LMFDB - Embedded newform 2160.3.c.q.1889.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,3,Mod(1889,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, 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("2160.1889"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.4
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.q.1889.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.52542 + 2.12616i) q^{5} -0.343255i q^{7} -3.68323i q^{11} -5.50147i q^{13} -6.36803 q^{17} +2.62792 q^{19} +12.2810 q^{23} +(15.9589 - 19.2435i) q^{25} +24.0167i q^{29} +1.04710 q^{31} +(0.729815 + 1.55337i) q^{35} -36.2550i q^{37} -8.06180i q^{41} +19.3113i q^{43} -38.7705 q^{47} +48.8822 q^{49} -3.24292 q^{53} +(7.83113 + 16.6682i) q^{55} +51.5904i q^{59} +28.7343 q^{61} +(11.6970 + 24.8965i) q^{65} +77.8013i q^{67} +87.5582i q^{71} -108.427i q^{73} -1.26429 q^{77} -78.1862 q^{79} -62.1805 q^{83} +(28.8180 - 13.5395i) q^{85} +35.2651i q^{89} -1.88841 q^{91} +(-11.8925 + 5.58738i) q^{95} -65.0096i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) $ 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.52542	+	2.12616i	−0.905084	+	0.425232i
$5$	−4.52542	+	2.12616i	−0.905084	+	0.425232i
$6$		0			0
$7$		−	0.343255i		−	0.0490364i	−0.999699	−	0.0245182i	$-0.992195\pi$
$7$		−	0.343255i		−	0.0490364i	0.999699	−	0.0245182i	$-0.00780517\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.68323i		−	0.334839i	−0.985886	−	0.167419i	$-0.946457\pi$
$11$		−	3.68323i		−	0.334839i	0.985886	−	0.167419i	$-0.0535434\pi$
$12$		0			0
$13$		−	5.50147i		−	0.423190i	−0.977357	−	0.211595i	$-0.932134\pi$
$13$		−	5.50147i		−	0.423190i	0.977357	−	0.211595i	$-0.0678658\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.36803			−0.374590			−0.187295	−	0.982304i	$-0.559972\pi$
$17$	−6.36803			−0.374590			−0.187295	+	0.982304i	$0.559972\pi$
$18$		0			0
$19$	2.62792			0.138312			0.0691558	−	0.997606i	$-0.477969\pi$
$19$	2.62792			0.138312			0.0691558	+	0.997606i	$0.477969\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	12.2810			0.533959			0.266979	−	0.963702i	$-0.413974\pi$
$23$	12.2810			0.533959			0.266979	+	0.963702i	$0.413974\pi$
$24$		0			0
$25$	15.9589	−	19.2435i	0.638355	−	0.769742i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			24.0167i			0.828162i	0.910240	+	0.414081i	$0.135897\pi$
$29$			24.0167i			0.828162i	−0.910240	+	0.414081i	$0.864103\pi$
$30$		0			0
$31$	1.04710			0.0337774			0.0168887	−	0.999857i	$-0.494624\pi$
$31$	1.04710			0.0337774			0.0168887	+	0.999857i	$0.494624\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.729815	+	1.55337i	0.0208519	+	0.0443821i
$36$		0			0
$37$		−	36.2550i		−	0.979864i	−0.871761	−	0.489932i	$-0.837022\pi$
$37$		−	36.2550i		−	0.979864i	0.871761	−	0.489932i	$-0.162978\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.06180i		−	0.196629i	−0.995155	−	0.0983147i	$-0.968655\pi$
$41$		−	8.06180i		−	0.196629i	0.995155	−	0.0983147i	$-0.0313452\pi$
$42$		0			0
$43$			19.3113i			0.449100i	0.974463	+	0.224550i	$0.0720912\pi$
$43$			19.3113i			0.449100i	−0.974463	+	0.224550i	$0.927909\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−38.7705			−0.824904			−0.412452	−	0.910979i	$-0.635328\pi$
$47$	−38.7705			−0.824904			−0.412452	+	0.910979i	$0.635328\pi$
$48$		0			0
$49$	48.8822			0.997595
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.24292			−0.0611871			−0.0305936	−	0.999532i	$-0.509740\pi$
$53$	−3.24292			−0.0611871			−0.0305936	+	0.999532i	$0.509740\pi$
$54$		0			0
$55$	7.83113	+	16.6682i	0.142384	+	0.303057i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			51.5904i			0.874413i	0.899361	+	0.437206i	$0.144032\pi$
$59$			51.5904i			0.874413i	−0.899361	+	0.437206i	$0.855968\pi$
$60$		0			0
$61$	28.7343			0.471054			0.235527	−	0.971868i	$-0.424318\pi$
$61$	28.7343			0.471054			0.235527	+	0.971868i	$0.424318\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	11.6970	+	24.8965i	0.179954	+	0.383023i
$66$		0			0
$67$			77.8013i			1.16121i	0.814184	+	0.580607i	$0.197185\pi$
$67$			77.8013i			1.16121i	−0.814184	+	0.580607i	$0.802815\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			87.5582i			1.23321i	0.787271	+	0.616607i	$0.211493\pi$
$71$			87.5582i			1.23321i	−0.787271	+	0.616607i	$0.788507\pi$
$72$		0			0
$73$		−	108.427i		−	1.48530i	−0.669681	−	0.742649i	$-0.733569\pi$
$73$		−	108.427i		−	1.48530i	0.669681	−	0.742649i	$-0.266431\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.26429			−0.0164193
$78$		0			0
$79$	−78.1862			−0.989699			−0.494849	−	0.868979i	$-0.664777\pi$
$79$	−78.1862			−0.989699			−0.494849	+	0.868979i	$0.664777\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−62.1805			−0.749163			−0.374581	−	0.927194i	$-0.622214\pi$
$83$	−62.1805			−0.749163			−0.374581	+	0.927194i	$0.622214\pi$
$84$		0			0
$85$	28.8180	−	13.5395i	0.339036	−	0.159288i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			35.2651i			0.396237i	0.980178	+	0.198119i	$0.0634831\pi$
$89$			35.2651i			0.396237i	−0.980178	+	0.198119i	$0.936517\pi$
$90$		0			0
$91$	−1.88841			−0.0207517
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.8925	+	5.58738i	−0.125184	+	0.0588146i
$96$		0			0
$97$		−	65.0096i		−	0.670202i	−0.942182	−	0.335101i	$-0.891230\pi$
$97$		−	65.0096i		−	0.670202i	0.942182	−	0.335101i	$-0.108770\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			31.8095i			0.314945i	0.987523	+	0.157473i	$0.0503346\pi$
$101$			31.8095i			0.314945i	−0.987523	+	0.157473i	$0.949665\pi$
$102$		0			0
$103$			105.375i			1.02306i	0.859266	+	0.511529i	$0.170921\pi$
$103$			105.375i			1.02306i	−0.859266	+	0.511529i	$0.829079\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.48988			−0.0326157			−0.0163078	−	0.999867i	$-0.505191\pi$
$107$	−3.48988			−0.0326157			−0.0163078	+	0.999867i	$0.505191\pi$
$108$		0			0
$109$	−111.004			−1.01839			−0.509195	−	0.860651i	$-0.670057\pi$
$109$	−111.004			−1.01839			−0.509195	+	0.860651i	$0.670057\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−38.1865			−0.337933			−0.168967	−	0.985622i	$-0.554043\pi$
$113$	−38.1865			−0.337933			−0.168967	+	0.985622i	$0.554043\pi$
$114$		0			0
$115$	−55.5769	+	26.1115i	−0.483277	+	0.227056i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.18586i			0.0183686i
$120$		0			0
$121$	107.434			0.887883
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−31.3058	+	121.016i	−0.250446	+	0.968130i
$126$		0			0
$127$			138.765i			1.09264i	0.837577	+	0.546320i	$0.183972\pi$
$127$			138.765i			1.09264i	−0.837577	+	0.546320i	$0.816028\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			131.550i			1.00420i	0.864811	+	0.502098i	$0.167438\pi$
$131$			131.550i			1.00420i	−0.864811	+	0.502098i	$0.832562\pi$
$132$		0			0
$133$		−	0.902047i		−	0.00678231i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−152.211			−1.11103			−0.555513	−	0.831508i	$-0.687478\pi$
$137$	−152.211			−1.11103			−0.555513	+	0.831508i	$0.687478\pi$
$138$		0			0
$139$	−64.1907			−0.461803			−0.230902	−	0.972977i	$-0.574168\pi$
$139$	−64.1907			−0.461803			−0.230902	+	0.972977i	$0.574168\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−20.2632			−0.141700
$144$		0			0
$145$	−51.0634	−	108.686i	−0.352161	−	0.749556i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			233.886i			1.56970i	0.619683	+	0.784852i	$0.287261\pi$
$149$			233.886i			1.56970i	−0.619683	+	0.784852i	$0.712739\pi$
$150$		0			0
$151$	−173.707			−1.15038			−0.575189	−	0.818021i	$-0.695071\pi$
$151$	−173.707			−1.15038			−0.575189	+	0.818021i	$0.695071\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.73857	+	2.22630i	−0.0305714	+	0.0143632i
$156$		0			0
$157$			121.059i			0.771079i	0.922691	+	0.385539i	$0.125985\pi$
$157$			121.059i			0.771079i	−0.922691	+	0.385539i	$0.874015\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.21553i		−	0.0261834i
$162$		0			0
$163$			93.2845i			0.572297i	0.958185	+	0.286149i	$0.0923751\pi$
$163$			93.2845i			0.572297i	−0.958185	+	0.286149i	$0.907625\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.15499			−0.0188922			−0.00944608	−	0.999955i	$-0.503007\pi$
$167$	−3.15499			−0.0188922			−0.00944608	+	0.999955i	$0.503007\pi$
$168$		0			0
$169$	138.734			0.820910
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−163.435			−0.944708			−0.472354	−	0.881409i	$-0.656596\pi$
$173$	−163.435			−0.944708			−0.472354	+	0.881409i	$0.656596\pi$
$174$		0			0
$175$	−6.60544	−	5.47796i	−0.0377454	−	0.0313027i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			39.4592i			0.220442i	0.993907	+	0.110221i	$0.0351559\pi$
$179$			39.4592i			0.220442i	−0.993907	+	0.110221i	$0.964844\pi$
$180$		0			0
$181$	−278.211			−1.53708			−0.768538	−	0.639804i	$-0.779015\pi$
$181$	−278.211			−1.53708			−0.768538	+	0.639804i	$0.779015\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	77.0839	+	164.069i	0.416669	+	0.886859i
$186$		0			0
$187$			23.4549i			0.125427i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			318.694i			1.66855i	0.551347	+	0.834276i	$0.314114\pi$
$191$			318.694i			1.66855i	−0.551347	+	0.834276i	$0.685886\pi$
$192$		0			0
$193$		−	5.67533i		−	0.0294059i	−0.999892	−	0.0147029i	$-0.995320\pi$
$193$		−	5.67533i		−	0.0294059i	0.999892	−	0.0147029i	$-0.00468026\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−45.3319			−0.230111			−0.115056	−	0.993359i	$-0.536705\pi$
$197$	−45.3319			−0.230111			−0.115056	+	0.993359i	$0.536705\pi$
$198$		0			0
$199$	177.557			0.892245			0.446123	−	0.894972i	$-0.352805\pi$
$199$	177.557			0.892245			0.446123	+	0.894972i	$0.352805\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.24385			0.0406101
$204$		0			0
$205$	17.1407	+	36.4831i	0.0836131	+	0.177966i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	9.67923i		−	0.0463121i
$210$		0			0
$211$	−62.1257			−0.294435			−0.147217	−	0.989104i	$-0.547032\pi$
$211$	−62.1257			−0.294435			−0.147217	+	0.989104i	$0.547032\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−41.0589	−	87.3918i	−0.190972	−	0.406473i
$216$		0			0
$217$		−	0.359422i		−	0.00165632i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			35.0335i			0.158523i
$222$		0			0
$223$		−	242.729i		−	1.08847i	−0.838933	−	0.544235i	$-0.816820\pi$
$223$		−	242.729i		−	1.08847i	0.838933	−	0.544235i	$-0.183180\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−190.244			−0.838079			−0.419039	−	0.907968i	$-0.637633\pi$
$227$	−190.244			−0.838079			−0.419039	+	0.907968i	$0.637633\pi$
$228$		0			0
$229$	70.8762			0.309503			0.154752	−	0.987953i	$-0.450542\pi$
$229$	70.8762			0.309503			0.154752	+	0.987953i	$0.450542\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	200.325			0.859763			0.429881	−	0.902885i	$-0.358555\pi$
$233$	200.325			0.859763			0.429881	+	0.902885i	$0.358555\pi$
$234$		0			0
$235$	175.453	−	82.4323i	0.746608	−	0.350776i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			96.9317i			0.405572i	0.979223	+	0.202786i	$0.0649996\pi$
$239$			96.9317i			0.405572i	−0.979223	+	0.202786i	$0.935000\pi$
$240$		0			0
$241$	218.474			0.906532			0.453266	−	0.891375i	$-0.350259\pi$
$241$	218.474			0.906532			0.453266	+	0.891375i	$0.350259\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−221.212	+	103.931i	−0.902908	+	0.424210i
$246$		0			0
$247$		−	14.4574i		−	0.0585321i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			429.990i			1.71311i	0.516057	+	0.856554i	$0.327399\pi$
$251$			429.990i			1.71311i	−0.516057	+	0.856554i	$0.672601\pi$
$252$		0			0
$253$		−	45.2339i		−	0.178790i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	250.541			0.974866			0.487433	−	0.873160i	$-0.337933\pi$
$257$	250.541			0.974866			0.487433	+	0.873160i	$0.337933\pi$
$258$		0			0
$259$	−12.4447			−0.0480490
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−235.274			−0.894578			−0.447289	−	0.894389i	$-0.647610\pi$
$263$	−235.274			−0.894578			−0.447289	+	0.894389i	$0.647610\pi$
$264$		0			0
$265$	14.6756	−	6.89496i	0.0553795	−	0.0260187i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			362.657i			1.34817i	0.738656	+	0.674083i	$0.235461\pi$
$269$			362.657i			1.34817i	−0.738656	+	0.674083i	$0.764539\pi$
$270$		0			0
$271$	435.221			1.60598			0.802992	−	0.595990i	$-0.203240\pi$
$271$	435.221			1.60598			0.802992	+	0.595990i	$0.203240\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−70.8783	−	58.7802i	−0.257739	−	0.213746i
$276$		0			0
$277$			340.335i			1.22865i	0.789055	+	0.614323i	$0.210571\pi$
$277$			340.335i			1.22865i	−0.789055	+	0.614323i	$0.789429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	394.063i		−	1.40236i	−0.712985	−	0.701179i	$-0.752657\pi$
$281$		−	394.063i		−	1.40236i	0.712985	−	0.701179i	$-0.247343\pi$
$282$		0			0
$283$		−	454.014i		−	1.60429i	−0.597129	−	0.802145i	$-0.703692\pi$
$283$		−	454.014i		−	1.60429i	0.597129	−	0.802145i	$-0.296308\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.76725			−0.00964200
$288$		0			0
$289$	−248.448			−0.859682
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−98.6447			−0.336671			−0.168336	−	0.985730i	$-0.553839\pi$
$293$	−98.6447			−0.336671			−0.168336	+	0.985730i	$0.553839\pi$
$294$		0			0
$295$	−109.689	−	233.468i	−0.371828	−	0.791417i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	67.5638i		−	0.225966i
$300$		0			0
$301$	6.62870			0.0220223
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−130.035	+	61.0937i	−0.426343	+	0.200307i
$306$		0			0
$307$			56.4148i			0.183762i	0.995770	+	0.0918809i	$0.0292879\pi$
$307$			56.4148i			0.183762i	−0.995770	+	0.0918809i	$0.970712\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			348.198i			1.11961i	0.828626	+	0.559803i	$0.189123\pi$
$311$			348.198i			1.11961i	−0.828626	+	0.559803i	$0.810877\pi$
$312$		0			0
$313$			220.046i			0.703023i	0.936184	+	0.351512i	$0.114332\pi$
$313$			220.046i			0.703023i	−0.936184	+	0.351512i	$0.885668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−270.223			−0.852439			−0.426220	−	0.904620i	$-0.640155\pi$
$317$	−270.223			−0.852439			−0.426220	+	0.904620i	$0.640155\pi$
$318$		0			0
$319$	88.4589			0.277301
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.7347			−0.0518102
$324$		0			0
$325$	−105.868	−	87.7973i	−0.325747	−	0.270145i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			13.3082i			0.0404503i
$330$		0			0
$331$	128.278			0.387546			0.193773	−	0.981046i	$-0.437927\pi$
$331$	128.278			0.387546			0.193773	+	0.981046i	$0.437927\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−165.418	−	352.084i	−0.493785	−	1.05100i
$336$		0			0
$337$			308.043i			0.914073i	0.889448	+	0.457037i	$0.151089\pi$
$337$			308.043i			0.914073i	−0.889448	+	0.457037i	$0.848911\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	3.85671i		−	0.0113100i
$342$		0			0
$343$		−	33.5985i		−	0.0979549i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	147.805			0.425952			0.212976	−	0.977057i	$-0.431684\pi$
$347$	147.805			0.425952			0.212976	+	0.977057i	$0.431684\pi$
$348$		0			0
$349$	182.397			0.522627			0.261314	−	0.965254i	$-0.415844\pi$
$349$	182.397			0.522627			0.261314	+	0.965254i	$0.415844\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−208.438			−0.590475			−0.295238	−	0.955424i	$-0.595399\pi$
$353$	−208.438			−0.590475			−0.295238	+	0.955424i	$0.595399\pi$
$354$		0			0
$355$	−186.163	−	396.238i	−0.524402	−	1.11616i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			44.2454i			0.123246i	0.998099	+	0.0616231i	$0.0196277\pi$
$359$			44.2454i			0.123246i	−0.998099	+	0.0616231i	$0.980372\pi$
$360$		0			0
$361$	−354.094			−0.980870
$362$		0			0
$363$		0			0
$364$		0			0
$365$	230.533	+	490.677i	0.631596	+	1.34432i
$366$		0			0
$367$		−	281.769i		−	0.767763i	−0.923382	−	0.383881i	$-0.874587\pi$
$367$		−	281.769i		−	0.767763i	0.923382	−	0.383881i	$-0.125413\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.11315i			0.00300040i
$372$		0			0
$373$		−	26.4362i		−	0.0708745i	−0.999372	−	0.0354372i	$-0.988718\pi$
$373$		−	26.4362i		−	0.0708745i	0.999372	−	0.0354372i	$-0.0112824\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	132.127			0.350470
$378$		0			0
$379$	547.809			1.44541			0.722703	−	0.691158i	$-0.242899\pi$
$379$	547.809			1.44541			0.722703	+	0.691158i	$0.242899\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	685.561			1.78998			0.894988	−	0.446089i	$-0.147184\pi$
$383$	685.561			1.78998			0.894988	+	0.446089i	$0.147184\pi$
$384$		0			0
$385$	5.72142	−	2.68807i	0.0148608	−	0.00698201i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			447.088i			1.14933i	0.818390	+	0.574663i	$0.194867\pi$
$389$			447.088i			1.14933i	−0.818390	+	0.574663i	$0.805133\pi$
$390$		0			0
$391$	−78.2061			−0.200016
$392$		0			0
$393$		0			0
$394$		0			0
$395$	353.826	−	166.236i	0.895761	−	0.420852i
$396$		0			0
$397$			346.488i			0.872766i	0.899761	+	0.436383i	$0.143741\pi$
$397$			346.488i			0.872766i	−0.899761	+	0.436383i	$0.856259\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	650.283i		−	1.62165i	−0.585286	−	0.810827i	$-0.699018\pi$
$401$		−	650.283i		−	1.62165i	0.585286	−	0.810827i	$-0.300982\pi$
$402$		0			0
$403$		−	5.76059i		−	0.0142943i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−133.535			−0.328096
$408$		0			0
$409$	−213.767			−0.522658			−0.261329	−	0.965250i	$-0.584161\pi$
$409$	−213.767			−0.522658			−0.261329	+	0.965250i	$0.584161\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	17.7086			0.0428781
$414$		0			0
$415$	281.393	−	132.206i	0.678055	−	0.318568i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	248.334i		−	0.592683i	−0.955082	−	0.296342i	$-0.904233\pi$
$419$		−	248.334i		−	0.592683i	0.955082	−	0.296342i	$-0.0957667\pi$
$420$		0			0
$421$	93.8004			0.222804			0.111402	−	0.993775i	$-0.464466\pi$
$421$	93.8004			0.222804			0.111402	+	0.993775i	$0.464466\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−101.627	+	122.544i	−0.239122	+	0.288338i
$426$		0			0
$427$		−	9.86318i		−	0.0230988i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			125.023i			0.290077i	0.989426	+	0.145038i	$0.0463306\pi$
$431$			125.023i			0.290077i	−0.989426	+	0.145038i	$0.953669\pi$
$432$		0			0
$433$			221.389i			0.511291i	0.966771	+	0.255645i	$0.0822880\pi$
$433$			221.389i			0.511291i	−0.966771	+	0.255645i	$0.917712\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	32.2736			0.0738527
$438$		0			0
$439$	−436.727			−0.994823			−0.497412	−	0.867515i	$-0.665716\pi$
$439$	−436.727			−0.994823			−0.497412	+	0.867515i	$0.665716\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−755.517			−1.70546			−0.852729	−	0.522354i	$-0.825054\pi$
$443$	−755.517			−1.70546			−0.852729	+	0.522354i	$0.825054\pi$
$444$		0			0
$445$	−74.9793	−	159.590i	−0.168493	−	0.358628i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	217.633i		−	0.484706i	−0.970188	−	0.242353i	$-0.922081\pi$
$449$		−	217.633i		−	0.484706i	0.970188	−	0.242353i	$-0.0779191\pi$
$450$		0			0
$451$	−29.6934			−0.0658391
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.54583	−	4.01505i	0.0187821	−	0.00882430i
$456$		0			0
$457$			654.376i			1.43189i	0.698154	+	0.715947i	$0.254005\pi$
$457$			654.376i			1.43189i	−0.698154	+	0.715947i	$0.745995\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	418.815i		−	0.908492i	−0.890876	−	0.454246i	$-0.849909\pi$
$461$		−	418.815i		−	0.908492i	0.890876	−	0.454246i	$-0.150091\pi$
$462$		0			0
$463$		−	282.863i		−	0.610935i	−0.952203	−	0.305467i	$-0.901187\pi$
$463$		−	282.863i		−	0.610935i	0.952203	−	0.305467i	$-0.0988127\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	537.024			1.14994			0.574972	−	0.818173i	$-0.305013\pi$
$467$	537.024			1.14994			0.574972	+	0.818173i	$0.305013\pi$
$468$		0			0
$469$	26.7057			0.0569418
$470$		0			0
$471$		0			0
$472$		0			0
$473$	71.1279			0.150376
$474$		0			0
$475$	41.9387	−	50.5705i	0.0882920	−	0.106464i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	611.445i		−	1.27650i	−0.769828	−	0.638252i	$-0.779658\pi$
$479$		−	611.445i		−	1.27650i	0.769828	−	0.638252i	$-0.220342\pi$
$480$		0			0
$481$	−199.455			−0.414668
$482$		0			0
$483$		0			0
$484$		0			0
$485$	138.221	+	294.196i	0.284991	+	0.606589i
$486$		0			0
$487$			706.656i			1.45104i	0.688201	+	0.725520i	$0.258401\pi$
$487$			706.656i			1.45104i	−0.688201	+	0.725520i	$0.741599\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			628.043i			1.27911i	0.768745	+	0.639555i	$0.220881\pi$
$491$			628.043i			1.27911i	−0.768745	+	0.639555i	$0.779119\pi$
$492$		0			0
$493$		−	152.939i		−	0.310221i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	30.0548			0.0604724
$498$		0			0
$499$	−483.558			−0.969055			−0.484527	−	0.874776i	$-0.661008\pi$
$499$	−483.558			−0.969055			−0.484527	+	0.874776i	$0.661008\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−365.346			−0.726335			−0.363167	−	0.931724i	$-0.618305\pi$
$503$	−365.346			−0.726335			−0.363167	+	0.931724i	$0.618305\pi$
$504$		0			0
$505$	−67.6320	−	143.951i	−0.133925	−	0.285052i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			234.721i			0.461141i	0.973056	+	0.230570i	$0.0740592\pi$
$509$			234.721i			0.461141i	−0.973056	+	0.230570i	$0.925941\pi$
$510$		0			0
$511$	−37.2180			−0.0728337
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−224.044	−	476.866i	−0.435037	−	0.925953i
$516$		0			0
$517$			142.801i			0.276210i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	982.201i		−	1.88522i	−0.333891	−	0.942612i	$-0.608362\pi$
$521$		−	982.201i		−	1.88522i	0.333891	−	0.942612i	$-0.391638\pi$
$522$		0			0
$523$		−	110.800i		−	0.211855i	−0.994374	−	0.105927i	$-0.966219\pi$
$523$		−	110.800i		−	0.211855i	0.994374	−	0.105927i	$-0.0337811\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.66797			−0.0126527
$528$		0			0
$529$	−378.176			−0.714888
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−44.3517			−0.0832115
$534$		0			0
$535$	15.7932	−	7.42004i	0.0295199	−	0.0138692i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	180.044i		−	0.334034i
$540$		0			0
$541$	−54.4450			−0.100638			−0.0503189	−	0.998733i	$-0.516024\pi$
$541$	−54.4450			−0.100638			−0.0503189	+	0.998733i	$0.516024\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	502.342	−	236.013i	0.921729	−	0.433052i
$546$		0			0
$547$			807.086i			1.47548i	0.675087	+	0.737738i	$0.264106\pi$
$547$			807.086i			1.47548i	−0.675087	+	0.737738i	$0.735894\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			63.1140i			0.114544i
$552$		0			0
$553$			26.8378i			0.0485313i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−90.5746			−0.162612			−0.0813058	−	0.996689i	$-0.525909\pi$
$557$	−90.5746			−0.162612			−0.0813058	+	0.996689i	$0.525909\pi$
$558$		0			0
$559$	106.241			0.190055
$560$		0			0
$561$		0			0
$562$		0			0
$563$	933.320			1.65776			0.828881	−	0.559425i	$-0.188978\pi$
$563$	933.320			1.65776			0.828881	+	0.559425i	$0.188978\pi$
$564$		0			0
$565$	172.810	−	81.1905i	0.305858	−	0.143700i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1022.25i		−	1.79658i	−0.439404	−	0.898289i	$-0.644810\pi$
$569$		−	1022.25i		−	1.79658i	0.439404	−	0.898289i	$-0.355190\pi$
$570$		0			0
$571$	−102.773			−0.179987			−0.0899936	−	0.995942i	$-0.528685\pi$
$571$	−102.773			−0.179987			−0.0899936	+	0.995942i	$0.528685\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	195.992	−	236.331i	0.340855	−	0.411010i
$576$		0			0
$577$			354.062i			0.613625i	0.951770	+	0.306812i	$0.0992624\pi$
$577$			354.062i			0.613625i	−0.951770	+	0.306812i	$0.900738\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			21.3438i			0.0367363i
$582$		0			0
$583$			11.9444i			0.0204878i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−462.839			−0.788482			−0.394241	−	0.919007i	$-0.628992\pi$
$587$	−462.839			−0.788482			−0.394241	+	0.919007i	$0.628992\pi$
$588$		0			0
$589$	2.75170			0.00467181
$590$		0			0
$591$		0			0
$592$		0			0
$593$	955.037			1.61052			0.805259	−	0.592923i	$-0.202026\pi$
$593$	955.037			1.61052			0.805259	+	0.592923i	$0.202026\pi$
$594$		0			0
$595$	−4.64749	−	9.89193i	−0.00781090	−	0.0166251i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	156.127i		−	0.260645i	−0.991472	−	0.130323i	$-0.958399\pi$
$599$		−	156.127i		−	0.260645i	0.991472	−	0.130323i	$-0.0416013\pi$
$600$		0			0
$601$	287.914			0.479059			0.239530	−	0.970889i	$-0.423007\pi$
$601$	287.914			0.479059			0.239530	+	0.970889i	$0.423007\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−486.183	+	228.422i	−0.803609	+	0.377556i
$606$		0			0
$607$			651.496i			1.07331i	0.843803	+	0.536653i	$0.180311\pi$
$607$			651.496i			1.07331i	−0.843803	+	0.536653i	$0.819689\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			213.295i			0.349091i
$612$		0			0
$613$		−	769.780i		−	1.25576i	−0.778311	−	0.627879i	$-0.783923\pi$
$613$		−	769.780i		−	1.25576i	0.778311	−	0.627879i	$-0.216077\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	852.034			1.38093			0.690465	−	0.723365i	$-0.257406\pi$
$617$	852.034			1.38093			0.690465	+	0.723365i	$0.257406\pi$
$618$		0			0
$619$	202.420			0.327012			0.163506	−	0.986542i	$-0.447720\pi$
$619$	202.420			0.327012			0.163506	+	0.986542i	$0.447720\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.1049			0.0194301
$624$		0			0
$625$	−115.628	−	614.211i	−0.185005	−	0.982738i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			230.873i			0.367047i
$630$		0			0
$631$	−829.830			−1.31510			−0.657551	−	0.753410i	$-0.728408\pi$
$631$	−829.830			−1.31510			−0.657551	+	0.753410i	$0.728408\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−295.037	−	627.971i	−0.464625	−	0.988931i
$636$		0			0
$637$		−	268.924i		−	0.422172i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1138.98i		−	1.77688i	−0.458992	−	0.888440i	$-0.651789\pi$
$641$		−	1138.98i		−	1.77688i	0.458992	−	0.888440i	$-0.348211\pi$
$642$		0			0
$643$			322.082i			0.500905i	0.968129	+	0.250452i	$0.0805793\pi$
$643$			322.082i			0.500905i	−0.968129	+	0.250452i	$0.919421\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	148.507			0.229531			0.114766	−	0.993393i	$-0.463388\pi$
$647$	148.507			0.229531			0.114766	+	0.993393i	$0.463388\pi$
$648$		0			0
$649$	190.019			0.292787
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−332.895			−0.509794			−0.254897	−	0.966968i	$-0.582041\pi$
$653$	−332.895			−0.509794			−0.254897	+	0.966968i	$0.582041\pi$
$654$		0			0
$655$	−279.696	−	595.317i	−0.427016	−	0.908881i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	718.648i		−	1.09051i	−0.838269	−	0.545257i	$-0.816432\pi$
$659$		−	718.648i		−	1.09051i	0.838269	−	0.545257i	$-0.183568\pi$
$660$		0			0
$661$	585.033			0.885072			0.442536	−	0.896751i	$-0.354079\pi$
$661$	585.033			0.885072			0.442536	+	0.896751i	$0.354079\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.91790	+	4.08214i	0.00288406	+	0.00613856i
$666$		0			0
$667$			294.950i			0.442204i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	105.835i		−	0.157727i
$672$		0			0
$673$		−	417.638i		−	0.620561i	−0.950645	−	0.310281i	$-0.899577\pi$
$673$		−	417.638i		−	0.620561i	0.950645	−	0.310281i	$-0.100423\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1076.48			−1.59007			−0.795036	−	0.606562i	$-0.792548\pi$
$677$	−1076.48			−1.59007			−0.795036	+	0.606562i	$0.792548\pi$
$678$		0			0
$679$	−22.3149			−0.0328643
$680$		0			0
$681$		0			0
$682$		0			0
$683$	333.571			0.488391			0.244195	−	0.969726i	$-0.421476\pi$
$683$	333.571			0.488391			0.244195	+	0.969726i	$0.421476\pi$
$684$		0			0
$685$	688.817	−	323.624i	1.00557	−	0.472444i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			17.8408i			0.0258938i
$690$		0			0
$691$	366.106			0.529820			0.264910	−	0.964273i	$-0.414658\pi$
$691$	366.106			0.529820			0.264910	+	0.964273i	$0.414658\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	290.490	−	136.480i	0.417971	−	0.196374i
$696$		0			0
$697$			51.3378i			0.0736554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			188.763i			0.269277i	0.990895	+	0.134639i	$0.0429873\pi$
$701$			188.763i			0.269277i	−0.990895	+	0.134639i	$0.957013\pi$
$702$		0			0
$703$		−	95.2752i		−	0.135527i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.9188			0.0154438
$708$		0			0
$709$	101.543			0.143220			0.0716102	−	0.997433i	$-0.477186\pi$
$709$	101.543			0.143220			0.0716102	+	0.997433i	$0.477186\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.8595			0.0180357
$714$		0			0
$715$	91.6993	−	43.0827i	0.128251	−	0.0602555i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	856.985i		−	1.19191i	−0.803017	−	0.595956i	$-0.796773\pi$
$719$		−	856.985i		−	1.19191i	0.803017	−	0.595956i	$-0.203227\pi$
$720$		0			0
$721$	36.1705			0.0501671
$722$		0			0
$723$		0			0
$724$		0			0
$725$	462.166	+	383.280i	0.637471	+	0.528661i
$726$		0			0
$727$			910.786i			1.25280i	0.779502	+	0.626400i	$0.215472\pi$
$727$			910.786i			1.25280i	−0.779502	+	0.626400i	$0.784528\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	122.975i		−	0.168228i
$732$		0			0
$733$		−	925.303i		−	1.26235i	−0.775640	−	0.631175i	$-0.782573\pi$
$733$		−	925.303i		−	1.26235i	0.775640	−	0.631175i	$-0.217427\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	286.560			0.388819
$738$		0			0
$739$	−935.453			−1.26584			−0.632918	−	0.774219i	$-0.718143\pi$
$739$	−935.453			−1.26584			−0.632918	+	0.774219i	$0.718143\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1014.92			1.36597			0.682985	−	0.730433i	$-0.260682\pi$
$743$	1014.92			1.36597			0.682985	+	0.730433i	$0.260682\pi$
$744$		0			0
$745$	−497.279	−	1058.43i	−0.667489	−	1.42071i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1.19792i			0.00159936i
$750$		0			0
$751$	813.709			1.08350			0.541750	−	0.840540i	$-0.317762\pi$
$751$	813.709			1.08350			0.541750	+	0.840540i	$0.317762\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	786.097	−	369.329i	1.04119	−	0.489178i
$756$		0			0
$757$			137.031i			0.181018i	0.995896	+	0.0905092i	$0.0288495\pi$
$757$			137.031i			0.181018i	−0.995896	+	0.0905092i	$0.971151\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	703.254i		−	0.924118i	−0.886849	−	0.462059i	$-0.847111\pi$
$761$		−	703.254i		−	0.924118i	0.886849	−	0.462059i	$-0.152889\pi$
$762$		0			0
$763$			38.1028i			0.0499382i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	283.823			0.370043
$768$		0			0
$769$	515.478			0.670322			0.335161	−	0.942161i	$-0.391209\pi$
$769$	515.478			0.670322			0.335161	+	0.942161i	$0.391209\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	405.636			0.524756			0.262378	−	0.964965i	$-0.415493\pi$
$773$	405.636			0.524756			0.262378	+	0.964965i	$0.415493\pi$
$774$		0			0
$775$	16.7105	−	20.1499i	0.0215620	−	0.0259999i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	21.1858i		−	0.0271961i
$780$		0			0
$781$	322.497			0.412928
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−257.392	−	547.845i	−0.327887	−	0.697891i
$786$		0			0
$787$			701.437i			0.891279i	0.895212	+	0.445640i	$0.147024\pi$
$787$			701.437i			0.891279i	−0.895212	+	0.445640i	$0.852976\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			13.1077i			0.0165710i
$792$		0			0
$793$		−	158.081i		−	0.199345i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	867.533			1.08850			0.544249	−	0.838924i	$-0.316815\pi$
$797$	867.533			1.08850			0.544249	+	0.838924i	$0.316815\pi$
$798$		0			0
$799$	246.892			0.309001
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−399.360			−0.497335
$804$		0			0
$805$	8.96289	+	19.0770i	0.0111340	+	0.0236982i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	348.682i		−	0.431003i	−0.976503	−	0.215502i	$-0.930861\pi$
$809$		−	348.682i		−	0.431003i	0.976503	−	0.215502i	$-0.0691387\pi$
$810$		0			0
$811$	−591.992			−0.729953			−0.364977	−	0.931017i	$-0.618923\pi$
$811$	−591.992			−0.729953			−0.364977	+	0.931017i	$0.618923\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−198.338	−	422.151i	−0.243359	−	0.517977i
$816$		0			0
$817$			50.7486i			0.0621158i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	285.989i		−	0.348342i	−0.984715	−	0.174171i	$-0.944275\pi$
$821$		−	285.989i		−	0.348342i	0.984715	−	0.174171i	$-0.0557246\pi$
$822$		0			0
$823$			510.682i			0.620513i	0.950653	+	0.310256i	$0.100415\pi$
$823$			510.682i			0.620513i	−0.950653	+	0.310256i	$0.899585\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	454.533			0.549617			0.274809	−	0.961499i	$-0.411386\pi$
$827$	454.533			0.549617			0.274809	+	0.961499i	$0.411386\pi$
$828$		0			0
$829$	741.467			0.894411			0.447205	−	0.894431i	$-0.352419\pi$
$829$	741.467			0.894411			0.447205	+	0.894431i	$0.352419\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−311.283			−0.373689
$834$		0			0
$835$	14.2777	−	6.70802i	0.0170990	−	0.00803355i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			284.263i			0.338811i	0.985546	+	0.169406i	$0.0541848\pi$
$839$			284.263i			0.338811i	−0.985546	+	0.169406i	$0.945815\pi$
$840$		0			0
$841$	264.198			0.314148
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−627.829	+	294.970i	−0.742993	+	0.349077i
$846$		0			0
$847$		−	36.8772i		−	0.0435386i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	445.249i		−	0.523207i
$852$		0			0
$853$		−	1145.18i		−	1.34253i	−0.741218	−	0.671264i	$-0.765752\pi$
$853$		−	1145.18i		−	1.34253i	0.741218	−	0.671264i	$-0.234248\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−174.584			−0.203715			−0.101858	−	0.994799i	$-0.532479\pi$
$857$	−174.584			−0.203715			−0.101858	+	0.994799i	$0.532479\pi$
$858$		0			0
$859$	642.095			0.747491			0.373746	−	0.927531i	$-0.378073\pi$
$859$	642.095			0.747491			0.373746	+	0.927531i	$0.378073\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1175.06			−1.36160			−0.680801	−	0.732468i	$-0.738368\pi$
$863$	−1175.06			−1.36160			−0.680801	+	0.732468i	$0.738368\pi$
$864$		0			0
$865$	739.610	−	347.488i	0.855041	−	0.401720i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			287.977i			0.331389i
$870$		0			0
$871$	428.022			0.491414
$872$		0			0
$873$		0			0
$874$		0			0
$875$	41.5394	+	10.7459i	0.0474736	+	0.0122810i
$876$		0			0
$877$			790.089i			0.900899i	0.892802	+	0.450450i	$0.148736\pi$
$877$			790.089i			0.900899i	−0.892802	+	0.450450i	$0.851264\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	982.888i		−	1.11565i	−0.829958	−	0.557825i	$-0.811636\pi$
$881$		−	982.888i		−	1.11565i	0.829958	−	0.557825i	$-0.188364\pi$
$882$		0			0
$883$		−	289.110i		−	0.327418i	−0.986509	−	0.163709i	$-0.947654\pi$
$883$		−	289.110i		−	0.327418i	0.986509	−	0.163709i	$-0.0523457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	174.200			0.196392			0.0981960	−	0.995167i	$-0.468693\pi$
$887$	174.200			0.196392			0.0981960	+	0.995167i	$0.468693\pi$
$888$		0			0
$889$	47.6318			0.0535791
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−101.886			−0.114094
$894$		0			0
$895$	−83.8966	−	178.569i	−0.0937392	−	0.199519i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			25.1479i			0.0279732i
$900$		0			0
$901$	20.6510			0.0229201
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1259.02	−	591.521i	1.39118	−	0.653614i
$906$		0			0
$907$			814.675i			0.898208i	0.893479	+	0.449104i	$0.148257\pi$
$907$			814.675i			0.898208i	−0.893479	+	0.449104i	$0.851743\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	205.317i		−	0.225376i	−0.993630	−	0.112688i	$-0.964054\pi$
$911$		−	205.317i		−	0.225376i	0.993630	−	0.112688i	$-0.0359460\pi$
$912$		0			0
$913$			229.025i			0.250849i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	45.1550			0.0492421
$918$		0			0
$919$	−299.910			−0.326344			−0.163172	−	0.986598i	$-0.552172\pi$
$919$	−299.910			−0.326344			−0.163172	+	0.986598i	$0.552172\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	481.699			0.521884
$924$		0			0
$925$	−697.674	−	578.588i	−0.754242	−	0.625501i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	625.007i		−	0.672774i	−0.941724	−	0.336387i	$-0.890795\pi$
$929$		−	625.007i		−	0.672774i	0.941724	−	0.336387i	$-0.109205\pi$
$930$		0			0
$931$	128.459			0.137979
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−49.8689	−	106.143i	−0.0533357	−	0.113522i
$936$		0			0
$937$			112.805i			0.120389i	0.998187	+	0.0601946i	$0.0191721\pi$
$937$			112.805i			0.120389i	−0.998187	+	0.0601946i	$0.980828\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			669.751i			0.711744i	0.934535	+	0.355872i	$0.115816\pi$
$941$			669.751i			0.711744i	−0.934535	+	0.355872i	$0.884184\pi$
$942$		0			0
$943$		−	99.0074i		−	0.104992i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−739.023			−0.780383			−0.390191	−	0.920734i	$-0.627591\pi$
$947$	−739.023			−0.780383			−0.390191	+	0.920734i	$0.627591\pi$
$948$		0			0
$949$	−596.506			−0.628563
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1043.04			1.09448			0.547241	−	0.836975i	$-0.315678\pi$
$953$	1043.04			1.09448			0.547241	+	0.836975i	$0.315678\pi$
$954$		0			0
$955$	−677.594	−	1442.22i	−0.709522	−	1.51018i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			52.2470i			0.0544808i
$960$		0			0
$961$	−959.904			−0.998859
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.0667	+	25.6833i	0.0125043	+	0.0266148i
$966$		0			0
$967$			1198.42i			1.23931i	0.784873	+	0.619657i	$0.212728\pi$
$967$			1198.42i			1.23931i	−0.784873	+	0.619657i	$0.787272\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			528.245i			0.544021i	0.962294	+	0.272011i	$0.0876886\pi$
$971$			528.245i			0.544021i	−0.962294	+	0.272011i	$0.912311\pi$
$972$		0			0
$973$			22.0338i			0.0226452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	252.143			0.258079			0.129040	−	0.991639i	$-0.458811\pi$
$977$	252.143			0.258079			0.129040	+	0.991639i	$0.458811\pi$
$978$		0			0
$979$	129.889			0.132676
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−532.362			−0.541569			−0.270784	−	0.962640i	$-0.587283\pi$
$983$	−532.362			−0.541569			−0.270784	+	0.962640i	$0.587283\pi$
$984$		0			0
$985$	205.146	−	96.3829i	0.208270	−	0.0978506i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			237.163i			0.239801i
$990$		0			0
$991$	−1560.13			−1.57429			−0.787147	−	0.616765i	$-0.788443\pi$
$991$	−1560.13			−1.57429			−0.787147	+	0.616765i	$0.788443\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−803.519	+	377.514i	−0.807557	+	0.379411i
$996$		0			0
$997$			1466.05i			1.47046i	0.677819	+	0.735229i	$0.262925\pi$
$997$			1466.05i			1.47046i	−0.677819	+	0.735229i	$0.737075\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.q.1889.4		24
3.2	odd	2	inner	2160.3.c.q.1889.21		24
4.3	odd	2		1080.3.c.c.809.4	yes	24
5.4	even	2	inner	2160.3.c.q.1889.22		24
12.11	even	2		1080.3.c.c.809.21	yes	24
15.14	odd	2	inner	2160.3.c.q.1889.3		24
20.19	odd	2		1080.3.c.c.809.22	yes	24
60.59	even	2		1080.3.c.c.809.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.c.809.3	✓	24	60.59	even	2
1080.3.c.c.809.4	yes	24	4.3	odd	2
1080.3.c.c.809.21	yes	24	12.11	even	2
1080.3.c.c.809.22	yes	24	20.19	odd	2
2160.3.c.q.1889.3		24	15.14	odd	2	inner
2160.3.c.q.1889.4		24	1.1	even	1	trivial
2160.3.c.q.1889.21		24	3.2	odd	2	inner
2160.3.c.q.1889.22		24	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.52542 + 2.12616i) q^{5} -0.343255i q^{7} -3.68323i q^{11} -5.50147i q^{13} -6.36803 q^{17} +2.62792 q^{19} +12.2810 q^{23} +(15.9589 - 19.2435i) q^{25} +24.0167i q^{29} +1.04710 q^{31} +(0.729815 + 1.55337i) q^{35} -36.2550i q^{37} -8.06180i q^{41} +19.3113i q^{43} -38.7705 q^{47} +48.8822 q^{49} -3.24292 q^{53} +(7.83113 + 16.6682i) q^{55} +51.5904i q^{59} +28.7343 q^{61} +(11.6970 + 24.8965i) q^{65} +77.8013i q^{67} +87.5582i q^{71} -108.427i q^{73} -1.26429 q^{77} -78.1862 q^{79} -62.1805 q^{83} +(28.8180 - 13.5395i) q^{85} +35.2651i q^{89} -1.88841 q^{91} +(-11.8925 + 5.58738i) q^{95} -65.0096i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) \) 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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