LMFDB - Embedded newform 567.4.c.c.566.4

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			566.4
Character	$\chi$	$=$	567.566
Dual form			567.4.c.c.566.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0979055i q^{2} +7.99041 q^{4} -18.1269 q^{5} +(5.25594 - 17.7588i) q^{7} +1.56555i q^{8} +O(q^{10})$	\(q+0.0979055i q^{2} +7.99041 q^{4} -18.1269 q^{5} +(5.25594 - 17.7588i) q^{7} +1.56555i q^{8} -1.77473i q^{10} -37.0124i q^{11} -18.8993i q^{13} +(1.73869 + 0.514586i) q^{14} +63.7700 q^{16} -62.5187 q^{17} +70.2680i q^{19} -144.842 q^{20} +3.62372 q^{22} +162.458i q^{23} +203.586 q^{25} +1.85034 q^{26} +(41.9972 - 141.900i) q^{28} -95.2645i q^{29} +127.722i q^{31} +18.7678i q^{32} -6.12092i q^{34} +(-95.2742 + 321.913i) q^{35} -378.832 q^{37} -6.87963 q^{38} -28.3786i q^{40} -198.387 q^{41} -321.384 q^{43} -295.744i q^{44} -15.9055 q^{46} -158.777 q^{47} +(-287.750 - 186.678i) q^{49} +19.9322i q^{50} -151.013i q^{52} +191.780i q^{53} +670.922i q^{55} +(27.8023 + 8.22844i) q^{56} +9.32693 q^{58} +213.835 q^{59} +220.466i q^{61} -12.5047 q^{62} +508.323 q^{64} +342.586i q^{65} -136.589 q^{67} -499.550 q^{68} +(-31.5170 - 9.32787i) q^{70} -458.924i q^{71} -967.869i q^{73} -37.0898i q^{74} +561.470i q^{76} +(-657.296 - 194.535i) q^{77} -596.637 q^{79} -1155.96 q^{80} -19.4232i q^{82} -361.430 q^{83} +1133.27 q^{85} -31.4653i q^{86} +57.9448 q^{88} +35.4987 q^{89} +(-335.629 - 99.3336i) q^{91} +1298.10i q^{92} -15.5452i q^{94} -1273.74i q^{95} +1329.79i q^{97} +(18.2769 - 28.1723i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			0.0979055i			0.0346148i	0.999850	+	0.0173074i	$0.00550940\pi$
$2$			0.0979055i			0.0346148i	−0.999850	+	0.0173074i	$0.994491\pi$
$3$		0			0
$3$		0			0
$4$	7.99041			0.998802
$5$	−18.1269			−1.62132			−0.810661	−	0.585515i	$-0.800892\pi$
$5$	−18.1269			−1.62132			−0.810661	+	0.585515i	$0.800892\pi$
$6$		0			0
$7$	5.25594	−	17.7588i	0.283794	−	0.958885i
$7$	5.25594	−	17.7588i	0.283794	−	0.958885i
$8$			1.56555i			0.0691882i
$9$		0			0
$10$		−	1.77473i		−	0.0561218i
$11$		−	37.0124i		−	1.01451i	−0.861795	−	0.507257i	$-0.830659\pi$
$11$		−	37.0124i		−	1.01451i	0.861795	−	0.507257i	$-0.169341\pi$
$12$		0			0
$13$		−	18.8993i		−	0.403209i	−0.979467	−	0.201605i	$-0.935384\pi$
$13$		−	18.8993i		−	0.403209i	0.979467	−	0.201605i	$-0.0646156\pi$
$14$	1.73869	+	0.514586i	0.0331917	+	0.00982349i
$15$		0			0
$16$	63.7700			0.996407
$17$	−62.5187			−0.891941			−0.445971	−	0.895048i	$-0.647141\pi$
$17$	−62.5187			−0.891941			−0.445971	+	0.895048i	$0.647141\pi$
$18$		0			0
$19$			70.2680i			0.848452i	0.905556	+	0.424226i	$0.139454\pi$
$19$			70.2680i			0.848452i	−0.905556	+	0.424226i	$0.860546\pi$
$20$	−144.842			−1.61938
$21$		0			0
$22$	3.62372			0.0351173
$23$			162.458i			1.47282i	0.676538	+	0.736408i	$0.263479\pi$
$23$			162.458i			1.47282i	−0.676538	+	0.736408i	$0.736521\pi$
$24$		0			0
$25$	203.586			1.62869
$26$	1.85034			0.0139570
$27$		0			0
$28$	41.9972	−	141.900i	0.283454	−	0.957736i
$29$		−	95.2645i		−	0.610006i	−0.952351	−	0.305003i	$-0.901342\pi$
$29$		−	95.2645i		−	0.610006i	0.952351	−	0.305003i	$-0.0986575\pi$
$30$		0			0
$31$			127.722i			0.739987i	0.929034	+	0.369994i	$0.120640\pi$
$31$			127.722i			0.739987i	−0.929034	+	0.369994i	$0.879360\pi$
$32$			18.7678i			0.103679i
$33$		0			0
$34$		−	6.12092i		−	0.0308744i
$35$	−95.2742	+	321.913i	−0.460122	+	1.55466i
$36$		0			0
$37$	−378.832			−1.68323			−0.841617	−	0.540075i	$-0.818396\pi$
$37$	−378.832			−1.68323			−0.841617	+	0.540075i	$0.818396\pi$
$38$	−6.87963			−0.0293690
$39$		0			0
$40$		−	28.3786i		−	0.112176i
$41$	−198.387			−0.755681			−0.377840	−	0.925871i	$-0.623333\pi$
$41$	−198.387			−0.755681			−0.377840	+	0.925871i	$0.623333\pi$
$42$		0			0
$43$	−321.384			−1.13978			−0.569892	−	0.821720i	$-0.693015\pi$
$43$	−321.384			−1.13978			−0.569892	+	0.821720i	$0.693015\pi$
$44$		−	295.744i		−	1.01330i
$45$		0			0
$46$	−15.9055			−0.0509813
$47$	−158.777			−0.492766			−0.246383	−	0.969173i	$-0.579242\pi$
$47$	−158.777			−0.492766			−0.246383	+	0.969173i	$0.579242\pi$
$48$		0			0
$49$	−287.750	−	186.678i	−0.838922	−	0.544252i
$50$			19.9322i			0.0563768i
$51$		0			0
$52$		−	151.013i		−	0.402726i
$53$			191.780i			0.497039i	0.968627	+	0.248519i	$0.0799440\pi$
$53$			191.780i			0.497039i	−0.968627	+	0.248519i	$0.920056\pi$
$54$		0			0
$55$			670.922i			1.64486i
$56$	27.8023	+	8.22844i	0.0663435	+	0.0196352i
$57$		0			0
$58$	9.32693			0.0211153
$59$	213.835			0.471848			0.235924	−	0.971772i	$-0.424188\pi$
$59$	213.835			0.471848			0.235924	+	0.971772i	$0.424188\pi$
$60$		0			0
$61$			220.466i			0.462751i	0.972865	+	0.231375i	$0.0743226\pi$
$61$			220.466i			0.462751i	−0.972865	+	0.231375i	$0.925677\pi$
$62$	−12.5047			−0.0256145
$63$		0			0
$64$	508.323			0.992818
$65$			342.586i			0.653732i
$66$		0			0
$67$	−136.589			−0.249060			−0.124530	−	0.992216i	$-0.539742\pi$
$67$	−136.589			−0.249060			−0.124530	+	0.992216i	$0.539742\pi$
$68$	−499.550			−0.890873
$69$		0			0
$70$	−31.5170	−	9.32787i	−0.0538144	−	0.0159270i
$71$		−	458.924i		−	0.767102i	−0.923520	−	0.383551i	$-0.874701\pi$
$71$		−	458.924i		−	0.767102i	0.923520	−	0.383551i	$-0.125299\pi$
$72$		0			0
$73$		−	967.869i		−	1.55179i	−0.630863	−	0.775894i	$-0.717299\pi$
$73$		−	967.869i		−	1.55179i	0.630863	−	0.775894i	$-0.282701\pi$
$74$		−	37.0898i		−	0.0582648i
$75$		0			0
$76$			561.470i			0.847435i
$77$	−657.296	−	194.535i	−0.972803	−	0.287913i
$78$		0			0
$79$	−596.637			−0.849707			−0.424853	−	0.905262i	$-0.639674\pi$
$79$	−596.637			−0.849707			−0.424853	+	0.905262i	$0.639674\pi$
$80$	−1155.96			−1.61550
$81$		0			0
$82$		−	19.4232i		−	0.0261578i
$83$	−361.430			−0.477976			−0.238988	−	0.971022i	$-0.576816\pi$
$83$	−361.430			−0.477976			−0.238988	+	0.971022i	$0.576816\pi$
$84$		0			0
$85$	1133.27			1.44613
$86$		−	31.4653i		−	0.0394534i
$87$		0			0
$88$	57.9448			0.0701924
$89$	35.4987			0.0422792			0.0211396	−	0.999777i	$-0.493271\pi$
$89$	35.4987			0.0422792			0.0211396	+	0.999777i	$0.493271\pi$
$90$		0			0
$91$	−335.629	−	99.3336i	−0.386631	−	0.114428i
$92$			1298.10i			1.47105i
$93$		0			0
$94$		−	15.5452i		−	0.0170570i
$95$		−	1273.74i		−	1.37561i
$96$		0			0
$97$			1329.79i			1.39195i	0.718064	+	0.695977i	$0.245028\pi$
$97$			1329.79i			1.39195i	−0.718064	+	0.695977i	$0.754972\pi$
$98$	18.2769	−	28.1723i	0.0188392	−	0.0290391i
$99$		0			0
$100$	1626.74			1.62674
$101$	−914.982			−0.901427			−0.450714	−	0.892669i	$-0.648830\pi$
$101$	−914.982			−0.901427			−0.450714	+	0.892669i	$0.648830\pi$
$102$		0			0
$103$		−	833.547i		−	0.797397i	−0.917082	−	0.398698i	$-0.869462\pi$
$103$		−	833.547i		−	0.797397i	0.917082	−	0.398698i	$-0.130538\pi$
$104$	29.5878			0.0278973
$105$		0			0
$106$	−18.7764			−0.0172049
$107$			1686.33i			1.52358i	0.647822	+	0.761792i	$0.275680\pi$
$107$			1686.33i			1.52358i	−0.647822	+	0.761792i	$0.724320\pi$
$108$		0			0
$109$	641.258			0.563499			0.281750	−	0.959488i	$-0.409085\pi$
$109$	641.258			0.563499			0.281750	+	0.959488i	$0.409085\pi$
$110$	−65.6870			−0.0569364
$111$		0			0
$112$	335.172	−	1132.48i	0.282775	−	0.955440i
$113$			1437.95i			1.19709i	0.801090	+	0.598543i	$0.204254\pi$
$113$			1437.95i			1.19709i	−0.801090	+	0.598543i	$0.795746\pi$
$114$		0			0
$115$		−	2944.86i		−	2.38791i
$116$		−	761.203i		−	0.609275i
$117$		0			0
$118$			20.9357i			0.0163329i
$119$	−328.595	+	1110.26i	−0.253128	+	0.855269i
$120$		0			0
$121$	−38.9185			−0.0292400
$122$	−21.5849			−0.0160180
$123$		0			0
$124$			1020.55i			0.739101i
$125$	−1424.52			−1.01931
$126$		0			0
$127$	−1845.03			−1.28914			−0.644568	−	0.764547i	$-0.722963\pi$
$127$	−1845.03			−1.28914			−0.644568	+	0.764547i	$0.722963\pi$
$128$			199.910i			0.138045i
$129$		0			0
$130$	−33.5411			−0.0226288
$131$	−2198.44			−1.46625			−0.733125	−	0.680094i	$-0.761939\pi$
$131$	−2198.44			−1.46625			−0.733125	+	0.680094i	$0.761939\pi$
$132$		0			0
$133$	1247.88	+	369.325i	0.813568	+	0.240786i
$134$		−	13.3729i		−	0.00862119i
$135$		0			0
$136$		−	97.8761i		−	0.0617118i
$137$		−	2158.87i		−	1.34631i	−0.739501	−	0.673156i	$-0.764938\pi$
$137$		−	2158.87i		−	1.34631i	0.739501	−	0.673156i	$-0.235062\pi$
$138$		0			0
$139$			2911.62i			1.77670i	0.459172	+	0.888348i	$0.348146\pi$
$139$			2911.62i			1.77670i	−0.459172	+	0.888348i	$0.651854\pi$
$140$	−761.280	+	2572.22i	−0.459571	+	1.55280i
$141$		0			0
$142$	44.9312			0.0265531
$143$	−699.508			−0.409062
$144$		0			0
$145$			1726.85i			0.989017i
$146$	94.7598			0.0537149
$147$		0			0
$148$	−3027.03			−1.68122
$149$		−	266.321i		−	0.146429i	−0.997316	−	0.0732144i	$-0.976674\pi$
$149$		−	266.321i		−	0.146429i	0.997316	−	0.0732144i	$-0.0233257\pi$
$150$		0			0
$151$	−935.515			−0.504180			−0.252090	−	0.967704i	$-0.581118\pi$
$151$	−935.515			−0.504180			−0.252090	+	0.967704i	$0.581118\pi$
$152$	−110.008			−0.0587029
$153$		0			0
$154$	19.0461	−	64.3529i	0.00996607	−	0.0336734i
$155$		−	2315.22i		−	1.19976i
$156$		0			0
$157$			1007.76i			0.512281i	0.966640	+	0.256141i	$0.0824510\pi$
$157$			1007.76i			0.512281i	−0.966640	+	0.256141i	$0.917549\pi$
$158$		−	58.4140i		−	0.0294125i
$159$		0			0
$160$		−	340.204i		−	0.168097i
$161$	2885.05	+	853.868i	1.41226	+	0.417976i
$162$		0			0
$163$	−1801.27			−0.865561			−0.432780	−	0.901499i	$-0.642467\pi$
$163$	−1801.27			−0.865561			−0.432780	+	0.901499i	$0.642467\pi$
$164$	−1585.20			−0.754775
$165$		0			0
$166$		−	35.3860i		−	0.0165451i
$167$	1635.80			0.757975			0.378987	−	0.925402i	$-0.376272\pi$
$167$	1635.80			0.757975			0.378987	+	0.925402i	$0.376272\pi$
$168$		0			0
$169$	1839.82			0.837422
$170$			110.954i			0.0500574i
$171$		0			0
$172$	−2568.00			−1.13842
$173$	799.517			0.351365			0.175683	−	0.984447i	$-0.443787\pi$
$173$	799.517			0.351365			0.175683	+	0.984447i	$0.443787\pi$
$174$		0			0
$175$	1070.04	−	3615.44i	0.462212	−	1.56172i
$176$		−	2360.28i		−	1.01087i
$177$		0			0
$178$			3.47552i			0.00146349i
$179$		−	2940.19i		−	1.22771i	−0.789418	−	0.613856i	$-0.789618\pi$
$179$		−	2940.19i		−	1.22771i	0.789418	−	0.613856i	$-0.210382\pi$
$180$		0			0
$181$		−	376.924i		−	0.154788i	−0.997001	−	0.0773938i	$-0.975340\pi$
$181$		−	376.924i		−	0.154788i	0.997001	−	0.0773938i	$-0.0246599\pi$
$182$	9.72531	−	32.8599i	0.00396092	−	0.0133832i
$183$		0			0
$184$	−254.336			−0.101901
$185$	6867.07			2.72906
$186$		0			0
$187$			2313.97i			0.904888i
$188$	−1268.69			−0.492176
$189$		0			0
$190$	124.707			0.0476167
$191$		−	1831.46i		−	0.693822i	−0.937898	−	0.346911i	$-0.887231\pi$
$191$		−	1831.46i		−	0.693822i	0.937898	−	0.346911i	$-0.112769\pi$
$192$		0			0
$193$	−380.016			−0.141731			−0.0708657	−	0.997486i	$-0.522576\pi$
$193$	−380.016			−0.141731			−0.0708657	+	0.997486i	$0.522576\pi$
$194$	−130.194			−0.0481822
$195$		0			0
$196$	−2299.24	−	1491.64i	−0.837916	−	0.543600i
$197$			3579.56i			1.29458i	0.762243	+	0.647291i	$0.224098\pi$
$197$			3579.56i			1.29458i	−0.762243	+	0.647291i	$0.775902\pi$
$198$		0			0
$199$		−	881.304i		−	0.313939i	−0.987603	−	0.156970i	$-0.949827\pi$
$199$		−	881.304i		−	0.313939i	0.987603	−	0.156970i	$-0.0501725\pi$
$200$			318.724i			0.112686i
$201$		0			0
$202$		−	89.5818i		−	0.0312028i
$203$	−1691.78	−	500.705i	−0.584926	−	0.173116i
$204$		0			0
$205$	3596.16			1.22520
$206$	81.6089			0.0276018
$207$		0			0
$208$		−	1205.21i		−	0.401760i
$209$	2600.79			0.860767
$210$		0			0
$211$	837.162			0.273140			0.136570	−	0.990630i	$-0.456392\pi$
$211$	837.162			0.273140			0.136570	+	0.990630i	$0.456392\pi$
$212$			1532.40i			0.496443i
$213$		0			0
$214$	−165.101			−0.0527386
$215$	5825.72			1.84796
$216$		0			0
$217$	2268.20	+	671.301i	0.709563	+	0.210004i
$218$			62.7828i			0.0195054i
$219$		0			0
$220$			5360.94i			1.64288i
$221$			1181.56i			0.359639i
$222$		0			0
$223$		−	1133.71i		−	0.340445i	−0.985406	−	0.170222i	$-0.945551\pi$
$223$		−	1133.71i		−	0.340445i	0.985406	−	0.170222i	$-0.0544486\pi$
$224$	333.294	+	98.6427i	0.0994159	+	0.0294234i
$225$		0			0
$226$	−140.783			−0.0414370
$227$	3505.91			1.02509			0.512545	−	0.858660i	$-0.328703\pi$
$227$	3505.91			1.02509			0.512545	+	0.858660i	$0.328703\pi$
$228$		0			0
$229$		−	3142.45i		−	0.906808i	−0.891305	−	0.453404i	$-0.850209\pi$
$229$		−	3142.45i		−	0.906808i	0.891305	−	0.453404i	$-0.149791\pi$
$230$	288.318			0.0826571
$231$		0			0
$232$	149.141			0.0422052
$233$		−	6882.00i		−	1.93500i	−0.252872	−	0.967500i	$-0.581375\pi$
$233$		−	6882.00i		−	1.93500i	0.252872	−	0.967500i	$-0.418625\pi$
$234$		0			0
$235$	2878.14			0.798933
$236$	1708.63			0.471282
$237$		0			0
$238$	−108.700	−	32.1712i	−0.0296050	−	0.00876198i
$239$		−	1196.43i		−	0.323811i	−0.986806	−	0.161906i	$-0.948236\pi$
$239$		−	1196.43i		−	0.323811i	0.986806	−	0.161906i	$-0.0517640\pi$
$240$		0			0
$241$		−	4377.00i		−	1.16991i	−0.811067	−	0.584953i	$-0.801113\pi$
$241$		−	4377.00i		−	1.16991i	0.811067	−	0.584953i	$-0.198887\pi$
$242$		−	3.81033i		−	0.00101214i
$243$		0			0
$244$			1761.62i			0.462196i
$245$	5216.03	+	3383.91i	1.36016	+	0.882408i
$246$		0			0
$247$	1328.02			0.342104
$248$	−199.956			−0.0511984
$249$		0			0
$250$		−	139.469i		−	0.0352831i
$251$	5787.19			1.45532			0.727658	−	0.685940i	$-0.240609\pi$
$251$	5787.19			1.45532			0.727658	+	0.685940i	$0.240609\pi$
$252$		0			0
$253$	6012.95			1.49419
$254$		−	180.639i		−	0.0446232i
$255$		0			0
$256$	4047.01			0.988040
$257$	2148.89			0.521573			0.260786	−	0.965397i	$-0.416018\pi$
$257$	2148.89			0.521573			0.260786	+	0.965397i	$0.416018\pi$
$258$		0			0
$259$	−1991.12	+	6727.61i	−0.477692	+	1.61403i
$260$			2737.41i			0.652949i
$261$		0			0
$262$		−	215.240i		−	0.0507540i
$263$			1220.76i			0.286218i	0.989707	+	0.143109i	$0.0457099\pi$
$263$			1220.76i			0.286218i	−0.989707	+	0.143109i	$0.954290\pi$
$264$		0			0
$265$		−	3476.39i		−	0.805860i
$266$	−36.1589	+	122.174i	−0.00833476	+	0.0281615i
$267$		0			0
$268$	−1091.41			−0.248762
$269$	−4430.56			−1.00422			−0.502112	−	0.864803i	$-0.667444\pi$
$269$	−4430.56			−1.00422			−0.502112	+	0.864803i	$0.667444\pi$
$270$		0			0
$271$			111.259i			0.0249391i	0.999922	+	0.0124695i	$0.00396928\pi$
$271$			111.259i			0.0249391i	−0.999922	+	0.0124695i	$0.996031\pi$
$272$	−3986.82			−0.888737
$273$		0			0
$274$	211.365			0.0466024
$275$		−	7535.21i		−	1.65233i
$276$		0			0
$277$	−3058.65			−0.663453			−0.331727	−	0.943376i	$-0.607631\pi$
$277$	−3058.65			−0.663453			−0.331727	+	0.943376i	$0.607631\pi$
$278$	−285.064			−0.0615000
$279$		0			0
$280$	−503.971	−	149.156i	−0.107564	−	0.0318350i
$281$			4762.02i			1.01096i	0.862840	+	0.505478i	$0.168684\pi$
$281$			4762.02i			1.01096i	−0.862840	+	0.505478i	$0.831316\pi$
$282$		0			0
$283$			4119.44i			0.865284i	0.901566	+	0.432642i	$0.142419\pi$
$283$			4119.44i			0.865284i	−0.901566	+	0.432642i	$0.857581\pi$
$284$		−	3666.99i		−	0.766183i
$285$		0			0
$286$		−	68.4857i		−	0.0141596i
$287$	−1042.71	+	3523.12i	−0.214458	+	0.724611i
$288$		0			0
$289$	−1004.42			−0.204440
$290$	−169.069			−0.0342347
$291$		0			0
$292$		−	7733.68i		−	1.54993i
$293$	2803.40			0.558963			0.279482	−	0.960151i	$-0.409837\pi$
$293$	2803.40			0.558963			0.279482	+	0.960151i	$0.409837\pi$
$294$		0			0
$295$	−3876.18			−0.765017
$296$		−	593.081i		−	0.116460i
$297$		0			0
$298$	26.0743			0.00506861
$299$	3070.33			0.593853
$300$		0			0
$301$	−1689.18	+	5707.40i	−0.323464	+	1.09292i
$302$		−	91.5921i		−	0.0174521i
$303$		0			0
$304$			4480.99i			0.845403i
$305$		−	3996.38i		−	0.750268i
$306$		0			0
$307$		−	3959.31i		−	0.736058i	−0.929814	−	0.368029i	$-0.880033\pi$
$307$		−	3959.31i		−	0.736058i	0.929814	−	0.368029i	$-0.119967\pi$
$308$	−5252.07	−	1554.42i	−0.971638	−	0.287568i
$309$		0			0
$310$	226.672			0.0415294
$311$	4984.50			0.908827			0.454413	−	0.890791i	$-0.349849\pi$
$311$	4984.50			0.908827			0.454413	+	0.890791i	$0.349849\pi$
$312$		0			0
$313$		−	807.951i		−	0.145904i	−0.997335	−	0.0729522i	$-0.976758\pi$
$313$		−	807.951i		−	0.145904i	0.997335	−	0.0729522i	$-0.0232421\pi$
$314$	−98.6654			−0.0177325
$315$		0			0
$316$	−4767.37			−0.848689
$317$		−	2380.58i		−	0.421789i	−0.977509	−	0.210894i	$-0.932362\pi$
$317$		−	2380.58i		−	0.421789i	0.977509	−	0.210894i	$-0.0676376\pi$
$318$		0			0
$319$	−3525.97			−0.618860
$320$	−9214.34			−1.60968
$321$		0			0
$322$	−83.5984	+	282.463i	−0.0144682	+	0.0488852i
$323$		−	4393.06i		−	0.756769i
$324$		0			0
$325$		−	3847.63i		−	0.656702i
$326$		−	176.354i		−	0.0299612i
$327$		0			0
$328$		−	310.586i		−	0.0522842i
$329$	−834.523	+	2819.69i	−0.139844	+	0.472506i
$330$		0			0
$331$	−7455.49			−1.23804			−0.619019	−	0.785376i	$-0.712470\pi$
$331$	−7455.49			−1.23804			−0.619019	+	0.785376i	$0.712470\pi$
$332$	−2887.97			−0.477404
$333$		0			0
$334$			160.154i			0.0262372i
$335$	2475.95			0.403807
$336$		0			0
$337$	−1956.52			−0.316256			−0.158128	−	0.987419i	$-0.550546\pi$
$337$	−1956.52			−0.316256			−0.158128	+	0.987419i	$0.550546\pi$
$338$			180.128i			0.0289872i
$339$		0			0
$340$	9055.31			1.44439
$341$	4727.31			0.750728
$342$		0			0
$343$	−4827.58	+	4128.93i	−0.759956	+	0.649974i
$344$		−	503.143i		−	0.0788595i
$345$		0			0
$346$			78.2772i			0.0121624i
$347$		−	1376.05i		−	0.212882i	−0.994319	−	0.106441i	$-0.966054\pi$
$347$		−	1376.05i		−	0.212882i	0.994319	−	0.106441i	$-0.0339456\pi$
$348$		0			0
$349$		−	9350.82i		−	1.43421i	−0.696967	−	0.717103i	$-0.745468\pi$
$349$		−	9350.82i		−	1.43421i	0.696967	−	0.717103i	$-0.254532\pi$
$350$	353.972	+	104.762i	0.0540588	+	0.0159994i
$351$		0			0
$352$	694.643			0.105184
$353$	−677.671			−0.102178			−0.0510890	−	0.998694i	$-0.516269\pi$
$353$	−677.671			−0.102178			−0.0510890	+	0.998694i	$0.516269\pi$
$354$		0			0
$355$			8318.88i			1.24372i
$356$	283.649			0.0422286
$357$		0			0
$358$	287.861			0.0424970
$359$		−	1662.36i		−	0.244390i	−0.992506	−	0.122195i	$-0.961007\pi$
$359$		−	1662.36i		−	0.244390i	0.992506	−	0.122195i	$-0.0389933\pi$
$360$		0			0
$361$	1921.41			0.280129
$362$	36.9030			0.00535795
$363$		0			0
$364$	−2681.81	−	793.716i	−0.386168	−	0.114291i
$365$			17544.5i			2.51595i
$366$		0			0
$367$			3063.89i			0.435787i	0.975973	+	0.217893i	$0.0699185\pi$
$367$			3063.89i			0.435787i	−0.975973	+	0.217893i	$0.930082\pi$
$368$			10359.9i			1.46752i
$369$		0			0
$370$			672.324i			0.0944661i
$371$	3405.79	+	1007.99i	0.476603	+	0.141057i
$372$		0			0
$373$	−7800.12			−1.08277			−0.541387	−	0.840773i	$-0.682101\pi$
$373$	−7800.12			−1.08277			−0.541387	+	0.840773i	$0.682101\pi$
$374$	−226.550			−0.0313225
$375$		0			0
$376$		−	248.573i		−	0.0340936i
$377$	−1800.43			−0.245960
$378$		0			0
$379$	−6855.89			−0.929191			−0.464595	−	0.885523i	$-0.653800\pi$
$379$	−6855.89			−0.929191			−0.464595	+	0.885523i	$0.653800\pi$
$380$		−	10177.7i		−	1.37397i
$381$		0			0
$382$	179.310			0.0240165
$383$	11022.9			1.47061			0.735307	−	0.677734i	$-0.237038\pi$
$383$	11022.9			1.47061			0.735307	+	0.677734i	$0.237038\pi$
$384$		0			0
$385$	11914.8	+	3526.33i	1.57723	+	0.466801i
$386$		−	37.2057i		−	0.00490601i
$387$		0			0
$388$			10625.6i			1.39029i
$389$		−	10924.3i		−	1.42386i	−0.702249	−	0.711931i	$-0.747821\pi$
$389$		−	10924.3i		−	1.42386i	0.702249	−	0.711931i	$-0.252179\pi$
$390$		0			0
$391$		−	10156.6i		−	1.31367i
$392$	292.255	−	450.487i	0.0376558	−	0.0580435i
$393$		0			0
$394$	−350.458			−0.0448118
$395$	10815.2			1.37765
$396$		0			0
$397$			7203.22i			0.910627i	0.890331	+	0.455314i	$0.150473\pi$
$397$			7203.22i			0.910627i	−0.890331	+	0.455314i	$0.849527\pi$
$398$	86.2845			0.0108670
$399$		0			0
$400$	12982.7			1.62284
$401$		−	12278.1i		−	1.52902i	−0.644612	−	0.764510i	$-0.722981\pi$
$401$		−	12278.1i		−	1.52902i	0.644612	−	0.764510i	$-0.277019\pi$
$402$		0			0
$403$	2413.86			0.298370
$404$	−7311.09			−0.900347
$405$		0			0
$406$	49.0218	−	165.635i	0.00599239	−	0.0202471i
$407$			14021.5i			1.70766i
$408$		0			0
$409$		−	849.278i		−	0.102675i	−0.998681	−	0.0513376i	$-0.983652\pi$
$409$		−	849.278i		−	0.102675i	0.998681	−	0.0513376i	$-0.0163484\pi$
$410$			352.084i			0.0424102i
$411$		0			0
$412$		−	6660.39i		−	0.796441i
$413$	1123.91	−	3797.46i	0.133908	−	0.452448i
$414$		0			0
$415$	6551.61			0.774954
$416$	354.699			0.0418042
$417$		0			0
$418$			254.632i			0.0297953i
$419$	−6219.48			−0.725159			−0.362579	−	0.931953i	$-0.618104\pi$
$419$	−6219.48			−0.725159			−0.362579	+	0.931953i	$0.618104\pi$
$420$		0			0
$421$	2024.13			0.234323			0.117162	−	0.993113i	$-0.462620\pi$
$421$	2024.13			0.234323			0.117162	+	0.993113i	$0.462620\pi$
$422$			81.9628i			0.00945471i
$423$		0			0
$424$	−300.242			−0.0343892
$425$	−12727.9			−1.45269
$426$		0			0
$427$	3915.21	+	1158.76i	0.443725	+	0.131326i
$428$			13474.5i			1.52176i
$429$		0			0
$430$			570.370i			0.0639667i
$431$			6519.65i			0.728633i	0.931275	+	0.364316i	$0.118697\pi$
$431$			6519.65i			0.728633i	−0.931275	+	0.364316i	$0.881303\pi$
$432$		0			0
$433$		−	17767.9i		−	1.97199i	−0.166772	−	0.985996i	$-0.553334\pi$
$433$		−	17767.9i		−	1.97199i	0.166772	−	0.985996i	$-0.446666\pi$
$434$	−65.7241	+	222.069i	−0.00726926	+	0.0245614i
$435$		0			0
$436$	5123.92			0.562824
$437$	−11415.6			−1.24961
$438$		0			0
$439$		−	1478.94i		−	0.160788i	−0.996763	−	0.0803942i	$-0.974382\pi$
$439$		−	1478.94i		−	0.160788i	0.996763	−	0.0803942i	$-0.0256179\pi$
$440$	−1050.36			−0.113805
$441$		0			0
$442$	−115.681			−0.0124488
$443$		−	15570.2i		−	1.66989i	−0.550333	−	0.834945i	$-0.685499\pi$
$443$		−	15570.2i		−	1.66989i	0.550333	−	0.834945i	$-0.314501\pi$
$444$		0			0
$445$	−643.483			−0.0685483
$446$	110.997			0.0117844
$447$		0			0
$448$	2671.72	−	9027.21i	0.281756	−	0.951999i
$449$			6120.30i			0.643284i	0.946861	+	0.321642i	$0.104235\pi$
$449$			6120.30i			0.643284i	−0.946861	+	0.321642i	$0.895765\pi$
$450$		0			0
$451$			7342.80i			0.766649i
$452$			11489.8i			1.19565i
$453$		0			0
$454$			343.248i			0.0354833i
$455$	6083.92	+	1800.61i	0.626854	+	0.185525i
$456$		0			0
$457$	6996.45			0.716149			0.358074	−	0.933693i	$-0.383433\pi$
$457$	6996.45			0.716149			0.358074	+	0.933693i	$0.383433\pi$
$458$	307.663			0.0313890
$459$		0			0
$460$		−	23530.6i		−	2.38505i
$461$	3666.35			0.370410			0.185205	−	0.982700i	$-0.440705\pi$
$461$	3666.35			0.370410			0.185205	+	0.982700i	$0.440705\pi$
$462$		0			0
$463$	−7027.41			−0.705381			−0.352691	−	0.935740i	$-0.614733\pi$
$463$	−7027.41			−0.705381			−0.352691	+	0.935740i	$0.614733\pi$
$464$		−	6075.02i		−	0.607814i
$465$		0			0
$466$	673.786			0.0669797
$467$	−13367.8			−1.32460			−0.662301	−	0.749238i	$-0.730420\pi$
$467$	−13367.8			−1.32460			−0.662301	+	0.749238i	$0.730420\pi$
$468$		0			0
$469$	−717.906	+	2425.66i	−0.0706819	+	0.238820i
$470$			281.786i			0.0276549i
$471$		0			0
$472$			334.770i			0.0326463i
$473$			11895.2i			1.15633i
$474$		0			0
$475$			14305.6i			1.38186i
$476$	−2625.61	+	8871.41i	−0.252825	+	0.854245i
$477$		0			0
$478$	117.138			0.0112087
$479$	−6994.40			−0.667186			−0.333593	−	0.942717i	$-0.608261\pi$
$479$	−6994.40			−0.667186			−0.333593	+	0.942717i	$0.608261\pi$
$480$		0			0
$481$			7159.66i			0.678695i
$482$	428.533			0.0404961
$483$		0			0
$484$	−310.975			−0.0292050
$485$		−	24105.0i		−	2.25681i
$486$		0			0
$487$	−8233.24			−0.766086			−0.383043	−	0.923731i	$-0.625124\pi$
$487$	−8233.24			−0.766086			−0.383043	+	0.923731i	$0.625124\pi$
$488$	−345.151			−0.0320169
$489$		0			0
$490$	−331.304	+	510.678i	−0.0305444	+	0.0470818i
$491$		−	14925.5i		−	1.37185i	−0.727671	−	0.685927i	$-0.759397\pi$
$491$		−	14925.5i		−	1.37185i	0.727671	−	0.685927i	$-0.240603\pi$
$492$		0			0
$493$			5955.81i			0.544090i
$494$			130.020i			0.0118419i
$495$		0			0
$496$			8144.86i			0.737329i
$497$	−8149.94	−	2412.08i	−0.735563	−	0.217699i
$498$		0			0
$499$	8330.93			0.747382			0.373691	−	0.927553i	$-0.378092\pi$
$499$	8330.93			0.747382			0.373691	+	0.927553i	$0.378092\pi$
$500$	−11382.5			−1.01808
$501$		0			0
$502$			566.598i			0.0503755i
$503$	−15511.0			−1.37495			−0.687475	−	0.726208i	$-0.741281\pi$
$503$	−15511.0			−1.37495			−0.687475	+	0.726208i	$0.741281\pi$
$504$		0			0
$505$	16585.8			1.46150
$506$			588.701i			0.0517212i
$507$		0			0
$508$	−14742.6			−1.28759
$509$	−3632.29			−0.316304			−0.158152	−	0.987415i	$-0.550554\pi$
$509$	−3632.29			−0.316304			−0.158152	+	0.987415i	$0.550554\pi$
$510$		0			0
$511$	−17188.2	−	5087.06i	−1.48799	−	0.440388i
$512$			1995.51i			0.172246i
$513$		0			0
$514$			210.388i			0.0180542i
$515$			15109.7i			1.29284i
$516$		0			0
$517$			5876.72i			0.499919i
$518$	−658.670	−	194.942i	−0.0558693	−	0.0165352i
$519$		0			0
$520$	−536.336			−0.0452306
$521$	531.760			0.0447156			0.0223578	−	0.999750i	$-0.492883\pi$
$521$	531.760			0.0447156			0.0223578	+	0.999750i	$0.492883\pi$
$522$		0			0
$523$			14041.2i			1.17395i	0.809604	+	0.586977i	$0.199682\pi$
$523$			14041.2i			1.17395i	−0.809604	+	0.586977i	$0.800318\pi$
$524$	−17566.5			−1.46449
$525$		0			0
$526$	−119.519			−0.00990739
$527$		−	7985.03i		−	0.660025i
$528$		0			0
$529$	−14225.5			−1.16918
$530$	340.358			0.0278947
$531$		0			0
$532$	9971.04	+	2951.06i	0.812593	+	0.240497i
$533$			3749.38i			0.304697i
$534$		0			0
$535$		−	30568.0i		−	2.47022i
$536$		−	213.838i		−	0.0172320i
$537$		0			0
$538$		−	433.777i		−	0.0347611i
$539$	−6909.42	+	10650.3i	−0.552152	+	0.851098i
$540$		0			0
$541$	−11065.4			−0.879372			−0.439686	−	0.898151i	$-0.644910\pi$
$541$	−11065.4			−0.879372			−0.439686	+	0.898151i	$0.644910\pi$
$542$	−10.8928			−0.000863261
$543$		0			0
$544$		−	1173.34i		−	0.0924753i
$545$	−11624.1			−0.913614
$546$		0			0
$547$	−12048.6			−0.941790			−0.470895	−	0.882189i	$-0.656069\pi$
$547$	−12048.6			−0.941790			−0.470895	+	0.882189i	$0.656069\pi$
$548$		−	17250.3i		−	1.34470i
$549$		0			0
$550$	737.739			0.0571950
$551$	6694.05			0.517561
$552$		0			0
$553$	−3135.89	+	10595.5i	−0.241142	+	0.814771i
$554$		−	299.459i		−	0.0229653i
$555$		0			0
$556$			23265.1i			1.77457i
$557$		−	2546.80i		−	0.193737i	−0.995297	−	0.0968684i	$-0.969117\pi$
$557$		−	2546.80i		−	0.193737i	0.995297	−	0.0968684i	$-0.0308826\pi$
$558$		0			0
$559$			6073.94i			0.459571i
$560$	−6075.64	+	20528.4i	−0.458469	+	1.54908i
$561$		0			0
$562$	−466.228			−0.0349941
$563$	−23072.5			−1.72716			−0.863578	−	0.504216i	$-0.831782\pi$
$563$	−23072.5			−1.72716			−0.863578	+	0.504216i	$0.831782\pi$
$564$		0			0
$565$		−	26065.6i		−	1.94086i
$566$	−403.316			−0.0299517
$567$		0			0
$568$	718.468			0.0530744
$569$			6317.82i			0.465478i	0.972539	+	0.232739i	$0.0747687\pi$
$569$			6317.82i			0.465478i	−0.972539	+	0.232739i	$0.925231\pi$
$570$		0			0
$571$	18552.1			1.35969			0.679845	−	0.733356i	$-0.262047\pi$
$571$	18552.1			1.35969			0.679845	+	0.733356i	$0.262047\pi$
$572$	−5589.36			−0.408571
$573$		0			0
$574$	−344.933	−	102.087i	−0.0250823	−	0.00742342i
$575$			33074.1i			2.39876i
$576$		0			0
$577$		−	2684.61i		−	0.193695i	−0.995299	−	0.0968473i	$-0.969124\pi$
$577$		−	2684.61i		−	0.193695i	0.995299	−	0.0968473i	$-0.0308758\pi$
$578$		−	98.3379i		−	0.00707667i
$579$		0			0
$580$			13798.3i			0.987832i
$581$	−1899.65	+	6418.56i	−0.135647	+	0.458325i
$582$		0			0
$583$	7098.25			0.504253
$584$	1515.25			0.107365
$585$		0			0
$586$			274.468i			0.0193484i
$587$	18248.3			1.28312			0.641558	−	0.767075i	$-0.278289\pi$
$587$	18248.3			1.28312			0.641558	+	0.767075i	$0.278289\pi$
$588$		0			0
$589$	−8974.79			−0.627844
$590$		−	379.500i		−	0.0264809i
$591$		0			0
$592$	−24158.1			−1.67719
$593$	−10331.8			−0.715473			−0.357736	−	0.933823i	$-0.616451\pi$
$593$	−10331.8			−0.715473			−0.357736	+	0.933823i	$0.616451\pi$
$594$		0			0
$595$	5956.41	−	20125.6i	0.410402	−	1.38667i
$596$		−	2128.02i		−	0.146253i
$597$		0			0
$598$			300.603i			0.0205561i
$599$		−	3987.38i		−	0.271987i	−0.990710	−	0.135993i	$-0.956577\pi$
$599$		−	3987.38i		−	0.271987i	0.990710	−	0.135993i	$-0.0434226\pi$
$600$		0			0
$601$			18400.9i			1.24890i	0.781066	+	0.624448i	$0.214676\pi$
$601$			18400.9i			1.24890i	−0.781066	+	0.624448i	$0.785324\pi$
$602$	−558.786	−	165.380i	−0.0378313	−	0.0111966i
$603$		0			0
$604$	−7475.15			−0.503576
$605$	705.473			0.0474075
$606$		0			0
$607$			14175.4i			0.947874i	0.880559	+	0.473937i	$0.157168\pi$
$607$			14175.4i			0.947874i	−0.880559	+	0.473937i	$0.842832\pi$
$608$	−1318.78			−0.0879663
$609$		0			0
$610$	391.267			0.0259704
$611$			3000.77i			0.198688i
$612$		0			0
$613$	−2164.43			−0.142611			−0.0713054	−	0.997455i	$-0.522716\pi$
$613$	−2164.43			−0.142611			−0.0713054	+	0.997455i	$0.522716\pi$
$614$	387.638			0.0254785
$615$		0			0
$616$	304.554	−	1029.03i	0.0199202	−	0.0673065i
$617$			3232.84i			0.210939i	0.994423	+	0.105469i	$0.0336345\pi$
$617$			3232.84i			0.210939i	−0.994423	+	0.105469i	$0.966365\pi$
$618$		0			0
$619$		−	21873.9i		−	1.42033i	−0.704034	−	0.710166i	$-0.748620\pi$
$619$		−	21873.9i		−	1.42033i	0.704034	−	0.710166i	$-0.251380\pi$
$620$		−	18499.5i		−	1.19832i
$621$		0			0
$622$			488.010i			0.0314589i
$623$	186.579	−	630.414i	0.0119986	−	0.0405409i
$624$		0			0
$625$	374.001			0.0239361
$626$	79.1029			0.00505046
$627$		0			0
$628$			8052.43i			0.511667i
$629$	23684.1			1.50135
$630$		0			0
$631$	−9979.57			−0.629604			−0.314802	−	0.949157i	$-0.601938\pi$
$631$	−9979.57			−0.629604			−0.314802	+	0.949157i	$0.601938\pi$
$632$		−	934.064i		−	0.0587897i
$633$		0			0
$634$	233.072			0.0146001
$635$	33444.8			2.09010
$636$		0			0
$637$	−3528.09	+	5438.27i	−0.219447	+	0.338261i
$638$		−	345.212i		−	0.0214217i
$639$		0			0
$640$		−	3623.76i		−	0.223815i
$641$		−	15802.4i		−	0.973727i	−0.873478	−	0.486864i	$-0.838141\pi$
$641$		−	15802.4i		−	0.973727i	0.873478	−	0.486864i	$-0.161859\pi$
$642$		0			0
$643$		−	12679.7i		−	0.777663i	−0.921309	−	0.388831i	$-0.872879\pi$
$643$		−	12679.7i		−	0.777663i	0.921309	−	0.388831i	$-0.127121\pi$
$644$	23052.8	+	6822.76i	1.41057	+	0.417476i
$645$		0			0
$646$	430.105			0.0261954
$647$	20710.7			1.25846			0.629230	−	0.777219i	$-0.283370\pi$
$647$	20710.7			1.25846			0.629230	+	0.777219i	$0.283370\pi$
$648$		0			0
$649$		−	7914.57i		−	0.478696i
$650$	376.704			0.0227316
$651$		0			0
$652$	−14392.9			−0.864523
$653$		−	1565.54i		−	0.0938200i	−0.998899	−	0.0469100i	$-0.985063\pi$
$653$		−	1565.54i		−	0.0938200i	0.998899	−	0.0469100i	$-0.0149374\pi$
$654$		0			0
$655$	39851.0			2.37727
$656$	−12651.2			−0.752966
$657$		0			0
$658$	−276.063	−	81.7044i	−0.0163557	−	0.00484069i
$659$			17875.7i			1.05666i	0.849040	+	0.528328i	$0.177181\pi$
$659$			17875.7i			1.05666i	−0.849040	+	0.528328i	$0.822819\pi$
$660$		0			0
$661$		−	25301.0i		−	1.48880i	−0.667735	−	0.744399i	$-0.732736\pi$
$661$		−	25301.0i		−	1.48880i	0.667735	−	0.744399i	$-0.267264\pi$
$662$		−	729.933i		−	0.0428545i
$663$		0			0
$664$		−	565.836i		−	0.0330703i
$665$	−22620.2	−	6694.72i	−1.31906	−	0.390391i
$666$		0			0
$667$	15476.4			0.898427
$668$	13070.7			0.757067
$669$		0			0
$670$			242.409i			0.0139777i
$671$	8159.98			0.469467
$672$		0			0
$673$	33548.8			1.92156			0.960781	−	0.277310i	$-0.0894429\pi$
$673$	33548.8			1.92156			0.960781	+	0.277310i	$0.0894429\pi$
$674$		−	191.554i		−	0.0109472i
$675$		0			0
$676$	14700.9			0.836419
$677$	−32268.4			−1.83187			−0.915935	−	0.401327i	$-0.868549\pi$
$677$	−32268.4			−1.83187			−0.915935	+	0.401327i	$0.868549\pi$
$678$		0			0
$679$	23615.4	+	6989.29i	1.33472	+	0.395028i
$680$			1774.19i			0.100055i
$681$		0			0
$682$			462.830i			0.0259863i
$683$		−	14942.5i		−	0.837126i	−0.908188	−	0.418563i	$-0.862534\pi$
$683$		−	14942.5i		−	0.837126i	0.908188	−	0.418563i	$-0.137466\pi$
$684$		0			0
$685$			39133.7i			2.18281i
$686$	−404.245	−	472.647i	−0.0224987	−	0.0263058i
$687$		0			0
$688$	−20494.7			−1.13569
$689$	3624.51			0.200411
$690$		0			0
$691$			11240.3i			0.618817i	0.950929	+	0.309409i	$0.100131\pi$
$691$			11240.3i			0.618817i	−0.950929	+	0.309409i	$0.899869\pi$
$692$	6388.47			0.350944
$693$		0			0
$694$	134.723			0.00736889
$695$		−	52778.8i		−	2.88060i
$696$		0			0
$697$	12402.9			0.674023
$698$	915.497			0.0496448
$699$		0			0
$700$	8550.03	−	28888.9i	0.461658	−	1.55985i
$701$			19409.2i			1.04576i	0.852407	+	0.522879i	$0.175142\pi$
$701$			19409.2i			1.04576i	−0.852407	+	0.522879i	$0.824858\pi$
$702$		0			0
$703$		−	26619.8i		−	1.42814i
$704$		−	18814.3i		−	1.00723i
$705$		0			0
$706$		−	66.3478i		−	0.00353687i
$707$	−4809.09	+	16249.0i	−0.255820	+	0.864365i
$708$		0			0
$709$	−1732.22			−0.0917556			−0.0458778	−	0.998947i	$-0.514608\pi$
$709$	−1732.22			−0.0917556			−0.0458778	+	0.998947i	$0.514608\pi$
$710$	−814.465			−0.0430512
$711$		0			0
$712$			55.5750i			0.00292522i
$713$	−20749.5			−1.08986
$714$		0			0
$715$	12679.9			0.663221
$716$		−	23493.4i		−	1.22624i
$717$		0			0
$718$	162.754			0.00845952
$719$	36946.9			1.91639			0.958197	−	0.286109i	$-0.0923620\pi$
$719$	36946.9			1.91639			0.958197	+	0.286109i	$0.0923620\pi$
$720$		0			0
$721$	−14802.8	−	4381.08i	−0.764612	−	0.226297i
$722$			188.116i			0.00969663i
$723$		0			0
$724$		−	3011.78i		−	0.154602i
$725$		−	19394.5i		−	0.993510i
$726$		0			0
$727$			2050.75i			0.104619i	0.998631	+	0.0523096i	$0.0166583\pi$
$727$			2050.75i			0.104619i	−0.998631	+	0.0523096i	$0.983342\pi$
$728$	155.512	−	525.444i	0.00791710	−	0.0267503i
$729$		0			0
$730$	−1717.70			−0.0870892
$731$	20092.5			1.01662
$732$		0			0
$733$			31357.7i			1.58012i	0.613033	+	0.790058i	$0.289949\pi$
$733$			31357.7i			1.58012i	−0.613033	+	0.790058i	$0.710051\pi$
$734$	−299.972			−0.0150847
$735$		0			0
$736$	−3048.98			−0.152700
$737$			5055.50i			0.252675i
$738$		0			0
$739$	1233.93			0.0614219			0.0307109	−	0.999528i	$-0.490223\pi$
$739$	1233.93			0.0614219			0.0307109	+	0.999528i	$0.490223\pi$
$740$	54870.7			2.72579
$741$		0			0
$742$	−98.6874	+	333.446i	−0.00488266	+	0.0164975i
$743$			30242.1i			1.49324i	0.665253	+	0.746618i	$0.268324\pi$
$743$			30242.1i			1.49324i	−0.665253	+	0.746618i	$0.731676\pi$
$744$		0			0
$745$			4827.59i			0.237408i
$746$		−	763.675i		−	0.0374801i
$747$		0			0
$748$			18489.6i			0.903803i
$749$	29947.2	+	8863.24i	1.46094	+	0.432384i
$750$		0			0
$751$	5493.40			0.266920			0.133460	−	0.991054i	$-0.457391\pi$
$751$	5493.40			0.266920			0.133460	+	0.991054i	$0.457391\pi$
$752$	−10125.2			−0.490996
$753$		0			0
$754$		−	176.272i		−	0.00851387i
$755$	16958.0			0.817438
$756$		0			0
$757$	−31971.7			−1.53505			−0.767523	−	0.641021i	$-0.778511\pi$
$757$	−31971.7			−1.53505			−0.767523	+	0.641021i	$0.778511\pi$
$758$		−	671.229i		−	0.0321638i
$759$		0			0
$760$	1994.11			0.0951763
$761$	16441.0			0.783162			0.391581	−	0.920144i	$-0.371928\pi$
$761$	16441.0			0.783162			0.391581	+	0.920144i	$0.371928\pi$
$762$		0			0
$763$	3370.42	−	11388.0i	0.159918	−	0.540331i
$764$		−	14634.2i		−	0.692991i
$765$		0			0
$766$			1079.21i			0.0509051i
$767$		−	4041.34i		−	0.190253i
$768$		0			0
$769$			1409.80i			0.0661102i	0.999454	+	0.0330551i	$0.0105237\pi$
$769$			1409.80i			0.0661102i	−0.999454	+	0.0330551i	$0.989476\pi$
$770$	−345.247	+	1166.52i	−0.0161582	+	0.0545955i
$771$		0			0
$772$	−3036.49			−0.141562
$773$	−9575.08			−0.445526			−0.222763	−	0.974873i	$-0.571508\pi$
$773$	−9575.08			−0.445526			−0.222763	+	0.974873i	$0.571508\pi$
$774$		0			0
$775$			26002.5i			1.20521i
$776$	−2081.85			−0.0963067
$777$		0			0
$778$	1069.55			0.0492868
$779$		−	13940.3i		−	0.641159i
$780$		0			0
$781$	−16985.9			−0.778236
$782$	994.391			0.0454723
$783$		0			0
$784$	−18349.8	−	11904.5i	−0.835907	−	0.542297i
$785$		−	18267.6i		−	0.830573i
$786$		0			0
$787$			1138.13i			0.0515499i	0.999668	+	0.0257750i	$0.00820534\pi$
$787$			1138.13i			0.0515499i	−0.999668	+	0.0257750i	$0.991795\pi$
$788$			28602.1i			1.29303i
$789$		0			0
$790$			1058.87i			0.0476871i
$791$	25536.2	+	7557.78i	1.14787	+	0.339726i
$792$		0			0
$793$	4166.65			0.186585
$794$	−705.235			−0.0315212
$795$		0			0
$796$		−	7041.98i		−	0.313563i
$797$	−21371.4			−0.949829			−0.474914	−	0.880032i	$-0.657521\pi$
$797$	−21371.4			−0.949829			−0.474914	+	0.880032i	$0.657521\pi$
$798$		0			0
$799$	9926.53			0.439519
$800$			3820.87i			0.168860i
$801$		0			0
$802$	1202.09			0.0529267
$803$	−35823.2			−1.57431
$804$		0			0
$805$	−52297.2	−	15478.0i	−2.28973	−	0.677675i
$806$			236.330i			0.0103280i
$807$		0			0
$808$		−	1432.45i		−	0.0623681i
$809$			16824.0i			0.731150i	0.930782	+	0.365575i	$0.119128\pi$
$809$			16824.0i			0.731150i	−0.930782	+	0.365575i	$0.880872\pi$
$810$		0			0
$811$		−	32964.6i		−	1.42731i	−0.700500	−	0.713653i	$-0.747039\pi$
$811$		−	32964.6i		−	1.42731i	0.700500	−	0.713653i	$-0.252961\pi$
$812$	−13518.1	−	4000.84i	−0.584225	−	0.172909i
$813$		0			0
$814$	−1372.78			−0.0591105
$815$	32651.5			1.40335
$816$		0			0
$817$		−	22583.0i		−	0.967051i
$818$	83.1491			0.00355408
$819$		0			0
$820$	28734.8			1.22373
$821$			21634.0i			0.919648i	0.888010	+	0.459824i	$0.152087\pi$
$821$			21634.0i			0.919648i	−0.888010	+	0.459824i	$0.847913\pi$
$822$		0			0
$823$	−24388.9			−1.03298			−0.516492	−	0.856292i	$-0.672762\pi$
$823$	−24388.9			−1.03298			−0.516492	+	0.856292i	$0.672762\pi$
$824$	1304.96			0.0551704
$825$		0			0
$826$	371.793	+	110.037i	0.0156614	+	0.00463519i
$827$		−	29015.7i		−	1.22004i	−0.792385	−	0.610022i	$-0.791161\pi$
$827$		−	29015.7i		−	1.22004i	0.792385	−	0.610022i	$-0.208839\pi$
$828$		0			0
$829$			17685.2i			0.740931i	0.928846	+	0.370466i	$0.120802\pi$
$829$			17685.2i			0.740931i	−0.928846	+	0.370466i	$0.879198\pi$
$830$			641.439i			0.0268249i
$831$		0			0
$832$		−	9606.94i		−	0.400313i
$833$	17989.8	+	11670.9i	0.748269	+	0.485441i
$834$		0			0
$835$	−29652.0			−1.22892
$836$	20781.4			0.859735
$837$		0			0
$838$		−	608.922i		−	0.0251013i
$839$	37751.1			1.55341			0.776706	−	0.629863i	$-0.216889\pi$
$839$	37751.1			1.55341			0.776706	+	0.629863i	$0.216889\pi$
$840$		0			0
$841$	15313.7			0.627892
$842$			198.174i			0.00811107i
$843$		0			0
$844$	6689.27			0.272813
$845$	−33350.3			−1.35773
$846$		0			0
$847$	−204.553	+	691.145i	−0.00829815	+	0.0280378i
$848$			12229.8i			0.495253i
$849$		0			0
$850$		−	1246.13i		−	0.0502848i
$851$		−	61544.2i		−	2.47909i
$852$		0			0
$853$			22651.0i			0.909209i	0.890693	+	0.454605i	$0.150219\pi$
$853$			22651.0i			0.909209i	−0.890693	+	0.454605i	$0.849781\pi$
$854$	−113.449	+	383.321i	−0.00454583	+	0.0153595i
$855$		0			0
$856$	−2640.03			−0.105414
$857$	−12160.4			−0.484703			−0.242351	−	0.970189i	$-0.577919\pi$
$857$	−12160.4			−0.484703			−0.242351	+	0.970189i	$0.577919\pi$
$858$		0			0
$859$		−	14075.1i		−	0.559063i	−0.960137	−	0.279531i	$-0.909821\pi$
$859$		−	14075.1i		−	0.559063i	0.960137	−	0.279531i	$-0.0901791\pi$
$860$	46549.9			1.84574
$861$		0			0
$862$	−638.310			−0.0252215
$863$		−	8687.35i		−	0.342666i	−0.985213	−	0.171333i	$-0.945193\pi$
$863$		−	8687.35i		−	0.342666i	0.985213	−	0.171333i	$-0.0548074\pi$
$864$		0			0
$865$	−14492.8			−0.569676
$866$	1739.58			0.0682601
$867$		0			0
$868$	18123.8	+	5363.98i	0.708713	+	0.209753i
$869$			22083.0i			0.862040i
$870$		0			0
$871$			2581.44i			0.100423i
$872$			1003.92i			0.0389875i
$873$		0			0
$874$		−	1117.65i		−	0.0432551i
$875$	−7487.21	+	25297.8i	−0.289273	+	0.977397i
$876$		0			0
$877$	3208.37			0.123533			0.0617667	−	0.998091i	$-0.480327\pi$
$877$	3208.37			0.123533			0.0617667	+	0.998091i	$0.480327\pi$
$878$	144.797			0.00556566
$879$		0			0
$880$			42784.7i			1.63895i
$881$	−36469.9			−1.39467			−0.697334	−	0.716746i	$-0.745631\pi$
$881$	−36469.9			−1.39467			−0.697334	+	0.716746i	$0.745631\pi$
$882$		0			0
$883$	4262.90			0.162467			0.0812333	−	0.996695i	$-0.474114\pi$
$883$	4262.90			0.162467			0.0812333	+	0.996695i	$0.474114\pi$
$884$			9441.14i			0.359208i
$885$		0			0
$886$	1524.41			0.0578030
$887$	−32578.7			−1.23324			−0.616620	−	0.787261i	$-0.711499\pi$
$887$	−32578.7			−1.23324			−0.616620	+	0.787261i	$0.711499\pi$
$888$		0			0
$889$	−9697.39	+	32765.6i	−0.365849	+	1.23613i
$890$		−	63.0005i		−	0.00237279i
$891$		0			0
$892$		−	9058.85i		−	0.340037i
$893$		−	11156.9i		−	0.418089i
$894$		0			0
$895$			53296.7i			1.99052i
$896$	3550.17	+	1050.72i	0.132369	+	0.0391763i
$897$		0			0
$898$	−599.211			−0.0222672
$899$	12167.4			0.451397
$900$		0			0
$901$		−	11989.9i		−	0.443329i
$902$	−718.901			−0.0265374
$903$		0			0
$904$	−2251.18			−0.0828243
$905$			6832.48i			0.250961i
$906$		0			0
$907$	4842.00			0.177261			0.0886307	−	0.996065i	$-0.471751\pi$
$907$	4842.00			0.177261			0.0886307	+	0.996065i	$0.471751\pi$
$908$	28013.7			1.02386
$909$		0			0
$910$	−176.290	+	595.650i	−0.00642193	+	0.0216985i
$911$			35456.2i			1.28948i	0.764402	+	0.644740i	$0.223034\pi$
$911$			35456.2i			1.28948i	−0.764402	+	0.644740i	$0.776966\pi$
$912$		0			0
$913$			13377.4i			0.484914i
$914$			684.991i			0.0247894i
$915$		0			0
$916$		−	25109.5i		−	0.905722i
$917$	−11554.9	+	39041.7i	−0.416113	+	1.40597i
$918$		0			0
$919$	16541.1			0.593732			0.296866	−	0.954919i	$-0.404059\pi$
$919$	16541.1			0.593732			0.296866	+	0.954919i	$0.404059\pi$
$920$	4610.32			0.165215
$921$		0			0
$922$			358.956i			0.0128217i
$923$	−8673.33			−0.309303
$924$		0			0
$925$	−77124.9			−2.74146
$926$		−	688.023i		−	0.0244167i
$927$		0			0
$928$	1787.91			0.0632446
$929$	40698.6			1.43733			0.718664	−	0.695358i	$-0.244754\pi$
$929$	40698.6			1.43733			0.718664	+	0.695358i	$0.244754\pi$
$930$		0			0
$931$	13117.5	−	20219.6i	0.461772	−	0.711785i
$932$		−	54990.0i		−	1.93268i
$933$		0			0
$934$		−	1308.78i		−	0.0458509i
$935$		−	41945.1i		−	1.46712i
$936$		0			0
$937$		−	13937.5i		−	0.485933i	−0.970035	−	0.242967i	$-0.921879\pi$
$937$		−	13937.5i		−	0.485933i	0.970035	−	0.242967i	$-0.0781205\pi$
$938$	−237.486	−	70.2870i	−0.00826673	−	0.00244664i
$939$		0			0
$940$	22997.6			0.797976
$941$	−32383.2			−1.12185			−0.560925	−	0.827867i	$-0.689554\pi$
$941$	−32383.2			−1.12185			−0.560925	+	0.827867i	$0.689554\pi$
$942$		0			0
$943$		−	32229.6i		−	1.11298i
$944$	13636.3			0.470152
$945$		0			0
$946$	−1164.61			−0.0400261
$947$		−	7506.46i		−	0.257579i	−0.991672	−	0.128789i	$-0.958891\pi$
$947$		−	7506.46i		−	0.257579i	0.991672	−	0.128789i	$-0.0411091\pi$
$948$		0			0
$949$	−18292.0			−0.625695
$950$	−1400.60			−0.0478330
$951$		0			0
$952$	−1738.16	−	514.431i	−0.0591746	−	0.0175135i
$953$		−	469.935i		−	0.0159735i	−0.999968	−	0.00798673i	$-0.997458\pi$
$953$		−	469.935i		−	0.0159735i	0.999968	−	0.00798673i	$-0.00254228\pi$
$954$		0			0
$955$			33198.8i			1.12491i
$956$		−	9560.00i		−	0.323423i
$957$		0			0
$958$		−	684.790i		−	0.0230945i
$959$	−38339.0	−	11346.9i	−1.29096	−	0.382076i
$960$		0			0
$961$	13478.0			0.452419
$962$	−700.970			−0.0234929
$963$		0			0
$964$		−	34974.1i		−	1.16850i
$965$	6888.53			0.229792
$966$		0			0
$967$	32016.5			1.06472			0.532359	−	0.846519i	$-0.321306\pi$
$967$	32016.5			1.06472			0.532359	+	0.846519i	$0.321306\pi$
$968$		−	60.9288i		−	0.00202306i
$969$		0			0
$970$	2360.01			0.0781189
$971$	−2588.27			−0.0855424			−0.0427712	−	0.999085i	$-0.513619\pi$
$971$	−2588.27			−0.0855424			−0.0427712	+	0.999085i	$0.513619\pi$
$972$		0			0
$973$	51706.9	+	15303.3i	1.70365	+	0.504216i
$974$		−	806.080i		−	0.0265179i
$975$		0			0
$976$			14059.1i			0.461088i
$977$		−	36668.7i		−	1.20075i	−0.799717	−	0.600376i	$-0.795017\pi$
$977$		−	36668.7i		−	1.20075i	0.799717	−	0.600376i	$-0.204983\pi$
$978$		0			0
$979$		−	1313.89i		−	0.0428929i
$980$	41678.2	+	27038.8i	1.35853	+	0.881351i
$981$		0			0
$982$	1461.29			0.0474865
$983$	−9010.66			−0.292366			−0.146183	−	0.989258i	$-0.546699\pi$
$983$	−9010.66			−0.292366			−0.146183	+	0.989258i	$0.546699\pi$
$984$		0			0
$985$		−	64886.4i		−	2.09894i
$986$	−583.107			−0.0188336
$987$		0			0
$988$	10611.4			0.341694
$989$		−	52211.4i		−	1.67869i
$990$		0			0
$991$	13431.5			0.430540			0.215270	−	0.976555i	$-0.430937\pi$
$991$	13431.5			0.430540			0.215270	+	0.976555i	$0.430937\pi$
$992$	−2397.07			−0.0767209
$993$		0			0
$994$	236.156	−	797.924i	0.00753562	−	0.0254614i
$995$			15975.3i			0.508997i
$996$		0			0
$997$		−	15763.6i		−	0.500741i	−0.968150	−	0.250371i	$-0.919447\pi$
$997$		−	15763.6i		−	0.500741i	0.968150	−	0.250371i	$-0.0805525\pi$
$998$			815.644i			0.0258705i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.c.c.566.4		44
3.2	odd	2	inner	567.4.c.c.566.41		44
7.6	odd	2	inner	567.4.c.c.566.42		44
9.2	odd	6		63.4.o.a.41.12	yes	44
9.4	even	3		63.4.o.a.20.11	✓	44
9.5	odd	6		189.4.o.a.62.11		44
9.7	even	3		189.4.o.a.125.12		44
21.20	even	2	inner	567.4.c.c.566.3		44
63.13	odd	6		63.4.o.a.20.12	yes	44
63.20	even	6		63.4.o.a.41.11	yes	44
63.34	odd	6		189.4.o.a.125.11		44
63.41	even	6		189.4.o.a.62.12		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.11	✓	44	9.4	even	3
63.4.o.a.20.12	yes	44	63.13	odd	6
63.4.o.a.41.11	yes	44	63.20	even	6
63.4.o.a.41.12	yes	44	9.2	odd	6
189.4.o.a.62.11		44	9.5	odd	6
189.4.o.a.62.12		44	63.41	even	6
189.4.o.a.125.11		44	63.34	odd	6
189.4.o.a.125.12		44	9.7	even	3
567.4.c.c.566.3		44	21.20	even	2	inner
567.4.c.c.566.4		44	1.1	even	1	trivial
567.4.c.c.566.41		44	3.2	odd	2	inner
567.4.c.c.566.42		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)

\(f(q)\)	\(=\)	\(q+0.0979055i q^{2} +7.99041 q^{4} -18.1269 q^{5} +(5.25594 - 17.7588i) q^{7} +1.56555i q^{8} +O(q^{10})\)	\(q+0.0979055i q^{2} +7.99041 q^{4} -18.1269 q^{5} +(5.25594 - 17.7588i) q^{7} +1.56555i q^{8} -1.77473i q^{10} -37.0124i q^{11} -18.8993i q^{13} +(1.73869 + 0.514586i) q^{14} +63.7700 q^{16} -62.5187 q^{17} +70.2680i q^{19} -144.842 q^{20} +3.62372 q^{22} +162.458i q^{23} +203.586 q^{25} +1.85034 q^{26} +(41.9972 - 141.900i) q^{28} -95.2645i q^{29} +127.722i q^{31} +18.7678i q^{32} -6.12092i q^{34} +(-95.2742 + 321.913i) q^{35} -378.832 q^{37} -6.87963 q^{38} -28.3786i q^{40} -198.387 q^{41} -321.384 q^{43} -295.744i q^{44} -15.9055 q^{46} -158.777 q^{47} +(-287.750 - 186.678i) q^{49} +19.9322i q^{50} -151.013i q^{52} +191.780i q^{53} +670.922i q^{55} +(27.8023 + 8.22844i) q^{56} +9.32693 q^{58} +213.835 q^{59} +220.466i q^{61} -12.5047 q^{62} +508.323 q^{64} +342.586i q^{65} -136.589 q^{67} -499.550 q^{68} +(-31.5170 - 9.32787i) q^{70} -458.924i q^{71} -967.869i q^{73} -37.0898i q^{74} +561.470i q^{76} +(-657.296 - 194.535i) q^{77} -596.637 q^{79} -1155.96 q^{80} -19.4232i q^{82} -361.430 q^{83} +1133.27 q^{85} -31.4653i q^{86} +57.9448 q^{88} +35.4987 q^{89} +(-335.629 - 99.3336i) q^{91} +1298.10i q^{92} -15.5452i q^{94} -1273.74i q^{95} +1329.79i q^{97} +(18.2769 - 28.1723i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) 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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