LMFDB - Embedded newform 69.7.d.a.22.3

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			22.3
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.6749 q^{2} -15.5885 q^{3} +123.003 q^{4} -37.2861i q^{5} +213.170 q^{6} -322.993i q^{7} -806.853 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-13.6749 q^{2} -15.5885 q^{3} +123.003 q^{4} -37.2861i q^{5} +213.170 q^{6} -322.993i q^{7} -806.853 q^{8} +243.000 q^{9} +509.883i q^{10} +2211.16i q^{11} -1917.42 q^{12} -449.620 q^{13} +4416.89i q^{14} +581.232i q^{15} +3161.46 q^{16} +5356.12i q^{17} -3323.00 q^{18} -2647.39i q^{19} -4586.28i q^{20} +5034.96i q^{21} -30237.4i q^{22} +(-4323.93 - 11372.8i) q^{23} +12577.6 q^{24} +14234.7 q^{25} +6148.50 q^{26} -3788.00 q^{27} -39729.0i q^{28} +3292.80 q^{29} -7948.29i q^{30} +21814.2 q^{31} +8405.97 q^{32} -34468.6i q^{33} -73244.3i q^{34} -12043.1 q^{35} +29889.6 q^{36} -36999.4i q^{37} +36202.8i q^{38} +7008.88 q^{39} +30084.4i q^{40} -84193.7 q^{41} -68852.6i q^{42} -23651.2i q^{43} +271978. i q^{44} -9060.51i q^{45} +(59129.3 + 155521. i) q^{46} +65868.0 q^{47} -49282.3 q^{48} +13324.4 q^{49} -194659. q^{50} -83493.6i q^{51} -55304.3 q^{52} -183208. i q^{53} +51800.4 q^{54} +82445.5 q^{55} +260608. i q^{56} +41268.8i q^{57} -45028.6 q^{58} +283187. q^{59} +71493.0i q^{60} -36008.5i q^{61} -298307. q^{62} -78487.3i q^{63} -317284. q^{64} +16764.5i q^{65} +471354. i q^{66} -154256. i q^{67} +658816. i q^{68} +(67403.4 + 177284. i) q^{69} +164689. q^{70} -415789. q^{71} -196065. q^{72} -44631.0 q^{73} +505963. i q^{74} -221898. q^{75} -325636. i q^{76} +714190. q^{77} -95845.6 q^{78} -751803. i q^{79} -117878. i q^{80} +59049.0 q^{81} +1.15134e6 q^{82} -988237. i q^{83} +619313. i q^{84} +199709. q^{85} +323427. i q^{86} -51329.6 q^{87} -1.78408e6i q^{88} -410195. i q^{89} +123902. i q^{90} +145224. i q^{91} +(-531854. - 1.39888e6i) q^{92} -340050. q^{93} -900737. q^{94} -98710.9 q^{95} -131036. q^{96} -1.20225e6i q^{97} -182210. q^{98} +537312. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−13.6749			−1.70936			−0.854680	−	0.519154i	$-0.826247\pi$
$2$	−13.6749			−1.70936			−0.854680	+	0.519154i	$0.826247\pi$
$3$	−15.5885			−0.577350
$3$	−15.5885			−0.577350
$4$	123.003			1.92191
$5$		−	37.2861i		−	0.298289i	−0.988815	−	0.149144i	$-0.952348\pi$
$5$		−	37.2861i		−	0.298289i	0.988815	−	0.149144i	$-0.0476519\pi$
$6$	213.170			0.986900
$7$		−	322.993i		−	0.941671i	−0.882221	−	0.470835i	$-0.843953\pi$
$7$		−	322.993i		−	0.941671i	0.882221	−	0.470835i	$-0.156047\pi$
$8$	−806.853			−1.57588
$9$	243.000			0.333333
$10$			509.883i			0.509883i
$11$			2211.16i			1.66128i	0.556811	+	0.830639i	$0.312025\pi$
$11$			2211.16i			1.66128i	−0.556811	+	0.830639i	$0.687975\pi$
$12$	−1917.42			−1.10962
$13$	−449.620			−0.204652			−0.102326	−	0.994751i	$-0.532628\pi$
$13$	−449.620			−0.204652			−0.102326	+	0.994751i	$0.532628\pi$
$14$			4416.89i			1.60966i
$15$			581.232i			0.172217i
$16$	3161.46			0.771841
$17$			5356.12i			1.09019i	0.838373	+	0.545096i	$0.183507\pi$
$17$			5356.12i			1.09019i	−0.838373	+	0.545096i	$0.816493\pi$
$18$	−3323.00			−0.569787
$19$		−	2647.39i		−	0.385974i	−0.981201	−	0.192987i	$-0.938183\pi$
$19$		−	2647.39i		−	0.385974i	0.981201	−	0.192987i	$-0.0618175\pi$
$20$		−	4586.28i		−	0.573285i
$21$			5034.96i			0.543674i
$22$		−	30237.4i		−	2.83972i
$23$	−4323.93	−	11372.8i	−0.355382	−	0.934721i
$23$	−4323.93	−	11372.8i	−0.355382	−	0.934721i
$24$	12577.6			0.909837
$25$	14234.7			0.911024
$26$	6148.50			0.349823
$27$	−3788.00			−0.192450
$28$		−	39729.0i		−	1.80981i
$29$	3292.80			0.135012			0.0675058	−	0.997719i	$-0.478496\pi$
$29$	3292.80			0.135012			0.0675058	+	0.997719i	$0.478496\pi$
$30$		−	7948.29i		−	0.294381i
$31$	21814.2			0.732243			0.366121	−	0.930567i	$-0.380686\pi$
$31$	21814.2			0.732243			0.366121	+	0.930567i	$0.380686\pi$
$32$	8405.97			0.256530
$33$		−	34468.6i		−	0.959139i
$34$		−	73244.3i		−	1.86353i
$35$	−12043.1			−0.280890
$36$	29889.6			0.640638
$37$		−	36999.4i		−	0.730449i	−0.930920	−	0.365224i	$-0.880992\pi$
$37$		−	36999.4i		−	0.730449i	0.930920	−	0.365224i	$-0.119008\pi$
$38$			36202.8i			0.659768i
$39$	7008.88			0.118156
$40$			30084.4i			0.470068i
$41$	−84193.7			−1.22160			−0.610799	−	0.791786i	$-0.709152\pi$
$41$	−84193.7			−1.22160			−0.610799	+	0.791786i	$0.709152\pi$
$42$		−	68852.6i		−	0.929335i
$43$		−	23651.2i		−	0.297473i	−0.988877	−	0.148736i	$-0.952479\pi$
$43$		−	23651.2i		−	0.297473i	0.988877	−	0.148736i	$-0.0475206\pi$
$44$			271978.i			3.19283i
$45$		−	9060.51i		−	0.0994295i
$46$	59129.3	+	155521.i	0.607476	+	1.59778i
$47$	65868.0			0.634426			0.317213	−	0.948354i	$-0.397253\pi$
$47$	65868.0			0.634426			0.317213	+	0.948354i	$0.397253\pi$
$48$	−49282.3			−0.445623
$49$	13324.4			0.113256
$50$	−194659.			−1.55727
$51$		−	83493.6i		−	0.629423i
$52$	−55304.3			−0.393323
$53$		−	183208.i		−	1.23060i	−0.788294	−	0.615299i	$-0.789035\pi$
$53$		−	183208.i		−	1.23060i	0.788294	−	0.615299i	$-0.210965\pi$
$54$	51800.4			0.328967
$55$	82445.5			0.495540
$56$			260608.i			1.48396i
$57$			41268.8i			0.222842i
$58$	−45028.6			−0.230783
$59$	283187.			1.37885			0.689426	−	0.724357i	$-0.257863\pi$
$59$	283187.			1.37885			0.689426	+	0.724357i	$0.257863\pi$
$60$			71493.0i			0.330986i
$61$		−	36008.5i		−	0.158641i	−0.996849	−	0.0793204i	$-0.974725\pi$
$61$		−	36008.5i		−	0.158641i	0.996849	−	0.0793204i	$-0.0252750\pi$
$62$	−298307.			−1.25167
$63$		−	78487.3i		−	0.313890i
$64$	−317284.			−1.21034
$65$			16764.5i			0.0610452i
$66$			471354.i			1.63952i
$67$		−	154256.i		−	0.512883i	−0.966560	−	0.256442i	$-0.917450\pi$
$67$		−	154256.i		−	0.512883i	0.966560	−	0.256442i	$-0.0825502\pi$
$68$			658816.i			2.09526i
$69$	67403.4	+	177284.i	0.205180	+	0.539662i
$70$	164689.			0.480142
$71$	−415789.			−1.16171			−0.580856	−	0.814007i	$-0.697282\pi$
$71$	−415789.			−1.16171			−0.580856	+	0.814007i	$0.697282\pi$
$72$	−196065.			−0.525295
$73$	−44631.0			−0.114728			−0.0573638	−	0.998353i	$-0.518269\pi$
$73$	−44631.0			−0.114728			−0.0573638	+	0.998353i	$0.518269\pi$
$74$			505963.i			1.24860i
$75$	−221898.			−0.525980
$76$		−	325636.i		−	0.741808i
$77$	714190.			1.56438
$78$	−95845.6			−0.201971
$79$		−	751803.i		−	1.52483i	−0.647086	−	0.762417i	$-0.724012\pi$
$79$		−	751803.i		−	1.52483i	0.647086	−	0.762417i	$-0.275988\pi$
$80$		−	117878.i		−	0.230231i
$81$	59049.0			0.111111
$82$	1.15134e6			2.08815
$83$		−	988237.i		−	1.72833i	−0.503208	−	0.864166i	$-0.667847\pi$
$83$		−	988237.i		−	1.72833i	0.503208	−	0.864166i	$-0.332153\pi$
$84$			619313.i			1.04489i
$85$	199709.			0.325192
$86$			323427.i			0.508488i
$87$	−51329.6			−0.0779490
$88$		−	1.78408e6i		−	2.61798i
$89$		−	410195.i		−	0.581863i	−0.956744	−	0.290932i	$-0.906035\pi$
$89$		−	410195.i		−	0.581863i	0.956744	−	0.290932i	$-0.0939652\pi$
$90$			123902.i			0.169961i
$91$			145224.i			0.192714i
$92$	−531854.	−	1.39888e6i	−0.683014	−	1.79645i
$93$	−340050.			−0.422760
$94$	−900737.			−1.08446
$95$	−98710.9			−0.115132
$96$	−131036.			−0.148108
$97$		−	1.20225e6i		−	1.31728i	−0.752458	−	0.658640i	$-0.771132\pi$
$97$		−	1.20225e6i		−	1.31728i	0.752458	−	0.658640i	$-0.228868\pi$
$98$	−182210.			−0.193595
$99$			537312.i			0.553759i
$100$	1.75091e6			1.75091
$101$	1.05470e6			1.02368			0.511841	−	0.859080i	$-0.328964\pi$
$101$	1.05470e6			1.02368			0.511841	+	0.859080i	$0.328964\pi$
$102$			1.14177e6i			1.07591i
$103$			1.03009e6i			0.942675i	0.881953	+	0.471338i	$0.156229\pi$
$103$			1.03009e6i			0.942675i	−0.881953	+	0.471338i	$0.843771\pi$
$104$	362777.			0.322507
$105$	187734.			0.162172
$106$			2.50534e6i			2.10353i
$107$		−	1.68790e6i		−	1.37783i	−0.724842	−	0.688915i	$-0.758087\pi$
$107$		−	1.68790e6i		−	1.37783i	0.724842	−	0.688915i	$-0.241913\pi$
$108$	−465933.			−0.369873
$109$			1.83556e6i			1.41739i	0.705515	+	0.708695i	$0.250716\pi$
$109$			1.83556e6i			1.41739i	−0.705515	+	0.708695i	$0.749284\pi$
$110$	−1.12743e6			−0.847057
$111$			576764.i			0.421725i
$112$		−	1.02113e6i		−	0.726820i
$113$			949159.i			0.657815i	0.944362	+	0.328908i	$0.106680\pi$
$113$			949159.i			0.657815i	−0.944362	+	0.328908i	$0.893320\pi$
$114$		−	564346.i		−	0.380917i
$115$	−424045.	+	161222.i	−0.278817	+	0.106006i
$116$	405022.			0.259481
$117$	−109258.			−0.0682172
$118$	−3.87255e6			−2.35695
$119$	1.72999e6			1.02660
$120$		−	468969.i		−	0.271394i
$121$	−3.11767e6			−1.75985
$122$			492412.i			0.271175i
$123$	1.31245e6			0.705290
$124$	2.68321e6			1.40731
$125$		−	1.11335e6i		−	0.570037i
$126$			1.07331e6i			0.536552i
$127$	1.28898e6			0.629267			0.314634	−	0.949213i	$-0.398118\pi$
$127$	1.28898e6			0.629267			0.314634	+	0.949213i	$0.398118\pi$
$128$	3.80084e6			1.81238
$129$			368685.i			0.171746i
$130$		−	229253.i		−	0.104348i
$131$	−4.14126e6			−1.84213			−0.921063	−	0.389415i	$-0.872677\pi$
$131$	−4.14126e6			−1.84213			−0.921063	+	0.389415i	$0.872677\pi$
$132$		−	4.23972e6i		−	1.84338i
$133$	−855090.			−0.363460
$134$			2.10944e6i			0.876703i
$135$			141239.i			0.0574057i
$136$		−	4.32160e6i		−	1.71802i
$137$		−	1.53664e6i		−	0.597599i	−0.954316	−	0.298800i	$-0.903414\pi$
$137$		−	1.53664e6i		−	0.597599i	0.954316	−	0.298800i	$-0.0965862\pi$
$138$	−921734.	−	2.42433e6i	−0.350726	−	0.922476i
$139$	2.86467e6			1.06667			0.533334	−	0.845904i	$-0.320939\pi$
$139$	2.86467e6			1.06667			0.533334	+	0.845904i	$0.320939\pi$
$140$	−1.48134e6			−0.539846
$141$	−1.02678e6			−0.366286
$142$	5.68587e6			1.98578
$143$		−	994181.i		−	0.339983i
$144$	768235.			0.257280
$145$		−	122775.i		−	0.0402724i
$146$	610324.			0.196111
$147$	−207707.			−0.0653882
$148$		−	4.55102e6i		−	1.40386i
$149$			2.26609e6i			0.685043i	0.939510	+	0.342522i	$0.111281\pi$
$149$			2.26609e6i			0.685043i	−0.939510	+	0.342522i	$0.888719\pi$
$150$	3.03443e6			0.899089
$151$	1.59103e6			0.462112			0.231056	−	0.972941i	$-0.425782\pi$
$151$	1.59103e6			0.462112			0.231056	+	0.972941i	$0.425782\pi$
$152$			2.13606e6i			0.608250i
$153$			1.30154e6i			0.363398i
$154$	−9.76647e6			−2.67409
$155$		−	813367.i		−	0.218420i
$156$	862109.			0.227085
$157$			2.27500e6i			0.587872i	0.955825	+	0.293936i	$0.0949652\pi$
$157$			2.27500e6i			0.587872i	−0.955825	+	0.293936i	$0.905035\pi$
$158$			1.02808e7i			2.60649i
$159$			2.85592e6i			0.710486i
$160$		−	313426.i		−	0.0765199i
$161$	−3.67332e6	+	1.39660e6i	−0.880200	+	0.334653i
$162$	−807488.			−0.189929
$163$	7.17414e6			1.65656			0.828280	−	0.560314i	$-0.189320\pi$
$163$	7.17414e6			1.65656			0.828280	+	0.560314i	$0.189320\pi$
$164$	−1.03560e7			−2.34781
$165$	−1.28520e6			−0.286100
$166$			1.35140e7i			2.95434i
$167$	3.11636e6			0.669112			0.334556	−	0.942376i	$-0.391414\pi$
$167$	3.11636e6			0.669112			0.334556	+	0.942376i	$0.391414\pi$
$168$		−	4.06248e6i		−	0.856767i
$169$	−4.62465e6			−0.958118
$170$	−2.73099e6			−0.555870
$171$		−	643317.i		−	0.128658i
$172$		−	2.90915e6i		−	0.571717i
$173$	7.18746e6			1.38815			0.694077	−	0.719901i	$-0.255813\pi$
$173$	7.18746e6			1.38815			0.694077	+	0.719901i	$0.255813\pi$
$174$	701927.			0.133243
$175$		−	4.59773e6i		−	0.857885i
$176$			6.99050e6i			1.28224i
$177$	−4.41445e6			−0.796080
$178$			5.60938e6i			0.994614i
$179$	−2.51654e6			−0.438777			−0.219389	−	0.975638i	$-0.570406\pi$
$179$	−2.51654e6			−0.438777			−0.219389	+	0.975638i	$0.570406\pi$
$180$		−	1.11447e6i		−	0.191095i
$181$		−	5.09223e6i		−	0.858761i	−0.903124	−	0.429380i	$-0.858732\pi$
$181$		−	5.09223e6i		−	0.858761i	0.903124	−	0.429380i	$-0.141268\pi$
$182$		−	1.98592e6i		−	0.329419i
$183$			561316.i			0.0915914i
$184$	3.48878e6	+	9.17614e6i	0.560041	+	1.47301i
$185$	−1.37956e6			−0.217884
$186$	4.65015e6			0.722650
$187$	−1.18432e7			−1.81111
$188$	8.10193e6			1.21931
$189$			1.22350e6i			0.181225i
$190$	1.34986e6			0.196801
$191$		−	1.03443e7i		−	1.48457i	−0.670083	−	0.742286i	$-0.733742\pi$
$191$		−	1.03443e7i		−	1.48457i	0.670083	−	0.742286i	$-0.266258\pi$
$192$	4.94597e6			0.698792
$193$	−375366.			−0.0522136			−0.0261068	−	0.999659i	$-0.508311\pi$
$193$	−375366.			−0.0522136			−0.0261068	+	0.999659i	$0.508311\pi$
$194$			1.64406e7i			2.25171i
$195$		−	261333.i		−	0.0352445i
$196$	1.63894e6			0.217668
$197$	1.02944e7			1.34649			0.673246	−	0.739418i	$-0.264899\pi$
$197$	1.02944e7			1.34649			0.673246	+	0.739418i	$0.264899\pi$
$198$		−	7.34768e6i		−	0.946575i
$199$			609008.i			0.0772795i	0.999253	+	0.0386397i	$0.0123025\pi$
$199$			609008.i			0.0772795i	−0.999253	+	0.0386397i	$0.987698\pi$
$200$	−1.14853e7			−1.43567
$201$			2.40462e6i			0.296113i
$202$	−1.44229e7			−1.74984
$203$		−	1.06355e6i		−	0.127136i
$204$		−	1.02699e7i		−	1.20970i
$205$			3.13925e6i			0.364389i
$206$		−	1.40863e7i		−	1.61137i
$207$	−1.05072e6	−	2.76358e6i	−0.118461	−	0.311574i
$208$	−1.42145e6			−0.157958
$209$	5.85381e6			0.641210
$210$	−2.56724e6			−0.277210
$211$	1.63949e7			1.74527			0.872634	−	0.488375i	$-0.162410\pi$
$211$	1.63949e7			1.74527			0.872634	+	0.488375i	$0.162410\pi$
$212$		−	2.25350e7i		−	2.36510i
$213$	6.48151e6			0.670714
$214$			2.30819e7i			2.35521i
$215$	−881859.			−0.0887327
$216$	3.05635e6			0.303279
$217$		−	7.04585e6i		−	0.689532i
$218$		−	2.51011e7i		−	2.42283i
$219$	695728.			0.0662380
$220$	1.01410e7			0.952386
$221$		−	2.40821e6i		−	0.223110i
$222$		−	7.88718e6i		−	0.720880i
$223$	−5.00948e6			−0.451729			−0.225865	−	0.974159i	$-0.572521\pi$
$223$	−5.00948e6			−0.451729			−0.225865	+	0.974159i	$0.572521\pi$
$224$		−	2.71507e6i		−	0.241567i
$225$	3.45904e6			0.303675
$226$		−	1.29796e7i		−	1.12444i
$227$		−	1.80090e7i		−	1.53962i	−0.638276	−	0.769808i	$-0.720352\pi$
$227$		−	1.80090e7i		−	1.53962i	0.638276	−	0.769808i	$-0.279648\pi$
$228$			5.07616e6i			0.428283i
$229$			1.75626e7i			1.46245i	0.682135	+	0.731227i	$0.261052\pi$
$229$			1.75626e7i			1.46245i	−0.682135	+	0.731227i	$0.738948\pi$
$230$	5.79877e6	−	2.20470e6i	0.476598	−	0.181203i
$231$	−1.11331e7			−0.903194
$232$	−2.65680e6			−0.212763
$233$	−9.02808e6			−0.713720			−0.356860	−	0.934158i	$-0.616153\pi$
$233$	−9.02808e6			−0.713720			−0.356860	+	0.934158i	$0.616153\pi$
$234$	1.49408e6			0.116608
$235$		−	2.45596e6i		−	0.189242i
$236$	3.48327e7			2.65003
$237$			1.17194e7i			0.880364i
$238$	−2.36574e7			−1.75483
$239$	−3.96922e6			−0.290744			−0.145372	−	0.989377i	$-0.546438\pi$
$239$	−3.96922e6			−0.290744			−0.145372	+	0.989377i	$0.546438\pi$
$240$			1.83754e6i			0.132924i
$241$			629815.i			0.0449948i	0.999747	+	0.0224974i	$0.00716175\pi$
$241$			629815.i			0.0449948i	−0.999747	+	0.0224974i	$0.992838\pi$
$242$	4.26338e7			3.00821
$243$	−920483.			−0.0641500
$244$		−	4.42913e6i		−	0.304894i
$245$		−	496816.i		−	0.0337829i
$246$	−1.79476e7			−1.20559
$247$			1.19032e6i			0.0789901i
$248$	−1.76009e7			−1.15393
$249$			1.54051e7i			0.997852i
$250$			1.52250e7i			0.974398i
$251$		−	1.95221e7i		−	1.23454i	−0.786750	−	0.617271i	$-0.788238\pi$
$251$		−	1.95221e7i		−	1.23454i	0.786750	−	0.617271i	$-0.211762\pi$
$252$		−	9.65414e6i		−	0.603270i
$253$	2.51470e7	−	9.56091e6i	1.55283	−	0.590388i
$254$	−1.76267e7			−1.07564
$255$	−3.11315e6			−0.187750
$256$	−3.16699e7			−1.88767
$257$	−2.29451e7			−1.35173			−0.675865	−	0.737026i	$-0.736230\pi$
$257$	−2.29451e7			−1.35173			−0.675865	+	0.737026i	$0.736230\pi$
$258$		−	5.04173e6i		−	0.293576i
$259$	−1.19506e7			−0.687842
$260$			2.06208e6i			0.117324i
$261$	800150.			0.0450039
$262$	5.66313e7			3.14886
$263$			1.91287e7i			1.05152i	0.850633	+	0.525760i	$0.176219\pi$
$263$			1.91287e7i			1.05152i	−0.850633	+	0.525760i	$0.823781\pi$
$264$			2.78111e7i			1.51149i
$265$	−6.83109e6			−0.367073
$266$	1.16933e7			0.621285
$267$			6.39431e6i			0.335939i
$268$		−	1.89739e7i		−	0.985718i
$269$	1.19114e6			0.0611938			0.0305969	−	0.999532i	$-0.490259\pi$
$269$	1.19114e6			0.0611938			0.0305969	+	0.999532i	$0.490259\pi$
$270$		−	1.93143e6i		−	0.0981270i
$271$	−2.17557e6			−0.109311			−0.0546555	−	0.998505i	$-0.517406\pi$
$271$	−2.17557e6			−0.109311			−0.0546555	+	0.998505i	$0.517406\pi$
$272$			1.69332e7i			0.841455i
$273$		−	2.26382e6i		−	0.111264i
$274$			2.10134e7i			1.02151i
$275$			3.14753e7i			1.51346i
$276$	8.29079e6	+	2.18063e7i	0.394338	+	1.03718i
$277$	−1.09170e7			−0.513644			−0.256822	−	0.966459i	$-0.582675\pi$
$277$	−1.09170e7			−0.513644			−0.256822	+	0.966459i	$0.582675\pi$
$278$	−3.91740e7			−1.82332
$279$	5.30086e6			0.244081
$280$	9.71705e6			0.442650
$281$			3.02627e7i			1.36392i	0.731389	+	0.681960i	$0.238872\pi$
$281$			3.02627e7i			1.36392i	−0.731389	+	0.681960i	$0.761128\pi$
$282$	1.40411e7			0.626115
$283$		−	1.25781e7i		−	0.554952i	−0.960733	−	0.277476i	$-0.910502\pi$
$283$		−	1.25781e7i		−	0.554952i	0.960733	−	0.277476i	$-0.0894979\pi$
$284$	−5.11431e7			−2.23271
$285$	1.53875e6			0.0664712
$286$			1.35953e7i			0.581154i
$287$			2.71940e7i			1.15034i
$288$	2.04265e6			0.0855100
$289$	−4.55042e6			−0.188520
$290$			1.67894e6i			0.0688401i
$291$			1.87412e7i			0.760532i
$292$	−5.48972e6			−0.220497
$293$			3.33132e7i			1.32438i	0.749334	+	0.662192i	$0.230374\pi$
$293$			3.33132e7i			1.32438i	−0.749334	+	0.662192i	$0.769626\pi$
$294$	2.84037e6			0.111772
$295$		−	1.05589e7i		−	0.411295i
$296$			2.98531e7i			1.15110i
$297$		−	8.37587e6i		−	0.319713i
$298$		−	3.09885e7i		−	1.17099i
$299$	1.94412e6	+	5.11341e6i	0.0727295	+	0.191292i
$300$	−2.72940e7			−1.01089
$301$	−7.63916e6			−0.280121
$302$	−2.17571e7			−0.789915
$303$	−1.64411e7			−0.591023
$304$		−	8.36963e6i		−	0.297910i
$305$	−1.34261e6			−0.0473208
$306$		−	1.77984e7i		−	0.621178i
$307$	−1.60265e7			−0.553889			−0.276944	−	0.960886i	$-0.589322\pi$
$307$	−1.60265e7			−0.553889			−0.276944	+	0.960886i	$0.589322\pi$
$308$	8.78472e7			3.00660
$309$		−	1.60575e7i		−	0.544254i
$310$			1.11227e7i			0.373358i
$311$	5.76398e7			1.91620			0.958101	−	0.286431i	$-0.0924690\pi$
$311$	5.76398e7			1.91620			0.958101	+	0.286431i	$0.0924690\pi$
$312$	−5.65513e6			−0.186200
$313$		−	2.12438e7i		−	0.692785i	−0.938090	−	0.346393i	$-0.887406\pi$
$313$		−	2.12438e7i		−	0.692785i	0.938090	−	0.346393i	$-0.112594\pi$
$314$		−	3.11104e7i		−	1.00489i
$315$	−2.92648e6			−0.0936299
$316$		−	9.24737e7i		−	2.93060i
$317$	−1.49707e7			−0.469963			−0.234982	−	0.972000i	$-0.575503\pi$
$317$	−1.49707e7			−0.469963			−0.234982	+	0.972000i	$0.575503\pi$
$318$		−	3.90544e7i		−	1.21448i
$319$			7.28091e6i			0.224292i
$320$			1.18303e7i			0.361032i
$321$			2.63118e7i			0.795491i
$322$	5.02323e7	−	1.90983e7i	1.50458	−	0.572042i
$323$	1.41797e7			0.420786
$324$	7.26318e6			0.213546
$325$	−6.40022e6			−0.186442
$326$	−9.81056e7			−2.83166
$327$		−	2.86136e7i		−	0.818331i
$328$	6.79320e7			1.92510
$329$		−	2.12749e7i		−	0.597420i
$330$	1.75749e7			0.489049
$331$	7.01864e7			1.93539			0.967696	−	0.252118i	$-0.0811272\pi$
$331$	7.01864e7			1.93539			0.967696	+	0.252118i	$0.0811272\pi$
$332$		−	1.21556e8i		−	3.32170i
$333$		−	8.99086e6i		−	0.243483i
$334$	−4.26159e7			−1.14375
$335$	−5.75161e6			−0.152987
$336$			1.59178e7i			0.419630i
$337$			5.70254e7i			1.48997i	0.667079	+	0.744987i	$0.267544\pi$
$337$			5.70254e7i			1.48997i	−0.667079	+	0.744987i	$0.732456\pi$
$338$	6.32416e7			1.63777
$339$		−	1.47959e7i		−	0.379790i
$340$	2.45647e7			0.624991
$341$			4.82348e7i			1.21646i
$342$			8.79728e6i			0.219923i
$343$		−	4.23035e7i		−	1.04832i
$344$			1.90830e7i			0.468782i
$345$	6.61021e6	−	2.51321e6i	0.160975	−	0.0612028i
$346$	−9.82878e7			−2.37285
$347$	−4.93406e7			−1.18091			−0.590454	−	0.807071i	$-0.701051\pi$
$347$	−4.93406e7			−1.18091			−0.590454	+	0.807071i	$0.701051\pi$
$348$	−6.31367e6			−0.149811
$349$	−4.04362e7			−0.951248			−0.475624	−	0.879649i	$-0.657778\pi$
$349$	−4.04362e7			−0.951248			−0.475624	+	0.879649i	$0.657778\pi$
$350$			6.28734e7i			1.46643i
$351$	1.70316e6			0.0393852
$352$			1.85870e7i			0.426168i
$353$	−7.10708e7			−1.61572			−0.807862	−	0.589371i	$-0.799375\pi$
$353$	−7.10708e7			−1.61572			−0.807862	+	0.589371i	$0.799375\pi$
$354$	6.03671e7			1.36079
$355$			1.55031e7i			0.346525i
$356$		−	5.04551e7i		−	1.11829i
$357$	−2.69679e7			−0.592709
$358$	3.44134e7			0.750029
$359$			9.17063e7i			1.98206i	0.133654	+	0.991028i	$0.457329\pi$
$359$			9.17063e7i			1.98206i	−0.133654	+	0.991028i	$0.542671\pi$
$360$			7.31050e6i			0.156689i
$361$	4.00372e7			0.851024
$362$			6.96357e7i			1.46793i
$363$	4.85997e7			1.01605
$364$			1.78629e7i			0.370381i
$365$			1.66411e6i			0.0342219i
$366$		−	7.67594e6i		−	0.156563i
$367$		−	5.00464e7i		−	1.01245i	−0.862401	−	0.506226i	$-0.831040\pi$
$367$		−	5.00464e7i		−	1.01245i	0.862401	−	0.506226i	$-0.168960\pi$
$368$	−1.36699e7	−	3.59545e7i	−0.274298	−	0.721456i
$369$	−2.04591e7			−0.407199
$370$	1.88654e7			0.372443
$371$	−5.91748e7			−1.15882
$372$	−4.18270e7			−0.812509
$373$		−	4.27907e7i		−	0.824560i	−0.911057	−	0.412280i	$-0.864732\pi$
$373$		−	4.27907e7i		−	0.824560i	0.911057	−	0.412280i	$-0.135268\pi$
$374$	1.61955e8			3.09585
$375$			1.73555e7i			0.329111i
$376$	−5.31458e7			−0.999782
$377$	−1.48051e6			−0.0276303
$378$		−	1.67312e7i		−	0.309778i
$379$		−	5.51532e7i		−	1.01310i	−0.862210	−	0.506551i	$-0.830920\pi$
$379$		−	5.51532e7i		−	1.01310i	0.862210	−	0.506551i	$-0.169080\pi$
$380$	−1.21417e7			−0.221273
$381$	−2.00932e7			−0.363308
$382$			1.41457e8i			2.53767i
$383$		−	7.95541e7i		−	1.41601i	−0.706208	−	0.708005i	$-0.749596\pi$
$383$		−	7.95541e7i		−	1.41601i	0.706208	−	0.708005i	$-0.250404\pi$
$384$	−5.92493e7			−1.04638
$385$		−	2.66293e7i		−	0.466636i
$386$	5.13309e6			0.0892519
$387$		−	5.74723e6i		−	0.0991575i
$388$		−	1.47879e8i		−	2.53170i
$389$			5.54580e7i			0.942140i	0.882096	+	0.471070i	$0.156132\pi$
$389$			5.54580e7i			0.942140i	−0.882096	+	0.471070i	$0.843868\pi$
$390$			3.57370e6i			0.0602455i
$391$	6.09138e7	−	2.31595e7i	1.01903	−	0.387435i
$392$	−1.07509e7			−0.178478
$393$	6.45559e7			1.06355
$394$	−1.40775e8			−2.30164
$395$	−2.80318e7			−0.454841
$396$			6.60908e7i			1.06428i
$397$	2.45447e7			0.392271			0.196136	−	0.980577i	$-0.437161\pi$
$397$	2.45447e7			0.392271			0.196136	+	0.980577i	$0.437161\pi$
$398$		−	8.32812e6i		−	0.132098i
$399$	1.33295e7			0.209844
$400$	4.50026e7			0.703166
$401$			5.32006e7i			0.825056i	0.910945	+	0.412528i	$0.135354\pi$
$401$			5.32006e7i			0.825056i	−0.910945	+	0.412528i	$0.864646\pi$
$402$		−	3.28829e7i		−	0.506165i
$403$	−9.80811e6			−0.149855
$404$	1.29731e8			1.96743
$405$		−	2.20171e6i		−	0.0331432i
$406$			1.45439e7i			0.217322i
$407$	8.18117e7			1.21348
$408$			6.73671e7i			0.991898i
$409$	1.24784e8			1.82385			0.911924	−	0.410360i	$-0.134597\pi$
$409$	1.24784e8			1.82385			0.911924	+	0.410360i	$0.134597\pi$
$410$		−	4.29289e7i		−	0.622872i
$411$			2.39538e7i			0.345024i
$412$			1.26703e8i			1.81174i
$413$		−	9.14675e7i		−	1.29842i
$414$	1.43684e7	+	3.77916e7i	0.202492	+	0.532592i
$415$	−3.68475e7			−0.515541
$416$	−3.77949e6			−0.0524993
$417$	−4.46557e7			−0.615842
$418$	−8.00502e7			−1.09606
$419$		−	1.38099e8i		−	1.87737i	−0.344775	−	0.938685i	$-0.612045\pi$
$419$		−	1.38099e8i		−	1.87737i	0.344775	−	0.938685i	$-0.387955\pi$
$420$	2.30918e7			0.311680
$421$		−	1.35095e8i		−	1.81048i	−0.424905	−	0.905238i	$-0.639693\pi$
$421$		−	1.35095e8i		−	1.81048i	0.424905	−	0.905238i	$-0.360307\pi$
$422$	−2.24199e8			−2.98329
$423$	1.60059e7			0.211475
$424$			1.47822e8i			1.93928i
$425$			7.62430e7i			0.993192i
$426$	−8.86340e7			−1.14649
$427$	−1.16305e7			−0.149388
$428$		−	2.07616e8i		−	2.64807i
$429$			1.54978e7i			0.196289i
$430$	1.20593e7			0.151676
$431$		−	1.04449e8i		−	1.30459i	−0.757966	−	0.652294i	$-0.773807\pi$
$431$		−	1.04449e8i		−	1.30459i	0.757966	−	0.652294i	$-0.226193\pi$
$432$	−1.19756e7			−0.148541
$433$		−	7.45093e7i		−	0.917797i	−0.888489	−	0.458899i	$-0.848244\pi$
$433$		−	7.45093e7i		−	0.917797i	0.888489	−	0.458899i	$-0.151756\pi$
$434$			9.63512e7i			1.17866i
$435$			1.91388e6i			0.0232513i
$436$			2.25779e8i			2.72410i
$437$	−3.01081e7	+	1.14471e7i	−0.360778	+	0.137168i
$438$	−9.51400e6			−0.113225
$439$	7.03400e7			0.831397			0.415699	−	0.909502i	$-0.363537\pi$
$439$	7.03400e7			0.831397			0.415699	+	0.909502i	$0.363537\pi$
$440$	−6.65214e7			−0.780914
$441$	3.23784e6			0.0377519
$442$			3.29321e7i			0.381375i
$443$	4.84579e7			0.557383			0.278691	−	0.960381i	$-0.410099\pi$
$443$	4.84579e7			0.557383			0.278691	+	0.960381i	$0.410099\pi$
$444$			7.09434e7i			0.810519i
$445$	−1.52946e7			−0.173563
$446$	6.85041e7			0.772168
$447$		−	3.53248e7i		−	0.395510i
$448$			1.02481e8i			1.13975i
$449$	5.38849e7			0.595289			0.297644	−	0.954677i	$-0.403799\pi$
$449$	5.38849e7			0.595289			0.297644	+	0.954677i	$0.403799\pi$
$450$	−4.73020e7			−0.519090
$451$		−	1.86166e8i		−	2.02941i
$452$			1.16749e8i			1.26426i
$453$	−2.48017e7			−0.266800
$454$			2.46271e8i			2.63176i
$455$	5.41483e6			0.0574845
$456$		−	3.32978e7i		−	0.351173i
$457$			1.72326e8i			1.80552i	0.430141	+	0.902762i	$0.358464\pi$
$457$			1.72326e8i			1.80552i	−0.430141	+	0.902762i	$0.641536\pi$
$458$		−	2.40166e8i		−	2.49986i
$459$		−	2.02889e7i		−	0.209808i
$460$	−5.21586e7	+	1.98308e7i	−0.535862	+	0.203735i
$461$	1.09790e8			1.12063			0.560315	−	0.828280i	$-0.310680\pi$
$461$	1.09790e8			1.12063			0.560315	+	0.828280i	$0.310680\pi$
$462$	1.52244e8			1.54388
$463$	−8.40623e7			−0.846951			−0.423475	−	0.905908i	$-0.639190\pi$
$463$	−8.40623e7			−0.846951			−0.423475	+	0.905908i	$0.639190\pi$
$464$	1.04100e7			0.104207
$465$			1.26791e7i			0.126105i
$466$	1.23458e8			1.22000
$467$			9.26107e7i			0.909307i	0.890668	+	0.454654i	$0.150237\pi$
$467$			9.26107e7i			0.909307i	−0.890668	+	0.454654i	$0.849763\pi$
$468$	−1.34390e7			−0.131108
$469$	−4.98237e7			−0.482967
$470$			3.35849e7i			0.323483i
$471$		−	3.54638e7i		−	0.339408i
$472$	−2.28490e8			−2.17291
$473$	5.22965e7			0.494185
$474$		−	1.60262e8i		−	1.50486i
$475$		−	3.76850e7i		−	0.351631i
$476$	2.12793e8			1.97304
$477$		−	4.45195e7i		−	0.410199i
$478$	5.42786e7			0.496987
$479$		−	4.47728e7i		−	0.407387i	−0.979035	−	0.203694i	$-0.934705\pi$
$479$		−	4.47728e7i		−	0.407387i	0.979035	−	0.203694i	$-0.0652946\pi$
$480$			4.88582e6i			0.0441788i
$481$			1.66357e7i			0.149487i
$482$		−	8.61266e6i		−	0.0769123i
$483$	5.72614e7	−	2.17708e7i	0.508184	−	0.193212i
$484$	−3.83482e8			−3.38227
$485$	−4.48270e7			−0.392930
$486$	1.25875e7			0.109656
$487$	3.68023e7			0.318631			0.159316	−	0.987228i	$-0.449071\pi$
$487$	3.68023e7			0.318631			0.159316	+	0.987228i	$0.449071\pi$
$488$			2.90535e7i			0.250000i
$489$	−1.11834e8			−0.956416
$490$			6.79390e6i			0.0577472i
$491$	−1.41972e8			−1.19938			−0.599691	−	0.800232i	$-0.704710\pi$
$491$	−1.41972e8			−1.19938			−0.599691	+	0.800232i	$0.704710\pi$
$492$	1.61435e8			1.35551
$493$			1.76366e7i			0.147189i
$494$		−	1.62775e7i		−	0.135023i
$495$	2.00343e7			0.165180
$496$	6.89648e7			0.565175
$497$			1.34297e8i			1.09395i
$498$		−	2.10663e8i		−	1.70569i
$499$	−4.54773e7			−0.366010			−0.183005	−	0.983112i	$-0.558583\pi$
$499$	−4.54773e7			−0.366010			−0.183005	+	0.983112i	$0.558583\pi$
$500$		−	1.36945e8i		−	1.09556i
$501$	−4.85793e7			−0.386312
$502$			2.66963e8i			2.11028i
$503$		−	6.18309e7i		−	0.485850i	−0.970045	−	0.242925i	$-0.921893\pi$
$503$		−	6.18309e7i		−	0.485850i	0.970045	−	0.242925i	$-0.0781068\pi$
$504$			6.33277e7i			0.494655i
$505$		−	3.93256e7i		−	0.305352i
$506$	−3.43882e8	+	1.30744e8i	−2.65435	+	1.00919i
$507$	7.20912e7			0.553170
$508$	1.58548e8			1.20940
$509$	−2.87994e7			−0.218389			−0.109194	−	0.994020i	$-0.534827\pi$
$509$	−2.87994e7			−0.218389			−0.109194	+	0.994020i	$0.534827\pi$
$510$	4.25719e7			0.320932
$511$			1.44155e7i			0.108036i
$512$	1.89828e8			1.41433
$513$			1.00283e7i			0.0742807i
$514$	3.13771e8			2.31059
$515$	3.84079e7			0.281189
$516$			4.53492e7i			0.330081i
$517$			1.45645e8i			1.05396i
$518$	1.63423e8			1.17577
$519$	−1.12041e8			−0.801451
$520$		−	1.35265e7i		−	0.0962002i
$521$			8.76991e7i			0.620129i	0.950716	+	0.310064i	$0.100351\pi$
$521$			8.76991e7i			0.620129i	−0.950716	+	0.310064i	$0.899649\pi$
$522$	−1.09420e7			−0.0769278
$523$		−	2.05677e7i		−	0.143774i	−0.997413	−	0.0718871i	$-0.977098\pi$
$523$		−	2.05677e7i		−	0.143774i	0.997413	−	0.0718871i	$-0.0229021\pi$
$524$	−5.09386e8			−3.54041
$525$			7.16715e7i			0.495300i
$526$		−	2.61582e8i		−	1.79743i
$527$			1.16840e8i			0.798285i
$528$		−	1.08971e8i		−	0.740303i
$529$	−1.10643e8	+	9.83500e7i	−0.747407	+	0.664366i
$530$	9.34144e7			0.627460
$531$	6.88145e7			0.459617
$532$	−1.05178e8			−0.698539
$533$	3.78551e7			0.250002
$534$		−	8.74415e7i		−	0.574241i
$535$	−6.29352e7			−0.410991
$536$			1.24462e8i			0.808245i
$537$	3.92289e7			0.253328
$538$	−1.62888e7			−0.104602
$539$			2.94625e7i			0.188149i
$540$			1.73728e7i			0.110329i
$541$	1.59395e8			1.00666			0.503329	−	0.864095i	$-0.332108\pi$
$541$	1.59395e8			1.00666			0.503329	+	0.864095i	$0.332108\pi$
$542$	2.97506e7			0.186852
$543$			7.93800e7i			0.495806i
$544$			4.50234e7i			0.279667i
$545$	6.84409e7			0.422791
$546$			3.09575e7i			0.190190i
$547$	−2.45660e7			−0.150097			−0.0750486	−	0.997180i	$-0.523911\pi$
$547$	−2.45660e7			−0.150097			−0.0750486	+	0.997180i	$0.523911\pi$
$548$		−	1.89010e8i		−	1.14853i
$549$		−	8.75006e6i		−	0.0528803i
$550$		−	4.30422e8i		−	2.58706i
$551$		−	8.71733e6i		−	0.0521109i
$552$	−5.43846e7	−	1.43042e8i	−0.323340	−	0.850444i
$553$	−2.42827e8			−1.43589
$554$	1.49288e8			0.878003
$555$	2.15053e7			0.125796
$556$	3.52361e8			2.05005
$557$			4.07722e7i			0.235938i	0.993017	+	0.117969i	$0.0376384\pi$
$557$			4.07722e7i			0.235938i	−0.993017	+	0.117969i	$0.962362\pi$
$558$	−7.24887e7			−0.417222
$559$			1.06340e7i			0.0608782i
$560$	−3.80739e7			−0.216802
$561$	1.84618e8			1.04565
$562$		−	4.13839e8i		−	2.33143i
$563$		−	1.75318e8i		−	0.982429i	−0.871039	−	0.491215i	$-0.836553\pi$
$563$		−	1.75318e8i		−	0.982429i	0.871039	−	0.491215i	$-0.163447\pi$
$564$	−1.26297e8			−0.703970
$565$	3.53904e7			0.196219
$566$			1.72004e8i			0.948613i
$567$		−	1.90724e7i		−	0.104630i
$568$	3.35481e8			1.83072
$569$			9.94027e7i			0.539587i	0.962918	+	0.269793i	$0.0869554\pi$
$569$			9.94027e7i			0.539587i	−0.962918	+	0.269793i	$0.913045\pi$
$570$	−2.10422e7			−0.113623
$571$		−	1.33988e7i		−	0.0719712i	−0.999352	−	0.0359856i	$-0.988543\pi$
$571$		−	1.33988e7i		−	0.0719712i	0.999352	−	0.0359856i	$-0.0114571\pi$
$572$		−	1.22287e8i		−	0.653419i
$573$			1.61252e8i			0.857118i
$574$		−	3.71875e8i		−	1.96635i
$575$	−6.15501e7	−	1.61888e8i	−0.323761	−	0.851553i
$576$	−7.71001e7			−0.403448
$577$	−5.21549e7			−0.271499			−0.135749	−	0.990743i	$-0.543344\pi$
$577$	−5.21549e7			−0.271499			−0.135749	+	0.990743i	$0.543344\pi$
$578$	6.22264e7			0.322249
$579$	5.85138e6			0.0301455
$580$		−	1.51017e7i		−	0.0774001i
$581$	−3.19194e8			−1.62752
$582$		−	2.56283e8i		−	1.30002i
$583$	4.05102e8			2.04436
$584$	3.60106e7			0.180797
$585$			4.07378e6i			0.0203484i
$586$		−	4.55555e8i		−	2.26385i
$587$	2.48417e8			1.22820			0.614098	−	0.789230i	$-0.289520\pi$
$587$	2.48417e8			1.22820			0.614098	+	0.789230i	$0.289520\pi$
$588$	−2.55485e7			−0.125671
$589$		−	5.77509e7i		−	0.282626i
$590$			1.44392e8i			0.703052i
$591$	−1.60474e8			−0.777398
$592$		−	1.16972e8i		−	0.563790i
$593$	−3.98370e8			−1.91039			−0.955197	−	0.295971i	$-0.904357\pi$
$593$	−3.98370e8			−1.91039			−0.955197	+	0.295971i	$0.904357\pi$
$594$			1.14539e8i			0.546505i
$595$		−	6.45045e7i		−	0.306224i
$596$			2.78735e8i			1.31659i
$597$		−	9.49350e6i		−	0.0446173i
$598$	−2.65857e7	−	6.99253e7i	−0.124321	−	0.326987i
$599$	2.13213e8			0.992050			0.496025	−	0.868308i	$-0.334792\pi$
$599$	2.13213e8			0.992050			0.496025	+	0.868308i	$0.334792\pi$
$600$	1.79039e8			0.828884
$601$	5.00614e6			0.0230611			0.0115305	−	0.999934i	$-0.496330\pi$
$601$	5.00614e6			0.0230611			0.0115305	+	0.999934i	$0.496330\pi$
$602$	1.04465e8			0.478828
$603$		−	3.74843e7i		−	0.170961i
$604$	1.95700e8			0.888139
$605$			1.16246e8i			0.524942i
$606$	2.24831e8			1.01027
$607$	−2.11822e8			−0.947122			−0.473561	−	0.880761i	$-0.657032\pi$
$607$	−2.11822e8			−0.947122			−0.473561	+	0.880761i	$0.657032\pi$
$608$		−	2.22539e7i		−	0.0990138i
$609$			1.65791e7i			0.0734023i
$610$	1.83601e7			0.0808883
$611$	−2.96155e7			−0.129836
$612$			1.60092e8i			0.698419i
$613$			2.61252e8i			1.13417i	0.823660	+	0.567084i	$0.191929\pi$
$613$			2.61252e8i			1.13417i	−0.823660	+	0.567084i	$0.808071\pi$
$614$	2.19160e8			0.946796
$615$		−	4.89361e7i		−	0.210380i
$616$	−5.76246e8			−2.46528
$617$		−	592013.i		−	0.00252044i	−0.999999	−	0.00126022i	$-0.999599\pi$
$617$		−	592013.i		−	0.00252044i	0.999999	−	0.00126022i	$-0.000401140\pi$
$618$			2.19584e8i			0.930326i
$619$			1.03326e8i			0.435652i	0.975988	+	0.217826i	$0.0698965\pi$
$619$			1.03326e8i			0.435652i	−0.975988	+	0.217826i	$0.930104\pi$
$620$		−	1.00046e8i		−	0.419784i
$621$	1.63790e7	+	4.30799e7i	0.0683933	+	0.179887i
$622$	−7.88218e8			−3.27548
$623$	−1.32490e8			−0.547924
$624$	2.21583e7			0.0911974
$625$	1.80905e8			0.740989
$626$			2.90506e8i			1.18422i
$627$	−9.12519e7			−0.370203
$628$			2.79831e8i			1.12984i
$629$	1.98173e8			0.796330
$630$	4.00193e7			0.160047
$631$			2.30118e8i			0.915930i	0.888970	+	0.457965i	$0.151422\pi$
$631$			2.30118e8i			0.915930i	−0.888970	+	0.457965i	$0.848578\pi$
$632$			6.06594e8i			2.40296i
$633$	−2.55572e8			−1.00763
$634$	2.04723e8			0.803337
$635$		−	4.80610e7i		−	0.187703i
$636$			3.51286e8i			1.36549i
$637$	−5.99092e6			−0.0231780
$638$		−	9.95656e7i		−	0.383396i
$639$	−1.01037e8			−0.387237
$640$		−	1.41719e8i		−	0.540613i
$641$		−	1.68567e8i		−	0.640026i	−0.947413	−	0.320013i	$-0.896313\pi$
$641$		−	1.68567e8i		−	0.640026i	0.947413	−	0.320013i	$-0.103687\pi$
$642$		−	3.59811e8i		−	1.35978i
$643$			4.56236e8i			1.71615i	0.513521	+	0.858077i	$0.328341\pi$
$643$			4.56236e8i			1.71615i	−0.513521	+	0.858077i	$0.671659\pi$
$644$	−4.51828e8	+	1.71785e8i	−1.69167	+	0.643174i
$645$	1.37468e7			0.0512298
$646$	−1.93906e8			−0.719274
$647$	1.41351e8			0.521897			0.260949	−	0.965353i	$-0.415965\pi$
$647$	1.41351e8			0.521897			0.260949	+	0.965353i	$0.415965\pi$
$648$	−4.76439e7			−0.175098
$649$			6.26172e8i			2.29066i
$650$	8.75223e7			0.318698
$651$			1.09834e8i			0.398101i
$652$	8.82438e8			3.18377
$653$	6.26888e7			0.225139			0.112569	−	0.993644i	$-0.464092\pi$
$653$	6.26888e7			0.225139			0.112569	+	0.993644i	$0.464092\pi$
$654$			3.91288e8i			1.39882i
$655$			1.54411e8i			0.549485i
$656$	−2.66175e8			−0.942879
$657$	−1.08453e7			−0.0382425
$658$			2.90932e8i			1.02121i
$659$		−	4.33761e8i		−	1.51563i	−0.652468	−	0.757816i	$-0.726266\pi$
$659$		−	4.33761e8i		−	1.51563i	0.652468	−	0.757816i	$-0.273734\pi$
$660$	−1.58083e8			−0.549860
$661$			1.04892e8i			0.363192i	0.983373	+	0.181596i	$0.0581264\pi$
$661$			1.04892e8i			0.363192i	−0.983373	+	0.181596i	$0.941874\pi$
$662$	−9.59791e8			−3.30828
$663$			3.75404e7i			0.128812i
$664$			7.97362e8i			2.72365i
$665$			3.18829e7i			0.108416i
$666$			1.22949e8i			0.416200i
$667$	−1.42378e7	−	3.74482e7i	−0.0479807	−	0.126198i
$668$	3.83321e8			1.28598
$669$	7.80901e7			0.260806
$670$	7.86527e7			0.261510
$671$	7.96205e7			0.263547
$672$			4.23238e7i			0.139469i
$673$	−3.60124e8			−1.18143			−0.590713	−	0.806882i	$-0.701153\pi$
$673$	−3.60124e8			−1.18143			−0.590713	+	0.806882i	$0.701153\pi$
$674$		−	7.79816e8i		−	2.54690i
$675$	−5.39212e7			−0.175327
$676$	−5.68844e8			−1.84142
$677$		−	2.26507e8i		−	0.729987i	−0.931010	−	0.364994i	$-0.881071\pi$
$677$		−	2.26507e8i		−	0.729987i	0.931010	−	0.364994i	$-0.118929\pi$
$678$			2.02333e8i			0.649198i
$679$	−3.88317e8			−1.24044
$680$	−1.61135e8			−0.512465
$681$			2.80733e8i			0.888898i
$682$		−	6.59605e8i		−	2.07937i
$683$	4.40752e8			1.38335			0.691674	−	0.722209i	$-0.256873\pi$
$683$	4.40752e8			1.38335			0.691674	+	0.722209i	$0.256873\pi$
$684$		−	7.91296e7i		−	0.247269i
$685$	−5.72952e7			−0.178257
$686$			5.78496e8i			1.79196i
$687$		−	2.73774e8i		−	0.844348i
$688$		−	7.47722e7i		−	0.229602i
$689$			8.23737e7i			0.251844i
$690$	−9.03939e7	+	3.43678e7i	−0.275164	+	0.104618i
$691$	−1.00558e8			−0.304776			−0.152388	−	0.988321i	$-0.548696\pi$
$691$	−1.00558e8			−0.304776			−0.152388	+	0.988321i	$0.548696\pi$
$692$	8.84076e8			2.66791
$693$	1.73548e8			0.521459
$694$	6.74728e8			2.01860
$695$		−	1.06812e8i		−	0.318175i
$696$	4.14155e7			0.122839
$697$		−	4.50951e8i		−	1.33178i
$698$	5.52960e8			1.62603
$699$	1.40734e8			0.412066
$700$		−	5.65532e8i		−	1.64878i
$701$		−	1.87714e8i		−	0.544933i	−0.962165	−	0.272466i	$-0.912161\pi$
$701$		−	1.87714e8i		−	0.544933i	0.962165	−	0.272466i	$-0.0878393\pi$
$702$	−2.32905e7			−0.0673235
$703$	−9.79520e7			−0.281934
$704$		−	7.01567e8i		−	2.01072i
$705$			3.82846e7i			0.109259i
$706$	9.71886e8			2.76186
$707$		−	3.40661e8i		−	0.963971i
$708$	−5.42988e8			−1.53000
$709$		−	4.15837e8i		−	1.16677i	−0.812196	−	0.583384i	$-0.801728\pi$
$709$		−	4.15837e8i		−	1.16677i	0.812196	−	0.583384i	$-0.198272\pi$
$710$		−	2.12004e8i		−	0.592337i
$711$		−	1.82688e8i		−	0.508278i
$712$			3.30967e8i			0.916949i
$713$	−9.43233e7	−	2.48088e8i	−0.260226	−	0.684443i
$714$	3.68782e8			1.01315
$715$	−3.70691e7			−0.101413
$716$	−3.09540e8			−0.843293
$717$	6.18740e7			0.167861
$718$		−	1.25407e9i		−	3.38805i
$719$	−4.55072e7			−0.122432			−0.0612158	−	0.998125i	$-0.519498\pi$
$719$	−4.55072e7			−0.122432			−0.0612158	+	0.998125i	$0.519498\pi$
$720$		−	2.86445e7i		−	0.0767438i
$721$	3.32711e8			0.887690
$722$	−5.47504e8			−1.45471
$723$		−	9.81785e6i		−	0.0259778i
$724$		−	6.26357e8i		−	1.65046i
$725$	4.68721e7			0.122999
$726$	−6.64596e8			−1.73679
$727$			2.63013e8i			0.684502i	0.939609	+	0.342251i	$0.111189\pi$
$727$			2.63013e8i			0.684502i	−0.939609	+	0.342251i	$0.888811\pi$
$728$		−	1.17174e8i		−	0.303696i
$729$	1.43489e7			0.0370370
$730$		−	2.27566e7i		−	0.0584976i
$731$	1.26678e8			0.324302
$732$			6.90433e7i			0.176031i
$733$			3.66525e8i			0.930662i	0.885137	+	0.465331i	$0.154065\pi$
$733$			3.66525e8i			0.930662i	−0.885137	+	0.465331i	$0.845935\pi$
$734$			6.84379e8i			1.73065i
$735$			7.74459e6i			0.0195046i
$736$	−3.63468e7	−	9.55991e7i	−0.0911661	−	0.239784i
$737$	3.41086e8			0.852042
$738$	2.79776e8			0.696050
$739$	−2.84036e8			−0.703785			−0.351892	−	0.936040i	$-0.614462\pi$
$739$	−2.84036e8			−0.703785			−0.351892	+	0.936040i	$0.614462\pi$
$740$	−1.69690e8			−0.418755
$741$		−	1.85552e7i		−	0.0456050i
$742$	8.09209e8			1.98084
$743$			4.52986e8i			1.10438i	0.833718	+	0.552190i	$0.186208\pi$
$743$			4.52986e8i			1.10438i	−0.833718	+	0.552190i	$0.813792\pi$
$744$	2.74371e8			0.666222
$745$	8.44935e7			0.204341
$746$			5.85157e8i			1.40947i
$747$		−	2.40142e8i		−	0.576110i
$748$	−1.45675e9			−3.48080
$749$	−5.45180e8			−1.29746
$750$		−	2.37334e8i		−	0.562569i
$751$		−	2.97297e8i		−	0.701893i	−0.936396	−	0.350946i	$-0.885860\pi$
$751$		−	2.97297e8i		−	0.701893i	0.936396	−	0.350946i	$-0.114140\pi$
$752$	2.08239e8			0.489676
$753$			3.04320e8i			0.712764i
$754$	2.02458e7			0.0472302
$755$		−	5.93232e7i		−	0.137843i
$756$			1.50493e8i			0.348298i
$757$		−	1.73511e8i		−	0.399982i	−0.979798	−	0.199991i	$-0.935909\pi$
$757$		−	1.73511e8i		−	0.399982i	0.979798	−	0.199991i	$-0.0640912\pi$
$758$			7.54214e8i			1.73176i
$759$	−3.92003e8	+	1.49040e8i	−0.896528	+	0.340861i
$760$	7.96452e7			0.181434
$761$	−9.85680e7			−0.223657			−0.111828	−	0.993728i	$-0.535671\pi$
$761$	−9.85680e7			−0.223657			−0.111828	+	0.993728i	$0.535671\pi$
$762$	2.74772e8			0.621024
$763$	5.92874e8			1.33472
$764$		−	1.27238e9i		−	2.85322i
$765$	4.85292e7			0.108397
$766$			1.08789e9i			2.42047i
$767$	−1.27326e8			−0.282184
$768$	4.93685e8			1.08985
$769$		−	1.01130e8i		−	0.222382i	−0.993799	−	0.111191i	$-0.964533\pi$
$769$		−	1.01130e8i		−	0.222382i	0.993799	−	0.111191i	$-0.0354665\pi$
$770$			3.64153e8i			0.797649i
$771$	3.57678e8			0.780421
$772$	−4.61710e7			−0.100350
$773$		−	7.09878e8i		−	1.53690i	−0.639910	−	0.768449i	$-0.721029\pi$
$773$		−	7.09878e8i		−	1.53690i	0.639910	−	0.768449i	$-0.278971\pi$
$774$			7.85927e7i			0.169496i
$775$	3.10520e8			0.667090
$776$			9.70036e8i			2.07588i
$777$	1.86291e8			0.397126
$778$		−	7.58382e8i		−	1.61046i
$779$			2.22894e8i			0.471504i
$780$		−	3.21447e7i		−	0.0677369i
$781$		−	9.19377e8i		−	1.92993i
$782$	−8.32989e8	+	3.16703e8i	−1.74188	+	0.662266i
$783$	−1.24731e7			−0.0259830
$784$	4.21246e7			0.0874154
$785$	8.48258e7			0.175355
$786$	−8.82795e8			−1.81799
$787$			4.84756e8i			0.994488i	0.867611	+	0.497244i	$0.165655\pi$
$787$			4.84756e8i			0.994488i	−0.867611	+	0.497244i	$0.834345\pi$
$788$	1.26624e9			2.58784
$789$		−	2.98187e8i		−	0.607095i
$790$	3.83331e8			0.777487
$791$	3.06572e8			0.619445
$792$		−	4.33532e8i		−	0.872661i
$793$			1.61901e7i			0.0324661i
$794$	−3.35646e8			−0.670533
$795$	1.06486e8			0.211930
$796$			7.49096e7i			0.148525i
$797$			3.25900e8i			0.643738i	0.946784	+	0.321869i	$0.104311\pi$
$797$			3.25900e8i			0.643738i	−0.946784	+	0.321869i	$0.895689\pi$
$798$	−1.82280e8			−0.358699
$799$			3.52797e8i			0.691646i
$800$	1.19657e8			0.233705
$801$		−	9.96775e7i		−	0.193954i
$802$		−	7.27512e8i		−	1.41032i
$803$		−	9.86863e7i		−	0.190594i
$804$			2.95774e8i			0.569105i
$805$	5.20737e7	+	1.36964e8i	0.0998231	+	0.262554i
$806$	1.34125e8			0.256156
$807$	−1.85681e7			−0.0353303
$808$	−8.50988e8			−1.61320
$809$	2.20633e8			0.416701			0.208350	−	0.978054i	$-0.433191\pi$
$809$	2.20633e8			0.416701			0.208350	+	0.978054i	$0.433191\pi$
$810$			3.01081e7i			0.0566536i
$811$	−7.54112e8			−1.41375			−0.706876	−	0.707337i	$-0.749896\pi$
$811$	−7.54112e8			−1.41375			−0.706876	+	0.707337i	$0.749896\pi$
$812$		−	1.30819e8i		−	0.244345i
$813$	3.39137e7			0.0631108
$814$	−1.11877e9			−2.07427
$815$		−	2.67496e8i		−	0.494133i
$816$		−	2.63962e8i		−	0.485814i
$817$	−6.26139e7			−0.114817
$818$	−1.70641e9			−3.11761
$819$			3.52894e7i			0.0642382i
$820$			3.86136e8i			0.700324i
$821$	−1.56017e8			−0.281931			−0.140966	−	0.990014i	$-0.545021\pi$
$821$	−1.56017e8			−0.281931			−0.140966	+	0.990014i	$0.545021\pi$
$822$		−	3.27566e8i		−	0.589771i
$823$	8.59518e8			1.54190			0.770949	−	0.636897i	$-0.219782\pi$
$823$	8.59518e8			1.54190			0.770949	+	0.636897i	$0.219782\pi$
$824$		−	8.31129e8i		−	1.48555i
$825$		−	4.90652e8i		−	0.873799i
$826$			1.25081e9i			2.21948i
$827$			4.79370e8i			0.847529i	0.905772	+	0.423764i	$0.139292\pi$
$827$			4.79370e8i			0.847529i	−0.905772	+	0.423764i	$0.860708\pi$
$828$	−1.29241e8	−	3.39927e8i	−0.227671	−	0.598818i
$829$	3.33479e8			0.585336			0.292668	−	0.956214i	$-0.405457\pi$
$829$	3.33479e8			0.585336			0.292668	+	0.956214i	$0.405457\pi$
$830$	5.03885e8			0.881246
$831$	1.70178e8			0.296552
$832$	1.42657e8			0.247699
$833$			7.13672e7i			0.123471i
$834$	6.10662e8			1.05270
$835$		−	1.16197e8i		−	0.199588i
$836$	7.20034e8			1.23235
$837$	−8.26322e7			−0.140920
$838$			1.88849e9i			3.20910i
$839$		−	1.23665e8i		−	0.209392i	−0.994504	−	0.104696i	$-0.966613\pi$
$839$		−	1.23665e8i		−	0.209392i	0.994504	−	0.104696i	$-0.0333869\pi$
$840$	−1.51474e8			−0.255564
$841$	−5.83981e8			−0.981772
$842$			1.84741e9i			3.09476i
$843$		−	4.71749e8i		−	0.787460i
$844$	2.01662e9			3.35426
$845$			1.72435e8i			0.285796i
$846$	−2.18879e8			−0.361487
$847$			1.00699e9i			1.65720i
$848$		−	5.79204e8i		−	0.949825i
$849$			1.96073e8i			0.320402i
$850$		−	1.04261e9i		−	1.69772i
$851$	−4.20785e8	+	1.59983e8i	−0.682766	+	0.259588i
$852$	7.97243e8			1.28906
$853$	−1.12289e9			−1.80921			−0.904604	−	0.426253i	$-0.859833\pi$
$853$	−1.12289e9			−1.80921			−0.904604	+	0.426253i	$0.859833\pi$
$854$	1.59046e8			0.255357
$855$	−2.39867e7			−0.0383772
$856$			1.36189e9i			2.17130i
$857$	−159596.			−0.000253559			−0.000126780	−	1.00000i	$-0.500040\pi$
$857$	−159596.			−0.000253559			−0.000126780		1.00000i	$0.500040\pi$
$858$		−	2.11930e8i		−	0.335529i
$859$	−1.05263e9			−1.66073			−0.830363	−	0.557223i	$-0.811867\pi$
$859$	−1.05263e9			−1.66073			−0.830363	+	0.557223i	$0.811867\pi$
$860$	−1.08471e8			−0.170537
$861$		−	4.23913e8i		−	0.664151i
$862$			1.42833e9i			2.23001i
$863$	−7.06006e8			−1.09844			−0.549220	−	0.835678i	$-0.685075\pi$
$863$	−7.06006e8			−1.09844			−0.549220	+	0.835678i	$0.685075\pi$
$864$	−3.18418e7			−0.0493692
$865$		−	2.67992e8i		−	0.414070i
$866$			1.01891e9i			1.56885i
$867$	7.09340e7			0.108842
$868$		−	8.66657e8i		−	1.32522i
$869$	1.66236e9			2.53317
$870$		−	2.61721e7i		−	0.0397448i
$871$			6.93567e7i			0.104962i
$872$		−	1.48103e9i		−	2.23364i
$873$		−	2.92146e8i		−	0.439094i
$874$	4.11726e8	−	1.56538e8i	0.616699	−	0.234470i
$875$	−3.59605e8			−0.536787
$876$	8.55763e7			0.127304
$877$	−2.66251e8			−0.394723			−0.197361	−	0.980331i	$-0.563237\pi$
$877$	−2.66251e8			−0.394723			−0.197361	+	0.980331i	$0.563237\pi$
$878$	−9.61891e8			−1.42116
$879$		−	5.19302e8i		−	0.764634i
$880$	2.60648e8			0.382478
$881$			1.01423e8i			0.148323i	0.997246	+	0.0741616i	$0.0236281\pi$
$881$			1.01423e8i			0.148323i	−0.997246	+	0.0741616i	$0.976372\pi$
$882$	−4.42770e7			−0.0645317
$883$	2.80171e8			0.406950			0.203475	−	0.979080i	$-0.434776\pi$
$883$	2.80171e8			0.406950			0.203475	+	0.979080i	$0.434776\pi$
$884$		−	2.96217e8i		−	0.428798i
$885$			1.64597e8i			0.237462i
$886$	−6.62656e8			−0.952768
$887$	−8.88986e8			−1.27387			−0.636934	−	0.770919i	$-0.719798\pi$
$887$	−8.88986e8			−1.27387			−0.636934	+	0.770919i	$0.719798\pi$
$888$		−	4.65364e8i		−	0.664590i
$889$		−	4.16332e8i		−	0.592563i
$890$	2.09152e8			0.296682
$891$			1.30567e8i			0.184586i
$892$	−6.16179e8			−0.868185
$893$		−	1.74378e8i		−	0.244872i
$894$			4.83063e8i			0.676069i
$895$			9.38318e7i			0.130882i
$896$		−	1.22765e9i		−	1.70667i
$897$	−3.03059e7	−	7.97102e7i	−0.0419904	−	0.110443i
$898$	−7.36869e8			−1.01756
$899$	7.18299e7			0.0988612
$900$	4.25471e8			0.583637
$901$	9.81281e8			1.34159
$902$			2.54580e9i			3.46900i
$903$	1.19083e8			0.161728
$904$		−	7.65832e8i		−	1.03664i
$905$	−1.89869e8			−0.256159
$906$	3.39160e8			0.456058
$907$		−	9.24962e8i		−	1.23966i	−0.784737	−	0.619829i	$-0.787202\pi$
$907$		−	9.24962e8i		−	1.23966i	0.784737	−	0.619829i	$-0.212798\pi$
$908$		−	2.21515e9i		−	2.95901i
$909$	2.56292e8			0.341227
$910$	−7.40472e7			−0.0982618
$911$			4.97349e6i			0.00657819i	0.999995	+	0.00328909i	$0.00104695\pi$
$911$			4.97349e6i			0.00657819i	−0.999995	+	0.00328909i	$0.998953\pi$
$912$			1.30470e8i			0.171999i
$913$	2.18515e9			2.87124
$914$		−	2.35654e9i		−	3.08629i
$915$	2.09293e7			0.0273207
$916$			2.16024e9i			2.81071i
$917$			1.33760e9i			1.73468i
$918$			2.77449e8i			0.358637i
$919$			1.20270e9i			1.54957i	0.632226	+	0.774784i	$0.282141\pi$
$919$			1.20270e9i			1.54957i	−0.632226	+	0.774784i	$0.717859\pi$
$920$	3.42142e8	−	1.30083e8i	0.439383	−	0.167054i
$921$	2.49828e8			0.319788
$922$	−1.50137e9			−1.91556
$923$	1.86947e8			0.237746
$924$	−1.36940e9			−1.73586
$925$		−	5.26677e8i		−	0.665456i
$926$	1.14954e9			1.44774
$927$			2.50311e8i			0.314225i
$928$	2.76792e7			0.0346345
$929$	1.05714e9			1.31851			0.659257	−	0.751917i	$-0.270871\pi$
$929$	1.05714e9			1.31851			0.659257	+	0.751917i	$0.270871\pi$
$930$		−	1.73386e8i		−	0.215558i
$931$		−	3.52750e7i		−	0.0437137i
$932$	−1.11048e9			−1.37171
$933$	−8.98515e8			−1.10632
$934$		−	1.26644e9i		−	1.55433i
$935$			4.41588e8i			0.540234i
$936$	8.81548e7			0.107502
$937$		−	2.38223e8i		−	0.289578i	−0.989463	−	0.144789i	$-0.953750\pi$
$937$		−	2.38223e8i		−	0.289578i	0.989463	−	0.144789i	$-0.0462503\pi$
$938$	6.81334e8			0.825566
$939$			3.31158e8i			0.399980i
$940$		−	3.02089e8i		−	0.363707i
$941$		−	1.34052e9i		−	1.60880i	−0.594086	−	0.804402i	$-0.702486\pi$
$941$		−	1.34052e9i		−	1.60880i	0.594086	−	0.804402i	$-0.297514\pi$
$942$			4.84963e8i			0.580171i
$943$	3.64048e8	+	9.57515e8i	0.434134	+	1.14185i
$944$	8.95285e8			1.06425
$945$	4.56194e7			0.0540572
$946$	−7.15149e8			−0.844740
$947$	−2.81671e8			−0.331659			−0.165830	−	0.986154i	$-0.553030\pi$
$947$	−2.81671e8			−0.331659			−0.165830	+	0.986154i	$0.553030\pi$
$948$			1.44152e9i			1.69198i
$949$	2.00670e7			0.0234792
$950$			5.15338e8i			0.601065i
$951$	2.33370e8			0.271334
$952$	−1.39585e9			−1.61781
$953$			2.21424e8i			0.255826i	0.991785	+	0.127913i	$0.0408279\pi$
$953$			2.21424e8i			0.255826i	−0.991785	+	0.127913i	$0.959172\pi$
$954$			6.08799e8i			0.701178i
$955$	−3.85698e8			−0.442831
$956$	−4.88224e8			−0.558786
$957$		−	1.13498e8i		−	0.129495i
$958$			6.12262e8i			0.696372i
$959$	−4.96324e8			−0.562742
$960$		−	1.84416e8i		−	0.208442i
$961$	−4.11643e8			−0.463821
$962$		−	2.27491e8i		−	0.255528i
$963$		−	4.10160e8i		−	0.459277i
$964$			7.74689e7i			0.0864761i
$965$			1.39959e7i			0.0155747i
$966$	−7.83043e8	+	2.97714e8i	−0.868669	+	0.330269i
$967$	−6.19509e8			−0.685123			−0.342561	−	0.939495i	$-0.611294\pi$
$967$	−6.19509e8			−0.685123			−0.342561	+	0.939495i	$0.611294\pi$
$968$	2.51550e9			2.77331
$969$	−2.21040e8			−0.242941
$970$	6.13005e8			0.671659
$971$		−	1.55871e9i		−	1.70258i	−0.524699	−	0.851288i	$-0.675822\pi$
$971$		−	1.55871e9i		−	1.70258i	0.524699	−	0.851288i	$-0.324178\pi$
$972$	−1.13222e8			−0.123291
$973$		−	9.25268e8i		−	1.00445i
$974$	−5.03268e8			−0.544656
$975$	9.97696e7			0.107643
$976$		−	1.13839e8i		−	0.122446i
$977$			9.07105e8i			0.972689i	0.873767	+	0.486345i	$0.161670\pi$
$977$			9.07105e8i			0.972689i	−0.873767	+	0.486345i	$0.838330\pi$
$978$	1.52932e9			1.63486
$979$	9.07008e8			0.966636
$980$		−	6.11096e7i		−	0.0649278i
$981$			4.46042e8i			0.472464i
$982$	1.94145e9			2.05018
$983$			1.29142e9i			1.35958i	0.733406	+	0.679791i	$0.237930\pi$
$983$			1.29142e9i			1.35958i	−0.733406	+	0.679791i	$0.762070\pi$
$984$	−1.05895e9			−1.11146
$985$		−	3.83839e8i		−	0.401643i
$986$		−	2.41179e8i		−	0.251598i
$987$			3.31643e8i			0.344921i
$988$			1.46412e8i			0.151812i
$989$	−2.68979e8	+	1.02266e8i	−0.278054	+	0.105716i
$990$	−2.73966e8			−0.282352
$991$	−1.49144e9			−1.53245			−0.766224	−	0.642574i	$-0.777867\pi$
$991$	−1.49144e9			−1.53245			−0.766224	+	0.642574i	$0.777867\pi$
$992$	1.83370e8			0.187842
$993$	−1.09410e9			−1.11740
$994$		−	1.83650e9i		−	1.86996i
$995$	2.27075e7			0.0230516
$996$			1.89487e9i			1.91779i
$997$	−3.90316e8			−0.393850			−0.196925	−	0.980419i	$-0.563096\pi$
$997$	−3.90316e8			−0.393850			−0.196925	+	0.980419i	$0.563096\pi$
$998$	6.21897e8			0.625644
$999$			1.40154e8i			0.140575i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.3	✓	24
3.2	odd	2		207.7.d.e.91.22		24
23.22	odd	2	inner	69.7.d.a.22.4	yes	24
69.68	even	2		207.7.d.e.91.21		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.3	✓	24	1.1	even	1	trivial
69.7.d.a.22.4	yes	24	23.22	odd	2	inner
207.7.d.e.91.21		24	69.68	even	2
207.7.d.e.91.22		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-13.6749 q^{2} -15.5885 q^{3} +123.003 q^{4} -37.2861i q^{5} +213.170 q^{6} -322.993i q^{7} -806.853 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-13.6749 q^{2} -15.5885 q^{3} +123.003 q^{4} -37.2861i q^{5} +213.170 q^{6} -322.993i q^{7} -806.853 q^{8} +243.000 q^{9} +509.883i q^{10} +2211.16i q^{11} -1917.42 q^{12} -449.620 q^{13} +4416.89i q^{14} +581.232i q^{15} +3161.46 q^{16} +5356.12i q^{17} -3323.00 q^{18} -2647.39i q^{19} -4586.28i q^{20} +5034.96i q^{21} -30237.4i q^{22} +(-4323.93 - 11372.8i) q^{23} +12577.6 q^{24} +14234.7 q^{25} +6148.50 q^{26} -3788.00 q^{27} -39729.0i q^{28} +3292.80 q^{29} -7948.29i q^{30} +21814.2 q^{31} +8405.97 q^{32} -34468.6i q^{33} -73244.3i q^{34} -12043.1 q^{35} +29889.6 q^{36} -36999.4i q^{37} +36202.8i q^{38} +7008.88 q^{39} +30084.4i q^{40} -84193.7 q^{41} -68852.6i q^{42} -23651.2i q^{43} +271978. i q^{44} -9060.51i q^{45} +(59129.3 + 155521. i) q^{46} +65868.0 q^{47} -49282.3 q^{48} +13324.4 q^{49} -194659. q^{50} -83493.6i q^{51} -55304.3 q^{52} -183208. i q^{53} +51800.4 q^{54} +82445.5 q^{55} +260608. i q^{56} +41268.8i q^{57} -45028.6 q^{58} +283187. q^{59} +71493.0i q^{60} -36008.5i q^{61} -298307. q^{62} -78487.3i q^{63} -317284. q^{64} +16764.5i q^{65} +471354. i q^{66} -154256. i q^{67} +658816. i q^{68} +(67403.4 + 177284. i) q^{69} +164689. q^{70} -415789. q^{71} -196065. q^{72} -44631.0 q^{73} +505963. i q^{74} -221898. q^{75} -325636. i q^{76} +714190. q^{77} -95845.6 q^{78} -751803. i q^{79} -117878. i q^{80} +59049.0 q^{81} +1.15134e6 q^{82} -988237. i q^{83} +619313. i q^{84} +199709. q^{85} +323427. i q^{86} -51329.6 q^{87} -1.78408e6i q^{88} -410195. i q^{89} +123902. i q^{90} +145224. i q^{91} +(-531854. - 1.39888e6i) q^{92} -340050. q^{93} -900737. q^{94} -98710.9 q^{95} -131036. q^{96} -1.20225e6i q^{97} -182210. q^{98} +537312. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 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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