LMFDB - Embedded newform 69.7.d.a.22.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,7,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	69.d (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:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.8
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.53823 q^{2} -15.5885 q^{3} +8.90142 q^{4} +233.104i q^{5} +133.098 q^{6} +155.980i q^{7} +470.445 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-8.53823 q^{2} -15.5885 q^{3} +8.90142 q^{4} +233.104i q^{5} +133.098 q^{6} +155.980i q^{7} +470.445 q^{8} +243.000 q^{9} -1990.29i q^{10} +880.164i q^{11} -138.759 q^{12} +3100.64 q^{13} -1331.79i q^{14} -3633.73i q^{15} -4586.46 q^{16} +5586.40i q^{17} -2074.79 q^{18} +9608.40i q^{19} +2074.95i q^{20} -2431.49i q^{21} -7515.05i q^{22} +(12069.8 + 1534.74i) q^{23} -7333.50 q^{24} -38712.4 q^{25} -26474.0 q^{26} -3788.00 q^{27} +1388.44i q^{28} +14991.2 q^{29} +31025.6i q^{30} -49729.0 q^{31} +9051.77 q^{32} -13720.4i q^{33} -47698.0i q^{34} -36359.5 q^{35} +2163.04 q^{36} -7101.92i q^{37} -82038.7i q^{38} -48334.2 q^{39} +109662. i q^{40} +39666.9 q^{41} +20760.6i q^{42} +97163.8i q^{43} +7834.71i q^{44} +56644.2i q^{45} +(-103055. - 13104.0i) q^{46} -14422.3 q^{47} +71495.8 q^{48} +93319.3 q^{49} +330536. q^{50} -87083.4i q^{51} +27600.1 q^{52} -196331. i q^{53} +32342.8 q^{54} -205170. q^{55} +73379.9i q^{56} -149780. i q^{57} -127998. q^{58} +177846. q^{59} -32345.3i q^{60} -343123. i q^{61} +424597. q^{62} +37903.1i q^{63} +216247. q^{64} +722772. i q^{65} +117148. i q^{66} -300398. i q^{67} +49726.9i q^{68} +(-188150. - 23924.2i) q^{69} +310446. q^{70} -104094. q^{71} +114318. q^{72} -466983. q^{73} +60637.8i q^{74} +603467. q^{75} +85528.3i q^{76} -137288. q^{77} +412689. q^{78} +780636. i q^{79} -1.06912e6i q^{80} +59049.0 q^{81} -338685. q^{82} +608861. i q^{83} -21643.7i q^{84} -1.30221e6 q^{85} -829607. i q^{86} -233689. q^{87} +414068. i q^{88} -346420. i q^{89} -483642. i q^{90} +483638. i q^{91} +(107438. + 13661.4i) q^{92} +775198. q^{93} +123141. q^{94} -2.23975e6 q^{95} -141103. q^{96} -691811. i q^{97} -796782. q^{98} +213880. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

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 $

$2$	−8.53823			−1.06728			−0.533640	−	0.845712i	$-0.679176\pi$
$2$	−8.53823			−1.06728			−0.533640	+	0.845712i	$0.679176\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	8.90142			0.139085
$5$			233.104i			1.86483i	0.361388	+	0.932415i	$0.382303\pi$
$5$			233.104i			1.86483i	−0.361388	+	0.932415i	$0.617697\pi$
$6$	133.098			0.616194
$7$			155.980i			0.454752i	0.973807	+	0.227376i	$0.0730146\pi$
$7$			155.980i			0.454752i	−0.973807	+	0.227376i	$0.926985\pi$
$8$	470.445			0.918837
$9$	243.000			0.333333
$10$		−	1990.29i		−	1.99029i
$11$			880.164i			0.661280i	0.943757	+	0.330640i	$0.107265\pi$
$11$			880.164i			0.661280i	−0.943757	+	0.330640i	$0.892735\pi$
$12$	−138.759			−0.0803005
$13$	3100.64			1.41131			0.705654	−	0.708557i	$-0.250653\pi$
$13$	3100.64			1.41131			0.705654	+	0.708557i	$0.250653\pi$
$14$		−	1331.79i		−	0.485347i
$15$		−	3633.73i		−	1.07666i
$16$	−4586.46			−1.11974
$17$			5586.40i			1.13707i	0.822661	+	0.568533i	$0.192489\pi$
$17$			5586.40i			1.13707i	−0.822661	+	0.568533i	$0.807511\pi$
$18$	−2074.79			−0.355760
$19$			9608.40i			1.40084i	0.713729	+	0.700422i	$0.247005\pi$
$19$			9608.40i			1.40084i	−0.713729	+	0.700422i	$0.752995\pi$
$20$			2074.95i			0.259369i
$21$		−	2431.49i		−	0.262551i
$22$		−	7515.05i		−	0.705771i
$23$	12069.8	+	1534.74i	0.992013	+	0.126140i
$23$	12069.8	+	1534.74i	0.992013	+	0.126140i
$24$	−7333.50			−0.530491
$25$	−38712.4			−2.47759
$26$	−26474.0			−1.50626
$27$	−3788.00			−0.192450
$28$			1388.44i			0.0632490i
$29$	14991.2			0.614669			0.307334	−	0.951602i	$-0.400563\pi$
$29$	14991.2			0.614669			0.307334	+	0.951602i	$0.400563\pi$
$30$			31025.6i			1.14910i
$31$	−49729.0			−1.66926			−0.834630	−	0.550810i	$-0.814319\pi$
$31$	−49729.0			−1.66926			−0.834630	+	0.550810i	$0.814319\pi$
$32$	9051.77			0.276238
$33$		−	13720.4i		−	0.381790i
$34$		−	47698.0i		−	1.21357i
$35$	−36359.5			−0.848035
$36$	2163.04			0.0463615
$37$		−	7101.92i		−	0.140207i	−0.997540	−	0.0701036i	$-0.977667\pi$
$37$		−	7101.92i		−	0.140207i	0.997540	−	0.0701036i	$-0.0223330\pi$
$38$		−	82038.7i		−	1.49509i
$39$	−48334.2			−0.814819
$40$			109662.i			1.71348i
$41$	39666.9			0.575541			0.287770	−	0.957699i	$-0.407086\pi$
$41$	39666.9			0.575541			0.287770	+	0.957699i	$0.407086\pi$
$42$			20760.6i			0.280215i
$43$			97163.8i			1.22208i	0.791600	+	0.611039i	$0.209248\pi$
$43$			97163.8i			1.22208i	−0.791600	+	0.611039i	$0.790752\pi$
$44$			7834.71i			0.0919739i
$45$			56644.2i			0.621610i
$46$	−103055.	−	13104.0i	−1.05875	−	0.134626i
$47$	−14422.3			−0.138913			−0.0694564	−	0.997585i	$-0.522126\pi$
$47$	−14422.3			−0.138913			−0.0694564	+	0.997585i	$0.522126\pi$
$48$	71495.8			0.646482
$49$	93319.3			0.793201
$50$	330536.			2.64428
$51$		−	87083.4i		−	0.656485i
$52$	27600.1			0.196291
$53$		−	196331.i		−	1.31874i	−0.751816	−	0.659372i	$-0.770822\pi$
$53$		−	196331.i		−	1.31874i	0.751816	−	0.659372i	$-0.229178\pi$
$54$	32342.8			0.205398
$55$	−205170.			−1.23318
$56$			73379.9i			0.417843i
$57$		−	149780.i		−	0.808778i
$58$	−127998.			−0.656023
$59$	177846.			0.865942			0.432971	−	0.901408i	$-0.357465\pi$
$59$	177846.			0.865942			0.432971	+	0.901408i	$0.357465\pi$
$60$		−	32345.3i		−	0.149747i
$61$		−	343123.i		−	1.51168i	−0.654756	−	0.755840i	$-0.727229\pi$
$61$		−	343123.i		−	1.51168i	0.654756	−	0.755840i	$-0.272771\pi$
$62$	424597.			1.78157
$63$			37903.1i			0.151584i
$64$	216247.			0.824917
$65$			722772.i			2.63185i
$66$			117148.i			0.407477i
$67$		−	300398.i		−	0.998785i	−0.866376	−	0.499393i	$-0.833556\pi$
$67$		−	300398.i		−	0.998785i	0.866376	−	0.499393i	$-0.166444\pi$
$68$			49726.9i			0.158148i
$69$	−188150.	−	23924.2i	−0.572739	−	0.0728267i
$70$	310446.			0.905090
$71$	−104094.			−0.290838			−0.145419	−	0.989370i	$-0.546453\pi$
$71$	−104094.			−0.290838			−0.145419	+	0.989370i	$0.546453\pi$
$72$	114318.			0.306279
$73$	−466983.			−1.20042			−0.600208	−	0.799844i	$-0.704916\pi$
$73$	−466983.			−1.20042			−0.600208	+	0.799844i	$0.704916\pi$
$74$			60637.8i			0.149640i
$75$	603467.			1.43044
$76$			85528.3i			0.194836i
$77$	−137288.			−0.300718
$78$	412689.			0.869639
$79$			780636.i			1.58332i	0.610965	+	0.791658i	$0.290782\pi$
$79$			780636.i			1.58332i	−0.610965	+	0.791658i	$0.709218\pi$
$80$		−	1.06912e6i		−	2.08813i
$81$	59049.0			0.111111
$82$	−338685.			−0.614263
$83$			608861.i			1.06484i	0.846481	+	0.532420i	$0.178717\pi$
$83$			608861.i			1.06484i	−0.846481	+	0.532420i	$0.821283\pi$
$84$		−	21643.7i		−	0.0365168i
$85$	−1.30221e6			−2.12043
$86$		−	829607.i		−	1.30430i
$87$	−233689.			−0.354879
$88$			414068.i			0.607609i
$89$		−	346420.i		−	0.491398i	−0.969346	−	0.245699i	$-0.920983\pi$
$89$		−	346420.i		−	0.491398i	0.969346	−	0.245699i	$-0.0790174\pi$
$90$		−	483642.i		−	0.663432i
$91$			483638.i			0.641795i
$92$	107438.	+	13661.4i	0.137974	+	0.0175441i
$93$	775198.			0.963748
$94$	123141.			0.148259
$95$	−2.23975e6			−2.61234
$96$	−141103.			−0.159486
$97$		−	691811.i		−	0.758005i	−0.925396	−	0.379002i	$-0.876267\pi$
$97$		−	691811.i		−	0.758005i	0.925396	−	0.379002i	$-0.123733\pi$
$98$	−796782.			−0.846567
$99$			213880.i			0.220427i
$100$	−344595.			−0.344595
$101$	−193600.			−0.187906			−0.0939530	−	0.995577i	$-0.529950\pi$
$101$	−193600.			−0.187906			−0.0939530	+	0.995577i	$0.529950\pi$
$102$			743538.i			0.700653i
$103$		−	925480.i		−	0.846945i	−0.905909	−	0.423472i	$-0.860811\pi$
$103$		−	925480.i		−	0.846945i	0.905909	−	0.423472i	$-0.139189\pi$
$104$	1.45868e6			1.29676
$105$	566789.			0.489613
$106$			1.67632e6i			1.40747i
$107$			835236.i			0.681801i	0.940099	+	0.340901i	$0.110732\pi$
$107$			835236.i			0.681801i	−0.940099	+	0.340901i	$0.889268\pi$
$108$	−33718.5			−0.0267668
$109$		−	1.68824e6i		−	1.30363i	−0.758377	−	0.651817i	$-0.774007\pi$
$109$		−	1.68824e6i		−	1.30363i	0.758377	−	0.651817i	$-0.225993\pi$
$110$	1.75179e6			1.31614
$111$			110708.i			0.0809487i
$112$		−	715395.i		−	0.509204i
$113$			562684.i			0.389968i	0.980806	+	0.194984i	$0.0624655\pi$
$113$			562684.i			0.389968i	−0.980806	+	0.194984i	$0.937534\pi$
$114$			1.27886e6i			0.863192i
$115$	−357754.	+	2.81352e6i	−0.235229	+	1.84994i
$116$	133443.			0.0854910
$117$	753456.			0.470436
$118$	−1.51849e6			−0.924202
$119$	−871366.			−0.517083
$120$		−	1.70947e6i		−	0.989276i
$121$	996872.			0.562708
$122$			2.92966e6i			1.61339i
$123$	−618345.			−0.332289
$124$	−442658.			−0.232169
$125$		−	5.38176e6i		−	2.75546i
$126$		−	323626.i		−	0.161782i
$127$	−232896.			−0.113698			−0.0568489	−	0.998383i	$-0.518105\pi$
$127$	−232896.			−0.113698			−0.0568489	+	0.998383i	$0.518105\pi$
$128$	−2.42568e6			−1.15665
$129$		−	1.51463e6i		−	0.705567i
$130$		−	6.17119e6i		−	2.80892i
$131$	1.02283e6			0.454976			0.227488	−	0.973781i	$-0.426949\pi$
$131$	1.02283e6			0.454976			0.227488	+	0.973781i	$0.426949\pi$
$132$		−	122131.i		−	0.0531012i
$133$	−1.49872e6			−0.637037
$134$			2.56487e6i			1.06598i
$135$		−	882996.i		−	0.358887i
$136$			2.62809e6i			1.04478i
$137$			1.41970e6i			0.552121i	0.961140	+	0.276060i	$0.0890290\pi$
$137$			1.41970e6i			0.552121i	−0.961140	+	0.276060i	$0.910971\pi$
$138$	1.60647e6	+	204271.i	0.611272	+	0.0777264i
$139$	−487470.			−0.181511			−0.0907556	−	0.995873i	$-0.528928\pi$
$139$	−487470.			−0.181511			−0.0907556	+	0.995873i	$0.528928\pi$
$140$	−323651.			−0.117949
$141$	224822.			0.0802014
$142$	888781.			0.310406
$143$			2.72908e6i			0.933270i
$144$	−1.11451e6			−0.373247
$145$			3.49450e6i			1.14625i
$146$	3.98721e6			1.28118
$147$	−1.45470e6			−0.457955
$148$		−	63217.1i		−	0.0195007i
$149$			3.65532e6i			1.10501i	0.833509	+	0.552506i	$0.186328\pi$
$149$			3.65532e6i			1.10501i	−0.833509	+	0.552506i	$0.813672\pi$
$150$	−5.15254e6			−1.52668
$151$	5.75177e6			1.67059			0.835296	−	0.549801i	$-0.185296\pi$
$151$	5.75177e6			1.67059			0.835296	+	0.549801i	$0.185296\pi$
$152$			4.52022e6i			1.28715i
$153$			1.35750e6i			0.379022i
$154$	1.17220e6			0.320950
$155$		−	1.15920e7i		−	3.11289i
$156$	−430243.			−0.113329
$157$			20743.8i			0.00536031i	0.999996	+	0.00268016i	$0.000853121\pi$
$157$			20743.8i			0.00536031i	−0.999996	+	0.00268016i	$0.999147\pi$
$158$		−	6.66526e6i		−	1.68984i
$159$			3.06049e6i			0.761378i
$160$			2.11000e6i			0.515138i
$161$	−239389.	+	1.88265e6i	−0.0573622	+	0.451119i
$162$	−504174.			−0.118587
$163$	−4.04646e6			−0.934356			−0.467178	−	0.884163i	$-0.654729\pi$
$163$	−4.04646e6			−0.934356			−0.467178	+	0.884163i	$0.654729\pi$
$164$	353091.			0.0800489
$165$	3.19828e6			0.711975
$166$		−	5.19860e6i		−	1.13648i
$167$	−1.52335e6			−0.327077			−0.163539	−	0.986537i	$-0.552291\pi$
$167$	−1.52335e6			−0.327077			−0.163539	+	0.986537i	$0.552291\pi$
$168$		−	1.14388e6i		−	0.241242i
$169$	4.78718e6			0.991790
$170$	1.11186e7			2.26310
$171$			2.33484e6i			0.466948i
$172$			864895.i			0.169972i
$173$	6.58097e6			1.27102			0.635509	−	0.772094i	$-0.280790\pi$
$173$	6.58097e6			1.27102			0.635509	+	0.772094i	$0.280790\pi$
$174$	1.99529e6			0.378755
$175$		−	6.03836e6i		−	1.12669i
$176$		−	4.03683e6i		−	0.740462i
$177$	−2.77235e6			−0.499952
$178$			2.95781e6i			0.524458i
$179$	−3.00610e6			−0.524136			−0.262068	−	0.965049i	$-0.584405\pi$
$179$	−3.00610e6			−0.524136			−0.262068	+	0.965049i	$0.584405\pi$
$180$			504214.i			0.0864564i
$181$			2.95903e6i			0.499014i	0.968373	+	0.249507i	$0.0802686\pi$
$181$			2.95903e6i			0.499014i	−0.968373	+	0.249507i	$0.919731\pi$
$182$		−	4.12941e6i		−	0.684974i
$183$			5.34876e6i			0.872769i
$184$	5.67818e6	+	722010.i	0.911498	+	0.115902i
$185$	1.65548e6			0.261463
$186$	−6.61882e6			−1.02859
$187$	−4.91695e6			−0.751919
$188$	−128379.			−0.0193206
$189$		−	590851.i		−	0.0875170i
$190$	1.91235e7			2.78809
$191$		−	907350.i		−	0.130219i	−0.997878	−	0.0651095i	$-0.979260\pi$
$191$		−	907350.i		−	0.130219i	0.997878	−	0.0651095i	$-0.0207397\pi$
$192$	−3.37096e6			−0.476266
$193$	7.22908e6			1.00557			0.502783	−	0.864412i	$-0.332309\pi$
$193$	7.22908e6			1.00557			0.502783	+	0.864412i	$0.332309\pi$
$194$			5.90684e6i			0.809003i
$195$		−	1.12669e7i		−	1.51950i
$196$	830674.			0.110322
$197$	−4.23662e6			−0.554141			−0.277071	−	0.960850i	$-0.589364\pi$
$197$	−4.23662e6			−0.554141			−0.277071	+	0.960850i	$0.589364\pi$
$198$		−	1.82616e6i		−	0.235257i
$199$			1.50761e6i			0.191307i	0.995415	+	0.0956535i	$0.0304941\pi$
$199$			1.50761e6i			0.191307i	−0.995415	+	0.0956535i	$0.969506\pi$
$200$	−1.82120e7			−2.27650
$201$			4.68274e6i			0.576649i
$202$	1.65300e6			0.200548
$203$			2.33832e6i			0.279522i
$204$		−	775166.i		−	0.0913070i
$205$			9.24650e6i			1.07329i
$206$			7.90196e6i			0.903927i
$207$	2.93297e6	+	372942.i	0.330671	+	0.0420465i
$208$	−1.42210e7			−1.58030
$209$	−8.45697e6			−0.926351
$210$	−4.83937e6			−0.522554
$211$	−3.83645e6			−0.408396			−0.204198	−	0.978930i	$-0.565459\pi$
$211$	−3.83645e6			−0.408396			−0.204198	+	0.978930i	$0.565459\pi$
$212$		−	1.74762e6i		−	0.183417i
$213$	1.62267e6			0.167916
$214$		−	7.13144e6i		−	0.727672i
$215$	−2.26493e7			−2.27897
$216$	−1.78204e6			−0.176830
$217$		−	7.75671e6i		−	0.759099i
$218$			1.44146e7i			1.39134i
$219$	7.27954e6			0.693061
$220$	−1.82630e6			−0.171516
$221$			1.73214e7i			1.60475i
$222$		−	945250.i		−	0.0863949i
$223$	−1.17209e7			−1.05693			−0.528466	−	0.848954i	$-0.677233\pi$
$223$	−1.17209e7			−1.05693			−0.528466	+	0.848954i	$0.677233\pi$
$224$			1.41189e6i			0.125620i
$225$	−9.40711e6			−0.825865
$226$		−	4.80433e6i		−	0.416205i
$227$		−	319941.i		−	0.0273522i	−0.999906	−	0.0136761i	$-0.995647\pi$
$227$		−	319941.i		−	0.0273522i	0.999906	−	0.0136761i	$-0.00435337\pi$
$228$		−	1.33325e6i		−	0.112489i
$229$		−	1.58296e7i		−	1.31814i	−0.752079	−	0.659072i	$-0.770949\pi$
$229$		−	1.58296e7i		−	1.31814i	0.752079	−	0.659072i	$-0.229051\pi$
$230$	3.05459e6	−	2.40225e7i	0.251055	−	1.97440i
$231$	2.14011e6			0.173620
$232$	7.05251e6			0.564781
$233$	−1.39552e7			−1.10323			−0.551616	−	0.834098i	$-0.685989\pi$
$233$	−1.39552e7			−1.10323			−0.551616	+	0.834098i	$0.685989\pi$
$234$	−6.43319e6			−0.502086
$235$		−	3.36190e6i		−	0.259049i
$236$	1.58308e6			0.120439
$237$		−	1.21689e7i		−	0.914128i
$238$	7.43993e6			0.551871
$239$	1.19906e7			0.878309			0.439154	−	0.898412i	$-0.355278\pi$
$239$	1.19906e7			0.878309			0.439154	+	0.898412i	$0.355278\pi$
$240$			1.66659e7i			1.20558i
$241$		−	2.51859e7i		−	1.79931i	−0.436598	−	0.899657i	$-0.643817\pi$
$241$		−	2.51859e7i		−	1.79931i	0.436598	−	0.899657i	$-0.356183\pi$
$242$	−8.51152e6			−0.600567
$243$	−920483.			−0.0641500
$244$		−	3.05428e6i		−	0.210252i
$245$			2.17531e7i			1.47919i
$246$	5.27957e6			0.354645
$247$			2.97922e7i			1.97702i
$248$	−2.33947e7			−1.53378
$249$		−	9.49121e6i		−	0.614785i
$250$			4.59507e7i			2.94085i
$251$		−	3.11906e7i		−	1.97243i	−0.165457	−	0.986217i	$-0.552910\pi$
$251$		−	3.11906e7i		−	1.97243i	0.165457	−	0.986217i	$-0.447090\pi$
$252$			337391.i			0.0210830i
$253$	−1.35082e6	+	1.06234e7i	−0.0834136	+	0.655998i
$254$	1.98852e6			0.121347
$255$	2.02995e7			1.22423
$256$	6.87122e6			0.409556
$257$	1.48329e7			0.873829			0.436914	−	0.899503i	$-0.356071\pi$
$257$	1.48329e7			0.873829			0.436914	+	0.899503i	$0.356071\pi$
$258$			1.29323e7i			0.753037i
$259$	1.10776e6			0.0637595
$260$			6.43369e6i			0.366050i
$261$	3.64285e6			0.204890
$262$	−8.73313e6			−0.485586
$263$		−	5.01534e6i		−	0.275698i	−0.990453	−	0.137849i	$-0.955981\pi$
$263$		−	5.01534e6i		−	0.275698i	0.990453	−	0.137849i	$-0.0440188\pi$
$264$		−	6.45469e6i		−	0.350803i
$265$	4.57655e7			2.45924
$266$	1.27964e7			0.679896
$267$			5.40015e6i			0.283708i
$268$		−	2.67396e6i		−	0.138916i
$269$	3.20238e7			1.64519			0.822596	−	0.568627i	$-0.192525\pi$
$269$	3.20238e7			1.64519			0.822596	+	0.568627i	$0.192525\pi$
$270$			7.53923e6i			0.383032i
$271$	4.61616e6			0.231939			0.115969	−	0.993253i	$-0.463003\pi$
$271$	4.61616e6			0.231939			0.115969	+	0.993253i	$0.463003\pi$
$272$		−	2.56218e7i		−	1.27322i
$273$		−	7.53917e6i		−	0.370540i
$274$		−	1.21217e7i		−	0.589267i
$275$		−	3.40733e7i		−	1.63838i
$276$	−1.67480e6	−	212960.i	−0.0796591	−	0.0101291i
$277$	−1.97766e6			−0.0930490			−0.0465245	−	0.998917i	$-0.514815\pi$
$277$	−1.97766e6			−0.0930490			−0.0465245	+	0.998917i	$0.514815\pi$
$278$	4.16213e6			0.193723
$279$	−1.20841e7			−0.556420
$280$	−1.71051e7			−0.779206
$281$			4.00608e7i			1.80551i	0.430154	+	0.902756i	$0.358459\pi$
$281$			4.00608e7i			1.80551i	−0.430154	+	0.902756i	$0.641541\pi$
$282$	−1.91958e6			−0.0855972
$283$		−	1.57392e7i		−	0.694422i	−0.937787	−	0.347211i	$-0.887129\pi$
$283$		−	1.57392e7i		−	0.694422i	0.937787	−	0.347211i	$-0.112871\pi$
$284$	−926586.			−0.0404511
$285$	3.49143e7			1.50823
$286$		−	2.33015e7i		−	0.996060i
$287$			6.18723e6i			0.261728i
$288$	2.19958e6			0.0920794
$289$	−7.07033e6			−0.292918
$290$		−	2.98368e7i		−	1.22337i
$291$			1.07843e7i			0.437634i
$292$	−4.15681e6			−0.166960
$293$			1.98649e7i			0.789740i	0.918737	+	0.394870i	$0.129210\pi$
$293$			1.98649e7i			0.789740i	−0.918737	+	0.394870i	$0.870790\pi$
$294$	1.24206e7			0.488766
$295$			4.14567e7i			1.61484i
$296$		−	3.34106e6i		−	0.128828i
$297$		−	3.33406e6i		−	0.127263i
$298$		−	3.12100e7i		−	1.17936i
$299$	3.74242e7	+	4.75868e6i	1.40004	+	0.178022i
$300$	5.37171e6			0.198952
$301$	−1.51556e7			−0.555742
$302$	−4.91099e7			−1.78299
$303$	3.01792e6			0.108488
$304$		−	4.40685e7i		−	1.56858i
$305$	7.99832e7			2.81903
$306$		−	1.15906e7i		−	0.404522i
$307$	−4.37453e7			−1.51188			−0.755938	−	0.654643i	$-0.772819\pi$
$307$	−4.37453e7			−1.51188			−0.755938	+	0.654643i	$0.772819\pi$
$308$	−1.22206e6			−0.0418253
$309$			1.44268e7i			0.488984i
$310$			9.89753e7i			3.32232i
$311$	559668.			0.0186058			0.00930292	−	0.999957i	$-0.497039\pi$
$311$	559668.			0.0186058			0.00930292	+	0.999957i	$0.497039\pi$
$312$	−2.27386e7			−0.748686
$313$		−	4.87636e7i		−	1.59024i	−0.606451	−	0.795121i	$-0.707408\pi$
$313$		−	4.87636e7i		−	1.59024i	0.606451	−	0.795121i	$-0.292592\pi$
$314$		−	177116.i		−	0.00572095i
$315$	−8.83536e6			−0.282678
$316$			6.94877e6i			0.220215i
$317$	3.15059e7			0.989042			0.494521	−	0.869166i	$-0.335344\pi$
$317$	3.15059e7			0.989042			0.494521	+	0.869166i	$0.335344\pi$
$318$		−	2.61312e7i		−	0.812603i
$319$			1.31947e7i			0.406469i
$320$			5.04080e7i			1.53833i
$321$		−	1.30200e7i		−	0.393638i
$322$	2.04396e6	−	1.60745e7i	0.0612215	−	0.481470i
$323$	−5.36764e7			−1.59285
$324$	525620.			0.0154538
$325$	−1.20033e8			−3.49665
$326$	3.45496e7			0.997219
$327$			2.63171e7i			0.752653i
$328$	1.86611e7			0.528828
$329$		−	2.24960e6i		−	0.0631708i
$330$	−2.73076e7			−0.759876
$331$	1.24947e7			0.344541			0.172270	−	0.985050i	$-0.444890\pi$
$331$	1.24947e7			0.344541			0.172270	+	0.985050i	$0.444890\pi$
$332$			5.41973e6i			0.148103i
$333$		−	1.72577e6i		−	0.0467358i
$334$	1.30067e7			0.349083
$335$	7.00239e7			1.86257
$336$			1.11519e7i			0.293989i
$337$			1.79945e7i			0.470166i	0.971975	+	0.235083i	$0.0755361\pi$
$337$			1.79945e7i			0.470166i	−0.971975	+	0.235083i	$0.924464\pi$
$338$	−4.08741e7			−1.05852
$339$		−	8.77138e6i		−	0.225148i
$340$	−1.15915e7			−0.294920
$341$		−	4.37696e7i		−	1.10385i
$342$		−	1.99354e7i		−	0.498364i
$343$			3.29068e7i			0.815461i
$344$			4.57102e7i			1.12289i
$345$	5.57683e6	−	4.38584e7i	0.135810	−	1.06806i
$346$	−5.61899e7			−1.35653
$347$	3.46526e7			0.829369			0.414685	−	0.909965i	$-0.363892\pi$
$347$	3.46526e7			0.829369			0.414685	+	0.909965i	$0.363892\pi$
$348$	−2.08016e6			−0.0493583
$349$	−9.71946e6			−0.228647			−0.114324	−	0.993444i	$-0.536470\pi$
$349$	−9.71946e6			−0.228647			−0.114324	+	0.993444i	$0.536470\pi$
$350$			5.15569e7i			1.20249i
$351$	−1.17452e7			−0.271606
$352$			7.96705e6i			0.182671i
$353$	−5.53376e7			−1.25805			−0.629023	−	0.777387i	$-0.716545\pi$
$353$	−5.53376e7			−1.25805			−0.629023	+	0.777387i	$0.716545\pi$
$354$	2.36710e7			0.533588
$355$		−	2.42648e7i		−	0.542364i
$356$		−	3.08363e6i		−	0.0683458i
$357$	1.35833e7			0.298538
$358$	2.56668e7			0.559400
$359$		−	2.79634e7i		−	0.604374i	−0.953249	−	0.302187i	$-0.902283\pi$
$359$		−	2.79634e7i		−	0.604374i	0.953249	−	0.302187i	$-0.0977167\pi$
$360$			2.66480e7i			0.571158i
$361$	−4.52754e7			−0.962366
$362$		−	2.52648e7i		−	0.532588i
$363$	−1.55397e7			−0.324880
$364$			4.30506e6i			0.0892638i
$365$		−	1.08855e8i		−	2.23857i
$366$		−	4.56689e7i		−	0.931488i
$367$		−	8.28559e7i		−	1.67620i	−0.545518	−	0.838099i	$-0.683667\pi$
$367$		−	8.28559e7i		−	1.67620i	0.545518	−	0.838099i	$-0.316333\pi$
$368$	−5.53577e7	−	7.03902e6i	−1.11080	−	0.141244i
$369$	9.63905e6			0.191847
$370$	−1.41349e7			−0.279054
$371$	3.06236e7			0.599702
$372$	6.90036e6			0.134043
$373$		−	1.22033e7i		−	0.235153i	−0.993064	−	0.117576i	$-0.962488\pi$
$373$		−	1.22033e7i		−	0.235153i	0.993064	−	0.117576i	$-0.0375125\pi$
$374$	4.19821e7			0.802508
$375$			8.38934e7i			1.59087i
$376$	−6.78491e6			−0.127638
$377$	4.64822e7			0.867487
$378$			5.04482e6i			0.0934051i
$379$			9.11449e7i			1.67423i	0.547028	+	0.837114i	$0.315759\pi$
$379$			9.11449e7i			1.67423i	−0.547028	+	0.837114i	$0.684241\pi$
$380$	−1.99370e7			−0.363336
$381$	3.63050e6			0.0656434
$382$			7.74716e6i			0.138980i
$383$			5.39033e7i			0.959442i	0.877421	+	0.479721i	$0.159262\pi$
$383$			5.39033e7i			0.959442i	−0.877421	+	0.479721i	$0.840738\pi$
$384$	3.78126e7			0.667795
$385$		−	3.20023e7i		−	0.560789i
$386$	−6.17235e7			−1.07322
$387$			2.36108e7i			0.407359i
$388$		−	6.15809e6i		−	0.105427i
$389$		−	3.68792e6i		−	0.0626517i	−0.999509	−	0.0313258i	$-0.990027\pi$
$389$		−	3.68792e6i		−	0.0626517i	0.999509	−	0.0313258i	$-0.00997295\pi$
$390$			9.61994e7i			1.62173i
$391$	−8.57368e6	+	6.74269e7i	−0.143429	+	1.12798i
$392$	4.39015e7			0.728822
$393$	−1.59443e7			−0.262680
$394$	3.61732e7			0.591423
$395$	−1.81969e8			−2.95262
$396$			1.90383e6i			0.0306580i
$397$	4.27706e7			0.683556			0.341778	−	0.939781i	$-0.388971\pi$
$397$	4.27706e7			0.683556			0.341778	+	0.939781i	$0.388971\pi$
$398$		−	1.28724e7i		−	0.204178i
$399$	2.33627e7			0.367793
$400$	1.77553e8			2.77426
$401$			1.03595e8i			1.60659i	0.595581	+	0.803295i	$0.296922\pi$
$401$			1.03595e8i			1.60659i	−0.595581	+	0.803295i	$0.703078\pi$
$402$		−	3.99823e7i		−	0.615445i
$403$	−1.54192e8			−2.35584
$404$	−1.72331e6			−0.0261348
$405$			1.37645e7i			0.207203i
$406$		−	1.99651e7i		−	0.298328i
$407$	6.25086e6			0.0927163
$408$		−	4.09679e7i		−	0.603203i
$409$	3.94971e7			0.577292			0.288646	−	0.957436i	$-0.406795\pi$
$409$	3.94971e7			0.577292			0.288646	+	0.957436i	$0.406795\pi$
$410$		−	7.89487e7i		−	1.14550i
$411$		−	2.21309e7i		−	0.318767i
$412$		−	8.23808e6i		−	0.117797i
$413$			2.77404e7i			0.393789i
$414$	−2.50423e7	−	3.18426e6i	−0.352918	−	0.0448754i
$415$	−1.41928e8			−1.98574
$416$	2.80663e7			0.389857
$417$	7.59890e6			0.104796
$418$	7.22075e7			0.988675
$419$		−	4.99743e7i		−	0.679368i	−0.940540	−	0.339684i	$-0.889680\pi$
$419$		−	4.99743e7i		−	0.679368i	0.940540	−	0.339684i	$-0.110320\pi$
$420$	5.04522e6			0.0680977
$421$			2.83213e7i			0.379549i	0.981828	+	0.189774i	$0.0607756\pi$
$421$			2.83213e7i			0.379549i	−0.981828	+	0.189774i	$0.939224\pi$
$422$	3.27565e7			0.435873
$423$	−3.50463e6			−0.0463043
$424$		−	9.23628e7i		−	1.21171i
$425$		−	2.16263e8i		−	2.81719i
$426$	−1.38547e7			−0.179213
$427$	5.35202e7			0.687439
$428$			7.43478e6i			0.0948281i
$429$		−	4.25421e7i		−	0.538824i
$430$	1.93385e8			2.43230
$431$			4.75872e7i			0.594372i	0.954820	+	0.297186i	$0.0960481\pi$
$431$			4.75872e7i			0.594372i	−0.954820	+	0.297186i	$0.903952\pi$
$432$	1.73735e7			0.215494
$433$			9.11510e7i			1.12279i	0.827549	+	0.561394i	$0.189735\pi$
$433$			9.11510e7i			1.12279i	−0.827549	+	0.561394i	$0.810265\pi$
$434$			6.62286e7i			0.810171i
$435$		−	5.44738e7i		−	0.661790i
$436$		−	1.50278e7i		−	0.181315i
$437$	−1.47464e7	+	1.15972e8i	−0.176702	+	1.38966i
$438$	−6.21544e7			−0.739689
$439$	6.55753e7			0.775080			0.387540	−	0.921853i	$-0.373325\pi$
$439$	6.55753e7			0.775080			0.387540	+	0.921853i	$0.373325\pi$
$440$	−9.65209e7			−1.13309
$441$	2.26766e7			0.264400
$442$		−	1.47895e8i		−	1.71272i
$443$	−8.27131e7			−0.951400			−0.475700	−	0.879608i	$-0.657805\pi$
$443$	−8.27131e7			−0.951400			−0.475700	+	0.879608i	$0.657805\pi$
$444$			985458.i			0.0112587i
$445$	8.07518e7			0.916373
$446$	1.00076e8			1.12804
$447$		−	5.69808e7i		−	0.637979i
$448$			3.37302e7i			0.375132i
$449$	6.56635e7			0.725412			0.362706	−	0.931904i	$-0.381853\pi$
$449$	6.56635e7			0.725412			0.362706	+	0.931904i	$0.381853\pi$
$450$	8.03201e7			0.881428
$451$			3.49133e7i			0.380594i
$452$			5.00868e6i			0.0542386i
$453$	−8.96612e7			−0.964517
$454$			2.73173e6i			0.0291924i
$455$	−1.12738e8			−1.19684
$456$		−	7.04632e7i		−	0.743135i
$457$			1.41479e8i			1.48233i	0.671325	+	0.741163i	$0.265726\pi$
$457$			1.41479e8i			1.48233i	−0.671325	+	0.741163i	$0.734274\pi$
$458$			1.35157e8i			1.40683i
$459$		−	2.11613e7i		−	0.218828i
$460$	−3.18452e6	+	2.50443e7i	−0.0327167	+	0.257298i
$461$	6.07075e7			0.619641			0.309820	−	0.950795i	$-0.399731\pi$
$461$	6.07075e7			0.619641			0.309820	+	0.950795i	$0.399731\pi$
$462$	−1.82727e7			−0.185301
$463$	−1.44944e8			−1.46035			−0.730176	−	0.683259i	$-0.760562\pi$
$463$	−1.44944e8			−1.46035			−0.730176	+	0.683259i	$0.760562\pi$
$464$	−6.87563e7			−0.688270
$465$			1.80702e8i			1.79723i
$466$	1.19152e8			1.17746
$467$			4.86379e7i			0.477556i	0.971074	+	0.238778i	$0.0767468\pi$
$467$			4.86379e7i			0.477556i	−0.971074	+	0.238778i	$0.923253\pi$
$468$	6.70683e6			0.0654304
$469$	4.68560e7			0.454199
$470$			2.87047e7i			0.276477i
$471$		−	323364.i		−	0.00309478i
$472$	8.36668e7			0.795660
$473$	−8.55201e7			−0.808136
$474$			1.03901e8i			0.975629i
$475$		−	3.71964e8i		−	3.47072i
$476$	−7.75639e6			−0.0719182
$477$		−	4.77084e7i		−	0.439582i
$478$	−1.02379e8			−0.937401
$479$			1.16190e8i			1.05721i	0.848867	+	0.528607i	$0.177285\pi$
$479$			1.16190e8i			1.05721i	−0.848867	+	0.528607i	$0.822715\pi$
$480$		−	3.28917e7i		−	0.297415i
$481$		−	2.20205e7i		−	0.197876i
$482$			2.15043e8i			1.92037i
$483$	3.73170e6	−	2.93476e7i	0.0331181	−	0.260454i
$484$	8.87357e6			0.0782641
$485$	1.61264e8			1.41355
$486$	7.85930e6			0.0684660
$487$	3.04632e7			0.263748			0.131874	−	0.991266i	$-0.457901\pi$
$487$	3.04632e7			0.263748			0.131874	+	0.991266i	$0.457901\pi$
$488$		−	1.61420e8i		−	1.38899i
$489$	6.30781e7			0.539451
$490$		−	1.85733e8i		−	1.57870i
$491$	−8.17805e7			−0.690885			−0.345442	−	0.938440i	$-0.612271\pi$
$491$	−8.17805e7			−0.690885			−0.345442	+	0.938440i	$0.612271\pi$
$492$	−5.50415e6			−0.0462163
$493$			8.37467e7i			0.698919i
$494$		−	2.54373e8i		−	2.11004i
$495$	−4.98562e7			−0.411059
$496$	2.28080e8			1.86914
$497$		−	1.62366e7i		−	0.132259i
$498$			8.10381e7i			0.656147i
$499$	2.46779e8			1.98613			0.993063	−	0.117585i	$-0.0375154\pi$
$499$	2.46779e8			1.98613			0.993063	+	0.117585i	$0.0375154\pi$
$500$		−	4.79053e7i		−	0.383242i
$501$	2.37467e7			0.188838
$502$			2.66313e8i			2.10514i
$503$			4.84475e7i			0.380686i	0.981718	+	0.190343i	$0.0609600\pi$
$503$			4.84475e7i			0.380686i	−0.981718	+	0.190343i	$0.939040\pi$
$504$			1.78313e7i			0.139281i
$505$		−	4.51288e7i		−	0.350413i
$506$	1.15336e7	−	9.07052e7i	0.0890256	−	0.700133i
$507$	−7.46248e7			−0.572610
$508$	−2.07311e6			−0.0158136
$509$	−1.85245e8			−1.40473			−0.702365	−	0.711817i	$-0.747873\pi$
$509$	−1.85245e8			−1.40473			−0.702365	+	0.711817i	$0.747873\pi$
$510$	−1.73322e8			−1.30660
$511$		−	7.28399e7i		−	0.545892i
$512$	9.65755e7			0.719544
$513$		−	3.63966e7i		−	0.269593i
$514$	−1.26647e8			−0.932619
$515$	2.15733e8			1.57941
$516$		−	1.34824e7i		−	0.0981336i
$517$		−	1.26940e7i		−	0.0918603i
$518$	−9.45828e6			−0.0680492
$519$	−1.02587e8			−0.733822
$520$			3.40024e8i			2.41824i
$521$		−	6.41523e7i		−	0.453627i	−0.973938	−	0.226814i	$-0.927169\pi$
$521$		−	6.41523e7i		−	0.453627i	0.973938	−	0.226814i	$-0.0728309\pi$
$522$	−3.11035e7			−0.218674
$523$		−	1.79460e8i		−	1.25448i	−0.778826	−	0.627240i	$-0.784185\pi$
$523$		−	1.79460e8i		−	1.25448i	0.778826	−	0.627240i	$-0.215815\pi$
$524$	9.10461e6			0.0632801
$525$			9.41286e7i			0.650495i
$526$			4.28222e7i			0.294246i
$527$		−	2.77806e8i		−	1.89806i
$528$			6.29280e7i			0.427506i
$529$	1.43325e8	+	3.70481e7i	0.968178	+	0.250264i
$530$	−3.90756e8			−2.62469
$531$	4.32167e7			0.288647
$532$	−1.33407e7			−0.0886020
$533$	1.22993e8			0.812265
$534$		−	4.61078e7i		−	0.302796i
$535$	−1.94697e8			−1.27144
$536$		−	1.41320e8i		−	0.917721i
$537$	4.68605e7			0.302610
$538$	−2.73427e8			−1.75588
$539$			8.21363e7i			0.524528i
$540$		−	7.85992e6i		−	0.0499156i
$541$	−6.40754e7			−0.404669			−0.202334	−	0.979316i	$-0.564853\pi$
$541$	−6.40754e7			−0.404669			−0.202334	+	0.979316i	$0.564853\pi$
$542$	−3.94139e7			−0.247543
$543$		−	4.61266e7i		−	0.288106i
$544$			5.05669e7i			0.314101i
$545$	3.93536e8			2.43106
$546$			6.43712e7i			0.395470i
$547$	−2.23997e8			−1.36861			−0.684307	−	0.729194i	$-0.739895\pi$
$547$	−2.23997e8			−1.36861			−0.684307	+	0.729194i	$0.739895\pi$
$548$			1.26373e7i			0.0767915i
$549$		−	8.33788e7i		−	0.503894i
$550$			2.90926e8i			1.74861i
$551$			1.44041e8i			0.861056i
$552$	−8.85140e7	−	1.12550e7i	−0.526253	−	0.0669159i
$553$	−1.21764e8			−0.720016
$554$	1.68857e7			0.0993092
$555$	−2.58065e7			−0.150956
$556$	−4.33917e6			−0.0252454
$557$		−	7.97533e6i		−	0.0461512i	−0.999734	−	0.0230756i	$-0.992654\pi$
$557$		−	7.97533e6i		−	0.0461512i	0.999734	−	0.0230756i	$-0.00734584\pi$
$558$	1.03177e8			0.593856
$559$			3.01270e8i			1.72473i
$560$	1.66761e8			0.949579
$561$	7.66477e7			0.434121
$562$		−	3.42048e8i		−	1.92698i
$563$			1.94553e8i			1.09022i	0.838366	+	0.545108i	$0.183511\pi$
$563$			1.94553e8i			1.09022i	−0.838366	+	0.545108i	$0.816489\pi$
$564$	2.00124e6			0.0111548
$565$	−1.31164e8			−0.727225
$566$			1.34385e8i			0.741142i
$567$			9.21045e6i			0.0505280i
$568$	−4.89706e7			−0.267233
$569$			1.28517e8i			0.697626i	0.937192	+	0.348813i	$0.113415\pi$
$569$			1.28517e8i			0.697626i	−0.937192	+	0.348813i	$0.886585\pi$
$570$	−2.98106e8			−1.60971
$571$			4.75561e7i			0.255445i	0.991810	+	0.127723i	$0.0407667\pi$
$571$			4.75561e7i			0.255445i	−0.991810	+	0.127723i	$0.959233\pi$
$572$			2.42926e7i			0.129804i
$573$			1.41442e7i			0.0751820i
$574$		−	5.28280e7i		−	0.279337i
$575$	−4.67252e8	−	5.94135e7i	−2.45780	−	0.312523i
$576$	5.25480e7			0.274972
$577$	2.94461e7			0.153285			0.0766425	−	0.997059i	$-0.475580\pi$
$577$	2.94461e7			0.153285			0.0766425	+	0.997059i	$0.475580\pi$
$578$	6.03681e7			0.312625
$579$	−1.12690e8			−0.580564
$580$			3.11060e7i			0.159426i
$581$	−9.49701e7			−0.484237
$582$		−	9.20785e7i		−	0.467078i
$583$	1.72803e8			0.872060
$584$	−2.19689e8			−1.10299
$585$			1.75634e8i			0.877283i
$586$		−	1.69611e8i		−	0.842873i
$587$	1.53077e8			0.756824			0.378412	−	0.925637i	$-0.376470\pi$
$587$	1.53077e8			0.756824			0.378412	+	0.925637i	$0.376470\pi$
$588$	−1.29489e7			−0.0636945
$589$		−	4.77815e8i		−	2.33838i
$590$		−	3.53967e8i		−	1.72348i
$591$	6.60423e7			0.319934
$592$			3.25726e7i			0.156996i
$593$	−2.42998e7			−0.116530			−0.0582651	−	0.998301i	$-0.518557\pi$
$593$	−2.42998e7			−0.116530			−0.0582651	+	0.998301i	$0.518557\pi$
$594$			2.84670e7i			0.135826i
$595$		−	2.03119e8i		−	0.964271i
$596$			3.25375e7i			0.153690i
$597$		−	2.35014e7i		−	0.110451i
$598$	−3.19536e8	−	4.06307e7i	−1.49423	−	0.189999i
$599$	−3.22739e7			−0.150166			−0.0750828	−	0.997177i	$-0.523922\pi$
$599$	−3.22739e7			−0.150166			−0.0750828	+	0.997177i	$0.523922\pi$
$600$	2.83898e8			1.31434
$601$	9.65355e7			0.444697			0.222348	−	0.974967i	$-0.428628\pi$
$601$	9.65355e7			0.444697			0.222348	+	0.974967i	$0.428628\pi$
$602$	1.29402e8			0.593132
$603$		−	7.29966e7i		−	0.332928i
$604$	5.11989e7			0.232354
$605$			2.32375e8i			1.04936i
$606$	−2.57677e7			−0.115786
$607$	1.84789e8			0.826248			0.413124	−	0.910675i	$-0.364438\pi$
$607$	1.84789e8			0.826248			0.413124	+	0.910675i	$0.364438\pi$
$608$			8.69730e7i			0.386967i
$609$		−	3.64508e7i		−	0.161382i
$610$	−6.82916e8			−3.00869
$611$	−4.47186e7			−0.196049
$612$			1.20836e7i			0.0527161i
$613$			3.35927e8i			1.45836i	0.684324	+	0.729178i	$0.260097\pi$
$613$			3.35927e8i			1.45836i	−0.684324	+	0.729178i	$0.739903\pi$
$614$	3.73507e8			1.61359
$615$		−	1.44139e8i		−	0.619662i
$616$	−6.45863e7			−0.276311
$617$		−	5.50327e7i		−	0.234296i	−0.993114	−	0.117148i	$-0.962625\pi$
$617$		−	5.50327e7i		−	0.234296i	0.993114	−	0.117148i	$-0.0373752\pi$
$618$		−	1.23179e8i		−	0.521882i
$619$			2.90917e7i			0.122658i	0.998118	+	0.0613291i	$0.0195339\pi$
$619$			2.90917e7i			0.122658i	−0.998118	+	0.0613291i	$0.980466\pi$
$620$		−	1.03185e8i		−	0.432955i
$621$	−4.57204e7	−	5.81359e6i	−0.190913	−	0.0242756i
$622$	−4.77857e6			−0.0198576
$623$	5.40345e7			0.223464
$624$	2.21683e8			0.912386
$625$	6.49628e8			2.66088
$626$			4.16355e8i			1.69723i
$627$	1.31831e8			0.534829
$628$			184650.i			0.000745537i
$629$	3.96742e7			0.159425
$630$	7.54384e7			0.301697
$631$			1.47047e8i			0.585285i	0.956222	+	0.292642i	$0.0945345\pi$
$631$			1.47047e8i			0.585285i	−0.956222	+	0.292642i	$0.905465\pi$
$632$			3.67246e8i			1.45481i
$633$	5.98043e7			0.235788
$634$	−2.69005e8			−1.05558
$635$		−	5.42891e7i		−	0.212027i
$636$			2.72427e7i			0.105896i
$637$	2.89350e8			1.11945
$638$		−	1.12659e8i		−	0.433815i
$639$	−2.52949e7			−0.0969461
$640$		−	5.65436e8i		−	2.15697i
$641$		−	3.77045e8i		−	1.43159i	−0.698309	−	0.715796i	$-0.746064\pi$
$641$		−	3.77045e8i		−	1.43159i	0.698309	−	0.715796i	$-0.253936\pi$
$642$			1.11168e8i			0.420122i
$643$			2.19355e8i			0.825116i	0.910931	+	0.412558i	$0.135365\pi$
$643$			2.19355e8i			0.825116i	−0.910931	+	0.412558i	$0.864635\pi$
$644$	−2.13090e6	+	1.67582e7i	−0.00797820	+	0.0627438i
$645$	3.53067e8			1.31576
$646$	4.58301e8			1.70002
$647$	1.87104e8			0.690829			0.345415	−	0.938450i	$-0.387738\pi$
$647$	1.87104e8			0.690829			0.345415	+	0.938450i	$0.387738\pi$
$648$	2.77793e7			0.102093
$649$			1.56534e8i			0.572630i
$650$	1.02487e9			3.73190
$651$			1.20915e8i			0.438266i
$652$	−3.60192e7			−0.129955
$653$	4.22342e7			0.151679			0.0758394	−	0.997120i	$-0.475836\pi$
$653$	4.22342e7			0.151679			0.0758394	+	0.997120i	$0.475836\pi$
$654$		−	2.24702e8i		−	0.803291i
$655$			2.38425e8i			0.848453i
$656$	−1.81930e8			−0.644456
$657$	−1.13477e8			−0.400139
$658$			1.92076e7i			0.0674209i
$659$			3.47285e8i			1.21347i	0.794904	+	0.606736i	$0.207521\pi$
$659$			3.47285e8i			1.21347i	−0.794904	+	0.606736i	$0.792479\pi$
$660$	2.84692e7			0.0990247
$661$		−	4.59695e7i		−	0.159172i	−0.996828	−	0.0795858i	$-0.974640\pi$
$661$		−	4.59695e7i		−	0.159172i	0.996828	−	0.0795858i	$-0.0253598\pi$
$662$	−1.06682e8			−0.367721
$663$		−	2.70015e8i		−	0.926503i
$664$			2.86435e8i			0.978414i
$665$		−	3.49356e8i		−	1.18797i
$666$			1.47350e7i			0.0498801i
$667$	1.80941e8	+	2.30075e7i	0.609759	+	0.0775341i
$668$	−1.35600e7			−0.0454914
$669$	1.82711e8			0.610221
$670$	−5.97880e8			−1.98788
$671$	3.02004e8			0.999645
$672$		−	2.20093e7i		−	0.0725266i
$673$	3.08900e8			1.01338			0.506690	−	0.862128i	$-0.330869\pi$
$673$	3.08900e8			1.01338			0.506690	+	0.862128i	$0.330869\pi$
$674$		−	1.53642e8i		−	0.501798i
$675$	1.46642e8			0.476813
$676$	4.26127e7			0.137943
$677$		−	4.53814e8i		−	1.46255i	−0.682080	−	0.731277i	$-0.738925\pi$
$677$		−	4.53814e8i		−	1.46255i	0.682080	−	0.731277i	$-0.261075\pi$
$678$			7.48921e7i			0.240296i
$679$	1.07909e8			0.344704
$680$	−6.12619e8			−1.94833
$681$			4.98738e6i			0.0157918i
$682$			3.73715e8i			1.17812i
$683$	−5.29830e8			−1.66293			−0.831466	−	0.555576i	$-0.812498\pi$
$683$	−5.29830e8			−1.66293			−0.831466	+	0.555576i	$0.812498\pi$
$684$			2.07834e7i			0.0649453i
$685$	−3.30937e8			−1.02961
$686$		−	2.80966e8i		−	0.870325i
$687$			2.46759e8i			0.761031i
$688$		−	4.45637e8i		−	1.36841i
$689$		−	6.08752e8i		−	1.86116i
$690$	−4.76163e7	+	3.74474e8i	−0.144947	+	1.13992i
$691$	−1.82674e8			−0.553659			−0.276830	−	0.960919i	$-0.589284\pi$
$691$	−1.82674e8			−0.553659			−0.276830	+	0.960919i	$0.589284\pi$
$692$	5.85799e7			0.176779
$693$	−3.33610e7			−0.100239
$694$	−2.95872e8			−0.885168
$695$		−	1.13631e8i		−	0.338488i
$696$	−1.09938e8			−0.326076
$697$			2.21595e8i			0.654428i
$698$	8.29870e7			0.244030
$699$	2.17539e8			0.636951
$700$		−	5.37499e7i		−	0.156705i
$701$			3.20737e8i			0.931098i	0.885022	+	0.465549i	$0.154143\pi$
$701$			3.20737e8i			0.931098i	−0.885022	+	0.465549i	$0.845857\pi$
$702$	1.00283e8			0.289880
$703$	6.82381e7			0.196409
$704$			1.90333e8i			0.545501i
$705$			5.24069e7i			0.149562i
$706$	4.72485e8			1.34269
$707$		−	3.01977e7i		−	0.0854506i
$708$	−2.46778e7			−0.0695356
$709$		−	6.61878e8i		−	1.85712i	−0.371186	−	0.928559i	$-0.621049\pi$
$709$		−	6.61878e8i		−	1.85712i	0.371186	−	0.928559i	$-0.378951\pi$
$710$			2.07178e8i			0.578854i
$711$			1.89695e8i			0.527772i
$712$		−	1.62971e8i		−	0.451514i
$713$	−6.00219e8	−	7.63210e7i	−1.65593	−	0.210560i
$714$	−1.15977e8			−0.318623
$715$	−6.36158e8			−1.74039
$716$	−2.67585e7			−0.0728993
$717$	−1.86915e8			−0.507092
$718$			2.38758e8i			0.645036i
$719$	−3.51124e8			−0.944656			−0.472328	−	0.881423i	$-0.656586\pi$
$719$	−3.51124e8			−0.944656			−0.472328	+	0.881423i	$0.656586\pi$
$720$		−	2.59796e8i		−	0.696042i
$721$	1.44356e8			0.385150
$722$	3.86572e8			1.02711
$723$			3.92610e8i			1.03883i
$724$			2.63395e7i			0.0694052i
$725$	−5.80344e8			−1.52290
$726$	1.32682e8			0.346737
$727$		−	6.22025e8i		−	1.61884i	−0.587229	−	0.809421i	$-0.699781\pi$
$727$		−	6.22025e8i		−	1.61884i	0.587229	−	0.809421i	$-0.300219\pi$
$728$			2.27525e8i			0.589705i
$729$	1.43489e7			0.0370370
$730$			9.29433e8i			2.38918i
$731$	−5.42796e8			−1.38958
$732$			4.76115e7i			0.121389i
$733$			4.80767e7i			0.122074i	0.998136	+	0.0610369i	$0.0194407\pi$
$733$			4.80767e7i			0.122074i	−0.998136	+	0.0610369i	$0.980559\pi$
$734$			7.07443e8i			1.78897i
$735$		−	3.39097e8i		−	0.854008i
$736$	1.09253e8	+	1.38921e7i	0.274032	+	0.0348446i
$737$	2.64399e8			0.660477
$738$	−8.23004e7			−0.204754
$739$	−2.92154e8			−0.723901			−0.361951	−	0.932197i	$-0.617889\pi$
$739$	−2.92154e8			−0.723901			−0.361951	+	0.932197i	$0.617889\pi$
$740$	1.47362e7			0.0363655
$741$		−	4.64415e8i		−	1.14144i
$742$	−2.61472e8			−0.640049
$743$		−	1.89488e8i		−	0.461971i	−0.972957	−	0.230985i	$-0.925805\pi$
$743$		−	1.89488e8i		−	0.461971i	0.972957	−	0.230985i	$-0.0741949\pi$
$744$	3.64687e8			0.885528
$745$	−8.52070e8			−2.06066
$746$			1.04194e8i			0.250973i
$747$			1.47953e8i			0.354946i
$748$	−4.37678e7			−0.104580
$749$	−1.30280e8			−0.310050
$750$		−	7.16301e8i		−	1.69790i
$751$		−	1.03335e7i		−	0.0243965i	−0.999926	−	0.0121982i	$-0.996117\pi$
$751$		−	1.03335e7i		−	0.0243965i	0.999926	−	0.0121982i	$-0.00388291\pi$
$752$	6.61474e7			0.155546
$753$			4.86213e8i			1.13879i
$754$	−3.96876e8			−0.925851
$755$			1.34076e9i			3.11537i
$756$		−	5.25941e6i		−	0.0121723i
$757$			7.76020e7i			0.178890i	0.995992	+	0.0894448i	$0.0285093\pi$
$757$			7.76020e7i			0.178890i	−0.995992	+	0.0894448i	$0.971491\pi$
$758$		−	7.78216e8i		−	1.78687i
$759$	2.10573e7	−	1.65603e8i	0.0481589	−	0.378741i
$760$	−1.05368e9			−2.40031
$761$	6.53691e8			1.48326			0.741632	−	0.670807i	$-0.234052\pi$
$761$	6.53691e8			1.48326			0.741632	+	0.670807i	$0.234052\pi$
$762$	−3.09980e7			−0.0700598
$763$	2.63332e8			0.592830
$764$		−	8.07670e6i		−	0.0181115i
$765$	−3.16438e8			−0.706812
$766$		−	4.60239e8i		−	1.02399i
$767$	5.51438e8			1.22211
$768$	−1.07112e8			−0.236458
$769$			9.05773e8i			1.99177i	0.0906001	+	0.995887i	$0.471121\pi$
$769$			9.05773e8i			1.99177i	−0.0906001	+	0.995887i	$0.528879\pi$
$770$			2.73243e8i			0.598518i
$771$	−2.31222e8			−0.504505
$772$	6.43490e7			0.139859
$773$		−	4.32370e7i		−	0.0936088i	−0.998904	−	0.0468044i	$-0.985096\pi$
$773$		−	4.32370e7i		−	0.0936088i	0.998904	−	0.0468044i	$-0.0149037\pi$
$774$		−	2.01594e8i		−	0.434766i
$775$	1.92513e9			4.13575
$776$		−	3.25459e8i		−	0.696483i
$777$	−1.72682e7			−0.0368116
$778$			3.14883e7i			0.0668668i
$779$			3.81135e8i			0.806244i
$780$		−	1.00291e8i		−	0.211339i
$781$		−	9.16200e7i		−	0.192326i
$782$	7.32041e7	−	5.75706e8i	0.153079	−	1.20387i
$783$	−5.67864e7			−0.118293
$784$	−4.28005e8			−0.888179
$785$	−4.83547e6			−0.00999608
$786$	1.36136e8			0.280353
$787$			3.38601e8i			0.694647i	0.937745	+	0.347324i	$0.112909\pi$
$787$			3.38601e8i			0.694647i	−0.937745	+	0.347324i	$0.887091\pi$
$788$	−3.77119e7			−0.0770725
$789$			7.81814e7i			0.159174i
$790$	1.55370e9			3.15127
$791$	−8.77674e7			−0.177339
$792$			1.00619e8i			0.202536i
$793$		−	1.06390e9i		−	2.13345i
$794$	−3.65186e8			−0.729545
$795$	−7.13413e8			−1.41984
$796$			1.34199e7i			0.0266079i
$797$			6.36108e8i			1.25648i	0.778019	+	0.628241i	$0.216225\pi$
$797$			6.36108e8i			1.25648i	−0.778019	+	0.628241i	$0.783775\pi$
$798$	−1.99476e8			−0.392538
$799$		−	8.05690e7i		−	0.157953i
$800$	−3.50416e8			−0.684406
$801$		−	8.41801e7i		−	0.163799i
$802$		−	8.84517e8i		−	1.71468i
$803$		−	4.11021e8i		−	0.793812i
$804$			4.16830e7i			0.0802030i
$805$	−4.38853e8	−	5.58024e7i	−0.841261	−	0.106971i
$806$	1.31653e9			2.51434
$807$	−4.99202e8			−0.949852
$808$	−9.10779e7			−0.172655
$809$	−8.30004e8			−1.56760			−0.783799	−	0.621015i	$-0.786721\pi$
$809$	−8.30004e8			−1.56760			−0.783799	+	0.621015i	$0.786721\pi$
$810$		−	1.17525e8i		−	0.221144i
$811$	5.71545e8			1.07149			0.535745	−	0.844380i	$-0.320031\pi$
$811$	5.71545e8			1.07149			0.535745	+	0.844380i	$0.320031\pi$
$812$			2.08144e7i			0.0388772i
$813$	−7.19588e7			−0.133910
$814$	−5.33713e7			−0.0989542
$815$		−	9.43246e8i		−	1.74242i
$816$			3.99404e8i			0.735093i
$817$	−9.33588e8			−1.71194
$818$	−3.37236e8			−0.616132
$819$			1.17524e8i			0.213932i
$820$			8.23069e7i			0.149278i
$821$	9.72456e8			1.75728			0.878639	−	0.477486i	$-0.158452\pi$
$821$	9.72456e8			1.75728			0.878639	+	0.477486i	$0.158452\pi$
$822$			1.88959e8i			0.340213i
$823$	5.80764e8			1.04184			0.520919	−	0.853606i	$-0.325589\pi$
$823$	5.80764e8			1.04184			0.520919	+	0.853606i	$0.325589\pi$
$824$		−	4.35387e8i		−	0.778204i
$825$			5.31150e8i			0.945922i
$826$		−	2.36854e8i		−	0.420282i
$827$			3.44734e7i			0.0609491i	0.999536	+	0.0304746i	$0.00970186\pi$
$827$			3.44734e7i			0.0609491i	−0.999536	+	0.0304746i	$0.990298\pi$
$828$	2.61075e7	+	3.31971e6i	0.0459912	+	0.00584803i
$829$	−1.40901e8			−0.247315			−0.123657	−	0.992325i	$-0.539462\pi$
$829$	−1.40901e8			−0.247315			−0.123657	+	0.992325i	$0.539462\pi$
$830$	1.21181e9			2.11934
$831$	3.08286e7			0.0537218
$832$	6.70505e8			1.16421
$833$			5.21319e8i			0.901921i
$834$	−6.48812e7			−0.111846
$835$		−	3.55099e8i		−	0.609944i
$836$	−7.52790e7			−0.128841
$837$	1.88373e8			0.321249
$838$			4.26693e8i			0.725075i
$839$		−	2.68059e8i		−	0.453884i	−0.973908	−	0.226942i	$-0.927127\pi$
$839$		−	2.68059e8i		−	0.453884i	0.973908	−	0.226942i	$-0.0728728\pi$
$840$	2.66643e8			0.449875
$841$	−3.70088e8			−0.622182
$842$		−	2.41814e8i		−	0.405084i
$843$		−	6.24485e8i		−	1.04241i
$844$	−3.41498e7			−0.0568017
$845$			1.11591e9i			1.84952i
$846$	2.99233e7			0.0494196
$847$			1.55492e8i			0.255893i
$848$			9.00462e8i			1.47665i
$849$			2.45350e8i			0.400925i
$850$			1.84650e9i			3.00672i
$851$	1.08996e7	−	8.57189e7i	0.0176857	−	0.139087i
$852$	1.44440e7			0.0233545
$853$	−3.66983e8			−0.591287			−0.295644	−	0.955298i	$-0.595534\pi$
$853$	−3.66983e8			−0.591287			−0.295644	+	0.955298i	$0.595534\pi$
$854$	−4.56968e8			−0.733690
$855$	−5.44260e8			−0.870780
$856$			3.92932e8i			0.626464i
$857$	4.09620e8			0.650786			0.325393	−	0.945579i	$-0.394503\pi$
$857$	4.09620e8			0.650786			0.325393	+	0.945579i	$0.394503\pi$
$858$			3.63234e8i			0.575075i
$859$	1.21318e9			1.91401			0.957006	−	0.290068i	$-0.0936780\pi$
$859$	1.21318e9			1.91401			0.957006	+	0.290068i	$0.0936780\pi$
$860$	−2.01610e8			−0.316970
$861$		−	9.64494e7i		−	0.151109i
$862$		−	4.06310e8i		−	0.634360i
$863$	−1.25360e7			−0.0195042			−0.00975208	−	0.999952i	$-0.503104\pi$
$863$	−1.25360e7			−0.0195042			−0.00975208	+	0.999952i	$0.503104\pi$
$864$	−3.42881e7			−0.0531621
$865$			1.53405e9i			2.37023i
$866$		−	7.78269e8i		−	1.19833i
$867$	1.10215e8			0.169116
$868$		−	6.90457e7i		−	0.105579i
$869$	−6.87088e8			−1.04702
$870$			4.65110e8i			0.706314i
$871$		−	9.31426e8i		−	1.40959i
$872$		−	7.94225e8i		−	1.19783i
$873$		−	1.68110e8i		−	0.252668i
$874$	1.25908e8	−	9.90192e8i	0.188590	−	1.48315i
$875$	8.39447e8			1.25305
$876$	6.47982e7			0.0963941
$877$	−2.44906e8			−0.363078			−0.181539	−	0.983384i	$-0.558108\pi$
$877$	−2.44906e8			−0.363078			−0.181539	+	0.983384i	$0.558108\pi$
$878$	−5.59897e8			−0.827226
$879$		−	3.09664e8i		−	0.455957i
$880$	9.41002e8			1.38084
$881$		−	1.16588e9i		−	1.70501i	−0.522719	−	0.852505i	$-0.675082\pi$
$881$		−	1.16588e9i		−	1.70501i	0.522719	−	0.852505i	$-0.324918\pi$
$882$	−1.93618e8			−0.282189
$883$	4.90699e8			0.712743			0.356371	−	0.934344i	$-0.384014\pi$
$883$	4.90699e8			0.712743			0.356371	+	0.934344i	$0.384014\pi$
$884$			1.54185e8i			0.223196i
$885$		−	6.46245e8i		−	0.932326i
$886$	7.06224e8			1.01541
$887$	−4.95552e8			−0.710099			−0.355049	−	0.934848i	$-0.615536\pi$
$887$	−4.95552e8			−0.710099			−0.355049	+	0.934848i	$0.615536\pi$
$888$			5.20820e7i			0.0743787i
$889$		−	3.63272e7i		−	0.0517042i
$890$	−6.89478e8			−0.978026
$891$			5.19728e7i			0.0734756i
$892$	−1.04333e8			−0.147003
$893$		−	1.38576e8i		−	0.194595i
$894$			4.86516e8i			0.680901i
$895$		−	7.00733e8i		−	0.977426i
$896$		−	3.78357e8i		−	0.525991i
$897$	−5.83385e8	−	7.41805e7i	−0.808311	−	0.102781i
$898$	−5.60650e8			−0.774217
$899$	−7.45495e8			−1.02604
$900$	−8.37366e7			−0.114865
$901$	1.09678e9			1.49950
$902$		−	2.98098e8i		−	0.406200i
$903$	2.36252e8			0.320858
$904$			2.64712e8i			0.358317i
$905$	−6.89760e8			−0.930577
$906$	7.65548e8			1.02941
$907$			4.26244e7i			0.0571263i	0.999592	+	0.0285631i	$0.00909317\pi$
$907$			4.26244e7i			0.0571263i	−0.999592	+	0.0285631i	$0.990907\pi$
$908$		−	2.84792e6i		−	0.00380427i
$909$	−4.70447e7			−0.0626353
$910$	9.62582e8			1.27736
$911$			8.15116e8i			1.07811i	0.842270	+	0.539056i	$0.181219\pi$
$911$			8.15116e8i			1.07811i	−0.842270	+	0.539056i	$0.818781\pi$
$912$			6.86960e8i			0.905621i
$913$	−5.35898e8			−0.704157
$914$		−	1.20798e9i		−	1.58206i
$915$	−1.24682e9			−1.62757
$916$		−	1.40906e8i		−	0.183334i
$917$			1.59540e8i			0.206901i
$918$			1.80680e8i			0.233551i
$919$		−	1.26859e8i		−	0.163446i	−0.996655	−	0.0817231i	$-0.973958\pi$
$919$		−	1.26859e8i		−	0.163446i	0.996655	−	0.0817231i	$-0.0260423\pi$
$920$	−1.68303e8	+	1.32361e9i	−0.216137	+	1.69979i
$921$	6.81921e8			0.872882
$922$	−5.18335e8			−0.661330
$923$	−3.22759e8			−0.410462
$924$	1.90500e7			0.0241479
$925$			2.74932e8i			0.347377i
$926$	1.23757e9			1.55860
$927$		−	2.24892e8i		−	0.282315i
$928$	1.35697e8			0.169795
$929$	8.35978e8			1.04267			0.521336	−	0.853352i	$-0.325434\pi$
$929$	8.35978e8			1.04267			0.521336	+	0.853352i	$0.325434\pi$
$930$		−	1.54287e9i		−	1.91814i
$931$			8.96649e8i			1.11115i
$932$	−1.24221e8			−0.153443
$933$	−8.72436e6			−0.0107421
$934$		−	4.15282e8i		−	0.509685i
$935$		−	1.14616e9i		−	1.40220i
$936$	3.54459e8			0.432254
$937$		−	7.49615e8i		−	0.911212i	−0.890181	−	0.455606i	$-0.849423\pi$
$937$		−	7.49615e8i		−	0.911212i	0.890181	−	0.455606i	$-0.150577\pi$
$938$	−4.00067e8			−0.484758
$939$			7.60150e8i			0.918126i
$940$		−	2.99257e7i		−	0.0360297i
$941$			1.21290e9i			1.45565i	0.685762	+	0.727826i	$0.259469\pi$
$941$			1.21290e9i			1.45565i	−0.685762	+	0.727826i	$0.740531\pi$
$942$			2.76096e6i			0.00330299i
$943$	4.78772e8	+	6.08783e7i	0.570944	+	0.0725985i
$944$	−8.15684e8			−0.969630
$945$	1.37730e8			0.163204
$946$	7.30190e8			0.862507
$947$	1.02846e9			1.21098			0.605492	−	0.795851i	$-0.292976\pi$
$947$	1.02846e9			1.21098			0.605492	+	0.795851i	$0.292976\pi$
$948$		−	1.08321e8i		−	0.127141i
$949$	−1.44795e9			−1.69416
$950$			3.17592e9i			3.70423i
$951$	−4.91129e8			−0.571023
$952$	−4.09929e8			−0.475115
$953$		−	5.65303e8i		−	0.653135i	−0.945174	−	0.326567i	$-0.894108\pi$
$953$		−	5.65303e8i		−	0.653135i	0.945174	−	0.326567i	$-0.105892\pi$
$954$			4.07345e8i			0.469156i
$955$	2.11507e8			0.242837
$956$	1.06733e8			0.122159
$957$		−	2.05685e8i		−	0.234675i
$958$		−	9.92059e8i		−	1.12834i
$959$	−2.21444e8			−0.251078
$960$		−	7.85783e8i		−	0.888155i
$961$	1.58547e9			1.78643
$962$			1.88016e8i			0.211189i
$963$			2.02962e8i			0.227267i
$964$		−	2.24190e8i		−	0.250257i
$965$			1.68513e9i			1.87521i
$966$	−3.18621e7	+	2.50576e8i	−0.0353462	+	0.277977i
$967$	−1.45567e8			−0.160984			−0.0804920	−	0.996755i	$-0.525649\pi$
$967$	−1.45567e8			−0.160984			−0.0804920	+	0.996755i	$0.525649\pi$
$968$	4.68973e8			0.517037
$969$	8.36732e8			0.919634
$970$	−1.37691e9			−1.50865
$971$		−	1.00039e9i		−	1.09272i	−0.837549	−	0.546362i	$-0.816012\pi$
$971$		−	1.00039e9i		−	1.09272i	0.837549	−	0.546362i	$-0.183988\pi$
$972$	−8.19360e6			−0.00892228
$973$		−	7.60355e7i		−	0.0825425i
$974$	−2.60102e8			−0.281493
$975$	1.87114e9			2.01879
$976$			1.57372e9i			1.69269i
$977$		−	7.45733e7i		−	0.0799650i	−0.999200	−	0.0399825i	$-0.987270\pi$
$977$		−	7.45733e7i		−	0.0799650i	0.999200	−	0.0399825i	$-0.0127302\pi$
$978$	−5.38575e8			−0.575745
$979$	3.04907e8			0.324952
$980$			1.93633e8i			0.205732i
$981$		−	4.10243e8i		−	0.434544i
$982$	6.98261e8			0.737367
$983$		−	5.50359e8i		−	0.579409i	−0.957116	−	0.289704i	$-0.906443\pi$
$983$		−	5.50359e8i		−	0.579409i	0.957116	−	0.289704i	$-0.0935570\pi$
$984$	−2.90897e8			−0.305319
$985$		−	9.87572e8i		−	1.03338i
$986$		−	7.15049e8i		−	0.745942i
$987$			3.50677e7i			0.0364717i
$988$			2.65193e8i			0.274974i
$989$	−1.49121e8	+	1.17275e9i	−0.154152	+	1.21232i
$990$	4.25684e8			0.438714
$991$	−1.52359e9			−1.56547			−0.782737	−	0.622353i	$-0.786177\pi$
$991$	−1.52359e9			−1.56547			−0.782737	+	0.622353i	$0.786177\pi$
$992$	−4.50135e8			−0.461114
$993$	−1.94772e8			−0.198921
$994$			1.38632e8i			0.141158i
$995$	−3.51431e8			−0.356755
$996$		−	8.44852e7i		−	0.0855072i
$997$	−1.45542e9			−1.46860			−0.734300	−	0.678825i	$-0.762489\pi$
$997$	−1.45542e9			−1.46860			−0.734300	+	0.678825i	$0.762489\pi$
$998$	−2.10706e9			−2.11975
$999$			2.69020e7i			0.0269829i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.8	yes	24
3.2	odd	2		207.7.d.e.91.17		24
23.22	odd	2	inner	69.7.d.a.22.7	✓	24
69.68	even	2		207.7.d.e.91.18		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.7	✓	24	23.22	odd	2	inner
69.7.d.a.22.8	yes	24	1.1	even	1	trivial
207.7.d.e.91.17		24	3.2	odd	2
207.7.d.e.91.18		24	69.68	even	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.53823 q^{2} -15.5885 q^{3} +8.90142 q^{4} +233.104i q^{5} +133.098 q^{6} +155.980i q^{7} +470.445 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-8.53823 q^{2} -15.5885 q^{3} +8.90142 q^{4} +233.104i q^{5} +133.098 q^{6} +155.980i q^{7} +470.445 q^{8} +243.000 q^{9} -1990.29i q^{10} +880.164i q^{11} -138.759 q^{12} +3100.64 q^{13} -1331.79i q^{14} -3633.73i q^{15} -4586.46 q^{16} +5586.40i q^{17} -2074.79 q^{18} +9608.40i q^{19} +2074.95i q^{20} -2431.49i q^{21} -7515.05i q^{22} +(12069.8 + 1534.74i) q^{23} -7333.50 q^{24} -38712.4 q^{25} -26474.0 q^{26} -3788.00 q^{27} +1388.44i q^{28} +14991.2 q^{29} +31025.6i q^{30} -49729.0 q^{31} +9051.77 q^{32} -13720.4i q^{33} -47698.0i q^{34} -36359.5 q^{35} +2163.04 q^{36} -7101.92i q^{37} -82038.7i q^{38} -48334.2 q^{39} +109662. i q^{40} +39666.9 q^{41} +20760.6i q^{42} +97163.8i q^{43} +7834.71i q^{44} +56644.2i q^{45} +(-103055. - 13104.0i) q^{46} -14422.3 q^{47} +71495.8 q^{48} +93319.3 q^{49} +330536. q^{50} -87083.4i q^{51} +27600.1 q^{52} -196331. i q^{53} +32342.8 q^{54} -205170. q^{55} +73379.9i q^{56} -149780. i q^{57} -127998. q^{58} +177846. q^{59} -32345.3i q^{60} -343123. i q^{61} +424597. q^{62} +37903.1i q^{63} +216247. q^{64} +722772. i q^{65} +117148. i q^{66} -300398. i q^{67} +49726.9i q^{68} +(-188150. - 23924.2i) q^{69} +310446. q^{70} -104094. q^{71} +114318. q^{72} -466983. q^{73} +60637.8i q^{74} +603467. q^{75} +85528.3i q^{76} -137288. q^{77} +412689. q^{78} +780636. i q^{79} -1.06912e6i q^{80} +59049.0 q^{81} -338685. q^{82} +608861. i q^{83} -21643.7i q^{84} -1.30221e6 q^{85} -829607. i q^{86} -233689. q^{87} +414068. i q^{88} -346420. i q^{89} -483642. i q^{90} +483638. i q^{91} +(107438. + 13661.4i) q^{92} +775198. q^{93} +123141. q^{94} -2.23975e6 q^{95} -141103. q^{96} -691811. i q^{97} -796782. q^{98} +213880. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

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Database

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