LMFDB - Embedded newform 2420.4.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2420,4,Mod(1,2420)] 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("2420.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2420.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-15,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$142.784622214$
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:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.67648$ of defining polynomial
Character	$\chi$	$=$	2420.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.86946 q^{3} -5.00000 q^{5} -15.2554 q^{7} +20.1894 q^{9} -30.1909 q^{13} -34.3473 q^{15} -74.4793 q^{17} -30.6126 q^{19} -104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} -23.5336 q^{29} -272.083 q^{31} +76.2770 q^{35} +292.415 q^{37} -207.395 q^{39} +127.493 q^{41} +466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} -511.632 q^{51} +235.761 q^{53} -210.292 q^{57} +167.162 q^{59} +363.291 q^{61} -307.998 q^{63} +150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} +372.861 q^{73} +171.736 q^{75} +300.829 q^{79} -866.502 q^{81} +1218.18 q^{83} +372.396 q^{85} -161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} +153.063 q^{95} -1389.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 15 q^{5} + 5 q^{7} + 24 q^{9} - 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} - 25 q^{35} + 317 q^{37} - 554 q^{39}+ \cdots - 1980 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 - 15 * q^5 + 5 * q^7 + 24 * q^9 - 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 - 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 + 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 + 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

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	$-0.635120\pi$
$7$	−15.2554	−0.823715	−0.411857	+	0.911248i	$0.635120\pi$
$8$	0	0
$9$	20.1894	0.747756
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−30.1909	−0.644110	−0.322055	−	0.946721i	$-0.604374\pi$
$13$	−30.1909	−0.644110	−0.322055	+	0.946721i	$0.604374\pi$
$14$	0	0
$15$	−34.3473	−0.591229
$16$	0	0
$17$	−74.4793	−1.06258	−0.531290	−	0.847190i	$-0.678293\pi$
$17$	−74.4793	−1.06258	−0.531290	+	0.847190i	$0.678293\pi$
$18$	0	0
$19$	−30.6126	−0.369632	−0.184816	−	0.982773i	$-0.559169\pi$
$19$	−30.6126	−0.369632	−0.184816	+	0.982773i	$0.559169\pi$
$20$	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	$-0.524006\pi$
$29$	−23.5336	−0.150692	−0.0753462	+	0.997157i	$0.524006\pi$
$30$	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
$33$	0	0
$34$	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	$-0.421929\pi$
$41$	127.493	0.485634	0.242817	+	0.970072i	$0.421929\pi$
$42$	0	0
$43$	466.210	1.65341	0.826703	−	0.562639i	$-0.190214\pi$
$43$	466.210	1.65341	0.826703	+	0.562639i	$0.190214\pi$
$44$	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
$55$	0	0
$56$	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	$-0.375488\pi$
$61$	363.291	0.762535	0.381267	+	0.924465i	$0.375488\pi$
$62$	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	$-0.403379\pi$
$73$	372.861	0.597810	0.298905	+	0.954283i	$0.403379\pi$
$74$	0	0
$75$	171.736	0.264405
$76$	0	0
$77$	0	0
$78$	0	0
$79$	300.829	0.428430	0.214215	−	0.976787i	$-0.431281\pi$
$79$	300.829	0.428430	0.214215	+	0.976787i	$0.431281\pi$
$80$	0	0
$81$	−866.502	−1.18862
$82$	0	0
$83$	1218.18	1.61100	0.805498	−	0.592599i	$-0.201898\pi$
$83$	1218.18	1.61100	0.805498	+	0.592599i	$0.201898\pi$
$84$	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
$99$	0	0
$100$	0	0
$101$	321.478	0.316715	0.158358	−	0.987382i	$-0.449380\pi$
$101$	321.478	0.316715	0.158358	+	0.987382i	$0.449380\pi$
$102$	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	$-0.372047\pi$
$107$	866.060	0.782478	0.391239	+	0.920289i	$0.372047\pi$
$108$	0	0
$109$	1407.61	1.23693	0.618463	−	0.785814i	$-0.287756\pi$
$109$	1407.61	1.23693	0.618463	+	0.785814i	$0.287756\pi$
$110$	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
$121$	0	0
$122$	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	$-0.455521\pi$
$127$	398.684	0.278563	0.139282	+	0.990253i	$0.455521\pi$
$128$	0	0
$129$	3202.61	2.18585
$130$	0	0
$131$	−1504.91	−1.00370	−0.501849	−	0.864955i	$-0.667347\pi$
$131$	−1504.91	−1.00370	−0.501849	+	0.864955i	$0.667347\pi$
$132$	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	$-0.660294\pi$
$139$	−1581.63	−0.965125	−0.482563	+	0.875861i	$0.660294\pi$
$140$	0	0
$141$	2956.74	1.76597
$142$	0	0
$143$	0	0
$144$	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	$-0.313062\pi$
$149$	2015.57	1.10820	0.554101	+	0.832449i	$0.313062\pi$
$150$	0	0
$151$	−2275.41	−1.22629	−0.613146	−	0.789969i	$-0.710096\pi$
$151$	−2275.41	−1.22629	−0.613146	+	0.789969i	$0.710096\pi$
$152$	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
$165$	0	0
$166$	0	0
$167$	−1047.55	−0.485401	−0.242701	−	0.970101i	$-0.578033\pi$
$167$	−1047.55	−0.485401	−0.242701	+	0.970101i	$0.578033\pi$
$168$	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	$-0.305794\pi$
$173$	2607.51	1.14593	0.572963	+	0.819581i	$0.305794\pi$
$174$	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
$187$	0	0
$188$	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	$-0.361806\pi$
$193$	2255.66	0.841275	0.420638	+	0.907229i	$0.361806\pi$
$194$	0	0
$195$	1036.97	0.380817
$196$	0	0
$197$	−3284.67	−1.18793	−0.593967	−	0.804490i	$-0.702439\pi$
$197$	−3284.67	−1.18793	−0.593967	+	0.804490i	$0.702439\pi$
$198$	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
$209$	0	0
$210$	0	0
$211$	4649.74	1.51707	0.758533	−	0.651634i	$-0.225916\pi$
$211$	4649.74	1.51707	0.758533	+	0.651634i	$0.225916\pi$
$212$	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	$-0.175019\pi$
$227$	5832.01	1.70522	0.852608	+	0.522551i	$0.175019\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	919.503	0.258535	0.129267	−	0.991610i	$-0.458737\pi$
$233$	919.503	0.258535	0.129267	+	0.991610i	$0.458737\pi$
$234$	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	$-0.731764\pi$
$239$	−4917.54	−1.33092	−0.665459	+	0.746434i	$0.731764\pi$
$240$	0	0
$241$	−2908.06	−0.777282	−0.388641	−	0.921389i	$-0.627055\pi$
$241$	−2908.06	−0.777282	−0.388641	+	0.921389i	$0.627055\pi$
$242$	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
$253$	0	0
$254$	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	$-0.637238\pi$
$263$	−3564.90	−0.835822	−0.417911	+	0.908488i	$0.637238\pi$
$264$	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	$-0.211015\pi$
$271$	7032.65	1.57639	0.788197	+	0.615423i	$0.211015\pi$
$272$	0	0
$273$	3163.89	0.701419
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7403.56	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$277$	−7403.56	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$278$	0	0
$279$	−5493.19	−1.17874
$280$	0	0
$281$	−5709.48	−1.21210	−0.606048	−	0.795428i	$-0.707246\pi$
$281$	−5709.48	−1.21210	−0.606048	+	0.795428i	$0.707246\pi$
$282$	0	0
$283$	3030.73	0.636602	0.318301	−	0.947990i	$-0.396888\pi$
$283$	3030.73	0.636602	0.318301	+	0.947990i	$0.396888\pi$
$284$	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	$-0.582140\pi$
$293$	−2559.81	−0.510395	−0.255197	+	0.966889i	$0.582140\pi$
$294$	0	0
$295$	−835.811	−0.164959
$296$	0	0
$297$	0	0
$298$	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	$-0.650461\pi$
$307$	−4897.98	−0.910561	−0.455281	+	0.890348i	$0.650461\pi$
$308$	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
$319$	0	0
$320$	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	$-0.385166\pi$
$337$	4367.51	0.705975	0.352987	+	0.935628i	$0.385166\pi$
$338$	0	0
$339$	−2930.45	−0.469499
$340$	0	0
$341$	0	0
$342$	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	$-0.760088\pi$
$347$	−9426.38	−1.45831	−0.729157	+	0.684347i	$0.760088\pi$
$348$	0	0
$349$	8924.60	1.36883	0.684417	−	0.729091i	$-0.260057\pi$
$349$	8924.60	1.36883	0.684417	+	0.729091i	$0.260057\pi$
$350$	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	$-0.115376\pi$
$359$	12720.2	1.87005	0.935026	+	0.354580i	$0.115376\pi$
$360$	0	0
$361$	−5921.87	−0.863372
$362$	0	0
$363$	0	0
$364$	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	$-0.764320\pi$
$373$	−10635.6	−1.47638	−0.738192	+	0.674591i	$0.764320\pi$
$374$	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
$385$	0	0
$386$	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
$407$	0	0
$408$	0	0
$409$	334.332	0.0404197	0.0202098	−	0.999796i	$-0.493567\pi$
$409$	334.332	0.0404197	0.0202098	+	0.999796i	$0.493567\pi$
$410$	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
$429$	0	0
$430$	0	0
$431$	10394.2	1.16165	0.580823	−	0.814030i	$-0.302731\pi$
$431$	10394.2	1.16165	0.580823	+	0.814030i	$0.302731\pi$
$432$	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	$-0.634472\pi$
$439$	−7542.46	−0.820005	−0.410002	+	0.912084i	$0.634472\pi$
$440$	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
$451$	0	0
$452$	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	$-0.217310\pi$
$457$	15159.8	1.55175	0.775873	+	0.630890i	$0.217310\pi$
$458$	0	0
$459$	3484.51	0.354342
$460$	0	0
$461$	−11761.3	−1.18824	−0.594118	−	0.804378i	$-0.702499\pi$
$461$	−11761.3	−1.18824	−0.594118	+	0.804378i	$0.702499\pi$
$462$	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
$473$	0	0
$474$	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	$-0.567812\pi$
$479$	−4432.99	−0.422857	−0.211428	+	0.977393i	$0.567812\pi$
$480$	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	$-0.571122\pi$
$491$	−4821.55	−0.443164	−0.221582	+	0.975142i	$0.571122\pi$
$492$	0	0
$493$	1752.77	0.160123
$494$	0	0
$495$	0	0
$496$	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	$-0.367032\pi$
$503$	9153.19	0.811373	0.405687	+	0.914012i	$0.367032\pi$
$504$	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
$517$	0	0
$518$	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	$-0.676841\pi$
$523$	−12616.5	−1.05484	−0.527420	+	0.849605i	$0.676841\pi$
$524$	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
$539$	0	0
$540$	0	0
$541$	21440.2	1.70386	0.851929	−	0.523657i	$-0.175433\pi$
$541$	21440.2	1.70386	0.851929	+	0.523657i	$0.175433\pi$
$542$	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	$-0.167166\pi$
$547$	22138.5	1.73048	0.865240	+	0.501358i	$0.167166\pi$
$548$	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	$-0.257279\pi$
$557$	18160.8	1.38151	0.690753	+	0.723090i	$0.257279\pi$
$558$	0	0
$559$	−14075.3	−1.06498
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11886.6	−0.889806	−0.444903	−	0.895579i	$-0.646762\pi$
$563$	−11886.6	−0.889806	−0.444903	+	0.895579i	$0.646762\pi$
$564$	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	$-0.244100\pi$
$569$	19547.3	1.44018	0.720092	+	0.693879i	$0.244100\pi$
$570$	0	0
$571$	6953.38	0.509614	0.254807	−	0.966992i	$-0.417988\pi$
$571$	6953.38	0.509614	0.254807	+	0.966992i	$0.417988\pi$
$572$	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
$583$	0	0
$584$	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	$-0.484435\pi$
$593$	1411.66	0.0977573	0.0488786	+	0.998805i	$0.484435\pi$
$594$	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	$-0.629913\pi$
$601$	−11695.5	−0.793793	−0.396897	+	0.917863i	$0.629913\pi$
$602$	0	0
$603$	12341.7	0.833489
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.57260	−0.000506363 0	−0.000253181	−	1.00000i	$-0.500081\pi$
$607$	−7.57260	−0.000506363 0	−0.000253181		1.00000i	$0.500081\pi$
$608$	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	$-0.272537\pi$
$613$	19891.5	1.31062	0.655311	+	0.755359i	$0.272537\pi$
$614$	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
$627$	0	0
$628$	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
$649$	0	0
$650$	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	$-0.696121\pi$
$659$	−19552.3	−1.15576	−0.577882	+	0.816120i	$0.696121\pi$
$660$	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
$671$	0	0
$672$	0	0
$673$	−24272.9	−1.39027	−0.695135	−	0.718879i	$-0.744656\pi$
$673$	−24272.9	−1.39027	−0.695135	+	0.718879i	$0.744656\pi$
$674$	0	0
$675$	−1169.62	−0.0666946
$676$	0	0
$677$	18258.3	1.03652	0.518261	−	0.855223i	$-0.326580\pi$
$677$	18258.3	1.03652	0.518261	+	0.855223i	$0.326580\pi$
$678$	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
$693$	0	0
$694$	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	$-0.380358\pi$
$701$	13625.9	0.734158	0.367079	+	0.930190i	$0.380358\pi$
$702$	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
$715$	0	0
$716$	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	$-0.729768\pi$
$733$	−26226.1	−1.32153	−0.660765	+	0.750593i	$0.729768\pi$
$734$	0	0
$735$	3787.56	0.190077
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−37686.2	−1.87592	−0.937962	−	0.346737i	$-0.887290\pi$
$739$	−37686.2	−1.87592	−0.937962	+	0.346737i	$0.887290\pi$
$740$	0	0
$741$	6348.89	0.314754
$742$	0	0
$743$	5461.18	0.269652	0.134826	−	0.990869i	$-0.456952\pi$
$743$	5461.18	0.269652	0.134826	+	0.990869i	$0.456952\pi$
$744$	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
$759$	0	0
$760$	0	0
$761$	22985.6	1.09491	0.547456	−	0.836835i	$-0.315596\pi$
$761$	22985.6	1.09491	0.547456	+	0.836835i	$0.315596\pi$
$762$	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	$-0.472235\pi$
$769$	3715.50	0.174232	0.0871159	+	0.996198i	$0.472235\pi$
$770$	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
$781$	0	0
$782$	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	$-0.539159\pi$
$787$	−5418.55	−0.245426	−0.122713	+	0.992442i	$0.539159\pi$
$788$	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
$803$	0	0
$804$	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	$-0.220803\pi$
$809$	35385.4	1.53780	0.768902	+	0.639366i	$0.220803\pi$
$810$	0	0
$811$	−11033.5	−0.477731	−0.238865	−	0.971053i	$-0.576775\pi$
$811$	−11033.5	−0.477731	−0.238865	+	0.971053i	$0.576775\pi$
$812$	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	$-0.496957\pi$
$821$	449.705	0.0191167	0.00955834	+	0.999954i	$0.496957\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−33096.0	−1.39161	−0.695804	−	0.718232i	$-0.744952\pi$
$827$	−33096.0	−1.39161	−0.695804	+	0.718232i	$0.744952\pi$
$828$	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
$847$	0	0
$848$	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	$-0.277344\pi$
$853$	32079.3	1.28766	0.643830	+	0.765169i	$0.277344\pi$
$854$	0	0
$855$	3090.25	0.123608
$856$	0	0
$857$	−15049.5	−0.599862	−0.299931	−	0.953961i	$-0.596964\pi$
$857$	−15049.5	−0.599862	−0.299931	+	0.953961i	$0.596964\pi$
$858$	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
$869$	0	0
$870$	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	$-0.309412\pi$
$877$	29275.7	1.12722	0.563609	+	0.826041i	$0.309412\pi$
$878$	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	$-0.481306\pi$
$887$	3101.20	0.117393	0.0586967	+	0.998276i	$0.481306\pi$
$888$	0	0
$889$	−6082.09	−0.229457
$890$	0	0
$891$	0	0
$892$	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
$913$	0	0
$914$	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	$-0.740366\pi$
$919$	−38188.9	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$920$	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
$935$	0	0
$936$	0	0
$937$	−6058.27	−0.211222	−0.105611	−	0.994408i	$-0.533680\pi$
$937$	−6058.27	−0.211222	−0.105611	+	0.994408i	$0.533680\pi$
$938$	0	0
$939$	−17494.7	−0.608007
$940$	0	0
$941$	−32923.7	−1.14058	−0.570288	−	0.821445i	$-0.693168\pi$
$941$	−32923.7	−1.14058	−0.570288	+	0.821445i	$0.693168\pi$
$942$	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	$-0.549718\pi$
$953$	−9152.99	−0.311117	−0.155558	+	0.987827i	$0.549718\pi$
$954$	0	0
$955$	−20567.3	−0.696901
$956$	0	0
$957$	0	0
$958$	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	$-0.524033\pi$
$967$	−4536.39	−0.150859	−0.0754294	+	0.997151i	$0.524033\pi$
$968$	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
$979$	0	0
$980$	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	$-0.245571\pi$
$997$	45135.4	1.43375	0.716877	+	0.697200i	$0.245571\pi$
$998$	0	0
$999$	−13680.6	−0.433270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.4.a.i.1.3		3
11.10	odd	2		220.4.a.f.1.3	✓	3
33.32	even	2		1980.4.a.l.1.3		3
44.43	even	2		880.4.a.w.1.1		3
55.32	even	4		1100.4.b.h.749.2		6
55.43	even	4		1100.4.b.h.749.5		6
55.54	odd	2		1100.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.f.1.3	✓	3	11.10	odd	2
880.4.a.w.1.1		3	44.43	even	2
1100.4.a.i.1.1		3	55.54	odd	2
1100.4.b.h.749.2		6	55.32	even	4
1100.4.b.h.749.5		6	55.43	even	4
1980.4.a.l.1.3		3	33.32	even	2
2420.4.a.i.1.3		3	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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2420.a (trivial)

\(f(q)\)	\(=\)	\(q+6.86946 q^{3} -5.00000 q^{5} -15.2554 q^{7} +20.1894 q^{9} -30.1909 q^{13} -34.3473 q^{15} -74.4793 q^{17} -30.6126 q^{19} -104.796 q^{21} +111.339 q^{23} +25.0000 q^{25} -46.7850 q^{27} -23.5336 q^{29} -272.083 q^{31} +76.2770 q^{35} +292.415 q^{37} -207.395 q^{39} +127.493 q^{41} +466.210 q^{43} -100.947 q^{45} +430.418 q^{47} -110.273 q^{49} -511.632 q^{51} +235.761 q^{53} -210.292 q^{57} +167.162 q^{59} +363.291 q^{61} -307.998 q^{63} +150.954 q^{65} +611.297 q^{67} +764.836 q^{69} -315.466 q^{71} +372.861 q^{73} +171.736 q^{75} +300.829 q^{79} -866.502 q^{81} +1218.18 q^{83} +372.396 q^{85} -161.663 q^{87} +73.6581 q^{89} +460.574 q^{91} -1869.06 q^{93} +153.063 q^{95} -1389.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 15 q^{5} + 5 q^{7} + 24 q^{9} - 2 q^{13} + 15 q^{15} - 77 q^{17} - 171 q^{19} - 259 q^{21} + 222 q^{23} + 75 q^{25} + 111 q^{27} - 55 q^{29} + 181 q^{31} - 25 q^{35} + 317 q^{37} - 554 q^{39}+ \cdots - 1980 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 - 15 * q^5 + 5 * q^7 + 24 * q^9 - 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 - 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 + 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 + 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

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