LMFDB - Embedded newform 189.3.b.c.134.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(134,189)]

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([1, 0]))

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

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

chi := DirichletCharacter("189.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.b (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:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.3
Root			$2.92812i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.c.134.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.928117i q^{2} +3.13860 q^{4} -4.06404i q^{5} -2.64575 q^{7} -6.62545i q^{8} +O(q^{10})$	\(q-0.928117i q^{2} +3.13860 q^{4} -4.06404i q^{5} -2.64575 q^{7} -6.62545i q^{8} -3.77190 q^{10} -9.90533i q^{11} +9.26916 q^{13} +2.45557i q^{14} +6.40520 q^{16} +7.63929i q^{17} -13.1169 q^{19} -12.7554i q^{20} -9.19331 q^{22} -33.2491i q^{23} +8.48361 q^{25} -8.60286i q^{26} -8.30395 q^{28} +12.8183i q^{29} +22.1741 q^{31} -32.4466i q^{32} +7.09016 q^{34} +10.7524i q^{35} -56.7478 q^{37} +12.1740i q^{38} -26.9261 q^{40} +31.1738i q^{41} +55.1493 q^{43} -31.0889i q^{44} -30.8590 q^{46} +56.4863i q^{47} +7.00000 q^{49} -7.87378i q^{50} +29.0922 q^{52} +83.3823i q^{53} -40.2556 q^{55} +17.5293i q^{56} +11.8969 q^{58} +112.380i q^{59} +39.7968 q^{61} -20.5801i q^{62} -4.49342 q^{64} -37.6702i q^{65} +93.9405 q^{67} +23.9767i q^{68} +9.97951 q^{70} +11.1120i q^{71} -39.2693 q^{73} +52.6686i q^{74} -41.1687 q^{76} +26.2070i q^{77} +44.3997 q^{79} -26.0310i q^{80} +28.9329 q^{82} -118.691i q^{83} +31.0464 q^{85} -51.1850i q^{86} -65.6273 q^{88} +110.348i q^{89} -24.5239 q^{91} -104.355i q^{92} +52.4259 q^{94} +53.3076i q^{95} -138.581 q^{97} -6.49682i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4}+O(q^{10}) $ 8 * q - 12 * q^4	$ 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) $ 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

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 $

$2$	− 0.928117i	− 0.464058i	−0.972709	−	0.232029i	$-0.925464\pi$
$2$	− 0.928117i	− 0.464058i	0.972709	−	0.232029i	$-0.0745365\pi$
$3$	0	0
$3$	0	0
$4$	3.13860	0.784650
$5$	− 4.06404i	− 0.812807i	−0.913694	−	0.406404i	$-0.866783\pi$
$5$	− 4.06404i	− 0.812807i	0.913694	−	0.406404i	$-0.133217\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	− 6.62545i	− 0.828182i
$9$	0	0
$10$	−3.77190	−0.377190
$11$	− 9.90533i	− 0.900485i	−0.892906	−	0.450242i	$-0.851338\pi$
$11$	− 9.90533i	− 0.900485i	0.892906	−	0.450242i	$-0.148662\pi$
$12$	0	0
$13$	9.26916	0.713012	0.356506	−	0.934293i	$-0.383968\pi$
$13$	9.26916	0.713012	0.356506	+	0.934293i	$0.383968\pi$
$14$	2.45557i	0.175398i
$15$	0	0
$16$	6.40520	0.400325
$17$	7.63929i	0.449370i	0.974431	+	0.224685i	$0.0721353\pi$
$17$	7.63929i	0.449370i	−0.974431	+	0.224685i	$0.927865\pi$
$18$	0	0
$19$	−13.1169	−0.690364	−0.345182	−	0.938536i	$-0.612183\pi$
$19$	−13.1169	−0.690364	−0.345182	+	0.938536i	$0.612183\pi$
$20$	− 12.7554i	− 0.637769i
$21$	0	0
$22$	−9.19331	−0.417878
$23$	− 33.2491i	− 1.44561i	−0.691052	−	0.722806i	$-0.742852\pi$
$23$	− 33.2491i	− 1.44561i	0.691052	−	0.722806i	$-0.257148\pi$
$24$	0	0
$25$	8.48361	0.339344
$26$	− 8.60286i	− 0.330879i
$27$	0	0
$28$	−8.30395	−0.296570
$29$	12.8183i	0.442011i	0.975273	+	0.221005i	$0.0709339\pi$
$29$	12.8183i	0.442011i	−0.975273	+	0.221005i	$0.929066\pi$
$30$	0	0
$31$	22.1741	0.715292	0.357646	−	0.933857i	$-0.383579\pi$
$31$	22.1741	0.715292	0.357646	+	0.933857i	$0.383579\pi$
$32$	− 32.4466i	− 1.01396i
$33$	0	0
$34$	7.09016	0.208534
$35$	10.7524i	0.307212i
$36$	0	0
$37$	−56.7478	−1.53372	−0.766862	−	0.641812i	$-0.778183\pi$
$37$	−56.7478	−1.53372	−0.766862	+	0.641812i	$0.778183\pi$
$38$	12.1740i	0.320369i
$39$	0	0
$40$	−26.9261	−0.673152
$41$	31.1738i	0.760336i	0.924918	+	0.380168i	$0.124134\pi$
$41$	31.1738i	0.760336i	−0.924918	+	0.380168i	$0.875866\pi$
$42$	0	0
$43$	55.1493	1.28254	0.641271	−	0.767314i	$-0.278407\pi$
$43$	55.1493	1.28254	0.641271	+	0.767314i	$0.278407\pi$
$44$	− 31.0889i	− 0.706565i
$45$	0	0
$46$	−30.8590	−0.670848
$47$	56.4863i	1.20184i	0.799310	+	0.600918i	$0.205198\pi$
$47$	56.4863i	1.20184i	−0.799310	+	0.600918i	$0.794802\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 7.87378i	− 0.157476i
$51$	0	0
$52$	29.0922	0.559465
$53$	83.3823i	1.57325i	0.617431	+	0.786625i	$0.288174\pi$
$53$	83.3823i	1.57325i	−0.617431	+	0.786625i	$0.711826\pi$
$54$	0	0
$55$	−40.2556	−0.731920
$56$	17.5293i	0.313023i
$57$	0	0
$58$	11.8969	0.205119
$59$	112.380i	1.90475i	0.304924	+	0.952377i	$0.401369\pi$
$59$	112.380i	1.90475i	−0.304924	+	0.952377i	$0.598631\pi$
$60$	0	0
$61$	39.7968	0.652407	0.326203	−	0.945300i	$-0.394231\pi$
$61$	39.7968	0.652407	0.326203	+	0.945300i	$0.394231\pi$
$62$	− 20.5801i	− 0.331937i
$63$	0	0
$64$	−4.49342	−0.0702097
$65$	− 37.6702i	− 0.579541i
$66$	0	0
$67$	93.9405	1.40210	0.701049	−	0.713114i	$-0.252716\pi$
$67$	93.9405	1.40210	0.701049	+	0.713114i	$0.252716\pi$
$68$	23.9767i	0.352598i
$69$	0	0
$70$	9.97951	0.142564
$71$	11.1120i	0.156506i	0.996934	+	0.0782532i	$0.0249343\pi$
$71$	11.1120i	0.156506i	−0.996934	+	0.0782532i	$0.975066\pi$
$72$	0	0
$73$	−39.2693	−0.537936	−0.268968	−	0.963149i	$-0.586683\pi$
$73$	−39.2693	−0.537936	−0.268968	+	0.963149i	$0.586683\pi$
$74$	52.6686i	0.711738i
$75$	0	0
$76$	−41.1687	−0.541694
$77$	26.2070i	0.340351i
$78$	0	0
$79$	44.3997	0.562022	0.281011	−	0.959705i	$-0.409330\pi$
$79$	44.3997	0.562022	0.281011	+	0.959705i	$0.409330\pi$
$80$	− 26.0310i	− 0.325387i
$81$	0	0
$82$	28.9329	0.352840
$83$	− 118.691i	− 1.43001i	−0.699121	−	0.715004i	$-0.746425\pi$
$83$	− 118.691i	− 1.43001i	0.699121	−	0.715004i	$-0.253575\pi$
$84$	0	0
$85$	31.0464	0.365251
$86$	− 51.1850i	− 0.595174i
$87$	0	0
$88$	−65.6273	−0.745765
$89$	110.348i	1.23987i	0.784654	+	0.619934i	$0.212840\pi$
$89$	110.348i	1.23987i	−0.784654	+	0.619934i	$0.787160\pi$
$90$	0	0
$91$	−24.5239	−0.269493
$92$	− 104.355i	− 1.13430i
$93$	0	0
$94$	52.4259	0.557722
$95$	53.3076i	0.561133i
$96$	0	0
$97$	−138.581	−1.42867	−0.714337	−	0.699802i	$-0.753271\pi$
$97$	−138.581	−1.42867	−0.714337	+	0.699802i	$0.753271\pi$
$98$	− 6.49682i	− 0.0662941i
$99$	0	0
$100$	26.6267	0.266267
$101$	112.445i	1.11332i	0.830741	+	0.556659i	$0.187917\pi$
$101$	112.445i	1.11332i	−0.830741	+	0.556659i	$0.812083\pi$
$102$	0	0
$103$	−137.105	−1.33111	−0.665556	−	0.746348i	$-0.731806\pi$
$103$	−137.105	−1.33111	−0.665556	+	0.746348i	$0.731806\pi$
$104$	− 61.4124i	− 0.590504i
$105$	0	0
$106$	77.3885	0.730080
$107$	− 191.559i	− 1.79027i	−0.445795	−	0.895135i	$-0.647079\pi$
$107$	− 191.559i	− 1.79027i	0.445795	−	0.895135i	$-0.352921\pi$
$108$	0	0
$109$	−36.1672	−0.331809	−0.165905	−	0.986142i	$-0.553054\pi$
$109$	−36.1672	−0.331809	−0.165905	+	0.986142i	$0.553054\pi$
$110$	37.3619i	0.339654i
$111$	0	0
$112$	−16.9466	−0.151309
$113$	− 15.6708i	− 0.138680i	−0.997593	−	0.0693399i	$-0.977911\pi$
$113$	− 15.6708i	− 0.138680i	0.997593	−	0.0693399i	$-0.0220893\pi$
$114$	0	0
$115$	−135.125	−1.17500
$116$	40.2316i	0.346824i
$117$	0	0
$118$	104.302	0.883917
$119$	− 20.2117i	− 0.169846i
$120$	0	0
$121$	22.8844	0.189127
$122$	− 36.9361i	− 0.302755i
$123$	0	0
$124$	69.5955	0.561254
$125$	− 136.079i	− 1.08863i
$126$	0	0
$127$	17.6053	0.138624	0.0693122	−	0.997595i	$-0.477920\pi$
$127$	17.6053	0.138624	0.0693122	+	0.997595i	$0.477920\pi$
$128$	− 125.616i	− 0.981375i
$129$	0	0
$130$	−34.9623	−0.268941
$131$	45.4693i	0.347094i	0.984826	+	0.173547i	$0.0555228\pi$
$131$	45.4693i	0.347094i	−0.984826	+	0.173547i	$0.944477\pi$
$132$	0	0
$133$	34.7041	0.260933
$134$	− 87.1878i	− 0.650655i
$135$	0	0
$136$	50.6138	0.372160
$137$	79.4453i	0.579893i	0.957043	+	0.289946i	$0.0936375\pi$
$137$	79.4453i	0.579893i	−0.957043	+	0.289946i	$0.906363\pi$
$138$	0	0
$139$	204.211	1.46914	0.734572	−	0.678531i	$-0.237383\pi$
$139$	204.211	1.46914	0.734572	+	0.678531i	$0.237383\pi$
$140$	33.7476i	0.241054i
$141$	0	0
$142$	10.3132	0.0726282
$143$	− 91.8141i	− 0.642057i
$144$	0	0
$145$	52.0941	0.359270
$146$	36.4465i	0.249634i
$147$	0	0
$148$	−178.109	−1.20344
$149$	− 189.416i	− 1.27125i	−0.771998	−	0.635625i	$-0.780743\pi$
$149$	− 189.416i	− 1.27125i	0.771998	−	0.635625i	$-0.219257\pi$
$150$	0	0
$151$	−65.0282	−0.430651	−0.215325	−	0.976542i	$-0.569081\pi$
$151$	−65.0282	−0.430651	−0.215325	+	0.976542i	$0.569081\pi$
$152$	86.9055i	0.571747i
$153$	0	0
$154$	24.3232	0.157943
$155$	− 90.1161i	− 0.581394i
$156$	0	0
$157$	3.12356	0.0198953	0.00994763	−	0.999951i	$-0.496834\pi$
$157$	3.12356	0.0198953	0.00994763	+	0.999951i	$0.496834\pi$
$158$	− 41.2081i	− 0.260811i
$159$	0	0
$160$	−131.864	−0.824151
$161$	87.9687i	0.546390i
$162$	0	0
$163$	−45.5693	−0.279566	−0.139783	−	0.990182i	$-0.544641\pi$
$163$	−45.5693	−0.279566	−0.139783	+	0.990182i	$0.544641\pi$
$164$	97.8419i	0.596597i
$165$	0	0
$166$	−110.159	−0.663607
$167$	− 237.883i	− 1.42445i	−0.701953	−	0.712224i	$-0.747688\pi$
$167$	− 237.883i	− 1.42445i	0.701953	−	0.712224i	$-0.252312\pi$
$168$	0	0
$169$	−83.0827	−0.491614
$170$	− 28.8147i	− 0.169498i
$171$	0	0
$172$	173.092	1.00635
$173$	− 102.292i	− 0.591283i	−0.955299	−	0.295641i	$-0.904467\pi$
$173$	− 102.292i	− 0.591283i	0.955299	−	0.295641i	$-0.0955333\pi$
$174$	0	0
$175$	−22.4455	−0.128260
$176$	− 63.4456i	− 0.360487i
$177$	0	0
$178$	102.416	0.575371
$179$	40.8818i	0.228390i	0.993458	+	0.114195i	$0.0364289\pi$
$179$	40.8818i	0.228390i	−0.993458	+	0.114195i	$0.963571\pi$
$180$	0	0
$181$	249.507	1.37849	0.689245	−	0.724528i	$-0.257942\pi$
$181$	249.507	1.37849	0.689245	+	0.724528i	$0.257942\pi$
$182$	22.7610i	0.125061i
$183$	0	0
$184$	−220.290	−1.19723
$185$	230.625i	1.24662i
$186$	0	0
$187$	75.6697	0.404651
$188$	177.288i	0.943021i
$189$	0	0
$190$	49.4757	0.260398
$191$	− 47.2999i	− 0.247643i	−0.992304	−	0.123822i	$-0.960485\pi$
$191$	− 47.2999i	− 0.247643i	0.992304	−	0.123822i	$-0.0395151\pi$
$192$	0	0
$193$	142.456	0.738113	0.369057	−	0.929407i	$-0.379681\pi$
$193$	142.456	0.738113	0.369057	+	0.929407i	$0.379681\pi$
$194$	128.620i	0.662988i
$195$	0	0
$196$	21.9702	0.112093
$197$	360.091i	1.82787i	0.405855	+	0.913937i	$0.366974\pi$
$197$	360.091i	1.82787i	−0.405855	+	0.913937i	$0.633026\pi$
$198$	0	0
$199$	108.657	0.546016	0.273008	−	0.962012i	$-0.411982\pi$
$199$	108.657	0.546016	0.273008	+	0.962012i	$0.411982\pi$
$200$	− 56.2078i	− 0.281039i
$201$	0	0
$202$	104.362	0.516644
$203$	− 33.9141i	− 0.167064i
$204$	0	0
$205$	126.691	0.618006
$206$	127.249i	0.617714i
$207$	0	0
$208$	59.3708	0.285437
$209$	129.927i	0.621662i
$210$	0	0
$211$	−288.934	−1.36936	−0.684679	−	0.728845i	$-0.740057\pi$
$211$	−288.934	−1.36936	−0.684679	+	0.728845i	$0.740057\pi$
$212$	261.704i	1.23445i
$213$	0	0
$214$	−177.789	−0.830790
$215$	− 224.129i	− 1.04246i
$216$	0	0
$217$	−58.6670	−0.270355
$218$	33.5674i	0.153979i
$219$	0	0
$220$	−126.346	−0.574301
$221$	70.8098i	0.320406i
$222$	0	0
$223$	−422.378	−1.89407	−0.947035	−	0.321131i	$-0.895937\pi$
$223$	−422.378	−1.89407	−0.947035	+	0.321131i	$0.895937\pi$
$224$	85.8456i	0.383239i
$225$	0	0
$226$	−14.5443	−0.0643555
$227$	173.001i	0.762120i	0.924550	+	0.381060i	$0.124441\pi$
$227$	173.001i	0.762120i	−0.924550	+	0.381060i	$0.875559\pi$
$228$	0	0
$229$	349.672	1.52695	0.763475	−	0.645837i	$-0.223491\pi$
$229$	349.672	1.52695	0.763475	+	0.645837i	$0.223491\pi$
$230$	125.412i	0.545270i
$231$	0	0
$232$	84.9272	0.366065
$233$	− 144.452i	− 0.619965i	−0.950742	−	0.309982i	$-0.899677\pi$
$233$	− 144.452i	− 0.619965i	0.950742	−	0.309982i	$-0.100323\pi$
$234$	0	0
$235$	229.562	0.976861
$236$	352.717i	1.49456i
$237$	0	0
$238$	−18.7588	−0.0788185
$239$	− 67.4483i	− 0.282211i	−0.989995	−	0.141105i	$-0.954934\pi$
$239$	− 67.4483i	− 0.282211i	0.989995	−	0.141105i	$-0.0450656\pi$
$240$	0	0
$241$	−370.831	−1.53872	−0.769359	−	0.638816i	$-0.779424\pi$
$241$	−370.831	−1.53872	−0.769359	+	0.638816i	$0.779424\pi$
$242$	− 21.2394i	− 0.0877661i
$243$	0	0
$244$	124.906	0.511911
$245$	− 28.4483i	− 0.116115i
$246$	0	0
$247$	−121.583	−0.492238
$248$	− 146.913i	− 0.592392i
$249$	0	0
$250$	−126.297	−0.505187
$251$	115.752i	0.461162i	0.973053	+	0.230581i	$0.0740627\pi$
$251$	115.752i	0.461162i	−0.973053	+	0.230581i	$0.925937\pi$
$252$	0	0
$253$	−329.343	−1.30175
$254$	− 16.3398i	− 0.0643298i
$255$	0	0
$256$	−134.560	−0.525625
$257$	9.25028i	0.0359933i	0.999838	+	0.0179967i	$0.00572882\pi$
$257$	9.25028i	0.0359933i	−0.999838	+	0.0179967i	$0.994271\pi$
$258$	0	0
$259$	150.141	0.579693
$260$	− 118.232i	− 0.454737i
$261$	0	0
$262$	42.2008	0.161072
$263$	− 398.775i	− 1.51625i	−0.652107	−	0.758127i	$-0.726115\pi$
$263$	− 398.775i	− 1.51625i	0.652107	−	0.758127i	$-0.273885\pi$
$264$	0	0
$265$	338.869	1.27875
$266$	− 32.2094i	− 0.121088i
$267$	0	0
$268$	294.842	1.10016
$269$	47.1757i	0.175374i	0.996148	+	0.0876872i	$0.0279476\pi$
$269$	47.1757i	0.175374i	−0.996148	+	0.0876872i	$0.972052\pi$
$270$	0	0
$271$	−222.577	−0.821316	−0.410658	−	0.911789i	$-0.634701\pi$
$271$	−222.577	−0.821316	−0.410658	+	0.911789i	$0.634701\pi$
$272$	48.9312i	0.179894i
$273$	0	0
$274$	73.7345	0.269104
$275$	− 84.0330i	− 0.305574i
$276$	0	0
$277$	−317.973	−1.14792	−0.573958	−	0.818884i	$-0.694593\pi$
$277$	−317.973	−1.14792	−0.573958	+	0.818884i	$0.694593\pi$
$278$	− 189.532i	− 0.681768i
$279$	0	0
$280$	71.2397	0.254428
$281$	255.084i	0.907772i	0.891060	+	0.453886i	$0.149963\pi$
$281$	255.084i	0.907772i	−0.891060	+	0.453886i	$0.850037\pi$
$282$	0	0
$283$	152.722	0.539654	0.269827	−	0.962909i	$-0.413034\pi$
$283$	152.722	0.539654	0.269827	+	0.962909i	$0.413034\pi$
$284$	34.8760i	0.122803i
$285$	0	0
$286$	−85.2142	−0.297952
$287$	− 82.4780i	− 0.287380i
$288$	0	0
$289$	230.641	0.798066
$290$	− 48.3494i	− 0.166722i
$291$	0	0
$292$	−123.251	−0.422091
$293$	− 42.7363i	− 0.145858i	−0.997337	−	0.0729289i	$-0.976765\pi$
$293$	− 42.7363i	− 0.145858i	0.997337	−	0.0729289i	$-0.0232346\pi$
$294$	0	0
$295$	456.718	1.54820
$296$	375.980i	1.27020i
$297$	0	0
$298$	−175.800	−0.589934
$299$	− 308.191i	− 1.03074i
$300$	0	0
$301$	−145.911	−0.484755
$302$	60.3538i	0.199847i
$303$	0	0
$304$	−84.0165	−0.276370
$305$	− 161.736i	− 0.530281i
$306$	0	0
$307$	−293.330	−0.955473	−0.477736	−	0.878503i	$-0.658543\pi$
$307$	−293.330	−0.955473	−0.477736	+	0.878503i	$0.658543\pi$
$308$	82.2534i	0.267057i
$309$	0	0
$310$	−83.6383	−0.269801
$311$	− 389.727i	− 1.25314i	−0.779365	−	0.626571i	$-0.784458\pi$
$311$	− 389.727i	− 1.25314i	0.779365	−	0.626571i	$-0.215542\pi$
$312$	0	0
$313$	286.670	0.915879	0.457939	−	0.888983i	$-0.348588\pi$
$313$	286.670	0.915879	0.457939	+	0.888983i	$0.348588\pi$
$314$	− 2.89902i	− 0.00923256i
$315$	0	0
$316$	139.353	0.440990
$317$	378.651i	1.19448i	0.802062	+	0.597241i	$0.203736\pi$
$317$	378.651i	1.19448i	−0.802062	+	0.597241i	$0.796264\pi$
$318$	0	0
$319$	126.970	0.398024
$320$	18.2614i	0.0570670i
$321$	0	0
$322$	81.6453	0.253557
$323$	− 100.204i	− 0.310229i
$324$	0	0
$325$	78.6359	0.241957
$326$	42.2937i	0.129735i
$327$	0	0
$328$	206.540	0.629696
$329$	− 149.449i	− 0.454251i
$330$	0	0
$331$	556.761	1.68206	0.841029	−	0.540990i	$-0.181950\pi$
$331$	556.761	1.68206	0.841029	+	0.540990i	$0.181950\pi$
$332$	− 372.522i	− 1.12205i
$333$	0	0
$334$	−220.783	−0.661027
$335$	− 381.778i	− 1.13963i
$336$	0	0
$337$	223.664	0.663691	0.331845	−	0.943334i	$-0.392329\pi$
$337$	223.664	0.663691	0.331845	+	0.943334i	$0.392329\pi$
$338$	77.1105i	0.228137i
$339$	0	0
$340$	97.4421	0.286594
$341$	− 219.641i	− 0.644109i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	− 365.389i	− 1.06218i
$345$	0	0
$346$	−94.9389	−0.274390
$347$	307.274i	0.885516i	0.896641	+	0.442758i	$0.146000\pi$
$347$	307.274i	0.885516i	−0.896641	+	0.442758i	$0.854000\pi$
$348$	0	0
$349$	−622.822	−1.78459	−0.892294	−	0.451454i	$-0.850906\pi$
$349$	−622.822	−1.78459	−0.892294	+	0.451454i	$0.850906\pi$
$350$	20.8321i	0.0595202i
$351$	0	0
$352$	−321.394	−0.913052
$353$	− 399.205i	− 1.13089i	−0.824785	−	0.565446i	$-0.808704\pi$
$353$	− 399.205i	− 1.13089i	0.824785	−	0.565446i	$-0.191296\pi$
$354$	0	0
$355$	45.1594	0.127210
$356$	346.339i	0.972861i
$357$	0	0
$358$	37.9431	0.105986
$359$	− 79.0523i	− 0.220201i	−0.993920	−	0.110101i	$-0.964883\pi$
$359$	− 79.0523i	− 0.220201i	0.993920	−	0.110101i	$-0.0351173\pi$
$360$	0	0
$361$	−188.947	−0.523398
$362$	− 231.571i	− 0.639700i
$363$	0	0
$364$	−76.9707	−0.211458
$365$	159.592i	0.437238i
$366$	0	0
$367$	405.086	1.10378	0.551889	−	0.833918i	$-0.313907\pi$
$367$	405.086	1.10378	0.551889	+	0.833918i	$0.313907\pi$
$368$	− 212.967i	− 0.578714i
$369$	0	0
$370$	214.047	0.578506
$371$	− 220.609i	− 0.594633i
$372$	0	0
$373$	−526.946	−1.41272	−0.706362	−	0.707851i	$-0.749665\pi$
$373$	−526.946	−1.41272	−0.706362	+	0.707851i	$0.749665\pi$
$374$	− 70.2304i	− 0.187782i
$375$	0	0
$376$	374.247	0.995339
$377$	118.815i	0.315159i
$378$	0	0
$379$	−669.614	−1.76679	−0.883396	−	0.468627i	$-0.844749\pi$
$379$	−669.614	−1.76679	−0.883396	+	0.468627i	$0.844749\pi$
$380$	167.311i	0.440293i
$381$	0	0
$382$	−43.8998	−0.114921
$383$	− 597.621i	− 1.56037i	−0.625550	−	0.780184i	$-0.715125\pi$
$383$	− 597.621i	− 1.56037i	0.625550	−	0.780184i	$-0.284875\pi$
$384$	0	0
$385$	106.506	0.276640
$386$	− 132.216i	− 0.342528i
$387$	0	0
$388$	−434.951	−1.12101
$389$	84.8552i	0.218137i	0.994034	+	0.109068i	$0.0347867\pi$
$389$	84.8552i	0.218137i	−0.994034	+	0.109068i	$0.965213\pi$
$390$	0	0
$391$	253.999	0.649615
$392$	− 46.3782i	− 0.118312i
$393$	0	0
$394$	334.207	0.848241
$395$	− 180.442i	− 0.456815i
$396$	0	0
$397$	−244.891	−0.616854	−0.308427	−	0.951248i	$-0.599803\pi$
$397$	−244.891	−0.616854	−0.308427	+	0.951248i	$0.599803\pi$
$398$	− 100.846i	− 0.253383i
$399$	0	0
$400$	54.3392	0.135848
$401$	632.401i	1.57706i	0.614997	+	0.788530i	$0.289157\pi$
$401$	632.401i	1.57706i	−0.614997	+	0.788530i	$0.710843\pi$
$402$	0	0
$403$	205.535	0.510012
$404$	352.920i	0.873564i
$405$	0	0
$406$	−31.4762	−0.0775277
$407$	562.106i	1.38110i
$408$	0	0
$409$	179.987	0.440067	0.220034	−	0.975492i	$-0.429383\pi$
$409$	179.987	0.440067	0.220034	+	0.975492i	$0.429383\pi$
$410$	− 117.584i	− 0.286791i
$411$	0	0
$412$	−430.316	−1.04446
$413$	− 297.331i	− 0.719929i
$414$	0	0
$415$	−482.363	−1.16232
$416$	− 300.753i	− 0.722963i
$417$	0	0
$418$	120.588	0.288488
$419$	− 144.434i	− 0.344711i	−0.985035	−	0.172355i	$-0.944862\pi$
$419$	− 144.434i	− 0.344711i	0.985035	−	0.172355i	$-0.0551378\pi$
$420$	0	0
$421$	446.140	1.05971	0.529857	−	0.848087i	$-0.322246\pi$
$421$	446.140	1.05971	0.529857	+	0.848087i	$0.322246\pi$
$422$	268.165i	0.635462i
$423$	0	0
$424$	552.445	1.30294
$425$	64.8088i	0.152491i
$426$	0	0
$427$	−105.292	−0.246587
$428$	− 601.227i	− 1.40474i
$429$	0	0
$430$	−208.018	−0.483762
$431$	210.603i	0.488637i	0.969695	+	0.244318i	$0.0785642\pi$
$431$	210.603i	0.488637i	−0.969695	+	0.244318i	$0.921436\pi$
$432$	0	0
$433$	−33.8210	−0.0781086	−0.0390543	−	0.999237i	$-0.512435\pi$
$433$	−33.8210	−0.0781086	−0.0390543	+	0.999237i	$0.512435\pi$
$434$	54.4499i	0.125460i
$435$	0	0
$436$	−113.514	−0.260354
$437$	436.125i	0.997998i
$438$	0	0
$439$	−146.724	−0.334224	−0.167112	−	0.985938i	$-0.553444\pi$
$439$	−146.724	−0.334224	−0.167112	+	0.985938i	$0.553444\pi$
$440$	266.712i	0.606163i
$441$	0	0
$442$	65.7198	0.148687
$443$	668.625i	1.50931i	0.656121	+	0.754656i	$0.272196\pi$
$443$	668.625i	1.50931i	−0.656121	+	0.754656i	$0.727804\pi$
$444$	0	0
$445$	448.459	1.00777
$446$	392.016i	0.878959i
$447$	0	0
$448$	11.8885	0.0265368
$449$	700.572i	1.56029i	0.625597	+	0.780147i	$0.284856\pi$
$449$	700.572i	1.56029i	−0.625597	+	0.780147i	$0.715144\pi$
$450$	0	0
$451$	308.786	0.684671
$452$	− 49.1844i	− 0.108815i
$453$	0	0
$454$	160.565	0.353668
$455$	99.6660i	0.219046i
$456$	0	0
$457$	−314.633	−0.688474	−0.344237	−	0.938883i	$-0.611862\pi$
$457$	−314.633	−0.688474	−0.344237	+	0.938883i	$0.611862\pi$
$458$	− 324.536i	− 0.708594i
$459$	0	0
$460$	−424.104	−0.921966
$461$	738.731i	1.60245i	0.598361	+	0.801227i	$0.295819\pi$
$461$	738.731i	1.60245i	−0.598361	+	0.801227i	$0.704181\pi$
$462$	0	0
$463$	−166.595	−0.359816	−0.179908	−	0.983683i	$-0.557580\pi$
$463$	−166.595	−0.359816	−0.179908	+	0.983683i	$0.557580\pi$
$464$	82.1039i	0.176948i
$465$	0	0
$466$	−134.068	−0.287700
$467$	298.169i	0.638479i	0.947674	+	0.319239i	$0.103427\pi$
$467$	298.169i	0.638479i	−0.947674	+	0.319239i	$0.896573\pi$
$468$	0	0
$469$	−248.543	−0.529943
$470$	− 213.061i	− 0.453321i
$471$	0	0
$472$	744.571	1.57748
$473$	− 546.272i	− 1.15491i
$474$	0	0
$475$	−111.279	−0.234271
$476$	− 63.4363i	− 0.133270i
$477$	0	0
$478$	−62.5999	−0.130962
$479$	− 192.842i	− 0.402593i	−0.979530	−	0.201297i	$-0.935484\pi$
$479$	− 192.842i	− 0.402593i	0.979530	−	0.201297i	$-0.0645155\pi$
$480$	0	0
$481$	−526.005	−1.09356
$482$	344.175i	0.714055i
$483$	0	0
$484$	71.8249	0.148399
$485$	563.199i	1.16124i
$486$	0	0
$487$	−220.343	−0.452451	−0.226225	−	0.974075i	$-0.572639\pi$
$487$	−220.343	−0.452451	−0.226225	+	0.974075i	$0.572639\pi$
$488$	− 263.672i	− 0.540311i
$489$	0	0
$490$	−26.4033	−0.0538843
$491$	800.437i	1.63022i	0.579308	+	0.815108i	$0.303323\pi$
$491$	800.437i	1.63022i	−0.579308	+	0.815108i	$0.696677\pi$
$492$	0	0
$493$	−97.9229	−0.198627
$494$	112.843i	0.228427i
$495$	0	0
$496$	142.029	0.286349
$497$	− 29.3995i	− 0.0591539i
$498$	0	0
$499$	158.791	0.318219	0.159110	−	0.987261i	$-0.449138\pi$
$499$	158.791	0.318219	0.159110	+	0.987261i	$0.449138\pi$
$500$	− 427.096i	− 0.854192i
$501$	0	0
$502$	107.431	0.214006
$503$	− 233.322i	− 0.463861i	−0.972732	−	0.231931i	$-0.925496\pi$
$503$	− 233.322i	− 0.463861i	0.972732	−	0.231931i	$-0.0745042\pi$
$504$	0	0
$505$	456.981	0.904912
$506$	305.669i	0.604088i
$507$	0	0
$508$	55.2560	0.108772
$509$	− 584.878i	− 1.14907i	−0.818479	−	0.574536i	$-0.805182\pi$
$509$	− 584.878i	− 1.14907i	0.818479	−	0.574536i	$-0.194818\pi$
$510$	0	0
$511$	103.897	0.203321
$512$	− 377.576i	− 0.737454i
$513$	0	0
$514$	8.58534	0.0167030
$515$	557.198i	1.08194i
$516$	0	0
$517$	559.516	1.08224
$518$	− 139.348i	− 0.269012i
$519$	0	0
$520$	−249.582	−0.479966
$521$	593.891i	1.13991i	0.821677	+	0.569953i	$0.193039\pi$
$521$	593.891i	1.13991i	−0.821677	+	0.569953i	$0.806961\pi$
$522$	0	0
$523$	252.241	0.482296	0.241148	−	0.970488i	$-0.422476\pi$
$523$	252.241	0.482296	0.241148	+	0.970488i	$0.422476\pi$
$524$	142.710i	0.272347i
$525$	0	0
$526$	−370.110	−0.703630
$527$	169.394i	0.321431i
$528$	0	0
$529$	−576.500	−1.08979
$530$	− 314.510i	− 0.593414i
$531$	0	0
$532$	108.922	0.204741
$533$	288.955i	0.542129i
$534$	0	0
$535$	−778.502	−1.45514
$536$	− 622.398i	− 1.16119i
$537$	0	0
$538$	43.7846	0.0813840
$539$	− 69.3373i	− 0.128641i
$540$	0	0
$541$	257.889	0.476689	0.238345	−	0.971181i	$-0.423395\pi$
$541$	257.889	0.476689	0.238345	+	0.971181i	$0.423395\pi$
$542$	206.577i	0.381139i
$543$	0	0
$544$	247.869	0.455642
$545$	146.985i	0.269697i
$546$	0	0
$547$	440.386	0.805092	0.402546	−	0.915400i	$-0.368125\pi$
$547$	440.386	0.805092	0.402546	+	0.915400i	$0.368125\pi$
$548$	249.347i	0.455013i
$549$	0	0
$550$	−77.9924	−0.141804
$551$	− 168.137i	− 0.305148i
$552$	0	0
$553$	−117.471	−0.212424
$554$	295.116i	0.532700i
$555$	0	0
$556$	640.936	1.15276
$557$	− 340.135i	− 0.610655i	−0.952247	−	0.305328i	$-0.901234\pi$
$557$	− 340.135i	− 0.610655i	0.952247	−	0.305328i	$-0.0987659\pi$
$558$	0	0
$559$	511.188	0.914468
$560$	68.8715i	0.122985i
$561$	0	0
$562$	236.748	0.421259
$563$	− 338.788i	− 0.601756i	−0.953663	−	0.300878i	$-0.902720\pi$
$563$	− 338.788i	− 0.601756i	0.953663	−	0.300878i	$-0.0972796\pi$
$564$	0	0
$565$	−63.6867	−0.112720
$566$	− 141.744i	− 0.250431i
$567$	0	0
$568$	73.6218	0.129616
$569$	− 906.528i	− 1.59320i	−0.604510	−	0.796598i	$-0.706631\pi$
$569$	− 906.528i	− 1.59320i	0.604510	−	0.796598i	$-0.293369\pi$
$570$	0	0
$571$	−20.2541	−0.0354713	−0.0177356	−	0.999843i	$-0.505646\pi$
$571$	−20.2541	−0.0354713	−0.0177356	+	0.999843i	$0.505646\pi$
$572$	− 288.168i	− 0.503790i
$573$	0	0
$574$	−76.5492	−0.133361
$575$	− 282.072i	− 0.490560i
$576$	0	0
$577$	425.266	0.737029	0.368515	−	0.929622i	$-0.379866\pi$
$577$	425.266	0.737029	0.368515	+	0.929622i	$0.379866\pi$
$578$	− 214.062i	− 0.370349i
$579$	0	0
$580$	163.503	0.281901
$581$	314.026i	0.540492i
$582$	0	0
$583$	825.929	1.41669
$584$	260.177i	0.445509i
$585$	0	0
$586$	−39.6643	−0.0676865
$587$	391.132i	0.666324i	0.942870	+	0.333162i	$0.108116\pi$
$587$	391.132i	0.666324i	−0.942870	+	0.333162i	$0.891884\pi$
$588$	0	0
$589$	−290.855	−0.493812
$590$	− 423.888i	− 0.718454i
$591$	0	0
$592$	−363.481	−0.613989
$593$	− 517.310i	− 0.872360i	−0.899859	−	0.436180i	$-0.856331\pi$
$593$	− 517.310i	− 0.872360i	0.899859	−	0.436180i	$-0.143669\pi$
$594$	0	0
$595$	−82.1410	−0.138052
$596$	− 594.502i	− 0.997486i
$597$	0	0
$598$	−286.037	−0.478323
$599$	120.359i	0.200933i	0.994940	+	0.100467i	$0.0320336\pi$
$599$	120.359i	0.200933i	−0.994940	+	0.100467i	$0.967966\pi$
$600$	0	0
$601$	−676.370	−1.12541	−0.562704	−	0.826658i	$-0.690239\pi$
$601$	−676.370	−1.12541	−0.562704	+	0.826658i	$0.690239\pi$
$602$	135.423i	0.224955i
$603$	0	0
$604$	−204.098	−0.337910
$605$	− 93.0030i	− 0.153724i
$606$	0	0
$607$	−894.229	−1.47319	−0.736597	−	0.676332i	$-0.763569\pi$
$607$	−894.229	−1.47319	−0.736597	+	0.676332i	$0.763569\pi$
$608$	425.599i	0.699998i
$609$	0	0
$610$	−150.110	−0.246081
$611$	523.581i	0.856924i
$612$	0	0
$613$	508.965	0.830286	0.415143	−	0.909756i	$-0.363732\pi$
$613$	508.965	0.830286	0.415143	+	0.909756i	$0.363732\pi$
$614$	272.245i	0.443395i
$615$	0	0
$616$	173.634	0.281873
$617$	376.858i	0.610791i	0.952226	+	0.305395i	$0.0987887\pi$
$617$	376.858i	0.610791i	−0.952226	+	0.305395i	$0.901211\pi$
$618$	0	0
$619$	−1138.53	−1.83930	−0.919652	−	0.392735i	$-0.871529\pi$
$619$	−1138.53	−1.83930	−0.919652	+	0.392735i	$0.871529\pi$
$620$	− 282.838i	− 0.456191i
$621$	0	0
$622$	−361.712	−0.581531
$623$	− 291.954i	− 0.468626i
$624$	0	0
$625$	−340.938	−0.545501
$626$	− 266.063i	− 0.425021i
$627$	0	0
$628$	9.80359	0.0156108
$629$	− 433.513i	− 0.689210i
$630$	0	0
$631$	192.810	0.305563	0.152782	−	0.988260i	$-0.451177\pi$
$631$	192.810	0.305563	0.152782	+	0.988260i	$0.451177\pi$
$632$	− 294.168i	− 0.465456i
$633$	0	0
$634$	351.432	0.554309
$635$	− 71.5486i	− 0.112675i
$636$	0	0
$637$	64.8841	0.101859
$638$	− 117.843i	− 0.184706i
$639$	0	0
$640$	−510.508	−0.797668
$641$	− 413.151i	− 0.644541i	−0.946648	−	0.322270i	$-0.895554\pi$
$641$	− 413.151i	− 0.644541i	0.946648	−	0.322270i	$-0.104446\pi$
$642$	0	0
$643$	−798.451	−1.24176	−0.620880	−	0.783906i	$-0.713224\pi$
$643$	−798.451	−1.24176	−0.620880	+	0.783906i	$0.713224\pi$
$644$	276.099i	0.428724i
$645$	0	0
$646$	−93.0010	−0.143964
$647$	0.503971i	0 0.000778936i	1.00000		0.000389468i	$0.000123971\pi$
$647$	0.503971i	0 0.000778936i	−1.00000		0.000389468i	$0.999876\pi$
$648$	0	0
$649$	1113.17	1.71520
$650$	− 72.9833i	− 0.112282i
$651$	0	0
$652$	−143.024	−0.219362
$653$	− 191.537i	− 0.293318i	−0.989187	−	0.146659i	$-0.953148\pi$
$653$	− 191.537i	− 0.293318i	0.989187	−	0.146659i	$-0.0468521\pi$
$654$	0	0
$655$	184.789	0.282120
$656$	199.674i	0.304381i
$657$	0	0
$658$	−138.706	−0.210799
$659$	429.019i	0.651015i	0.945539	+	0.325507i	$0.105535\pi$
$659$	429.019i	0.651015i	−0.945539	+	0.325507i	$0.894465\pi$
$660$	0	0
$661$	451.763	0.683455	0.341727	−	0.939799i	$-0.388988\pi$
$661$	451.763	0.683455	0.341727	+	0.939799i	$0.388988\pi$
$662$	− 516.739i	− 0.780573i
$663$	0	0
$664$	−786.379	−1.18431
$665$	− 141.039i	− 0.212088i
$666$	0	0
$667$	426.197	0.638976
$668$	− 746.619i	− 1.11769i
$669$	0	0
$670$	−354.334	−0.528857
$671$	− 394.201i	− 0.587482i
$672$	0	0
$673$	273.821	0.406867	0.203433	−	0.979089i	$-0.434790\pi$
$673$	273.821	0.406867	0.203433	+	0.979089i	$0.434790\pi$
$674$	− 207.586i	− 0.307991i
$675$	0	0
$676$	−260.763	−0.385745
$677$	− 487.829i	− 0.720575i	−0.932841	−	0.360288i	$-0.882679\pi$
$677$	− 487.829i	− 0.720575i	0.932841	−	0.360288i	$-0.117321\pi$
$678$	0	0
$679$	366.652	0.539988
$680$	− 205.696i	− 0.302495i
$681$	0	0
$682$	−203.853	−0.298904
$683$	− 55.8655i	− 0.0817943i	−0.999163	−	0.0408972i	$-0.986978\pi$
$683$	− 55.8655i	− 0.0817943i	0.999163	−	0.0408972i	$-0.0130216\pi$
$684$	0	0
$685$	322.869	0.471341
$686$	17.1890i	0.0250568i
$687$	0	0
$688$	353.242	0.513434
$689$	772.884i	1.12175i
$690$	0	0
$691$	496.097	0.717940	0.358970	−	0.933349i	$-0.383128\pi$
$691$	496.097	0.717940	0.358970	+	0.933349i	$0.383128\pi$
$692$	− 321.053i	− 0.463950i
$693$	0	0
$694$	285.186	0.410931
$695$	− 829.921i	− 1.19413i
$696$	0	0
$697$	−238.146	−0.341672
$698$	578.051i	0.828154i
$699$	0	0
$700$	−70.4475	−0.100639
$701$	− 829.247i	− 1.18295i	−0.806324	−	0.591474i	$-0.798546\pi$
$701$	− 829.247i	− 1.18295i	0.806324	−	0.591474i	$-0.201454\pi$
$702$	0	0
$703$	744.356	1.05883
$704$	44.5088i	0.0632228i
$705$	0	0
$706$	−370.509	−0.524800
$707$	− 297.502i	− 0.420794i
$708$	0	0
$709$	−664.220	−0.936840	−0.468420	−	0.883506i	$-0.655177\pi$
$709$	−664.220	−0.936840	−0.468420	+	0.883506i	$0.655177\pi$
$710$	− 41.9132i	− 0.0590327i
$711$	0	0
$712$	731.107	1.02684
$713$	− 737.266i	− 1.03403i
$714$	0	0
$715$	−373.136	−0.521868
$716$	128.312i	0.179206i
$717$	0	0
$718$	−73.3698	−0.102186
$719$	609.105i	0.847156i	0.905860	+	0.423578i	$0.139226\pi$
$719$	609.105i	0.847156i	−0.905860	+	0.423578i	$0.860774\pi$
$720$	0	0
$721$	362.745	0.503113
$722$	175.365i	0.242887i
$723$	0	0
$724$	783.101	1.08163
$725$	108.746i	0.149994i
$726$	0	0
$727$	1203.15	1.65495	0.827475	−	0.561503i	$-0.189777\pi$
$727$	1203.15	1.65495	0.827475	+	0.561503i	$0.189777\pi$
$728$	162.482i	0.223189i
$729$	0	0
$730$	148.120	0.202904
$731$	421.302i	0.576336i
$732$	0	0
$733$	308.100	0.420328	0.210164	−	0.977666i	$-0.432600\pi$
$733$	308.100	0.420328	0.210164	+	0.977666i	$0.432600\pi$
$734$	− 375.967i	− 0.512217i
$735$	0	0
$736$	−1078.82	−1.46579
$737$	− 930.512i	− 1.26257i
$738$	0	0
$739$	−620.860	−0.840136	−0.420068	−	0.907493i	$-0.637994\pi$
$739$	−620.860	−0.840136	−0.420068	+	0.907493i	$0.637994\pi$
$740$	723.840i	0.978162i
$741$	0	0
$742$	−204.751	−0.275944
$743$	− 359.485i	− 0.483829i	−0.970298	−	0.241914i	$-0.922225\pi$
$743$	− 359.485i	− 0.483829i	0.970298	−	0.241914i	$-0.0777753\pi$
$744$	0	0
$745$	−769.794	−1.03328
$746$	489.068i	0.655587i
$747$	0	0
$748$	237.497	0.317509
$749$	506.817i	0.676659i
$750$	0	0
$751$	−350.315	−0.466465	−0.233233	−	0.972421i	$-0.574930\pi$
$751$	−350.315	−0.466465	−0.233233	+	0.972421i	$0.574930\pi$
$752$	361.806i	0.481125i
$753$	0	0
$754$	110.274	0.146252
$755$	264.277i	0.350036i
$756$	0	0
$757$	−384.649	−0.508123	−0.254061	−	0.967188i	$-0.581767\pi$
$757$	−384.649	−0.508123	−0.254061	+	0.967188i	$0.581767\pi$
$758$	621.480i	0.819895i
$759$	0	0
$760$	353.187	0.464720
$761$	129.147i	0.169707i	0.996393	+	0.0848535i	$0.0270422\pi$
$761$	129.147i	0.169707i	−0.996393	+	0.0848535i	$0.972958\pi$
$762$	0	0
$763$	95.6894	0.125412
$764$	− 148.455i	− 0.194313i
$765$	0	0
$766$	−554.662	−0.724102
$767$	1041.67i	1.35811i
$768$	0	0
$769$	−25.3603	−0.0329783	−0.0164892	−	0.999864i	$-0.505249\pi$
$769$	−25.3603	−0.0329783	−0.0164892	+	0.999864i	$0.505249\pi$
$770$	− 98.8504i	− 0.128377i
$771$	0	0
$772$	447.112	0.579160
$773$	906.013i	1.17207i	0.810284	+	0.586037i	$0.199313\pi$
$773$	906.013i	1.17207i	−0.810284	+	0.586037i	$0.800687\pi$
$774$	0	0
$775$	188.116	0.242730
$776$	918.164i	1.18320i
$777$	0	0
$778$	78.7555	0.101228
$779$	− 408.903i	− 0.524908i
$780$	0	0
$781$	110.068	0.140932
$782$	− 235.741i	− 0.301459i
$783$	0	0
$784$	44.8364	0.0571893
$785$	− 12.6942i	− 0.0161710i
$786$	0	0
$787$	−428.209	−0.544104	−0.272052	−	0.962283i	$-0.587702\pi$
$787$	−428.209	−0.544104	−0.272052	+	0.962283i	$0.587702\pi$
$788$	1130.18i	1.43424i
$789$	0	0
$790$	−167.471	−0.211989
$791$	41.4611i	0.0524160i
$792$	0	0
$793$	368.883	0.465174
$794$	227.288i	0.286256i
$795$	0	0
$796$	341.031	0.428431
$797$	10.0897i	0.0126596i	0.999980	+	0.00632979i	$0.00201485\pi$
$797$	10.0897i	0.0126596i	−0.999980	+	0.00632979i	$0.997985\pi$
$798$	0	0
$799$	−431.516	−0.540070
$800$	− 275.264i	− 0.344080i
$801$	0	0
$802$	586.942	0.731848
$803$	388.976i	0.484403i
$804$	0	0
$805$	357.508	0.444109
$806$	− 190.760i	− 0.236675i
$807$	0	0
$808$	744.999	0.922029
$809$	123.474i	0.152626i	0.997084	+	0.0763129i	$0.0243148\pi$
$809$	123.474i	0.152626i	−0.997084	+	0.0763129i	$0.975685\pi$
$810$	0	0
$811$	332.493	0.409979	0.204990	−	0.978764i	$-0.434284\pi$
$811$	332.493	0.409979	0.204990	+	0.978764i	$0.434284\pi$
$812$	− 106.443i	− 0.131087i
$813$	0	0
$814$	521.700	0.640909
$815$	185.195i	0.227234i
$816$	0	0
$817$	−723.389	−0.885421
$818$	− 167.049i	− 0.204217i
$819$	0	0
$820$	397.633	0.484918
$821$	− 235.797i	− 0.287207i	−0.989635	−	0.143603i	$-0.954131\pi$
$821$	− 235.797i	− 0.287207i	0.989635	−	0.143603i	$-0.0458689\pi$
$822$	0	0
$823$	−577.522	−0.701728	−0.350864	−	0.936426i	$-0.614112\pi$
$823$	−577.522	−0.701728	−0.350864	+	0.936426i	$0.614112\pi$
$824$	908.380i	1.10240i
$825$	0	0
$826$	−275.958	−0.334089
$827$	820.765i	0.992461i	0.868191	+	0.496231i	$0.165283\pi$
$827$	820.765i	0.992461i	−0.868191	+	0.496231i	$0.834717\pi$
$828$	0	0
$829$	421.387	0.508308	0.254154	−	0.967164i	$-0.418203\pi$
$829$	421.387	0.508308	0.254154	+	0.967164i	$0.418203\pi$
$830$	447.689i	0.539384i
$831$	0	0
$832$	−41.6502	−0.0500604
$833$	53.4751i	0.0641957i
$834$	0	0
$835$	−966.764	−1.15780
$836$	407.790i	0.487787i
$837$	0	0
$838$	−134.051	−0.159966
$839$	− 1428.70i	− 1.70287i	−0.524463	−	0.851433i	$-0.675734\pi$
$839$	− 1428.70i	− 1.70287i	0.524463	−	0.851433i	$-0.324266\pi$
$840$	0	0
$841$	676.691	0.804626
$842$	− 414.070i	− 0.491769i
$843$	0	0
$844$	−906.849	−1.07447
$845$	337.651i	0.399587i
$846$	0	0
$847$	−60.5464	−0.0714834
$848$	534.080i	0.629812i
$849$	0	0
$850$	60.1501	0.0707649
$851$	1886.81i	2.21717i
$852$	0	0
$853$	1021.97	1.19809	0.599043	−	0.800717i	$-0.295548\pi$
$853$	1021.97	1.19809	0.599043	+	0.800717i	$0.295548\pi$
$854$	97.7237i	0.114431i
$855$	0	0
$856$	−1269.16	−1.48267
$857$	− 179.243i	− 0.209152i	−0.994517	−	0.104576i	$-0.966652\pi$
$857$	− 179.243i	− 0.209152i	0.994517	−	0.104576i	$-0.0333485\pi$
$858$	0	0
$859$	−1152.39	−1.34155	−0.670775	−	0.741661i	$-0.734038\pi$
$859$	−1152.39	−1.34155	−0.670775	+	0.741661i	$0.734038\pi$
$860$	− 703.450i	− 0.817966i
$861$	0	0
$862$	195.464	0.226756
$863$	− 437.138i	− 0.506533i	−0.967397	−	0.253267i	$-0.918495\pi$
$863$	− 437.138i	− 0.506533i	0.967397	−	0.253267i	$-0.0815050\pi$
$864$	0	0
$865$	−415.718	−0.480599
$866$	31.3899i	0.0362469i
$867$	0	0
$868$	−184.132	−0.212134
$869$	− 439.794i	− 0.506092i
$870$	0	0
$871$	870.749	0.999712
$872$	239.624i	0.274798i
$873$	0	0
$874$	404.775	0.463129
$875$	360.030i	0.411463i
$876$	0	0
$877$	521.611	0.594767	0.297384	−	0.954758i	$-0.403886\pi$
$877$	521.611	0.594767	0.297384	+	0.954758i	$0.403886\pi$
$878$	136.177i	0.155099i
$879$	0	0
$880$	−257.845	−0.293006
$881$	− 256.994i	− 0.291708i	−0.989306	−	0.145854i	$-0.953407\pi$
$881$	− 256.994i	− 0.291708i	0.989306	−	0.145854i	$-0.0465929\pi$
$882$	0	0
$883$	−803.300	−0.909739	−0.454870	−	0.890558i	$-0.650314\pi$
$883$	−803.300	−0.909739	−0.454870	+	0.890558i	$0.650314\pi$
$884$	222.244i	0.251407i
$885$	0	0
$886$	620.562	0.700408
$887$	444.272i	0.500870i	0.968133	+	0.250435i	$0.0805736\pi$
$887$	444.272i	0.500870i	−0.968133	+	0.250435i	$0.919426\pi$
$888$	0	0
$889$	−46.5793	−0.0523951
$890$	− 416.222i	− 0.467665i
$891$	0	0
$892$	−1325.67	−1.48618
$893$	− 740.926i	− 0.829704i
$894$	0	0
$895$	166.145	0.185637
$896$	332.349i	0.370925i
$897$	0	0
$898$	650.212	0.724067
$899$	284.234i	0.316167i
$900$	0	0
$901$	−636.982	−0.706972
$902$	− 286.590i	− 0.317727i
$903$	0	0
$904$	−103.826	−0.114852
$905$	− 1014.00i	− 1.12045i
$906$	0	0
$907$	1138.11	1.25480	0.627402	−	0.778696i	$-0.284118\pi$
$907$	1138.11	1.25480	0.627402	+	0.778696i	$0.284118\pi$
$908$	542.981i	0.597997i
$909$	0	0
$910$	92.5017	0.101650
$911$	− 1227.80i	− 1.34775i	−0.738845	−	0.673875i	$-0.764628\pi$
$911$	− 1227.80i	− 1.34775i	0.738845	−	0.673875i	$-0.235372\pi$
$912$	0	0
$913$	−1175.67	−1.28770
$914$	292.016i	0.319492i
$915$	0	0
$916$	1097.48	1.19812
$917$	− 120.300i	− 0.131189i
$918$	0	0
$919$	217.976	0.237189	0.118594	−	0.992943i	$-0.462161\pi$
$919$	217.976	0.237189	0.118594	+	0.992943i	$0.462161\pi$
$920$	895.267i	0.973116i
$921$	0	0
$922$	685.629	0.743632
$923$	102.999i	0.111591i
$924$	0	0
$925$	−481.426	−0.520461
$926$	154.619i	0.166976i
$927$	0	0
$928$	415.911	0.448180
$929$	346.871i	0.373381i	0.982419	+	0.186691i	$0.0597762\pi$
$929$	346.871i	0.373381i	−0.982419	+	0.186691i	$0.940224\pi$
$930$	0	0
$931$	−91.8184	−0.0986234
$932$	− 453.376i	− 0.486455i
$933$	0	0
$934$	276.736	0.296291
$935$	− 307.525i	− 0.328903i
$936$	0	0
$937$	−1212.61	−1.29414	−0.647070	−	0.762430i	$-0.724006\pi$
$937$	−1212.61	−1.29414	−0.647070	+	0.762430i	$0.724006\pi$
$938$	230.677i	0.245924i
$939$	0	0
$940$	720.504	0.766494
$941$	− 1671.07i	− 1.77585i	−0.459991	−	0.887924i	$-0.652147\pi$
$941$	− 1671.07i	− 1.77585i	0.459991	−	0.887924i	$-0.347853\pi$
$942$	0	0
$943$	1036.50	1.09915
$944$	719.819i	0.762520i
$945$	0	0
$946$	−507.004	−0.535946
$947$	− 1003.65i	− 1.05982i	−0.848055	−	0.529909i	$-0.822226\pi$
$947$	− 1003.65i	− 1.05982i	0.848055	−	0.529909i	$-0.177774\pi$
$948$	0	0
$949$	−363.994	−0.383555
$950$	103.280i	0.108715i
$951$	0	0
$952$	−133.912	−0.140663
$953$	653.487i	0.685715i	0.939387	+	0.342858i	$0.111395\pi$
$953$	653.487i	0.685715i	−0.939387	+	0.342858i	$0.888605\pi$
$954$	0	0
$955$	−192.228	−0.201286
$956$	− 211.693i	− 0.221436i
$957$	0	0
$958$	−178.980	−0.186827
$959$	− 210.192i	− 0.219179i
$960$	0	0
$961$	−469.311	−0.488357
$962$	488.194i	0.507478i
$963$	0	0
$964$	−1163.89	−1.20736
$965$	− 578.946i	− 0.599944i
$966$	0	0
$967$	−406.102	−0.419960	−0.209980	−	0.977706i	$-0.567340\pi$
$967$	−406.102	−0.419960	−0.209980	+	0.977706i	$0.567340\pi$
$968$	− 151.619i	− 0.156632i
$969$	0	0
$970$	522.715	0.538881
$971$	1662.22i	1.71187i	0.517086	+	0.855934i	$0.327017\pi$
$971$	1662.22i	1.71187i	−0.517086	+	0.855934i	$0.672983\pi$
$972$	0	0
$973$	−540.291	−0.555284
$974$	204.504i	0.209964i
$975$	0	0
$976$	254.907	0.261175
$977$	− 66.0118i	− 0.0675658i	−0.999429	−	0.0337829i	$-0.989245\pi$
$977$	− 66.0118i	− 0.0675658i	0.999429	−	0.0337829i	$-0.0107555\pi$
$978$	0	0
$979$	1093.04	1.11648
$980$	− 89.2877i	− 0.0911099i
$981$	0	0
$982$	742.899	0.756516
$983$	549.034i	0.558529i	0.960214	+	0.279265i	$0.0900907\pi$
$983$	549.034i	0.558529i	−0.960214	+	0.279265i	$0.909909\pi$
$984$	0	0
$985$	1463.42	1.48571
$986$	90.8839i	0.0921743i
$987$	0	0
$988$	−381.599	−0.386234
$989$	− 1833.66i	− 1.85406i
$990$	0	0
$991$	−1072.89	−1.08263	−0.541317	−	0.840819i	$-0.682074\pi$
$991$	−1072.89	−1.08263	−0.541317	+	0.840819i	$0.682074\pi$
$992$	− 719.472i	− 0.725275i
$993$	0	0
$994$	−27.2862	−0.0274509
$995$	− 441.586i	− 0.443805i
$996$	0	0
$997$	−238.717	−0.239436	−0.119718	−	0.992808i	$-0.538199\pi$
$997$	−238.717	−0.239436	−0.119718	+	0.992808i	$0.538199\pi$
$998$	− 147.377i	− 0.147672i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.c.134.3	✓	8
3.2	odd	2	inner	189.3.b.c.134.6	yes	8
4.3	odd	2		3024.3.d.j.1457.4		8
9.2	odd	6		567.3.r.e.134.6		16
9.4	even	3		567.3.r.e.512.6		16
9.5	odd	6		567.3.r.e.512.3		16
9.7	even	3		567.3.r.e.134.3		16
12.11	even	2		3024.3.d.j.1457.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.c.134.3	✓	8	1.1	even	1	trivial
189.3.b.c.134.6	yes	8	3.2	odd	2	inner
567.3.r.e.134.3		16	9.7	even	3
567.3.r.e.134.6		16	9.2	odd	6
567.3.r.e.512.3		16	9.5	odd	6
567.3.r.e.512.6		16	9.4	even	3
3024.3.d.j.1457.4		8	4.3	odd	2
3024.3.d.j.1457.5		8	12.11	even	2

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.928117i q^{2} +3.13860 q^{4} -4.06404i q^{5} -2.64575 q^{7} -6.62545i q^{8} +O(q^{10})\)	\(q-0.928117i q^{2} +3.13860 q^{4} -4.06404i q^{5} -2.64575 q^{7} -6.62545i q^{8} -3.77190 q^{10} -9.90533i q^{11} +9.26916 q^{13} +2.45557i q^{14} +6.40520 q^{16} +7.63929i q^{17} -13.1169 q^{19} -12.7554i q^{20} -9.19331 q^{22} -33.2491i q^{23} +8.48361 q^{25} -8.60286i q^{26} -8.30395 q^{28} +12.8183i q^{29} +22.1741 q^{31} -32.4466i q^{32} +7.09016 q^{34} +10.7524i q^{35} -56.7478 q^{37} +12.1740i q^{38} -26.9261 q^{40} +31.1738i q^{41} +55.1493 q^{43} -31.0889i q^{44} -30.8590 q^{46} +56.4863i q^{47} +7.00000 q^{49} -7.87378i q^{50} +29.0922 q^{52} +83.3823i q^{53} -40.2556 q^{55} +17.5293i q^{56} +11.8969 q^{58} +112.380i q^{59} +39.7968 q^{61} -20.5801i q^{62} -4.49342 q^{64} -37.6702i q^{65} +93.9405 q^{67} +23.9767i q^{68} +9.97951 q^{70} +11.1120i q^{71} -39.2693 q^{73} +52.6686i q^{74} -41.1687 q^{76} +26.2070i q^{77} +44.3997 q^{79} -26.0310i q^{80} +28.9329 q^{82} -118.691i q^{83} +31.0464 q^{85} -51.1850i q^{86} -65.6273 q^{88} +110.348i q^{89} -24.5239 q^{91} -104.355i q^{92} +52.4259 q^{94} +53.3076i q^{95} -138.581 q^{97} -6.49682i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4}+O(q^{10}) \) 8 * q - 12 * q^4	\( 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) \) 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more