LMFDB - Embedded newform 69.7.d.a.22.15

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			22.15
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.88321 q^{2} -15.5885 q^{3} -55.6871 q^{4} -133.948i q^{5} -44.9448 q^{6} +59.3150i q^{7} -345.083 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+2.88321 q^{2} -15.5885 q^{3} -55.6871 q^{4} -133.948i q^{5} -44.9448 q^{6} +59.3150i q^{7} -345.083 q^{8} +243.000 q^{9} -386.201i q^{10} +578.563i q^{11} +868.076 q^{12} +943.370 q^{13} +171.018i q^{14} +2088.05i q^{15} +2569.03 q^{16} +5432.92i q^{17} +700.620 q^{18} +3909.12i q^{19} +7459.19i q^{20} -924.630i q^{21} +1668.12i q^{22} +(-4480.54 + 11312.0i) q^{23} +5379.31 q^{24} -2317.15 q^{25} +2719.93 q^{26} -3788.00 q^{27} -3303.08i q^{28} +11255.2 q^{29} +6020.28i q^{30} +3515.19 q^{31} +29492.4 q^{32} -9018.91i q^{33} +15664.2i q^{34} +7945.15 q^{35} -13532.0 q^{36} +8099.73i q^{37} +11270.8i q^{38} -14705.7 q^{39} +46223.3i q^{40} -13414.0 q^{41} -2665.90i q^{42} +66345.2i q^{43} -32218.5i q^{44} -32549.4i q^{45} +(-12918.3 + 32614.8i) q^{46} -25836.4 q^{47} -40047.2 q^{48} +114131. q^{49} -6680.84 q^{50} -84690.8i q^{51} -52533.5 q^{52} -60990.7i q^{53} -10921.6 q^{54} +77497.6 q^{55} -20468.6i q^{56} -60937.2i q^{57} +32451.1 q^{58} -76808.7 q^{59} -116277. i q^{60} +18606.1i q^{61} +10135.0 q^{62} +14413.5i q^{63} -79385.2 q^{64} -126363. i q^{65} -26003.4i q^{66} -24517.2i q^{67} -302543. i q^{68} +(69844.8 - 176336. i) q^{69} +22907.5 q^{70} -511739. q^{71} -83855.2 q^{72} -104789. q^{73} +23353.2i q^{74} +36120.9 q^{75} -217688. i q^{76} -34317.5 q^{77} -42399.6 q^{78} +699697. i q^{79} -344117. i q^{80} +59049.0 q^{81} -38675.4 q^{82} -288024. i q^{83} +51489.9i q^{84} +727730. q^{85} +191287. i q^{86} -175451. q^{87} -199652. i q^{88} -90597.8i q^{89} -93846.9i q^{90} +55956.0i q^{91} +(249509. - 629931. i) q^{92} -54796.4 q^{93} -74491.7 q^{94} +523620. q^{95} -459740. q^{96} +1.69461e6i q^{97} +329063. q^{98} +140591. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.88321			0.360401			0.180201	−	0.983630i	$-0.442325\pi$
$2$	2.88321			0.360401			0.180201	+	0.983630i	$0.442325\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	−55.6871			−0.870111
$5$		−	133.948i		−	1.07159i	−0.844349	−	0.535793i	$-0.820013\pi$
$5$		−	133.948i		−	1.07159i	0.844349	−	0.535793i	$-0.179987\pi$
$6$	−44.9448			−0.208078
$7$			59.3150i			0.172930i	0.996255	+	0.0864650i	$0.0275571\pi$
$7$			59.3150i			0.172930i	−0.996255	+	0.0864650i	$0.972443\pi$
$8$	−345.083			−0.673990
$9$	243.000			0.333333
$10$		−	386.201i		−	0.386201i
$11$			578.563i			0.434683i	0.976096	+	0.217341i	$0.0697385\pi$
$11$			578.563i			0.434683i	−0.976096	+	0.217341i	$0.930261\pi$
$12$	868.076			0.502359
$13$	943.370			0.429390			0.214695	−	0.976681i	$-0.431124\pi$
$13$	943.370			0.429390			0.214695	+	0.976681i	$0.431124\pi$
$14$			171.018i			0.0623242i
$15$			2088.05i			0.618681i
$16$	2569.03			0.627204
$17$			5432.92i			1.10582i	0.833240	+	0.552912i	$0.186484\pi$
$17$			5432.92i			1.10582i	−0.833240	+	0.552912i	$0.813516\pi$
$18$	700.620			0.120134
$19$			3909.12i			0.569926i	0.958539	+	0.284963i	$0.0919813\pi$
$19$			3909.12i			0.569926i	−0.958539	+	0.284963i	$0.908019\pi$
$20$			7459.19i			0.932399i
$21$		−	924.630i		−	0.0998412i
$22$			1668.12i			0.156660i
$23$	−4480.54	+	11312.0i	−0.368254	+	0.929725i
$23$	−4480.54	+	11312.0i	−0.368254	+	0.929725i
$24$	5379.31			0.389128
$25$	−2317.15			−0.148298
$26$	2719.93			0.154753
$27$	−3788.00			−0.192450
$28$		−	3303.08i		−	0.150468i
$29$	11255.2			0.461487			0.230743	−	0.973015i	$-0.425884\pi$
$29$	11255.2			0.461487			0.230743	+	0.973015i	$0.425884\pi$
$30$			6020.28i			0.222973i
$31$	3515.19			0.117995			0.0589975	−	0.998258i	$-0.481210\pi$
$31$	3515.19			0.117995			0.0589975	+	0.998258i	$0.481210\pi$
$32$	29492.4			0.900035
$33$		−	9018.91i		−	0.250964i
$34$			15664.2i			0.398540i
$35$	7945.15			0.185310
$36$	−13532.0			−0.290037
$37$			8099.73i			0.159906i	0.996799	+	0.0799532i	$0.0254771\pi$
$37$			8099.73i			0.159906i	−0.996799	+	0.0799532i	$0.974523\pi$
$38$			11270.8i			0.205402i
$39$	−14705.7			−0.247908
$40$			46223.3i			0.722239i
$41$	−13414.0			−0.194629			−0.0973144	−	0.995254i	$-0.531025\pi$
$41$	−13414.0			−0.194629			−0.0973144	+	0.995254i	$0.531025\pi$
$42$		−	2665.90i		−	0.0359829i
$43$			66345.2i			0.834457i	0.908802	+	0.417229i	$0.136999\pi$
$43$			66345.2i			0.834457i	−0.908802	+	0.417229i	$0.863001\pi$
$44$		−	32218.5i		−	0.378222i
$45$		−	32549.4i		−	0.357196i
$46$	−12918.3	+	32614.8i	−0.132719	+	0.335074i
$47$	−25836.4			−0.248850			−0.124425	−	0.992229i	$-0.539709\pi$
$47$	−25836.4			−0.248850			−0.124425	+	0.992229i	$0.539709\pi$
$48$	−40047.2			−0.362117
$49$	114131.			0.970095
$50$	−6680.84			−0.0534467
$51$		−	84690.8i		−	0.638448i
$52$	−52533.5			−0.373617
$53$		−	60990.7i		−	0.409672i	−0.978796	−	0.204836i	$-0.934334\pi$
$53$		−	60990.7i		−	0.409672i	0.978796	−	0.204836i	$-0.0656661\pi$
$54$	−10921.6			−0.0693592
$55$	77497.6			0.465800
$56$		−	20468.6i		−	0.116553i
$57$		−	60937.2i		−	0.329047i
$58$	32451.1			0.166320
$59$	−76808.7			−0.373985			−0.186993	−	0.982361i	$-0.559874\pi$
$59$	−76808.7			−0.373985			−0.186993	+	0.982361i	$0.559874\pi$
$60$		−	116277.i		−	0.538321i
$61$			18606.1i			0.0819721i	0.999160	+	0.0409861i	$0.0130499\pi$
$61$			18606.1i			0.0819721i	−0.999160	+	0.0409861i	$0.986950\pi$
$62$	10135.0			0.0425255
$63$			14413.5i			0.0576434i
$64$	−79385.2			−0.302830
$65$		−	126363.i		−	0.460129i
$66$		−	26003.4i		−	0.0904478i
$67$		−	24517.2i		−	0.0815167i	−0.999169	−	0.0407583i	$-0.987023\pi$
$67$		−	24517.2i		−	0.0815167i	0.999169	−	0.0407583i	$-0.0129774\pi$
$68$		−	302543.i		−	0.962190i
$69$	69844.8	−	176336.i	0.212611	−	0.536777i
$70$	22907.5			0.0667858
$71$	−511739.			−1.42980			−0.714898	−	0.699229i	$-0.753527\pi$
$71$	−511739.			−1.42980			−0.714898	+	0.699229i	$0.753527\pi$
$72$	−83855.2			−0.224663
$73$	−104789.			−0.269369			−0.134685	−	0.990889i	$-0.543002\pi$
$73$	−104789.			−0.269369			−0.134685	+	0.990889i	$0.543002\pi$
$74$			23353.2i			0.0576304i
$75$	36120.9			0.0856198
$76$		−	217688.i		−	0.495899i
$77$	−34317.5			−0.0751698
$78$	−42399.6			−0.0893465
$79$			699697.i			1.41915i	0.704630	+	0.709575i	$0.251113\pi$
$79$			699697.i			1.41915i	−0.704630	+	0.709575i	$0.748887\pi$
$80$		−	344117.i		−	0.672104i
$81$	59049.0			0.111111
$82$	−38675.4			−0.0701444
$83$		−	288024.i		−	0.503725i	−0.967763	−	0.251863i	$-0.918957\pi$
$83$		−	288024.i		−	0.503725i	0.967763	−	0.251863i	$-0.0810431\pi$
$84$			51489.9i			0.0868729i
$85$	727730.			1.18499
$86$			191287.i			0.300739i
$87$	−175451.			−0.266439
$88$		−	199652.i		−	0.292972i
$89$		−	90597.8i		−	0.128513i	−0.997933	−	0.0642566i	$-0.979532\pi$
$89$		−	90597.8i		−	0.128513i	0.997933	−	0.0642566i	$-0.0204676\pi$
$90$		−	93846.9i		−	0.128734i
$91$			55956.0i			0.0742545i
$92$	249509.	−	629931.i	0.320422	−	0.808964i
$93$	−54796.4			−0.0681245
$94$	−74491.7			−0.0896859
$95$	523620.			0.610725
$96$	−459740.			−0.519636
$97$			1.69461e6i			1.85676i	0.371634	+	0.928379i	$0.378798\pi$
$97$			1.69461e6i			1.85676i	−0.371634	+	0.928379i	$0.621202\pi$
$98$	329063.			0.349623
$99$			140591.i			0.144894i
$100$	129036.			0.129036
$101$	219524.			0.213068			0.106534	−	0.994309i	$-0.466025\pi$
$101$	219524.			0.213068			0.106534	+	0.994309i	$0.466025\pi$
$102$		−	244181.i		−	0.230097i
$103$			1.67613e6i			1.53389i	0.641711	+	0.766947i	$0.278225\pi$
$103$			1.67613e6i			1.53389i	−0.641711	+	0.766947i	$0.721775\pi$
$104$	−325541.			−0.289405
$105$	−123853.			−0.106989
$106$		−	175849.i		−	0.147646i
$107$			1.73015e6i			1.41232i	0.708054	+	0.706159i	$0.249574\pi$
$107$			1.73015e6i			1.41232i	−0.708054	+	0.706159i	$0.750426\pi$
$108$	210942.			0.167453
$109$			1.72020e6i			1.32831i	0.747595	+	0.664155i	$0.231209\pi$
$109$			1.72020e6i			1.32831i	−0.747595	+	0.664155i	$0.768791\pi$
$110$	223442.			0.167875
$111$		−	126262.i		−	0.0923220i
$112$			152382.i			0.108462i
$113$			226700.i			0.157114i	0.996910	+	0.0785571i	$0.0250313\pi$
$113$			226700.i			0.157114i	−0.996910	+	0.0785571i	$0.974969\pi$
$114$		−	175695.i		−	0.118589i
$115$	1.51522e6	+	600161.i	0.996281	+	0.394616i
$116$	−626769.			−0.401545
$117$	229239.			0.143130
$118$	−221456.			−0.134785
$119$	−322253.			−0.191230
$120$		−	720550.i		−	0.416985i
$121$	1.43683e6			0.811051
$122$			53645.3i			0.0295428i
$123$	209104.			0.112369
$124$	−195751.			−0.102669
$125$		−	1.78256e6i		−	0.912673i
$126$			41557.3i			0.0207747i
$127$	−745176.			−0.363787			−0.181894	−	0.983318i	$-0.558223\pi$
$127$	−745176.			−0.363787			−0.181894	+	0.983318i	$0.558223\pi$
$128$	−2.11639e6			−1.00918
$129$		−	1.03422e6i		−	0.481774i
$130$		−	364330.i		−	0.165831i
$131$	721322.			0.320860			0.160430	−	0.987047i	$-0.448712\pi$
$131$	721322.			0.320860			0.160430	+	0.987047i	$0.448712\pi$
$132$			502237.i			0.218367i
$133$	−231870.			−0.0985573
$134$		−	70688.2i		−	0.0293787i
$135$			507396.i			0.206227i
$136$		−	1.87481e6i		−	0.745315i
$137$		−	2.81334e6i		−	1.09411i	−0.837097	−	0.547055i	$-0.815749\pi$
$137$		−	2.81334e6i		−	1.09411i	0.837097	−	0.547055i	$-0.184251\pi$
$138$	201377.	−	508414.i	0.0766254	−	0.193455i
$139$	2.52591e6			0.940533			0.470267	−	0.882524i	$-0.344158\pi$
$139$	2.52591e6			0.940533			0.470267	+	0.882524i	$0.344158\pi$
$140$	−442442.			−0.161240
$141$	402749.			0.143674
$142$	−1.47545e6			−0.515300
$143$			545799.i			0.186649i
$144$	624274.			0.209068
$145$		−	1.50762e6i		−	0.494523i
$146$	−302129.			−0.0970810
$147$	−1.77912e6			−0.560085
$148$		−	451051.i		−	0.139136i
$149$			4.50361e6i			1.36145i	0.732539	+	0.680725i	$0.238335\pi$
$149$			4.50361e6i			1.36145i	−0.732539	+	0.680725i	$0.761665\pi$
$150$	104144.			0.0308575
$151$	−5.14332e6			−1.49387			−0.746935	−	0.664897i	$-0.768476\pi$
$151$	−5.14332e6			−1.49387			−0.746935	+	0.664897i	$0.768476\pi$
$152$		−	1.34897e6i		−	0.384125i
$153$			1.32020e6i			0.368608i
$154$	−98944.5			−0.0270913
$155$		−	470854.i		−	0.126442i
$156$	818917.			0.215708
$157$		−	6.18385e6i		−	1.59794i	−0.601372	−	0.798969i	$-0.705379\pi$
$157$		−	6.18385e6i		−	1.59794i	0.601372	−	0.798969i	$-0.294621\pi$
$158$			2.01737e6i			0.511463i
$159$			950751.i			0.236524i
$160$		−	3.95045e6i		−	0.964466i
$161$	−670970.	−	265764.i	−0.160777	−	0.0636822i
$162$	170251.			0.0400446
$163$	−1.21979e6			−0.281658			−0.140829	−	0.990034i	$-0.544977\pi$
$163$	−1.21979e6			−0.281658			−0.140829	+	0.990034i	$0.544977\pi$
$164$	746987.			0.169349
$165$	−1.20807e6			−0.268930
$166$		−	830432.i		−	0.181543i
$167$	5.49073e6			1.17891			0.589455	−	0.807801i	$-0.299343\pi$
$167$	5.49073e6			1.17891			0.589455	+	0.807801i	$0.299343\pi$
$168$			319074.i			0.0672920i
$169$	−3.93686e6			−0.815624
$170$	2.09820e6			0.427071
$171$			949917.i			0.189975i
$172$		−	3.69457e6i		−	0.726070i
$173$	−3.24115e6			−0.625980			−0.312990	−	0.949756i	$-0.601331\pi$
$173$	−3.24115e6			−0.625980			−0.312990	+	0.949756i	$0.601331\pi$
$174$	−505862.			−0.0960251
$175$		−	137442.i		−	0.0256452i
$176$			1.48634e6i			0.272635i
$177$	1.19733e6			0.215921
$178$		−	261212.i		−	0.0463163i
$179$	6.13192e6			1.06915			0.534573	−	0.845122i	$-0.320472\pi$
$179$	6.13192e6			1.06915			0.534573	+	0.845122i	$0.320472\pi$
$180$			1.81258e6i			0.310800i
$181$		−	3.65220e6i		−	0.615912i	−0.951401	−	0.307956i	$-0.900355\pi$
$181$		−	3.65220e6i		−	0.615912i	0.951401	−	0.307956i	$-0.0996450\pi$
$182$			161333.i			0.0267614i
$183$		−	290041.i		−	0.0473266i
$184$	1.54616e6	−	3.90357e6i	0.248199	−	0.626626i
$185$	1.08495e6			0.171353
$186$	−157989.			−0.0245521
$187$	−3.14328e6			−0.480683
$188$	1.43875e6			0.216527
$189$		−	224685.i		−	0.0332804i
$190$	1.50971e6			0.220106
$191$		−	1.83792e6i		−	0.263771i	−0.991265	−	0.131885i	$-0.957897\pi$
$191$		−	1.83792e6i		−	0.263771i	0.991265	−	0.131885i	$-0.0421031\pi$
$192$	1.23749e6			0.174839
$193$	649119.			0.0902927			0.0451463	−	0.998980i	$-0.485625\pi$
$193$	649119.			0.0902927			0.0451463	+	0.998980i	$0.485625\pi$
$194$			4.88593e6i			0.669178i
$195$			1.96980e6i			0.265655i
$196$	−6.35561e6			−0.844091
$197$	−1.26065e7			−1.64890			−0.824450	−	0.565935i	$-0.808515\pi$
$197$	−1.26065e7			−1.64890			−0.824450	+	0.565935i	$0.808515\pi$
$198$			405353.i			0.0522201i
$199$			3.26083e6i			0.413779i	0.978364	+	0.206889i	$0.0663340\pi$
$199$			3.26083e6i			0.413779i	−0.978364	+	0.206889i	$0.933666\pi$
$200$	799610.			0.0999513
$201$			382185.i			0.0470637i
$202$	632933.			0.0767899
$203$			667602.i			0.0798049i
$204$			4.71618e6i			0.555521i
$205$			1.79678e6i			0.208562i
$206$			4.83262e6i			0.552817i
$207$	−1.08877e6	+	2.74881e6i	−0.122751	+	0.309908i
$208$	2.42354e6			0.269315
$209$	−2.26167e6			−0.247737
$210$	−357093.			−0.0385588
$211$	7.77976e6			0.828169			0.414084	−	0.910239i	$-0.364102\pi$
$211$	7.77976e6			0.828169			0.414084	+	0.910239i	$0.364102\pi$
$212$			3.39640e6i			0.356460i
$213$	7.97723e6			0.825493
$214$			4.98838e6i			0.509001i
$215$	8.88683e6			0.894193
$216$	1.30717e6			0.129709
$217$			208504.i			0.0204049i
$218$			4.95970e6i			0.478724i
$219$	1.63350e6			0.155520
$220$	−4.31561e6			−0.405298
$221$			5.12525e6i			0.474830i
$222$		−	364041.i		−	0.0332729i
$223$	1.33038e7			1.19966			0.599832	−	0.800126i	$-0.295234\pi$
$223$	1.33038e7			1.19966			0.599832	+	0.800126i	$0.295234\pi$
$224$			1.74934e6i			0.155643i
$225$	−563068.			−0.0494326
$226$			653622.i			0.0566241i
$227$		−	1.61060e6i		−	0.137693i	−0.997627	−	0.0688464i	$-0.978068\pi$
$227$		−	1.61060e6i		−	0.137693i	0.997627	−	0.0688464i	$-0.0219318\pi$
$228$			3.39342e6i			0.286307i
$229$			1.58708e7i			1.32158i	0.750572	+	0.660789i	$0.229778\pi$
$229$			1.58708e7i			1.32158i	−0.750572	+	0.660789i	$0.770222\pi$
$230$	4.36869e6	+	1.73039e6i	0.359061	+	0.142220i
$231$	534957.			0.0433993
$232$	−3.88398e6			−0.311037
$233$	−7.28829e6			−0.576180			−0.288090	−	0.957603i	$-0.593020\pi$
$233$	−7.28829e6			−0.576180			−0.288090	+	0.957603i	$0.593020\pi$
$234$	660944.			0.0515842
$235$			3.46074e6i			0.266665i
$236$	4.27726e6			0.325409
$237$		−	1.09072e7i		−	0.819347i
$238$	−929124.			−0.0689196
$239$	−7.01963e6			−0.514186			−0.257093	−	0.966387i	$-0.582765\pi$
$239$	−7.01963e6			−0.514186			−0.257093	+	0.966387i	$0.582765\pi$
$240$			5.36425e6i			0.388039i
$241$		−	2.59499e7i		−	1.85389i	−0.375197	−	0.926945i	$-0.622425\pi$
$241$		−	2.59499e7i		−	1.85389i	0.375197	−	0.926945i	$-0.377575\pi$
$242$	4.14267e6			0.292304
$243$	−920483.			−0.0641500
$244$		−	1.03612e6i		−	0.0713248i
$245$		−	1.52876e7i		−	1.03954i
$246$	602890.			0.0404979
$247$			3.68775e6i			0.244721i
$248$	−1.21303e6			−0.0795275
$249$			4.48984e6i			0.290826i
$250$		−	5.13950e6i		−	0.328928i
$251$		−	881038.i		−	0.0557152i	−0.999612	−	0.0278576i	$-0.991132\pi$
$251$		−	881038.i		−	0.0557152i	0.999612	−	0.0278576i	$-0.00886849\pi$
$252$		−	802649.i		−	0.0501561i
$253$	−6.54469e6	−	2.59228e6i	−0.404136	−	0.160074i
$254$	−2.14850e6			−0.131109
$255$	−1.13442e7			−0.684152
$256$	−1.02136e6			−0.0608776
$257$	−5.39464e6			−0.317807			−0.158903	−	0.987294i	$-0.550796\pi$
$257$	−5.39464e6			−0.317807			−0.158903	+	0.987294i	$0.550796\pi$
$258$		−	2.98187e6i		−	0.173632i
$259$	−480436.			−0.0276526
$260$			7.03678e6i			0.400363i
$261$	2.73501e6			0.153829
$262$	2.07972e6			0.115638
$263$		−	1.83758e7i		−	1.01013i	−0.863080	−	0.505067i	$-0.831468\pi$
$263$		−	1.83758e7i		−	1.01013i	0.863080	−	0.505067i	$-0.168532\pi$
$264$			3.11227e6i			0.169147i
$265$	−8.16961e6			−0.438999
$266$	−668529.			−0.0355202
$267$			1.41228e6i			0.0741971i
$268$			1.36529e6i			0.0709286i
$269$	2.33084e6			0.119745			0.0598723	−	0.998206i	$-0.480931\pi$
$269$	2.33084e6			0.119745			0.0598723	+	0.998206i	$0.480931\pi$
$270$			1.46293e6i			0.0743244i
$271$	−2.55701e7			−1.28477			−0.642384	−	0.766383i	$-0.722055\pi$
$271$	−2.55701e7			−1.28477			−0.642384	+	0.766383i	$0.722055\pi$
$272$			1.39573e7i			0.693578i
$273$		−	872268.i		−	0.0428708i
$274$		−	8.11145e6i		−	0.394318i
$275$		−	1.34062e6i		−	0.0644626i
$276$	−3.88945e6	+	9.81965e6i	−0.184996	+	0.467056i
$277$	2.84046e7			1.33644			0.668220	−	0.743964i	$-0.267056\pi$
$277$	2.84046e7			1.33644			0.668220	+	0.743964i	$0.267056\pi$
$278$	7.28274e6			0.338969
$279$	854191.			0.0393317
$280$	−2.74173e6			−0.124897
$281$		−	2.93377e7i		−	1.32223i	−0.750284	−	0.661115i	$-0.770083\pi$
$281$		−	2.93377e7i		−	1.32223i	0.750284	−	0.661115i	$-0.229917\pi$
$282$	1.16121e6			0.0517802
$283$			9.30377e6i			0.410487i	0.978711	+	0.205244i	$0.0657987\pi$
$283$			9.30377e6i			0.410487i	−0.978711	+	0.205244i	$0.934201\pi$
$284$	2.84973e7			1.24408
$285$	−8.16243e6			−0.352602
$286$			1.57365e6i			0.0672684i
$287$		−	795652.i		−	0.0336572i
$288$	7.16664e6			0.300012
$289$	−5.37900e6			−0.222848
$290$		−	4.34677e6i		−	0.178227i
$291$		−	2.64164e7i		−	1.07200i
$292$	5.83541e6			0.234381
$293$		−	2.23704e7i		−	0.889345i	−0.895693	−	0.444673i	$-0.853320\pi$
$293$		−	2.23704e7i		−	0.889345i	0.895693	−	0.444673i	$-0.146680\pi$
$294$	−5.12958e6			−0.201855
$295$			1.02884e7i			0.400758i
$296$		−	2.79508e6i		−	0.107775i
$297$		−	2.19159e6i		−	0.0836548i
$298$			1.29848e7i			0.490668i
$299$	−4.22681e6	+	1.06714e7i	−0.158125	+	0.399215i
$300$	−2.01147e6			−0.0744987
$301$	−3.93527e6			−0.144303
$302$	−1.48293e7			−0.538393
$303$	−3.42204e6			−0.123015
$304$			1.00426e7i			0.357460i
$305$	2.49226e6			0.0878402
$306$			3.80641e6i			0.132847i
$307$	2.43258e7			0.840723			0.420361	−	0.907357i	$-0.361903\pi$
$307$	2.43258e7			0.840723			0.420361	+	0.907357i	$0.361903\pi$
$308$	1.91104e6			0.0654060
$309$		−	2.61282e7i		−	0.885594i
$310$		−	1.35757e6i		−	0.0455698i
$311$	−3.22419e7			−1.07186			−0.535932	−	0.844261i	$-0.680040\pi$
$311$	−3.22419e7			−1.07186			−0.535932	+	0.844261i	$0.680040\pi$
$312$	5.07468e6			0.167088
$313$			2.99672e7i			0.977265i	0.872490	+	0.488633i	$0.162504\pi$
$313$			2.99672e7i			0.977265i	−0.872490	+	0.488633i	$0.837496\pi$
$314$		−	1.78293e7i		−	0.575899i
$315$	1.93067e6			0.0617699
$316$		−	3.89641e7i		−	1.23482i
$317$	−3.88340e7			−1.21909			−0.609543	−	0.792753i	$-0.708647\pi$
$317$	−3.88340e7			−1.21909			−0.609543	+	0.792753i	$0.708647\pi$
$318$			2.74121e6i			0.0852436i
$319$			6.51184e6i			0.200600i
$320$			1.06335e7i			0.324509i
$321$		−	2.69704e7i		−	0.815402i
$322$	−1.93455e6	−	766252.i	−0.0579444	−	0.0229511i
$323$	−2.12379e7			−0.630238
$324$	−3.28827e6			−0.0966790
$325$	−2.18593e6			−0.0636776
$326$	−3.51691e6			−0.101510
$327$		−	2.68153e7i		−	0.766900i
$328$	4.62895e6			0.131178
$329$		−	1.53249e6i		−	0.0430337i
$330$	−3.48311e6			−0.0969227
$331$	2.73385e7			0.753860			0.376930	−	0.926242i	$-0.376980\pi$
$331$	2.73385e7			0.753860			0.376930	+	0.926242i	$0.376980\pi$
$332$			1.60392e7i			0.438297i
$333$			1.96824e6i			0.0533021i
$334$	1.58309e7			0.424880
$335$	−3.28404e6			−0.0873522
$336$		−	2.37540e6i		−	0.0626208i
$337$			2.42977e7i			0.634857i	0.948282	+	0.317428i	$0.102819\pi$
$337$			2.42977e7i			0.634857i	−0.948282	+	0.317428i	$0.897181\pi$
$338$	−1.13508e7			−0.293952
$339$		−	3.53390e6i		−	0.0907099i
$340$	−4.05252e7			−1.03107
$341$			2.03376e6i			0.0512904i
$342$			2.73881e6i			0.0684673i
$343$			1.37480e7i			0.340689i
$344$		−	2.28946e7i		−	0.562416i
$345$	−2.36199e7	−	9.35559e6i	−0.575203	−	0.227832i
$346$	−9.34491e6			−0.225604
$347$	6.67195e7			1.59685			0.798426	−	0.602093i	$-0.205666\pi$
$347$	6.67195e7			1.59685			0.798426	+	0.602093i	$0.205666\pi$
$348$	9.77037e6			0.231832
$349$	4.82883e7			1.13597			0.567983	−	0.823040i	$-0.307724\pi$
$349$	4.82883e7			1.13597			0.567983	+	0.823040i	$0.307724\pi$
$350$		−	396274.i		−	0.00924255i
$351$	−3.57348e6			−0.0826362
$352$			1.70632e7i			0.391230i
$353$	5.50097e7			1.25059			0.625295	−	0.780388i	$-0.284978\pi$
$353$	5.50097e7			1.25059			0.625295	+	0.780388i	$0.284978\pi$
$354$	3.45215e6			0.0778180
$355$			6.85466e7i			1.53215i
$356$			5.04513e6i			0.111821i
$357$	5.02343e6			0.110407
$358$	1.76796e7			0.385322
$359$			2.47715e7i			0.535389i	0.963504	+	0.267695i	$0.0862619\pi$
$359$			2.47715e7i			0.535389i	−0.963504	+	0.267695i	$0.913738\pi$
$360$			1.12323e7i			0.240746i
$361$	3.17646e7			0.675184
$362$		−	1.05301e7i		−	0.221975i
$363$	−2.23979e7			−0.468260
$364$		−	3.11603e6i		−	0.0646096i
$365$			1.40363e7i			0.288653i
$366$		−	836248.i		−	0.0170566i
$367$			9.76701e7i			1.97589i	0.154798	+	0.987946i	$0.450527\pi$
$367$			9.76701e7i			1.97589i	−0.154798	+	0.987946i	$0.549473\pi$
$368$	−1.15106e7	+	2.90608e7i	−0.230970	+	0.583128i
$369$	−3.25960e6			−0.0648762
$370$	3.12813e6			0.0617560
$371$	3.61767e6			0.0708446
$372$	3.05145e6			0.0592758
$373$			7.18558e7i			1.38464i	0.721593	+	0.692318i	$0.243410\pi$
$373$			7.18558e7i			1.38464i	−0.721593	+	0.692318i	$0.756590\pi$
$374$	−9.06275e6			−0.173239
$375$			2.77874e7i			0.526932i
$376$	8.91570e6			0.167723
$377$	1.06178e7			0.198158
$378$		−	647814.i		−	0.0119943i
$379$		−	8.89305e6i		−	0.163355i	−0.996659	−	0.0816777i	$-0.973972\pi$
$379$		−	8.89305e6i		−	0.163355i	0.996659	−	0.0816777i	$-0.0260278\pi$
$380$	−2.91589e7			−0.531399
$381$	1.16161e7			0.210033
$382$		−	5.29911e6i		−	0.0950632i
$383$			3.07656e7i			0.547607i	0.961786	+	0.273803i	$0.0882817\pi$
$383$			3.07656e7i			0.547607i	−0.961786	+	0.273803i	$0.911718\pi$
$384$	3.29913e7			0.582648
$385$			4.59677e6i			0.0805509i
$386$	1.87155e6			0.0325416
$387$			1.61219e7i			0.278152i
$388$		−	9.43681e7i		−	1.61559i
$389$			1.67646e7i			0.284802i	0.989809	+	0.142401i	$0.0454823\pi$
$389$			1.67646e7i			0.284802i	−0.989809	+	0.142401i	$0.954518\pi$
$390$			5.67935e6i			0.0957425i
$391$	−6.14570e7	−	2.43424e7i	−1.02811	−	0.407224i
$392$	−3.93846e7			−0.653835
$393$	−1.12443e7			−0.185248
$394$	−3.63470e7			−0.594265
$395$	9.37232e7			1.52074
$396$		−	7.82910e6i		−	0.126074i
$397$	2.13788e7			0.341674			0.170837	−	0.985299i	$-0.445353\pi$
$397$	2.13788e7			0.341674			0.170837	+	0.985299i	$0.445353\pi$
$398$			9.40164e6i			0.149126i
$399$	3.61449e6			0.0569021
$400$	−5.95283e6			−0.0930130
$401$		−	6.98073e7i		−	1.08260i	−0.840830	−	0.541300i	$-0.817932\pi$
$401$		−	6.98073e7i		−	1.08260i	0.840830	−	0.541300i	$-0.182068\pi$
$402$			1.10192e6i			0.0169618i
$403$	3.31612e6			0.0506659
$404$	−1.22247e7			−0.185393
$405$		−	7.90951e6i		−	0.119065i
$406$			1.92484e6i			0.0287618i
$407$	−4.68621e6			−0.0695086
$408$			2.92253e7i			0.430308i
$409$	−5.80792e7			−0.848888			−0.424444	−	0.905454i	$-0.639530\pi$
$409$	−5.80792e7			−0.848888			−0.424444	+	0.905454i	$0.639530\pi$
$410$			5.18050e6i			0.0751658i
$411$			4.38556e7i			0.631684i
$412$		−	9.33387e7i		−	1.33466i
$413$		−	4.55591e6i		−	0.0646733i
$414$	−3.13916e6	+	7.92539e6i	−0.0442397	+	0.111691i
$415$	−3.85803e7			−0.539785
$416$	2.78222e7			0.386466
$417$	−3.93751e7			−0.543017
$418$	−6.52088e6			−0.0892847
$419$		−	2.02760e7i		−	0.275639i	−0.990457	−	0.137819i	$-0.955991\pi$
$419$		−	2.02760e7i		−	0.275639i	0.990457	−	0.137819i	$-0.0440093\pi$
$420$	6.89699e6			0.0930919
$421$		−	9.25488e7i		−	1.24029i	−0.784486	−	0.620147i	$-0.787073\pi$
$421$		−	9.25488e7i		−	1.24029i	0.784486	−	0.620147i	$-0.212927\pi$
$422$	2.24307e7			0.298473
$423$	−6.27824e6			−0.0829501
$424$			2.10469e7i			0.276115i
$425$		−	1.25889e7i		−	0.163991i
$426$	2.30000e7			0.297508
$427$	−1.10362e6			−0.0141754
$428$		−	9.63470e7i		−	1.22887i
$429$		−	8.50816e6i		−	0.107762i
$430$	2.56226e7			0.322268
$431$			9.59109e7i			1.19794i	0.800770	+	0.598971i	$0.204424\pi$
$431$			9.59109e7i			1.19794i	−0.800770	+	0.598971i	$0.795576\pi$
$432$	−9.73147e6			−0.120706
$433$		−	1.00838e8i		−	1.24211i	−0.783767	−	0.621055i	$-0.786705\pi$
$433$		−	1.00838e8i		−	1.24211i	0.783767	−	0.621055i	$-0.213295\pi$
$434$			601159.i			0.00735394i
$435$			2.35014e7i			0.285513i
$436$		−	9.57930e7i		−	1.15578i
$437$	−4.42199e7	−	1.75150e7i	−0.529875	−	0.209877i
$438$	4.70973e6			0.0560498
$439$	9.86541e7			1.16606			0.583031	−	0.812450i	$-0.301867\pi$
$439$	9.86541e7			1.16606			0.583031	+	0.812450i	$0.301867\pi$
$440$	−2.67431e7			−0.313945
$441$	2.77338e7			0.323365
$442$			1.47772e7i			0.171129i
$443$	5.76044e6			0.0662589			0.0331294	−	0.999451i	$-0.489453\pi$
$443$	5.76044e6			0.0662589			0.0331294	+	0.999451i	$0.489453\pi$
$444$			7.03119e6i			0.0803304i
$445$	−1.21354e7			−0.137713
$446$	3.83575e7			0.432361
$447$		−	7.02043e7i		−	0.786033i
$448$		−	4.70873e6i		−	0.0523685i
$449$	−1.48365e8			−1.63905			−0.819523	−	0.573046i	$-0.805762\pi$
$449$	−1.48365e8			−1.63905			−0.819523	+	0.573046i	$0.805762\pi$
$450$	−1.62344e6			−0.0178156
$451$		−	7.76085e6i		−	0.0846018i
$452$		−	1.26242e7i		−	0.136707i
$453$	8.01765e7			0.862487
$454$		−	4.64371e6i		−	0.0496246i
$455$	7.49521e6			0.0795701
$456$			2.10284e7i			0.221774i
$457$		−	4.09189e7i		−	0.428721i	−0.976755	−	0.214361i	$-0.931233\pi$
$457$		−	4.09189e7i		−	0.428721i	0.976755	−	0.214361i	$-0.0687668\pi$
$458$			4.57589e7i			0.476298i
$459$		−	2.05799e7i		−	0.212816i
$460$	−8.43782e7	−	3.34213e7i	−0.866875	−	0.343360i
$461$	−1.53056e8			−1.56223			−0.781117	−	0.624384i	$-0.785350\pi$
$461$	−1.53056e8			−1.56223			−0.781117	+	0.624384i	$0.785350\pi$
$462$	1.54239e6			0.0156412
$463$	−1.76680e7			−0.178010			−0.0890051	−	0.996031i	$-0.528369\pi$
$463$	−1.76680e7			−0.178010			−0.0890051	+	0.996031i	$0.528369\pi$
$464$	2.89149e7			0.289446
$465$			7.33988e6i			0.0730013i
$466$	−2.10137e7			−0.207656
$467$		−	1.15744e7i		−	0.113645i	−0.998384	−	0.0568224i	$-0.981903\pi$
$467$		−	1.15744e7i		−	0.113645i	0.998384	−	0.0568224i	$-0.0180969\pi$
$468$	−1.27657e7			−0.124539
$469$	1.45424e6			0.0140967
$470$			9.97804e6i			0.0961062i
$471$			9.63967e7i			0.922570i
$472$	2.65054e7			0.252062
$473$	−3.83849e7			−0.362724
$474$		−	3.14477e7i		−	0.295294i
$475$		−	9.05804e6i		−	0.0845188i
$476$	1.79454e7			0.166392
$477$		−	1.48207e7i		−	0.136557i
$478$	−2.02391e7			−0.185313
$479$		−	7.45450e7i		−	0.678285i	−0.940735	−	0.339142i	$-0.889863\pi$
$479$		−	7.45450e7i		−	0.678285i	0.940735	−	0.339142i	$-0.110137\pi$
$480$			6.15814e7i			0.556835i
$481$			7.64105e6i			0.0686622i
$482$		−	7.48189e7i		−	0.668144i
$483$	1.04594e7	+	4.14284e6i	0.0928249	+	0.0367669i
$484$	−8.00127e7			−0.705704
$485$	2.26991e8			1.98968
$486$	−2.65394e6			−0.0231197
$487$	−1.21827e8			−1.05477			−0.527384	−	0.849627i	$-0.676827\pi$
$487$	−1.21827e8			−1.05477			−0.527384	+	0.849627i	$0.676827\pi$
$488$		−	6.42065e6i		−	0.0552484i
$489$	1.90146e7			0.162615
$490$		−	4.40774e7i		−	0.374652i
$491$	5.85887e6			0.0494960			0.0247480	−	0.999694i	$-0.492122\pi$
$491$	5.85887e6			0.0494960			0.0247480	+	0.999694i	$0.492122\pi$
$492$	−1.16444e7			−0.0977735
$493$			6.11485e7i			0.510323i
$494$			1.06326e7i			0.0881976i
$495$	1.88319e7			0.155267
$496$	9.03062e6			0.0740070
$497$		−	3.03538e7i		−	0.247255i
$498$			1.29452e7i			0.104814i
$499$	1.83935e8			1.48034			0.740171	−	0.672419i	$-0.234744\pi$
$499$	1.83935e8			1.48034			0.740171	+	0.672419i	$0.234744\pi$
$500$			9.92658e7i			0.794126i
$501$	−8.55920e7			−0.680644
$502$		−	2.54022e6i		−	0.0200798i
$503$			2.26141e8i			1.77695i	0.458922	+	0.888477i	$0.348236\pi$
$503$			2.26141e8i			1.77695i	−0.458922	+	0.888477i	$0.651764\pi$
$504$		−	4.97387e6i		−	0.0388511i
$505$		−	2.94049e7i		−	0.228321i
$506$	−1.88697e7	−	7.47408e6i	−0.145651	−	0.0576907i
$507$	6.13696e7			0.470901
$508$	4.14967e7			0.316535
$509$	−1.34998e8			−1.02370			−0.511851	−	0.859074i	$-0.671040\pi$
$509$	−1.34998e8			−1.02370			−0.511851	+	0.859074i	$0.671040\pi$
$510$	−3.27077e7			−0.246569
$511$		−	6.21558e6i		−	0.0465821i
$512$	1.32504e8			0.987235
$513$		−	1.48077e7i		−	0.109682i
$514$	−1.55539e7			−0.114538
$515$	2.24514e8			1.64370
$516$			5.75927e7i			0.419197i
$517$		−	1.49480e7i		−	0.108171i
$518$	−1.38520e6			−0.00996603
$519$	5.05245e7			0.361410
$520$			4.36057e7i			0.310122i
$521$			3.97398e7i			0.281004i	0.990080	+	0.140502i	$0.0448716\pi$
$521$			3.97398e7i			0.281004i	−0.990080	+	0.140502i	$0.955128\pi$
$522$	7.88562e6			0.0554401
$523$			9.74182e7i			0.680981i	0.940248	+	0.340491i	$0.110593\pi$
$523$			9.74182e7i			0.680981i	−0.940248	+	0.340491i	$0.889407\pi$
$524$	−4.01683e7			−0.279184
$525$			2.14251e6i			0.0148062i
$526$		−	5.29813e7i		−	0.364054i
$527$			1.90977e7i			0.130482i
$528$		−	2.31698e7i		−	0.157406i
$529$	−1.07885e8	−	1.01368e8i	−0.728778	−	0.684750i
$530$	−2.35547e7			−0.158216
$531$	−1.86645e7			−0.124662
$532$	1.29122e7			0.0857558
$533$	−1.26544e7			−0.0835717
$534$			4.07190e6i			0.0267407i
$535$	2.31751e8			1.51342
$536$			8.46047e6i			0.0549414i
$537$	−9.55871e7			−0.617272
$538$	6.72031e6			0.0431561
$539$			6.60318e7i			0.421684i
$540$		−	2.82554e7i		−	0.179440i
$541$	−5.61803e7			−0.354807			−0.177404	−	0.984138i	$-0.556770\pi$
$541$	−5.61803e7			−0.354807			−0.177404	+	0.984138i	$0.556770\pi$
$542$	−7.37240e7			−0.463032
$543$			5.69322e7i			0.355597i
$544$			1.60229e8i			0.995281i
$545$	2.30418e8			1.42340
$546$		−	2.51493e6i		−	0.0154507i
$547$	−2.26908e8			−1.38640			−0.693200	−	0.720745i	$-0.743800\pi$
$547$	−2.26908e8			−1.38640			−0.693200	+	0.720745i	$0.743800\pi$
$548$			1.56667e8i			0.951996i
$549$			4.52129e6i			0.0273240i
$550$		−	3.86529e6i		−	0.0232324i
$551$			4.39980e7i			0.263013i
$552$	−2.41022e7	+	6.08506e7i	−0.143298	+	0.361782i
$553$	−4.15025e7			−0.245414
$554$	8.18964e7			0.481654
$555$	−1.69126e7			−0.0989310
$556$	−1.40661e8			−0.818368
$557$			1.89140e7i			0.109450i	0.998501	+	0.0547252i	$0.0174283\pi$
$557$			1.89140e7i			0.109450i	−0.998501	+	0.0547252i	$0.982572\pi$
$558$	2.46281e6			0.0141752
$559$			6.25881e7i			0.358308i
$560$	2.04113e7			0.116227
$561$	4.89989e7			0.277522
$562$		−	8.45868e7i		−	0.476534i
$563$			2.57543e8i			1.44320i	0.692312	+	0.721598i	$0.256592\pi$
$563$			2.57543e8i			1.44320i	−0.692312	+	0.721598i	$0.743408\pi$
$564$	−2.24279e7			−0.125012
$565$	3.03660e7			0.168361
$566$			2.68247e7i			0.147940i
$567$			3.50249e6i			0.0192145i
$568$	1.76593e8			0.963668
$569$		−	1.72019e8i		−	0.933769i	−0.884318	−	0.466884i	$-0.845376\pi$
$569$		−	1.72019e8i		−	0.933769i	0.884318	−	0.466884i	$-0.154624\pi$
$570$	−2.35340e7			−0.127078
$571$		−	1.82638e8i		−	0.981031i	−0.871432	−	0.490516i	$-0.836808\pi$
$571$		−	1.82638e8i		−	0.981031i	0.871432	−	0.490516i	$-0.163192\pi$
$572$		−	3.03940e7i		−	0.162405i
$573$			2.86503e7i			0.152288i
$574$		−	2.29403e6i		−	0.0121301i
$575$	1.03821e7	−	2.62116e7i	0.0546113	−	0.137876i
$576$	−1.92906e7			−0.100943
$577$	−4.64583e6			−0.0241844			−0.0120922	−	0.999927i	$-0.503849\pi$
$577$	−4.64583e6			−0.0241844			−0.0120922	+	0.999927i	$0.503849\pi$
$578$	−1.55088e7			−0.0803145
$579$	−1.01188e7			−0.0521305
$580$			8.39547e7i			0.430290i
$581$	1.70841e7			0.0871093
$582$		−	7.61640e7i		−	0.386350i
$583$	3.52870e7			0.178077
$584$	3.61610e7			0.181552
$585$		−	3.07062e7i		−	0.153376i
$586$		−	6.44985e7i		−	0.320521i
$587$	−3.52293e8			−1.74177			−0.870883	−	0.491490i	$-0.836452\pi$
$587$	−3.52293e8			−1.74177			−0.870883	+	0.491490i	$0.836452\pi$
$588$	9.90742e7			0.487336
$589$			1.37413e7i			0.0672484i
$590$			2.96636e7i			0.144434i
$591$	1.96515e8			0.951993
$592$			2.08084e7i			0.100294i
$593$	2.14097e7			0.102671			0.0513353	−	0.998681i	$-0.483652\pi$
$593$	2.14097e7			0.102671			0.0513353	+	0.998681i	$0.483652\pi$
$594$		−	6.31882e6i		−	0.0301493i
$595$			4.31653e7i			0.204920i
$596$		−	2.50793e8i		−	1.18461i
$597$		−	5.08312e7i		−	0.238895i
$598$	−1.21868e7	+	3.07678e7i	−0.0569883	+	0.143877i
$599$	−2.67976e8			−1.24686			−0.623428	−	0.781881i	$-0.714260\pi$
$599$	−2.67976e8			−1.24686			−0.623428	+	0.781881i	$0.714260\pi$
$600$	−1.24647e7			−0.0577069
$601$	−1.13179e8			−0.521365			−0.260682	−	0.965425i	$-0.583947\pi$
$601$	−1.13179e8			−0.521365			−0.260682	+	0.965425i	$0.583947\pi$
$602$	−1.13462e7			−0.0520069
$603$		−	5.95768e6i		−	0.0271722i
$604$	2.86417e8			1.29983
$605$		−	1.92460e8i		−	0.869111i
$606$	−9.86646e6			−0.0443346
$607$	3.82018e8			1.70812			0.854059	−	0.520176i	$-0.174134\pi$
$607$	3.82018e8			1.70812			0.854059	+	0.520176i	$0.174134\pi$
$608$			1.15289e8i			0.512954i
$609$		−	1.04069e7i		−	0.0460754i
$610$	7.18570e6			0.0316577
$611$	−2.43733e7			−0.106854
$612$		−	7.35180e7i		−	0.320730i
$613$			7.07824e7i			0.307287i	0.988126	+	0.153643i	$0.0491007\pi$
$613$			7.07824e7i			0.307287i	−0.988126	+	0.153643i	$0.950899\pi$
$614$	7.01365e7			0.302997
$615$		−	2.80091e7i		−	0.120413i
$616$	1.18424e7			0.0506637
$617$		−	1.84723e8i		−	0.786441i	−0.919444	−	0.393221i	$-0.871361\pi$
$617$		−	1.84723e8i		−	0.786441i	0.919444	−	0.393221i	$-0.128639\pi$
$618$		−	7.53332e7i		−	0.319169i
$619$			5.75005e7i			0.242438i	0.992626	+	0.121219i	$0.0386803\pi$
$619$			5.75005e7i			0.242438i	−0.992626	+	0.121219i	$0.961320\pi$
$620$			2.62205e7i			0.110018i
$621$	1.69723e7	−	4.28497e7i	0.0708705	−	0.178926i
$622$	−9.29601e7			−0.386301
$623$	5.37381e6			0.0222238
$624$	−3.77793e7			−0.155489
$625$	−2.74977e8			−1.12631
$626$			8.64016e7i			0.352208i
$627$	3.52560e7			0.143031
$628$			3.44361e8i			1.39038i
$629$	−4.40052e7			−0.176828
$630$	5.56653e6			0.0222619
$631$		−	1.50039e8i		−	0.597195i	−0.954379	−	0.298597i	$-0.903481\pi$
$631$		−	1.50039e8i		−	0.597195i	0.954379	−	0.298597i	$-0.0965188\pi$
$632$		−	2.41453e8i		−	0.956493i
$633$	−1.21274e8			−0.478143
$634$	−1.11967e8			−0.439360
$635$			9.98150e7i			0.389830i
$636$		−	5.29446e7i		−	0.205802i
$637$	1.07668e8			0.416549
$638$			1.87750e7i			0.0722966i
$639$	−1.24353e8			−0.476598
$640$			2.83488e8i			1.08142i
$641$		−	4.57545e8i		−	1.73724i	−0.495481	−	0.868619i	$-0.665008\pi$
$641$		−	4.57545e8i		−	1.73724i	0.495481	−	0.868619i	$-0.334992\pi$
$642$		−	7.77612e7i		−	0.293872i
$643$			2.92745e8i			1.10118i	0.834777	+	0.550588i	$0.185596\pi$
$643$			2.92745e8i			1.10118i	−0.834777	+	0.550588i	$0.814404\pi$
$644$	3.73643e7	+	1.47996e7i	0.139894	+	0.0554106i
$645$	−1.38532e8			−0.516263
$646$	−6.12334e7			−0.227139
$647$	−3.52954e8			−1.30318			−0.651592	−	0.758570i	$-0.725898\pi$
$647$	−3.52954e8			−1.30318			−0.651592	+	0.758570i	$0.725898\pi$
$648$	−2.03768e7			−0.0748878
$649$		−	4.44387e7i		−	0.162565i
$650$	−6.30250e6			−0.0229495
$651$		−	3.25025e6i		−	0.0117808i
$652$	6.79265e7			0.245074
$653$	2.79192e8			1.00268			0.501342	−	0.865249i	$-0.332840\pi$
$653$	2.79192e8			1.00268			0.501342	+	0.865249i	$0.332840\pi$
$654$		−	7.73140e7i		−	0.276392i
$655$		−	9.66198e7i		−	0.343829i
$656$	−3.44610e7			−0.122072
$657$	−2.54638e7			−0.0897898
$658$		−	4.41848e6i		−	0.0155094i
$659$		−	3.74057e8i		−	1.30702i	−0.756919	−	0.653509i	$-0.773296\pi$
$659$		−	3.74057e8i		−	1.30702i	0.756919	−	0.653509i	$-0.226704\pi$
$660$	6.72738e7			0.233999
$661$		−	1.99796e8i		−	0.691802i	−0.938271	−	0.345901i	$-0.887573\pi$
$661$		−	1.99796e8i		−	0.691802i	0.938271	−	0.345901i	$-0.112427\pi$
$662$	7.88226e7			0.271692
$663$		−	7.98947e7i		−	0.274143i
$664$			9.93920e7i			0.339506i
$665$			3.10586e7i			0.105613i
$666$			5.67483e6i			0.0192101i
$667$	−5.04294e7	+	1.27318e8i	−0.169944	+	0.429056i
$668$	−3.05763e8			−1.02578
$669$	−2.07385e8			−0.692627
$670$	−9.46857e6			−0.0314818
$671$	−1.07648e7			−0.0356319
$672$		−	2.72695e7i		−	0.0898606i
$673$	4.50528e8			1.47801			0.739004	−	0.673701i	$-0.235296\pi$
$673$	4.50528e8			1.47801			0.739004	+	0.673701i	$0.235296\pi$
$674$			7.00554e7i			0.228803i
$675$	8.77737e6			0.0285399
$676$	2.19232e8			0.709684
$677$		−	3.07185e8i		−	0.989999i	−0.868893	−	0.494999i	$-0.835168\pi$
$677$		−	3.07185e8i		−	0.989999i	0.868893	−	0.494999i	$-0.164832\pi$
$678$		−	1.01890e7i		−	0.0326920i
$679$	−1.00516e8			−0.321089
$680$	−2.51127e8			−0.798669
$681$			2.51068e7i			0.0794970i
$682$			5.86375e6i			0.0184851i
$683$	3.65060e8			1.14578			0.572890	−	0.819632i	$-0.305822\pi$
$683$	3.65060e8			1.14578			0.572890	+	0.819632i	$0.305822\pi$
$684$		−	5.28981e7i		−	0.165300i
$685$	−3.76842e8			−1.17243
$686$			3.96384e7i			0.122785i
$687$		−	2.47401e8i		−	0.763013i
$688$			1.70443e8i			0.523375i
$689$		−	5.75368e7i		−	0.175909i
$690$	−6.81012e7	−	2.69741e7i	−0.207304	−	0.0821108i
$691$	−1.40121e8			−0.424688			−0.212344	−	0.977195i	$-0.568110\pi$
$691$	−1.40121e8			−0.424688			−0.212344	+	0.977195i	$0.568110\pi$
$692$	1.80490e8			0.544673
$693$	−8.33915e6			−0.0250566
$694$	1.92366e8			0.575507
$695$		−	3.38342e8i		−	1.00786i
$696$	6.05452e7			0.179578
$697$		−	7.28772e7i		−	0.215225i
$698$	1.39225e8			0.409404
$699$	1.13613e8			0.332658
$700$			7.65375e6i			0.0223141i
$701$		−	1.00190e8i		−	0.290850i	−0.989369	−	0.145425i	$-0.953545\pi$
$701$		−	1.00190e8i		−	0.290850i	0.989369	−	0.145425i	$-0.0464550\pi$
$702$	−1.03031e7			−0.0297822
$703$	−3.16629e7			−0.0911348
$704$		−	4.59293e7i		−	0.131635i
$705$		−	5.39476e7i		−	0.153959i
$706$	1.58605e8			0.450714
$707$			1.30211e7i			0.0368458i
$708$	−6.66758e7			−0.187875
$709$			2.00481e8i			0.562517i	0.959632	+	0.281258i	$0.0907518\pi$
$709$			2.00481e8i			0.562517i	−0.959632	+	0.281258i	$0.909248\pi$
$710$			1.97634e8i			0.552188i
$711$			1.70026e8i			0.473050i
$712$			3.12637e7i			0.0866166i
$713$	−1.57500e7	+	3.97637e7i	−0.0434521	+	0.109703i
$714$	1.44836e7			0.0397908
$715$	7.31089e7			0.200010
$716$	−3.41469e8			−0.930276
$717$	1.09425e8			0.296866
$718$			7.14215e7i			0.192955i
$719$	5.03409e8			1.35436			0.677180	−	0.735817i	$-0.263202\pi$
$719$	5.03409e8			1.35436			0.677180	+	0.735817i	$0.263202\pi$
$720$		−	8.36204e7i		−	0.224035i
$721$	−9.94195e7			−0.265256
$722$	9.15841e7			0.243337
$723$			4.04518e8i			1.07034i
$724$			2.03380e8i			0.535912i
$725$	−2.60800e7			−0.0684375
$726$	−6.45778e7			−0.168762
$727$		−	2.36133e8i		−	0.614546i	−0.951621	−	0.307273i	$-0.900584\pi$
$727$		−	2.36133e8i		−	0.614546i	0.951621	−	0.307273i	$-0.0994164\pi$
$728$		−	1.93095e7i		−	0.0500468i
$729$	1.43489e7			0.0370370
$730$			4.04697e7i			0.104031i
$731$	−3.60448e8			−0.922763
$732$			1.61515e7i			0.0411794i
$733$			5.20274e8i			1.32105i	0.750803	+	0.660526i	$0.229667\pi$
$733$			5.20274e8i			1.32105i	−0.750803	+	0.660526i	$0.770333\pi$
$734$			2.81603e8i			0.712114i
$735$			2.38310e8i			0.600179i
$736$	−1.32142e8	+	3.33617e8i	−0.331441	+	0.836786i
$737$	1.41847e7			0.0354339
$738$	−9.39812e6			−0.0233815
$739$	−4.39693e8			−1.08947			−0.544736	−	0.838607i	$-0.683370\pi$
$739$	−4.39693e8			−1.08947			−0.544736	+	0.838607i	$0.683370\pi$
$740$	−6.04175e7			−0.149097
$741$		−	5.74863e7i		−	0.141289i
$742$	1.04305e7			0.0255325
$743$			3.38209e8i			0.824554i	0.911059	+	0.412277i	$0.135266\pi$
$743$			3.38209e8i			0.824554i	−0.911059	+	0.412277i	$0.864734\pi$
$744$	1.89093e7			0.0459152
$745$	6.03251e8			1.45891
$746$			2.07175e8i			0.499024i
$747$		−	6.99897e7i		−	0.167908i
$748$	1.75040e8			0.418248
$749$	−1.02624e8			−0.244232
$750$			8.01169e7i			0.189907i
$751$		−	8.04152e8i		−	1.89853i	−0.314472	−	0.949267i	$-0.601827\pi$
$751$		−	8.04152e8i		−	1.89853i	0.314472	−	0.949267i	$-0.398173\pi$
$752$	−6.63744e7			−0.156080
$753$			1.37340e7i			0.0321672i
$754$	3.06134e7			0.0714163
$755$			6.88940e8i			1.60081i
$756$			1.25121e7i			0.0289576i
$757$			4.21509e7i			0.0971670i	0.998819	+	0.0485835i	$0.0154707\pi$
$757$			4.21509e7i			0.0971670i	−0.998819	+	0.0485835i	$0.984529\pi$
$758$		−	2.56405e7i		−	0.0588734i
$759$	1.02022e8	+	4.04096e7i	0.233328	+	0.0924186i
$760$	−1.80692e8			−0.411623
$761$	5.67488e8			1.28766			0.643832	−	0.765167i	$-0.277344\pi$
$761$	5.67488e8			1.28766			0.643832	+	0.765167i	$0.277344\pi$
$762$	3.34918e7			0.0756960
$763$	−1.02034e8			−0.229705
$764$			1.02348e8i			0.229510i
$765$	1.76838e8			0.394996
$766$			8.87036e7i			0.197358i
$767$	−7.24591e7			−0.160586
$768$	1.59214e7			0.0351477
$769$			4.44661e7i			0.0977800i	0.998804	+	0.0488900i	$0.0155684\pi$
$769$			4.44661e7i			0.0977800i	−0.998804	+	0.0488900i	$0.984432\pi$
$770$			1.32534e7i			0.0290306i
$771$	8.40942e7			0.183486
$772$	−3.61476e7			−0.0785646
$773$			6.81005e8i			1.47439i	0.675681	+	0.737194i	$0.263850\pi$
$773$			6.81005e8i			1.47439i	−0.675681	+	0.737194i	$0.736150\pi$
$774$			4.64827e7i			0.100246i
$775$	−8.14524e6			−0.0174984
$776$		−	5.84782e8i		−	1.25144i
$777$	7.48925e6			0.0159652
$778$			4.83358e7i			0.102643i
$779$		−	5.24370e7i		−	0.110924i
$780$		−	1.09693e8i		−	0.231150i
$781$		−	2.96073e8i		−	0.621508i
$782$	−1.77193e8	−	7.01843e7i	−0.370533	−	0.146764i
$783$	−4.26346e7			−0.0888132
$784$	2.93205e8			0.608448
$785$	−8.28316e8			−1.71233
$786$	−3.24196e7			−0.0667637
$787$		−	5.36571e8i		−	1.10079i	−0.834905	−	0.550394i	$-0.814478\pi$
$787$		−	5.36571e8i		−	1.10079i	0.834905	−	0.550394i	$-0.185522\pi$
$788$	7.02017e8			1.43473
$789$			2.86451e8i			0.583202i
$790$	2.70224e8			0.548077
$791$	−1.34467e7			−0.0271698
$792$		−	4.85155e7i		−	0.0976573i
$793$			1.75524e7i			0.0351980i
$794$	6.16397e7			0.123140
$795$	1.27352e8			0.253456
$796$		−	1.81586e8i		−	0.360034i
$797$			7.42677e8i			1.46698i	0.679699	+	0.733491i	$0.262110\pi$
$797$			7.42677e8i			1.46698i	−0.679699	+	0.733491i	$0.737890\pi$
$798$	1.04213e7			0.0205076
$799$		−	1.40367e8i		−	0.275185i
$800$	−6.83383e7			−0.133473
$801$		−	2.20153e7i		−	0.0428377i
$802$		−	2.01269e8i		−	0.390170i
$803$		−	6.06272e7i		−	0.117090i
$804$		−	2.12828e7i		−	0.0409506i
$805$	−3.55986e7	+	8.98752e7i	−0.0682410	+	0.172287i
$806$	9.56108e6			0.0182600
$807$	−3.63342e7			−0.0691346
$808$	−7.57540e7			−0.143606
$809$	6.69319e8			1.26412			0.632059	−	0.774920i	$-0.282210\pi$
$809$	6.69319e8			1.26412			0.632059	+	0.774920i	$0.282210\pi$
$810$		−	2.28048e7i		−	0.0429112i
$811$	−1.60228e8			−0.300384			−0.150192	−	0.988657i	$-0.547989\pi$
$811$	−1.60228e8			−0.300384			−0.150192	+	0.988657i	$0.547989\pi$
$812$		−	3.71768e7i		−	0.0694391i
$813$	3.98599e8			0.741762
$814$	−1.35113e7			−0.0250510
$815$			1.63389e8i			0.301821i
$816$		−	2.17573e8i		−	0.400437i
$817$	−2.59351e8			−0.475579
$818$	−1.67454e8			−0.305940
$819$			1.35973e7i			0.0247515i
$820$		−	1.00058e8i		−	0.181472i
$821$	−7.01161e7			−0.126703			−0.0633517	−	0.997991i	$-0.520179\pi$
$821$	−7.01161e7			−0.126703			−0.0633517	+	0.997991i	$0.520179\pi$
$822$			1.26445e8i			0.227660i
$823$	−7.75298e8			−1.39081			−0.695407	−	0.718616i	$-0.744776\pi$
$823$	−7.75298e8			−1.39081			−0.695407	+	0.718616i	$0.744776\pi$
$824$		−	5.78403e8i		−	1.03383i
$825$			2.08982e7i			0.0372175i
$826$		−	1.31356e7i		−	0.0233083i
$827$			7.09941e8i			1.25518i	0.778544	+	0.627590i	$0.215958\pi$
$827$			7.09941e8i			1.25518i	−0.778544	+	0.627590i	$0.784042\pi$
$828$	6.06306e7	−	1.53073e8i	0.106807	−	0.269655i
$829$	5.98520e8			1.05055			0.525273	−	0.850934i	$-0.323963\pi$
$829$	5.98520e8			1.05055			0.525273	+	0.850934i	$0.323963\pi$
$830$	−1.11235e8			−0.194539
$831$	−4.42784e8			−0.771594
$832$	−7.48896e7			−0.130032
$833$			6.20063e8i			1.07275i
$834$	−1.13527e8			−0.195704
$835$		−	7.35474e8i		−	1.26330i
$836$	1.25946e8			0.215559
$837$	−1.33155e7			−0.0227082
$838$		−	5.84599e7i		−	0.0993405i
$839$		−	7.07760e8i		−	1.19840i	−0.800601	−	0.599198i	$-0.795486\pi$
$839$		−	7.07760e8i		−	1.19840i	0.800601	−	0.599198i	$-0.204514\pi$
$840$	4.27394e7			0.0721092
$841$	−4.68144e8			−0.787030
$842$		−	2.66837e8i		−	0.447003i
$843$			4.57330e8i			0.763390i
$844$	−4.33232e8			−0.720599
$845$			5.27336e8i			0.874012i
$846$	−1.81015e7			−0.0298953
$847$			8.52253e7i			0.140255i
$848$		−	1.56687e8i		−	0.256948i
$849$		−	1.45031e8i		−	0.236995i
$850$		−	3.62964e7i		−	0.0591027i
$851$	−9.16239e7	−	3.62912e7i	−0.148669	−	0.0588861i
$852$	−4.44229e8			−0.718270
$853$	5.67257e8			0.913972			0.456986	−	0.889474i	$-0.348929\pi$
$853$	5.67257e8			0.913972			0.456986	+	0.889474i	$0.348929\pi$
$854$	−3.18197e6			−0.00510885
$855$	1.27240e8			0.203575
$856$		−	5.97045e8i		−	0.951888i
$857$	6.81847e8			1.08329			0.541645	−	0.840607i	$-0.317802\pi$
$857$	6.81847e8			1.08329			0.541645	+	0.840607i	$0.317802\pi$
$858$		−	2.45308e7i		−	0.0388374i
$859$	−2.53119e8			−0.399342			−0.199671	−	0.979863i	$-0.563987\pi$
$859$	−2.53119e8			−0.399342			−0.199671	+	0.979863i	$0.563987\pi$
$860$	−4.94882e8			−0.778047
$861$			1.24030e7i			0.0194320i
$862$			2.76531e8i			0.431740i
$863$	2.00680e8			0.312228			0.156114	−	0.987739i	$-0.450103\pi$
$863$	2.00680e8			0.312228			0.156114	+	0.987739i	$0.450103\pi$
$864$	−1.11717e8			−0.173212
$865$			4.34147e8i			0.670792i
$866$		−	2.90737e8i		−	0.447658i
$867$	8.38503e7			0.128661
$868$		−	1.16110e7i		−	0.0177545i
$869$	−4.04819e8			−0.616881
$870$			6.77594e7i			0.102899i
$871$		−	2.31288e7i		−	0.0350024i
$872$		−	5.93612e8i		−	0.895268i
$873$			4.11791e8i			0.618920i
$874$	−1.27495e8	−	5.04994e7i	−0.190967	−	0.0756401i
$875$	1.05733e8			0.157829
$876$	−9.09651e7			−0.135320
$877$	−5.93741e8			−0.880234			−0.440117	−	0.897940i	$-0.645063\pi$
$877$	−5.93741e8			−0.880234			−0.440117	+	0.897940i	$0.645063\pi$
$878$	2.84440e8			0.420250
$879$			3.48720e8i			0.513464i
$880$	1.99093e8			0.292152
$881$		−	2.69314e8i		−	0.393851i	−0.980418	−	0.196925i	$-0.936904\pi$
$881$		−	2.69314e8i		−	0.393851i	0.980418	−	0.196925i	$-0.0630957\pi$
$882$	7.99623e7			0.116541
$883$	5.61291e8			0.815278			0.407639	−	0.913143i	$-0.366352\pi$
$883$	5.61291e8			0.815278			0.407639	+	0.913143i	$0.366352\pi$
$884$		−	2.85410e8i		−	0.413155i
$885$		−	1.60380e8i		−	0.231378i
$886$	1.66085e7			0.0238798
$887$	8.59983e8			1.23231			0.616154	−	0.787626i	$-0.288690\pi$
$887$	8.59983e8			1.23231			0.616154	+	0.787626i	$0.288690\pi$
$888$			4.35710e7i			0.0622241i
$889$		−	4.42001e7i		−	0.0629097i
$890$	−3.49889e7			−0.0496319
$891$			3.41636e7i			0.0482981i
$892$	−7.40848e8			−1.04384
$893$		−	1.00998e8i		−	0.141826i
$894$		−	2.02414e8i		−	0.283287i
$895$		−	8.21360e8i		−	1.14568i
$896$		−	1.25534e8i		−	0.174517i
$897$	6.58895e7	−	1.66350e8i	0.0912933	−	0.230487i
$898$	−4.27766e8			−0.590714
$899$	3.95642e7			0.0544531
$900$	3.13557e7			0.0430119
$901$	3.31357e8			0.453025
$902$		−	2.23762e7i		−	0.0304906i
$903$	6.13447e7			0.0833132
$904$		−	7.82302e7i		−	0.105893i
$905$	−4.89206e8			−0.660003
$906$	2.31166e8			0.310841
$907$		−	7.14901e8i		−	0.958129i	−0.877780	−	0.479065i	$-0.840976\pi$
$907$		−	7.14901e8i		−	0.958129i	0.877780	−	0.479065i	$-0.159024\pi$
$908$			8.96899e7i			0.119808i
$909$	5.33443e7			0.0710226
$910$	2.16103e7			0.0286771
$911$		−	5.76443e8i		−	0.762432i	−0.924486	−	0.381216i	$-0.875505\pi$
$911$		−	5.76443e8i		−	0.762432i	0.924486	−	0.381216i	$-0.124495\pi$
$912$		−	1.56549e8i		−	0.206380i
$913$	1.66640e8			0.218961
$914$		−	1.17978e8i		−	0.154512i
$915$	−3.88505e7			−0.0507146
$916$		−	8.83799e8i		−	1.14992i
$917$			4.27852e7i			0.0554863i
$918$		−	5.93360e7i		−	0.0766991i
$919$		−	1.49428e9i		−	1.92524i	−0.270858	−	0.962619i	$-0.587307\pi$
$919$		−	1.49428e9i		−	1.92524i	0.270858	−	0.962619i	$-0.412693\pi$
$920$	−5.22876e8	−	2.07106e8i	−0.671484	−	0.265967i
$921$	−3.79202e8			−0.485391
$922$	−4.41291e8			−0.563031
$923$	−4.82760e8			−0.613940
$924$	−2.97902e7			−0.0377622
$925$		−	1.87683e7i		−	0.0237138i
$926$	−5.09406e7			−0.0641551
$927$			4.07299e8i			0.511298i
$928$	3.31942e8			0.415354
$929$	−8.50023e8			−1.06019			−0.530095	−	0.847938i	$-0.677844\pi$
$929$	−8.50023e8			−1.06019			−0.530095	+	0.847938i	$0.677844\pi$
$930$			2.11624e7i			0.0263097i
$931$			4.46151e8i			0.552882i
$932$	4.05864e8			0.501340
$933$	5.02601e8			0.618841
$934$		−	3.33715e7i		−	0.0409577i
$935$			4.21038e8i			0.515094i
$936$	−7.91064e7			−0.0964682
$937$			6.39691e8i			0.777592i	0.921324	+	0.388796i	$0.127109\pi$
$937$			6.39691e8i			0.777592i	−0.921324	+	0.388796i	$0.872891\pi$
$938$	4.19287e6			0.00508046
$939$		−	4.67142e8i		−	0.564224i
$940$		−	1.92719e8i		−	0.232028i
$941$			4.61469e8i			0.553826i	0.960895	+	0.276913i	$0.0893114\pi$
$941$			4.61469e8i			0.553826i	−0.960895	+	0.276913i	$0.910689\pi$
$942$			2.77932e8i			0.332495i
$943$	6.01021e7	−	1.51739e8i	0.0716728	−	0.180951i
$944$	−1.97324e8			−0.234565
$945$	−3.00962e7			−0.0356628
$946$	−1.10672e8			−0.130726
$947$	6.26119e8			0.737237			0.368619	−	0.929581i	$-0.379831\pi$
$947$	6.26119e8			0.737237			0.368619	+	0.929581i	$0.379831\pi$
$948$			6.07390e8i			0.712923i
$949$	−9.88551e7			−0.115665
$950$		−	2.61162e7i		−	0.0304607i
$951$	6.05363e8			0.703840
$952$	1.11204e8			0.128887
$953$			4.99599e8i			0.577222i	0.957446	+	0.288611i	$0.0931934\pi$
$953$			4.99599e8i			0.577222i	−0.957446	+	0.288611i	$0.906807\pi$
$954$		−	4.27313e7i		−	0.0492154i
$955$	−2.46186e8			−0.282653
$956$	3.90903e8			0.447399
$957$		−	1.01510e8i		−	0.115817i
$958$		−	2.14929e8i		−	0.244455i
$959$	1.66873e8			0.189204
$960$		−	1.65760e8i		−	0.187355i
$961$	−8.75147e8			−0.986077
$962$			2.20307e7i			0.0247459i
$963$			4.20426e8i			0.470772i
$964$			1.44507e9i			1.61309i
$965$		−	8.69484e7i		−	0.0967564i
$966$	3.01566e7	+	1.19447e7i	0.0334542	+	0.0132508i
$967$	5.93301e8			0.656138			0.328069	−	0.944654i	$-0.393602\pi$
$967$	5.93301e8			0.656138			0.328069	+	0.944654i	$0.393602\pi$
$968$	−4.95824e8			−0.546640
$969$	3.31067e8			0.363868
$970$	6.54462e8			0.717082
$971$			8.71183e8i			0.951594i	0.879555	+	0.475797i	$0.157840\pi$
$971$			8.71183e8i			0.951594i	−0.879555	+	0.475797i	$0.842160\pi$
$972$	5.12590e7			0.0558176
$973$			1.49825e8i			0.162647i
$974$	−3.51253e8			−0.380140
$975$	3.40753e7			0.0367643
$976$			4.77996e7i			0.0514133i
$977$			1.19611e9i			1.28259i	0.767296	+	0.641293i	$0.221602\pi$
$977$			1.19611e9i			1.28259i	−0.767296	+	0.641293i	$0.778398\pi$
$978$	5.48231e7			0.0586067
$979$	5.24165e7			0.0558625
$980$			8.51323e8i			0.904516i
$981$			4.18009e8i			0.442770i
$982$	1.68924e7			0.0178384
$983$		−	6.05873e7i		−	0.0637854i	−0.999491	−	0.0318927i	$-0.989847\pi$
$983$		−	6.05873e7i		−	0.0637854i	0.999491	−	0.0318927i	$-0.0101535\pi$
$984$	−7.21581e7			−0.0757356
$985$			1.68861e9i			1.76694i
$986$			1.76304e8i			0.183921i
$987$			2.38891e7i			0.0248455i
$988$		−	2.05360e8i		−	0.212934i
$989$	−7.50494e8	−	2.97263e8i	−0.775816	−	0.307292i
$990$	5.42963e7			0.0559583
$991$	−2.72089e7			−0.0279570			−0.0139785	−	0.999902i	$-0.504450\pi$
$991$	−2.72089e7			−0.0279570			−0.0139785	+	0.999902i	$0.504450\pi$
$992$	1.03671e8			0.106200
$993$	−4.26165e8			−0.435241
$994$		−	8.75164e7i		−	0.0891108i
$995$	4.36782e8			0.443400
$996$		−	2.50026e8i		−	0.253051i
$997$	1.32840e9			1.34043			0.670215	−	0.742167i	$-0.266202\pi$
$997$	1.32840e9			1.34043			0.670215	+	0.742167i	$0.266202\pi$
$998$	5.30322e8			0.533517
$999$		−	3.06818e7i		−	0.0307740i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.15	✓	24
3.2	odd	2		207.7.d.e.91.9		24
23.22	odd	2	inner	69.7.d.a.22.16	yes	24
69.68	even	2		207.7.d.e.91.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.15	✓	24	1.1	even	1	trivial
69.7.d.a.22.16	yes	24	23.22	odd	2	inner
207.7.d.e.91.9		24	3.2	odd	2
207.7.d.e.91.10		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)

\(f(q)\)	\(=\)	\(q+2.88321 q^{2} -15.5885 q^{3} -55.6871 q^{4} -133.948i q^{5} -44.9448 q^{6} +59.3150i q^{7} -345.083 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+2.88321 q^{2} -15.5885 q^{3} -55.6871 q^{4} -133.948i q^{5} -44.9448 q^{6} +59.3150i q^{7} -345.083 q^{8} +243.000 q^{9} -386.201i q^{10} +578.563i q^{11} +868.076 q^{12} +943.370 q^{13} +171.018i q^{14} +2088.05i q^{15} +2569.03 q^{16} +5432.92i q^{17} +700.620 q^{18} +3909.12i q^{19} +7459.19i q^{20} -924.630i q^{21} +1668.12i q^{22} +(-4480.54 + 11312.0i) q^{23} +5379.31 q^{24} -2317.15 q^{25} +2719.93 q^{26} -3788.00 q^{27} -3303.08i q^{28} +11255.2 q^{29} +6020.28i q^{30} +3515.19 q^{31} +29492.4 q^{32} -9018.91i q^{33} +15664.2i q^{34} +7945.15 q^{35} -13532.0 q^{36} +8099.73i q^{37} +11270.8i q^{38} -14705.7 q^{39} +46223.3i q^{40} -13414.0 q^{41} -2665.90i q^{42} +66345.2i q^{43} -32218.5i q^{44} -32549.4i q^{45} +(-12918.3 + 32614.8i) q^{46} -25836.4 q^{47} -40047.2 q^{48} +114131. q^{49} -6680.84 q^{50} -84690.8i q^{51} -52533.5 q^{52} -60990.7i q^{53} -10921.6 q^{54} +77497.6 q^{55} -20468.6i q^{56} -60937.2i q^{57} +32451.1 q^{58} -76808.7 q^{59} -116277. i q^{60} +18606.1i q^{61} +10135.0 q^{62} +14413.5i q^{63} -79385.2 q^{64} -126363. i q^{65} -26003.4i q^{66} -24517.2i q^{67} -302543. i q^{68} +(69844.8 - 176336. i) q^{69} +22907.5 q^{70} -511739. q^{71} -83855.2 q^{72} -104789. q^{73} +23353.2i q^{74} +36120.9 q^{75} -217688. i q^{76} -34317.5 q^{77} -42399.6 q^{78} +699697. i q^{79} -344117. i q^{80} +59049.0 q^{81} -38675.4 q^{82} -288024. i q^{83} +51489.9i q^{84} +727730. q^{85} +191287. i q^{86} -175451. q^{87} -199652. i q^{88} -90597.8i q^{89} -93846.9i q^{90} +55956.0i q^{91} +(249509. - 629931. i) q^{92} -54796.4 q^{93} -74491.7 q^{94} +523620. q^{95} -459740. q^{96} +1.69461e6i q^{97} +329063. q^{98} +140591. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 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

Varieties

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Database

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