LMFDB - Embedded newform 189.3.d.d.55.8

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:	8.0.81622204416.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 165x^{4} + 434x^{2} + 961 $ x^8 + 14x^6 + 165x^4 + 434*x^2 + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.8
Root			$1.67650 + 2.90379i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.d.55.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35300 q^{2} +7.24264 q^{4} +2.40558i q^{5} +(2.24264 + 6.63103i) q^{7} +10.8726 q^{8} +O(q^{10})$	\(q+3.35300 q^{2} +7.24264 q^{4} +2.40558i q^{5} +(2.24264 + 6.63103i) q^{7} +10.8726 q^{8} +8.06591i q^{10} +3.35300 q^{11} -16.3059i q^{13} +(7.51958 + 22.2339i) q^{14} +7.48528 q^{16} -24.6395i q^{17} +2.15232i q^{19} +17.4227i q^{20} +11.2426 q^{22} -33.4313 q^{23} +19.2132 q^{25} -54.6737i q^{26} +(16.2426 + 48.0262i) q^{28} -41.0496 q^{29} +32.9811i q^{31} -18.3922 q^{32} -82.6162i q^{34} +(-15.9514 + 5.39484i) q^{35} +37.4853 q^{37} +7.21673i q^{38} +26.1548i q^{40} +57.6630i q^{41} +7.69848 q^{43} +24.2846 q^{44} -112.095 q^{46} -89.5192i q^{47} +(-38.9411 + 29.7420i) q^{49} +64.4220 q^{50} -118.098i q^{52} -18.4909 q^{53} +8.06591i q^{55} +(24.3833 + 72.0965i) q^{56} -137.640 q^{58} +13.8498i q^{59} -24.7411i q^{61} +110.586i q^{62} -91.6102 q^{64} +39.2251 q^{65} +99.1543 q^{67} -178.455i q^{68} +(-53.4853 + 18.0889i) q^{70} +64.7181 q^{71} -50.7007i q^{73} +125.688 q^{74} +15.5885i q^{76} +(7.51958 + 22.2339i) q^{77} +18.2426 q^{79} +18.0064i q^{80} +193.344i q^{82} +70.2746i q^{83} +59.2721 q^{85} +25.8131 q^{86} +36.4558 q^{88} +36.0836i q^{89} +(108.125 - 36.5683i) q^{91} -242.131 q^{92} -300.158i q^{94} -5.17756 q^{95} +143.013i q^{97} +(-130.570 + 99.7252i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4} - 16 q^{7}+O(q^{10}) $ 8 * q + 24 * q^4 - 16 * q^7	$ 8 q + 24 q^{4} - 16 q^{7} - 8 q^{16} + 56 q^{22} - 16 q^{25} + 96 q^{28} + 232 q^{37} - 176 q^{43} - 184 q^{46} - 40 q^{49} - 592 q^{58} - 88 q^{64} + 352 q^{67} - 360 q^{70} + 112 q^{79} + 576 q^{85} + 88 q^{88} + 288 q^{91}+O(q^{100}) $ 8 * q + 24 * q^4 - 16 * q^7 - 8 * q^16 + 56 * q^22 - 16 * q^25 + 96 * q^28 + 232 * q^37 - 176 * q^43 - 184 * q^46 - 40 * q^49 - 592 * q^58 - 88 * q^64 + 352 * q^67 - 360 * q^70 + 112 * q^79 + 576 * q^85 + 88 * q^88 + 288 * 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 $

$2$	3.35300			1.67650			0.838251	−	0.545284i	$-0.183578\pi$
$2$	3.35300			1.67650			0.838251	+	0.545284i	$0.183578\pi$
$3$		0			0
$3$		0			0
$4$	7.24264			1.81066
$5$			2.40558i			0.481115i	0.970635	+	0.240558i	$0.0773303\pi$
$5$			2.40558i			0.481115i	−0.970635	+	0.240558i	$0.922670\pi$
$6$		0			0
$7$	2.24264	+	6.63103i	0.320377	+	0.947290i
$7$	2.24264	+	6.63103i	0.320377	+	0.947290i
$8$	10.8726			1.35907
$9$		0			0
$10$			8.06591i			0.806591i
$11$	3.35300			0.304819			0.152409	−	0.988317i	$-0.451297\pi$
$11$	3.35300			0.304819			0.152409	+	0.988317i	$0.451297\pi$
$12$		0			0
$13$		−	16.3059i		−	1.25430i	−0.778899	−	0.627150i	$-0.784221\pi$
$13$		−	16.3059i		−	1.25430i	0.778899	−	0.627150i	$-0.215779\pi$
$14$	7.51958	+	22.2339i	0.537113	+	1.58813i
$15$		0			0
$16$	7.48528			0.467830
$17$		−	24.6395i		−	1.44938i	−0.689075	−	0.724690i	$-0.741983\pi$
$17$		−	24.6395i		−	1.44938i	0.689075	−	0.724690i	$-0.258017\pi$
$18$		0			0
$19$			2.15232i			0.113280i	0.998395	+	0.0566399i	$0.0180387\pi$
$19$			2.15232i			0.113280i	−0.998395	+	0.0566399i	$0.981961\pi$
$20$			17.4227i			0.871136i
$21$		0			0
$22$	11.2426			0.511029
$23$	−33.4313			−1.45354			−0.726768	−	0.686883i	$-0.758979\pi$
$23$	−33.4313			−1.45354			−0.726768	+	0.686883i	$0.758979\pi$
$24$		0			0
$25$	19.2132			0.768528
$26$		−	54.6737i		−	2.10284i
$27$		0			0
$28$	16.2426	+	48.0262i	0.580094	+	1.71522i
$29$	−41.0496			−1.41550			−0.707752	−	0.706461i	$-0.750291\pi$
$29$	−41.0496			−1.41550			−0.707752	+	0.706461i	$0.750291\pi$
$30$		0			0
$31$			32.9811i			1.06391i	0.846774	+	0.531953i	$0.178542\pi$
$31$			32.9811i			1.06391i	−0.846774	+	0.531953i	$0.821458\pi$
$32$	−18.3922			−0.574755
$33$		0			0
$34$		−	82.6162i		−	2.42989i
$35$	−15.9514	+	5.39484i	−0.455756	+	0.154138i
$36$		0			0
$37$	37.4853			1.01312			0.506558	−	0.862206i	$-0.330918\pi$
$37$	37.4853			1.01312			0.506558	+	0.862206i	$0.330918\pi$
$38$			7.21673i			0.189914i
$39$		0			0
$40$			26.1548i			0.653871i
$41$			57.6630i			1.40641i	0.710985	+	0.703207i	$0.248250\pi$
$41$			57.6630i			1.40641i	−0.710985	+	0.703207i	$0.751750\pi$
$42$		0			0
$43$	7.69848			0.179035			0.0895173	−	0.995985i	$-0.471468\pi$
$43$	7.69848			0.179035			0.0895173	+	0.995985i	$0.471468\pi$
$44$	24.2846			0.551923
$45$		0			0
$46$	−112.095			−2.43686
$47$		−	89.5192i		−	1.90466i	−0.305064	−	0.952332i	$-0.598678\pi$
$47$		−	89.5192i		−	1.90466i	0.305064	−	0.952332i	$-0.401322\pi$
$48$		0			0
$49$	−38.9411	+	29.7420i	−0.794717	+	0.606980i
$50$	64.4220			1.28844
$51$		0			0
$52$		−	118.098i		−	2.27111i
$53$	−18.4909			−0.348884			−0.174442	−	0.984667i	$-0.555812\pi$
$53$	−18.4909			−0.348884			−0.174442	+	0.984667i	$0.555812\pi$
$54$		0			0
$55$			8.06591i			0.146653i
$56$	24.3833	+	72.0965i	0.435416	+	1.28744i
$57$		0			0
$58$	−137.640			−2.37310
$59$			13.8498i			0.234742i	0.993088	+	0.117371i	$0.0374466\pi$
$59$			13.8498i			0.234742i	−0.993088	+	0.117371i	$0.962553\pi$
$60$		0			0
$61$		−	24.7411i		−	0.405592i	−0.979221	−	0.202796i	$-0.934997\pi$
$61$		−	24.7411i		−	0.405592i	0.979221	−	0.202796i	$-0.0650028\pi$
$62$			110.586i			1.78364i
$63$		0			0
$64$	−91.6102			−1.43141
$65$	39.2251			0.603463
$66$		0			0
$67$	99.1543			1.47992			0.739958	−	0.672653i	$-0.234845\pi$
$67$	99.1543			1.47992			0.739958	+	0.672653i	$0.234845\pi$
$68$		−	178.455i		−	2.62433i
$69$		0			0
$70$	−53.4853	+	18.0889i	−0.764075	+	0.258413i
$71$	64.7181			0.911522			0.455761	−	0.890102i	$-0.349367\pi$
$71$	64.7181			0.911522			0.455761	+	0.890102i	$0.349367\pi$
$72$		0			0
$73$		−	50.7007i		−	0.694530i	−0.937767	−	0.347265i	$-0.887110\pi$
$73$		−	50.7007i		−	0.694530i	0.937767	−	0.347265i	$-0.112890\pi$
$74$	125.688			1.69849
$75$		0			0
$76$			15.5885i			0.205111i
$77$	7.51958	+	22.2339i	0.0976569	+	0.288752i
$78$		0			0
$79$	18.2426			0.230920			0.115460	−	0.993312i	$-0.463166\pi$
$79$	18.2426			0.230920			0.115460	+	0.993312i	$0.463166\pi$
$80$			18.0064i			0.225080i
$81$		0			0
$82$			193.344i			2.35786i
$83$			70.2746i			0.846682i	0.905970	+	0.423341i	$0.139143\pi$
$83$			70.2746i			0.846682i	−0.905970	+	0.423341i	$0.860857\pi$
$84$		0			0
$85$	59.2721			0.697319
$86$	25.8131			0.300152
$87$		0			0
$88$	36.4558			0.414271
$89$			36.0836i			0.405434i	0.979237	+	0.202717i	$0.0649772\pi$
$89$			36.0836i			0.405434i	−0.979237	+	0.202717i	$0.935023\pi$
$90$		0			0
$91$	108.125	−	36.5683i	1.18819	−	0.401849i
$92$	−242.131			−2.63186
$93$		0			0
$94$		−	300.158i		−	3.19317i
$95$	−5.17756			−0.0545007
$96$		0			0
$97$			143.013i			1.47436i	0.675696	+	0.737180i	$0.263843\pi$
$97$			143.013i			1.47436i	−0.675696	+	0.737180i	$0.736157\pi$
$98$	−130.570	+	99.7252i	−1.33234	+	1.01760i
$99$		0			0
$100$	139.154			1.39154
$101$		−	153.232i		−	1.51714i	−0.651589	−	0.758572i	$-0.725897\pi$
$101$		−	153.232i		−	1.51714i	0.651589	−	0.758572i	$-0.274103\pi$
$102$		0			0
$103$			47.6780i			0.462893i	0.972848	+	0.231447i	$0.0743459\pi$
$103$			47.6780i			0.462893i	−0.972848	+	0.231447i	$0.925654\pi$
$104$		−	177.287i		−	1.70469i
$105$		0			0
$106$	−62.0000			−0.584906
$107$	−0.813575			−0.00760351			−0.00380175	−	0.999993i	$-0.501210\pi$
$107$	−0.813575			−0.00760351			−0.00380175	+	0.999993i	$0.501210\pi$
$108$		0			0
$109$	−135.184			−1.24022			−0.620109	−	0.784516i	$-0.712912\pi$
$109$	−135.184			−1.24022			−0.620109	+	0.784516i	$0.712912\pi$
$110$			27.0450i			0.245864i
$111$		0			0
$112$	16.7868	+	49.6351i	0.149882	+	0.443171i
$113$	−56.7050			−0.501814			−0.250907	−	0.968011i	$-0.580729\pi$
$113$	−56.7050			−0.501814			−0.250907	+	0.968011i	$0.580729\pi$
$114$		0			0
$115$		−	80.4216i		−	0.699319i
$116$	−297.308			−2.56300
$117$		0			0
$118$			46.4383i			0.393545i
$119$	163.385	−	55.2574i	1.37298	−	0.464348i
$120$		0			0
$121$	−109.757			−0.907086
$122$		−	82.9570i		−	0.679975i
$123$		0			0
$124$			238.870i			1.92637i
$125$			106.358i			0.850866i
$126$		0			0
$127$	63.5807			0.500636			0.250318	−	0.968164i	$-0.419465\pi$
$127$	63.5807			0.500636			0.250318	+	0.968164i	$0.419465\pi$
$128$	−233.601			−1.82501
$129$		0			0
$130$	131.522			1.01171
$131$			124.365i			0.949348i	0.880162	+	0.474674i	$0.157434\pi$
$131$			124.365i			0.949348i	−0.880162	+	0.474674i	$0.842566\pi$
$132$		0			0
$133$	−14.2721	+	4.82687i	−0.107309	+	0.0362923i
$134$	332.465			2.48108
$135$		0			0
$136$		−	267.895i		−	1.96981i
$137$	171.916			1.25486			0.627429	−	0.778674i	$-0.284107\pi$
$137$	171.916			1.25486			0.627429	+	0.778674i	$0.284107\pi$
$138$		0			0
$139$			139.795i			1.00572i	0.864368	+	0.502860i	$0.167719\pi$
$139$			139.795i			1.00572i	−0.864368	+	0.502860i	$0.832281\pi$
$140$	−115.531	+	39.0729i	−0.825219	+	0.279092i
$141$		0			0
$142$	217.000			1.52817
$143$		−	54.6737i		−	0.382334i
$144$		0			0
$145$		−	98.7480i		−	0.681021i
$146$		−	170.000i		−	1.16438i
$147$		0			0
$148$	271.492			1.83441
$149$	199.356			1.33796			0.668979	−	0.743281i	$-0.266732\pi$
$149$	199.356			1.33796			0.668979	+	0.743281i	$0.266732\pi$
$150$		0			0
$151$	62.6030			0.414590			0.207295	−	0.978279i	$-0.433534\pi$
$151$	62.6030			0.414590			0.207295	+	0.978279i	$0.433534\pi$
$152$			23.4013i			0.153956i
$153$		0			0
$154$	25.2132	+	74.5503i	0.163722	+	0.484093i
$155$	−79.3385			−0.511861
$156$		0			0
$157$			13.0880i			0.0833629i	0.999131	+	0.0416815i	$0.0132715\pi$
$157$			13.0880i			0.0833629i	−0.999131	+	0.0416815i	$0.986729\pi$
$158$	61.1677			0.387137
$159$		0			0
$160$		−	44.2438i		−	0.276524i
$161$	−74.9745	−	221.684i	−0.465680	−	1.37692i
$162$		0			0
$163$	139.948			0.858578			0.429289	−	0.903167i	$-0.358764\pi$
$163$	139.948			0.858578			0.429289	+	0.903167i	$0.358764\pi$
$164$			417.632i			2.54654i
$165$		0			0
$166$			235.631i			1.41946i
$167$			256.017i			1.53303i	0.642223	+	0.766517i	$0.278012\pi$
$167$			256.017i			1.53303i	−0.642223	+	0.766517i	$0.721988\pi$
$168$		0			0
$169$	−96.8823			−0.573268
$170$	198.740			1.16906
$171$		0			0
$172$	55.7574			0.324171
$173$			167.594i			0.968753i	0.874860	+	0.484376i	$0.160953\pi$
$173$			167.594i			0.968753i	−0.874860	+	0.484376i	$0.839047\pi$
$174$		0			0
$175$	43.0883	+	127.403i	0.246219	+	0.728019i
$176$	25.0982			0.142603
$177$		0			0
$178$			120.989i			0.679711i
$179$	102.636			0.573386			0.286693	−	0.958023i	$-0.407444\pi$
$179$	102.636			0.573386			0.286693	+	0.958023i	$0.407444\pi$
$180$		0			0
$181$		−	262.762i		−	1.45172i	−0.687841	−	0.725861i	$-0.741441\pi$
$181$		−	262.762i		−	1.45172i	0.687841	−	0.725861i	$-0.258559\pi$
$182$	362.543	−	122.614i	1.99200	−	0.673701i
$183$		0			0
$184$	−363.485			−1.97546
$185$			90.1737i			0.487425i
$186$		0			0
$187$		−	82.6162i		−	0.441798i
$188$		−	648.355i		−	3.44870i
$189$		0			0
$190$	−17.3604			−0.0913705
$191$	−217.353			−1.13797			−0.568987	−	0.822346i	$-0.692665\pi$
$191$	−217.353			−1.13797			−0.568987	+	0.822346i	$0.692665\pi$
$192$		0			0
$193$	−251.706			−1.30417			−0.652087	−	0.758144i	$-0.726106\pi$
$193$	−251.706			−1.30417			−0.652087	+	0.758144i	$0.726106\pi$
$194$			479.523i			2.47177i
$195$		0			0
$196$	−282.037	+	215.411i	−1.43896	+	1.09904i
$197$	197.926			1.00470			0.502350	−	0.864664i	$-0.332469\pi$
$197$	197.926			1.00470			0.502350	+	0.864664i	$0.332469\pi$
$198$		0			0
$199$		−	80.4639i		−	0.404341i	−0.979350	−	0.202171i	$-0.935200\pi$
$199$		−	80.4639i		−	0.404341i	0.979350	−	0.202171i	$-0.0647995\pi$
$200$	208.897			1.04449
$201$		0			0
$202$		−	513.786i		−	2.54350i
$203$	−92.0596	−	272.201i	−0.453495	−	1.34089i
$204$		0			0
$205$	−138.713			−0.676648
$206$			159.865i			0.776042i
$207$		0			0
$208$		−	122.054i		−	0.586799i
$209$			7.21673i			0.0345298i
$210$		0			0
$211$	−165.044			−0.782198			−0.391099	−	0.920349i	$-0.627905\pi$
$211$	−165.044			−0.782198			−0.391099	+	0.920349i	$0.627905\pi$
$212$	−133.923			−0.631711
$213$		0			0
$214$	−2.72792			−0.0127473
$215$			18.5193i			0.0861362i
$216$		0			0
$217$	−218.698	+	73.9647i	−1.00783	+	0.340851i
$218$	−453.272			−2.07923
$219$		0			0
$220$			58.4185i			0.265539i
$221$	−401.768			−1.81796
$222$		0			0
$223$		−	3.06496i		−	0.0137442i	−0.999976	−	0.00687210i	$-0.997813\pi$
$223$		−	3.06496i		−	0.0137442i	0.999976	−	0.00687210i	$-0.00218747\pi$
$224$	−41.2470	−	121.959i	−0.184139	−	0.544460i
$225$		0			0
$226$	−190.132			−0.841292
$227$		−	221.171i		−	0.974323i	−0.873312	−	0.487162i	$-0.838032\pi$
$227$		−	221.171i		−	0.974323i	0.873312	−	0.487162i	$-0.161968\pi$
$228$		0			0
$229$		−	195.713i		−	0.854642i	−0.904100	−	0.427321i	$-0.859457\pi$
$229$		−	195.713i		−	0.854642i	0.904100	−	0.427321i	$-0.140543\pi$
$230$		−	269.654i		−	1.17241i
$231$		0			0
$232$	−446.316			−1.92378
$233$	−281.455			−1.20796			−0.603981	−	0.796999i	$-0.706420\pi$
$233$	−281.455			−1.20796			−0.603981	+	0.796999i	$0.706420\pi$
$234$		0			0
$235$	215.345			0.916363
$236$			100.309i			0.425038i
$237$		0			0
$238$	547.831	−	185.278i	2.30181	−	0.778481i
$239$	35.9708			0.150505			0.0752527	−	0.997164i	$-0.476024\pi$
$239$	35.9708			0.150505			0.0752527	+	0.997164i	$0.476024\pi$
$240$		0			0
$241$			148.146i			0.614712i	0.951595	+	0.307356i	$0.0994443\pi$
$241$			148.146i			0.614712i	−0.951595	+	0.307356i	$0.900556\pi$
$242$	−368.017			−1.52073
$243$		0			0
$244$		−	179.191i		−	0.734388i
$245$	−71.5467	−	93.6758i	−0.292027	−	0.382350i
$246$		0			0
$247$	35.0955			0.142087
$248$			358.590i			1.44593i
$249$		0			0
$250$			356.620i			1.42648i
$251$			73.1930i			0.291606i	0.989314	+	0.145803i	$0.0465765\pi$
$251$			73.1930i			0.291606i	−0.989314	+	0.145803i	$0.953423\pi$
$252$		0			0
$253$	−112.095			−0.443065
$254$	213.187			0.839317
$255$		0			0
$256$	−416.823			−1.62822
$257$		−	82.4441i		−	0.320794i	−0.987053	−	0.160397i	$-0.948723\pi$
$257$		−	82.4441i		−	0.320794i	0.987053	−	0.160397i	$-0.0512775\pi$
$258$		0			0
$259$	84.0660	+	248.566i	0.324579	+	0.959714i
$260$	284.093			1.09267
$261$		0			0
$262$			416.995i			1.59158i
$263$	3.35300			0.0127491			0.00637453	−	0.999980i	$-0.497971\pi$
$263$	3.35300			0.0127491			0.00637453	+	0.999980i	$0.497971\pi$
$264$		0			0
$265$		−	44.4812i		−	0.167854i
$266$	−47.8543	+	16.1845i	−0.179904	+	0.0608441i
$267$		0			0
$268$	718.139			2.67962
$269$		−	177.217i		−	0.658797i	−0.944191	−	0.329399i	$-0.893154\pi$
$269$		−	177.217i		−	0.658797i	0.944191	−	0.329399i	$-0.106846\pi$
$270$		0			0
$271$		−	387.775i		−	1.43091i	−0.698661	−	0.715453i	$-0.746220\pi$
$271$		−	387.775i		−	1.43091i	0.698661	−	0.715453i	$-0.253780\pi$
$272$		−	184.433i		−	0.678063i
$273$		0			0
$274$	576.434			2.10377
$275$	64.4220			0.234262
$276$		0			0
$277$	−301.500			−1.08845			−0.544223	−	0.838941i	$-0.683175\pi$
$277$	−301.500			−1.08845			−0.544223	+	0.838941i	$0.683175\pi$
$278$			468.733i			1.68609i
$279$		0			0
$280$	−173.434	+	58.6559i	−0.619406	+	0.209485i
$281$	−335.917			−1.19543			−0.597716	−	0.801708i	$-0.703925\pi$
$281$	−335.917			−1.19543			−0.597716	+	0.801708i	$0.703925\pi$
$282$		0			0
$283$			220.106i			0.777759i	0.921289	+	0.388880i	$0.127138\pi$
$283$			220.106i			0.777759i	−0.921289	+	0.388880i	$0.872862\pi$
$284$	468.730			1.65046
$285$		0			0
$286$		−	183.321i		−	0.640984i
$287$	−382.365	+	129.317i	−1.33228	+	0.450583i
$288$		0			0
$289$	−318.103			−1.10070
$290$		−	331.103i		−	1.14173i
$291$		0			0
$292$		−	367.207i		−	1.25756i
$293$		−	148.420i		−	0.506554i	−0.967394	−	0.253277i	$-0.918492\pi$
$293$		−	148.420i		−	0.506554i	0.967394	−	0.253277i	$-0.0815085\pi$
$294$		0			0
$295$	−33.3167			−0.112938
$296$	407.562			1.37690
$297$		0			0
$298$	668.441			2.24309
$299$			545.128i			1.82317i
$300$		0			0
$301$	17.2649	+	51.0489i	0.0573586	+	0.169598i
$302$	209.908			0.695060
$303$		0			0
$304$			16.1107i			0.0529957i
$305$	59.5166			0.195136
$306$		0			0
$307$			380.274i			1.23868i	0.785124	+	0.619339i	$0.212599\pi$
$307$			380.274i			1.23868i	−0.785124	+	0.619339i	$0.787401\pi$
$308$	54.4617	+	161.032i	0.176824	+	0.522831i
$309$		0			0
$310$	−266.022			−0.858136
$311$		−	540.175i		−	1.73690i	−0.495779	−	0.868449i	$-0.665117\pi$
$311$		−	540.175i		−	1.73690i	0.495779	−	0.868449i	$-0.334883\pi$
$312$		0			0
$313$		−	296.508i		−	0.947309i	−0.880711	−	0.473655i	$-0.842935\pi$
$313$		−	296.508i		−	0.947309i	0.880711	−	0.473655i	$-0.157065\pi$
$314$			43.8841i			0.139758i
$315$		0			0
$316$	132.125			0.418117
$317$	219.300			0.691799			0.345900	−	0.938272i	$-0.387574\pi$
$317$	219.300			0.691799			0.345900	+	0.938272i	$0.387574\pi$
$318$		0			0
$319$	−137.640			−0.431472
$320$		−	220.375i		−	0.688673i
$321$		0			0
$322$	−251.390	−	743.308i	−0.780714	−	2.30841i
$323$	53.0319			0.164185
$324$		0			0
$325$		−	313.289i		−	0.963965i
$326$	469.247			1.43941
$327$		0			0
$328$			626.946i			1.91142i
$329$	593.604	−	200.759i	1.80427	−	0.610211i
$330$		0			0
$331$	296.853			0.896836			0.448418	−	0.893824i	$-0.351988\pi$
$331$	296.853			0.896836			0.448418	+	0.893824i	$0.351988\pi$
$332$			508.974i			1.53305i
$333$		0			0
$334$			858.426i			2.57014i
$335$			238.523i			0.712010i
$336$		0			0
$337$	−82.7645			−0.245592			−0.122796	−	0.992432i	$-0.539186\pi$
$337$	−82.7645			−0.245592			−0.122796	+	0.992432i	$0.539186\pi$
$338$	−324.847			−0.961085
$339$		0			0
$340$	429.286			1.26261
$341$			110.586i			0.324298i
$342$		0			0
$343$	−284.551	−	191.519i	−0.829596	−	0.558365i
$344$	83.7025			0.243321
$345$		0			0
$346$			561.944i			1.62412i
$347$	−75.2706			−0.216918			−0.108459	−	0.994101i	$-0.534592\pi$
$347$	−75.2706			−0.216918			−0.108459	+	0.994101i	$0.534592\pi$
$348$		0			0
$349$			89.5743i			0.256660i	0.991732	+	0.128330i	$0.0409616\pi$
$349$			89.5743i			0.256660i	−0.991732	+	0.128330i	$0.959038\pi$
$350$	144.475	+	427.184i	0.412787	+	1.22053i
$351$		0			0
$352$	−61.6690			−0.175196
$353$			249.967i			0.708123i	0.935222	+	0.354062i	$0.115200\pi$
$353$			249.967i			0.708123i	−0.935222	+	0.354062i	$0.884800\pi$
$354$		0			0
$355$			155.684i			0.438547i
$356$			261.341i			0.734104i
$357$		0			0
$358$	344.139			0.961283
$359$	109.539			0.305124			0.152562	−	0.988294i	$-0.451248\pi$
$359$	109.539			0.305124			0.152562	+	0.988294i	$0.451248\pi$
$360$		0			0
$361$	356.368			0.987168
$362$		−	881.042i		−	2.43382i
$363$		0			0
$364$	783.110	−	264.851i	2.15140	−	0.727612i
$365$	121.964			0.334149
$366$		0			0
$367$		−	425.277i		−	1.15879i	−0.815045	−	0.579397i	$-0.803288\pi$
$367$		−	425.277i		−	1.15879i	0.815045	−	0.579397i	$-0.196712\pi$
$368$	−250.243			−0.680008
$369$		0			0
$370$			302.353i			0.817170i
$371$	−41.4684	−	122.614i	−0.111775	−	0.330495i
$372$		0			0
$373$	161.624			0.433310			0.216655	−	0.976248i	$-0.430485\pi$
$373$	161.624			0.433310			0.216655	+	0.976248i	$0.430485\pi$
$374$		−	277.013i		−	0.740675i
$375$		0			0
$376$		−	973.305i		−	2.58858i
$377$			669.351i			1.77547i
$378$		0			0
$379$	591.227			1.55997			0.779983	−	0.625800i	$-0.215227\pi$
$379$	591.227			1.55997			0.779983	+	0.625800i	$0.215227\pi$
$380$	−37.4992			−0.0986822
$381$		0			0
$382$	−728.786			−1.90782
$383$		−	480.690i		−	1.25507i	−0.778590	−	0.627533i	$-0.784065\pi$
$383$		−	480.690i		−	1.25507i	0.778590	−	0.627533i	$-0.215935\pi$
$384$		0			0
$385$	−53.4853	+	18.0889i	−0.138923	+	0.0469842i
$386$	−843.970			−2.18645
$387$		0			0
$388$			1035.79i			2.66956i
$389$	−414.786			−1.06629			−0.533143	−	0.846025i	$-0.678989\pi$
$389$	−414.786			−1.06629			−0.533143	+	0.846025i	$0.678989\pi$
$390$		0			0
$391$			823.730i			2.10673i
$392$	−423.391	+	323.373i	−1.08008	+	0.824931i
$393$		0			0
$394$	663.647			1.68438
$395$			43.8841i			0.111099i
$396$		0			0
$397$		−	34.0044i		−	0.0856535i	−0.999083	−	0.0428267i	$-0.986364\pi$
$397$		−	34.0044i		−	0.0856535i	0.999083	−	0.0428267i	$-0.0136363\pi$
$398$		−	269.796i		−	0.677879i
$399$		0			0
$400$	143.816			0.359541
$401$	−14.0282			−0.0349830			−0.0174915	−	0.999847i	$-0.505568\pi$
$401$	−14.0282			−0.0349830			−0.0174915	+	0.999847i	$0.505568\pi$
$402$		0			0
$403$	537.786			1.33446
$404$		−	1109.80i		−	2.74703i
$405$		0			0
$406$	−308.676	−	912.692i	−0.760286	−	2.24801i
$407$	125.688			0.308817
$408$		0			0
$409$		−	550.439i		−	1.34582i	−0.739726	−	0.672908i	$-0.765045\pi$
$409$		−	550.439i		−	1.34582i	0.739726	−	0.672908i	$-0.234955\pi$
$410$	−465.105			−1.13440
$411$		0			0
$412$			345.315i			0.838142i
$413$	−91.8382	+	31.0600i	−0.222369	+	0.0752059i
$414$		0			0
$415$	−169.051			−0.407351
$416$			299.901i			0.720916i
$417$		0			0
$418$			24.1977i			0.0578893i
$419$			224.674i			0.536214i	0.963389	+	0.268107i	$0.0863980\pi$
$419$			224.674i			0.536214i	−0.963389	+	0.268107i	$0.913602\pi$
$420$		0			0
$421$	−744.044			−1.76732			−0.883662	−	0.468125i	$-0.844930\pi$
$421$	−744.044			−1.76732			−0.883662	+	0.468125i	$0.844930\pi$
$422$	−553.392			−1.31136
$423$		0			0
$424$	−201.044			−0.474160
$425$		−	473.403i		−	1.11389i
$426$		0			0
$427$	164.059	−	55.4854i	0.384213	−	0.129942i
$428$	−5.89243			−0.0137674
$429$		0			0
$430$			62.0953i			0.144408i
$431$	668.678			1.55146			0.775728	−	0.631067i	$-0.217383\pi$
$431$	668.678			1.55146			0.775728	+	0.631067i	$0.217383\pi$
$432$		0			0
$433$			310.941i			0.718109i	0.933317	+	0.359054i	$0.116901\pi$
$433$			310.941i			0.718109i	−0.933317	+	0.359054i	$0.883099\pi$
$434$	−733.297	+	248.004i	−1.68962	+	0.571438i
$435$		0			0
$436$	−979.087			−2.24561
$437$		−	71.9548i		−	0.164656i
$438$		0			0
$439$		−	79.5301i		−	0.181162i	−0.995889	−	0.0905810i	$-0.971128\pi$
$439$		−	79.5301i		−	0.181162i	0.995889	−	0.0905810i	$-0.0288724\pi$
$440$			87.6973i			0.199312i
$441$		0			0
$442$	−1347.13			−3.04781
$443$	293.931			0.663501			0.331750	−	0.943367i	$-0.392361\pi$
$443$	293.931			0.663501			0.331750	+	0.943367i	$0.392361\pi$
$444$		0			0
$445$	−86.8019			−0.195061
$446$		−	10.2768i		−	0.0230422i
$447$		0			0
$448$	−205.449	−	607.470i	−0.458591	−	1.35596i
$449$	277.214			0.617402			0.308701	−	0.951159i	$-0.400106\pi$
$449$	277.214			0.617402			0.308701	+	0.951159i	$0.400106\pi$
$450$		0			0
$451$			193.344i			0.428701i
$452$	−410.694			−0.908614
$453$		0			0
$454$		−	741.589i		−	1.63346i
$455$	87.9677	+	260.103i	0.193336	+	0.571654i
$456$		0			0
$457$	−264.220			−0.578163			−0.289081	−	0.957305i	$-0.593350\pi$
$457$	−264.220			−0.578163			−0.289081	+	0.957305i	$0.593350\pi$
$458$		−	656.227i		−	1.43281i
$459$		0			0
$460$		−	582.465i		−	1.26623i
$461$			739.555i			1.60424i	0.597163	+	0.802120i	$0.296295\pi$
$461$			739.555i			1.60424i	−0.597163	+	0.802120i	$0.703705\pi$
$462$		0			0
$463$	581.691			1.25635			0.628176	−	0.778071i	$-0.283802\pi$
$463$	581.691			1.25635			0.628176	+	0.778071i	$0.283802\pi$
$464$	−307.268			−0.662216
$465$		0			0
$466$	−943.720			−2.02515
$467$		−	347.729i		−	0.744602i	−0.928112	−	0.372301i	$-0.878569\pi$
$467$		−	347.729i		−	0.744602i	0.928112	−	0.372301i	$-0.121431\pi$
$468$		0			0
$469$	222.368	+	657.495i	0.474131	+	1.40191i
$470$	722.054			1.53628
$471$		0			0
$472$			150.583i			0.319031i
$473$	25.8131			0.0545731
$474$		0			0
$475$			41.3529i			0.0870587i
$476$	1183.34	−	400.210i	2.48601	−	0.840777i
$477$		0			0
$478$	120.610			0.252323
$479$			195.948i			0.409078i	0.978858	+	0.204539i	$0.0655695\pi$
$479$			195.948i			0.409078i	−0.978858	+	0.204539i	$0.934431\pi$
$480$		0			0
$481$		−	611.231i		−	1.27075i
$482$			496.733i			1.03057i
$483$		0			0
$484$	−794.933			−1.64242
$485$	−344.028			−0.709337
$486$		0			0
$487$	−511.889			−1.05111			−0.525554	−	0.850760i	$-0.676142\pi$
$487$	−511.889			−1.05111			−0.525554	+	0.850760i	$0.676142\pi$
$488$		−	269.000i		−	0.551229i
$489$		0			0
$490$	−239.897	−	314.096i	−0.489585	−	0.641011i
$491$	9.24544			0.0188298			0.00941491	−	0.999956i	$-0.497003\pi$
$491$	9.24544			0.0188298			0.00941491	+	0.999956i	$0.497003\pi$
$492$		0			0
$493$			1011.44i			2.05160i
$494$	117.675			0.238209
$495$		0			0
$496$			246.873i			0.497727i
$497$	145.139	+	429.147i	0.292031	+	0.863476i
$498$		0			0
$499$	−19.1400			−0.0383568			−0.0191784	−	0.999816i	$-0.506105\pi$
$499$	−19.1400			−0.0383568			−0.0191784	+	0.999816i	$0.506105\pi$
$500$			770.314i			1.54063i
$501$		0			0
$502$			245.417i			0.488878i
$503$		−	437.832i		−	0.870441i	−0.900324	−	0.435221i	$-0.856670\pi$
$503$		−	437.832i		−	0.870441i	0.900324	−	0.435221i	$-0.143330\pi$
$504$		0			0
$505$	368.610			0.729921
$506$	−375.857			−0.742800
$507$		0			0
$508$	460.492			0.906481
$509$			47.6695i			0.0936532i	0.998903	+	0.0468266i	$0.0149108\pi$
$509$			47.6695i			0.0936532i	−0.998903	+	0.0468266i	$0.985089\pi$
$510$		0			0
$511$	336.198	−	113.704i	0.657922	−	0.222512i
$512$	−463.208			−0.904703
$513$		0			0
$514$		−	276.435i		−	0.537812i
$515$	−114.693			−0.222705
$516$		0			0
$517$		−	300.158i		−	0.580577i
$518$	281.874	+	833.443i	0.544158	+	1.60896i
$519$		0			0
$520$	426.478			0.820150
$521$		−	588.358i		−	1.12929i	−0.825336	−	0.564643i	$-0.809014\pi$
$521$		−	588.358i		−	1.12929i	0.825336	−	0.564643i	$-0.190986\pi$
$522$		0			0
$523$		−	521.551i		−	0.997230i	−0.866824	−	0.498615i	$-0.833842\pi$
$523$		−	521.551i		−	0.997230i	0.866824	−	0.498615i	$-0.166158\pi$
$524$			900.728i			1.71895i
$525$		0			0
$526$	11.2426			0.0213738
$527$	812.636			1.54200
$528$		0			0
$529$	588.655			1.11277
$530$		−	149.146i		−	0.281407i
$531$		0			0
$532$	−103.368	+	34.9593i	−0.194300	+	0.0657130i
$533$	940.247			1.76407
$534$		0			0
$535$		−	1.95712i		−	0.00365816i
$536$	1078.06			2.01131
$537$		0			0
$538$		−	594.208i		−	1.10448i
$539$	−130.570	+	99.7252i	−0.242244	+	0.185019i
$540$		0			0
$541$	342.955			0.633929			0.316964	−	0.948437i	$-0.397336\pi$
$541$	342.955			0.633929			0.316964	+	0.948437i	$0.397336\pi$
$542$		−	1300.21i		−	2.39892i
$543$		0			0
$544$			453.173i			0.833039i
$545$		−	325.195i		−	0.596688i
$546$		0			0
$547$	674.823			1.23368			0.616840	−	0.787089i	$-0.288413\pi$
$547$	674.823			1.23368			0.616840	+	0.787089i	$0.288413\pi$
$548$	1245.12			2.27212
$549$		0			0
$550$	216.007			0.392740
$551$		−	88.3518i		−	0.160348i
$552$		0			0
$553$	40.9117	+	120.968i	0.0739814	+	0.218748i
$554$	−1010.93			−1.82478
$555$		0			0
$556$			1012.48i			1.82102i
$557$	957.530			1.71908			0.859542	−	0.511065i	$-0.170749\pi$
$557$	957.530			1.71908			0.859542	+	0.511065i	$0.170749\pi$
$558$		0			0
$559$		−	125.531i		−	0.224563i
$560$	−119.401	+	40.3819i	−0.213216	+	0.0721106i
$561$		0			0
$562$	−1126.33			−2.00415
$563$			651.999i			1.15808i	0.815299	+	0.579040i	$0.196573\pi$
$563$			651.999i			1.15808i	−0.815299	+	0.579040i	$0.803427\pi$
$564$		0			0
$565$		−	136.408i		−	0.241430i
$566$			738.016i			1.30392i
$567$		0			0
$568$	703.653			1.23883
$569$	−156.111			−0.274360			−0.137180	−	0.990546i	$-0.543804\pi$
$569$	−156.111			−0.274360			−0.137180	+	0.990546i	$0.543804\pi$
$570$		0			0
$571$	144.294			0.252705			0.126352	−	0.991985i	$-0.459673\pi$
$571$	144.294			0.252705			0.126352	+	0.991985i	$0.459673\pi$
$572$		−	395.982i		−	0.692277i
$573$		0			0
$574$	−1282.07	+	433.602i	−2.23358	+	0.755404i
$575$	−642.323			−1.11708
$576$		0			0
$577$			575.270i			0.997001i	0.866889	+	0.498500i	$0.166116\pi$
$577$			575.270i			0.997001i	−0.866889	+	0.498500i	$0.833884\pi$
$578$	−1066.60			−1.84533
$579$		0			0
$580$		−	715.196i		−	1.23310i
$581$	−465.993	+	157.601i	−0.802053	+	0.271258i
$582$		0			0
$583$	−62.0000			−0.106346
$584$		−	551.248i		−	0.943918i
$585$		0			0
$586$		−	497.654i		−	0.849239i
$587$		−	734.372i		−	1.25106i	−0.780200	−	0.625530i	$-0.784883\pi$
$587$		−	734.372i		−	1.25106i	0.780200	−	0.625530i	$-0.215117\pi$
$588$		0			0
$589$	−70.9857			−0.120519
$590$	−111.711			−0.189341
$591$		0			0
$592$	280.588			0.473966
$593$			434.542i			0.732786i	0.930460	+	0.366393i	$0.119407\pi$
$593$			434.542i			0.732786i	−0.930460	+	0.366393i	$0.880593\pi$
$594$		0			0
$595$	132.926	+	393.035i	0.223405	+	0.660563i
$596$	1443.86			2.42259
$597$		0			0
$598$			1827.82i			3.05655i
$599$	−806.842			−1.34698			−0.673491	−	0.739196i	$-0.735206\pi$
$599$	−806.842			−1.34698			−0.673491	+	0.739196i	$0.735206\pi$
$600$		0			0
$601$		−	189.757i		−	0.315736i	−0.987460	−	0.157868i	$-0.949538\pi$
$601$		−	189.757i		−	0.315736i	0.987460	−	0.157868i	$-0.0504620\pi$
$602$	57.8894	+	171.167i	0.0961618	+	0.284331i
$603$		0			0
$604$	453.411			0.750681
$605$		−	264.030i		−	0.436413i
$606$		0			0
$607$		−	220.802i		−	0.363760i	−0.983321	−	0.181880i	$-0.941782\pi$
$607$		−	220.802i		−	0.363760i	0.983321	−	0.181880i	$-0.0582182\pi$
$608$		−	39.5858i		−	0.0651082i
$609$		0			0
$610$	199.559			0.327146
$611$	−1459.69			−2.38902
$612$		0			0
$613$	−787.801			−1.28516			−0.642578	−	0.766220i	$-0.722135\pi$
$613$	−787.801			−1.28516			−0.642578	+	0.766220i	$0.722135\pi$
$614$			1275.06i			2.07665i
$615$		0			0
$616$	81.7574	+	241.740i	0.132723	+	0.392435i
$617$	−853.757			−1.38372			−0.691861	−	0.722030i	$-0.743209\pi$
$617$	−853.757			−1.38372			−0.691861	+	0.722030i	$0.743209\pi$
$618$		0			0
$619$		−	1025.51i		−	1.65672i	−0.560196	−	0.828360i	$-0.689274\pi$
$619$		−	1025.51i		−	1.65672i	0.560196	−	0.828360i	$-0.310726\pi$
$620$	−574.620			−0.926807
$621$		0			0
$622$		−	1811.21i		−	2.91191i
$623$	−239.272	+	80.9226i	−0.384064	+	0.129892i
$624$		0			0
$625$	224.477			0.359164
$626$		−	994.192i		−	1.58817i
$627$		0			0
$628$			94.7915i			0.150942i
$629$		−	923.617i		−	1.46839i
$630$		0			0
$631$	−777.492			−1.23216			−0.616079	−	0.787684i	$-0.711280\pi$
$631$	−777.492			−1.23216			−0.616079	+	0.787684i	$0.711280\pi$
$632$	198.345			0.313837
$633$		0			0
$634$	735.315			1.15980
$635$			152.948i			0.240863i
$636$		0			0
$637$	484.971	+	634.970i	0.761335	+	0.996813i
$638$	−461.506			−0.723364
$639$		0			0
$640$		−	561.944i		−	0.878038i
$641$	−867.492			−1.35334			−0.676671	−	0.736286i	$-0.736578\pi$
$641$	−867.492			−1.35334			−0.676671	+	0.736286i	$0.736578\pi$
$642$		0			0
$643$			411.841i			0.640500i	0.947333	+	0.320250i	$0.103767\pi$
$643$			411.841i			0.640500i	−0.947333	+	0.320250i	$0.896233\pi$
$644$	−543.013	−	1605.58i	−0.843188	−	2.49314i
$645$		0			0
$646$	177.816			0.275257
$647$			1037.63i			1.60376i	0.597484	+	0.801881i	$0.296167\pi$
$647$			1037.63i			1.60376i	−0.597484	+	0.801881i	$0.703833\pi$
$648$		0			0
$649$			46.4383i			0.0715537i
$650$		−	1050.46i		−	1.61609i
$651$		0			0
$652$	1013.60			1.55459
$653$	204.901			0.313784			0.156892	−	0.987616i	$-0.449852\pi$
$653$	204.901			0.313784			0.156892	+	0.987616i	$0.449852\pi$
$654$		0			0
$655$	−299.169			−0.456746
$656$			431.624i			0.657963i
$657$		0			0
$658$	1990.36	−	673.147i	3.02486	−	1.02302i
$659$	30.1291			0.0457195			0.0228597	−	0.999739i	$-0.492723\pi$
$659$	30.1291			0.0457195			0.0228597	+	0.999739i	$0.492723\pi$
$660$		0			0
$661$		−	210.194i		−	0.317993i	−0.987279	−	0.158997i	$-0.949174\pi$
$661$		−	210.194i		−	0.317993i	0.987279	−	0.158997i	$-0.0508259\pi$
$662$	995.349			1.50355
$663$		0			0
$664$			764.067i			1.15070i
$665$	−11.6114	−	34.3326i	−0.0174608	−	0.0516279i
$666$		0			0
$667$	1372.34			2.05749
$668$			1854.24i			2.77581i
$669$		0			0
$670$			799.770i			1.19369i
$671$		−	82.9570i		−	0.123632i
$672$		0			0
$673$	−375.574			−0.558059			−0.279029	−	0.960283i	$-0.590013\pi$
$673$	−375.574			−0.558059			−0.279029	+	0.960283i	$0.590013\pi$
$674$	−277.510			−0.411736
$675$		0			0
$676$	−701.683			−1.03799
$677$		−	432.933i		−	0.639487i	−0.947504	−	0.319744i	$-0.896403\pi$
$677$		−	432.933i		−	0.639487i	0.947504	−	0.319744i	$-0.103597\pi$
$678$		0			0
$679$	−948.323	+	320.727i	−1.39665	+	0.472351i
$680$	644.441			0.947707
$681$		0			0
$682$			370.794i			0.543687i
$683$	1073.06			1.57110			0.785549	−	0.618799i	$-0.212381\pi$
$683$	1073.06			1.57110			0.785549	+	0.618799i	$0.212381\pi$
$684$		0			0
$685$			413.556i			0.603731i
$686$	−954.102	−	642.164i	−1.39082	−	0.936100i
$687$		0			0
$688$	57.6253			0.0837577
$689$			301.510i			0.437606i
$690$		0			0
$691$			132.315i			0.191483i	0.995406	+	0.0957415i	$0.0305222\pi$
$691$			132.315i			0.191483i	−0.995406	+	0.0957415i	$0.969478\pi$
$692$			1213.82i			1.75408i
$693$		0			0
$694$	−252.383			−0.363664
$695$	−336.288			−0.483867
$696$		0			0
$697$	1420.79			2.03843
$698$			300.343i			0.430291i
$699$		0			0
$700$	312.073	+	922.737i	0.445819	+	1.31820i
$701$	−1079.34			−1.53972			−0.769861	−	0.638212i	$-0.779674\pi$
$701$	−1079.34			−1.53972			−0.769861	+	0.638212i	$0.779674\pi$
$702$		0			0
$703$			80.6802i			0.114766i
$704$	−307.169			−0.436320
$705$		0			0
$706$			838.142i			1.18717i
$707$	1016.08	−	343.643i	1.43718	−	0.486058i
$708$		0			0
$709$	254.317			0.358698			0.179349	−	0.983786i	$-0.442601\pi$
$709$	254.317			0.358698			0.179349	+	0.983786i	$0.442601\pi$
$710$			522.010i			0.735225i
$711$		0			0
$712$			392.323i			0.551015i
$713$		−	1102.60i		−	1.54643i
$714$		0			0
$715$	131.522			0.183947
$716$	743.356			1.03821
$717$		0			0
$718$	367.286			0.511541
$719$		−	37.0214i		−	0.0514901i	−0.999669	−	0.0257451i	$-0.991804\pi$
$719$		−	37.0214i		−	0.0514901i	0.999669	−	0.0257451i	$-0.00819581\pi$
$720$		0			0
$721$	−316.154	+	106.925i	−0.438494	+	0.148300i
$722$	1194.90			1.65499
$723$		0			0
$724$		−	1903.09i		−	2.62858i
$725$	−788.695			−1.08786
$726$		0			0
$727$			254.680i			0.350316i	0.984540	+	0.175158i	$0.0560437\pi$
$727$			254.680i			0.350316i	−0.984540	+	0.175158i	$0.943956\pi$
$728$	1175.60	−	397.592i	1.61483	−	0.546143i
$729$		0			0
$730$	408.947			0.560202
$731$		−	189.686i		−	0.259489i
$732$		0			0
$733$		−	1287.25i		−	1.75614i	−0.478535	−	0.878068i	$-0.658832\pi$
$733$		−	1287.25i		−	1.75614i	0.478535	−	0.878068i	$-0.341168\pi$
$734$		−	1425.96i		−	1.94272i
$735$		0			0
$736$	614.875			0.835428
$737$	332.465			0.451106
$738$		0			0
$739$	−820.073			−1.10971			−0.554853	−	0.831948i	$-0.687226\pi$
$739$	−820.073			−1.10971			−0.554853	+	0.831948i	$0.687226\pi$
$740$			653.096i			0.882562i
$741$		0			0
$742$	−139.044	−	411.124i	−0.187390	−	0.554075i
$743$	−331.331			−0.445937			−0.222969	−	0.974826i	$-0.571575\pi$
$743$	−331.331			−0.445937			−0.222969	+	0.974826i	$0.571575\pi$
$744$		0			0
$745$			479.565i			0.643712i
$746$	541.928			0.726444
$747$		0			0
$748$		−	598.359i		−	0.799946i
$749$	−1.82456	−	5.39484i	−0.00243599	−	0.00720273i
$750$		0			0
$751$	−1159.26			−1.54362			−0.771808	−	0.635855i	$-0.780648\pi$
$751$	−1159.26			−1.54362			−0.771808	+	0.635855i	$0.780648\pi$
$752$		−	670.076i		−	0.891059i
$753$		0			0
$754$			2244.34i			2.97657i
$755$			150.596i			0.199465i
$756$		0			0
$757$	501.632			0.662658			0.331329	−	0.943515i	$-0.392503\pi$
$757$	501.632			0.662658			0.331329	+	0.943515i	$0.392503\pi$
$758$	1982.39			2.61529
$759$		0			0
$760$	−56.2935			−0.0740704
$761$			434.472i			0.570922i	0.958390	+	0.285461i	$0.0921467\pi$
$761$			434.472i			0.570922i	−0.958390	+	0.285461i	$0.907853\pi$
$762$		0			0
$763$	−303.169	−	896.408i	−0.397338	−	1.17485i
$764$	−1574.21			−2.06048
$765$		0			0
$766$		−	1611.76i		−	2.10412i
$767$	225.833			0.294437
$768$		0			0
$769$		−	1388.65i		−	1.80579i	−0.429865	−	0.902893i	$-0.641439\pi$
$769$		−	1388.65i		−	1.80579i	0.429865	−	0.902893i	$-0.358561\pi$
$770$	−179.336	+	60.6523i	−0.232904	+	0.0787692i
$771$		0			0
$772$	−1823.01			−2.36142
$773$			1266.61i			1.63856i	0.573395	+	0.819279i	$0.305626\pi$
$773$			1266.61i			1.63856i	−0.573395	+	0.819279i	$0.694374\pi$
$774$		0			0
$775$			633.672i			0.817641i
$776$			1554.92i			2.00376i
$777$		0			0
$778$	−1390.78			−1.78763
$779$	−124.109			−0.159318
$780$		0			0
$781$	217.000			0.277849
$782$			2761.97i			3.53193i
$783$		0			0
$784$	−291.485	+	222.628i	−0.371792	+	0.283964i
$785$	−31.4841			−0.0401072
$786$		0			0
$787$		−	64.0524i		−	0.0813880i	−0.999172	−	0.0406940i	$-0.987043\pi$
$787$		−	64.0524i		−	0.0813880i	0.999172	−	0.0406940i	$-0.0129569\pi$
$788$	1433.51			1.81917
$789$		0			0
$790$			147.143i			0.186258i
$791$	−127.169	−	376.012i	−0.160770	−	0.475363i
$792$		0			0
$793$	−403.426			−0.508733
$794$		−	114.017i		−	0.143598i
$795$		0			0
$796$		−	582.771i		−	0.732124i
$797$		−	223.152i		−	0.279990i	−0.990152	−	0.139995i	$-0.955291\pi$
$797$		−	223.152i		−	0.279990i	0.990152	−	0.139995i	$-0.0447087\pi$
$798$		0			0
$799$	−2205.70			−2.76058
$800$	−353.373			−0.441716
$801$		0			0
$802$	−47.0366			−0.0586491
$803$		−	170.000i		−	0.211706i
$804$		0			0
$805$	533.278	−	180.357i	0.662458	−	0.224046i
$806$	1803.20			2.23722
$807$		0			0
$808$		−	1666.02i		−	2.06191i
$809$	673.436			0.832431			0.416215	−	0.909266i	$-0.363356\pi$
$809$	673.436			0.832431			0.416215	+	0.909266i	$0.363356\pi$
$810$		0			0
$811$			1588.83i			1.95909i	0.201216	+	0.979547i	$0.435511\pi$
$811$			1588.83i			1.95909i	−0.201216	+	0.979547i	$0.564489\pi$
$812$	−666.754	−	1971.46i	−0.821126	−	2.42790i
$813$		0			0
$814$	421.434			0.517732
$815$			336.656i			0.413075i
$816$		0			0
$817$			16.5696i			0.0202810i
$818$		−	1845.62i		−	2.25626i
$819$		0			0
$820$	−1004.65			−1.22518
$821$	896.165			1.09155			0.545776	−	0.837931i	$-0.316235\pi$
$821$	896.165			1.09155			0.545776	+	0.837931i	$0.316235\pi$
$822$		0			0
$823$	−717.367			−0.871648			−0.435824	−	0.900032i	$-0.643543\pi$
$823$	−717.367			−0.871648			−0.435824	+	0.900032i	$0.643543\pi$
$824$			518.383i			0.629106i
$825$		0			0
$826$	−307.934	+	104.144i	−0.372801	+	0.126083i
$827$	1049.32			1.26882			0.634412	−	0.772995i	$-0.281242\pi$
$827$	1049.32			1.26882			0.634412	+	0.772995i	$0.281242\pi$
$828$		0			0
$829$		−	778.289i		−	0.938828i	−0.882978	−	0.469414i	$-0.844465\pi$
$829$		−	778.289i		−	0.938828i	0.882978	−	0.469414i	$-0.155535\pi$
$830$	−566.828			−0.682926
$831$		0			0
$832$			1493.79i			1.79542i
$833$	732.827	+	959.488i	0.879745	+	1.15185i
$834$		0			0
$835$	−615.868			−0.737566
$836$			52.2682i			0.0625217i
$837$		0			0
$838$			753.331i			0.898963i
$839$		−	709.733i		−	0.845927i	−0.906147	−	0.422964i	$-0.860990\pi$
$839$		−	709.733i		−	0.845927i	0.906147	−	0.422964i	$-0.139010\pi$
$840$		0			0
$841$	844.072			1.00365
$842$	−2494.78			−2.96292
$843$		0			0
$844$	−1195.35			−1.41629
$845$		−	233.058i		−	0.275808i
$846$		0			0
$847$	−246.146	−	727.804i	−0.290610	−	0.859273i
$848$	−138.409			−0.163219
$849$		0			0
$850$		−	1587.32i		−	1.86744i
$851$	−1253.18			−1.47260
$852$		0			0
$853$			884.992i			1.03751i	0.854924	+	0.518753i	$0.173603\pi$
$853$			884.992i			1.03751i	−0.854924	+	0.518753i	$0.826397\pi$
$854$	550.090	−	186.043i	0.644134	−	0.217849i
$855$		0			0
$856$	−8.84567			−0.0103337
$857$		−	55.1866i		−	0.0643951i	−0.999482	−	0.0321976i	$-0.989749\pi$
$857$		−	55.1866i		−	0.0643951i	0.999482	−	0.0321976i	$-0.0102506\pi$
$858$		0			0
$859$			724.528i			0.843455i	0.906723	+	0.421728i	$0.138576\pi$
$859$			724.528i			0.843455i	−0.906723	+	0.421728i	$0.861424\pi$
$860$			134.129i			0.155963i
$861$		0			0
$862$	2242.08			2.60102
$863$	−623.288			−0.722234			−0.361117	−	0.932520i	$-0.617605\pi$
$863$	−623.288			−0.722234			−0.361117	+	0.932520i	$0.617605\pi$
$864$		0			0
$865$	−403.161			−0.466082
$866$			1042.59i			1.20391i
$867$		0			0
$868$	−1583.95	+	535.700i	−1.82483	+	0.617166i
$869$	61.1677			0.0703886
$870$		0			0
$871$		−	1616.80i		−	1.85626i
$872$	−1469.80			−1.68555
$873$		0			0
$874$		−	241.265i		−	0.276047i
$875$	−705.265	+	238.523i	−0.806017	+	0.272598i
$876$		0			0
$877$	473.691			0.540127			0.270063	−	0.962843i	$-0.412955\pi$
$877$	473.691			0.540127			0.270063	+	0.962843i	$0.412955\pi$
$878$		−	266.665i		−	0.303719i
$879$		0			0
$880$			60.3756i			0.0686086i
$881$		−	1267.03i		−	1.43817i	−0.694921	−	0.719087i	$-0.744560\pi$
$881$		−	1267.03i		−	1.43817i	0.694921	−	0.719087i	$-0.255440\pi$
$882$		0			0
$883$	764.676			0.865998			0.432999	−	0.901394i	$-0.357455\pi$
$883$	764.676			0.865998			0.432999	+	0.901394i	$0.357455\pi$
$884$	−2909.86			−3.29170
$885$		0			0
$886$	985.551			1.11236
$887$			1671.40i			1.88433i	0.335155	+	0.942163i	$0.391211\pi$
$887$			1671.40i			1.88433i	−0.335155	+	0.942163i	$0.608789\pi$
$888$		0			0
$889$	142.589	+	421.606i	0.160392	+	0.474247i
$890$	−291.047			−0.327019
$891$		0			0
$892$		−	22.1984i		−	0.0248861i
$893$	192.674			0.215760
$894$		0			0
$895$			246.899i			0.275865i
$896$	−523.882	−	1549.01i	−0.584690	−	1.72881i
$897$		0			0
$898$	929.499			1.03508
$899$		−	1353.86i		−	1.50596i
$900$		0			0
$901$			455.605i			0.505666i
$902$			648.285i			0.718719i
$903$		0			0
$904$	−616.530			−0.682002
$905$	632.094			0.698446
$906$		0			0
$907$	996.264			1.09842			0.549208	−	0.835685i	$-0.314929\pi$
$907$	996.264			1.09842			0.549208	+	0.835685i	$0.314929\pi$
$908$		−	1601.86i		−	1.76417i
$909$		0			0
$910$	294.956	+	872.125i	0.324128	+	0.958380i
$911$	583.201			0.640177			0.320089	−	0.947388i	$-0.396287\pi$
$911$	583.201			0.640177			0.320089	+	0.947388i	$0.396287\pi$
$912$		0			0
$913$			235.631i			0.258084i
$914$	−885.932			−0.969291
$915$		0			0
$916$		−	1417.48i		−	1.54747i
$917$	−824.666	+	278.905i	−0.899308	+	0.304150i
$918$		0			0
$919$	−1382.66			−1.50453			−0.752263	−	0.658862i	$-0.771038\pi$
$919$	−1382.66			−1.50453			−0.752263	+	0.658862i	$0.771038\pi$
$920$		−	874.392i		−	0.950426i
$921$		0			0
$922$			2479.73i			2.68951i
$923$		−	1055.29i		−	1.14332i
$924$		0			0
$925$	720.212			0.778608
$926$	1950.41			2.10628
$927$		0			0
$928$	754.992			0.813569
$929$			281.399i			0.302905i	0.988465	+	0.151453i	$0.0483951\pi$
$929$			281.399i			0.302905i	−0.988465	+	0.151453i	$0.951605\pi$
$930$		0			0
$931$	−64.0143	−	83.8136i	−0.0687586	−	0.0900254i
$932$	−2038.48			−2.18721
$933$		0			0
$934$		−	1165.94i		−	1.24833i
$935$	198.740			0.212556
$936$		0			0
$937$		−	384.694i		−	0.410560i	−0.978703	−	0.205280i	$-0.934190\pi$
$937$		−	384.694i		−	0.410560i	0.978703	−	0.205280i	$-0.0658105\pi$
$938$	745.599	+	2204.59i	0.794882	+	2.35030i
$939$		0			0
$940$	1559.67			1.65922
$941$			549.001i			0.583423i	0.956506	+	0.291712i	$0.0942248\pi$
$941$			549.001i			0.583423i	−0.956506	+	0.291712i	$0.905775\pi$
$942$		0			0
$943$		−	1927.75i		−	2.04428i
$944$			103.669i			0.109819i
$945$		0			0
$946$	86.5513			0.0914919
$947$	1315.14			1.38874			0.694372	−	0.719616i	$-0.255682\pi$
$947$	1315.14			1.38874			0.694372	+	0.719616i	$0.255682\pi$
$948$		0			0
$949$	−826.721			−0.871149
$950$			138.656i			0.145954i
$951$		0			0
$952$	1776.42	−	600.791i	1.86598	−	0.631083i
$953$	673.335			0.706542			0.353271	−	0.935521i	$-0.385069\pi$
$953$	673.335			0.706542			0.353271	+	0.935521i	$0.385069\pi$
$954$		0			0
$955$		−	522.859i		−	0.547497i
$956$	260.523			0.272514
$957$		0			0
$958$			657.015i			0.685820i
$959$	385.545	+	1139.98i	0.402028	+	1.18871i
$960$		0			0
$961$	−126.751			−0.131895
$962$		−	2049.46i		−	2.13042i
$963$		0			0
$964$			1072.97i			1.11304i
$965$		−	605.497i		−	0.627458i
$966$		0			0
$967$	−1511.93			−1.56352			−0.781761	−	0.623579i	$-0.785678\pi$
$967$	−1511.93			−1.56352			−0.781761	+	0.623579i	$0.785678\pi$
$968$	−1193.35			−1.23280
$969$		0			0
$970$	−1153.53			−1.18921
$971$		−	252.073i		−	0.259601i	−0.991540	−	0.129801i	$-0.958566\pi$
$971$		−	252.073i		−	0.259601i	0.991540	−	0.129801i	$-0.0414337\pi$
$972$		0			0
$973$	−926.985	+	313.510i	−0.952708	+	0.322210i
$974$	−1716.37			−1.76218
$975$		0			0
$976$		−	185.194i		−	0.189748i
$977$	588.155			0.602001			0.301000	−	0.953624i	$-0.402679\pi$
$977$	588.155			0.602001			0.301000	+	0.953624i	$0.402679\pi$
$978$		0			0
$979$			120.989i			0.123584i
$980$	−518.187	−	678.460i	−0.528763	−	0.692307i
$981$		0			0
$982$	31.0000			0.0315682
$983$			1323.68i			1.34658i	0.739380	+	0.673288i	$0.235119\pi$
$983$			1323.68i			1.34658i	−0.739380	+	0.673288i	$0.764881\pi$
$984$		0			0
$985$			476.126i			0.483377i
$986$			3391.36i			3.43952i
$987$		0			0
$988$	254.184			0.257271
$989$	−257.371			−0.260233
$990$		0			0
$991$	−674.419			−0.680544			−0.340272	−	0.940327i	$-0.610519\pi$
$991$	−674.419			−0.680544			−0.340272	+	0.940327i	$0.610519\pi$
$992$		−	606.594i		−	0.611486i
$993$		0			0
$994$	486.653	+	1438.93i	0.489591	+	1.44762i
$995$	193.562			0.194535
$996$		0			0
$997$			717.365i			0.719523i	0.933044	+	0.359762i	$0.117142\pi$
$997$			717.365i			0.719523i	−0.933044	+	0.359762i	$0.882858\pi$
$998$	−64.1767			−0.0643053
$999$		0			0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+3.35300 q^{2} +7.24264 q^{4} +2.40558i q^{5} +(2.24264 + 6.63103i) q^{7} +10.8726 q^{8} +O(q^{10})\)	\(q+3.35300 q^{2} +7.24264 q^{4} +2.40558i q^{5} +(2.24264 + 6.63103i) q^{7} +10.8726 q^{8} +8.06591i q^{10} +3.35300 q^{11} -16.3059i q^{13} +(7.51958 + 22.2339i) q^{14} +7.48528 q^{16} -24.6395i q^{17} +2.15232i q^{19} +17.4227i q^{20} +11.2426 q^{22} -33.4313 q^{23} +19.2132 q^{25} -54.6737i q^{26} +(16.2426 + 48.0262i) q^{28} -41.0496 q^{29} +32.9811i q^{31} -18.3922 q^{32} -82.6162i q^{34} +(-15.9514 + 5.39484i) q^{35} +37.4853 q^{37} +7.21673i q^{38} +26.1548i q^{40} +57.6630i q^{41} +7.69848 q^{43} +24.2846 q^{44} -112.095 q^{46} -89.5192i q^{47} +(-38.9411 + 29.7420i) q^{49} +64.4220 q^{50} -118.098i q^{52} -18.4909 q^{53} +8.06591i q^{55} +(24.3833 + 72.0965i) q^{56} -137.640 q^{58} +13.8498i q^{59} -24.7411i q^{61} +110.586i q^{62} -91.6102 q^{64} +39.2251 q^{65} +99.1543 q^{67} -178.455i q^{68} +(-53.4853 + 18.0889i) q^{70} +64.7181 q^{71} -50.7007i q^{73} +125.688 q^{74} +15.5885i q^{76} +(7.51958 + 22.2339i) q^{77} +18.2426 q^{79} +18.0064i q^{80} +193.344i q^{82} +70.2746i q^{83} +59.2721 q^{85} +25.8131 q^{86} +36.4558 q^{88} +36.0836i q^{89} +(108.125 - 36.5683i) q^{91} -242.131 q^{92} -300.158i q^{94} -5.17756 q^{95} +143.013i q^{97} +(-130.570 + 99.7252i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4} - 16 q^{7}+O(q^{10}) \) 8 * q + 24 * q^4 - 16 * q^7	\( 8 q + 24 q^{4} - 16 q^{7} - 8 q^{16} + 56 q^{22} - 16 q^{25} + 96 q^{28} + 232 q^{37} - 176 q^{43} - 184 q^{46} - 40 q^{49} - 592 q^{58} - 88 q^{64} + 352 q^{67} - 360 q^{70} + 112 q^{79} + 576 q^{85} + 88 q^{88} + 288 q^{91}+O(q^{100}) \) 8 * q + 24 * q^4 - 16 * q^7 - 8 * q^16 + 56 * q^22 - 16 * q^25 + 96 * q^28 + 232 * q^37 - 176 * q^43 - 184 * q^46 - 40 * q^49 - 592 * q^58 - 88 * q^64 + 352 * q^67 - 360 * q^70 + 112 * q^79 + 576 * q^85 + 88 * q^88 + 288 * q^91

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