LMFDB - Embedded newform 162.3.b.a.161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,3,Mod(161,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-8,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.41418028264$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.3.b.a.161.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} -8.34847 q^{7} +2.82843i q^{8} -7.34847 q^{10} -0.953512i q^{11} -9.69694 q^{13} +11.8065i q^{14} +4.00000 q^{16} -18.8776i q^{17} -24.6969 q^{19} +10.3923i q^{20} -1.34847 q^{22} +0.953512i q^{23} -2.00000 q^{25} +13.7135i q^{26} +16.6969 q^{28} -13.6814i q^{29} +3.04541 q^{31} -5.65685i q^{32} -26.6969 q^{34} +43.3799i q^{35} +46.6969 q^{37} +34.9267i q^{38} +14.6969 q^{40} -10.9172i q^{41} +45.0454 q^{43} +1.90702i q^{44} +1.34847 q^{46} -45.2869i q^{47} +20.6969 q^{49} +2.82843i q^{50} +19.3939 q^{52} -94.3879i q^{53} -4.95459 q^{55} -23.6130i q^{56} -19.3485 q^{58} -18.7813i q^{59} +13.0908 q^{61} -4.30686i q^{62} -8.00000 q^{64} +50.3868i q^{65} +75.0454 q^{67} +37.7552i q^{68} +61.3485 q^{70} +18.0204i q^{71} -7.90918 q^{73} -66.0394i q^{74} +49.3939 q^{76} +7.96036i q^{77} -43.7423 q^{79} -20.7846i q^{80} -15.4393 q^{82} +130.332i q^{83} -98.0908 q^{85} -63.7038i q^{86} +2.69694 q^{88} +145.300i q^{89} +80.9546 q^{91} -1.90702i q^{92} -64.0454 q^{94} +128.329i q^{95} -109.879 q^{97} -29.2699i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58}+ \cdots - 28 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 40 * q^19 + 24 * q^22 - 8 * q^25 + 8 * q^28 - 76 * q^31 - 48 * q^34 + 128 * q^37 + 92 * q^43 - 24 * q^46 + 24 * q^49 - 40 * q^52 - 108 * q^55 - 48 * q^58 - 124 * q^61 - 32 * q^64 + 212 * q^67 + 216 * q^70 - 208 * q^73 + 80 * q^76 - 28 * q^79 + 144 * q^82 - 216 * q^85 - 48 * q^88 + 412 * q^91 - 168 * q^94 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 5.19615i	− 1.03923i	−0.854400	−	0.519615i	$-0.826075\pi$
$5$	− 5.19615i	− 1.03923i	0.854400	−	0.519615i	$-0.173925\pi$
$6$	0	0
$7$	−8.34847	−1.19264	−0.596319	−	0.802747i	$-0.703371\pi$
$7$	−8.34847	−1.19264	−0.596319	+	0.802747i	$0.703371\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−7.34847	−0.734847
$11$	− 0.953512i	− 0.0866829i	−0.999060	−	0.0433414i	$-0.986200\pi$
$11$	− 0.953512i	− 0.0866829i	0.999060	−	0.0433414i	$-0.0138003\pi$
$12$	0	0
$13$	−9.69694	−0.745918	−0.372959	−	0.927848i	$-0.621657\pi$
$13$	−9.69694	−0.745918	−0.372959	+	0.927848i	$0.621657\pi$
$14$	11.8065i	0.843323i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 18.8776i	− 1.11045i	−0.831701	−	0.555223i	$-0.812633\pi$
$17$	− 18.8776i	− 1.11045i	0.831701	−	0.555223i	$-0.187367\pi$
$18$	0	0
$19$	−24.6969	−1.29984	−0.649919	−	0.760003i	$-0.725197\pi$
$19$	−24.6969	−1.29984	−0.649919	+	0.760003i	$0.725197\pi$
$20$	10.3923i	0.519615i
$21$	0	0
$22$	−1.34847	−0.0612941
$23$	0.953512i	0.0414570i	0.999785	+	0.0207285i	$0.00659856\pi$
$23$	0.953512i	0.0414570i	−0.999785	+	0.0207285i	$0.993401\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	13.7135i	0.527444i
$27$	0	0
$28$	16.6969	0.596319
$29$	− 13.6814i	− 0.471774i	−0.971781	−	0.235887i	$-0.924201\pi$
$29$	− 13.6814i	− 0.471774i	0.971781	−	0.235887i	$-0.0757995\pi$
$30$	0	0
$31$	3.04541	0.0982390	0.0491195	−	0.998793i	$-0.484358\pi$
$31$	3.04541	0.0982390	0.0491195	+	0.998793i	$0.484358\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−26.6969	−0.785204
$35$	43.3799i	1.23943i
$36$	0	0
$37$	46.6969	1.26208	0.631040	−	0.775751i	$-0.282628\pi$
$37$	46.6969	1.26208	0.631040	+	0.775751i	$0.282628\pi$
$38$	34.9267i	0.919125i
$39$	0	0
$40$	14.6969	0.367423
$41$	− 10.9172i	− 0.266274i	−0.991098	−	0.133137i	$-0.957495\pi$
$41$	− 10.9172i	− 0.266274i	0.991098	−	0.133137i	$-0.0425050\pi$
$42$	0	0
$43$	45.0454	1.04757	0.523784	−	0.851851i	$-0.324520\pi$
$43$	45.0454	1.04757	0.523784	+	0.851851i	$0.324520\pi$
$44$	1.90702i	0.0433414i
$45$	0	0
$46$	1.34847	0.0293145
$47$	− 45.2869i	− 0.963552i	−0.876294	−	0.481776i	$-0.839992\pi$
$47$	− 45.2869i	− 0.963552i	0.876294	−	0.481776i	$-0.160008\pi$
$48$	0	0
$49$	20.6969	0.422386
$50$	2.82843i	0.0565685i
$51$	0	0
$52$	19.3939	0.372959
$53$	− 94.3879i	− 1.78090i	−0.455077	−	0.890452i	$-0.650388\pi$
$53$	− 94.3879i	− 1.78090i	0.455077	−	0.890452i	$-0.349612\pi$
$54$	0	0
$55$	−4.95459	−0.0900835
$56$	− 23.6130i	− 0.421661i
$57$	0	0
$58$	−19.3485	−0.333594
$59$	− 18.7813i	− 0.318326i	−0.987252	−	0.159163i	$-0.949120\pi$
$59$	− 18.7813i	− 0.318326i	0.987252	−	0.159163i	$-0.0508796\pi$
$60$	0	0
$61$	13.0908	0.214604	0.107302	−	0.994226i	$-0.465779\pi$
$61$	13.0908	0.214604	0.107302	+	0.994226i	$0.465779\pi$
$62$	− 4.30686i	− 0.0694654i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	50.3868i	0.775181i
$66$	0	0
$67$	75.0454	1.12008	0.560040	−	0.828465i	$-0.310786\pi$
$67$	75.0454	1.12008	0.560040	+	0.828465i	$0.310786\pi$
$68$	37.7552i	0.555223i
$69$	0	0
$70$	61.3485	0.876407
$71$	18.0204i	0.253808i	0.991915	+	0.126904i	$0.0405041\pi$
$71$	18.0204i	0.253808i	−0.991915	+	0.126904i	$0.959496\pi$
$72$	0	0
$73$	−7.90918	−0.108345	−0.0541725	−	0.998532i	$-0.517252\pi$
$73$	−7.90918	−0.108345	−0.0541725	+	0.998532i	$0.517252\pi$
$74$	− 66.0394i	− 0.892425i
$75$	0	0
$76$	49.3939	0.649919
$77$	7.96036i	0.103381i
$78$	0	0
$79$	−43.7423	−0.553701	−0.276850	−	0.960913i	$-0.589291\pi$
$79$	−43.7423	−0.553701	−0.276850	+	0.960913i	$0.589291\pi$
$80$	− 20.7846i	− 0.259808i
$81$	0	0
$82$	−15.4393	−0.188284
$83$	130.332i	1.57027i	0.619325	+	0.785135i	$0.287406\pi$
$83$	130.332i	1.57027i	−0.619325	+	0.785135i	$0.712594\pi$
$84$	0	0
$85$	−98.0908	−1.15401
$86$	− 63.7038i	− 0.740742i
$87$	0	0
$88$	2.69694	0.0306470
$89$	145.300i	1.63258i	0.577642	+	0.816290i	$0.303973\pi$
$89$	145.300i	1.63258i	−0.577642	+	0.816290i	$0.696027\pi$
$90$	0	0
$91$	80.9546	0.889611
$92$	− 1.90702i	− 0.0207285i
$93$	0	0
$94$	−64.0454	−0.681334
$95$	128.329i	1.35083i
$96$	0	0
$97$	−109.879	−1.13277	−0.566384	−	0.824141i	$-0.691658\pi$
$97$	−109.879	−1.13277	−0.566384	+	0.824141i	$0.691658\pi$
$98$	− 29.2699i	− 0.298672i
$99$	0	0
$100$	4.00000	0.0400000
$101$	− 147.539i	− 1.46078i	−0.683030	−	0.730391i	$-0.739338\pi$
$101$	− 147.539i	− 1.46078i	0.683030	−	0.730391i	$-0.260662\pi$
$102$	0	0
$103$	−103.136	−1.00132	−0.500661	−	0.865643i	$-0.666910\pi$
$103$	−103.136	−1.00132	−0.500661	+	0.865643i	$0.666910\pi$
$104$	− 27.4271i	− 0.263722i
$105$	0	0
$106$	−133.485	−1.25929
$107$	36.0408i	0.336830i	0.985716	+	0.168415i	$0.0538649\pi$
$107$	36.0408i	0.336830i	−0.985716	+	0.168415i	$0.946135\pi$
$108$	0	0
$109$	−148.272	−1.36030	−0.680149	−	0.733074i	$-0.738085\pi$
$109$	−148.272	−1.36030	−0.680149	+	0.733074i	$0.738085\pi$
$110$	7.00685i	0.0636987i
$111$	0	0
$112$	−33.3939	−0.298160
$113$	− 171.088i	− 1.51405i	−0.653385	−	0.757025i	$-0.726652\pi$
$113$	− 171.088i	− 1.51405i	0.653385	−	0.757025i	$-0.273348\pi$
$114$	0	0
$115$	4.95459	0.0430834
$116$	27.3629i	0.235887i
$117$	0	0
$118$	−26.5607	−0.225091
$119$	157.599i	1.32436i
$120$	0	0
$121$	120.091	0.992486
$122$	− 18.5132i	− 0.151748i
$123$	0	0
$124$	−6.09082	−0.0491195
$125$	− 119.512i	− 0.956092i
$126$	0	0
$127$	−78.0908	−0.614888	−0.307444	−	0.951566i	$-0.599474\pi$
$127$	−78.0908	−0.614888	−0.307444	+	0.951566i	$0.599474\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	71.2577	0.548136
$131$	234.063i	1.78674i	0.449323	+	0.893369i	$0.351665\pi$
$131$	234.063i	1.78674i	−0.449323	+	0.893369i	$0.648335\pi$
$132$	0	0
$133$	206.182	1.55024
$134$	− 106.130i	− 0.792017i
$135$	0	0
$136$	53.3939	0.392602
$137$	− 149.831i	− 1.09366i	−0.837244	−	0.546829i	$-0.815835\pi$
$137$	− 149.831i	− 1.09366i	0.837244	−	0.546829i	$-0.184165\pi$
$138$	0	0
$139$	−84.5301	−0.608130	−0.304065	−	0.952651i	$-0.598344\pi$
$139$	−84.5301	−0.608130	−0.304065	+	0.952651i	$0.598344\pi$
$140$	− 86.7598i	− 0.619713i
$141$	0	0
$142$	25.4847	0.179470
$143$	9.24614i	0.0646584i
$144$	0	0
$145$	−71.0908	−0.490281
$146$	11.1853i	0.0766115i
$147$	0	0
$148$	−93.3939	−0.631040
$149$	115.505i	0.775200i	0.921828	+	0.387600i	$0.126696\pi$
$149$	115.505i	0.775200i	−0.921828	+	0.387600i	$0.873304\pi$
$150$	0	0
$151$	−64.7730	−0.428960	−0.214480	−	0.976728i	$-0.568806\pi$
$151$	−64.7730	−0.428960	−0.214480	+	0.976728i	$0.568806\pi$
$152$	− 69.8535i	− 0.459562i
$153$	0	0
$154$	11.2577	0.0731016
$155$	− 15.8244i	− 0.102093i
$156$	0	0
$157$	−20.8184	−0.132601	−0.0663005	−	0.997800i	$-0.521120\pi$
$157$	−20.8184	−0.132601	−0.0663005	+	0.997800i	$0.521120\pi$
$158$	61.8610i	0.391525i
$159$	0	0
$160$	−29.3939	−0.183712
$161$	− 7.96036i	− 0.0494433i
$162$	0	0
$163$	133.060	0.816320	0.408160	−	0.912910i	$-0.366171\pi$
$163$	133.060	0.816320	0.408160	+	0.912910i	$0.366171\pi$
$164$	21.8344i	0.133137i
$165$	0	0
$166$	184.318	1.11035
$167$	− 294.510i	− 1.76353i	−0.471688	−	0.881765i	$-0.656355\pi$
$167$	− 294.510i	− 1.76353i	0.471688	−	0.881765i	$-0.343645\pi$
$168$	0	0
$169$	−74.9694	−0.443606
$170$	138.721i	0.816008i
$171$	0	0
$172$	−90.0908	−0.523784
$173$	69.2644i	0.400372i	0.979758	+	0.200186i	$0.0641546\pi$
$173$	69.2644i	0.400372i	−0.979758	+	0.200186i	$0.935845\pi$
$174$	0	0
$175$	16.6969	0.0954111
$176$	− 3.81405i	− 0.0216707i
$177$	0	0
$178$	205.485	1.15441
$179$	47.4829i	0.265268i	0.991165	+	0.132634i	$0.0423435\pi$
$179$	47.4829i	0.265268i	−0.991165	+	0.132634i	$0.957657\pi$
$180$	0	0
$181$	242.879	1.34187	0.670935	−	0.741516i	$-0.265893\pi$
$181$	242.879	1.34187	0.670935	+	0.741516i	$0.265893\pi$
$182$	− 114.487i	− 0.629050i
$183$	0	0
$184$	−2.69694	−0.0146573
$185$	− 242.644i	− 1.31159i
$186$	0	0
$187$	−18.0000	−0.0962567
$188$	90.5739i	0.481776i
$189$	0	0
$190$	181.485	0.955183
$191$	− 7.53177i	− 0.0394333i	−0.999806	−	0.0197167i	$-0.993724\pi$
$191$	− 7.53177i	− 0.0394333i	0.999806	−	0.0197167i	$-0.00627642\pi$
$192$	0	0
$193$	345.454	1.78992	0.894959	−	0.446149i	$-0.147205\pi$
$193$	345.454	1.78992	0.894959	+	0.446149i	$0.147205\pi$
$194$	155.392i	0.800988i
$195$	0	0
$196$	−41.3939	−0.211193
$197$	77.2247i	0.392004i	0.980604	+	0.196002i	$0.0627959\pi$
$197$	77.2247i	0.392004i	−0.980604	+	0.196002i	$0.937204\pi$
$198$	0	0
$199$	153.485	0.771280	0.385640	−	0.922649i	$-0.373981\pi$
$199$	153.485	0.771280	0.385640	+	0.922649i	$0.373981\pi$
$200$	− 5.65685i	− 0.0282843i
$201$	0	0
$202$	−208.652	−1.03293
$203$	114.219i	0.562655i
$204$	0	0
$205$	−56.7276	−0.276720
$206$	145.857i	0.708042i
$207$	0	0
$208$	−38.7878	−0.186480
$209$	23.5488i	0.112674i
$210$	0	0
$211$	−51.5607	−0.244364	−0.122182	−	0.992508i	$-0.538989\pi$
$211$	−51.5607	−0.244364	−0.122182	+	0.992508i	$0.538989\pi$
$212$	188.776i	0.890452i
$213$	0	0
$214$	50.9694	0.238175
$215$	− 234.063i	− 1.08866i
$216$	0	0
$217$	−25.4245	−0.117164
$218$	209.689i	0.961876i
$219$	0	0
$220$	9.90918	0.0450417
$221$	183.055i	0.828302i
$222$	0	0
$223$	313.227	1.40461	0.702303	−	0.711878i	$-0.252155\pi$
$223$	313.227	1.40461	0.702303	+	0.711878i	$0.252155\pi$
$224$	47.2261i	0.210831i
$225$	0	0
$226$	−241.955	−1.07060
$227$	76.2712i	0.335997i	0.985787	+	0.167998i	$0.0537303\pi$
$227$	76.2712i	0.335997i	−0.985787	+	0.167998i	$0.946270\pi$
$228$	0	0
$229$	−121.545	−0.530764	−0.265382	−	0.964143i	$-0.585498\pi$
$229$	−121.545	−0.530764	−0.265382	+	0.964143i	$0.585498\pi$
$230$	− 7.00685i	− 0.0304646i
$231$	0	0
$232$	38.6969	0.166797
$233$	− 151.021i	− 0.648157i	−0.946030	−	0.324079i	$-0.894946\pi$
$233$	− 151.021i	− 0.648157i	0.946030	−	0.324079i	$-0.105054\pi$
$234$	0	0
$235$	−235.318	−1.00135
$236$	37.5625i	0.159163i
$237$	0	0
$238$	222.879	0.936465
$239$	− 87.7133i	− 0.367001i	−0.983020	−	0.183501i	$-0.941257\pi$
$239$	− 87.7133i	− 0.367001i	0.983020	−	0.183501i	$-0.0587430\pi$
$240$	0	0
$241$	201.788	0.837294	0.418647	−	0.908149i	$-0.362505\pi$
$241$	201.788	0.837294	0.418647	+	0.908149i	$0.362505\pi$
$242$	− 169.834i	− 0.701794i
$243$	0	0
$244$	−26.1816	−0.107302
$245$	− 107.544i	− 0.438957i
$246$	0	0
$247$	239.485	0.969574
$248$	8.61371i	0.0347327i
$249$	0	0
$250$	−169.015	−0.676059
$251$	52.6261i	0.209666i	0.994490	+	0.104833i	$0.0334307\pi$
$251$	52.6261i	0.209666i	−0.994490	+	0.104833i	$0.966569\pi$
$252$	0	0
$253$	0.909185	0.00359362
$254$	110.437i	0.434792i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 80.7065i	− 0.314033i	−0.987596	−	0.157017i	$-0.949812\pi$
$257$	− 80.7065i	− 0.314033i	0.987596	−	0.157017i	$-0.0501876\pi$
$258$	0	0
$259$	−389.848	−1.50520
$260$	− 100.774i	− 0.387591i
$261$	0	0
$262$	331.015	1.26342
$263$	− 463.743i	− 1.76328i	−0.471920	−	0.881641i	$-0.656439\pi$
$263$	− 463.743i	− 1.76328i	0.471920	−	0.881641i	$-0.343561\pi$
$264$	0	0
$265$	−490.454	−1.85077
$266$	− 291.585i	− 1.09618i
$267$	0	0
$268$	−150.091	−0.560040
$269$	− 43.4762i	− 0.161622i	−0.996729	−	0.0808109i	$-0.974249\pi$
$269$	− 43.4762i	− 0.161622i	0.996729	−	0.0808109i	$-0.0257510\pi$
$270$	0	0
$271$	−342.636	−1.26434	−0.632169	−	0.774830i	$-0.717835\pi$
$271$	−342.636	−1.26434	−0.632169	+	0.774830i	$0.717835\pi$
$272$	− 75.5103i	− 0.277612i
$273$	0	0
$274$	−211.893	−0.773333
$275$	1.90702i	0.00693463i
$276$	0	0
$277$	−49.0000	−0.176895	−0.0884477	−	0.996081i	$-0.528191\pi$
$277$	−49.0000	−0.176895	−0.0884477	+	0.996081i	$0.528191\pi$
$278$	119.544i	0.430013i
$279$	0	0
$280$	−122.697	−0.438203
$281$	20.6450i	0.0734697i	0.999325	+	0.0367349i	$0.0116957\pi$
$281$	20.6450i	0.0734697i	−0.999325	+	0.0367349i	$0.988304\pi$
$282$	0	0
$283$	53.4087	0.188723	0.0943616	−	0.995538i	$-0.469919\pi$
$283$	53.4087	0.188723	0.0943616	+	0.995538i	$0.469919\pi$
$284$	− 36.0408i	− 0.126904i
$285$	0	0
$286$	13.0760	0.0457204
$287$	91.1421i	0.317568i
$288$	0	0
$289$	−67.3633	−0.233091
$290$	100.538i	0.346681i
$291$	0	0
$292$	15.8184	0.0541725
$293$	− 14.9239i	− 0.0509349i	−0.999676	−	0.0254674i	$-0.991893\pi$
$293$	− 14.9239i	− 0.0509349i	0.999676	−	0.0254674i	$-0.00810741\pi$
$294$	0	0
$295$	−97.5903	−0.330815
$296$	132.079i	0.446212i
$297$	0	0
$298$	163.348	0.548149
$299$	− 9.24614i	− 0.0309236i
$300$	0	0
$301$	−376.060	−1.24937
$302$	91.6028i	0.303321i
$303$	0	0
$304$	−98.7878	−0.324960
$305$	− 68.0219i	− 0.223023i
$306$	0	0
$307$	65.9092	0.214688	0.107344	−	0.994222i	$-0.465765\pi$
$307$	65.9092	0.214688	0.107344	+	0.994222i	$0.465765\pi$
$308$	− 15.9207i	− 0.0516907i
$309$	0	0
$310$	−22.3791	−0.0721906
$311$	− 250.176i	− 0.804425i	−0.915546	−	0.402213i	$-0.868241\pi$
$311$	− 250.176i	− 0.804425i	0.915546	−	0.402213i	$-0.131759\pi$
$312$	0	0
$313$	−426.394	−1.36228	−0.681140	−	0.732153i	$-0.738516\pi$
$313$	−426.394	−1.36228	−0.681140	+	0.732153i	$0.738516\pi$
$314$	29.4416i	0.0937631i
$315$	0	0
$316$	87.4847	0.276850
$317$	463.979i	1.46366i	0.681489	+	0.731829i	$0.261333\pi$
$317$	463.979i	1.46366i	−0.681489	+	0.731829i	$0.738667\pi$
$318$	0	0
$319$	−13.0454	−0.0408947
$320$	41.5692i	0.129904i
$321$	0	0
$322$	−11.2577	−0.0349617
$323$	466.219i	1.44340i
$324$	0	0
$325$	19.3939	0.0596735
$326$	− 188.176i	− 0.577226i
$327$	0	0
$328$	30.8786	0.0941420
$329$	378.077i	1.14917i
$330$	0	0
$331$	472.803	1.42841	0.714203	−	0.699938i	$-0.246789\pi$
$331$	472.803	1.42841	0.714203	+	0.699938i	$0.246789\pi$
$332$	− 260.665i	− 0.785135i
$333$	0	0
$334$	−416.499	−1.24700
$335$	− 389.947i	− 1.16402i
$336$	0	0
$337$	305.606	0.906843	0.453422	−	0.891296i	$-0.350203\pi$
$337$	305.606	0.906843	0.453422	+	0.891296i	$0.350203\pi$
$338$	106.023i	0.313677i
$339$	0	0
$340$	196.182	0.577005
$341$	− 2.90383i	− 0.00851564i
$342$	0	0
$343$	236.287	0.688884
$344$	127.408i	0.370371i
$345$	0	0
$346$	97.9546	0.283106
$347$	133.675i	0.385229i	0.981274	+	0.192615i	$0.0616967\pi$
$347$	133.675i	0.385229i	−0.981274	+	0.192615i	$0.938303\pi$
$348$	0	0
$349$	−98.7571	−0.282972	−0.141486	−	0.989940i	$-0.545188\pi$
$349$	−98.7571	−0.282972	−0.141486	+	0.989940i	$0.545188\pi$
$350$	− 23.6130i	− 0.0674658i
$351$	0	0
$352$	−5.39388	−0.0153235
$353$	326.115i	0.923839i	0.886922	+	0.461919i	$0.152839\pi$
$353$	326.115i	0.923839i	−0.886922	+	0.461919i	$0.847161\pi$
$354$	0	0
$355$	93.6367	0.263765
$356$	− 290.599i	− 0.816290i
$357$	0	0
$358$	67.1510	0.187573
$359$	− 418.736i	− 1.16639i	−0.812331	−	0.583197i	$-0.801801\pi$
$359$	− 418.736i	− 1.16639i	0.812331	−	0.583197i	$-0.198199\pi$
$360$	0	0
$361$	248.939	0.689581
$362$	− 343.482i	− 0.948846i
$363$	0	0
$364$	−161.909	−0.444805
$365$	41.0973i	0.112595i
$366$	0	0
$367$	187.227	0.510155	0.255078	−	0.966921i	$-0.417899\pi$
$367$	187.227	0.510155	0.255078	+	0.966921i	$0.417899\pi$
$368$	3.81405i	0.0103643i
$369$	0	0
$370$	−343.151	−0.927435
$371$	787.995i	2.12398i
$372$	0	0
$373$	451.030	1.20919	0.604597	−	0.796531i	$-0.293334\pi$
$373$	451.030	1.20919	0.604597	+	0.796531i	$0.293334\pi$
$374$	25.4558i	0.0680638i
$375$	0	0
$376$	128.091	0.340667
$377$	132.668i	0.351905i
$378$	0	0
$379$	−489.666	−1.29200	−0.645998	−	0.763339i	$-0.723558\pi$
$379$	−489.666	−1.29200	−0.645998	+	0.763339i	$0.723558\pi$
$380$	− 256.658i	− 0.675416i
$381$	0	0
$382$	−10.6515	−0.0278836
$383$	103.056i	0.269076i	0.990908	+	0.134538i	$0.0429550\pi$
$383$	103.056i	0.269076i	−0.990908	+	0.134538i	$0.957045\pi$
$384$	0	0
$385$	41.3633	0.107437
$386$	− 488.546i	− 1.26566i
$387$	0	0
$388$	219.757	0.566384
$389$	− 34.2734i	− 0.0881064i	−0.999029	−	0.0440532i	$-0.985973\pi$
$389$	− 34.2734i	− 0.0881064i	0.999029	−	0.0440532i	$-0.0140271\pi$
$390$	0	0
$391$	18.0000	0.0460358
$392$	58.5398i	0.149336i
$393$	0	0
$394$	109.212	0.277188
$395$	227.292i	0.575423i
$396$	0	0
$397$	8.27245	0.0208374	0.0104187	−	0.999946i	$-0.496684\pi$
$397$	8.27245	0.0208374	0.0104187	+	0.999946i	$0.496684\pi$
$398$	− 217.060i	− 0.545377i
$399$	0	0
$400$	−8.00000	−0.0200000
$401$	414.117i	1.03271i	0.856374	+	0.516356i	$0.172712\pi$
$401$	414.117i	1.03271i	−0.856374	+	0.516356i	$0.827288\pi$
$402$	0	0
$403$	−29.5311	−0.0732782
$404$	295.078i	0.730391i
$405$	0	0
$406$	161.530	0.397857
$407$	− 44.5261i	− 0.109401i
$408$	0	0
$409$	−326.212	−0.797585	−0.398792	−	0.917041i	$-0.630571\pi$
$409$	−326.212	−0.797585	−0.398792	+	0.917041i	$0.630571\pi$
$410$	80.2249i	0.195670i
$411$	0	0
$412$	206.272	0.500661
$413$	156.795i	0.379648i
$414$	0	0
$415$	677.227	1.63187
$416$	54.8542i	0.131861i
$417$	0	0
$418$	33.3031	0.0796724
$419$	− 540.775i	− 1.29063i	−0.763915	−	0.645317i	$-0.776725\pi$
$419$	− 540.775i	− 1.29063i	0.763915	−	0.645317i	$-0.223275\pi$
$420$	0	0
$421$	283.697	0.673864	0.336932	−	0.941529i	$-0.390611\pi$
$421$	283.697	0.673864	0.336932	+	0.941529i	$0.390611\pi$
$422$	72.9179i	0.172791i
$423$	0	0
$424$	266.969	0.629645
$425$	37.7552i	0.0888357i
$426$	0	0
$427$	−109.288	−0.255944
$428$	− 72.0816i	− 0.168415i
$429$	0	0
$430$	−331.015	−0.769802
$431$	− 257.429i	− 0.597282i	−0.954365	−	0.298641i	$-0.903467\pi$
$431$	− 257.429i	− 0.597282i	0.954365	−	0.298641i	$-0.0965334\pi$
$432$	0	0
$433$	476.272	1.09994	0.549968	−	0.835186i	$-0.314640\pi$
$433$	476.272	1.09994	0.549968	+	0.835186i	$0.314640\pi$
$434$	35.9557i	0.0828471i
$435$	0	0
$436$	296.545	0.680149
$437$	− 23.5488i	− 0.0538875i
$438$	0	0
$439$	−557.863	−1.27076	−0.635379	−	0.772200i	$-0.719156\pi$
$439$	−557.863	−1.27076	−0.635379	+	0.772200i	$0.719156\pi$
$440$	− 14.0137i	− 0.0318493i
$441$	0	0
$442$	258.879	0.585698
$443$	− 831.847i	− 1.87776i	−0.344248	−	0.938879i	$-0.611866\pi$
$443$	− 831.847i	− 1.87776i	0.344248	−	0.938879i	$-0.388134\pi$
$444$	0	0
$445$	754.999	1.69663
$446$	− 442.970i	− 0.993206i
$447$	0	0
$448$	66.7878	0.149080
$449$	729.927i	1.62567i	0.582492	+	0.812836i	$0.302078\pi$
$449$	729.927i	1.62567i	−0.582492	+	0.812836i	$0.697922\pi$
$450$	0	0
$451$	−10.4097	−0.0230814
$452$	342.175i	0.757025i
$453$	0	0
$454$	107.864	0.237585
$455$	− 420.652i	− 0.924511i
$456$	0	0
$457$	709.636	1.55281	0.776407	−	0.630232i	$-0.217040\pi$
$457$	709.636	1.55281	0.776407	+	0.630232i	$0.217040\pi$
$458$	171.890i	0.375307i
$459$	0	0
$460$	−9.90918	−0.0215417
$461$	9.20285i	0.0199628i	0.999950	+	0.00998140i	$0.00317723\pi$
$461$	9.20285i	0.0199628i	−0.999950	+	0.00998140i	$0.996823\pi$
$462$	0	0
$463$	−55.1975	−0.119217	−0.0596085	−	0.998222i	$-0.518985\pi$
$463$	−55.1975	−0.119217	−0.0596085	+	0.998222i	$0.518985\pi$
$464$	− 54.7257i	− 0.117943i
$465$	0	0
$466$	−213.576	−0.458317
$467$	− 625.811i	− 1.34007i	−0.742331	−	0.670033i	$-0.766280\pi$
$467$	− 625.811i	− 1.34007i	0.742331	−	0.670033i	$-0.233720\pi$
$468$	0	0
$469$	−626.514	−1.33585
$470$	332.790i	0.708063i
$471$	0	0
$472$	53.1214	0.112545
$473$	− 42.9513i	− 0.0908062i
$474$	0	0
$475$	49.3939	0.103987
$476$	− 315.198i	− 0.662180i
$477$	0	0
$478$	−124.045	−0.259509
$479$	− 309.294i	− 0.645708i	−0.946449	−	0.322854i	$-0.895358\pi$
$479$	− 309.294i	− 0.645708i	0.946449	−	0.322854i	$-0.104642\pi$
$480$	0	0
$481$	−452.817	−0.941408
$482$	− 285.371i	− 0.592056i
$483$	0	0
$484$	−240.182	−0.496243
$485$	570.946i	1.17721i
$486$	0	0
$487$	−28.3337	−0.0581800	−0.0290900	−	0.999577i	$-0.509261\pi$
$487$	−28.3337	−0.0581800	−0.0290900	+	0.999577i	$0.509261\pi$
$488$	37.0264i	0.0758738i
$489$	0	0
$490$	−152.091	−0.310389
$491$	949.697i	1.93421i	0.254379	+	0.967105i	$0.418129\pi$
$491$	949.697i	1.93421i	−0.254379	+	0.967105i	$0.581871\pi$
$492$	0	0
$493$	−258.272	−0.523879
$494$	− 338.682i	− 0.685592i
$495$	0	0
$496$	12.1816	0.0245597
$497$	− 150.443i	− 0.302702i
$498$	0	0
$499$	−560.226	−1.12270	−0.561349	−	0.827579i	$-0.689717\pi$
$499$	−560.226	−1.12270	−0.561349	+	0.827579i	$0.689717\pi$
$500$	239.023i	0.478046i
$501$	0	0
$502$	74.4245	0.148256
$503$	− 897.832i	− 1.78495i	−0.451094	−	0.892477i	$-0.648966\pi$
$503$	− 897.832i	− 1.78495i	0.451094	−	0.892477i	$-0.351034\pi$
$504$	0	0
$505$	−766.635	−1.51809
$506$	− 1.28578i	− 0.00254107i
$507$	0	0
$508$	156.182	0.307444
$509$	− 196.823i	− 0.386685i	−0.981131	−	0.193343i	$-0.938067\pi$
$509$	− 196.823i	− 0.386685i	0.981131	−	0.193343i	$-0.0619329\pi$
$510$	0	0
$511$	66.0296	0.129216
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−114.136	−0.222055
$515$	535.912i	1.04060i
$516$	0	0
$517$	−43.1816	−0.0835235
$518$	551.328i	1.06434i
$519$	0	0
$520$	−142.515	−0.274068
$521$	− 375.837i	− 0.721377i	−0.932686	−	0.360688i	$-0.882542\pi$
$521$	− 375.837i	− 0.721377i	0.932686	−	0.360688i	$-0.117458\pi$
$522$	0	0
$523$	91.1827	0.174345	0.0871727	−	0.996193i	$-0.472217\pi$
$523$	91.1827	0.174345	0.0871727	+	0.996193i	$0.472217\pi$
$524$	− 468.126i	− 0.893369i
$525$	0	0
$526$	−655.832	−1.24683
$527$	− 57.4899i	− 0.109089i
$528$	0	0
$529$	528.091	0.998281
$530$	693.607i	1.30869i
$531$	0	0
$532$	−412.363	−0.775119
$533$	105.864i	0.198618i
$534$	0	0
$535$	187.273	0.350044
$536$	212.260i	0.396008i
$537$	0	0
$538$	−61.4847	−0.114284
$539$	− 19.7348i	− 0.0366137i
$540$	0	0
$541$	−38.8490	−0.0718096	−0.0359048	−	0.999355i	$-0.511431\pi$
$541$	−38.8490	−0.0718096	−0.0359048	+	0.999355i	$0.511431\pi$
$542$	484.560i	0.894022i
$543$	0	0
$544$	−106.788	−0.196301
$545$	770.446i	1.41366i
$546$	0	0
$547$	−466.044	−0.852001	−0.426000	−	0.904723i	$-0.640078\pi$
$547$	−466.044	−0.852001	−0.426000	+	0.904723i	$0.640078\pi$
$548$	299.662i	0.546829i
$549$	0	0
$550$	2.69694	0.00490352
$551$	337.890i	0.613230i
$552$	0	0
$553$	365.182	0.660365
$554$	69.2965i	0.125084i
$555$	0	0
$556$	169.060	0.304065
$557$	695.042i	1.24783i	0.781492	+	0.623916i	$0.214459\pi$
$557$	695.042i	1.24783i	−0.781492	+	0.623916i	$0.785541\pi$
$558$	0	0
$559$	−436.803	−0.781400
$560$	173.520i	0.309857i
$561$	0	0
$562$	29.1964	0.0519509
$563$	− 547.074i	− 0.971713i	−0.874039	−	0.485857i	$-0.838508\pi$
$563$	− 547.074i	− 0.971713i	0.874039	−	0.485857i	$-0.161492\pi$
$564$	0	0
$565$	−888.998	−1.57345
$566$	− 75.5313i	− 0.133447i
$567$	0	0
$568$	−50.9694	−0.0897348
$569$	− 249.362i	− 0.438247i	−0.975697	−	0.219123i	$-0.929680\pi$
$569$	− 249.362i	− 0.438247i	0.975697	−	0.219123i	$-0.0703197\pi$
$570$	0	0
$571$	73.8332	0.129305	0.0646525	−	0.997908i	$-0.479406\pi$
$571$	73.8332	0.129305	0.0646525	+	0.997908i	$0.479406\pi$
$572$	− 18.4923i	− 0.0323292i
$573$	0	0
$574$	128.894	0.224555
$575$	− 1.90702i	− 0.00331656i
$576$	0	0
$577$	−43.9092	−0.0760991	−0.0380496	−	0.999276i	$-0.512114\pi$
$577$	−43.9092	−0.0760991	−0.0380496	+	0.999276i	$0.512114\pi$
$578$	95.2660i	0.164820i
$579$	0	0
$580$	142.182	0.245141
$581$	− 1088.08i	− 1.87276i
$582$	0	0
$583$	−90.0000	−0.154374
$584$	− 22.3706i	− 0.0383057i
$585$	0	0
$586$	−21.1056	−0.0360164
$587$	440.387i	0.750234i	0.926978	+	0.375117i	$0.122397\pi$
$587$	440.387i	0.750234i	−0.926978	+	0.375117i	$0.877603\pi$
$588$	0	0
$589$	−75.2122	−0.127695
$590$	138.014i	0.233921i
$591$	0	0
$592$	186.788	0.315520
$593$	− 347.232i	− 0.585551i	−0.956181	−	0.292776i	$-0.905421\pi$
$593$	− 347.232i	− 0.585551i	0.956181	−	0.292776i	$-0.0945789\pi$
$594$	0	0
$595$	818.908	1.37632
$596$	− 231.010i	− 0.387600i
$597$	0	0
$598$	−13.0760	−0.0218663
$599$	− 789.911i	− 1.31872i	−0.751829	−	0.659359i	$-0.770828\pi$
$599$	− 789.911i	− 1.31872i	0.751829	−	0.659359i	$-0.229172\pi$
$600$	0	0
$601$	−706.909	−1.17622	−0.588111	−	0.808780i	$-0.700128\pi$
$601$	−706.909	−1.17622	−0.588111	+	0.808780i	$0.700128\pi$
$602$	531.829i	0.883438i
$603$	0	0
$604$	129.546	0.214480
$605$	− 624.010i	− 1.03142i
$606$	0	0
$607$	−1193.26	−1.96583	−0.982913	−	0.184070i	$-0.941073\pi$
$607$	−1193.26	−1.96583	−0.982913	+	0.184070i	$0.941073\pi$
$608$	139.707i	0.229781i
$609$	0	0
$610$	−96.1975	−0.157701
$611$	439.145i	0.718731i
$612$	0	0
$613$	629.181	1.02640	0.513198	−	0.858270i	$-0.328461\pi$
$613$	629.181	1.02640	0.513198	+	0.858270i	$0.328461\pi$
$614$	− 93.2097i	− 0.151807i
$615$	0	0
$616$	−22.5153	−0.0365508
$617$	192.730i	0.312366i	0.987728	+	0.156183i	$0.0499189\pi$
$617$	192.730i	0.312366i	−0.987728	+	0.156183i	$0.950081\pi$
$618$	0	0
$619$	−152.955	−0.247100	−0.123550	−	0.992338i	$-0.539428\pi$
$619$	−152.955	−0.247100	−0.123550	+	0.992338i	$0.539428\pi$
$620$	31.6488i	0.0510465i
$621$	0	0
$622$	−353.803	−0.568814
$623$	− 1213.03i	− 1.94708i
$624$	0	0
$625$	−671.000	−1.07360
$626$	603.012i	0.963278i
$627$	0	0
$628$	41.6367	0.0663005
$629$	− 881.525i	− 1.40147i
$630$	0	0
$631$	44.8786	0.0711229	0.0355615	−	0.999367i	$-0.488678\pi$
$631$	44.8786	0.0711229	0.0355615	+	0.999367i	$0.488678\pi$
$632$	− 123.722i	− 0.195763i
$633$	0	0
$634$	656.166	1.03496
$635$	405.772i	0.639011i
$636$	0	0
$637$	−200.697	−0.315066
$638$	18.4490i	0.0289169i
$639$	0	0
$640$	58.7878	0.0918559
$641$	241.455i	0.376685i	0.982103	+	0.188342i	$0.0603115\pi$
$641$	241.455i	0.376685i	−0.982103	+	0.188342i	$0.939689\pi$
$642$	0	0
$643$	791.409	1.23081	0.615403	−	0.788212i	$-0.288993\pi$
$643$	791.409	1.23081	0.615403	+	0.788212i	$0.288993\pi$
$644$	15.9207i	0.0247216i
$645$	0	0
$646$	659.333	1.02064
$647$	294.028i	0.454448i	0.973842	+	0.227224i	$0.0729650\pi$
$647$	294.028i	0.454448i	−0.973842	+	0.227224i	$0.927035\pi$
$648$	0	0
$649$	−17.9082	−0.0275935
$650$	− 27.4271i	− 0.0421955i
$651$	0	0
$652$	−266.120	−0.408160
$653$	− 768.313i	− 1.17659i	−0.808647	−	0.588295i	$-0.799799\pi$
$653$	− 768.313i	− 1.17659i	0.808647	−	0.588295i	$-0.200201\pi$
$654$	0	0
$655$	1216.23	1.85683
$656$	− 43.6689i	− 0.0665684i
$657$	0	0
$658$	534.681	0.812585
$659$	430.939i	0.653928i	0.945037	+	0.326964i	$0.106026\pi$
$659$	430.939i	0.653928i	−0.945037	+	0.326964i	$0.893974\pi$
$660$	0	0
$661$	1012.27	1.53142	0.765712	−	0.643183i	$-0.222387\pi$
$661$	1012.27	1.53142	0.765712	+	0.643183i	$0.222387\pi$
$662$	− 668.644i	− 1.01004i
$663$	0	0
$664$	−368.636	−0.555174
$665$	− 1071.35i	− 1.61105i
$666$	0	0
$667$	13.0454	0.0195583
$668$	589.019i	0.881765i
$669$	0	0
$670$	−551.469	−0.823088
$671$	− 12.4822i	− 0.0186025i
$672$	0	0
$673$	563.211	0.836867	0.418433	−	0.908248i	$-0.362579\pi$
$673$	563.211	0.836867	0.418433	+	0.908248i	$0.362579\pi$
$674$	− 432.192i	− 0.641235i
$675$	0	0
$676$	149.939	0.221803
$677$	350.136i	0.517187i	0.965986	+	0.258594i	$0.0832591\pi$
$677$	350.136i	0.517187i	−0.965986	+	0.258594i	$0.916741\pi$
$678$	0	0
$679$	917.318	1.35098
$680$	− 277.443i	− 0.408004i
$681$	0	0
$682$	−4.10664	−0.00602146
$683$	502.818i	0.736190i	0.929788	+	0.368095i	$0.119990\pi$
$683$	502.818i	0.736190i	−0.929788	+	0.368095i	$0.880010\pi$
$684$	0	0
$685$	−778.546	−1.13656
$686$	− 334.161i	− 0.487115i
$687$	0	0
$688$	180.182	0.261892
$689$	915.274i	1.32841i
$690$	0	0
$691$	376.319	0.544600	0.272300	−	0.962212i	$-0.412216\pi$
$691$	376.319	0.544600	0.272300	+	0.962212i	$0.412216\pi$
$692$	− 138.529i	− 0.200186i
$693$	0	0
$694$	189.044	0.272398
$695$	439.231i	0.631987i
$696$	0	0
$697$	−206.091	−0.295683
$698$	139.664i	0.200091i
$699$	0	0
$700$	−33.3939	−0.0477055
$701$	489.681i	0.698546i	0.937021	+	0.349273i	$0.113571\pi$
$701$	489.681i	0.698546i	−0.937021	+	0.349273i	$0.886429\pi$
$702$	0	0
$703$	−1153.27	−1.64050
$704$	7.62809i	0.0108354i
$705$	0	0
$706$	461.196	0.653253
$707$	1231.72i	1.74218i
$708$	0	0
$709$	474.029	0.668588	0.334294	−	0.942469i	$-0.391502\pi$
$709$	474.029	0.668588	0.334294	+	0.942469i	$0.391502\pi$
$710$	− 132.422i	− 0.186510i
$711$	0	0
$712$	−410.969	−0.577204
$713$	2.90383i	0.00407270i
$714$	0	0
$715$	48.0444	0.0671949
$716$	− 94.9659i	− 0.132634i
$717$	0	0
$718$	−592.182	−0.824766
$719$	− 108.122i	− 0.150379i	−0.997169	−	0.0751894i	$-0.976044\pi$
$719$	− 108.122i	− 0.150379i	0.997169	−	0.0751894i	$-0.0239561\pi$
$720$	0	0
$721$	861.030	1.19422
$722$	− 352.053i	− 0.487607i
$723$	0	0
$724$	−485.757	−0.670935
$725$	27.3629i	0.0377419i
$726$	0	0
$727$	−444.591	−0.611542	−0.305771	−	0.952105i	$-0.598914\pi$
$727$	−444.591	−0.611542	−0.305771	+	0.952105i	$0.598914\pi$
$728$	228.974i	0.314525i
$729$	0	0
$730$	58.1204	0.0796170
$731$	− 850.349i	− 1.16327i
$732$	0	0
$733$	−716.362	−0.977302	−0.488651	−	0.872479i	$-0.662511\pi$
$733$	−716.362	−0.977302	−0.488651	+	0.872479i	$0.662511\pi$
$734$	− 264.779i	− 0.360734i
$735$	0	0
$736$	5.39388	0.00732864
$737$	− 71.5567i	− 0.0970918i
$738$	0	0
$739$	933.362	1.26301	0.631504	−	0.775373i	$-0.282438\pi$
$739$	933.362	1.26301	0.631504	+	0.775373i	$0.282438\pi$
$740$	485.289i	0.655796i
$741$	0	0
$742$	1114.39	1.50188
$743$	− 15.9110i	− 0.0214145i	−0.999943	−	0.0107073i	$-0.996592\pi$
$743$	− 15.9110i	− 0.0214145i	0.999943	−	0.0107073i	$-0.00340829\pi$
$744$	0	0
$745$	600.181	0.805612
$746$	− 637.852i	− 0.855030i
$747$	0	0
$748$	36.0000	0.0481283
$749$	− 300.885i	− 0.401716i
$750$	0	0
$751$	809.831	1.07834	0.539169	−	0.842198i	$-0.318739\pi$
$751$	809.831	1.07834	0.539169	+	0.842198i	$0.318739\pi$
$752$	− 181.148i	− 0.240888i
$753$	0	0
$754$	187.621	0.248834
$755$	336.570i	0.445788i
$756$	0	0
$757$	689.637	0.911013	0.455506	−	0.890232i	$-0.349458\pi$
$757$	689.637	0.911013	0.455506	+	0.890232i	$0.349458\pi$
$758$	692.493i	0.913579i
$759$	0	0
$760$	−362.969	−0.477591
$761$	− 953.082i	− 1.25241i	−0.779659	−	0.626204i	$-0.784608\pi$
$761$	− 953.082i	− 1.25241i	0.779659	−	0.626204i	$-0.215392\pi$
$762$	0	0
$763$	1237.85	1.62234
$764$	15.0635i	0.0197167i
$765$	0	0
$766$	145.743	0.190266
$767$	182.121i	0.237446i
$768$	0	0
$769$	−656.696	−0.853961	−0.426980	−	0.904261i	$-0.640423\pi$
$769$	−656.696	−0.853961	−0.426980	+	0.904261i	$0.640423\pi$
$770$	− 58.4965i	− 0.0759695i
$771$	0	0
$772$	−690.908	−0.894959
$773$	− 278.021i	− 0.359665i	−0.983697	−	0.179832i	$-0.942445\pi$
$773$	− 278.021i	− 0.359665i	0.983697	−	0.179832i	$-0.0575555\pi$
$774$	0	0
$775$	−6.09082	−0.00785912
$776$	− 310.784i	− 0.400494i
$777$	0	0
$778$	−48.4699	−0.0623006
$779$	269.622i	0.346113i
$780$	0	0
$781$	17.1827	0.0220008
$782$	− 25.4558i	− 0.0325522i
$783$	0	0
$784$	82.7878	0.105597
$785$	108.175i	0.137803i
$786$	0	0
$787$	821.954	1.04441	0.522207	−	0.852819i	$-0.325109\pi$
$787$	821.954	1.04441	0.522207	+	0.852819i	$0.325109\pi$
$788$	− 154.449i	− 0.196002i
$789$	0	0
$790$	321.439	0.406885
$791$	1428.32i	1.80572i
$792$	0	0
$793$	−126.941	−0.160077
$794$	− 11.6990i	− 0.0147343i
$795$	0	0
$796$	−306.969	−0.385640
$797$	1322.51i	1.65937i	0.558235	+	0.829683i	$0.311479\pi$
$797$	1322.51i	1.65937i	−0.558235	+	0.829683i	$0.688521\pi$
$798$	0	0
$799$	−854.908	−1.06997
$800$	11.3137i	0.0141421i
$801$	0	0
$802$	585.650	0.730238
$803$	7.54150i	0.00939166i
$804$	0	0
$805$	−41.3633	−0.0513829
$806$	41.7633i	0.0518155i
$807$	0	0
$808$	417.303	0.516464
$809$	235.681i	0.291324i	0.989334	+	0.145662i	$0.0465311\pi$
$809$	235.681i	0.291324i	−0.989334	+	0.145662i	$0.953469\pi$
$810$	0	0
$811$	−587.362	−0.724244	−0.362122	−	0.932131i	$-0.617948\pi$
$811$	−587.362	−0.724244	−0.362122	+	0.932131i	$0.617948\pi$
$812$	− 228.438i	− 0.281328i
$813$	0	0
$814$	−62.9694	−0.0773580
$815$	− 691.401i	− 0.848345i
$816$	0	0
$817$	−1112.48	−1.36167
$818$	461.334i	0.563978i
$819$	0	0
$820$	113.455	0.138360
$821$	943.913i	1.14971i	0.818255	+	0.574856i	$0.194942\pi$
$821$	943.913i	1.14971i	−0.818255	+	0.574856i	$0.805058\pi$
$822$	0	0
$823$	−1615.74	−1.96323	−0.981617	−	0.190859i	$-0.938873\pi$
$823$	−1615.74	−1.96323	−0.981617	+	0.190859i	$0.938873\pi$
$824$	− 291.713i	− 0.354021i
$825$	0	0
$826$	221.741	0.268452
$827$	582.354i	0.704177i	0.935967	+	0.352088i	$0.114528\pi$
$827$	582.354i	0.704177i	−0.935967	+	0.352088i	$0.885472\pi$
$828$	0	0
$829$	877.121	1.05805	0.529024	−	0.848607i	$-0.322558\pi$
$829$	877.121	1.05805	0.529024	+	0.848607i	$0.322558\pi$
$830$	− 957.744i	− 1.15391i
$831$	0	0
$832$	77.5755	0.0932398
$833$	− 390.708i	− 0.469038i
$834$	0	0
$835$	−1530.32	−1.83271
$836$	− 47.0976i	− 0.0563369i
$837$	0	0
$838$	−764.772	−0.912616
$839$	1137.12i	1.35533i	0.735370	+	0.677666i	$0.237009\pi$
$839$	1137.12i	1.35533i	−0.735370	+	0.677666i	$0.762991\pi$
$840$	0	0
$841$	653.818	0.777430
$842$	− 401.208i	− 0.476494i
$843$	0	0
$844$	103.121	0.122182
$845$	389.552i	0.461009i
$846$	0	0
$847$	−1002.57	−1.18368
$848$	− 377.552i	− 0.445226i
$849$	0	0
$850$	53.3939	0.0628163
$851$	44.5261i	0.0523221i
$852$	0	0
$853$	−319.817	−0.374932	−0.187466	−	0.982271i	$-0.560028\pi$
$853$	−319.817	−0.374932	−0.187466	+	0.982271i	$0.560028\pi$
$854$	154.557i	0.180980i
$855$	0	0
$856$	−101.939	−0.119087
$857$	− 797.968i	− 0.931118i	−0.885017	−	0.465559i	$-0.845853\pi$
$857$	− 797.968i	− 0.931118i	0.885017	−	0.465559i	$-0.154147\pi$
$858$	0	0
$859$	−467.802	−0.544588	−0.272294	−	0.962214i	$-0.587782\pi$
$859$	−467.802	−0.544588	−0.272294	+	0.962214i	$0.587782\pi$
$860$	468.126i	0.544332i
$861$	0	0
$862$	−364.059	−0.422342
$863$	− 1304.85i	− 1.51199i	−0.654578	−	0.755994i	$-0.727154\pi$
$863$	− 1304.85i	− 1.51199i	0.654578	−	0.755994i	$-0.272846\pi$
$864$	0	0
$865$	359.908	0.416079
$866$	− 673.551i	− 0.777772i
$867$	0	0
$868$	50.8490	0.0585818
$869$	41.7088i	0.0479964i
$870$	0	0
$871$	−727.711	−0.835489
$872$	− 419.378i	− 0.480938i
$873$	0	0
$874$	−33.3031	−0.0381042
$875$	997.738i	1.14027i
$876$	0	0
$877$	−373.756	−0.426176	−0.213088	−	0.977033i	$-0.568352\pi$
$877$	−373.756	−0.426176	−0.213088	+	0.977033i	$0.568352\pi$
$878$	788.937i	0.898562i
$879$	0	0
$880$	−19.8184	−0.0225209
$881$	229.979i	0.261043i	0.991445	+	0.130522i	$0.0416652\pi$
$881$	229.979i	0.261043i	−0.991445	+	0.130522i	$0.958335\pi$
$882$	0	0
$883$	−1381.79	−1.56488	−0.782439	−	0.622728i	$-0.786024\pi$
$883$	−1381.79	−1.56488	−0.782439	+	0.622728i	$0.786024\pi$
$884$	− 366.110i	− 0.414151i
$885$	0	0
$886$	−1176.41	−1.32778
$887$	876.180i	0.987802i	0.869518	+	0.493901i	$0.164429\pi$
$887$	876.180i	0.987802i	−0.869518	+	0.493901i	$0.835571\pi$
$888$	0	0
$889$	651.939	0.733339
$890$	− 1067.73i	− 1.19970i
$891$	0	0
$892$	−626.454	−0.702303
$893$	1118.45i	1.25246i
$894$	0	0
$895$	246.729	0.275674
$896$	− 94.4521i	− 0.105415i
$897$	0	0
$898$	1032.27	1.14952
$899$	− 41.6655i	− 0.0463465i
$900$	0	0
$901$	−1781.82	−1.97760
$902$	14.7215i	0.0163210i
$903$	0	0
$904$	483.909	0.535298
$905$	− 1262.03i	− 1.39451i
$906$	0	0
$907$	1180.07	1.30107	0.650537	−	0.759475i	$-0.274544\pi$
$907$	1180.07	1.30107	0.650537	+	0.759475i	$0.274544\pi$
$908$	− 152.542i	− 0.167998i
$909$	0	0
$910$	−594.892	−0.653728
$911$	− 1270.32i	− 1.39442i	−0.716867	−	0.697210i	$-0.754424\pi$
$911$	− 1270.32i	− 1.39442i	0.716867	−	0.697210i	$-0.245576\pi$
$912$	0	0
$913$	124.273	0.136116
$914$	− 1003.58i	− 1.09800i
$915$	0	0
$916$	243.090	0.265382
$917$	− 1954.07i	− 2.13093i
$918$	0	0
$919$	1316.63	1.43268	0.716340	−	0.697751i	$-0.245816\pi$
$919$	1316.63	1.43268	0.716340	+	0.697751i	$0.245816\pi$
$920$	14.0137i	0.0152323i
$921$	0	0
$922$	13.0148	0.0141158
$923$	− 174.743i	− 0.189320i
$924$	0	0
$925$	−93.3939	−0.100966
$926$	78.0610i	0.0842991i
$927$	0	0
$928$	−77.3939	−0.0833986
$929$	− 627.492i	− 0.675449i	−0.941245	−	0.337724i	$-0.890343\pi$
$929$	− 627.492i	− 0.675449i	0.941245	−	0.337724i	$-0.109657\pi$
$930$	0	0
$931$	−511.151	−0.549034
$932$	302.041i	0.324079i
$933$	0	0
$934$	−885.031	−0.947570
$935$	93.5307i	0.100033i
$936$	0	0
$937$	469.789	0.501375	0.250688	−	0.968068i	$-0.419343\pi$
$937$	469.789	0.501375	0.250688	+	0.968068i	$0.419343\pi$
$938$	886.025i	0.944590i
$939$	0	0
$940$	470.636	0.500676
$941$	930.670i	0.989022i	0.869171	+	0.494511i	$0.164653\pi$
$941$	930.670i	0.989022i	−0.869171	+	0.494511i	$0.835347\pi$
$942$	0	0
$943$	10.4097	0.0110389
$944$	− 75.1250i	− 0.0795816i
$945$	0	0
$946$	−60.7423	−0.0642097
$947$	3.63113i	0.00383435i	0.999998	+	0.00191717i	$0.000610256\pi$
$947$	3.63113i	0.00383435i	−0.999998	+	0.00191717i	$0.999390\pi$
$948$	0	0
$949$	76.6949	0.0808165
$950$	− 69.8535i	− 0.0735300i
$951$	0	0
$952$	−445.757	−0.468232
$953$	− 719.641i	− 0.755132i	−0.925983	−	0.377566i	$-0.876761\pi$
$953$	− 719.641i	− 0.755132i	0.925983	−	0.377566i	$-0.123239\pi$
$954$	0	0
$955$	−39.1362	−0.0409803
$956$	175.427i	0.183501i
$957$	0	0
$958$	−437.408	−0.456584
$959$	1250.86i	1.30434i
$960$	0	0
$961$	−951.725	−0.990349
$962$	640.380i	0.665676i
$963$	0	0
$964$	−403.576	−0.418647
$965$	− 1795.03i	− 1.86014i
$966$	0	0
$967$	33.7740	0.0349266	0.0174633	−	0.999848i	$-0.494441\pi$
$967$	33.7740	0.0349266	0.0174633	+	0.999848i	$0.494441\pi$
$968$	339.668i	0.350897i
$969$	0	0
$970$	807.439	0.832412
$971$	970.472i	0.999456i	0.866182	+	0.499728i	$0.166567\pi$
$971$	970.472i	0.999456i	−0.866182	+	0.499728i	$0.833433\pi$
$972$	0	0
$973$	705.697	0.725279
$974$	40.0699i	0.0411395i
$975$	0	0
$976$	52.3633	0.0536509
$977$	− 1570.30i	− 1.60727i	−0.595123	−	0.803635i	$-0.702897\pi$
$977$	− 1570.30i	− 1.60727i	0.595123	−	0.803635i	$-0.297103\pi$
$978$	0	0
$979$	138.545	0.141517
$980$	215.089i	0.219478i
$981$	0	0
$982$	1343.07	1.36769
$983$	775.878i	0.789296i	0.918832	+	0.394648i	$0.129134\pi$
$983$	775.878i	0.789296i	−0.918832	+	0.394648i	$0.870866\pi$
$984$	0	0
$985$	401.271	0.407382
$986$	365.252i	0.370439i
$987$	0	0
$988$	−478.969	−0.484787
$989$	42.9513i	0.0434290i
$990$	0	0
$991$	870.454	0.878359	0.439180	−	0.898399i	$-0.355269\pi$
$991$	870.454	0.878359	0.439180	+	0.898399i	$0.355269\pi$
$992$	− 17.2274i	− 0.0173664i
$993$	0	0
$994$	−212.758	−0.214042
$995$	− 797.530i	− 0.801538i
$996$	0	0
$997$	1245.00	1.24874	0.624372	−	0.781127i	$-0.285355\pi$
$997$	1245.00	1.24874	0.624372	+	0.781127i	$0.285355\pi$
$998$	792.279i	0.793867i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.b.a.161.1		4
3.2	odd	2	inner	162.3.b.a.161.4		4
4.3	odd	2		1296.3.e.g.161.2		4
9.2	odd	6		18.3.d.a.5.2	✓	4
9.4	even	3		18.3.d.a.11.2	yes	4
9.5	odd	6		54.3.d.a.35.1		4
9.7	even	3		54.3.d.a.17.1		4
12.11	even	2		1296.3.e.g.161.4		4
36.7	odd	6		432.3.q.d.17.1		4
36.11	even	6		144.3.q.c.113.2		4
36.23	even	6		432.3.q.d.305.1		4
36.31	odd	6		144.3.q.c.65.2		4
45.2	even	12		450.3.k.a.149.4		8
45.4	even	6		450.3.i.b.101.1		4
45.7	odd	12		1350.3.k.a.449.1		8
45.13	odd	12		450.3.k.a.299.4		8
45.14	odd	6		1350.3.i.b.251.2		4
45.22	odd	12		450.3.k.a.299.1		8
45.23	even	12		1350.3.k.a.899.1		8
45.29	odd	6		450.3.i.b.401.1		4
45.32	even	12		1350.3.k.a.899.4		8
45.34	even	6		1350.3.i.b.1151.2		4
45.38	even	12		450.3.k.a.149.1		8
45.43	odd	12		1350.3.k.a.449.4		8
72.5	odd	6		1728.3.q.d.1601.2		4
72.11	even	6		576.3.q.e.257.1		4
72.13	even	6		576.3.q.f.65.2		4
72.29	odd	6		576.3.q.f.257.2		4
72.43	odd	6		1728.3.q.c.449.1		4
72.59	even	6		1728.3.q.c.1601.1		4
72.61	even	6		1728.3.q.d.449.2		4
72.67	odd	6		576.3.q.e.65.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.2	✓	4	9.2	odd	6
18.3.d.a.11.2	yes	4	9.4	even	3
54.3.d.a.17.1		4	9.7	even	3
54.3.d.a.35.1		4	9.5	odd	6
144.3.q.c.65.2		4	36.31	odd	6
144.3.q.c.113.2		4	36.11	even	6
162.3.b.a.161.1		4	1.1	even	1	trivial
162.3.b.a.161.4		4	3.2	odd	2	inner
432.3.q.d.17.1		4	36.7	odd	6
432.3.q.d.305.1		4	36.23	even	6
450.3.i.b.101.1		4	45.4	even	6
450.3.i.b.401.1		4	45.29	odd	6
450.3.k.a.149.1		8	45.38	even	12
450.3.k.a.149.4		8	45.2	even	12
450.3.k.a.299.1		8	45.22	odd	12
450.3.k.a.299.4		8	45.13	odd	12
576.3.q.e.65.1		4	72.67	odd	6
576.3.q.e.257.1		4	72.11	even	6
576.3.q.f.65.2		4	72.13	even	6
576.3.q.f.257.2		4	72.29	odd	6
1296.3.e.g.161.2		4	4.3	odd	2
1296.3.e.g.161.4		4	12.11	even	2
1350.3.i.b.251.2		4	45.14	odd	6
1350.3.i.b.1151.2		4	45.34	even	6
1350.3.k.a.449.1		8	45.7	odd	12
1350.3.k.a.449.4		8	45.43	odd	12
1350.3.k.a.899.1		8	45.23	even	12
1350.3.k.a.899.4		8	45.32	even	12
1728.3.q.c.449.1		4	72.43	odd	6
1728.3.q.c.1601.1		4	72.59	even	6
1728.3.q.d.449.2		4	72.61	even	6
1728.3.q.d.1601.2		4	72.5	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -5.19615i q^{5} -8.34847 q^{7} +2.82843i q^{8} -7.34847 q^{10} -0.953512i q^{11} -9.69694 q^{13} +11.8065i q^{14} +4.00000 q^{16} -18.8776i q^{17} -24.6969 q^{19} +10.3923i q^{20} -1.34847 q^{22} +0.953512i q^{23} -2.00000 q^{25} +13.7135i q^{26} +16.6969 q^{28} -13.6814i q^{29} +3.04541 q^{31} -5.65685i q^{32} -26.6969 q^{34} +43.3799i q^{35} +46.6969 q^{37} +34.9267i q^{38} +14.6969 q^{40} -10.9172i q^{41} +45.0454 q^{43} +1.90702i q^{44} +1.34847 q^{46} -45.2869i q^{47} +20.6969 q^{49} +2.82843i q^{50} +19.3939 q^{52} -94.3879i q^{53} -4.95459 q^{55} -23.6130i q^{56} -19.3485 q^{58} -18.7813i q^{59} +13.0908 q^{61} -4.30686i q^{62} -8.00000 q^{64} +50.3868i q^{65} +75.0454 q^{67} +37.7552i q^{68} +61.3485 q^{70} +18.0204i q^{71} -7.90918 q^{73} -66.0394i q^{74} +49.3939 q^{76} +7.96036i q^{77} -43.7423 q^{79} -20.7846i q^{80} -15.4393 q^{82} +130.332i q^{83} -98.0908 q^{85} -63.7038i q^{86} +2.69694 q^{88} +145.300i q^{89} +80.9546 q^{91} -1.90702i q^{92} -64.0454 q^{94} +128.329i q^{95} -109.879 q^{97} -29.2699i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58}+ \cdots - 28 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 40 * q^19 + 24 * q^22 - 8 * q^25 + 8 * q^28 - 76 * q^31 - 48 * q^34 + 128 * q^37 + 92 * q^43 - 24 * q^46 + 24 * q^49 - 40 * q^52 - 108 * q^55 - 48 * q^58 - 124 * q^61 - 32 * q^64 + 212 * q^67 + 216 * q^70 - 208 * q^73 + 80 * q^76 - 28 * q^79 + 144 * q^82 - 216 * q^85 - 48 * q^88 + 412 * q^91 - 168 * q^94 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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