LMFDB - Embedded newform 189.3.d.e.55.7

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			55.7
Root			$0.238746 - 0.413520i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.e.55.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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	$-0.121726\pi$
$2$	3.71106			1.85553			0.927766	+	0.373162i	$0.121726\pi$
$3$		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	$-0.654879\pi$
$11$	−10.2871			−0.935189			−0.467594	+	0.883943i	$0.654879\pi$
$12$		0			0
$13$			6.42775i			0.494443i	0.968959	+	0.247221i	$0.0795175\pi$
$13$			6.42775i			0.494443i	−0.968959	+	0.247221i	$0.920483\pi$
$14$	−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	$0.0416856\pi$
$19$			4.96224i			0.261170i	−0.991437	+	0.130585i	$0.958314\pi$
$20$		−	80.7690i		−	4.03845i
$21$		0			0
$22$	−38.1760			−1.73527
$23$	28.8424			1.25402			0.627009	−	0.779012i	$-0.284279\pi$
$23$	28.8424			1.25402			0.627009	+	0.779012i	$0.284279\pi$
$24$		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	$-0.511082\pi$
$29$	−2.01883			−0.0696149			−0.0348075	+	0.999394i	$0.511082\pi$
$30$		0			0
$31$			16.3522i			0.527491i	0.964592	+	0.263746i	$0.0849579\pi$
$31$			16.3522i			0.527491i	−0.964592	+	0.263746i	$0.915042\pi$
$32$	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	$-0.635218\pi$
$53$	−43.6867			−0.824277			−0.412138	+	0.911121i	$0.635218\pi$
$54$		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	$-0.655073\pi$
$61$		−	107.806i		−	1.76732i	0.468133	−	0.883658i	$-0.344927\pi$
$62$			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	$-0.311922\pi$
$71$	79.1051			1.11416			0.557078	+	0.830460i	$0.311922\pi$
$72$		0			0
$73$		−	38.5665i		−	0.528308i	−0.964480	−	0.264154i	$-0.914907\pi$
$73$		−	38.5665i		−	0.528308i	0.964480	−	0.264154i	$-0.0850928\pi$
$74$	−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	$0.241614\pi$
$97$			133.517i			1.37647i	−0.725489	+	0.688233i	$0.758386\pi$
$98$	−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	$0.397587\pi$
$103$			195.429i			1.89737i	−0.316217	+	0.948687i	$0.602413\pi$
$104$			137.684i			1.32389i
$105$		0			0
$106$	−162.124			−1.52947
$107$	103.910			0.971120			0.485560	−	0.874203i	$-0.338616\pi$
$107$	103.910			0.971120			0.485560	+	0.874203i	$0.338616\pi$
$108$		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	$-0.652896\pi$
$113$	−104.429			−0.924153			−0.462077	+	0.886840i	$0.652896\pi$
$114$		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	$-0.319907\pi$
$137$	146.884			1.07215			0.536073	+	0.844172i	$0.319907\pi$
$138$		0			0
$139$		−	96.0822i		−	0.691239i	−0.938375	−	0.345619i	$-0.887669\pi$
$139$		−	96.0822i		−	0.691239i	0.938375	−	0.345619i	$-0.112331\pi$
$140$	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	$-0.875791\pi$
$149$	−275.599			−1.84966			−0.924828	+	0.380387i	$0.875791\pi$
$150$		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	$-0.906880\pi$
$157$		−	90.5542i		−	0.576779i	0.957513	−	0.288389i	$-0.0931197\pi$
$158$	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	$-0.354162\pi$
$179$	158.344			0.884602			0.442301	+	0.896867i	$0.354162\pi$
$180$		0			0
$181$			285.418i			1.57690i	0.615102	+	0.788448i	$0.289115\pi$
$181$			285.418i			1.57690i	−0.615102	+	0.788448i	$0.710885\pi$
$182$	−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	$-0.656631\pi$
$191$	−180.477			−0.944903			−0.472452	+	0.881357i	$0.656631\pi$
$192$		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	$-0.604208\pi$
$197$	−126.696			−0.643125			−0.321563	+	0.946888i	$0.604208\pi$
$198$		0			0
$199$			14.3210i			0.0719649i	0.999352	+	0.0359825i	$0.0114560\pi$
$199$			14.3210i			0.0719649i	−0.999352	+	0.0359825i	$0.988544\pi$
$200$	−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	$0.142502\pi$
$223$			193.064i			0.865758i	−0.901452	+	0.432879i	$0.857498\pi$
$224$	−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	$0.302712\pi$
$229$			372.810i			1.62799i	−0.580872	+	0.813995i	$0.697288\pi$
$230$		−	884.690i		−	3.84648i
$231$		0			0
$232$	−43.2440			−0.186396
$233$	−177.612			−0.762282			−0.381141	−	0.924517i	$-0.624469\pi$
$233$	−177.612			−0.762282			−0.381141	+	0.924517i	$0.624469\pi$
$234$		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	$-0.490716\pi$
$239$	13.9389			0.0583218			0.0291609	+	0.999575i	$0.490716\pi$
$240$		0			0
$241$		−	183.808i		−	0.762689i	−0.924433	−	0.381344i	$-0.875461\pi$
$241$		−	183.808i		−	0.762689i	0.924433	−	0.381344i	$-0.124539\pi$
$242$	−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	$-0.450593\pi$
$263$	81.3168			0.309189			0.154595	+	0.987978i	$0.450593\pi$
$264$		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	$0.356940\pi$
$271$			488.175i			1.80138i	−0.434458	+	0.900692i	$0.643060\pi$
$272$			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	$-0.564833\pi$
$281$	−113.677			−0.404546			−0.202273	+	0.979329i	$0.564833\pi$
$282$		0			0
$283$		−	65.1774i		−	0.230309i	−0.993348	−	0.115154i	$-0.963264\pi$
$283$		−	65.1774i		−	0.230309i	0.993348	−	0.115154i	$-0.0367363\pi$
$284$	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	$0.196802\pi$
$307$			355.892i			1.15926i	−0.814881	+	0.579628i	$0.803198\pi$
$308$	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	$0.00653726\pi$
$313$			12.8555i			0.0410719i	−0.999789	+	0.0205360i	$0.993463\pi$
$314$		−	336.053i		−	1.07023i
$315$		0			0
$316$	980.288			3.10218
$317$	−268.236			−0.846169			−0.423085	−	0.906090i	$-0.639053\pi$
$317$	−268.236			−0.846169			−0.423085	+	0.906090i	$0.639053\pi$
$318$		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	$-0.476687\pi$
$347$	50.7822			0.146346			0.0731732	+	0.997319i	$0.476687\pi$
$348$		0			0
$349$		−	129.352i		−	0.370637i	−0.982679	−	0.185318i	$-0.940668\pi$
$349$		−	129.352i		−	0.370637i	0.982679	−	0.185318i	$-0.0593317\pi$
$350$	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	$-0.392105\pi$
$359$	238.740			0.665014			0.332507	+	0.943101i	$0.392105\pi$
$360$		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	$0.0247220\pi$
$367$			56.9499i			0.155177i	−0.996985	+	0.0775885i	$0.975278\pi$
$368$	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	$-0.456985\pi$
$389$	104.815			0.269448			0.134724	+	0.990883i	$0.456985\pi$
$390$		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	$-0.728323\pi$
$397$		−	598.347i		−	1.50717i	0.657350	−	0.753585i	$-0.271677\pi$
$398$			53.1462i			0.133533i
$399$		0			0
$400$	−1750.14			−4.37535
$401$	289.730			0.722520			0.361260	−	0.932465i	$-0.382347\pi$
$401$	289.730			0.722520			0.361260	+	0.932465i	$0.382347\pi$
$402$		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	$-0.964912\pi$
$409$		−	89.9885i		−	0.220021i	0.993930	−	0.110010i	$-0.0350884\pi$
$410$	−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	$-0.729882\pi$
$431$	−569.812			−1.32207			−0.661035	+	0.750356i	$0.729882\pi$
$432$		0			0
$433$		−	478.816i		−	1.10581i	−0.833244	−	0.552906i	$-0.813519\pi$
$433$		−	478.816i		−	1.10581i	0.833244	−	0.552906i	$-0.186481\pi$
$434$	−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	$0.145079\pi$
$439$			386.462i			0.880324i	−0.897918	+	0.440162i	$0.854921\pi$
$440$			1821.29i			4.13929i
$441$		0			0
$442$	−371.844			−0.841276
$443$	−128.002			−0.288944			−0.144472	−	0.989509i	$-0.546148\pi$
$443$	−128.002			−0.288944			−0.144472	+	0.989509i	$0.546148\pi$
$444$		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	$-0.501384\pi$
$449$	−3.90398			−0.00869483			−0.00434741	+	0.999991i	$0.501384\pi$
$450$		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	$-0.278692\pi$
$491$	629.055			1.28117			0.640586	+	0.767887i	$0.278692\pi$
$492$		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	$0.0398429\pi$
$523$			130.586i			0.249687i	−0.992176	+	0.124843i	$0.960157\pi$
$524$		−	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	$-0.353556\pi$
$557$	494.626			0.888018			0.444009	+	0.896022i	$0.353556\pi$
$558$		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	$-0.846124\pi$
$569$	−1007.60			−1.77082			−0.885412	+	0.464807i	$0.846124\pi$
$570$		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	$-0.949018\pi$
$577$		−	184.039i		−	0.318959i	0.987201	−	0.159480i	$-0.0509816\pi$
$578$	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	$-0.649093\pi$
$599$	−540.836			−0.902898			−0.451449	+	0.892297i	$0.649093\pi$
$600$		0			0
$601$		−	10.8243i		−	0.0180105i	−0.999959	−	0.00900524i	$-0.997134\pi$
$601$		−	10.8243i		−	0.0180105i	0.999959	−	0.00900524i	$-0.00286650\pi$
$602$	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	$-0.700521\pi$
$607$		−	980.978i		−	1.61611i	0.589108	−	0.808055i	$-0.299479\pi$
$608$			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	$-0.332886\pi$
$617$	618.501			1.00243			0.501216	+	0.865322i	$0.332886\pi$
$618$		0			0
$619$			153.366i			0.247765i	0.992297	+	0.123882i	$0.0395345\pi$
$619$			153.366i			0.247765i	−0.992297	+	0.123882i	$0.960465\pi$
$620$	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	$-0.209864\pi$
$641$	1013.31			1.58083			0.790416	+	0.612570i	$0.209864\pi$
$642$		0			0
$643$		−	752.150i		−	1.16975i	−0.811123	−	0.584875i	$-0.801143\pi$
$643$		−	752.150i		−	1.16975i	0.811123	−	0.584875i	$-0.198857\pi$
$644$	−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	$-0.632887\pi$
$653$	−529.525			−0.810911			−0.405455	+	0.914115i	$0.632887\pi$
$654$		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	$-0.403819\pi$
$659$	392.215			0.595168			0.297584	+	0.954696i	$0.403819\pi$
$660$		0			0
$661$			1212.04i			1.83365i	0.399288	+	0.916825i	$0.369257\pi$
$661$			1212.04i			1.83365i	−0.399288	+	0.916825i	$0.630743\pi$
$662$	−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	$-0.552304\pi$
$683$	−223.451			−0.327161			−0.163580	+	0.986530i	$0.552304\pi$
$684$		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	$0.0683111\pi$
$691$			294.314i			0.425924i	−0.977060	+	0.212962i	$0.931689\pi$
$692$		−	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	$-0.155146\pi$
$701$	1238.74			1.76710			0.883552	+	0.468334i	$0.155146\pi$
$702$		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	$-0.877648\pi$
$727$		−	545.228i		−	0.749969i	0.927031	−	0.374985i	$-0.122352\pi$
$728$	−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	$0.0244813\pi$
$733$			112.640i			0.153669i	−0.997044	+	0.0768346i	$0.975519\pi$
$734$			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	$-0.596048\pi$
$743$	−441.617			−0.594370			−0.297185	+	0.954820i	$0.596048\pi$
$744$		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	$-0.854328\pi$
$769$		−	679.542i		−	0.883670i	0.897096	−	0.441835i	$-0.145672\pi$
$770$	−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	$0.178679\pi$
$787$			837.871i			1.06464i	−0.846544	+	0.532319i	$0.821321\pi$
$788$	−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	$-0.791033\pi$
$809$	−1281.68			−1.58428			−0.792140	+	0.610339i	$0.791033\pi$
$810$		0			0
$811$		−	13.7553i		−	0.0169610i	−0.999964	−	0.00848048i	$-0.997301\pi$
$811$		−	13.7553i		−	0.0169610i	0.999964	−	0.00848048i	$-0.00269945\pi$
$812$	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	$-0.590721\pi$
$821$	−461.671			−0.562328			−0.281164	+	0.959660i	$0.590721\pi$
$822$		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	$-0.485812\pi$
$827$	73.7018			0.0891194			0.0445597	+	0.999007i	$0.485812\pi$
$828$		0			0
$829$		−	134.854i		−	0.162670i	−0.996687	−	0.0813352i	$-0.974082\pi$
$829$		−	134.854i		−	0.162670i	0.996687	−	0.0813352i	$-0.0259184\pi$
$830$	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	$-0.995227\pi$
$853$		−	25.5820i		−	0.0299907i	0.999888	−	0.0149953i	$-0.00477334\pi$
$854$	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	$0.122967\pi$
$859$			647.301i			0.753552i	−0.926304	+	0.376776i	$0.877033\pi$
$860$			315.968i			0.367404i
$861$		0			0
$862$	−2114.61			−2.45314
$863$	−709.719			−0.822386			−0.411193	−	0.911548i	$-0.634888\pi$
$863$	−709.719			−0.822386			−0.411193	+	0.911548i	$0.634888\pi$
$864$		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	$-0.809771\pi$
$911$	−1506.20			−1.65335			−0.826676	+	0.562679i	$0.809771\pi$
$912$		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	$-0.808252\pi$
$937$		−	1061.84i		−	1.13323i	0.823982	−	0.566617i	$-0.191748\pi$
$938$	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	$-0.367076\pi$
$947$	768.131			0.811121			0.405560	+	0.914068i	$0.367076\pi$
$948$		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	$-0.542048\pi$
$953$	−251.046			−0.263427			−0.131714	+	0.991288i	$0.542048\pi$
$954$		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	$-0.554981\pi$
$977$	−335.837			−0.343743			−0.171872	+	0.985119i	$0.554981\pi$
$978$		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	$-0.808120\pi$
$997$		−	1130.51i		−	1.13392i	0.823747	−	0.566958i	$-0.191880\pi$
$998$	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.55.7	yes	8
3.2	odd	2	inner	189.3.d.e.55.2	yes	8
7.6	odd	2	inner	189.3.d.e.55.8	yes	8
21.20	even	2	inner	189.3.d.e.55.1	✓	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

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