LMFDB - Embedded newform 72.8.f.a.35.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (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:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.13
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.319783 - 11.3092i) q^{2} +(-127.795 + 7.23297i) q^{4} +412.177 q^{5} -359.164i q^{7} +(122.666 + 1442.95i) q^{8} +O(q^{10})$	\(q+(-0.319783 - 11.3092i) q^{2} +(-127.795 + 7.23297i) q^{4} +412.177 q^{5} -359.164i q^{7} +(122.666 + 1442.95i) q^{8} +(-131.807 - 4661.38i) q^{10} +3819.02i q^{11} +14141.7i q^{13} +(-4061.85 + 114.854i) q^{14} +(16279.4 - 1848.68i) q^{16} +3331.48i q^{17} +34327.6 q^{19} +(-52674.3 + 2981.26i) q^{20} +(43190.0 - 1221.26i) q^{22} +69918.3 q^{23} +91764.7 q^{25} +(159932. - 4522.29i) q^{26} +(2597.82 + 45899.5i) q^{28} -186332. q^{29} -189759. i q^{31} +(-26112.9 - 183515. i) q^{32} +(37676.4 - 1065.35i) q^{34} -148039. i q^{35} +190024. i q^{37} +(-10977.4 - 388217. i) q^{38} +(50560.0 + 594751. i) q^{40} -159461. i q^{41} +757231. q^{43} +(-27622.9 - 488054. i) q^{44} +(-22358.7 - 790720. i) q^{46} +235539. q^{47} +694544. q^{49} +(-29344.8 - 1.03778e6i) q^{50} +(-102287. - 1.80725e6i) q^{52} +732030. q^{53} +1.57411e6i q^{55} +(518255. - 44057.1i) q^{56} +(59585.9 + 2.10727e6i) q^{58} +1.98198e6i q^{59} -1.37932e6i q^{61} +(-2.14602e6 + 60681.5i) q^{62} +(-2.06706e6 + 354001. i) q^{64} +5.82890e6i q^{65} +2.28147e6 q^{67} +(-24096.5 - 425749. i) q^{68} +(-1.67420e6 + 47340.3i) q^{70} -5.12011e6 q^{71} -521725. q^{73} +(2.14901e6 - 60766.2i) q^{74} +(-4.38691e6 + 248290. i) q^{76} +1.37165e6 q^{77} -3.85831e6i q^{79} +(6.70998e6 - 761983. i) q^{80} +(-1.80337e6 + 50992.8i) q^{82} +4.75271e6i q^{83} +1.37316e6i q^{85} +(-242150. - 8.56367e6i) q^{86} +(-5.51066e6 + 468463. i) q^{88} +7.08773e6i q^{89} +5.07920e6 q^{91} +(-8.93525e6 + 505717. i) q^{92} +(-75321.4 - 2.66376e6i) q^{94} +1.41490e7 q^{95} -1.34983e7 q^{97} +(-222103. - 7.85473e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−0.319783	−	11.3092i	−0.0282651	−	0.999600i
$2$	−0.319783	−	11.3092i	−0.0282651	−	0.999600i
$3$		0			0
$3$		0			0
$4$	−127.795	+	7.23297i	−0.998402	+	0.0565076i
$5$	412.177			1.47465			0.737324	−	0.675539i	$-0.236089\pi$
$5$	412.177			1.47465			0.737324	+	0.675539i	$0.236089\pi$
$6$		0			0
$7$		−	359.164i		−	0.395776i	−0.980225	−	0.197888i	$-0.936592\pi$
$7$		−	359.164i		−	0.395776i	0.980225	−	0.197888i	$-0.0634082\pi$
$8$	122.666	+	1442.95i	0.0847049	+	0.996406i
$9$		0			0
$10$	−131.807	−	4661.38i	−0.0416810	−	1.47406i
$11$			3819.02i			0.865123i	0.901604	+	0.432561i	$0.142390\pi$
$11$			3819.02i			0.865123i	−0.901604	+	0.432561i	$0.857610\pi$
$12$		0			0
$13$			14141.7i			1.78526i	0.450791	+	0.892630i	$0.351142\pi$
$13$			14141.7i			1.78526i	−0.450791	+	0.892630i	$0.648858\pi$
$14$	−4061.85	+	114.854i	−0.395618	+	0.0111866i
$15$		0			0
$16$	16279.4	−	1848.68i	0.993614	−	0.112835i
$17$			3331.48i			0.164462i	0.996613	+	0.0822312i	$0.0262046\pi$
$17$			3331.48i			0.164462i	−0.996613	+	0.0822312i	$0.973795\pi$
$18$		0			0
$19$	34327.6			1.14817			0.574084	−	0.818796i	$-0.305358\pi$
$19$	34327.6			1.14817			0.574084	+	0.818796i	$0.305358\pi$
$20$	−52674.3	+	2981.26i	−1.47229	+	0.0833288i
$21$		0			0
$22$	43190.0	−	1221.26i	0.864777	−	0.0244528i
$23$	69918.3			1.19824			0.599120	−	0.800659i	$-0.295517\pi$
$23$	69918.3			1.19824			0.599120	+	0.800659i	$0.295517\pi$
$24$		0			0
$25$	91764.7			1.17459
$26$	159932.	−	4522.29i	1.78455	−	0.0504605i
$27$		0			0
$28$	2597.82	+	45899.5i	0.0223643	+	0.395144i
$29$	−186332.			−1.41872			−0.709358	−	0.704848i	$-0.751015\pi$
$29$	−186332.			−1.41872			−0.709358	+	0.704848i	$0.751015\pi$
$30$		0			0
$31$		−	189759.i		−	1.14403i	−0.820245	−	0.572013i	$-0.806163\pi$
$31$		−	189759.i		−	1.14403i	0.820245	−	0.572013i	$-0.193837\pi$
$32$	−26112.9	−	183515.i	−0.140874	−	0.990028i
$33$		0			0
$34$	37676.4	−	1065.35i	0.164397	−	0.00464854i
$35$		−	148039.i		−	0.583630i
$36$		0			0
$37$			190024.i			0.616739i	0.951267	+	0.308369i	$0.0997833\pi$
$37$			190024.i			0.616739i	−0.951267	+	0.308369i	$0.900217\pi$
$38$	−10977.4	−	388217.i	−0.0324530	−	1.14771i
$39$		0			0
$40$	50560.0	+	594751.i	0.124910	+	1.46935i
$41$		−	159461.i		−	0.361335i	−0.983544	−	0.180668i	$-0.942174\pi$
$41$		−	159461.i		−	0.361335i	0.983544	−	0.180668i	$-0.0578258\pi$
$42$		0			0
$43$	757231.			1.45241			0.726205	−	0.687478i	$-0.241282\pi$
$43$	757231.			1.45241			0.726205	+	0.687478i	$0.241282\pi$
$44$	−27622.9	−	488054.i	−0.0488860	−	0.863741i
$45$		0			0
$46$	−22358.7	−	790720.i	−0.0338683	−	1.19776i
$47$	235539.			0.330918			0.165459	−	0.986217i	$-0.447089\pi$
$47$	235539.			0.330918			0.165459	+	0.986217i	$0.447089\pi$
$48$		0			0
$49$	694544.			0.843361
$50$	−29344.8	−	1.03778e6i	−0.0331998	−	1.17412i
$51$		0			0
$52$	−102287.	−	1.80725e6i	−0.100881	−	1.78241i
$53$	732030.			0.675404			0.337702	−	0.941253i	$-0.390350\pi$
$53$	732030.			0.675404			0.337702	+	0.941253i	$0.390350\pi$
$54$		0			0
$55$			1.57411e6i			1.27575i
$56$	518255.	−	44057.1i	0.394354	−	0.0335242i
$57$		0			0
$58$	59585.9	+	2.10727e6i	0.0401001	+	1.41815i
$59$			1.98198e6i			1.25637i	0.778064	+	0.628185i	$0.216202\pi$
$59$			1.98198e6i			1.25637i	−0.778064	+	0.628185i	$0.783798\pi$
$60$		0			0
$61$		−	1.37932e6i		−	0.778055i	−0.921226	−	0.389028i	$-0.872811\pi$
$61$		−	1.37932e6i		−	0.778055i	0.921226	−	0.389028i	$-0.127189\pi$
$62$	−2.14602e6	+	60681.5i	−1.14357	+	0.0323360i
$63$		0			0
$64$	−2.06706e6	+	354001.i	−0.985650	+	0.168801i
$65$			5.82890e6i			2.63263i
$66$		0			0
$67$	2.28147e6			0.926731			0.463366	−	0.886167i	$-0.346642\pi$
$67$	2.28147e6			0.926731			0.463366	+	0.886167i	$0.346642\pi$
$68$	−24096.5	−	425749.i	−0.00929337	−	0.164200i
$69$		0			0
$70$	−1.67420e6	+	47340.3i	−0.583397	+	0.0164964i
$71$	−5.12011e6			−1.69776			−0.848878	−	0.528589i	$-0.822721\pi$
$71$	−5.12011e6			−1.69776			−0.848878	+	0.528589i	$0.822721\pi$
$72$		0			0
$73$	−521725.			−0.156968			−0.0784841	−	0.996915i	$-0.525008\pi$
$73$	−521725.			−0.156968			−0.0784841	+	0.996915i	$0.525008\pi$
$74$	2.14901e6	−	60766.2i	0.616493	−	0.0174322i
$75$		0			0
$76$	−4.38691e6	+	248290.i	−1.14633	+	0.0648801i
$77$	1.37165e6			0.342395
$78$		0			0
$79$		−	3.85831e6i		−	0.880446i	−0.897888	−	0.440223i	$-0.854899\pi$
$79$		−	3.85831e6i		−	0.880446i	0.897888	−	0.440223i	$-0.145101\pi$
$80$	6.70998e6	−	761983.i	1.46523	−	0.166391i
$81$		0			0
$82$	−1.80337e6	+	50992.8i	−0.361191	+	0.0102132i
$83$			4.75271e6i			0.912363i	0.889887	+	0.456182i	$0.150783\pi$
$83$			4.75271e6i			0.912363i	−0.889887	+	0.456182i	$0.849217\pi$
$84$		0			0
$85$			1.37316e6i			0.242524i
$86$	−242150.	−	8.56367e6i	−0.0410525	−	1.45183i
$87$		0			0
$88$	−5.51066e6	+	468463.i	−0.862014	+	0.0732801i
$89$			7.08773e6i			1.06572i	0.846204	+	0.532859i	$0.178882\pi$
$89$			7.08773e6i			1.06572i	−0.846204	+	0.532859i	$0.821118\pi$
$90$		0			0
$91$	5.07920e6			0.706563
$92$	−8.93525e6	+	505717.i	−1.19632	+	0.0677096i
$93$		0			0
$94$	−75321.4	−	2.66376e6i	−0.00935343	−	0.330786i
$95$	1.41490e7			1.69314
$96$		0			0
$97$	−1.34983e7			−1.50169			−0.750844	−	0.660480i	$-0.770353\pi$
$97$	−1.34983e7			−1.50169			−0.750844	+	0.660480i	$0.770353\pi$
$98$	−222103.	−	7.85473e6i	−0.0238377	−	0.843024i
$99$		0			0
$100$	−1.17271e7	+	663731.i	−1.17271	+	0.0663731i
$101$	−1.25843e7			−1.21535			−0.607677	−	0.794184i	$-0.707899\pi$
$101$	−1.25843e7			−1.21535			−0.607677	+	0.794184i	$0.707899\pi$
$102$		0			0
$103$			8.44045e6i			0.761089i	0.924763	+	0.380544i	$0.124263\pi$
$103$			8.44045e6i			0.761089i	−0.924763	+	0.380544i	$0.875737\pi$
$104$	−2.04058e7	+	1.73471e6i	−1.77884	+	0.151220i
$105$		0			0
$106$	−234091.	−	8.27867e6i	−0.0190903	−	0.675134i
$107$		−	1.12718e7i		−	0.889505i	−0.895654	−	0.444752i	$-0.853292\pi$
$107$		−	1.12718e7i		−	0.889505i	0.895654	−	0.444752i	$-0.146708\pi$
$108$		0			0
$109$			7.83959e6i			0.579830i	0.957052	+	0.289915i	$0.0936270\pi$
$109$			7.83959e6i			0.579830i	−0.957052	+	0.289915i	$0.906373\pi$
$110$	1.78019e7	−	503374.i	1.27524	−	0.0360592i
$111$		0			0
$112$	−663979.	−	5.84696e6i	−0.0446572	−	0.393248i
$113$			1.34305e7i			0.875623i	0.899067	+	0.437811i	$0.144246\pi$
$113$			1.34305e7i			0.875623i	−0.899067	+	0.437811i	$0.855754\pi$
$114$		0			0
$115$	2.88187e7			1.76698
$116$	2.38125e7	−	1.34774e6i	1.41645	−	0.0801682i
$117$		0			0
$118$	2.24146e7	−	633804.i	1.25587	−	0.0355114i
$119$	1.19655e6			0.0650903
$120$		0			0
$121$	4.90224e6			0.251562
$122$	−1.55990e7	+	441083.i	−0.777744	+	0.0219918i
$123$		0			0
$124$	1.37252e6	+	2.42503e7i	0.0646461	+	1.14220i
$125$	5.62197e6			0.257456
$126$		0			0
$127$		−	2.93022e7i		−	1.26937i	−0.772771	−	0.634684i	$-0.781130\pi$
$127$		−	2.93022e7i		−	1.26937i	0.772771	−	0.634684i	$-0.218870\pi$
$128$	4.66448e6	+	2.32635e7i	0.196593	+	0.980485i
$129$		0			0
$130$	6.59201e7	−	1.86398e6i	2.63158	−	0.0744115i
$131$		−	4.27923e7i		−	1.66309i	−0.555457	−	0.831546i	$-0.687456\pi$
$131$		−	4.27923e7i		−	1.66309i	0.555457	−	0.831546i	$-0.312544\pi$
$132$		0			0
$133$		−	1.23292e7i		−	0.454417i
$134$	−729576.	−	2.58016e7i	−0.0261941	−	0.926361i
$135$		0			0
$136$	−4.80717e6	+	408659.i	−0.163871	+	0.0139308i
$137$		−	3.57159e6i		−	0.118670i	−0.998238	−	0.0593348i	$-0.981102\pi$
$137$		−	3.57159e6i		−	0.118670i	0.998238	−	0.0593348i	$-0.0188979\pi$
$138$		0			0
$139$	4.65242e7			1.46936			0.734678	−	0.678417i	$-0.237333\pi$
$139$	4.65242e7			1.46936			0.734678	+	0.678417i	$0.237333\pi$
$140$	1.07076e6	+	1.89187e7i	0.0329795	+	0.582698i
$141$		0			0
$142$	1.63732e6	+	5.79043e7i	0.0479872	+	1.69708i
$143$	−5.40076e7			−1.54447
$144$		0			0
$145$	−7.68019e7			−2.09211
$146$	166839.	+	5.90029e6i	0.00443672	+	0.156906i
$147$		0			0
$148$	−1.37443e6	−	2.42841e7i	−0.0348504	−	0.615753i
$149$	3.43823e7			0.851496			0.425748	−	0.904842i	$-0.360011\pi$
$149$	3.43823e7			0.851496			0.425748	+	0.904842i	$0.360011\pi$
$150$		0			0
$151$			7.33709e7i			1.73422i	0.498116	+	0.867110i	$0.334025\pi$
$151$			7.33709e7i			1.73422i	−0.498116	+	0.867110i	$0.665975\pi$
$152$	4.21082e6	+	4.95330e7i	0.0972554	+	1.14404i
$153$		0			0
$154$	−438631.	−	1.55123e7i	−0.00967781	−	0.342258i
$155$		−	7.82141e7i		−	1.68704i
$156$		0			0
$157$		−	3.78154e7i		−	0.779867i	−0.920843	−	0.389933i	$-0.872498\pi$
$157$		−	3.78154e7i		−	0.779867i	0.920843	−	0.389933i	$-0.127502\pi$
$158$	−4.36344e7	+	1.23382e6i	−0.880094	+	0.0248859i
$159$		0			0
$160$	−1.07631e7	−	7.56407e7i	−0.207740	−	1.45994i
$161$		−	2.51121e7i		−	0.474234i
$162$		0			0
$163$	−1.97142e7			−0.356552			−0.178276	−	0.983981i	$-0.557052\pi$
$163$	−1.97142e7			−0.356552			−0.178276	+	0.983981i	$0.557052\pi$
$164$	1.15337e6	+	2.03784e7i	0.0204182	+	0.360758i
$165$		0			0
$166$	5.37493e7	−	1.51983e6i	0.911999	−	0.0257880i
$167$	8.21858e7			1.36549			0.682746	−	0.730656i	$-0.260786\pi$
$167$	8.21858e7			1.36549			0.682746	+	0.730656i	$0.260786\pi$
$168$		0			0
$169$	−1.37240e8			−2.18715
$170$	1.55293e7	−	439113.i	0.242427	−	0.00685496i
$171$		0			0
$172$	−9.67708e7	+	5.47703e6i	−1.45009	+	0.0820721i
$173$	2.64838e7			0.388883			0.194441	−	0.980914i	$-0.437711\pi$
$173$	2.64838e7			0.388883			0.194441	+	0.980914i	$0.437711\pi$
$174$		0			0
$175$		−	3.29586e7i		−	0.464874i
$176$	7.06015e6	+	6.21713e7i	0.0976157	+	0.859598i
$177$		0			0
$178$	8.01565e7	−	2.26653e6i	1.06529	−	0.0301226i
$179$		−	9.25036e7i		−	1.20552i	−0.797924	−	0.602758i	$-0.794068\pi$
$179$		−	9.25036e7i		−	1.20552i	0.797924	−	0.602758i	$-0.205932\pi$
$180$		0			0
$181$			7.91658e7i			0.992344i	0.868224	+	0.496172i	$0.165262\pi$
$181$			7.91658e7i			0.992344i	−0.868224	+	0.496172i	$0.834738\pi$
$182$	−1.62424e6	−	5.74417e7i	−0.0199710	−	0.706280i
$183$		0			0
$184$	8.57659e6	+	1.00889e8i	0.101497	+	1.19393i
$185$			7.83233e7i			0.909473i
$186$		0			0
$187$	−1.27230e7			−0.142280
$188$	−3.01008e7	+	1.70365e6i	−0.330390	+	0.0186994i
$189$		0			0
$190$	−4.52461e6	−	1.60014e8i	−0.0478568	−	1.69247i
$191$	−1.02924e8			−1.06881			−0.534403	−	0.845230i	$-0.679464\pi$
$191$	−1.02924e8			−1.06881			−0.534403	+	0.845230i	$0.679464\pi$
$192$		0			0
$193$	−5.98055e7			−0.598812			−0.299406	−	0.954126i	$-0.596789\pi$
$193$	−5.98055e7			−0.598812			−0.299406	+	0.954126i	$0.596789\pi$
$194$	4.31654e6	+	1.52655e8i	0.0424453	+	1.50109i
$195$		0			0
$196$	−8.87596e7	+	5.02362e6i	−0.842014	+	0.0476563i
$197$	−1.68356e8			−1.56890			−0.784452	−	0.620190i	$-0.787055\pi$
$197$	−1.68356e8			−1.56890			−0.784452	+	0.620190i	$0.787055\pi$
$198$		0			0
$199$		−	9.24295e7i		−	0.831428i	−0.909495	−	0.415714i	$-0.863532\pi$
$199$		−	9.24295e7i		−	0.831428i	0.909495	−	0.415714i	$-0.136468\pi$
$200$	1.12564e7	+	1.32412e8i	0.0994934	+	1.17037i
$201$		0			0
$202$	4.02423e6	+	1.42318e8i	0.0343521	+	1.21487i
$203$			6.69239e7i			0.561494i
$204$		0			0
$205$		−	6.57260e7i		−	0.532842i
$206$	9.54546e7	−	2.69911e6i	0.760785	−	0.0215122i
$207$		0			0
$208$	2.61436e7	+	2.30219e8i	0.201439	+	1.77386i
$209$			1.31098e8i			0.993306i
$210$		0			0
$211$	1.21026e8			0.886928			0.443464	−	0.896292i	$-0.353749\pi$
$211$	1.21026e8			0.886928			0.443464	+	0.896292i	$0.353749\pi$
$212$	−9.35502e7	+	5.29475e6i	−0.674325	+	0.0381654i
$213$		0			0
$214$	−1.27474e8	+	3.60451e6i	−0.889150	+	0.0251419i
$215$	3.12113e8			2.14179
$216$		0			0
$217$	−6.81544e7			−0.452778
$218$	8.86594e7	−	2.50696e6i	0.579598	−	0.0163889i
$219$		0			0
$220$	−1.13855e7	−	2.01164e8i	−0.0720896	−	1.27371i
$221$	−4.71130e7			−0.293608
$222$		0			0
$223$		−	6.34758e7i		−	0.383302i	−0.981463	−	0.191651i	$-0.938616\pi$
$223$		−	6.34758e7i		−	0.383302i	0.981463	−	0.191651i	$-0.0613842\pi$
$224$	−6.59120e7	+	9.37882e6i	−0.391829	+	0.0557545i
$225$		0			0
$226$	1.51888e8	−	4.29484e6i	0.875273	−	0.0247495i
$227$			1.62893e8i			0.924296i	0.886803	+	0.462148i	$0.152921\pi$
$227$			1.62893e8i			0.924296i	−0.886803	+	0.462148i	$0.847079\pi$
$228$		0			0
$229$			2.85285e7i			0.156984i	0.996915	+	0.0784918i	$0.0250105\pi$
$229$			2.85285e7i			0.156984i	−0.996915	+	0.0784918i	$0.974990\pi$
$230$	−9.21573e6	−	3.25916e8i	−0.0499439	−	1.76628i
$231$		0			0
$232$	−2.28566e7	−	2.68869e8i	−0.120172	−	1.41362i
$233$		−	8.67347e7i		−	0.449207i	−0.974450	−	0.224604i	$-0.927891\pi$
$233$		−	8.67347e7i		−	0.449207i	0.974450	−	0.224604i	$-0.0721088\pi$
$234$		0			0
$235$	9.70838e7			0.487988
$236$	−1.43356e7	−	2.53288e8i	−0.0709944	−	1.25436i
$237$		0			0
$238$	−382636.	−	1.35320e7i	−0.00183978	−	0.0650642i
$239$	−3.35437e8			−1.58935			−0.794673	−	0.607038i	$-0.792358\pi$
$239$	−3.35437e8			−1.58935			−0.794673	+	0.607038i	$0.792358\pi$
$240$		0			0
$241$	4.29134e8			1.97484			0.987422	−	0.158107i	$-0.0505391\pi$
$241$	4.29134e8			1.97484			0.987422	+	0.158107i	$0.0505391\pi$
$242$	−1.56765e6	−	5.54403e7i	−0.00711043	−	0.251462i
$243$		0			0
$244$	9.97657e6	+	1.76271e8i	0.0439660	+	0.776812i
$245$	2.86275e8			1.24366
$246$		0			0
$247$			4.85452e8i			2.04978i
$248$	2.73812e8	−	2.32769e7i	1.13991	−	0.0969045i
$249$		0			0
$250$	−1.79781e6	−	6.35799e7i	−0.00727702	−	0.257354i
$251$		−	1.57080e8i		−	0.626992i	−0.949590	−	0.313496i	$-0.898500\pi$
$251$		−	1.57080e8i		−	0.626992i	0.949590	−	0.313496i	$-0.101500\pi$
$252$		0			0
$253$			2.67020e8i			1.03662i
$254$	−3.31385e8	+	9.37035e6i	−1.26886	+	0.0358788i
$255$		0			0
$256$	2.61600e8	−	6.01907e7i	0.974537	−	0.224228i
$257$		−	4.69735e7i		−	0.172618i	−0.996268	−	0.0863092i	$-0.972493\pi$
$257$		−	4.69735e7i		−	0.172618i	0.996268	−	0.0863092i	$-0.0275073\pi$
$258$		0			0
$259$	6.82496e7			0.244090
$260$	−4.21602e7	−	7.44907e8i	−0.148763	−	2.62842i
$261$		0			0
$262$	−4.83946e8	+	1.36842e7i	−1.66243	+	0.0470074i
$263$	−1.00330e8			−0.340082			−0.170041	−	0.985437i	$-0.554390\pi$
$263$	−1.00330e8			−0.340082			−0.170041	+	0.985437i	$0.554390\pi$
$264$		0			0
$265$	3.01726e8			0.995983
$266$	−1.39433e8	+	3.94267e6i	−0.454236	+	0.0128441i
$267$		0			0
$268$	−2.91562e8	+	1.65018e7i	−0.925250	+	0.0523673i
$269$	2.70778e8			0.848164			0.424082	−	0.905624i	$-0.360597\pi$
$269$	2.70778e8			0.848164			0.424082	+	0.905624i	$0.360597\pi$
$270$		0			0
$271$		−	6.20621e8i		−	1.89424i	−0.320885	−	0.947118i	$-0.603980\pi$
$271$		−	6.20621e8i		−	1.89424i	0.320885	−	0.947118i	$-0.396020\pi$
$272$	6.15885e6	+	5.42345e7i	0.0185570	+	0.163412i
$273$		0			0
$274$	−4.03918e7	+	1.14213e6i	−0.118622	+	0.00335420i
$275$			3.50452e8i			1.01616i
$276$		0			0
$277$		−	5.93506e8i		−	1.67782i	−0.544268	−	0.838911i	$-0.683193\pi$
$277$		−	5.93506e8i		−	1.67782i	0.544268	−	0.838911i	$-0.316807\pi$
$278$	−1.48776e7	−	5.26150e8i	−0.0415314	−	1.46877i
$279$		0			0
$280$	2.13613e8	−	1.81593e7i	0.581533	−	0.0494363i
$281$			2.49386e8i			0.670503i	0.942129	+	0.335251i	$0.108821\pi$
$281$			2.49386e8i			0.670503i	−0.942129	+	0.335251i	$0.891179\pi$
$282$		0			0
$283$	−2.41236e8			−0.632688			−0.316344	−	0.948645i	$-0.602455\pi$
$283$	−2.41236e8			−0.632688			−0.316344	+	0.948645i	$0.602455\pi$
$284$	6.54327e8	−	3.70336e7i	1.69504	−	0.0959360i
$285$		0			0
$286$	1.72707e7	+	6.10783e8i	0.0436545	+	1.54385i
$287$	−5.72725e7			−0.143008
$288$		0			0
$289$	3.99240e8			0.972952
$290$	2.45599e7	+	8.68567e8i	0.0591336	+	2.09127i
$291$		0			0
$292$	6.66741e7	−	3.77362e6i	0.156717	−	0.00886989i
$293$	−4.41368e8			−1.02509			−0.512547	−	0.858659i	$-0.671298\pi$
$293$	−4.41368e8			−1.02509			−0.512547	+	0.858659i	$0.671298\pi$
$294$		0			0
$295$			8.16927e8i			1.85271i
$296$	−2.74194e8	+	2.33094e7i	−0.614522	+	0.0522408i
$297$		0			0
$298$	−1.09949e7	−	3.88836e8i	−0.0240676	−	0.851156i
$299$			9.88767e8i			2.13917i
$300$		0			0
$301$		−	2.71970e8i		−	0.574829i
$302$	8.29765e8	−	2.34627e7i	1.73353	−	0.0490179i
$303$		0			0
$304$	5.58831e8	−	6.34607e7i	1.14084	−	0.129553i
$305$		−	5.68523e8i		−	1.14736i
$306$		0			0
$307$	−4.02987e8			−0.794889			−0.397445	−	0.917626i	$-0.630103\pi$
$307$	−4.02987e8			−0.794889			−0.397445	+	0.917626i	$0.630103\pi$
$308$	−1.75291e8	+	9.92113e6i	−0.341848	+	0.0193479i
$309$		0			0
$310$	−8.84538e8	+	2.50115e7i	−1.68636	+	0.0476842i
$311$	1.04195e8			0.196420			0.0982099	−	0.995166i	$-0.468688\pi$
$311$	1.04195e8			0.196420			0.0982099	+	0.995166i	$0.468688\pi$
$312$		0			0
$313$	6.28900e8			1.15925			0.579624	−	0.814884i	$-0.303200\pi$
$313$	6.28900e8			1.15925			0.579624	+	0.814884i	$0.303200\pi$
$314$	−4.27662e8	+	1.20927e7i	−0.779555	+	0.0220430i
$315$		0			0
$316$	2.79071e7	+	4.93075e8i	0.0497519	+	0.879039i
$317$	−5.51759e8			−0.972841			−0.486421	−	0.873725i	$-0.661698\pi$
$317$	−5.51759e8			−0.972841			−0.486421	+	0.873725i	$0.661698\pi$
$318$		0			0
$319$		−	7.11608e8i		−	1.22736i
$320$	−8.51993e8	+	1.45911e8i	−1.45349	+	0.248922i
$321$		0			0
$322$	−2.83998e8	+	8.03043e6i	−0.474045	+	0.0134043i
$323$			1.14362e8i			0.188830i
$324$		0			0
$325$			1.29771e9i			2.09694i
$326$	6.30426e6	+	2.22952e8i	0.0100780	+	0.356409i
$327$		0			0
$328$	2.30094e8	−	1.95604e7i	0.360037	−	0.0306068i
$329$		−	8.45972e7i		−	0.130969i
$330$		0			0
$331$	−8.03548e8			−1.21791			−0.608953	−	0.793206i	$-0.708410\pi$
$331$	−8.03548e8			−1.21791			−0.608953	+	0.793206i	$0.708410\pi$
$332$	−3.43762e7	−	6.07375e8i	−0.0515554	−	0.910906i
$333$		0			0
$334$	−2.62816e7	−	9.29455e8i	−0.0385957	−	1.36495i
$335$	9.40371e8			1.36660
$336$		0			0
$337$	−8.74822e8			−1.24513			−0.622565	−	0.782568i	$-0.713910\pi$
$337$	−8.74822e8			−1.24513			−0.622565	+	0.782568i	$0.713910\pi$
$338$	4.38871e7	+	1.55208e9i	0.0618199	+	2.18628i
$339$		0			0
$340$	−9.93202e6	−	1.75484e8i	−0.0137044	−	0.242137i
$341$	7.24693e8			0.989723
$342$		0			0
$343$		−	5.45242e8i		−	0.729558i
$344$	9.28864e7	+	1.09265e9i	0.123026	+	1.44719i
$345$		0			0
$346$	−8.46905e6	−	2.99510e8i	−0.0109918	−	0.388727i
$347$		−	3.16428e8i		−	0.406557i	−0.979121	−	0.203279i	$-0.934840\pi$
$347$		−	3.16428e8i		−	0.406557i	0.979121	−	0.203279i	$-0.0651597\pi$
$348$		0			0
$349$			4.41239e8i			0.555629i	0.960635	+	0.277814i	$0.0896101\pi$
$349$			4.41239e8i			0.555629i	−0.960635	+	0.277814i	$0.910390\pi$
$350$	−3.72735e8	+	1.05396e7i	−0.464688	+	0.0131397i
$351$		0			0
$352$	7.00849e8	−	9.97259e7i	0.856496	−	0.121873i
$353$		−	3.79064e8i		−	0.458670i	−0.973348	−	0.229335i	$-0.926345\pi$
$353$		−	3.79064e8i		−	0.458670i	0.973348	−	0.229335i	$-0.0736552\pi$
$354$		0			0
$355$	−2.11039e9			−2.50359
$356$	−5.12653e7	−	9.05780e8i	−0.0602211	−	1.06402i
$357$		0			0
$358$	−1.04614e9	+	2.95811e7i	−1.20503	+	0.0340740i
$359$	−4.43347e8			−0.505723			−0.252862	−	0.967502i	$-0.581372\pi$
$359$	−4.43347e8			−0.505723			−0.252862	+	0.967502i	$0.581372\pi$
$360$		0			0
$361$	2.84510e8			0.318289
$362$	8.95300e8	−	2.53158e7i	0.991948	−	0.0280487i
$363$		0			0
$364$	−6.49099e8	+	3.67377e7i	−0.705434	+	0.0399261i
$365$	−2.15043e8			−0.231473
$366$		0			0
$367$			2.76323e8i			0.291800i	0.989299	+	0.145900i	$0.0466078\pi$
$367$			2.76323e8i			0.291800i	−0.989299	+	0.145900i	$0.953392\pi$
$368$	1.13823e9	−	1.29257e8i	1.19059	−	0.135203i
$369$		0			0
$370$	8.85773e8	−	2.50464e7i	0.909110	−	0.0257063i
$371$		−	2.62919e8i		−	0.267309i
$372$		0			0
$373$		−	1.23995e9i		−	1.23715i	−0.785726	−	0.618575i	$-0.787710\pi$
$373$		−	1.23995e9i		−	1.23715i	0.785726	−	0.618575i	$-0.212290\pi$
$374$	4.06860e6	+	1.43887e8i	0.00402156	+	0.142223i
$375$		0			0
$376$	2.88926e7	+	3.39871e8i	0.0280304	+	0.329729i
$377$		−	2.63507e9i		−	2.53278i
$378$		0			0
$379$	−5.00434e8			−0.472183			−0.236091	−	0.971731i	$-0.575866\pi$
$379$	−5.00434e8			−0.472183			−0.236091	+	0.971731i	$0.575866\pi$
$380$	−1.80818e9	+	1.02339e8i	−1.69044	+	0.0956754i
$381$		0			0
$382$	3.29133e7	+	1.16398e9i	0.0302099	+	1.06838i
$383$	−4.63975e8			−0.421987			−0.210993	−	0.977487i	$-0.567670\pi$
$383$	−4.63975e8			−0.421987			−0.210993	+	0.977487i	$0.567670\pi$
$384$		0			0
$385$	5.65364e8			0.504912
$386$	1.91248e7	+	6.76352e8i	0.0169255	+	0.598573i
$387$		0			0
$388$	1.72503e9	−	9.76331e7i	1.49929	−	0.0848567i
$389$	−2.74898e8			−0.236781			−0.118391	−	0.992967i	$-0.537774\pi$
$389$	−2.74898e8			−0.236781			−0.118391	+	0.992967i	$0.537774\pi$
$390$		0			0
$391$			2.32932e8i			0.197065i
$392$	8.51968e7	+	1.00219e9i	0.0714368	+	0.840330i
$393$		0			0
$394$	5.38373e7	+	1.90397e9i	0.0443452	+	1.56828i
$395$		−	1.59031e9i		−	1.29835i
$396$		0			0
$397$		−	1.28191e9i		−	1.02823i	−0.857721	−	0.514116i	$-0.828120\pi$
$397$		−	1.28191e9i		−	1.02823i	0.857721	−	0.514116i	$-0.171880\pi$
$398$	−1.04530e9	+	2.95573e7i	−0.831096	+	0.0235004i
$399$		0			0
$400$	1.49387e9	−	1.69644e8i	1.16709	−	0.132534i
$401$			2.19451e8i			0.169954i	0.996383	+	0.0849772i	$0.0270817\pi$
$401$			2.19451e8i			0.169954i	−0.996383	+	0.0849772i	$0.972918\pi$
$402$		0			0
$403$	2.68352e9			2.04238
$404$	1.60821e9	−	9.10216e7i	1.21341	−	0.0686767i
$405$		0			0
$406$	7.56855e8	−	2.14011e7i	0.561269	−	0.0158707i
$407$	−7.25704e8			−0.533555
$408$		0			0
$409$	−1.87644e8			−0.135614			−0.0678068	−	0.997698i	$-0.521600\pi$
$409$	−1.87644e8			−0.135614			−0.0678068	+	0.997698i	$0.521600\pi$
$410$	−7.43308e8	+	2.10180e7i	−0.532629	+	0.0150608i
$411$		0			0
$412$	−6.10495e7	−	1.07865e9i	−0.0430073	−	0.759873i
$413$	7.11856e8			0.497241
$414$		0			0
$415$			1.95896e9i			1.34542i
$416$	2.59523e9	−	3.69283e8i	1.76746	−	0.251497i
$417$		0			0
$418$	1.48261e9	−	4.19228e7i	0.992909	−	0.0280759i
$419$			7.92192e8i			0.526116i	0.964780	+	0.263058i	$0.0847310\pi$
$419$			7.92192e8i			0.526116i	−0.964780	+	0.263058i	$0.915269\pi$
$420$		0			0
$421$		−	5.56076e7i		−	0.0363201i	−0.999835	−	0.0181601i	$-0.994219\pi$
$421$		−	5.56076e7i		−	0.0363201i	0.999835	−	0.0181601i	$-0.00578084\pi$
$422$	−3.87019e7	−	1.36870e9i	−0.0250691	−	0.886574i
$423$		0			0
$424$	8.97951e7	+	1.05628e9i	0.0572100	+	0.672977i
$425$			3.05713e8i			0.193176i
$426$		0			0
$427$	−4.95402e8			−0.307936
$428$	8.15282e7	+	1.44048e9i	0.0502637	+	0.888084i
$429$		0			0
$430$	−9.98084e7	−	3.52975e9i	−0.0605379	−	2.14094i
$431$	−8.21789e8			−0.494413			−0.247206	−	0.968963i	$-0.579513\pi$
$431$	−8.21789e8			−0.494413			−0.247206	+	0.968963i	$0.579513\pi$
$432$		0			0
$433$	2.34831e8			0.139011			0.0695054	−	0.997582i	$-0.477858\pi$
$433$	2.34831e8			0.139011			0.0695054	+	0.997582i	$0.477858\pi$
$434$	2.17946e7	+	7.70771e8i	0.0127978	+	0.452597i
$435$		0			0
$436$	−5.67035e7	−	1.00186e9i	−0.0327648	−	0.578903i
$437$	2.40013e9			1.37578
$438$		0			0
$439$		−	2.08919e9i		−	1.17856i	−0.807928	−	0.589281i	$-0.799411\pi$
$439$		−	2.08919e9i		−	1.17856i	0.807928	−	0.589281i	$-0.200589\pi$
$440$	−2.27137e9	+	1.93090e8i	−1.27117	+	0.108062i
$441$		0			0
$442$	1.50659e7	+	5.32810e8i	0.00829885	+	0.293491i
$443$		−	4.20790e8i		−	0.229960i	−0.993368	−	0.114980i	$-0.963320\pi$
$443$		−	4.20790e8i		−	0.229960i	0.993368	−	0.114980i	$-0.0366804\pi$
$444$		0			0
$445$			2.92140e9i			1.57156i
$446$	−7.17860e8	+	2.02985e7i	−0.383149	+	0.0108341i
$447$		0			0
$448$	1.27144e8	+	7.42412e8i	0.0668073	+	0.390097i
$449$			1.20517e9i			0.628329i	0.949369	+	0.314164i	$0.101724\pi$
$449$			1.20517e9i			0.628329i	−0.949369	+	0.314164i	$0.898276\pi$
$450$		0			0
$451$	6.08984e8			0.312599
$452$	−9.71422e7	−	1.71635e9i	−0.0494793	−	0.874224i
$453$		0			0
$454$	1.84218e9	−	5.20902e7i	0.923926	−	0.0261253i
$455$	2.09353e9			1.04193
$456$		0			0
$457$	1.17565e8			0.0576198			0.0288099	−	0.999585i	$-0.490828\pi$
$457$	1.17565e8			0.0576198			0.0288099	+	0.999585i	$0.490828\pi$
$458$	3.22634e8	−	9.12292e6i	0.156921	−	0.00443715i
$459$		0			0
$460$	−3.68290e9	+	2.08445e8i	−1.76416	+	0.0998478i
$461$	−1.15104e9			−0.547190			−0.273595	−	0.961845i	$-0.588213\pi$
$461$	−1.15104e9			−0.547190			−0.273595	+	0.961845i	$0.588213\pi$
$462$		0			0
$463$			2.00264e9i			0.937712i	0.883275	+	0.468856i	$0.155334\pi$
$463$			2.00264e9i			0.937712i	−0.883275	+	0.468856i	$0.844666\pi$
$464$	−3.03338e9	+	3.44469e8i	−1.40966	+	0.160080i
$465$		0			0
$466$	−9.80899e8	+	2.77362e7i	−0.449028	+	0.0126969i
$467$		−	1.12001e9i		−	0.508876i	−0.967089	−	0.254438i	$-0.918110\pi$
$467$		−	1.12001e9i		−	0.508876i	0.967089	−	0.254438i	$-0.0818905\pi$
$468$		0			0
$469$		−	8.19423e8i		−	0.366778i
$470$	−3.10457e7	−	1.09794e9i	−0.0137930	−	0.487793i
$471$		0			0
$472$	−2.85990e9	+	2.43121e8i	−1.25186	+	0.106421i
$473$			2.89188e9i			1.25651i
$474$		0			0
$475$	3.15006e9			1.34862
$476$	−1.52913e8	+	8.65460e6i	−0.0649862	+	0.00367809i
$477$		0			0
$478$	1.07267e8	+	3.79352e9i	0.0449230	+	1.58871i
$479$	1.11855e9			0.465031			0.232515	−	0.972593i	$-0.425304\pi$
$479$	1.11855e9			0.465031			0.232515	+	0.972593i	$0.425304\pi$
$480$		0			0
$481$	−2.68726e9			−1.10104
$482$	−1.37229e8	−	4.85315e9i	−0.0558191	−	1.97405i
$483$		0			0
$484$	−6.26484e8	+	3.54577e7i	−0.251160	+	0.0142152i
$485$	−5.56371e9			−2.21446
$486$		0			0
$487$		−	3.33703e9i		−	1.30921i	−0.755972	−	0.654604i	$-0.772835\pi$
$487$		−	3.33703e9i		−	1.30921i	0.755972	−	0.654604i	$-0.227165\pi$
$488$	1.99029e9	−	1.69195e8i	0.775259	−	0.0659051i
$489$		0			0
$490$	−9.15458e7	−	3.23754e9i	−0.0351522	−	1.24316i
$491$		−	4.90001e8i		−	0.186815i	−0.995628	−	0.0934075i	$-0.970224\pi$
$491$		−	4.90001e8i		−	0.186815i	0.995628	−	0.0934075i	$-0.0297759\pi$
$492$		0			0
$493$		−	6.20764e8i		−	0.233325i
$494$	5.49006e9	−	1.55239e8i	2.04896	−	0.0579371i
$495$		0			0
$496$	−3.50803e8	−	3.08915e9i	−0.129086	−	1.13672i
$497$			1.83896e9i			0.671931i
$498$		0			0
$499$	1.65302e9			0.595563			0.297781	−	0.954634i	$-0.403753\pi$
$499$	1.65302e9			0.595563			0.297781	+	0.954634i	$0.403753\pi$
$500$	−7.18463e8	+	4.06635e7i	−0.257045	+	0.0145482i
$501$		0			0
$502$	−1.77644e9	+	5.02314e7i	−0.626741	+	0.0177220i
$503$	1.28708e7			0.00450938			0.00225469	−	0.999997i	$-0.499282\pi$
$503$	1.28708e7			0.00450938			0.00225469	+	0.999997i	$0.499282\pi$
$504$		0			0
$505$	−5.18694e9			−1.79222
$506$	3.01978e9	−	8.53883e7i	1.03621	−	0.0293003i
$507$		0			0
$508$	2.11942e8	+	3.74469e9i	0.0717289	+	1.26734i
$509$	5.13852e8			0.172713			0.0863567	−	0.996264i	$-0.472478\pi$
$509$	5.13852e8			0.172713			0.0863567	+	0.996264i	$0.472478\pi$
$510$		0			0
$511$			1.87385e8i			0.0621243i
$512$	−7.64363e8	−	2.93924e9i	−0.251684	−	0.967810i
$513$		0			0
$514$	−5.31232e8	+	1.50213e7i	−0.172549	+	0.00487907i
$515$			3.47896e9i			1.12234i
$516$		0			0
$517$			8.99530e8i			0.286285i
$518$	−2.18250e7	−	7.71847e8i	−0.00689923	−	0.243993i
$519$		0			0
$520$	−8.41081e9	+	7.15006e8i	−2.62317	+	0.222997i
$521$			4.90297e9i			1.51889i	0.650570	+	0.759446i	$0.274530\pi$
$521$			4.90297e9i			1.51889i	−0.650570	+	0.759446i	$0.725470\pi$
$522$		0			0
$523$	2.66052e8			0.0813224			0.0406612	−	0.999173i	$-0.487054\pi$
$523$	2.66052e8			0.0813224			0.0406612	+	0.999173i	$0.487054\pi$
$524$	3.09515e8	+	5.46866e9i	0.0939772	+	1.66043i
$525$		0			0
$526$	3.20836e7	+	1.13465e9i	0.00961244	+	0.339946i
$527$	6.32178e8			0.188149
$528$		0			0
$529$	1.48375e9			0.435778
$530$	−9.64868e7	−	3.41228e9i	−0.0281515	−	0.995586i
$531$		0			0
$532$	8.91768e7	+	1.57562e9i	0.0256780	+	0.453691i
$533$	2.25505e9			0.645077
$534$		0			0
$535$		−	4.64596e9i		−	1.31171i
$536$	2.79859e8	+	3.29205e9i	0.0784986	+	0.923400i
$537$		0			0
$538$	−8.65900e7	−	3.06227e9i	−0.0239734	−	0.847825i
$539$			2.65248e9i			0.729611i
$540$		0			0
$541$		−	2.26607e9i		−	0.615295i	−0.951500	−	0.307647i	$-0.900458\pi$
$541$		−	2.26607e9i		−	0.615295i	0.951500	−	0.307647i	$-0.0995417\pi$
$542$	−7.01872e9	+	1.98464e8i	−1.89348	+	0.0535407i
$543$		0			0
$544$	6.11378e8	−	8.69949e7i	0.162822	−	0.0231685i
$545$			3.23130e9i			0.855045i
$546$		0			0
$547$	6.06164e9			1.58356			0.791780	−	0.610806i	$-0.209155\pi$
$547$	6.06164e9			1.58356			0.791780	+	0.610806i	$0.209155\pi$
$548$	2.58332e7	+	4.56433e8i	0.00670573	+	0.118480i
$549$		0			0
$550$	3.96332e9	−	1.12068e8i	1.01576	−	0.0287219i
$551$	−6.39634e9			−1.62892
$552$		0			0
$553$	−1.38577e9			−0.348459
$554$	−6.71207e9	+	1.89793e8i	−1.67715	+	0.0474238i
$555$		0			0
$556$	−5.94558e9	+	3.36508e8i	−1.46701	+	0.0830297i
$557$	2.14513e9			0.525968			0.262984	−	0.964800i	$-0.415293\pi$
$557$	2.14513e9			0.525968			0.262984	+	0.964800i	$0.415293\pi$
$558$		0			0
$559$			1.07086e10i			2.59293i
$560$	−2.73677e8	−	2.40998e9i	−0.0658537	−	0.579903i
$561$		0			0
$562$	2.82036e9	−	7.97494e7i	0.670235	−	0.0189518i
$563$		−	3.66627e9i		−	0.865855i	−0.901429	−	0.432927i	$-0.857481\pi$
$563$		−	3.66627e9i		−	0.865855i	0.901429	−	0.432927i	$-0.142519\pi$
$564$		0			0
$565$			5.53573e9i			1.29124i
$566$	7.71431e7	+	2.72818e9i	0.0178830	+	0.632435i
$567$		0			0
$568$	−6.28063e8	−	7.38807e9i	−0.143808	−	1.69165i
$569$			4.28882e9i			0.975989i	0.872847	+	0.487994i	$0.162271\pi$
$569$			4.28882e9i			0.975989i	−0.872847	+	0.487994i	$0.837729\pi$
$570$		0			0
$571$	2.79269e9			0.627763			0.313881	−	0.949462i	$-0.398371\pi$
$571$	2.79269e9			0.627763			0.313881	+	0.949462i	$0.398371\pi$
$572$	6.90193e9	−	3.90635e8i	1.54200	−	0.0872741i
$573$		0			0
$574$	1.83148e7	+	6.47706e8i	0.00404212	+	0.142951i
$575$	6.41604e9			1.40744
$576$		0			0
$577$	−1.42684e9			−0.309214			−0.154607	−	0.987976i	$-0.549411\pi$
$577$	−1.42684e9			−0.309214			−0.154607	+	0.987976i	$0.549411\pi$
$578$	−1.27670e8	−	4.51508e9i	−0.0275006	−	0.972563i
$579$		0			0
$580$	9.81494e9	−	5.55506e8i	2.08877	−	0.118220i
$581$	1.70700e9			0.361092
$582$		0			0
$583$			2.79564e9i			0.584307i
$584$	−6.39978e7	−	7.52824e8i	−0.0132960	−	0.156404i
$585$		0			0
$586$	1.41142e8	+	4.99151e9i	0.0289744	+	1.02469i
$587$		−	8.26290e9i		−	1.68616i	−0.537787	−	0.843081i	$-0.680740\pi$
$587$		−	8.26290e9i		−	1.68616i	0.537787	−	0.843081i	$-0.319260\pi$
$588$		0			0
$589$		−	6.51395e9i		−	1.31353i
$590$	9.23878e9	−	2.61239e8i	1.85196	−	0.0523668i
$591$		0			0
$592$	3.51293e8	+	3.09346e9i	0.0695894	+	0.612800i
$593$		−	4.44985e9i		−	0.876302i	−0.898901	−	0.438151i	$-0.855634\pi$
$593$		−	4.44985e9i		−	0.876302i	0.898901	−	0.438151i	$-0.144366\pi$
$594$		0			0
$595$	4.93190e8			0.0959852
$596$	−4.39390e9	+	2.48686e8i	−0.850136	+	0.0481160i
$597$		0			0
$598$	1.11822e10	−	3.16191e8i	2.13831	−	0.0604637i
$599$	−5.97431e9			−1.13578			−0.567890	−	0.823105i	$-0.692240\pi$
$599$	−5.97431e9			−1.13578			−0.567890	+	0.823105i	$0.692240\pi$
$600$		0			0
$601$	5.81457e9			1.09259			0.546295	−	0.837593i	$-0.316038\pi$
$601$	5.81457e9			1.09259			0.546295	+	0.837593i	$0.316038\pi$
$602$	−3.07576e9	+	8.69713e7i	−0.574599	+	0.0162476i
$603$		0			0
$604$	−5.30689e8	−	9.37646e9i	−0.0979965	−	1.73145i
$605$	2.02059e9			0.370966
$606$		0			0
$607$		−	3.54687e9i		−	0.643703i	−0.946790	−	0.321851i	$-0.895695\pi$
$607$		−	3.54687e9i		−	0.643703i	0.946790	−	0.321851i	$-0.104305\pi$
$608$	−8.96394e8	−	6.29963e9i	−0.161747	−	1.13672i
$609$		0			0
$610$	−6.42954e9	+	1.81804e8i	−1.14690	+	0.0324302i
$611$			3.33094e9i			0.590775i
$612$		0			0
$613$		−	8.92550e9i		−	1.56502i	−0.622635	−	0.782512i	$-0.713938\pi$
$613$		−	8.92550e9i		−	1.56502i	0.622635	−	0.782512i	$-0.286062\pi$
$614$	1.28868e8	+	4.55746e9i	0.0224676	+	0.794572i
$615$		0			0
$616$	1.68255e8	+	1.97923e9i	0.0290025	+	0.341164i
$617$		−	6.93844e9i		−	1.18923i	−0.804012	−	0.594613i	$-0.797305\pi$
$617$		−	6.93844e9i		−	1.18923i	0.804012	−	0.594613i	$-0.202695\pi$
$618$		0			0
$619$	−1.41395e9			−0.239617			−0.119808	−	0.992797i	$-0.538228\pi$
$619$	−1.41395e9			−0.239617			−0.119808	+	0.992797i	$0.538228\pi$
$620$	5.65720e8	+	9.99541e9i	0.0953302	+	1.68434i
$621$		0			0
$622$	−3.33197e7	−	1.17836e9i	−0.00555182	−	0.196341i
$623$	2.54566e9			0.421786
$624$		0			0
$625$	−4.85187e9			−0.794931
$626$	−2.01111e8	−	7.11234e9i	−0.0327662	−	1.15878i
$627$		0			0
$628$	2.73518e8	+	4.83264e9i	0.0440683	+	0.778620i
$629$	−6.33060e8			−0.101430
$630$		0			0
$631$			7.00062e8i			0.110926i	0.998461	+	0.0554631i	$0.0176635\pi$
$631$			7.00062e8i			0.110926i	−0.998461	+	0.0554631i	$0.982336\pi$
$632$	5.56736e9	−	4.73283e8i	0.877282	−	0.0745781i
$633$		0			0
$634$	1.76443e8	+	6.23995e9i	0.0274974	+	0.972453i
$635$		−	1.20777e10i		−	1.87187i
$636$		0			0
$637$			9.82207e9i			1.50562i
$638$	−8.04771e9	+	2.27560e8i	−1.22687	+	0.0346915i
$639$		0			0
$640$	1.92259e9	+	9.58869e9i	0.289905	+	1.44587i
$641$		−	1.19021e10i		−	1.78493i	−0.451117	−	0.892465i	$-0.648974\pi$
$641$		−	1.19021e10i		−	1.78493i	0.451117	−	0.892465i	$-0.351026\pi$
$642$		0			0
$643$	−2.57955e9			−0.382653			−0.191326	−	0.981526i	$-0.561279\pi$
$643$	−2.57955e9			−0.382653			−0.191326	+	0.981526i	$0.561279\pi$
$644$	1.81635e8	+	3.20922e9i	0.0267978	+	0.473477i
$645$		0			0
$646$	1.29334e9	−	3.65709e7i	0.188755	−	0.00533730i
$647$	−5.01688e9			−0.728230			−0.364115	−	0.931354i	$-0.618629\pi$
$647$	−5.01688e9			−0.728230			−0.364115	+	0.931354i	$0.618629\pi$
$648$		0			0
$649$	−7.56924e9			−1.08692
$650$	1.46761e10	−	4.14986e8i	2.09611	−	0.0592703i
$651$		0			0
$652$	2.51939e9	−	1.42592e8i	0.355982	−	0.0201479i
$653$	−9.90200e9			−1.39164			−0.695820	−	0.718216i	$-0.744959\pi$
$653$	−9.90200e9			−1.39164			−0.695820	+	0.718216i	$0.744959\pi$
$654$		0			0
$655$		−	1.76380e10i		−	2.45247i
$656$	−2.94792e8	−	2.59592e9i	−0.0407711	−	0.359028i
$657$		0			0
$658$	−9.56725e8	+	2.70527e7i	−0.130917	+	0.00370186i
$659$			3.56641e9i			0.485437i	0.970097	+	0.242719i	$0.0780391\pi$
$659$			3.56641e9i			0.485437i	−0.970097	+	0.242719i	$0.921961\pi$
$660$		0			0
$661$			1.03477e10i			1.39360i	0.717265	+	0.696800i	$0.245394\pi$
$661$			1.03477e10i			1.39360i	−0.717265	+	0.696800i	$0.754606\pi$
$662$	2.56961e8	+	9.08748e9i	0.0344242	+	1.21742i
$663$		0			0
$664$	−6.85792e9	+	5.82995e8i	−0.909085	+	0.0772816i
$665$		−	5.08182e9i		−	0.670106i
$666$		0			0
$667$	−1.30281e10			−1.69996
$668$	−1.05030e10	+	5.94447e8i	−1.36331	+	0.0771606i
$669$		0			0
$670$	−3.00714e8	−	1.06348e10i	−0.0386271	−	1.36606i
$671$	5.26765e9			0.673113
$672$		0			0
$673$	9.52836e9			1.20494			0.602470	−	0.798142i	$-0.294183\pi$
$673$	9.52836e9			1.20494			0.602470	+	0.798142i	$0.294183\pi$
$674$	2.79753e8	+	9.89352e9i	0.0351937	+	1.24463i
$675$		0			0
$676$	1.75387e10	−	9.92655e8i	2.18366	−	0.123590i
$677$	1.90058e8			0.0235411			0.0117705	−	0.999931i	$-0.496253\pi$
$677$	1.90058e8			0.0235411			0.0117705	+	0.999931i	$0.496253\pi$
$678$		0			0
$679$			4.84812e9i			0.594332i
$680$	−1.98140e9	+	1.68440e8i	−0.241653	+	0.0205430i
$681$		0			0
$682$	−2.31744e8	−	8.19569e9i	−0.0279746	−	0.989327i
$683$			3.28062e9i			0.393988i	0.980405	+	0.196994i	$0.0631180\pi$
$683$			3.28062e9i			0.393988i	−0.980405	+	0.196994i	$0.936882\pi$
$684$		0			0
$685$		−	1.47213e9i		−	0.174996i
$686$	−6.16624e9	+	1.74359e8i	−0.729267	+	0.0206210i
$687$		0			0
$688$	1.23273e10	−	1.39988e9i	1.44313	−	0.163882i
$689$			1.03522e10i			1.20577i
$690$		0			0
$691$	3.49476e9			0.402944			0.201472	−	0.979494i	$-0.435428\pi$
$691$	3.49476e9			0.402944			0.201472	+	0.979494i	$0.435428\pi$
$692$	−3.38451e9	+	1.91556e8i	−0.388261	+	0.0219748i
$693$		0			0
$694$	−3.57854e9	+	1.01188e8i	−0.406395	+	0.0114914i
$695$	1.91762e10			2.16678
$696$		0			0
$697$	5.31241e8			0.0594260
$698$	4.99005e9	−	1.41101e8i	0.555407	−	0.0157049i
$699$		0			0
$700$	2.38388e8	+	4.21195e9i	0.0262689	+	0.464131i
$701$	1.22907e10			1.34761			0.673805	−	0.738909i	$-0.264659\pi$
$701$	1.22907e10			1.34761			0.673805	+	0.738909i	$0.264659\pi$
$702$		0			0
$703$			6.52304e9i			0.708120i
$704$	−1.35194e9	−	7.89414e9i	−0.146034	−	0.852709i
$705$		0			0
$706$	−4.28690e9	+	1.21218e8i	−0.458487	+	0.0129643i
$707$			4.51981e9i			0.481008i
$708$		0			0
$709$		−	8.80574e9i		−	0.927906i	−0.885860	−	0.463953i	$-0.846431\pi$
$709$		−	8.80574e9i		−	0.927906i	0.885860	−	0.463953i	$-0.153569\pi$
$710$	6.74867e8	+	2.38668e10i	0.0707642	+	2.50259i
$711$		0			0
$712$	−1.02272e10	+	8.69422e8i	−1.06189	+	0.0902715i
$713$		−	1.32676e10i		−	1.37082i
$714$		0			0
$715$	−2.22607e10			−2.27755
$716$	6.69075e8	+	1.18215e10i	0.0681208	+	1.20359i
$717$		0			0
$718$	1.41775e8	+	5.01389e9i	0.0142943	+	0.505521i
$719$	−6.43721e9			−0.645872			−0.322936	−	0.946421i	$-0.604670\pi$
$719$	−6.43721e9			−0.645872			−0.322936	+	0.946421i	$0.604670\pi$
$720$		0			0
$721$	3.03150e9			0.301221
$722$	−9.09813e7	−	3.21757e9i	−0.00899647	−	0.318162i
$723$		0			0
$724$	−5.72603e8	−	1.01170e10i	−0.0560749	−	0.990759i
$725$	−1.70987e10			−1.66641
$726$		0			0
$727$			1.15462e10i			1.11447i	0.830354	+	0.557236i	$0.188138\pi$
$727$			1.15462e10i			1.11447i	−0.830354	+	0.557236i	$0.811862\pi$
$728$	6.23044e8	+	7.32904e9i	0.0598493	+	0.704023i
$729$		0			0
$730$	6.87671e7	+	2.43196e9i	0.00654260	+	0.231381i
$731$			2.52270e9i			0.238867i
$732$		0			0
$733$			4.00798e9i			0.375890i	0.982180	+	0.187945i	$0.0601828\pi$
$733$			4.00798e9i			0.375890i	−0.982180	+	0.187945i	$0.939817\pi$
$734$	3.12499e9	−	8.83633e7i	0.291684	−	0.00824775i
$735$		0			0
$736$	−1.82577e9	−	1.28311e10i	−0.168801	−	1.18629i
$737$			8.71300e9i			0.801736i
$738$		0			0
$739$	−1.11260e10			−1.01411			−0.507055	−	0.861914i	$-0.669266\pi$
$739$	−1.11260e10			−1.01411			−0.507055	+	0.861914i	$0.669266\pi$
$740$	−5.66510e8	−	1.00094e10i	−0.0513921	−	0.908020i
$741$		0			0
$742$	−2.97340e9	+	8.40769e7i	−0.267202	+	0.00755550i
$743$	−3.41895e9			−0.305796			−0.152898	−	0.988242i	$-0.548861\pi$
$743$	−3.41895e9			−0.305796			−0.152898	+	0.988242i	$0.548861\pi$
$744$		0			0
$745$	1.41716e10			1.25566
$746$	−1.40228e10	+	3.96514e8i	−1.23666	+	0.0349681i
$747$		0			0
$748$	1.62594e9	−	9.20251e7i	0.142053	−	0.00803990i
$749$	−4.04841e9			−0.352045
$750$		0			0
$751$		−	2.35287e9i		−	0.202702i	−0.994851	−	0.101351i	$-0.967683\pi$
$751$		−	2.35287e9i		−	0.202702i	0.994851	−	0.101351i	$-0.0323165\pi$
$752$	3.83443e9	−	4.35437e8i	0.328805	−	0.0373390i
$753$		0			0
$754$	−2.98005e10	+	8.42649e8i	−2.53176	+	0.0715891i
$755$			3.02418e10i			2.55737i
$756$		0			0
$757$			2.65364e9i			0.222334i	0.993802	+	0.111167i	$0.0354589\pi$
$757$			2.65364e9i			0.222334i	−0.993802	+	0.111167i	$0.964541\pi$
$758$	1.60030e8	+	5.65951e9i	0.0133463	+	0.471994i
$759$		0			0
$760$	1.73560e9	+	2.04163e10i	0.143418	+	1.68706i
$761$		−	4.75292e9i		−	0.390944i	−0.980709	−	0.195472i	$-0.937376\pi$
$761$		−	4.75292e9i		−	0.390944i	0.980709	−	0.195472i	$-0.0626238\pi$
$762$		0			0
$763$	2.81570e9			0.229483
$764$	1.31532e10	−	7.44445e8i	1.06710	−	0.0603956i
$765$		0			0
$766$	1.48371e8	+	5.24718e9i	0.0119275	+	0.421818i
$767$	−2.80287e10			−2.24295
$768$		0			0
$769$	−1.27901e10			−1.01422			−0.507108	−	0.861883i	$-0.669285\pi$
$769$	−1.27901e10			−1.01422			−0.507108	+	0.861883i	$0.669285\pi$
$770$	−1.80794e8	−	6.39381e9i	−0.0142714	−	0.504710i
$771$		0			0
$772$	7.64287e9	−	4.32571e8i	0.597855	−	0.0338374i
$773$	1.92520e10			1.49916			0.749579	−	0.661915i	$-0.230256\pi$
$773$	1.92520e10			1.49916			0.749579	+	0.661915i	$0.230256\pi$
$774$		0			0
$775$		−	1.74131e10i		−	1.34376i
$776$	−1.65579e9	−	1.94774e10i	−0.127200	−	1.49629i
$777$		0			0
$778$	8.79075e7	+	3.10887e9i	0.00669264	+	0.236687i
$779$		−	5.47390e9i		−	0.414873i
$780$		0			0
$781$		−	1.95538e10i		−	1.46877i
$782$	2.63427e9	−	7.44876e7i	0.196987	−	0.00557006i
$783$		0			0
$784$	1.13067e10	−	1.28399e9i	0.837976	−	0.0951603i
$785$		−	1.55866e10i		−	1.15003i
$786$		0			0
$787$	−7.18441e9			−0.525388			−0.262694	−	0.964879i	$-0.584611\pi$
$787$	−7.18441e9			−0.525388			−0.262694	+	0.964879i	$0.584611\pi$
$788$	2.15151e10	−	1.21771e9i	1.56640	−	0.0886549i
$789$		0			0
$790$	−1.79851e10	+	5.08553e8i	−1.29783	+	0.0366979i
$791$	4.82374e9			0.346550
$792$		0			0
$793$	1.95060e10			1.38903
$794$	−1.44974e10	+	4.09933e8i	−1.02782	+	0.0290630i
$795$		0			0
$796$	6.68539e8	+	1.18121e10i	0.0469820	+	0.830100i
$797$	2.37227e10			1.65981			0.829907	−	0.557901i	$-0.188393\pi$
$797$	2.37227e10			1.65981			0.829907	+	0.557901i	$0.188393\pi$
$798$		0			0
$799$			7.84695e8i			0.0544236i
$800$	−2.39625e9	−	1.68402e10i	−0.165469	−	1.16287i
$801$		0			0
$802$	2.48181e9	−	7.01767e7i	0.169887	−	0.00480377i
$803$		−	1.99248e9i		−	0.135797i
$804$		0			0
$805$		−	1.03506e10i		−	0.699329i
$806$	−8.58143e8	−	3.03484e10i	−0.0577281	−	2.04157i
$807$		0			0
$808$	−1.54366e9	−	1.81585e10i	−0.102946	−	1.21099i
$809$			5.26128e9i			0.349359i	0.984625	+	0.174679i	$0.0558889\pi$
$809$			5.26128e9i			0.349359i	−0.984625	+	0.174679i	$0.944111\pi$
$810$		0			0
$811$	6.25968e9			0.412078			0.206039	−	0.978544i	$-0.433943\pi$
$811$	6.25968e9			0.412078			0.206039	+	0.978544i	$0.433943\pi$
$812$	−4.84058e8	−	8.55257e9i	−0.0317286	−	0.560597i
$813$		0			0
$814$	2.32068e8	+	8.20713e9i	0.0150810	+	0.533342i
$815$	−8.12574e9			−0.525789
$816$		0			0
$817$	2.59939e10			1.66761
$818$	6.00054e7	+	2.12210e9i	0.00383313	+	0.135559i
$819$		0			0
$820$	4.75394e8	+	8.39949e9i	0.0301096	+	0.531991i
$821$	4.64242e9			0.292781			0.146391	−	0.989227i	$-0.453234\pi$
$821$	4.64242e9			0.292781			0.146391	+	0.989227i	$0.453234\pi$
$822$		0			0
$823$			1.47371e10i			0.921540i	0.887520	+	0.460770i	$0.152427\pi$
$823$			1.47371e10i			0.921540i	−0.887520	+	0.460770i	$0.847573\pi$
$824$	−1.21791e10	+	1.03535e9i	−0.758354	+	0.0644679i
$825$		0			0
$826$	−2.27639e8	−	8.05052e9i	−0.0140546	−	0.497043i
$827$			5.88827e9i			0.362008i	0.983482	+	0.181004i	$0.0579348\pi$
$827$			5.88827e9i			0.362008i	−0.983482	+	0.181004i	$0.942065\pi$
$828$		0			0
$829$			3.00855e10i			1.83407i	0.398807	+	0.917035i	$0.369424\pi$
$829$			3.00855e10i			1.83407i	−0.398807	+	0.917035i	$0.630576\pi$
$830$	2.21542e10	−	6.26440e8i	1.34488	−	0.0380283i
$831$		0			0
$832$	−5.00619e9	−	2.92318e10i	−0.301353	−	1.75964i
$833$			2.31386e9i			0.138701i
$834$		0			0
$835$	3.38751e10			2.01362
$836$	−9.48226e8	−	1.67537e10i	−0.0561293	−	0.991719i
$837$		0			0
$838$	8.95905e9	−	2.53329e8i	0.525906	−	0.0148707i
$839$	2.18944e10			1.27987			0.639937	−	0.768428i	$-0.278961\pi$
$839$	2.18944e10			1.27987			0.639937	+	0.768428i	$0.278961\pi$
$840$		0			0
$841$	1.74699e10			1.01276
$842$	−6.28877e8	+	1.77824e7i	−0.0363056	+	0.00102659i
$843$		0			0
$844$	−1.54665e10	+	8.75374e8i	−0.885511	+	0.0501182i
$845$	−5.65673e10			−3.22528
$846$		0			0
$847$		−	1.76071e9i		−	0.0995623i
$848$	1.19170e10	−	1.35329e9i	0.671091	−	0.0762089i
$849$		0			0
$850$	3.45736e9	−	9.77617e7i	0.193098	−	0.00546012i
$851$			1.32861e10i			0.739001i
$852$		0			0
$853$			1.74899e10i			0.964862i	0.875934	+	0.482431i	$0.160246\pi$
$853$			1.74899e10i			0.964862i	−0.875934	+	0.482431i	$0.839754\pi$
$854$	1.58421e8	+	5.60259e9i	0.00870382	+	0.307812i
$855$		0			0
$856$	1.62646e10	−	1.38266e9i	0.886308	−	0.0753454i
$857$			1.86981e10i			1.01476i	0.861721	+	0.507382i	$0.169387\pi$
$857$			1.86981e10i			1.01476i	−0.861721	+	0.507382i	$0.830613\pi$
$858$		0			0
$859$	−3.23322e10			−1.74044			−0.870221	−	0.492662i	$-0.836024\pi$
$859$	−3.23322e10			−1.74044			−0.870221	+	0.492662i	$0.836024\pi$
$860$	−3.98867e10	+	2.25750e9i	−2.13837	+	0.121028i
$861$		0			0
$862$	2.62794e8	+	9.29376e9i	0.0139746	+	0.494215i
$863$	1.19116e9			0.0630859			0.0315429	−	0.999502i	$-0.489958\pi$
$863$	1.19116e9			0.0630859			0.0315429	+	0.999502i	$0.489958\pi$
$864$		0			0
$865$	1.09160e10			0.573465
$866$	−7.50950e7	−	2.65575e9i	−0.00392915	−	0.138955i
$867$		0			0
$868$	8.70983e9	−	4.92959e8i	0.452054	−	0.0255854i
$869$	1.47350e10			0.761694
$870$		0			0
$871$			3.22640e10i			1.65445i
$872$	−1.13121e10	+	9.61649e8i	−0.577746	+	0.0491144i
$873$		0			0
$874$	−7.67519e8	−	2.71435e10i	−0.0388865	−	1.37523i
$875$		−	2.01921e9i		−	0.101895i
$876$		0			0
$877$		−	2.08862e10i		−	1.04559i	−0.852459	−	0.522794i	$-0.824890\pi$
$877$		−	2.08862e10i		−	1.04559i	0.852459	−	0.522794i	$-0.175110\pi$
$878$	−2.36271e10	+	6.68088e8i	−1.17809	+	0.0333122i
$879$		0			0
$880$	2.91003e9	+	2.56256e10i	0.143949	+	1.26761i
$881$		−	4.96861e9i		−	0.244804i	−0.992481	−	0.122402i	$-0.960940\pi$
$881$		−	4.96861e9i		−	0.244804i	0.992481	−	0.122402i	$-0.0390598\pi$
$882$		0			0
$883$	2.57384e10			1.25811			0.629056	−	0.777360i	$-0.283442\pi$
$883$	2.57384e10			1.25811			0.629056	+	0.777360i	$0.283442\pi$
$884$	6.02083e9	−	3.40767e8i	0.293139	−	0.0165911i
$885$		0			0
$886$	−4.75879e9	+	1.34561e8i	−0.229868	+	0.00649984i
$887$	3.14294e10			1.51218			0.756089	−	0.654468i	$-0.227108\pi$
$887$	3.14294e10			1.51218			0.756089	+	0.654468i	$0.227108\pi$
$888$		0			0
$889$	−1.05243e10			−0.502386
$890$	3.30387e10	−	9.34213e8i	1.57093	−	0.0444202i
$891$		0			0
$892$	4.59118e8	+	8.11192e9i	0.0216595	+	0.382690i
$893$	8.08549e9			0.379950
$894$		0			0
$895$		−	3.81278e10i		−	1.77771i
$896$	8.35542e9	−	1.67531e9i	0.388052	−	0.0778068i
$897$		0			0
$898$	1.36295e10	−	3.85393e8i	0.628078	−	0.0177598i
$899$			3.53582e10i			1.62305i
$900$		0			0
$901$			2.43875e9i			0.111079i
$902$	−1.94743e8	−	6.88712e9i	−0.00883564	−	0.312474i
$903$		0			0
$904$	−1.93795e10	+	1.64746e9i	−0.872476	+	0.0741695i
$905$			3.26303e10i			1.46336i
$906$		0			0
$907$	2.28352e10			1.01620			0.508099	−	0.861299i	$-0.330348\pi$
$907$	2.28352e10			1.01620			0.508099	+	0.861299i	$0.330348\pi$
$908$	−1.17820e9	−	2.08169e10i	−0.0522297	−	0.922819i
$909$		0			0
$910$	−6.69475e8	−	2.36761e10i	−0.0294503	−	1.04152i
$911$	−1.67176e10			−0.732589			−0.366295	−	0.930499i	$-0.619374\pi$
$911$	−1.67176e10			−0.732589			−0.366295	+	0.930499i	$0.619374\pi$
$912$		0			0
$913$	−1.81507e10			−0.789307
$914$	−3.75953e7	−	1.32957e9i	−0.00162863	−	0.0575968i
$915$		0			0
$916$	−2.06346e8	−	3.64581e9i	−0.00887076	−	0.156733i
$917$	−1.53694e10			−0.658211
$918$		0			0
$919$		−	3.46269e10i		−	1.47167i	−0.677162	−	0.735834i	$-0.736790\pi$
$919$		−	3.46269e10i		−	1.47167i	0.677162	−	0.735834i	$-0.263210\pi$
$920$	3.53507e9	+	4.15840e10i	0.149672	+	1.76063i
$921$		0			0
$922$	3.68084e8	+	1.30174e10i	0.0154664	+	0.546972i
$923$		−	7.24073e10i		−	3.03093i
$924$		0			0
$925$			1.74375e10i			0.724414i
$926$	2.26482e10	−	6.40410e8i	0.937338	−	0.0265045i
$927$		0			0
$928$	4.86569e9	+	3.41949e10i	0.199860	+	1.40457i
$929$		−	1.05754e10i		−	0.432753i	−0.976310	−	0.216377i	$-0.930576\pi$
$929$		−	1.05754e10i		−	0.432753i	0.976310	−	0.216377i	$-0.0694239\pi$
$930$		0			0
$931$	2.38420e10			0.968320
$932$	6.27349e8	+	1.10843e10i	0.0253836	+	0.448490i
$933$		0			0
$934$	−1.26664e10	+	3.58159e8i	−0.508673	+	0.0143834i
$935$	−5.24413e9			−0.209813
$936$		0			0
$937$	−3.29022e10			−1.30658			−0.653290	−	0.757107i	$-0.726612\pi$
$937$	−3.29022e10			−1.30658			−0.653290	+	0.757107i	$0.726612\pi$
$938$	−9.26701e9	+	2.62037e8i	−0.366631	+	0.0103670i
$939$		0			0
$940$	−1.24069e10	+	7.02204e8i	−0.487208	+	0.0275750i
$941$	−3.87420e9			−0.151572			−0.0757859	−	0.997124i	$-0.524147\pi$
$941$	−3.87420e9			−0.151572			−0.0757859	+	0.997124i	$0.524147\pi$
$942$		0			0
$943$		−	1.11492e10i		−	0.432966i
$944$	3.66405e9	+	3.22654e10i	0.141762	+	1.24835i
$945$		0			0
$946$	3.27049e10	−	9.24775e8i	1.25601	−	0.0355154i
$947$			3.64475e10i			1.39458i	0.716791	+	0.697288i	$0.245610\pi$
$947$			3.64475e10i			1.39458i	−0.716791	+	0.697288i	$0.754390\pi$
$948$		0			0
$949$		−	7.37811e9i		−	0.280229i
$950$	−1.00733e9	−	3.56246e10i	−0.0381190	−	1.34809i
$951$		0			0
$952$	1.46776e8	+	1.72656e9i	0.00551346	+	0.0648563i
$953$		−	4.75252e10i		−	1.77868i	−0.457242	−	0.889342i	$-0.651163\pi$
$953$		−	4.75252e10i		−	1.77868i	0.457242	−	0.889342i	$-0.348837\pi$
$954$		0			0
$955$	−4.24228e10			−1.57611
$956$	4.28673e10	−	2.42620e9i	1.58681	−	0.0898100i
$957$		0			0
$958$	−3.57693e8	−	1.26499e10i	−0.0131441	−	0.464845i
$959$	−1.28278e9			−0.0469666
$960$		0			0
$961$	−8.49574e9			−0.308794
$962$	8.59341e8	+	3.03908e10i	0.0311209	+	1.10060i
$963$		0			0
$964$	−5.48413e10	+	3.10391e9i	−1.97169	+	0.111594i
$965$	−2.46504e10			−0.883037
$966$		0			0
$967$			5.17956e10i			1.84205i	0.389508	+	0.921023i	$0.372645\pi$
$967$			5.17956e10i			1.84205i	−0.389508	+	0.921023i	$0.627355\pi$
$968$	6.01337e8	+	7.07368e9i	0.0213086	+	0.250658i
$969$		0			0
$970$	1.77918e9	+	6.29210e10i	0.0625919	+	2.21358i
$971$		−	4.27726e10i		−	1.49933i	−0.661816	−	0.749667i	$-0.730214\pi$
$971$		−	4.27726e10i		−	1.49933i	0.661816	−	0.749667i	$-0.269786\pi$
$972$		0			0
$973$		−	1.67098e10i		−	0.581535i
$974$	−3.77391e10	+	1.06712e9i	−1.30868	+	0.0370048i
$975$		0			0
$976$	−2.54992e9	−	2.24544e10i	−0.0877915	−	0.773086i
$977$			4.75207e10i			1.63024i	0.579291	+	0.815121i	$0.303330\pi$
$977$			4.75207e10i			1.63024i	−0.579291	+	0.815121i	$0.696670\pi$
$978$		0			0
$979$	−2.70682e10			−0.921977
$980$	−3.65847e10	+	2.07062e9i	−1.24167	+	0.0702763i
$981$		0			0
$982$	−5.54151e9	+	1.56694e8i	−0.186740	+	0.00528034i
$983$	−3.39823e10			−1.14108			−0.570540	−	0.821270i	$-0.693266\pi$
$983$	−3.39823e10			−1.14108			−0.570540	+	0.821270i	$0.693266\pi$
$984$		0			0
$985$	−6.93923e10			−2.31358
$986$	−7.02033e9	+	1.98510e8i	−0.233232	+	0.00659496i
$987$		0			0
$988$	−3.51126e9	−	6.20385e10i	−0.115828	−	2.04650i
$989$	5.29444e10			1.74033
$990$		0			0
$991$			6.63551e9i			0.216579i	0.994119	+	0.108289i	$0.0345373\pi$
$991$			6.63551e9i			0.216579i	−0.994119	+	0.108289i	$0.965463\pi$
$992$	−3.48236e10	+	4.95516e9i	−1.13262	+	0.161163i
$993$		0			0
$994$	2.07971e10	−	5.88067e8i	0.671662	−	0.0189922i
$995$		−	3.80973e10i		−	1.22606i
$996$		0			0
$997$		−	5.33996e10i		−	1.70650i	−0.521506	−	0.853248i	$-0.674630\pi$
$997$		−	5.33996e10i		−	1.70650i	0.521506	−	0.853248i	$-0.325370\pi$
$998$	−5.28609e8	−	1.86944e10i	−0.0168336	−	0.595325i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.13	✓	28
3.2	odd	2	inner	72.8.f.a.35.16	yes	28
4.3	odd	2		288.8.f.a.143.25		28
8.3	odd	2	inner	72.8.f.a.35.15	yes	28
8.5	even	2		288.8.f.a.143.4		28
12.11	even	2		288.8.f.a.143.3		28
24.5	odd	2		288.8.f.a.143.26		28
24.11	even	2	inner	72.8.f.a.35.14	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.13	✓	28	1.1	even	1	trivial
72.8.f.a.35.14	yes	28	24.11	even	2	inner
72.8.f.a.35.15	yes	28	8.3	odd	2	inner
72.8.f.a.35.16	yes	28	3.2	odd	2	inner
288.8.f.a.143.3		28	12.11	even	2
288.8.f.a.143.4		28	8.5	even	2
288.8.f.a.143.25		28	4.3	odd	2
288.8.f.a.143.26		28	24.5	odd	2

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.319783 - 11.3092i) q^{2} +(-127.795 + 7.23297i) q^{4} +412.177 q^{5} -359.164i q^{7} +(122.666 + 1442.95i) q^{8} +O(q^{10})\)	\(q+(-0.319783 - 11.3092i) q^{2} +(-127.795 + 7.23297i) q^{4} +412.177 q^{5} -359.164i q^{7} +(122.666 + 1442.95i) q^{8} +(-131.807 - 4661.38i) q^{10} +3819.02i q^{11} +14141.7i q^{13} +(-4061.85 + 114.854i) q^{14} +(16279.4 - 1848.68i) q^{16} +3331.48i q^{17} +34327.6 q^{19} +(-52674.3 + 2981.26i) q^{20} +(43190.0 - 1221.26i) q^{22} +69918.3 q^{23} +91764.7 q^{25} +(159932. - 4522.29i) q^{26} +(2597.82 + 45899.5i) q^{28} -186332. q^{29} -189759. i q^{31} +(-26112.9 - 183515. i) q^{32} +(37676.4 - 1065.35i) q^{34} -148039. i q^{35} +190024. i q^{37} +(-10977.4 - 388217. i) q^{38} +(50560.0 + 594751. i) q^{40} -159461. i q^{41} +757231. q^{43} +(-27622.9 - 488054. i) q^{44} +(-22358.7 - 790720. i) q^{46} +235539. q^{47} +694544. q^{49} +(-29344.8 - 1.03778e6i) q^{50} +(-102287. - 1.80725e6i) q^{52} +732030. q^{53} +1.57411e6i q^{55} +(518255. - 44057.1i) q^{56} +(59585.9 + 2.10727e6i) q^{58} +1.98198e6i q^{59} -1.37932e6i q^{61} +(-2.14602e6 + 60681.5i) q^{62} +(-2.06706e6 + 354001. i) q^{64} +5.82890e6i q^{65} +2.28147e6 q^{67} +(-24096.5 - 425749. i) q^{68} +(-1.67420e6 + 47340.3i) q^{70} -5.12011e6 q^{71} -521725. q^{73} +(2.14901e6 - 60766.2i) q^{74} +(-4.38691e6 + 248290. i) q^{76} +1.37165e6 q^{77} -3.85831e6i q^{79} +(6.70998e6 - 761983. i) q^{80} +(-1.80337e6 + 50992.8i) q^{82} +4.75271e6i q^{83} +1.37316e6i q^{85} +(-242150. - 8.56367e6i) q^{86} +(-5.51066e6 + 468463. i) q^{88} +7.08773e6i q^{89} +5.07920e6 q^{91} +(-8.93525e6 + 505717. i) q^{92} +(-75321.4 - 2.66376e6i) q^{94} +1.41490e7 q^{95} -1.34983e7 q^{97} +(-222103. - 7.85473e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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