LMFDB - Embedded newform 567.4.c.c.566.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(566,567)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("567.566");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			566.6
Character	$\chi$	$=$	567.566
Dual form			567.4.c.c.566.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.93216i q^{2} -16.3262 q^{4} +15.3130 q^{5} +(-18.2376 - 3.22314i) q^{7} -41.0664i q^{8} +O(q^{10})$	\(q+4.93216i q^{2} -16.3262 q^{4} +15.3130 q^{5} +(-18.2376 - 3.22314i) q^{7} -41.0664i q^{8} +75.5262i q^{10} -42.6147i q^{11} +28.0353i q^{13} +(15.8970 - 89.9510i) q^{14} +71.9364 q^{16} +82.0161 q^{17} -113.268i q^{19} -250.004 q^{20} +210.183 q^{22} +29.1567i q^{23} +109.488 q^{25} -138.275 q^{26} +(297.752 + 52.6217i) q^{28} +19.1262i q^{29} -112.017i q^{31} +26.2707i q^{32} +404.517i q^{34} +(-279.273 - 49.3559i) q^{35} +69.2948 q^{37} +558.655 q^{38} -628.850i q^{40} +484.465 q^{41} +389.141 q^{43} +695.738i q^{44} -143.806 q^{46} +106.889 q^{47} +(322.223 + 117.565i) q^{49} +540.012i q^{50} -457.711i q^{52} -207.332i q^{53} -652.559i q^{55} +(-132.363 + 748.954i) q^{56} -94.3337 q^{58} +195.637 q^{59} +515.261i q^{61} +552.486 q^{62} +445.920 q^{64} +429.304i q^{65} -442.628 q^{67} -1339.01 q^{68} +(243.431 - 1377.42i) q^{70} +640.082i q^{71} -528.567i q^{73} +341.773i q^{74} +1849.24i q^{76} +(-137.353 + 777.192i) q^{77} -41.4271 q^{79} +1101.56 q^{80} +2389.46i q^{82} -1109.10 q^{83} +1255.91 q^{85} +1919.31i q^{86} -1750.03 q^{88} +121.109 q^{89} +(90.3615 - 511.297i) q^{91} -476.019i q^{92} +527.196i q^{94} -1734.47i q^{95} -864.627i q^{97} +(-579.849 + 1589.26i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) $ 44 * q - 156 * q^4 - 10 * q^7	$ 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) $ 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$-1$

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 $

$2$			4.93216i			1.74378i	0.489699	+	0.871892i	$0.337107\pi$
$2$			4.93216i			1.74378i	−0.489699	+	0.871892i	$0.662893\pi$
$3$		0			0
$3$		0			0
$4$	−16.3262			−2.04078
$5$	15.3130			1.36964			0.684818	−	0.728714i	$-0.259882\pi$
$5$	15.3130			1.36964			0.684818	+	0.728714i	$0.259882\pi$
$6$		0			0
$7$	−18.2376	−	3.22314i	−0.984740	−	0.174033i
$7$	−18.2376	−	3.22314i	−0.984740	−	0.174033i
$8$		−	41.0664i		−	1.81490i
$9$		0			0
$10$			75.5262i			2.38835i
$11$		−	42.6147i		−	1.16807i	−0.811727	−	0.584037i	$-0.801472\pi$
$11$		−	42.6147i		−	1.16807i	0.811727	−	0.584037i	$-0.198528\pi$
$12$		0			0
$13$			28.0353i			0.598122i	0.954234	+	0.299061i	$0.0966734\pi$
$13$			28.0353i			0.598122i	−0.954234	+	0.299061i	$0.903327\pi$
$14$	15.8970	−	89.9510i	0.303476	−	1.71717i
$15$		0			0
$16$	71.9364			1.12401
$17$	82.0161			1.17011			0.585053	−	0.810995i	$-0.301074\pi$
$17$	82.0161			1.17011			0.585053	+	0.810995i	$0.301074\pi$
$18$		0			0
$19$		−	113.268i		−	1.36765i	−0.729645	−	0.683827i	$-0.760314\pi$
$19$		−	113.268i		−	1.36765i	0.729645	−	0.683827i	$-0.239686\pi$
$20$	−250.004			−2.79513
$21$		0			0
$22$	210.183			2.03687
$23$			29.1567i			0.264330i	0.991228	+	0.132165i	$0.0421929\pi$
$23$			29.1567i			0.264330i	−0.991228	+	0.132165i	$0.957807\pi$
$24$		0			0
$25$	109.488			0.875902
$26$	−138.275			−1.04300
$27$		0			0
$28$	297.752	+	52.6217i	2.00964	+	0.355163i
$29$			19.1262i			0.122471i	0.998123	+	0.0612353i	$0.0195040\pi$
$29$			19.1262i			0.122471i	−0.998123	+	0.0612353i	$0.980496\pi$
$30$		0			0
$31$		−	112.017i		−	0.648995i	−0.945887	−	0.324497i	$-0.894805\pi$
$31$		−	112.017i		−	0.648995i	0.945887	−	0.324497i	$-0.105195\pi$
$32$			26.2707i			0.145126i
$33$		0			0
$34$			404.517i			2.04041i
$35$	−279.273	−	49.3559i	−1.34873	−	0.238362i
$36$		0			0
$37$	69.2948			0.307892			0.153946	−	0.988079i	$-0.450802\pi$
$37$	69.2948			0.307892			0.153946	+	0.988079i	$0.450802\pi$
$38$	558.655			2.38489
$39$		0			0
$40$		−	628.850i		−	2.48575i
$41$	484.465			1.84538			0.922692	−	0.385539i	$-0.125984\pi$
$41$	484.465			1.84538			0.922692	+	0.385539i	$0.125984\pi$
$42$		0			0
$43$	389.141			1.38008			0.690040	−	0.723771i	$-0.257593\pi$
$43$	389.141			1.38008			0.690040	+	0.723771i	$0.257593\pi$
$44$			695.738i			2.38378i
$45$		0			0
$46$	−143.806			−0.460934
$47$	106.889			0.331733			0.165866	−	0.986148i	$-0.446958\pi$
$47$	106.889			0.331733			0.165866	+	0.986148i	$0.446958\pi$
$48$		0			0
$49$	322.223	+	117.565i	0.939425	+	0.342754i
$50$			540.012i			1.52738i
$51$		0			0
$52$		−	457.711i		−	1.22064i
$53$		−	207.332i		−	0.537345i	−0.963232	−	0.268672i	$-0.913415\pi$
$53$		−	207.332i		−	0.537345i	0.963232	−	0.268672i	$-0.0865849\pi$
$54$		0			0
$55$		−	652.559i		−	1.59984i
$56$	−132.363	+	748.954i	−0.315852	+	1.78720i
$57$		0			0
$58$	−94.3337			−0.213562
$59$	195.637			0.431691			0.215845	−	0.976428i	$-0.430749\pi$
$59$	195.637			0.431691			0.215845	+	0.976428i	$0.430749\pi$
$60$		0			0
$61$			515.261i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$61$			515.261i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$62$	552.486			1.13171
$63$		0			0
$64$	445.920			0.870937
$65$			429.304i			0.819209i
$66$		0			0
$67$	−442.628			−0.807098			−0.403549	−	0.914958i	$-0.632224\pi$
$67$	−442.628			−0.807098			−0.403549	+	0.914958i	$0.632224\pi$
$68$	−1339.01			−2.38793
$69$		0			0
$70$	243.431	−	1377.42i	0.415651	−	2.35190i
$71$			640.082i			1.06991i	0.844880	+	0.534956i	$0.179672\pi$
$71$			640.082i			1.06991i	−0.844880	+	0.534956i	$0.820328\pi$
$72$		0			0
$73$		−	528.567i		−	0.847453i	−0.905790	−	0.423727i	$-0.860722\pi$
$73$		−	528.567i		−	0.847453i	0.905790	−	0.423727i	$-0.139278\pi$
$74$			341.773i			0.536896i
$75$		0			0
$76$			1849.24i			2.79108i
$77$	−137.353	+	777.192i	−0.203283	+	1.15025i
$78$		0			0
$79$	−41.4271			−0.0589989			−0.0294994	−	0.999565i	$-0.509391\pi$
$79$	−41.4271			−0.0589989			−0.0294994	+	0.999565i	$0.509391\pi$
$80$	1101.56			1.53948
$81$		0			0
$82$			2389.46i			3.21795i
$83$	−1109.10			−1.46674			−0.733368	−	0.679832i	$-0.762053\pi$
$83$	−1109.10			−1.46674			−0.733368	+	0.679832i	$0.762053\pi$
$84$		0			0
$85$	1255.91			1.60262
$86$			1919.31i			2.40656i
$87$		0			0
$88$	−1750.03			−2.11993
$89$	121.109			0.144242			0.0721208	−	0.997396i	$-0.477023\pi$
$89$	121.109			0.144242			0.0721208	+	0.997396i	$0.477023\pi$
$90$		0			0
$91$	90.3615	−	511.297i	0.104093	−	0.588995i
$92$		−	476.019i		−	0.539439i
$93$		0			0
$94$			527.196i			0.578470i
$95$		−	1734.47i		−	1.87319i
$96$		0			0
$97$		−	864.627i		−	0.905047i	−0.891752	−	0.452524i	$-0.850524\pi$
$97$		−	864.627i		−	0.905047i	0.891752	−	0.452524i	$-0.149476\pi$
$98$	−579.849	+	1589.26i	−0.597689	+	1.63815i
$99$		0			0
$100$	−1787.52			−1.78752
$101$	186.372			0.183611			0.0918056	−	0.995777i	$-0.470736\pi$
$101$	186.372			0.183611			0.0918056	+	0.995777i	$0.470736\pi$
$102$		0			0
$103$		−	841.456i		−	0.804963i	−0.915428	−	0.402481i	$-0.868148\pi$
$103$		−	841.456i		−	0.804963i	0.915428	−	0.402481i	$-0.131852\pi$
$104$	1151.31			1.08553
$105$		0			0
$106$	1022.60			0.937013
$107$		−	644.664i		−	0.582449i	−0.956655	−	0.291224i	$-0.905937\pi$
$107$		−	644.664i		−	0.582449i	0.956655	−	0.291224i	$-0.0940626\pi$
$108$		0			0
$109$	804.517			0.706961			0.353481	−	0.935442i	$-0.384998\pi$
$109$	804.517			0.706961			0.353481	+	0.935442i	$0.384998\pi$
$110$	3218.53			2.78977
$111$		0			0
$112$	−1311.95	−	231.861i	−1.10685	−	0.195614i
$113$		−	601.855i		−	0.501042i	−0.968111	−	0.250521i	$-0.919398\pi$
$113$		−	601.855i		−	0.501042i	0.968111	−	0.250521i	$-0.0806020\pi$
$114$		0			0
$115$			446.476i			0.362036i
$116$		−	312.259i		−	0.249936i
$117$		0			0
$118$			964.914i			0.752776i
$119$	−1495.78	−	264.349i	−1.15225	−	0.203637i
$120$		0			0
$121$	−485.014			−0.364398
$122$	−2541.35			−1.88593
$123$		0			0
$124$			1828.82i			1.32446i
$125$	−237.539			−0.169969
$126$		0			0
$127$	−940.735			−0.657297			−0.328649	−	0.944452i	$-0.606593\pi$
$127$	−940.735			−0.657297			−0.328649	+	0.944452i	$0.606593\pi$
$128$			2409.52i			1.66385i
$129$		0			0
$130$	−2117.40			−1.42852
$131$	−276.025			−0.184095			−0.0920473	−	0.995755i	$-0.529341\pi$
$131$	−276.025			−0.184095			−0.0920473	+	0.995755i	$0.529341\pi$
$132$		0			0
$133$	−365.077	+	2065.74i	−0.238017	+	1.34678i
$134$		−	2183.11i		−	1.40740i
$135$		0			0
$136$		−	3368.11i		−	2.12362i
$137$		−	1831.31i		−	1.14204i	−0.820937	−	0.571018i	$-0.806549\pi$
$137$		−	1831.31i		−	1.14204i	0.820937	−	0.571018i	$-0.193451\pi$
$138$		0			0
$139$			1514.22i			0.923986i	0.886884	+	0.461993i	$0.152865\pi$
$139$			1514.22i			0.923986i	−0.886884	+	0.461993i	$0.847135\pi$
$140$	4559.48	+	805.796i	2.75247	+	0.486444i
$141$		0			0
$142$	−3156.99			−1.86570
$143$	1194.72			0.698651
$144$		0			0
$145$			292.880i			0.167740i
$146$	2606.98			1.47778
$147$		0			0
$148$	−1131.32			−0.628339
$149$			73.2458i			0.0402720i	0.999797	+	0.0201360i	$0.00640992\pi$
$149$			73.2458i			0.0402720i	−0.999797	+	0.0201360i	$0.993590\pi$
$150$		0			0
$151$	−97.5320			−0.0525632			−0.0262816	−	0.999655i	$-0.508367\pi$
$151$	−97.5320			−0.0525632			−0.0262816	+	0.999655i	$0.508367\pi$
$152$	−4651.50			−2.48215
$153$		0			0
$154$	−3833.24	−	677.448i	−2.00579	−	0.354482i
$155$		−	1715.31i		−	0.888886i
$156$		0			0
$157$		−	2018.73i		−	1.02619i	−0.858331	−	0.513096i	$-0.828499\pi$
$157$		−	2018.73i		−	1.02619i	0.858331	−	0.513096i	$-0.171501\pi$
$158$		−	204.325i		−	0.102881i
$159$		0			0
$160$			402.282i			0.198770i
$161$	93.9759	−	531.749i	0.0460021	−	0.260296i
$162$		0			0
$163$	3529.01			1.69579			0.847893	−	0.530167i	$-0.177871\pi$
$163$	3529.01			1.69579			0.847893	+	0.530167i	$0.177871\pi$
$164$	−7909.50			−3.76602
$165$		0			0
$166$		−	5470.24i		−	2.55767i
$167$	−2450.93			−1.13568			−0.567841	−	0.823138i	$-0.692221\pi$
$167$	−2450.93			−1.13568			−0.567841	+	0.823138i	$0.692221\pi$
$168$		0			0
$169$	1411.02			0.642250
$170$			6194.36i			2.79462i
$171$		0			0
$172$	−6353.21			−2.81644
$173$	919.707			0.404185			0.202093	−	0.979366i	$-0.435226\pi$
$173$	919.707			0.404185			0.202093	+	0.979366i	$0.435226\pi$
$174$		0			0
$175$	−1996.80	−	352.894i	−0.862536	−	0.152436i
$176$		−	3065.55i		−	1.31292i
$177$		0			0
$178$			597.328i			0.251526i
$179$		−	4046.51i		−	1.68966i	−0.535031	−	0.844832i	$-0.679700\pi$
$179$		−	4046.51i		−	1.68966i	0.535031	−	0.844832i	$-0.320300\pi$
$180$		0			0
$181$			4462.98i			1.83277i	0.400301	+	0.916384i	$0.368906\pi$
$181$			4462.98i			1.83277i	−0.400301	+	0.916384i	$0.631094\pi$
$182$	2521.80	+	445.678i	1.02708	+	0.181516i
$183$		0			0
$184$	1197.36			0.479731
$185$	1061.11			0.421699
$186$		0			0
$187$		−	3495.09i		−	1.36677i
$188$	−1745.10			−0.676993
$189$		0			0
$190$	8554.69			3.26643
$191$			3827.06i			1.44982i	0.688842	+	0.724911i	$0.258119\pi$
$191$			3827.06i			1.44982i	−0.688842	+	0.724911i	$0.741881\pi$
$192$		0			0
$193$	4407.52			1.64384			0.821918	−	0.569605i	$-0.192904\pi$
$193$	4407.52			1.64384			0.821918	+	0.569605i	$0.192904\pi$
$194$	4264.48			1.57821
$195$		0			0
$196$	−5260.69	−	1919.39i	−1.91716	−	0.699487i
$197$			1514.82i			0.547852i	0.961751	+	0.273926i	$0.0883223\pi$
$197$			1514.82i			0.547852i	−0.961751	+	0.273926i	$0.911678\pi$
$198$		0			0
$199$		−	2622.01i		−	0.934017i	−0.884253	−	0.467009i	$-0.845332\pi$
$199$		−	2622.01i		−	0.934017i	0.884253	−	0.467009i	$-0.154668\pi$
$200$		−	4496.27i		−	1.58967i
$201$		0			0
$202$			919.219i			0.320178i
$203$	61.6464	−	348.817i	0.0213139	−	0.120602i
$204$		0			0
$205$	7418.61			2.52750
$206$	4150.20			1.40368
$207$		0			0
$208$			2016.76i			0.672293i
$209$	−4826.87			−1.59752
$210$		0			0
$211$	3160.65			1.03122			0.515611	−	0.856823i	$-0.327565\pi$
$211$	3160.65			1.03122			0.515611	+	0.856823i	$0.327565\pi$
$212$			3384.96i			1.09660i
$213$		0			0
$214$	3179.59			1.01566
$215$	5958.91			1.89021
$216$		0			0
$217$	−361.046	+	2042.92i	−0.112946	+	0.639091i
$218$			3968.01i			1.23279i
$219$		0			0
$220$			10653.8i			3.26492i
$221$			2299.34i			0.699867i
$222$		0			0
$223$			953.529i			0.286337i	0.989698	+	0.143168i	$0.0457290\pi$
$223$			953.529i			0.286337i	−0.989698	+	0.143168i	$0.954271\pi$
$224$	84.6739	−	479.115i	0.0252567	−	0.142912i
$225$		0			0
$226$	2968.45			0.873709
$227$	4279.37			1.25124			0.625620	−	0.780128i	$-0.284846\pi$
$227$	4279.37			1.25124			0.625620	+	0.780128i	$0.284846\pi$
$228$		0			0
$229$			3998.61i			1.15387i	0.816791	+	0.576934i	$0.195751\pi$
$229$			3998.61i			1.15387i	−0.816791	+	0.576934i	$0.804249\pi$
$230$	−2202.09			−0.631312
$231$		0			0
$232$	785.445			0.222272
$233$		−	843.996i		−	0.237305i	−0.992936	−	0.118652i	$-0.962143\pi$
$233$		−	843.996i		−	0.237305i	0.992936	−	0.118652i	$-0.0378574\pi$
$234$		0			0
$235$	1636.80			0.454353
$236$	−3194.02			−0.880987
$237$		0			0
$238$	1303.81	−	7377.43i	0.355099	−	2.00928i
$239$		−	1917.00i		−	0.518830i	−0.965766	−	0.259415i	$-0.916470\pi$
$239$		−	1917.00i		−	0.518830i	0.965766	−	0.259415i	$-0.0835297\pi$
$240$		0			0
$241$			1083.75i			0.289669i	0.989456	+	0.144834i	$0.0462650\pi$
$241$			1083.75i			0.289669i	−0.989456	+	0.144834i	$0.953735\pi$
$242$		−	2392.17i		−	0.635432i
$243$		0			0
$244$		−	8412.28i		−	2.20714i
$245$	4934.20	+	1800.27i	1.28667	+	0.469449i
$246$		0			0
$247$	3175.49			0.818024
$248$	−4600.14			−1.17786
$249$		0			0
$250$		−	1171.58i		−	0.296389i
$251$	1984.35			0.499007			0.249504	−	0.968374i	$-0.419733\pi$
$251$	1984.35			0.499007			0.249504	+	0.968374i	$0.419733\pi$
$252$		0			0
$253$	1242.50			0.308757
$254$		−	4639.86i		−	1.14618i
$255$		0			0
$256$	−8316.77			−2.03046
$257$	2831.63			0.687285			0.343642	−	0.939101i	$-0.388339\pi$
$257$	2831.63			0.687285			0.343642	+	0.939101i	$0.388339\pi$
$258$		0			0
$259$	−1263.77	−	223.346i	−0.303193	−	0.0535833i
$260$		−	7008.92i		−	1.67183i
$261$		0			0
$262$		−	1361.40i		−	0.321021i
$263$		−	6394.75i		−	1.49930i	−0.661832	−	0.749652i	$-0.730221\pi$
$263$		−	6394.75i		−	1.49930i	0.661832	−	0.749652i	$-0.269779\pi$
$264$		0			0
$265$		−	3174.88i		−	0.735967i
$266$	−10188.6	−	1800.62i	−2.34850	−	0.415050i
$267$		0			0
$268$	7226.45			1.64711
$269$	−7580.29			−1.71814			−0.859068	−	0.511862i	$-0.828956\pi$
$269$	−7580.29			−1.71814			−0.859068	+	0.511862i	$0.828956\pi$
$270$		0			0
$271$		−	56.4463i		−	0.0126526i	−0.999980	−	0.00632632i	$-0.997986\pi$
$271$		−	56.4463i		−	0.0126526i	0.999980	−	0.00632632i	$-0.00201374\pi$
$272$	5899.94			1.31521
$273$		0			0
$274$	9032.30			1.99146
$275$		−	4665.79i		−	1.02312i
$276$		0			0
$277$	−4217.11			−0.914734			−0.457367	−	0.889278i	$-0.651208\pi$
$277$	−4217.11			−0.914734			−0.457367	+	0.889278i	$0.651208\pi$
$278$	−7468.36			−1.61123
$279$		0			0
$280$	−2026.87	+	11468.7i	−0.432602	+	2.44781i
$281$		−	2615.42i		−	0.555242i	−0.960691	−	0.277621i	$-0.910454\pi$
$281$		−	2615.42i		−	0.555242i	0.960691	−	0.277621i	$-0.0895459\pi$
$282$		0			0
$283$		−	445.316i		−	0.0935381i	−0.998906	−	0.0467691i	$-0.985108\pi$
$283$		−	445.316i		−	0.0935381i	0.998906	−	0.0467691i	$-0.0148925\pi$
$284$		−	10450.1i		−	2.18346i
$285$		0			0
$286$			5892.53i			1.21830i
$287$	−8835.50	−	1561.50i	−1.81722	−	0.321158i
$288$		0			0
$289$	1813.63			0.369150
$290$	−1444.53			−0.292503
$291$		0			0
$292$			8629.52i			1.72947i
$293$	−355.408			−0.0708639			−0.0354320	−	0.999372i	$-0.511281\pi$
$293$	−355.408			−0.0708639			−0.0354320	+	0.999372i	$0.511281\pi$
$294$		0			0
$295$	2995.79			0.591259
$296$		−	2845.69i		−	0.558791i
$297$		0			0
$298$	−361.260			−0.0702256
$299$	−817.415			−0.158102
$300$		0			0
$301$	−7097.01	−	1254.25i	−1.35902	−	0.240179i
$302$		−	481.044i		−	0.0916588i
$303$		0			0
$304$		−	8148.07i		−	1.53725i
$305$			7890.19i			1.48128i
$306$		0			0
$307$		−	6232.15i		−	1.15859i	−0.815118	−	0.579295i	$-0.803328\pi$
$307$		−	6232.15i		−	1.15859i	0.815118	−	0.579295i	$-0.196672\pi$
$308$	2242.46	−	12688.6i	0.414857	−	2.34741i
$309$		0			0
$310$	8460.21			1.55003
$311$	−3102.30			−0.565645			−0.282822	−	0.959172i	$-0.591271\pi$
$311$	−3102.30			−0.565645			−0.282822	+	0.959172i	$0.591271\pi$
$312$		0			0
$313$			1224.78i			0.221178i	0.993866	+	0.110589i	$0.0352738\pi$
$313$			1224.78i			0.221178i	−0.993866	+	0.110589i	$0.964726\pi$
$314$	9956.70			1.78946
$315$		0			0
$316$	676.349			0.120404
$317$		−	7578.16i		−	1.34269i	−0.741146	−	0.671343i	$-0.765718\pi$
$317$		−	7578.16i		−	1.34269i	0.741146	−	0.671343i	$-0.234282\pi$
$318$		0			0
$319$	815.058			0.143055
$320$	6828.37			1.19287
$321$		0			0
$322$	2622.67	+	463.505i	0.453900	+	0.0802177i
$323$		−	9289.78i		−	1.60030i
$324$		0			0
$325$			3069.52i			0.523896i
$326$			17405.6i			2.95708i
$327$		0			0
$328$		−	19895.2i		−	3.34918i
$329$	−1949.41	−	344.519i	−0.326670	−	0.0577324i
$330$		0			0
$331$	−5297.00			−0.879605			−0.439803	−	0.898094i	$-0.644952\pi$
$331$	−5297.00			−0.879605			−0.439803	+	0.898094i	$0.644952\pi$
$332$	18107.4			2.99329
$333$		0			0
$334$		−	12088.4i		−	1.98038i
$335$	−6777.95			−1.10543
$336$		0			0
$337$	−269.836			−0.0436170			−0.0218085	−	0.999762i	$-0.506942\pi$
$337$	−269.836			−0.0436170			−0.0218085	+	0.999762i	$0.506942\pi$
$338$			6959.40i			1.11994i
$339$		0			0
$340$	−20504.3			−3.27060
$341$	−4773.57			−0.758074
$342$		0			0
$343$	−5497.66	−	3182.67i	−0.865439	−	0.501015i
$344$		−	15980.6i		−	2.50470i
$345$		0			0
$346$			4536.15i			0.704811i
$347$			2094.05i			0.323961i	0.986794	+	0.161981i	$0.0517882\pi$
$347$			2094.05i			0.323961i	−0.986794	+	0.161981i	$0.948212\pi$
$348$		0			0
$349$		−	12163.4i		−	1.86559i	−0.360412	−	0.932793i	$-0.617364\pi$
$349$		−	12163.4i		−	1.86559i	0.360412	−	0.932793i	$-0.382636\pi$
$350$	1740.53	−	9848.54i	0.265815	−	1.50408i
$351$		0			0
$352$	1119.52			0.169518
$353$	6010.36			0.906231			0.453115	−	0.891452i	$-0.350313\pi$
$353$	6010.36			0.906231			0.453115	+	0.891452i	$0.350313\pi$
$354$		0			0
$355$			9801.57i			1.46539i
$356$	−1977.25			−0.294366
$357$		0			0
$358$	19958.0			2.94641
$359$		−	1865.30i		−	0.274225i	−0.990555	−	0.137112i	$-0.956218\pi$
$359$		−	1865.30i		−	0.274225i	0.990555	−	0.137112i	$-0.0437822\pi$
$360$		0			0
$361$	−5970.59			−0.870476
$362$	−22012.2			−3.19595
$363$		0			0
$364$	−1475.26	+	8347.56i	−0.212431	+	1.20201i
$365$		−	8093.94i		−	1.16070i
$366$		0			0
$367$		−	2543.59i		−	0.361783i	−0.983503	−	0.180891i	$-0.942102\pi$
$367$		−	2543.59i		−	0.361783i	0.983503	−	0.180891i	$-0.0578983\pi$
$368$			2097.43i			0.297108i
$369$		0			0
$370$			5233.57i			0.735352i
$371$	−668.260	+	3781.25i	−0.0935157	+	0.529145i
$372$		0			0
$373$	−7341.53			−1.01912			−0.509558	−	0.860436i	$-0.670191\pi$
$373$	−7341.53			−1.01912			−0.509558	+	0.860436i	$0.670191\pi$
$374$	17238.4			2.38335
$375$		0			0
$376$		−	4389.57i		−	0.602060i
$377$	−536.209			−0.0732524
$378$		0			0
$379$	9824.37			1.33151			0.665757	−	0.746168i	$-0.268109\pi$
$379$	9824.37			1.33151			0.665757	+	0.746168i	$0.268109\pi$
$380$			28317.4i			3.82276i
$381$		0			0
$382$	−18875.7			−2.52818
$383$	−1822.93			−0.243205			−0.121602	−	0.992579i	$-0.538803\pi$
$383$	−1822.93			−0.243205			−0.121602	+	0.992579i	$0.538803\pi$
$384$		0			0
$385$	−2103.29	+	11901.1i	−0.278424	+	1.57542i
$386$			21738.6i			2.86650i
$387$		0			0
$388$			14116.1i			1.84700i
$389$			12588.3i			1.64075i	0.571825	+	0.820375i	$0.306235\pi$
$389$			12588.3i			1.64075i	−0.571825	+	0.820375i	$0.693765\pi$
$390$		0			0
$391$			2391.32i			0.309294i
$392$	4827.96	−	13232.5i	0.622064	−	1.70496i
$393$		0			0
$394$	−7471.36			−0.955335
$395$	−634.373			−0.0808070
$396$		0			0
$397$			5809.38i			0.734419i	0.930138	+	0.367209i	$0.119687\pi$
$397$			5809.38i			0.734419i	−0.930138	+	0.367209i	$0.880313\pi$
$398$	12932.2			1.62872
$399$		0			0
$400$	7876.15			0.984519
$401$			5251.47i			0.653980i	0.945028	+	0.326990i	$0.106034\pi$
$401$			5251.47i			0.653980i	−0.945028	+	0.326990i	$0.893966\pi$
$402$		0			0
$403$	3140.43			0.388178
$404$	−3042.76			−0.374710
$405$		0			0
$406$	1720.42	+	304.050i	0.210303	+	0.0371669i
$407$		−	2952.98i		−	0.359640i
$408$		0			0
$409$		−	3573.01i		−	0.431966i	−0.976397	−	0.215983i	$-0.930704\pi$
$409$		−	3573.01i		−	0.431966i	0.976397	−	0.215983i	$-0.0692956\pi$
$410$			36589.8i			4.40742i
$411$		0			0
$412$			13737.8i			1.64275i
$413$	−3567.96	−	630.564i	−0.425103	−	0.0751284i
$414$		0			0
$415$	−16983.6			−2.00889
$416$	−736.505			−0.0868032
$417$		0			0
$418$		−	23806.9i		−	2.78573i
$419$	2868.34			0.334434			0.167217	−	0.985920i	$-0.446522\pi$
$419$	2868.34			0.334434			0.167217	+	0.985920i	$0.446522\pi$
$420$		0			0
$421$	−7976.93			−0.923449			−0.461724	−	0.887023i	$-0.652769\pi$
$421$	−7976.93			−0.923449			−0.461724	+	0.887023i	$0.652769\pi$
$422$			15588.8i			1.79823i
$423$		0			0
$424$	−8514.39			−0.975225
$425$	8979.76			1.02490
$426$		0			0
$427$	1660.76	−	9397.15i	0.188219	−	1.06501i
$428$			10524.9i			1.18865i
$429$		0			0
$430$			29390.3i			3.29611i
$431$			11462.7i			1.28106i	0.767933	+	0.640531i	$0.221286\pi$
$431$			11462.7i			1.28106i	−0.767933	+	0.640531i	$0.778714\pi$
$432$		0			0
$433$			11847.0i			1.31485i	0.753521	+	0.657424i	$0.228354\pi$
$433$			11847.0i			1.31485i	−0.753521	+	0.657424i	$0.771646\pi$
$434$	−10076.0	−	1780.74i	−1.11444	−	0.196954i
$435$		0			0
$436$	−13134.7			−1.44275
$437$	3302.51			0.361512
$438$		0			0
$439$		−	4444.08i		−	0.483154i	−0.970382	−	0.241577i	$-0.922335\pi$
$439$		−	4444.08i		−	0.483154i	0.970382	−	0.241577i	$-0.0776646\pi$
$440$	−26798.3			−2.90354
$441$		0			0
$442$	−11340.7			−1.22042
$443$		−	12954.8i		−	1.38939i	−0.719305	−	0.694694i	$-0.755540\pi$
$443$		−	12954.8i		−	1.38939i	0.719305	−	0.694694i	$-0.244460\pi$
$444$		0			0
$445$	1854.54			0.197558
$446$	−4702.96			−0.499309
$447$		0			0
$448$	−8132.52	−	1437.26i	−0.857646	−	0.151572i
$449$		−	4587.25i		−	0.482151i	−0.970506	−	0.241075i	$-0.922500\pi$
$449$		−	4587.25i		−	0.482151i	0.970506	−	0.241075i	$-0.0775001\pi$
$450$		0			0
$451$		−	20645.3i		−	2.15555i
$452$			9826.04i			1.02252i
$453$		0			0
$454$			21106.5i			2.18189i
$455$	1383.71	−	7829.49i	0.142569	−	0.806708i
$456$		0			0
$457$	−16199.2			−1.65813			−0.829065	−	0.559153i	$-0.811127\pi$
$457$	−16199.2			−1.65813			−0.829065	+	0.559153i	$0.811127\pi$
$458$	−19721.8			−2.01210
$459$		0			0
$460$		−	7289.28i		−	0.738835i
$461$	3665.18			0.370291			0.185146	−	0.982711i	$-0.440724\pi$
$461$	3665.18			0.370291			0.185146	+	0.982711i	$0.440724\pi$
$462$		0			0
$463$	−3827.66			−0.384204			−0.192102	−	0.981375i	$-0.561530\pi$
$463$	−3827.66			−0.384204			−0.192102	+	0.981375i	$0.561530\pi$
$464$			1375.87i			0.137658i
$465$		0			0
$466$	4162.73			0.413808
$467$	16957.8			1.68033			0.840166	−	0.542330i	$-0.182458\pi$
$467$	16957.8			1.68033			0.840166	+	0.542330i	$0.182458\pi$
$468$		0			0
$469$	8072.48	+	1426.65i	0.794781	+	0.140462i
$470$			8072.95i			0.792293i
$471$		0			0
$472$		−	8034.11i		−	0.783474i
$473$		−	16583.1i		−	1.61204i
$474$		0			0
$475$		−	12401.4i		−	1.19793i
$476$	24420.5	+	4315.83i	2.35149	+	0.415579i
$477$		0			0
$478$	9454.95			0.904727
$479$	−3997.47			−0.381313			−0.190657	−	0.981657i	$-0.561062\pi$
$479$	−3997.47			−0.381313			−0.190657	+	0.981657i	$0.561062\pi$
$480$		0			0
$481$			1942.70i			0.184157i
$482$	−5345.21			−0.505120
$483$		0			0
$484$	7918.46			0.743657
$485$		−	13240.0i		−	1.23958i
$486$		0			0
$487$	−6968.26			−0.648382			−0.324191	−	0.945992i	$-0.605092\pi$
$487$	−6968.26			−0.648382			−0.324191	+	0.945992i	$0.605092\pi$
$488$	21159.9			1.96284
$489$		0			0
$490$	−8879.22	+	24336.3i	−0.818617	+	2.24367i
$491$			14925.0i			1.37180i	0.727694	+	0.685902i	$0.240592\pi$
$491$			14925.0i			1.37180i	−0.727694	+	0.685902i	$0.759408\pi$
$492$		0			0
$493$			1568.66i			0.143304i
$494$			15662.1i			1.42646i
$495$		0			0
$496$		−	8058.09i		−	0.729474i
$497$	2063.07	−	11673.6i	0.186200	−	1.05359i
$498$		0			0
$499$	1100.96			0.0987686			0.0493843	−	0.998780i	$-0.484274\pi$
$499$	1100.96			0.0987686			0.0493843	+	0.998780i	$0.484274\pi$
$500$	3878.11			0.346869
$501$		0			0
$502$			9787.12i			0.870160i
$503$	11379.2			1.00869			0.504346	−	0.863502i	$-0.331734\pi$
$503$	11379.2			1.00869			0.504346	+	0.863502i	$0.331734\pi$
$504$		0			0
$505$	2853.92			0.251481
$506$			6128.23i			0.538405i
$507$		0			0
$508$	15358.7			1.34140
$509$	−16353.2			−1.42405			−0.712026	−	0.702153i	$-0.752222\pi$
$509$	−16353.2			−1.42405			−0.712026	+	0.702153i	$0.752222\pi$
$510$		0			0
$511$	−1703.64	+	9639.81i	−0.147485	+	0.834521i
$512$		−	21743.5i		−	1.87683i
$513$		0			0
$514$			13966.1i			1.19848i
$515$		−	12885.2i		−	1.10251i
$516$		0			0
$517$		−	4555.06i		−	0.387488i
$518$	1101.58	−	6233.14i	0.0934377	−	0.528703i
$519$		0			0
$520$	17630.0			1.48678
$521$	−7065.14			−0.594106			−0.297053	−	0.954861i	$-0.596004\pi$
$521$	−7065.14			−0.594106			−0.297053	+	0.954861i	$0.596004\pi$
$522$		0			0
$523$			14893.4i			1.24520i	0.782539	+	0.622602i	$0.213924\pi$
$523$			14893.4i			1.24520i	−0.782539	+	0.622602i	$0.786076\pi$
$524$	4506.45			0.375697
$525$		0			0
$526$	31540.0			2.61446
$527$		−	9187.19i		−	0.759393i
$528$		0			0
$529$	11316.9			0.930130
$530$	15659.0			1.28337
$531$		0			0
$532$	5960.34	−	33725.7i	0.485740	−	2.74849i
$533$			13582.1i			1.10376i
$534$		0			0
$535$		−	9871.73i		−	0.797743i
$536$			18177.1i			1.46480i
$537$		0			0
$538$		−	37387.2i		−	2.99606i
$539$	5009.99	−	13731.4i	0.400363	−	1.09732i
$540$		0			0
$541$	−3448.82			−0.274079			−0.137039	−	0.990566i	$-0.543759\pi$
$541$	−3448.82			−0.274079			−0.137039	+	0.990566i	$0.543759\pi$
$542$	278.402			0.0220635
$543$		0			0
$544$			2154.62i			0.169813i
$545$	12319.6			0.968280
$546$		0			0
$547$	14928.2			1.16688			0.583442	−	0.812155i	$-0.301706\pi$
$547$	14928.2			1.16688			0.583442	+	0.812155i	$0.301706\pi$
$548$			29898.4i			2.33065i
$549$		0			0
$550$	23012.4			1.78410
$551$	2166.38			0.167497
$552$		0			0
$553$	755.532	+	133.525i	0.0580986	+	0.0102678i
$554$		−	20799.5i		−	1.59510i
$555$		0			0
$556$		−	24721.5i		−	1.88565i
$557$		−	11054.3i		−	0.840909i	−0.907314	−	0.420455i	$-0.861871\pi$
$557$		−	11054.3i		−	0.840909i	0.907314	−	0.420455i	$-0.138129\pi$
$558$		0			0
$559$			10909.7i			0.825456i
$560$	−20089.9	−	3550.48i	−1.51599	−	0.267920i
$561$		0			0
$562$	12899.7			0.968222
$563$	−25624.7			−1.91821			−0.959107	−	0.283045i	$-0.908655\pi$
$563$	−25624.7			−1.91821			−0.959107	+	0.283045i	$0.908655\pi$
$564$		0			0
$565$		−	9216.21i		−	0.686245i
$566$	2196.37			0.163110
$567$		0			0
$568$	26285.9			1.94178
$569$		−	993.428i		−	0.0731928i	−0.999330	−	0.0365964i	$-0.988348\pi$
$569$		−	993.428i		−	0.0731928i	0.999330	−	0.0365964i	$-0.0116516\pi$
$570$		0			0
$571$	−16575.7			−1.21484			−0.607418	−	0.794382i	$-0.707795\pi$
$571$	−16575.7			−1.21484			−0.607418	+	0.794382i	$0.707795\pi$
$572$	−19505.2			−1.42579
$573$		0			0
$574$	7701.56	−	43578.1i	0.560029	−	3.16884i
$575$			3192.30i			0.231527i
$576$		0			0
$577$			10749.5i			0.775579i	0.921748	+	0.387789i	$0.126761\pi$
$577$			10749.5i			0.775579i	−0.921748	+	0.387789i	$0.873239\pi$
$578$			8945.14i			0.643718i
$579$		0			0
$580$		−	4781.63i		−	0.342321i
$581$	20227.3	+	3574.77i	1.44435	+	0.255260i
$582$		0			0
$583$	−8835.40			−0.627659
$584$	−21706.4			−1.53804
$585$		0			0
$586$		−	1752.93i		−	0.123571i
$587$	−26688.1			−1.87655			−0.938277	−	0.345884i	$-0.887579\pi$
$587$	−26688.1			−1.87655			−0.938277	+	0.345884i	$0.887579\pi$
$588$		0			0
$589$	−12687.9			−0.887600
$590$			14775.7i			1.03103i
$591$		0			0
$592$	4984.81			0.346072
$593$	−5486.54			−0.379941			−0.189971	−	0.981790i	$-0.560839\pi$
$593$	−5486.54			−0.379941			−0.189971	+	0.981790i	$0.560839\pi$
$594$		0			0
$595$	−22904.9	−	4047.97i	−1.57816	−	0.278909i
$596$		−	1195.83i		−	0.0821863i
$597$		0			0
$598$		−	4031.63i		−	0.275695i
$599$			17346.7i			1.18325i	0.806212	+	0.591626i	$0.201514\pi$
$599$			17346.7i			1.18325i	−0.806212	+	0.591626i	$0.798486\pi$
$600$		0			0
$601$			10380.2i			0.704519i	0.935902	+	0.352259i	$0.114586\pi$
$601$			10380.2i			0.704519i	−0.935902	+	0.352259i	$0.885414\pi$
$602$	6186.19	−	35003.6i	0.418821	−	2.36984i
$603$		0			0
$604$	1592.33			0.107270
$605$	−7427.02			−0.499093
$606$		0			0
$607$			23528.1i			1.57327i	0.617419	+	0.786635i	$0.288178\pi$
$607$			23528.1i			1.57327i	−0.617419	+	0.786635i	$0.711822\pi$
$608$	2975.62			0.198482
$609$		0			0
$610$	−38915.7			−2.58303
$611$			2996.68i			0.198417i
$612$		0			0
$613$	10216.7			0.673164			0.336582	−	0.941654i	$-0.390729\pi$
$613$	10216.7			0.673164			0.336582	+	0.941654i	$0.390729\pi$
$614$	30738.0			2.02033
$615$		0			0
$616$	31916.5	+	5640.60i	2.08758	+	0.368938i
$617$		−	1833.79i		−	0.119653i	−0.998209	−	0.0598264i	$-0.980945\pi$
$617$		−	1833.79i		−	0.119653i	0.998209	−	0.0598264i	$-0.0190547\pi$
$618$		0			0
$619$			3792.08i			0.246230i	0.992392	+	0.123115i	$0.0392884\pi$
$619$			3792.08i			0.246230i	−0.992392	+	0.123115i	$0.960712\pi$
$620$			28004.7i			1.81402i
$621$		0			0
$622$		−	15301.1i		−	0.986362i
$623$	−2208.74	−	390.350i	−0.142040	−	0.0251028i
$624$		0			0
$625$	−17323.4			−1.10870
$626$	−6040.83			−0.385687
$627$		0			0
$628$			32958.3i			2.09423i
$629$	5683.28			0.360266
$630$		0			0
$631$	−484.095			−0.0305413			−0.0152706	−	0.999883i	$-0.504861\pi$
$631$	−484.095			−0.0305413			−0.0152706	+	0.999883i	$0.504861\pi$
$632$			1701.26i			0.107077i
$633$		0			0
$634$	37376.7			2.34136
$635$	−14405.5			−0.900258
$636$		0			0
$637$	−3295.96	+	9033.61i	−0.205009	+	0.561891i
$638$			4020.00i			0.249457i
$639$		0			0
$640$			36896.9i			2.27887i
$641$			17910.8i			1.10364i	0.833963	+	0.551821i	$0.186067\pi$
$641$			17910.8i			1.10364i	−0.833963	+	0.551821i	$0.813933\pi$
$642$		0			0
$643$			17669.0i			1.08366i	0.840487	+	0.541832i	$0.182269\pi$
$643$			17669.0i			1.08366i	−0.840487	+	0.541832i	$0.817731\pi$
$644$	−1534.27	+	8681.46i	−0.0938802	+	0.531207i
$645$		0			0
$646$	45818.7			2.79058
$647$	11975.6			0.727680			0.363840	−	0.931461i	$-0.381465\pi$
$647$	11975.6			0.727680			0.363840	+	0.931461i	$0.381465\pi$
$648$		0			0
$649$		−	8337.01i		−	0.504247i
$650$	−15139.4			−0.913562
$651$		0			0
$652$	−57615.4			−3.46073
$653$			15949.2i			0.955806i	0.878413	+	0.477903i	$0.158603\pi$
$653$			15949.2i			0.955806i	−0.878413	+	0.477903i	$0.841397\pi$
$654$		0			0
$655$	−4226.76			−0.252142
$656$	34850.7			2.07422
$657$		0			0
$658$	1699.23	−	9614.82i	0.100673	−	0.569642i
$659$		−	2560.92i		−	0.151380i	−0.997131	−	0.0756899i	$-0.975884\pi$
$659$		−	2560.92i		−	0.151380i	0.997131	−	0.0756899i	$-0.0241159\pi$
$660$		0			0
$661$		−	18689.3i		−	1.09975i	−0.835249	−	0.549873i	$-0.814676\pi$
$661$		−	18689.3i		−	1.09975i	0.835249	−	0.549873i	$-0.185324\pi$
$662$		−	26125.7i		−	1.53384i
$663$		0			0
$664$			45546.6i			2.66197i
$665$	−5590.43	+	31632.6i	−0.325996	+	1.84460i
$666$		0			0
$667$	−557.657			−0.0323727
$668$	40014.6			2.31768
$669$		0			0
$670$		−	33430.0i		−	1.92763i
$671$	21957.7			1.26329
$672$		0			0
$673$	−21756.4			−1.24614			−0.623068	−	0.782168i	$-0.714114\pi$
$673$	−21756.4			−1.24614			−0.623068	+	0.782168i	$0.714114\pi$
$674$		−	1330.88i		−	0.0760586i
$675$		0			0
$676$	−23036.7			−1.31069
$677$	−19524.1			−1.10838			−0.554188	−	0.832391i	$-0.686971\pi$
$677$	−19524.1			−1.10838			−0.554188	+	0.832391i	$0.686971\pi$
$678$		0			0
$679$	−2786.81	+	15768.8i	−0.157508	+	0.891236i
$680$		−	51575.8i		−	2.90859i
$681$		0			0
$682$		−	23544.0i		−	1.32192i
$683$		−	7420.21i		−	0.415705i	−0.978160	−	0.207852i	$-0.933353\pi$
$683$		−	7420.21i		−	0.415705i	0.978160	−	0.207852i	$-0.0666473\pi$
$684$		0			0
$685$		−	28042.8i		−	1.56417i
$686$	15697.5	−	27115.3i	0.873661	−	1.50914i
$687$		0			0
$688$	27993.4			1.55122
$689$	5812.62			0.321398
$690$		0			0
$691$			20343.5i			1.11998i	0.828501	+	0.559988i	$0.189194\pi$
$691$			20343.5i			1.11998i	−0.828501	+	0.559988i	$0.810806\pi$
$692$	−15015.4			−0.824853
$693$		0			0
$694$	−10328.2			−0.564918
$695$			23187.2i			1.26552i
$696$		0			0
$697$	39733.9			2.15930
$698$	59991.7			3.25318
$699$		0			0
$700$	32600.2	+	5761.43i	1.76025	+	0.311088i
$701$			11939.2i			0.643275i	0.946863	+	0.321638i	$0.104233\pi$
$701$			11939.2i			0.643275i	−0.946863	+	0.321638i	$0.895767\pi$
$702$		0			0
$703$		−	7848.86i		−	0.421089i
$704$		−	19002.7i		−	1.01732i
$705$		0			0
$706$			29644.1i			1.58027i
$707$	−3398.99	−	600.703i	−0.180809	−	0.0319544i
$708$		0			0
$709$	18172.6			0.962602			0.481301	−	0.876555i	$-0.340164\pi$
$709$	18172.6			0.962602			0.481301	+	0.876555i	$0.340164\pi$
$710$	−48343.0			−2.55532
$711$		0			0
$712$		−	4973.50i		−	0.261784i
$713$	3266.04			0.171549
$714$		0			0
$715$	18294.7			0.956898
$716$			66064.2i			3.44824i
$717$		0			0
$718$	9199.97			0.478189
$719$	−15645.6			−0.811520			−0.405760	−	0.913980i	$-0.632993\pi$
$719$	−15645.6			−0.811520			−0.405760	+	0.913980i	$0.632993\pi$
$720$		0			0
$721$	−2712.13	+	15346.2i	−0.140090	+	0.792679i
$722$		−	29447.9i		−	1.51792i
$723$		0			0
$724$		−	72863.8i		−	3.74028i
$725$			2094.09i			0.107272i
$726$		0			0
$727$			35180.5i			1.79473i	0.441286	+	0.897367i	$0.354522\pi$
$727$			35180.5i			1.79473i	−0.441286	+	0.897367i	$0.645478\pi$
$728$	−20997.1	−	3710.82i	−1.06896	−	0.188918i
$729$		0			0
$730$	39920.7			2.02401
$731$	31915.8			1.61484
$732$		0			0
$733$		−	22559.5i		−	1.13677i	−0.822762	−	0.568387i	$-0.807568\pi$
$733$		−	22559.5i		−	1.13677i	0.822762	−	0.568387i	$-0.192432\pi$
$734$	12545.4			0.630871
$735$		0			0
$736$	−765.965			−0.0383612
$737$			18862.4i			0.942751i
$738$		0			0
$739$	37974.5			1.89028			0.945138	−	0.326671i	$-0.105927\pi$
$739$	37974.5			1.89028			0.945138	+	0.326671i	$0.105927\pi$
$740$	−17323.9			−0.860596
$741$		0			0
$742$	−18649.7	−	3295.97i	−0.922714	−	0.163071i
$743$		−	5405.57i		−	0.266906i	−0.991055	−	0.133453i	$-0.957393\pi$
$743$		−	5405.57i		−	0.266906i	0.991055	−	0.133453i	$-0.0426065\pi$
$744$		0			0
$745$			1121.61i			0.0551580i
$746$		−	36209.6i		−	1.77712i
$747$		0			0
$748$			57061.7i			2.78928i
$749$	−2077.84	+	11757.1i	−0.101365	+	0.573560i
$750$		0			0
$751$	5281.64			0.256631			0.128315	−	0.991733i	$-0.459043\pi$
$751$	5281.64			0.256631			0.128315	+	0.991733i	$0.459043\pi$
$752$	7689.24			0.372869
$753$		0			0
$754$		−	2644.67i		−	0.127736i
$755$	−1493.51			−0.0719924
$756$		0			0
$757$	−17876.6			−0.858302			−0.429151	−	0.903233i	$-0.641187\pi$
$757$	−17876.6			−0.858302			−0.429151	+	0.903233i	$0.641187\pi$
$758$			48455.4i			2.32187i
$759$		0			0
$760$	−71228.4			−3.39964
$761$	−31430.6			−1.49719			−0.748594	−	0.663029i	$-0.769271\pi$
$761$	−31430.6			−1.49719			−0.748594	+	0.663029i	$0.769271\pi$
$762$		0			0
$763$	−14672.5	−	2593.07i	−0.696173	−	0.123035i
$764$		−	62481.5i		−	2.95877i
$765$		0			0
$766$		−	8990.99i		−	0.424096i
$767$			5484.74i			0.258204i
$768$		0			0
$769$		−	16043.9i		−	0.752351i	−0.926548	−	0.376176i	$-0.877239\pi$
$769$		−	16043.9i		−	0.752351i	0.926548	−	0.376176i	$-0.122761\pi$
$770$	−58698.3	−	10373.8i	−2.74720	−	0.485512i
$771$		0			0
$772$	−71958.3			−3.35471
$773$	−9079.52			−0.422468			−0.211234	−	0.977436i	$-0.567748\pi$
$773$	−9079.52			−0.422468			−0.211234	+	0.977436i	$0.567748\pi$
$774$		0			0
$775$		−	12264.5i		−	0.568456i
$776$	−35507.1			−1.64257
$777$		0			0
$778$	−62087.6			−2.86111
$779$		−	54874.3i		−	2.52384i
$780$		0			0
$781$	27276.9			1.24974
$782$	−11794.4			−0.539342
$783$		0			0
$784$	23179.5	+	8457.18i	1.05592	+	0.385258i
$785$		−	30912.8i		−	1.40551i
$786$		0			0
$787$			31731.8i			1.43725i	0.695397	+	0.718625i	$0.255228\pi$
$787$			31731.8i			1.43725i	−0.695397	+	0.718625i	$0.744772\pi$
$788$		−	24731.4i		−	1.11805i
$789$		0			0
$790$		−	3128.83i		−	0.140910i
$791$	−1939.86	+	10976.4i	−0.0871979	+	0.493396i
$792$		0			0
$793$	−14445.5			−0.646878
$794$	−28652.8			−1.28067
$795$		0			0
$796$			42807.6i			1.90612i
$797$	25315.0			1.12510			0.562550	−	0.826763i	$-0.309820\pi$
$797$	25315.0			1.12510			0.562550	+	0.826763i	$0.309820\pi$
$798$		0			0
$799$	8766.65			0.388162
$800$			2876.32i			0.127116i
$801$		0			0
$802$	−25901.1			−1.14040
$803$	−22524.7			−0.989889
$804$		0			0
$805$	1439.05	−	8142.67i	0.0630061	−	0.356511i
$806$			15489.1i			0.676899i
$807$		0			0
$808$		−	7653.64i		−	0.333235i
$809$		−	5229.08i		−	0.227249i	−0.993524	−	0.113625i	$-0.963754\pi$
$809$		−	5229.08i		−	0.227249i	0.993524	−	0.113625i	$-0.0362461\pi$
$810$		0			0
$811$		−	13240.3i		−	0.573281i	−0.958038	−	0.286641i	$-0.907461\pi$
$811$		−	13240.3i		−	0.573281i	0.958038	−	0.286641i	$-0.0925386\pi$
$812$	−1006.45	+	5694.87i	−0.0434971	+	0.246122i
$813$		0			0
$814$	14564.6			0.627135
$815$	54039.7			2.32261
$816$		0			0
$817$		−	44077.1i		−	1.88747i
$818$	17622.7			0.753255
$819$		0			0
$820$	−121118.			−5.15808
$821$			14336.5i			0.609437i	0.952442	+	0.304718i	$0.0985624\pi$
$821$			14336.5i			0.609437i	−0.952442	+	0.304718i	$0.901438\pi$
$822$		0			0
$823$	−7027.74			−0.297657			−0.148828	−	0.988863i	$-0.547550\pi$
$823$	−7027.74			−0.297657			−0.148828	+	0.988863i	$0.547550\pi$
$824$	−34555.6			−1.46092
$825$		0			0
$826$	3110.05	−	17597.7i	0.131008	−	0.741288i
$827$		−	25184.9i		−	1.05896i	−0.848321	−	0.529482i	$-0.822386\pi$
$827$		−	25184.9i		−	1.05896i	0.848321	−	0.529482i	$-0.177614\pi$
$828$		0			0
$829$		−	31846.9i		−	1.33425i	−0.744948	−	0.667123i	$-0.767526\pi$
$829$		−	31846.9i		−	1.33425i	0.744948	−	0.667123i	$-0.232474\pi$
$830$		−	83765.8i		−	3.50308i
$831$		0			0
$832$			12501.5i			0.520927i
$833$	26427.4	+	9642.20i	1.09923	+	0.401059i
$834$		0			0
$835$	−37531.1			−1.55547
$836$	78804.7			3.26019
$837$		0			0
$838$			14147.1i			0.583180i
$839$	11560.1			0.475683			0.237841	−	0.971304i	$-0.423560\pi$
$839$	11560.1			0.475683			0.237841	+	0.971304i	$0.423560\pi$
$840$		0			0
$841$	24023.2			0.985001
$842$		−	39343.5i		−	1.61029i
$843$		0			0
$844$	−51601.5			−2.10450
$845$	21607.0			0.879649
$846$		0			0
$847$	8845.51	+	1563.27i	0.358837	+	0.0634173i
$848$		−	14914.7i		−	0.603979i
$849$		0			0
$850$			44289.6i			1.78720i
$851$			2020.40i			0.0813849i
$852$		0			0
$853$			6138.65i			0.246405i	0.992382	+	0.123202i	$0.0393164\pi$
$853$			6138.65i			0.246405i	−0.992382	+	0.123202i	$0.960684\pi$
$854$	46348.3	+	8191.12i	1.85715	+	0.328214i
$855$		0			0
$856$	−26474.0			−1.05708
$857$	−7977.46			−0.317975			−0.158988	−	0.987281i	$-0.550823\pi$
$857$	−7977.46			−0.317975			−0.158988	+	0.987281i	$0.550823\pi$
$858$		0			0
$859$			34507.1i			1.37063i	0.728249	+	0.685313i	$0.240334\pi$
$859$			34507.1i			1.37063i	−0.728249	+	0.685313i	$0.759666\pi$
$860$	−97286.7			−3.85750
$861$		0			0
$862$	−56535.8			−2.23389
$863$			14772.1i			0.582674i	0.956621	+	0.291337i	$0.0941000\pi$
$863$			14772.1i			0.582674i	−0.956621	+	0.291337i	$0.905900\pi$
$864$		0			0
$865$	14083.5			0.553586
$866$	−58431.2			−2.29281
$867$		0			0
$868$	5894.52	−	33353.3i	0.230499	−	1.30424i
$869$			1765.40i			0.0689151i
$870$		0			0
$871$		−	12409.2i		−	0.482743i
$872$		−	33038.6i		−	1.28306i
$873$		0			0
$874$			16288.5i			0.630398i
$875$	4332.14	+	765.619i	0.167375	+	0.0295802i
$876$		0			0
$877$	−46472.1			−1.78934			−0.894671	−	0.446726i	$-0.852590\pi$
$877$	−46472.1			−1.78934			−0.894671	+	0.446726i	$0.852590\pi$
$878$	21919.0			0.842516
$879$		0			0
$880$		−	46942.7i		−	1.79823i
$881$	−14050.3			−0.537304			−0.268652	−	0.963237i	$-0.586578\pi$
$881$	−14050.3			−0.537304			−0.268652	+	0.963237i	$0.586578\pi$
$882$		0			0
$883$	−24097.5			−0.918400			−0.459200	−	0.888333i	$-0.651864\pi$
$883$	−24097.5			−0.918400			−0.459200	+	0.888333i	$0.651864\pi$
$884$		−	37539.6i		−	1.42827i
$885$		0			0
$886$	63895.0			2.42279
$887$	−4298.21			−0.162705			−0.0813527	−	0.996685i	$-0.525924\pi$
$887$	−4298.21			−0.162705			−0.0813527	+	0.996685i	$0.525924\pi$
$888$		0			0
$889$	17156.8	+	3032.12i	0.647267	+	0.114391i
$890$			9146.89i			0.344499i
$891$		0			0
$892$		−	15567.6i		−	0.584350i
$893$		−	12107.1i		−	0.453695i
$894$		0			0
$895$		−	61964.1i		−	2.31423i
$896$	7766.19	−	43943.9i	0.289565	−	1.63846i
$897$		0			0
$898$	22625.1			0.840766
$899$	2142.46			0.0794828
$900$		0			0
$901$		−	17004.6i		−	0.628751i
$902$	101826.			3.75880
$903$		0			0
$904$	−24716.0			−0.909340
$905$			68341.6i			2.51022i
$906$		0			0
$907$	33876.5			1.24019			0.620095	−	0.784527i	$-0.287094\pi$
$907$	33876.5			1.24019			0.620095	+	0.784527i	$0.287094\pi$
$908$	−69866.0			−2.55351
$909$		0			0
$910$	38616.3	+	6824.66i	1.40672	+	0.248610i
$911$		−	4592.67i		−	0.167027i	−0.996507	−	0.0835137i	$-0.973386\pi$
$911$		−	4592.67i		−	0.167027i	0.996507	−	0.0835137i	$-0.0266142\pi$
$912$		0			0
$913$			47263.8i			1.71326i
$914$		−	79897.0i		−	2.89142i
$915$		0			0
$916$		−	65282.3i		−	2.35479i
$917$	5034.04	+	889.665i	0.181285	+	0.0320385i
$918$		0			0
$919$	28868.8			1.03623			0.518114	−	0.855312i	$-0.326634\pi$
$919$	28868.8			1.03623			0.518114	+	0.855312i	$0.326634\pi$
$920$	18335.2			0.657057
$921$		0			0
$922$			18077.2i			0.645708i
$923$	−17944.9			−0.639938
$924$		0			0
$925$	7586.93			0.269683
$926$		−	18878.6i		−	0.669968i
$927$		0			0
$928$	−502.458			−0.0177737
$929$	24122.9			0.851935			0.425968	−	0.904738i	$-0.359934\pi$
$929$	24122.9			0.851935			0.425968	+	0.904738i	$0.359934\pi$
$930$		0			0
$931$	13316.3	−	36497.5i	0.468769	−	1.28481i
$932$			13779.3i			0.484287i
$933$		0			0
$934$			83638.8i			2.93013i
$935$		−	53520.3i		−	1.87198i
$936$		0			0
$937$			26745.1i			0.932470i	0.884661	+	0.466235i	$0.154390\pi$
$937$			26745.1i			0.932470i	−0.884661	+	0.466235i	$0.845610\pi$
$938$	−7036.47	+	39814.8i	−0.244935	+	1.38593i
$939$		0			0
$940$	−26722.8			−0.927234
$941$	−40194.7			−1.39247			−0.696233	−	0.717816i	$-0.745142\pi$
$941$	−40194.7			−1.39247			−0.696233	+	0.717816i	$0.745142\pi$
$942$		0			0
$943$			14125.4i			0.487790i
$944$	14073.4			0.485223
$945$		0			0
$946$	81790.7			2.81104
$947$		−	4995.99i		−	0.171434i	−0.996320	−	0.0857169i	$-0.972682\pi$
$947$		−	4995.99i		−	0.171434i	0.996320	−	0.0857169i	$-0.0273181\pi$
$948$		0			0
$949$	14818.5			0.506881
$950$	61165.9			2.08893
$951$		0			0
$952$	−10855.9	+	61426.3i	−0.369580	+	2.09122i
$953$			31444.5i			1.06882i	0.845224	+	0.534412i	$0.179467\pi$
$953$			31444.5i			1.06882i	−0.845224	+	0.534412i	$0.820533\pi$
$954$		0			0
$955$			58603.7i			1.98573i
$956$			31297.4i			1.05882i
$957$		0			0
$958$		−	19716.2i		−	0.664928i
$959$	−5902.55	+	33398.7i	−0.198752	+	1.12461i
$960$		0			0
$961$	17243.2			0.578806
$962$	−9581.71			−0.321130
$963$		0			0
$964$		−	17693.5i		−	0.591151i
$965$	67492.4			2.25146
$966$		0			0
$967$	−36695.5			−1.22032			−0.610158	−	0.792279i	$-0.708894\pi$
$967$	−36695.5			−1.22032			−0.610158	+	0.792279i	$0.708894\pi$
$968$			19917.8i			0.661345i
$969$		0			0
$970$	65302.0			2.16157
$971$	−22268.7			−0.735978			−0.367989	−	0.929830i	$-0.619954\pi$
$971$	−22268.7			−0.735978			−0.367989	+	0.929830i	$0.619954\pi$
$972$		0			0
$973$	4880.52	−	27615.7i	0.160804	−	0.909886i
$974$		−	34368.6i		−	1.13064i
$975$		0			0
$976$			37066.0i			1.21563i
$977$			24557.6i			0.804163i	0.915604	+	0.402081i	$0.131713\pi$
$977$			24557.6i			0.804163i	−0.915604	+	0.402081i	$0.868287\pi$
$978$		0			0
$979$		−	5161.02i		−	0.168485i
$980$	−80556.9	−	29391.6i	−2.62581	−	0.958042i
$981$		0			0
$982$	−73612.5			−2.39213
$983$	−39309.1			−1.27545			−0.637724	−	0.770265i	$-0.720124\pi$
$983$	−39309.1			−1.27545			−0.637724	+	0.770265i	$0.720124\pi$
$984$		0			0
$985$			23196.5i			0.750357i
$986$	−7736.88			−0.249891
$987$		0			0
$988$	−51843.9			−1.66941
$989$			11346.1i			0.364796i
$990$		0			0
$991$	42829.0			1.37286			0.686431	−	0.727195i	$-0.259176\pi$
$991$	42829.0			1.37286			0.686431	+	0.727195i	$0.259176\pi$
$992$	2942.76			0.0941862
$993$		0			0
$994$	57576.0	+	10175.4i	1.83722	+	0.324692i
$995$		−	40150.8i		−	1.27926i
$996$		0			0
$997$		−	35623.4i		−	1.13160i	−0.824543	−	0.565799i	$-0.808568\pi$
$997$		−	35623.4i		−	1.13160i	0.824543	−	0.565799i	$-0.191432\pi$
$998$			5430.09i			0.172231i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.c.c.566.6		44
3.2	odd	2	inner	567.4.c.c.566.39		44
7.6	odd	2	inner	567.4.c.c.566.40		44
9.2	odd	6		189.4.o.a.125.2		44
9.4	even	3		189.4.o.a.62.1		44
9.5	odd	6		63.4.o.a.20.21	✓	44
9.7	even	3		63.4.o.a.41.22	yes	44
21.20	even	2	inner	567.4.c.c.566.5		44
63.13	odd	6		189.4.o.a.62.2		44
63.20	even	6		189.4.o.a.125.1		44
63.34	odd	6		63.4.o.a.41.21	yes	44
63.41	even	6		63.4.o.a.20.22	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.21	✓	44	9.5	odd	6
63.4.o.a.20.22	yes	44	63.41	even	6
63.4.o.a.41.21	yes	44	63.34	odd	6
63.4.o.a.41.22	yes	44	9.7	even	3
189.4.o.a.62.1		44	9.4	even	3
189.4.o.a.62.2		44	63.13	odd	6
189.4.o.a.125.1		44	63.20	even	6
189.4.o.a.125.2		44	9.2	odd	6
567.4.c.c.566.5		44	21.20	even	2	inner
567.4.c.c.566.6		44	1.1	even	1	trivial
567.4.c.c.566.39		44	3.2	odd	2	inner
567.4.c.c.566.40		44	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.93216i q^{2} -16.3262 q^{4} +15.3130 q^{5} +(-18.2376 - 3.22314i) q^{7} -41.0664i q^{8} +O(q^{10})\)	\(q+4.93216i q^{2} -16.3262 q^{4} +15.3130 q^{5} +(-18.2376 - 3.22314i) q^{7} -41.0664i q^{8} +75.5262i q^{10} -42.6147i q^{11} +28.0353i q^{13} +(15.8970 - 89.9510i) q^{14} +71.9364 q^{16} +82.0161 q^{17} -113.268i q^{19} -250.004 q^{20} +210.183 q^{22} +29.1567i q^{23} +109.488 q^{25} -138.275 q^{26} +(297.752 + 52.6217i) q^{28} +19.1262i q^{29} -112.017i q^{31} +26.2707i q^{32} +404.517i q^{34} +(-279.273 - 49.3559i) q^{35} +69.2948 q^{37} +558.655 q^{38} -628.850i q^{40} +484.465 q^{41} +389.141 q^{43} +695.738i q^{44} -143.806 q^{46} +106.889 q^{47} +(322.223 + 117.565i) q^{49} +540.012i q^{50} -457.711i q^{52} -207.332i q^{53} -652.559i q^{55} +(-132.363 + 748.954i) q^{56} -94.3337 q^{58} +195.637 q^{59} +515.261i q^{61} +552.486 q^{62} +445.920 q^{64} +429.304i q^{65} -442.628 q^{67} -1339.01 q^{68} +(243.431 - 1377.42i) q^{70} +640.082i q^{71} -528.567i q^{73} +341.773i q^{74} +1849.24i q^{76} +(-137.353 + 777.192i) q^{77} -41.4271 q^{79} +1101.56 q^{80} +2389.46i q^{82} -1109.10 q^{83} +1255.91 q^{85} +1919.31i q^{86} -1750.03 q^{88} +121.109 q^{89} +(90.3615 - 511.297i) q^{91} -476.019i q^{92} +527.196i q^{94} -1734.47i q^{95} -864.627i q^{97} +(-579.849 + 1589.26i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) 44 * q - 156 * q^4 - 10 * q^7	\( 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) \) 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

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