LMFDB - Embedded newform 189.3.d.e.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(55,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("189.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.d (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:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2355463701504.14
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 79x^{4} + 18x^{2} + 4 $ x^8 + 9x^6 + 79x^4 + 18*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.1
Root			$-0.238746 + 0.413520i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.e.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.71106 q^{2} +9.77200 q^{4} -8.26535i q^{5} +(-2.77200 - 6.42775i) q^{7} -21.4203 q^{8} +30.6732i q^{10} +10.2871 q^{11} -6.42775i q^{13} +(10.2871 + 23.8538i) q^{14} +40.4040 q^{16} +15.5885i q^{17} -4.96224i q^{19} -80.7690i q^{20} -38.1760 q^{22} -28.8424 q^{23} -43.3160 q^{25} +23.8538i q^{26} +(-27.0880 - 62.8120i) q^{28} +2.01883 q^{29} -16.3522i q^{31} -64.2608 q^{32} -57.8498i q^{34} +(-53.1276 + 22.9116i) q^{35} -33.2280 q^{37} +18.4152i q^{38} +177.046i q^{40} -39.4423i q^{41} -3.91199 q^{43} +100.525 q^{44} +107.036 q^{46} -41.3267i q^{47} +(-33.6320 + 35.6355i) q^{49} +160.749 q^{50} -62.8120i q^{52} +43.6867 q^{53} -85.0263i q^{55} +(59.3770 + 137.684i) q^{56} -7.49202 q^{58} -30.2347i q^{59} +107.806i q^{61} +60.6842i q^{62} +76.8600 q^{64} -53.1276 q^{65} -36.5800 q^{67} +152.330i q^{68} +(197.160 - 85.0263i) q^{70} -79.1051 q^{71} +38.5665i q^{73} +123.311 q^{74} -48.4910i q^{76} +(-28.5158 - 66.1228i) q^{77} +100.316 q^{79} -333.953i q^{80} +146.373i q^{82} +52.2040i q^{83} +128.844 q^{85} +14.5177 q^{86} -220.352 q^{88} +71.5614i q^{89} +(-41.3160 + 17.8177i) q^{91} -281.848 q^{92} +153.366i q^{94} -41.0146 q^{95} -133.517i q^{97} +(124.811 - 132.246i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4} + 12 q^{7} + 84 q^{16} - 32 q^{22} - 244 q^{25} - 80 q^{28} - 300 q^{37} - 168 q^{43} + 412 q^{46} - 64 q^{49} + 316 q^{58} + 444 q^{64} + 220 q^{67} + 552 q^{70} + 700 q^{79} + 108 q^{85}+ \cdots - 228 q^{91}+O(q^{100}) $ 8 * q + 44 * q^4 + 12 * q^7 + 84 * q^16 - 32 * q^22 - 244 * q^25 - 80 * q^28 - 300 * q^37 - 168 * q^43 + 412 * q^46 - 64 * q^49 + 316 * q^58 + 444 * q^64 + 220 * q^67 + 552 * q^70 + 700 * q^79 + 108 * q^85 - 1216 * q^88 - 228 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$1$	$-1$

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 $

$2$	−3.71106			−1.85553			−0.927766	−	0.373162i	$-0.878274\pi$
$2$	−3.71106			−1.85553			−0.927766	+	0.373162i	$0.878274\pi$
$3$		0			0
$3$		0			0
$4$	9.77200			2.44300
$5$		−	8.26535i		−	1.65307i	−0.562885	−	0.826535i	$-0.690309\pi$
$5$		−	8.26535i		−	1.65307i	0.562885	−	0.826535i	$-0.309691\pi$
$6$		0			0
$7$	−2.77200	−	6.42775i	−0.396000	−	0.918250i
$7$	−2.77200	−	6.42775i	−0.396000	−	0.918250i
$8$	−21.4203			−2.67753
$9$		0			0
$10$			30.6732i			3.06732i
$11$	10.2871			0.935189			0.467594	−	0.883943i	$-0.345121\pi$
$11$	10.2871			0.935189			0.467594	+	0.883943i	$0.345121\pi$
$12$		0			0
$13$		−	6.42775i		−	0.494443i	−0.968959	−	0.247221i	$-0.920483\pi$
$13$		−	6.42775i		−	0.494443i	0.968959	−	0.247221i	$-0.0795175\pi$
$14$	10.2871	+	23.8538i	0.734791	+	1.70384i
$15$		0			0
$16$	40.4040			2.52525
$17$			15.5885i			0.916968i	0.888703	+	0.458484i	$0.151607\pi$
$17$			15.5885i			0.916968i	−0.888703	+	0.458484i	$0.848393\pi$
$18$		0			0
$19$		−	4.96224i		−	0.261170i	−0.991437	−	0.130585i	$-0.958314\pi$
$19$		−	4.96224i		−	0.261170i	0.991437	−	0.130585i	$-0.0416856\pi$
$20$		−	80.7690i		−	4.03845i
$21$		0			0
$22$	−38.1760			−1.73527
$23$	−28.8424			−1.25402			−0.627009	−	0.779012i	$-0.715721\pi$
$23$	−28.8424			−1.25402			−0.627009	+	0.779012i	$0.715721\pi$
$24$		0			0
$25$	−43.3160			−1.73264
$26$			23.8538i			0.917454i
$27$		0			0
$28$	−27.0880	−	62.8120i	−0.967429	−	2.24329i
$29$	2.01883			0.0696149			0.0348075	−	0.999394i	$-0.488918\pi$
$29$	2.01883			0.0696149			0.0348075	+	0.999394i	$0.488918\pi$
$30$		0			0
$31$		−	16.3522i		−	0.527491i	−0.964592	−	0.263746i	$-0.915042\pi$
$31$		−	16.3522i		−	0.527491i	0.964592	−	0.263746i	$-0.0849579\pi$
$32$	−64.2608			−2.00815
$33$		0			0
$34$		−	57.8498i		−	1.70146i
$35$	−53.1276	+	22.9116i	−1.51793	+	0.654616i
$36$		0			0
$37$	−33.2280			−0.898054			−0.449027	−	0.893518i	$-0.648229\pi$
$37$	−33.2280			−0.898054			−0.449027	+	0.893518i	$0.648229\pi$
$38$			18.4152i			0.484610i
$39$		0			0
$40$			177.046i			4.42615i
$41$		−	39.4423i		−	0.962006i	−0.876719	−	0.481003i	$-0.840273\pi$
$41$		−	39.4423i		−	0.962006i	0.876719	−	0.481003i	$-0.159727\pi$
$42$		0			0
$43$	−3.91199			−0.0909766			−0.0454883	−	0.998965i	$-0.514484\pi$
$43$	−3.91199			−0.0909766			−0.0454883	+	0.998965i	$0.514484\pi$
$44$	100.525			2.28467
$45$		0			0
$46$	107.036			2.32687
$47$		−	41.3267i		−	0.879293i	−0.898171	−	0.439646i	$-0.855104\pi$
$47$		−	41.3267i		−	0.879293i	0.898171	−	0.439646i	$-0.144896\pi$
$48$		0			0
$49$	−33.6320	+	35.6355i	−0.686368	+	0.727255i
$50$	160.749			3.21497
$51$		0			0
$52$		−	62.8120i		−	1.20792i
$53$	43.6867			0.824277			0.412138	−	0.911121i	$-0.364782\pi$
$53$	43.6867			0.824277			0.412138	+	0.911121i	$0.364782\pi$
$54$		0			0
$55$		−	85.0263i		−	1.54593i
$56$	59.3770	+	137.684i	1.06030	+	2.45865i
$57$		0			0
$58$	−7.49202			−0.129173
$59$		−	30.2347i		−	0.512452i	−0.966617	−	0.256226i	$-0.917521\pi$
$59$		−	30.2347i		−	0.512452i	0.966617	−	0.256226i	$-0.0824791\pi$
$60$		0			0
$61$			107.806i			1.76732i	0.468133	+	0.883658i	$0.344927\pi$
$61$			107.806i			1.76732i	−0.468133	+	0.883658i	$0.655073\pi$
$62$			60.6842i			0.978777i
$63$		0			0
$64$	76.8600			1.20094
$65$	−53.1276			−0.817348
$66$		0			0
$67$	−36.5800			−0.545971			−0.272985	−	0.962018i	$-0.588011\pi$
$67$	−36.5800			−0.545971			−0.272985	+	0.962018i	$0.588011\pi$
$68$			152.330i			2.24015i
$69$		0			0
$70$	197.160	−	85.0263i	2.81657	−	1.21466i
$71$	−79.1051			−1.11416			−0.557078	−	0.830460i	$-0.688078\pi$
$71$	−79.1051			−1.11416			−0.557078	+	0.830460i	$0.688078\pi$
$72$		0			0
$73$			38.5665i			0.528308i	0.964480	+	0.264154i	$0.0850928\pi$
$73$			38.5665i			0.528308i	−0.964480	+	0.264154i	$0.914907\pi$
$74$	123.311			1.66637
$75$		0			0
$76$		−	48.4910i		−	0.638039i
$77$	−28.5158	−	66.1228i	−0.370335	−	0.858738i
$78$		0			0
$79$	100.316			1.26982			0.634911	−	0.772585i	$-0.281037\pi$
$79$	100.316			1.26982			0.634911	+	0.772585i	$0.281037\pi$
$80$		−	333.953i		−	4.17442i
$81$		0			0
$82$			146.373i			1.78503i
$83$			52.2040i			0.628964i	0.949263	+	0.314482i	$0.101831\pi$
$83$			52.2040i			0.628964i	−0.949263	+	0.314482i	$0.898169\pi$
$84$		0			0
$85$	128.844			1.51581
$86$	14.5177			0.168810
$87$		0			0
$88$	−220.352			−2.50400
$89$			71.5614i			0.804061i	0.915626	+	0.402030i	$0.131695\pi$
$89$			71.5614i			0.804061i	−0.915626	+	0.402030i	$0.868305\pi$
$90$		0			0
$91$	−41.3160	+	17.8177i	−0.454022	+	0.195799i
$92$	−281.848			−3.06357
$93$		0			0
$94$			153.366i			1.63156i
$95$	−41.0146			−0.431733
$96$		0			0
$97$		−	133.517i		−	1.37647i	−0.725489	−	0.688233i	$-0.758386\pi$
$97$		−	133.517i		−	1.37647i	0.725489	−	0.688233i	$-0.241614\pi$
$98$	124.811	−	132.246i	1.27358	−	1.34944i
$99$		0			0
$100$	−423.284			−4.23284
$101$		−	10.8772i		−	0.107696i	−0.998549	−	0.0538478i	$-0.982851\pi$
$101$		−	10.8772i		−	0.107696i	0.998549	−	0.0538478i	$-0.0171486\pi$
$102$		0			0
$103$		−	195.429i		−	1.89737i	−0.316217	−	0.948687i	$-0.602413\pi$
$103$		−	195.429i		−	1.89737i	0.316217	−	0.948687i	$-0.397587\pi$
$104$			137.684i			1.32389i
$105$		0			0
$106$	−162.124			−1.52947
$107$	−103.910			−0.971120			−0.485560	−	0.874203i	$-0.661384\pi$
$107$	−103.910			−0.971120			−0.485560	+	0.874203i	$0.661384\pi$
$108$		0			0
$109$	132.036			1.21134			0.605670	−	0.795716i	$-0.292905\pi$
$109$	132.036			1.21134			0.605670	+	0.795716i	$0.292905\pi$
$110$			315.538i			2.86853i
$111$		0			0
$112$	−112.000	−	259.707i	−1.00000	−	2.31881i
$113$	104.429			0.924153			0.462077	−	0.886840i	$-0.347104\pi$
$113$	104.429			0.924153			0.462077	+	0.886840i	$0.347104\pi$
$114$		0			0
$115$			238.393i			2.07298i
$116$	19.7280			0.170069
$117$		0			0
$118$			112.203i			0.950871i
$119$	100.199	−	43.2112i	0.842006	−	0.363120i
$120$		0			0
$121$	−15.1760			−0.125422
$122$		−	400.076i		−	3.27931i
$123$		0			0
$124$		−	159.794i		−	1.28866i
$125$			151.388i			1.21111i
$126$		0			0
$127$	145.580			1.14630			0.573150	−	0.819451i	$-0.305721\pi$
$127$	145.580			1.14630			0.573150	+	0.819451i	$0.305721\pi$
$128$	−28.1892			−0.220228
$129$		0			0
$130$	197.160			1.51662
$131$		−	222.222i		−	1.69635i	−0.529714	−	0.848176i	$-0.677701\pi$
$131$		−	222.222i		−	1.69635i	0.529714	−	0.848176i	$-0.322299\pi$
$132$		0			0
$133$	−31.8960	+	13.7553i	−0.239820	+	0.103424i
$134$	135.751			1.01307
$135$		0			0
$136$		−	333.909i		−	2.45521i
$137$	−146.884			−1.07215			−0.536073	−	0.844172i	$-0.680093\pi$
$137$	−146.884			−1.07215			−0.536073	+	0.844172i	$0.680093\pi$
$138$		0			0
$139$			96.0822i			0.691239i	0.938375	+	0.345619i	$0.112331\pi$
$139$			96.0822i			0.691239i	−0.938375	+	0.345619i	$0.887669\pi$
$140$	−519.163	+	223.892i	−3.70831	+	1.59923i
$141$		0			0
$142$	293.564			2.06735
$143$		−	66.1228i		−	0.462397i
$144$		0			0
$145$		−	16.6864i		−	0.115078i
$146$		−	143.123i		−	0.980293i
$147$		0			0
$148$	−324.704			−2.19395
$149$	275.599			1.84966			0.924828	−	0.380387i	$-0.124209\pi$
$149$	275.599			1.84966			0.924828	+	0.380387i	$0.124209\pi$
$150$		0			0
$151$	−191.652			−1.26922			−0.634610	−	0.772833i	$-0.718839\pi$
$151$	−191.652			−1.26922			−0.634610	+	0.772833i	$0.718839\pi$
$152$			106.292i			0.699293i
$153$		0			0
$154$	105.824	+	245.386i	0.687169	+	1.59342i
$155$	−135.157			−0.871980
$156$		0			0
$157$			90.5542i			0.576779i	0.957513	+	0.288389i	$0.0931197\pi$
$157$			90.5542i			0.576779i	−0.957513	+	0.288389i	$0.906880\pi$
$158$	−372.279			−2.35620
$159$		0			0
$160$			531.138i			3.31961i
$161$	79.9512	+	185.392i	0.496591	+	1.15150i
$162$		0			0
$163$	189.320			1.16147			0.580737	−	0.814092i	$-0.302765\pi$
$163$	189.320			1.16147			0.580737	+	0.814092i	$0.302765\pi$
$164$		−	385.430i		−	2.35018i
$165$		0			0
$166$		−	193.732i		−	1.16706i
$167$			72.5037i			0.434154i	0.976155	+	0.217077i	$0.0696522\pi$
$167$			72.5037i			0.434154i	−0.976155	+	0.217077i	$0.930348\pi$
$168$		0			0
$169$	127.684			0.755527
$170$	−478.149			−2.81264
$171$		0			0
$172$	−38.2280			−0.222256
$173$		−	172.415i		−	0.996620i	−0.866999	−	0.498310i	$-0.833954\pi$
$173$		−	172.415i		−	0.996620i	0.866999	−	0.498310i	$-0.166046\pi$
$174$		0			0
$175$	120.072	+	278.425i	0.686126	+	1.59100i
$176$	415.639			2.36159
$177$		0			0
$178$		−	265.569i		−	1.49196i
$179$	−158.344			−0.884602			−0.442301	−	0.896867i	$-0.645838\pi$
$179$	−158.344			−0.884602			−0.442301	+	0.896867i	$0.645838\pi$
$180$		0			0
$181$		−	285.418i		−	1.57690i	−0.615102	−	0.788448i	$-0.710885\pi$
$181$		−	285.418i		−	1.57690i	0.615102	−	0.788448i	$-0.289115\pi$
$182$	153.326	−	66.1228i	0.842453	−	0.363312i
$183$		0			0
$184$	617.812			3.35767
$185$			274.641i			1.48455i
$186$		0			0
$187$			160.360i			0.857538i
$188$		−	403.845i		−	2.14811i
$189$		0			0
$190$	152.208			0.801094
$191$	180.477			0.944903			0.472452	−	0.881357i	$-0.343369\pi$
$191$	180.477			0.944903			0.472452	+	0.881357i	$0.343369\pi$
$192$		0			0
$193$	119.880			0.621140			0.310570	−	0.950550i	$-0.399480\pi$
$193$	119.880			0.621140			0.310570	+	0.950550i	$0.399480\pi$
$194$			495.491i			2.55408i
$195$		0			0
$196$	−328.652	+	348.230i	−1.67680	+	1.77668i
$197$	126.696			0.643125			0.321563	−	0.946888i	$-0.395792\pi$
$197$	126.696			0.643125			0.321563	+	0.946888i	$0.395792\pi$
$198$		0			0
$199$		−	14.3210i		−	0.0719649i	−0.999352	−	0.0359825i	$-0.988544\pi$
$199$		−	14.3210i		−	0.0719649i	0.999352	−	0.0359825i	$-0.0114560\pi$
$200$	927.841			4.63920
$201$		0			0
$202$			40.3662i			0.199833i
$203$	−5.59621	−	12.9766i	−0.0275675	−	0.0639239i
$204$		0			0
$205$	−326.004			−1.59026
$206$			725.251i			3.52064i
$207$		0			0
$208$		−	259.707i		−	1.24859i
$209$		−	51.0469i		−	0.244244i
$210$		0			0
$211$	−332.124			−1.57405			−0.787024	−	0.616923i	$-0.788379\pi$
$211$	−332.124			−1.57405			−0.787024	+	0.616923i	$0.788379\pi$
$212$	426.906			2.01371
$213$		0			0
$214$	385.616			1.80194
$215$			32.3340i			0.150391i
$216$		0			0
$217$	−105.108	+	45.3284i	−0.484369	+	0.208887i
$218$	−489.994			−2.24768
$219$		0			0
$220$		−	830.877i		−	3.77671i
$221$	100.199			0.453388
$222$		0			0
$223$		−	193.064i		−	0.865758i	−0.901452	−	0.432879i	$-0.857498\pi$
$223$		−	193.064i		−	0.865758i	0.901452	−	0.432879i	$-0.142502\pi$
$224$	178.131	+	413.053i	0.795228	+	1.84399i
$225$		0			0
$226$	−387.544			−1.71480
$227$			23.8538i			0.105083i	0.998619	+	0.0525414i	$0.0167322\pi$
$227$			23.8538i			0.105083i	−0.998619	+	0.0525414i	$0.983268\pi$
$228$		0			0
$229$		−	372.810i		−	1.62799i	−0.580872	−	0.813995i	$-0.697288\pi$
$229$		−	372.810i		−	1.62799i	0.580872	−	0.813995i	$-0.302712\pi$
$230$		−	884.690i		−	3.84648i
$231$		0			0
$232$	−43.2440			−0.186396
$233$	177.612			0.762282			0.381141	−	0.924517i	$-0.375531\pi$
$233$	177.612			0.762282			0.381141	+	0.924517i	$0.375531\pi$
$234$		0			0
$235$	−341.580			−1.45353
$236$		−	295.453i		−	1.25192i
$237$		0			0
$238$	−371.844	+	160.360i	−1.56237	+	0.673780i
$239$	−13.9389			−0.0583218			−0.0291609	−	0.999575i	$-0.509284\pi$
$239$	−13.9389			−0.0583218			−0.0291609	+	0.999575i	$0.509284\pi$
$240$		0			0
$241$			183.808i			0.762689i	0.924433	+	0.381344i	$0.124539\pi$
$241$			183.808i			0.762689i	−0.924433	+	0.381344i	$0.875461\pi$
$242$	56.3192			0.232724
$243$		0			0
$244$			1053.48i			4.31755i
$245$	294.540	+	277.980i	1.20220	+	1.13461i
$246$		0			0
$247$	−31.8960			−0.129134
$248$			350.269i			1.41238i
$249$		0			0
$250$		−	561.811i		−	2.24725i
$251$		−	191.988i		−	0.764891i	−0.923978	−	0.382445i	$-0.875082\pi$
$251$		−	191.988i		−	0.764891i	0.923978	−	0.382445i	$-0.124918\pi$
$252$		0			0
$253$	−296.704			−1.17274
$254$	−540.257			−2.12700
$255$		0			0
$256$	−202.828			−0.792297
$257$		−	146.462i		−	0.569892i	−0.958544	−	0.284946i	$-0.908024\pi$
$257$		−	146.462i		−	0.569892i	0.958544	−	0.284946i	$-0.0919756\pi$
$258$		0			0
$259$	92.1081	+	213.581i	0.355630	+	0.824638i
$260$	−519.163			−1.99678
$261$		0			0
$262$			824.681i			3.14764i
$263$	−81.3168			−0.309189			−0.154595	−	0.987978i	$-0.549407\pi$
$263$	−81.3168			−0.309189			−0.154595	+	0.987978i	$0.549407\pi$
$264$		0			0
$265$		−	361.086i		−	1.36259i
$266$	118.368	−	51.0469i	0.444993	−	0.191906i
$267$		0			0
$268$	−357.460			−1.33381
$269$		−	391.811i		−	1.45655i	−0.685287	−	0.728273i	$-0.740323\pi$
$269$		−	391.811i		−	1.45655i	0.685287	−	0.728273i	$-0.259677\pi$
$270$		0			0
$271$		−	488.175i		−	1.80138i	−0.434458	−	0.900692i	$-0.643060\pi$
$271$		−	488.175i		−	1.80138i	0.434458	−	0.900692i	$-0.356940\pi$
$272$			629.836i			2.31557i
$273$		0			0
$274$	545.096			1.98940
$275$	−445.595			−1.62035
$276$		0			0
$277$	376.004			1.35742			0.678708	−	0.734408i	$-0.262540\pi$
$277$	376.004			1.35742			0.678708	+	0.734408i	$0.262540\pi$
$278$		−	356.567i		−	1.28262i
$279$		0			0
$280$	1138.01	−	490.772i	4.06432	−	1.75276i
$281$	113.677			0.404546			0.202273	−	0.979329i	$-0.435167\pi$
$281$	113.677			0.404546			0.202273	+	0.979329i	$0.435167\pi$
$282$		0			0
$283$			65.1774i			0.230309i	0.993348	+	0.115154i	$0.0367363\pi$
$283$			65.1774i			0.230309i	−0.993348	+	0.115154i	$0.963264\pi$
$284$	−773.015			−2.72188
$285$		0			0
$286$			245.386i			0.857993i
$287$	−253.525	+	109.334i	−0.883363	+	0.380955i
$288$		0			0
$289$	46.0000			0.159170
$290$			61.9242i			0.213532i
$291$		0			0
$292$			376.872i			1.29066i
$293$			389.926i			1.33081i	0.746484	+	0.665403i	$0.231740\pi$
$293$			389.926i			1.33081i	−0.746484	+	0.665403i	$0.768260\pi$
$294$		0			0
$295$	−249.900			−0.847119
$296$	711.753			2.40457
$297$		0			0
$298$	−1022.76			−3.43209
$299$			185.392i			0.620040i
$300$		0			0
$301$	10.8441	+	25.1453i	0.0360267	+	0.0835393i
$302$	711.233			2.35508
$303$		0			0
$304$		−	200.494i		−	0.659521i
$305$	891.057			2.92150
$306$		0			0
$307$		−	355.892i		−	1.15926i	−0.814881	−	0.579628i	$-0.803198\pi$
$307$		−	355.892i		−	1.15926i	0.814881	−	0.579628i	$-0.196802\pi$
$308$	−278.656	−	646.152i	−0.904729	−	2.09790i
$309$		0			0
$310$	501.576			1.61799
$311$			90.9188i			0.292344i	0.989259	+	0.146172i	$0.0466952\pi$
$311$			90.9188i			0.292344i	−0.989259	+	0.146172i	$0.953305\pi$
$312$		0			0
$313$		−	12.8555i		−	0.0410719i	−0.999789	−	0.0205360i	$-0.993463\pi$
$313$		−	12.8555i		−	0.0410719i	0.999789	−	0.0205360i	$-0.00653726\pi$
$314$		−	336.053i		−	1.07023i
$315$		0			0
$316$	980.288			3.10218
$317$	268.236			0.846169			0.423085	−	0.906090i	$-0.360947\pi$
$317$	268.236			0.846169			0.423085	+	0.906090i	$0.360947\pi$
$318$		0			0
$319$	20.7679			0.0651031
$320$		−	635.275i		−	1.98523i
$321$		0			0
$322$	−296.704	−	688.001i	−0.921441	−	2.13665i
$323$	77.3536			0.239485
$324$		0			0
$325$			278.425i			0.856691i
$326$	−702.579			−2.15515
$327$		0			0
$328$			844.864i			2.57581i
$329$	−265.638	+	114.558i	−0.807411	+	0.348200i
$330$		0			0
$331$	−86.5440			−0.261462			−0.130731	−	0.991418i	$-0.541732\pi$
$331$	−86.5440			−0.261462			−0.130731	+	0.991418i	$0.541732\pi$
$332$			510.138i			1.53656i
$333$		0			0
$334$		−	269.066i		−	0.805586i
$335$			302.347i			0.902528i
$336$		0			0
$337$	−77.2800			−0.229317			−0.114659	−	0.993405i	$-0.536577\pi$
$337$	−77.2800			−0.229317			−0.114659	+	0.993405i	$0.536577\pi$
$338$	−473.844			−1.40190
$339$		0			0
$340$	1259.06			3.70313
$341$		−	168.217i		−	0.493304i
$342$		0			0
$343$	322.284	+	117.397i	0.939604	+	0.342264i
$344$	83.7959			0.243593
$345$		0			0
$346$			639.844i			1.84926i
$347$	−50.7822			−0.146346			−0.0731732	−	0.997319i	$-0.523313\pi$
$347$	−50.7822			−0.146346			−0.0731732	+	0.997319i	$0.523313\pi$
$348$		0			0
$349$			129.352i			0.370637i	0.982679	+	0.185318i	$0.0593317\pi$
$349$			129.352i			0.370637i	−0.982679	+	0.185318i	$0.940668\pi$
$350$	−445.595	−	1033.25i	−1.27313	−	2.95215i
$351$		0			0
$352$	−661.056			−1.87800
$353$			323.589i			0.916682i	0.888777	+	0.458341i	$0.151556\pi$
$353$			323.589i			0.916682i	−0.888777	+	0.458341i	$0.848444\pi$
$354$		0			0
$355$			653.831i			1.84178i
$356$			699.298i			1.96432i
$357$		0			0
$358$	587.624			1.64141
$359$	−238.740			−0.665014			−0.332507	−	0.943101i	$-0.607895\pi$
$359$	−238.740			−0.665014			−0.332507	+	0.943101i	$0.607895\pi$
$360$		0			0
$361$	336.376			0.931790
$362$			1059.20i			2.92598i
$363$		0			0
$364$	−403.740	+	174.115i	−1.10918	+	0.478338i
$365$	318.766			0.873331
$366$		0			0
$367$		−	56.9499i		−	0.155177i	−0.996985	−	0.0775885i	$-0.975278\pi$
$367$		−	56.9499i		−	0.155177i	0.996985	−	0.0775885i	$-0.0247220\pi$
$368$	−1165.35			−3.16671
$369$		0			0
$370$		−	1019.21i		−	2.75462i
$371$	−121.100	−	280.807i	−0.326414	−	0.756892i
$372$		0			0
$373$	−367.060			−0.984076			−0.492038	−	0.870574i	$-0.663748\pi$
$373$	−367.060			−0.984076			−0.492038	+	0.870574i	$0.663748\pi$
$374$		−	595.105i		−	1.59119i
$375$		0			0
$376$			885.230i			2.35434i
$377$		−	12.9766i		−	0.0344206i
$378$		0			0
$379$	51.0839			0.134786			0.0673930	−	0.997727i	$-0.478532\pi$
$379$	51.0839			0.134786			0.0673930	+	0.997727i	$0.478532\pi$
$380$	−400.795			−1.05472
$381$		0			0
$382$	−669.760			−1.75330
$383$		−	461.703i		−	1.20549i	−0.797934	−	0.602745i	$-0.794074\pi$
$383$		−	461.703i		−	1.20549i	0.797934	−	0.602745i	$-0.205926\pi$
$384$		0			0
$385$	−546.528	+	235.693i	−1.41955	+	0.612190i
$386$	−444.883			−1.15255
$387$		0			0
$388$		−	1304.73i		−	3.36271i
$389$	−104.815			−0.269448			−0.134724	−	0.990883i	$-0.543015\pi$
$389$	−104.815			−0.269448			−0.134724	+	0.990883i	$0.543015\pi$
$390$		0			0
$391$		−	449.609i		−	1.14989i
$392$	720.407	−	763.322i	1.83777	−	1.94725i
$393$		0			0
$394$	−470.176			−1.19334
$395$		−	829.147i		−	2.09911i
$396$		0			0
$397$			598.347i			1.50717i	0.657350	+	0.753585i	$0.271677\pi$
$397$			598.347i			1.50717i	−0.657350	+	0.753585i	$0.728323\pi$
$398$			53.1462i			0.133533i
$399$		0			0
$400$	−1750.14			−4.37535
$401$	−289.730			−0.722520			−0.361260	−	0.932465i	$-0.617653\pi$
$401$	−289.730			−0.722520			−0.361260	+	0.932465i	$0.617653\pi$
$402$		0			0
$403$	−105.108			−0.260814
$404$		−	106.292i		−	0.263100i
$405$		0			0
$406$	20.7679	+	48.1569i	0.0511525	+	0.118613i
$407$	−341.819			−0.839850
$408$		0			0
$409$			89.9885i			0.220021i	0.993930	+	0.110010i	$0.0350884\pi$
$409$			89.9885i			0.220021i	−0.993930	+	0.110010i	$0.964912\pi$
$410$	1209.82			2.95079
$411$		0			0
$412$		−	1909.74i		−	4.63528i
$413$	−194.341	+	83.8106i	−0.470559	+	0.202931i
$414$		0			0
$415$	431.484			1.03972
$416$			413.053i			0.992915i
$417$		0			0
$418$			189.438i			0.453202i
$419$			661.311i			1.57831i	0.614196	+	0.789154i	$0.289481\pi$
$419$			661.311i			1.57831i	−0.614196	+	0.789154i	$0.710519\pi$
$420$		0			0
$421$	135.056			0.320798			0.160399	−	0.987052i	$-0.448722\pi$
$421$	135.056			0.320798			0.160399	+	0.987052i	$0.448722\pi$
$422$	1232.53			2.92070
$423$		0			0
$424$	−935.780			−2.20703
$425$		−	675.230i		−	1.58878i
$426$		0			0
$427$	692.952	−	298.839i	1.62284	−	0.699858i
$428$	−1015.41			−2.37245
$429$		0			0
$430$		−	119.994i		−	0.279055i
$431$	569.812			1.32207			0.661035	−	0.750356i	$-0.270118\pi$
$431$	569.812			1.32207			0.661035	+	0.750356i	$0.270118\pi$
$432$		0			0
$433$			478.816i			1.10581i	0.833244	+	0.552906i	$0.186481\pi$
$433$			478.816i			1.10581i	−0.833244	+	0.552906i	$0.813519\pi$
$434$	390.063	−	168.217i	0.898762	−	0.387596i
$435$		0			0
$436$	1290.26			2.95930
$437$			143.123i			0.327512i
$438$		0			0
$439$		−	386.462i		−	0.880324i	−0.897918	−	0.440162i	$-0.854921\pi$
$439$		−	386.462i		−	0.880324i	0.897918	−	0.440162i	$-0.145079\pi$
$440$			1821.29i			4.13929i
$441$		0			0
$442$	−371.844			−0.841276
$443$	128.002			0.288944			0.144472	−	0.989509i	$-0.453852\pi$
$443$	128.002			0.288944			0.144472	+	0.989509i	$0.453852\pi$
$444$		0			0
$445$	591.480			1.32917
$446$			716.474i			1.60644i
$447$		0			0
$448$	−213.056	−	494.037i	−0.475572	−	1.10276i
$449$	3.90398			0.00869483			0.00434741	−	0.999991i	$-0.498616\pi$
$449$	3.90398			0.00869483			0.00434741	+	0.999991i	$0.498616\pi$
$450$		0			0
$451$		−	405.746i		−	0.899658i
$452$	1020.48			2.25771
$453$		0			0
$454$		−	88.5230i		−	0.194985i
$455$	147.270	+	341.491i	0.323670	+	0.750530i
$456$		0			0
$457$	376.932			0.824797			0.412398	−	0.911004i	$-0.364691\pi$
$457$	376.932			0.824797			0.412398	+	0.911004i	$0.364691\pi$
$458$			1383.52i			3.02079i
$459$		0			0
$460$			2329.57i			5.06429i
$461$			331.341i			0.718745i	0.933194	+	0.359372i	$0.117009\pi$
$461$			331.341i			0.718745i	−0.933194	+	0.359372i	$0.882991\pi$
$462$		0			0
$463$	−27.5800			−0.0595681			−0.0297840	−	0.999556i	$-0.509482\pi$
$463$	−27.5800			−0.0595681			−0.0297840	+	0.999556i	$0.509482\pi$
$464$	81.5690			0.175795
$465$		0			0
$466$	−659.128			−1.41444
$467$		−	447.784i		−	0.958852i	−0.877582	−	0.479426i	$-0.840845\pi$
$467$		−	447.784i		−	0.958852i	0.877582	−	0.479426i	$-0.159155\pi$
$468$		0			0
$469$	101.400	+	235.127i	0.216204	+	0.501338i
$470$	1267.63			2.69708
$471$		0			0
$472$			647.635i			1.37211i
$473$	−40.2430			−0.0850803
$474$		0			0
$475$			214.944i			0.452514i
$476$	979.142	−	422.260i	2.05702	−	0.887101i
$477$		0			0
$478$	51.7282			0.108218
$479$		−	714.457i		−	1.49156i	−0.666192	−	0.745780i	$-0.732077\pi$
$479$		−	714.457i		−	1.49156i	0.666192	−	0.745780i	$-0.267923\pi$
$480$		0			0
$481$			213.581i			0.444036i
$482$		−	682.123i		−	1.41519i
$483$		0			0
$484$	−148.300			−0.306405
$485$	−1103.57			−2.27540
$486$		0			0
$487$	103.824			0.213191			0.106595	−	0.994302i	$-0.466005\pi$
$487$	103.824			0.213191			0.106595	+	0.994302i	$0.466005\pi$
$488$		−	2309.24i		−	4.73205i
$489$		0			0
$490$	−1093.06	−	1031.60i	−2.23073	−	2.10531i
$491$	−629.055			−1.28117			−0.640586	−	0.767887i	$-0.721308\pi$
$491$	−629.055			−1.28117			−0.640586	+	0.767887i	$0.721308\pi$
$492$		0			0
$493$			31.4705i			0.0638347i
$494$	118.368			0.239612
$495$		0			0
$496$		−	660.696i		−	1.33205i
$497$	219.279	+	508.468i	0.441206	+	1.02307i
$498$		0			0
$499$	184.140			0.369018			0.184509	−	0.982831i	$-0.440931\pi$
$499$	184.140			0.369018			0.184509	+	0.982831i	$0.440931\pi$
$500$			1479.37i			2.95873i
$501$		0			0
$502$			712.478i			1.41928i
$503$			224.619i			0.446559i	0.974754	+	0.223280i	$0.0716763\pi$
$503$			224.619i			0.446559i	−0.974754	+	0.223280i	$0.928324\pi$
$504$		0			0
$505$	−89.9042			−0.178028
$506$	1101.09			2.17606
$507$		0			0
$508$	1422.61			2.80041
$509$			643.540i			1.26432i	0.774837	+	0.632161i	$0.217832\pi$
$509$			643.540i			1.26432i	−0.774837	+	0.632161i	$0.782168\pi$
$510$		0			0
$511$	247.896	−	106.906i	0.485119	−	0.209210i
$512$	865.465			1.69036
$513$		0			0
$514$			543.531i			1.05745i
$515$	−1615.29			−3.13649
$516$		0			0
$517$		−	425.132i		−	0.822305i
$518$	−341.819	−	792.614i	−0.659882	−	1.53014i
$519$		0			0
$520$	1138.01			2.18848
$521$		−	1.37190i		−	0.00263321i	−0.999999	−	0.00131660i	$-0.999581\pi$
$521$		−	1.37190i		−	0.00263321i	0.999999	−	0.00131660i	$-0.000419088\pi$
$522$		0			0
$523$		−	130.586i		−	0.249687i	−0.992176	−	0.124843i	$-0.960157\pi$
$523$		−	130.586i		−	0.249687i	0.992176	−	0.124843i	$-0.0398429\pi$
$524$		−	2171.56i		−	4.14419i
$525$		0			0
$526$	301.772			0.573711
$527$	254.906			0.483693
$528$		0			0
$529$	302.884			0.572560
$530$			1340.01i			2.52832i
$531$		0			0
$532$	−311.688	+	134.417i	−0.585880	+	0.252664i
$533$	−253.525			−0.475657
$534$		0			0
$535$			858.851i			1.60533i
$536$	783.554			1.46185
$537$		0			0
$538$			1454.04i			2.70267i
$539$	−345.975	+	366.585i	−0.641883	+	0.680121i
$540$		0			0
$541$	−825.484			−1.52585			−0.762924	−	0.646488i	$-0.776237\pi$
$541$	−825.484			−1.52585			−0.762924	+	0.646488i	$0.776237\pi$
$542$			1811.65i			3.34253i
$543$		0			0
$544$		−	1001.73i		−	1.84141i
$545$		−	1091.32i		−	2.00243i
$546$		0			0
$547$	700.004			1.27972			0.639858	−	0.768494i	$-0.278993\pi$
$547$	700.004			1.27972			0.639858	+	0.768494i	$0.278993\pi$
$548$	−1435.35			−2.61925
$549$		0			0
$550$	1653.63			3.00660
$551$		−	10.0179i		−	0.0181814i
$552$		0			0
$553$	−278.076	−	644.806i	−0.502850	−	1.16602i
$554$	−1395.38			−2.51873
$555$		0			0
$556$			938.915i			1.68870i
$557$	−494.626			−0.888018			−0.444009	−	0.896022i	$-0.646444\pi$
$557$	−494.626			−0.888018			−0.444009	+	0.896022i	$0.646444\pi$
$558$		0			0
$559$			25.1453i			0.0449827i
$560$	−2146.57	+	925.719i	−3.83316	+	1.65307i
$561$		0			0
$562$	−421.864			−0.750648
$563$		−	64.8828i		−	0.115245i	−0.998338	−	0.0576224i	$-0.981648\pi$
$563$		−	64.8828i		−	0.115245i	0.998338	−	0.0576224i	$-0.0183519\pi$
$564$		0			0
$565$		−	863.145i		−	1.52769i
$566$		−	241.877i		−	0.427345i
$567$		0			0
$568$	1694.45			2.98319
$569$	1007.60			1.77082			0.885412	−	0.464807i	$-0.153876\pi$
$569$	1007.60			1.77082			0.885412	+	0.464807i	$0.153876\pi$
$570$		0			0
$571$	−849.328			−1.48744			−0.743720	−	0.668491i	$-0.766940\pi$
$571$	−849.328			−1.48744			−0.743720	+	0.668491i	$0.766940\pi$
$572$		−	646.152i		−	1.12964i
$573$		0			0
$574$	940.848	−	405.746i	1.63911	−	0.706874i
$575$	1249.34			2.17276
$576$		0			0
$577$			184.039i			0.318959i	0.987201	+	0.159480i	$0.0509816\pi$
$577$			184.039i			0.318959i	−0.987201	+	0.159480i	$0.949018\pi$
$578$	−170.709			−0.295344
$579$		0			0
$580$		−	163.059i		−	0.281136i
$581$	335.554	−	144.710i	0.577546	−	0.249070i
$582$		0			0
$583$	449.408			0.770854
$584$		−	826.105i		−	1.41456i
$585$		0			0
$586$		−	1447.04i		−	2.46935i
$587$		−	413.912i		−	0.705131i	−0.935787	−	0.352566i	$-0.885309\pi$
$587$		−	413.912i		−	0.705131i	0.935787	−	0.352566i	$-0.114691\pi$
$588$		0			0
$589$	−81.1436			−0.137765
$590$	927.396			1.57186
$591$		0			0
$592$	−1342.54			−2.26781
$593$		−	230.273i		−	0.388318i	−0.980970	−	0.194159i	$-0.937802\pi$
$593$		−	230.273i		−	0.388318i	0.980970	−	0.194159i	$-0.0621978\pi$
$594$		0			0
$595$	−357.156	−	828.178i	−0.600262	−	1.39190i
$596$	2693.15			4.51871
$597$		0			0
$598$		−	688.001i		−	1.15050i
$599$	540.836			0.902898			0.451449	−	0.892297i	$-0.350907\pi$
$599$	540.836			0.902898			0.451449	+	0.892297i	$0.350907\pi$
$600$		0			0
$601$			10.8243i			0.0180105i	0.999959	+	0.00900524i	$0.00286650\pi$
$601$			10.8243i			0.0180105i	−0.999959	+	0.00900524i	$0.997134\pi$
$602$	−40.2430	−	93.3159i	−0.0668488	−	0.155010i
$603$		0			0
$604$	−1872.82			−3.10070
$605$			125.435i			0.207331i
$606$		0			0
$607$			980.978i			1.61611i	0.589108	+	0.808055i	$0.299479\pi$
$607$			980.978i			1.61611i	−0.589108	+	0.808055i	$0.700521\pi$
$608$			318.877i			0.524469i
$609$		0			0
$610$	−3306.77			−5.42093
$611$	−265.638			−0.434760
$612$		0			0
$613$	−498.640			−0.813442			−0.406721	−	0.913552i	$-0.633328\pi$
$613$	−498.640			−0.813442			−0.406721	+	0.913552i	$0.633328\pi$
$614$			1320.74i			2.15104i
$615$		0			0
$616$	610.816	+	1416.37i	0.991585	+	2.29930i
$617$	−618.501			−1.00243			−0.501216	−	0.865322i	$-0.667114\pi$
$617$	−618.501			−1.00243			−0.501216	+	0.865322i	$0.667114\pi$
$618$		0			0
$619$		−	153.366i		−	0.247765i	−0.992297	−	0.123882i	$-0.960465\pi$
$619$		−	153.366i		−	0.247765i	0.992297	−	0.123882i	$-0.0395345\pi$
$620$	−1320.75			−2.13025
$621$		0			0
$622$		−	337.406i		−	0.542453i
$623$	459.979	−	198.368i	0.738329	−	0.318408i
$624$		0			0
$625$	168.376			0.269402
$626$			47.7076i			0.0762102i
$627$		0			0
$628$			884.896i			1.40907i
$629$		−	517.973i		−	0.823487i
$630$		0			0
$631$	−521.260			−0.826085			−0.413043	−	0.910712i	$-0.635534\pi$
$631$	−521.260			−0.826085			−0.413043	+	0.910712i	$0.635534\pi$
$632$	−2148.80			−3.39999
$633$		0			0
$634$	−995.440			−1.57009
$635$		−	1203.27i		−	1.89491i
$636$		0			0
$637$	229.056	+	216.178i	0.359586	+	0.339369i
$638$	−77.0710			−0.120801
$639$		0			0
$640$			232.994i			0.364052i
$641$	−1013.31			−1.58083			−0.790416	−	0.612570i	$-0.790136\pi$
$641$	−1013.31			−1.58083			−0.790416	+	0.612570i	$0.790136\pi$
$642$		0			0
$643$			752.150i			1.16975i	0.811123	+	0.584875i	$0.198857\pi$
$643$			752.150i			1.16975i	−0.811123	+	0.584875i	$0.801143\pi$
$644$	781.283	+	1811.65i	1.21317	+	2.81312i
$645$		0			0
$646$	−287.064			−0.444372
$647$			1042.89i			1.61188i	0.591995	+	0.805942i	$0.298341\pi$
$647$			1042.89i			1.61188i	−0.591995	+	0.805942i	$0.701659\pi$
$648$		0			0
$649$		−	311.026i		−	0.479240i
$650$		−	1033.25i		−	1.58962i
$651$		0			0
$652$	1850.04			2.83748
$653$	529.525			0.810911			0.405455	−	0.914115i	$-0.367113\pi$
$653$	529.525			0.810911			0.405455	+	0.914115i	$0.367113\pi$
$654$		0			0
$655$	−1836.74			−2.80419
$656$		−	1593.63i		−	2.42931i
$657$		0			0
$658$	985.800	−	425.132i	1.49818	−	0.646097i
$659$	−392.215			−0.595168			−0.297584	−	0.954696i	$-0.596181\pi$
$659$	−392.215			−0.595168			−0.297584	+	0.954696i	$0.596181\pi$
$660$		0			0
$661$		−	1212.04i		−	1.83365i	−0.399288	−	0.916825i	$-0.630743\pi$
$661$		−	1212.04i		−	1.83365i	0.399288	−	0.916825i	$-0.369257\pi$
$662$	321.170			0.485152
$663$		0			0
$664$		−	1118.22i		−	1.68407i
$665$	113.693	+	263.632i	0.170966	+	0.396439i
$666$		0			0
$667$	−58.2280			−0.0872983
$668$			708.506i			1.06064i
$669$		0			0
$670$		−	1122.03i		−	1.67467i
$671$			1109.01i			1.65277i
$672$		0			0
$673$	376.932			0.560077			0.280039	−	0.959989i	$-0.409653\pi$
$673$	376.932			0.560077			0.280039	+	0.959989i	$0.409653\pi$
$674$	286.791			0.425506
$675$		0			0
$676$	1247.73			1.84575
$677$		−	485.473i		−	0.717095i	−0.933511	−	0.358548i	$-0.883272\pi$
$677$		−	485.473i		−	0.717095i	0.933511	−	0.358548i	$-0.116728\pi$
$678$		0			0
$679$	−858.216	+	370.110i	−1.26394	+	0.545081i
$680$	−2759.87			−4.05864
$681$		0			0
$682$			624.263i			0.915341i
$683$	223.451			0.327161			0.163580	−	0.986530i	$-0.447696\pi$
$683$	223.451			0.327161			0.163580	+	0.986530i	$0.447696\pi$
$684$		0			0
$685$			1214.05i			1.77233i
$686$	−1196.02	−	435.666i	−1.74347	−	0.635082i
$687$		0			0
$688$	−158.060			−0.229739
$689$		−	280.807i		−	0.407557i
$690$		0			0
$691$		−	294.314i		−	0.425924i	−0.977060	−	0.212962i	$-0.931689\pi$
$691$		−	294.314i		−	0.425924i	0.977060	−	0.212962i	$-0.0683111\pi$
$692$		−	1684.84i		−	2.43474i
$693$		0			0
$694$	188.456			0.271550
$695$	794.153			1.14267
$696$		0			0
$697$	614.844			0.882129
$698$		−	480.035i		−	0.687729i
$699$		0			0
$700$	1173.34	+	2720.77i	1.67621	+	3.88681i
$701$	−1238.74			−1.76710			−0.883552	−	0.468334i	$-0.844854\pi$
$701$	−1238.74			−1.76710			−0.883552	+	0.468334i	$0.844854\pi$
$702$		0			0
$703$			164.885i			0.234545i
$704$	790.665			1.12310
$705$		0			0
$706$		−	1200.86i		−	1.70093i
$707$	−69.9163	+	30.1517i	−0.0988914	+	0.0426475i
$708$		0			0
$709$	−323.340			−0.456051			−0.228026	−	0.973655i	$-0.573227\pi$
$709$	−323.340			−0.456051			−0.228026	+	0.973655i	$0.573227\pi$
$710$		−	2426.41i		−	3.41748i
$711$		0			0
$712$		−	1532.87i		−	2.15290i
$713$			471.638i			0.661483i
$714$		0			0
$715$	−546.528			−0.764375
$716$	−1547.34			−2.16108
$717$		0			0
$718$	885.980			1.23396
$719$			533.479i			0.741973i	0.928638	+	0.370987i	$0.120980\pi$
$719$			533.479i			0.741973i	−0.928638	+	0.370987i	$0.879020\pi$
$720$		0			0
$721$	−1256.17	+	541.731i	−1.74226	+	0.751360i
$722$	−1248.31			−1.72897
$723$		0			0
$724$		−	2789.11i		−	3.85236i
$725$	−87.4478			−0.120618
$726$		0			0
$727$			545.228i			0.749969i	0.927031	+	0.374985i	$0.122352\pi$
$727$			545.228i			0.749969i	−0.927031	+	0.374985i	$0.877648\pi$
$728$	885.000	−	381.661i	1.21566	−	0.524259i
$729$		0			0
$730$	−1182.96			−1.62049
$731$		−	60.9819i		−	0.0834226i
$732$		0			0
$733$		−	112.640i		−	0.153669i	−0.997044	−	0.0768346i	$-0.975519\pi$
$733$		−	112.640i		−	0.153669i	0.997044	−	0.0768346i	$-0.0244813\pi$
$734$			211.345i			0.287936i
$735$		0			0
$736$	1853.44			2.51826
$737$	−376.302			−0.510586
$738$		0			0
$739$	764.528			1.03454			0.517272	−	0.855821i	$-0.326948\pi$
$739$	764.528			1.03454			0.517272	+	0.855821i	$0.326948\pi$
$740$			2683.79i			3.62675i
$741$		0			0
$742$	449.408	+	1042.09i	0.605671	+	1.40444i
$743$	441.617			0.594370			0.297185	−	0.954820i	$-0.403952\pi$
$743$	441.617			0.594370			0.297185	+	0.954820i	$0.403952\pi$
$744$		0			0
$745$		−	2277.92i		−	3.05761i
$746$	1362.18			1.82598
$747$		0			0
$748$			1567.04i			2.09497i
$749$	288.038	+	667.907i	0.384564	+	0.891731i
$750$		0			0
$751$	−799.656			−1.06479			−0.532394	−	0.846497i	$-0.678708\pi$
$751$	−799.656			−1.06479			−0.532394	+	0.846497i	$0.678708\pi$
$752$		−	1669.77i		−	2.22043i
$753$		0			0
$754$			48.1569i			0.0638685i
$755$			1584.07i			2.09811i
$756$		0			0
$757$	1403.41			1.85391			0.926957	−	0.375169i	$-0.122415\pi$
$757$	1403.41			1.85391			0.926957	+	0.375169i	$0.122415\pi$
$758$	−189.576			−0.250100
$759$		0			0
$760$	878.544			1.15598
$761$			40.5164i			0.0532410i	0.999646	+	0.0266205i	$0.00847457\pi$
$761$			40.5164i			0.0532410i	−0.999646	+	0.0266205i	$0.991525\pi$
$762$		0			0
$763$	−366.004	−	848.695i	−0.479691	−	1.11231i
$764$	1763.62			2.30840
$765$		0			0
$766$			1713.41i			2.23682i
$767$	−194.341			−0.253378
$768$		0			0
$769$			679.542i			0.883670i	0.897096	+	0.441835i	$0.145672\pi$
$769$			679.542i			0.883670i	−0.897096	+	0.441835i	$0.854328\pi$
$770$	2028.20	−	874.672i	2.63403	−	1.13594i
$771$		0			0
$772$	1171.47			1.51745
$773$			737.501i			0.954076i	0.878883	+	0.477038i	$0.158290\pi$
$773$			737.501i			0.954076i	−0.878883	+	0.477038i	$0.841710\pi$
$774$		0			0
$775$			708.313i			0.913952i
$776$			2859.98i			3.68554i
$777$		0			0
$778$	388.976			0.499969
$779$	−195.722			−0.251248
$780$		0			0
$781$	−813.760			−1.04195
$782$			1668.53i			2.13367i
$783$		0			0
$784$	−1358.87	+	1439.82i	−1.73325	+	1.83650i
$785$	748.462			0.953455
$786$		0			0
$787$		−	837.871i		−	1.06464i	−0.846544	−	0.532319i	$-0.821321\pi$
$787$		−	837.871i		−	1.06464i	0.846544	−	0.532319i	$-0.178679\pi$
$788$	1238.07			1.57116
$789$		0			0
$790$			3077.02i			3.89496i
$791$	−289.478	−	671.246i	−0.365965	−	0.848604i
$792$		0			0
$793$	692.952			0.873836
$794$		−	2220.50i		−	2.79660i
$795$		0			0
$796$		−	139.945i		−	0.175810i
$797$		−	1387.72i		−	1.74118i	−0.492010	−	0.870589i	$-0.663738\pi$
$797$		−	1387.72i		−	1.74118i	0.492010	−	0.870589i	$-0.336262\pi$
$798$		0			0
$799$	644.220			0.806283
$800$	2783.52			3.47940
$801$		0			0
$802$	1075.21			1.34066
$803$			396.737i			0.494068i
$804$		0			0
$805$	1532.33	−	660.825i	1.90351	−	0.820900i
$806$	390.063			0.483949
$807$		0			0
$808$			232.994i			0.288358i
$809$	1281.68			1.58428			0.792140	−	0.610339i	$-0.208967\pi$
$809$	1281.68			1.58428			0.792140	+	0.610339i	$0.208967\pi$
$810$		0			0
$811$			13.7553i			0.0169610i	0.999964	+	0.00848048i	$0.00269945\pi$
$811$			13.7553i			0.0169610i	−0.999964	+	0.00848048i	$0.997301\pi$
$812$	−54.6862	−	126.807i	−0.0673475	−	0.156166i
$813$		0			0
$814$	1268.51			1.55837
$815$		−	1564.80i		−	1.92000i
$816$		0			0
$817$			19.4122i			0.0237604i
$818$		−	333.953i		−	0.408256i
$819$		0			0
$820$	−3185.71			−3.88502
$821$	461.671			0.562328			0.281164	−	0.959660i	$-0.409279\pi$
$821$	461.671			0.562328			0.281164	+	0.959660i	$0.409279\pi$
$822$		0			0
$823$	1057.27			1.28465			0.642326	−	0.766432i	$-0.277970\pi$
$823$	1057.27			1.28465			0.642326	+	0.766432i	$0.277970\pi$
$824$			4186.15i			5.08028i
$825$		0			0
$826$	721.212	−	311.026i	0.873138	−	0.376545i
$827$	−73.7018			−0.0891194			−0.0445597	−	0.999007i	$-0.514188\pi$
$827$	−73.7018			−0.0891194			−0.0445597	+	0.999007i	$0.514188\pi$
$828$		0			0
$829$			134.854i			0.162670i	0.996687	+	0.0813352i	$0.0259184\pi$
$829$			134.854i			0.162670i	−0.996687	+	0.0813352i	$0.974082\pi$
$830$	−1601.27			−1.92924
$831$		0			0
$832$		−	494.037i		−	0.593795i
$833$	−555.502	−	524.271i	−0.666869	−	0.629377i
$834$		0			0
$835$	599.268			0.717686
$836$		−	498.831i		−	0.596687i
$837$		0			0
$838$		−	2454.17i		−	2.92860i
$839$			936.299i			1.11597i	0.829851	+	0.557985i	$0.188425\pi$
$839$			936.299i			1.11597i	−0.829851	+	0.557985i	$0.811575\pi$
$840$		0			0
$841$	−836.924			−0.995154
$842$	−501.202			−0.595252
$843$		0			0
$844$	−3245.52			−3.84540
$845$		−	1055.35i		−	1.24894i
$846$		0			0
$847$	42.0679	+	97.5477i	0.0496670	+	0.115168i
$848$	1765.12			2.08151
$849$		0			0
$850$			2505.82i			2.94802i
$851$	958.375			1.12618
$852$		0			0
$853$			25.5820i			0.0299907i	0.999888	+	0.0149953i	$0.00477334\pi$
$853$			25.5820i			0.0299907i	−0.999888	+	0.0149953i	$0.995227\pi$
$854$	−2571.59	+	1109.01i	−3.01123	+	1.29861i
$855$		0			0
$856$	2225.78			2.60021
$857$		−	553.564i		−	0.645932i	−0.946411	−	0.322966i	$-0.895320\pi$
$857$		−	553.564i		−	0.645932i	0.946411	−	0.322966i	$-0.104680\pi$
$858$		0			0
$859$		−	647.301i		−	0.753552i	−0.926304	−	0.376776i	$-0.877033\pi$
$859$		−	647.301i		−	0.753552i	0.926304	−	0.376776i	$-0.122967\pi$
$860$			315.968i			0.367404i
$861$		0			0
$862$	−2114.61			−2.45314
$863$	709.719			0.822386			0.411193	−	0.911548i	$-0.365112\pi$
$863$	709.719			0.822386			0.411193	+	0.911548i	$0.365112\pi$
$864$		0			0
$865$	−1425.07			−1.64748
$866$		−	1776.92i		−	2.05187i
$867$		0			0
$868$	−1027.12	+	442.949i	−1.18331	+	0.510310i
$869$	1031.96			1.18752
$870$		0			0
$871$			235.127i			0.269951i
$872$	−2828.25			−3.24340
$873$		0			0
$874$		−	531.138i		−	0.607710i
$875$	973.086	−	419.648i	1.11210	−	0.479598i
$876$		0			0
$877$	−367.996			−0.419608			−0.209804	−	0.977744i	$-0.567283\pi$
$877$	−367.996			−0.419608			−0.209804	+	0.977744i	$0.567283\pi$
$878$			1434.19i			1.63347i
$879$		0			0
$880$		−	3435.40i		−	3.90387i
$881$			1183.91i			1.34383i	0.740629	+	0.671914i	$0.234528\pi$
$881$			1183.91i			1.34383i	−0.740629	+	0.671914i	$0.765472\pi$
$882$		0			0
$883$	845.784			0.957853			0.478927	−	0.877855i	$-0.341026\pi$
$883$	845.784			0.957853			0.478927	+	0.877855i	$0.341026\pi$
$884$	979.142			1.10763
$885$		0			0
$886$	−475.024			−0.536145
$887$			174.598i			0.196840i	0.995145	+	0.0984202i	$0.0313789\pi$
$887$			174.598i			0.196840i	−0.995145	+	0.0984202i	$0.968621\pi$
$888$		0			0
$889$	−403.548	−	935.752i	−0.453935	−	1.05259i
$890$	−2195.02			−2.46632
$891$		0			0
$892$		−	1886.62i		−	2.11505i
$893$	−205.073			−0.229645
$894$		0			0
$895$			1308.77i			1.46231i
$896$	78.1405	+	181.193i	0.0872104	+	0.202225i
$897$		0			0
$898$	−14.4879			−0.0161335
$899$		−	33.0124i		−	0.0367213i
$900$		0			0
$901$			681.008i			0.755835i
$902$			1505.75i			1.66934i
$903$		0			0
$904$	−2236.90			−2.47445
$905$	−2359.08			−2.60672
$906$		0			0
$907$	1055.10			1.16329			0.581643	−	0.813444i	$-0.302410\pi$
$907$	1055.10			1.16329			0.581643	+	0.813444i	$0.302410\pi$
$908$			233.099i			0.256717i
$909$		0			0
$910$	−546.528	−	1267.30i	−0.600580	−	1.39263i
$911$	1506.20			1.65335			0.826676	−	0.562679i	$-0.190229\pi$
$911$	1506.20			1.65335			0.826676	+	0.562679i	$0.190229\pi$
$912$		0			0
$913$			537.027i			0.588200i
$914$	−1398.82			−1.53044
$915$		0			0
$916$		−	3643.10i		−	3.97718i
$917$	−1428.39	+	616.000i	−1.55768	+	0.671756i
$918$		0			0
$919$	−69.2280			−0.0753297			−0.0376649	−	0.999290i	$-0.511992\pi$
$919$	−69.2280			−0.0753297			−0.0376649	+	0.999290i	$0.511992\pi$
$920$		−	5106.43i		−	5.55047i
$921$		0			0
$922$		−	1229.63i		−	1.33365i
$923$			508.468i			0.550886i
$924$		0			0
$925$	1439.30			1.55600
$926$	102.351			0.110531
$927$		0			0
$928$	−129.732			−0.139797
$929$		−	1289.69i		−	1.38826i	−0.719850	−	0.694129i	$-0.755790\pi$
$929$		−	1289.69i		−	1.38826i	0.719850	−	0.694129i	$-0.244210\pi$
$930$		0			0
$931$	176.832	+	166.890i	0.189937	+	0.179259i
$932$	1735.62			1.86225
$933$		0			0
$934$			1661.75i			1.77918i
$935$	1325.43			1.41757
$936$		0			0
$937$			1061.84i			1.13323i	0.823982	+	0.566617i	$0.191748\pi$
$937$			1061.84i			1.13323i	−0.823982	+	0.566617i	$0.808252\pi$
$938$	−376.302	−	872.573i	−0.401174	−	0.930248i
$939$		0			0
$940$	−3337.92			−3.55098
$941$			653.558i			0.694536i	0.937766	+	0.347268i	$0.112891\pi$
$941$			653.558i			0.694536i	−0.937766	+	0.347268i	$0.887109\pi$
$942$		0			0
$943$			1137.61i			1.20637i
$944$		−	1221.60i		−	1.29407i
$945$		0			0
$946$	149.344			0.157869
$947$	−768.131			−0.811121			−0.405560	−	0.914068i	$-0.632924\pi$
$947$	−768.131			−0.811121			−0.405560	+	0.914068i	$0.632924\pi$
$948$		0			0
$949$	247.896			0.261218
$950$		−	797.672i		−	0.839655i
$951$		0			0
$952$	−2146.28	+	925.596i	−2.25450	+	0.972265i
$953$	251.046			0.263427			0.131714	−	0.991288i	$-0.457952\pi$
$953$	251.046			0.263427			0.131714	+	0.991288i	$0.457952\pi$
$954$		0			0
$955$		−	1491.70i		−	1.56199i
$956$	−136.211			−0.142480
$957$		0			0
$958$			2651.40i			2.76764i
$959$	407.163	+	944.134i	0.424570	+	0.984499i
$960$		0			0
$961$	693.605			0.721753
$962$		−	792.614i		−	0.823923i
$963$		0			0
$964$			1796.17i			1.86325i
$965$		−	990.851i		−	1.02679i
$966$		0			0
$967$	−474.640			−0.490838			−0.245419	−	0.969417i	$-0.578926\pi$
$967$	−474.640			−0.490838			−0.245419	+	0.969417i	$0.578926\pi$
$968$	325.074			0.335821
$969$		0			0
$970$	4095.41			4.22207
$971$			1205.96i			1.24198i	0.783818	+	0.620991i	$0.213270\pi$
$971$			1205.96i			1.24198i	−0.783818	+	0.620991i	$0.786730\pi$
$972$		0			0
$973$	617.592	−	266.340i	0.634730	−	0.273731i
$974$	−385.298			−0.395583
$975$		0			0
$976$			4355.81i			4.46292i
$977$	335.837			0.343743			0.171872	−	0.985119i	$-0.445019\pi$
$977$	335.837			0.343743			0.171872	+	0.985119i	$0.445019\pi$
$978$		0			0
$979$			736.158i			0.751949i
$980$	2878.24	+	2716.42i	2.93698	+	2.77186i
$981$		0			0
$982$	2334.46			2.37725
$983$			1260.83i			1.28263i	0.767276	+	0.641317i	$0.221612\pi$
$983$			1260.83i			1.28263i	−0.767276	+	0.641317i	$0.778388\pi$
$984$		0			0
$985$		−	1047.18i		−	1.06313i
$986$		−	116.789i		−	0.118447i
$987$		0			0
$988$	−311.688			−0.315474
$989$	112.831			0.114086
$990$		0			0
$991$	505.836			0.510430			0.255215	−	0.966884i	$-0.417854\pi$
$991$	505.836			0.510430			0.255215	+	0.966884i	$0.417854\pi$
$992$			1050.81i			1.05928i
$993$		0			0
$994$	−813.760	−	1886.96i	−0.818672	−	1.89835i
$995$	−118.368			−0.118963
$996$		0			0
$997$			1130.51i			1.13392i	0.823747	+	0.566958i	$0.191880\pi$
$997$			1130.51i			1.13392i	−0.823747	+	0.566958i	$0.808120\pi$
$998$	−683.355			−0.684725
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.d.e.55.1	✓	8
3.2	odd	2	inner	189.3.d.e.55.8	yes	8
7.6	odd	2	inner	189.3.d.e.55.2	yes	8
21.20	even	2	inner	189.3.d.e.55.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.d.e.55.1	✓	8	1.1	even	1	trivial
189.3.d.e.55.2	yes	8	7.6	odd	2	inner
189.3.d.e.55.7	yes	8	21.20	even	2	inner
189.3.d.e.55.8	yes	8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.71106 q^{2} +9.77200 q^{4} -8.26535i q^{5} +(-2.77200 - 6.42775i) q^{7} -21.4203 q^{8} +30.6732i q^{10} +10.2871 q^{11} -6.42775i q^{13} +(10.2871 + 23.8538i) q^{14} +40.4040 q^{16} +15.5885i q^{17} -4.96224i q^{19} -80.7690i q^{20} -38.1760 q^{22} -28.8424 q^{23} -43.3160 q^{25} +23.8538i q^{26} +(-27.0880 - 62.8120i) q^{28} +2.01883 q^{29} -16.3522i q^{31} -64.2608 q^{32} -57.8498i q^{34} +(-53.1276 + 22.9116i) q^{35} -33.2280 q^{37} +18.4152i q^{38} +177.046i q^{40} -39.4423i q^{41} -3.91199 q^{43} +100.525 q^{44} +107.036 q^{46} -41.3267i q^{47} +(-33.6320 + 35.6355i) q^{49} +160.749 q^{50} -62.8120i q^{52} +43.6867 q^{53} -85.0263i q^{55} +(59.3770 + 137.684i) q^{56} -7.49202 q^{58} -30.2347i q^{59} +107.806i q^{61} +60.6842i q^{62} +76.8600 q^{64} -53.1276 q^{65} -36.5800 q^{67} +152.330i q^{68} +(197.160 - 85.0263i) q^{70} -79.1051 q^{71} +38.5665i q^{73} +123.311 q^{74} -48.4910i q^{76} +(-28.5158 - 66.1228i) q^{77} +100.316 q^{79} -333.953i q^{80} +146.373i q^{82} +52.2040i q^{83} +128.844 q^{85} +14.5177 q^{86} -220.352 q^{88} +71.5614i q^{89} +(-41.3160 + 17.8177i) q^{91} -281.848 q^{92} +153.366i q^{94} -41.0146 q^{95} -133.517i q^{97} +(124.811 - 132.246i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4} + 12 q^{7} + 84 q^{16} - 32 q^{22} - 244 q^{25} - 80 q^{28} - 300 q^{37} - 168 q^{43} + 412 q^{46} - 64 q^{49} + 316 q^{58} + 444 q^{64} + 220 q^{67} + 552 q^{70} + 700 q^{79} + 108 q^{85}+ \cdots - 228 q^{91}+O(q^{100}) \) 8 * q + 44 * q^4 + 12 * q^7 + 84 * q^16 - 32 * q^22 - 244 * q^25 - 80 * q^28 - 300 * q^37 - 168 * q^43 + 412 * q^46 - 64 * q^49 + 316 * q^58 + 444 * q^64 + 220 * q^67 + 552 * q^70 + 700 * q^79 + 108 * q^85 - 1216 * q^88 - 228 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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