LMFDB - Embedded newform 567.4.c.c.566.10

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			566.10
Character	$\chi$	$=$	567.566
Dual form			567.4.c.c.566.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.90742i q^{2} -7.26794 q^{4} +11.6534 q^{5} +(-14.6430 - 11.3394i) q^{7} +2.86048i q^{8} +O(q^{10})$	\(q+3.90742i q^{2} -7.26794 q^{4} +11.6534 q^{5} +(-14.6430 - 11.3394i) q^{7} +2.86048i q^{8} +45.5347i q^{10} +47.6890i q^{11} -69.5200i q^{13} +(44.3077 - 57.2166i) q^{14} -69.3206 q^{16} -25.6583 q^{17} +41.1363i q^{19} -84.6961 q^{20} -186.341 q^{22} +168.137i q^{23} +10.8016 q^{25} +271.644 q^{26} +(106.425 + 82.4138i) q^{28} -32.5956i q^{29} +171.472i q^{31} -247.981i q^{32} -100.258i q^{34} +(-170.641 - 132.142i) q^{35} -3.86757 q^{37} -160.737 q^{38} +33.3343i q^{40} -337.467 q^{41} -414.701 q^{43} -346.601i q^{44} -656.982 q^{46} -148.771 q^{47} +(85.8377 + 332.086i) q^{49} +42.2064i q^{50} +505.267i q^{52} -59.7358i q^{53} +555.739i q^{55} +(32.4360 - 41.8861i) q^{56} +127.365 q^{58} -564.112 q^{59} +660.091i q^{61} -670.014 q^{62} +414.401 q^{64} -810.144i q^{65} -291.978 q^{67} +186.483 q^{68} +(516.335 - 666.767i) q^{70} -3.05665i q^{71} -506.596i q^{73} -15.1122i q^{74} -298.976i q^{76} +(540.763 - 698.312i) q^{77} +193.286 q^{79} -807.820 q^{80} -1318.62i q^{82} +432.493 q^{83} -299.007 q^{85} -1620.41i q^{86} -136.413 q^{88} +924.591 q^{89} +(-788.312 + 1017.98i) q^{91} -1222.01i q^{92} -581.311i q^{94} +479.378i q^{95} -1089.35i q^{97} +(-1297.60 + 335.404i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			3.90742i			1.38148i	0.723102	+	0.690741i	$0.242715\pi$
$2$			3.90742i			1.38148i	−0.723102	+	0.690741i	$0.757285\pi$
$3$		0			0
$3$		0			0
$4$	−7.26794			−0.908492
$5$	11.6534			1.04231			0.521156	−	0.853462i	$-0.325501\pi$
$5$	11.6534			1.04231			0.521156	+	0.853462i	$0.325501\pi$
$6$		0			0
$7$	−14.6430	−	11.3394i	−0.790650	−	0.612268i
$7$	−14.6430	−	11.3394i	−0.790650	−	0.612268i
$8$			2.86048i			0.126416i
$9$		0			0
$10$			45.5347i			1.43993i
$11$			47.6890i			1.30716i	0.756857	+	0.653581i	$0.226734\pi$
$11$			47.6890i			1.30716i	−0.756857	+	0.653581i	$0.773266\pi$
$12$		0			0
$13$		−	69.5200i		−	1.48318i	−0.670852	−	0.741591i	$-0.734071\pi$
$13$		−	69.5200i		−	1.48318i	0.670852	−	0.741591i	$-0.265929\pi$
$14$	44.3077	−	57.2166i	0.845837	−	1.09227i
$15$		0			0
$16$	−69.3206			−1.08313
$17$	−25.6583			−0.366062			−0.183031	−	0.983107i	$-0.558591\pi$
$17$	−25.6583			−0.366062			−0.183031	+	0.983107i	$0.558591\pi$
$18$		0			0
$19$			41.1363i			0.496701i	0.968670	+	0.248351i	$0.0798885\pi$
$19$			41.1363i			0.496701i	−0.968670	+	0.248351i	$0.920111\pi$
$20$	−84.6961			−0.946932
$21$		0			0
$22$	−186.341			−1.80582
$23$			168.137i			1.52430i	0.647398	+	0.762152i	$0.275857\pi$
$23$			168.137i			1.52430i	−0.647398	+	0.762152i	$0.724143\pi$
$24$		0			0
$25$	10.8016			0.0864128
$26$	271.644			2.04899
$27$		0			0
$28$	106.425	+	82.4138i	0.718300	+	0.556241i
$29$		−	32.5956i		−	0.208719i	−0.994540	−	0.104360i	$-0.966721\pi$
$29$		−	32.5956i		−	0.208719i	0.994540	−	0.104360i	$-0.0332793\pi$
$30$		0			0
$31$			171.472i			0.993461i	0.867905	+	0.496731i	$0.165466\pi$
$31$			171.472i			0.993461i	−0.867905	+	0.496731i	$0.834534\pi$
$32$		−	247.981i		−	1.36991i
$33$		0			0
$34$		−	100.258i		−	0.505708i
$35$	−170.641	−	132.142i	−0.824104	−	0.638174i
$36$		0			0
$37$	−3.86757			−0.0171844			−0.00859222	−	0.999963i	$-0.502735\pi$
$37$	−3.86757			−0.0171844			−0.00859222	+	0.999963i	$0.502735\pi$
$38$	−160.737			−0.686184
$39$		0			0
$40$			33.3343i			0.131765i
$41$	−337.467			−1.28545			−0.642725	−	0.766097i	$-0.722196\pi$
$41$	−337.467			−1.28545			−0.642725	+	0.766097i	$0.722196\pi$
$42$		0			0
$43$	−414.701			−1.47073			−0.735364	−	0.677672i	$-0.762989\pi$
$43$	−414.701			−1.47073			−0.735364	+	0.677672i	$0.762989\pi$
$44$		−	346.601i		−	1.18755i
$45$		0			0
$46$	−656.982			−2.10580
$47$	−148.771			−0.461713			−0.230856	−	0.972988i	$-0.574153\pi$
$47$	−148.771			−0.461713			−0.230856	+	0.972988i	$0.574153\pi$
$48$		0			0
$49$	85.8377	+	332.086i	0.250256	+	0.968180i
$50$			42.2064i			0.119378i
$51$		0			0
$52$			505.267i			1.34746i
$53$		−	59.7358i		−	0.154818i	−0.996999	−	0.0774089i	$-0.975335\pi$
$53$		−	59.7358i		−	0.154818i	0.996999	−	0.0774089i	$-0.0246647\pi$
$54$		0			0
$55$			555.739i			1.36247i
$56$	32.4360	−	41.8861i	0.0774007	−	0.0999512i
$57$		0			0
$58$	127.365			0.288342
$59$	−564.112			−1.24476			−0.622382	−	0.782714i	$-0.713835\pi$
$59$	−564.112			−1.24476			−0.622382	+	0.782714i	$0.713835\pi$
$60$		0			0
$61$			660.091i			1.38551i	0.721174	+	0.692754i	$0.243603\pi$
$61$			660.091i			1.38551i	−0.721174	+	0.692754i	$0.756397\pi$
$62$	−670.014			−1.37245
$63$		0			0
$64$	414.401			0.809377
$65$		−	810.144i		−	1.54594i
$66$		0			0
$67$	−291.978			−0.532401			−0.266200	−	0.963918i	$-0.585768\pi$
$67$	−291.978			−0.532401			−0.266200	+	0.963918i	$0.585768\pi$
$68$	186.483			0.332565
$69$		0			0
$70$	516.335	−	666.767i	0.881626	−	1.13848i
$71$		−	3.05665i		−	0.00510927i	−0.999997	−	0.00255464i	$-0.999187\pi$
$71$		−	3.05665i		−	0.00510927i	0.999997	−	0.00255464i	$-0.000813166\pi$
$72$		0			0
$73$		−	506.596i		−	0.812227i	−0.913823	−	0.406114i	$-0.866884\pi$
$73$		−	506.596i		−	0.812227i	0.913823	−	0.406114i	$-0.133116\pi$
$74$		−	15.1122i		−	0.0237400i
$75$		0			0
$76$		−	298.976i		−	0.451249i
$77$	540.763	−	698.312i	0.800333	−	1.03351i
$78$		0			0
$79$	193.286			0.275270			0.137635	−	0.990483i	$-0.456050\pi$
$79$	193.286			0.275270			0.137635	+	0.990483i	$0.456050\pi$
$80$	−807.820			−1.12896
$81$		0			0
$82$		−	1318.62i		−	1.77583i
$83$	432.493			0.571955			0.285977	−	0.958236i	$-0.407682\pi$
$83$	432.493			0.571955			0.285977	+	0.958236i	$0.407682\pi$
$84$		0			0
$85$	−299.007			−0.381551
$86$		−	1620.41i		−	2.03178i
$87$		0			0
$88$	−136.413			−0.165247
$89$	924.591			1.10120			0.550598	−	0.834771i	$-0.314400\pi$
$89$	924.591			1.10120			0.550598	+	0.834771i	$0.314400\pi$
$90$		0			0
$91$	−788.312	+	1017.98i	−0.908105	+	1.17268i
$92$		−	1222.01i		−	1.38482i
$93$		0			0
$94$		−	581.311i		−	0.637848i
$95$			479.378i			0.517717i
$96$		0			0
$97$		−	1089.35i		−	1.14027i	−0.821551	−	0.570135i	$-0.806891\pi$
$97$		−	1089.35i		−	1.14027i	0.821551	−	0.570135i	$-0.193109\pi$
$98$	−1297.60	+	335.404i	−1.33752	+	0.345724i
$99$		0			0
$100$	−78.5053			−0.0785053
$101$	337.598			0.332597			0.166298	−	0.986075i	$-0.446819\pi$
$101$	337.598			0.332597			0.166298	+	0.986075i	$0.446819\pi$
$102$		0			0
$103$		−	212.071i		−	0.202873i	−0.994842	−	0.101437i	$-0.967656\pi$
$103$		−	212.071i		−	0.202873i	0.994842	−	0.101437i	$-0.0323439\pi$
$104$	198.860			0.187499
$105$		0			0
$106$	233.413			0.213878
$107$		−	399.253i		−	0.360722i	−0.983600	−	0.180361i	$-0.942273\pi$
$107$		−	399.253i		−	0.360722i	0.983600	−	0.180361i	$-0.0577266\pi$
$108$		0			0
$109$	1847.45			1.62343			0.811714	−	0.584055i	$-0.198535\pi$
$109$	1847.45			1.62343			0.811714	+	0.584055i	$0.198535\pi$
$110$	−2171.51			−1.88223
$111$		0			0
$112$	1015.06	+	786.051i	0.856380	+	0.663168i
$113$			1272.16i			1.05906i	0.848290	+	0.529532i	$0.177633\pi$
$113$			1272.16i			1.05906i	−0.848290	+	0.529532i	$0.822367\pi$
$114$		0			0
$115$			1959.37i			1.58880i
$116$			236.903i			0.189620i
$117$		0			0
$118$		−	2204.22i		−	1.71962i
$119$	375.716	+	290.949i	0.289427	+	0.224128i
$120$		0			0
$121$	−943.241			−0.708671
$122$	−2579.25			−1.91405
$123$		0			0
$124$		−	1246.25i		−	0.902552i
$125$	−1330.80			−0.952242
$126$		0			0
$127$	−1835.56			−1.28252			−0.641259	−	0.767325i	$-0.721588\pi$
$127$	−1835.56			−1.28252			−0.641259	+	0.767325i	$0.721588\pi$
$128$		−	364.608i		−	0.251774i
$129$		0			0
$130$	3165.57			2.13569
$131$	−1347.40			−0.898646			−0.449323	−	0.893369i	$-0.648335\pi$
$131$	−1347.40			−0.898646			−0.449323	+	0.893369i	$0.648335\pi$
$132$		0			0
$133$	466.460	−	602.362i	0.304114	−	0.392717i
$134$		−	1140.88i		−	0.735502i
$135$		0			0
$136$		−	73.3951i		−	0.0462763i
$137$			1113.62i			0.694474i	0.937777	+	0.347237i	$0.112880\pi$
$137$			1113.62i			0.694474i	−0.937777	+	0.347237i	$0.887120\pi$
$138$		0			0
$139$		−	1316.61i		−	0.803408i	−0.915769	−	0.401704i	$-0.868418\pi$
$139$		−	1316.61i		−	0.803408i	0.915769	−	0.401704i	$-0.131582\pi$
$140$	1240.21	+	960.400i	0.748692	+	0.579776i
$141$		0			0
$142$	11.9436			0.00705836
$143$	3315.34			1.93876
$144$		0			0
$145$		−	379.850i		−	0.217550i
$146$	1979.48			1.12208
$147$		0			0
$148$	28.1092			0.0156119
$149$		−	612.224i		−	0.336613i	−0.985735	−	0.168307i	$-0.946170\pi$
$149$		−	612.224i		−	0.336613i	0.985735	−	0.168307i	$-0.0538299\pi$
$150$		0			0
$151$	2902.18			1.56408			0.782040	−	0.623228i	$-0.214179\pi$
$151$	2902.18			1.56408			0.782040	+	0.623228i	$0.214179\pi$
$152$	−117.670			−0.0627912
$153$		0			0
$154$	2728.60	+	2112.99i	1.42777	+	1.10565i
$155$			1998.23i			1.03550i
$156$		0			0
$157$		−	1343.42i		−	0.682906i	−0.939899	−	0.341453i	$-0.889081\pi$
$157$		−	1343.42i		−	0.682906i	0.939899	−	0.341453i	$-0.110919\pi$
$158$			755.250i			0.380281i
$159$		0			0
$160$		−	2889.82i		−	1.42788i
$161$	1906.57	−	2462.04i	0.933283	−	1.20519i
$162$		0			0
$163$	974.519			0.468284			0.234142	−	0.972202i	$-0.424772\pi$
$163$	974.519			0.468284			0.234142	+	0.972202i	$0.424772\pi$
$164$	2452.69			1.16782
$165$		0			0
$166$			1689.93i			0.790145i
$167$	2842.93			1.31732			0.658661	−	0.752440i	$-0.271123\pi$
$167$	2842.93			1.31732			0.658661	+	0.752440i	$0.271123\pi$
$168$		0			0
$169$	−2636.03			−1.19983
$170$		−	1168.34i		−	0.527106i
$171$		0			0
$172$	3014.02			1.33615
$173$	−3981.82			−1.74990			−0.874949	−	0.484215i	$-0.839105\pi$
$173$	−3981.82			−1.74990			−0.874949	+	0.484215i	$0.839105\pi$
$174$		0			0
$175$	−158.168	−	122.483i	−0.0683223	−	0.0529078i
$176$		−	3305.83i		−	1.41583i
$177$		0			0
$178$			3612.76i			1.52128i
$179$			2465.66i			1.02956i	0.857321	+	0.514782i	$0.172127\pi$
$179$			2465.66i			1.02956i	−0.857321	+	0.514782i	$0.827873\pi$
$180$		0			0
$181$			2517.90i			1.03400i	0.855986	+	0.516999i	$0.172951\pi$
$181$			2517.90i			1.03400i	−0.855986	+	0.516999i	$0.827049\pi$
$182$	−3977.69	−	3080.27i	−1.62003	−	1.25453i
$183$		0			0
$184$	−480.952			−0.192697
$185$	−45.0703			−0.0179115
$186$		0			0
$187$		−	1223.62i		−	0.478503i
$188$	1081.26			0.419462
$189$		0			0
$190$	−1873.13			−0.715217
$191$			461.101i			0.174681i	0.996179	+	0.0873405i	$0.0278368\pi$
$191$			461.101i			0.174681i	−0.996179	+	0.0873405i	$0.972163\pi$
$192$		0			0
$193$	−2562.28			−0.955630			−0.477815	−	0.878460i	$-0.658571\pi$
$193$	−2562.28			−0.955630			−0.477815	+	0.878460i	$0.658571\pi$
$194$	4256.53			1.57526
$195$		0			0
$196$	−623.863	−	2413.58i	−0.227355	−	0.879584i
$197$			3764.93i			1.36162i	0.732458	+	0.680812i	$0.238373\pi$
$197$			3764.93i			1.36162i	−0.732458	+	0.680812i	$0.761627\pi$
$198$		0			0
$199$			958.945i			0.341597i	0.985306	+	0.170799i	$0.0546347\pi$
$199$			958.945i			0.341597i	−0.985306	+	0.170799i	$0.945365\pi$
$200$			30.8977i			0.0109240i
$201$		0			0
$202$			1319.14i			0.459476i
$203$	−369.613	+	477.299i	−0.127792	+	0.165024i
$204$		0			0
$205$	−3932.63			−1.33984
$206$	828.650			0.280266
$207$		0			0
$208$			4819.17i			1.60649i
$209$	−1961.75			−0.649269
$210$		0			0
$211$	−4036.36			−1.31694			−0.658471	−	0.752606i	$-0.728796\pi$
$211$	−4036.36			−1.31694			−0.658471	+	0.752606i	$0.728796\pi$
$212$			434.156i			0.140651i
$213$		0			0
$214$	1560.05			0.498331
$215$	−4832.68			−1.53296
$216$		0			0
$217$	1944.38	−	2510.87i	0.608265	−	0.785481i
$218$			7218.77i			2.24274i
$219$		0			0
$220$		−	4039.07i		−	1.23779i
$221$			1783.77i			0.542937i
$222$		0			0
$223$			5853.39i			1.75772i	0.477079	+	0.878860i	$0.341696\pi$
$223$			5853.39i			1.75772i	−0.477079	+	0.878860i	$0.658304\pi$
$224$	−2811.95	+	3631.20i	−0.838754	+	1.08312i
$225$		0			0
$226$	−4970.85			−1.46308
$227$	6076.75			1.77678			0.888388	−	0.459094i	$-0.151826\pi$
$227$	6076.75			1.77678			0.888388	+	0.459094i	$0.151826\pi$
$228$		0			0
$229$			940.297i			0.271339i	0.990754	+	0.135669i	$0.0433185\pi$
$229$			940.297i			0.271339i	−0.990754	+	0.135669i	$0.956682\pi$
$230$	−7656.07			−2.19490
$231$		0			0
$232$	93.2390			0.0263855
$233$		−	3314.66i		−	0.931976i	−0.884791	−	0.465988i	$-0.845699\pi$
$233$		−	3314.66i		−	0.931976i	0.884791	−	0.465988i	$-0.154301\pi$
$234$		0			0
$235$	−1733.69			−0.481248
$236$	4099.93			1.13086
$237$		0			0
$238$	−1136.86	+	1468.08i	−0.309629	+	0.399839i
$239$			2556.27i			0.691848i	0.938263	+	0.345924i	$0.112434\pi$
$239$			2556.27i			0.691848i	−0.938263	+	0.345924i	$0.887566\pi$
$240$		0			0
$241$			2079.32i			0.555772i	0.960614	+	0.277886i	$0.0896337\pi$
$241$			2079.32i			0.555772i	−0.960614	+	0.277886i	$0.910366\pi$
$242$		−	3685.64i		−	0.979016i
$243$		0			0
$244$		−	4797.50i		−	1.25872i
$245$	1000.30	+	3869.93i	0.260844	+	1.00914i
$246$		0			0
$247$	2859.80			0.736699
$248$	−490.492			−0.125590
$249$		0			0
$250$		−	5199.99i		−	1.31551i
$251$	4718.74			1.18663			0.593316	−	0.804970i	$-0.297819\pi$
$251$	4718.74			1.18663			0.593316	+	0.804970i	$0.297819\pi$
$252$		0			0
$253$	−8018.29			−1.99251
$254$		−	7172.31i		−	1.77178i
$255$		0			0
$256$	4739.88			1.15720
$257$	4802.90			1.16575			0.582873	−	0.812563i	$-0.301929\pi$
$257$	4802.90			1.16575			0.582873	+	0.812563i	$0.301929\pi$
$258$		0			0
$259$	56.6330	+	43.8557i	0.0135869	+	0.0105215i
$260$			5888.08i			1.40447i
$261$		0			0
$262$		−	5264.85i		−	1.24146i
$263$		−	2695.07i		−	0.631883i	−0.948779	−	0.315941i	$-0.897680\pi$
$263$		−	2695.07i		−	0.631883i	0.948779	−	0.315941i	$-0.102320\pi$
$264$		0			0
$265$		−	696.125i		−	0.161368i
$266$	2353.68	+	1822.66i	0.542532	+	0.420128i
$267$		0			0
$268$	2122.08			0.483682
$269$	−4182.34			−0.947961			−0.473981	−	0.880535i	$-0.657183\pi$
$269$	−4182.34			−0.947961			−0.473981	+	0.880535i	$0.657183\pi$
$270$		0			0
$271$		−	654.173i		−	0.146635i	−0.997309	−	0.0733176i	$-0.976641\pi$
$271$		−	654.173i		−	0.146635i	0.997309	−	0.0733176i	$-0.0233587\pi$
$272$	1778.65			0.396495
$273$		0			0
$274$	−4351.38			−0.959403
$275$			515.117i			0.112955i
$276$		0			0
$277$	−3657.50			−0.793350			−0.396675	−	0.917959i	$-0.629836\pi$
$277$	−3657.50			−0.793350			−0.396675	+	0.917959i	$0.629836\pi$
$278$	5144.57			1.10989
$279$		0			0
$280$	377.989	−	488.115i	0.0806756	−	0.104180i
$281$		−	2832.80i		−	0.601390i	−0.953720	−	0.300695i	$-0.902781\pi$
$281$		−	2832.80i		−	0.601390i	0.953720	−	0.300695i	$-0.0972187\pi$
$282$		0			0
$283$			9097.18i			1.91085i	0.295231	+	0.955426i	$0.404603\pi$
$283$			9097.18i			1.91085i	−0.295231	+	0.955426i	$0.595397\pi$
$284$			22.2156i			0.00464173i
$285$		0			0
$286$			12954.4i			2.67836i
$287$	4941.54	+	3826.66i	1.01634	+	0.787040i
$288$		0			0
$289$	−4254.65			−0.865998
$290$	1484.23			0.300542
$291$		0			0
$292$			3681.91i			0.737902i
$293$	3076.89			0.613494			0.306747	−	0.951791i	$-0.400759\pi$
$293$	3076.89			0.613494			0.306747	+	0.951791i	$0.400759\pi$
$294$		0			0
$295$	−6573.81			−1.29743
$296$		−	11.0631i		−	0.00217239i
$297$		0			0
$298$	2392.22			0.465025
$299$	11688.9			2.26082
$300$		0			0
$301$	6072.49	+	4702.45i	1.16283	+	0.900480i
$302$			11340.0i			2.16075i
$303$		0			0
$304$		−	2851.60i		−	0.537994i
$305$			7692.30i			1.44413i
$306$		0			0
$307$			3006.74i			0.558970i	0.960150	+	0.279485i	$0.0901637\pi$
$307$			3006.74i			0.558970i	−0.960150	+	0.279485i	$0.909836\pi$
$308$	−3930.23	+	5075.29i	−0.727096	+	0.938933i
$309$		0			0
$310$	−7807.93			−1.43052
$311$	−4465.06			−0.814118			−0.407059	−	0.913402i	$-0.633446\pi$
$311$	−4465.06			−0.814118			−0.407059	+	0.913402i	$0.633446\pi$
$312$		0			0
$313$		−	6937.22i		−	1.25276i	−0.779517	−	0.626382i	$-0.784535\pi$
$313$		−	6937.22i		−	1.25276i	0.779517	−	0.626382i	$-0.215465\pi$
$314$	5249.29			0.943423
$315$		0			0
$316$	−1404.79			−0.250081
$317$			5697.23i			1.00943i	0.863287	+	0.504713i	$0.168402\pi$
$317$			5697.23i			1.00943i	−0.863287	+	0.504713i	$0.831598\pi$
$318$		0			0
$319$	1554.45			0.272830
$320$	4829.18			0.843623
$321$		0			0
$322$	9620.22	+	7449.76i	1.66495	+	1.28931i
$323$		−	1055.49i		−	0.181824i
$324$		0			0
$325$		−	750.927i		−	0.128166i
$326$			3807.86i			0.646925i
$327$		0			0
$328$		−	965.315i		−	0.162502i
$329$	2178.46	+	1686.97i	0.365053	+	0.282692i
$330$		0			0
$331$	7856.62			1.30465			0.652325	−	0.757940i	$-0.273794\pi$
$331$	7856.62			1.30465			0.652325	+	0.757940i	$0.273794\pi$
$332$	−3143.33			−0.519617
$333$		0			0
$334$			11108.5i			1.81986i
$335$	−3402.54			−0.554927
$336$		0			0
$337$	−4895.60			−0.791336			−0.395668	−	0.918394i	$-0.629487\pi$
$337$	−4895.60			−0.791336			−0.395668	+	0.918394i	$0.629487\pi$
$338$		−	10300.1i		−	1.65755i
$339$		0			0
$340$	2173.16			0.346636
$341$	−8177.34			−1.29861
$342$		0			0
$343$	2508.71	−	5836.09i	0.394921	−	0.918715i
$344$		−	1186.24i		−	0.185924i
$345$		0			0
$346$		−	15558.7i		−	2.41745i
$347$			2097.12i			0.324436i	0.986755	+	0.162218i	$0.0518647\pi$
$347$			2097.12i			0.324436i	−0.986755	+	0.162218i	$0.948135\pi$
$348$		0			0
$349$		−	8458.86i		−	1.29740i	−0.761045	−	0.648700i	$-0.775313\pi$
$349$		−	8458.86i		−	1.29740i	0.761045	−	0.648700i	$-0.224687\pi$
$350$	478.593	−	618.030i	0.0730911	−	0.0943860i
$351$		0			0
$352$	11826.0			1.79070
$353$	−632.890			−0.0954258			−0.0477129	−	0.998861i	$-0.515193\pi$
$353$	−632.890			−0.0954258			−0.0477129	+	0.998861i	$0.515193\pi$
$354$		0			0
$355$		−	35.6204i		−	0.00532545i
$356$	−6719.87			−1.00043
$357$		0			0
$358$	−9634.36			−1.42232
$359$			7038.13i			1.03470i	0.855773	+	0.517351i	$0.173082\pi$
$359$			7038.13i			1.03470i	−0.855773	+	0.517351i	$0.826918\pi$
$360$		0			0
$361$	5166.80			0.753288
$362$	−9838.48			−1.42845
$363$		0			0
$364$	5729.41	−	7398.65i	0.825007	−	1.06537i
$365$		−	5903.56i		−	0.846594i
$366$		0			0
$367$			683.183i			0.0971712i	0.998819	+	0.0485856i	$0.0154714\pi$
$367$			683.183i			0.0971712i	−0.998819	+	0.0485856i	$0.984529\pi$
$368$		−	11655.4i		−	1.65103i
$369$		0			0
$370$		−	176.109i		−	0.0247445i
$371$	−677.366	+	874.714i	−0.0947900	+	0.122407i
$372$		0			0
$373$	9047.99			1.25600			0.627999	−	0.778214i	$-0.283874\pi$
$373$	9047.99			1.25600			0.627999	+	0.778214i	$0.283874\pi$
$374$	4781.20			0.661043
$375$		0			0
$376$		−	425.556i		−	0.0583681i
$377$	−2266.05			−0.309569
$378$		0			0
$379$	6790.65			0.920349			0.460175	−	0.887828i	$-0.347787\pi$
$379$	6790.65			0.920349			0.460175	+	0.887828i	$0.347787\pi$
$380$		−	3484.09i		−	0.470342i
$381$		0			0
$382$	−1801.71			−0.241319
$383$	2341.19			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$383$	2341.19			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$384$		0			0
$385$	6301.72	−	8137.71i	0.834196	−	1.07724i
$386$		−	10011.9i		−	1.32019i
$387$		0			0
$388$			7917.29i			1.03593i
$389$		−	1758.06i		−	0.229145i	−0.993415	−	0.114572i	$-0.963450\pi$
$389$		−	1758.06i		−	0.229145i	0.993415	−	0.114572i	$-0.0365498\pi$
$390$		0			0
$391$		−	4314.12i		−	0.557990i
$392$	−949.923	+	245.537i	−0.122394	+	0.0316364i
$393$		0			0
$394$	−14711.2			−1.88106
$395$	2252.44			0.286918
$396$		0			0
$397$			1120.19i			0.141614i	0.997490	+	0.0708071i	$0.0225575\pi$
$397$			1120.19i			0.141614i	−0.997490	+	0.0708071i	$0.977443\pi$
$398$	−3747.00			−0.471910
$399$		0			0
$400$	−748.773			−0.0935966
$401$		−	1764.88i		−	0.219785i	−0.993943	−	0.109893i	$-0.964949\pi$
$401$		−	1764.88i		−	0.219785i	0.993943	−	0.109893i	$-0.0350507\pi$
$402$		0			0
$403$	11920.7			1.47349
$404$	−2453.64			−0.302162
$405$		0			0
$406$	−1865.01	−	1444.24i	−0.227977	−	0.176542i
$407$		−	184.440i		−	0.0224628i
$408$		0			0
$409$			6776.10i			0.819210i	0.912263	+	0.409605i	$0.134333\pi$
$409$			6776.10i			0.819210i	−0.912263	+	0.409605i	$0.865667\pi$
$410$		−	15366.4i		−	1.85096i
$411$		0			0
$412$			1541.32i			0.184309i
$413$	8260.31	+	6396.67i	0.984173	+	0.762129i
$414$		0			0
$415$	5040.01			0.596155
$416$	−17239.6			−2.03183
$417$		0			0
$418$		−	7665.39i		−	0.896953i
$419$	10707.1			1.24839			0.624197	−	0.781267i	$-0.285426\pi$
$419$	10707.1			1.24839			0.624197	+	0.781267i	$0.285426\pi$
$420$		0			0
$421$	−2849.72			−0.329898			−0.164949	−	0.986302i	$-0.552746\pi$
$421$	−2849.72			−0.329898			−0.164949	+	0.986302i	$0.552746\pi$
$422$		−	15771.8i		−	1.81933i
$423$		0			0
$424$	170.873			0.0195715
$425$	−277.151			−0.0316325
$426$		0			0
$427$	7485.01	−	9665.75i	0.848303	−	1.09545i
$428$			2901.75i			0.327713i
$429$		0			0
$430$		−	18883.3i		−	2.11775i
$431$		−	10469.5i		−	1.17007i	−0.811010	−	0.585033i	$-0.801082\pi$
$431$		−	10469.5i		−	1.17007i	0.811010	−	0.585033i	$-0.198918\pi$
$432$		0			0
$433$			5079.76i			0.563782i	0.959446	+	0.281891i	$0.0909617\pi$
$433$			5079.76i			0.563782i	−0.959446	+	0.281891i	$0.909038\pi$
$434$	9811.04	+	7597.53i	1.08513	+	0.840307i
$435$		0			0
$436$	−13427.2			−1.47487
$437$	−6916.54			−0.757124
$438$		0			0
$439$			7502.14i			0.815621i	0.913067	+	0.407811i	$0.133708\pi$
$439$			7502.14i			0.815621i	−0.913067	+	0.407811i	$0.866292\pi$
$440$	−1589.68			−0.172238
$441$		0			0
$442$	−6969.93			−0.750058
$443$		−	2758.00i		−	0.295794i	−0.989003	−	0.147897i	$-0.952750\pi$
$443$		−	2758.00i		−	0.295794i	0.989003	−	0.147897i	$-0.0472504\pi$
$444$		0			0
$445$	10774.6			1.14779
$446$	−22871.6			−2.42826
$447$		0			0
$448$	−6068.09	−	4699.04i	−0.639934	−	0.495556i
$449$			8061.59i			0.847327i	0.905820	+	0.423663i	$0.139256\pi$
$449$			8061.59i			0.847327i	−0.905820	+	0.423663i	$0.860744\pi$
$450$		0			0
$451$		−	16093.4i		−	1.68029i
$452$		−	9245.94i		−	0.962152i
$453$		0			0
$454$			23744.4i			2.45458i
$455$	−9186.52	+	11863.0i	−0.946529	+	1.22230i
$456$		0			0
$457$	−216.632			−0.0221742			−0.0110871	−	0.999939i	$-0.503529\pi$
$457$	−216.632			−0.0221742			−0.0110871	+	0.999939i	$0.503529\pi$
$458$	−3674.14			−0.374850
$459$		0			0
$460$		−	14240.6i		−	1.44341i
$461$	−1188.86			−0.120110			−0.0600552	−	0.998195i	$-0.519128\pi$
$461$	−1188.86			−0.120110			−0.0600552	+	0.998195i	$0.519128\pi$
$462$		0			0
$463$	−4794.34			−0.481235			−0.240617	−	0.970620i	$-0.577350\pi$
$463$	−4794.34			−0.481235			−0.240617	+	0.970620i	$0.577350\pi$
$464$			2259.55i			0.226071i
$465$		0			0
$466$	12951.8			1.28751
$467$	2953.79			0.292687			0.146344	−	0.989234i	$-0.453249\pi$
$467$	2953.79			0.292687			0.146344	+	0.989234i	$0.453249\pi$
$468$		0			0
$469$	4275.46	+	3310.85i	0.420943	+	0.325972i
$470$		−	6774.25i		−	0.664836i
$471$		0			0
$472$		−	1613.63i		−	0.157359i
$473$		−	19776.7i		−	1.92248i
$474$		0			0
$475$			444.338i			0.0429213i
$476$	−2730.68	−	2114.60i	−0.262942	−	0.203619i
$477$		0			0
$478$	−9988.44			−0.955775
$479$	−109.238			−0.0104201			−0.00521003	−	0.999986i	$-0.501658\pi$
$479$	−109.238			−0.0104201			−0.00521003	+	0.999986i	$0.501658\pi$
$480$		0			0
$481$			268.873i			0.0254877i
$482$	−8124.79			−0.767788
$483$		0			0
$484$	6855.42			0.643822
$485$		−	12694.6i		−	1.18852i
$486$		0			0
$487$	−3614.94			−0.336363			−0.168182	−	0.985756i	$-0.553789\pi$
$487$	−3614.94			−0.336363			−0.168182	+	0.985756i	$0.553789\pi$
$488$	−1888.18			−0.175151
$489$		0			0
$490$	−15121.4	+	3908.60i	−1.39412	+	0.360352i
$491$			18799.1i			1.72788i	0.503594	+	0.863940i	$0.332011\pi$
$491$			18799.1i			1.72788i	−0.503594	+	0.863940i	$0.667989\pi$
$492$		0			0
$493$			836.349i			0.0764042i
$494$			11174.4i			1.01774i
$495$		0			0
$496$		−	11886.5i		−	1.07605i
$497$	−34.6605	+	44.7587i	−0.00312824	+	0.00403965i
$498$		0			0
$499$	3241.77			0.290825			0.145413	−	0.989371i	$-0.453549\pi$
$499$	3241.77			0.290825			0.145413	+	0.989371i	$0.453549\pi$
$500$	9672.16			0.865105
$501$		0			0
$502$			18438.1i			1.63931i
$503$	−4737.83			−0.419979			−0.209989	−	0.977704i	$-0.567343\pi$
$503$	−4737.83			−0.419979			−0.209989	+	0.977704i	$0.567343\pi$
$504$		0			0
$505$	3934.16			0.346669
$506$		−	31330.8i		−	2.75262i
$507$		0			0
$508$	13340.7			1.16516
$509$	2544.77			0.221601			0.110800	−	0.993843i	$-0.464659\pi$
$509$	2544.77			0.221601			0.110800	+	0.993843i	$0.464659\pi$
$510$		0			0
$511$	−5744.48	+	7418.11i	−0.497301	+	0.642188i
$512$			15603.9i			1.34687i
$513$		0			0
$514$			18766.9i			1.61046i
$515$		−	2471.35i		−	0.211457i
$516$		0			0
$517$		−	7094.75i		−	0.603533i
$518$	−171.363	+	221.289i	−0.0145352	+	0.0187700i
$519$		0			0
$520$	2317.40			0.195432
$521$	7623.86			0.641089			0.320545	−	0.947233i	$-0.396134\pi$
$521$	7623.86			0.641089			0.320545	+	0.947233i	$0.396134\pi$
$522$		0			0
$523$		−	20941.4i		−	1.75087i	−0.483335	−	0.875435i	$-0.660575\pi$
$523$		−	20941.4i		−	1.75087i	0.483335	−	0.875435i	$-0.339425\pi$
$524$	9792.79			0.816413
$525$		0			0
$526$	10530.8			0.872935
$527$		−	4399.69i		−	0.363669i
$528$		0			0
$529$	−16103.1			−1.32350
$530$	2720.05			0.222927
$531$		0			0
$532$	−3390.20	+	4377.93i	−0.276286	+	0.356780i
$533$			23460.7i			1.90656i
$534$		0			0
$535$		−	4652.65i		−	0.375985i
$536$		−	835.198i		−	0.0673042i
$537$		0			0
$538$		−	16342.1i		−	1.30959i
$539$	−15836.8	+	4093.52i	−1.26557	+	0.327125i
$540$		0			0
$541$	−6660.29			−0.529294			−0.264647	−	0.964345i	$-0.585255\pi$
$541$	−6660.29			−0.529294			−0.264647	+	0.964345i	$0.585255\pi$
$542$	2556.13			0.202574
$543$		0			0
$544$			6362.78i			0.501474i
$545$	21529.1			1.69212
$546$		0			0
$547$	21559.6			1.68523			0.842616	−	0.538515i	$-0.181015\pi$
$547$	21559.6			1.68523			0.842616	+	0.538515i	$0.181015\pi$
$548$		−	8093.71i		−	0.630924i
$549$		0			0
$550$	−2012.78			−0.156046
$551$	1340.86			0.103671
$552$		0			0
$553$	−2830.30	−	2191.74i	−0.217643	−	0.168539i
$554$		−	14291.4i		−	1.09600i
$555$		0			0
$556$			9569.07i			0.729890i
$557$		−	15772.0i		−	1.19979i	−0.800079	−	0.599895i	$-0.795209\pi$
$557$		−	15772.0i		−	1.19979i	0.800079	−	0.599895i	$-0.204791\pi$
$558$		0			0
$559$			28830.0i			2.18136i
$560$	11828.9	+	9160.16i	0.892615	+	0.691228i
$561$		0			0
$562$	11068.9			0.830810
$563$	11684.5			0.874673			0.437337	−	0.899298i	$-0.355922\pi$
$563$	11684.5			0.874673			0.437337	+	0.899298i	$0.355922\pi$
$564$		0			0
$565$			14824.9i			1.10387i
$566$	−35546.5			−2.63981
$567$		0			0
$568$	8.74349			0.000645896
$569$			6707.58i			0.494194i	0.968991	+	0.247097i	$0.0794766\pi$
$569$			6707.58i			0.494194i	−0.968991	+	0.247097i	$0.920523\pi$
$570$		0			0
$571$	−18572.0			−1.36115			−0.680573	−	0.732681i	$-0.738269\pi$
$571$	−18572.0			−1.36115			−0.680573	+	0.732681i	$0.738269\pi$
$572$	−24095.7			−1.76135
$573$		0			0
$574$	−14952.4	+	19308.7i	−1.08728	+	1.40406i
$575$			1816.15i			0.131719i
$576$		0			0
$577$			8147.17i			0.587818i	0.955833	+	0.293909i	$0.0949563\pi$
$577$			8147.17i			0.587818i	−0.955833	+	0.293909i	$0.905044\pi$
$578$		−	16624.7i		−	1.19636i
$579$		0			0
$580$			2760.72i			0.197643i
$581$	−6333.01	−	4904.19i	−0.452216	−	0.350190i
$582$		0			0
$583$	2848.74			0.202372
$584$	1449.11			0.102679
$585$		0			0
$586$			12022.7i			0.847531i
$587$	4140.09			0.291107			0.145553	−	0.989350i	$-0.453504\pi$
$587$	4140.09			0.291107			0.145553	+	0.989350i	$0.453504\pi$
$588$		0			0
$589$	−7053.74			−0.493454
$590$		−	25686.7i		−	1.79238i
$591$		0			0
$592$	268.102			0.0186130
$593$	14381.9			0.995945			0.497973	−	0.867193i	$-0.334078\pi$
$593$	14381.9			0.995945			0.497973	+	0.867193i	$0.334078\pi$
$594$		0			0
$595$	4378.37	+	3390.54i	0.301673	+	0.233611i
$596$			4449.61i			0.305811i
$597$		0			0
$598$			45673.4i			3.12328i
$599$			13879.2i			0.946724i	0.880868	+	0.473362i	$0.156960\pi$
$599$			13879.2i			0.946724i	−0.880868	+	0.473362i	$0.843040\pi$
$600$		0			0
$601$		−	14404.6i		−	0.977661i	−0.872379	−	0.488831i	$-0.837424\pi$
$601$		−	14404.6i		−	0.977661i	0.872379	−	0.488831i	$-0.162576\pi$
$602$	−18374.4	+	23727.8i	−1.24400	+	1.60643i
$603$		0			0
$604$	−21092.9			−1.42095
$605$	−10992.0			−0.738656
$606$		0			0
$607$		−	19888.2i		−	1.32988i	−0.746898	−	0.664938i	$-0.768458\pi$
$607$		−	19888.2i		−	1.32988i	0.746898	−	0.664938i	$-0.231542\pi$
$608$	10201.0			0.680438
$609$		0			0
$610$	−30057.1			−1.99504
$611$			10342.6i			0.684804i
$612$		0			0
$613$	8096.44			0.533462			0.266731	−	0.963771i	$-0.414057\pi$
$613$	8096.44			0.533462			0.266731	+	0.963771i	$0.414057\pi$
$614$	−11748.6			−0.772207
$615$		0			0
$616$	1997.51	+	1546.84i	0.130652	+	0.101175i
$617$		−	9405.05i		−	0.613668i	−0.951763	−	0.306834i	$-0.900730\pi$
$617$		−	9405.05i		−	0.613668i	0.951763	−	0.306834i	$-0.0992696\pi$
$618$		0			0
$619$		−	539.359i		−	0.0350221i	−0.999847	−	0.0175110i	$-0.994426\pi$
$619$		−	539.359i		−	0.0350221i	0.999847	−	0.0175110i	$-0.00557422\pi$
$620$		−	14523.0i		−	0.940740i
$621$		0			0
$622$		−	17446.9i		−	1.12469i
$623$	−13538.8	−	10484.3i	−0.870661	−	0.674227i
$624$		0			0
$625$	−16858.5			−1.07895
$626$	27106.6			1.73067
$627$		0			0
$628$			9763.87i			0.620415i
$629$	99.2353			0.00629057
$630$		0			0
$631$	−14096.7			−0.889349			−0.444675	−	0.895692i	$-0.646681\pi$
$631$	−14096.7			−0.889349			−0.444675	+	0.895692i	$0.646681\pi$
$632$			552.890i			0.0347987i
$633$		0			0
$634$	−22261.5			−1.39450
$635$	−21390.5			−1.33678
$636$		0			0
$637$	23086.6	−	5967.44i	1.43599	−	0.371175i
$638$			6073.90i			0.376909i
$639$		0			0
$640$		−	4248.92i		−	0.262427i
$641$		−	21909.0i		−	1.35000i	−0.737816	−	0.675002i	$-0.764143\pi$
$641$		−	21909.0i		−	1.35000i	0.737816	−	0.675002i	$-0.235857\pi$
$642$		0			0
$643$		−	16772.8i		−	1.02870i	−0.857580	−	0.514350i	$-0.828033\pi$
$643$		−	16772.8i		−	1.02870i	0.857580	−	0.514350i	$-0.171967\pi$
$644$	−13856.8	+	17893.9i	−0.847880	+	1.09491i
$645$		0			0
$646$	4124.24			0.251186
$647$	7088.41			0.430717			0.215359	−	0.976535i	$-0.430908\pi$
$647$	7088.41			0.430717			0.215359	+	0.976535i	$0.430908\pi$
$648$		0			0
$649$		−	26901.9i		−	1.62711i
$650$	2934.19			0.177059
$651$		0			0
$652$	−7082.75			−0.425432
$653$			3656.74i			0.219141i	0.993979	+	0.109571i	$0.0349476\pi$
$653$			3656.74i			0.219141i	−0.993979	+	0.109571i	$0.965052\pi$
$654$		0			0
$655$	−15701.7			−0.936669
$656$	23393.4			1.39231
$657$		0			0
$658$	−6591.70	+	8512.17i	−0.390534	+	0.504314i
$659$			7783.32i			0.460084i	0.973181	+	0.230042i	$0.0738863\pi$
$659$			7783.32i			0.460084i	−0.973181	+	0.230042i	$0.926114\pi$
$660$		0			0
$661$			7962.64i			0.468549i	0.972171	+	0.234274i	$0.0752714\pi$
$661$			7962.64i			0.468549i	−0.972171	+	0.234274i	$0.924729\pi$
$662$			30699.1i			1.80235i
$663$		0			0
$664$			1237.14i			0.0723045i
$665$	5435.84	−	7019.56i	0.316982	−	0.409333i
$666$		0			0
$667$	5480.53			0.318151
$668$	−20662.2			−1.19678
$669$		0			0
$670$		−	13295.2i		−	0.766622i
$671$	−31479.1			−1.81108
$672$		0			0
$673$	989.331			0.0566655			0.0283328	−	0.999599i	$-0.490980\pi$
$673$	989.331			0.0566655			0.0283328	+	0.999599i	$0.490980\pi$
$674$		−	19129.2i		−	1.09322i
$675$		0			0
$676$	19158.5			1.09004
$677$	22166.7			1.25840			0.629198	−	0.777245i	$-0.283384\pi$
$677$	22166.7			1.25840			0.629198	+	0.777245i	$0.283384\pi$
$678$		0			0
$679$	−12352.5	+	15951.3i	−0.698151	+	0.901555i
$680$		−	855.301i		−	0.0482343i
$681$		0			0
$682$		−	31952.3i		−	1.79401i
$683$		−	2458.98i		−	0.137760i	−0.997625	−	0.0688802i	$-0.978057\pi$
$683$		−	2458.98i		−	0.137760i	0.997625	−	0.0688802i	$-0.0219426\pi$
$684$		0			0
$685$			12977.4i			0.723858i
$686$	22804.1	+	9802.60i	1.26919	+	0.545576i
$687$		0			0
$688$	28747.3			1.59300
$689$	−4152.83			−0.229623
$690$		0			0
$691$			24420.0i			1.34440i	0.740369	+	0.672200i	$0.234651\pi$
$691$			24420.0i			1.34440i	−0.740369	+	0.672200i	$0.765349\pi$
$692$	28939.6			1.58977
$693$		0			0
$694$	−8194.32			−0.448202
$695$		−	15343.0i		−	0.837402i
$696$		0			0
$697$	8658.83			0.470555
$698$	33052.3			1.79233
$699$		0			0
$700$	1149.56	+	890.200i	0.0620703	+	0.0480663i
$701$		−	6521.09i		−	0.351353i	−0.984448	−	0.175676i	$-0.943789\pi$
$701$		−	6521.09i		−	0.351353i	0.984448	−	0.175676i	$-0.0562112\pi$
$702$		0			0
$703$		−	159.098i		−	0.00853553i
$704$			19762.4i			1.05799i
$705$		0			0
$706$		−	2472.97i		−	0.131829i
$707$	−4943.47	−	3828.15i	−0.262968	−	0.203638i
$708$		0			0
$709$	−10062.9			−0.533034			−0.266517	−	0.963830i	$-0.585873\pi$
$709$	−10062.9			−0.533034			−0.266517	+	0.963830i	$0.585873\pi$
$710$	139.184			0.00735701
$711$		0			0
$712$			2644.77i			0.139209i
$713$	−28830.8			−1.51434
$714$		0			0
$715$	38635.0			2.02079
$716$		−	17920.2i		−	0.935351i
$717$		0			0
$718$	−27500.9			−1.42942
$719$	−3449.87			−0.178941			−0.0894704	−	0.995989i	$-0.528517\pi$
$719$	−3449.87			−0.178941			−0.0894704	+	0.995989i	$0.528517\pi$
$720$		0			0
$721$	−2404.75	+	3105.36i	−0.124213	+	0.160402i
$722$			20188.9i			1.04065i
$723$		0			0
$724$		−	18299.9i		−	0.939380i
$725$		−	352.085i		−	0.0180360i
$726$		0			0
$727$		−	21151.9i		−	1.07906i	−0.841965	−	0.539532i	$-0.818601\pi$
$727$		−	21151.9i		−	1.07906i	0.841965	−	0.539532i	$-0.181399\pi$
$728$	−2911.92	−	2254.95i	−0.148246	−	0.114799i
$729$		0			0
$730$	23067.7			1.16955
$731$	10640.5			0.538378
$732$		0			0
$733$			10223.0i			0.515138i	0.966260	+	0.257569i	$0.0829214\pi$
$733$			10223.0i			0.515138i	−0.966260	+	0.257569i	$0.917079\pi$
$734$	−2669.48			−0.134240
$735$		0			0
$736$	41694.8			2.08817
$737$		−	13924.2i		−	0.695934i
$738$		0			0
$739$	−38602.2			−1.92152			−0.960761	−	0.277378i	$-0.910534\pi$
$739$	−38602.2			−1.92152			−0.960761	+	0.277378i	$0.910534\pi$
$740$	327.568			0.0162725
$741$		0			0
$742$	−3417.88	−	2646.75i	−0.169103	−	0.130951i
$743$			14937.8i			0.737570i	0.929515	+	0.368785i	$0.120226\pi$
$743$			14937.8i			0.737570i	−0.929515	+	0.368785i	$0.879774\pi$
$744$		0			0
$745$		−	7134.49i		−	0.350856i
$746$			35354.3i			1.73514i
$747$		0			0
$748$			8893.20i			0.434716i
$749$	−4527.28	+	5846.28i	−0.220859	+	0.285205i
$750$		0			0
$751$	−24660.3			−1.19823			−0.599113	−	0.800664i	$-0.704480\pi$
$751$	−24660.3			−1.19823			−0.599113	+	0.800664i	$0.704480\pi$
$752$	10312.9			0.500097
$753$		0			0
$754$		−	8854.40i		−	0.427663i
$755$	33820.2			1.63026
$756$		0			0
$757$	31078.1			1.49214			0.746071	−	0.665867i	$-0.231938\pi$
$757$	31078.1			1.49214			0.746071	+	0.665867i	$0.231938\pi$
$758$			26533.9i			1.27145i
$759$		0			0
$760$	−1371.25			−0.0654480
$761$	−3727.45			−0.177556			−0.0887780	−	0.996051i	$-0.528296\pi$
$761$	−3727.45			−0.177556			−0.0887780	+	0.996051i	$0.528296\pi$
$762$		0			0
$763$	−27052.3	−	20948.9i	−1.28356	−	0.993973i
$764$		−	3351.25i		−	0.158696i
$765$		0			0
$766$			9148.01i			0.431503i
$767$			39217.0i			1.84621i
$768$		0			0
$769$		−	6080.76i		−	0.285147i	−0.989784	−	0.142573i	$-0.954462\pi$
$769$		−	6080.76i		−	0.285147i	0.989784	−	0.142573i	$-0.0455377\pi$
$770$	31797.5	+	24623.5i	1.48818	+	1.15243i
$771$		0			0
$772$	18622.5			0.868183
$773$	−34364.4			−1.59897			−0.799483	−	0.600688i	$-0.794893\pi$
$773$	−34364.4			−1.59897			−0.799483	+	0.600688i	$0.794893\pi$
$774$		0			0
$775$			1852.17i			0.0858478i
$776$	3116.05			0.144149
$777$		0			0
$778$	6869.50			0.316560
$779$		−	13882.1i		−	0.638485i
$780$		0			0
$781$	145.769			0.00667864
$782$	16857.1			0.770854
$783$		0			0
$784$	−5950.32	−	23020.4i	−0.271061	−	1.04867i
$785$		−	15655.4i		−	0.711801i
$786$		0			0
$787$			3905.23i			0.176882i	0.996081	+	0.0884411i	$0.0281885\pi$
$787$			3905.23i			0.176882i	−0.996081	+	0.0884411i	$0.971811\pi$
$788$		−	27363.3i		−	1.23702i
$789$		0			0
$790$			8801.22i			0.396371i
$791$	14425.4	−	18628.2i	0.648431	−	0.837350i
$792$		0			0
$793$	45889.5			2.05496
$794$	−4377.06			−0.195638
$795$		0			0
$796$		−	6969.55i		−	0.310338i
$797$	4522.48			0.200997			0.100499	−	0.994937i	$-0.467956\pi$
$797$	4522.48			0.200997			0.100499	+	0.994937i	$0.467956\pi$
$798$		0			0
$799$	3817.22			0.169016
$800$		−	2678.59i		−	0.118378i
$801$		0			0
$802$	6896.13			0.303629
$803$	24159.1			1.06171
$804$		0			0
$805$	22218.0	−	28691.1i	0.972771	−	1.25618i
$806$			46579.4i			2.03559i
$807$		0			0
$808$			965.691i			0.0420457i
$809$			20951.2i			0.910513i	0.890360	+	0.455256i	$0.150452\pi$
$809$			20951.2i			0.910513i	−0.890360	+	0.455256i	$0.849548\pi$
$810$		0			0
$811$		−	22578.2i		−	0.977592i	−0.872398	−	0.488796i	$-0.837436\pi$
$811$		−	22578.2i		−	0.977592i	0.872398	−	0.488796i	$-0.162564\pi$
$812$	2686.33	−	3468.98i	0.116098	−	0.149923i
$813$		0			0
$814$	720.686			0.0310320
$815$	11356.5			0.488097
$816$		0			0
$817$		−	17059.3i		−	0.730513i
$818$	−26477.1			−1.13172
$819$		0			0
$820$	28582.1			1.21723
$821$		−	40522.8i		−	1.72260i	−0.508097	−	0.861300i	$-0.669651\pi$
$821$		−	40522.8i		−	1.72260i	0.508097	−	0.861300i	$-0.330349\pi$
$822$		0			0
$823$	−31646.2			−1.34036			−0.670181	−	0.742198i	$-0.733783\pi$
$823$	−31646.2			−1.34036			−0.670181	+	0.742198i	$0.733783\pi$
$824$	606.624			0.0256465
$825$		0			0
$826$	−24994.5	+	32276.5i	−1.05287	+	1.35962i
$827$			19924.2i			0.837766i	0.908040	+	0.418883i	$0.137578\pi$
$827$			19924.2i			0.837766i	−0.908040	+	0.418883i	$0.862422\pi$
$828$		0			0
$829$			27088.6i			1.13489i	0.823410	+	0.567447i	$0.192069\pi$
$829$			27088.6i			1.13489i	−0.823410	+	0.567447i	$0.807931\pi$
$830$			19693.4i			0.823577i
$831$		0			0
$832$		−	28809.2i		−	1.20045i
$833$	−2202.45	−	8520.76i	−0.0916092	−	0.354414i
$834$		0			0
$835$	33129.8			1.37306
$836$	14257.9			0.589856
$837$		0			0
$838$			41837.2i			1.72463i
$839$	−5571.84			−0.229274			−0.114637	−	0.993407i	$-0.536571\pi$
$839$	−5571.84			−0.229274			−0.114637	+	0.993407i	$0.536571\pi$
$840$		0			0
$841$	23326.5			0.956436
$842$		−	11135.1i		−	0.455748i
$843$		0			0
$844$	29336.0			1.19643
$845$	−30718.7			−1.25060
$846$		0			0
$847$	13811.9	+	10695.7i	0.560311	+	0.433896i
$848$			4140.92i			0.167688i
$849$		0			0
$850$		−	1082.95i		−	0.0436997i
$851$		−	650.281i		−	0.0261943i
$852$		0			0
$853$			17056.2i			0.684634i	0.939585	+	0.342317i	$0.111212\pi$
$853$			17056.2i			0.684634i	−0.939585	+	0.342317i	$0.888788\pi$
$854$	37768.1	+	29247.1i	1.51335	+	1.17191i
$855$		0			0
$856$	1142.05			0.0456012
$857$	−6191.24			−0.246778			−0.123389	−	0.992358i	$-0.539376\pi$
$857$	−6191.24			−0.246778			−0.123389	+	0.992358i	$0.539376\pi$
$858$		0			0
$859$			4787.20i			0.190148i	0.995470	+	0.0950740i	$0.0303088\pi$
$859$			4787.20i			0.190148i	−0.995470	+	0.0950740i	$0.969691\pi$
$860$	35123.6			1.39268
$861$		0			0
$862$	40908.8			1.61642
$863$			20578.3i			0.811694i	0.913941	+	0.405847i	$0.133024\pi$
$863$			20578.3i			0.811694i	−0.913941	+	0.405847i	$0.866976\pi$
$864$		0			0
$865$	−46401.8			−1.82394
$866$	−19848.8			−0.778855
$867$		0			0
$868$	−14131.7	+	18248.9i	−0.552604	+	0.713603i
$869$			9217.61i			0.359823i
$870$		0			0
$871$			20298.3i			0.789648i
$872$			5284.59i			0.205228i
$873$		0			0
$874$		−	27025.8i		−	1.04595i
$875$	19487.0	+	15090.4i	0.752891	+	0.583027i
$876$		0			0
$877$	36650.2			1.41116			0.705581	−	0.708630i	$-0.250686\pi$
$877$	36650.2			1.41116			0.705581	+	0.708630i	$0.250686\pi$
$878$	−29314.0			−1.12677
$879$		0			0
$880$		−	38524.1i		−	1.47574i
$881$	−13272.5			−0.507561			−0.253780	−	0.967262i	$-0.581674\pi$
$881$	−13272.5			−0.507561			−0.253780	+	0.967262i	$0.581674\pi$
$882$		0			0
$883$	7209.05			0.274750			0.137375	−	0.990519i	$-0.456134\pi$
$883$	7209.05			0.274750			0.137375	+	0.990519i	$0.456134\pi$
$884$		−	12964.3i		−	0.493254i
$885$		0			0
$886$	10776.7			0.408634
$887$	24148.9			0.914137			0.457068	−	0.889432i	$-0.348899\pi$
$887$	24148.9			0.914137			0.457068	+	0.889432i	$0.348899\pi$
$888$		0			0
$889$	26878.2	+	20814.1i	1.01402	+	0.785245i
$890$			42101.0i			1.58565i
$891$		0			0
$892$		−	42542.0i		−	1.59688i
$893$		−	6119.90i		−	0.229333i
$894$		0			0
$895$			28733.3i			1.07313i
$896$	−4134.42	+	5338.97i	−0.154153	+	0.199065i
$897$		0			0
$898$	−31500.0			−1.17057
$899$	5589.24			0.207354
$900$		0			0
$901$			1532.72i			0.0566730i
$902$	62883.9			2.32129
$903$		0			0
$904$	−3638.97			−0.133883
$905$			29342.0i			1.07775i
$906$		0			0
$907$	−29231.2			−1.07013			−0.535064	−	0.844812i	$-0.679713\pi$
$907$	−29231.2			−1.07013			−0.535064	+	0.844812i	$0.679713\pi$
$908$	−44165.4			−1.61419
$909$		0			0
$910$	−46353.6	−	35895.6i	−1.68858	−	1.30761i
$911$			33712.2i			1.22605i	0.790062	+	0.613027i	$0.210048\pi$
$911$			33712.2i			1.22605i	−0.790062	+	0.613027i	$0.789952\pi$
$912$		0			0
$913$			20625.2i			0.747637i
$914$		−	846.471i		−	0.0306332i
$915$		0			0
$916$		−	6834.02i		−	0.246509i
$917$	19730.0	+	15278.6i	0.710515	+	0.550212i
$918$		0			0
$919$	12924.3			0.463912			0.231956	−	0.972726i	$-0.425488\pi$
$919$	12924.3			0.463912			0.231956	+	0.972726i	$0.425488\pi$
$920$	−5604.72			−0.200850
$921$		0			0
$922$		−	4645.39i		−	0.165930i
$923$	−212.499			−0.00757798
$924$		0			0
$925$	−41.7759			−0.00148495
$926$		−	18733.5i		−	0.664817i
$927$		0			0
$928$	−8083.09			−0.285927
$929$	29383.5			1.03772			0.518859	−	0.854860i	$-0.326357\pi$
$929$	29383.5			1.03772			0.518859	+	0.854860i	$0.326357\pi$
$930$		0			0
$931$	−13660.8	+	3531.05i	−0.480896	+	0.124302i
$932$			24090.7i			0.846693i
$933$		0			0
$934$			11541.7i			0.404342i
$935$		−	14259.3i		−	0.498749i
$936$		0			0
$937$		−	42385.1i		−	1.47776i	−0.673837	−	0.738880i	$-0.735355\pi$
$937$		−	42385.1i		−	1.47776i	0.673837	−	0.738880i	$-0.264645\pi$
$938$	−12936.9	+	16706.0i	−0.450324	+	0.581525i
$939$		0			0
$940$	12600.3			0.437210
$941$	−33740.5			−1.16887			−0.584436	−	0.811440i	$-0.698684\pi$
$941$	−33740.5			−1.16887			−0.584436	+	0.811440i	$0.698684\pi$
$942$		0			0
$943$		−	56740.7i		−	1.95942i
$944$	39104.5			1.34825
$945$		0			0
$946$	77275.8			2.65587
$947$		−	30903.7i		−	1.06044i	−0.847861	−	0.530219i	$-0.822110\pi$
$947$		−	30903.7i		−	1.06044i	0.847861	−	0.530219i	$-0.177890\pi$
$948$		0			0
$949$	−35218.6			−1.20468
$950$	−1736.22			−0.0592951
$951$		0			0
$952$	−832.253	+	1074.73i	−0.0283335	+	0.0365884i
$953$			14796.1i			0.502931i	0.967866	+	0.251465i	$0.0809125\pi$
$953$			14796.1i			0.502931i	−0.967866	+	0.251465i	$0.919088\pi$
$954$		0			0
$955$			5373.39i			0.182072i
$956$		−	18578.8i		−	0.628538i
$957$		0			0
$958$		−	426.838i		−	0.0143951i
$959$	12627.7	−	16306.8i	0.425204	−	0.549086i
$960$		0			0
$961$	388.306			0.0130343
$962$	−1050.60			−0.0352107
$963$		0			0
$964$		−	15112.4i		−	0.504914i
$965$	−29859.2			−0.996064
$966$		0			0
$967$	18171.9			0.604311			0.302156	−	0.953259i	$-0.402294\pi$
$967$	18171.9			0.604311			0.302156	+	0.953259i	$0.402294\pi$
$968$		−	2698.12i		−	0.0895876i
$969$		0			0
$970$	49603.0			1.64191
$971$	−36450.4			−1.20468			−0.602342	−	0.798238i	$-0.705766\pi$
$971$	−36450.4			−1.20468			−0.602342	+	0.798238i	$0.705766\pi$
$972$		0			0
$973$	−14929.6	+	19279.2i	−0.491901	+	0.635215i
$974$		−	14125.1i		−	0.464679i
$975$		0			0
$976$		−	45757.9i		−	1.50069i
$977$			5045.43i			0.165218i	0.996582	+	0.0826088i	$0.0263252\pi$
$977$			5045.43i			0.165218i	−0.996582	+	0.0826088i	$0.973675\pi$
$978$		0			0
$979$			44092.8i			1.43944i
$980$	−7270.12	−	28126.4i	−0.236975	−	0.916800i
$981$		0			0
$982$	−73455.8			−2.38704
$983$	48929.7			1.58760			0.793802	−	0.608176i	$-0.208099\pi$
$983$	48929.7			1.58760			0.793802	+	0.608176i	$0.208099\pi$
$984$		0			0
$985$			43874.2i			1.41924i
$986$	−3267.97			−0.105551
$987$		0			0
$988$	−20784.8			−0.669285
$989$		−	69726.6i		−	2.24184i
$990$		0			0
$991$	−41118.3			−1.31803			−0.659014	−	0.752131i	$-0.729026\pi$
$991$	−41118.3			−1.31803			−0.659014	+	0.752131i	$0.729026\pi$
$992$	42521.8			1.36096
$993$		0			0
$994$	−174.891	−	135.433i	−0.00558070	−	0.00432161i
$995$			11175.0i			0.356050i
$996$		0			0
$997$			23917.3i			0.759749i	0.925038	+	0.379874i	$0.124033\pi$
$997$			23917.3i			0.759749i	−0.925038	+	0.379874i	$0.875967\pi$
$998$			12667.0i			0.401770i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.c.c.566.10		44
3.2	odd	2	inner	567.4.c.c.566.35		44
7.6	odd	2	inner	567.4.c.c.566.36		44
9.2	odd	6		63.4.o.a.41.3	yes	44
9.4	even	3		63.4.o.a.20.4	yes	44
9.5	odd	6		189.4.o.a.62.20		44
9.7	even	3		189.4.o.a.125.19		44
21.20	even	2	inner	567.4.c.c.566.9		44
63.13	odd	6		63.4.o.a.20.3	✓	44
63.20	even	6		63.4.o.a.41.4	yes	44
63.34	odd	6		189.4.o.a.125.20		44
63.41	even	6		189.4.o.a.62.19		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.3	✓	44	63.13	odd	6
63.4.o.a.20.4	yes	44	9.4	even	3
63.4.o.a.41.3	yes	44	9.2	odd	6
63.4.o.a.41.4	yes	44	63.20	even	6
189.4.o.a.62.19		44	63.41	even	6
189.4.o.a.62.20		44	9.5	odd	6
189.4.o.a.125.19		44	9.7	even	3
189.4.o.a.125.20		44	63.34	odd	6
567.4.c.c.566.9		44	21.20	even	2	inner
567.4.c.c.566.10		44	1.1	even	1	trivial
567.4.c.c.566.35		44	3.2	odd	2	inner
567.4.c.c.566.36		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)

\(f(q)\)	\(=\)	\(q+3.90742i q^{2} -7.26794 q^{4} +11.6534 q^{5} +(-14.6430 - 11.3394i) q^{7} +2.86048i q^{8} +O(q^{10})\)	\(q+3.90742i q^{2} -7.26794 q^{4} +11.6534 q^{5} +(-14.6430 - 11.3394i) q^{7} +2.86048i q^{8} +45.5347i q^{10} +47.6890i q^{11} -69.5200i q^{13} +(44.3077 - 57.2166i) q^{14} -69.3206 q^{16} -25.6583 q^{17} +41.1363i q^{19} -84.6961 q^{20} -186.341 q^{22} +168.137i q^{23} +10.8016 q^{25} +271.644 q^{26} +(106.425 + 82.4138i) q^{28} -32.5956i q^{29} +171.472i q^{31} -247.981i q^{32} -100.258i q^{34} +(-170.641 - 132.142i) q^{35} -3.86757 q^{37} -160.737 q^{38} +33.3343i q^{40} -337.467 q^{41} -414.701 q^{43} -346.601i q^{44} -656.982 q^{46} -148.771 q^{47} +(85.8377 + 332.086i) q^{49} +42.2064i q^{50} +505.267i q^{52} -59.7358i q^{53} +555.739i q^{55} +(32.4360 - 41.8861i) q^{56} +127.365 q^{58} -564.112 q^{59} +660.091i q^{61} -670.014 q^{62} +414.401 q^{64} -810.144i q^{65} -291.978 q^{67} +186.483 q^{68} +(516.335 - 666.767i) q^{70} -3.05665i q^{71} -506.596i q^{73} -15.1122i q^{74} -298.976i q^{76} +(540.763 - 698.312i) q^{77} +193.286 q^{79} -807.820 q^{80} -1318.62i q^{82} +432.493 q^{83} -299.007 q^{85} -1620.41i q^{86} -136.413 q^{88} +924.591 q^{89} +(-788.312 + 1017.98i) q^{91} -1222.01i q^{92} -581.311i q^{94} +479.378i q^{95} -1089.35i q^{97} +(-1297.60 + 335.404i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) 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

Introduction

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