LMFDB - Embedded newform 420.3.p.a.349.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [420,3,Mod(349,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 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("420.349"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	420.p (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$11.4441711031$
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} + 88 x^{14} + 3876 x^{12} + 102922 x^{10} + 1866070 x^{8} + 23190492 x^{6} + 203608845 x^{4} + \cdots + 3839661225 $ x^16 + 88x^14 + 3876x^12 + 102922x^10 + 1866070x^8 + 23190492x^6 + 203608845x^4 + 1131190758*x^2 + 3839661225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.3
Root			$-1.73205 + 4.99238i$ of defining polynomial
Character	$\chi$	$=$	420.349
Dual form			420.3.p.a.349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{3} +(-0.275940 - 4.99238i) q^{5} +(6.91828 + 1.06651i) q^{7} +3.00000 q^{9} +3.07378 q^{11} -17.8525 q^{13} +(0.477941 + 8.64706i) q^{15} +24.0048 q^{17} -32.7193i q^{19} +(-11.9828 - 1.84726i) q^{21} +24.3154i q^{23} +(-24.8477 + 2.75519i) q^{25} -5.19615 q^{27} -18.6560 q^{29} -44.2221i q^{31} -5.32394 q^{33} +(3.41542 - 34.8330i) q^{35} -37.2443i q^{37} +30.9215 q^{39} -57.4609i q^{41} -59.9230i q^{43} +(-0.827819 - 14.9771i) q^{45} -17.6682 q^{47} +(46.7251 + 14.7569i) q^{49} -41.5775 q^{51} +18.8050i q^{53} +(-0.848177 - 15.3455i) q^{55} +56.6714i q^{57} -85.2600i q^{59} +75.7642i q^{61} +(20.7548 + 3.19954i) q^{63} +(4.92622 + 89.1266i) q^{65} -29.6009i q^{67} -42.1154i q^{69} +47.5533 q^{71} +11.0469 q^{73} +(43.0375 - 4.77213i) q^{75} +(21.2653 + 3.27823i) q^{77} +40.4910 q^{79} +9.00000 q^{81} -90.4984 q^{83} +(-6.62387 - 119.841i) q^{85} +32.3132 q^{87} +111.158i q^{89} +(-123.509 - 19.0400i) q^{91} +76.5950i q^{93} +(-163.347 + 9.02854i) q^{95} -26.3778 q^{97} +9.22134 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} - 24 q^{11} - 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} + 76 q^{35} + 72 q^{39} - 88 q^{49} + 24 q^{51} + 152 q^{65} + 168 q^{71} + 16 q^{79} + 144 q^{81} - 416 q^{85} - 568 q^{91} + 136 q^{95}+ \cdots - 72 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 24 * q^11 - 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 + 76 * q^35 + 72 * q^39 - 88 * q^49 + 24 * q^51 + 152 * q^65 + 168 * q^71 + 16 * q^79 + 144 * q^81 - 416 * q^85 - 568 * q^91 + 136 * q^95 - 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	−0.275940	−	4.99238i	−0.0551879	−	0.998476i
$5$	−0.275940	−	4.99238i	−0.0551879	−	0.998476i
$6$		0			0
$7$	6.91828	+	1.06651i	0.988325	+	0.152359i
$7$	6.91828	+	1.06651i	0.988325	+	0.152359i
$8$		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
$13$	−17.8525			−1.37327			−0.686636	−	0.727001i	$-0.740913\pi$
$13$	−17.8525			−1.37327			−0.686636	+	0.727001i	$0.740913\pi$
$14$		0			0
$15$	0.477941	+	8.64706i	0.0318628	+	0.576470i
$16$		0			0
$17$	24.0048			1.41205			0.706024	−	0.708188i	$-0.250487\pi$
$17$	24.0048			1.41205			0.706024	+	0.708188i	$0.250487\pi$
$18$		0			0
$19$		−	32.7193i		−	1.72207i	−0.508548	−	0.861033i	$-0.669818\pi$
$19$		−	32.7193i		−	1.72207i	0.508548	−	0.861033i	$-0.330182\pi$
$20$		0			0
$21$	−11.9828	−	1.84726i	−0.570610	−	0.0879647i
$22$		0			0
$23$			24.3154i			1.05719i	0.848874	+	0.528595i	$0.177281\pi$
$23$			24.3154i			1.05719i	−0.848874	+	0.528595i	$0.822719\pi$
$24$		0			0
$25$	−24.8477	+	2.75519i	−0.993909	+	0.110208i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	−18.6560			−0.643312			−0.321656	−	0.946857i	$-0.604239\pi$
$29$	−18.6560			−0.643312			−0.321656	+	0.946857i	$0.604239\pi$
$30$		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
$33$	−5.32394			−0.161332
$34$		0			0
$35$	3.41542	−	34.8330i	0.0975835	−	0.995227i
$36$		0			0
$37$		−	37.2443i		−	1.00660i	−0.864111	−	0.503301i	$-0.832119\pi$
$37$		−	37.2443i		−	1.00660i	0.864111	−	0.503301i	$-0.167881\pi$
$38$		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
$43$		−	59.9230i		−	1.39356i	−0.717285	−	0.696780i	$-0.754616\pi$
$43$		−	59.9230i		−	1.39356i	0.717285	−	0.696780i	$-0.245384\pi$
$44$		0			0
$45$	−0.827819	−	14.9771i	−0.0183960	−	0.332825i
$46$		0			0
$47$	−17.6682			−0.375919			−0.187959	−	0.982177i	$-0.560187\pi$
$47$	−17.6682			−0.375919			−0.187959	+	0.982177i	$0.560187\pi$
$48$		0			0
$49$	46.7251	+	14.7569i	0.953573	+	0.301161i
$50$		0			0
$51$	−41.5775			−0.815246
$52$		0			0
$53$			18.8050i			0.354811i	0.984138	+	0.177405i	$0.0567704\pi$
$53$			18.8050i			0.354811i	−0.984138	+	0.177405i	$0.943230\pi$
$54$		0			0
$55$	−0.848177	−	15.3455i	−0.0154214	−	0.279009i
$56$		0			0
$57$			56.6714i			0.994236i
$58$		0			0
$59$		−	85.2600i		−	1.44508i	−0.691327	−	0.722542i	$-0.742974\pi$
$59$		−	85.2600i		−	1.44508i	0.691327	−	0.722542i	$-0.257026\pi$
$60$		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
$63$	20.7548	+	3.19954i	0.329442	+	0.0507864i
$64$		0			0
$65$	4.92622	+	89.1266i	0.0757880	+	1.37118i
$66$		0			0
$67$		−	29.6009i		−	0.441804i	−0.975296	−	0.220902i	$-0.929100\pi$
$67$		−	29.6009i		−	0.441804i	0.975296	−	0.220902i	$-0.0709001\pi$
$68$		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
$73$	11.0469			0.151328			0.0756640	−	0.997133i	$-0.475892\pi$
$73$	11.0469			0.151328			0.0756640	+	0.997133i	$0.475892\pi$
$74$		0			0
$75$	43.0375	−	4.77213i	0.573833	−	0.0636284i
$76$		0			0
$77$	21.2653	+	3.27823i	0.276172	+	0.0425744i
$78$		0			0
$79$	40.4910			0.512545			0.256272	−	0.966605i	$-0.417506\pi$
$79$	40.4910			0.512545			0.256272	+	0.966605i	$0.417506\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−90.4984			−1.09034			−0.545171	−	0.838325i	$-0.683535\pi$
$83$	−90.4984			−1.09034			−0.545171	+	0.838325i	$0.683535\pi$
$84$		0			0
$85$	−6.62387	−	119.841i	−0.0779279	−	1.40990i
$86$		0			0
$87$	32.3132			0.371416
$88$		0			0
$89$			111.158i			1.24896i	0.781040	+	0.624481i	$0.214689\pi$
$89$			111.158i			1.24896i	−0.781040	+	0.624481i	$0.785311\pi$
$90$		0			0
$91$	−123.509	−	19.0400i	−1.35724	−	0.209231i
$92$		0			0
$93$			76.5950i			0.823602i
$94$		0			0
$95$	−163.347	+	9.02854i	−1.71944	+	0.0950373i
$96$		0			0
$97$	−26.3778			−0.271936			−0.135968	−	0.990713i	$-0.543414\pi$
$97$	−26.3778			−0.271936			−0.135968	+	0.990713i	$0.543414\pi$
$98$		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
$103$	−64.0140			−0.621495			−0.310747	−	0.950493i	$-0.600579\pi$
$103$	−64.0140			−0.621495			−0.310747	+	0.950493i	$0.600579\pi$
$104$		0			0
$105$	−5.91568	+	60.3325i	−0.0563398	+	0.574595i
$106$		0			0
$107$			111.016i			1.03753i	0.854917	+	0.518765i	$0.173608\pi$
$107$			111.016i			1.03753i	−0.854917	+	0.518765i	$0.826392\pi$
$108$		0			0
$109$	146.751			1.34634			0.673171	−	0.739487i	$-0.264932\pi$
$109$	146.751			1.34634			0.673171	+	0.739487i	$0.264932\pi$
$110$		0			0
$111$			64.5090i			0.581162i
$112$		0			0
$113$			0.298184i			0.00263880i	0.999999	+	0.00131940i	$0.000419978\pi$
$113$			0.298184i			0.00263880i	−0.999999	+	0.00131940i	$0.999580\pi$
$114$		0			0
$115$	121.392	−	6.70957i	1.05558	−	0.0583441i
$116$		0			0
$117$	−53.5576			−0.457757
$118$		0			0
$119$	166.072	+	25.6015i	1.39556	+	0.215138i
$120$		0			0
$121$	−111.552			−0.921916
$122$		0			0
$123$			99.5252i			0.809148i
$124$		0			0
$125$	20.6114	+	123.289i	0.164891	+	0.986312i
$126$		0			0
$127$		−	175.074i		−	1.37854i	−0.724506	−	0.689268i	$-0.757932\pi$
$127$		−	175.074i		−	1.37854i	0.724506	−	0.689268i	$-0.242068\pi$
$128$		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
$133$	34.8956	−	226.361i	0.262373	−	1.70196i
$134$		0			0
$135$	1.43382	+	25.9412i	0.0106209	+	0.192157i
$136$		0			0
$137$			90.0204i			0.657083i	0.944490	+	0.328541i	$0.106557\pi$
$137$			90.0204i			0.657083i	−0.944490	+	0.328541i	$0.893443\pi$
$138$		0			0
$139$			81.8640i			0.588950i	0.955659	+	0.294475i	$0.0951448\pi$
$139$			81.8640i			0.588950i	−0.955659	+	0.294475i	$0.904855\pi$
$140$		0			0
$141$	30.6022			0.217037
$142$		0			0
$143$	−54.8748			−0.383740
$144$		0			0
$145$	5.14794	+	93.1380i	0.0355030	+	0.642331i
$146$		0			0
$147$	−80.9302	−	25.5597i	−0.550546	−	0.173875i
$148$		0			0
$149$	250.370			1.68034			0.840168	−	0.542327i	$-0.182456\pi$
$149$	250.370			1.68034			0.840168	+	0.542327i	$0.182456\pi$
$150$		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
$153$	72.0144			0.470682
$154$		0			0
$155$	−220.774	+	12.2026i	−1.42435	+	0.0787267i
$156$		0			0
$157$	−282.120			−1.79694			−0.898471	−	0.439033i	$-0.855321\pi$
$157$	−282.120			−1.79694			−0.898471	+	0.439033i	$0.855321\pi$
$158$		0			0
$159$		−	32.5712i		−	0.204850i
$160$		0			0
$161$	−25.9327	+	168.220i	−0.161073	+	1.04485i
$162$		0			0
$163$			168.152i			1.03161i	0.856707	+	0.515804i	$0.172507\pi$
$163$			168.152i			1.03161i	−0.856707	+	0.515804i	$0.827493\pi$
$164$		0			0
$165$	1.46909	+	26.5791i	0.00890355	+	0.161086i
$166$		0			0
$167$	293.030			1.75467			0.877334	−	0.479880i	$-0.159320\pi$
$167$	293.030			1.75467			0.877334	+	0.479880i	$0.159320\pi$
$168$		0			0
$169$	149.713			0.885875
$170$		0			0
$171$		−	98.1578i		−	0.574022i
$172$		0			0
$173$	291.545			1.68523			0.842617	−	0.538514i	$-0.181014\pi$
$173$	291.545			1.68523			0.842617	+	0.538514i	$0.181014\pi$
$174$		0			0
$175$	−174.842	−	7.43929i	−0.999096	−	0.0425102i
$176$		0			0
$177$			147.675i			0.834320i
$178$		0			0
$179$	38.6361			0.215844			0.107922	−	0.994159i	$-0.465580\pi$
$179$	38.6361			0.215844			0.107922	+	0.994159i	$0.465580\pi$
$180$		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
$183$		−	131.228i		−	0.717090i
$184$		0			0
$185$	−185.938	+	10.2772i	−1.00507	+	0.0555523i
$186$		0			0
$187$	73.7855			0.394575
$188$		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
$193$		−	208.116i		−	1.07832i	−0.842203	−	0.539160i	$-0.818742\pi$
$193$		−	208.116i		−	1.07832i	0.842203	−	0.539160i	$-0.181258\pi$
$194$		0			0
$195$	−8.53246	−	154.372i	−0.0437562	−	0.791651i
$196$		0			0
$197$		−	249.955i		−	1.26881i	−0.773002	−	0.634404i	$-0.781246\pi$
$197$		−	249.955i		−	1.26881i	0.773002	−	0.634404i	$-0.218754\pi$
$198$		0			0
$199$		−	85.1816i		−	0.428048i	−0.976828	−	0.214024i	$-0.931343\pi$
$199$		−	85.1816i		−	0.428048i	0.976828	−	0.214024i	$-0.0686571\pi$
$200$		0			0
$201$			51.2702i			0.255076i
$202$		0			0
$203$	−129.068	−	19.8969i	−0.635801	−	0.0980145i
$204$		0			0
$205$	−286.867	+	15.8557i	−1.39935	+	0.0773451i
$206$		0			0
$207$			72.9461i			0.352397i
$208$		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
$213$	−82.3648			−0.386689
$214$		0			0
$215$	−299.159	+	16.5351i	−1.39144	+	0.0769076i
$216$		0			0
$217$	47.1636	−	305.941i	0.217344	−	1.40987i
$218$		0			0
$219$	−19.1339			−0.0873692
$220$		0			0
$221$	−428.546			−1.93912
$222$		0			0
$223$	234.568			1.05187			0.525937	−	0.850524i	$-0.323715\pi$
$223$	234.568			1.05187			0.525937	+	0.850524i	$0.323715\pi$
$224$		0			0
$225$	−74.5431	+	8.26557i	−0.331303	+	0.0367359i
$226$		0			0
$227$	318.302			1.40221			0.701106	−	0.713057i	$-0.252690\pi$
$227$	318.302			1.40221			0.701106	+	0.713057i	$0.252690\pi$
$228$		0			0
$229$			2.89432i			0.0126389i	0.999980	+	0.00631947i	$0.00201156\pi$
$229$			2.89432i			0.0126389i	−0.999980	+	0.00631947i	$0.997988\pi$
$230$		0			0
$231$	−36.8325	−	5.67806i	−0.159448	−	0.0245804i
$232$		0			0
$233$			328.322i			1.40911i	0.709652	+	0.704553i	$0.248852\pi$
$233$			328.322i			1.40911i	−0.709652	+	0.704553i	$0.751148\pi$
$234$		0			0
$235$	4.87535	+	88.2063i	0.0207462	+	0.375346i
$236$		0			0
$237$	−70.1326			−0.295918
$238$		0			0
$239$	356.042			1.48972			0.744858	−	0.667223i	$-0.232517\pi$
$239$	356.042			1.48972			0.744858	+	0.667223i	$0.232517\pi$
$240$		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
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	60.7787	−	237.341i	0.248076	−	0.968740i
$246$		0			0
$247$			584.122i			2.36487i
$248$		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
$253$			74.7401i			0.295415i
$254$		0			0
$255$	11.4729	+	207.571i	0.0449917	+	0.814003i
$256$		0			0
$257$	−79.6126			−0.309777			−0.154888	−	0.987932i	$-0.549502\pi$
$257$	−79.6126			−0.309777			−0.154888	+	0.987932i	$0.549502\pi$
$258$		0			0
$259$	39.7216	−	257.666i	0.153365	−	0.994851i
$260$		0			0
$261$	−55.9681			−0.214437
$262$		0			0
$263$			197.790i			0.752052i	0.926609	+	0.376026i	$0.122710\pi$
$263$			197.790i			0.752052i	−0.926609	+	0.376026i	$0.877290\pi$
$264$		0			0
$265$	93.8816	−	5.18904i	0.354270	−	0.0195813i
$266$		0			0
$267$		−	192.531i		−	0.721089i
$268$		0			0
$269$			38.5253i			0.143217i	0.997433	+	0.0716084i	$0.0228132\pi$
$269$			38.5253i			0.143217i	−0.997433	+	0.0716084i	$0.977187\pi$
$270$		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
$273$	213.923	+	32.9782i	0.783602	+	0.120799i
$274$		0			0
$275$	−76.3764	+	8.46885i	−0.277732	+	0.0307958i
$276$		0			0
$277$			204.467i			0.738147i	0.929400	+	0.369074i	$0.120325\pi$
$277$			204.467i			0.738147i	−0.929400	+	0.369074i	$0.879675\pi$
$278$		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
$283$	−47.9753			−0.169524			−0.0847619	−	0.996401i	$-0.527013\pi$
$283$	−47.9753			−0.169524			−0.0847619	+	0.996401i	$0.527013\pi$
$284$		0			0
$285$	282.925	−	15.6379i	0.992720	−	0.0548698i
$286$		0			0
$287$	61.2829	−	397.531i	0.213529	−	1.38512i
$288$		0			0
$289$	287.230			0.993877
$290$		0			0
$291$	45.6877			0.157002
$292$		0			0
$293$	−339.798			−1.15972			−0.579859	−	0.814717i	$-0.696893\pi$
$293$	−339.798			−1.15972			−0.579859	+	0.814717i	$0.696893\pi$
$294$		0			0
$295$	−425.650	+	23.5266i	−1.44288	+	0.0797512i
$296$		0			0
$297$	−15.9718			−0.0537772
$298$		0			0
$299$		−	434.091i		−	1.45181i
$300$		0			0
$301$	63.9088	−	414.564i	0.212322	−	1.37729i
$302$		0			0
$303$		−	192.860i		−	0.636503i
$304$		0			0
$305$	378.244	−	20.9064i	1.24014	−	0.0685454i
$306$		0			0
$307$	−394.637			−1.28546			−0.642731	−	0.766092i	$-0.722199\pi$
$307$	−394.637			−1.28546			−0.642731	+	0.766092i	$0.722199\pi$
$308$		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
$313$	101.725			0.325000			0.162500	−	0.986709i	$-0.448044\pi$
$313$	101.725			0.325000			0.162500	+	0.986709i	$0.448044\pi$
$314$		0			0
$315$	10.2463	−	104.499i	0.0325278	−	0.331742i
$316$		0			0
$317$			475.720i			1.50069i	0.661045	+	0.750346i	$0.270113\pi$
$317$			475.720i			1.50069i	−0.661045	+	0.750346i	$0.729887\pi$
$318$		0			0
$319$	−57.3446			−0.179763
$320$		0			0
$321$		−	192.285i		−	0.599019i
$322$		0			0
$323$		−	785.419i		−	2.43164i
$324$		0			0
$325$	443.595	−	49.1871i	1.36491	−	0.151345i
$326$		0			0
$327$	−254.181			−0.777311
$328$		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
$333$		−	111.733i		−	0.335534i
$334$		0			0
$335$	−147.779	+	8.16805i	−0.441131	+	0.0243823i
$336$		0			0
$337$		−	80.4587i		−	0.238750i	−0.992849	−	0.119375i	$-0.961911\pi$
$337$		−	80.4587i		−	0.238750i	0.992849	−	0.119375i	$-0.0380891\pi$
$338$		0			0
$339$		−	0.516470i		−	0.00152351i
$340$		0			0
$341$		−	135.929i		−	0.398619i
$342$		0			0
$343$	307.519	+	151.925i	0.896556	+	0.442931i
$344$		0			0
$345$	−210.256	+	11.6213i	−0.609439	+	0.0336850i
$346$		0			0
$347$		−	197.979i		−	0.570546i	−0.958446	−	0.285273i	$-0.907916\pi$
$347$		−	197.979i		−	0.570546i	0.958446	−	0.285273i	$-0.0920843\pi$
$348$		0			0
$349$			116.788i			0.334637i	0.985903	+	0.167318i	$0.0535108\pi$
$349$			116.788i			0.334637i	−0.985903	+	0.167318i	$0.946489\pi$
$350$		0			0
$351$	92.7645			0.264286
$352$		0			0
$353$	−439.644			−1.24545			−0.622725	−	0.782441i	$-0.713974\pi$
$353$	−439.644			−1.24545			−0.622725	+	0.782441i	$0.713974\pi$
$354$		0			0
$355$	−13.1219	−	237.404i	−0.0369630	−	0.668745i
$356$		0			0
$357$	−287.645	−	44.3431i	−0.805728	−	0.124210i
$358$		0			0
$359$	−456.067			−1.27038			−0.635191	−	0.772355i	$-0.719078\pi$
$359$	−456.067			−1.27038			−0.635191	+	0.772355i	$0.719078\pi$
$360$		0			0
$361$	−709.551			−1.96551
$362$		0			0
$363$	193.214			0.532269
$364$		0			0
$365$	−3.04829	−	55.1505i	−0.00835147	−	0.151097i
$366$		0			0
$367$	14.4035			0.0392465			0.0196233	−	0.999807i	$-0.493753\pi$
$367$	14.4035			0.0392465			0.0196233	+	0.999807i	$0.493753\pi$
$368$		0			0
$369$		−	172.383i		−	0.467162i
$370$		0			0
$371$	−20.0558	+	130.098i	−0.0540587	+	0.350669i
$372$		0			0
$373$			357.070i			0.957293i	0.878008	+	0.478647i	$0.158873\pi$
$373$			357.070i			0.957293i	−0.878008	+	0.478647i	$0.841127\pi$
$374$		0			0
$375$	−35.7000	−	213.543i	−0.0952001	−	0.569447i
$376$		0			0
$377$	333.058			0.883442
$378$		0			0
$379$	5.90977			0.0155931			0.00779653	−	0.999970i	$-0.497518\pi$
$379$	5.90977			0.0155931			0.00779653	+	0.999970i	$0.497518\pi$
$380$		0			0
$381$			303.237i			0.795898i
$382$		0			0
$383$	334.550			0.873500			0.436750	−	0.899583i	$-0.356130\pi$
$383$	334.550			0.873500			0.436750	+	0.899583i	$0.356130\pi$
$384$		0			0
$385$	10.4983	−	107.069i	0.0272682	−	0.278101i
$386$		0			0
$387$		−	179.769i		−	0.464520i
$388$		0			0
$389$	297.061			0.763653			0.381826	−	0.924234i	$-0.375295\pi$
$389$	297.061			0.763653			0.381826	+	0.924234i	$0.375295\pi$
$390$		0			0
$391$			583.685i			1.49280i
$392$		0			0
$393$			116.819i			0.297250i
$394$		0			0
$395$	−11.1731	−	202.147i	−0.0282863	−	0.511764i
$396$		0			0
$397$	−189.999			−0.478586			−0.239293	−	0.970947i	$-0.576916\pi$
$397$	−189.999			−0.478586			−0.239293	+	0.970947i	$0.576916\pi$
$398$		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
$403$			789.477i			1.95900i
$404$		0			0
$405$	−2.48346	−	44.9314i	−0.00613199	−	0.110942i
$406$		0			0
$407$		−	114.481i		−	0.281280i
$408$		0			0
$409$		−	229.455i		−	0.561015i	−0.959852	−	0.280508i	$-0.909497\pi$
$409$		−	229.455i		−	0.561015i	0.959852	−	0.280508i	$-0.0905028\pi$
$410$		0			0
$411$		−	155.920i		−	0.379367i
$412$		0			0
$413$	90.9310	−	589.852i	0.220172	−	1.42821i
$414$		0			0
$415$	24.9721	+	451.802i	0.0601737	+	1.08868i
$416$		0			0
$417$		−	141.793i		−	0.340030i
$418$		0			0
$419$			171.731i			0.409859i	0.978777	+	0.204929i	$0.0656965\pi$
$419$			171.731i			0.409859i	−0.978777	+	0.204929i	$0.934304\pi$
$420$		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
$423$	−53.0046			−0.125306
$424$		0			0
$425$	−596.464	+	66.1378i	−1.40345	+	0.155618i
$426$		0			0
$427$	−80.8037	+	524.158i	−0.189236	+	1.22754i
$428$		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
$433$	−255.445			−0.589941			−0.294971	−	0.955506i	$-0.595310\pi$
$433$	−255.445			−0.589941			−0.294971	+	0.955506i	$0.595310\pi$
$434$		0			0
$435$	−8.91649	−	161.320i	−0.0204977	−	0.370850i
$436$		0			0
$437$	795.581			1.82055
$438$		0			0
$439$			570.966i			1.30061i	0.759675	+	0.650303i	$0.225358\pi$
$439$			570.966i			1.30061i	−0.759675	+	0.650303i	$0.774642\pi$
$440$		0			0
$441$	140.175	+	44.2707i	0.317858	+	0.100387i
$442$		0			0
$443$			234.490i			0.529323i	0.964341	+	0.264661i	$0.0852602\pi$
$443$			234.490i			0.529323i	−0.964341	+	0.264661i	$0.914740\pi$
$444$		0			0
$445$	554.941	−	30.6728i	1.24706	−	0.0689276i
$446$		0			0
$447$	−433.654			−0.970142
$448$		0			0
$449$	−327.098			−0.728503			−0.364251	−	0.931301i	$-0.618675\pi$
$449$	−327.098			−0.728503			−0.364251	+	0.931301i	$0.618675\pi$
$450$		0			0
$451$		−	176.622i		−	0.391624i
$452$		0			0
$453$	−161.639			−0.356818
$454$		0			0
$455$	−60.9739	+	621.857i	−0.134009	+	1.36672i
$456$		0			0
$457$		−	57.4776i		−	0.125772i	−0.998021	−	0.0628858i	$-0.979970\pi$
$457$		−	57.4776i		−	0.125772i	0.998021	−	0.0628858i	$-0.0200304\pi$
$458$		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
$463$		−	235.364i		−	0.508345i	−0.967159	−	0.254172i	$-0.918197\pi$
$463$		−	235.364i		−	0.508345i	0.967159	−	0.254172i	$-0.0818030\pi$
$464$		0			0
$465$	382.391	−	21.1356i	0.822347	−	0.0454529i
$466$		0			0
$467$	551.433			1.18080			0.590399	−	0.807111i	$-0.298970\pi$
$467$	551.433			1.18080			0.590399	+	0.807111i	$0.298970\pi$
$468$		0			0
$469$	31.5698	−	204.787i	0.0673130	−	0.436646i
$470$		0			0
$471$	488.646			1.03746
$472$		0			0
$473$		−	184.190i		−	0.389409i
$474$		0			0
$475$	90.1478	+	812.999i	0.189785	+	1.71158i
$476$		0			0
$477$			56.4149i			0.118270i
$478$		0			0
$479$			863.232i			1.80215i	0.433659	+	0.901077i	$0.357222\pi$
$479$			863.232i			1.80215i	−0.433659	+	0.901077i	$0.642778\pi$
$480$		0			0
$481$			664.905i			1.38234i
$482$		0			0
$483$	44.9167	−	291.366i	0.0929953	−	0.603243i
$484$		0			0
$485$	7.27868	+	131.688i	0.0150076	+	0.271522i
$486$		0			0
$487$			588.387i			1.20819i	0.796913	+	0.604094i	$0.206465\pi$
$487$			588.387i			1.20819i	−0.796913	+	0.604094i	$0.793535\pi$
$488$		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
$493$	−447.834			−0.908386
$494$		0			0
$495$	−2.54453	−	46.0364i	−0.00514047	−	0.0930029i
$496$		0			0
$497$	328.987	+	50.7164i	0.661946	+	0.102045i
$498$		0			0
$499$	−499.204			−1.00041			−0.500204	−	0.865908i	$-0.666742\pi$
$499$	−499.204			−1.00041			−0.500204	+	0.865908i	$0.666742\pi$
$500$		0			0
$501$	−507.542			−1.01306
$502$		0			0
$503$	−809.915			−1.61017			−0.805085	−	0.593160i	$-0.797880\pi$
$503$	−809.915			−1.61017			−0.805085	+	0.593160i	$0.797880\pi$
$504$		0			0
$505$	555.892	−	30.7253i	1.10078	−	0.0608422i
$506$		0			0
$507$	−259.310			−0.511460
$508$		0			0
$509$			172.031i			0.337978i	0.985618	+	0.168989i	$0.0540502\pi$
$509$			172.031i			0.337978i	−0.985618	+	0.168989i	$0.945950\pi$
$510$		0			0
$511$	76.4258	+	11.7817i	0.149561	+	0.0230562i
$512$		0			0
$513$			170.014i			0.331412i
$514$		0			0
$515$	17.6640	+	319.582i	0.0342990	+	0.620548i
$516$		0			0
$517$	−54.3081			−0.105045
$518$		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
$523$	111.743			0.213657			0.106828	−	0.994277i	$-0.465930\pi$
$523$	111.743			0.213657			0.106828	+	0.994277i	$0.465930\pi$
$524$		0			0
$525$	302.835	+	12.8852i	0.576828	+	0.0245433i
$526$		0			0
$527$		−	1061.54i		−	2.01431i
$528$		0			0
$529$	−62.2368			−0.117650
$530$		0			0
$531$		−	255.780i		−	0.481695i
$532$		0			0
$533$			1025.82i			1.92462i
$534$		0			0
$535$	554.233	−	30.6336i	1.03595	−	0.0572592i
$536$		0			0
$537$	−66.9197			−0.124618
$538$		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
$543$		−	472.437i		−	0.870050i
$544$		0			0
$545$	−40.4945	−	732.638i	−0.0743018	−	1.34429i
$546$		0			0
$547$			241.126i			0.440816i	0.975408	+	0.220408i	$0.0707389\pi$
$547$			241.126i			0.440816i	−0.975408	+	0.220408i	$0.929261\pi$
$548$		0			0
$549$			227.293i			0.414012i
$550$		0			0
$551$			610.412i			1.10783i
$552$		0			0
$553$	280.128	+	43.1843i	0.506561	+	0.0780910i
$554$		0			0
$555$	322.053	−	17.8006i	0.580277	−	0.0320731i
$556$		0			0
$557$		−	533.112i		−	0.957114i	−0.878056	−	0.478557i	$-0.841160\pi$
$557$		−	533.112i		−	0.957114i	0.878056	−	0.478557i	$-0.158840\pi$
$558$		0			0
$559$			1069.78i			1.91374i
$560$		0			0
$561$	−127.800			−0.227808
$562$		0			0
$563$	−965.363			−1.71468			−0.857338	−	0.514754i	$-0.827883\pi$
$563$	−965.363			−1.71468			−0.857338	+	0.514754i	$0.827883\pi$
$564$		0			0
$565$	1.48865	−	0.0822807i	0.00263477	−	0.000145630i
$566$		0			0
$567$	62.2645	+	9.59863i	0.109814	+	0.0169288i
$568$		0			0
$569$	−599.641			−1.05385			−0.526926	−	0.849911i	$-0.676655\pi$
$569$	−599.641			−1.05385			−0.526926	+	0.849911i	$0.676655\pi$
$570$		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
$573$	32.3201			0.0564051
$574$		0			0
$575$	−66.9934	−	604.181i	−0.116510	−	1.05075i
$576$		0			0
$577$	268.952			0.466122			0.233061	−	0.972462i	$-0.425126\pi$
$577$	268.952			0.466122			0.233061	+	0.972462i	$0.425126\pi$
$578$		0			0
$579$			360.467i			0.622568i
$580$		0			0
$581$	−626.093	−	96.5179i	−1.07761	−	0.166124i
$582$		0			0
$583$			57.8024i			0.0991464i
$584$		0			0
$585$	14.7787	+	267.380i	0.0252627	+	0.457060i
$586$		0			0
$587$	784.787			1.33695			0.668473	−	0.743736i	$-0.266948\pi$
$587$	784.787			1.33695			0.668473	+	0.743736i	$0.266948\pi$
$588$		0			0
$589$	−1446.92			−2.45656
$590$		0			0
$591$			432.935i			0.732546i
$592$		0			0
$593$	45.1151			0.0760794			0.0380397	−	0.999276i	$-0.487889\pi$
$593$	45.1151			0.0760794			0.0380397	+	0.999276i	$0.487889\pi$
$594$		0			0
$595$	81.9865	−	836.158i	0.137792	−	1.40531i
$596$		0			0
$597$			147.539i			0.247134i
$598$		0			0
$599$	75.0608			0.125310			0.0626551	−	0.998035i	$-0.480043\pi$
$599$	75.0608			0.125310			0.0626551	+	0.998035i	$0.480043\pi$
$600$		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
$603$		−	88.8027i		−	0.147268i
$604$		0			0
$605$	30.7816	+	556.909i	0.0508786	+	0.920511i
$606$		0			0
$607$	−244.933			−0.403515			−0.201757	−	0.979436i	$-0.564665\pi$
$607$	−244.933			−0.403515			−0.201757	+	0.979436i	$0.564665\pi$
$608$		0			0
$609$	223.552	+	34.4625i	0.367080	+	0.0565887i
$610$		0			0
$611$	315.422			0.516239
$612$		0			0
$613$		−	448.080i		−	0.730963i	−0.930819	−	0.365481i	$-0.880904\pi$
$613$		−	448.080i		−	0.730963i	0.930819	−	0.365481i	$-0.119096\pi$
$614$		0			0
$615$	496.868	−	27.4629i	0.807915	−	0.0446552i
$616$		0			0
$617$			617.239i			1.00039i	0.865913	+	0.500194i	$0.166738\pi$
$617$			617.239i			1.00039i	−0.865913	+	0.500194i	$0.833262\pi$
$618$		0			0
$619$		−	960.005i		−	1.55090i	−0.631411	−	0.775449i	$-0.717524\pi$
$619$		−	960.005i		−	1.55090i	0.631411	−	0.775449i	$-0.282476\pi$
$620$		0			0
$621$		−	126.346i		−	0.203456i
$622$		0			0
$623$	−118.551	+	769.019i	−0.190291	+	1.23438i
$624$		0			0
$625$	609.818	−	136.920i	0.975709	−	0.219073i
$626$		0			0
$627$			174.196i			0.277824i
$628$		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
$633$	−111.348			−0.175905
$634$		0			0
$635$	−874.037	+	48.3099i	−1.37644	+	0.0760786i
$636$		0			0
$637$	−834.161	−	263.448i	−1.30952	−	0.413576i
$638$		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
$643$	624.036			0.970507			0.485254	−	0.874373i	$-0.338727\pi$
$643$	624.036			0.970507			0.485254	+	0.874373i	$0.338727\pi$
$644$		0			0
$645$	518.158	−	28.6397i	0.803346	−	0.0444026i
$646$		0			0
$647$	157.813			0.243914			0.121957	−	0.992535i	$-0.461083\pi$
$647$	157.813			0.243914			0.121957	+	0.992535i	$0.461083\pi$
$648$		0			0
$649$		−	262.070i		−	0.403806i
$650$		0			0
$651$	−81.6897	+	529.905i	−0.125483	+	0.813987i
$652$		0			0
$653$		−	671.329i		−	1.02807i	−0.857770	−	0.514034i	$-0.828150\pi$
$653$		−	671.329i		−	1.02807i	0.857770	−	0.514034i	$-0.171850\pi$
$654$		0			0
$655$	−336.714	+	18.6109i	−0.514068	+	0.0284136i
$656$		0			0
$657$	33.1408			0.0504426
$658$		0			0
$659$	−499.820			−0.758452			−0.379226	−	0.925304i	$-0.623810\pi$
$659$	−499.820			−0.758452			−0.379226	+	0.925304i	$0.623810\pi$
$660$		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
$663$	742.264			1.11955
$664$		0			0
$665$	−1139.71	−	111.750i	−1.71385	−	0.168045i
$666$		0			0
$667$		−	453.628i		−	0.680102i
$668$		0			0
$669$	−406.283			−0.607299
$670$		0			0
$671$			232.883i			0.347068i
$672$		0			0
$673$		−	420.271i		−	0.624474i	−0.950004	−	0.312237i	$-0.898922\pi$
$673$		−	420.271i		−	0.624474i	0.950004	−	0.312237i	$-0.101078\pi$
$674$		0			0
$675$	129.113	−	14.3164i	0.191278	−	0.0212095i
$676$		0			0
$677$	471.364			0.696254			0.348127	−	0.937447i	$-0.386818\pi$
$677$	471.364			0.696254			0.348127	+	0.937447i	$0.386818\pi$
$678$		0			0
$679$	−182.489	−	28.1323i	−0.268761	−	0.0414320i
$680$		0			0
$681$	−551.316			−0.809568
$682$		0			0
$683$		−	390.212i		−	0.571320i	−0.958331	−	0.285660i	$-0.907787\pi$
$683$		−	390.212i		−	0.571320i	0.958331	−	0.285660i	$-0.0922129\pi$
$684$		0			0
$685$	449.416	−	24.8402i	0.656081	−	0.0362630i
$686$		0			0
$687$		−	5.01311i		−	0.00729710i
$688$		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
$693$	63.7958	+	9.83470i	0.0920574	+	0.0141915i
$694$		0			0
$695$	408.696	−	22.5895i	0.588052	−	0.0325029i
$696$		0			0
$697$		−	1379.34i		−	1.97896i
$698$		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
$703$	−1218.61			−1.73344
$704$		0			0
$705$	−8.44436	−	152.778i	−0.0119778	−	0.216706i
$706$		0			0
$707$	−118.754	+	770.337i	−0.167969	+	1.08959i
$708$		0			0
$709$	485.352			0.684559			0.342279	−	0.939598i	$-0.388801\pi$
$709$	485.352			0.684559			0.342279	+	0.939598i	$0.388801\pi$
$710$		0			0
$711$	121.473			0.170848
$712$		0			0
$713$	1075.28			1.50810
$714$		0			0
$715$	15.1421	+	273.956i	0.0211778	+	0.383155i
$716$		0			0
$717$	−616.683			−0.860088
$718$		0			0
$719$		−	25.1312i		−	0.0349529i	−0.999847	−	0.0174765i	$-0.994437\pi$
$719$		−	25.1312i		−	0.0349529i	0.999847	−	0.0174765i	$-0.00556322\pi$
$720$		0			0
$721$	−442.866	−	68.2718i	−0.614239	−	0.0946905i
$722$		0			0
$723$			330.377i			0.456953i
$724$		0			0
$725$	463.560	−	51.4009i	0.639393	−	0.0708978i
$726$		0			0
$727$	−332.426			−0.457258			−0.228629	−	0.973514i	$-0.573424\pi$
$727$	−332.426			−0.457258			−0.228629	+	0.973514i	$0.573424\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	1438.44i		−	1.96777i
$732$		0			0
$733$	1410.78			1.92467			0.962334	−	0.271870i	$-0.0876420\pi$
$733$	1410.78			1.92467			0.962334	+	0.271870i	$0.0876420\pi$
$734$		0			0
$735$	−105.272	+	411.087i	−0.143227	+	0.559303i
$736$		0			0
$737$		−	90.9866i		−	0.123455i
$738$		0			0
$739$	−928.931			−1.25701			−0.628505	−	0.777805i	$-0.716333\pi$
$739$	−928.931			−1.25701			−0.628505	+	0.777805i	$0.716333\pi$
$740$		0			0
$741$		−	1011.73i		−	1.36536i
$742$		0			0
$743$			451.882i			0.608186i	0.952642	+	0.304093i	$0.0983534\pi$
$743$			451.882i			0.608186i	−0.952642	+	0.304093i	$0.901647\pi$
$744$		0			0
$745$	−69.0870	−	1249.94i	−0.0927342	−	1.67777i
$746$		0			0
$747$	−271.495			−0.363447
$748$		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
$753$			642.932i			0.853828i
$754$		0			0
$755$	−25.7513	−	465.899i	−0.0341076	−	0.617085i
$756$		0			0
$757$			1119.68i			1.47910i	0.673104	+	0.739548i	$0.264961\pi$
$757$			1119.68i			1.47910i	−0.673104	+	0.739548i	$0.735039\pi$
$758$		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
$763$	1015.27	+	156.512i	1.33062	+	0.205128i
$764$		0			0
$765$	−19.8716	−	359.523i	−0.0259760	−	0.469965i
$766$		0			0
$767$			1522.11i			1.98449i
$768$		0			0
$769$			1347.17i			1.75185i	0.482446	+	0.875926i	$0.339749\pi$
$769$			1347.17i			1.75185i	−0.482446	+	0.875926i	$0.660251\pi$
$770$		0			0
$771$	137.893			0.178850
$772$		0			0
$773$	−508.563			−0.657908			−0.328954	−	0.944346i	$-0.606696\pi$
$773$	−508.563			−0.657908			−0.328954	+	0.944346i	$0.606696\pi$
$774$		0			0
$775$	121.840	+	1098.82i	0.157213	+	1.41783i
$776$		0			0
$777$	−68.7998	+	446.291i	−0.0885455	+	0.574377i
$778$		0			0
$779$	−1880.08			−2.41345
$780$		0			0
$781$	146.169			0.187156
$782$		0			0
$783$	96.9396			0.123805
$784$		0			0
$785$	77.8480	+	1408.45i	0.0991695	+	1.79420i
$786$		0			0
$787$	275.176			0.349652			0.174826	−	0.984599i	$-0.444064\pi$
$787$	275.176			0.349652			0.174826	+	0.984599i	$0.444064\pi$
$788$		0			0
$789$		−	342.582i		−	0.434197i
$790$		0			0
$791$	−0.318018	+	2.06292i	−0.000402045	+	0.00260799i
$792$		0			0
$793$		−	1352.58i		−	1.70565i
$794$		0			0
$795$	−162.608	+	8.98768i	−0.204538	+	0.0113053i
$796$		0			0
$797$	2.56622			0.00321985			0.00160992	−	0.999999i	$-0.499488\pi$
$797$	2.56622			0.00321985			0.00160992	+	0.999999i	$0.499488\pi$
$798$		0			0
$799$	−424.121			−0.530815
$800$		0			0
$801$			333.473i			0.416321i
$802$		0			0
$803$	33.9559			0.0422862
$804$		0			0
$805$	846.976	+	83.0472i	1.05214	+	0.103164i
$806$		0			0
$807$		−	66.7278i		−	0.0826863i
$808$		0			0
$809$	−1019.40			−1.26007			−0.630036	−	0.776566i	$-0.716960\pi$
$809$	−1019.40			−1.26007			−0.630036	+	0.776566i	$0.716960\pi$
$810$		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
$813$			324.213i			0.398786i
$814$		0			0
$815$	839.479	−	46.3998i	1.03004	−	0.0569323i
$816$		0			0
$817$	−1960.64			−2.39980
$818$		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
$823$		−	64.0535i		−	0.0778293i	−0.999243	−	0.0389147i	$-0.987610\pi$
$823$		−	64.0535i		−	0.0778293i	0.999243	−	0.0389147i	$-0.0123900\pi$
$824$		0			0
$825$	132.288	−	14.6685i	0.160349	−	0.0177800i
$826$		0			0
$827$		−	1423.82i		−	1.72167i	−0.508888	−	0.860833i	$-0.669943\pi$
$827$		−	1423.82i		−	1.72167i	0.508888	−	0.860833i	$-0.330057\pi$
$828$		0			0
$829$		−	871.031i		−	1.05070i	−0.850886	−	0.525350i	$-0.823934\pi$
$829$		−	871.031i		−	1.05070i	0.850886	−	0.525350i	$-0.176066\pi$
$830$		0			0
$831$		−	354.147i		−	0.426169i
$832$		0			0
$833$	1121.63	+	354.236i	1.34649	+	0.425254i
$834$		0			0
$835$	−80.8584	−	1462.91i	−0.0968365	−	1.75199i
$836$		0			0
$837$			229.785i			0.274534i
$838$		0			0
$839$		−	1346.65i		−	1.60506i	−0.596611	−	0.802530i	$-0.703487\pi$
$839$		−	1346.65i		−	1.60506i	0.596611	−	0.802530i	$-0.296513\pi$
$840$		0			0
$841$	−492.952			−0.586150
$842$		0			0
$843$	312.952			0.371236
$844$		0			0
$845$	−41.3117	−	747.424i	−0.0488896	−	0.884525i
$846$		0			0
$847$	−771.747	−	118.972i	−0.911153	−	0.140463i
$848$		0			0
$849$	83.0956			0.0978747
$850$		0			0
$851$	905.608			1.06417
$852$		0			0
$853$	−1553.19			−1.82086			−0.910429	−	0.413666i	$-0.864248\pi$
$853$	−1553.19			−1.82086			−0.910429	+	0.413666i	$0.864248\pi$
$854$		0			0
$855$	−490.041	+	27.0856i	−0.573147	+	0.0316791i
$856$		0			0
$857$	916.383			1.06929			0.534646	−	0.845076i	$-0.320445\pi$
$857$	916.383			1.06929			0.534646	+	0.845076i	$0.320445\pi$
$858$		0			0
$859$		−	220.008i		−	0.256121i	−0.991766	−	0.128061i	$-0.959125\pi$
$859$		−	220.008i		−	0.256121i	0.991766	−	0.128061i	$-0.0408752\pi$
$860$		0			0
$861$	−106.145	+	688.543i	−0.123281	+	0.799702i
$862$		0			0
$863$		−	1143.68i		−	1.32523i	−0.748959	−	0.662616i	$-0.769446\pi$
$863$		−	1143.68i		−	1.32523i	0.748959	−	0.662616i	$-0.230554\pi$
$864$		0			0
$865$	−80.4489	−	1455.51i	−0.0930045	−	1.68267i
$866$		0			0
$867$	−497.498			−0.573815
$868$		0			0
$869$	124.461			0.143223
$870$		0			0
$871$			528.451i			0.606717i
$872$		0			0
$873$	−79.1335			−0.0906454
$874$		0			0
$875$	11.1060	+	874.930i	0.0126926	+	0.999919i
$876$		0			0
$877$		−	711.003i		−	0.810722i	−0.914157	−	0.405361i	$-0.867146\pi$
$877$		−	711.003i		−	0.810722i	0.914157	−	0.405361i	$-0.132854\pi$
$878$		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
$883$			203.330i			0.230271i	0.993350	+	0.115136i	$0.0367303\pi$
$883$			203.330i			0.230271i	−0.993350	+	0.115136i	$0.963270\pi$
$884$		0			0
$885$	737.248	−	40.7493i	0.833048	−	0.0460444i
$886$		0			0
$887$	840.574			0.947659			0.473830	−	0.880617i	$-0.342871\pi$
$887$	840.574			0.947659			0.473830	+	0.880617i	$0.342871\pi$
$888$		0			0
$889$	186.719	−	1211.21i	0.210033	−	1.36244i
$890$		0			0
$891$	27.6640			0.0310483
$892$		0			0
$893$			578.090i			0.647357i
$894$		0			0
$895$	−10.6612	−	192.886i	−0.0119120	−	0.215515i
$896$		0			0
$897$			751.867i			0.838202i
$898$		0			0
$899$			825.010i			0.917697i
$900$		0			0
$901$			451.410i			0.501010i
$902$		0			0
$903$	−110.693	+	718.046i	−0.122584	+	0.795179i
$904$		0			0
$905$	1361.73	−	75.2658i	1.50467	−	0.0831666i
$906$		0			0
$907$		−	611.612i		−	0.674324i	−0.941447	−	0.337162i	$-0.890533\pi$
$907$		−	611.612i		−	0.674324i	0.941447	−	0.337162i	$-0.109467\pi$
$908$		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
$913$	−278.172			−0.304679
$914$		0			0
$915$	−655.138	+	36.2109i	−0.715997	+	0.0395747i
$916$		0			0
$917$	71.9318	−	466.608i	0.0784426	−	0.508842i
$918$		0			0
$919$	204.249			0.222251			0.111126	−	0.993806i	$-0.464554\pi$
$919$	204.249			0.222251			0.111126	+	0.993806i	$0.464554\pi$
$920$		0			0
$921$	683.531			0.742162
$922$		0			0
$923$	−848.948			−0.919770
$924$		0			0
$925$	102.615	+	925.436i	0.110935	+	1.00047i
$926$		0			0
$927$	−192.042			−0.207165
$928$		0			0
$929$			607.668i			0.654110i	0.945005	+	0.327055i	$0.106056\pi$
$929$			607.668i			0.654110i	−0.945005	+	0.327055i	$0.893944\pi$
$930$		0			0
$931$	482.835	−	1528.81i	0.518619	−	1.64212i
$932$		0			0
$933$		−	590.793i		−	0.633219i
$934$		0			0
$935$	−20.3603	−	368.365i	−0.0217758	−	0.393973i
$936$		0			0
$937$	1226.32			1.30878			0.654389	−	0.756158i	$-0.272926\pi$
$937$	1226.32			1.30878			0.654389	+	0.756158i	$0.272926\pi$
$938$		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
$943$	1397.18			1.48164
$944$		0			0
$945$	−17.7471	+	180.997i	−0.0187799	+	0.191532i
$946$		0			0
$947$			845.061i			0.892356i	0.894944	+	0.446178i	$0.147215\pi$
$947$			845.061i			0.892356i	−0.894944	+	0.446178i	$0.852785\pi$
$948$		0			0
$949$	−197.216			−0.207814
$950$		0			0
$951$		−	823.970i		−	0.866425i
$952$		0			0
$953$			367.141i			0.385248i	0.981273	+	0.192624i	$0.0616997\pi$
$953$			367.141i			0.385248i	−0.981273	+	0.192624i	$0.938300\pi$
$954$		0			0
$955$	5.14904	+	93.1580i	0.00539167	+	0.0975477i
$956$		0			0
$957$	99.3237			0.103786
$958$		0			0
$959$	−96.0080	+	622.786i	−0.100113	+	0.649412i
$960$		0			0
$961$	−994.598			−1.03496
$962$		0			0
$963$			333.047i			0.345844i
$964$		0			0
$965$	−1038.99	+	57.4273i	−1.07668	+	0.0595102i
$966$		0			0
$967$		−	1013.20i		−	1.04777i	−0.851789	−	0.523886i	$-0.824482\pi$
$967$		−	1013.20i		−	1.04777i	0.851789	−	0.523886i	$-0.175518\pi$
$968$		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
$973$	−87.3092	+	566.358i	−0.0897320	+	0.582074i
$974$		0			0
$975$	−768.328	+	85.1946i	−0.788029	+	0.0873791i
$976$		0			0
$977$		−	867.534i		−	0.887957i	−0.896037	−	0.443978i	$-0.853567\pi$
$977$		−	867.534i		−	0.887957i	0.896037	−	0.443978i	$-0.146433\pi$
$978$		0			0
$979$			341.674i			0.349003i
$980$		0			0
$981$	440.254			0.448781
$982$		0			0
$983$	−1757.82			−1.78822			−0.894109	−	0.447849i	$-0.852190\pi$
$983$	−1757.82			−1.78822			−0.894109	+	0.447849i	$0.852190\pi$
$984$		0			0
$985$	−1247.87	+	68.9725i	−1.26687	+	0.0700228i
$986$		0			0
$987$	211.714	+	32.6377i	0.214503	+	0.0330676i
$988$		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
$993$	−488.259			−0.491701
$994$		0			0
$995$	−425.259	+	23.5050i	−0.427396	+	0.0236231i
$996$		0			0
$997$	1605.98			1.61081			0.805406	−	0.592724i	$-0.201947\pi$
$997$	1605.98			1.61081			0.805406	+	0.592724i	$0.201947\pi$
$998$		0			0
$999$			193.527i			0.193721i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.3.p.a.349.3	✓	16
3.2	odd	2		1260.3.p.e.1189.10		16
4.3	odd	2		1680.3.bd.b.769.11		16
5.2	odd	4		2100.3.j.g.601.12		16
5.3	odd	4		2100.3.j.g.601.5		16
5.4	even	2	inner	420.3.p.a.349.13	yes	16
7.6	odd	2	inner	420.3.p.a.349.14	yes	16
15.14	odd	2		1260.3.p.e.1189.8		16
20.19	odd	2		1680.3.bd.b.769.5		16
21.20	even	2		1260.3.p.e.1189.7		16
28.27	even	2		1680.3.bd.b.769.6		16
35.13	even	4		2100.3.j.g.601.13		16
35.27	even	4		2100.3.j.g.601.4		16
35.34	odd	2	inner	420.3.p.a.349.4	yes	16
105.104	even	2		1260.3.p.e.1189.9		16
140.139	even	2		1680.3.bd.b.769.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.p.a.349.3	✓	16	1.1	even	1	trivial
420.3.p.a.349.4	yes	16	35.34	odd	2	inner
420.3.p.a.349.13	yes	16	5.4	even	2	inner
420.3.p.a.349.14	yes	16	7.6	odd	2	inner
1260.3.p.e.1189.7		16	21.20	even	2
1260.3.p.e.1189.8		16	15.14	odd	2
1260.3.p.e.1189.9		16	105.104	even	2
1260.3.p.e.1189.10		16	3.2	odd	2
1680.3.bd.b.769.5		16	20.19	odd	2
1680.3.bd.b.769.6		16	28.27	even	2
1680.3.bd.b.769.11		16	4.3	odd	2
1680.3.bd.b.769.12		16	140.139	even	2
2100.3.j.g.601.4		16	35.27	even	4
2100.3.j.g.601.5		16	5.3	odd	4
2100.3.j.g.601.12		16	5.2	odd	4
2100.3.j.g.601.13		16	35.13	even	4

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	420.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(-0.275940 - 4.99238i) q^{5} +(6.91828 + 1.06651i) q^{7} +3.00000 q^{9} +3.07378 q^{11} -17.8525 q^{13} +(0.477941 + 8.64706i) q^{15} +24.0048 q^{17} -32.7193i q^{19} +(-11.9828 - 1.84726i) q^{21} +24.3154i q^{23} +(-24.8477 + 2.75519i) q^{25} -5.19615 q^{27} -18.6560 q^{29} -44.2221i q^{31} -5.32394 q^{33} +(3.41542 - 34.8330i) q^{35} -37.2443i q^{37} +30.9215 q^{39} -57.4609i q^{41} -59.9230i q^{43} +(-0.827819 - 14.9771i) q^{45} -17.6682 q^{47} +(46.7251 + 14.7569i) q^{49} -41.5775 q^{51} +18.8050i q^{53} +(-0.848177 - 15.3455i) q^{55} +56.6714i q^{57} -85.2600i q^{59} +75.7642i q^{61} +(20.7548 + 3.19954i) q^{63} +(4.92622 + 89.1266i) q^{65} -29.6009i q^{67} -42.1154i q^{69} +47.5533 q^{71} +11.0469 q^{73} +(43.0375 - 4.77213i) q^{75} +(21.2653 + 3.27823i) q^{77} +40.4910 q^{79} +9.00000 q^{81} -90.4984 q^{83} +(-6.62387 - 119.841i) q^{85} +32.3132 q^{87} +111.158i q^{89} +(-123.509 - 19.0400i) q^{91} +76.5950i q^{93} +(-163.347 + 9.02854i) q^{95} -26.3778 q^{97} +9.22134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} - 24 q^{11} - 24 q^{15} - 12 q^{21} - 48 q^{25} - 32 q^{29} + 76 q^{35} + 72 q^{39} - 88 q^{49} + 24 q^{51} + 152 q^{65} + 168 q^{71} + 16 q^{79} + 144 q^{81} - 416 q^{85} - 568 q^{91} + 136 q^{95}+ \cdots - 72 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 24 * q^11 - 24 * q^15 - 12 * q^21 - 48 * q^25 - 32 * q^29 + 76 * q^35 + 72 * q^39 - 88 * q^49 + 24 * q^51 + 152 * q^65 + 168 * q^71 + 16 * q^79 + 144 * q^81 - 416 * q^85 - 568 * q^91 + 136 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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