LMFDB - Embedded newform 69.7.d.a.22.13

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			22.13
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.368531 q^{2} +15.5885 q^{3} -63.8642 q^{4} -113.928i q^{5} -5.74483 q^{6} +569.901i q^{7} +47.1219 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-0.368531 q^{2} +15.5885 q^{3} -63.8642 q^{4} -113.928i q^{5} -5.74483 q^{6} +569.901i q^{7} +47.1219 q^{8} +243.000 q^{9} +41.9859i q^{10} -814.298i q^{11} -995.544 q^{12} +3513.83 q^{13} -210.026i q^{14} -1775.96i q^{15} +4069.94 q^{16} -7484.26i q^{17} -89.5530 q^{18} +4283.93i q^{19} +7275.90i q^{20} +8883.88i q^{21} +300.094i q^{22} +(12048.1 + 1696.83i) q^{23} +734.558 q^{24} +2645.50 q^{25} -1294.96 q^{26} +3788.00 q^{27} -36396.3i q^{28} -3314.46 q^{29} +654.495i q^{30} +34401.9 q^{31} -4515.70 q^{32} -12693.6i q^{33} +2758.18i q^{34} +64927.5 q^{35} -15519.0 q^{36} -7160.07i q^{37} -1578.76i q^{38} +54775.2 q^{39} -5368.49i q^{40} +77969.3 q^{41} -3273.99i q^{42} +113275. i q^{43} +52004.5i q^{44} -27684.4i q^{45} +(-4440.10 - 625.334i) q^{46} -62809.9 q^{47} +63444.1 q^{48} -207138. q^{49} -974.947 q^{50} -116668. i q^{51} -224408. q^{52} -135215. i q^{53} -1395.99 q^{54} -92771.0 q^{55} +26854.8i q^{56} +66779.9i q^{57} +1221.48 q^{58} -101668. q^{59} +113420. i q^{60} -38234.1i q^{61} -12678.2 q^{62} +138486. i q^{63} -258812. q^{64} -400322. i q^{65} +4678.00i q^{66} +229273. i q^{67} +477976. i q^{68} +(187811. + 26451.0i) q^{69} -23927.8 q^{70} -64522.4 q^{71} +11450.6 q^{72} -53631.6 q^{73} +2638.71i q^{74} +41239.2 q^{75} -273590. i q^{76} +464069. q^{77} -20186.4 q^{78} -458737. i q^{79} -463679. i q^{80} +59049.0 q^{81} -28734.1 q^{82} +1.13063e6i q^{83} -567362. i q^{84} -852664. q^{85} -41745.4i q^{86} -51667.4 q^{87} -38371.3i q^{88} -829443. i q^{89} +10202.6i q^{90} +2.00254e6i q^{91} +(-769442. - 108367. i) q^{92} +536273. q^{93} +23147.4 q^{94} +488058. q^{95} -70392.8 q^{96} -525418. i q^{97} +76337.0 q^{98} -197874. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.368531			−0.0460664			−0.0230332	−	0.999735i	$-0.507332\pi$
$2$	−0.368531			−0.0460664			−0.0230332	+	0.999735i	$0.507332\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−63.8642			−0.997878
$5$		−	113.928i		−	0.911421i	−0.890128	−	0.455711i	$-0.849385\pi$
$5$		−	113.928i		−	0.911421i	0.890128	−	0.455711i	$-0.150615\pi$
$6$	−5.74483			−0.0265964
$7$			569.901i			1.66152i	0.556631	+	0.830760i	$0.312094\pi$
$7$			569.901i			1.66152i	−0.556631	+	0.830760i	$0.687906\pi$
$8$	47.1219			0.0920350
$9$	243.000			0.333333
$10$			41.9859i			0.0419859i
$11$		−	814.298i		−	0.611794i	−0.952065	−	0.305897i	$-0.901044\pi$
$11$		−	814.298i		−	0.611794i	0.952065	−	0.305897i	$-0.0989563\pi$
$12$	−995.544			−0.576125
$13$	3513.83			1.59938			0.799689	−	0.600415i	$-0.204998\pi$
$13$	3513.83			1.59938			0.799689	+	0.600415i	$0.204998\pi$
$14$		−	210.026i		−	0.0765402i
$15$		−	1775.96i		−	0.526209i
$16$	4069.94			0.993638
$17$		−	7484.26i		−	1.52336i	−0.647954	−	0.761679i	$-0.724375\pi$
$17$		−	7484.26i		−	1.52336i	0.647954	−	0.761679i	$-0.275625\pi$
$18$	−89.5530			−0.0153555
$19$			4283.93i			0.624571i	0.949988	+	0.312285i	$0.101095\pi$
$19$			4283.93i			0.624571i	−0.949988	+	0.312285i	$0.898905\pi$
$20$			7275.90i			0.909487i
$21$			8883.88i			0.959279i
$22$			300.094i			0.0281831i
$23$	12048.1	+	1696.83i	0.990227	+	0.139462i
$23$	12048.1	+	1696.83i	0.990227	+	0.139462i
$24$	734.558			0.0531364
$25$	2645.50			0.169312
$26$	−1294.96			−0.0736775
$27$	3788.00			0.192450
$28$		−	36396.3i		−	1.65799i
$29$	−3314.46			−0.135900			−0.0679500	−	0.997689i	$-0.521646\pi$
$29$	−3314.46			−0.135900			−0.0679500	+	0.997689i	$0.521646\pi$
$30$			654.495i			0.0242405i
$31$	34401.9			1.15478			0.577388	−	0.816470i	$-0.304072\pi$
$31$	34401.9			1.15478			0.577388	+	0.816470i	$0.304072\pi$
$32$	−4515.70			−0.137808
$33$		−	12693.6i		−	0.353219i
$34$			2758.18i			0.0701756i
$35$	64927.5			1.51434
$36$	−15519.0			−0.332626
$37$		−	7160.07i		−	0.141355i	−0.997499	−	0.0706776i	$-0.977484\pi$
$37$		−	7160.07i		−	0.141355i	0.997499	−	0.0706776i	$-0.0225162\pi$
$38$		−	1578.76i		−	0.0287717i
$39$	54775.2			0.923401
$40$		−	5368.49i		−	0.0838826i
$41$	77969.3			1.13129			0.565643	−	0.824650i	$-0.308628\pi$
$41$	77969.3			1.13129			0.565643	+	0.824650i	$0.308628\pi$
$42$		−	3273.99i		−	0.0441905i
$43$			113275.i			1.42472i	0.701815	+	0.712359i	$0.252373\pi$
$43$			113275.i			1.42472i	−0.701815	+	0.712359i	$0.747627\pi$
$44$			52004.5i			0.610496i
$45$		−	27684.4i		−	0.303807i
$46$	−4440.10	−	625.334i	−0.0456162	−	0.00642449i
$47$	−62809.9			−0.604971			−0.302485	−	0.953154i	$-0.597816\pi$
$47$	−62809.9			−0.604971			−0.302485	+	0.953154i	$0.597816\pi$
$48$	63444.1			0.573677
$49$	−207138.			−1.76065
$50$	−974.947			−0.00779958
$51$		−	116668.i		−	0.879511i
$52$	−224408.			−1.59598
$53$		−	135215.i		−	0.908235i	−0.890942	−	0.454118i	$-0.849955\pi$
$53$		−	135215.i		−	0.908235i	0.890942	−	0.454118i	$-0.150045\pi$
$54$	−1395.99			−0.00886548
$55$	−92771.0			−0.557602
$56$			26854.8i			0.152918i
$57$			66779.9i			0.360596i
$58$	1221.48			0.00626042
$59$	−101668.			−0.495025			−0.247513	−	0.968885i	$-0.579613\pi$
$59$	−101668.			−0.495025			−0.247513	+	0.968885i	$0.579613\pi$
$60$			113420.i			0.525093i
$61$		−	38234.1i		−	0.168446i	−0.996447	−	0.0842231i	$-0.973159\pi$
$61$		−	38234.1i		−	0.168446i	0.996447	−	0.0842231i	$-0.0268409\pi$
$62$	−12678.2			−0.0531963
$63$			138486.i			0.553840i
$64$	−258812.			−0.987290
$65$		−	400322.i		−	1.45771i
$66$			4678.00i			0.0162715i
$67$			229273.i			0.762306i	0.924512	+	0.381153i	$0.124473\pi$
$67$			229273.i			0.762306i	−0.924512	+	0.381153i	$0.875527\pi$
$68$			477976.i			1.52013i
$69$	187811.	+	26451.0i	0.571708	+	0.0805182i
$70$	−23927.8			−0.0697603
$71$	−64522.4			−0.180275			−0.0901375	−	0.995929i	$-0.528731\pi$
$71$	−64522.4			−0.180275			−0.0901375	+	0.995929i	$0.528731\pi$
$72$	11450.6			0.0306783
$73$	−53631.6			−0.137864			−0.0689322	−	0.997621i	$-0.521959\pi$
$73$	−53631.6			−0.137864			−0.0689322	+	0.997621i	$0.521959\pi$
$74$			2638.71i			0.00651172i
$75$	41239.2			0.0977522
$76$		−	273590.i		−	0.623245i
$77$	464069.			1.01651
$78$	−20186.4			−0.0425377
$79$		−	458737.i		−	0.930427i	−0.885199	−	0.465213i	$-0.845978\pi$
$79$		−	458737.i		−	0.930427i	0.885199	−	0.465213i	$-0.154022\pi$
$80$		−	463679.i		−	0.905623i
$81$	59049.0			0.111111
$82$	−28734.1			−0.0521142
$83$			1.13063e6i			1.97736i	0.150055	+	0.988678i	$0.452055\pi$
$83$			1.13063e6i			1.97736i	−0.150055	+	0.988678i	$0.547945\pi$
$84$		−	567362.i		−	0.957243i
$85$	−852664.			−1.38842
$86$		−	41745.4i		−	0.0656316i
$87$	−51667.4			−0.0784619
$88$		−	38371.3i		−	0.0563065i
$89$		−	829443.i		−	1.17657i	−0.808655	−	0.588284i	$-0.799804\pi$
$89$		−	829443.i		−	1.17657i	0.808655	−	0.588284i	$-0.200196\pi$
$90$			10202.6i			0.0139953i
$91$			2.00254e6i			2.65740i
$92$	−769442.	−	108367.i	−0.988126	−	0.139166i
$93$	536273.			0.666710
$94$	23147.4			0.0278688
$95$	488058.			0.569247
$96$	−70392.8			−0.0795637
$97$		−	525418.i		−	0.575691i	−0.957677	−	0.287846i	$-0.907061\pi$
$97$		−	525418.i		−	0.575691i	0.957677	−	0.287846i	$-0.0929390\pi$
$98$	76337.0			0.0811067
$99$		−	197874.i		−	0.203931i
$100$	−168952.			−0.168952
$101$	1.66810e6			1.61904			0.809521	−	0.587090i	$-0.199727\pi$
$101$	1.66810e6			1.61904			0.809521	+	0.587090i	$0.199727\pi$
$102$			42995.8i			0.0405159i
$103$		−	226535.i		−	0.207311i	−0.994613	−	0.103656i	$-0.966946\pi$
$103$		−	226535.i		−	0.207311i	0.994613	−	0.103656i	$-0.0330540\pi$
$104$	165578.			0.147199
$105$	1.01212e6			0.874307
$106$			49831.0i			0.0418391i
$107$		−	742387.i		−	0.606009i	−0.952989	−	0.303004i	$-0.902010\pi$
$107$		−	742387.i		−	0.606009i	0.952989	−	0.303004i	$-0.0979897\pi$
$108$	−241917.			−0.192042
$109$		−	2.09107e6i		−	1.61469i	−0.590080	−	0.807344i	$-0.700904\pi$
$109$		−	2.09107e6i		−	1.61469i	0.590080	−	0.807344i	$-0.299096\pi$
$110$	34189.0			0.0256867
$111$		−	111614.i		−	0.0816115i
$112$			2.31947e6i			1.65095i
$113$			2.20924e6i			1.53111i	0.643368	+	0.765557i	$0.277536\pi$
$113$			2.20924e6i			1.53111i	−0.643368	+	0.765557i	$0.722464\pi$
$114$		−	24610.5i		−	0.0166114i
$115$	193316.	−	1.37261e6i	0.127108	−	0.902514i
$116$	211676.			0.135612
$117$	853861.			0.533126
$118$	37467.7			0.0228040
$119$	4.26529e6			2.53109
$120$		−	83686.4i		−	0.0484297i
$121$	1.10848e6			0.625708
$122$			14090.5i			0.00775971i
$123$	1.21542e6			0.653148
$124$	−2.19705e6			−1.15232
$125$		−	2.08151e6i		−	1.06574i
$126$		−	51036.4i		−	0.0255134i
$127$	−723004.			−0.352963			−0.176482	−	0.984304i	$-0.556472\pi$
$127$	−723004.			−0.352963			−0.176482	+	0.984304i	$0.556472\pi$
$128$	384385.			0.183289
$129$			1.76578e6i			0.822561i
$130$			147531.i			0.0671512i
$131$	−3.35181e6			−1.49096			−0.745479	−	0.666529i	$-0.767779\pi$
$131$	−3.35181e6			−1.49096			−0.745479	+	0.666529i	$0.767779\pi$
$132$			810669.i			0.352470i
$133$	−2.44142e6			−1.03774
$134$		−	84494.3i		−	0.0351167i
$135$		−	431557.i		−	0.175403i
$136$		−	352673.i		−	0.140202i
$137$		−	546016.i		−	0.212346i	−0.994348	−	0.106173i	$-0.966140\pi$
$137$		−	546016.i		−	0.212346i	0.994348	−	0.106173i	$-0.0338597\pi$
$138$	−69214.3	−	9748.00i	−0.0263365	−	0.00370918i
$139$	1.07985e6			0.402086			0.201043	−	0.979582i	$-0.435567\pi$
$139$	1.07985e6			0.402086			0.201043	+	0.979582i	$0.435567\pi$
$140$	−4.14654e6			−1.51113
$141$	−979109.			−0.349280
$142$	23778.5			0.00830461
$143$		−	2.86131e6i		−	0.978489i
$144$	988996.			0.331213
$145$			377609.i			0.123862i
$146$	19764.9			0.00635091
$147$	−3.22897e6			−1.01651
$148$			457272.i			0.141055i
$149$			4.51099e6i			1.36368i	0.731501	+	0.681840i	$0.238820\pi$
$149$			4.51099e6i			1.36368i	−0.731501	+	0.681840i	$0.761180\pi$
$150$	−15197.9			−0.00450309
$151$	3.08008e6			0.894605			0.447302	−	0.894383i	$-0.352385\pi$
$151$	3.08008e6			0.894605			0.447302	+	0.894383i	$0.352385\pi$
$152$			201867.i			0.0574824i
$153$		−	1.81868e6i		−	0.507786i
$154$	−171024.			−0.0468268
$155$		−	3.91933e6i		−	1.05249i
$156$	−3.49817e6			−0.921441
$157$			5.92399e6i			1.53079i	0.643562	+	0.765394i	$0.277456\pi$
$157$			5.92399e6i			1.53079i	−0.643562	+	0.765394i	$0.722544\pi$
$158$			169059.i			0.0428614i
$159$		−	2.10780e6i		−	0.524370i
$160$			514463.i			0.125601i
$161$	−967025.	+	6.86623e6i	−0.231718	+	1.64528i
$162$	−21761.4			−0.00511849
$163$	917099.			0.211765			0.105882	−	0.994379i	$-0.466233\pi$
$163$	917099.			0.211765			0.105882	+	0.994379i	$0.466233\pi$
$164$	−4.97945e6			−1.12888
$165$	−1.44616e6			−0.321932
$166$		−	416671.i		−	0.0910896i
$167$	−7.58083e6			−1.62767			−0.813837	−	0.581094i	$-0.802625\pi$
$167$	−7.58083e6			−1.62767			−0.813837	+	0.581094i	$0.802625\pi$
$168$			418626.i			0.0882872i
$169$	7.52020e6			1.55801
$170$	314233.			0.0639595
$171$			1.04100e6i			0.208190i
$172$		−	7.23422e6i		−	1.42169i
$173$	5.62249e6			1.08590			0.542951	−	0.839765i	$-0.317307\pi$
$173$	5.62249e6			1.08590			0.542951	+	0.839765i	$0.317307\pi$
$174$	19041.0			0.00361445
$175$			1.50767e6i			0.281315i
$176$		−	3.31414e6i		−	0.607902i
$177$	−1.58484e6			−0.285803
$178$			305676.i			0.0542002i
$179$	656490.			0.114464			0.0572320	−	0.998361i	$-0.481773\pi$
$179$	656490.			0.114464			0.0572320	+	0.998361i	$0.481773\pi$
$180$			1.76804e6i			0.303162i
$181$		−	9.20126e6i		−	1.55171i	−0.630909	−	0.775857i	$-0.717318\pi$
$181$		−	9.20126e6i		−	1.55171i	0.630909	−	0.775857i	$-0.282682\pi$
$182$		−	737997.i		−	0.122417i
$183$		−	596011.i		−	0.0972525i
$184$	567729.	+	79957.9i	0.0911356	+	0.0128353i
$185$	−815730.			−0.128834
$186$	−197633.			−0.0307129
$187$	−6.09442e6			−0.931982
$188$	4.01130e6			0.603687
$189$			2.15878e6i			0.319760i
$190$	−179865.			−0.0262231
$191$			9.24960e6i			1.32746i	0.747971	+	0.663732i	$0.231028\pi$
$191$			9.24960e6i			1.32746i	−0.747971	+	0.663732i	$0.768972\pi$
$192$	−4.03448e6			−0.570012
$193$	−1.56196e6			−0.217269			−0.108634	−	0.994082i	$-0.534648\pi$
$193$	−1.56196e6			−0.217269			−0.108634	+	0.994082i	$0.534648\pi$
$194$			193633.i			0.0265200i
$195$		−	6.24041e6i		−	0.841607i
$196$	1.32287e7			1.75691
$197$	−1.12002e7			−1.46496			−0.732482	−	0.680786i	$-0.761638\pi$
$197$	−1.12002e7			−1.46496			−0.732482	+	0.680786i	$0.761638\pi$
$198$			72922.8i			0.00939438i
$199$			4.87456e6i			0.618552i	0.950972	+	0.309276i	$0.100087\pi$
$199$			4.87456e6i			0.618552i	−0.950972	+	0.309276i	$0.899913\pi$
$200$	124661.			0.0155826
$201$			3.57402e6i			0.440117i
$202$	−614747.			−0.0745834
$203$		−	1.88892e6i		−	0.225800i
$204$			7.45091e6i			0.877645i
$205$		−	8.88286e6i		−	1.03108i
$206$			83485.1i			0.00955008i
$207$	2.92769e6	+	412330.i	0.330076	+	0.0464872i
$208$	1.43011e7			1.58920
$209$	3.48840e6			0.382109
$210$	−372997.			−0.0402762
$211$	1.01176e7			1.07704			0.538518	−	0.842614i	$-0.318984\pi$
$211$	1.01176e7			1.07704			0.538518	+	0.842614i	$0.318984\pi$
$212$			8.63542e6i			0.906308i
$213$	−1.00580e6			−0.104082
$214$			273593.i			0.0279166i
$215$	1.29052e7			1.29852
$216$	178498.			0.0177121
$217$			1.96057e7i			1.91868i
$218$			770624.i			0.0743829i
$219$	−836034.			−0.0795960
$220$	5.92475e6			0.556419
$221$		−	2.62984e7i		−	2.43642i
$222$			41133.4i			0.00375955i
$223$	−1.27139e7			−1.14647			−0.573236	−	0.819390i	$-0.694312\pi$
$223$	−1.27139e7			−1.14647			−0.573236	+	0.819390i	$0.694312\pi$
$224$		−	2.57350e6i		−	0.228971i
$225$	642855.			0.0564372
$226$		−	814173.i		−	0.0705328i
$227$		−	1.69641e7i		−	1.45028i	−0.688599	−	0.725142i	$-0.741774\pi$
$227$		−	1.69641e7i		−	1.45028i	0.688599	−	0.725142i	$-0.258226\pi$
$228$		−	4.26484e6i		−	0.359831i
$229$		−	1.22935e7i		−	1.02369i	−0.859077	−	0.511847i	$-0.828962\pi$
$229$		−	1.22935e7i		−	1.02369i	0.859077	−	0.511847i	$-0.171038\pi$
$230$	−71242.8	+	505850.i	−0.00585542	+	0.0415756i
$231$	7.23413e6			0.586881
$232$	−156184.			−0.0125075
$233$	−171237.			−0.0135372			−0.00676861	−	0.999977i	$-0.502155\pi$
$233$	−171237.			−0.0135372			−0.00676861	+	0.999977i	$0.502155\pi$
$234$	−314674.			−0.0245592
$235$			7.15578e6i			0.551383i
$236$	6.49293e6			0.493975
$237$		−	7.15100e6i		−	0.537182i
$238$	−1.57189e6			−0.116598
$239$	−543012.			−0.0397755			−0.0198877	−	0.999802i	$-0.506331\pi$
$239$	−543012.			−0.0397755			−0.0198877	+	0.999802i	$0.506331\pi$
$240$		−	7.22804e6i		−	0.522862i
$241$		−	3.84804e6i		−	0.274909i	−0.990508	−	0.137455i	$-0.956108\pi$
$241$		−	3.84804e6i		−	0.274909i	0.990508	−	0.137455i	$-0.0438921\pi$
$242$	−408509.			−0.0288241
$243$	920483.			0.0641500
$244$			2.44179e6i			0.168089i
$245$			2.35988e7i			1.60469i
$246$	−447920.			−0.0300882
$247$			1.50530e7i			0.998924i
$248$	1.62108e6			0.106280
$249$			1.76247e7i			1.14163i
$250$			767103.i			0.0490946i
$251$			1.18733e7i			0.750845i	0.926854	+	0.375422i	$0.122502\pi$
$251$			1.18733e7i			0.750845i	−0.926854	+	0.375422i	$0.877498\pi$
$252$		−	8.84430e6i		−	0.552665i
$253$	1.38172e6	−	9.81074e6i	0.0853218	−	0.605815i
$254$	266449.			0.0162597
$255$	−1.32917e7			−0.801605
$256$	1.64223e7			0.978846
$257$	−1.02407e7			−0.603296			−0.301648	−	0.953419i	$-0.597537\pi$
$257$	−1.02407e7			−0.603296			−0.301648	+	0.953419i	$0.597537\pi$
$258$		−	650746.i		−	0.0378924i
$259$	4.08053e6			0.234865
$260$			2.55663e7i			1.45461i
$261$	−805415.			−0.0453000
$262$	1.23525e6			0.0686831
$263$			2.04038e6i			0.112162i	0.998426	+	0.0560809i	$0.0178605\pi$
$263$			2.04038e6i			0.112162i	−0.998426	+	0.0560809i	$0.982140\pi$
$264$		−	598149.i		−	0.0325085i
$265$	−1.54048e7			−0.827785
$266$	899738.			0.0478048
$267$		−	1.29297e7i		−	0.679291i
$268$		−	1.46424e7i		−	0.760688i
$269$	−3.09947e7			−1.59232			−0.796161	−	0.605085i	$-0.793139\pi$
$269$	−3.09947e7			−1.59232			−0.796161	+	0.605085i	$0.793139\pi$
$270$			159042.i			0.00808018i
$271$	−4.55560e6			−0.228896			−0.114448	−	0.993429i	$-0.536510\pi$
$271$	−4.55560e6			−0.228896			−0.114448	+	0.993429i	$0.536510\pi$
$272$		−	3.04605e7i		−	1.51367i
$273$			3.12165e7i			1.53425i
$274$			201224.i			0.00978200i
$275$		−	2.15422e6i		−	0.103584i
$276$	−1.19944e7	−	1.68927e6i	−0.570495	−	0.0803473i
$277$	−1.18273e7			−0.556475			−0.278237	−	0.960512i	$-0.589750\pi$
$277$	−1.18273e7			−0.556475			−0.278237	+	0.960512i	$0.589750\pi$
$278$	−397958.			−0.0185226
$279$	8.35966e6			0.384925
$280$	3.05951e6			0.139373
$281$		−	1.70316e7i		−	0.767601i	−0.923416	−	0.383801i	$-0.874615\pi$
$281$		−	1.70316e7i		−	0.767601i	0.923416	−	0.383801i	$-0.125385\pi$
$282$	360832.			0.0160901
$283$		−	3.65196e7i		−	1.61126i	−0.592418	−	0.805631i	$-0.701826\pi$
$283$		−	3.65196e7i		−	1.61126i	0.592418	−	0.805631i	$-0.298174\pi$
$284$	4.12067e6			0.179892
$285$	7.60807e6			0.328655
$286$			1.05448e6i			0.0450755i
$287$			4.44348e7i			1.87965i
$288$	−1.09732e6			−0.0459361
$289$	−3.18766e7			−1.32062
$290$		−	139161.i		−	0.00570588i
$291$		−	8.19045e6i		−	0.332375i
$292$	3.42514e6			0.137572
$293$		−	1.85690e7i		−	0.738220i	−0.929386	−	0.369110i	$-0.879663\pi$
$293$		−	1.85690e7i		−	0.738220i	0.929386	−	0.369110i	$-0.120337\pi$
$294$	1.18998e6			0.0468270
$295$			1.15828e7i			0.451177i
$296$		−	337396.i		−	0.0130096i
$297$		−	3.08456e6i		−	0.117740i
$298$		−	1.66244e6i		−	0.0628198i
$299$	4.23350e7	+	5.96237e6i	1.58375	+	0.223052i
$300$	−2.63371e6			−0.0975447
$301$	−6.45556e7			−2.36720
$302$	−1.13511e6			−0.0412112
$303$	2.60031e7			0.934755
$304$			1.74354e7i			0.620597i
$305$	−4.35592e6			−0.153525
$306$			670238.i			0.0233919i
$307$	−3.39160e7			−1.17217			−0.586084	−	0.810250i	$-0.699331\pi$
$307$	−3.39160e7			−1.17217			−0.586084	+	0.810250i	$0.699331\pi$
$308$	−2.96374e7			−1.01435
$309$		−	3.53133e6i		−	0.119691i
$310$			1.44439e6i			0.0484842i
$311$	−3.91451e7			−1.30136			−0.650679	−	0.759353i	$-0.725516\pi$
$311$	−3.91451e7			−1.30136			−0.650679	+	0.759353i	$0.725516\pi$
$312$	2.58111e6			0.0849852
$313$			4.03448e7i			1.31569i	0.753152	+	0.657847i	$0.228533\pi$
$313$			4.03448e7i			1.31569i	−0.753152	+	0.657847i	$0.771467\pi$
$314$		−	2.18317e6i		−	0.0705179i
$315$	1.57774e7			0.504781
$316$			2.92968e7i			0.928452i
$317$	3.59652e7			1.12903			0.564514	−	0.825423i	$-0.309063\pi$
$317$	3.59652e7			1.12903			0.564514	+	0.825423i	$0.309063\pi$
$318$			776789.i			0.0241558i
$319$			2.69896e6i			0.0831428i
$320$			2.94858e7i			0.899837i
$321$		−	1.15727e7i		−	0.349879i
$322$	356379.	−	2.53042e6i	0.0106744	−	0.0757922i
$323$	3.20621e7			0.951445
$324$	−3.77112e6			−0.110875
$325$	9.29583e6			0.270793
$326$	−337979.			−0.00975523
$327$		−	3.25965e7i		−	0.932241i
$328$	3.67406e6			0.104118
$329$		−	3.57954e7i		−	1.00517i
$330$	532954.			0.0148302
$331$	−2.80276e7			−0.772863			−0.386431	−	0.922318i	$-0.626292\pi$
$331$	−2.80276e7			−0.772863			−0.386431	+	0.922318i	$0.626292\pi$
$332$		−	7.22065e7i		−	1.97316i
$333$		−	1.73990e6i		−	0.0471184i
$334$	2.79377e6			0.0749810
$335$	2.61206e7			0.694781
$336$			3.61569e7i			0.953176i
$337$			3.06197e7i			0.800038i	0.916507	+	0.400019i	$0.130997\pi$
$337$			3.06197e7i			0.800038i	−0.916507	+	0.400019i	$0.869003\pi$
$338$	−2.77143e6			−0.0717717
$339$			3.44386e7i			0.883989i
$340$	5.44547e7			1.38547
$341$		−	2.80134e7i		−	0.706485i
$342$		−	383639.i		−	0.00959057i
$343$		−	5.10002e7i		−	1.26383i
$344$			5.33774e6i			0.131124i
$345$	3.01349e6	−	2.13969e7i	0.0733860	−	0.521067i
$346$	−2.07206e6			−0.0500235
$347$	7.04261e7			1.68556			0.842782	−	0.538255i	$-0.180916\pi$
$347$	7.04261e7			1.68556			0.842782	+	0.538255i	$0.180916\pi$
$348$	3.29969e6			0.0782954
$349$	−6.68006e7			−1.57146			−0.785731	−	0.618568i	$-0.787713\pi$
$349$	−6.68006e7			−1.57146			−0.785731	+	0.618568i	$0.787713\pi$
$350$		−	555624.i		−	0.0129592i
$351$	1.33104e7			0.307800
$352$			3.67713e6i			0.0843103i
$353$	−7.85575e6			−0.178593			−0.0892963	−	0.996005i	$-0.528462\pi$
$353$	−7.85575e6			−0.178593			−0.0892963	+	0.996005i	$0.528462\pi$
$354$	584064.			0.0131659
$355$			7.35088e6i			0.164306i
$356$			5.29717e7i			1.17407i
$357$	6.64893e7			1.46133
$358$	−241937.			−0.00527294
$359$		−	8.88485e6i		−	0.192029i	−0.995380	−	0.0960145i	$-0.969390\pi$
$359$		−	8.88485e6i		−	0.192029i	0.995380	−	0.0960145i	$-0.0306095\pi$
$360$		−	1.30454e6i		−	0.0279609i
$361$	2.86938e7			0.609911
$362$			3.39095e6i			0.0714818i
$363$	1.72795e7			0.361253
$364$		−	1.27890e8i		−	2.65176i
$365$			6.11012e6i			0.125652i
$366$			219648.i			0.00448007i
$367$			8.43071e7i			1.70556i	0.522274	+	0.852778i	$0.325084\pi$
$367$			8.43071e7i			1.70556i	−0.522274	+	0.852778i	$0.674916\pi$
$368$	4.90351e7	+	6.90600e6i	0.983928	+	0.138574i
$369$	1.89465e7			0.377095
$370$	300622.			0.00593492
$371$	7.70594e7			1.50905
$372$	−3.42486e7			−0.665295
$373$			6.67702e7i			1.28664i	0.765599	+	0.643318i	$0.222443\pi$
$373$			6.67702e7i			1.28664i	−0.765599	+	0.643318i	$0.777557\pi$
$374$	2.24598e6			0.0429330
$375$		−	3.24476e7i		−	0.615303i
$376$	−2.95972e6			−0.0556785
$377$	−1.16465e7			−0.217355
$378$		−	795579.i		−	0.0147302i
$379$		−	5.19954e7i		−	0.955097i	−0.878605	−	0.477549i	$-0.841525\pi$
$379$		−	5.19954e7i		−	0.955097i	0.878605	−	0.477549i	$-0.158475\pi$
$380$	−3.11694e7			−0.568039
$381$	−1.12705e7			−0.203783
$382$		−	3.40876e6i		−	0.0611514i
$383$			3.43487e7i			0.611384i	0.952130	+	0.305692i	$0.0988878\pi$
$383$			3.43487e7i			0.611384i	−0.952130	+	0.305692i	$0.901112\pi$
$384$	5.99197e6			0.105822
$385$		−	5.28703e7i		−	0.926467i
$386$	575629.			0.0100088
$387$			2.75258e7i			0.474906i
$388$			3.35554e7i			0.574470i
$389$			7.76864e7i			1.31976i	0.751369	+	0.659882i	$0.229394\pi$
$389$			7.76864e7i			1.31976i	−0.751369	+	0.659882i	$0.770606\pi$
$390$			2.29978e6i			0.0387698i
$391$	1.26995e7	−	9.01711e7i	0.212450	−	1.50847i
$392$	−9.76076e6			−0.162041
$393$	−5.22496e7			−0.860805
$394$	4.12762e6			0.0674856
$395$	−5.22628e7			−0.848011
$396$			1.26371e7i			0.203499i
$397$	−9.93649e7			−1.58804			−0.794020	−	0.607892i	$-0.792015\pi$
$397$	−9.93649e7			−1.58804			−0.794020	+	0.607892i	$0.792015\pi$
$398$		−	1.79643e6i		−	0.0284945i
$399$	−3.80579e7			−0.599138
$400$	1.07670e7			0.168235
$401$			3.38580e7i			0.525083i	0.964921	+	0.262542i	$0.0845607\pi$
$401$			3.38580e7i			0.525083i	−0.964921	+	0.262542i	$0.915439\pi$
$402$		−	1.31714e6i		−	0.0202746i
$403$	1.20883e8			1.84692
$404$	−1.06532e8			−1.61561
$405$		−	6.72731e6i		−	0.101269i
$406$			696124.i			0.0104018i
$407$	−5.83043e6			−0.0864803
$408$		−	5.49762e6i		−	0.0809458i
$409$	4.05687e7			0.592955			0.296477	−	0.955040i	$-0.404188\pi$
$409$	4.05687e7			0.592955			0.296477	+	0.955040i	$0.404188\pi$
$410$			3.27361e6i			0.0474980i
$411$		−	8.51155e6i		−	0.122598i
$412$			1.44675e7i			0.206871i
$413$		−	5.79406e7i		−	0.822494i
$414$	−1.07894e6	−	151956.i	−0.0152054	−	0.00214150i
$415$	1.28810e8			1.80220
$416$	−1.58674e7			−0.220407
$417$	1.68332e7			0.232144
$418$	−1.28558e6			−0.0176024
$419$		−	8.70221e7i		−	1.18301i	−0.806302	−	0.591504i	$-0.798534\pi$
$419$		−	8.70221e7i		−	1.18301i	0.806302	−	0.591504i	$-0.201466\pi$
$420$	−6.46382e7			−0.872452
$421$			6.05889e7i			0.811983i	0.913877	+	0.405991i	$0.133074\pi$
$421$			6.05889e7i			0.811983i	−0.913877	+	0.405991i	$0.866926\pi$
$422$	−3.72865e6			−0.0496152
$423$	−1.52628e7			−0.201657
$424$		−	6.37161e6i		−	0.0835894i
$425$		−	1.97996e7i		−	0.257922i
$426$	370670.			0.00479467
$427$	2.17897e7			0.279877
$428$			4.74119e7i			0.604723i
$429$		−	4.46033e7i		−	0.564931i
$430$	−4.75595e6			−0.0598180
$431$		−	5.02127e7i		−	0.627165i	−0.949561	−	0.313583i	$-0.898471\pi$
$431$		−	5.02127e7i		−	0.627165i	0.949561	−	0.313583i	$-0.101529\pi$
$432$	1.54169e7			0.191226
$433$			1.09950e8i			1.35435i	0.735820	+	0.677177i	$0.236797\pi$
$433$			1.09950e8i			1.35435i	−0.735820	+	0.677177i	$0.763203\pi$
$434$		−	7.22531e6i		−	0.0883867i
$435$			5.88634e6i			0.0715118i
$436$			1.33544e8i			1.61126i
$437$	−7.26910e6	+	5.16132e7i	−0.0871037	+	0.618467i
$438$	308104.			0.00366670
$439$	−3.55005e7			−0.419606			−0.209803	−	0.977744i	$-0.567282\pi$
$439$	−3.55005e7			−0.419606			−0.209803	+	0.977744i	$0.567282\pi$
$440$	−4.37155e6			−0.0513189
$441$	−5.03347e7			−0.586883
$442$			9.69179e6i			0.112237i
$443$	−9.78683e7			−1.12572			−0.562861	−	0.826552i	$-0.690299\pi$
$443$	−9.78683e7			−1.12572			−0.562861	+	0.826552i	$0.690299\pi$
$444$			7.12816e6i			0.0814383i
$445$	−9.44965e7			−1.07235
$446$	4.68546e6			0.0528138
$447$			7.03193e7i			0.787322i
$448$		−	1.47497e8i		−	1.64040i
$449$	−8.49424e7			−0.938394			−0.469197	−	0.883093i	$-0.655457\pi$
$449$	−8.49424e7			−0.938394			−0.469197	+	0.883093i	$0.655457\pi$
$450$	−236912.			−0.00259986
$451$		−	6.34903e7i		−	0.692114i
$452$		−	1.41091e8i		−	1.52786i
$453$	4.80137e7			0.516500
$454$			6.25180e6i			0.0668093i
$455$	2.28144e8			2.42201
$456$			3.14680e6i			0.0331875i
$457$		−	8.16735e7i		−	0.855722i	−0.903844	−	0.427861i	$-0.859267\pi$
$457$		−	8.16735e7i		−	0.855722i	0.903844	−	0.427861i	$-0.140733\pi$
$458$			4.53055e6i			0.0471579i
$459$		−	2.83503e7i		−	0.293170i
$460$	−1.23460e7	+	8.76607e7i	−0.126839	+	0.900599i
$461$	−1.10947e8			−1.13243			−0.566215	−	0.824258i	$-0.691593\pi$
$461$	−1.10947e8			−1.13243			−0.566215	+	0.824258i	$0.691593\pi$
$462$	−2.66600e6			−0.0270355
$463$	1.53700e8			1.54857			0.774287	−	0.632835i	$-0.218109\pi$
$463$	1.53700e8			1.54857			0.774287	+	0.632835i	$0.218109\pi$
$464$	−1.34897e7			−0.135035
$465$		−	6.10963e7i		−	0.607653i
$466$	63106.1			0.000623611
$467$			2.80988e7i			0.275890i	0.990440	+	0.137945i	$0.0440498\pi$
$467$			2.80988e7i			0.275890i	−0.990440	+	0.137945i	$0.955950\pi$
$468$	−5.45311e7			−0.531994
$469$	−1.30663e8			−1.26659
$470$		−	2.63713e6i		−	0.0254002i
$471$			9.23458e7i			0.883801i
$472$	−4.79078e6			−0.0455597
$473$	9.22396e7			0.871634
$474$			2.63536e6i			0.0247460i
$475$			1.13331e7i			0.105747i
$476$	−2.72399e8			−2.52572
$477$		−	3.28573e7i		−	0.302745i
$478$	200117.			0.00183231
$479$			2.05854e8i			1.87307i	0.350578	+	0.936534i	$0.385985\pi$
$479$			2.05854e8i			1.87307i	−0.350578	+	0.936534i	$0.614015\pi$
$480$			8.01969e6i			0.0725160i
$481$		−	2.51593e7i		−	0.226080i
$482$			1.41812e6i			0.0126641i
$483$	−1.50744e7	+	1.07034e8i	−0.133783	+	0.949904i
$484$	−7.07922e7			−0.624380
$485$	−5.98596e7			−0.524697
$486$	−339226.			−0.00295516
$487$	7.83541e7			0.678383			0.339191	−	0.940717i	$-0.389847\pi$
$487$	7.83541e7			0.678383			0.339191	+	0.940717i	$0.389847\pi$
$488$		−	1.80166e6i		−	0.0155029i
$489$	1.42962e7			0.122262
$490$		−	8.69689e6i		−	0.0739223i
$491$	−9.63485e7			−0.813955			−0.406978	−	0.913438i	$-0.633417\pi$
$491$	−9.63485e7			−0.813955			−0.406978	+	0.913438i	$0.633417\pi$
$492$	−7.76219e7			−0.651762
$493$			2.48063e7i			0.207024i
$494$		−	5.54750e6i		−	0.0460168i
$495$	−2.25434e7			−0.185867
$496$	1.40014e8			1.14743
$497$		−	3.67714e7i		−	0.299530i
$498$		−	6.49525e6i		−	0.0525906i
$499$	−3.11543e7			−0.250736			−0.125368	−	0.992110i	$-0.540011\pi$
$499$	−3.11543e7			−0.250736			−0.125368	+	0.992110i	$0.540011\pi$
$500$			1.32934e8i			1.06347i
$501$	−1.18173e8			−0.939738
$502$		−	4.37568e6i		−	0.0345887i
$503$		−	5.43218e7i		−	0.426845i	−0.976960	−	0.213423i	$-0.931539\pi$
$503$		−	5.43218e7i		−	0.426845i	0.976960	−	0.213423i	$-0.0684611\pi$
$504$			6.52573e6i			0.0509727i
$505$		−	1.90043e8i		−	1.47563i
$506$	−509208.	+	3.61556e6i	−0.00393047	+	0.0279077i
$507$	1.17228e8			0.899516
$508$	4.61740e7			0.352214
$509$	2.23459e8			1.69451			0.847255	−	0.531187i	$-0.178254\pi$
$509$	2.23459e8			1.69451			0.847255	+	0.531187i	$0.178254\pi$
$510$	4.89841e6			0.0369270
$511$		−	3.05647e7i		−	0.229064i
$512$	−3.06528e7			−0.228381
$513$			1.62275e7i			0.120199i
$514$	3.77402e6			0.0277917
$515$	−2.58086e7			−0.188948
$516$		−	1.12770e8i		−	0.820816i
$517$			5.11459e7i			0.370117i
$518$	−1.50380e6			−0.0108194
$519$	8.76459e7			0.626945
$520$		−	1.88640e7i		−	0.134160i
$521$			1.32853e7i			0.0939415i	0.998896	+	0.0469707i	$0.0149568\pi$
$521$			1.32853e7i			0.0939415i	−0.998896	+	0.0469707i	$0.985043\pi$
$522$	296820.			0.00208681
$523$		−	9.24267e7i		−	0.646089i	−0.946384	−	0.323044i	$-0.895294\pi$
$523$		−	9.24267e7i		−	0.646089i	0.946384	−	0.323044i	$-0.104706\pi$
$524$	2.14061e8			1.48779
$525$			2.35023e7i			0.162417i
$526$		−	751945.i		−	0.00516689i
$527$		−	2.57473e8i		−	1.75914i
$528$		−	5.16624e7i		−	0.350972i
$529$	1.42277e8	+	4.08871e7i	0.961101	+	0.276197i
$530$	5.67713e6			0.0381330
$531$	−2.47053e7			−0.165008
$532$	1.55919e8			1.03553
$533$	2.73971e8			1.80935
$534$			4.76501e6i			0.0312925i
$535$	−8.45784e7			−0.552329
$536$			1.08038e7i			0.0701588i
$537$	1.02337e7			0.0660859
$538$	1.14225e7			0.0733525
$539$			1.68672e8i			1.07715i
$540$			2.75611e7i			0.175031i
$541$	5.01668e7			0.316829			0.158414	−	0.987373i	$-0.449362\pi$
$541$	5.01668e7			0.316829			0.158414	+	0.987373i	$0.449362\pi$
$542$	1.67888e6			0.0105444
$543$		−	1.43433e8i		−	0.895883i
$544$			3.37967e7i			0.209931i
$545$	−2.38231e8			−1.47166
$546$		−	1.15042e7i		−	0.0706773i
$547$	8.48201e7			0.518247			0.259124	−	0.965844i	$-0.416566\pi$
$547$	8.48201e7			0.518247			0.259124	+	0.965844i	$0.416566\pi$
$548$			3.48709e7i			0.211895i
$549$		−	9.29089e6i		−	0.0561488i
$550$			793897.i			0.00477173i
$551$		−	1.41989e7i		−	0.0848791i
$552$	8.85003e6	+	1.24642e6i	0.0526171	+	0.00741049i
$553$	2.61435e8			1.54592
$554$	4.35872e6			0.0256348
$555$	−1.27160e7			−0.0743824
$556$	−6.89637e7			−0.401233
$557$			1.80275e8i			1.04321i	0.853189	+	0.521603i	$0.174666\pi$
$557$			1.80275e8i			1.04321i	−0.853189	+	0.521603i	$0.825334\pi$
$558$	−3.08080e6			−0.0177321
$559$			3.98029e8i			2.27866i
$560$	2.64251e8			1.50471
$561$	−9.50026e7			−0.538080
$562$			6.27666e6i			0.0353606i
$563$		−	3.03781e8i		−	1.70230i	−0.524925	−	0.851149i	$-0.675907\pi$
$563$		−	3.03781e8i		−	1.70230i	0.524925	−	0.851149i	$-0.324093\pi$
$564$	6.25300e7			0.348539
$565$	2.51693e8			1.39549
$566$			1.34586e7i			0.0742250i
$567$			3.36521e7i			0.184613i
$568$	−3.04042e6			−0.0165916
$569$		−	1.02127e8i		−	0.554376i	−0.960816	−	0.277188i	$-0.910598\pi$
$569$		−	1.02127e8i		−	0.554376i	0.960816	−	0.277188i	$-0.0894025\pi$
$570$	−2.80381e6			−0.0151399
$571$		−	2.29011e7i		−	0.123012i	−0.998107	−	0.0615059i	$-0.980410\pi$
$571$		−	2.29011e7i		−	0.123012i	0.998107	−	0.0615059i	$-0.0195903\pi$
$572$			1.82735e8i			0.976413i
$573$			1.44187e8i			0.766411i
$574$		−	1.63756e7i		−	0.0865888i
$575$	3.18732e7	+	4.48895e6i	0.167657	+	0.0236125i
$576$	−6.28913e7			−0.329097
$577$	−2.38699e8			−1.24257			−0.621287	−	0.783583i	$-0.713390\pi$
$577$	−2.38699e8			−1.24257			−0.621287	+	0.783583i	$0.713390\pi$
$578$	1.17475e7			0.0608362
$579$	−2.43485e7			−0.125440
$580$		−	2.41157e7i		−	0.123599i
$581$	−6.44345e8			−3.28541
$582$			3.01844e6i			0.0153113i
$583$	−1.10106e8			−0.555653
$584$	−2.52722e6			−0.0126883
$585$		−	9.72784e7i		−	0.485902i
$586$			6.84325e6i			0.0340071i
$587$	1.94478e8			0.961514			0.480757	−	0.876854i	$-0.340362\pi$
$587$	1.94478e8			0.961514			0.480757	+	0.876854i	$0.340362\pi$
$588$	2.06215e8			1.01435
$589$			1.47375e8i			0.721239i
$590$		−	4.26861e6i		−	0.0207841i
$591$	−1.74594e8			−0.845798
$592$		−	2.91411e7i		−	0.140456i
$593$	1.04386e8			0.500586			0.250293	−	0.968170i	$-0.419473\pi$
$593$	1.04386e8			0.500586			0.250293	+	0.968170i	$0.419473\pi$
$594$			1.13675e6i			0.00542385i
$595$		−	4.85934e8i		−	2.30689i
$596$		−	2.88091e8i		−	1.36079i
$597$			7.59869e7i			0.357121i
$598$	−1.56018e7	−	2.19732e6i	−0.0729575	−	0.0102752i
$599$	−8.10305e7			−0.377023			−0.188512	−	0.982071i	$-0.560366\pi$
$599$	−8.10305e7			−0.377023			−0.188512	+	0.982071i	$0.560366\pi$
$600$	1.94327e6			0.00899662
$601$	−2.22644e7			−0.102562			−0.0512812	−	0.998684i	$-0.516330\pi$
$601$	−2.22644e7			−0.102562			−0.0512812	+	0.998684i	$0.516330\pi$
$602$	2.37907e7			0.109048
$603$			5.57134e7i			0.254102i
$604$	−1.96707e8			−0.892707
$605$		−	1.26286e8i		−	0.570283i
$606$	−9.58296e6			−0.0430608
$607$	−3.25433e8			−1.45511			−0.727554	−	0.686050i	$-0.759343\pi$
$607$	−3.25433e8			−1.45511			−0.727554	+	0.686050i	$0.759343\pi$
$608$		−	1.93450e7i		−	0.0860710i
$609$		−	2.94453e7i		−	0.130366i
$610$	1.60529e6			0.00707236
$611$	−2.20703e8			−0.967576
$612$			1.16148e8i			0.506709i
$613$			5.64281e7i			0.244971i	0.992470	+	0.122485i	$0.0390864\pi$
$613$			5.64281e7i			0.244971i	−0.992470	+	0.122485i	$0.960914\pi$
$614$	1.24991e7			0.0539975
$615$		−	1.38470e8i		−	0.595293i
$616$	2.18678e7			0.0935543
$617$			1.30168e8i			0.554176i	0.960845	+	0.277088i	$0.0893694\pi$
$617$			1.30168e8i			0.554176i	−0.960845	+	0.277088i	$0.910631\pi$
$618$			1.30140e6i			0.00551374i
$619$			9.93172e7i			0.418748i	0.977836	+	0.209374i	$0.0671426\pi$
$619$			9.93172e7i			0.418748i	−0.977836	+	0.209374i	$0.932857\pi$
$620$			2.50305e8i			1.05025i
$621$	4.56381e7	+	6.42758e6i	0.190569	+	0.0268394i
$622$	1.44262e7			0.0599488
$623$	4.72701e8			1.95489
$624$	2.22932e8			0.917526
$625$	−1.95806e8			−0.802022
$626$		−	1.48683e7i		−	0.0606092i
$627$	5.43787e7			0.220611
$628$		−	3.78331e8i		−	1.52754i
$629$	−5.35878e7			−0.215335
$630$	−5.81445e6			−0.0232534
$631$			2.00601e8i			0.798444i	0.916854	+	0.399222i	$0.130720\pi$
$631$			2.00601e8i			0.798444i	−0.916854	+	0.399222i	$0.869280\pi$
$632$		−	2.16166e7i		−	0.0856318i
$633$	1.57718e8			0.621828
$634$	−1.32543e7			−0.0520103
$635$			8.23701e7i			0.321698i
$636$			1.34613e8i			0.523257i
$637$	−7.27850e8			−2.81594
$638$		−	994651.i		−	0.00383009i
$639$	−1.56789e7			−0.0600916
$640$		−	4.37921e7i		−	0.167054i
$641$			4.64796e8i			1.76477i	0.470528	+	0.882385i	$0.344063\pi$
$641$			4.64796e8i			1.76477i	−0.470528	+	0.882385i	$0.655937\pi$
$642$			4.26489e6i			0.0161177i
$643$			8.47536e7i			0.318805i	0.987214	+	0.159402i	$0.0509568\pi$
$643$			8.47536e7i			0.318805i	−0.987214	+	0.159402i	$0.949043\pi$
$644$	6.17583e7	−	4.38506e8i	0.231227	−	1.64179i
$645$	2.01172e8			0.749700
$646$	−1.18159e7			−0.0438296
$647$	−1.91959e8			−0.708752			−0.354376	−	0.935103i	$-0.615307\pi$
$647$	−1.91959e8			−0.708752			−0.354376	+	0.935103i	$0.615307\pi$
$648$	2.78250e6			0.0102261
$649$			8.27879e7i			0.302854i
$650$	−3.42580e6			−0.0124745
$651$			3.05623e8i			1.10775i
$652$	−5.85698e7			−0.211315
$653$	−1.15990e8			−0.416564			−0.208282	−	0.978069i	$-0.566787\pi$
$653$	−1.15990e8			−0.416564			−0.208282	+	0.978069i	$0.566787\pi$
$654$			1.20128e7i			0.0429450i
$655$			3.81864e8i			1.35889i
$656$	3.17331e8			1.12409
$657$	−1.30325e7			−0.0459548
$658$			1.31917e7i			0.0463046i
$659$		−	1.84207e8i		−	0.643649i	−0.946799	−	0.321824i	$-0.895704\pi$
$659$		−	1.84207e8i		−	0.643649i	0.946799	−	0.321824i	$-0.104296\pi$
$660$	9.23576e7			0.321248
$661$			8.44401e6i			0.0292378i	0.999893	+	0.0146189i	$0.00465350\pi$
$661$			8.44401e6i			0.0292378i	−0.999893	+	0.0146189i	$0.995346\pi$
$662$	1.03291e7			0.0356030
$663$		−	4.09952e8i		−	1.40667i
$664$			5.32773e7i			0.181986i
$665$			2.78145e8i			0.945815i
$666$			641206.i			0.00217057i
$667$	−3.99330e7	−	5.62408e6i	−0.134572	−	0.0189528i
$668$	4.84143e8			1.62422
$669$	−1.98190e8			−0.661916
$670$	−9.62624e6			−0.0320061
$671$	−3.11339e7			−0.103054
$672$		−	4.01170e7i		−	0.132197i
$673$	−4.84325e8			−1.58888			−0.794441	−	0.607341i	$-0.792236\pi$
$673$	−4.84325e8			−1.58888			−0.794441	+	0.607341i	$0.792236\pi$
$674$		−	1.12843e7i		−	0.0368549i
$675$	1.00211e7			0.0325841
$676$	−4.80272e8			−1.55470
$677$		−	2.68560e8i		−	0.865517i	−0.901510	−	0.432759i	$-0.857540\pi$
$677$		−	2.68560e8i		−	0.865517i	0.901510	−	0.432759i	$-0.142460\pi$
$678$		−	1.26917e7i		−	0.0407222i
$679$	2.99436e8			0.956522
$680$	−4.01792e7			−0.127783
$681$		−	2.64444e8i		−	0.837322i
$682$			1.03238e7i			0.0325452i
$683$	−4.25685e8			−1.33606			−0.668030	−	0.744134i	$-0.732862\pi$
$683$	−4.25685e8			−1.33606			−0.668030	+	0.744134i	$0.732862\pi$
$684$		−	6.64823e7i		−	0.207748i
$685$	−6.22063e7			−0.193536
$686$			1.87951e7i			0.0582201i
$687$		−	1.91637e8i		−	0.591030i
$688$			4.61023e8i			1.41565i
$689$		−	4.75124e8i		−	1.45261i
$690$	−1.11057e6	+	7.88542e6i	−0.00338063	+	0.0240037i
$691$	1.62850e8			0.493577			0.246788	−	0.969069i	$-0.420625\pi$
$691$	1.62850e8			0.493577			0.246788	+	0.969069i	$0.420625\pi$
$692$	−3.59076e8			−1.08360
$693$	1.12769e8			0.338836
$694$	−2.59542e7			−0.0776478
$695$		−	1.23025e8i		−	0.366470i
$696$	−2.43467e6			−0.00722124
$697$		−	5.83543e8i		−	1.72335i
$698$	2.46181e7			0.0723916
$699$	−2.66932e6			−0.00781572
$700$		−	9.62862e7i		−	0.280718i
$701$			2.15840e7i			0.0626581i	0.999509	+	0.0313291i	$0.00997398\pi$
$701$			2.15840e7i			0.0626581i	−0.999509	+	0.0313291i	$0.990026\pi$
$702$	−4.90529e6			−0.0141792
$703$	3.06732e7			0.0882864
$704$			2.10750e8i			0.604018i
$705$			1.11548e8i			0.318341i
$706$	2.89509e6			0.00822711
$707$			9.50653e8i			2.69007i
$708$	1.01215e8			0.285197
$709$		−	4.64513e8i		−	1.30334i	−0.758501	−	0.651672i	$-0.774068\pi$
$709$		−	4.64513e8i		−	1.30334i	0.758501	−	0.651672i	$-0.225932\pi$
$710$		−	2.70903e6i		−	0.00756900i
$711$		−	1.11473e8i		−	0.310142i
$712$		−	3.90850e7i		−	0.108285i
$713$	4.14478e8	+	5.83742e7i	1.14349	+	0.161047i
$714$	−2.45034e7			−0.0673180
$715$	−3.25982e8			−0.891816
$716$	−4.19262e7			−0.114221
$717$	−8.46471e6			−0.0229644
$718$			3.27434e6i			0.00884608i
$719$	−3.55461e8			−0.956324			−0.478162	−	0.878272i	$-0.658697\pi$
$719$	−3.55461e8			−0.956324			−0.478162	+	0.878272i	$0.658697\pi$
$720$		−	1.12674e8i		−	0.301874i
$721$	1.29102e8			0.344452
$722$	−1.05746e7			−0.0280964
$723$		−	5.99851e7i		−	0.158719i
$724$			5.87631e8i			1.54842i
$725$	−8.76840e6			−0.0230094
$726$	−6.36803e6			−0.0166416
$727$		−	9.14852e7i		−	0.238094i	−0.992889	−	0.119047i	$-0.962016\pi$
$727$		−	9.14852e7i		−	0.238094i	0.992889	−	0.119047i	$-0.0379839\pi$
$728$			9.43634e7i			0.244573i
$729$	1.43489e7			0.0370370
$730$		−	2.25177e6i		−	0.00578835i
$731$	8.47780e8			2.17036
$732$			3.80637e7i			0.0970461i
$733$			5.88821e8i			1.49510i	0.664203	+	0.747552i	$0.268771\pi$
$733$			5.88821e8i			1.49510i	−0.664203	+	0.747552i	$0.731229\pi$
$734$		−	3.10698e7i		−	0.0785688i
$735$			3.67869e8i			0.926469i
$736$	−5.44056e7	−	7.66238e6i	−0.136462	−	0.0192190i
$737$	1.86697e8			0.466374
$738$	−6.98239e6			−0.0173714
$739$	5.03241e8			1.24693			0.623466	−	0.781850i	$-0.285724\pi$
$739$	5.03241e8			1.24693			0.623466	+	0.781850i	$0.285724\pi$
$740$	5.20959e7			0.128561
$741$			2.34653e8i			0.576729i
$742$	−2.83988e7			−0.0695165
$743$		−	5.26147e8i		−	1.28275i	−0.767230	−	0.641373i	$-0.778365\pi$
$743$		−	5.26147e8i		−	1.28275i	0.767230	−	0.641373i	$-0.221635\pi$
$744$	2.52702e7			0.0613606
$745$	5.13926e8			1.24289
$746$		−	2.46069e7i		−	0.0592707i
$747$			2.74742e8i			0.659118i
$748$	3.89215e8			0.930004
$749$	4.23087e8			1.00690
$750$			1.19579e7i			0.0283448i
$751$		−	2.74856e8i		−	0.648911i	−0.945901	−	0.324456i	$-0.894819\pi$
$751$		−	2.74856e8i		−	0.648911i	0.945901	−	0.324456i	$-0.105181\pi$
$752$	−2.55632e8			−0.601122
$753$			1.85086e8i			0.433500i
$754$	4.29208e6			0.0100128
$755$		−	3.50906e8i		−	0.815362i
$756$		−	1.37869e8i		−	0.319081i
$757$		−	2.98024e8i		−	0.687010i	−0.939151	−	0.343505i	$-0.888386\pi$
$757$		−	2.98024e8i		−	0.687010i	0.939151	−	0.343505i	$-0.111614\pi$
$758$			1.91619e7i			0.0439979i
$759$	2.15390e7	−	1.52934e8i	0.0492606	−	0.349768i
$760$	2.29982e7			0.0523906
$761$	−1.44743e8			−0.328431			−0.164216	−	0.986424i	$-0.552509\pi$
$761$	−1.44743e8			−0.328431			−0.164216	+	0.986424i	$0.552509\pi$
$762$	4.15353e6			0.00938756
$763$	1.19170e9			2.68284
$764$		−	5.90718e8i		−	1.32465i
$765$	−2.07197e8			−0.462807
$766$		−	1.26586e7i		−	0.0281642i
$767$	−3.57244e8			−0.791732
$768$	2.55999e8			0.565137
$769$		−	4.15230e8i		−	0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$769$		−	4.15230e8i		−	0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$770$			1.94844e7i			0.0426790i
$771$	−1.59637e8			−0.348313
$772$	9.97530e7			0.216807
$773$			3.40961e7i			0.0738187i	0.999319	+	0.0369094i	$0.0117513\pi$
$773$			3.40961e7i			0.0738187i	−0.999319	+	0.0369094i	$0.988249\pi$
$774$		−	1.01441e7i		−	0.0218772i
$775$	9.10101e7			0.195517
$776$		−	2.47587e7i		−	0.0529837i
$777$	6.36092e7			0.135599
$778$		−	2.86299e7i		−	0.0607968i
$779$			3.34015e8i			0.706568i
$780$			3.98539e8i			0.839821i
$781$			5.25404e7i			0.110291i
$782$	−4.68016e6	+	3.32308e7i	−0.00978680	+	0.0694898i
$783$	−1.25552e7			−0.0261540
$784$	−8.43042e8			−1.74945
$785$	6.74906e8			1.39519
$786$	1.92556e7			0.0396542
$787$		−	6.07623e8i		−	1.24655i	−0.782003	−	0.623275i	$-0.785802\pi$
$787$		−	6.07623e8i		−	1.24655i	0.782003	−	0.623275i	$-0.214198\pi$
$788$	7.15292e8			1.46186
$789$			3.18065e7i			0.0647566i
$790$	1.92605e7			0.0390648
$791$	−1.25905e9			−2.54398
$792$		−	9.32422e6i		−	0.0187688i
$793$		−	1.34348e8i		−	0.269409i
$794$	3.66190e7			0.0731552
$795$	−2.40136e8			−0.477922
$796$		−	3.11310e8i		−	0.617240i
$797$			2.55908e8i			0.505485i	0.967534	+	0.252743i	$0.0813326\pi$
$797$			2.55908e8i			0.505485i	−0.967534	+	0.252743i	$0.918667\pi$
$798$	1.40255e7			0.0276001
$799$			4.70085e8i			0.921587i
$800$	−1.19463e7			−0.0233326
$801$		−	2.01555e8i		−	0.392189i
$802$		−	1.24777e7i		−	0.0241887i
$803$			4.36721e7i			0.0843446i
$804$		−	2.28252e8i		−	0.439183i
$805$	7.82253e8	+	1.10171e8i	1.49955	+	0.211193i
$806$	−4.45490e7			−0.0850810
$807$	−4.83160e8			−0.919327
$808$	7.86041e7			0.149009
$809$	−1.94709e8			−0.367740			−0.183870	−	0.982951i	$-0.558862\pi$
$809$	−1.94709e8			−0.367740			−0.183870	+	0.982951i	$0.558862\pi$
$810$			2.47922e6i			0.00466510i
$811$	4.20154e8			0.787674			0.393837	−	0.919180i	$-0.371148\pi$
$811$	4.20154e8			0.787674			0.393837	+	0.919180i	$0.371148\pi$
$812$			1.20634e8i			0.225321i
$813$	−7.10148e7			−0.132153
$814$	2.14869e6			0.00398383
$815$		−	1.04483e8i		−	0.193007i
$816$		−	4.74832e8i		−	0.873916i
$817$	−4.85263e8			−0.889837
$818$	−1.49508e7			−0.0273153
$819$			4.86617e8i			0.885799i
$820$			5.67297e8i			1.02889i
$821$	−5.01126e8			−0.905560			−0.452780	−	0.891622i	$-0.649568\pi$
$821$	−5.01126e8			−0.905560			−0.452780	+	0.891622i	$0.649568\pi$
$822$			3.13677e6i			0.00564764i
$823$	8.26839e8			1.48327			0.741637	−	0.670801i	$-0.234050\pi$
$823$	8.26839e8			1.48327			0.741637	+	0.670801i	$0.234050\pi$
$824$		−	1.06748e7i		−	0.0190799i
$825$		−	3.35810e7i		−	0.0598042i
$826$			2.13529e7i			0.0378893i
$827$			3.01216e8i			0.532551i	0.963897	+	0.266276i	$0.0857931\pi$
$827$			3.01216e8i			0.532551i	−0.963897	+	0.266276i	$0.914207\pi$
$828$	−1.86974e8	−	2.63331e7i	−0.329375	−	0.0463886i
$829$	4.68607e8			0.822517			0.411259	−	0.911519i	$-0.365089\pi$
$829$	4.68607e8			0.822517			0.411259	+	0.911519i	$0.365089\pi$
$830$	−4.74703e7			−0.0830210
$831$	−1.84369e8			−0.321281
$832$	−9.09422e8			−1.57905
$833$			1.55028e9i			2.68210i
$834$	−6.20355e6			−0.0106941
$835$			8.63666e8i			1.48350i
$836$	−2.22784e8			−0.381298
$837$	1.30314e8			0.222237
$838$			3.20704e7i			0.0544969i
$839$		−	1.01082e9i		−	1.71154i	−0.517353	−	0.855772i	$-0.673082\pi$
$839$		−	1.01082e9i		−	1.71154i	0.517353	−	0.855772i	$-0.326918\pi$
$840$	4.76930e7			0.0804668
$841$	−5.83838e8			−0.981531
$842$		−	2.23289e7i		−	0.0374051i
$843$		−	2.65496e8i		−	0.443175i
$844$	−6.46153e8			−1.07475
$845$		−	8.56759e8i		−	1.42000i
$846$	5.62481e6			0.00928960
$847$			6.31724e8i			1.03963i
$848$		−	5.50319e8i		−	0.902457i
$849$		−	5.69284e8i		−	0.930263i
$850$			7.29676e6i			0.0118816i
$851$	1.21494e7	−	8.62652e7i	0.0197136	−	0.139974i
$852$	6.42349e7			0.103861
$853$	1.99330e8			0.321163			0.160581	−	0.987023i	$-0.448663\pi$
$853$	1.99330e8			0.321163			0.160581	+	0.987023i	$0.448663\pi$
$854$	−8.03017e6			−0.0128929
$855$	1.18598e8			0.189749
$856$		−	3.49827e7i		−	0.0557740i
$857$	6.74308e8			1.07131			0.535656	−	0.844437i	$-0.320065\pi$
$857$	6.74308e8			1.07131			0.535656	+	0.844437i	$0.320065\pi$
$858$			1.64377e7i			0.0260243i
$859$	−2.08702e8			−0.329266			−0.164633	−	0.986355i	$-0.552644\pi$
$859$	−2.08702e8			−0.329266			−0.164633	+	0.986355i	$0.552644\pi$
$860$	−8.24177e8			−1.29576
$861$			6.92670e8i			1.08522i
$862$			1.85049e7i			0.0288912i
$863$	−8.99089e8			−1.39885			−0.699424	−	0.714707i	$-0.746560\pi$
$863$	−8.99089e8			−1.39885			−0.699424	+	0.714707i	$0.746560\pi$
$864$	−1.71055e7			−0.0265212
$865$		−	6.40557e8i		−	0.989713i
$866$		−	4.05200e7i		−	0.0623902i
$867$	−4.96907e8			−0.762461
$868$		−	1.25210e9i		−	1.91461i
$869$	−3.73548e8			−0.569230
$870$		−	2.16930e6i		−	0.00329429i
$871$			8.05628e8i			1.21921i
$872$		−	9.85352e7i		−	0.148608i
$873$		−	1.27677e8i		−	0.191897i
$874$	2.67889e6	−	1.90211e7i	0.00401255	−	0.0284905i
$875$	1.18626e9			1.77074
$876$	5.33926e7			0.0794271
$877$	3.34022e8			0.495195			0.247598	−	0.968863i	$-0.420359\pi$
$877$	3.34022e8			0.495195			0.247598	+	0.968863i	$0.420359\pi$
$878$	1.30831e7			0.0193297
$879$		−	2.89462e8i		−	0.426211i
$880$	−3.77573e8			−0.554055
$881$			3.45526e8i			0.505304i	0.967557	+	0.252652i	$0.0813028\pi$
$881$			3.45526e8i			0.505304i	−0.967557	+	0.252652i	$0.918697\pi$
$882$	1.85499e7			0.0270356
$883$	7.01907e7			0.101952			0.0509762	−	0.998700i	$-0.483767\pi$
$883$	7.01907e7			0.101952			0.0509762	+	0.998700i	$0.483767\pi$
$884$			1.67953e9i			2.43125i
$885$			1.80558e8i			0.260487i
$886$	3.60675e7			0.0518579
$887$	−5.64947e8			−0.809537			−0.404769	−	0.914419i	$-0.632648\pi$
$887$	−5.64947e8			−0.809537			−0.404769	+	0.914419i	$0.632648\pi$
$888$		−	5.25949e6i		−	0.00751111i
$889$		−	4.12041e8i		−	0.586455i
$890$	3.48249e7			0.0493992
$891$		−	4.80835e7i		−	0.0679771i
$892$	8.11962e8			1.14404
$893$		−	2.69073e8i		−	0.377847i
$894$		−	2.59149e7i		−	0.0362691i
$895$		−	7.47924e7i		−	0.104325i
$896$			2.19062e8i			0.304539i
$897$	6.59937e8	+	9.29442e7i	0.914377	+	0.128779i
$898$	3.13039e7			0.0432284
$899$	−1.14024e8			−0.156934
$900$	−4.10554e7			−0.0563175
$901$	−1.01199e9			−1.38357
$902$			2.33981e7i			0.0318832i
$903$	−1.00632e9			−1.36670
$904$			1.04104e8i			0.140916i
$905$	−1.04828e9			−1.41426
$906$	−1.76945e7			−0.0237933
$907$		−	6.29881e8i		−	0.844184i	−0.906553	−	0.422092i	$-0.861296\pi$
$907$		−	6.29881e8i		−	0.844184i	0.906553	−	0.422092i	$-0.138704\pi$
$908$			1.08340e9i			1.44721i
$909$	4.05349e8			0.539681
$910$	−8.40782e7			−0.111573
$911$			4.77257e8i			0.631244i	0.948885	+	0.315622i	$0.102213\pi$
$911$			4.77257e8i			0.631244i	−0.948885	+	0.315622i	$0.897787\pi$
$912$			2.71790e8i			0.358302i
$913$	9.20666e8			1.20973
$914$			3.00992e7i			0.0394200i
$915$	−6.79021e7			−0.0886380
$916$			7.85116e8i			1.02152i
$917$		−	1.91020e9i		−	2.47726i
$918$			1.04480e7i			0.0135053i
$919$		−	1.23534e9i		−	1.59163i	−0.605541	−	0.795814i	$-0.707043\pi$
$919$		−	1.23534e9i		−	1.59163i	0.605541	−	0.795814i	$-0.292957\pi$
$920$	9.10941e6	−	6.46801e7i	0.0116984	−	0.0830629i
$921$	−5.28698e8			−0.676751
$922$	4.08872e7			0.0521669
$923$	−2.26721e8			−0.288328
$924$	−4.62002e8			−0.585636
$925$		−	1.89419e7i		−	0.0239331i
$926$	−5.66434e7			−0.0713372
$927$		−	5.50479e7i		−	0.0691038i
$928$	1.49671e7			0.0187281
$929$	5.67410e8			0.707701			0.353850	−	0.935302i	$-0.384872\pi$
$929$	5.67410e8			0.707701			0.353850	+	0.935302i	$0.384872\pi$
$930$			2.25159e7i			0.0279924i
$931$		−	8.87367e8i		−	1.09965i
$932$	1.09359e7			0.0135085
$933$	−6.10212e8			−0.751339
$934$		−	1.03553e7i		−	0.0127093i
$935$			6.94322e8i			0.849428i
$936$	4.02356e7			0.0490662
$937$			1.36840e9i			1.66339i	0.555237	+	0.831693i	$0.312628\pi$
$937$			1.36840e9i			1.66339i	−0.555237	+	0.831693i	$0.687372\pi$
$938$	4.81534e7			0.0583470
$939$			6.28913e8i			0.759616i
$940$		−	4.56998e8i		−	0.550213i
$941$		−	6.47091e8i		−	0.776599i	−0.921533	−	0.388299i	$-0.873063\pi$
$941$		−	6.47091e8i		−	0.776599i	0.921533	−	0.388299i	$-0.126937\pi$
$942$		−	3.40323e7i		−	0.0407135i
$943$	9.39382e8	+	1.32301e8i	1.12023	+	0.157771i
$944$	−4.13782e8			−0.491876
$945$	2.45945e8			0.291436
$946$	−3.39932e7			−0.0401530
$947$	−1.95225e8			−0.229871			−0.114936	−	0.993373i	$-0.536666\pi$
$947$	−1.95225e8			−0.229871			−0.114936	+	0.993373i	$0.536666\pi$
$948$			4.56693e8i			0.536042i
$949$	−1.88452e8			−0.220497
$950$		−	4.17661e6i		−	0.00487139i
$951$	5.60642e8			0.651845
$952$	2.00989e8			0.232949
$953$		−	5.88380e8i		−	0.679797i	−0.940462	−	0.339898i	$-0.889607\pi$
$953$		−	5.88380e8i		−	0.679797i	0.940462	−	0.339898i	$-0.110393\pi$
$954$			1.21089e7i			0.0139464i
$955$	1.05378e9			1.20988
$956$	3.46790e7			0.0396911
$957$			4.20726e7i			0.0480025i
$958$		−	7.58637e7i		−	0.0862854i
$959$	3.11175e8			0.352817
$960$			4.59639e8i			0.519521i
$961$	2.95988e8			0.333506
$962$			9.27197e6i			0.0104147i
$963$		−	1.80400e8i		−	0.202003i
$964$			2.45752e8i			0.274326i
$965$			1.77950e8i			0.198023i
$966$	5.55540e6	−	3.94453e7i	0.00616288	−	0.0437586i
$967$	−1.85929e8			−0.205621			−0.102811	−	0.994701i	$-0.532784\pi$
$967$	−1.85929e8			−0.205621			−0.102811	+	0.994701i	$0.532784\pi$
$968$	5.22337e7			0.0575870
$969$	4.99798e8			0.549317
$970$	2.20601e7			0.0241709
$971$			2.12025e8i			0.231595i	0.993273	+	0.115798i	$0.0369424\pi$
$971$			2.12025e8i			0.231595i	−0.993273	+	0.115798i	$0.963058\pi$
$972$	−5.87859e7			−0.0640139
$973$			6.15408e8i			0.668074i
$974$	−2.88759e7			−0.0312506
$975$	1.44908e8			0.156343
$976$		−	1.55611e8i		−	0.167375i
$977$			4.00645e8i			0.429612i	0.976657	+	0.214806i	$0.0689119\pi$
$977$			4.00645e8i			0.429612i	−0.976657	+	0.214806i	$0.931088\pi$
$978$	−5.26858e6			−0.00563218
$979$	−6.75414e8			−0.719817
$980$		−	1.50712e9i		−	1.60129i
$981$		−	5.08130e8i		−	0.538230i
$982$	3.55074e7			0.0374960
$983$		−	1.40422e9i		−	1.47834i	−0.673517	−	0.739172i	$-0.735217\pi$
$983$		−	1.40422e9i		−	1.47834i	0.673517	−	0.739172i	$-0.264783\pi$
$984$	5.72730e7			0.0601125
$985$			1.27601e9i			1.33520i
$986$		−	9.14189e6i		−	0.00953686i
$987$		−	5.57995e8i		−	0.580335i
$988$		−	9.61348e8i		−	0.996804i
$989$	−1.92208e8	+	1.36475e9i	−0.198693	+	1.41079i
$990$	8.30793e6			0.00856223
$991$	−1.62423e8			−0.166888			−0.0834441	−	0.996512i	$-0.526592\pi$
$991$	−1.62423e8			−0.166888			−0.0834441	+	0.996512i	$0.526592\pi$
$992$	−1.55349e8			−0.159138
$993$	−4.36908e8			−0.446213
$994$			1.35514e7i			0.0137983i
$995$	5.55347e8			0.563761
$996$		−	1.12559e9i		−	1.13920i
$997$	1.73006e9			1.74572			0.872861	−	0.487970i	$-0.162262\pi$
$997$	1.73006e9			1.74572			0.872861	+	0.487970i	$0.162262\pi$
$998$	1.14813e7			0.0115505
$999$		−	2.71223e7i		−	0.0272038i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.13	✓	24
3.2	odd	2		207.7.d.e.91.11		24
23.22	odd	2	inner	69.7.d.a.22.14	yes	24
69.68	even	2		207.7.d.e.91.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.13	✓	24	1.1	even	1	trivial
69.7.d.a.22.14	yes	24	23.22	odd	2	inner
207.7.d.e.91.11		24	3.2	odd	2
207.7.d.e.91.12		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)

\(f(q)\)	\(=\)	\(q-0.368531 q^{2} +15.5885 q^{3} -63.8642 q^{4} -113.928i q^{5} -5.74483 q^{6} +569.901i q^{7} +47.1219 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-0.368531 q^{2} +15.5885 q^{3} -63.8642 q^{4} -113.928i q^{5} -5.74483 q^{6} +569.901i q^{7} +47.1219 q^{8} +243.000 q^{9} +41.9859i q^{10} -814.298i q^{11} -995.544 q^{12} +3513.83 q^{13} -210.026i q^{14} -1775.96i q^{15} +4069.94 q^{16} -7484.26i q^{17} -89.5530 q^{18} +4283.93i q^{19} +7275.90i q^{20} +8883.88i q^{21} +300.094i q^{22} +(12048.1 + 1696.83i) q^{23} +734.558 q^{24} +2645.50 q^{25} -1294.96 q^{26} +3788.00 q^{27} -36396.3i q^{28} -3314.46 q^{29} +654.495i q^{30} +34401.9 q^{31} -4515.70 q^{32} -12693.6i q^{33} +2758.18i q^{34} +64927.5 q^{35} -15519.0 q^{36} -7160.07i q^{37} -1578.76i q^{38} +54775.2 q^{39} -5368.49i q^{40} +77969.3 q^{41} -3273.99i q^{42} +113275. i q^{43} +52004.5i q^{44} -27684.4i q^{45} +(-4440.10 - 625.334i) q^{46} -62809.9 q^{47} +63444.1 q^{48} -207138. q^{49} -974.947 q^{50} -116668. i q^{51} -224408. q^{52} -135215. i q^{53} -1395.99 q^{54} -92771.0 q^{55} +26854.8i q^{56} +66779.9i q^{57} +1221.48 q^{58} -101668. q^{59} +113420. i q^{60} -38234.1i q^{61} -12678.2 q^{62} +138486. i q^{63} -258812. q^{64} -400322. i q^{65} +4678.00i q^{66} +229273. i q^{67} +477976. i q^{68} +(187811. + 26451.0i) q^{69} -23927.8 q^{70} -64522.4 q^{71} +11450.6 q^{72} -53631.6 q^{73} +2638.71i q^{74} +41239.2 q^{75} -273590. i q^{76} +464069. q^{77} -20186.4 q^{78} -458737. i q^{79} -463679. i q^{80} +59049.0 q^{81} -28734.1 q^{82} +1.13063e6i q^{83} -567362. i q^{84} -852664. q^{85} -41745.4i q^{86} -51667.4 q^{87} -38371.3i q^{88} -829443. i q^{89} +10202.6i q^{90} +2.00254e6i q^{91} +(-769442. - 108367. i) q^{92} +536273. q^{93} +23147.4 q^{94} +488058. q^{95} -70392.8 q^{96} -525418. i q^{97} +76337.0 q^{98} -197874. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 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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