LMFDB - Embedded newform 192.4.c.d.191.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,4,Mod(191,192)] 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("192.191"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.26525057735983104.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{44}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.11
Root			$2.59708 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.4.c.d.191.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(5.19416 - 0.143987i) q^{3} -7.96714i q^{5} -25.6706i q^{7} +(26.9585 - 1.49578i) q^{9} -23.3310 q^{11} -55.6133 q^{13} +(-1.14716 - 41.3826i) q^{15} -115.419i q^{17} +82.4772i q^{19} +(-3.69622 - 133.337i) q^{21} +28.1798 q^{23} +61.5246 q^{25} +(139.811 - 11.6510i) q^{27} -105.588i q^{29} +226.081i q^{31} +(-121.185 + 3.35936i) q^{33} -204.521 q^{35} +295.889 q^{37} +(-288.864 + 8.00756i) q^{39} -446.806i q^{41} -90.6582i q^{43} +(-11.9171 - 214.782i) q^{45} +446.007 q^{47} -315.978 q^{49} +(-16.6187 - 599.502i) q^{51} +332.395i q^{53} +185.882i q^{55} +(11.8756 + 428.400i) q^{57} +261.090 q^{59} -232.663 q^{61} +(-38.3974 - 692.041i) q^{63} +443.079i q^{65} +636.950i q^{67} +(146.370 - 4.05751i) q^{69} -449.162 q^{71} +354.630 q^{73} +(319.569 - 8.85872i) q^{75} +598.921i q^{77} +368.362i q^{79} +(724.525 - 80.6479i) q^{81} -913.178 q^{83} -919.557 q^{85} +(-15.2032 - 548.440i) q^{87} +1069.69i q^{89} +1427.62i q^{91} +(32.5526 + 1174.30i) q^{93} +657.108 q^{95} +521.111 q^{97} +(-628.971 + 34.8980i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) $ 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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			0
$2$		0			0
$3$	5.19416	−	0.143987i	0.999616	−	0.0277102i
$3$	5.19416	−	0.143987i	0.999616	−	0.0277102i
$4$		0			0
$5$		−	7.96714i		−	0.712603i	−0.934371	−	0.356301i	$-0.884038\pi$
$5$		−	7.96714i		−	0.712603i	0.934371	−	0.356301i	$-0.115962\pi$
$6$		0			0
$7$		−	25.6706i		−	1.38608i	−0.720899	−	0.693040i	$-0.756271\pi$
$7$		−	25.6706i		−	1.38608i	0.720899	−	0.693040i	$-0.243729\pi$
$8$		0			0
$9$	26.9585	−	1.49578i	0.998464	−	0.0553992i
$10$		0			0
$11$	−23.3310			−0.639507			−0.319753	−	0.947501i	$-0.603600\pi$
$11$	−23.3310			−0.639507			−0.319753	+	0.947501i	$0.603600\pi$
$12$		0			0
$13$	−55.6133			−1.18649			−0.593244	−	0.805023i	$-0.702153\pi$
$13$	−55.6133			−1.18649			−0.593244	+	0.805023i	$0.702153\pi$
$14$		0			0
$15$	−1.14716	−	41.3826i	−0.0197464	−	0.712329i
$16$		0			0
$17$		−	115.419i		−	1.64665i	−0.567567	−	0.823327i	$-0.692115\pi$
$17$		−	115.419i		−	1.64665i	0.567567	−	0.823327i	$-0.307885\pi$
$18$		0			0
$19$			82.4772i			0.995872i	0.867214	+	0.497936i	$0.165909\pi$
$19$			82.4772i			0.995872i	−0.867214	+	0.497936i	$0.834091\pi$
$20$		0			0
$21$	−3.69622	−	133.337i	−0.0384086	−	1.38555i
$22$		0			0
$23$	28.1798			0.255474			0.127737	−	0.991808i	$-0.459229\pi$
$23$	28.1798			0.255474			0.127737	+	0.991808i	$0.459229\pi$
$24$		0			0
$25$	61.5246			0.492197
$26$		0			0
$27$	139.811	−	11.6510i	0.996546	−	0.0830456i
$28$		0			0
$29$		−	105.588i		−	0.676110i	−0.941126	−	0.338055i	$-0.890231\pi$
$29$		−	105.588i		−	0.676110i	0.941126	−	0.338055i	$-0.109769\pi$
$30$		0			0
$31$			226.081i			1.30985i	0.755695	+	0.654924i	$0.227299\pi$
$31$			226.081i			1.30985i	−0.755695	+	0.654924i	$0.772701\pi$
$32$		0			0
$33$	−121.185	+	3.35936i	−0.639261	+	0.0177209i
$34$		0			0
$35$	−204.521			−0.987725
$36$		0			0
$37$	295.889			1.31470			0.657350	−	0.753586i	$-0.271677\pi$
$37$	295.889			1.31470			0.657350	+	0.753586i	$0.271677\pi$
$38$		0			0
$39$	−288.864	+	8.00756i	−1.18603	+	0.0328779i
$40$		0			0
$41$		−	446.806i		−	1.70194i	−0.525217	−	0.850968i	$-0.676016\pi$
$41$		−	446.806i		−	1.70194i	0.525217	−	0.850968i	$-0.323984\pi$
$42$		0			0
$43$		−	90.6582i		−	0.321517i	−0.986994	−	0.160759i	$-0.948606\pi$
$43$		−	90.6582i		−	0.321517i	0.986994	−	0.160759i	$-0.0513941\pi$
$44$		0			0
$45$	−11.9171	−	214.782i	−0.0394776	−	0.711509i
$46$		0			0
$47$	446.007			1.38419			0.692093	−	0.721808i	$-0.256689\pi$
$47$	446.007			1.38419			0.692093	+	0.721808i	$0.256689\pi$
$48$		0			0
$49$	−315.978			−0.921218
$50$		0			0
$51$	−16.6187	−	599.502i	−0.0456292	−	1.64602i
$52$		0			0
$53$			332.395i			0.861471i	0.902478	+	0.430736i	$0.141746\pi$
$53$			332.395i			0.861471i	−0.902478	+	0.430736i	$0.858254\pi$
$54$		0			0
$55$			185.882i			0.455714i
$56$		0			0
$57$	11.8756	+	428.400i	0.0275958	+	0.995490i
$58$		0			0
$59$	261.090			0.576118			0.288059	−	0.957613i	$-0.406990\pi$
$59$	261.090			0.576118			0.288059	+	0.957613i	$0.406990\pi$
$60$		0			0
$61$	−232.663			−0.488351			−0.244175	−	0.969731i	$-0.578517\pi$
$61$	−232.663			−0.488351			−0.244175	+	0.969731i	$0.578517\pi$
$62$		0			0
$63$	−38.3974	−	692.041i	−0.0767877	−	1.38395i
$64$		0			0
$65$			443.079i			0.845495i
$66$		0			0
$67$			636.950i			1.16143i	0.814107	+	0.580715i	$0.197227\pi$
$67$			636.950i			1.16143i	−0.814107	+	0.580715i	$0.802773\pi$
$68$		0			0
$69$	146.370	−	4.05751i	0.255375	−	0.00707923i
$70$		0			0
$71$	−449.162			−0.750786			−0.375393	−	0.926866i	$-0.622492\pi$
$71$	−449.162			−0.750786			−0.375393	+	0.926866i	$0.622492\pi$
$72$		0			0
$73$	354.630			0.568580			0.284290	−	0.958738i	$-0.408242\pi$
$73$	354.630			0.568580			0.284290	+	0.958738i	$0.408242\pi$
$74$		0			0
$75$	319.569	−	8.85872i	0.492008	−	0.0136389i
$76$		0			0
$77$			598.921i			0.886408i
$78$		0			0
$79$			368.362i			0.524607i	0.964985	+	0.262303i	$0.0844821\pi$
$79$			368.362i			0.524607i	−0.964985	+	0.262303i	$0.915518\pi$
$80$		0			0
$81$	724.525	−	80.6479i	0.993862	−	0.110628i
$82$		0			0
$83$	−913.178			−1.20764			−0.603821	−	0.797120i	$-0.706356\pi$
$83$	−913.178			−1.20764			−0.603821	+	0.797120i	$0.706356\pi$
$84$		0			0
$85$	−919.557			−1.17341
$86$		0			0
$87$	−15.2032	−	548.440i	−0.0187351	−	0.675850i
$88$		0			0
$89$			1069.69i			1.27401i	0.770861	+	0.637003i	$0.219826\pi$
$89$			1069.69i			1.27401i	−0.770861	+	0.637003i	$0.780174\pi$
$90$		0			0
$91$			1427.62i			1.64457i
$92$		0			0
$93$	32.5526	+	1174.30i	0.0362962	+	1.30934i
$94$		0			0
$95$	657.108			0.709661
$96$		0			0
$97$	521.111			0.545472			0.272736	−	0.962089i	$-0.412071\pi$
$97$	521.111			0.545472			0.272736	+	0.962089i	$0.412071\pi$
$98$		0			0
$99$	−628.971	+	34.8980i	−0.638525	+	0.0354281i
$100$		0			0
$101$			215.416i			0.212224i	0.994354	+	0.106112i	$0.0338403\pi$
$101$			215.416i			0.212224i	−0.994354	+	0.106112i	$0.966160\pi$
$102$		0			0
$103$			564.822i			0.540326i	0.962815	+	0.270163i	$0.0870776\pi$
$103$			564.822i			0.540326i	−0.962815	+	0.270163i	$0.912922\pi$
$104$		0			0
$105$	−1062.31	+	29.4483i	−0.987345	+	0.0273701i
$106$		0			0
$107$	1824.55			1.64846			0.824232	−	0.566252i	$-0.191607\pi$
$107$	1824.55			1.64846			0.824232	+	0.566252i	$0.191607\pi$
$108$		0			0
$109$	875.800			0.769600			0.384800	−	0.923000i	$-0.374270\pi$
$109$	875.800			0.769600			0.384800	+	0.923000i	$0.374270\pi$
$110$		0			0
$111$	1536.89	−	42.6041i	1.31419	−	0.0364306i
$112$		0			0
$113$			415.887i			0.346224i	0.984902	+	0.173112i	$0.0553823\pi$
$113$			415.887i			0.346224i	−0.984902	+	0.173112i	$0.944618\pi$
$114$		0			0
$115$		−	224.512i		−	0.182051i
$116$		0			0
$117$	−1499.25	+	83.1851i	−1.18467	+	0.0657305i
$118$		0			0
$119$	−2962.86			−2.28240
$120$		0			0
$121$	−786.662			−0.591031
$122$		0			0
$123$	−64.3341	−	2320.78i	−0.0471610	−	1.70128i
$124$		0			0
$125$		−	1486.07i		−	1.06334i
$126$		0			0
$127$			625.490i			0.437033i	0.975833	+	0.218517i	$0.0701218\pi$
$127$			625.490i			0.437033i	−0.975833	+	0.218517i	$0.929878\pi$
$128$		0			0
$129$	−13.0536	−	470.893i	−0.00890932	−	0.321394i
$130$		0			0
$131$	−777.904			−0.518823			−0.259412	−	0.965767i	$-0.583529\pi$
$131$	−777.904			−0.518823			−0.259412	+	0.965767i	$0.583529\pi$
$132$		0			0
$133$	2117.24			1.38036
$134$		0			0
$135$	−92.8249	−	1113.90i	−0.0591785	−	0.710141i
$136$		0			0
$137$		−	201.060i		−	0.125385i	−0.998033	−	0.0626925i	$-0.980031\pi$
$137$		−	201.060i		−	0.125385i	0.998033	−	0.0626925i	$-0.0199687\pi$
$138$		0			0
$139$		−	1317.85i		−	0.804165i	−0.915603	−	0.402083i	$-0.868286\pi$
$139$		−	1317.85i		−	0.804165i	0.915603	−	0.402083i	$-0.131714\pi$
$140$		0			0
$141$	2316.63	−	64.2190i	1.38365	−	0.0383561i
$142$		0			0
$143$	1297.52			0.758767
$144$		0			0
$145$	−841.234			−0.481798
$146$		0			0
$147$	−1641.24	+	45.4965i	−0.920864	+	0.0255272i
$148$		0			0
$149$		−	1489.20i		−	0.818793i	−0.912357	−	0.409397i	$-0.865739\pi$
$149$		−	1489.20i		−	0.818793i	0.912357	−	0.409397i	$-0.134261\pi$
$150$		0			0
$151$			838.137i			0.451699i	0.974162	+	0.225850i	$0.0725158\pi$
$151$			838.137i			0.451699i	−0.974162	+	0.225850i	$0.927484\pi$
$152$		0			0
$153$	−172.641	−	3111.52i	−0.0912233	−	1.64413i
$154$		0			0
$155$	1801.22			0.933401
$156$		0			0
$157$	−1934.33			−0.983290			−0.491645	−	0.870796i	$-0.663604\pi$
$157$	−1934.33			−0.983290			−0.491645	+	0.870796i	$0.663604\pi$
$158$		0			0
$159$	47.8604	+	1726.51i	0.0238716	+	0.861140i
$160$		0			0
$161$		−	723.391i		−	0.354107i
$162$		0			0
$163$		−	822.129i		−	0.395056i	−0.980297	−	0.197528i	$-0.936709\pi$
$163$		−	822.129i		−	0.395056i	0.980297	−	0.197528i	$-0.0632913\pi$
$164$		0			0
$165$	26.7645	+	965.499i	0.0126279	+	0.455539i
$166$		0			0
$167$	1671.73			0.774626			0.387313	−	0.921948i	$-0.373403\pi$
$167$	1671.73			0.774626			0.387313	+	0.921948i	$0.373403\pi$
$168$		0			0
$169$	895.838			0.407755
$170$		0			0
$171$	123.368	+	2223.46i	0.0551705	+	0.994343i
$172$		0			0
$173$		−	1617.78i		−	0.710970i	−0.934682	−	0.355485i	$-0.884316\pi$
$173$		−	1617.78i		−	0.710970i	0.934682	−	0.355485i	$-0.115684\pi$
$174$		0			0
$175$		−	1579.37i		−	0.682225i
$176$		0			0
$177$	1356.14	−	37.5934i	0.575897	−	0.0159644i
$178$		0			0
$179$	785.554			0.328017			0.164009	−	0.986459i	$-0.447558\pi$
$179$	785.554			0.328017			0.164009	+	0.986459i	$0.447558\pi$
$180$		0			0
$181$	−869.337			−0.357002			−0.178501	−	0.983940i	$-0.557125\pi$
$181$	−869.337			−0.357002			−0.178501	+	0.983940i	$0.557125\pi$
$182$		0			0
$183$	−1208.49	+	33.5003i	−0.488163	+	0.0135323i
$184$		0			0
$185$		−	2357.39i		−	0.936859i
$186$		0			0
$187$			2692.84i			1.05305i
$188$		0			0
$189$	−299.087	−	3589.04i	−0.115108	−	1.38129i
$190$		0			0
$191$	−2376.75			−0.900395			−0.450198	−	0.892929i	$-0.648647\pi$
$191$	−2376.75			−0.900395			−0.450198	+	0.892929i	$0.648647\pi$
$192$		0			0
$193$	−1410.28			−0.525979			−0.262989	−	0.964799i	$-0.584708\pi$
$193$	−1410.28			−0.525979			−0.262989	+	0.964799i	$0.584708\pi$
$194$		0			0
$195$	63.7974	+	2301.42i	0.0234289	+	0.845170i
$196$		0			0
$197$		−	2259.53i		−	0.817184i	−0.912717	−	0.408592i	$-0.866020\pi$
$197$		−	2259.53i		−	0.817184i	0.912717	−	0.408592i	$-0.133980\pi$
$198$		0			0
$199$		−	516.478i		−	0.183981i	−0.995760	−	0.0919904i	$-0.970677\pi$
$199$		−	516.478i		−	0.183981i	0.995760	−	0.0919904i	$-0.0293229\pi$
$200$		0			0
$201$	91.7122	+	3308.42i	0.0321835	+	1.16098i
$202$		0			0
$203$	−2710.50			−0.937142
$204$		0			0
$205$	−3559.77			−1.21280
$206$		0			0
$207$	759.686	−	42.1507i	0.255081	−	0.0141530i
$208$		0			0
$209$		−	1924.28i		−	0.636867i
$210$		0			0
$211$			513.467i			0.167529i	0.996486	+	0.0837643i	$0.0266943\pi$
$211$			513.467i			0.167529i	−0.996486	+	0.0837643i	$0.973306\pi$
$212$		0			0
$213$	−2333.02	+	64.6734i	−0.750497	+	0.0208044i
$214$		0			0
$215$	−722.287			−0.229114
$216$		0			0
$217$	5803.62			1.81555
$218$		0			0
$219$	1842.01	−	51.0620i	0.568362	−	0.0157555i
$220$		0			0
$221$			6418.81i			1.95374i
$222$		0			0
$223$		−	4633.52i		−	1.39140i	−0.718330	−	0.695702i	$-0.755093\pi$
$223$		−	4633.52i		−	1.39140i	0.718330	−	0.695702i	$-0.244907\pi$
$224$		0			0
$225$	1658.61	−	92.0272i	0.491441	−	0.0272673i
$226$		0			0
$227$	−1245.64			−0.364211			−0.182105	−	0.983279i	$-0.558291\pi$
$227$	−1245.64			−0.364211			−0.182105	+	0.983279i	$0.558291\pi$
$228$		0			0
$229$	6053.22			1.74676			0.873381	−	0.487038i	$-0.161923\pi$
$229$	6053.22			1.74676			0.873381	+	0.487038i	$0.161923\pi$
$230$		0			0
$231$	86.2366	+	3110.89i	0.0245626	+	0.886067i
$232$		0			0
$233$			2938.67i			0.826260i	0.910672	+	0.413130i	$0.135564\pi$
$233$			2938.67i			0.826260i	−0.910672	+	0.413130i	$0.864436\pi$
$234$		0			0
$235$		−	3553.40i		−	0.986375i
$236$		0			0
$237$	53.0391	+	1913.33i	0.0145370	+	0.524405i
$238$		0			0
$239$	6245.16			1.69023			0.845117	−	0.534582i	$-0.179531\pi$
$239$	6245.16			1.69023			0.845117	+	0.534582i	$0.179531\pi$
$240$		0			0
$241$	−3913.67			−1.04607			−0.523033	−	0.852312i	$-0.675200\pi$
$241$	−3913.67			−1.04607			−0.523033	+	0.852312i	$0.675200\pi$
$242$		0			0
$243$	3751.69	−	523.220i	0.990415	−	0.138126i
$244$		0			0
$245$			2517.44i			0.656463i
$246$		0			0
$247$		−	4586.83i		−	1.18159i
$248$		0			0
$249$	−4743.19	+	131.485i	−1.20718	+	0.0334640i
$250$		0			0
$251$	−2382.51			−0.599134			−0.299567	−	0.954075i	$-0.596842\pi$
$251$	−2382.51			−0.599134			−0.299567	+	0.954075i	$0.596842\pi$
$252$		0			0
$253$	−657.464			−0.163377
$254$		0			0
$255$	−4776.32	+	132.404i	−1.17296	+	0.0325155i
$256$		0			0
$257$		−	956.287i		−	0.232107i	−0.993243	−	0.116054i	$-0.962976\pi$
$257$		−	956.287i		−	0.232107i	0.993243	−	0.116054i	$-0.0370245\pi$
$258$		0			0
$259$		−	7595.64i		−	1.82228i
$260$		0			0
$261$	−157.936	−	2846.50i	−0.0374559	−	0.675071i
$262$		0			0
$263$	3154.32			0.739558			0.369779	−	0.929120i	$-0.379433\pi$
$263$	3154.32			0.739558			0.369779	+	0.929120i	$0.379433\pi$
$264$		0			0
$265$	2648.24			0.613887
$266$		0			0
$267$	154.021	+	5556.12i	0.0353030	+	1.27352i
$268$		0			0
$269$			1303.93i			0.295547i	0.989021	+	0.147774i	$0.0472107\pi$
$269$			1303.93i			0.295547i	−0.989021	+	0.147774i	$0.952789\pi$
$270$		0			0
$271$			5921.03i			1.32722i	0.748078	+	0.663611i	$0.230977\pi$
$271$			5921.03i			1.32722i	−0.748078	+	0.663611i	$0.769023\pi$
$272$		0			0
$273$	205.559	+	7415.31i	0.0455713	+	1.64394i
$274$		0			0
$275$	−1435.43			−0.314763
$276$		0			0
$277$	2409.31			0.522604			0.261302	−	0.965257i	$-0.415848\pi$
$277$	2409.31			0.522604			0.261302	+	0.965257i	$0.415848\pi$
$278$		0			0
$279$	338.166	+	6094.80i	0.0725645	+	1.30784i
$280$		0			0
$281$			5183.15i			1.10036i	0.835046	+	0.550180i	$0.185441\pi$
$281$			5183.15i			1.10036i	−0.835046	+	0.550180i	$0.814559\pi$
$282$		0			0
$283$		−	6710.86i		−	1.40961i	−0.709402	−	0.704804i	$-0.751035\pi$
$283$		−	6710.86i		−	1.40961i	0.709402	−	0.704804i	$-0.248965\pi$
$284$		0			0
$285$	3413.12	−	94.6147i	0.709389	−	0.0196649i
$286$		0			0
$287$	−11469.8			−2.35902
$288$		0			0
$289$	−8408.46			−1.71147
$290$		0			0
$291$	2706.73	−	75.0329i	0.545262	−	0.0151151i
$292$		0			0
$293$			9713.30i			1.93671i	0.249570	+	0.968357i	$0.419711\pi$
$293$			9713.30i			1.93671i	−0.249570	+	0.968357i	$0.580289\pi$
$294$		0			0
$295$		−	2080.14i		−	0.410543i
$296$		0			0
$297$	−3261.95	+	271.829i	−0.637298	+	0.0531082i
$298$		0			0
$299$	−1567.17			−0.303116
$300$		0			0
$301$	−2327.25			−0.445649
$302$		0			0
$303$	31.0170	+	1118.90i	0.00588079	+	0.212143i
$304$		0			0
$305$			1853.66i			0.348000i
$306$		0			0
$307$			9022.51i			1.67734i	0.544644	+	0.838668i	$0.316665\pi$
$307$			9022.51i			1.67734i	−0.544644	+	0.838668i	$0.683335\pi$
$308$		0			0
$309$	81.3268	+	2933.77i	0.0149725	+	0.540118i
$310$		0			0
$311$	7350.41			1.34020			0.670102	−	0.742269i	$-0.266250\pi$
$311$	7350.41			1.34020			0.670102	+	0.742269i	$0.266250\pi$
$312$		0			0
$313$	−1810.71			−0.326988			−0.163494	−	0.986544i	$-0.552276\pi$
$313$	−1810.71			−0.326988			−0.163494	+	0.986544i	$0.552276\pi$
$314$		0			0
$315$	−5513.59	+	305.918i	−0.986208	+	0.0547191i
$316$		0			0
$317$		−	3825.09i		−	0.677724i	−0.940836	−	0.338862i	$-0.889958\pi$
$317$		−	3825.09i		−	0.677724i	0.940836	−	0.338862i	$-0.110042\pi$
$318$		0			0
$319$			2463.48i			0.432377i
$320$		0			0
$321$	9476.99	−	262.710i	1.64783	−	0.0456793i
$322$		0			0
$323$	9519.41			1.63986
$324$		0			0
$325$	−3421.59			−0.583986
$326$		0			0
$327$	4549.04	−	126.103i	0.769305	−	0.0213258i
$328$		0			0
$329$		−	11449.2i		−	1.91859i
$330$		0			0
$331$		−	1632.70i		−	0.271122i	−0.990769	−	0.135561i	$-0.956716\pi$
$331$		−	1632.70i		−	0.271122i	0.990769	−	0.135561i	$-0.0432837\pi$
$332$		0			0
$333$	7976.74	−	442.584i	1.31268	−	0.0728332i
$334$		0			0
$335$	5074.67			0.827639
$336$		0			0
$337$	7685.23			1.24226			0.621129	−	0.783708i	$-0.286674\pi$
$337$	7685.23			1.24226			0.621129	+	0.783708i	$0.286674\pi$
$338$		0			0
$339$	59.8821	+	2160.18i	0.00959395	+	0.346091i
$340$		0			0
$341$		−	5274.70i		−	0.837657i
$342$		0			0
$343$		−	693.675i		−	0.109198i
$344$		0			0
$345$	−32.3268	−	1166.15i	−0.00504468	−	0.181981i
$346$		0			0
$347$	−8052.38			−1.24575			−0.622874	−	0.782322i	$-0.714035\pi$
$347$	−8052.38			−1.24575			−0.622874	+	0.782322i	$0.714035\pi$
$348$		0			0
$349$	−639.800			−0.0981309			−0.0490655	−	0.998796i	$-0.515624\pi$
$349$	−639.800			−0.0981309			−0.0490655	+	0.998796i	$0.515624\pi$
$350$		0			0
$351$	−7775.38	+	647.949i	−1.18239	+	0.0985326i
$352$		0			0
$353$			7770.76i			1.17166i	0.810434	+	0.585829i	$0.199231\pi$
$353$			7770.76i			1.17166i	−0.810434	+	0.585829i	$0.800769\pi$
$354$		0			0
$355$			3578.54i			0.535012i
$356$		0			0
$357$	−15389.6	+	426.612i	−2.28152	+	0.0632457i
$358$		0			0
$359$	1804.28			0.265255			0.132627	−	0.991166i	$-0.457659\pi$
$359$	1804.28			0.265255			0.132627	+	0.991166i	$0.457659\pi$
$360$		0			0
$361$	56.5105			0.00823888
$362$		0			0
$363$	−4086.05	+	113.269i	−0.590804	+	0.0163776i
$364$		0			0
$365$		−	2825.39i		−	0.405172i
$366$		0			0
$367$		−	13240.2i		−	1.88319i	−0.336746	−	0.941596i	$-0.609326\pi$
$367$		−	13240.2i		−	1.88319i	0.336746	−	0.941596i	$-0.390674\pi$
$368$		0			0
$369$	−668.323	−	12045.2i	−0.0942859	−	1.69932i
$370$		0			0
$371$	8532.77			1.19407
$372$		0			0
$373$	4963.96			0.689073			0.344537	−	0.938773i	$-0.388036\pi$
$373$	4963.96			0.689073			0.344537	+	0.938773i	$0.388036\pi$
$374$		0			0
$375$	−213.974	−	7718.87i	−0.0294655	−	1.06294i
$376$		0			0
$377$			5872.09i			0.802196i
$378$		0			0
$379$			5469.16i			0.741245i	0.928784	+	0.370622i	$0.120856\pi$
$379$			5469.16i			0.741245i	−0.928784	+	0.370622i	$0.879144\pi$
$380$		0			0
$381$	90.0621	+	3248.89i	0.0121103	+	0.436865i
$382$		0			0
$383$	−14696.1			−1.96067			−0.980333	−	0.197349i	$-0.936767\pi$
$383$	−14696.1			−1.96067			−0.980333	+	0.197349i	$0.936767\pi$
$384$		0			0
$385$	4771.69			0.631657
$386$		0			0
$387$	−135.604	−	2444.01i	−0.0178118	−	0.321024i
$388$		0			0
$389$		−	13333.2i		−	1.73785i	−0.494947	−	0.868923i	$-0.664813\pi$
$389$		−	13333.2i		−	1.73785i	0.494947	−	0.868923i	$-0.335187\pi$
$390$		0			0
$391$		−	3252.47i		−	0.420677i
$392$		0			0
$393$	−4040.56	+	112.008i	−0.518624	+	0.0143767i
$394$		0			0
$395$	2934.79			0.373836
$396$		0			0
$397$	−8605.86			−1.08795			−0.543975	−	0.839102i	$-0.683081\pi$
$397$	−8605.86			−1.08795			−0.543975	+	0.839102i	$0.683081\pi$
$398$		0			0
$399$	10997.3	−	304.854i	1.37983	−	0.0382500i
$400$		0			0
$401$		−	8801.23i		−	1.09604i	−0.836465	−	0.548020i	$-0.815382\pi$
$401$		−	8801.23i		−	1.09604i	0.836465	−	0.548020i	$-0.184618\pi$
$402$		0			0
$403$		−	12573.1i		−	1.55412i
$404$		0			0
$405$	−642.534	−	5772.40i	−0.0788340	−	0.708229i
$406$		0			0
$407$	−6903.40			−0.840759
$408$		0			0
$409$	5221.12			0.631218			0.315609	−	0.948889i	$-0.397791\pi$
$409$	5221.12			0.631218			0.315609	+	0.948889i	$0.397791\pi$
$410$		0			0
$411$	−28.9500	−	1044.34i	−0.00347444	−	0.125337i
$412$		0			0
$413$		−	6702.32i		−	0.798546i
$414$		0			0
$415$			7275.42i			0.860569i
$416$		0			0
$417$	−189.753	−	6845.15i	−0.0222836	−	0.803857i
$418$		0			0
$419$	−3566.56			−0.415842			−0.207921	−	0.978146i	$-0.566670\pi$
$419$	−3566.56			−0.415842			−0.207921	+	0.978146i	$0.566670\pi$
$420$		0			0
$421$	209.057			0.0242015			0.0121007	−	0.999927i	$-0.496148\pi$
$421$	209.057			0.0242015			0.0121007	+	0.999927i	$0.496148\pi$
$422$		0			0
$423$	12023.7	−	667.127i	1.38206	−	0.0766828i
$424$		0			0
$425$		−	7101.09i		−	0.810479i
$426$		0			0
$427$			5972.58i			0.676893i
$428$		0			0
$429$	6739.50	−	186.825i	0.758476	−	0.0210256i
$430$		0			0
$431$	1783.54			0.199328			0.0996639	−	0.995021i	$-0.468223\pi$
$431$	1783.54			0.199328			0.0996639	+	0.995021i	$0.468223\pi$
$432$		0			0
$433$	−8790.64			−0.975638			−0.487819	−	0.872945i	$-0.662207\pi$
$433$	−8790.64			−0.975638			−0.487819	+	0.872945i	$0.662207\pi$
$434$		0			0
$435$	−4369.50	+	121.126i	−0.481613	+	0.0133507i
$436$		0			0
$437$			2324.19i			0.254419i
$438$		0			0
$439$			15961.9i			1.73535i	0.497129	+	0.867677i	$0.334388\pi$
$439$			15961.9i			1.73535i	−0.497129	+	0.867677i	$0.665612\pi$
$440$		0			0
$441$	−8518.30	+	472.632i	−0.919803	+	0.0510347i
$442$		0			0
$443$	4876.45			0.522996			0.261498	−	0.965204i	$-0.415784\pi$
$443$	4876.45			0.522996			0.261498	+	0.965204i	$0.415784\pi$
$444$		0			0
$445$	8522.35			0.907861
$446$		0			0
$447$	−214.425	−	7735.15i	−0.0226889	−	0.818479i
$448$		0			0
$449$			520.654i			0.0547242i	0.999626	+	0.0273621i	$0.00871072\pi$
$449$			520.654i			0.0547242i	−0.999626	+	0.0273621i	$0.991289\pi$
$450$		0			0
$451$			10424.5i			1.08840i
$452$		0			0
$453$	120.680	+	4353.42i	0.0125167	+	0.451526i
$454$		0			0
$455$	11374.1			1.17192
$456$		0			0
$457$	3684.65			0.377157			0.188578	−	0.982058i	$-0.439612\pi$
$457$	3684.65			0.377157			0.188578	+	0.982058i	$0.439612\pi$
$458$		0			0
$459$	−1344.74	−	16136.8i	−0.136747	−	1.64097i
$460$		0			0
$461$		−	110.224i		−	0.0111359i	−0.999984	−	0.00556794i	$-0.998228\pi$
$461$		−	110.224i		−	0.0111359i	0.999984	−	0.00556794i	$-0.00177234\pi$
$462$		0			0
$463$			10287.0i			1.03256i	0.856418	+	0.516282i	$0.172685\pi$
$463$			10287.0i			1.03256i	−0.856418	+	0.516282i	$0.827315\pi$
$464$		0			0
$465$	9355.80	−	259.351i	0.933043	−	0.0258648i
$466$		0			0
$467$	4811.83			0.476799			0.238399	−	0.971167i	$-0.423377\pi$
$467$	4811.83			0.476799			0.238399	+	0.971167i	$0.423377\pi$
$468$		0			0
$469$	16350.9			1.60984
$470$		0			0
$471$	−10047.2	+	278.518i	−0.982912	+	0.0272472i
$472$		0			0
$473$			2115.15i			0.205613i
$474$		0			0
$475$			5074.38i			0.490165i
$476$		0			0
$477$	497.189	+	8960.88i	0.0477248	+	0.860148i
$478$		0			0
$479$	−14954.9			−1.42653			−0.713265	−	0.700894i	$-0.752784\pi$
$479$	−14954.9			−1.42653			−0.713265	+	0.700894i	$0.752784\pi$
$480$		0			0
$481$	−16455.4			−1.55988
$482$		0			0
$483$	−104.159	−	3757.41i	−0.00981238	−	0.353971i
$484$		0			0
$485$		−	4151.76i		−	0.388705i
$486$		0			0
$487$			18724.9i			1.74231i	0.491010	+	0.871154i	$0.336628\pi$
$487$			18724.9i			1.74231i	−0.491010	+	0.871154i	$0.663372\pi$
$488$		0			0
$489$	−118.375	−	4270.27i	−0.0109471	−	0.394904i
$490$		0			0
$491$	−14320.6			−1.31625			−0.658127	−	0.752907i	$-0.728651\pi$
$491$	−14320.6			−1.31625			−0.658127	+	0.752907i	$0.728651\pi$
$492$		0			0
$493$	−12186.8			−1.11332
$494$		0			0
$495$	278.038	+	5011.10i	0.0252462	+	0.455015i
$496$		0			0
$497$			11530.3i			1.04065i
$498$		0			0
$499$		−	9558.56i		−	0.857515i	−0.903420	−	0.428757i	$-0.858951\pi$
$499$		−	9558.56i		−	0.857515i	0.903420	−	0.428757i	$-0.141049\pi$
$500$		0			0
$501$	8683.24	−	240.707i	0.774329	−	0.0214651i
$502$		0			0
$503$	−11851.2			−1.05053			−0.525266	−	0.850938i	$-0.676034\pi$
$503$	−11851.2			−1.05053			−0.525266	+	0.850938i	$0.676034\pi$
$504$		0			0
$505$	1716.25			0.151232
$506$		0			0
$507$	4653.12	−	128.989i	0.407598	−	0.0112990i
$508$		0			0
$509$		−	2421.32i		−	0.210851i	−0.994427	−	0.105426i	$-0.966380\pi$
$509$		−	2421.32i		−	0.210851i	0.994427	−	0.105426i	$-0.0336205\pi$
$510$		0			0
$511$		−	9103.56i		−	0.788098i
$512$		0			0
$513$	960.939	+	11531.3i	0.0827027	+	0.992432i
$514$		0			0
$515$	4500.02			0.385038
$516$		0			0
$517$	−10405.8			−0.885197
$518$		0			0
$519$	−232.939	−	8403.02i	−0.0197011	−	0.710697i
$520$		0			0
$521$			14301.2i			1.20258i	0.799030	+	0.601291i	$0.205347\pi$
$521$			14301.2i			1.20258i	−0.799030	+	0.601291i	$0.794653\pi$
$522$		0			0
$523$			17793.9i			1.48771i	0.668340	+	0.743856i	$0.267005\pi$
$523$			17793.9i			1.48771i	−0.668340	+	0.743856i	$0.732995\pi$
$524$		0			0
$525$	−227.408	−	8203.51i	−0.0189046	−	0.681963i
$526$		0			0
$527$	26093.9			2.15687
$528$		0			0
$529$	−11372.9			−0.934733
$530$		0			0
$531$	7038.59	−	390.532i	0.575233	−	0.0319165i
$532$		0			0
$533$			24848.4i			2.01933i
$534$		0			0
$535$		−	14536.4i		−	1.17470i
$536$		0			0
$537$	4080.29	−	113.109i	0.327891	−	0.00908943i
$538$		0			0
$539$	7372.09			0.589125
$540$		0			0
$541$	−112.850			−0.00896822			−0.00448411	−	0.999990i	$-0.501427\pi$
$541$	−112.850			−0.00896822			−0.00448411	+	0.999990i	$0.501427\pi$
$542$		0			0
$543$	−4515.47	+	125.173i	−0.356865	+	0.00989260i
$544$		0			0
$545$		−	6977.63i		−	0.548420i
$546$		0			0
$547$			4625.21i			0.361535i	0.983526	+	0.180768i	$0.0578582\pi$
$547$			4625.21i			0.361535i	−0.983526	+	0.180768i	$0.942142\pi$
$548$		0			0
$549$	−6272.24	+	348.011i	−0.487601	+	0.0270542i
$550$		0			0
$551$	8708.59			0.673319
$552$		0			0
$553$	9456.05			0.727147
$554$		0			0
$555$	−339.433	−	12244.7i	−0.0259606	−	0.936499i
$556$		0			0
$557$			13195.7i			1.00381i	0.864923	+	0.501905i	$0.167367\pi$
$557$			13195.7i			1.00381i	−0.864923	+	0.501905i	$0.832633\pi$
$558$		0			0
$559$			5041.80i			0.381477i
$560$		0			0
$561$	387.732	+	13987.0i	0.0291802	+	1.05264i
$562$		0			0
$563$	−17336.0			−1.29773			−0.648867	−	0.760902i	$-0.724757\pi$
$563$	−17336.0			−1.29773			−0.648867	+	0.760902i	$0.724757\pi$
$564$		0			0
$565$	3313.43			0.246720
$566$		0			0
$567$	−2070.28	−	18599.0i	−0.153340	−	1.37757i
$568$		0			0
$569$		−	12853.5i		−	0.947004i	−0.880793	−	0.473502i	$-0.842990\pi$
$569$		−	12853.5i		−	0.947004i	0.880793	−	0.473502i	$-0.157010\pi$
$570$		0			0
$571$		−	16704.7i		−	1.22429i	−0.790746	−	0.612144i	$-0.790307\pi$
$571$		−	16704.7i		−	1.22429i	0.790746	−	0.612144i	$-0.209693\pi$
$572$		0			0
$573$	−12345.2	+	342.220i	−0.900050	+	0.0249502i
$574$		0			0
$575$	1733.75			0.125743
$576$		0			0
$577$	21248.7			1.53309			0.766547	−	0.642188i	$-0.221973\pi$
$577$	21248.7			1.53309			0.766547	+	0.642188i	$0.221973\pi$
$578$		0			0
$579$	−7325.19	+	203.061i	−0.525777	+	0.0145750i
$580$		0			0
$581$			23441.8i			1.67389i
$582$		0			0
$583$		−	7755.12i		−	0.550917i
$584$		0			0
$585$	662.748	+	11944.8i	0.0468397	+	0.844197i
$586$		0			0
$587$	19358.6			1.36118			0.680592	−	0.732663i	$-0.261723\pi$
$587$	19358.6			1.36118			0.680592	+	0.732663i	$0.261723\pi$
$588$		0			0
$589$	−18646.5			−1.30444
$590$		0			0
$591$	−325.343	−	11736.4i	−0.0226443	−	0.816870i
$592$		0			0
$593$			4834.94i			0.334818i	0.985888	+	0.167409i	$0.0535401\pi$
$593$			4834.94i			0.334818i	−0.985888	+	0.167409i	$0.946460\pi$
$594$		0			0
$595$			23605.5i			1.62644i
$596$		0			0
$597$	−74.3659	−	2682.67i	−0.00509815	−	0.183910i
$598$		0			0
$599$	−11168.9			−0.761848			−0.380924	−	0.924606i	$-0.624394\pi$
$599$	−11168.9			−0.761848			−0.380924	+	0.924606i	$0.624394\pi$
$600$		0			0
$601$	18001.2			1.22177			0.610886	−	0.791719i	$-0.290813\pi$
$601$	18001.2			1.22177			0.610886	+	0.791719i	$0.290813\pi$
$602$		0			0
$603$	952.735	+	17171.2i	0.0643423	+	1.15965i
$604$		0			0
$605$			6267.45i			0.421170i
$606$		0			0
$607$			1849.50i			0.123672i	0.998086	+	0.0618360i	$0.0196956\pi$
$607$			1849.50i			0.123672i	−0.998086	+	0.0618360i	$0.980304\pi$
$608$		0			0
$609$	−14078.8	+	390.276i	−0.936782	+	0.0259684i
$610$		0			0
$611$	−24803.9			−1.64232
$612$		0			0
$613$	14440.9			0.951490			0.475745	−	0.879583i	$-0.342179\pi$
$613$	14440.9			0.951490			0.475745	+	0.879583i	$0.342179\pi$
$614$		0			0
$615$	−18490.0	+	512.559i	−1.21234	+	0.0336071i
$616$		0			0
$617$			13836.0i			0.902780i	0.892327	+	0.451390i	$0.149072\pi$
$617$			13836.0i			0.902780i	−0.892327	+	0.451390i	$0.850928\pi$
$618$		0			0
$619$		−	317.423i		−	0.0206112i	−0.999947	−	0.0103056i	$-0.996720\pi$
$619$		−	317.423i		−	0.0206112i	0.999947	−	0.0103056i	$-0.00328043\pi$
$620$		0			0
$621$	3939.86	−	328.322i	0.254591	−	0.0212159i
$622$		0			0
$623$	27459.5			1.76588
$624$		0			0
$625$	−4149.14			−0.265545
$626$		0			0
$627$	−277.070	−	9995.01i	−0.0176477	−	0.636622i
$628$		0			0
$629$		−	34151.1i		−	2.16486i
$630$		0			0
$631$			12633.5i			0.797036i	0.917160	+	0.398518i	$0.130475\pi$
$631$			12633.5i			0.797036i	−0.917160	+	0.398518i	$0.869525\pi$
$632$		0			0
$633$	73.9323	+	2667.03i	0.00464225	+	0.167464i
$634$		0			0
$635$	4983.36			0.311431
$636$		0			0
$637$	17572.6			1.09301
$638$		0			0
$639$	−12108.8	+	671.847i	−0.749633	+	0.0415929i
$640$		0			0
$641$		−	8902.34i		−	0.548551i	−0.961651	−	0.274276i	$-0.911562\pi$
$641$		−	8902.34i		−	0.548551i	0.961651	−	0.274276i	$-0.0884381\pi$
$642$		0			0
$643$			11374.0i			0.697587i	0.937200	+	0.348794i	$0.113409\pi$
$643$			11374.0i			0.697587i	−0.937200	+	0.348794i	$0.886591\pi$
$644$		0			0
$645$	−3751.67	+	104.000i	−0.229026	+	0.00634880i
$646$		0			0
$647$	−12659.4			−0.769229			−0.384614	−	0.923077i	$-0.625666\pi$
$647$	−12659.4			−0.769229			−0.384614	+	0.923077i	$0.625666\pi$
$648$		0			0
$649$	−6091.49			−0.368431
$650$		0			0
$651$	30144.9	−	835.643i	1.81486	−	0.0503094i
$652$		0			0
$653$			11847.4i			0.709992i	0.934868	+	0.354996i	$0.115518\pi$
$653$			11847.4i			0.709992i	−0.934868	+	0.354996i	$0.884482\pi$
$654$		0			0
$655$			6197.67i			0.369715i
$656$		0			0
$657$	9560.32	−	530.448i	0.567707	−	0.0314989i
$658$		0			0
$659$	−14621.4			−0.864295			−0.432148	−	0.901803i	$-0.642244\pi$
$659$	−14621.4			−0.864295			−0.432148	+	0.901803i	$0.642244\pi$
$660$		0			0
$661$	−16648.8			−0.979673			−0.489836	−	0.871814i	$-0.662943\pi$
$661$	−16648.8			−0.979673			−0.489836	+	0.871814i	$0.662943\pi$
$662$		0			0
$663$	924.222	+	33340.3i	0.0541385	+	1.95299i
$664$		0			0
$665$		−	16868.3i		−	0.983647i
$666$		0			0
$667$		−	2975.44i		−	0.172728i
$668$		0			0
$669$	−667.164	−	24067.2i	−0.0385561	−	1.39087i
$670$		0			0
$671$	5428.26			0.312304
$672$		0			0
$673$	−5387.78			−0.308594			−0.154297	−	0.988025i	$-0.549311\pi$
$673$	−5387.78			−0.308594			−0.154297	+	0.988025i	$0.549311\pi$
$674$		0			0
$675$	8601.85	−	716.822i	0.490497	−	0.0408748i
$676$		0			0
$677$		−	19495.1i		−	1.10673i	−0.832938	−	0.553366i	$-0.813343\pi$
$677$		−	19495.1i		−	1.10673i	0.832938	−	0.553366i	$-0.186657\pi$
$678$		0			0
$679$		−	13377.2i		−	0.756068i
$680$		0			0
$681$	−6470.03	+	179.355i	−0.364071	+	0.0100924i
$682$		0			0
$683$	186.998			0.0104763			0.00523814	−	0.999986i	$-0.498333\pi$
$683$	186.998			0.0104763			0.00523814	+	0.999986i	$0.498333\pi$
$684$		0			0
$685$	−1601.88			−0.0893497
$686$		0			0
$687$	31441.4	−	871.583i	1.74609	−	0.0484031i
$688$		0			0
$689$		−	18485.6i		−	1.02213i
$690$		0			0
$691$		−	28857.1i		−	1.58868i	−0.607477	−	0.794338i	$-0.707818\pi$
$691$		−	28857.1i		−	1.58868i	0.607477	−	0.794338i	$-0.292182\pi$
$692$		0			0
$693$	895.853	+	16146.0i	0.0491062	+	0.885046i
$694$		0			0
$695$	−10499.5			−0.573051
$696$		0			0
$697$	−51569.7			−2.80250
$698$		0			0
$699$	423.129	+	15263.9i	0.0228958	+	0.825942i
$700$		0			0
$701$			24163.6i			1.30192i	0.759111	+	0.650961i	$0.225634\pi$
$701$			24163.6i			1.30192i	−0.759111	+	0.650961i	$0.774366\pi$
$702$		0			0
$703$			24404.1i			1.30927i
$704$		0			0
$705$	−511.642	−	18456.9i	−0.0273327	−	0.985996i
$706$		0			0
$707$	5529.84			0.294160
$708$		0			0
$709$	9401.09			0.497977			0.248988	−	0.968506i	$-0.419902\pi$
$709$	9401.09			0.497977			0.248988	+	0.968506i	$0.419902\pi$
$710$		0			0
$711$	550.987	+	9930.49i	0.0290628	+	0.523801i
$712$		0			0
$713$			6370.90i			0.334631i
$714$		0			0
$715$		−	10337.5i		−	0.540700i
$716$		0			0
$717$	32438.3	−	899.219i	1.68958	−	0.0468367i
$718$		0			0
$719$	2696.45			0.139862			0.0699309	−	0.997552i	$-0.477722\pi$
$719$	2696.45			0.139862			0.0699309	+	0.997552i	$0.477722\pi$
$720$		0			0
$721$	14499.3			0.748935
$722$		0			0
$723$	−20328.2	+	563.517i	−1.04566	+	0.0289867i
$724$		0			0
$725$		−	6496.26i		−	0.332779i
$726$		0			0
$727$		−	21059.4i		−	1.07434i	−0.843472	−	0.537172i	$-0.819492\pi$
$727$		−	21059.4i		−	1.07434i	0.843472	−	0.537172i	$-0.180508\pi$
$728$		0			0
$729$	19411.5	−	3257.88i	0.986207	−	0.165517i
$730$		0			0
$731$	−10463.6			−0.529428
$732$		0			0
$733$	23866.2			1.20262			0.601309	−	0.799017i	$-0.294646\pi$
$733$	23866.2			1.20262			0.601309	+	0.799017i	$0.294646\pi$
$734$		0			0
$735$	362.477	+	13076.0i	0.0181907	+	0.656211i
$736$		0			0
$737$		−	14860.7i		−	0.742743i
$738$		0			0
$739$		−	12335.5i		−	0.614033i	−0.951704	−	0.307016i	$-0.900669\pi$
$739$		−	12335.5i		−	0.614033i	0.951704	−	0.307016i	$-0.0993306\pi$
$740$		0			0
$741$	−660.442	−	23824.7i	−0.0327421	−	1.18114i
$742$		0			0
$743$	−12201.5			−0.602461			−0.301231	−	0.953551i	$-0.597397\pi$
$743$	−12201.5			−0.602461			−0.301231	+	0.953551i	$0.597397\pi$
$744$		0			0
$745$	−11864.7			−0.583474
$746$		0			0
$747$	−24617.9	+	1365.91i	−1.20579	+	0.0669024i
$748$		0			0
$749$		−	46837.2i		−	2.28490i
$750$		0			0
$751$		−	20108.9i		−	0.977074i	−0.872543	−	0.488537i	$-0.837531\pi$
$751$		−	20108.9i		−	0.977074i	0.872543	−	0.488537i	$-0.162469\pi$
$752$		0			0
$753$	−12375.1	+	343.049i	−0.598904	+	0.0166021i
$754$		0			0
$755$	6677.56			0.321882
$756$		0			0
$757$	−27880.4			−1.33861			−0.669307	−	0.742986i	$-0.733409\pi$
$757$	−27880.4			−1.33861			−0.669307	+	0.742986i	$0.733409\pi$
$758$		0			0
$759$	−3414.97	+	94.6659i	−0.163314	+	0.00452721i
$760$		0			0
$761$		−	10517.2i		−	0.500982i	−0.968119	−	0.250491i	$-0.919408\pi$
$761$		−	10517.2i		−	0.500982i	0.968119	−	0.250491i	$-0.0805920\pi$
$762$		0			0
$763$		−	22482.3i		−	1.06673i
$764$		0			0
$765$	−24789.9	+	1375.45i	−1.17161	+	0.0650060i
$766$		0			0
$767$	−14520.0			−0.683557
$768$		0			0
$769$	27544.2			1.29164			0.645818	−	0.763491i	$-0.276516\pi$
$769$	27544.2			1.29164			0.645818	+	0.763491i	$0.276516\pi$
$770$		0			0
$771$	−137.693	−	4967.11i	−0.00643174	−	0.232018i
$772$		0			0
$773$			5541.74i			0.257856i	0.991654	+	0.128928i	$0.0411536\pi$
$773$			5541.74i			0.257856i	−0.991654	+	0.128928i	$0.958846\pi$
$774$		0			0
$775$			13909.5i			0.644703i
$776$		0			0
$777$	−1093.67	−	39453.0i	−0.0504957	−	1.82158i
$778$		0			0
$779$	36851.3			1.69491
$780$		0			0
$781$	10479.4			0.480132
$782$		0			0
$783$	−1230.20	−	14762.4i	−0.0561479	−	0.673774i
$784$		0			0
$785$			15411.1i			0.700695i
$786$		0			0
$787$		−	5307.17i		−	0.240381i	−0.992751	−	0.120191i	$-0.961649\pi$
$787$		−	5307.17i		−	0.240381i	0.992751	−	0.120191i	$-0.0383506\pi$
$788$		0			0
$789$	16384.0	−	454.180i	0.739274	−	0.0204933i
$790$		0			0
$791$	10676.0			0.479894
$792$		0			0
$793$	12939.1			0.579422
$794$		0			0
$795$	13755.4	−	381.311i	0.613651	−	0.0170109i
$796$		0			0
$797$		−	36068.1i		−	1.60301i	−0.597990	−	0.801503i	$-0.704034\pi$
$797$		−	36068.1i		−	1.60301i	0.597990	−	0.801503i	$-0.295966\pi$
$798$		0			0
$799$		−	51477.5i		−	2.27928i
$800$		0			0
$801$	1600.01	+	28837.2i	0.0705789	+	1.27205i
$802$		0			0
$803$	−8273.90			−0.363611
$804$		0			0
$805$	−5763.36			−0.252337
$806$		0			0
$807$	187.749	+	6772.84i	0.00818968	+	0.295434i
$808$		0			0
$809$		−	2017.07i		−	0.0876593i	−0.999039	−	0.0438296i	$-0.986044\pi$
$809$		−	2017.07i		−	0.0876593i	0.999039	−	0.0438296i	$-0.0139559\pi$
$810$		0			0
$811$			21742.6i			0.941413i	0.882290	+	0.470706i	$0.156001\pi$
$811$			21742.6i			0.941413i	−0.882290	+	0.470706i	$0.843999\pi$
$812$		0			0
$813$	852.549	+	30754.8i	0.0367776	+	1.32671i
$814$		0			0
$815$	−6550.02			−0.281518
$816$		0			0
$817$	7477.23			0.320190
$818$		0			0
$819$	2135.41	+	38486.7i	0.0911077	+	1.64204i
$820$		0			0
$821$			14525.9i			0.617488i	0.951145	+	0.308744i	$0.0999086\pi$
$821$			14525.9i			0.617488i	−0.951145	+	0.308744i	$0.900091\pi$
$822$		0			0
$823$		−	4606.47i		−	0.195105i	−0.995230	−	0.0975524i	$-0.968899\pi$
$823$		−	4606.47i		−	0.195105i	0.995230	−	0.0975524i	$-0.0311014\pi$
$824$		0			0
$825$	−7455.87	+	206.683i	−0.314643	+	0.00872216i
$826$		0			0
$827$	17196.7			0.723079			0.361540	−	0.932357i	$-0.382251\pi$
$827$	17196.7			0.723079			0.361540	+	0.932357i	$0.382251\pi$
$828$		0			0
$829$	−4939.98			−0.206963			−0.103482	−	0.994631i	$-0.532998\pi$
$829$	−4939.98			−0.206963			−0.103482	+	0.994631i	$0.532998\pi$
$830$		0			0
$831$	12514.3	−	346.908i	0.522403	−	0.0144815i
$832$		0			0
$833$			36469.7i			1.51693i
$834$		0			0
$835$		−	13318.9i		−	0.552001i
$836$		0			0
$837$	2634.06	+	31608.7i	0.108777	+	1.30532i
$838$		0			0
$839$	−799.731			−0.0329080			−0.0164540	−	0.999865i	$-0.505238\pi$
$839$	−799.731			−0.0329080			−0.0164540	+	0.999865i	$0.505238\pi$
$840$		0			0
$841$	13240.2			0.542876
$842$		0			0
$843$	746.304	+	26922.1i	0.0304912	+	1.09994i
$844$		0			0
$845$		−	7137.27i		−	0.290567i
$846$		0			0
$847$			20194.1i			0.819216i
$848$		0			0
$849$	−966.273	−	34857.2i	−0.0390605	−	1.40907i
$850$		0			0
$851$	8338.09			0.335871
$852$		0			0
$853$	−44383.8			−1.78156			−0.890781	−	0.454433i	$-0.849842\pi$
$853$	−44383.8			−1.78156			−0.890781	+	0.454433i	$0.849842\pi$
$854$		0			0
$855$	17714.7	−	982.887i	0.708571	−	0.0393146i
$856$		0			0
$857$			17339.6i			0.691143i	0.938393	+	0.345571i	$0.112315\pi$
$857$			17339.6i			0.691143i	−0.938393	+	0.345571i	$0.887685\pi$
$858$		0			0
$859$			1410.15i			0.0560113i	0.999608	+	0.0280057i	$0.00891565\pi$
$859$			1410.15i			0.0560113i	−0.999608	+	0.0280057i	$0.991084\pi$
$860$		0			0
$861$	−59575.8	+	1651.49i	−2.35811	+	0.0653690i
$862$		0			0
$863$	20435.8			0.806077			0.403038	−	0.915183i	$-0.367954\pi$
$863$	20435.8			0.806077			0.403038	+	0.915183i	$0.367954\pi$
$864$		0			0
$865$	−12889.1			−0.506639
$866$		0			0
$867$	−43674.8	+	1210.70i	−1.71081	+	0.0474252i
$868$		0			0
$869$		−	8594.26i		−	0.335489i
$870$		0			0
$871$		−	35422.9i		−	1.37802i
$872$		0			0
$873$	14048.4	−	779.466i	0.544634	−	0.0302187i
$874$		0			0
$875$	−38148.2			−1.47388
$876$		0			0
$877$	17187.5			0.661778			0.330889	−	0.943670i	$-0.392651\pi$
$877$	17187.5			0.661778			0.330889	+	0.943670i	$0.392651\pi$
$878$		0			0
$879$	1398.58	+	50452.4i	0.0536668	+	1.93597i
$880$		0			0
$881$		−	7846.32i		−	0.300056i	−0.988682	−	0.150028i	$-0.952064\pi$
$881$		−	7846.32i		−	0.300056i	0.988682	−	0.150028i	$-0.0479364\pi$
$882$		0			0
$883$			40114.6i			1.52884i	0.644719	+	0.764420i	$0.276974\pi$
$883$			40114.6i			1.52884i	−0.644719	+	0.764420i	$0.723026\pi$
$884$		0			0
$885$	−299.512	−	10804.6i	−0.0113762	−	0.410386i
$886$		0			0
$887$	−27083.6			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$887$	−27083.6			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$888$		0			0
$889$	16056.7			0.605763
$890$		0			0
$891$	−16903.9	+	1881.60i	−0.635581	+	0.0707475i
$892$		0			0
$893$			36785.4i			1.37847i
$894$		0			0
$895$		−	6258.62i		−	0.233746i
$896$		0			0
$897$	−8140.13	+	225.651i	−0.303000	+	0.00839942i
$898$		0			0
$899$	23871.4			0.885601
$900$		0			0
$901$	38364.6			1.41855
$902$		0			0
$903$	−12088.1	+	335.092i	−0.445478	+	0.0123490i
$904$		0			0
$905$			6926.13i			0.254400i
$906$		0			0
$907$		−	26105.1i		−	0.955686i	−0.878445	−	0.477843i	$-0.841419\pi$
$907$		−	26105.1i		−	0.955686i	0.878445	−	0.477843i	$-0.158581\pi$
$908$		0			0
$909$	322.214	+	5807.29i	0.0117571	+	0.211899i
$910$		0			0
$911$	−26451.2			−0.961982			−0.480991	−	0.876725i	$-0.659723\pi$
$911$	−26451.2			−0.961982			−0.480991	+	0.876725i	$0.659723\pi$
$912$		0			0
$913$	21305.4			0.772295
$914$		0			0
$915$	266.901	+	9628.18i	0.00964316	+	0.347866i
$916$		0			0
$917$			19969.2i			0.719130i
$918$		0			0
$919$			2092.33i			0.0751028i	0.999295	+	0.0375514i	$0.0119558\pi$
$919$			2092.33i			0.0751028i	−0.999295	+	0.0375514i	$0.988044\pi$
$920$		0			0
$921$	1299.12	+	46864.3i	0.0464793	+	1.67669i
$922$		0			0
$923$	24979.4			0.890799
$924$		0			0
$925$	18204.5			0.647091
$926$		0			0
$927$	844.848	+	15226.8i	0.0299336	+	0.539496i
$928$		0			0
$929$		−	339.581i		−	0.0119928i	−0.999982	−	0.00599639i	$-0.998091\pi$
$929$		−	339.581i		−	0.0119928i	0.999982	−	0.00599639i	$-0.00190872\pi$
$930$		0			0
$931$		−	26061.0i		−	0.917415i
$932$		0			0
$933$	38179.2	−	1058.36i	1.33969	−	0.0371374i
$934$		0			0
$935$	21454.2			0.750404
$936$		0			0
$937$	−36120.7			−1.25935			−0.629675	−	0.776858i	$-0.716812\pi$
$937$	−36120.7			−1.25935			−0.629675	+	0.776858i	$0.716812\pi$
$938$		0			0
$939$	−9405.09	+	260.717i	−0.326862	+	0.00906090i
$940$		0			0
$941$			26179.8i			0.906947i	0.891270	+	0.453473i	$0.149815\pi$
$941$			26179.8i			0.906947i	−0.891270	+	0.453473i	$0.850185\pi$
$942$		0			0
$943$		−	12590.9i		−	0.434800i
$944$		0			0
$945$	−28594.4	+	2382.87i	−0.984313	+	0.0820261i
$946$		0			0
$947$	19655.3			0.674458			0.337229	−	0.941423i	$-0.390510\pi$
$947$	19655.3			0.674458			0.337229	+	0.941423i	$0.390510\pi$
$948$		0			0
$949$	−19722.2			−0.674614
$950$		0			0
$951$	−550.761	−	19868.1i	−0.0187799	−	0.677463i
$952$		0			0
$953$			47634.4i			1.61913i	0.587031	+	0.809565i	$0.300297\pi$
$953$			47634.4i			1.61913i	−0.587031	+	0.809565i	$0.699703\pi$
$954$		0			0
$955$			18935.9i			0.641624i
$956$		0			0
$957$	354.707	+	12795.7i	0.0119813	+	0.432211i
$958$		0			0
$959$	−5161.33			−0.173794
$960$		0			0
$961$	−21321.5			−0.715701
$962$		0			0
$963$	49187.1	−	2729.12i	1.64593	−	0.0913235i
$964$		0			0
$965$			11235.9i			0.374814i
$966$		0			0
$967$			40981.6i			1.36285i	0.731887	+	0.681427i	$0.238640\pi$
$967$			40981.6i			1.36285i	−0.731887	+	0.681427i	$0.761360\pi$
$968$		0			0
$969$	49445.3	−	1370.67i	1.63923	−	0.0454408i
$970$		0			0
$971$	−22380.5			−0.739675			−0.369837	−	0.929097i	$-0.620587\pi$
$971$	−22380.5			−0.739675			−0.369837	+	0.929097i	$0.620587\pi$
$972$		0			0
$973$	−33830.1			−1.11464
$974$		0			0
$975$	−17772.3	+	492.663i	−0.583762	+	0.0161824i
$976$		0			0
$977$			47920.9i			1.56922i	0.619991	+	0.784609i	$0.287136\pi$
$977$			47920.9i			1.56922i	−0.619991	+	0.784609i	$0.712864\pi$
$978$		0			0
$979$		−	24956.9i		−	0.814736i
$980$		0			0
$981$	23610.3	−	1310.00i	0.768419	−	0.0426352i
$982$		0			0
$983$	19453.4			0.631197			0.315599	−	0.948893i	$-0.397795\pi$
$983$	19453.4			0.631197			0.315599	+	0.948893i	$0.397795\pi$
$984$		0			0
$985$	−18002.0			−0.582327
$986$		0			0
$987$	−1648.54	−	59469.2i	−0.0531646	−	1.91786i
$988$		0			0
$989$		−	2554.73i		−	0.0821392i
$990$		0			0
$991$			22843.1i			0.732226i	0.930570	+	0.366113i	$0.119312\pi$
$991$			22843.1i			0.732226i	−0.930570	+	0.366113i	$0.880688\pi$
$992$		0			0
$993$	−235.087	−	8480.52i	−0.00751286	−	0.271018i
$994$		0			0
$995$	−4114.86			−0.131105
$996$		0			0
$997$	21789.5			0.692155			0.346078	−	0.938206i	$-0.387513\pi$
$997$	21789.5			0.692155			0.346078	+	0.938206i	$0.387513\pi$
$998$		0			0
$999$	41368.7	−	3447.40i	1.31016	−	0.109180i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.4.c.d.191.11		12
3.2	odd	2	inner	192.4.c.d.191.1		12
4.3	odd	2	inner	192.4.c.d.191.2		12
8.3	odd	2		96.4.c.a.95.11	yes	12
8.5	even	2		96.4.c.a.95.2	yes	12
12.11	even	2	inner	192.4.c.d.191.12		12
16.3	odd	4		768.4.f.i.383.5		12
16.5	even	4		768.4.f.i.383.6		12
16.11	odd	4		768.4.f.d.383.8		12
16.13	even	4		768.4.f.d.383.7		12
24.5	odd	2		96.4.c.a.95.12	yes	12
24.11	even	2		96.4.c.a.95.1	✓	12
48.5	odd	4		768.4.f.i.383.7		12
48.11	even	4		768.4.f.d.383.5		12
48.29	odd	4		768.4.f.d.383.6		12
48.35	even	4		768.4.f.i.383.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.4.c.a.95.1	✓	12	24.11	even	2
96.4.c.a.95.2	yes	12	8.5	even	2
96.4.c.a.95.11	yes	12	8.3	odd	2
96.4.c.a.95.12	yes	12	24.5	odd	2
192.4.c.d.191.1		12	3.2	odd	2	inner
192.4.c.d.191.2		12	4.3	odd	2	inner
192.4.c.d.191.11		12	1.1	even	1	trivial
192.4.c.d.191.12		12	12.11	even	2	inner
768.4.f.d.383.5		12	48.11	even	4
768.4.f.d.383.6		12	48.29	odd	4
768.4.f.d.383.7		12	16.13	even	4
768.4.f.d.383.8		12	16.11	odd	4
768.4.f.i.383.5		12	16.3	odd	4
768.4.f.i.383.6		12	16.5	even	4
768.4.f.i.383.7		12	48.5	odd	4
768.4.f.i.383.8		12	48.35	even	4

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.19416 - 0.143987i) q^{3} -7.96714i q^{5} -25.6706i q^{7} +(26.9585 - 1.49578i) q^{9} -23.3310 q^{11} -55.6133 q^{13} +(-1.14716 - 41.3826i) q^{15} -115.419i q^{17} +82.4772i q^{19} +(-3.69622 - 133.337i) q^{21} +28.1798 q^{23} +61.5246 q^{25} +(139.811 - 11.6510i) q^{27} -105.588i q^{29} +226.081i q^{31} +(-121.185 + 3.35936i) q^{33} -204.521 q^{35} +295.889 q^{37} +(-288.864 + 8.00756i) q^{39} -446.806i q^{41} -90.6582i q^{43} +(-11.9171 - 214.782i) q^{45} +446.007 q^{47} -315.978 q^{49} +(-16.6187 - 599.502i) q^{51} +332.395i q^{53} +185.882i q^{55} +(11.8756 + 428.400i) q^{57} +261.090 q^{59} -232.663 q^{61} +(-38.3974 - 692.041i) q^{63} +443.079i q^{65} +636.950i q^{67} +(146.370 - 4.05751i) q^{69} -449.162 q^{71} +354.630 q^{73} +(319.569 - 8.85872i) q^{75} +598.921i q^{77} +368.362i q^{79} +(724.525 - 80.6479i) q^{81} -913.178 q^{83} -919.557 q^{85} +(-15.2032 - 548.440i) q^{87} +1069.69i q^{89} +1427.62i q^{91} +(32.5526 + 1174.30i) q^{93} +657.108 q^{95} +521.111 q^{97} +(-628.971 + 34.8980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100}) \) 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more