.00	−0.346509	−0.173254	+	0.984877i	$0.555428\pi$
$608$	−855.000	−0.0570310
$609$	0	0
$610$	−12888.0	−0.855442
$611$	1848.00	0.122360
$612$	0	0
$613$	10082.0	0.664287	0.332144	−	0.943229i	$-0.392228\pi$
$613$	10082.0	0.664287	0.332144	+	0.943229i	$0.392228\pi$
$614$	−4260.00	−0.279999
$615$	0	0
$616$	−12474.0	−0.815896
$617$	12174.0	0.794338	0.397169	−	0.917745i	$-0.369993\pi$
$617$	12174.0	0.794338	0.397169	+	0.917745i	$0.369993\pi$
$618$	0	0
$619$	7490.00	0.486347	0.243173	−	0.969983i	$-0.421812\pi$
$619$	7490.00	0.486347	0.243173	+	0.969983i	$0.421812\pi$
$620$	672.000	0.0435293
$621$	0	0
$622$	19683.0	1.26884
$623$	1848.00	0.118842
$624$	0	0
$625$	−17639.0	−1.12890
$626$	−4449.00	−0.284054
$627$	0	0
$628$	2846.00	0.180840
$629$	−23250.0	−1.47383
$630$	0	0
$631$	11072.0	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$631$	11072.0	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$632$	714.000	0.0449389
$633$	0	0
$634$	3717.00	0.232841
$635$	−13800.0	−0.862419
$636$	0	0
$637$	−2442.00	−0.151893
$638$	40338.0	2.50313
$639$	0	0
$640$	19908.0	1.22958
$641$	18894.0	1.16422	0.582112	−	0.813108i	$-0.302226\pi$
$641$	18894.0	1.16422	0.582112	+	0.813108i	$0.302226\pi$
$642$	0	0
$643$	−19834.0	−1.21645	−0.608224	−	0.793765i	$-0.708118\pi$
$643$	−19834.0	−1.21645	−0.608224	+	0.793765i	$0.708118\pi$
$644$	−2013.00	−0.123173
$645$	0	0
$646$	5301.00	0.322856
$647$	−3375.00	−0.205077	−0.102539	−	0.994729i	$-0.532697\pi$
$647$	−3375.00	−0.205077	−0.102539	+	0.994729i	$0.532697\pi$
$648$	0	0
$649$	−10530.0	−0.636885
$650$	627.000	0.0378353
$651$	0	0
$652$	−1600.00	−0.0961056
$653$	24948.0	1.49509	0.747543	−	0.664214i	$-0.231234\pi$
$653$	24948.0	1.49509	0.747543	+	0.664214i	$0.231234\pi$
$654$	0	0
$655$	17424.0	1.03941
$656$	17040.0	1.01418
$657$	0	0
$658$	5544.00	0.328461
$659$	9879.00	0.583962	0.291981	−	0.956424i	$-0.405686\pi$
$659$	9879.00	0.583962	0.291981	+	0.956424i	$0.405686\pi$
$660$	0	0
$661$	−14155.0	−0.832928	−0.416464	−	0.909152i	$-0.636731\pi$
$661$	−14155.0	−0.832928	−0.416464	+	0.909152i	$0.636731\pi$
$662$	−26697.0	−1.56738
$663$	0	0
$664$	4914.00	0.287199
$665$	2508.00	0.146250
$666$	0	0
$667$	−45567.0	−2.64522
$668$	2004.00	0.116073
$669$	0	0
$670$	−34596.0	−1.99487
$671$	−19332.0	−1.11223
$672$	0	0
$673$	8948.00	0.512511	0.256256	−	0.966609i	$-0.417511\pi$
$673$	8948.00	0.512511	0.256256	+	0.966609i	$0.417511\pi$
$674$	17448.0	0.997139
$675$	0	0
$676$	−2076.00	−0.118116
$677$	11511.0	0.653477	0.326738	−	0.945115i	$-0.394050\pi$
$677$	11511.0	0.653477	0.326738	+	0.945115i	$0.394050\pi$
$678$	0	0
$679$	8338.00	0.471256
$680$	−23436.0	−1.32166
$681$	0	0
$682$	9072.00	0.509362
$683$	10476.0	0.586900	0.293450	−	0.955974i	$-0.405197\pi$
$683$	10476.0	0.586900	0.293450	+	0.955974i	$0.405197\pi$
$684$	0	0
$685$	20268.0	1.13051
$686$	−18645.0	−1.03771
$687$	0	0
$688$	13916.0	0.771137
$689$	−4785.00	−0.264578
$690$	0	0
$691$	30098.0	1.65699	0.828496	−	0.559995i	$-0.189197\pi$
$691$	30098.0	1.65699	0.828496	+	0.559995i	$0.189197\pi$
$692$	462.000	0.0253795
$693$	0	0
$694$	4734.00	0.258934
$695$	25728.0	1.40420
$696$	0	0
$697$	−22320.0	−1.21296
$698$	4974.00	0.269726
$699$	0	0
$700$	209.000	0.0112849
$701$	14700.0	0.792028	0.396014	−	0.918245i	$-0.370393\pi$
$701$	14700.0	0.792028	0.396014	+	0.918245i	$0.370393\pi$
$702$	0	0
$703$	−4750.00	−0.254836
$704$	23382.0	1.25176
$705$	0	0
$706$	34101.0	1.81786
$707$	7986.00	0.424815
$708$	0	0
$709$	31178.0	1.65150	0.825751	−	0.564035i	$-0.190752\pi$
$709$	31178.0	1.65150	0.825751	+	0.564035i	$0.190752\pi$
$710$	8856.00	0.468112
$711$	0	0
$712$	−3528.00	−0.185699
$713$	−10248.0	−0.538276
$714$	0	0
$715$	7128.00	0.372828
$716$	−720.000	−0.0375805
$717$	0	0
$718$	−7659.00	−0.398094
$719$	33285.0	1.72645	0.863227	−	0.504815i	$-0.168439\pi$
$719$	33285.0	1.72645	0.863227	+	0.504815i	$0.168439\pi$
$720$	0	0
$721$	22.0000	0.00113637
$722$	1083.00	0.0558242
$723$	0	0
$724$	−2338.00	−0.120015
$725$	4731.00	0.242352
$726$	0	0
$727$	−34729.0	−1.77170	−0.885851	−	0.463970i	$-0.846425\pi$
$727$	−34729.0	−1.77170	−0.885851	+	0.463970i	$0.846425\pi$
$728$	−2541.00	−0.129362
$729$	0	0
$730$	12708.0	0.644307
$731$	−18228.0	−0.922280
$732$	0	0
$733$	4196.00	0.211436	0.105718	−	0.994396i	$-0.466286\pi$
$733$	4196.00	0.211436	0.105718	+	0.994396i	$0.466286\pi$
$734$	−588.000	−0.0295688
$735$	0	0
$736$	8235.00	0.412427
$737$	−51894.0	−2.59368
$738$	0	0
$739$	−10744.0	−0.534810	−0.267405	−	0.963584i	$-0.586166\pi$
$739$	−10744.0	−0.534810	−0.267405	+	0.963584i	$0.586166\pi$
$740$	−3000.00	−0.149030
$741$	0	0
$742$	−14355.0	−0.710227
$743$	2208.00	0.109022	0.0545112	−	0.998513i	$-0.482640\pi$
$743$	2208.00	0.109022	0.0545112	+	0.998513i	$0.482640\pi$
$744$	0	0
$745$	36000.0	1.77039
$746$	28059.0	1.37710
$747$	0	0
$748$	5022.00	0.245485
$749$	−15543.0	−0.758249
$750$	0	0
$751$	13160.0	0.639434	0.319717	−	0.947513i	$-0.396412\pi$
$751$	13160.0	0.639434	0.319717	+	0.947513i	$0.396412\pi$
$752$	−11928.0	−0.578417
$753$	0	0
$754$	8217.00	0.396877
$755$	−12072.0	−0.581914
$756$	0	0
$757$	758.000	0.0363936	0.0181968	−	0.999834i	$-0.494207\pi$
$757$	758.000	0.0363936	0.0181968	+	0.999834i	$0.494207\pi$
$758$	11481.0	0.550143
$759$	0	0
$760$	−4788.00	−0.228525
$761$	−4851.00	−0.231076	−0.115538	−	0.993303i	$-0.536859\pi$
$761$	−4851.00	−0.231076	−0.115538	+	0.993303i	$0.536859\pi$
$762$	0	0
$763$	4279.00	0.203028
$764$	−2871.00	−0.135954
$765$	0	0
$766$	−17082.0	−0.805741
$767$	−2145.00	−0.100980
$768$	0	0
$769$	−33091.0	−1.55175	−0.775873	−	0.630890i	$-0.782690\pi$
$769$	−33091.0	−1.55175	−0.775873	+	0.630890i	$0.782690\pi$
$770$	21384.0	1.00081
$771$	0	0
$772$	1658.00	0.0772963
$773$	−42357.0	−1.97086	−0.985430	−	0.170079i	$-0.945598\pi$
$773$	−42357.0	−1.97086	−0.985430	+	0.170079i	$0.945598\pi$
$774$	0	0
$775$	1064.00	0.0493161
$776$	−15918.0	−0.736370
$777$	0	0
$778$	−3870.00	−0.178337
$779$	−4560.00	−0.209729
$780$	0	0
$781$	13284.0	0.608629
$782$	−51057.0	−2.33478
$783$	0	0
$784$	15762.0	0.718021
$785$	34152.0	1.55279
$786$	0	0
$787$	−39877.0	−1.80618	−0.903089	−	0.429454i	$-0.858706\pi$
$787$	−39877.0	−1.80618	−0.903089	+	0.429454i	$0.858706\pi$
$788$	4176.00	0.188787
$789$	0	0
$790$	−1224.00	−0.0551240
$791$	−3762.00	−0.169104
$792$	0	0
$793$	−3938.00	−0.176346
$794$	19608.0	0.876400
$795$	0	0
$796$	−241.000	−0.0107312
$797$	30033.0	1.33478	0.667392	−	0.744706i	$-0.267410\pi$
$797$	30033.0	1.33478	0.667392	+	0.744706i	$0.267410\pi$
$798$	0	0
$799$	15624.0	0.691786
$800$	−855.000	−0.0377860
$801$	0	0
$802$	−6984.00	−0.307498
$803$	19062.0	0.837713
$804$	0	0
$805$	−24156.0	−1.05762
$806$	1848.00	0.0807606
$807$	0	0
$808$	−15246.0	−0.663802
$809$	−585.000	−0.0254234	−0.0127117	−	0.999919i	$-0.504046\pi$
$809$	−585.000	−0.0254234	−0.0127117	+	0.999919i	$0.504046\pi$
$810$	0	0
$811$	28361.0	1.22798	0.613989	−	0.789315i	$-0.289564\pi$
$811$	28361.0	1.22798	0.613989	+	0.789315i	$0.289564\pi$
$812$	2739.00	0.118374
$813$	0	0
$814$	−40500.0	−1.74389
$815$	−19200.0	−0.825211
$816$	0	0
$817$	−3724.00	−0.159469
$818$	−20028.0	−0.856067
$819$	0	0
$820$	−2880.00	−0.122651
$821$	−25068.0	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$821$	−25068.0	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$822$	0	0
$823$	10901.0	0.461707	0.230854	−	0.972989i	$-0.425848\pi$
$823$	10901.0	0.461707	0.230854	+	0.972989i	$0.425848\pi$
$824$	−42.0000	−0.00177565
$825$	0	0
$826$	−6435.00	−0.271068
$827$	−12027.0	−0.505707	−0.252854	−	0.967505i	$-0.581369\pi$
$827$	−12027.0	−0.505707	−0.252854	+	0.967505i	$0.581369\pi$
$828$	0	0
$829$	−19339.0	−0.810219	−0.405109	−	0.914268i	$-0.632767\pi$
$829$	−19339.0	−0.810219	−0.405109	+	0.914268i	$0.632767\pi$
$830$	−8424.00	−0.352291
$831$	0	0
$832$	4763.00	0.198470
$833$	−20646.0	−0.858753
$834$	0	0
$835$	24048.0	0.996665
$836$	1026.00	0.0424461
$837$	0	0
$838$	24408.0	1.00616
$839$	13188.0	0.542670	0.271335	−	0.962485i	$-0.412535\pi$
$839$	13188.0	0.542670	0.271335	+	0.962485i	$0.412535\pi$
$840$	0	0
$841$	37612.0	1.54217
$842$	−25995.0	−1.06395
$843$	0	0
$844$	−745.000	−0.0303838
$845$	−24912.0	−1.01420
$846$	0	0
$847$	17435.0	0.707289
$848$	30885.0	1.25070
$849$	0	0
$850$	5301.00	0.213909
$851$	45750.0	1.84288
$852$	0	0
$853$	−4678.00	−0.187775	−0.0938873	−	0.995583i	$-0.529929\pi$
$853$	−4678.00	−0.187775	−0.0938873	+	0.995583i	$0.529929\pi$
$854$	−11814.0	−0.473380
$855$	0	0
$856$	29673.0	1.18482
$857$	−15252.0	−0.607933	−0.303966	−	0.952683i	$-0.598311\pi$
$857$	−15252.0	−0.607933	−0.303966	+	0.952683i	$0.598311\pi$
$858$	0	0
$859$	−610.000	−0.0242293	−0.0121146	−	0.999927i	$-0.503856\pi$
$859$	−610.000	−0.0242293	−0.0121146	+	0.999927i	$0.503856\pi$
$860$	−2352.00	−0.0932588
$861$	0	0
$862$	−2250.00	−0.0889041
$863$	−774.000	−0.0305299	−0.0152649	−	0.999883i	$-0.504859\pi$
$863$	−774.000	−0.0305299	−0.0152649	+	0.999883i	$0.504859\pi$
$864$	0	0
$865$	5544.00	0.217921
$866$	−14574.0	−0.571876
$867$	0	0
$868$	616.000	0.0240880
$869$	−1836.00	−0.0716709
$870$	0	0
$871$	−10571.0	−0.411234
$872$	−8169.00	−0.317245
$873$	0	0
$874$	−10431.0	−0.403700
$875$	−13992.0	−0.540590
$876$	0	0
$877$	−31039.0	−1.19511	−0.597556	−	0.801827i	$-0.703861\pi$
$877$	−31039.0	−1.19511	−0.597556	+	0.801827i	$0.703861\pi$
$878$	19500.0	0.749537
$879$	0	0
$880$	−46008.0	−1.76242
$881$	−33678.0	−1.28790	−0.643950	−	0.765067i	$-0.722706\pi$
$881$	−33678.0	−1.28790	−0.643950	+	0.765067i	$0.722706\pi$
$882$	0	0
$883$	−42982.0	−1.63812	−0.819060	−	0.573708i	$-0.805504\pi$
$883$	−42982.0	−1.63812	−0.819060	+	0.573708i	$0.805504\pi$
$884$	1023.00	0.0389222
$885$	0	0
$886$	−10458.0	−0.396550
$887$	−4494.00	−0.170117	−0.0850585	−	0.996376i	$-0.527108\pi$
$887$	−4494.00	−0.170117	−0.0850585	+	0.996376i	$0.527108\pi$
$888$	0	0
$889$	−12650.0	−0.477241
$890$	6048.00	0.227786
$891$	0	0
$892$	−1978.00	−0.0742470
$893$	3192.00	0.119615
$894$	0	0
$895$	−8640.00	−0.322685
$896$	18249.0	0.680420
$897$	0	0
$898$	45090.0	1.67558
$899$	13944.0	0.517306
$900$	0	0
$901$	−40455.0	−1.49584
$902$	−38880.0	−1.43521
$903$	0	0
$904$	7182.00	0.264236
$905$	−28056.0	−1.03051
$906$	0	0
$907$	−23839.0	−0.872724	−0.436362	−	0.899771i	$-0.643733\pi$
$907$	−23839.0	−0.872724	−0.436362	+	0.899771i	$0.643733\pi$
$908$	−5355.00	−0.195718
$909$	0	0
$910$	4356.00	0.158681
$911$	10332.0	0.375757	0.187878	−	0.982192i	$-0.439839\pi$
$911$	10332.0	0.375757	0.187878	+	0.982192i	$0.439839\pi$
$912$	0	0
$913$	−12636.0	−0.458040
$914$	−8877.00	−0.321253
$915$	0	0
$916$	−6370.00	−0.229772
$917$	15972.0	0.575182
$918$	0	0
$919$	−14371.0	−0.515838	−0.257919	−	0.966166i	$-0.583037\pi$
$919$	−14371.0	−0.515838	−0.257919	+	0.966166i	$0.583037\pi$
$920$	46116.0	1.65261
$921$	0	0
$922$	468.000	0.0167167
$923$	2706.00	0.0964995
$924$	0	0
$925$	−4750.00	−0.168842
$926$	13452.0	0.477387
$927$	0	0
$928$	−11205.0	−0.396360
$929$	−26889.0	−0.949623	−0.474811	−	0.880088i	$-0.657484\pi$
$929$	−26889.0	−0.949623	−0.474811	+	0.880088i	$0.657484\pi$
$930$	0	0
$931$	−4218.00	−0.148485
$932$	2838.00	0.0997444
$933$	0	0
$934$	−26298.0	−0.921303
$935$	60264.0	2.10785
$936$	0	0
$937$	785.000	0.0273691	0.0136845	−	0.999906i	$-0.495644\pi$
$937$	785.000	0.0273691	0.0136845	+	0.999906i	$0.495644\pi$
$938$	−31713.0	−1.10391
$939$	0	0
$940$	2016.00	0.0699518
$941$	18141.0	0.628459	0.314229	−	0.949347i	$-0.398254\pi$
$941$	18141.0	0.628459	0.314229	+	0.949347i	$0.398254\pi$
$942$	0	0
$943$	43920.0	1.51668
$944$	13845.0	0.477348
$945$	0	0
$946$	−31752.0	−1.09128
$947$	−23100.0	−0.792660	−0.396330	−	0.918108i	$-0.629716\pi$
$947$	−23100.0	−0.792660	−0.396330	+	0.918108i	$0.629716\pi$
$948$	0	0
$949$	3883.00	0.132821
$950$	1083.00	0.0369865
$951$	0	0
$952$	−21483.0	−0.731374
$953$	−45690.0	−1.55304	−0.776519	−	0.630094i	$-0.783016\pi$
$953$	−45690.0	−1.55304	−0.776519	+	0.630094i	$0.783016\pi$
$954$	0	0
$955$	−34452.0	−1.16737
$956$	369.000	0.0124836
$957$	0	0
$958$	56988.0	1.92192
$959$	18579.0	0.625597
$960$	0	0
$961$	−26655.0	−0.894733
$962$	−8250.00	−0.276498
$963$	0	0
$964$	6608.00	0.220777
$965$	19896.0	0.663705
$966$	0	0
$967$	21584.0	0.717781	0.358891	−	0.933380i	$-0.383155\pi$
$967$	21584.0	0.717781	0.358891	+	0.933380i	$0.383155\pi$
$968$	−33285.0	−1.10519
$969$	0	0
$970$	27288.0	0.903263
$971$	50556.0	1.67087	0.835437	−	0.549586i	$-0.185214\pi$
$971$	50556.0	1.67087	0.835437	+	0.549586i	$0.185214\pi$
$972$	0	0
$973$	23584.0	0.777049
$974$	−22350.0	−0.735257
$975$	0	0
$976$	25418.0	0.833617
$977$	−8568.00	−0.280568	−0.140284	−	0.990111i	$-0.544802\pi$
$977$	−8568.00	−0.280568	−0.140284	+	0.990111i	$0.544802\pi$
$978$	0	0
$979$	9072.00	0.296162
$980$	−2664.00	−0.0868351
$981$	0	0
$982$	−18540.0	−0.602480
$983$	−29706.0	−0.963860	−0.481930	−	0.876210i	$-0.660064\pi$
$983$	−29706.0	−0.963860	−0.481930	+	0.876210i	$0.660064\pi$
$984$	0	0
$985$	50112.0	1.62102
$986$	69471.0	2.24382
$987$	0	0
$988$	209.000	0.00672993
$989$	35868.0	1.15322
$990$	0	0
$991$	30512.0	0.978048	0.489024	−	0.872270i	$-0.337353\pi$
$991$	30512.0	0.978048	0.489024	+	0.872270i	$0.337353\pi$
$992$	−2520.00	−0.0806553
$993$	0	0
$994$	8118.00	0.259042
$995$	−2892.00	−0.0921433
$996$	0	0
$997$	47756.0	1.51700	0.758499	−	0.651674i	$-0.225933\pi$
$997$	47756.0	1.51700	0.758499	+	0.651674i	$0.225933\pi$
$998$	7728.00	0.245116
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.4.a.d.1.1		1
3.2	odd	2		19.4.a.a.1.1	✓	1
12.11	even	2		304.4.a.b.1.1		1
15.2	even	4		475.4.b.c.324.1		2
15.8	even	4		475.4.b.c.324.2		2
15.14	odd	2		475.4.a.e.1.1		1
21.20	even	2		931.4.a.a.1.1		1
24.5	odd	2		1216.4.a.f.1.1		1
24.11	even	2		1216.4.a.a.1.1		1
33.32	even	2		2299.4.a.b.1.1		1
57.56	even	2		361.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.a.1.1	✓	1	3.2	odd	2
171.4.a.d.1.1		1	1.1	even	1	trivial
304.4.a.b.1.1		1	12.11	even	2
361.4.a.b.1.1		1	57.56	even	2
475.4.a.e.1.1		1	15.14	odd	2
475.4.b.c.324.1		2	15.2	even	4
475.4.b.c.324.2		2	15.8	even	4
931.4.a.a.1.1		1	21.20	even	2
1216.4.a.a.1.1		1	24.11	even	2
1216.4.a.f.1.1		1	24.5	odd	2
2299.4.a.b.1.1		1	33.32	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more