LMFDB - Embedded newform 1536.2.c.m.1535.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1535,1536)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1536.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.3
Root			$0.281691 - 1.38588i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.m.1535.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.51022 + 0.848071i) q^{3} -0.936426i q^{5} -2.20837i q^{7} +(1.56155 - 2.56155i) q^{9} +O(q^{10})$	\(q+(-1.51022 + 0.848071i) q^{3} -0.936426i q^{5} -2.20837i q^{7} +(1.56155 - 2.56155i) q^{9} -1.69614 q^{11} -4.27156 q^{13} +(0.794156 + 1.41421i) q^{15} -1.12311i q^{17} +4.34475i q^{19} +(1.87285 + 3.33513i) q^{21} -7.24517 q^{23} +4.12311 q^{25} +(-0.185917 + 5.19283i) q^{27} -5.73384i q^{29} +5.03680i q^{31} +(2.56155 - 1.43845i) q^{33} -2.06798 q^{35} +9.06897 q^{37} +(6.45101 - 3.62258i) q^{39} +4.00000i q^{41} +7.73704i q^{43} +(-2.39871 - 1.46228i) q^{45} +5.65685 q^{47} +2.12311 q^{49} +(0.952473 + 1.69614i) q^{51} +14.2770i q^{53} +1.58831i q^{55} +(-3.68466 - 6.56155i) q^{57} -6.41273 q^{59} -8.01726 q^{61} +(-5.65685 - 3.44849i) q^{63} +4.00000i q^{65} +10.3857i q^{67} +(10.9418 - 6.14441i) q^{69} +1.58831 q^{71} -6.24621 q^{73} +(-6.22681 + 3.49668i) q^{75} +3.74571i q^{77} -10.6937i q^{79} +(-4.12311 - 8.00000i) q^{81} -13.7779 q^{83} -1.05171 q^{85} +(4.86270 + 8.65938i) q^{87} +13.1231i q^{89} +9.43318i q^{91} +(-4.27156 - 7.60669i) q^{93} +4.06854 q^{95} -7.12311 q^{97} +(-2.64861 + 4.34475i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} + 8 q^{33} - 32 q^{49} + 40 q^{57} + 32 q^{73} - 48 q^{97}+O(q^{100}) $ 16 * q - 8 * q^9 + 8 * q^33 - 32 * q^49 + 40 * q^57 + 32 * q^73 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.51022	+	0.848071i	−0.871928	+	0.489634i
$3$	−1.51022	+	0.848071i	−0.871928	+	0.489634i
$4$		0			0
$5$		−	0.936426i		−	0.418783i	−0.977832	−	0.209391i	$-0.932852\pi$
$5$		−	0.936426i		−	0.418783i	0.977832	−	0.209391i	$-0.0671482\pi$
$6$		0			0
$7$		−	2.20837i		−	0.834685i	−0.908749	−	0.417343i	$-0.862962\pi$
$7$		−	2.20837i		−	0.834685i	0.908749	−	0.417343i	$-0.137038\pi$
$8$		0			0
$9$	1.56155	−	2.56155i	0.520518	−	0.853851i
$10$		0			0
$11$	−1.69614			−0.511406			−0.255703	−	0.966755i	$-0.582307\pi$
$11$	−1.69614			−0.511406			−0.255703	+	0.966755i	$0.582307\pi$
$12$		0			0
$13$	−4.27156			−1.18472			−0.592359	−	0.805674i	$-0.701803\pi$
$13$	−4.27156			−1.18472			−0.592359	+	0.805674i	$0.701803\pi$
$14$		0			0
$15$	0.794156	+	1.41421i	0.205050	+	0.365148i
$16$		0			0
$17$		−	1.12311i		−	0.272393i	−0.990682	−	0.136197i	$-0.956512\pi$
$17$		−	1.12311i		−	0.272393i	0.990682	−	0.136197i	$-0.0434879\pi$
$18$		0			0
$19$			4.34475i			0.996755i	0.866960	+	0.498378i	$0.166071\pi$
$19$			4.34475i			0.996755i	−0.866960	+	0.498378i	$0.833929\pi$
$20$		0			0
$21$	1.87285	+	3.33513i	0.408690	+	0.727785i
$22$		0			0
$23$	−7.24517			−1.51072			−0.755361	−	0.655309i	$-0.772538\pi$
$23$	−7.24517			−1.51072			−0.755361	+	0.655309i	$0.772538\pi$
$24$		0			0
$25$	4.12311			0.824621
$26$		0			0
$27$	−0.185917	+	5.19283i	−0.0357798	+	0.999360i
$28$		0			0
$29$		−	5.73384i		−	1.06475i	−0.846510	−	0.532373i	$-0.821300\pi$
$29$		−	5.73384i		−	1.06475i	0.846510	−	0.532373i	$-0.178700\pi$
$30$		0			0
$31$			5.03680i			0.904635i	0.891857	+	0.452318i	$0.149403\pi$
$31$			5.03680i			0.904635i	−0.891857	+	0.452318i	$0.850597\pi$
$32$		0			0
$33$	2.56155	−	1.43845i	0.445909	−	0.250402i
$34$		0			0
$35$	−2.06798			−0.349552
$36$		0			0
$37$	9.06897			1.49093			0.745465	−	0.666545i	$-0.232228\pi$
$37$	9.06897			1.49093			0.745465	+	0.666545i	$0.232228\pi$
$38$		0			0
$39$	6.45101	−	3.62258i	1.03299	−	0.580077i
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$			7.73704i			1.17989i	0.807445	+	0.589944i	$0.200850\pi$
$43$			7.73704i			1.17989i	−0.807445	+	0.589944i	$0.799150\pi$
$44$		0			0
$45$	−2.39871	−	1.46228i	−0.357578	−	0.217984i
$46$		0			0
$47$	5.65685			0.825137			0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685			0.825137			0.412568	+	0.910927i	$0.364632\pi$
$48$		0			0
$49$	2.12311			0.303301
$50$		0			0
$51$	0.952473	+	1.69614i	0.133373	+	0.237507i
$52$		0			0
$53$			14.2770i			1.96109i	0.196293	+	0.980545i	$0.437110\pi$
$53$			14.2770i			1.96109i	−0.196293	+	0.980545i	$0.562890\pi$
$54$		0			0
$55$			1.58831i			0.214168i
$56$		0			0
$57$	−3.68466	−	6.56155i	−0.488045	−	0.869099i
$58$		0			0
$59$	−6.41273			−0.834866			−0.417433	−	0.908708i	$-0.637070\pi$
$59$	−6.41273			−0.834866			−0.417433	+	0.908708i	$0.637070\pi$
$60$		0			0
$61$	−8.01726			−1.02651			−0.513253	−	0.858238i	$-0.671560\pi$
$61$	−8.01726			−1.02651			−0.513253	+	0.858238i	$0.671560\pi$
$62$		0			0
$63$	−5.65685	−	3.44849i	−0.712697	−	0.434468i
$64$		0			0
$65$			4.00000i			0.496139i
$66$		0			0
$67$			10.3857i			1.26881i	0.773001	+	0.634405i	$0.218755\pi$
$67$			10.3857i			1.26881i	−0.773001	+	0.634405i	$0.781245\pi$
$68$		0			0
$69$	10.9418	−	6.14441i	1.31724	−	0.739700i
$70$		0			0
$71$	1.58831			0.188498			0.0942489	−	0.995549i	$-0.469955\pi$
$71$	1.58831			0.188498			0.0942489	+	0.995549i	$0.469955\pi$
$72$		0			0
$73$	−6.24621			−0.731064			−0.365532	−	0.930799i	$-0.619113\pi$
$73$	−6.24621			−0.731064			−0.365532	+	0.930799i	$0.619113\pi$
$74$		0			0
$75$	−6.22681	+	3.49668i	−0.719010	+	0.403762i
$76$		0			0
$77$			3.74571i			0.426863i
$78$		0			0
$79$		−	10.6937i		−	1.20313i	−0.798824	−	0.601565i	$-0.794544\pi$
$79$		−	10.6937i		−	1.20313i	0.798824	−	0.601565i	$-0.205456\pi$
$80$		0			0
$81$	−4.12311	−	8.00000i	−0.458123	−	0.888889i
$82$		0			0
$83$	−13.7779			−1.51232			−0.756162	−	0.654384i	$-0.772928\pi$
$83$	−13.7779			−1.51232			−0.756162	+	0.654384i	$0.772928\pi$
$84$		0			0
$85$	−1.05171			−0.114074
$86$		0			0
$87$	4.86270	+	8.65938i	0.521336	+	0.928383i
$88$		0			0
$89$			13.1231i			1.39105i	0.718504	+	0.695523i	$0.244827\pi$
$89$			13.1231i			1.39105i	−0.718504	+	0.695523i	$0.755173\pi$
$90$		0			0
$91$			9.43318i			0.988866i
$92$		0			0
$93$	−4.27156	−	7.60669i	−0.442940	−	0.788777i
$94$		0			0
$95$	4.06854			0.417424
$96$		0			0
$97$	−7.12311			−0.723242			−0.361621	−	0.932325i	$-0.617777\pi$
$97$	−7.12311			−0.723242			−0.361621	+	0.932325i	$0.617777\pi$
$98$		0			0
$99$	−2.64861	+	4.34475i	−0.266196	+	0.436664i
$100$		0			0
$101$		−	13.2252i		−	1.31596i	−0.753035	−	0.657981i	$-0.771411\pi$
$101$		−	13.2252i		−	1.31596i	0.753035	−	0.657981i	$-0.228589\pi$
$102$		0			0
$103$			7.86522i			0.774983i	0.921873	+	0.387492i	$0.126658\pi$
$103$			7.86522i			0.774983i	−0.921873	+	0.387492i	$0.873342\pi$
$104$		0			0
$105$	3.12311	−	1.75379i	0.304784	−	0.171152i
$106$		0			0
$107$	2.27678			0.220105			0.110052	−	0.993926i	$-0.464898\pi$
$107$	2.27678			0.220105			0.110052	+	0.993926i	$0.464898\pi$
$108$		0			0
$109$	−9.06897			−0.868650			−0.434325	−	0.900756i	$-0.643013\pi$
$109$	−9.06897			−0.868650			−0.434325	+	0.900756i	$0.643013\pi$
$110$		0			0
$111$	−13.6962	+	7.69113i	−1.29998	+	0.730009i
$112$		0			0
$113$			20.4924i			1.92776i	0.266328	+	0.963882i	$0.414190\pi$
$113$			20.4924i			1.92776i	−0.266328	+	0.963882i	$0.585810\pi$
$114$		0			0
$115$			6.78456i			0.632664i
$116$		0			0
$117$	−6.67026	+	10.9418i	−0.616666	+	1.01157i
$118$		0			0
$119$	−2.48023			−0.227362
$120$		0			0
$121$	−8.12311			−0.738464
$122$		0			0
$123$	−3.39228	−	6.04090i	−0.305872	−	0.544689i
$124$		0			0
$125$		−	8.54312i		−	0.764120i
$126$		0			0
$127$			1.86017i			0.165064i	0.996588	+	0.0825319i	$0.0263006\pi$
$127$			1.86017i			0.165064i	−0.996588	+	0.0825319i	$0.973699\pi$
$128$		0			0
$129$	−6.56155	−	11.6847i	−0.577713	−	1.02878i
$130$		0			0
$131$	5.66906			0.495308			0.247654	−	0.968849i	$-0.420340\pi$
$131$	5.66906			0.495308			0.247654	+	0.968849i	$0.420340\pi$
$132$		0			0
$133$	9.59482			0.831977
$134$		0			0
$135$	4.86270	+	0.174098i	0.418514	+	0.0149839i
$136$		0			0
$137$			4.00000i			0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$			4.00000i			0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$		0			0
$139$		−	5.08842i		−	0.431594i	−0.976438	−	0.215797i	$-0.930765\pi$
$139$		−	5.08842i		−	0.431594i	0.976438	−	0.215797i	$-0.0692350\pi$
$140$		0			0
$141$	−8.54312	+	4.79741i	−0.719460	+	0.404015i
$142$		0			0
$143$	7.24517			0.605871
$144$		0			0
$145$	−5.36932			−0.445897
$146$		0			0
$147$	−3.20636	+	1.80054i	−0.264457	+	0.148506i
$148$		0			0
$149$			16.1498i			1.32304i	0.749926	+	0.661522i	$0.230089\pi$
$149$			16.1498i			1.32304i	−0.749926	+	0.661522i	$0.769911\pi$
$150$		0			0
$151$		−	6.62511i		−	0.539144i	−0.962980	−	0.269572i	$-0.913118\pi$
$151$		−	6.62511i		−	0.539144i	0.962980	−	0.269572i	$-0.0868822\pi$
$152$		0			0
$153$	−2.87689	−	1.75379i	−0.232583	−	0.141785i
$154$		0			0
$155$	4.71659			0.378846
$156$		0			0
$157$	12.8147			1.02272			0.511361	−	0.859366i	$-0.329141\pi$
$157$	12.8147			1.02272			0.511361	+	0.859366i	$0.329141\pi$
$158$		0			0
$159$	−12.1079	−	21.5614i	−0.960216	−	1.70993i
$160$		0			0
$161$			16.0000i			1.26098i
$162$		0			0
$163$		−	4.34475i		−	0.340307i	−0.985418	−	0.170154i	$-0.945574\pi$
$163$		−	4.34475i		−	0.340307i	0.985418	−	0.170154i	$-0.0544264\pi$
$164$		0			0
$165$	−1.34700	−	2.39871i	−0.104864	−	0.186739i
$166$		0			0
$167$	−18.5589			−1.43613			−0.718064	−	0.695977i	$-0.754972\pi$
$167$	−18.5589			−1.43613			−0.718064	+	0.695977i	$0.754972\pi$
$168$		0			0
$169$	5.24621			0.403555
$170$		0			0
$171$	11.1293	+	6.78456i	0.851080	+	0.518829i
$172$		0			0
$173$		−	0.936426i		−	0.0711952i	−0.999366	−	0.0355976i	$-0.988667\pi$
$173$		−	0.936426i		−	0.0711952i	0.999366	−	0.0355976i	$-0.0113335\pi$
$174$		0			0
$175$		−	9.10534i		−	0.688299i
$176$		0			0
$177$	9.68466	−	5.43845i	0.727944	−	0.408779i
$178$		0			0
$179$	18.4945			1.38235			0.691173	−	0.722690i	$-0.257094\pi$
$179$	18.4945			1.38235			0.691173	+	0.722690i	$0.257094\pi$
$180$		0			0
$181$	−5.32326			−0.395675			−0.197838	−	0.980235i	$-0.563392\pi$
$181$	−5.32326			−0.395675			−0.197838	+	0.980235i	$0.563392\pi$
$182$		0			0
$183$	12.1079	−	6.79921i	0.895039	−	0.502612i
$184$		0			0
$185$		−	8.49242i		−	0.624375i
$186$		0			0
$187$			1.90495i			0.139303i
$188$		0			0
$189$	11.4677	+	0.410574i	0.834151	+	0.0298648i
$190$		0			0
$191$	20.1472			1.45780			0.728900	−	0.684621i	$-0.240032\pi$
$191$	20.1472			1.45780			0.728900	+	0.684621i	$0.240032\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$	−3.39228	−	6.04090i	−0.242926	−	0.432598i
$196$		0			0
$197$		−	17.2015i		−	1.22556i	−0.790255	−	0.612779i	$-0.790052\pi$
$197$		−	17.2015i		−	1.22556i	0.790255	−	0.612779i	$-0.209948\pi$
$198$		0			0
$199$			7.86522i			0.557551i	0.960356	+	0.278775i	$0.0899285\pi$
$199$			7.86522i			0.557551i	−0.960356	+	0.278775i	$0.910071\pi$
$200$		0			0
$201$	−8.80776	−	15.6847i	−0.621252	−	1.10631i
$202$		0			0
$203$	−12.6624			−0.888728
$204$		0			0
$205$	3.74571			0.261611
$206$		0			0
$207$	−11.3137	+	18.5589i	−0.786357	+	1.28993i
$208$		0			0
$209$		−	7.36932i		−	0.509746i
$210$		0			0
$211$		−	23.9548i		−	1.64911i	−0.565778	−	0.824557i	$-0.691424\pi$
$211$		−	23.9548i		−	1.64911i	0.565778	−	0.824557i	$-0.308576\pi$
$212$		0			0
$213$	−2.39871	+	1.34700i	−0.164357	+	0.0922949i
$214$		0			0
$215$	7.24517			0.494116
$216$		0			0
$217$	11.1231			0.755086
$218$		0			0
$219$	9.43318	−	5.29723i	0.637435	−	0.357953i
$220$		0			0
$221$			4.79741i			0.322709i
$222$		0			0
$223$			26.4241i			1.76949i	0.466077	+	0.884744i	$0.345667\pi$
$223$			26.4241i			1.76949i	−0.466077	+	0.884744i	$0.654333\pi$
$224$		0			0
$225$	6.43845	−	10.5616i	0.429230	−	0.704104i
$226$		0			0
$227$	−6.99337			−0.464166			−0.232083	−	0.972696i	$-0.574554\pi$
$227$	−6.99337			−0.464166			−0.232083	+	0.972696i	$0.574554\pi$
$228$		0			0
$229$	4.27156			0.282273			0.141136	−	0.989990i	$-0.454924\pi$
$229$	4.27156			0.282273			0.141136	+	0.989990i	$0.454924\pi$
$230$		0			0
$231$	−3.17662	−	5.65685i	−0.209006	−	0.372194i
$232$		0			0
$233$			21.1231i			1.38382i	0.721983	+	0.691910i	$0.243231\pi$
$233$			21.1231i			1.38382i	−0.721983	+	0.691910i	$0.756769\pi$
$234$		0			0
$235$		−	5.29723i		−	0.345553i
$236$		0			0
$237$	9.06897	+	16.1498i	0.589093	+	1.04904i
$238$		0			0
$239$	2.48023			0.160433			0.0802164	−	0.996777i	$-0.474439\pi$
$239$	2.48023			0.160433			0.0802164	+	0.996777i	$0.474439\pi$
$240$		0			0
$241$	8.49242			0.547045			0.273523	−	0.961866i	$-0.411811\pi$
$241$	8.49242			0.547045			0.273523	+	0.961866i	$0.411811\pi$
$242$		0			0
$243$	13.0114	+	8.58511i	0.834680	+	0.550735i
$244$		0			0
$245$		−	1.98813i		−	0.127017i
$246$		0			0
$247$		−	18.5589i		−	1.18087i
$248$		0			0
$249$	20.8078	−	11.6847i	1.31864	−	0.740485i
$250$		0			0
$251$	−3.60109			−0.227299			−0.113649	−	0.993521i	$-0.536254\pi$
$251$	−3.60109			−0.227299			−0.113649	+	0.993521i	$0.536254\pi$
$252$		0			0
$253$	12.2888			0.772592
$254$		0			0
$255$	1.58831	−	0.891921i	0.0994639	−	0.0558542i
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		−	20.0276i		−	1.24446i
$260$		0			0
$261$	−14.6875	−	8.95369i	−0.909135	−	0.554220i
$262$		0			0
$263$	−27.3924			−1.68908			−0.844542	−	0.535489i	$-0.820127\pi$
$263$	−27.3924			−1.68908			−0.844542	+	0.535489i	$0.820127\pi$
$264$		0			0
$265$	13.3693			0.821271
$266$		0			0
$267$	−11.1293	−	19.8188i	−0.681103	−	1.21289i
$268$		0			0
$269$			14.2770i			0.870481i	0.900314	+	0.435241i	$0.143337\pi$
$269$			14.2770i			0.870481i	−0.900314	+	0.435241i	$0.856663\pi$
$270$		0			0
$271$		−	7.51703i		−	0.456627i	−0.973588	−	0.228313i	$-0.926679\pi$
$271$		−	7.51703i		−	0.456627i	0.973588	−	0.228313i	$-0.0733211\pi$
$272$		0			0
$273$	−8.00000	−	14.2462i	−0.484182	−	0.862220i
$274$		0			0
$275$	−6.99337			−0.421716
$276$		0			0
$277$	−25.1035			−1.50832			−0.754161	−	0.656689i	$-0.771956\pi$
$277$	−25.1035			−1.50832			−0.754161	+	0.656689i	$0.771956\pi$
$278$		0			0
$279$	12.9020	+	7.86522i	0.772424	+	0.470879i
$280$		0			0
$281$			15.3693i			0.916857i	0.888731	+	0.458428i	$0.151587\pi$
$281$			15.3693i			0.916857i	−0.888731	+	0.458428i	$0.848413\pi$
$282$		0			0
$283$		−	19.0752i		−	1.13390i	−0.823752	−	0.566950i	$-0.808123\pi$
$283$		−	19.0752i		−	1.13390i	0.823752	−	0.566950i	$-0.191877\pi$
$284$		0			0
$285$	−6.14441	+	3.45041i	−0.363964	+	0.204385i
$286$		0			0
$287$	8.83348			0.521424
$288$		0			0
$289$	15.7386			0.925802
$290$		0			0
$291$	10.7575	−	6.04090i	0.630615	−	0.354124i
$292$		0			0
$293$		−	7.60669i		−	0.444388i	−0.975003	−	0.222194i	$-0.928678\pi$
$293$		−	7.60669i		−	0.444388i	0.975003	−	0.222194i	$-0.0713218\pi$
$294$		0			0
$295$			6.00505i			0.349628i
$296$		0			0
$297$	0.315342	−	8.80776i	0.0182980	−	0.511078i
$298$		0			0
$299$	30.9481			1.78978
$300$		0			0
$301$	17.0862			0.984834
$302$		0			0
$303$	11.2159	+	19.9731i	0.644339	+	1.14742i
$304$		0			0
$305$			7.50758i			0.429883i
$306$		0			0
$307$			11.8730i			0.677627i	0.940854	+	0.338814i	$0.110026\pi$
$307$			11.8730i			0.677627i	−0.940854	+	0.338814i	$0.889974\pi$
$308$		0			0
$309$	−6.67026	−	11.8782i	−0.379458	−	0.675730i
$310$		0			0
$311$	−16.0786			−0.911736			−0.455868	−	0.890047i	$-0.650671\pi$
$311$	−16.0786			−0.911736			−0.455868	+	0.890047i	$0.650671\pi$
$312$		0			0
$313$	−21.6155			−1.22178			−0.610891	−	0.791715i	$-0.709189\pi$
$313$	−21.6155			−1.22178			−0.610891	+	0.791715i	$0.709189\pi$
$314$		0			0
$315$	−3.22925	+	5.29723i	−0.181948	+	0.298465i
$316$		0			0
$317$			6.78554i			0.381114i	0.981676	+	0.190557i	$0.0610294\pi$
$317$			6.78554i			0.381114i	−0.981676	+	0.190557i	$0.938971\pi$
$318$		0			0
$319$			9.72540i			0.544518i
$320$		0			0
$321$	−3.43845	+	1.93087i	−0.191915	+	0.107771i
$322$		0			0
$323$	4.87962			0.271509
$324$		0			0
$325$	−17.6121			−0.976943
$326$		0			0
$327$	13.6962	−	7.69113i	0.757400	−	0.425320i
$328$		0			0
$329$		−	12.4924i		−	0.688730i
$330$		0			0
$331$		−	15.2653i		−	0.839055i	−0.907743	−	0.419528i	$-0.862196\pi$
$331$		−	15.2653i		−	0.839055i	0.907743	−	0.419528i	$-0.137804\pi$
$332$		0			0
$333$	14.1617	−	23.2306i	0.776055	−	1.27303i
$334$		0			0
$335$	9.72540			0.531355
$336$		0			0
$337$	−1.12311			−0.0611795			−0.0305897	−	0.999532i	$-0.509739\pi$
$337$	−1.12311			−0.0611795			−0.0305897	+	0.999532i	$0.509739\pi$
$338$		0			0
$339$	−17.3790	−	30.9481i	−0.943899	−	1.68087i
$340$		0			0
$341$		−	8.54312i		−	0.462636i
$342$		0			0
$343$		−	20.1472i		−	1.08785i
$344$		0			0
$345$	−5.75379	−	10.2462i	−0.309774	−	0.551637i
$346$		0			0
$347$	31.1570			1.67259			0.836296	−	0.548278i	$-0.184716\pi$
$347$	31.1570			1.67259			0.836296	+	0.548278i	$0.184716\pi$
$348$		0			0
$349$	−3.21985			−0.172355			−0.0861774	−	0.996280i	$-0.527465\pi$
$349$	−3.21985			−0.172355			−0.0861774	+	0.996280i	$0.527465\pi$
$350$		0			0
$351$	0.794156	−	22.1815i	0.0423889	−	1.18396i
$352$		0			0
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$		−	1.48734i		−	0.0789396i
$356$		0			0
$357$	3.74571	−	2.10341i	0.198244	−	0.111324i
$358$		0			0
$359$	12.9020			0.680943			0.340471	−	0.940255i	$-0.389413\pi$
$359$	12.9020			0.680943			0.340471	+	0.940255i	$0.389413\pi$
$360$		0			0
$361$	0.123106			0.00647924
$362$		0			0
$363$	12.2677	−	6.88897i	0.643888	−	0.361577i
$364$		0			0
$365$			5.84912i			0.306157i
$366$		0			0
$367$		−	17.5906i		−	0.918223i	−0.888379	−	0.459111i	$-0.848168\pi$
$367$		−	17.5906i		−	0.918223i	0.888379	−	0.459111i	$-0.151832\pi$
$368$		0			0
$369$	10.2462	+	6.24621i	0.533396	+	0.325165i
$370$		0			0
$371$	31.5288			1.63689
$372$		0			0
$373$	−4.27156			−0.221173			−0.110586	−	0.993867i	$-0.535273\pi$
$373$	−4.27156			−0.221173			−0.110586	+	0.993867i	$0.535273\pi$
$374$		0			0
$375$	7.24517	+	12.9020i	0.374139	+	0.666257i
$376$		0			0
$377$			24.4924i			1.26142i
$378$		0			0
$379$		−	28.5083i		−	1.46437i	−0.681103	−	0.732187i	$-0.738499\pi$
$379$		−	28.5083i		−	1.46437i	0.681103	−	0.732187i	$-0.261501\pi$
$380$		0			0
$381$	−1.57756	−	2.80928i	−0.0808208	−	0.143924i
$382$		0			0
$383$	−12.0101			−0.613687			−0.306844	−	0.951760i	$-0.599273\pi$
$383$	−12.0101			−0.613687			−0.306844	+	0.951760i	$0.599273\pi$
$384$		0			0
$385$	3.50758			0.178763
$386$		0			0
$387$	19.8188	+	12.0818i	1.00745	+	0.614152i
$388$		0			0
$389$			21.7684i			1.10370i	0.833943	+	0.551850i	$0.186078\pi$
$389$			21.7684i			1.10370i	−0.833943	+	0.551850i	$0.813922\pi$
$390$		0			0
$391$			8.13709i			0.411510i
$392$		0			0
$393$	−8.56155	+	4.80776i	−0.431873	+	0.242520i
$394$		0			0
$395$	−10.0138			−0.503850
$396$		0			0
$397$	17.6121			0.883925			0.441963	−	0.897033i	$-0.354282\pi$
$397$	17.6121			0.883925			0.441963	+	0.897033i	$0.354282\pi$
$398$		0			0
$399$	−14.4903	+	8.13709i	−0.725424	+	0.407364i
$400$		0			0
$401$			1.12311i			0.0560852i	0.999607	+	0.0280426i	$0.00892741\pi$
$401$			1.12311i			0.0560852i	−0.999607	+	0.0280426i	$0.991073\pi$
$402$		0			0
$403$		−	21.5150i		−	1.07174i
$404$		0			0
$405$	−7.49141	+	3.86098i	−0.372251	+	0.191854i
$406$		0			0
$407$	−15.3823			−0.762470
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$	−3.39228	−	6.04090i	−0.167329	−	0.297975i
$412$		0			0
$413$			14.1617i			0.696850i
$414$		0			0
$415$			12.9020i			0.633335i
$416$		0			0
$417$	4.31534	+	7.68466i	0.211323	+	0.376319i
$418$		0			0
$419$	8.48071			0.414310			0.207155	−	0.978308i	$-0.433580\pi$
$419$	8.48071			0.414310			0.207155	+	0.978308i	$0.433580\pi$
$420$		0			0
$421$	−29.9009			−1.45728			−0.728641	−	0.684896i	$-0.759847\pi$
$421$	−29.9009			−1.45728			−0.728641	+	0.684896i	$0.759847\pi$
$422$		0			0
$423$	8.83348	−	14.4903i	0.429498	−	0.704544i
$424$		0			0
$425$		−	4.63068i		−	0.224621i
$426$		0			0
$427$			17.7051i			0.856809i
$428$		0			0
$429$	−10.9418	+	6.14441i	−0.528276	+	0.296655i
$430$		0			0
$431$	−13.7939			−0.664431			−0.332215	−	0.943204i	$-0.607796\pi$
$431$	−13.7939			−0.664431			−0.332215	+	0.943204i	$0.607796\pi$
$432$		0			0
$433$	−33.8617			−1.62729			−0.813646	−	0.581361i	$-0.802520\pi$
$433$	−33.8617			−1.62729			−0.813646	+	0.581361i	$0.802520\pi$
$434$		0			0
$435$	8.10887	−	4.55356i	0.388791	−	0.218326i
$436$		0			0
$437$		−	31.4785i		−	1.50582i
$438$		0			0
$439$			36.8459i			1.75856i	0.476307	+	0.879279i	$0.341975\pi$
$439$			36.8459i			1.75856i	−0.476307	+	0.879279i	$0.658025\pi$
$440$		0			0
$441$	3.31534	−	5.43845i	0.157873	−	0.258974i
$442$		0			0
$443$	−19.0752			−0.906288			−0.453144	−	0.891437i	$-0.649698\pi$
$443$	−19.0752			−0.906288			−0.453144	+	0.891437i	$0.649698\pi$
$444$		0			0
$445$	12.2888			0.582546
$446$		0			0
$447$	−13.6962	−	24.3898i	−0.647807	−	1.15360i
$448$		0			0
$449$			14.8769i			0.702084i	0.936360	+	0.351042i	$0.114173\pi$
$449$			14.8769i			0.702084i	−0.936360	+	0.351042i	$0.885827\pi$
$450$		0			0
$451$		−	6.78456i		−	0.319473i
$452$		0			0
$453$	5.61856	+	10.0054i	0.263983	+	0.470095i
$454$		0			0
$455$	8.83348			0.414120
$456$		0			0
$457$	−8.24621			−0.385741			−0.192871	−	0.981224i	$-0.561780\pi$
$457$	−8.24621			−0.385741			−0.192871	+	0.981224i	$0.561780\pi$
$458$		0			0
$459$	5.83209	+	0.208805i	0.272219	+	0.00974616i
$460$		0			0
$461$		−	5.73384i		−	0.267051i	−0.991045	−	0.133526i	$-0.957370\pi$
$461$		−	5.73384i		−	0.267051i	0.991045	−	0.133526i	$-0.0426299\pi$
$462$		0			0
$463$		−	23.9439i		−	1.11277i	−0.830926	−	0.556383i	$-0.812189\pi$
$463$		−	23.9439i		−	1.11277i	0.830926	−	0.556383i	$-0.187811\pi$
$464$		0			0
$465$	−7.12311	+	4.00000i	−0.330326	+	0.185496i
$466$		0			0
$467$	−36.0366			−1.66757			−0.833787	−	0.552087i	$-0.813832\pi$
$467$	−36.0366			−1.66757			−0.833787	+	0.552087i	$0.813832\pi$
$468$		0			0
$469$	22.9354			1.05906
$470$		0			0
$471$	−19.3530	+	10.8677i	−0.891741	+	0.500759i
$472$		0			0
$473$		−	13.1231i		−	0.603401i
$474$		0			0
$475$			17.9139i			0.821945i
$476$		0			0
$477$	36.5712	+	22.2942i	1.67448	+	1.02078i
$478$		0			0
$479$	−3.17662			−0.145144			−0.0725718	−	0.997363i	$-0.523121\pi$
$479$	−3.17662			−0.145144			−0.0725718	+	0.997363i	$0.523121\pi$
$480$		0			0
$481$	−38.7386			−1.76633
$482$		0			0
$483$	−13.5691	−	24.1636i	−0.617417	−	1.09948i
$484$		0			0
$485$			6.67026i			0.302881i
$486$		0			0
$487$			5.92872i			0.268656i	0.990937	+	0.134328i	$0.0428875\pi$
$487$			5.92872i			0.268656i	−0.990937	+	0.134328i	$0.957112\pi$
$488$		0			0
$489$	3.68466	+	6.56155i	0.166626	+	0.296724i
$490$		0			0
$491$	4.50778			0.203433			0.101717	−	0.994813i	$-0.467566\pi$
$491$	4.50778			0.203433			0.101717	+	0.994813i	$0.467566\pi$
$492$		0			0
$493$	−6.43971			−0.290030
$494$		0			0
$495$	4.06854	+	2.48023i	0.182867	+	0.111478i
$496$		0			0
$497$		−	3.50758i		−	0.157336i
$498$		0			0
$499$			8.48071i			0.379649i	0.981818	+	0.189824i	$0.0607918\pi$
$499$			8.48071i			0.379649i	−0.981818	+	0.189824i	$0.939208\pi$
$500$		0			0
$501$	28.0281	−	15.7392i	1.25220	−	0.703177i
$502$		0			0
$503$	9.72540			0.433634			0.216817	−	0.976212i	$-0.430433\pi$
$503$	9.72540			0.433634			0.216817	+	0.976212i	$0.430433\pi$
$504$		0			0
$505$	−12.3845			−0.551102
$506$		0			0
$507$	−7.92295	+	4.44916i	−0.351871	+	0.197594i
$508$		0			0
$509$		−	10.5312i		−	0.466789i	−0.972382	−	0.233395i	$-0.925017\pi$
$509$		−	10.5312i		−	0.466789i	0.972382	−	0.233395i	$-0.0749834\pi$
$510$		0			0
$511$			13.7939i			0.610208i
$512$		0			0
$513$	−22.5616	−	0.807764i	−0.996117	−	0.0356637i
$514$		0			0
$515$	7.36520			0.324550
$516$		0			0
$517$	−9.59482			−0.421980
$518$		0			0
$519$	0.794156	+	1.41421i	0.0348596	+	0.0620771i
$520$		0			0
$521$			32.4924i			1.42352i	0.702423	+	0.711759i	$0.252101\pi$
$521$			32.4924i			1.42352i	−0.702423	+	0.711759i	$0.747899\pi$
$522$		0			0
$523$		−	14.5216i		−	0.634985i	−0.948261	−	0.317493i	$-0.897159\pi$
$523$		−	14.5216i		−	0.634985i	0.948261	−	0.317493i	$-0.102841\pi$
$524$		0			0
$525$	7.72197	+	13.7511i	0.337014	+	0.600147i
$526$		0			0
$527$	5.65685			0.246416
$528$		0			0
$529$	29.4924			1.28228
$530$		0			0
$531$	−10.0138	+	16.4265i	−0.434563	+	0.712851i
$532$		0			0
$533$		−	17.0862i		−	0.740087i
$534$		0			0
$535$		−	2.13204i		−	0.0921760i
$536$		0			0
$537$	−27.9309	+	15.6847i	−1.20531	+	0.676843i
$538$		0			0
$539$	−3.60109			−0.155110
$540$		0			0
$541$	−17.6121			−0.757203			−0.378601	−	0.925560i	$-0.623595\pi$
$541$	−17.6121			−0.757203			−0.378601	+	0.925560i	$0.623595\pi$
$542$		0			0
$543$	8.03932	−	4.51450i	0.345000	−	0.193736i
$544$		0			0
$545$			8.49242i			0.363775i
$546$		0			0
$547$			11.1293i			0.475855i	0.971283	+	0.237928i	$0.0764681\pi$
$547$			11.1293i			0.475855i	−0.971283	+	0.237928i	$0.923532\pi$
$548$		0			0
$549$	−12.5194	+	20.5366i	−0.534314	+	0.876483i
$550$		0			0
$551$	24.9121			1.06129
$552$		0			0
$553$	−23.6155			−1.00423
$554$		0			0
$555$	7.20217	+	12.8255i	0.305715	+	0.544410i
$556$		0			0
$557$		−	15.0981i		−	0.639727i	−0.947464	−	0.319864i	$-0.896363\pi$
$557$		−	15.0981i		−	0.639727i	0.947464	−	0.319864i	$-0.103637\pi$
$558$		0			0
$559$		−	33.0492i		−	1.39783i
$560$		0			0
$561$	−1.61553	−	2.87689i	−0.0682077	−	0.121463i
$562$		0			0
$563$	25.8597			1.08986			0.544929	−	0.838482i	$-0.316557\pi$
$563$	25.8597			1.08986			0.544929	+	0.838482i	$0.316557\pi$
$564$		0			0
$565$	19.1896			0.807314
$566$		0			0
$567$	−17.6670	+	9.10534i	−0.741942	+	0.382388i
$568$		0			0
$569$		−	32.4924i		−	1.36215i	−0.732212	−	0.681077i	$-0.761512\pi$
$569$		−	32.4924i		−	1.36215i	0.732212	−	0.681077i	$-0.238488\pi$
$570$		0			0
$571$		−	8.48071i		−	0.354906i	−0.984129	−	0.177453i	$-0.943214\pi$
$571$		−	8.48071i		−	0.354906i	0.984129	−	0.177453i	$-0.0567858\pi$
$572$		0			0
$573$	−30.4268	+	17.0862i	−1.27110	+	0.713788i
$574$		0			0
$575$	−29.8726			−1.24577
$576$		0			0
$577$	23.8617			0.993377			0.496689	−	0.867929i	$-0.334549\pi$
$577$	23.8617			0.993377			0.496689	+	0.867929i	$0.334549\pi$
$578$		0			0
$579$	−6.04090	+	3.39228i	−0.251051	+	0.140978i
$580$		0			0
$581$			30.4268i			1.26231i
$582$		0			0
$583$		−	24.2157i		−	1.00291i
$584$		0			0
$585$	10.2462	+	6.24621i	0.423629	+	0.258249i
$586$		0			0
$587$	−44.8891			−1.85277			−0.926386	−	0.376576i	$-0.877102\pi$
$587$	−44.8891			−1.85277			−0.926386	+	0.376576i	$0.877102\pi$
$588$		0			0
$589$	−21.8836			−0.901700
$590$		0			0
$591$	14.5881	+	25.9781i	0.600074	+	1.06860i
$592$		0			0
$593$			32.9848i			1.35453i	0.735741	+	0.677263i	$0.236834\pi$
$593$			32.9848i			1.35453i	−0.735741	+	0.677263i	$0.763166\pi$
$594$		0			0
$595$			2.32255i			0.0952155i
$596$		0			0
$597$	−6.67026	−	11.8782i	−0.272996	−	0.486144i
$598$		0			0
$599$	−4.06854			−0.166236			−0.0831181	−	0.996540i	$-0.526488\pi$
$599$	−4.06854			−0.166236			−0.0831181	+	0.996540i	$0.526488\pi$
$600$		0			0
$601$	9.75379			0.397865			0.198933	−	0.980013i	$-0.436252\pi$
$601$	9.75379			0.397865			0.198933	+	0.980013i	$0.436252\pi$
$602$		0			0
$603$	26.6034	+	16.2177i	1.08337	+	0.660438i
$604$		0			0
$605$			7.60669i			0.309256i
$606$		0			0
$607$			5.03680i			0.204437i	0.994762	+	0.102219i	$0.0325941\pi$
$607$			5.03680i			0.204437i	−0.994762	+	0.102219i	$0.967406\pi$
$608$		0			0
$609$	19.1231	−	10.7386i	0.774907	−	0.435151i
$610$		0			0
$611$	−24.1636			−0.977554
$612$		0			0
$613$	−1.57756			−0.0637170			−0.0318585	−	0.999492i	$-0.510143\pi$
$613$	−1.57756			−0.0637170			−0.0318585	+	0.999492i	$0.510143\pi$
$614$		0			0
$615$	−5.65685	+	3.17662i	−0.228106	+	0.128094i
$616$		0			0
$617$		−	21.1231i		−	0.850384i	−0.905103	−	0.425192i	$-0.860207\pi$
$617$		−	21.1231i		−	0.850384i	0.905103	−	0.425192i	$-0.139793\pi$
$618$		0			0
$619$		−	1.69614i		−	0.0681737i	−0.999419	−	0.0340868i	$-0.989148\pi$
$619$		−	1.69614i		−	0.0681737i	0.999419	−	0.0340868i	$-0.0108523\pi$
$620$		0			0
$621$	1.34700	−	37.6229i	0.0540532	−	1.50975i
$622$		0			0
$623$	28.9807			1.16109
$624$		0			0
$625$	12.6155			0.504621
$626$		0			0
$627$	6.24970	+	11.1293i	0.249589	+	0.444462i
$628$		0			0
$629$		−	10.1854i		−	0.406119i
$630$		0			0
$631$		−	13.5221i		−	0.538305i	−0.963098	−	0.269153i	$-0.913256\pi$
$631$		−	13.5221i		−	0.538305i	0.963098	−	0.269153i	$-0.0867436\pi$
$632$		0			0
$633$	20.3153	+	36.1771i	0.807462	+	1.43791i
$634$		0			0
$635$	1.74192			0.0691258
$636$		0			0
$637$	−9.06897			−0.359326
$638$		0			0
$639$	2.48023	−	4.06854i	0.0981165	−	0.160949i
$640$		0			0
$641$		−	27.3693i		−	1.08102i	−0.841337	−	0.540512i	$-0.818231\pi$
$641$		−	27.3693i		−	1.08102i	0.841337	−	0.540512i	$-0.181769\pi$
$642$		0			0
$643$			29.9957i			1.18291i	0.806337	+	0.591457i	$0.201447\pi$
$643$			29.9957i			1.18291i	−0.806337	+	0.591457i	$0.798553\pi$
$644$		0			0
$645$	−10.9418	+	6.14441i	−0.430834	+	0.241936i
$646$		0			0
$647$	41.1863			1.61920			0.809600	−	0.586982i	$-0.199684\pi$
$647$	41.1863			1.61920			0.809600	+	0.586982i	$0.199684\pi$
$648$		0			0
$649$	10.8769			0.426955
$650$		0			0
$651$	−16.7984	+	9.43318i	−0.658380	+	0.369715i
$652$		0			0
$653$		−	32.4149i		−	1.26849i	−0.773131	−	0.634246i	$-0.781310\pi$
$653$		−	32.4149i		−	1.26849i	0.773131	−	0.634246i	$-0.218690\pi$
$654$		0			0
$655$		−	5.30866i		−	0.207426i
$656$		0			0
$657$	−9.75379	+	16.0000i	−0.380532	+	0.624219i
$658$		0			0
$659$	3.76412			0.146629			0.0733146	−	0.997309i	$-0.476642\pi$
$659$	3.76412			0.146629			0.0733146	+	0.997309i	$0.476642\pi$
$660$		0			0
$661$	27.2069			1.05823			0.529113	−	0.848551i	$-0.322525\pi$
$661$	27.2069			1.05823			0.529113	+	0.848551i	$0.322525\pi$
$662$		0			0
$663$	−4.06854	−	7.24517i	−0.158009	−	0.281379i
$664$		0			0
$665$		−	8.98485i		−	0.348417i
$666$		0			0
$667$			41.5426i			1.60854i
$668$		0			0
$669$	−22.4095	−	39.9063i	−0.866401	−	1.54287i
$670$		0			0
$671$	13.5984			0.524961
$672$		0			0
$673$	−29.3693			−1.13210			−0.566052	−	0.824370i	$-0.691530\pi$
$673$	−29.3693			−1.13210			−0.566052	+	0.824370i	$0.691530\pi$
$674$		0			0
$675$	−0.766556	+	21.4106i	−0.0295047	+	0.824093i
$676$		0			0
$677$			38.0335i			1.46174i	0.682514	+	0.730872i	$0.260886\pi$
$677$			38.0335i			1.46174i	−0.682514	+	0.730872i	$0.739114\pi$
$678$		0			0
$679$			15.7304i			0.603679i
$680$		0			0
$681$	10.5616	−	5.93087i	0.404720	−	0.227271i
$682$		0			0
$683$	48.5360			1.85718			0.928589	−	0.371111i	$-0.121023\pi$
$683$	48.5360			1.85718			0.928589	+	0.371111i	$0.121023\pi$
$684$		0			0
$685$	3.74571			0.143116
$686$		0			0
$687$	−6.45101	+	3.62258i	−0.246121	+	0.138210i
$688$		0			0
$689$		−	60.9848i		−	2.32334i
$690$		0			0
$691$			21.3062i			0.810525i	0.914200	+	0.405262i	$0.132820\pi$
$691$			21.3062i			0.810525i	−0.914200	+	0.405262i	$0.867180\pi$
$692$		0			0
$693$	9.59482	+	5.84912i	0.364477	+	0.222190i
$694$		0			0
$695$	−4.76493			−0.180744
$696$		0			0
$697$	4.49242			0.170163
$698$		0			0
$699$	−17.9139	−	31.9006i	−0.677565	−	1.20659i
$700$		0			0
$701$			9.47954i			0.358037i	0.983846	+	0.179019i	$0.0572923\pi$
$701$			9.47954i			0.358037i	−0.983846	+	0.179019i	$0.942708\pi$
$702$		0			0
$703$			39.4024i			1.48609i
$704$		0			0
$705$	4.49242	+	8.00000i	0.169194	+	0.301297i
$706$		0			0
$707$	−29.2062			−1.09841
$708$		0			0
$709$	24.0518			0.903284			0.451642	−	0.892199i	$-0.350838\pi$
$709$	24.0518			0.903284			0.451642	+	0.892199i	$0.350838\pi$
$710$		0			0
$711$	−27.3924	−	16.6987i	−1.02729	−	0.626250i
$712$		0			0
$713$		−	36.4924i		−	1.36665i
$714$		0			0
$715$		−	6.78456i		−	0.253728i
$716$		0			0
$717$	−3.74571	+	2.10341i	−0.139886	+	0.0785533i
$718$		0			0
$719$	−37.1177			−1.38426			−0.692129	−	0.721774i	$-0.743327\pi$
$719$	−37.1177			−1.38426			−0.692129	+	0.721774i	$0.743327\pi$
$720$		0			0
$721$	17.3693			0.646867
$722$		0			0
$723$	−12.8255	+	7.20217i	−0.476984	+	0.267852i
$724$		0			0
$725$		−	23.6412i		−	0.878013i
$726$		0			0
$727$			4.68860i			0.173891i	0.996213	+	0.0869453i	$0.0277105\pi$
$727$			4.68860i			0.173891i	−0.996213	+	0.0869453i	$0.972289\pi$
$728$		0			0
$729$	−26.9309	−	1.93087i	−0.997440	−	0.0715137i
$730$		0			0
$731$	8.68951			0.321393
$732$		0			0
$733$	−49.0906			−1.81320			−0.906600	−	0.421990i	$-0.861332\pi$
$733$	−49.0906			−1.81320			−0.906600	+	0.421990i	$0.861332\pi$
$734$		0			0
$735$	1.68608	+	3.00252i	0.0621919	+	0.110750i
$736$		0			0
$737$		−	17.6155i		−	0.648876i
$738$		0			0
$739$		−	17.1702i		−	0.631617i	−0.948823	−	0.315808i	$-0.897724\pi$
$739$		−	17.1702i		−	0.631617i	0.948823	−	0.315808i	$-0.102276\pi$
$740$		0			0
$741$	15.7392	+	28.0281i	0.578195	+	1.02964i
$742$		0			0
$743$	−24.9121			−0.913937			−0.456968	−	0.889483i	$-0.651065\pi$
$743$	−24.9121			−0.913937			−0.456968	+	0.889483i	$0.651065\pi$
$744$		0			0
$745$	15.1231			0.554068
$746$		0			0
$747$	−21.5150	+	35.2929i	−0.787192	+	1.29130i
$748$		0			0
$749$		−	5.02797i		−	0.183718i
$750$		0			0
$751$			32.7773i			1.19606i	0.801473	+	0.598031i	$0.204050\pi$
$751$			32.7773i			1.19606i	−0.801473	+	0.598031i	$0.795950\pi$
$752$		0			0
$753$	5.43845	−	3.05398i	0.198188	−	0.111293i
$754$		0			0
$755$	−6.20393			−0.225784
$756$		0			0
$757$	−3.21985			−0.117028			−0.0585138	−	0.998287i	$-0.518636\pi$
$757$	−3.21985			−0.117028			−0.0585138	+	0.998287i	$0.518636\pi$
$758$		0			0
$759$	−18.5589	+	10.4218i	−0.673644	+	0.378287i
$760$		0			0
$761$			20.9848i			0.760700i	0.924843	+	0.380350i	$0.124196\pi$
$761$			20.9848i			0.760700i	−0.924843	+	0.380350i	$0.875804\pi$
$762$		0			0
$763$			20.0276i			0.725049i
$764$		0			0
$765$	−1.64229	+	2.69400i	−0.0593773	+	0.0974018i
$766$		0			0
$767$	27.3924			0.989080
$768$		0			0
$769$	36.0000			1.29819			0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000			1.29819			0.649097	+	0.760706i	$0.275147\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			20.9472i			0.753419i	0.926331	+	0.376710i	$0.122945\pi$
$773$			20.9472i			0.753419i	−0.926331	+	0.376710i	$0.877055\pi$
$774$		0			0
$775$			20.7672i			0.745981i
$776$		0			0
$777$	16.9848	+	30.2462i	0.609328	+	1.08508i
$778$		0			0
$779$	−17.3790			−0.622668
$780$		0			0
$781$	−2.69400			−0.0963989
$782$		0			0
$783$	29.7748	+	1.06602i	1.06407	+	0.0380964i
$784$		0			0
$785$		−	12.0000i		−	0.428298i
$786$		0			0
$787$			26.6034i			0.948309i	0.880442	+	0.474154i	$0.157246\pi$
$787$			26.6034i			0.948309i	−0.880442	+	0.474154i	$0.842754\pi$
$788$		0			0
$789$	41.3686	−	23.2306i	1.47276	−	0.827033i
$790$		0			0
$791$	45.2548			1.60908
$792$		0			0
$793$	34.2462			1.21612
$794$		0			0
$795$	−20.1907	+	11.3381i	−0.716089	+	0.402122i
$796$		0			0
$797$			9.47954i			0.335783i	0.985806	+	0.167891i	$0.0536958\pi$
$797$			9.47954i			0.335783i	−0.985806	+	0.167891i	$0.946304\pi$
$798$		0			0
$799$		−	6.35324i		−	0.224762i
$800$		0			0
$801$	33.6155	+	20.4924i	1.18775	+	0.724064i
$802$		0			0
$803$	10.5945			0.373870
$804$		0			0
$805$	14.9828			0.528075
$806$		0			0
$807$	−12.1079	−	21.5614i	−0.426217	−	0.758997i
$808$		0			0
$809$		−	20.9848i		−	0.737788i	−0.929471	−	0.368894i	$-0.879737\pi$
$809$		−	20.9848i		−	0.737788i	0.929471	−	0.368894i	$-0.120263\pi$
$810$		0			0
$811$			42.0775i			1.47754i	0.673958	+	0.738770i	$0.264593\pi$
$811$			42.0775i			1.47754i	−0.673958	+	0.738770i	$0.735407\pi$
$812$		0			0
$813$	6.37497	+	11.3524i	0.223580	+	0.398146i
$814$		0			0
$815$	−4.06854			−0.142515
$816$		0			0
$817$	−33.6155			−1.17606
$818$		0			0
$819$	24.1636	+	14.7304i	0.844344	+	0.514722i
$820$		0			0
$821$			20.9472i			0.731063i	0.930799	+	0.365531i	$0.119113\pi$
$821$			20.9472i			0.731063i	−0.930799	+	0.365531i	$0.880887\pi$
$822$		0			0
$823$		−	45.6794i		−	1.59228i	−0.605111	−	0.796141i	$-0.706871\pi$
$823$		−	45.6794i		−	1.59228i	0.605111	−	0.796141i	$-0.293129\pi$
$824$		0			0
$825$	10.5616	−	5.93087i	0.367706	−	0.206486i
$826$		0			0
$827$	−39.2658			−1.36541			−0.682703	−	0.730696i	$-0.739196\pi$
$827$	−39.2658			−1.36541			−0.682703	+	0.730696i	$0.739196\pi$
$828$		0			0
$829$	−11.7630			−0.408545			−0.204272	−	0.978914i	$-0.565483\pi$
$829$	−11.7630			−0.408545			−0.204272	+	0.978914i	$0.565483\pi$
$830$		0			0
$831$	37.9119	−	21.2895i	1.31515	−	0.738526i
$832$		0			0
$833$		−	2.38447i		−	0.0826171i
$834$		0			0
$835$			17.3790i			0.601426i
$836$		0			0
$837$	−26.1552	−	0.936426i	−0.904056	−	0.0323676i
$838$		0			0
$839$	−24.2157			−0.836020			−0.418010	−	0.908442i	$-0.637272\pi$
$839$	−24.2157			−0.836020			−0.418010	+	0.908442i	$0.637272\pi$
$840$		0			0
$841$	−3.87689			−0.133686
$842$		0			0
$843$	−13.0343	−	23.2111i	−0.448924	−	0.799433i
$844$		0			0
$845$		−	4.91269i		−	0.169002i
$846$		0			0
$847$			17.9388i			0.616385i
$848$		0			0
$849$	16.1771	+	28.8078i	0.555196	+	0.988680i
$850$		0			0
$851$	−65.7062			−2.25238
$852$		0			0
$853$	37.3923			1.28029			0.640144	−	0.768255i	$-0.278875\pi$
$853$	37.3923			1.28029			0.640144	+	0.768255i	$0.278875\pi$
$854$		0			0
$855$	6.35324	−	10.4218i	0.217276	−	0.356418i
$856$		0			0
$857$		−	52.0000i		−	1.77629i	−0.459567	−	0.888143i	$-0.651995\pi$
$857$		−	52.0000i		−	1.77629i	0.459567	−	0.888143i	$-0.348005\pi$
$858$		0			0
$859$			28.0907i			0.958443i	0.877694	+	0.479222i	$0.159081\pi$
$859$			28.0907i			0.958443i	−0.877694	+	0.479222i	$0.840919\pi$
$860$		0			0
$861$	−13.3405	+	7.49141i	−0.454644	+	0.255307i
$862$		0			0
$863$	−54.0883			−1.84119			−0.920594	−	0.390522i	$-0.872295\pi$
$863$	−54.0883			−1.84119			−0.920594	+	0.390522i	$0.872295\pi$
$864$		0			0
$865$	−0.876894			−0.0298153
$866$		0			0
$867$	−23.7689	+	13.3475i	−0.807233	+	0.453304i
$868$		0			0
$869$			18.1379i			0.615287i
$870$		0			0
$871$		−	44.3629i		−	1.50318i
$872$		0			0
$873$	−11.1231	+	18.2462i	−0.376460	+	0.617541i
$874$		0			0
$875$	−18.8664			−0.637799
$876$		0			0
$877$	11.7630			0.397207			0.198604	−	0.980080i	$-0.436359\pi$
$877$	11.7630			0.397207			0.198604	+	0.980080i	$0.436359\pi$
$878$		0			0
$879$	6.45101	+	11.4878i	0.217587	+	0.387474i
$880$		0			0
$881$			19.5076i			0.657227i	0.944464	+	0.328613i	$0.106581\pi$
$881$			19.5076i			0.657227i	−0.944464	+	0.328613i	$0.893419\pi$
$882$		0			0
$883$		−	11.5469i		−	0.388585i	−0.980944	−	0.194293i	$-0.937759\pi$
$883$		−	11.5469i		−	0.388585i	0.980944	−	0.194293i	$-0.0622411\pi$
$884$		0			0
$885$	−5.09271	−	9.06897i	−0.171189	−	0.304850i
$886$		0			0
$887$	19.2553			0.646529			0.323264	−	0.946309i	$-0.395220\pi$
$887$	19.2553			0.646529			0.323264	+	0.946309i	$0.395220\pi$
$888$		0			0
$889$	4.10795			0.137776
$890$		0			0
$891$	6.99337	+	13.5691i	0.234287	+	0.454583i
$892$		0			0
$893$			24.5776i			0.822460i
$894$		0			0
$895$		−	17.3188i		−	0.578902i
$896$		0			0
$897$	−46.7386	+	26.2462i	−1.56056	+	0.876335i
$898$		0			0
$899$	28.8802			0.963208
$900$		0			0
$901$	16.0345			0.534188
$902$		0			0
$903$	−25.8040	+	14.4903i	−0.858705	+	0.482208i
$904$		0			0
$905$			4.98485i			0.165702i
$906$		0			0
$907$			37.1978i			1.23513i	0.786518	+	0.617567i	$0.211882\pi$
$907$			37.1978i			1.23513i	−0.786518	+	0.617567i	$0.788118\pi$
$908$		0			0
$909$	−33.8772	−	20.6519i	−1.12363	−	0.684981i
$910$		0			0
$911$	−34.6375			−1.14759			−0.573796	−	0.818998i	$-0.694530\pi$
$911$	−34.6375			−1.14759			−0.573796	+	0.818998i	$0.694530\pi$
$912$		0			0
$913$	23.3693			0.773412
$914$		0			0
$915$	−6.36696	−	11.3381i	−0.210485	−	0.374827i
$916$		0			0
$917$		−	12.5194i		−	0.413426i
$918$		0			0
$919$		−	2.20837i		−	0.0728474i	−0.999336	−	0.0364237i	$-0.988403\pi$
$919$		−	2.20837i		−	0.0728474i	0.999336	−	0.0364237i	$-0.0115966\pi$
$920$		0			0
$921$	−10.0691	−	17.9309i	−0.331789	−	0.590842i
$922$		0			0
$923$	−6.78456			−0.223317
$924$		0			0
$925$	37.3923			1.22945
$926$		0			0
$927$	20.1472	+	12.2820i	0.661720	+	0.403393i
$928$		0			0
$929$		−	30.8769i		−	1.01304i	−0.862229	−	0.506519i	$-0.830932\pi$
$929$		−	30.8769i		−	1.01304i	0.862229	−	0.506519i	$-0.169068\pi$
$930$		0			0
$931$			9.22437i			0.302317i
$932$		0			0
$933$	24.2824	−	13.6358i	0.794968	−	0.446417i
$934$		0			0
$935$	1.78384			0.0583378
$936$		0			0
$937$	−39.6155			−1.29418			−0.647091	−	0.762412i	$-0.724015\pi$
$937$	−39.6155			−1.29418			−0.647091	+	0.762412i	$0.724015\pi$
$938$		0			0
$939$	32.6443	−	18.3315i	1.06531	−	0.598226i
$940$		0			0
$941$		−	22.8201i		−	0.743913i	−0.928250	−	0.371957i	$-0.878687\pi$
$941$		−	22.8201i		−	0.743913i	0.928250	−	0.371957i	$-0.121313\pi$
$942$		0			0
$943$		−	28.9807i		−	0.943740i
$944$		0			0
$945$	0.384472	−	10.7386i	0.0125069	−	0.349328i
$946$		0			0
$947$	−10.5487			−0.342786			−0.171393	−	0.985203i	$-0.554827\pi$
$947$	−10.5487			−0.342786			−0.171393	+	0.985203i	$0.554827\pi$
$948$		0			0
$949$	26.6811			0.866104
$950$		0			0
$951$	−5.75462	−	10.2477i	−0.186606	−	0.332304i
$952$		0			0
$953$			24.4924i			0.793387i	0.917951	+	0.396694i	$0.129842\pi$
$953$			24.4924i			0.793387i	−0.917951	+	0.396694i	$0.870158\pi$
$954$		0			0
$955$		−	18.8664i		−	0.610501i
$956$		0			0
$957$	−8.24782	−	14.6875i	−0.266614	−	0.474780i
$958$		0			0
$959$	8.83348			0.285248
$960$		0			0
$961$	5.63068			0.181635
$962$		0			0
$963$	3.55531	−	5.83209i	0.114568	−	0.187937i
$964$		0			0
$965$		−	3.74571i		−	0.120579i
$966$		0			0
$967$			52.5763i			1.69074i	0.534181	+	0.845370i	$0.320620\pi$
$967$			52.5763i			1.69074i	−0.534181	+	0.845370i	$0.679380\pi$
$968$		0			0
$969$	−7.36932	+	4.13826i	−0.236737	+	0.132940i
$970$		0			0
$971$	−13.3603			−0.428753			−0.214377	−	0.976751i	$-0.568772\pi$
$971$	−13.3603			−0.428753			−0.214377	+	0.976751i	$0.568772\pi$
$972$		0			0
$973$	−11.2371			−0.360245
$974$		0			0
$975$	26.5982	−	14.9363i	0.851824	−	0.478344i
$976$		0			0
$977$		−	29.6155i		−	0.947485i	−0.880663	−	0.473742i	$-0.842903\pi$
$977$		−	29.6155i		−	0.947485i	0.880663	−	0.473742i	$-0.157097\pi$
$978$		0			0
$979$		−	22.2586i		−	0.711389i
$980$		0			0
$981$	−14.1617	+	23.2306i	−0.452147	+	0.741697i
$982$		0			0
$983$	−4.76493			−0.151978			−0.0759889	−	0.997109i	$-0.524211\pi$
$983$	−4.76493			−0.151978			−0.0759889	+	0.997109i	$0.524211\pi$
$984$		0			0
$985$	−16.1080			−0.513242
$986$		0			0
$987$	10.5945	+	18.8664i	0.337225	+	0.600523i
$988$		0			0
$989$		−	56.0561i		−	1.78248i
$990$		0			0
$991$		−	31.5372i		−	1.00181i	−0.865501	−	0.500907i	$-0.833000\pi$
$991$		−	31.5372i		−	1.00181i	0.865501	−	0.500907i	$-0.167000\pi$
$992$		0			0
$993$	12.9460	+	23.0540i	0.410830	+	0.731596i
$994$		0			0
$995$	7.36520			0.233493
$996$		0			0
$997$	−42.1897			−1.33616			−0.668081	−	0.744088i	$-0.732884\pi$
$997$	−42.1897			−1.33616			−0.668081	+	0.744088i	$0.732884\pi$
$998$		0			0
$999$	−1.68608	+	47.0936i	−0.0533451	+	1.48997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.m.1535.3	yes	16
3.2	odd	2	inner	1536.2.c.m.1535.16	yes	16
4.3	odd	2	inner	1536.2.c.m.1535.13	yes	16
8.3	odd	2	inner	1536.2.c.m.1535.4	yes	16
8.5	even	2	inner	1536.2.c.m.1535.14	yes	16
12.11	even	2	inner	1536.2.c.m.1535.2	yes	16
16.3	odd	4		1536.2.f.l.767.11		16
16.5	even	4		1536.2.f.l.767.12		16
16.11	odd	4		1536.2.f.l.767.6		16
16.13	even	4		1536.2.f.l.767.5		16
24.5	odd	2	inner	1536.2.c.m.1535.1	✓	16
24.11	even	2	inner	1536.2.c.m.1535.15	yes	16
48.5	odd	4		1536.2.f.l.767.9		16
48.11	even	4		1536.2.f.l.767.7		16
48.29	odd	4		1536.2.f.l.767.8		16
48.35	even	4		1536.2.f.l.767.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.m.1535.1	✓	16	24.5	odd	2	inner
1536.2.c.m.1535.2	yes	16	12.11	even	2	inner
1536.2.c.m.1535.3	yes	16	1.1	even	1	trivial
1536.2.c.m.1535.4	yes	16	8.3	odd	2	inner
1536.2.c.m.1535.13	yes	16	4.3	odd	2	inner
1536.2.c.m.1535.14	yes	16	8.5	even	2	inner
1536.2.c.m.1535.15	yes	16	24.11	even	2	inner
1536.2.c.m.1535.16	yes	16	3.2	odd	2	inner
1536.2.f.l.767.5		16	16.13	even	4
1536.2.f.l.767.6		16	16.11	odd	4
1536.2.f.l.767.7		16	48.11	even	4
1536.2.f.l.767.8		16	48.29	odd	4
1536.2.f.l.767.9		16	48.5	odd	4
1536.2.f.l.767.10		16	48.35	even	4
1536.2.f.l.767.11		16	16.3	odd	4
1536.2.f.l.767.12		16	16.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.51022 + 0.848071i) q^{3} -0.936426i q^{5} -2.20837i q^{7} +(1.56155 - 2.56155i) q^{9} +O(q^{10})\)	\(q+(-1.51022 + 0.848071i) q^{3} -0.936426i q^{5} -2.20837i q^{7} +(1.56155 - 2.56155i) q^{9} -1.69614 q^{11} -4.27156 q^{13} +(0.794156 + 1.41421i) q^{15} -1.12311i q^{17} +4.34475i q^{19} +(1.87285 + 3.33513i) q^{21} -7.24517 q^{23} +4.12311 q^{25} +(-0.185917 + 5.19283i) q^{27} -5.73384i q^{29} +5.03680i q^{31} +(2.56155 - 1.43845i) q^{33} -2.06798 q^{35} +9.06897 q^{37} +(6.45101 - 3.62258i) q^{39} +4.00000i q^{41} +7.73704i q^{43} +(-2.39871 - 1.46228i) q^{45} +5.65685 q^{47} +2.12311 q^{49} +(0.952473 + 1.69614i) q^{51} +14.2770i q^{53} +1.58831i q^{55} +(-3.68466 - 6.56155i) q^{57} -6.41273 q^{59} -8.01726 q^{61} +(-5.65685 - 3.44849i) q^{63} +4.00000i q^{65} +10.3857i q^{67} +(10.9418 - 6.14441i) q^{69} +1.58831 q^{71} -6.24621 q^{73} +(-6.22681 + 3.49668i) q^{75} +3.74571i q^{77} -10.6937i q^{79} +(-4.12311 - 8.00000i) q^{81} -13.7779 q^{83} -1.05171 q^{85} +(4.86270 + 8.65938i) q^{87} +13.1231i q^{89} +9.43318i q^{91} +(-4.27156 - 7.60669i) q^{93} +4.06854 q^{95} -7.12311 q^{97} +(-2.64861 + 4.34475i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} + 8 q^{33} - 32 q^{49} + 40 q^{57} + 32 q^{73} - 48 q^{97}+O(q^{100}) \) 16 * q - 8 * q^9 + 8 * q^33 - 32 * q^49 + 40 * q^57 + 32 * q^73 - 48 * q^97

Introduction

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