LMFDB - Embedded newform 2100.3.j.g.601.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,3,Mod(601,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.601"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$57.2208555157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 56 x^{14} + 2052 x^{12} - 43310 x^{10} + 663499 x^{8} - 6680748 x^{6} + 49052709 x^{4} + \cdots + 601475625 $ x^16 - 56x^14 + 2052x^12 - 43310x^10 + 663499x^8 - 6680748x^6 + 49052709x^4 - 213293925*x^2 + 601475625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.5
Root			$-4.32353 - 2.49619i$ of defining polynomial
Character	$\chi$	$=$	2100.601
Dual form			2100.3.j.g.601.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(1.06651 - 6.91828i) q^{7} -3.00000 q^{9} +3.07378 q^{11} -17.8525i q^{13} -24.0048i q^{17} +32.7193i q^{19} +(-11.9828 - 1.84726i) q^{21} -24.3154 q^{23} +5.19615i q^{27} +18.6560 q^{29} -44.2221i q^{31} -5.32394i q^{33} -37.2443 q^{37} -30.9215 q^{39} -57.4609i q^{41} +59.9230 q^{43} +17.6682i q^{47} +(-46.7251 - 14.7569i) q^{49} -41.5775 q^{51} -18.8050 q^{53} +56.6714 q^{57} +85.2600i q^{59} +75.7642i q^{61} +(-3.19954 + 20.7548i) q^{63} -29.6009 q^{67} +42.1154i q^{69} +47.5533 q^{71} +11.0469i q^{73} +(3.27823 - 21.2653i) q^{77} -40.4910 q^{79} +9.00000 q^{81} -90.4984i q^{83} -32.3132i q^{87} -111.158i q^{89} +(-123.509 - 19.0400i) q^{91} -76.5950 q^{93} +26.3778i q^{97} -9.22134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) $ 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.06651	−	6.91828i	0.152359	−	0.988325i
$7$	1.06651	−	6.91828i	0.152359	−	0.988325i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	3.07378			0.279435			0.139717	−	0.990191i	$-0.455381\pi$
$11$	3.07378			0.279435			0.139717	+	0.990191i	$0.455381\pi$
$12$		0			0
$13$		−	17.8525i		−	1.37327i	−0.727001	−	0.686636i	$-0.759087\pi$
$13$		−	17.8525i		−	1.37327i	0.727001	−	0.686636i	$-0.240913\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	24.0048i		−	1.41205i	−0.708188	−	0.706024i	$-0.750487\pi$
$17$		−	24.0048i		−	1.41205i	0.708188	−	0.706024i	$-0.249513\pi$
$18$		0			0
$19$			32.7193i			1.72207i	0.508548	+	0.861033i	$0.330182\pi$
$19$			32.7193i			1.72207i	−0.508548	+	0.861033i	$0.669818\pi$
$20$		0			0
$21$	−11.9828	−	1.84726i	−0.570610	−	0.0879647i
$22$		0			0
$23$	−24.3154			−1.05719			−0.528595	−	0.848874i	$-0.677281\pi$
$23$	−24.3154			−1.05719			−0.528595	+	0.848874i	$0.677281\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	18.6560			0.643312			0.321656	−	0.946857i	$-0.395761\pi$
$29$	18.6560			0.643312			0.321656	+	0.946857i	$0.395761\pi$
$30$		0			0
$31$		−	44.2221i		−	1.42652i	−0.700899	−	0.713260i	$-0.747218\pi$
$31$		−	44.2221i		−	1.42652i	0.700899	−	0.713260i	$-0.252782\pi$
$32$		0			0
$33$		−	5.32394i		−	0.161332i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−37.2443			−1.00660			−0.503301	−	0.864111i	$-0.667881\pi$
$37$	−37.2443			−1.00660			−0.503301	+	0.864111i	$0.667881\pi$
$38$		0			0
$39$	−30.9215			−0.792859
$40$		0			0
$41$		−	57.4609i		−	1.40149i	−0.713414	−	0.700743i	$-0.752852\pi$
$41$		−	57.4609i		−	1.40149i	0.713414	−	0.700743i	$-0.247148\pi$
$42$		0			0
$43$	59.9230			1.39356			0.696780	−	0.717285i	$-0.254616\pi$
$43$	59.9230			1.39356			0.696780	+	0.717285i	$0.254616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			17.6682i			0.375919i	0.982177	+	0.187959i	$0.0601873\pi$
$47$			17.6682i			0.375919i	−0.982177	+	0.187959i	$0.939813\pi$
$48$		0			0
$49$	−46.7251	−	14.7569i	−0.953573	−	0.301161i
$50$		0			0
$51$	−41.5775			−0.815246
$52$		0			0
$53$	−18.8050			−0.354811			−0.177405	−	0.984138i	$-0.556770\pi$
$53$	−18.8050			−0.354811			−0.177405	+	0.984138i	$0.556770\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	56.6714			0.994236
$58$		0			0
$59$			85.2600i			1.44508i	0.691327	+	0.722542i	$0.257026\pi$
$59$			85.2600i			1.44508i	−0.691327	+	0.722542i	$0.742974\pi$
$60$		0			0
$61$			75.7642i			1.24204i	0.783796	+	0.621018i	$0.213281\pi$
$61$			75.7642i			1.24204i	−0.783796	+	0.621018i	$0.786719\pi$
$62$		0			0
$63$	−3.19954	+	20.7548i	−0.0507864	+	0.329442i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−29.6009			−0.441804			−0.220902	−	0.975296i	$-0.570900\pi$
$67$	−29.6009			−0.441804			−0.220902	+	0.975296i	$0.570900\pi$
$68$		0			0
$69$			42.1154i			0.610369i
$70$		0			0
$71$	47.5533			0.669765			0.334883	−	0.942260i	$-0.391303\pi$
$71$	47.5533			0.669765			0.334883	+	0.942260i	$0.391303\pi$
$72$		0			0
$73$			11.0469i			0.151328i	0.997133	+	0.0756640i	$0.0241076\pi$
$73$			11.0469i			0.151328i	−0.997133	+	0.0756640i	$0.975892\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.27823	−	21.2653i	0.0425744	−	0.276172i
$78$		0			0
$79$	−40.4910			−0.512545			−0.256272	−	0.966605i	$-0.582494\pi$
$79$	−40.4910			−0.512545			−0.256272	+	0.966605i	$0.582494\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	90.4984i		−	1.09034i	−0.838325	−	0.545171i	$-0.816465\pi$
$83$		−	90.4984i		−	1.09034i	0.838325	−	0.545171i	$-0.183535\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	32.3132i		−	0.371416i
$88$		0			0
$89$		−	111.158i		−	1.24896i	−0.781040	−	0.624481i	$-0.785311\pi$
$89$		−	111.158i		−	1.24896i	0.781040	−	0.624481i	$-0.214689\pi$
$90$		0			0
$91$	−123.509	−	19.0400i	−1.35724	−	0.209231i
$92$		0			0
$93$	−76.5950			−0.823602
$94$		0			0
$95$		0			0
$96$		0			0
$97$			26.3778i			0.271936i	0.990713	+	0.135968i	$0.0434145\pi$
$97$			26.3778i			0.271936i	−0.990713	+	0.135968i	$0.956586\pi$
$98$		0			0
$99$	−9.22134			−0.0931448
$100$		0			0
$101$			111.348i			1.10246i	0.834355	+	0.551228i	$0.185841\pi$
$101$			111.348i			1.10246i	−0.834355	+	0.551228i	$0.814159\pi$
$102$		0			0
$103$		−	64.0140i		−	0.621495i	−0.950493	−	0.310747i	$-0.899421\pi$
$103$		−	64.0140i		−	0.621495i	0.950493	−	0.310747i	$-0.100579\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	111.016			1.03753			0.518765	−	0.854917i	$-0.326392\pi$
$107$	111.016			1.03753			0.518765	+	0.854917i	$0.326392\pi$
$108$		0			0
$109$	−146.751			−1.34634			−0.673171	−	0.739487i	$-0.735068\pi$
$109$	−146.751			−1.34634			−0.673171	+	0.739487i	$0.735068\pi$
$110$		0			0
$111$			64.5090i			0.581162i
$112$		0			0
$113$	−0.298184			−0.00263880			−0.00131940	−	0.999999i	$-0.500420\pi$
$113$	−0.298184			−0.00263880			−0.00131940	+	0.999999i	$0.500420\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			53.5576i			0.457757i
$118$		0			0
$119$	−166.072	−	25.6015i	−1.39556	−	0.215138i
$120$		0			0
$121$	−111.552			−0.921916
$122$		0			0
$123$	−99.5252			−0.809148
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−175.074			−1.37854			−0.689268	−	0.724506i	$-0.742068\pi$
$127$	−175.074			−1.37854			−0.689268	+	0.724506i	$0.742068\pi$
$128$		0			0
$129$		−	103.790i		−	0.804572i
$130$		0			0
$131$		−	67.4457i		−	0.514853i	−0.966298	−	0.257426i	$-0.917126\pi$
$131$		−	67.4457i		−	0.514853i	0.966298	−	0.257426i	$-0.0828744\pi$
$132$		0			0
$133$	226.361	+	34.8956i	1.70196	+	0.262373i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	90.0204			0.657083			0.328541	−	0.944490i	$-0.393443\pi$
$137$	90.0204			0.657083			0.328541	+	0.944490i	$0.393443\pi$
$138$		0			0
$139$		−	81.8640i		−	0.588950i	−0.955659	−	0.294475i	$-0.904855\pi$
$139$		−	81.8640i		−	0.588950i	0.955659	−	0.294475i	$-0.0951448\pi$
$140$		0			0
$141$	30.6022			0.217037
$142$		0			0
$143$		−	54.8748i		−	0.383740i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−25.5597	+	80.9302i	−0.173875	+	0.550546i
$148$		0			0
$149$	−250.370			−1.68034			−0.840168	−	0.542327i	$-0.817544\pi$
$149$	−250.370			−1.68034			−0.840168	+	0.542327i	$0.817544\pi$
$150$		0			0
$151$	93.3221			0.618027			0.309014	−	0.951058i	$-0.400001\pi$
$151$	93.3221			0.618027			0.309014	+	0.951058i	$0.400001\pi$
$152$		0			0
$153$			72.0144i			0.470682i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			282.120i			1.79694i	0.439033	+	0.898471i	$0.355321\pi$
$157$			282.120i			1.79694i	−0.439033	+	0.898471i	$0.644679\pi$
$158$		0			0
$159$			32.5712i			0.204850i
$160$		0			0
$161$	−25.9327	+	168.220i	−0.161073	+	1.04485i
$162$		0			0
$163$	−168.152			−1.03161			−0.515804	−	0.856707i	$-0.672507\pi$
$163$	−168.152			−1.03161			−0.515804	+	0.856707i	$0.672507\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	293.030i		−	1.75467i	−0.479880	−	0.877334i	$-0.659320\pi$
$167$		−	293.030i		−	1.75467i	0.479880	−	0.877334i	$-0.340680\pi$
$168$		0			0
$169$	−149.713			−0.885875
$170$		0			0
$171$		−	98.1578i		−	0.574022i
$172$		0			0
$173$			291.545i			1.68523i	0.538514	+	0.842617i	$0.318986\pi$
$173$			291.545i			1.68523i	−0.538514	+	0.842617i	$0.681014\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	147.675			0.834320
$178$		0			0
$179$	−38.6361			−0.215844			−0.107922	−	0.994159i	$-0.534420\pi$
$179$	−38.6361			−0.215844			−0.107922	+	0.994159i	$0.534420\pi$
$180$		0			0
$181$			272.762i			1.50697i	0.657465	+	0.753485i	$0.271629\pi$
$181$			272.762i			1.50697i	−0.657465	+	0.753485i	$0.728371\pi$
$182$		0			0
$183$	131.228			0.717090
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	73.7855i		−	0.394575i
$188$		0			0
$189$	35.9484	+	5.54177i	0.190203	+	0.0293216i
$190$		0			0
$191$	−18.6600			−0.0976966			−0.0488483	−	0.998806i	$-0.515555\pi$
$191$	−18.6600			−0.0976966			−0.0488483	+	0.998806i	$0.515555\pi$
$192$		0			0
$193$	208.116			1.07832			0.539160	−	0.842203i	$-0.318742\pi$
$193$	208.116			1.07832			0.539160	+	0.842203i	$0.318742\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−249.955			−1.26881			−0.634404	−	0.773002i	$-0.718754\pi$
$197$	−249.955			−1.26881			−0.634404	+	0.773002i	$0.718754\pi$
$198$		0			0
$199$			85.1816i			0.428048i	0.976828	+	0.214024i	$0.0686571\pi$
$199$			85.1816i			0.428048i	−0.976828	+	0.214024i	$0.931343\pi$
$200$		0			0
$201$			51.2702i			0.255076i
$202$		0			0
$203$	19.8969	−	129.068i	0.0980145	−	0.635801i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	72.9461			0.352397
$208$		0			0
$209$			100.572i			0.481205i
$210$		0			0
$211$	64.2866			0.304676			0.152338	−	0.988328i	$-0.451320\pi$
$211$	64.2866			0.304676			0.152338	+	0.988328i	$0.451320\pi$
$212$		0			0
$213$		−	82.3648i		−	0.386689i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−305.941	−	47.1636i	−1.40987	−	0.217344i
$218$		0			0
$219$	19.1339			0.0873692
$220$		0			0
$221$	−428.546			−1.93912
$222$		0			0
$223$			234.568i			1.05187i	0.850524	+	0.525937i	$0.176285\pi$
$223$			234.568i			1.05187i	−0.850524	+	0.525937i	$0.823715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	318.302i		−	1.40221i	−0.713057	−	0.701106i	$-0.752690\pi$
$227$		−	318.302i		−	1.40221i	0.713057	−	0.701106i	$-0.247310\pi$
$228$		0			0
$229$		−	2.89432i		−	0.0126389i	−0.999980	−	0.00631947i	$-0.997988\pi$
$229$		−	2.89432i		−	0.0126389i	0.999980	−	0.00631947i	$-0.00201156\pi$
$230$		0			0
$231$	−36.8325	−	5.67806i	−0.159448	−	0.0245804i
$232$		0			0
$233$	−328.322			−1.40911			−0.704553	−	0.709652i	$-0.748852\pi$
$233$	−328.322			−1.40911			−0.704553	+	0.709652i	$0.748852\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			70.1326i			0.295918i
$238$		0			0
$239$	−356.042			−1.48972			−0.744858	−	0.667223i	$-0.767483\pi$
$239$	−356.042			−1.48972			−0.744858	+	0.667223i	$0.767483\pi$
$240$		0			0
$241$		−	190.743i		−	0.791466i	−0.918366	−	0.395733i	$-0.870491\pi$
$241$		−	190.743i		−	0.791466i	0.918366	−	0.395733i	$-0.129509\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	584.122			2.36487
$248$		0			0
$249$	−156.748			−0.629509
$250$		0			0
$251$		−	371.197i		−	1.47887i	−0.673226	−	0.739436i	$-0.735092\pi$
$251$		−	371.197i		−	1.47887i	0.673226	−	0.739436i	$-0.264908\pi$
$252$		0			0
$253$	−74.7401			−0.295415
$254$		0			0
$255$		0			0
$256$		0			0
$257$			79.6126i			0.309777i	0.987932	+	0.154888i	$0.0495018\pi$
$257$			79.6126i			0.309777i	−0.987932	+	0.154888i	$0.950498\pi$
$258$		0			0
$259$	−39.7216	+	257.666i	−0.153365	+	0.994851i
$260$		0			0
$261$	−55.9681			−0.214437
$262$		0			0
$263$	−197.790			−0.752052			−0.376026	−	0.926609i	$-0.622710\pi$
$263$	−197.790			−0.752052			−0.376026	+	0.926609i	$0.622710\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−192.531			−0.721089
$268$		0			0
$269$		−	38.5253i		−	0.143217i	−0.997433	−	0.0716084i	$-0.977187\pi$
$269$		−	38.5253i		−	0.143217i	0.997433	−	0.0716084i	$-0.0228132\pi$
$270$		0			0
$271$		−	187.184i		−	0.690717i	−0.938471	−	0.345359i	$-0.887757\pi$
$271$		−	187.184i		−	0.690717i	0.938471	−	0.345359i	$-0.112243\pi$
$272$		0			0
$273$	−32.9782	+	213.923i	−0.120799	+	0.783602i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	204.467			0.738147			0.369074	−	0.929400i	$-0.379675\pi$
$277$	204.467			0.738147			0.369074	+	0.929400i	$0.379675\pi$
$278$		0			0
$279$			132.666i			0.475507i
$280$		0			0
$281$	−180.683			−0.643000			−0.321500	−	0.946910i	$-0.604187\pi$
$281$	−180.683			−0.643000			−0.321500	+	0.946910i	$0.604187\pi$
$282$		0			0
$283$		−	47.9753i		−	0.169524i	−0.996401	−	0.0847619i	$-0.972987\pi$
$283$		−	47.9753i		−	0.169524i	0.996401	−	0.0847619i	$-0.0270130\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−397.531	−	61.2829i	−1.38512	−	0.213529i
$288$		0			0
$289$	−287.230			−0.993877
$290$		0			0
$291$	45.6877			0.157002
$292$		0			0
$293$		−	339.798i		−	1.15972i	−0.814717	−	0.579859i	$-0.803107\pi$
$293$		−	339.798i		−	1.15972i	0.814717	−	0.579859i	$-0.196893\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			15.9718i			0.0537772i
$298$		0			0
$299$			434.091i			1.45181i
$300$		0			0
$301$	63.9088	−	414.564i	0.212322	−	1.37729i
$302$		0			0
$303$	192.860			0.636503
$304$		0			0
$305$		0			0
$306$		0			0
$307$			394.637i			1.28546i	0.766092	+	0.642731i	$0.222199\pi$
$307$			394.637i			1.28546i	−0.766092	+	0.642731i	$0.777801\pi$
$308$		0			0
$309$	−110.875			−0.358820
$310$		0			0
$311$			341.095i			1.09677i	0.836227	+	0.548384i	$0.184757\pi$
$311$			341.095i			1.09677i	−0.836227	+	0.548384i	$0.815243\pi$
$312$		0			0
$313$			101.725i			0.325000i	0.986709	+	0.162500i	$0.0519557\pi$
$313$			101.725i			0.325000i	−0.986709	+	0.162500i	$0.948044\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	475.720			1.50069			0.750346	−	0.661045i	$-0.229887\pi$
$317$	475.720			1.50069			0.750346	+	0.661045i	$0.229887\pi$
$318$		0			0
$319$	57.3446			0.179763
$320$		0			0
$321$		−	192.285i		−	0.599019i
$322$		0			0
$323$	785.419			2.43164
$324$		0			0
$325$		0			0
$326$		0			0
$327$			254.181i			0.777311i
$328$		0			0
$329$	122.233	+	18.8434i	0.371530	+	0.0572747i
$330$		0			0
$331$	281.896			0.851651			0.425825	−	0.904805i	$-0.359984\pi$
$331$	281.896			0.851651			0.425825	+	0.904805i	$0.359984\pi$
$332$		0			0
$333$	111.733			0.335534
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−80.4587			−0.238750			−0.119375	−	0.992849i	$-0.538089\pi$
$337$	−80.4587			−0.238750			−0.119375	+	0.992849i	$0.538089\pi$
$338$		0			0
$339$			0.516470i			0.00152351i
$340$		0			0
$341$		−	135.929i		−	0.398619i
$342$		0			0
$343$	−151.925	+	307.519i	−0.442931	+	0.896556i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−197.979			−0.570546			−0.285273	−	0.958446i	$-0.592084\pi$
$347$	−197.979			−0.570546			−0.285273	+	0.958446i	$0.592084\pi$
$348$		0			0
$349$		−	116.788i		−	0.334637i	−0.985903	−	0.167318i	$-0.946489\pi$
$349$		−	116.788i		−	0.334637i	0.985903	−	0.167318i	$-0.0535108\pi$
$350$		0			0
$351$	92.7645			0.264286
$352$		0			0
$353$		−	439.644i		−	1.24545i	−0.782441	−	0.622725i	$-0.786026\pi$
$353$		−	439.644i		−	1.24545i	0.782441	−	0.622725i	$-0.213974\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−44.3431	+	287.645i	−0.124210	+	0.805728i
$358$		0			0
$359$	456.067			1.27038			0.635191	−	0.772355i	$-0.280922\pi$
$359$	456.067			1.27038			0.635191	+	0.772355i	$0.280922\pi$
$360$		0			0
$361$	−709.551			−1.96551
$362$		0			0
$363$			193.214i			0.532269i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	14.4035i		−	0.0392465i	−0.999807	−	0.0196233i	$-0.993753\pi$
$367$		−	14.4035i		−	0.0392465i	0.999807	−	0.0196233i	$-0.00624668\pi$
$368$		0			0
$369$			172.383i			0.467162i
$370$		0			0
$371$	−20.0558	+	130.098i	−0.0540587	+	0.350669i
$372$		0			0
$373$	−357.070			−0.957293			−0.478647	−	0.878008i	$-0.658873\pi$
$373$	−357.070			−0.957293			−0.478647	+	0.878008i	$0.658873\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	333.058i		−	0.883442i
$378$		0			0
$379$	−5.90977			−0.0155931			−0.00779653	−	0.999970i	$-0.502482\pi$
$379$	−5.90977			−0.0155931			−0.00779653	+	0.999970i	$0.502482\pi$
$380$		0			0
$381$			303.237i			0.795898i
$382$		0			0
$383$			334.550i			0.873500i	0.899583	+	0.436750i	$0.143870\pi$
$383$			334.550i			0.873500i	−0.899583	+	0.436750i	$0.856130\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−179.769			−0.464520
$388$		0			0
$389$	−297.061			−0.763653			−0.381826	−	0.924234i	$-0.624705\pi$
$389$	−297.061			−0.763653			−0.381826	+	0.924234i	$0.624705\pi$
$390$		0			0
$391$			583.685i			1.49280i
$392$		0			0
$393$	−116.819			−0.297250
$394$		0			0
$395$		0			0
$396$		0			0
$397$			189.999i			0.478586i	0.970947	+	0.239293i	$0.0769156\pi$
$397$			189.999i			0.478586i	−0.970947	+	0.239293i	$0.923084\pi$
$398$		0			0
$399$	60.4409	−	392.069i	0.151481	−	0.982628i
$400$		0			0
$401$	224.908			0.560869			0.280434	−	0.959873i	$-0.409522\pi$
$401$	224.908			0.560869			0.280434	+	0.959873i	$0.409522\pi$
$402$		0			0
$403$	−789.477			−1.95900
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−114.481			−0.281280
$408$		0			0
$409$			229.455i			0.561015i	0.959852	+	0.280508i	$0.0905028\pi$
$409$			229.455i			0.561015i	−0.959852	+	0.280508i	$0.909497\pi$
$410$		0			0
$411$		−	155.920i		−	0.379367i
$412$		0			0
$413$	589.852	+	90.9310i	1.42821	+	0.220172i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−141.793			−0.340030
$418$		0			0
$419$		−	171.731i		−	0.409859i	−0.978777	−	0.204929i	$-0.934304\pi$
$419$		−	171.731i		−	0.409859i	0.978777	−	0.204929i	$-0.0656965\pi$
$420$		0			0
$421$	253.361			0.601809			0.300904	−	0.953654i	$-0.402712\pi$
$421$	253.361			0.601809			0.300904	+	0.953654i	$0.402712\pi$
$422$		0			0
$423$		−	53.0046i		−	0.125306i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	524.158	+	80.8037i	1.22754	+	0.189236i
$428$		0			0
$429$	−95.0459			−0.221552
$430$		0			0
$431$	66.9553			0.155349			0.0776743	−	0.996979i	$-0.475251\pi$
$431$	66.9553			0.155349			0.0776743	+	0.996979i	$0.475251\pi$
$432$		0			0
$433$		−	255.445i		−	0.589941i	−0.955506	−	0.294971i	$-0.904690\pi$
$433$		−	255.445i		−	0.589941i	0.955506	−	0.294971i	$-0.0953099\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	795.581i		−	1.82055i
$438$		0			0
$439$		−	570.966i		−	1.30061i	−0.759675	−	0.650303i	$-0.774642\pi$
$439$		−	570.966i		−	1.30061i	0.759675	−	0.650303i	$-0.225358\pi$
$440$		0			0
$441$	140.175	+	44.2707i	0.317858	+	0.100387i
$442$		0			0
$443$	−234.490			−0.529323			−0.264661	−	0.964341i	$-0.585260\pi$
$443$	−234.490			−0.529323			−0.264661	+	0.964341i	$0.585260\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			433.654i			0.970142i
$448$		0			0
$449$	327.098			0.728503			0.364251	−	0.931301i	$-0.381325\pi$
$449$	327.098			0.728503			0.364251	+	0.931301i	$0.381325\pi$
$450$		0			0
$451$		−	176.622i		−	0.391624i
$452$		0			0
$453$		−	161.639i		−	0.356818i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−57.4776			−0.125772			−0.0628858	−	0.998021i	$-0.520030\pi$
$457$	−57.4776			−0.125772			−0.0628858	+	0.998021i	$0.520030\pi$
$458$		0			0
$459$	124.733			0.271749
$460$		0			0
$461$		−	202.731i		−	0.439764i	−0.975526	−	0.219882i	$-0.929433\pi$
$461$		−	202.731i		−	0.439764i	0.975526	−	0.219882i	$-0.0705673\pi$
$462$		0			0
$463$	235.364			0.508345			0.254172	−	0.967159i	$-0.418197\pi$
$463$	235.364			0.508345			0.254172	+	0.967159i	$0.418197\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	551.433i		−	1.18080i	−0.807111	−	0.590399i	$-0.798970\pi$
$467$		−	551.433i		−	1.18080i	0.807111	−	0.590399i	$-0.201030\pi$
$468$		0			0
$469$	−31.5698	+	204.787i	−0.0673130	+	0.436646i
$470$		0			0
$471$	488.646			1.03746
$472$		0			0
$473$	184.190			0.389409
$474$		0			0
$475$		0			0
$476$		0			0
$477$	56.4149			0.118270
$478$		0			0
$479$		−	863.232i		−	1.80215i	−0.433659	−	0.901077i	$-0.642778\pi$
$479$		−	863.232i		−	1.80215i	0.433659	−	0.901077i	$-0.357222\pi$
$480$		0			0
$481$			664.905i			1.38234i
$482$		0			0
$483$	291.366	+	44.9167i	0.603243	+	0.0929953i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	588.387			1.20819			0.604094	−	0.796913i	$-0.293535\pi$
$487$	588.387			1.20819			0.604094	+	0.796913i	$0.293535\pi$
$488$		0			0
$489$			291.248i			0.595599i
$490$		0			0
$491$	765.548			1.55916			0.779580	−	0.626303i	$-0.215433\pi$
$491$	765.548			1.55916			0.779580	+	0.626303i	$0.215433\pi$
$492$		0			0
$493$		−	447.834i		−	0.908386i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	50.7164	−	328.987i	0.102045	−	0.661946i
$498$		0			0
$499$	499.204			1.00041			0.500204	−	0.865908i	$-0.333258\pi$
$499$	499.204			1.00041			0.500204	+	0.865908i	$0.333258\pi$
$500$		0			0
$501$	−507.542			−1.01306
$502$		0			0
$503$		−	809.915i		−	1.61017i	−0.593160	−	0.805085i	$-0.702120\pi$
$503$		−	809.915i		−	1.61017i	0.593160	−	0.805085i	$-0.297880\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			259.310i			0.511460i
$508$		0			0
$509$		−	172.031i		−	0.337978i	−0.985618	−	0.168989i	$-0.945950\pi$
$509$		−	172.031i		−	0.337978i	0.985618	−	0.168989i	$-0.0540502\pi$
$510$		0			0
$511$	76.4258	+	11.7817i	0.149561	+	0.0230562i
$512$		0			0
$513$	−170.014			−0.331412
$514$		0			0
$515$		0			0
$516$		0			0
$517$			54.3081i			0.105045i
$518$		0			0
$519$	504.971			0.972970
$520$		0			0
$521$			18.5895i			0.0356804i	0.999841	+	0.0178402i	$0.00567901\pi$
$521$			18.5895i			0.0356804i	−0.999841	+	0.0178402i	$0.994321\pi$
$522$		0			0
$523$			111.743i			0.213657i	0.994277	+	0.106828i	$0.0340696\pi$
$523$			111.743i			0.213657i	−0.994277	+	0.106828i	$0.965930\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1061.54			−2.01431
$528$		0			0
$529$	62.2368			0.117650
$530$		0			0
$531$		−	255.780i		−	0.481695i
$532$		0			0
$533$	−1025.82			−1.92462
$534$		0			0
$535$		0			0
$536$		0			0
$537$			66.9197i			0.124618i
$538$		0			0
$539$	−143.623	−	45.3594i	−0.266461	−	0.0841548i
$540$		0			0
$541$	−401.344			−0.741855			−0.370928	−	0.928662i	$-0.620960\pi$
$541$	−401.344			−0.741855			−0.370928	+	0.928662i	$0.620960\pi$
$542$		0			0
$543$	472.437			0.870050
$544$		0			0
$545$		0			0
$546$		0			0
$547$	241.126			0.440816			0.220408	−	0.975408i	$-0.429261\pi$
$547$	241.126			0.440816			0.220408	+	0.975408i	$0.429261\pi$
$548$		0			0
$549$		−	227.293i		−	0.414012i
$550$		0			0
$551$			610.412i			1.10783i
$552$		0			0
$553$	−43.1843	+	280.128i	−0.0780910	+	0.506561i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−533.112			−0.957114			−0.478557	−	0.878056i	$-0.658840\pi$
$557$	−533.112			−0.957114			−0.478557	+	0.878056i	$0.658840\pi$
$558$		0			0
$559$		−	1069.78i		−	1.91374i
$560$		0			0
$561$	−127.800			−0.227808
$562$		0			0
$563$		−	965.363i		−	1.71468i	−0.514754	−	0.857338i	$-0.672117\pi$
$563$		−	965.363i		−	1.71468i	0.514754	−	0.857338i	$-0.327883\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9.59863	−	62.2645i	0.0169288	−	0.109814i
$568$		0			0
$569$	599.641			1.05385			0.526926	−	0.849911i	$-0.323345\pi$
$569$	599.641			1.05385			0.526926	+	0.849911i	$0.323345\pi$
$570$		0			0
$571$	287.986			0.504353			0.252177	−	0.967681i	$-0.418854\pi$
$571$	287.986			0.504353			0.252177	+	0.967681i	$0.418854\pi$
$572$		0			0
$573$			32.3201i			0.0564051i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	268.952i		−	0.466122i	−0.972462	−	0.233061i	$-0.925126\pi$
$577$		−	268.952i		−	0.466122i	0.972462	−	0.233061i	$-0.0748741\pi$
$578$		0			0
$579$		−	360.467i		−	0.622568i
$580$		0			0
$581$	−626.093	−	96.5179i	−1.07761	−	0.166124i
$582$		0			0
$583$	−57.8024			−0.0991464
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	784.787i		−	1.33695i	−0.743736	−	0.668473i	$-0.766948\pi$
$587$		−	784.787i		−	1.33695i	0.743736	−	0.668473i	$-0.233052\pi$
$588$		0			0
$589$	1446.92			2.45656
$590$		0			0
$591$			432.935i			0.732546i
$592$		0			0
$593$			45.1151i			0.0760794i	0.999276	+	0.0380397i	$0.0121113\pi$
$593$			45.1151i			0.0760794i	−0.999276	+	0.0380397i	$0.987889\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	147.539			0.247134
$598$		0			0
$599$	−75.0608			−0.125310			−0.0626551	−	0.998035i	$-0.519957\pi$
$599$	−75.0608			−0.125310			−0.0626551	+	0.998035i	$0.519957\pi$
$600$		0			0
$601$			149.958i			0.249514i	0.992187	+	0.124757i	$0.0398151\pi$
$601$			149.958i			0.249514i	−0.992187	+	0.124757i	$0.960185\pi$
$602$		0			0
$603$	88.8027			0.147268
$604$		0			0
$605$		0			0
$606$		0			0
$607$			244.933i			0.403515i	0.979436	+	0.201757i	$0.0646652\pi$
$607$			244.933i			0.403515i	−0.979436	+	0.201757i	$0.935335\pi$
$608$		0			0
$609$	−223.552	−	34.4625i	−0.367080	−	0.0565887i
$610$		0			0
$611$	315.422			0.516239
$612$		0			0
$613$	448.080			0.730963			0.365481	−	0.930819i	$-0.380904\pi$
$613$	448.080			0.730963			0.365481	+	0.930819i	$0.380904\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	617.239			1.00039			0.500194	−	0.865913i	$-0.333262\pi$
$617$	617.239			1.00039			0.500194	+	0.865913i	$0.333262\pi$
$618$		0			0
$619$			960.005i			1.55090i	0.631411	+	0.775449i	$0.282476\pi$
$619$			960.005i			1.55090i	−0.631411	+	0.775449i	$0.717524\pi$
$620$		0			0
$621$		−	126.346i		−	0.203456i
$622$		0			0
$623$	−769.019	−	118.551i	−1.23438	−	0.190291i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	174.196			0.277824
$628$		0			0
$629$			894.042i			1.42137i
$630$		0			0
$631$	898.089			1.42328			0.711639	−	0.702545i	$-0.247953\pi$
$631$	898.089			1.42328			0.711639	+	0.702545i	$0.247953\pi$
$632$		0			0
$633$		−	111.348i		−	0.175905i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−263.448	+	834.161i	−0.413576	+	1.30952i
$638$		0			0
$639$	−142.660			−0.223255
$640$		0			0
$641$	724.165			1.12974			0.564871	−	0.825179i	$-0.308926\pi$
$641$	724.165			1.12974			0.564871	+	0.825179i	$0.308926\pi$
$642$		0			0
$643$			624.036i			0.970507i	0.874373	+	0.485254i	$0.161273\pi$
$643$			624.036i			0.970507i	−0.874373	+	0.485254i	$0.838727\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	157.813i		−	0.243914i	−0.992535	−	0.121957i	$-0.961083\pi$
$647$		−	157.813i		−	0.243914i	0.992535	−	0.121957i	$-0.0389170\pi$
$648$		0			0
$649$			262.070i			0.403806i
$650$		0			0
$651$	−81.6897	+	529.905i	−0.125483	+	0.813987i
$652$		0			0
$653$	671.329			1.02807			0.514034	−	0.857770i	$-0.328150\pi$
$653$	671.329			1.02807			0.514034	+	0.857770i	$0.328150\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	33.1408i		−	0.0504426i
$658$		0			0
$659$	499.820			0.758452			0.379226	−	0.925304i	$-0.376190\pi$
$659$	499.820			0.758452			0.379226	+	0.925304i	$0.376190\pi$
$660$		0			0
$661$		−	1061.68i		−	1.60617i	−0.595866	−	0.803084i	$-0.703191\pi$
$661$		−	1061.68i		−	1.60617i	0.595866	−	0.803084i	$-0.296809\pi$
$662$		0			0
$663$			742.264i			1.11955i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−453.628			−0.680102
$668$		0			0
$669$	406.283			0.607299
$670$		0			0
$671$			232.883i			0.347068i
$672$		0			0
$673$	420.271			0.624474			0.312237	−	0.950004i	$-0.398922\pi$
$673$	420.271			0.624474			0.312237	+	0.950004i	$0.398922\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	471.364i		−	0.696254i	−0.937447	−	0.348127i	$-0.886818\pi$
$677$		−	471.364i		−	0.696254i	0.937447	−	0.348127i	$-0.113182\pi$
$678$		0			0
$679$	182.489	+	28.1323i	0.268761	+	0.0414320i
$680$		0			0
$681$	−551.316			−0.809568
$682$		0			0
$683$	390.212			0.571320			0.285660	−	0.958331i	$-0.407787\pi$
$683$	390.212			0.571320			0.285660	+	0.958331i	$0.407787\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−5.01311			−0.00729710
$688$		0			0
$689$			335.717i			0.487252i
$690$		0			0
$691$		−	453.717i		−	0.656609i	−0.944572	−	0.328305i	$-0.893523\pi$
$691$		−	453.717i		−	0.656609i	0.944572	−	0.328305i	$-0.106477\pi$
$692$		0			0
$693$	−9.83470	+	63.7958i	−0.0141915	+	0.0920574i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1379.34			−1.97896
$698$		0			0
$699$			568.670i			0.813547i
$700$		0			0
$701$	242.952			0.346579			0.173290	−	0.984871i	$-0.444560\pi$
$701$	242.952			0.346579			0.173290	+	0.984871i	$0.444560\pi$
$702$		0			0
$703$		−	1218.61i		−	1.73344i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	770.337	+	118.754i	1.08959	+	0.167969i
$708$		0			0
$709$	−485.352			−0.684559			−0.342279	−	0.939598i	$-0.611199\pi$
$709$	−485.352			−0.684559			−0.342279	+	0.939598i	$0.611199\pi$
$710$		0			0
$711$	121.473			0.170848
$712$		0			0
$713$			1075.28i			1.50810i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			616.683i			0.860088i
$718$		0			0
$719$			25.1312i			0.0349529i	0.999847	+	0.0174765i	$0.00556322\pi$
$719$			25.1312i			0.0349529i	−0.999847	+	0.0174765i	$0.994437\pi$
$720$		0			0
$721$	−442.866	−	68.2718i	−0.614239	−	0.0946905i
$722$		0			0
$723$	−330.377			−0.456953
$724$		0			0
$725$		0			0
$726$		0			0
$727$			332.426i			0.457258i	0.973514	+	0.228629i	$0.0734242\pi$
$727$			332.426i			0.457258i	−0.973514	+	0.228629i	$0.926576\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	1438.44i		−	1.96777i
$732$		0			0
$733$			1410.78i			1.92467i	0.271870	+	0.962334i	$0.412358\pi$
$733$			1410.78i			1.92467i	−0.271870	+	0.962334i	$0.587642\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−90.9866			−0.123455
$738$		0			0
$739$	928.931			1.25701			0.628505	−	0.777805i	$-0.283667\pi$
$739$	928.931			1.25701			0.628505	+	0.777805i	$0.283667\pi$
$740$		0			0
$741$		−	1011.73i		−	1.36536i
$742$		0			0
$743$	−451.882			−0.608186			−0.304093	−	0.952642i	$-0.598353\pi$
$743$	−451.882			−0.608186			−0.304093	+	0.952642i	$0.598353\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			271.495i			0.363447i
$748$		0			0
$749$	118.400	−	768.038i	0.158077	−	1.02542i
$750$		0			0
$751$	650.483			0.866156			0.433078	−	0.901356i	$-0.357427\pi$
$751$	650.483			0.866156			0.433078	+	0.901356i	$0.357427\pi$
$752$		0			0
$753$	−642.932			−0.853828
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1119.68			1.47910			0.739548	−	0.673104i	$-0.235039\pi$
$757$	1119.68			1.47910			0.739548	+	0.673104i	$0.235039\pi$
$758$		0			0
$759$			129.454i			0.170558i
$760$		0			0
$761$		−	1165.04i		−	1.53093i	−0.643479	−	0.765463i	$-0.722510\pi$
$761$		−	1165.04i		−	1.53093i	0.643479	−	0.765463i	$-0.277490\pi$
$762$		0			0
$763$	−156.512	+	1015.27i	−0.205128	+	1.33062i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1522.11			1.98449
$768$		0			0
$769$		−	1347.17i		−	1.75185i	−0.482446	−	0.875926i	$-0.660251\pi$
$769$		−	1347.17i		−	1.75185i	0.482446	−	0.875926i	$-0.339749\pi$
$770$		0			0
$771$	137.893			0.178850
$772$		0			0
$773$		−	508.563i		−	0.657908i	−0.944346	−	0.328954i	$-0.893304\pi$
$773$		−	508.563i		−	0.657908i	0.944346	−	0.328954i	$-0.106696\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	446.291	+	68.7998i	0.574377	+	0.0885455i
$778$		0			0
$779$	1880.08			2.41345
$780$		0			0
$781$	146.169			0.187156
$782$		0			0
$783$			96.9396i			0.123805i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	275.176i		−	0.349652i	−0.984599	−	0.174826i	$-0.944064\pi$
$787$		−	275.176i		−	0.349652i	0.984599	−	0.174826i	$-0.0559364\pi$
$788$		0			0
$789$			342.582i			0.434197i
$790$		0			0
$791$	−0.318018	+	2.06292i	−0.000402045	+	0.00260799i
$792$		0			0
$793$	1352.58			1.70565
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	2.56622i		−	0.00321985i	−0.999999	−	0.00160992i	$-0.999488\pi$
$797$		−	2.56622i		−	0.00321985i	0.999999	−	0.00160992i	$-0.000512455\pi$
$798$		0			0
$799$	424.121			0.530815
$800$		0			0
$801$			333.473i			0.416321i
$802$		0			0
$803$			33.9559i			0.0422862i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−66.7278			−0.0826863
$808$		0			0
$809$	1019.40			1.26007			0.630036	−	0.776566i	$-0.283040\pi$
$809$	1019.40			1.26007			0.630036	+	0.776566i	$0.283040\pi$
$810$		0			0
$811$			461.638i			0.569221i	0.958643	+	0.284610i	$0.0918642\pi$
$811$			461.638i			0.569221i	−0.958643	+	0.284610i	$0.908136\pi$
$812$		0			0
$813$	−324.213			−0.398786
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1960.64i			2.39980i
$818$		0			0
$819$	370.526	+	57.1200i	0.452413	+	0.0697436i
$820$		0			0
$821$	−336.204			−0.409505			−0.204752	−	0.978814i	$-0.565639\pi$
$821$	−336.204			−0.409505			−0.204752	+	0.978814i	$0.565639\pi$
$822$		0			0
$823$	64.0535			0.0778293			0.0389147	−	0.999243i	$-0.487610\pi$
$823$	64.0535			0.0778293			0.0389147	+	0.999243i	$0.487610\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1423.82			−1.72167			−0.860833	−	0.508888i	$-0.830057\pi$
$827$	−1423.82			−1.72167			−0.860833	+	0.508888i	$0.830057\pi$
$828$		0			0
$829$			871.031i			1.05070i	0.850886	+	0.525350i	$0.176066\pi$
$829$			871.031i			1.05070i	−0.850886	+	0.525350i	$0.823934\pi$
$830$		0			0
$831$		−	354.147i		−	0.426169i
$832$		0			0
$833$	−354.236	+	1121.63i	−0.425254	+	1.34649i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	229.785			0.274534
$838$		0			0
$839$			1346.65i			1.60506i	0.596611	+	0.802530i	$0.296513\pi$
$839$			1346.65i			1.60506i	−0.596611	+	0.802530i	$0.703487\pi$
$840$		0			0
$841$	−492.952			−0.586150
$842$		0			0
$843$			312.952i			0.371236i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−118.972	+	771.747i	−0.140463	+	0.911153i
$848$		0			0
$849$	−83.0956			−0.0978747
$850$		0			0
$851$	905.608			1.06417
$852$		0			0
$853$		−	1553.19i		−	1.82086i	−0.413666	−	0.910429i	$-0.635752\pi$
$853$		−	1553.19i		−	1.82086i	0.413666	−	0.910429i	$-0.364248\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	916.383i		−	1.06929i	−0.845076	−	0.534646i	$-0.820445\pi$
$857$		−	916.383i		−	1.06929i	0.845076	−	0.534646i	$-0.179555\pi$
$858$		0			0
$859$			220.008i			0.256121i	0.991766	+	0.128061i	$0.0408752\pi$
$859$			220.008i			0.256121i	−0.991766	+	0.128061i	$0.959125\pi$
$860$		0			0
$861$	−106.145	+	688.543i	−0.123281	+	0.799702i
$862$		0			0
$863$	1143.68			1.32523			0.662616	−	0.748959i	$-0.269446\pi$
$863$	1143.68			1.32523			0.662616	+	0.748959i	$0.269446\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			497.498i			0.573815i
$868$		0			0
$869$	−124.461			−0.143223
$870$		0			0
$871$			528.451i			0.606717i
$872$		0			0
$873$		−	79.1335i		−	0.0906454i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−711.003			−0.810722			−0.405361	−	0.914157i	$-0.632854\pi$
$877$	−711.003			−0.810722			−0.405361	+	0.914157i	$0.632854\pi$
$878$		0			0
$879$	−588.547			−0.669564
$880$		0			0
$881$			804.482i			0.913146i	0.889686	+	0.456573i	$0.150923\pi$
$881$			804.482i			0.913146i	−0.889686	+	0.456573i	$0.849077\pi$
$882$		0			0
$883$	−203.330			−0.230271			−0.115136	−	0.993350i	$-0.536730\pi$
$883$	−203.330			−0.230271			−0.115136	+	0.993350i	$0.536730\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	840.574i		−	0.947659i	−0.880617	−	0.473830i	$-0.842871\pi$
$887$		−	840.574i		−	0.947659i	0.880617	−	0.473830i	$-0.157129\pi$
$888$		0			0
$889$	−186.719	+	1211.21i	−0.210033	+	1.36244i
$890$		0			0
$891$	27.6640			0.0310483
$892$		0			0
$893$	−578.090			−0.647357
$894$		0			0
$895$		0			0
$896$		0			0
$897$	751.867			0.838202
$898$		0			0
$899$		−	825.010i		−	0.917697i
$900$		0			0
$901$			451.410i			0.501010i
$902$		0			0
$903$	−718.046	−	110.693i	−0.795179	−	0.122584i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−611.612			−0.674324			−0.337162	−	0.941447i	$-0.609467\pi$
$907$	−611.612			−0.674324			−0.337162	+	0.941447i	$0.609467\pi$
$908$		0			0
$909$		−	334.044i		−	0.367485i
$910$		0			0
$911$	169.118			0.185640			0.0928198	−	0.995683i	$-0.470412\pi$
$911$	169.118			0.185640			0.0928198	+	0.995683i	$0.470412\pi$
$912$		0			0
$913$		−	278.172i		−	0.304679i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−466.608	−	71.9318i	−0.508842	−	0.0784426i
$918$		0			0
$919$	−204.249			−0.222251			−0.111126	−	0.993806i	$-0.535446\pi$
$919$	−204.249			−0.222251			−0.111126	+	0.993806i	$0.535446\pi$
$920$		0			0
$921$	683.531			0.742162
$922$		0			0
$923$		−	848.948i		−	0.919770i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			192.042i			0.207165i
$928$		0			0
$929$		−	607.668i		−	0.654110i	−0.945005	−	0.327055i	$-0.893944\pi$
$929$		−	607.668i		−	0.654110i	0.945005	−	0.327055i	$-0.106056\pi$
$930$		0			0
$931$	482.835	−	1528.81i	0.518619	−	1.64212i
$932$		0			0
$933$	590.793			0.633219
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1226.32i		−	1.30878i	−0.756158	−	0.654389i	$-0.772926\pi$
$937$		−	1226.32i		−	1.30878i	0.756158	−	0.654389i	$-0.227074\pi$
$938$		0			0
$939$	176.193			0.187639
$940$		0			0
$941$			1570.92i			1.66942i	0.550693	+	0.834708i	$0.314363\pi$
$941$			1570.92i			1.66942i	−0.550693	+	0.834708i	$0.685637\pi$
$942$		0			0
$943$			1397.18i			1.48164i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	845.061			0.892356			0.446178	−	0.894944i	$-0.352785\pi$
$947$	845.061			0.892356			0.446178	+	0.894944i	$0.352785\pi$
$948$		0			0
$949$	197.216			0.207814
$950$		0			0
$951$		−	823.970i		−	0.866425i
$952$		0			0
$953$	−367.141			−0.385248			−0.192624	−	0.981273i	$-0.561700\pi$
$953$	−367.141			−0.385248			−0.192624	+	0.981273i	$0.561700\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	99.3237i		−	0.103786i
$958$		0			0
$959$	96.0080	−	622.786i	0.100113	−	0.649412i
$960$		0			0
$961$	−994.598			−1.03496
$962$		0			0
$963$	−333.047			−0.345844
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1013.20			−1.04777			−0.523886	−	0.851789i	$-0.675518\pi$
$967$	−1013.20			−1.04777			−0.523886	+	0.851789i	$0.675518\pi$
$968$		0			0
$969$		−	1360.39i		−	1.40391i
$970$		0			0
$971$			659.717i			0.679420i	0.940530	+	0.339710i	$0.110329\pi$
$971$			659.717i			0.679420i	−0.940530	+	0.339710i	$0.889671\pi$
$972$		0			0
$973$	−566.358	−	87.3092i	−0.582074	−	0.0897320i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−867.534			−0.887957			−0.443978	−	0.896037i	$-0.646433\pi$
$977$	−867.534			−0.887957			−0.443978	+	0.896037i	$0.646433\pi$
$978$		0			0
$979$		−	341.674i		−	0.349003i
$980$		0			0
$981$	440.254			0.448781
$982$		0			0
$983$		−	1757.82i		−	1.78822i	−0.447849	−	0.894109i	$-0.647810\pi$
$983$		−	1757.82i		−	1.78822i	0.447849	−	0.894109i	$-0.352190\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	32.6377	−	211.714i	0.0330676	−	0.214503i
$988$		0			0
$989$	−1457.05			−1.47326
$990$		0			0
$991$	395.645			0.399238			0.199619	−	0.979874i	$-0.436030\pi$
$991$	395.645			0.399238			0.199619	+	0.979874i	$0.436030\pi$
$992$		0			0
$993$		−	488.259i		−	0.491701i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1605.98i		−	1.61081i	−0.592724	−	0.805406i	$-0.701947\pi$
$997$		−	1605.98i		−	1.61081i	0.592724	−	0.805406i	$-0.298053\pi$
$998$		0			0
$999$		−	193.527i		−	0.193721i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.j.g.601.5		16
5.2	odd	4		420.3.p.a.349.3	✓	16
5.3	odd	4		420.3.p.a.349.13	yes	16
5.4	even	2	inner	2100.3.j.g.601.12		16
7.6	odd	2	inner	2100.3.j.g.601.13		16
15.2	even	4		1260.3.p.e.1189.10		16
15.8	even	4		1260.3.p.e.1189.8		16
20.3	even	4		1680.3.bd.b.769.5		16
20.7	even	4		1680.3.bd.b.769.11		16
35.13	even	4		420.3.p.a.349.4	yes	16
35.27	even	4		420.3.p.a.349.14	yes	16
35.34	odd	2	inner	2100.3.j.g.601.4		16
105.62	odd	4		1260.3.p.e.1189.7		16
105.83	odd	4		1260.3.p.e.1189.9		16
140.27	odd	4		1680.3.bd.b.769.6		16
140.83	odd	4		1680.3.bd.b.769.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.p.a.349.3	✓	16	5.2	odd	4
420.3.p.a.349.4	yes	16	35.13	even	4
420.3.p.a.349.13	yes	16	5.3	odd	4
420.3.p.a.349.14	yes	16	35.27	even	4
1260.3.p.e.1189.7		16	105.62	odd	4
1260.3.p.e.1189.8		16	15.8	even	4
1260.3.p.e.1189.9		16	105.83	odd	4
1260.3.p.e.1189.10		16	15.2	even	4
1680.3.bd.b.769.5		16	20.3	even	4
1680.3.bd.b.769.6		16	140.27	odd	4
1680.3.bd.b.769.11		16	20.7	even	4
1680.3.bd.b.769.12		16	140.83	odd	4
2100.3.j.g.601.4		16	35.34	odd	2	inner
2100.3.j.g.601.5		16	1.1	even	1	trivial
2100.3.j.g.601.12		16	5.4	even	2	inner
2100.3.j.g.601.13		16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(1.06651 - 6.91828i) q^{7} -3.00000 q^{9} +3.07378 q^{11} -17.8525i q^{13} -24.0048i q^{17} +32.7193i q^{19} +(-11.9828 - 1.84726i) q^{21} -24.3154 q^{23} +5.19615i q^{27} +18.6560 q^{29} -44.2221i q^{31} -5.32394i q^{33} -37.2443 q^{37} -30.9215 q^{39} -57.4609i q^{41} +59.9230 q^{43} +17.6682i q^{47} +(-46.7251 - 14.7569i) q^{49} -41.5775 q^{51} -18.8050 q^{53} +56.6714 q^{57} +85.2600i q^{59} +75.7642i q^{61} +(-3.19954 + 20.7548i) q^{63} -29.6009 q^{67} +42.1154i q^{69} +47.5533 q^{71} +11.0469i q^{73} +(3.27823 - 21.2653i) q^{77} -40.4910 q^{79} +9.00000 q^{81} -90.4984i q^{83} -32.3132i q^{87} -111.158i q^{89} +(-123.509 - 19.0400i) q^{91} -76.5950 q^{93} +26.3778i q^{97} -9.22134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 24 q^{11} - 12 q^{21} + 32 q^{29} - 72 q^{39} + 88 q^{49} + 24 q^{51} + 168 q^{71} - 16 q^{79} + 144 q^{81} - 568 q^{91} + 72 q^{99}+O(q^{100}) \) 16 * q - 48 * q^9 - 24 * q^11 - 12 * q^21 + 32 * q^29 - 72 * q^39 + 88 * q^49 + 24 * q^51 + 168 * q^71 - 16 * q^79 + 144 * q^81 - 568 * q^91 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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