LMFDB - Embedded newform 220.4.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,4,Mod(1,220)] 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("220.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	220.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.9804202013$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.9192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 30 $ x^3 - x^2 - 18*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.67648$ of defining polynomial
Character	$\chi$	$=$	220.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.86946 q^{3} -5.00000 q^{5} +15.2554 q^{7} +20.1894 q^{9} +11.0000 q^{11} +30.1909 q^{13} -34.3473 q^{15} +74.4793 q^{17} +30.6126 q^{19} +104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} +23.5336 q^{29} -272.083 q^{31} +75.5640 q^{33} -76.2770 q^{35} +292.415 q^{37} +207.395 q^{39} -127.493 q^{41} -466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} +511.632 q^{51} +235.761 q^{53} -55.0000 q^{55} +210.292 q^{57} +167.162 q^{59} -363.291 q^{61} +307.998 q^{63} -150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} -372.861 q^{73} +171.736 q^{75} +167.809 q^{77} -300.829 q^{79} -866.502 q^{81} -1218.18 q^{83} -372.396 q^{85} +161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} -153.063 q^{95} -1389.82 q^{97} +222.084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 15 q^{5} - 5 q^{7} + 24 q^{9} + 33 q^{11} + 2 q^{13} + 15 q^{15} + 77 q^{17} + 171 q^{19} + 259 q^{21} + 222 q^{23} + 75 q^{25} + 111 q^{27} + 55 q^{29} + 181 q^{31} - 33 q^{33} + 25 q^{35}+ \cdots + 264 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 15 * q^5 - 5 * q^7 + 24 * q^9 + 33 * q^11 + 2 * q^13 + 15 * q^15 + 77 * q^17 + 171 * q^19 + 259 * q^21 + 222 * q^23 + 75 * q^25 + 111 * q^27 + 55 * q^29 + 181 * q^31 - 33 * q^33 + 25 * q^35 + 317 * q^37 + 554 * q^39 + 302 * q^41 - 188 * q^43 - 120 * q^45 + 662 * q^47 - 268 * q^49 + 833 * q^51 + 81 * q^53 - 165 * q^55 - 825 * q^57 - 42 * q^59 + 349 * q^61 - 266 * q^63 - 10 * q^65 + 152 * q^67 - 258 * q^69 + 927 * q^71 - 2378 * q^73 - 75 * q^75 - 55 * q^77 - 1146 * q^79 - 1533 * q^81 - 458 * q^83 - 385 * q^85 + 1087 * q^87 - 875 * q^89 + 1982 * q^91 - 3623 * q^93 - 855 * q^95 - 1980 * q^97 + 264 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	6.86946	1.32203	0.661014	−	0.750374i	$-0.270127\pi$
$3$	6.86946	1.32203	0.661014	+	0.750374i	$0.270127\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	15.2554	0.823715	0.411857	−	0.911248i	$-0.364880\pi$
$7$	15.2554	0.823715	0.411857	+	0.911248i	$0.364880\pi$
$8$	0	0
$9$	20.1894	0.747756
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	30.1909	0.644110	0.322055	−	0.946721i	$-0.395626\pi$
$13$	30.1909	0.644110	0.322055	+	0.946721i	$0.395626\pi$
$14$	0	0
$15$	−34.3473	−0.591229
$16$	0	0
$17$	74.4793	1.06258	0.531290	−	0.847190i	$-0.321707\pi$
$17$	74.4793	1.06258	0.531290	+	0.847190i	$0.321707\pi$
$18$	0	0
$19$	30.6126	0.369632	0.184816	−	0.982773i	$-0.440831\pi$
$19$	30.6126	0.369632	0.184816	+	0.982773i	$0.440831\pi$
$20$	0	0
$21$	104.796	1.08897
$22$	0	0
$23$	111.339	1.00938	0.504690	−	0.863301i	$-0.331607\pi$
$23$	111.339	1.00938	0.504690	+	0.863301i	$0.331607\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−46.7850	−0.333473
$28$	0	0
$29$	23.5336	0.150692	0.0753462	−	0.997157i	$-0.475994\pi$
$29$	23.5336	0.150692	0.0753462	+	0.997157i	$0.475994\pi$
$30$	0	0
$31$	−272.083	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$31$	−272.083	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$32$	0	0
$33$	75.5640	0.398606
$34$	0	0
$35$	−76.2770	−0.368376
$36$	0	0
$37$	292.415	1.29926	0.649632	−	0.760249i	$-0.274923\pi$
$37$	292.415	1.29926	0.649632	+	0.760249i	$0.274923\pi$
$38$	0	0
$39$	207.395	0.851532
$40$	0	0
$41$	−127.493	−0.485634	−0.242817	−	0.970072i	$-0.578071\pi$
$41$	−127.493	−0.485634	−0.242817	+	0.970072i	$0.578071\pi$
$42$	0	0
$43$	−466.210	−1.65341	−0.826703	−	0.562639i	$-0.809786\pi$
$43$	−466.210	−1.65341	−0.826703	+	0.562639i	$0.809786\pi$
$44$	0	0
$45$	−100.947	−0.334407
$46$	0	0
$47$	430.418	1.33581	0.667903	−	0.744248i	$-0.267192\pi$
$47$	430.418	1.33581	0.667903	+	0.744248i	$0.267192\pi$
$48$	0	0
$49$	−110.273	−0.321494
$50$	0	0
$51$	511.632	1.40476
$52$	0	0
$53$	235.761	0.611024	0.305512	−	0.952188i	$-0.401172\pi$
$53$	235.761	0.611024	0.305512	+	0.952188i	$0.401172\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	0	0
$57$	210.292	0.488664
$58$	0	0
$59$	167.162	0.368859	0.184429	−	0.982846i	$-0.440956\pi$
$59$	167.162	0.368859	0.184429	+	0.982846i	$0.440956\pi$
$60$	0	0
$61$	−363.291	−0.762535	−0.381267	−	0.924465i	$-0.624512\pi$
$61$	−363.291	−0.762535	−0.381267	+	0.924465i	$0.624512\pi$
$62$	0	0
$63$	307.998	0.615938
$64$	0	0
$65$	−150.954	−0.288055
$66$	0	0
$67$	611.297	1.11465	0.557327	−	0.830293i	$-0.311827\pi$
$67$	611.297	1.11465	0.557327	+	0.830293i	$0.311827\pi$
$68$	0	0
$69$	764.836	1.33443
$70$	0	0
$71$	−315.466	−0.527309	−0.263654	−	0.964617i	$-0.584928\pi$
$71$	−315.466	−0.527309	−0.263654	+	0.964617i	$0.584928\pi$
$72$	0	0
$73$	−372.861	−0.597810	−0.298905	−	0.954283i	$-0.596621\pi$
$73$	−372.861	−0.597810	−0.298905	+	0.954283i	$0.596621\pi$
$74$	0	0
$75$	171.736	0.264405
$76$	0	0
$77$	167.809	0.248359
$78$	0	0
$79$	−300.829	−0.428430	−0.214215	−	0.976787i	$-0.568719\pi$
$79$	−300.829	−0.428430	−0.214215	+	0.976787i	$0.568719\pi$
$80$	0	0
$81$	−866.502	−1.18862
$82$	0	0
$83$	−1218.18	−1.61100	−0.805498	−	0.592599i	$-0.798102\pi$
$83$	−1218.18	−1.61100	−0.805498	+	0.592599i	$0.798102\pi$
$84$	0	0
$85$	−372.396	−0.475201
$86$	0	0
$87$	161.663	0.199219
$88$	0	0
$89$	73.6581	0.0877275	0.0438637	−	0.999038i	$-0.486033\pi$
$89$	73.6581	0.0877275	0.0438637	+	0.999038i	$0.486033\pi$
$90$	0	0
$91$	460.574	0.530563
$92$	0	0
$93$	−1869.06	−2.08400
$94$	0	0
$95$	−153.063	−0.165305
$96$	0	0
$97$	−1389.82	−1.45480	−0.727398	−	0.686216i	$-0.759271\pi$
$97$	−1389.82	−1.45480	−0.727398	+	0.686216i	$0.759271\pi$
$98$	0	0
$99$	222.084	0.225457
$100$	0	0
$101$	−321.478	−0.316715	−0.158358	−	0.987382i	$-0.550620\pi$
$101$	−321.478	−0.316715	−0.158358	+	0.987382i	$0.550620\pi$
$102$	0	0
$103$	−566.635	−0.542060	−0.271030	−	0.962571i	$-0.587364\pi$
$103$	−566.635	−0.542060	−0.271030	+	0.962571i	$0.587364\pi$
$104$	0	0
$105$	−523.982	−0.487004
$106$	0	0
$107$	−866.060	−0.782478	−0.391239	−	0.920289i	$-0.627953\pi$
$107$	−866.060	−0.782478	−0.391239	+	0.920289i	$0.627953\pi$
$108$	0	0
$109$	−1407.61	−1.23693	−0.618463	−	0.785814i	$-0.712244\pi$
$109$	−1407.61	−1.23693	−0.618463	+	0.785814i	$0.712244\pi$
$110$	0	0
$111$	2008.73	1.71766
$112$	0	0
$113$	−426.591	−0.355135	−0.177568	−	0.984109i	$-0.556823\pi$
$113$	−426.591	−0.355135	−0.177568	+	0.984109i	$0.556823\pi$
$114$	0	0
$115$	−556.694	−0.451408
$116$	0	0
$117$	609.536	0.481638
$118$	0	0
$119$	1136.21	0.875263
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−875.805	−0.642022
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−398.684	−0.278563	−0.139282	−	0.990253i	$-0.544479\pi$
$127$	−398.684	−0.278563	−0.139282	+	0.990253i	$0.544479\pi$
$128$	0	0
$129$	−3202.61	−2.18585
$130$	0	0
$131$	1504.91	1.00370	0.501849	−	0.864955i	$-0.332653\pi$
$131$	1504.91	1.00370	0.501849	+	0.864955i	$0.332653\pi$
$132$	0	0
$133$	467.008	0.304472
$134$	0	0
$135$	233.925	0.149134
$136$	0	0
$137$	−189.347	−0.118080	−0.0590401	−	0.998256i	$-0.518804\pi$
$137$	−189.347	−0.118080	−0.0590401	+	0.998256i	$0.518804\pi$
$138$	0	0
$139$	1581.63	0.965125	0.482563	−	0.875861i	$-0.339706\pi$
$139$	1581.63	0.965125	0.482563	+	0.875861i	$0.339706\pi$
$140$	0	0
$141$	2956.74	1.76597
$142$	0	0
$143$	332.099	0.194207
$144$	0	0
$145$	−117.668	−0.0673917
$146$	0	0
$147$	−757.512	−0.425024
$148$	0	0
$149$	−2015.57	−1.10820	−0.554101	−	0.832449i	$-0.686938\pi$
$149$	−2015.57	−1.10820	−0.554101	+	0.832449i	$0.686938\pi$
$150$	0	0
$151$	2275.41	1.22629	0.613146	−	0.789969i	$-0.289904\pi$
$151$	2275.41	1.22629	0.613146	+	0.789969i	$0.289904\pi$
$152$	0	0
$153$	1503.69	0.794551
$154$	0	0
$155$	1360.41	0.704974
$156$	0	0
$157$	1028.52	0.522833	0.261416	−	0.965226i	$-0.415810\pi$
$157$	1028.52	0.522833	0.261416	+	0.965226i	$0.415810\pi$
$158$	0	0
$159$	1619.55	0.807790
$160$	0	0
$161$	1698.52	0.831441
$162$	0	0
$163$	−1518.33	−0.729602	−0.364801	−	0.931086i	$-0.618863\pi$
$163$	−1518.33	−0.729602	−0.364801	+	0.931086i	$0.618863\pi$
$164$	0	0
$165$	−377.820	−0.178262
$166$	0	0
$167$	1047.55	0.485401	0.242701	−	0.970101i	$-0.421967\pi$
$167$	1047.55	0.485401	0.242701	+	0.970101i	$0.421967\pi$
$168$	0	0
$169$	−1285.51	−0.585122
$170$	0	0
$171$	618.051	0.276395
$172$	0	0
$173$	−2607.51	−1.14593	−0.572963	−	0.819581i	$-0.694206\pi$
$173$	−2607.51	−1.14593	−0.572963	+	0.819581i	$0.694206\pi$
$174$	0	0
$175$	381.385	0.164743
$176$	0	0
$177$	1148.31	0.487641
$178$	0	0
$179$	3017.20	1.25987	0.629934	−	0.776649i	$-0.283082\pi$
$179$	3017.20	1.25987	0.629934	+	0.776649i	$0.283082\pi$
$180$	0	0
$181$	441.344	0.181242	0.0906210	−	0.995885i	$-0.471115\pi$
$181$	441.344	0.181242	0.0906210	+	0.995885i	$0.471115\pi$
$182$	0	0
$183$	−2495.61	−1.00809
$184$	0	0
$185$	−1462.08	−0.581048
$186$	0	0
$187$	819.272	0.320380
$188$	0	0
$189$	−713.724	−0.274687
$190$	0	0
$191$	4113.45	1.55832	0.779159	−	0.626826i	$-0.215646\pi$
$191$	4113.45	1.55832	0.779159	+	0.626826i	$0.215646\pi$
$192$	0	0
$193$	−2255.66	−0.841275	−0.420638	−	0.907229i	$-0.638194\pi$
$193$	−2255.66	−0.841275	−0.420638	+	0.907229i	$0.638194\pi$
$194$	0	0
$195$	−1036.97	−0.380817
$196$	0	0
$197$	3284.67	1.18793	0.593967	−	0.804490i	$-0.297561\pi$
$197$	3284.67	1.18793	0.593967	+	0.804490i	$0.297561\pi$
$198$	0	0
$199$	2296.22	0.817963	0.408982	−	0.912543i	$-0.365884\pi$
$199$	2296.22	0.817963	0.408982	+	0.912543i	$0.365884\pi$
$200$	0	0
$201$	4199.28	1.47360
$202$	0	0
$203$	359.015	0.124128
$204$	0	0
$205$	637.463	0.217182
$206$	0	0
$207$	2247.86	0.754770
$208$	0	0
$209$	336.739	0.111448
$210$	0	0
$211$	−4649.74	−1.51707	−0.758533	−	0.651634i	$-0.774084\pi$
$211$	−4649.74	−1.51707	−0.758533	+	0.651634i	$0.774084\pi$
$212$	0	0
$213$	−2167.08	−0.697117
$214$	0	0
$215$	2331.05	0.739425
$216$	0	0
$217$	−4150.73	−1.29848
$218$	0	0
$219$	−2561.35	−0.790321
$220$	0	0
$221$	2248.59	0.684419
$222$	0	0
$223$	689.598	0.207080	0.103540	−	0.994625i	$-0.466983\pi$
$223$	689.598	0.207080	0.103540	+	0.994625i	$0.466983\pi$
$224$	0	0
$225$	504.735	0.149551
$226$	0	0
$227$	−5832.01	−1.70522	−0.852608	−	0.522551i	$-0.824981\pi$
$227$	−5832.01	−1.70522	−0.852608	+	0.522551i	$0.824981\pi$
$228$	0	0
$229$	2877.35	0.830309	0.415154	−	0.909751i	$-0.363728\pi$
$229$	2877.35	0.830309	0.415154	+	0.909751i	$0.363728\pi$
$230$	0	0
$231$	1152.76	0.328338
$232$	0	0
$233$	−919.503	−0.258535	−0.129267	−	0.991610i	$-0.541263\pi$
$233$	−919.503	−0.258535	−0.129267	+	0.991610i	$0.541263\pi$
$234$	0	0
$235$	−2152.09	−0.597391
$236$	0	0
$237$	−2066.53	−0.566396
$238$	0	0
$239$	4917.54	1.33092	0.665459	−	0.746434i	$-0.268236\pi$
$239$	4917.54	1.33092	0.665459	+	0.746434i	$0.268236\pi$
$240$	0	0
$241$	2908.06	0.777282	0.388641	−	0.921389i	$-0.372945\pi$
$241$	2908.06	0.777282	0.388641	+	0.921389i	$0.372945\pi$
$242$	0	0
$243$	−4689.20	−1.23791
$244$	0	0
$245$	551.363	0.143777
$246$	0	0
$247$	924.221	0.238084
$248$	0	0
$249$	−8368.23	−2.12978
$250$	0	0
$251$	−720.590	−0.181208	−0.0906041	−	0.995887i	$-0.528880\pi$
$251$	−720.590	−0.181208	−0.0906041	+	0.995887i	$0.528880\pi$
$252$	0	0
$253$	1224.73	0.304339
$254$	0	0
$255$	−2558.16	−0.628228
$256$	0	0
$257$	4113.91	0.998516	0.499258	−	0.866453i	$-0.333606\pi$
$257$	4113.91	0.998516	0.499258	+	0.866453i	$0.333606\pi$
$258$	0	0
$259$	4460.91	1.07022
$260$	0	0
$261$	475.130	0.112681
$262$	0	0
$263$	3564.90	0.835822	0.417911	−	0.908488i	$-0.362762\pi$
$263$	3564.90	0.835822	0.417911	+	0.908488i	$0.362762\pi$
$264$	0	0
$265$	−1178.81	−0.273258
$266$	0	0
$267$	505.991	0.115978
$268$	0	0
$269$	4405.46	0.998533	0.499267	−	0.866448i	$-0.333603\pi$
$269$	4405.46	0.998533	0.499267	+	0.866448i	$0.333603\pi$
$270$	0	0
$271$	−7032.65	−1.57639	−0.788197	−	0.615423i	$-0.788985\pi$
$271$	−7032.65	−1.57639	−0.788197	+	0.615423i	$0.788985\pi$
$272$	0	0
$273$	3163.89	0.701419
$274$	0	0
$275$	275.000	0.0603023
$276$	0	0
$277$	7403.56	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$277$	7403.56	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$278$	0	0
$279$	−5493.19	−1.17874
$280$	0	0
$281$	5709.48	1.21210	0.606048	−	0.795428i	$-0.292754\pi$
$281$	5709.48	1.21210	0.606048	+	0.795428i	$0.292754\pi$
$282$	0	0
$283$	−3030.73	−0.636602	−0.318301	−	0.947990i	$-0.603112\pi$
$283$	−3030.73	−0.636602	−0.318301	+	0.947990i	$0.603112\pi$
$284$	0	0
$285$	−1051.46	−0.218537
$286$	0	0
$287$	−1944.95	−0.400024
$288$	0	0
$289$	634.160	0.129078
$290$	0	0
$291$	−9547.34	−1.92328
$292$	0	0
$293$	2559.81	0.510395	0.255197	−	0.966889i	$-0.417860\pi$
$293$	2559.81	0.510395	0.255197	+	0.966889i	$0.417860\pi$
$294$	0	0
$295$	−835.811	−0.164959
$296$	0	0
$297$	−514.635	−0.100546
$298$	0	0
$299$	3361.41	0.650152
$300$	0	0
$301$	−7112.23	−1.36193
$302$	0	0
$303$	−2208.38	−0.418707
$304$	0	0
$305$	1816.45	0.341016
$306$	0	0
$307$	4897.98	0.910561	0.455281	−	0.890348i	$-0.349539\pi$
$307$	4897.98	0.910561	0.455281	+	0.890348i	$0.349539\pi$
$308$	0	0
$309$	−3892.47	−0.716618
$310$	0	0
$311$	3220.87	0.587263	0.293631	−	0.955919i	$-0.405136\pi$
$311$	3220.87	0.587263	0.293631	+	0.955919i	$0.405136\pi$
$312$	0	0
$313$	−2546.74	−0.459905	−0.229952	−	0.973202i	$-0.573857\pi$
$313$	−2546.74	−0.459905	−0.229952	+	0.973202i	$0.573857\pi$
$314$	0	0
$315$	−1539.99	−0.275456
$316$	0	0
$317$	3054.11	0.541122	0.270561	−	0.962703i	$-0.412791\pi$
$317$	3054.11	0.541122	0.270561	+	0.962703i	$0.412791\pi$
$318$	0	0
$319$	258.870	0.0454355
$320$	0	0
$321$	−5949.36	−1.03446
$322$	0	0
$323$	2280.00	0.392764
$324$	0	0
$325$	754.771	0.128822
$326$	0	0
$327$	−9669.53	−1.63525
$328$	0	0
$329$	6566.20	1.10032
$330$	0	0
$331$	−8086.37	−1.34280	−0.671401	−	0.741095i	$-0.734307\pi$
$331$	−8086.37	−1.34280	−0.671401	+	0.741095i	$0.734307\pi$
$332$	0	0
$333$	5903.69	0.971533
$334$	0	0
$335$	−3056.48	−0.498488
$336$	0	0
$337$	−4367.51	−0.705975	−0.352987	−	0.935628i	$-0.614834\pi$
$337$	−4367.51	−0.705975	−0.352987	+	0.935628i	$0.614834\pi$
$338$	0	0
$339$	−2930.45	−0.469499
$340$	0	0
$341$	−2992.91	−0.475293
$342$	0	0
$343$	−6914.86	−1.08853
$344$	0	0
$345$	−3824.18	−0.596774
$346$	0	0
$347$	9426.38	1.45831	0.729157	−	0.684347i	$-0.239912\pi$
$347$	9426.38	1.45831	0.729157	+	0.684347i	$0.239912\pi$
$348$	0	0
$349$	−8924.60	−1.36883	−0.684417	−	0.729091i	$-0.739943\pi$
$349$	−8924.60	−1.36883	−0.684417	+	0.729091i	$0.739943\pi$
$350$	0	0
$351$	−1412.48	−0.214794
$352$	0	0
$353$	−9858.79	−1.48649	−0.743244	−	0.669020i	$-0.766714\pi$
$353$	−9858.79	−1.48649	−0.743244	+	0.669020i	$0.766714\pi$
$354$	0	0
$355$	1577.33	0.235820
$356$	0	0
$357$	7805.15	1.15712
$358$	0	0
$359$	−12720.2	−1.87005	−0.935026	−	0.354580i	$-0.884624\pi$
$359$	−12720.2	−1.87005	−0.935026	+	0.354580i	$0.884624\pi$
$360$	0	0
$361$	−5921.87	−0.863372
$362$	0	0
$363$	831.204	0.120184
$364$	0	0
$365$	1864.31	0.267349
$366$	0	0
$367$	−6175.96	−0.878427	−0.439213	−	0.898383i	$-0.644743\pi$
$367$	−6175.96	−0.878427	−0.439213	+	0.898383i	$0.644743\pi$
$368$	0	0
$369$	−2574.00	−0.363136
$370$	0	0
$371$	3596.63	0.503309
$372$	0	0
$373$	10635.6	1.47638	0.738192	−	0.674591i	$-0.235680\pi$
$373$	10635.6	1.47638	0.738192	+	0.674591i	$0.235680\pi$
$374$	0	0
$375$	−858.682	−0.118246
$376$	0	0
$377$	710.500	0.0970626
$378$	0	0
$379$	12614.8	1.70971	0.854855	−	0.518868i	$-0.173646\pi$
$379$	12614.8	1.70971	0.854855	+	0.518868i	$0.173646\pi$
$380$	0	0
$381$	−2738.75	−0.368268
$382$	0	0
$383$	11523.9	1.53745	0.768723	−	0.639582i	$-0.220893\pi$
$383$	11523.9	1.53745	0.768723	+	0.639582i	$0.220893\pi$
$384$	0	0
$385$	−839.047	−0.111070
$386$	0	0
$387$	−9412.52	−1.23634
$388$	0	0
$389$	−8729.61	−1.13781	−0.568906	−	0.822402i	$-0.692633\pi$
$389$	−8729.61	−1.13781	−0.568906	+	0.822402i	$0.692633\pi$
$390$	0	0
$391$	8292.43	1.07255
$392$	0	0
$393$	10337.9	1.32692
$394$	0	0
$395$	1504.15	0.191600
$396$	0	0
$397$	313.914	0.0396848	0.0198424	−	0.999803i	$-0.493684\pi$
$397$	313.914	0.0396848	0.0198424	+	0.999803i	$0.493684\pi$
$398$	0	0
$399$	3208.09	0.402520
$400$	0	0
$401$	−15699.2	−1.95506	−0.977531	−	0.210792i	$-0.932396\pi$
$401$	−15699.2	−1.95506	−0.977531	+	0.210792i	$0.932396\pi$
$402$	0	0
$403$	−8214.40	−1.01536
$404$	0	0
$405$	4332.51	0.531566
$406$	0	0
$407$	3216.57	0.391743
$408$	0	0
$409$	−334.332	−0.0404197	−0.0202098	−	0.999796i	$-0.506433\pi$
$409$	−334.332	−0.0404197	−0.0202098	+	0.999796i	$0.506433\pi$
$410$	0	0
$411$	−1300.71	−0.156105
$412$	0	0
$413$	2550.13	0.303834
$414$	0	0
$415$	6090.90	0.720459
$416$	0	0
$417$	10865.0	1.27592
$418$	0	0
$419$	6745.52	0.786493	0.393246	−	0.919433i	$-0.371352\pi$
$419$	6745.52	0.786493	0.393246	+	0.919433i	$0.371352\pi$
$420$	0	0
$421$	14779.7	1.71097	0.855484	−	0.517830i	$-0.173260\pi$
$421$	14779.7	1.71097	0.855484	+	0.517830i	$0.173260\pi$
$422$	0	0
$423$	8689.89	0.998858
$424$	0	0
$425$	1861.98	0.212516
$426$	0	0
$427$	−5542.15	−0.628111
$428$	0	0
$429$	2281.34	0.256746
$430$	0	0
$431$	−10394.2	−1.16165	−0.580823	−	0.814030i	$-0.697269\pi$
$431$	−10394.2	−1.16165	−0.580823	+	0.814030i	$0.697269\pi$
$432$	0	0
$433$	−5818.62	−0.645785	−0.322893	−	0.946436i	$-0.604655\pi$
$433$	−5818.62	−0.645785	−0.322893	+	0.946436i	$0.604655\pi$
$434$	0	0
$435$	−808.315	−0.0890937
$436$	0	0
$437$	3408.37	0.373099
$438$	0	0
$439$	7542.46	0.820005	0.410002	−	0.912084i	$-0.365528\pi$
$439$	7542.46	0.820005	0.410002	+	0.912084i	$0.365528\pi$
$440$	0	0
$441$	−2226.34	−0.240399
$442$	0	0
$443$	6599.99	0.707845	0.353922	−	0.935275i	$-0.384848\pi$
$443$	6599.99	0.707845	0.353922	+	0.935275i	$0.384848\pi$
$444$	0	0
$445$	−368.291	−0.0392329
$446$	0	0
$447$	−13845.9	−1.46507
$448$	0	0
$449$	10081.9	1.05968	0.529840	−	0.848098i	$-0.322252\pi$
$449$	10081.9	1.05968	0.529840	+	0.848098i	$0.322252\pi$
$450$	0	0
$451$	−1402.42	−0.146424
$452$	0	0
$453$	15630.8	1.62119
$454$	0	0
$455$	−2302.87	−0.237275
$456$	0	0
$457$	−15159.8	−1.55175	−0.775873	−	0.630890i	$-0.782690\pi$
$457$	−15159.8	−1.55175	−0.775873	+	0.630890i	$0.782690\pi$
$458$	0	0
$459$	−3484.51	−0.354342
$460$	0	0
$461$	11761.3	1.18824	0.594118	−	0.804378i	$-0.297501\pi$
$461$	11761.3	1.18824	0.594118	+	0.804378i	$0.297501\pi$
$462$	0	0
$463$	6693.19	0.671833	0.335917	−	0.941892i	$-0.390954\pi$
$463$	6693.19	0.671833	0.335917	+	0.941892i	$0.390954\pi$
$464$	0	0
$465$	9345.29	0.931995
$466$	0	0
$467$	19784.8	1.96046	0.980228	−	0.197873i	$-0.0634034\pi$
$467$	19784.8	1.96046	0.980228	+	0.197873i	$0.0634034\pi$
$468$	0	0
$469$	9325.58	0.918156
$470$	0	0
$471$	7065.37	0.691199
$472$	0	0
$473$	−5128.32	−0.498520
$474$	0	0
$475$	765.315	0.0739265
$476$	0	0
$477$	4759.88	0.456897
$478$	0	0
$479$	4432.99	0.422857	0.211428	−	0.977393i	$-0.432188\pi$
$479$	4432.99	0.422857	0.211428	+	0.977393i	$0.432188\pi$
$480$	0	0
$481$	8828.26	0.836870
$482$	0	0
$483$	11667.9	1.09919
$484$	0	0
$485$	6949.12	0.650605
$486$	0	0
$487$	−18994.9	−1.76743	−0.883716	−	0.468023i	$-0.844966\pi$
$487$	−18994.9	−1.76743	−0.883716	+	0.468023i	$0.844966\pi$
$488$	0	0
$489$	−10430.1	−0.964553
$490$	0	0
$491$	4821.55	0.443164	0.221582	−	0.975142i	$-0.428878\pi$
$491$	4821.55	0.443164	0.221582	+	0.975142i	$0.428878\pi$
$492$	0	0
$493$	1752.77	0.160123
$494$	0	0
$495$	−1110.42	−0.100827
$496$	0	0
$497$	−4812.56	−0.434352
$498$	0	0
$499$	17440.3	1.56460	0.782300	−	0.622902i	$-0.214047\pi$
$499$	17440.3	1.56460	0.782300	+	0.622902i	$0.214047\pi$
$500$	0	0
$501$	7196.11	0.641714
$502$	0	0
$503$	−9153.19	−0.811373	−0.405687	−	0.914012i	$-0.632968\pi$
$503$	−9153.19	−0.811373	−0.405687	+	0.914012i	$0.632968\pi$
$504$	0	0
$505$	1607.39	0.141639
$506$	0	0
$507$	−8830.77	−0.773547
$508$	0	0
$509$	−1383.24	−0.120454	−0.0602268	−	0.998185i	$-0.519182\pi$
$509$	−1383.24	−0.120454	−0.0602268	+	0.998185i	$0.519182\pi$
$510$	0	0
$511$	−5688.15	−0.492425
$512$	0	0
$513$	−1432.21	−0.123262
$514$	0	0
$515$	2833.17	0.242417
$516$	0	0
$517$	4734.60	0.402761
$518$	0	0
$519$	−17912.2	−1.51494
$520$	0	0
$521$	−22764.1	−1.91423	−0.957113	−	0.289713i	$-0.906440\pi$
$521$	−22764.1	−1.91423	−0.957113	+	0.289713i	$0.906440\pi$
$522$	0	0
$523$	12616.5	1.05484	0.527420	−	0.849605i	$-0.323159\pi$
$523$	12616.5	1.05484	0.527420	+	0.849605i	$0.323159\pi$
$524$	0	0
$525$	2619.91	0.217795
$526$	0	0
$527$	−20264.5	−1.67502
$528$	0	0
$529$	229.312	0.0188470
$530$	0	0
$531$	3374.91	0.275816
$532$	0	0
$533$	−3849.11	−0.312802
$534$	0	0
$535$	4330.30	0.349935
$536$	0	0
$537$	20726.5	1.66558
$538$	0	0
$539$	−1213.00	−0.0969342
$540$	0	0
$541$	−21440.2	−1.70386	−0.851929	−	0.523657i	$-0.824567\pi$
$541$	−21440.2	−1.70386	−0.851929	+	0.523657i	$0.824567\pi$
$542$	0	0
$543$	3031.79	0.239607
$544$	0	0
$545$	7038.06	0.553170
$546$	0	0
$547$	−22138.5	−1.73048	−0.865240	−	0.501358i	$-0.832834\pi$
$547$	−22138.5	−1.73048	−0.865240	+	0.501358i	$0.832834\pi$
$548$	0	0
$549$	−7334.63	−0.570190
$550$	0	0
$551$	720.425	0.0557008
$552$	0	0
$553$	−4589.27	−0.352904
$554$	0	0
$555$	−10043.7	−0.768162
$556$	0	0
$557$	−18160.8	−1.38151	−0.690753	−	0.723090i	$-0.742721\pi$
$557$	−18160.8	−1.38151	−0.690753	+	0.723090i	$0.742721\pi$
$558$	0	0
$559$	−14075.3	−1.06498
$560$	0	0
$561$	5627.95	0.423551
$562$	0	0
$563$	11886.6	0.889806	0.444903	−	0.895579i	$-0.353238\pi$
$563$	11886.6	0.889806	0.444903	+	0.895579i	$0.353238\pi$
$564$	0	0
$565$	2132.96	0.158821
$566$	0	0
$567$	−13218.8	−0.979081
$568$	0	0
$569$	−19547.3	−1.44018	−0.720092	−	0.693879i	$-0.755900\pi$
$569$	−19547.3	−1.44018	−0.720092	+	0.693879i	$0.755900\pi$
$570$	0	0
$571$	−6953.38	−0.509614	−0.254807	−	0.966992i	$-0.582012\pi$
$571$	−6953.38	−0.509614	−0.254807	+	0.966992i	$0.582012\pi$
$572$	0	0
$573$	28257.2	2.06014
$574$	0	0
$575$	2783.47	0.201876
$576$	0	0
$577$	−1267.04	−0.0914170	−0.0457085	−	0.998955i	$-0.514555\pi$
$577$	−1267.04	−0.0914170	−0.0457085	+	0.998955i	$0.514555\pi$
$578$	0	0
$579$	−15495.2	−1.11219
$580$	0	0
$581$	−18583.8	−1.32700
$582$	0	0
$583$	2593.37	0.184231
$584$	0	0
$585$	−3047.68	−0.215395
$586$	0	0
$587$	16816.0	1.18241	0.591203	−	0.806523i	$-0.298653\pi$
$587$	16816.0	1.18241	0.591203	+	0.806523i	$0.298653\pi$
$588$	0	0
$589$	−8329.16	−0.582677
$590$	0	0
$591$	22563.9	1.57048
$592$	0	0
$593$	−1411.66	−0.0977573	−0.0488786	−	0.998805i	$-0.515565\pi$
$593$	−1411.66	−0.0977573	−0.0488786	+	0.998805i	$0.515565\pi$
$594$	0	0
$595$	−5681.06	−0.391430
$596$	0	0
$597$	15773.8	1.08137
$598$	0	0
$599$	−4859.48	−0.331474	−0.165737	−	0.986170i	$-0.553000\pi$
$599$	−4859.48	−0.331474	−0.165737	+	0.986170i	$0.553000\pi$
$600$	0	0
$601$	11695.5	0.793793	0.396897	−	0.917863i	$-0.370087\pi$
$601$	11695.5	0.793793	0.396897	+	0.917863i	$0.370087\pi$
$602$	0	0
$603$	12341.7	0.833489
$604$	0	0
$605$	−605.000	−0.0406558
$606$	0	0
$607$	7.57260	0.000506363 0	0.000253181	−	1.00000i	$-0.499919\pi$
$607$	7.57260	0.000506363 0	0.000253181		1.00000i	$0.499919\pi$
$608$	0	0
$609$	2466.24	0.164100
$610$	0	0
$611$	12994.7	0.860407
$612$	0	0
$613$	−19891.5	−1.31062	−0.655311	−	0.755359i	$-0.727463\pi$
$613$	−19891.5	−1.31062	−0.655311	+	0.755359i	$0.727463\pi$
$614$	0	0
$615$	4379.02	0.287121
$616$	0	0
$617$	10779.1	0.703320	0.351660	−	0.936128i	$-0.385617\pi$
$617$	10779.1	0.703320	0.351660	+	0.936128i	$0.385617\pi$
$618$	0	0
$619$	−6513.73	−0.422955	−0.211477	−	0.977383i	$-0.567827\pi$
$619$	−6513.73	−0.422955	−0.211477	+	0.977383i	$0.567827\pi$
$620$	0	0
$621$	−5208.98	−0.336601
$622$	0	0
$623$	1123.68	0.0722624
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2313.21	0.147338
$628$	0	0
$629$	21778.9	1.38057
$630$	0	0
$631$	10913.6	0.688533	0.344266	−	0.938872i	$-0.388128\pi$
$631$	10913.6	0.688533	0.344266	+	0.938872i	$0.388128\pi$
$632$	0	0
$633$	−31941.2	−2.00560
$634$	0	0
$635$	1993.42	0.124577
$636$	0	0
$637$	−3329.22	−0.207078
$638$	0	0
$639$	−6369.08	−0.394298
$640$	0	0
$641$	7378.43	0.454650	0.227325	−	0.973819i	$-0.427002\pi$
$641$	7378.43	0.454650	0.227325	+	0.973819i	$0.427002\pi$
$642$	0	0
$643$	297.203	0.0182279	0.00911395	−	0.999958i	$-0.497099\pi$
$643$	297.203	0.0182279	0.00911395	+	0.999958i	$0.497099\pi$
$644$	0	0
$645$	16013.1	0.977541
$646$	0	0
$647$	−17665.5	−1.07342	−0.536709	−	0.843768i	$-0.680333\pi$
$647$	−17665.5	−1.07342	−0.536709	+	0.843768i	$0.680333\pi$
$648$	0	0
$649$	1838.78	0.111215
$650$	0	0
$651$	−28513.3	−1.71662
$652$	0	0
$653$	−28420.0	−1.70316	−0.851578	−	0.524228i	$-0.824354\pi$
$653$	−28420.0	−1.70316	−0.851578	+	0.524228i	$0.824354\pi$
$654$	0	0
$655$	−7524.54	−0.448867
$656$	0	0
$657$	−7527.86	−0.447016
$658$	0	0
$659$	19552.3	1.15576	0.577882	−	0.816120i	$-0.303879\pi$
$659$	19552.3	1.15576	0.577882	+	0.816120i	$0.303879\pi$
$660$	0	0
$661$	27600.3	1.62410	0.812049	−	0.583590i	$-0.198352\pi$
$661$	27600.3	1.62410	0.812049	+	0.583590i	$0.198352\pi$
$662$	0	0
$663$	15446.6	0.904821
$664$	0	0
$665$	−2335.04	−0.136164
$666$	0	0
$667$	2620.20	0.152106
$668$	0	0
$669$	4737.16	0.273766
$670$	0	0
$671$	−3996.20	−0.229913
$672$	0	0
$673$	24272.9	1.39027	0.695135	−	0.718879i	$-0.255344\pi$
$673$	24272.9	1.39027	0.695135	+	0.718879i	$0.255344\pi$
$674$	0	0
$675$	−1169.62	−0.0666946
$676$	0	0
$677$	−18258.3	−1.03652	−0.518261	−	0.855223i	$-0.673420\pi$
$677$	−18258.3	−1.03652	−0.518261	+	0.855223i	$0.673420\pi$
$678$	0	0
$679$	−21202.3	−1.19834
$680$	0	0
$681$	−40062.7	−2.25434
$682$	0	0
$683$	−2171.51	−0.121655	−0.0608277	−	0.998148i	$-0.519374\pi$
$683$	−2171.51	−0.121655	−0.0608277	+	0.998148i	$0.519374\pi$
$684$	0	0
$685$	946.734	0.0528071
$686$	0	0
$687$	19765.8	1.09769
$688$	0	0
$689$	7117.83	0.393567
$690$	0	0
$691$	−19495.9	−1.07331	−0.536657	−	0.843801i	$-0.680313\pi$
$691$	−19495.9	−1.07331	−0.536657	+	0.843801i	$0.680313\pi$
$692$	0	0
$693$	3387.98	0.185712
$694$	0	0
$695$	−7908.17	−0.431617
$696$	0	0
$697$	−9495.56	−0.516026
$698$	0	0
$699$	−6316.48	−0.341790
$700$	0	0
$701$	−13625.9	−0.734158	−0.367079	−	0.930190i	$-0.619642\pi$
$701$	−13625.9	−0.734158	−0.367079	+	0.930190i	$0.619642\pi$
$702$	0	0
$703$	8951.59	0.480250
$704$	0	0
$705$	−14783.7	−0.789767
$706$	0	0
$707$	−4904.28	−0.260883
$708$	0	0
$709$	6810.99	0.360779	0.180389	−	0.983595i	$-0.442264\pi$
$709$	6810.99	0.360779	0.180389	+	0.983595i	$0.442264\pi$
$710$	0	0
$711$	−6073.57	−0.320361
$712$	0	0
$713$	−30293.3	−1.59116
$714$	0	0
$715$	−1660.50	−0.0868518
$716$	0	0
$717$	33780.8	1.75951
$718$	0	0
$719$	13166.6	0.682934	0.341467	−	0.939894i	$-0.389076\pi$
$719$	13166.6	0.682934	0.341467	+	0.939894i	$0.389076\pi$
$720$	0	0
$721$	−8644.24	−0.446503
$722$	0	0
$723$	19976.8	1.02759
$724$	0	0
$725$	588.340	0.0301385
$726$	0	0
$727$	−8075.12	−0.411952	−0.205976	−	0.978557i	$-0.566037\pi$
$727$	−8075.12	−0.411952	−0.205976	+	0.978557i	$0.566037\pi$
$728$	0	0
$729$	−8816.71	−0.447935
$730$	0	0
$731$	−34723.0	−1.75688
$732$	0	0
$733$	26226.1	1.32153	0.660765	−	0.750593i	$-0.270232\pi$
$733$	26226.1	1.32153	0.660765	+	0.750593i	$0.270232\pi$
$734$	0	0
$735$	3787.56	0.190077
$736$	0	0
$737$	6724.26	0.336081
$738$	0	0
$739$	37686.2	1.87592	0.937962	−	0.346737i	$-0.112710\pi$
$739$	37686.2	1.87592	0.937962	+	0.346737i	$0.112710\pi$
$740$	0	0
$741$	6348.89	0.314754
$742$	0	0
$743$	−5461.18	−0.269652	−0.134826	−	0.990869i	$-0.543048\pi$
$743$	−5461.18	−0.269652	−0.134826	+	0.990869i	$0.543048\pi$
$744$	0	0
$745$	10077.9	0.495603
$746$	0	0
$747$	−24594.3	−1.20463
$748$	0	0
$749$	−13212.1	−0.644539
$750$	0	0
$751$	16325.5	0.793242	0.396621	−	0.917983i	$-0.370183\pi$
$751$	16325.5	0.793242	0.396621	+	0.917983i	$0.370183\pi$
$752$	0	0
$753$	−4950.06	−0.239562
$754$	0	0
$755$	−11377.0	−0.548415
$756$	0	0
$757$	15157.7	0.727763	0.363881	−	0.931445i	$-0.381451\pi$
$757$	15157.7	0.727763	0.363881	+	0.931445i	$0.381451\pi$
$758$	0	0
$759$	8413.20	0.402345
$760$	0	0
$761$	−22985.6	−1.09491	−0.547456	−	0.836835i	$-0.684404\pi$
$761$	−22985.6	−1.09491	−0.547456	+	0.836835i	$0.684404\pi$
$762$	0	0
$763$	−21473.7	−1.01887
$764$	0	0
$765$	−7518.46	−0.355334
$766$	0	0
$767$	5046.77	0.237586
$768$	0	0
$769$	−3715.50	−0.174232	−0.0871159	−	0.996198i	$-0.527765\pi$
$769$	−3715.50	−0.174232	−0.0871159	+	0.996198i	$0.527765\pi$
$770$	0	0
$771$	28260.3	1.32007
$772$	0	0
$773$	−11510.5	−0.535582	−0.267791	−	0.963477i	$-0.586294\pi$
$773$	−11510.5	−0.535582	−0.267791	+	0.963477i	$0.586294\pi$
$774$	0	0
$775$	−6802.06	−0.315274
$776$	0	0
$777$	30644.0	1.41486
$778$	0	0
$779$	−3902.88	−0.179506
$780$	0	0
$781$	−3470.13	−0.158990
$782$	0	0
$783$	−1101.02	−0.0502519
$784$	0	0
$785$	−5142.60	−0.233818
$786$	0	0
$787$	5418.55	0.245426	0.122713	−	0.992442i	$-0.460841\pi$
$787$	5418.55	0.245426	0.122713	+	0.992442i	$0.460841\pi$
$788$	0	0
$789$	24488.9	1.10498
$790$	0	0
$791$	−6507.82	−0.292530
$792$	0	0
$793$	−10968.1	−0.491157
$794$	0	0
$795$	−8097.75	−0.361255
$796$	0	0
$797$	11751.3	0.522275	0.261138	−	0.965302i	$-0.415902\pi$
$797$	11751.3	0.522275	0.261138	+	0.965302i	$0.415902\pi$
$798$	0	0
$799$	32057.2	1.41940
$800$	0	0
$801$	1487.11	0.0655988
$802$	0	0
$803$	−4101.48	−0.180246
$804$	0	0
$805$	−8492.59	−0.371832
$806$	0	0
$807$	30263.1	1.32009
$808$	0	0
$809$	−35385.4	−1.53780	−0.768902	−	0.639366i	$-0.779197\pi$
$809$	−35385.4	−1.53780	−0.768902	+	0.639366i	$0.779197\pi$
$810$	0	0
$811$	11033.5	0.477731	0.238865	−	0.971053i	$-0.423225\pi$
$811$	11033.5	0.477731	0.238865	+	0.971053i	$0.423225\pi$
$812$	0	0
$813$	−48310.4	−2.08404
$814$	0	0
$815$	7591.67	0.326288
$816$	0	0
$817$	−14271.9	−0.611152
$818$	0	0
$819$	9298.72	0.396732
$820$	0	0
$821$	−449.705	−0.0191167	−0.00955834	−	0.999954i	$-0.503043\pi$
$821$	−449.705	−0.0191167	−0.00955834	+	0.999954i	$0.503043\pi$
$822$	0	0
$823$	23306.1	0.987120	0.493560	−	0.869712i	$-0.335695\pi$
$823$	23306.1	0.987120	0.493560	+	0.869712i	$0.335695\pi$
$824$	0	0
$825$	1889.10	0.0797212
$826$	0	0
$827$	33096.0	1.39161	0.695804	−	0.718232i	$-0.255048\pi$
$827$	33096.0	1.39161	0.695804	+	0.718232i	$0.255048\pi$
$828$	0	0
$829$	42098.2	1.76373	0.881865	−	0.471502i	$-0.156288\pi$
$829$	42098.2	1.76373	0.881865	+	0.471502i	$0.156288\pi$
$830$	0	0
$831$	50858.4	2.12306
$832$	0	0
$833$	−8213.01	−0.341614
$834$	0	0
$835$	−5237.76	−0.217078
$836$	0	0
$837$	12729.4	0.525677
$838$	0	0
$839$	−452.609	−0.0186243	−0.00931215	−	0.999957i	$-0.502964\pi$
$839$	−452.609	−0.0186243	−0.00931215	+	0.999957i	$0.502964\pi$
$840$	0	0
$841$	−23835.2	−0.977292
$842$	0	0
$843$	39221.0	1.60242
$844$	0	0
$845$	6427.56	0.261674
$846$	0	0
$847$	1845.90	0.0748831
$848$	0	0
$849$	−20819.5	−0.841605
$850$	0	0
$851$	32557.1	1.31145
$852$	0	0
$853$	−32079.3	−1.28766	−0.643830	−	0.765169i	$-0.722656\pi$
$853$	−32079.3	−1.28766	−0.643830	+	0.765169i	$0.722656\pi$
$854$	0	0
$855$	−3090.25	−0.123608
$856$	0	0
$857$	15049.5	0.599862	0.299931	−	0.953961i	$-0.403036\pi$
$857$	15049.5	0.599862	0.299931	+	0.953961i	$0.403036\pi$
$858$	0	0
$859$	15862.2	0.630049	0.315025	−	0.949083i	$-0.397987\pi$
$859$	15862.2	0.630049	0.315025	+	0.949083i	$0.397987\pi$
$860$	0	0
$861$	−13360.8	−0.528843
$862$	0	0
$863$	26892.7	1.06076	0.530381	−	0.847759i	$-0.322049\pi$
$863$	26892.7	1.06076	0.530381	+	0.847759i	$0.322049\pi$
$864$	0	0
$865$	13037.5	0.512473
$866$	0	0
$867$	4356.33	0.170645
$868$	0	0
$869$	−3309.12	−0.129176
$870$	0	0
$871$	18455.6	0.717960
$872$	0	0
$873$	−28059.7	−1.08783
$874$	0	0
$875$	−1906.93	−0.0736753
$876$	0	0
$877$	−29275.7	−1.12722	−0.563609	−	0.826041i	$-0.690588\pi$
$877$	−29275.7	−1.12722	−0.563609	+	0.826041i	$0.690588\pi$
$878$	0	0
$879$	17584.5	0.674756
$880$	0	0
$881$	27090.9	1.03600	0.518000	−	0.855381i	$-0.326677\pi$
$881$	27090.9	1.03600	0.518000	+	0.855381i	$0.326677\pi$
$882$	0	0
$883$	21772.9	0.829805	0.414902	−	0.909866i	$-0.363816\pi$
$883$	21772.9	0.829805	0.414902	+	0.909866i	$0.363816\pi$
$884$	0	0
$885$	−5741.57	−0.218080
$886$	0	0
$887$	−3101.20	−0.117393	−0.0586967	−	0.998276i	$-0.518694\pi$
$887$	−3101.20	−0.117393	−0.0586967	+	0.998276i	$0.518694\pi$
$888$	0	0
$889$	−6082.09	−0.229457
$890$	0	0
$891$	−9531.52	−0.358381
$892$	0	0
$893$	13176.2	0.493757
$894$	0	0
$895$	−15086.0	−0.563430
$896$	0	0
$897$	23091.1	0.859519
$898$	0	0
$899$	−6403.08	−0.237547
$900$	0	0
$901$	17559.3	0.649262
$902$	0	0
$903$	−48857.2	−1.80051
$904$	0	0
$905$	−2206.72	−0.0810539
$906$	0	0
$907$	15615.2	0.571659	0.285829	−	0.958281i	$-0.407731\pi$
$907$	15615.2	0.571659	0.285829	+	0.958281i	$0.407731\pi$
$908$	0	0
$909$	−6490.46	−0.236826
$910$	0	0
$911$	14189.7	0.516053	0.258027	−	0.966138i	$-0.416928\pi$
$911$	14189.7	0.516053	0.258027	+	0.966138i	$0.416928\pi$
$912$	0	0
$913$	−13400.0	−0.485733
$914$	0	0
$915$	12478.0	0.450832
$916$	0	0
$917$	22958.0	0.826760
$918$	0	0
$919$	38188.9	1.37077	0.685384	−	0.728182i	$-0.259634\pi$
$919$	38188.9	1.37077	0.685384	+	0.728182i	$0.259634\pi$
$920$	0	0
$921$	33646.4	1.20379
$922$	0	0
$923$	−9524.19	−0.339645
$924$	0	0
$925$	7310.38	0.259853
$926$	0	0
$927$	−11440.0	−0.405329
$928$	0	0
$929$	−41832.9	−1.47739	−0.738693	−	0.674042i	$-0.764557\pi$
$929$	−41832.9	−1.47739	−0.738693	+	0.674042i	$0.764557\pi$
$930$	0	0
$931$	−3375.73	−0.118835
$932$	0	0
$933$	22125.6	0.776378
$934$	0	0
$935$	−4096.36	−0.143278
$936$	0	0
$937$	6058.27	0.211222	0.105611	−	0.994408i	$-0.466320\pi$
$937$	6058.27	0.211222	0.105611	+	0.994408i	$0.466320\pi$
$938$	0	0
$939$	−17494.7	−0.608007
$940$	0	0
$941$	32923.7	1.14058	0.570288	−	0.821445i	$-0.306832\pi$
$941$	32923.7	1.14058	0.570288	+	0.821445i	$0.306832\pi$
$942$	0	0
$943$	−14194.9	−0.490189
$944$	0	0
$945$	3568.62	0.122844
$946$	0	0
$947$	−3942.99	−0.135301	−0.0676505	−	0.997709i	$-0.521550\pi$
$947$	−3942.99	−0.135301	−0.0676505	+	0.997709i	$0.521550\pi$
$948$	0	0
$949$	−11257.0	−0.385056
$950$	0	0
$951$	20980.1	0.715379
$952$	0	0
$953$	9152.99	0.311117	0.155558	−	0.987827i	$-0.450282\pi$
$953$	9152.99	0.311117	0.155558	+	0.987827i	$0.450282\pi$
$954$	0	0
$955$	−20567.3	−0.696901
$956$	0	0
$957$	1778.29	0.0600669
$958$	0	0
$959$	−2888.56	−0.0972644
$960$	0	0
$961$	44237.9	1.48494
$962$	0	0
$963$	−17485.2	−0.585103
$964$	0	0
$965$	11278.3	0.376230
$966$	0	0
$967$	4536.39	0.150859	0.0754294	−	0.997151i	$-0.475967\pi$
$967$	4536.39	0.150859	0.0754294	+	0.997151i	$0.475967\pi$
$968$	0	0
$969$	15662.4	0.519245
$970$	0	0
$971$	−53683.7	−1.77424	−0.887122	−	0.461535i	$-0.847299\pi$
$971$	−53683.7	−1.77424	−0.887122	+	0.461535i	$0.847299\pi$
$972$	0	0
$973$	24128.5	0.794988
$974$	0	0
$975$	5184.87	0.170306
$976$	0	0
$977$	−39392.4	−1.28994	−0.644972	−	0.764206i	$-0.723131\pi$
$977$	−39392.4	−1.28994	−0.644972	+	0.764206i	$0.723131\pi$
$978$	0	0
$979$	810.239	0.0264508
$980$	0	0
$981$	−28418.9	−0.924919
$982$	0	0
$983$	−33348.6	−1.08205	−0.541026	−	0.841006i	$-0.681964\pi$
$983$	−33348.6	−1.08205	−0.541026	+	0.841006i	$0.681964\pi$
$984$	0	0
$985$	−16423.3	−0.531260
$986$	0	0
$987$	45106.2	1.45466
$988$	0	0
$989$	−51907.3	−1.66891
$990$	0	0
$991$	−696.027	−0.0223108	−0.0111554	−	0.999938i	$-0.503551\pi$
$991$	−696.027	−0.0223108	−0.0111554	+	0.999938i	$0.503551\pi$
$992$	0	0
$993$	−55549.0	−1.77522
$994$	0	0
$995$	−11481.1	−0.365804
$996$	0	0
$997$	−45135.4	−1.43375	−0.716877	−	0.697200i	$-0.754429\pi$
$997$	−45135.4	−1.43375	−0.716877	+	0.697200i	$0.754429\pi$
$998$	0	0
$999$	−13680.6	−0.433270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.4.a.f.1.3	✓	3
3.2	odd	2		1980.4.a.l.1.3		3
4.3	odd	2		880.4.a.w.1.1		3
5.2	odd	4		1100.4.b.h.749.2		6
5.3	odd	4		1100.4.b.h.749.5		6
5.4	even	2		1100.4.a.i.1.1		3
11.10	odd	2		2420.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.3	✓	3	1.1	even	1	trivial
880.4.a.w.1.1		3	4.3	odd	2
1100.4.a.i.1.1		3	5.4	even	2
1100.4.b.h.749.2		6	5.2	odd	4
1100.4.b.h.749.5		6	5.3	odd	4
1980.4.a.l.1.3		3	3.2	odd	2
2420.4.a.i.1.3		3	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)

\(f(q)\)	\(=\)	\(q+6.86946 q^{3} -5.00000 q^{5} +15.2554 q^{7} +20.1894 q^{9} +11.0000 q^{11} +30.1909 q^{13} -34.3473 q^{15} +74.4793 q^{17} +30.6126 q^{19} +104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} +23.5336 q^{29} -272.083 q^{31} +75.5640 q^{33} -76.2770 q^{35} +292.415 q^{37} +207.395 q^{39} -127.493 q^{41} -466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} +511.632 q^{51} +235.761 q^{53} -55.0000 q^{55} +210.292 q^{57} +167.162 q^{59} -363.291 q^{61} +307.998 q^{63} -150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} -372.861 q^{73} +171.736 q^{75} +167.809 q^{77} -300.829 q^{79} -866.502 q^{81} -1218.18 q^{83} -372.396 q^{85} +161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} -153.063 q^{95} -1389.82 q^{97} +222.084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 15 q^{5} - 5 q^{7} + 24 q^{9} + 33 q^{11} + 2 q^{13} + 15 q^{15} + 77 q^{17} + 171 q^{19} + 259 q^{21} + 222 q^{23} + 75 q^{25} + 111 q^{27} + 55 q^{29} + 181 q^{31} - 33 q^{33} + 25 q^{35}+ \cdots + 264 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 15 * q^5 - 5 * q^7 + 24 * q^9 + 33 * q^11 + 2 * q^13 + 15 * q^15 + 77 * q^17 + 171 * q^19 + 259 * q^21 + 222 * q^23 + 75 * q^25 + 111 * q^27 + 55 * q^29 + 181 * q^31 - 33 * q^33 + 25 * q^35 + 317 * q^37 + 554 * q^39 + 302 * q^41 - 188 * q^43 - 120 * q^45 + 662 * q^47 - 268 * q^49 + 833 * q^51 + 81 * q^53 - 165 * q^55 - 825 * q^57 - 42 * q^59 + 349 * q^61 - 266 * q^63 - 10 * q^65 + 152 * q^67 - 258 * q^69 + 927 * q^71 - 2378 * q^73 - 75 * q^75 - 55 * q^77 - 1146 * q^79 - 1533 * q^81 - 458 * q^83 - 385 * q^85 + 1087 * q^87 - 875 * q^89 + 1982 * q^91 - 3623 * q^93 - 855 * q^95 - 1980 * q^97 + 264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more