LMFDB - Embedded newform 405.3.d.b.404.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] 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("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.14
Character	$\chi$	$=$	405.404
Dual form			405.3.d.b.404.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.0969975 q^{2} -3.99059 q^{4} +(-4.26797 + 2.60469i) q^{5} -10.1824i q^{7} -0.775068 q^{8} +(-0.413983 + 0.252648i) q^{10} +1.78857i q^{11} +13.7380i q^{13} -0.987672i q^{14} +15.8872 q^{16} +13.0802 q^{17} -16.5731 q^{19} +(17.0317 - 10.3942i) q^{20} +0.173487i q^{22} +41.6700 q^{23} +(11.4312 - 22.2335i) q^{25} +1.33256i q^{26} +40.6340i q^{28} +44.1861i q^{29} +6.46522 q^{31} +4.64129 q^{32} +1.26875 q^{34} +(26.5221 + 43.4584i) q^{35} +20.6967i q^{37} -1.60755 q^{38} +(3.30797 - 2.01881i) q^{40} +56.9330i q^{41} -34.3308i q^{43} -7.13747i q^{44} +4.04188 q^{46} +60.1270 q^{47} -54.6821 q^{49} +(1.10880 - 2.15659i) q^{50} -54.8229i q^{52} +25.5290 q^{53} +(-4.65868 - 7.63359i) q^{55} +7.89208i q^{56} +4.28595i q^{58} -40.5638i q^{59} +69.2168 q^{61} +0.627110 q^{62} -63.0986 q^{64} +(-35.7833 - 58.6336i) q^{65} +60.0922i q^{67} -52.1979 q^{68} +(2.57258 + 4.21536i) q^{70} -91.1989i q^{71} -74.8762i q^{73} +2.00753i q^{74} +66.1365 q^{76} +18.2121 q^{77} -61.6522 q^{79} +(-67.8061 + 41.3812i) q^{80} +5.52236i q^{82} +58.5291 q^{83} +(-55.8261 + 34.0699i) q^{85} -3.33001i q^{86} -1.38627i q^{88} +66.7514i q^{89} +139.887 q^{91} -166.288 q^{92} +5.83217 q^{94} +(70.7335 - 43.1677i) q^{95} +158.078i q^{97} -5.30403 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.0969975			0.0484988			0.0242494	−	0.999706i	$-0.492280\pi$
$2$	0.0969975			0.0484988			0.0242494	+	0.999706i	$0.492280\pi$
$3$		0			0
$3$		0			0
$4$	−3.99059			−0.997648
$5$	−4.26797	+	2.60469i	−0.853595	+	0.520937i
$5$	−4.26797	+	2.60469i	−0.853595	+	0.520937i
$6$		0			0
$7$		−	10.1824i		−	1.45463i	−0.686301	−	0.727317i	$-0.740767\pi$
$7$		−	10.1824i		−	1.45463i	0.686301	−	0.727317i	$-0.259233\pi$
$8$	−0.775068			−0.0968835
$9$		0			0
$10$	−0.413983	+	0.252648i	−0.0413983	+	0.0252648i
$11$			1.78857i			0.162598i	0.996690	+	0.0812988i	$0.0259068\pi$
$11$			1.78857i			0.162598i	−0.996690	+	0.0812988i	$0.974093\pi$
$12$		0			0
$13$			13.7380i			1.05677i	0.849004	+	0.528386i	$0.177203\pi$
$13$			13.7380i			1.05677i	−0.849004	+	0.528386i	$0.822797\pi$
$14$		−	0.987672i		−	0.0705480i
$15$		0			0
$16$	15.8872			0.992949
$17$	13.0802			0.769426			0.384713	−	0.923036i	$-0.374300\pi$
$17$	13.0802			0.769426			0.384713	+	0.923036i	$0.374300\pi$
$18$		0			0
$19$	−16.5731			−0.872268			−0.436134	−	0.899882i	$-0.643653\pi$
$19$	−16.5731			−0.872268			−0.436134	+	0.899882i	$0.643653\pi$
$20$	17.0317	−	10.3942i	0.851587	−	0.519712i
$21$		0			0
$22$			0.173487i			0.00788579i
$23$	41.6700			1.81174			0.905869	−	0.423559i	$-0.139219\pi$
$23$	41.6700			1.81174			0.905869	+	0.423559i	$0.139219\pi$
$24$		0			0
$25$	11.4312	−	22.2335i	0.457248	−	0.889339i
$26$			1.33256i			0.0512521i
$27$		0			0
$28$			40.6340i			1.45121i
$29$			44.1861i			1.52366i	0.647777	+	0.761830i	$0.275699\pi$
$29$			44.1861i			1.52366i	−0.647777	+	0.761830i	$0.724301\pi$
$30$		0			0
$31$	6.46522			0.208555			0.104278	−	0.994548i	$-0.466747\pi$
$31$	6.46522			0.208555			0.104278	+	0.994548i	$0.466747\pi$
$32$	4.64129			0.145040
$33$		0			0
$34$	1.26875			0.0373162
$35$	26.5221	+	43.4584i	0.757774	+	1.24167i
$36$		0			0
$37$			20.6967i			0.559370i	0.960092	+	0.279685i	$0.0902300\pi$
$37$			20.6967i			0.559370i	−0.960092	+	0.279685i	$0.909770\pi$
$38$	−1.60755			−0.0423039
$39$		0			0
$40$	3.30797	−	2.01881i	0.0826992	−	0.0504702i
$41$			56.9330i			1.38861i	0.719681	+	0.694305i	$0.244288\pi$
$41$			56.9330i			1.38861i	−0.719681	+	0.694305i	$0.755712\pi$
$42$		0			0
$43$		−	34.3308i		−	0.798391i	−0.916866	−	0.399196i	$-0.869289\pi$
$43$		−	34.3308i		−	0.798391i	0.916866	−	0.399196i	$-0.130711\pi$
$44$		−	7.13747i		−	0.162215i
$45$		0			0
$46$	4.04188			0.0878670
$47$	60.1270			1.27930			0.639649	−	0.768667i	$-0.279080\pi$
$47$	60.1270			1.27930			0.639649	+	0.768667i	$0.279080\pi$
$48$		0			0
$49$	−54.6821			−1.11596
$50$	1.10880	−	2.15659i	0.0221760	−	0.0431319i
$51$		0			0
$52$		−	54.8229i		−	1.05429i
$53$	25.5290			0.481680			0.240840	−	0.970565i	$-0.422577\pi$
$53$	25.5290			0.481680			0.240840	+	0.970565i	$0.422577\pi$
$54$		0			0
$55$	−4.65868	−	7.63359i	−0.0847032	−	0.138793i
$56$			7.89208i			0.140930i
$57$		0			0
$58$			4.28595i			0.0738956i
$59$		−	40.5638i		−	0.687521i	−0.939057	−	0.343761i	$-0.888299\pi$
$59$		−	40.5638i		−	0.687521i	0.939057	−	0.343761i	$-0.111701\pi$
$60$		0			0
$61$	69.2168			1.13470			0.567351	−	0.823476i	$-0.307968\pi$
$61$	69.2168			1.13470			0.567351	+	0.823476i	$0.307968\pi$
$62$	0.627110			0.0101147
$63$		0			0
$64$	−63.0986			−0.985915
$65$	−35.7833	−	58.6336i	−0.550512	−	0.902055i
$66$		0			0
$67$			60.0922i			0.896898i	0.893808	+	0.448449i	$0.148023\pi$
$67$			60.0922i			0.896898i	−0.893808	+	0.448449i	$0.851977\pi$
$68$	−52.1979			−0.767616
$69$		0			0
$70$	2.57258	+	4.21536i	0.0367511	+	0.0602194i
$71$		−	91.1989i		−	1.28449i	−0.766499	−	0.642246i	$-0.778003\pi$
$71$		−	91.1989i		−	1.28449i	0.766499	−	0.642246i	$-0.221997\pi$
$72$		0			0
$73$		−	74.8762i		−	1.02570i	−0.858478	−	0.512850i	$-0.828590\pi$
$73$		−	74.8762i		−	1.02570i	0.858478	−	0.512850i	$-0.171410\pi$
$74$			2.00753i			0.0271288i
$75$		0			0
$76$	66.1365			0.870217
$77$	18.2121			0.236520
$78$		0			0
$79$	−61.6522			−0.780407			−0.390203	−	0.920729i	$-0.627595\pi$
$79$	−61.6522			−0.780407			−0.390203	+	0.920729i	$0.627595\pi$
$80$	−67.8061	+	41.3812i	−0.847576	+	0.517264i
$81$		0			0
$82$			5.52236i			0.0673459i
$83$	58.5291			0.705170			0.352585	−	0.935780i	$-0.385303\pi$
$83$	58.5291			0.705170			0.352585	+	0.935780i	$0.385303\pi$
$84$		0			0
$85$	−55.8261	+	34.0699i	−0.656778	+	0.400823i
$86$		−	3.33001i		−	0.0387210i
$87$		0			0
$88$		−	1.38627i		−	0.0157530i
$89$			66.7514i			0.750016i	0.927022	+	0.375008i	$0.122360\pi$
$89$			66.7514i			0.750016i	−0.927022	+	0.375008i	$0.877640\pi$
$90$		0			0
$91$	139.887			1.53722
$92$	−166.288			−1.80748
$93$		0			0
$94$	5.83217			0.0620443
$95$	70.7335	−	43.1677i	0.744564	−	0.454397i
$96$		0			0
$97$			158.078i			1.62967i	0.579694	+	0.814834i	$0.303172\pi$
$97$			158.078i			1.62967i	−0.579694	+	0.814834i	$0.696828\pi$
$98$	−5.30403			−0.0541228
$99$		0			0
$100$	−45.6173	+	88.7247i	−0.456173	+	0.887247i
$101$			95.7309i			0.947831i	0.880570	+	0.473915i	$0.157160\pi$
$101$			95.7309i			0.947831i	−0.880570	+	0.473915i	$0.842840\pi$
$102$		0			0
$103$		−	95.5944i		−	0.928101i	−0.885809	−	0.464051i	$-0.846396\pi$
$103$		−	95.5944i		−	0.928101i	0.885809	−	0.464051i	$-0.153604\pi$
$104$		−	10.6479i		−	0.102384i
$105$		0			0
$106$	2.47625			0.0233609
$107$	−60.3812			−0.564310			−0.282155	−	0.959369i	$-0.591049\pi$
$107$	−60.3812			−0.564310			−0.282155	+	0.959369i	$0.591049\pi$
$108$		0			0
$109$	80.8515			0.741757			0.370878	−	0.928681i	$-0.379057\pi$
$109$	80.8515			0.741757			0.370878	+	0.928681i	$0.379057\pi$
$110$	−0.451880	−	0.740440i	−0.00410800	−	0.00673127i
$111$		0			0
$112$		−	161.770i		−	1.44438i
$113$	−123.647			−1.09422			−0.547112	−	0.837060i	$-0.684273\pi$
$113$	−123.647			−1.09422			−0.547112	+	0.837060i	$0.684273\pi$
$114$		0			0
$115$	−177.846	+	108.537i	−1.54649	+	0.943802i
$116$		−	176.329i		−	1.52008i
$117$		0			0
$118$		−	3.93458i		−	0.0333439i
$119$		−	133.189i		−	1.11923i
$120$		0			0
$121$	117.801			0.973562
$122$	6.71386			0.0550317
$123$		0			0
$124$	−25.8000			−0.208065
$125$	9.12315	+	124.667i	0.0729852	+	0.997333i
$126$		0			0
$127$		−	151.814i		−	1.19539i	−0.801724	−	0.597694i	$-0.796084\pi$
$127$		−	151.814i		−	1.19539i	0.801724	−	0.597694i	$-0.203916\pi$
$128$	−24.6856			−0.192856
$129$		0			0
$130$	−3.47089	−	5.68731i	−0.0266992	−	0.0437486i
$131$			156.723i			1.19636i	0.801363	+	0.598178i	$0.204109\pi$
$131$			156.723i			1.19636i	−0.801363	+	0.598178i	$0.795891\pi$
$132$		0			0
$133$			168.755i			1.26883i
$134$			5.82879i			0.0434985i
$135$		0			0
$136$	−10.1381			−0.0745446
$137$	75.5554			0.551500			0.275750	−	0.961229i	$-0.411074\pi$
$137$	75.5554			0.551500			0.275750	+	0.961229i	$0.411074\pi$
$138$		0			0
$139$	95.5912			0.687707			0.343853	−	0.939023i	$-0.388268\pi$
$139$	95.5912			0.687707			0.343853	+	0.939023i	$0.388268\pi$
$140$	−105.839	−	173.425i	−0.755991	−	1.23875i
$141$		0			0
$142$		−	8.84607i		−	0.0622962i
$143$	−24.5715			−0.171829
$144$		0			0
$145$	−115.091	−	188.585i	−0.793731	−	1.30059i
$146$		−	7.26280i		−	0.0497452i
$147$		0			0
$148$		−	82.5920i		−	0.558054i
$149$		−	48.2700i		−	0.323960i	−0.986794	−	0.161980i	$-0.948212\pi$
$149$		−	48.2700i		−	0.323960i	0.986794	−	0.161980i	$-0.0517880\pi$
$150$		0			0
$151$	−22.8302			−0.151194			−0.0755968	−	0.997138i	$-0.524086\pi$
$151$	−22.8302			−0.151194			−0.0755968	+	0.997138i	$0.524086\pi$
$152$	12.8453			0.0845084
$153$		0			0
$154$	1.76652			0.0114709
$155$	−27.5934	+	16.8399i	−0.178022	+	0.108644i
$156$		0			0
$157$			117.218i			0.746612i	0.927708	+	0.373306i	$0.121776\pi$
$157$			117.218i			0.746612i	−0.927708	+	0.373306i	$0.878224\pi$
$158$	−5.98011			−0.0378488
$159$		0			0
$160$	−19.8089	+	12.0891i	−0.123806	+	0.0755569i
$161$		−	424.302i		−	2.63542i
$162$		0			0
$163$		−	91.1992i		−	0.559504i	−0.960072	−	0.279752i	$-0.909748\pi$
$163$		−	91.1992i		−	0.559504i	0.960072	−	0.279752i	$-0.0902523\pi$
$164$		−	227.196i		−	1.38534i
$165$		0			0
$166$	5.67718			0.0341999
$167$	−164.200			−0.983234			−0.491617	−	0.870811i	$-0.663594\pi$
$167$	−164.200			−0.983234			−0.491617	+	0.870811i	$0.663594\pi$
$168$		0			0
$169$	−19.7336			−0.116767
$170$	−5.41500	+	3.30470i	−0.0318529	+	0.0194394i
$171$		0			0
$172$			137.000i			0.796513i
$173$	−225.445			−1.30315			−0.651576	−	0.758583i	$-0.725892\pi$
$173$	−225.445			−1.30315			−0.651576	+	0.758583i	$0.725892\pi$
$174$		0			0
$175$	−226.391	−	116.398i	−1.29366	−	0.665129i
$176$			28.4154i			0.161451i
$177$		0			0
$178$			6.47473i			0.0363749i
$179$			349.517i			1.95261i	0.216395	+	0.976306i	$0.430570\pi$
$179$			349.517i			1.95261i	−0.216395	+	0.976306i	$0.569430\pi$
$180$		0			0
$181$	254.586			1.40655			0.703275	−	0.710918i	$-0.251720\pi$
$181$	254.586			1.40655			0.703275	+	0.710918i	$0.251720\pi$
$182$	13.5687			0.0745531
$183$		0			0
$184$	−32.2970			−0.175527
$185$	−53.9084	−	88.3330i	−0.291397	−	0.477475i
$186$		0			0
$187$			23.3950i			0.125107i
$188$	−239.942			−1.27629
$189$		0			0
$190$	6.86098	−	4.18716i	0.0361104	−	0.0220377i
$191$			268.329i			1.40487i	0.711750	+	0.702433i	$0.247903\pi$
$191$			268.329i			1.40487i	−0.711750	+	0.702433i	$0.752097\pi$
$192$		0			0
$193$			307.062i			1.59100i	0.605955	+	0.795499i	$0.292791\pi$
$193$			307.062i			1.59100i	−0.605955	+	0.795499i	$0.707209\pi$
$194$			15.3332i			0.0790369i
$195$		0			0
$196$	218.214			1.11334
$197$	−62.4666			−0.317089			−0.158545	−	0.987352i	$-0.550680\pi$
$197$	−62.4666			−0.317089			−0.158545	+	0.987352i	$0.550680\pi$
$198$		0			0
$199$	74.6147			0.374948			0.187474	−	0.982270i	$-0.439970\pi$
$199$	74.6147			0.374948			0.187474	+	0.982270i	$0.439970\pi$
$200$	−8.85996	+	17.2325i	−0.0442998	+	0.0861623i
$201$		0			0
$202$			9.28566i			0.0459686i
$203$	449.923			2.21637
$204$		0			0
$205$	−148.293	−	242.989i	−0.723379	−	1.18531i
$206$		−	9.27243i		−	0.0450118i
$207$		0			0
$208$			218.259i			1.04932i
$209$		−	29.6422i		−	0.141829i
$210$		0			0
$211$	163.538			0.775064			0.387532	−	0.921856i	$-0.373328\pi$
$211$	163.538			0.775064			0.387532	+	0.921856i	$0.373328\pi$
$212$	−101.876			−0.480547
$213$		0			0
$214$	−5.85682			−0.0273683
$215$	89.4211	+	146.523i	0.415912	+	0.681503i
$216$		0			0
$217$		−	65.8317i		−	0.303372i
$218$	7.84239			0.0359743
$219$		0			0
$220$	18.5909	+	30.4625i	0.0845040	+	0.138466i
$221$			179.697i			0.813108i
$222$		0			0
$223$			17.9722i			0.0805927i	0.999188	+	0.0402964i	$0.0128302\pi$
$223$			17.9722i			0.0805927i	−0.999188	+	0.0402964i	$0.987170\pi$
$224$		−	47.2597i		−	0.210981i
$225$		0			0
$226$	−11.9935			−0.0530685
$227$	32.0735			0.141293			0.0706464	−	0.997501i	$-0.477494\pi$
$227$	32.0735			0.141293			0.0706464	+	0.997501i	$0.477494\pi$
$228$		0			0
$229$	−160.655			−0.701549			−0.350775	−	0.936460i	$-0.614082\pi$
$229$	−160.655			−0.701549			−0.350775	+	0.936460i	$0.614082\pi$
$230$	−17.2507	+	10.5278i	−0.0750028	+	0.0457732i
$231$		0			0
$232$		−	34.2472i		−	0.147617i
$233$	345.174			1.48143			0.740717	−	0.671817i	$-0.234486\pi$
$233$	345.174			1.48143			0.740717	+	0.671817i	$0.234486\pi$
$234$		0			0
$235$	−256.620	+	156.612i	−1.09200	+	0.666434i
$236$			161.873i			0.685904i
$237$		0			0
$238$		−	12.9190i		−	0.0542815i
$239$		−	108.410i		−	0.453599i	−0.973941	−	0.226800i	$-0.927174\pi$
$239$		−	108.410i		−	0.453599i	0.973941	−	0.226800i	$-0.0728262\pi$
$240$		0			0
$241$	218.542			0.906813			0.453406	−	0.891304i	$-0.350209\pi$
$241$	218.542			0.906813			0.453406	+	0.891304i	$0.350209\pi$
$242$	11.4264			0.0472166
$243$		0			0
$244$	−276.216			−1.13203
$245$	233.382	−	142.430i	0.952579	−	0.581346i
$246$		0			0
$247$		−	227.682i		−	0.921789i
$248$	−5.01098			−0.0202056
$249$		0			0
$250$	0.884923	+	12.0924i	0.00353969	+	0.0483694i
$251$		−	178.976i		−	0.713053i	−0.934285	−	0.356527i	$-0.883961\pi$
$251$		−	178.976i		−	0.713053i	0.934285	−	0.356527i	$-0.116039\pi$
$252$		0			0
$253$			74.5298i			0.294584i
$254$		−	14.7256i		−	0.0579748i
$255$		0			0
$256$	250.000			0.976562
$257$	426.874			1.66099			0.830493	−	0.557028i	$-0.188059\pi$
$257$	426.874			1.66099			0.830493	+	0.557028i	$0.188059\pi$
$258$		0			0
$259$	210.743			0.813679
$260$	142.796	+	233.983i	0.549217	+	0.899933i
$261$		0			0
$262$			15.2017i			0.0580218i
$263$	388.671			1.47784			0.738919	−	0.673794i	$-0.235337\pi$
$263$	388.671			1.47784			0.738919	+	0.673794i	$0.235337\pi$
$264$		0			0
$265$	−108.957	+	66.4951i	−0.411159	+	0.250925i
$266$			16.3688i			0.0615368i
$267$		0			0
$268$		−	239.803i		−	0.894788i
$269$			57.7020i			0.214506i	0.994232	+	0.107253i	$0.0342054\pi$
$269$			57.7020i			0.214506i	−0.994232	+	0.107253i	$0.965795\pi$
$270$		0			0
$271$	−102.825			−0.379428			−0.189714	−	0.981839i	$-0.560756\pi$
$271$	−102.825			−0.379428			−0.189714	+	0.981839i	$0.560756\pi$
$272$	207.808			0.764001
$273$		0			0
$274$	7.32869			0.0267471
$275$	39.7662	+	20.4456i	0.144604	+	0.0743475i
$276$		0			0
$277$		−	506.084i		−	1.82702i	−0.406820	−	0.913508i	$-0.633362\pi$
$277$		−	506.084i		−	1.82702i	0.406820	−	0.913508i	$-0.366638\pi$
$278$	9.27211			0.0333529
$279$		0			0
$280$	−20.5564	−	33.6832i	−0.0734157	−	0.120297i
$281$		−	86.9438i		−	0.309408i	−0.987961	−	0.154704i	$-0.950558\pi$
$281$		−	86.9438i		−	0.309408i	0.987961	−	0.154704i	$-0.0494425\pi$
$282$		0			0
$283$		−	92.1515i		−	0.325624i	−0.986657	−	0.162812i	$-0.947944\pi$
$283$		−	92.1515i		−	0.325624i	0.986657	−	0.162812i	$-0.0520564\pi$
$284$			363.937i			1.28147i
$285$		0			0
$286$	−2.38338			−0.00833348
$287$	579.717			2.01992
$288$		0			0
$289$	−117.907			−0.407984
$290$	−11.1635	−	18.2923i	−0.0384950	−	0.0630769i
$291$		0			0
$292$			298.800i			1.02329i
$293$	−394.702			−1.34711			−0.673554	−	0.739138i	$-0.735233\pi$
$293$	−394.702			−1.34711			−0.673554	+	0.739138i	$0.735233\pi$
$294$		0			0
$295$	105.656	+	173.125i	0.358156	+	0.586865i
$296$		−	16.0413i		−	0.0541937i
$297$		0			0
$298$		−	4.68207i		−	0.0157117i
$299$			572.463i			1.91459i
$300$		0			0
$301$	−349.572			−1.16137
$302$	−2.21448			−0.00733271
$303$		0			0
$304$	−263.300			−0.866118
$305$	−295.416	+	180.288i	−0.968576	+	0.591109i
$306$		0			0
$307$			296.956i			0.967283i	0.875266	+	0.483642i	$0.160686\pi$
$307$			296.956i			0.967283i	−0.875266	+	0.483642i	$0.839314\pi$
$308$	−72.6769			−0.235964
$309$		0			0
$310$	−2.67649	+	1.63343i	−0.00863384	+	0.00526912i
$311$			539.359i			1.73427i	0.498070	+	0.867137i	$0.334042\pi$
$311$			539.359i			1.73427i	−0.498070	+	0.867137i	$0.665958\pi$
$312$		0			0
$313$		−	439.842i		−	1.40525i	−0.711562	−	0.702623i	$-0.752012\pi$
$313$		−	439.842i		−	1.40525i	0.711562	−	0.702623i	$-0.247988\pi$
$314$			11.3699i			0.0362098i
$315$		0			0
$316$	246.029			0.778571
$317$	−244.876			−0.772479			−0.386240	−	0.922398i	$-0.626226\pi$
$317$	−244.876			−0.772479			−0.386240	+	0.922398i	$0.626226\pi$
$318$		0			0
$319$	−79.0302			−0.247743
$320$	269.303	−	164.352i	0.841572	−	0.513600i
$321$		0			0
$322$		−	41.1562i		−	0.127814i
$323$	−216.780			−0.671146
$324$		0			0
$325$	305.444	+	157.042i	0.939828	+	0.483207i
$326$		−	8.84610i		−	0.0271353i
$327$		0			0
$328$		−	44.1269i		−	0.134533i
$329$		−	612.239i		−	1.86091i
$330$		0			0
$331$	361.313			1.09158			0.545790	−	0.837922i	$-0.316230\pi$
$331$	361.313			1.09158			0.545790	+	0.837922i	$0.316230\pi$
$332$	−233.566			−0.703511
$333$		0			0
$334$	−15.9270			−0.0476857
$335$	−156.521	−	256.472i	−0.467228	−	0.765587i
$336$		0			0
$337$			252.374i			0.748884i	0.927250	+	0.374442i	$0.122166\pi$
$337$			252.374i			0.748884i	−0.927250	+	0.374442i	$0.877834\pi$
$338$	−1.91411			−0.00566305
$339$		0			0
$340$	222.779	−	135.959i	0.655233	−	0.399880i
$341$			11.5635i			0.0339106i
$342$		0			0
$343$			57.8579i			0.168682i
$344$			26.6087i			0.0773509i
$345$		0			0
$346$	−21.8676			−0.0632013
$347$	107.993			0.311218			0.155609	−	0.987819i	$-0.450266\pi$
$347$	107.993			0.311218			0.155609	+	0.987819i	$0.450266\pi$
$348$		0			0
$349$	−1.71568			−0.00491599			−0.00245800	−	0.999997i	$-0.500782\pi$
$349$	−1.71568			−0.00491599			−0.00245800	+	0.999997i	$0.500782\pi$
$350$	−21.9594	−	11.2903i	−0.0627411	−	0.0322580i
$351$		0			0
$352$			8.30129i			0.0235832i
$353$	149.181			0.422610			0.211305	−	0.977420i	$-0.432229\pi$
$353$	149.181			0.422610			0.211305	+	0.977420i	$0.432229\pi$
$354$		0			0
$355$	237.545	+	389.234i	0.669139	+	1.09643i
$356$		−	266.378i		−	0.748252i
$357$		0			0
$358$			33.9023i			0.0946993i
$359$		−	108.011i		−	0.300865i	−0.988620	−	0.150433i	$-0.951933\pi$
$359$		−	108.011i		−	0.300865i	0.988620	−	0.150433i	$-0.0480666\pi$
$360$		0			0
$361$	−86.3325			−0.239148
$362$	24.6942			0.0682160
$363$		0			0
$364$	−558.231			−1.53360
$365$	195.029	+	319.569i	0.534326	+	0.875533i
$366$		0			0
$367$			479.393i			1.30625i	0.757251	+	0.653123i	$0.226542\pi$
$367$			479.393i			1.30625i	−0.757251	+	0.653123i	$0.773458\pi$
$368$	662.018			1.79896
$369$		0			0
$370$	−5.22898	−	8.56808i	−0.0141324	−	0.0231570i
$371$		−	259.948i		−	0.700668i
$372$		0			0
$373$			105.085i			0.281730i	0.990029	+	0.140865i	$0.0449883\pi$
$373$			105.085i			0.281730i	−0.990029	+	0.140865i	$0.955012\pi$
$374$			2.26926i			0.00606753i
$375$		0			0
$376$	−46.6025			−0.123943
$377$	−607.030			−1.61016
$378$		0			0
$379$	−175.029			−0.461817			−0.230909	−	0.972975i	$-0.574170\pi$
$379$	−175.029			−0.461817			−0.230909	+	0.972975i	$0.574170\pi$
$380$	−282.269	+	172.265i	−0.742812	+	0.453328i
$381$		0			0
$382$			26.0273i			0.0681343i
$383$	−311.741			−0.813944			−0.406972	−	0.913441i	$-0.633415\pi$
$383$	−311.741			−0.813944			−0.406972	+	0.913441i	$0.633415\pi$
$384$		0			0
$385$	−77.7286	+	47.4367i	−0.201892	+	0.123212i
$386$			29.7843i			0.0771614i
$387$		0			0
$388$		−	630.824i		−	1.62583i
$389$			11.7740i			0.0302674i	0.999885	+	0.0151337i	$0.00481739\pi$
$389$			11.7740i			0.0302674i	−0.999885	+	0.0151337i	$0.995183\pi$
$390$		0			0
$391$	545.053			1.39400
$392$	42.3824			0.108118
$393$		0			0
$394$	−6.05910			−0.0153784
$395$	263.130	−	160.585i	0.666151	−	0.406543i
$396$		0			0
$397$		−	261.770i		−	0.659369i	−0.944091	−	0.329685i	$-0.893058\pi$
$397$		−	261.770i		−	0.659369i	0.944091	−	0.329685i	$-0.106942\pi$
$398$	7.23745			0.0181845
$399$		0			0
$400$	181.610	−	353.227i	0.454024	−	0.883068i
$401$		−	330.403i		−	0.823947i	−0.911196	−	0.411973i	$-0.864840\pi$
$401$		−	330.403i		−	0.823947i	0.911196	−	0.411973i	$-0.135160\pi$
$402$		0			0
$403$			88.8194i			0.220396i
$404$		−	382.023i		−	0.945601i
$405$		0			0
$406$	43.6414			0.107491
$407$	−37.0176			−0.0909523
$408$		0			0
$409$	511.002			1.24939			0.624696	−	0.780868i	$-0.285223\pi$
$409$	511.002			1.24939			0.624696	+	0.780868i	$0.285223\pi$
$410$	−14.3840	−	23.5693i	−0.0350830	−	0.0574861i
$411$		0			0
$412$			381.478i			0.925918i
$413$	−413.038			−1.00009
$414$		0			0
$415$	−249.801	+	152.450i	−0.601929	+	0.367349i
$416$			63.7622i			0.153274i
$417$		0			0
$418$		−	2.87522i		−	0.00687852i
$419$		−	61.1283i		−	0.145891i	−0.997336	−	0.0729455i	$-0.976760\pi$
$419$		−	61.1283i		−	0.145891i	0.997336	−	0.0729455i	$-0.0232399\pi$
$420$		0			0
$421$	−618.492			−1.46910			−0.734551	−	0.678554i	$-0.762607\pi$
$421$	−618.492			−1.46910			−0.734551	+	0.678554i	$0.762607\pi$
$422$	15.8628			0.0375896
$423$		0			0
$424$	−19.7867			−0.0466668
$425$	149.523	−	290.819i	0.351819	−	0.684280i
$426$		0			0
$427$		−	704.797i		−	1.65058i
$428$	240.957			0.562983
$429$		0			0
$430$	8.67362	+	14.2124i	0.0201712	+	0.0330520i
$431$			344.839i			0.800089i	0.916496	+	0.400045i	$0.131005\pi$
$431$			344.839i			0.800089i	−0.916496	+	0.400045i	$0.868995\pi$
$432$		0			0
$433$			164.525i			0.379966i	0.981787	+	0.189983i	$0.0608433\pi$
$433$			164.525i			0.379966i	−0.981787	+	0.189983i	$0.939157\pi$
$434$		−	6.38551i		−	0.0147132i
$435$		0			0
$436$	−322.645			−0.740012
$437$	−690.600			−1.58032
$438$		0			0
$439$	−327.084			−0.745065			−0.372533	−	0.928019i	$-0.621511\pi$
$439$	−327.084			−0.745065			−0.372533	+	0.928019i	$0.621511\pi$
$440$	3.61079	+	5.91655i	0.00820634	+	0.0134467i
$441$		0			0
$442$			17.4301i			0.0394347i
$443$	−357.262			−0.806460			−0.403230	−	0.915099i	$-0.632113\pi$
$443$	−357.262			−0.806460			−0.403230	+	0.915099i	$0.632113\pi$
$444$		0			0
$445$	−173.867	−	284.893i	−0.390711	−	0.640210i
$446$			1.74326i			0.00390865i
$447$		0			0
$448$			642.497i			1.43415i
$449$		−	154.756i		−	0.344668i	−0.985039	−	0.172334i	$-0.944869\pi$
$449$		−	154.756i		−	0.344668i	0.985039	−	0.172334i	$-0.0551309\pi$
$450$		0			0
$451$	−101.829			−0.225785
$452$	493.426			1.09165
$453$		0			0
$454$	3.11105			0.00685252
$455$	−597.033	+	364.361i	−1.31216	+	0.800794i
$456$		0			0
$457$		−	154.754i		−	0.338631i	−0.985562	−	0.169316i	$-0.945844\pi$
$457$		−	154.754i		−	0.338631i	0.985562	−	0.169316i	$-0.0541557\pi$
$458$	−15.5831			−0.0340243
$459$		0			0
$460$	709.712	−	433.128i	1.54285	−	0.941582i
$461$		−	261.352i		−	0.566923i	−0.958984	−	0.283462i	$-0.908517\pi$
$461$		−	261.352i		−	0.566923i	0.958984	−	0.283462i	$-0.0914828\pi$
$462$		0			0
$463$		−	795.719i		−	1.71862i	−0.511458	−	0.859308i	$-0.670894\pi$
$463$		−	795.719i		−	1.71862i	0.511458	−	0.859308i	$-0.329106\pi$
$464$			701.993i			1.51292i
$465$		0			0
$466$	33.4810			0.0718477
$467$	427.874			0.916218			0.458109	−	0.888896i	$-0.348527\pi$
$467$	427.874			0.916218			0.458109	+	0.888896i	$0.348527\pi$
$468$		0			0
$469$	611.885			1.30466
$470$	−24.8915	+	15.1910i	−0.0529607	+	0.0323212i
$471$		0			0
$472$			31.4397i			0.0666094i
$473$	61.4032			0.129817
$474$		0			0
$475$	−189.451	+	368.478i	−0.398843	+	0.775742i
$476$			531.502i			1.11660i
$477$		0			0
$478$		−	10.5155i		−	0.0219990i
$479$			217.723i			0.454537i	0.973832	+	0.227269i	$0.0729796\pi$
$479$			217.723i			0.454537i	−0.973832	+	0.227269i	$0.927020\pi$
$480$		0			0
$481$	−284.332			−0.591127
$482$	21.1980			0.0439793
$483$		0			0
$484$	−470.096			−0.971272
$485$	−411.743	−	674.672i	−0.848955	−	1.39108i
$486$		0			0
$487$			106.417i			0.218515i	0.994013	+	0.109257i	$0.0348473\pi$
$487$			106.417i			0.218515i	−0.994013	+	0.109257i	$0.965153\pi$
$488$	−53.6477			−0.109934
$489$		0			0
$490$	22.6375	−	13.8153i	0.0461989	−	0.0281946i
$491$		−	608.064i		−	1.23842i	−0.785225	−	0.619210i	$-0.787453\pi$
$491$		−	608.064i		−	1.23842i	0.785225	−	0.619210i	$-0.212547\pi$
$492$		0			0
$493$			577.965i			1.17234i
$494$		−	22.0846i		−	0.0447056i
$495$		0			0
$496$	102.714			0.207085
$497$	−928.627			−1.86847
$498$		0			0
$499$	569.888			1.14206			0.571030	−	0.820929i	$-0.306544\pi$
$499$	569.888			1.14206			0.571030	+	0.820929i	$0.306544\pi$
$500$	−36.4068	−	497.494i	−0.0728135	−	0.994987i
$501$		0			0
$502$		−	17.3603i		−	0.0345822i
$503$	−811.764			−1.61385			−0.806923	−	0.590657i	$-0.798869\pi$
$503$	−811.764			−1.61385			−0.806923	+	0.590657i	$0.798869\pi$
$504$		0			0
$505$	−249.349	−	408.577i	−0.493760	−	0.809063i
$506$			7.22921i			0.0142870i
$507$		0			0
$508$			605.829i			1.19258i
$509$		−	338.453i		−	0.664938i	−0.943114	−	0.332469i	$-0.892118\pi$
$509$		−	338.453i		−	0.664938i	0.943114	−	0.332469i	$-0.107882\pi$
$510$		0			0
$511$	−762.422			−1.49202
$512$	122.992			0.240218
$513$		0			0
$514$	41.4057			0.0805558
$515$	248.994	+	407.995i	0.483483	+	0.792223i
$516$		0			0
$517$			107.542i			0.208011i
$518$	20.4415			0.0394624
$519$		0			0
$520$	27.7345	+	45.4450i	0.0533355	+	0.0873942i
$521$			90.5848i			0.173867i	0.996214	+	0.0869336i	$0.0277068\pi$
$521$			90.5848i			0.173867i	−0.996214	+	0.0869336i	$0.972293\pi$
$522$		0			0
$523$			350.782i			0.670712i	0.942091	+	0.335356i	$0.108857\pi$
$523$			350.782i			0.670712i	−0.942091	+	0.335356i	$0.891143\pi$
$524$		−	625.416i		−	1.19354i
$525$		0			0
$526$	37.7002			0.0716733
$527$	84.5666			0.160468
$528$		0			0
$529$	1207.38			2.28239
$530$	−10.5686	+	6.44986i	−0.0199407	+	0.0121696i
$531$		0			0
$532$		−	673.431i		−	1.26585i
$533$	−782.148			−1.46744
$534$		0			0
$535$	257.705	−	157.274i	0.481692	−	0.293970i
$536$		−	46.5755i		−	0.0868946i
$537$		0			0
$538$			5.59695i			0.0104033i
$539$		−	97.8031i		−	0.181453i
$540$		0			0
$541$	−845.632			−1.56309			−0.781545	−	0.623848i	$-0.785568\pi$
$541$	−845.632			−1.56309			−0.781545	+	0.623848i	$0.785568\pi$
$542$	−9.97377			−0.0184018
$543$		0			0
$544$	60.7092			0.111598
$545$	−345.072	+	210.593i	−0.633160	+	0.386409i
$546$		0			0
$547$			98.8139i			0.180647i	0.995912	+	0.0903235i	$0.0287901\pi$
$547$			98.8139i			0.180647i	−0.995912	+	0.0903235i	$0.971210\pi$
$548$	−301.511			−0.550202
$549$		0			0
$550$	3.85723	+	1.98317i	0.00701314	+	0.00360576i
$551$		−	732.301i		−	1.32904i
$552$		0			0
$553$			627.769i			1.13521i
$554$		−	49.0889i		−	0.0886081i
$555$		0			0
$556$	−381.465			−0.686089
$557$	−569.696			−1.02279			−0.511397	−	0.859345i	$-0.670872\pi$
$557$	−569.696			−1.02279			−0.511397	+	0.859345i	$0.670872\pi$
$558$		0			0
$559$	471.638			0.843717
$560$	421.361	+	690.432i	0.752431	+	1.23291i
$561$		0			0
$562$		−	8.43333i		−	0.0150059i
$563$	630.387			1.11969			0.559847	−	0.828596i	$-0.310860\pi$
$563$	630.387			1.11969			0.559847	+	0.828596i	$0.310860\pi$
$564$		0			0
$565$	527.723	−	322.062i	0.934024	−	0.570022i
$566$		−	8.93847i		−	0.0157924i
$567$		0			0
$568$			70.6853i			0.124446i
$569$		−	580.672i		−	1.02051i	−0.860022	−	0.510257i	$-0.829550\pi$
$569$		−	580.672i		−	1.02051i	0.860022	−	0.510257i	$-0.170450\pi$
$570$		0			0
$571$	−833.133			−1.45908			−0.729538	−	0.683940i	$-0.760265\pi$
$571$	−833.133			−1.45908			−0.729538	+	0.683940i	$0.760265\pi$
$572$	98.0548			0.171425
$573$		0			0
$574$	56.2311			0.0979637
$575$	476.338	−	926.468i	0.828414	−	1.61125i
$576$		0			0
$577$		−	454.896i		−	0.788382i	−0.919029	−	0.394191i	$-0.871025\pi$
$577$		−	454.896i		−	0.788382i	0.919029	−	0.394191i	$-0.128975\pi$
$578$	−11.4367			−0.0197867
$579$		0			0
$580$	459.281	+	752.567i	0.791864	+	1.29753i
$581$		−	595.969i		−	1.02576i
$582$		0			0
$583$			45.6606i			0.0783200i
$584$			58.0341i			0.0993734i
$585$		0			0
$586$	−38.2852			−0.0653330
$587$	82.5751			0.140673			0.0703365	−	0.997523i	$-0.477593\pi$
$587$	82.5751			0.140673			0.0703365	+	0.997523i	$0.477593\pi$
$588$		0			0
$589$	−107.149			−0.181916
$590$	10.2484	+	16.7927i	0.0173701	+	0.0284622i
$591$		0			0
$592$			328.812i			0.555426i
$593$	588.773			0.992872			0.496436	−	0.868073i	$-0.334642\pi$
$593$	588.773			0.992872			0.496436	+	0.868073i	$0.334642\pi$
$594$		0			0
$595$	346.915	+	568.446i	0.583051	+	0.955372i
$596$			192.626i			0.323198i
$597$		0			0
$598$			55.5275i			0.0928554i
$599$		−	301.226i		−	0.502881i	−0.967873	−	0.251441i	$-0.919096\pi$
$599$		−	301.226i		−	0.502881i	0.967873	−	0.251441i	$-0.0809043\pi$
$600$		0			0
$601$	6.66057			0.0110825			0.00554124	−	0.999985i	$-0.498236\pi$
$601$	6.66057			0.0110825			0.00554124	+	0.999985i	$0.498236\pi$
$602$	−33.9076			−0.0563249
$603$		0			0
$604$	91.1062			0.150838
$605$	−502.772	+	306.835i	−0.831027	+	0.507165i
$606$		0			0
$607$		−	404.517i		−	0.666420i	−0.942853	−	0.333210i	$-0.891868\pi$
$607$		−	404.517i		−	0.666420i	0.942853	−	0.333210i	$-0.108132\pi$
$608$	−76.9205			−0.126514
$609$		0			0
$610$	−28.6546	+	17.4875i	−0.0469748	+	0.0286681i
$611$			826.026i			1.35193i
$612$		0			0
$613$		−	27.7082i		−	0.0452010i	−0.999745	−	0.0226005i	$-0.992805\pi$
$613$		−	27.7082i		−	0.0452010i	0.999745	−	0.0226005i	$-0.00719458\pi$
$614$			28.8040i			0.0469120i
$615$		0			0
$616$	−14.1156			−0.0229149
$617$	−458.886			−0.743737			−0.371869	−	0.928285i	$-0.621283\pi$
$617$	−458.886			−0.743737			−0.371869	+	0.928285i	$0.621283\pi$
$618$		0			0
$619$	60.9561			0.0984751			0.0492375	−	0.998787i	$-0.484321\pi$
$619$	60.9561			0.0984751			0.0492375	+	0.998787i	$0.484321\pi$
$620$	110.114	−	67.2010i	0.177603	−	0.108389i
$621$		0			0
$622$			52.3165i			0.0841101i
$623$	679.693			1.09100
$624$		0			0
$625$	−363.655	−	508.311i	−0.581848	−	0.813298i
$626$		−	42.6636i		−	0.0681527i
$627$		0			0
$628$		−	467.770i		−	0.744856i
$629$			270.718i			0.430394i
$630$		0			0
$631$	−590.614			−0.935997			−0.467999	−	0.883729i	$-0.655025\pi$
$631$	−590.614			−0.935997			−0.467999	+	0.883729i	$0.655025\pi$
$632$	47.7846			0.0756085
$633$		0			0
$634$	−23.7524			−0.0374643
$635$	395.429	+	647.939i	0.622722	+	1.02038i
$636$		0			0
$637$		−	751.225i		−	1.17932i
$638$	−7.66573			−0.0120153
$639$		0			0
$640$	105.357	−	64.2982i	0.164621	−	0.100466i
$641$			278.929i			0.435147i	0.976044	+	0.217573i	$0.0698142\pi$
$641$			278.929i			0.435147i	−0.976044	+	0.217573i	$0.930186\pi$
$642$		0			0
$643$		−	740.066i		−	1.15096i	−0.817816	−	0.575479i	$-0.804816\pi$
$643$		−	740.066i		−	1.15096i	0.817816	−	0.575479i	$-0.195184\pi$
$644$			1693.22i			2.62922i
$645$		0			0
$646$	−21.0271			−0.0325497
$647$	71.8283			0.111017			0.0555087	−	0.998458i	$-0.482322\pi$
$647$	71.8283			0.111017			0.0555087	+	0.998458i	$0.482322\pi$
$648$		0			0
$649$	72.5513			0.111789
$650$	29.6273	+	15.2327i	0.0455805	+	0.0234350i
$651$		0			0
$652$			363.939i			0.558188i
$653$	−488.472			−0.748043			−0.374021	−	0.927420i	$-0.622021\pi$
$653$	−488.472			−0.748043			−0.374021	+	0.927420i	$0.622021\pi$
$654$		0			0
$655$	−408.214	−	668.889i	−0.623227	−	1.02120i
$656$			904.505i			1.37882i
$657$		0			0
$658$		−	59.3857i		−	0.0902518i
$659$			437.294i			0.663573i	0.943355	+	0.331786i	$0.107651\pi$
$659$			437.294i			0.663573i	−0.943355	+	0.331786i	$0.892349\pi$
$660$		0			0
$661$	−952.484			−1.44097			−0.720487	−	0.693468i	$-0.756082\pi$
$661$	−952.484			−1.44097			−0.720487	+	0.693468i	$0.756082\pi$
$662$	35.0465			0.0529403
$663$		0			0
$664$	−45.3640			−0.0683193
$665$	−439.553	−	720.240i	−0.660982	−	1.08307i
$666$		0			0
$667$			1841.23i			2.76047i
$668$	655.256			0.980922
$669$		0			0
$670$	−15.1822	−	24.8771i	−0.0226600	−	0.0371301i
$671$			123.799i			0.184500i
$672$		0			0
$673$		−	855.797i		−	1.27162i	−0.771847	−	0.635808i	$-0.780667\pi$
$673$		−	855.797i		−	1.27162i	0.771847	−	0.635808i	$-0.219333\pi$
$674$			24.4796i			0.0363199i
$675$		0			0
$676$	78.7487			0.116492
$677$	1038.97			1.53467			0.767333	−	0.641249i	$-0.221583\pi$
$677$	1038.97			1.53467			0.767333	+	0.641249i	$0.221583\pi$
$678$		0			0
$679$	1609.62			2.37057
$680$	43.2690	−	26.4065i	0.0636309	−	0.0388331i
$681$		0			0
$682$			1.12163i			0.00164462i
$683$	1308.84			1.91632			0.958158	−	0.286238i	$-0.0924049\pi$
$683$	1308.84			1.91632			0.958158	+	0.286238i	$0.0924049\pi$
$684$		0			0
$685$	−322.469	+	196.798i	−0.470757	+	0.287297i
$686$			5.61208i			0.00818087i
$687$		0			0
$688$		−	545.420i		−	0.792762i
$689$			350.719i			0.509026i
$690$		0			0
$691$	−318.933			−0.461553			−0.230776	−	0.973007i	$-0.574127\pi$
$691$	−318.933			−0.461553			−0.230776	+	0.973007i	$0.574127\pi$
$692$	899.660			1.30009
$693$		0			0
$694$	10.4750			0.0150937
$695$	−407.981	+	248.985i	−0.587023	+	0.358252i
$696$		0			0
$697$			744.698i			1.06843i
$698$	−0.166417			−0.000238420
$699$		0			0
$700$	903.434	+	464.495i	1.29062	+	0.663565i
$701$			823.878i			1.17529i	0.809119	+	0.587644i	$0.199945\pi$
$701$			823.878i			1.17529i	−0.809119	+	0.587644i	$0.800055\pi$
$702$		0			0
$703$		−	343.008i		−	0.487921i
$704$		−	112.856i		−	0.160307i
$705$		0			0
$706$	14.4702			0.0204960
$707$	974.774			1.37875
$708$		0			0
$709$	1062.07			1.49798			0.748989	−	0.662583i	$-0.230540\pi$
$709$	1062.07			1.49798			0.748989	+	0.662583i	$0.230540\pi$
$710$	23.0412	+	37.7548i	0.0324524	+	0.0531757i
$711$		0			0
$712$		−	51.7369i		−	0.0726642i
$713$	269.405			0.377848
$714$		0			0
$715$	104.871	−	64.0011i	0.146672	−	0.0895120i
$716$		−	1394.78i		−	1.94802i
$717$		0			0
$718$		−	10.4768i		−	0.0145916i
$719$		−	325.811i		−	0.453144i	−0.973994	−	0.226572i	$-0.927248\pi$
$719$		−	325.811i		−	0.453144i	0.973994	−	0.226572i	$-0.0727519\pi$
$720$		0			0
$721$	−973.385			−1.35005
$722$	−8.37404			−0.0115984
$723$		0			0
$724$	−1015.95			−1.40324
$725$	982.411	+	505.101i	1.35505	+	0.696691i
$726$		0			0
$727$			45.3290i			0.0623507i	0.999514	+	0.0311754i	$0.00992504\pi$
$727$			45.3290i			0.0623507i	−0.999514	+	0.0311754i	$0.990075\pi$
$728$	−108.422			−0.148931
$729$		0			0
$730$	18.9173	+	30.9975i	0.0259142	+	0.0424623i
$731$		−	449.055i		−	0.614303i
$732$		0			0
$733$			618.232i			0.843427i	0.906729	+	0.421714i	$0.138571\pi$
$733$			618.232i			0.843427i	−0.906729	+	0.421714i	$0.861429\pi$
$734$			46.4999i			0.0633514i
$735$		0			0
$736$	193.402			0.262775
$737$	−107.479			−0.145834
$738$		0			0
$739$	−62.7472			−0.0849082			−0.0424541	−	0.999098i	$-0.513518\pi$
$739$	−62.7472			−0.0849082			−0.0424541	+	0.999098i	$0.513518\pi$
$740$	215.126	+	352.501i	0.290711	+	0.476352i
$741$		0			0
$742$		−	25.2143i		−	0.0339815i
$743$	−273.724			−0.368404			−0.184202	−	0.982888i	$-0.558970\pi$
$743$	−273.724			−0.368404			−0.184202	+	0.982888i	$0.558970\pi$
$744$		0			0
$745$	125.728	+	206.015i	0.168763	+	0.276530i
$746$			10.1930i			0.0136635i
$747$		0			0
$748$		−	93.3598i		−	0.124813i
$749$			614.828i			0.820865i
$750$		0			0
$751$	414.271			0.551626			0.275813	−	0.961211i	$-0.411053\pi$
$751$	414.271			0.551626			0.275813	+	0.961211i	$0.411053\pi$
$752$	955.248			1.27028
$753$		0			0
$754$	−58.8805			−0.0780908
$755$	97.4389	−	59.4656i	0.129058	−	0.0787624i
$756$		0			0
$757$			93.2727i			0.123214i	0.998100	+	0.0616068i	$0.0196225\pi$
$757$			93.2727i			0.123214i	−0.998100	+	0.0616068i	$0.980378\pi$
$758$	−16.9774			−0.0223976
$759$		0			0
$760$	−54.8233	+	33.4579i	−0.0721359	+	0.0440236i
$761$			257.174i			0.337942i	0.985621	+	0.168971i	$0.0540444\pi$
$761$			257.174i			0.337942i	−0.985621	+	0.168971i	$0.945956\pi$
$762$		0			0
$763$		−	823.265i		−	1.07898i
$764$		−	1070.79i		−	1.40156i
$765$		0			0
$766$	−30.2381			−0.0394753
$767$	557.266			0.726553
$768$		0			0
$769$	−409.863			−0.532982			−0.266491	−	0.963837i	$-0.585864\pi$
$769$	−409.863			−0.532982			−0.266491	+	0.963837i	$0.585864\pi$
$770$	−7.53948	+	4.60125i	−0.00979154	+	0.00597564i
$771$		0			0
$772$		−	1225.36i		−	1.58726i
$773$	−27.9346			−0.0361379			−0.0180690	−	0.999837i	$-0.505752\pi$
$773$	−27.9346			−0.0361379			−0.0180690	+	0.999837i	$0.505752\pi$
$774$		0			0
$775$	73.9053	−	143.744i	0.0953616	−	0.185476i
$776$		−	122.521i		−	0.157888i
$777$		0			0
$778$			1.14205i			0.00146793i
$779$		−	943.556i		−	1.21124i
$780$		0			0
$781$	163.116			0.208855
$782$	52.8688			0.0676072
$783$		0			0
$784$	−868.745			−1.10809
$785$	−305.317	−	500.284i	−0.388938	−	0.637305i
$786$		0			0
$787$			1015.24i			1.29002i	0.764176	+	0.645008i	$0.223146\pi$
$787$			1015.24i			1.29002i	−0.764176	+	0.645008i	$0.776854\pi$
$788$	249.279			0.316343
$789$		0			0
$790$	25.5229	−	15.5763i	0.0323075	−	0.0197168i
$791$			1259.03i			1.59170i
$792$		0			0
$793$			950.903i			1.19912i
$794$		−	25.3910i		−	0.0319786i
$795$		0			0
$796$	−297.757			−0.374066
$797$	−505.204			−0.633882			−0.316941	−	0.948445i	$-0.602656\pi$
$797$	−505.204			−0.633882			−0.316941	+	0.948445i	$0.602656\pi$
$798$		0			0
$799$	786.475			0.984324
$800$	53.0555	−	103.192i	0.0663194	−	0.128990i
$801$		0			0
$802$		−	32.0482i		−	0.0399604i
$803$	133.922			0.166777
$804$		0			0
$805$	1105.17	+	1810.91i	1.37289	+	2.24958i
$806$			8.61526i			0.0106889i
$807$		0			0
$808$		−	74.1979i		−	0.0918291i
$809$			326.016i			0.402986i	0.979490	+	0.201493i	$0.0645794\pi$
$809$			326.016i			0.402986i	−0.979490	+	0.201493i	$0.935421\pi$
$810$		0			0
$811$	−176.933			−0.218167			−0.109083	−	0.994033i	$-0.534792\pi$
$811$	−176.933			−0.218167			−0.109083	+	0.994033i	$0.534792\pi$
$812$	−1795.46			−2.21115
$813$		0			0
$814$	−3.59061			−0.00441107
$815$	237.545	+	389.236i	0.291467	+	0.477590i
$816$		0			0
$817$			568.968i			0.696411i
$818$	49.5659			0.0605940
$819$		0			0
$820$	591.776	+	969.668i	0.721678	+	1.18252i
$821$		−	226.653i		−	0.276069i	−0.990427	−	0.138035i	$-0.955922\pi$
$821$		−	226.653i		−	0.276069i	0.990427	−	0.138035i	$-0.0440785\pi$
$822$		0			0
$823$		−	18.2804i		−	0.0222119i	−0.999938	−	0.0111059i	$-0.996465\pi$
$823$		−	18.2804i		−	0.0222119i	0.999938	−	0.0111059i	$-0.00353520\pi$
$824$			74.0922i			0.0899177i
$825$		0			0
$826$	−40.0637			−0.0485032
$827$	−1046.88			−1.26588			−0.632940	−	0.774201i	$-0.718152\pi$
$827$	−1046.88			−1.26588			−0.632940	+	0.774201i	$0.718152\pi$
$828$		0			0
$829$	1083.19			1.30662			0.653309	−	0.757091i	$-0.273380\pi$
$829$	1083.19			1.30662			0.653309	+	0.757091i	$0.273380\pi$
$830$	−24.2300	+	14.7873i	−0.0291928	+	0.0178160i
$831$		0			0
$832$		−	866.850i		−	1.04189i
$833$	−715.255			−0.858650
$834$		0			0
$835$	700.802	−	427.690i	0.839284	−	0.512204i
$836$			118.290i			0.141495i
$837$		0			0
$838$		−	5.92930i		−	0.00707553i
$839$			1285.11i			1.53172i	0.643010	+	0.765858i	$0.277686\pi$
$839$			1285.11i			1.53172i	−0.643010	+	0.765858i	$0.722314\pi$
$840$		0			0
$841$	−1111.41			−1.32154
$842$	−59.9922			−0.0712496
$843$		0			0
$844$	−652.615			−0.773241
$845$	84.2224	−	51.3998i	0.0996715	−	0.0608282i
$846$		0			0
$847$		−	1199.50i		−	1.41618i
$848$	405.584			0.478283
$849$		0			0
$850$	14.5034	−	28.2087i	0.0170628	−	0.0331868i
$851$			862.430i			1.01343i
$852$		0			0
$853$		−	193.821i		−	0.227223i	−0.993525	−	0.113611i	$-0.963758\pi$
$853$		−	193.821i		−	0.227223i	0.993525	−	0.113611i	$-0.0362419\pi$
$854$		−	68.3635i		−	0.0800510i
$855$		0			0
$856$	46.7995			0.0546723
$857$	30.1485			0.0351792			0.0175896	−	0.999845i	$-0.494401\pi$
$857$	30.1485			0.0351792			0.0175896	+	0.999845i	$0.494401\pi$
$858$		0			0
$859$	−773.804			−0.900820			−0.450410	−	0.892822i	$-0.648722\pi$
$859$	−773.804			−0.900820			−0.450410	+	0.892822i	$0.648722\pi$
$860$	−356.843	−	584.714i	−0.414934	−	0.679900i
$861$		0			0
$862$			33.4485i			0.0388034i
$863$	−421.375			−0.488267			−0.244134	−	0.969742i	$-0.578504\pi$
$863$	−421.375			−0.488267			−0.244134	+	0.969742i	$0.578504\pi$
$864$		0			0
$865$	962.195	−	587.215i	1.11236	−	0.678861i
$866$			15.9585i			0.0184279i
$867$		0			0
$868$			262.707i			0.302658i
$869$		−	110.269i		−	0.126892i
$870$		0			0
$871$	−825.548			−0.947817
$872$	−62.6654			−0.0718640
$873$		0			0
$874$	−66.9865			−0.0766436
$875$	1269.41	−	92.8960i	1.45076	−	0.106167i
$876$		0			0
$877$		−	893.177i		−	1.01845i	−0.860635	−	0.509223i	$-0.829933\pi$
$877$		−	893.177i		−	1.01845i	0.860635	−	0.509223i	$-0.170067\pi$
$878$	−31.7263			−0.0361348
$879$		0			0
$880$	−74.0133	−	121.276i	−0.0841060	−	0.137814i
$881$		−	718.024i		−	0.815011i	−0.913203	−	0.407505i	$-0.866399\pi$
$881$		−	718.024i		−	0.815011i	0.913203	−	0.407505i	$-0.133601\pi$
$882$		0			0
$883$		−	274.425i		−	0.310788i	−0.987853	−	0.155394i	$-0.950335\pi$
$883$		−	274.425i		−	0.310788i	0.987853	−	0.155394i	$-0.0496646\pi$
$884$		−	717.096i		−	0.811195i
$885$		0			0
$886$	−34.6535			−0.0391123
$887$	−239.686			−0.270221			−0.135110	−	0.990831i	$-0.543139\pi$
$887$	−239.686			−0.270221			−0.135110	+	0.990831i	$0.543139\pi$
$888$		0			0
$889$	−1545.84			−1.73885
$890$	−16.8646	−	27.6340i	−0.0189490	−	0.0310494i
$891$		0			0
$892$		−	71.7196i		−	0.0804032i
$893$	−996.490			−1.11589
$894$		0			0
$895$	−910.384	−	1491.73i	−1.01719	−	1.66674i
$896$			251.359i			0.280535i
$897$		0			0
$898$		−	15.0110i		−	0.0167160i
$899$			285.673i			0.317767i
$900$		0			0
$901$	333.926			0.370617
$902$	−9.87716			−0.0109503
$903$		0			0
$904$	95.8350			0.106012
$905$	−1086.57	+	663.116i	−1.20062	+	0.732725i
$906$		0			0
$907$			209.096i			0.230535i	0.993334	+	0.115268i	$0.0367726\pi$
$907$			209.096i			0.230535i	−0.993334	+	0.115268i	$0.963227\pi$
$908$	−127.992			−0.140960
$909$		0			0
$910$	−57.9107	+	35.3421i	−0.0636382	+	0.0388375i
$911$		−	1576.52i		−	1.73054i	−0.501304	−	0.865271i	$-0.667146\pi$
$911$		−	1576.52i		−	1.73054i	0.501304	−	0.865271i	$-0.332854\pi$
$912$		0			0
$913$			104.684i			0.114659i
$914$		−	15.0108i		−	0.0164232i
$915$		0			0
$916$	641.108			0.699899
$917$	1595.82			1.74026
$918$		0			0
$919$	−1192.75			−1.29788			−0.648939	−	0.760841i	$-0.724787\pi$
$919$	−1192.75			−1.29788			−0.648939	+	0.760841i	$0.724787\pi$
$920$	137.843	−	84.1237i	0.149829	−	0.0914388i
$921$		0			0
$922$		−	25.3505i		−	0.0274951i
$923$	1252.89			1.35741
$924$		0			0
$925$	460.159	+	236.588i	0.497470	+	0.255771i
$926$		−	77.1828i		−	0.0833508i
$927$		0			0
$928$			205.081i			0.220992i
$929$			1350.47i			1.45368i	0.686809	+	0.726838i	$0.259011\pi$
$929$			1350.47i			1.45368i	−0.686809	+	0.726838i	$0.740989\pi$
$930$		0			0
$931$	906.252			0.973418
$932$	−1377.45			−1.47795
$933$		0			0
$934$	41.5027			0.0444354
$935$	−60.9366	−	99.8492i	−0.0651729	−	0.106791i
$936$		0			0
$937$			559.923i			0.597570i	0.954320	+	0.298785i	$0.0965813\pi$
$937$			559.923i			0.597570i	−0.954320	+	0.298785i	$0.903419\pi$
$938$	59.3513			0.0632744
$939$		0			0
$940$	1024.07	−	624.974i	1.08943	−	0.664866i
$941$		−	1172.79i		−	1.24632i	−0.782093	−	0.623161i	$-0.785848\pi$
$941$		−	1172.79i		−	1.24632i	0.782093	−	0.623161i	$-0.214152\pi$
$942$		0			0
$943$			2372.40i			2.51580i
$944$		−	644.444i		−	0.682674i
$945$		0			0
$946$	5.95596			0.00629594
$947$	490.369			0.517813			0.258906	−	0.965902i	$-0.416638\pi$
$947$	490.369			0.517813			0.258906	+	0.965902i	$0.416638\pi$
$948$		0			0
$949$	1028.65			1.08393
$950$	−18.3762	+	35.7414i	−0.0193434	+	0.0376225i
$951$		0			0
$952$			103.230i			0.108435i
$953$	−607.867			−0.637846			−0.318923	−	0.947781i	$-0.603321\pi$
$953$	−607.867			−0.637846			−0.318923	+	0.947781i	$0.603321\pi$
$954$		0			0
$955$	−698.914	−	1145.22i	−0.731847	−	1.19919i
$956$			432.621i			0.452532i
$957$		0			0
$958$			21.1186i			0.0220445i
$959$		−	769.339i		−	0.802230i
$960$		0			0
$961$	−919.201			−0.956505
$962$	−27.5795			−0.0286689
$963$		0			0
$964$	−872.111			−0.904680
$965$	−799.802	−	1310.53i	−0.828810	−	1.35807i
$966$		0			0
$967$		−	107.415i		−	0.111081i	−0.998456	−	0.0555405i	$-0.982312\pi$
$967$		−	107.415i		−	0.111081i	0.998456	−	0.0555405i	$-0.0176882\pi$
$968$	−91.3038			−0.0943221
$969$		0			0
$970$	−39.9381	−	65.4415i	−0.0411733	−	0.0674655i
$971$			351.974i			0.362486i	0.983438	+	0.181243i	$0.0580120\pi$
$971$			351.974i			0.362486i	−0.983438	+	0.181243i	$0.941988\pi$
$972$		0			0
$973$		−	973.352i		−	1.00036i
$974$			10.3222i			0.0105977i
$975$		0			0
$976$	1099.66			1.12670
$977$	385.733			0.394814			0.197407	−	0.980322i	$-0.436748\pi$
$977$	385.733			0.394814			0.197407	+	0.980322i	$0.436748\pi$
$978$		0			0
$979$	−119.390			−0.121951
$980$	−931.332	+	568.379i	−0.950339	+	0.579979i
$981$		0			0
$982$		−	58.9808i		−	0.0600619i
$983$	49.1559			0.0500060			0.0250030	−	0.999687i	$-0.492040\pi$
$983$	49.1559			0.0500060			0.0250030	+	0.999687i	$0.492040\pi$
$984$		0			0
$985$	266.606	−	162.706i	0.270666	−	0.165184i
$986$			56.0612i			0.0568572i
$987$		0			0
$988$			908.585i			0.919620i
$989$		−	1430.56i		−	1.44647i
$990$		0			0
$991$	775.701			0.782746			0.391373	−	0.920232i	$-0.372000\pi$
$991$	775.701			0.782746			0.391373	+	0.920232i	$0.372000\pi$
$992$	30.0069			0.0302489
$993$		0			0
$994$	−90.0746			−0.0906183
$995$	−318.454	+	194.348i	−0.320054	+	0.195325i
$996$		0			0
$997$			725.889i			0.728073i	0.931385	+	0.364037i	$0.118602\pi$
$997$			725.889i			0.728073i	−0.931385	+	0.364037i	$0.881398\pi$
$998$	55.2778			0.0553885
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.b.404.14	yes	24
3.2	odd	2	inner	405.3.d.b.404.11	✓	24
5.4	even	2	inner	405.3.d.b.404.12	yes	24
9.2	odd	6		405.3.h.k.134.14		48
9.4	even	3		405.3.h.k.269.12		48
9.5	odd	6		405.3.h.k.269.13		48
9.7	even	3		405.3.h.k.134.11		48
15.14	odd	2	inner	405.3.d.b.404.13	yes	24
45.4	even	6		405.3.h.k.269.14		48
45.14	odd	6		405.3.h.k.269.11		48
45.29	odd	6		405.3.h.k.134.12		48
45.34	even	6		405.3.h.k.134.13		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.3.d.b.404.11	✓	24	3.2	odd	2	inner
405.3.d.b.404.12	yes	24	5.4	even	2	inner
405.3.d.b.404.13	yes	24	15.14	odd	2	inner
405.3.d.b.404.14	yes	24	1.1	even	1	trivial
405.3.h.k.134.11		48	9.7	even	3
405.3.h.k.134.12		48	45.29	odd	6
405.3.h.k.134.13		48	45.34	even	6
405.3.h.k.134.14		48	9.2	odd	6
405.3.h.k.269.11		48	45.14	odd	6
405.3.h.k.269.12		48	9.4	even	3
405.3.h.k.269.13		48	9.5	odd	6
405.3.h.k.269.14		48	45.4	even	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.0969975 q^{2} -3.99059 q^{4} +(-4.26797 + 2.60469i) q^{5} -10.1824i q^{7} -0.775068 q^{8} +(-0.413983 + 0.252648i) q^{10} +1.78857i q^{11} +13.7380i q^{13} -0.987672i q^{14} +15.8872 q^{16} +13.0802 q^{17} -16.5731 q^{19} +(17.0317 - 10.3942i) q^{20} +0.173487i q^{22} +41.6700 q^{23} +(11.4312 - 22.2335i) q^{25} +1.33256i q^{26} +40.6340i q^{28} +44.1861i q^{29} +6.46522 q^{31} +4.64129 q^{32} +1.26875 q^{34} +(26.5221 + 43.4584i) q^{35} +20.6967i q^{37} -1.60755 q^{38} +(3.30797 - 2.01881i) q^{40} +56.9330i q^{41} -34.3308i q^{43} -7.13747i q^{44} +4.04188 q^{46} +60.1270 q^{47} -54.6821 q^{49} +(1.10880 - 2.15659i) q^{50} -54.8229i q^{52} +25.5290 q^{53} +(-4.65868 - 7.63359i) q^{55} +7.89208i q^{56} +4.28595i q^{58} -40.5638i q^{59} +69.2168 q^{61} +0.627110 q^{62} -63.0986 q^{64} +(-35.7833 - 58.6336i) q^{65} +60.0922i q^{67} -52.1979 q^{68} +(2.57258 + 4.21536i) q^{70} -91.1989i q^{71} -74.8762i q^{73} +2.00753i q^{74} +66.1365 q^{76} +18.2121 q^{77} -61.6522 q^{79} +(-67.8061 + 41.3812i) q^{80} +5.52236i q^{82} +58.5291 q^{83} +(-55.8261 + 34.0699i) q^{85} -3.33001i q^{86} -1.38627i q^{88} +66.7514i q^{89} +139.887 q^{91} -166.288 q^{92} +5.83217 q^{94} +(70.7335 - 43.1677i) q^{95} +158.078i q^{97} -5.30403 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Introduction

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