LMFDB - Embedded newform 189.3.d.e.55.6

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			55.6
Root			$-1.48088 + 2.56496i$ of defining polynomial
Character	$\chi$	$=$	189.55
Dual form			189.3.d.e.55.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.28648 q^{2} +1.22800 q^{4} +6.53330i q^{5} +(5.77200 + 3.96030i) q^{7} -6.33813 q^{8} +14.9383i q^{10} +13.1976 q^{11} +3.96030i q^{13} +(13.1976 + 9.05516i) q^{14} -19.4040 q^{16} +15.5885i q^{17} -30.7795i q^{19} +8.02288i q^{20} +30.1760 q^{22} -1.76517 q^{23} -17.6840 q^{25} +9.05516i q^{26} +(7.08801 + 4.86324i) q^{28} +37.8276 q^{29} -57.5987i q^{31} -19.0144 q^{32} +35.6427i q^{34} +(-25.8738 + 37.7102i) q^{35} -41.7720 q^{37} -70.3767i q^{38} -41.4089i q^{40} -24.6436i q^{41} -38.0880 q^{43} +16.2066 q^{44} -4.03602 q^{46} +32.6665i q^{47} +(17.6320 + 45.7177i) q^{49} -40.4341 q^{50} +4.86324i q^{52} -7.38076 q^{53} +86.2237i q^{55} +(-36.5837 - 25.1009i) q^{56} +86.4920 q^{58} -59.8320i q^{59} -32.5854i q^{61} -131.698i q^{62} +34.1400 q^{64} -25.8738 q^{65} +91.5800 q^{67} +19.1426i q^{68} +(-59.1601 + 86.2237i) q^{70} -9.86847 q^{71} -23.7618i q^{73} -95.5109 q^{74} -37.7971i q^{76} +(76.1764 + 52.2664i) q^{77} +74.6840 q^{79} -126.772i q^{80} -56.3472i q^{82} +126.197i q^{83} -101.844 q^{85} -87.0875 q^{86} -83.6480 q^{88} +27.1655i q^{89} +(-15.6840 + 22.8589i) q^{91} -2.16762 q^{92} +74.6913i q^{94} +201.092 q^{95} +48.4266i q^{97} +(40.3153 + 104.533i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4} + 12 q^{7} + 84 q^{16} - 32 q^{22} - 244 q^{25} - 80 q^{28} - 300 q^{37} - 168 q^{43} + 412 q^{46} - 64 q^{49} + 316 q^{58} + 444 q^{64} + 220 q^{67} + 552 q^{70} + 700 q^{79} + 108 q^{85}+ \cdots - 228 q^{91}+O(q^{100}) $ 8 * q + 44 * q^4 + 12 * q^7 + 84 * q^16 - 32 * q^22 - 244 * q^25 - 80 * q^28 - 300 * q^37 - 168 * q^43 + 412 * q^46 - 64 * q^49 + 316 * q^58 + 444 * q^64 + 220 * q^67 + 552 * q^70 + 700 * q^79 + 108 * q^85 - 1216 * q^88 - 228 * q^91

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.28648			1.14324			0.571620	−	0.820518i	$-0.306315\pi$
$2$	2.28648			1.14324			0.571620	+	0.820518i	$0.306315\pi$
$3$		0			0
$3$		0			0
$4$	1.22800			0.307000
$5$			6.53330i			1.30666i	0.757073	+	0.653330i	$0.226629\pi$
$5$			6.53330i			1.30666i	−0.757073	+	0.653330i	$0.773371\pi$
$6$		0			0
$7$	5.77200	+	3.96030i	0.824572	+	0.565757i
$7$	5.77200	+	3.96030i	0.824572	+	0.565757i
$8$	−6.33813			−0.792266
$9$		0			0
$10$			14.9383i			1.49383i
$11$	13.1976			1.19978			0.599890	−	0.800083i	$-0.295211\pi$
$11$	13.1976			1.19978			0.599890	+	0.800083i	$0.295211\pi$
$12$		0			0
$13$			3.96030i			0.304639i	0.988331	+	0.152319i	$0.0486742\pi$
$13$			3.96030i			0.304639i	−0.988331	+	0.152319i	$0.951326\pi$
$14$	13.1976	+	9.05516i	0.942684	+	0.646797i
$15$		0			0
$16$	−19.4040			−1.21275
$17$			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
$19$		−	30.7795i		−	1.61997i	−0.586449	−	0.809986i	$-0.699475\pi$
$19$		−	30.7795i		−	1.61997i	0.586449	−	0.809986i	$-0.300525\pi$
$20$			8.02288i			0.401144i
$21$		0			0
$22$	30.1760			1.37164
$23$	−1.76517			−0.0767464			−0.0383732	−	0.999263i	$-0.512218\pi$
$23$	−1.76517			−0.0767464			−0.0383732	+	0.999263i	$0.512218\pi$
$24$		0			0
$25$	−17.6840			−0.707360
$26$			9.05516i			0.348275i
$27$		0			0
$28$	7.08801	+	4.86324i	0.253143	+	0.173687i
$29$	37.8276			1.30440			0.652199	−	0.758048i	$-0.273847\pi$
$29$	37.8276			1.30440			0.652199	+	0.758048i	$0.273847\pi$
$30$		0			0
$31$		−	57.5987i		−	1.85802i	−0.370053	−	0.929011i	$-0.620661\pi$
$31$		−	57.5987i		−	1.85802i	0.370053	−	0.929011i	$-0.379339\pi$
$32$	−19.0144			−0.594200
$33$		0			0
$34$			35.6427i			1.04832i
$35$	−25.8738	+	37.7102i	−0.739253	+	1.07743i
$36$		0			0
$37$	−41.7720			−1.12897			−0.564487	−	0.825442i	$-0.690926\pi$
$37$	−41.7720			−1.12897			−0.564487	+	0.825442i	$0.690926\pi$
$38$		−	70.3767i		−	1.85202i
$39$		0			0
$40$		−	41.4089i		−	1.03522i
$41$		−	24.6436i		−	0.601064i	−0.953772	−	0.300532i	$-0.902836\pi$
$41$		−	24.6436i		−	0.601064i	0.953772	−	0.300532i	$-0.0971642\pi$
$42$		0			0
$43$	−38.0880			−0.885768			−0.442884	−	0.896579i	$-0.646045\pi$
$43$	−38.0880			−0.885768			−0.442884	+	0.896579i	$0.646045\pi$
$44$	16.2066			0.368332
$45$		0			0
$46$	−4.03602			−0.0877397
$47$			32.6665i			0.695032i	0.937674	+	0.347516i	$0.112975\pi$
$47$			32.6665i			0.695032i	−0.937674	+	0.347516i	$0.887025\pi$
$48$		0			0
$49$	17.6320	+	45.7177i	0.359837	+	0.933015i
$50$	−40.4341			−0.808683
$51$		0			0
$52$			4.86324i			0.0935239i
$53$	−7.38076			−0.139260			−0.0696298	−	0.997573i	$-0.522182\pi$
$53$	−7.38076			−0.139260			−0.0696298	+	0.997573i	$0.522182\pi$
$54$		0			0
$55$			86.2237i			1.56770i
$56$	−36.5837	−	25.1009i	−0.653280	−	0.448231i
$57$		0			0
$58$	86.4920			1.49124
$59$		−	59.8320i		−	1.01410i	−0.861916	−	0.507051i	$-0.830736\pi$
$59$		−	59.8320i		−	1.01410i	0.861916	−	0.507051i	$-0.169264\pi$
$60$		0			0
$61$		−	32.5854i		−	0.534186i	−0.963671	−	0.267093i	$-0.913937\pi$
$61$		−	32.5854i		−	0.534186i	0.963671	−	0.267093i	$-0.0860631\pi$
$62$		−	131.698i		−	2.12417i
$63$		0			0
$64$	34.1400			0.533437
$65$	−25.8738			−0.398059
$66$		0			0
$67$	91.5800			1.36687			0.683433	−	0.730013i	$-0.260486\pi$
$67$	91.5800			1.36687			0.683433	+	0.730013i	$0.260486\pi$
$68$			19.1426i			0.281509i
$69$		0			0
$70$	−59.1601	+	86.2237i	−0.845144	+	1.23177i
$71$	−9.86847			−0.138992			−0.0694962	−	0.997582i	$-0.522139\pi$
$71$	−9.86847			−0.138992			−0.0694962	+	0.997582i	$0.522139\pi$
$72$		0			0
$73$		−	23.7618i		−	0.325504i	−0.986667	−	0.162752i	$-0.947963\pi$
$73$		−	23.7618i		−	0.325504i	0.986667	−	0.162752i	$-0.0520371\pi$
$74$	−95.5109			−1.29069
$75$		0			0
$76$		−	37.7971i		−	0.497331i
$77$	76.1764	+	52.2664i	0.989304	+	0.678784i
$78$		0			0
$79$	74.6840			0.945367			0.472684	−	0.881232i	$-0.343285\pi$
$79$	74.6840			0.945367			0.472684	+	0.881232i	$0.343285\pi$
$80$		−	126.772i		−	1.58465i
$81$		0			0
$82$		−	56.3472i		−	0.687161i
$83$			126.197i			1.52045i	0.649661	+	0.760224i	$0.274911\pi$
$83$			126.197i			1.52045i	−0.649661	+	0.760224i	$0.725089\pi$
$84$		0			0
$85$	−101.844			−1.19817
$86$	−87.0875			−1.01265
$87$		0			0
$88$	−83.6480			−0.950545
$89$			27.1655i			0.305230i	0.988286	+	0.152615i	$0.0487695\pi$
$89$			27.1655i			0.305230i	−0.988286	+	0.152615i	$0.951231\pi$
$90$		0			0
$91$	−15.6840	+	22.8589i	−0.172352	+	0.251196i
$92$	−2.16762			−0.0235611
$93$		0			0
$94$			74.6913i			0.794589i
$95$	201.092			2.11675
$96$		0			0
$97$			48.4266i			0.499243i	0.968343	+	0.249621i	$0.0803062\pi$
$97$			48.4266i			0.499243i	−0.968343	+	0.249621i	$0.919694\pi$
$98$	40.3153	+	104.533i	0.411380	+	1.06666i
$99$		0			0
$100$	−21.7159			−0.217159
$101$		−	158.864i		−	1.57291i	−0.617649	−	0.786454i	$-0.711915\pi$
$101$		−	158.864i		−	1.57291i	0.617649	−	0.786454i	$-0.288085\pi$
$102$		0			0
$103$		−	116.449i		−	1.13057i	−0.824895	−	0.565285i	$-0.808766\pi$
$103$		−	116.449i		−	1.13057i	0.824895	−	0.565285i	$-0.191234\pi$
$104$		−	25.1009i		−	0.241355i
$105$		0			0
$106$	−16.8760			−0.159207
$107$	64.0215			0.598332			0.299166	−	0.954201i	$-0.403292\pi$
$107$	64.0215			0.598332			0.299166	+	0.954201i	$0.403292\pi$
$108$		0			0
$109$	20.9640			0.192330			0.0961650	−	0.995365i	$-0.469342\pi$
$109$	20.9640			0.192330			0.0961650	+	0.995365i	$0.469342\pi$
$110$			197.149i			1.79226i
$111$		0			0
$112$	−112.000	−	76.8458i	−1.00000	−	0.686123i
$113$	−162.020			−1.43381			−0.716903	−	0.697173i	$-0.754441\pi$
$113$	−162.020			−1.43381			−0.716903	+	0.697173i	$0.754441\pi$
$114$		0			0
$115$		−	11.5324i		−	0.100281i
$116$	46.4522			0.400450
$117$		0			0
$118$		−	136.805i		−	1.15936i
$119$	−61.7350	+	89.9766i	−0.518782	+	0.756106i
$120$		0			0
$121$	53.1760			0.439471
$122$		−	74.5058i		−	0.610703i
$123$		0			0
$124$		−	70.7310i		−	0.570412i
$125$			47.7977i			0.382381i
$126$		0			0
$127$	17.4200			0.137165			0.0685826	−	0.997645i	$-0.478152\pi$
$127$	17.4200			0.137165			0.0685826	+	0.997645i	$0.478152\pi$
$128$	154.118			1.20405
$129$		0			0
$130$	−59.1601			−0.455077
$131$			147.744i			1.12782i	0.825837	+	0.563908i	$0.190703\pi$
$131$			147.744i			1.12782i	−0.825837	+	0.563908i	$0.809297\pi$
$132$		0			0
$133$	121.896	−	177.659i	0.916512	−	1.33578i
$134$	209.396			1.56266
$135$		0			0
$136$		−	98.8017i		−	0.726483i
$137$	−202.537			−1.47837			−0.739185	−	0.673503i	$-0.764789\pi$
$137$	−202.537			−1.47837			−0.739185	+	0.673503i	$0.764789\pi$
$138$		0			0
$139$			245.333i			1.76498i	0.470327	+	0.882492i	$0.344136\pi$
$139$			245.333i			1.76498i	−0.470327	+	0.882492i	$0.655864\pi$
$140$	−31.7730	+	46.3081i	−0.226950	+	0.330772i
$141$		0			0
$142$	−22.5641			−0.158902
$143$			52.2664i			0.365499i
$144$		0			0
$145$			247.139i			1.70441i
$146$		−	54.3310i		−	0.372130i
$147$		0			0
$148$	−51.2959			−0.346594
$149$	64.6252			0.433726			0.216863	−	0.976202i	$-0.430418\pi$
$149$	64.6252			0.433726			0.216863	+	0.976202i	$0.430418\pi$
$150$		0			0
$151$	158.652			1.05068			0.525338	−	0.850894i	$-0.323939\pi$
$151$	158.652			1.05068			0.525338	+	0.850894i	$0.323939\pi$
$152$			195.084i			1.28345i
$153$		0			0
$154$	174.176	+	119.506i	1.13101	+	0.776014i
$155$	376.309			2.42780
$156$		0			0
$157$			79.5546i			0.506717i	0.967372	+	0.253359i	$0.0815352\pi$
$157$			79.5546i			0.506717i	−0.967372	+	0.253359i	$0.918465\pi$
$158$	170.764			1.08078
$159$		0			0
$160$		−	124.227i		−	0.776417i
$161$	−10.1886	−	6.99060i	−0.0632829	−	0.0434199i
$162$		0			0
$163$	−323.320			−1.98356			−0.991779	−	0.127959i	$-0.959157\pi$
$163$	−323.320			−1.98356			−0.991779	+	0.127959i	$0.959157\pi$
$164$		−	30.2623i		−	0.184526i
$165$		0			0
$166$			288.548i			1.73824i
$167$		−	1.48958i		−	0.00891964i	−0.999990	−	0.00445982i	$-0.998580\pi$
$167$		−	1.48958i		−	0.00891964i	0.999990	−	0.00445982i	$-0.00141961\pi$
$168$		0			0
$169$	153.316			0.907195
$170$	−232.865			−1.36979
$171$		0			0
$172$	−46.7720			−0.271930
$173$		−	142.818i		−	0.825537i	−0.910836	−	0.412769i	$-0.864562\pi$
$173$		−	142.818i		−	0.825537i	0.910836	−	0.412769i	$-0.135438\pi$
$174$		0			0
$175$	−102.072	−	70.0340i	−0.583269	−	0.400194i
$176$	−256.086			−1.45503
$177$		0			0
$178$			62.1134i			0.348951i
$179$	−273.619			−1.52860			−0.764298	−	0.644863i	$-0.776914\pi$
$179$	−273.619			−1.52860			−0.764298	+	0.644863i	$0.776914\pi$
$180$		0			0
$181$		−	61.0045i		−	0.337042i	−0.985698	−	0.168521i	$-0.946101\pi$
$181$		−	61.0045i		−	0.337042i	0.985698	−	0.168521i	$-0.0538991\pi$
$182$	−35.8612	+	52.2664i	−0.197039	+	0.287178i
$183$		0			0
$184$	11.1879			0.0608036
$185$		−	272.909i		−	1.47518i
$186$		0			0
$187$			205.730i			1.10016i
$188$			40.1144i			0.213374i
$189$		0			0
$190$	459.792			2.41996
$191$	6.01805			0.0315081			0.0157540	−	0.999876i	$-0.494985\pi$
$191$	6.01805			0.0315081			0.0157540	+	0.999876i	$0.494985\pi$
$192$		0			0
$193$	−221.880			−1.14964			−0.574819	−	0.818281i	$-0.694927\pi$
$193$	−221.880			−1.14964			−0.574819	+	0.818281i	$0.694927\pi$
$194$			110.726i			0.570755i
$195$		0			0
$196$	21.6521	+	56.1413i	0.110470	+	0.286435i
$197$	−175.739			−0.892076			−0.446038	−	0.895014i	$-0.647165\pi$
$197$	−175.739			−0.892076			−0.446038	+	0.895014i	$0.647165\pi$
$198$		0			0
$199$			42.6604i			0.214374i	0.994239	+	0.107187i	$0.0341843\pi$
$199$			42.6604i			0.214374i	−0.994239	+	0.107187i	$0.965816\pi$
$200$	112.083			0.560417
$201$		0			0
$202$		−	363.239i		−	1.79821i
$203$	218.341	+	149.809i	1.07557	+	0.737973i
$204$		0			0
$205$	161.004			0.785386
$206$		−	266.258i		−	1.29251i
$207$		0			0
$208$		−	76.8458i		−	0.369451i
$209$		−	406.214i		−	1.94361i
$210$		0			0
$211$	−186.876			−0.885668			−0.442834	−	0.896604i	$-0.646027\pi$
$211$	−186.876			−0.885668			−0.442834	+	0.896604i	$0.646027\pi$
$212$	−9.06356			−0.0427526
$213$		0			0
$214$	146.384			0.684037
$215$		−	248.840i		−	1.15740i
$216$		0			0
$217$	228.108	−	332.460i	1.05119	−	1.53207i
$218$	47.9337			0.219880
$219$		0			0
$220$			105.883i			0.481284i
$221$	−61.7350			−0.279344
$222$		0			0
$223$		−	320.927i		−	1.43914i	−0.694422	−	0.719568i	$-0.744340\pi$
$223$		−	320.927i		−	1.43914i	0.694422	−	0.719568i	$-0.255660\pi$
$224$	−109.751	−	75.3027i	−0.489960	−	0.336173i
$225$		0			0
$226$	−370.456			−1.63919
$227$			9.05516i			0.0398906i	0.999801	+	0.0199453i	$0.00634920\pi$
$227$			9.05516i			0.0398906i	−0.999801	+	0.0199453i	$0.993651\pi$
$228$		0			0
$229$			229.698i			1.00305i	0.865144	+	0.501523i	$0.167227\pi$
$229$			229.698i			1.00305i	−0.865144	+	0.501523i	$0.832773\pi$
$230$		−	26.3686i		−	0.114646i
$231$		0			0
$232$	−239.756			−1.03343
$233$	−11.7525			−0.0504399			−0.0252199	−	0.999682i	$-0.508029\pi$
$233$	−11.7525			−0.0504399			−0.0252199	+	0.999682i	$0.508029\pi$
$234$		0			0
$235$	−213.420			−0.908170
$236$		−	73.4735i		−	0.311329i
$237$		0			0
$238$	−141.156	+	205.730i	−0.593092	+	0.864411i
$239$	−440.733			−1.84407			−0.922036	−	0.387104i	$-0.873475\pi$
$239$	−440.733			−1.84407			−0.922036	+	0.387104i	$0.873475\pi$
$240$		0			0
$241$		−	350.107i		−	1.45272i	−0.687312	−	0.726362i	$-0.741209\pi$
$241$		−	350.107i		−	1.45272i	0.687312	−	0.726362i	$-0.258791\pi$
$242$	121.586			0.502421
$243$		0			0
$244$		−	40.0148i		−	0.163995i
$245$	−298.688	+	115.195i	−1.21913	+	0.470184i
$246$		0			0
$247$	121.896			0.493506
$248$			365.068i			1.47205i
$249$		0			0
$250$			109.288i			0.437154i
$251$			207.576i			0.826996i	0.910505	+	0.413498i	$0.135693\pi$
$251$			207.576i			0.826996i	−0.910505	+	0.413498i	$0.864307\pi$
$252$		0			0
$253$	−23.2959			−0.0920788
$254$	39.8304			0.156813
$255$		0			0
$256$	215.828			0.843078
$257$		−	442.435i		−	1.72154i	−0.508996	−	0.860769i	$-0.669983\pi$
$257$		−	442.435i		−	1.72154i	0.508996	−	0.860769i	$-0.330017\pi$
$258$		0			0
$259$	−241.108	−	165.430i	−0.930919	−	0.638725i
$260$	−31.7730			−0.122204
$261$		0			0
$262$			337.814i			1.28937i
$263$	128.244			0.487621			0.243810	−	0.969823i	$-0.421603\pi$
$263$	128.244			0.487621			0.243810	+	0.969823i	$0.421603\pi$
$264$		0			0
$265$		−	48.2207i		−	0.181965i
$266$	278.713	−	406.214i	1.04779	−	1.52712i
$267$		0			0
$268$	112.460			0.419627
$269$		−	81.0391i		−	0.301261i	−0.988590	−	0.150630i	$-0.951870\pi$
$269$		−	81.0391i		−	0.301261i	0.988590	−	0.150630i	$-0.0481303\pi$
$270$		0			0
$271$		−	3.75443i		−	0.0138540i	−0.999976	−	0.00692700i	$-0.997795\pi$
$271$		−	3.75443i		−	0.0138540i	0.999976	−	0.00692700i	$-0.00220495\pi$
$272$		−	302.479i		−	1.11205i
$273$		0			0
$274$	−463.096			−1.69013
$275$	−233.386			−0.848676
$276$		0			0
$277$	−111.004			−0.400737			−0.200368	−	0.979721i	$-0.564214\pi$
$277$	−111.004			−0.400737			−0.200368	+	0.979721i	$0.564214\pi$
$278$			560.949i			2.01780i
$279$		0			0
$280$	163.992	−	239.012i	0.585685	−	0.853615i
$281$	47.1747			0.167882			0.0839408	−	0.996471i	$-0.473249\pi$
$281$	47.1747			0.167882			0.0839408	+	0.996471i	$0.473249\pi$
$282$		0			0
$283$		−	209.342i		−	0.739723i	−0.929087	−	0.369862i	$-0.879405\pi$
$283$		−	209.342i		−	0.739723i	0.929087	−	0.369862i	$-0.120595\pi$
$284$	−12.1185			−0.0426706
$285$		0			0
$286$			119.506i			0.417854i
$287$	97.5962	−	142.243i	0.340056	−	0.495620i
$288$		0			0
$289$	46.0000			0.159170
$290$			565.078i			1.94855i
$291$		0			0
$292$		−	29.1795i		−	0.0999297i
$293$			138.349i			0.472182i	0.971731	+	0.236091i	$0.0758663\pi$
$293$			138.349i			0.472182i	−0.971731	+	0.236091i	$0.924134\pi$
$294$		0			0
$295$	390.900			1.32509
$296$	264.756			0.894447
$297$		0			0
$298$	147.764			0.495853
$299$		−	6.99060i		−	0.0233799i
$300$		0			0
$301$	−219.844	−	150.840i	−0.730379	−	0.501130i
$302$	362.755			1.20118
$303$		0			0
$304$			597.245i			1.96462i
$305$	212.890			0.698000
$306$		0			0
$307$			422.295i			1.37555i	0.725922	+	0.687777i	$0.241413\pi$
$307$			422.295i			1.37555i	−0.725922	+	0.687777i	$0.758587\pi$
$308$	93.5445	+	64.1830i	0.303716	+	0.208386i
$309$		0			0
$310$	860.424			2.77556
$311$		−	71.8663i		−	0.231081i	−0.993303	−	0.115541i	$-0.963140\pi$
$311$		−	71.8663i		−	0.231081i	0.993303	−	0.115541i	$-0.0368601\pi$
$312$		0			0
$313$			7.92060i			0.0253054i	0.999920	+	0.0126527i	$0.00402759\pi$
$313$			7.92060i			0.0253054i	−0.999920	+	0.0126527i	$0.995972\pi$
$314$			181.900i			0.579300i
$315$		0			0
$316$	91.7118			0.290227
$317$	−360.624			−1.13761			−0.568807	−	0.822471i	$-0.692595\pi$
$317$	−360.624			−1.13761			−0.568807	+	0.822471i	$0.692595\pi$
$318$		0			0
$319$	499.232			1.56499
$320$			223.047i			0.697021i
$321$		0			0
$322$	−23.2959	−	15.9839i	−0.0723476	−	0.0496394i
$323$	479.805			1.48546
$324$		0			0
$325$		−	70.0340i		−	0.215489i
$326$	−739.265			−2.26769
$327$		0			0
$328$			156.194i			0.476203i
$329$	−129.369	+	188.551i	−0.393219	+	0.573104i
$330$		0			0
$331$	−69.4560			−0.209837			−0.104918	−	0.994481i	$-0.533458\pi$
$331$	−69.4560			−0.209837			−0.104918	+	0.994481i	$0.533458\pi$
$332$			154.970i			0.466777i
$333$		0			0
$334$		−	3.40590i		−	0.0101973i
$335$			598.320i			1.78603i
$336$		0			0
$337$	−162.720			−0.482849			−0.241424	−	0.970420i	$-0.577615\pi$
$337$	−162.720			−0.482849			−0.241424	+	0.970420i	$0.577615\pi$
$338$	350.554			1.03714
$339$		0			0
$340$	−125.064			−0.367836
$341$		−	760.163i		−	2.22922i
$342$		0			0
$343$	−79.2841	+	333.711i	−0.231149	+	0.972918i
$344$	241.407			0.701764
$345$		0			0
$346$		−	326.551i		−	0.943788i
$347$	89.8953			0.259064			0.129532	−	0.991575i	$-0.458652\pi$
$347$	89.8953			0.259064			0.129532	+	0.991575i	$0.458652\pi$
$348$		0			0
$349$			495.529i			1.41985i	0.704275	+	0.709927i	$0.251272\pi$
$349$			495.529i			1.41985i	−0.704275	+	0.709927i	$0.748728\pi$
$350$	−233.386	−	160.131i	−0.666817	−	0.457518i
$351$		0			0
$352$	−250.944			−0.712909
$353$			441.978i			1.25206i	0.779798	+	0.626031i	$0.215322\pi$
$353$			441.978i			1.25206i	−0.779798	+	0.626031i	$0.784678\pi$
$354$		0			0
$355$		−	64.4736i		−	0.181616i
$356$			33.3592i			0.0937055i
$357$		0			0
$358$	−625.624			−1.74755
$359$	518.272			1.44366			0.721828	−	0.692073i	$-0.243302\pi$
$359$	518.272			1.44366			0.721828	+	0.692073i	$0.243302\pi$
$360$		0			0
$361$	−586.376			−1.62431
$362$		−	139.486i		−	0.385320i
$363$		0			0
$364$	−19.2599	+	28.0707i	−0.0529119	+	0.0771172i
$365$	155.243			0.425323
$366$		0			0
$367$		−	134.096i		−	0.365384i	−0.983170	−	0.182692i	$-0.941519\pi$
$367$		−	134.096i		−	0.365384i	0.983170	−	0.182692i	$-0.0584811\pi$
$368$	34.2513			0.0930743
$369$		0			0
$370$		−	624.001i		−	1.68649i
$371$	−42.6018	−	29.2300i	−0.114830	−	0.0787872i
$372$		0			0
$373$	530.060			1.42107			0.710536	−	0.703660i	$-0.248452\pi$
$373$	530.060			1.42107			0.710536	+	0.703660i	$0.248452\pi$
$374$			470.398i			1.25775i
$375$		0			0
$376$		−	207.045i		−	0.550650i
$377$			149.809i			0.397370i
$378$		0			0
$379$	503.916			1.32959			0.664797	−	0.747024i	$-0.268518\pi$
$379$	503.916			1.32959			0.664797	+	0.747024i	$0.268518\pi$
$380$	246.940			0.649842
$381$		0			0
$382$	13.7601			0.0360213
$383$			85.8475i			0.224145i	0.993700	+	0.112072i	$0.0357489\pi$
$383$			85.8475i			0.224145i	−0.993700	+	0.112072i	$0.964251\pi$
$384$		0			0
$385$	−341.472	+	497.683i	−0.886940	+	1.29268i
$386$	−507.325			−1.31431
$387$		0			0
$388$			59.4677i			0.153267i
$389$	513.901			1.32108			0.660541	−	0.750790i	$-0.270327\pi$
$389$	513.901			1.32108			0.660541	+	0.750790i	$0.270327\pi$
$390$		0			0
$391$		−	27.5162i		−	0.0703740i
$392$	−111.754	−	289.765i	−0.285087	−	0.739197i
$393$		0			0
$394$	−401.824			−1.01986
$395$			487.933i			1.23527i
$396$		0			0
$397$		−	233.309i		−	0.587681i	−0.955854	−	0.293840i	$-0.905067\pi$
$397$		−	233.309i		−	0.587681i	0.955854	−	0.293840i	$-0.0949334\pi$
$398$			97.5422i			0.245081i
$399$		0			0
$400$	343.140			0.857851
$401$	−329.418			−0.821491			−0.410746	−	0.911750i	$-0.634732\pi$
$401$	−329.418			−0.821491			−0.410746	+	0.911750i	$0.634732\pi$
$402$		0			0
$403$	228.108			0.566025
$404$		−	195.084i		−	0.482882i
$405$		0			0
$406$	499.232	+	342.535i	1.22964	+	0.843681i
$407$	−551.289			−1.35452
$408$		0			0
$409$		−	55.4442i		−	0.135560i	−0.997700	−	0.0677802i	$-0.978408\pi$
$409$		−	55.4442i		−	0.135560i	0.997700	−	0.0677802i	$-0.0215917\pi$
$410$	368.133			0.897885
$411$		0			0
$412$		−	142.999i		−	0.347085i
$413$	236.953	−	345.350i	0.573735	−	0.836199i
$414$		0			0
$415$	−824.484			−1.98671
$416$		−	75.3027i		−	0.181016i
$417$		0			0
$418$		−	928.802i		−	2.22201i
$419$			454.130i			1.08384i	0.840429	+	0.541921i	$0.182303\pi$
$419$			454.130i			1.08384i	−0.840429	+	0.541921i	$0.817697\pi$
$420$		0			0
$421$	−275.056			−0.653340			−0.326670	−	0.945138i	$-0.605927\pi$
$421$	−275.056			−0.653340			−0.326670	+	0.945138i	$0.605927\pi$
$422$	−427.288			−1.01253
$423$		0			0
$424$	46.7802			0.110331
$425$		−	275.666i		−	0.648626i
$426$		0			0
$427$	129.048	−	188.083i	0.302220	−	0.440475i
$428$	78.6183			0.183688
$429$		0			0
$430$		−	568.969i		−	1.32318i
$431$	−312.004			−0.723907			−0.361954	−	0.932196i	$-0.617890\pi$
$431$	−312.004			−0.723907			−0.361954	+	0.932196i	$0.617890\pi$
$432$		0			0
$433$			77.1943i			0.178278i	0.996019	+	0.0891389i	$0.0284115\pi$
$433$			77.1943i			0.178278i	−0.996019	+	0.0891389i	$0.971588\pi$
$434$	521.565	−	760.163i	1.20176	−	1.75153i
$435$		0			0
$436$	25.7437			0.0590452
$437$			54.3310i			0.124327i
$438$		0			0
$439$		−	337.117i		−	0.767920i	−0.923350	−	0.383960i	$-0.874560\pi$
$439$		−	337.117i		−	0.767920i	0.923350	−	0.383960i	$-0.125440\pi$
$440$		−	546.497i		−	1.24204i
$441$		0			0
$442$	−141.156			−0.319357
$443$	136.027			0.307060			0.153530	−	0.988144i	$-0.450936\pi$
$443$	136.027			0.307060			0.153530	+	0.988144i	$0.450936\pi$
$444$		0			0
$445$	−177.480			−0.398832
$446$		−	733.794i		−	1.64528i
$447$		0			0
$448$	197.056	+	135.205i	0.439857	+	0.301796i
$449$	−178.227			−0.396941			−0.198471	−	0.980107i	$-0.563597\pi$
$449$	−178.227			−0.396941			−0.198471	+	0.980107i	$0.563597\pi$
$450$		0			0
$451$		−	325.236i		−	0.721144i
$452$	−198.960			−0.440178
$453$		0			0
$454$			20.7045i			0.0456045i
$455$	−149.344	−	102.468i	−0.328228	−	0.225205i
$456$		0			0
$457$	112.068			0.245225			0.122613	−	0.992455i	$-0.460873\pi$
$457$	112.068			0.245225			0.122613	+	0.992455i	$0.460873\pi$
$458$			525.199i			1.14672i
$459$		0			0
$460$		−	14.1617i		−	0.0307864i
$461$		−	38.6248i		−	0.0837848i	−0.999122	−	0.0418924i	$-0.986661\pi$
$461$		−	38.6248i		−	0.0837848i	0.999122	−	0.0418924i	$-0.0133387\pi$
$462$		0			0
$463$	100.580			0.217235			0.108618	−	0.994084i	$-0.465358\pi$
$463$	100.580			0.217235			0.108618	+	0.994084i	$0.465358\pi$
$464$	−734.006			−1.58191
$465$		0			0
$466$	−26.8719			−0.0576649
$467$		−	92.6161i		−	0.198322i	−0.995071	−	0.0991608i	$-0.968384\pi$
$467$		−	92.6161i		−	0.198322i	0.995071	−	0.0991608i	$-0.0316158\pi$
$468$		0			0
$469$	528.600	+	362.685i	1.12708	+	0.773315i
$470$	−487.981			−1.03826
$471$		0			0
$472$			379.223i			0.803438i
$473$	−502.669			−1.06273
$474$		0			0
$475$			544.304i			1.14590i
$476$	−75.8105	+	110.491i	−0.159266	+	0.232124i
$477$		0			0
$478$	−1007.73			−2.10822
$479$		−	551.672i		−	1.15172i	−0.817550	−	0.575858i	$-0.804668\pi$
$479$		−	551.672i		−	1.15172i	0.817550	−	0.575858i	$-0.195332\pi$
$480$		0			0
$481$		−	165.430i		−	0.343929i
$482$		−	800.512i		−	1.66081i
$483$		0			0
$484$	65.3000			0.134917
$485$	−316.385			−0.652341
$486$		0			0
$487$	172.176			0.353544			0.176772	−	0.984252i	$-0.443434\pi$
$487$	172.176			0.353544			0.176772	+	0.984252i	$0.443434\pi$
$488$			206.530i			0.423218i
$489$		0			0
$490$	−682.944	+	263.392i	−1.39376	+	0.537534i
$491$	602.470			1.22703			0.613513	−	0.789685i	$-0.289756\pi$
$491$	602.470			1.22703			0.613513	+	0.789685i	$0.289756\pi$
$492$		0			0
$493$			589.673i			1.19609i
$494$	278.713			0.564196
$495$		0			0
$496$			1117.64i			2.25332i
$497$	−56.9608	−	39.0821i	−0.114609	−	0.0786360i
$498$		0			0
$499$	226.860			0.454629			0.227315	−	0.973821i	$-0.427005\pi$
$499$	226.860			0.454629			0.227315	+	0.973821i	$0.427005\pi$
$500$			58.6954i			0.117391i
$501$		0			0
$502$			474.619i			0.945456i
$503$			269.015i			0.534821i	0.963583	+	0.267411i	$0.0861681\pi$
$503$			269.015i			0.534821i	−0.963583	+	0.267411i	$0.913832\pi$
$504$		0			0
$505$	1037.90			2.05526
$506$	−53.2657			−0.105268
$507$		0			0
$508$	21.3917			0.0421096
$509$		−	229.580i		−	0.451041i	−0.974238	−	0.225521i	$-0.927592\pi$
$509$		−	229.580i		−	0.451041i	0.974238	−	0.225521i	$-0.0724083\pi$
$510$		0			0
$511$	94.1040	−	137.153i	0.184156	−	0.268402i
$512$	−122.985			−0.240205
$513$		0			0
$514$		−	1011.62i		−	1.96813i
$515$	760.795			1.47727
$516$		0			0
$517$			431.119i			0.833885i
$518$	−551.289	−	378.252i	−1.06426	−	0.730216i
$519$		0			0
$520$	163.992			0.315369
$521$			531.379i			1.01992i	0.860197	+	0.509961i	$0.170340\pi$
$521$			531.379i			1.01992i	−0.860197	+	0.509961i	$0.829660\pi$
$522$		0			0
$523$		−	21.0530i		−	0.0402543i	−0.999797	−	0.0201271i	$-0.993593\pi$
$523$		−	21.0530i		−	0.0402543i	0.999797	−	0.0201271i	$-0.00640710\pi$
$524$			181.429i			0.346239i
$525$		0			0
$526$	293.228			0.557468
$527$	897.874			1.70375
$528$		0			0
$529$	−525.884			−0.994110
$530$		−	110.256i		−	0.208030i
$531$		0			0
$532$	149.688	−	218.165i	0.281369	−	0.410085i
$533$	97.5962			0.183107
$534$		0			0
$535$			418.271i			0.781816i
$536$	−580.446			−1.08292
$537$		0			0
$538$		−	185.294i		−	0.344414i
$539$	232.700	+	603.363i	0.431725	+	1.11941i
$540$		0			0
$541$	430.484			0.795720			0.397860	−	0.917446i	$-0.369753\pi$
$541$	430.484			0.795720			0.397860	+	0.917446i	$0.369753\pi$
$542$		−	8.58444i		−	0.0158385i
$543$		0			0
$544$		−	296.405i		−	0.544862i
$545$			136.964i			0.251310i
$546$		0			0
$547$	212.996			0.389389			0.194695	−	0.980864i	$-0.437628\pi$
$547$	212.996			0.389389			0.194695	+	0.980864i	$0.437628\pi$
$548$	−248.715			−0.453859
$549$		0			0
$550$	−533.632			−0.970241
$551$		−	1164.31i		−	2.11309i
$552$		0			0
$553$	431.076	+	295.771i	0.779523	+	0.534848i
$554$	−253.809			−0.458139
$555$		0			0
$556$			301.268i			0.541849i
$557$	−613.427			−1.10130			−0.550652	−	0.834735i	$-0.685621\pi$
$557$	−613.427			−1.10130			−0.550652	+	0.834735i	$0.685621\pi$
$558$		0			0
$559$		−	150.840i		−	0.269839i
$560$	502.056	−	731.729i	0.896529	−	1.30666i
$561$		0			0
$562$	107.864			0.191929
$563$			749.043i			1.33045i	0.746643	+	0.665225i	$0.231664\pi$
$563$			749.043i			1.33045i	−0.746643	+	0.665225i	$0.768336\pi$
$564$		0			0
$565$		−	1058.53i		−	1.87350i
$566$		−	478.656i		−	0.845682i
$567$		0			0
$568$	62.5476			0.110119
$569$	277.835			0.488287			0.244143	−	0.969739i	$-0.421493\pi$
$569$	277.835			0.488287			0.244143	+	0.969739i	$0.421493\pi$
$570$		0			0
$571$	637.328			1.11616			0.558081	−	0.829787i	$-0.311538\pi$
$571$	637.328			1.11616			0.558081	+	0.829787i	$0.311538\pi$
$572$			64.1830i			0.112208i
$573$		0			0
$574$	223.152	−	325.236i	0.388766	−	0.566613i
$575$	31.2152			0.0542873
$576$		0			0
$577$			89.6296i			0.155337i	0.996979	+	0.0776686i	$0.0247476\pi$
$577$			89.6296i			0.155337i	−0.996979	+	0.0776686i	$0.975252\pi$
$578$	105.178			0.181969
$579$		0			0
$580$			303.486i			0.523252i
$581$	−499.779	+	728.411i	−0.860205	+	1.25372i
$582$		0			0
$583$	−97.4081			−0.167081
$584$			150.605i			0.257886i
$585$		0			0
$586$			316.333i			0.539817i
$587$			1080.75i			1.84114i	0.390573	+	0.920572i	$0.372277\pi$
$587$			1080.75i			1.84114i	−0.390573	+	0.920572i	$0.627723\pi$
$588$		0			0
$589$	−1772.86			−3.00994
$590$	893.786			1.51489
$591$		0			0
$592$	810.544			1.36916
$593$		−	97.0849i		−	0.163718i	−0.996644	−	0.0818591i	$-0.973914\pi$
$593$		−	97.0849i		−	0.163718i	0.996644	−	0.0818591i	$-0.0260857\pi$
$594$		0			0
$595$	−587.844	−	403.333i	−0.987973	−	0.677871i
$596$	79.3596			0.133154
$597$		0			0
$598$		−	15.9839i		−	0.0267289i
$599$	−567.651			−0.947665			−0.473832	−	0.880615i	$-0.657130\pi$
$599$	−567.651			−0.947665			−0.473832	+	0.880615i	$0.657130\pi$
$600$		0			0
$601$		−	108.180i		−	0.179999i	−0.995942	−	0.0899997i	$-0.971313\pi$
$601$		−	108.180i		−	0.179999i	0.995942	−	0.0899997i	$-0.0286866\pi$
$602$	−502.669	−	344.893i	−0.834999	−	0.572912i
$603$		0			0
$604$	194.824			0.322557
$605$			347.415i			0.574239i
$606$		0			0
$607$			343.026i			0.565117i	0.959250	+	0.282558i	$0.0911831\pi$
$607$			343.026i			0.565117i	−0.959250	+	0.282558i	$0.908817\pi$
$608$			585.253i			0.962587i
$609$		0			0
$610$	486.769			0.797982
$611$	−129.369			−0.211734
$612$		0			0
$613$	526.640			0.859119			0.429560	−	0.903038i	$-0.358669\pi$
$613$	526.640			0.859119			0.429560	+	0.903038i	$0.358669\pi$
$614$			965.570i			1.57259i
$615$		0			0
$616$	−482.816	−	331.271i	−0.783793	−	0.537778i
$617$	1123.43			1.82080			0.910398	−	0.413734i	$-0.135776\pi$
$617$	1123.43			1.82080			0.910398	+	0.413734i	$0.135776\pi$
$618$		0			0
$619$		−	74.6913i		−	0.120665i	−0.998178	−	0.0603323i	$-0.980784\pi$
$619$		−	74.6913i		−	0.120665i	0.998178	−	0.0603323i	$-0.0192160\pi$
$620$	462.107			0.745334
$621$		0			0
$622$		−	164.321i		−	0.264182i
$623$	−107.583	+	156.799i	−0.172686	+	0.251684i
$624$		0			0
$625$	−754.376			−1.20700
$626$			18.1103i			0.0289302i
$627$		0			0
$628$			97.6929i			0.155562i
$629$		−	651.161i		−	1.03523i
$630$		0			0
$631$	−905.740			−1.43540			−0.717702	−	0.696350i	$-0.754806\pi$
$631$	−905.740			−1.43540			−0.717702	+	0.696350i	$0.754806\pi$
$632$	−473.357			−0.748983
$633$		0			0
$634$	−824.560			−1.30057
$635$			113.810i			0.179228i
$636$		0			0
$637$	−181.056	+	69.8281i	−0.284232	+	0.109620i
$638$	1141.49			1.78916
$639$		0			0
$640$			1006.90i			1.57328i
$641$	800.150			1.24828			0.624142	−	0.781311i	$-0.285449\pi$
$641$	800.150			1.24828			0.624142	+	0.781311i	$0.285449\pi$
$642$		0			0
$643$		−	1207.83i		−	1.87843i	−0.343333	−	0.939214i	$-0.611556\pi$
$643$		−	1207.83i		−	1.87843i	0.343333	−	0.939214i	$-0.388444\pi$
$644$	−12.5115	−	8.58444i	−0.0194278	−	0.0133299i
$645$		0			0
$646$	1097.06			1.69824
$647$		−	377.781i		−	0.583897i	−0.956434	−	0.291949i	$-0.905696\pi$
$647$		−	377.781i		−	0.583897i	0.956434	−	0.291949i	$-0.0943036\pi$
$648$		0			0
$649$		−	789.637i		−	1.21670i
$650$		−	160.131i		−	0.246356i
$651$		0			0
$652$	−397.036			−0.608952
$653$	728.674			1.11589			0.557944	−	0.829879i	$-0.311590\pi$
$653$	728.674			1.11589			0.557944	+	0.829879i	$0.311590\pi$
$654$		0			0
$655$	−965.256			−1.47367
$656$			478.185i			0.728941i
$657$		0			0
$658$	−295.800	+	431.119i	−0.449544	+	0.655195i
$659$	−813.274			−1.23410			−0.617052	−	0.786923i	$-0.711673\pi$
$659$	−813.274			−1.23410			−0.617052	+	0.786923i	$0.711673\pi$
$660$		0			0
$661$		−	505.192i		−	0.764285i	−0.924103	−	0.382142i	$-0.875186\pi$
$661$		−	505.192i		−	0.764285i	0.924103	−	0.382142i	$-0.124814\pi$
$662$	−158.810			−0.239894
$663$		0			0
$664$		−	799.855i		−	1.20460i
$665$	1160.70	+	796.383i	1.74541	+	1.19757i
$666$		0			0
$667$	−66.7720			−0.100108
$668$		−	1.82920i		−	0.00273833i
$669$		0			0
$670$			1368.05i			2.04186i
$671$		−	430.048i		−	0.640906i
$672$		0			0
$673$	112.068			0.166520			0.0832600	−	0.996528i	$-0.473467\pi$
$673$	112.068			0.166520			0.0832600	+	0.996528i	$0.473467\pi$
$674$	−372.056			−0.552012
$675$		0			0
$676$	188.272			0.278509
$677$			1053.59i			1.55626i	0.628105	+	0.778129i	$0.283831\pi$
$677$			1053.59i			1.55626i	−0.628105	+	0.778129i	$0.716169\pi$
$678$		0			0
$679$	−191.784	+	279.518i	−0.282450	+	0.411662i
$680$	645.501			0.949266
$681$		0			0
$682$		−	1738.10i		−	2.54853i
$683$	272.576			0.399087			0.199543	−	0.979889i	$-0.436054\pi$
$683$	272.576			0.399087			0.199543	+	0.979889i	$0.436054\pi$
$684$		0			0
$685$		−	1323.23i		−	1.93173i
$686$	−181.282	+	763.024i	−0.264259	+	1.11228i
$687$		0			0
$688$	739.060			1.07422
$689$		−	29.2300i		−	0.0424239i
$690$		0			0
$691$			891.908i			1.29075i	0.763866	+	0.645375i	$0.223299\pi$
$691$			891.908i			1.29075i	−0.763866	+	0.645375i	$0.776701\pi$
$692$		−	175.380i		−	0.253440i
$693$		0			0
$694$	205.544			0.296173
$695$	−1602.83			−2.30623
$696$		0			0
$697$	384.156			0.551156
$698$			1133.02i			1.62323i
$699$		0			0
$700$	−125.344	−	86.0016i	−0.179063	−	0.122859i
$701$	−1053.60			−1.50300			−0.751499	−	0.659734i	$-0.770669\pi$
$701$	−1053.60			−1.50300			−0.751499	+	0.659734i	$0.770669\pi$
$702$		0			0
$703$			1285.72i			1.82891i
$704$	450.565			0.640007
$705$		0			0
$706$			1010.57i			1.43141i
$707$	629.148	−	916.962i	0.889885	−	1.29698i
$708$		0			0
$709$	488.340			0.688773			0.344387	−	0.938828i	$-0.388087\pi$
$709$	488.340			0.688773			0.344387	+	0.938828i	$0.388087\pi$
$710$		−	147.418i		−	0.207631i
$711$		0			0
$712$		−	172.178i		−	0.241824i
$713$			101.671i			0.142596i
$714$		0			0
$715$	−341.472			−0.477583
$716$	−336.003			−0.469278
$717$		0			0
$718$	1185.02			1.65045
$719$		−	310.044i		−	0.431216i	−0.976480	−	0.215608i	$-0.930827\pi$
$719$		−	310.044i		−	0.431216i	0.976480	−	0.215608i	$-0.0691733\pi$
$720$		0			0
$721$	461.172	−	672.143i	0.639629	−	0.932237i
$722$	−1340.74			−1.85698
$723$		0			0
$724$		−	74.9135i		−	0.103472i
$725$	−668.942			−0.922679
$726$		0			0
$727$		−	606.623i		−	0.834420i	−0.908810	−	0.417210i	$-0.863008\pi$
$727$		−	606.623i		−	0.834420i	0.908810	−	0.417210i	$-0.136992\pi$
$728$	99.4072	−	144.883i	0.136548	−	0.199014i
$729$		0			0
$730$	354.960			0.486247
$731$		−	593.733i		−	0.812221i
$732$		0			0
$733$			1186.02i			1.61803i	0.587788	+	0.809015i	$0.299999\pi$
$733$			1186.02i			1.61803i	−0.587788	+	0.809015i	$0.700001\pi$
$734$		−	306.608i		−	0.417722i
$735$		0			0
$736$	33.5636			0.0456027
$737$	1208.63			1.63994
$738$		0			0
$739$	559.472			0.757066			0.378533	−	0.925588i	$-0.376429\pi$
$739$	559.472			0.757066			0.378533	+	0.925588i	$0.376429\pi$
$740$		−	335.132i		−	0.452881i
$741$		0			0
$742$	−97.4081	−	66.8339i	−0.131278	−	0.0900727i
$743$	−272.091			−0.366206			−0.183103	−	0.983094i	$-0.558614\pi$
$743$	−272.091			−0.366206			−0.183103	+	0.983094i	$0.558614\pi$
$744$		0			0
$745$			422.216i			0.566732i
$746$	1211.97			1.62463
$747$		0			0
$748$			252.636i			0.337748i
$749$	369.532	+	253.544i	0.493367	+	0.338511i
$750$		0			0
$751$	37.6562			0.0501414			0.0250707	−	0.999686i	$-0.492019\pi$
$751$	37.6562			0.0501414			0.0250707	+	0.999686i	$0.492019\pi$
$752$		−	633.861i		−	0.842900i
$753$		0			0
$754$			342.535i			0.454290i
$755$			1036.52i			1.37288i
$756$		0			0
$757$	369.588			0.488227			0.244113	−	0.969747i	$-0.421503\pi$
$757$	369.588			0.488227			0.244113	+	0.969747i	$0.421503\pi$
$758$	1152.19			1.52005
$759$		0			0
$760$	−1274.54			−1.67703
$761$		−	1232.17i		−	1.61914i	−0.587022	−	0.809571i	$-0.699700\pi$
$761$		−	1232.17i		−	1.61914i	0.587022	−	0.809571i	$-0.300300\pi$
$762$		0			0
$763$	121.004	+	83.0237i	0.158590	+	0.108812i
$764$	7.39015			0.00967297
$765$		0			0
$766$			196.289i			0.256252i
$767$	236.953			0.308934
$768$		0			0
$769$		−	80.3149i		−	0.104441i	−0.998636	−	0.0522203i	$-0.983370\pi$
$769$		−	80.3149i		−	0.104441i	0.998636	−	0.0522203i	$-0.0166298\pi$
$770$	−780.769	+	1137.94i	−1.01399	+	1.47785i
$771$		0			0
$772$	−272.468			−0.352938
$773$		−	638.774i		−	0.826357i	−0.910650	−	0.413178i	$-0.864419\pi$
$773$		−	638.774i		−	0.826357i	0.910650	−	0.413178i	$-0.135581\pi$
$774$		0			0
$775$			1018.57i			1.31429i
$776$		−	306.934i		−	0.395533i
$777$		0			0
$778$	1175.02			1.51031
$779$	−758.518			−0.973707
$780$		0			0
$781$	−130.240			−0.166760
$782$		−	62.9154i		−	0.0804545i
$783$		0			0
$784$	−342.132	−	887.108i	−0.436393	−	1.13151i
$785$	−519.754			−0.662107
$786$		0			0
$787$		−	25.1560i		−	0.0319644i	−0.999872	−	0.0159822i	$-0.994912\pi$
$787$		−	25.1560i		−	0.0319644i	0.999872	−	0.0159822i	$-0.00508750\pi$
$788$	−215.807			−0.273867
$789$		0			0
$790$			1115.65i			1.41221i
$791$	−935.180	−	641.649i	−1.18228	−	0.811187i
$792$		0			0
$793$	129.048			0.162734
$794$		−	533.457i		−	0.671861i
$795$		0			0
$796$			52.3869i			0.0658127i
$797$			92.1454i			0.115615i	0.998328	+	0.0578077i	$0.0184110\pi$
$797$			92.1454i			0.115615i	−0.998328	+	0.0578077i	$0.981589\pi$
$798$		0			0
$799$	−509.220			−0.637322
$800$	336.250			0.420313
$801$		0			0
$802$	−753.208			−0.939163
$803$		−	313.598i		−	0.390533i
$804$		0			0
$805$	45.6717	−	66.5649i	0.0567350	−	0.0826893i
$806$	521.565			0.647103
$807$		0			0
$808$			1006.90i			1.24616i
$809$	−906.892			−1.12100			−0.560502	−	0.828153i	$-0.689392\pi$
$809$	−906.892			−1.12100			−0.560502	+	0.828153i	$0.689392\pi$
$810$		0			0
$811$		−	177.659i		−	0.219062i	−0.993983	−	0.109531i	$-0.965065\pi$
$811$		−	177.659i		−	0.219062i	0.993983	−	0.109531i	$-0.0349349\pi$
$812$	268.122	+	183.965i	0.330200	+	0.226557i
$813$		0			0
$814$	−1260.51			−1.54854
$815$		−	2112.35i		−	2.59184i
$816$		0			0
$817$			1172.33i			1.43492i
$818$		−	126.772i		−	0.154978i
$819$		0			0
$820$	197.713			0.241113
$821$	−147.698			−0.179900			−0.0899498	−	0.995946i	$-0.528671\pi$
$821$	−147.698			−0.179900			−0.0899498	+	0.995946i	$0.528671\pi$
$822$		0			0
$823$	467.732			0.568325			0.284163	−	0.958776i	$-0.408284\pi$
$823$	467.732			0.568325			0.284163	+	0.958776i	$0.408284\pi$
$824$			738.068i			0.895713i
$825$		0			0
$826$	541.788	−	789.637i	0.655918	−	0.955977i
$827$	−52.2690			−0.0632031			−0.0316016	−	0.999501i	$-0.510061\pi$
$827$	−52.2690			−0.0632031			−0.0316016	+	0.999501i	$0.510061\pi$
$828$		0			0
$829$		−	1267.38i		−	1.52880i	−0.644742	−	0.764401i	$-0.723035\pi$
$829$		−	1267.38i		−	1.52880i	0.644742	−	0.764401i	$-0.276965\pi$
$830$	−1885.17			−2.27129
$831$		0			0
$832$			135.205i			0.162506i
$833$	−712.669	+	274.856i	−0.855545	+	0.329959i
$834$		0			0
$835$	9.73187			0.0116549
$836$		−	498.831i		−	0.596687i
$837$		0			0
$838$			1038.36i			1.23909i
$839$		−	1298.30i		−	1.54743i	−0.633531	−	0.773717i	$-0.718395\pi$
$839$		−	1298.30i		−	1.54743i	0.633531	−	0.773717i	$-0.281605\pi$
$840$		0			0
$841$	589.924			0.701456
$842$	−628.911			−0.746925
$843$		0			0
$844$	−229.483			−0.271900
$845$			1001.66i			1.18540i
$846$		0			0
$847$	306.932	+	210.593i	0.362376	+	0.248634i
$848$	143.216			0.168887
$849$		0			0
$850$		−	630.306i		−	0.741536i
$851$	73.7346			0.0866447
$852$		0			0
$853$		−	1200.05i		−	1.40686i	−0.710765	−	0.703430i	$-0.751651\pi$
$853$		−	1200.05i		−	1.40686i	0.710765	−	0.703430i	$-0.248349\pi$
$854$	295.066	−	430.048i	0.345510	−	0.503569i
$855$		0			0
$856$	−405.777			−0.474038
$857$			186.369i			0.217467i	0.994071	+	0.108733i	$0.0346794\pi$
$857$			186.369i			0.217467i	−0.994071	+	0.108733i	$0.965321\pi$
$858$		0			0
$859$		−	683.960i		−	0.796229i	−0.917336	−	0.398114i	$-0.869665\pi$
$859$		−	683.960i		−	0.796229i	0.917336	−	0.398114i	$-0.130335\pi$
$860$		−	305.575i		−	0.355320i
$861$		0			0
$862$	−713.392			−0.827601
$863$	−886.597			−1.02734			−0.513672	−	0.857987i	$-0.671715\pi$
$863$	−886.597			−1.02734			−0.513672	+	0.857987i	$0.671715\pi$
$864$		0			0
$865$	933.073			1.07870
$866$			176.503i			0.203814i
$867$		0			0
$868$	280.116	−	408.260i	0.322715	−	0.470345i
$869$	985.648			1.13423
$870$		0			0
$871$			362.685i			0.416400i
$872$	−132.872			−0.152377
$873$		0			0
$874$			124.227i			0.142136i
$875$	−189.293	+	275.888i	−0.216335	+	0.315301i
$876$		0			0
$877$	−855.004			−0.974919			−0.487460	−	0.873146i	$-0.662076\pi$
$877$	−855.004			−0.974919			−0.487460	+	0.873146i	$0.662076\pi$
$878$		−	770.812i		−	0.877918i
$879$		0			0
$880$		−	1673.09i		−	1.90123i
$881$		−	14.7781i		−	0.0167743i	−0.999965	−	0.00838713i	$-0.997330\pi$
$881$		−	14.7781i		−	0.0167743i	0.999965	−	0.00838713i	$-0.00266974\pi$
$882$		0			0
$883$	−623.784			−0.706438			−0.353219	−	0.935541i	$-0.614913\pi$
$883$	−623.784			−0.706438			−0.353219	+	0.935541i	$0.614913\pi$
$884$	−75.8105			−0.0857584
$885$		0			0
$886$	311.024			0.351043
$887$			810.939i			0.914250i	0.889403	+	0.457125i	$0.151121\pi$
$887$			810.939i			0.914250i	−0.889403	+	0.457125i	$0.848879\pi$
$888$		0			0
$889$	100.548	+	68.9884i	0.113102	+	0.0776022i
$890$	−405.805			−0.455961
$891$		0			0
$892$		−	394.098i		−	0.441814i
$893$	1005.46			1.12593
$894$		0			0
$895$		−	1787.63i		−	1.99736i
$896$	889.570	+	610.354i	0.992823	+	0.681199i
$897$		0			0
$898$	−407.512			−0.453800
$899$		−	2178.82i		−	2.42360i
$900$		0			0
$901$		−	115.055i		−	0.127697i
$902$		−	743.646i		−	0.824441i
$903$		0			0
$904$	1026.90			1.13596
$905$	398.561			0.440399
$906$		0			0
$907$	−440.100			−0.485226			−0.242613	−	0.970123i	$-0.578005\pi$
$907$	−440.100			−0.485226			−0.242613	+	0.970123i	$0.578005\pi$
$908$			11.1197i			0.0122464i
$909$		0			0
$910$	−341.472	−	234.292i	−0.375244	−	0.257463i
$911$	1396.74			1.53319			0.766597	−	0.642129i	$-0.221949\pi$
$911$	1396.74			1.53319			0.766597	+	0.642129i	$0.221949\pi$
$912$		0			0
$913$			1665.50i			1.82420i
$914$	256.241			0.280352
$915$		0			0
$916$			282.068i			0.307935i
$917$	−585.111	+	852.779i	−0.638071	+	0.929966i
$918$		0			0
$919$	−77.7720			−0.0846268			−0.0423134	−	0.999104i	$-0.513473\pi$
$919$	−77.7720			−0.0846268			−0.0423134	+	0.999104i	$0.513473\pi$
$920$			73.0937i			0.0794497i
$921$		0			0
$922$		−	88.3149i		−	0.0957863i
$923$		−	39.0821i		−	0.0423425i
$924$		0			0
$925$	738.696			0.798590
$926$	229.974			0.248352
$927$		0			0
$928$	−719.268			−0.775073
$929$			293.763i			0.316214i	0.987422	+	0.158107i	$0.0505391\pi$
$929$			293.763i			0.316214i	−0.987422	+	0.158107i	$0.949461\pi$
$930$		0			0
$931$	1407.17	−	542.704i	1.51146	−	0.582926i
$932$	−14.4320			−0.0154850
$933$		0			0
$934$		−	211.765i		−	0.226729i
$935$	−1344.09			−1.43753
$936$		0			0
$937$			800.758i			0.854597i	0.904111	+	0.427299i	$0.140535\pi$
$937$			800.758i			0.854597i	−0.904111	+	0.427299i	$0.859465\pi$
$938$	1208.63	+	829.272i	1.28852	+	0.884085i
$939$		0			0
$940$	−262.079			−0.278808
$941$			934.732i			0.993339i	0.867940	+	0.496670i	$0.165444\pi$
$941$			934.732i			0.993339i	−0.867940	+	0.496670i	$0.834556\pi$
$942$		0			0
$943$			43.5001i			0.0461295i
$944$			1160.98i			1.22985i
$945$		0			0
$946$	−1149.34			−1.21495
$947$	43.4796			0.0459130			0.0229565	−	0.999736i	$-0.492692\pi$
$947$	43.4796			0.0459130			0.0229565	+	0.999736i	$0.492692\pi$
$948$		0			0
$949$	94.1040			0.0991612
$950$			1244.54i			1.31004i
$951$		0			0
$952$	391.285	−	570.284i	0.411013	−	0.599037i
$953$	−467.247			−0.490291			−0.245145	−	0.969486i	$-0.578836\pi$
$953$	−467.247			−0.490291			−0.245145	+	0.969486i	$0.578836\pi$
$954$		0			0
$955$			39.3177i			0.0411704i
$956$	−541.219			−0.566129
$957$		0			0
$958$		−	1261.39i		−	1.31669i
$959$	−1169.04	−	802.106i	−1.21902	−	0.836399i
$960$		0			0
$961$	−2356.60			−2.45224
$962$		−	378.252i		−	0.393193i
$963$		0			0
$964$		−	429.930i		−	0.445986i
$965$		−	1449.61i		−	1.50219i
$966$		0			0
$967$	550.640			0.569431			0.284716	−	0.958612i	$-0.408101\pi$
$967$	550.640			0.569431			0.284716	+	0.958612i	$0.408101\pi$
$968$	−337.037			−0.348178
$969$		0			0
$970$	−723.409			−0.745783
$971$			1028.38i			1.05909i	0.848280	+	0.529547i	$0.177638\pi$
$971$			1028.38i			1.05909i	−0.848280	+	0.529547i	$0.822362\pi$
$972$		0			0
$973$	−971.592	+	1416.06i	−0.998553	+	1.45536i
$974$	393.677			0.404186
$975$		0			0
$976$			632.287i			0.647835i
$977$	1121.51			1.14791			0.573956	−	0.818886i	$-0.305408\pi$
$977$	1121.51			1.14791			0.573956	+	0.818886i	$0.305408\pi$
$978$		0			0
$979$			358.518i			0.366209i
$980$	−366.788	+	141.459i	−0.374273	+	0.144346i
$981$		0			0
$982$	1377.54			1.40279
$983$		−	884.975i		−	0.900279i	−0.892958	−	0.450140i	$-0.851374\pi$
$983$		−	884.975i		−	0.900279i	0.892958	−	0.450140i	$-0.148626\pi$
$984$		0			0
$985$		−	1148.16i		−	1.16564i
$986$			1348.28i			1.36742i
$987$		0			0
$988$	149.688			0.151506
$989$	67.2317			0.0679795
$990$		0			0
$991$	−886.836			−0.894890			−0.447445	−	0.894311i	$-0.647666\pi$
$991$	−886.836			−0.894890			−0.447445	+	0.894311i	$0.647666\pi$
$992$			1095.20i			1.10404i
$993$		0			0
$994$	−130.240	−	89.3605i	−0.131026	−	0.0898999i
$995$	−278.713			−0.280114
$996$		0			0
$997$			656.935i			0.658912i	0.944171	+	0.329456i	$0.106865\pi$
$997$			656.935i			0.658912i	−0.944171	+	0.329456i	$0.893135\pi$
$998$	518.711			0.519751
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.d.e.55.6	yes	8
3.2	odd	2	inner	189.3.d.e.55.3	✓	8
7.6	odd	2	inner	189.3.d.e.55.5	yes	8
21.20	even	2	inner	189.3.d.e.55.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.d.e.55.3	✓	8	3.2	odd	2	inner
189.3.d.e.55.4	yes	8	21.20	even	2	inner
189.3.d.e.55.5	yes	8	7.6	odd	2	inner
189.3.d.e.55.6	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)

\(f(q)\)	\(=\)	\(q+2.28648 q^{2} +1.22800 q^{4} +6.53330i q^{5} +(5.77200 + 3.96030i) q^{7} -6.33813 q^{8} +14.9383i q^{10} +13.1976 q^{11} +3.96030i q^{13} +(13.1976 + 9.05516i) q^{14} -19.4040 q^{16} +15.5885i q^{17} -30.7795i q^{19} +8.02288i q^{20} +30.1760 q^{22} -1.76517 q^{23} -17.6840 q^{25} +9.05516i q^{26} +(7.08801 + 4.86324i) q^{28} +37.8276 q^{29} -57.5987i q^{31} -19.0144 q^{32} +35.6427i q^{34} +(-25.8738 + 37.7102i) q^{35} -41.7720 q^{37} -70.3767i q^{38} -41.4089i q^{40} -24.6436i q^{41} -38.0880 q^{43} +16.2066 q^{44} -4.03602 q^{46} +32.6665i q^{47} +(17.6320 + 45.7177i) q^{49} -40.4341 q^{50} +4.86324i q^{52} -7.38076 q^{53} +86.2237i q^{55} +(-36.5837 - 25.1009i) q^{56} +86.4920 q^{58} -59.8320i q^{59} -32.5854i q^{61} -131.698i q^{62} +34.1400 q^{64} -25.8738 q^{65} +91.5800 q^{67} +19.1426i q^{68} +(-59.1601 + 86.2237i) q^{70} -9.86847 q^{71} -23.7618i q^{73} -95.5109 q^{74} -37.7971i q^{76} +(76.1764 + 52.2664i) q^{77} +74.6840 q^{79} -126.772i q^{80} -56.3472i q^{82} +126.197i q^{83} -101.844 q^{85} -87.0875 q^{86} -83.6480 q^{88} +27.1655i q^{89} +(-15.6840 + 22.8589i) q^{91} -2.16762 q^{92} +74.6913i q^{94} +201.092 q^{95} +48.4266i q^{97} +(40.3153 + 104.533i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4} + 12 q^{7} + 84 q^{16} - 32 q^{22} - 244 q^{25} - 80 q^{28} - 300 q^{37} - 168 q^{43} + 412 q^{46} - 64 q^{49} + 316 q^{58} + 444 q^{64} + 220 q^{67} + 552 q^{70} + 700 q^{79} + 108 q^{85}+ \cdots - 228 q^{91}+O(q^{100}) \) 8 * q + 44 * q^4 + 12 * q^7 + 84 * q^16 - 32 * q^22 - 244 * q^25 - 80 * q^28 - 300 * q^37 - 168 * q^43 + 412 * q^46 - 64 * q^49 + 316 * q^58 + 444 * q^64 + 220 * q^67 + 552 * q^70 + 700 * q^79 + 108 * q^85 - 1216 * q^88 - 228 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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