LMFDB - Embedded newform 112.6.i.f.81.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,6,Mod(65,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("112.65");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not 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:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 200 x^{8} - 198 x^{7} + 34197 x^{6} - 16185 x^{5} + 1170401 x^{4} + 2020497 x^{3} + \cdots + 13068225 $ x^10 + 200x^8 - 198x^7 + 34197x^6 - 16185x^5 + 1170401x^4 + 2020497x^3 + 33316924x^2 + 20977845x + 13068225
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.3
Root			$-0.319443 + 0.553292i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.6.i.f.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.13889 - 1.97261i) q^{3} +(-25.2555 - 43.7438i) q^{5} +(2.72289 - 129.613i) q^{7} +(118.906 + 205.951i) q^{9} +(96.0139 - 166.301i) q^{11} -656.798 q^{13} -115.053 q^{15} +(-1094.68 + 1896.05i) q^{17} +(-139.665 - 241.907i) q^{19} +(-252.575 - 152.986i) q^{21} +(-1856.80 - 3216.07i) q^{23} +(286.818 - 496.783i) q^{25} +1095.18 q^{27} -5276.51 q^{29} +(-3242.90 + 5616.87i) q^{31} +(-218.698 - 378.796i) q^{33} +(-5738.55 + 3154.34i) q^{35} +(-1475.04 - 2554.84i) q^{37} +(-748.019 + 1295.61i) q^{39} -2394.37 q^{41} -17261.5 q^{43} +(6006.06 - 10402.8i) q^{45} +(3860.38 + 6686.38i) q^{47} +(-16792.2 - 705.845i) q^{49} +(2493.44 + 4318.77i) q^{51} +(4248.63 - 7358.84i) q^{53} -9699.52 q^{55} -636.251 q^{57} +(-4467.51 + 7737.95i) q^{59} +(11194.8 + 19390.0i) q^{61} +(27017.7 - 14851.0i) q^{63} +(16587.8 + 28730.9i) q^{65} +(26347.5 - 45635.2i) q^{67} -8458.73 q^{69} +25479.0 q^{71} +(11454.1 - 19839.0i) q^{73} +(-653.306 - 1131.56i) q^{75} +(-21293.4 - 12897.5i) q^{77} +(-22959.5 - 39767.1i) q^{79} +(-27646.8 + 47885.7i) q^{81} +97180.7 q^{83} +110587. q^{85} +(-6009.35 + 10408.5i) q^{87} +(29565.7 + 51209.3i) q^{89} +(-1788.39 + 85129.8i) q^{91} +(7386.59 + 12793.9i) q^{93} +(-7054.63 + 12219.0i) q^{95} -73491.5 q^{97} +45666.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{3} + 81 q^{5} - 116 q^{7} - 390 q^{9} + 361 q^{11} - 684 q^{13} + 2098 q^{15} + 1809 q^{17} - 1277 q^{19} - 5253 q^{21} + 911 q^{23} - 3940 q^{25} - 9502 q^{27} + 10884 q^{29} - 2187 q^{31}+ \cdots - 499596 q^{99}+O(q^{100}) $ 10 * q + 5 * q^3 + 81 * q^5 - 116 * q^7 - 390 * q^9 + 361 * q^11 - 684 * q^13 + 2098 * q^15 + 1809 * q^17 - 1277 * q^19 - 5253 * q^21 + 911 * q^23 - 3940 * q^25 - 9502 * q^27 + 10884 * q^29 - 2187 * q^31 - 5553 * q^33 + 16845 * q^35 + 8181 * q^37 - 3422 * q^39 - 33156 * q^41 - 12664 * q^43 + 41310 * q^45 - 16101 * q^47 - 10070 * q^49 - 67865 * q^51 - 16047 * q^53 + 91258 * q^55 + 44694 * q^57 - 71027 * q^59 + 31093 * q^61 + 127978 * q^63 + 64370 * q^65 + 47981 * q^67 - 274498 * q^69 - 45024 * q^71 + 123333 * q^73 + 45460 * q^75 - 245825 * q^77 - 212481 * q^79 - 52917 * q^81 + 174920 * q^83 + 444282 * q^85 - 318070 * q^87 + 129045 * q^89 + 526488 * q^91 + 252835 * q^93 + 300417 * q^95 - 656548 * q^97 - 499596 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.13889	−	1.97261i	0.0730596	−	0.126543i	−0.827181	−	0.561935i	$-0.810057\pi$
$3$	1.13889	−	1.97261i	0.0730596	−	0.126543i	0.900241	+	0.435392i	$0.143390\pi$
$4$		0			0
$5$	−25.2555	−	43.7438i	−0.451784	−	0.782513i	0.546713	−	0.837320i	$-0.315879\pi$
$5$	−25.2555	−	43.7438i	−0.451784	−	0.782513i	−0.998497	+	0.0548068i	$0.982546\pi$
$6$		0			0
$7$	2.72289	−	129.613i	0.0210032	−	0.999779i
$7$	2.72289	−	129.613i	0.0210032	−	0.999779i
$8$		0			0
$9$	118.906	+	205.951i	0.489325	+	0.847535i
$10$		0			0
$11$	96.0139	−	166.301i	0.239250	−	0.414393i	−0.721249	−	0.692676i	$-0.756432\pi$
$11$	96.0139	−	166.301i	0.239250	−	0.414393i	0.960499	+	0.278282i	$0.0897651\pi$
$12$		0			0
$13$	−656.798			−1.07789			−0.538944	−	0.842342i	$-0.681177\pi$
$13$	−656.798			−1.07789			−0.538944	+	0.842342i	$0.681177\pi$
$14$		0			0
$15$	−115.053			−0.132029
$16$		0			0
$17$	−1094.68	+	1896.05i	−0.918685	+	1.59121i	−0.117271	+	0.993100i	$0.537415\pi$
$17$	−1094.68	+	1896.05i	−0.918685	+	1.59121i	−0.801414	+	0.598109i	$0.795919\pi$
$18$		0			0
$19$	−139.665	−	241.907i	−0.0887573	−	0.153732i	0.818229	−	0.574893i	$-0.194956\pi$
$19$	−139.665	−	241.907i	−0.0887573	−	0.153732i	−0.906986	+	0.421161i	$0.861623\pi$
$20$		0			0
$21$	−252.575	−	152.986i	−0.124981	−	0.0757013i
$22$		0			0
$23$	−1856.80	−	3216.07i	−0.731889	−	1.26767i	−0.956075	−	0.293123i	$-0.905306\pi$
$23$	−1856.80	−	3216.07i	−0.731889	−	1.26767i	0.224186	−	0.974546i	$-0.428028\pi$
$24$		0			0
$25$	286.818	−	496.783i	0.0917818	−	0.158971i
$26$		0			0
$27$	1095.18			0.289119
$28$		0			0
$29$	−5276.51			−1.16507			−0.582535	−	0.812806i	$-0.697939\pi$
$29$	−5276.51			−1.16507			−0.582535	+	0.812806i	$0.697939\pi$
$30$		0			0
$31$	−3242.90	+	5616.87i	−0.606079	+	1.04976i	0.385801	+	0.922582i	$0.373925\pi$
$31$	−3242.90	+	5616.87i	−0.606079	+	1.04976i	−0.991880	+	0.127177i	$0.959408\pi$
$32$		0			0
$33$	−218.698	−	378.796i	−0.0349590	−	0.0605508i
$34$		0			0
$35$	−5738.55	+	3154.34i	−0.791830	+	0.435249i
$36$		0			0
$37$	−1475.04	−	2554.84i	−0.177133	−	0.306803i	0.763764	−	0.645495i	$-0.223349\pi$
$37$	−1475.04	−	2554.84i	−0.177133	−	0.306803i	−0.940897	+	0.338692i	$0.890016\pi$
$38$		0			0
$39$	−748.019	+	1295.61i	−0.0787501	+	0.136399i
$40$		0			0
$41$	−2394.37			−0.222449			−0.111225	−	0.993795i	$-0.535477\pi$
$41$	−2394.37			−0.222449			−0.111225	+	0.993795i	$0.535477\pi$
$42$		0			0
$43$	−17261.5			−1.42366			−0.711832	−	0.702350i	$-0.752134\pi$
$43$	−17261.5			−1.42366			−0.711832	+	0.702350i	$0.752134\pi$
$44$		0			0
$45$	6006.06	−	10402.8i	0.442138	−	0.765806i
$46$		0			0
$47$	3860.38	+	6686.38i	0.254909	+	0.441516i	0.964871	−	0.262725i	$-0.0846210\pi$
$47$	3860.38	+	6686.38i	0.254909	+	0.441516i	−0.709962	+	0.704240i	$0.751288\pi$
$48$		0			0
$49$	−16792.2	−	705.845i	−0.999118	−	0.0419971i
$50$		0			0
$51$	2493.44	+	4318.77i	0.134238	+	0.232506i
$52$		0			0
$53$	4248.63	−	7358.84i	0.207759	−	0.359849i	−0.743249	−	0.669014i	$-0.766716\pi$
$53$	4248.63	−	7358.84i	0.207759	−	0.359849i	0.951008	+	0.309166i	$0.100050\pi$
$54$		0			0
$55$	−9699.52			−0.432358
$56$		0			0
$57$	−636.251			−0.0259383
$58$		0			0
$59$	−4467.51	+	7737.95i	−0.167084	+	0.289398i	−0.937393	−	0.348272i	$-0.886768\pi$
$59$	−4467.51	+	7737.95i	−0.167084	+	0.289398i	0.770309	+	0.637670i	$0.220102\pi$
$60$		0			0
$61$	11194.8	+	19390.0i	0.385205	+	0.667194i	0.991798	−	0.127818i	$-0.0407975\pi$
$61$	11194.8	+	19390.0i	0.385205	+	0.667194i	−0.606593	+	0.795013i	$0.707464\pi$
$62$		0			0
$63$	27017.7	−	14851.0i	0.857625	−	0.471416i
$64$		0			0
$65$	16587.8	+	28730.9i	0.486973	+	0.843462i
$66$		0			0
$67$	26347.5	−	45635.2i	0.717055	−	1.24198i	−0.245107	−	0.969496i	$-0.578823\pi$
$67$	26347.5	−	45635.2i	0.717055	−	1.24198i	0.962162	−	0.272479i	$-0.0878435\pi$
$68$		0			0
$69$	−8458.73			−0.213886
$70$		0			0
$71$	25479.0			0.599842			0.299921	−	0.953964i	$-0.403040\pi$
$71$	25479.0			0.599842			0.299921	+	0.953964i	$0.403040\pi$
$72$		0			0
$73$	11454.1	−	19839.0i	0.251566	−	0.435726i	−0.712391	−	0.701783i	$-0.752388\pi$
$73$	11454.1	−	19839.0i	0.251566	−	0.435726i	0.963957	+	0.266057i	$0.0857210\pi$
$74$		0			0
$75$	−653.306	−	1131.56i	−0.0134111	−	0.0232287i
$76$		0			0
$77$	−21293.4	−	12897.5i	−0.409277	−	0.247901i
$78$		0			0
$79$	−22959.5	−	39767.1i	−0.413900	−	0.716895i	0.581412	−	0.813609i	$-0.302500\pi$
$79$	−22959.5	−	39767.1i	−0.413900	−	0.716895i	−0.995312	+	0.0967135i	$0.969167\pi$
$80$		0			0
$81$	−27646.8	+	47885.7i	−0.468202	+	0.810949i
$82$		0			0
$83$	97180.7			1.54841			0.774203	−	0.632937i	$-0.218151\pi$
$83$	97180.7			1.54841			0.774203	+	0.632937i	$0.218151\pi$
$84$		0			0
$85$	110587.			1.66019
$86$		0			0
$87$	−6009.35	+	10408.5i	−0.0851196	+	0.147431i
$88$		0			0
$89$	29565.7	+	51209.3i	0.395652	+	0.685289i	0.993184	−	0.116556i	$-0.0371853\pi$
$89$	29565.7	+	51209.3i	0.395652	+	0.685289i	−0.597532	+	0.801845i	$0.703852\pi$
$90$		0			0
$91$	−1788.39	+	85129.8i	−0.0226391	+	1.07765i
$92$		0			0
$93$	7386.59	+	12793.9i	0.0885598	+	0.153390i
$94$		0			0
$95$	−7054.63	+	12219.0i	−0.0801983	+	0.138908i
$96$		0			0
$97$	−73491.5			−0.793063			−0.396531	−	0.918021i	$-0.629786\pi$
$97$	−73491.5			−0.793063			−0.396531	+	0.918021i	$0.629786\pi$
$98$		0			0
$99$	45666.5			0.468284
$100$		0			0
$101$	89429.8	−	154897.i	0.872326	−	1.51091i	0.0127420	−	0.999919i	$-0.495944\pi$
$101$	89429.8	−	154897.i	0.872326	−	1.51091i	0.859584	−	0.510994i	$-0.170723\pi$
$102$		0			0
$103$	4355.98	+	7544.78i	0.0404569	+	0.0700734i	0.885545	−	0.464554i	$-0.153785\pi$
$103$	4355.98	+	7544.78i	0.0404569	+	0.0700734i	−0.845088	+	0.534627i	$0.820452\pi$
$104$		0			0
$105$	−313.276	+	14912.3i	−0.00277302	+	0.132000i
$106$		0			0
$107$	−86359.4	−	149579.i	−0.729206	−	1.26302i	−0.957219	−	0.289364i	$-0.906556\pi$
$107$	−86359.4	−	149579.i	−0.729206	−	1.26302i	0.228013	−	0.973658i	$-0.426777\pi$
$108$		0			0
$109$	83153.2	−	144026.i	0.670367	−	1.16111i	−0.307433	−	0.951570i	$-0.599470\pi$
$109$	83153.2	−	144026.i	0.670367	−	1.16111i	0.977800	−	0.209540i	$-0.0671967\pi$
$110$		0			0
$111$	−6719.61			−0.0517651
$112$		0			0
$113$	29604.7			0.218104			0.109052	−	0.994036i	$-0.465218\pi$
$113$	29604.7			0.218104			0.109052	+	0.994036i	$0.465218\pi$
$114$		0			0
$115$	−93788.8	+	162447.i	−0.661312	+	1.14543i
$116$		0			0
$117$	−78097.2	−	135268.i	−0.527437	−	0.913548i
$118$		0			0
$119$	242772.	+	147048.i	1.57156	+	0.951903i
$120$		0			0
$121$	62088.2	+	107540.i	0.385519	+	0.667738i
$122$		0			0
$123$	−2726.91	+	4723.15i	−0.0162520	+	0.0281494i
$124$		0			0
$125$	−186822.			−1.06943
$126$		0			0
$127$	−333136.			−1.83279			−0.916395	−	0.400275i	$-0.868915\pi$
$127$	−333136.			−1.83279			−0.916395	+	0.400275i	$0.868915\pi$
$128$		0			0
$129$	−19658.9	+	34050.2i	−0.104012	+	0.180155i
$130$		0			0
$131$	−29114.0	−	50426.9i	−0.148226	−	0.256735i	0.782346	−	0.622844i	$-0.214023\pi$
$131$	−29114.0	−	50426.9i	−0.148226	−	0.256735i	−0.930572	+	0.366109i	$0.880690\pi$
$132$		0			0
$133$	−31734.6	+	17443.8i	−0.155562	+	0.0855088i
$134$		0			0
$135$	−27659.3	−	47907.4i	−0.130619	−	0.226239i
$136$		0			0
$137$	−163843.	+	283784.i	−0.745805	+	1.29177i	0.204012	+	0.978968i	$0.434602\pi$
$137$	−163843.	+	283784.i	−0.745805	+	1.29177i	−0.949818	+	0.312804i	$0.898732\pi$
$138$		0			0
$139$	−165074.			−0.724674			−0.362337	−	0.932047i	$-0.618021\pi$
$139$	−165074.			−0.724674			−0.362337	+	0.932047i	$0.618021\pi$
$140$		0			0
$141$	17586.2			0.0744943
$142$		0			0
$143$	−63061.8	+	109226.i	−0.257885	+	0.446670i
$144$		0			0
$145$	133261.	+	230815.i	0.526361	+	0.911683i
$146$		0			0
$147$	−20516.7	+	32320.5i	−0.0783096	+	0.123363i
$148$		0			0
$149$	−58994.0	−	102181.i	−0.217692	−	0.377053i	0.736410	−	0.676535i	$-0.236519\pi$
$149$	−58994.0	−	102181.i	−0.217692	−	0.377053i	−0.954102	+	0.299482i	$0.903186\pi$
$150$		0			0
$151$	207650.	−	359661.i	0.741123	−	1.28366i	−0.210862	−	0.977516i	$-0.567627\pi$
$151$	207650.	−	359661.i	0.741123	−	1.28366i	0.951984	−	0.306146i	$-0.0990397\pi$
$152$		0			0
$153$	−520658.			−1.79814
$154$		0			0
$155$	327604.			1.09527
$156$		0			0
$157$	226566.	−	392423.i	0.733575	−	1.27059i	−0.221770	−	0.975099i	$-0.571183\pi$
$157$	226566.	−	392423.i	0.733575	−	1.27059i	0.955346	−	0.295491i	$-0.0954832\pi$
$158$		0			0
$159$	−9677.42	−	16761.8i	−0.0303575	−	0.0525808i
$160$		0			0
$161$	−421901.	+	231909.i	−1.28276	+	0.705103i
$162$		0			0
$163$	−216167.	−	374412.i	−0.637265	−	1.10378i	−0.986030	−	0.166566i	$-0.946732\pi$
$163$	−216167.	−	374412.i	−0.637265	−	1.10378i	0.348765	−	0.937210i	$-0.386601\pi$
$164$		0			0
$165$	−11046.7	+	19133.4i	−0.0315879	+	0.0547119i
$166$		0			0
$167$	−343081.			−0.951932			−0.475966	−	0.879464i	$-0.657901\pi$
$167$	−343081.			−0.951932			−0.475966	+	0.879464i	$0.657901\pi$
$168$		0			0
$169$	60091.2			0.161843
$170$		0			0
$171$	33214.0	−	57528.3i	0.0868622	−	0.150450i
$172$		0			0
$173$	221752.	+	384086.i	0.563317	+	0.975694i	0.997204	+	0.0747267i	$0.0238084\pi$
$173$	221752.	+	384086.i	0.563317	+	0.975694i	−0.433887	+	0.900967i	$0.642858\pi$
$174$		0			0
$175$	−63608.7	−	38528.1i	−0.157008	−	0.0951004i
$176$		0			0
$177$	10176.0	+	17625.3i	0.0244142	+	0.0422866i
$178$		0			0
$179$	−47649.8	+	82531.8i	−0.111155	+	0.192526i	−0.916236	−	0.400639i	$-0.868788\pi$
$179$	−47649.8	+	82531.8i	−0.111155	+	0.192526i	0.805081	+	0.593164i	$0.202122\pi$
$180$		0			0
$181$	−78562.3			−0.178245			−0.0891226	−	0.996021i	$-0.528406\pi$
$181$	−78562.3			−0.178245			−0.0891226	+	0.996021i	$0.528406\pi$
$182$		0			0
$183$	50998.4			0.112572
$184$		0			0
$185$	−74505.8	+	129048.i	−0.160052	+	0.277218i
$186$		0			0
$187$	210210.	+	364094.i	0.439591	+	0.761394i
$188$		0			0
$189$	2982.05	−	141950.i	0.00607241	−	0.289055i
$190$		0			0
$191$	147381.	+	255271.i	0.292319	+	0.506311i	0.974358	−	0.225005i	$-0.0722398\pi$
$191$	147381.	+	255271.i	0.292319	+	0.506311i	−0.682039	+	0.731316i	$0.738907\pi$
$192$		0			0
$193$	437328.	−	757474.i	0.845111	−	1.46378i	−0.0404132	−	0.999183i	$-0.512867\pi$
$193$	437328.	−	757474.i	0.845111	−	1.46378i	0.885524	−	0.464593i	$-0.153799\pi$
$194$		0			0
$195$	75566.4			0.142312
$196$		0			0
$197$	338078.			0.620656			0.310328	−	0.950630i	$-0.399561\pi$
$197$	338078.			0.620656			0.310328	+	0.950630i	$0.399561\pi$
$198$		0			0
$199$	−176522.	+	305745.i	−0.315985	+	0.547302i	−0.979646	−	0.200731i	$-0.935668\pi$
$199$	−176522.	+	305745.i	−0.315985	+	0.547302i	0.663661	+	0.748033i	$0.269002\pi$
$200$		0			0
$201$	−60013.6	−	103947.i	−0.104775	−	0.181476i
$202$		0			0
$203$	−14367.4	+	683906.i	−0.0244702	+	1.16481i
$204$		0			0
$205$	60470.9	+	104739.i	0.100499	+	0.174069i
$206$		0			0
$207$	441569.	−	764819.i	0.716263	−	1.24060i
$208$		0			0
$209$	−53639.2			−0.0849408
$210$		0			0
$211$	−351876.			−0.544106			−0.272053	−	0.962282i	$-0.587703\pi$
$211$	−351876.			−0.544106			−0.272053	+	0.962282i	$0.587703\pi$
$212$		0			0
$213$	29017.7	−	50260.1i	0.0438242	−	0.0759057i
$214$		0			0
$215$	435948.	+	755084.i	0.643189	+	1.11404i
$216$		0			0
$217$	719190.	+	435617.i	1.03680	+	0.627993i
$218$		0			0
$219$	−26089.8	−	45188.8i	−0.0367587	−	0.0636679i
$220$		0			0
$221$	718987.	−	1.24532e6i	0.990240	−	1.71515i
$222$		0			0
$223$	−434832.			−0.585544			−0.292772	−	0.956182i	$-0.594578\pi$
$223$	−434832.			−0.585544			−0.292772	+	0.956182i	$0.594578\pi$
$224$		0			0
$225$	136417.			0.179644
$226$		0			0
$227$	535280.	−	927133.i	0.689472	−	1.19420i	−0.282537	−	0.959256i	$-0.591176\pi$
$227$	535280.	−	927133.i	0.689472	−	1.19420i	0.972009	−	0.234944i	$-0.0754907\pi$
$228$		0			0
$229$	−65933.2	−	114200.i	−0.0830836	−	0.143905i	0.821489	−	0.570224i	$-0.193144\pi$
$229$	−65933.2	−	114200.i	−0.0830836	−	0.143905i	−0.904573	+	0.426319i	$0.859810\pi$
$230$		0			0
$231$	−49692.4	+	27314.7i	−0.0612717	+	0.0336796i
$232$		0			0
$233$	−31331.5	−	54267.8i	−0.0378087	−	0.0654866i	0.846502	−	0.532386i	$-0.178704\pi$
$233$	−31331.5	−	54267.8i	−0.0378087	−	0.0654866i	−0.884311	+	0.466899i	$0.845371\pi$
$234$		0			0
$235$	194992.	−	337736.i	0.230328	−	0.398940i
$236$		0			0
$237$	−104593.			−0.120957
$238$		0			0
$239$	1.29337e6			1.46462			0.732312	−	0.680969i	$-0.238441\pi$
$239$	1.29337e6			1.46462			0.732312	+	0.680969i	$0.238441\pi$
$240$		0			0
$241$	−58141.1	+	100703.i	−0.0644823	+	0.111687i	−0.896464	−	0.443116i	$-0.853873\pi$
$241$	−58141.1	+	100703.i	−0.0644823	+	0.111687i	0.831982	+	0.554803i	$0.187206\pi$
$242$		0			0
$243$	196038.	+	339547.i	0.212973	+	0.368879i
$244$		0			0
$245$	393219.	+	752380.i	0.418522	+	0.800797i
$246$		0			0
$247$	91731.8	+	158884.i	0.0956704	+	0.165706i
$248$		0			0
$249$	110678.	−	191700.i	0.113126	−	0.195940i
$250$		0			0
$251$	−360974.			−0.361653			−0.180827	−	0.983515i	$-0.557877\pi$
$251$	−360974.			−0.361653			−0.180827	+	0.983515i	$0.557877\pi$
$252$		0			0
$253$	−713114.			−0.700419
$254$		0			0
$255$	125946.	−	218146.i	0.121293	−	0.210085i
$256$		0			0
$257$	966677.	+	1.67433e6i	0.912953	+	1.58128i	0.809870	+	0.586609i	$0.199537\pi$
$257$	966677.	+	1.67433e6i	0.912953	+	1.58128i	0.103083	+	0.994673i	$0.467129\pi$
$258$		0			0
$259$	−335158.	+	184228.i	−0.310456	+	0.170650i
$260$		0			0
$261$	−627409.	−	1.08670e6i	−0.570098	−	0.987438i
$262$		0			0
$263$	107394.	−	186012.i	0.0957396	−	0.165826i	−0.814177	−	0.580616i	$-0.802812\pi$
$263$	107394.	−	186012.i	0.0957396	−	0.165826i	0.909917	+	0.414790i	$0.136145\pi$
$264$		0			0
$265$	−429205.			−0.375449
$266$		0			0
$267$	134688.			0.115625
$268$		0			0
$269$	−951375.	+	1.64783e6i	−0.801625	+	1.38846i	0.116921	+	0.993141i	$0.462698\pi$
$269$	−951375.	+	1.64783e6i	−0.801625	+	1.38846i	−0.918546	+	0.395314i	$0.870636\pi$
$270$		0			0
$271$	−642462.	−	1.11278e6i	−0.531403	−	0.920418i	−0.999328	−	0.0366494i	$-0.988332\pi$
$271$	−642462.	−	1.11278e6i	−0.531403	−	0.920418i	0.467925	−	0.883768i	$-0.345002\pi$
$272$		0			0
$273$	165891.	+	100481.i	0.134715	+	0.0815975i
$274$		0			0
$275$	−55077.0	−	95396.2i	−0.0439176	−	0.0760675i
$276$		0			0
$277$	−713348.	+	1.23556e6i	−0.558602	+	0.967527i	0.439012	+	0.898481i	$0.355329\pi$
$277$	−713348.	+	1.23556e6i	−0.558602	+	0.967527i	−0.997614	+	0.0690453i	$0.978005\pi$
$278$		0			0
$279$	−1.54240e6			−1.18628
$280$		0			0
$281$	492636.			0.372186			0.186093	−	0.982532i	$-0.440417\pi$
$281$	492636.			0.372186			0.186093	+	0.982532i	$0.440417\pi$
$282$		0			0
$283$	1.00648e6	−	1.74327e6i	0.747031	−	1.29390i	−0.202208	−	0.979343i	$-0.564812\pi$
$283$	1.00648e6	−	1.74327e6i	0.747031	−	1.29390i	0.949240	−	0.314554i	$-0.101855\pi$
$284$		0			0
$285$	16068.8	+	27832.1i	0.0117185	+	0.0202971i
$286$		0			0
$287$	−6519.59	+	310341.i	−0.00467214	+	0.222400i
$288$		0			0
$289$	−1.68674e6	−	2.92152e6i	−1.18796	−	2.05762i
$290$		0			0
$291$	−83698.5	+	144970.i	−0.0579409	+	0.100357i
$292$		0			0
$293$	−58017.9			−0.0394814			−0.0197407	−	0.999805i	$-0.506284\pi$
$293$	−58017.9			−0.0394814			−0.0197407	+	0.999805i	$0.506284\pi$
$294$		0			0
$295$	451317.			0.301944
$296$		0			0
$297$	105152.	−	182129.i	0.0691717	−	0.119809i
$298$		0			0
$299$	1.21954e6	+	2.11231e6i	0.788895	+	1.36641i
$300$		0			0
$301$	−47001.1	+	2.23732e6i	−0.0299014	+	1.42335i
$302$		0			0
$303$	−203701.	−	352820.i	−0.127464	−	0.220773i
$304$		0			0
$305$	565461.	−	979407.i	0.348059	−	0.602856i
$306$		0			0
$307$	−3.08036e6			−1.86533			−0.932665	−	0.360744i	$-0.882523\pi$
$307$	−3.08036e6			−1.86533			−0.932665	+	0.360744i	$0.882523\pi$
$308$		0			0
$309$	19843.9			0.0118231
$310$		0			0
$311$	−280927.	+	486579.i	−0.164699	+	0.285268i	−0.936548	−	0.350538i	$-0.885999\pi$
$311$	−280927.	+	486579.i	−0.164699	+	0.285268i	0.771849	+	0.635806i	$0.219332\pi$
$312$		0			0
$313$	−1.48564e6	−	2.57320e6i	−0.857141	−	1.48461i	−0.874645	−	0.484765i	$-0.838905\pi$
$313$	−1.48564e6	−	2.57320e6i	−0.857141	−	1.48461i	0.0175037	−	0.999847i	$-0.494428\pi$
$314$		0			0
$315$	−1.33199e6	−	806790.i	−0.756351	−	0.458125i
$316$		0			0
$317$	375244.	+	649941.i	0.209732	+	0.363267i	0.951630	−	0.307246i	$-0.0994074\pi$
$317$	375244.	+	649941.i	0.209732	+	0.363267i	−0.741898	+	0.670513i	$0.766074\pi$
$318$		0			0
$319$	−506619.	+	877489.i	−0.278743	+	0.482798i
$320$		0			0
$321$	−393414.			−0.213102
$322$		0			0
$323$	611557.			0.326160
$324$		0			0
$325$	−188382.	+	326287.i	−0.0989305	+	0.171353i
$326$		0			0
$327$	−189404.	−	328058.i	−0.0979535	−	0.169660i
$328$		0			0
$329$	877155.	−	482150.i	0.446772	−	0.245580i
$330$		0			0
$331$	1.34338e6	+	2.32681e6i	0.673954	+	1.16732i	0.976773	+	0.214275i	$0.0687387\pi$
$331$	1.34338e6	+	2.32681e6i	0.673954	+	1.16732i	−0.302820	+	0.953048i	$0.597928\pi$
$332$		0			0
$333$	350782.	−	607572.i	0.173351	−	0.300253i
$334$		0			0
$335$	−2.66168e6			−1.29582
$336$		0			0
$337$	49365.1			0.0236780			0.0118390	−	0.999930i	$-0.496231\pi$
$337$	49365.1			0.0236780			0.0118390	+	0.999930i	$0.496231\pi$
$338$		0			0
$339$	33716.4	−	58398.5i	0.0159346	−	0.0275996i
$340$		0			0
$341$	622727.	+	1.07859e6i	0.290009	+	0.502310i
$342$		0			0
$343$	−137210.	+	2.17457e6i	−0.0629725	+	0.998015i
$344$		0			0
$345$	213630.	+	370017.i	0.0966304	+	0.167369i
$346$		0			0
$347$	−609492.	+	1.05567e6i	−0.271734	+	0.470657i	−0.969306	−	0.245857i	$-0.920930\pi$
$347$	−609492.	+	1.05567e6i	−0.271734	+	0.470657i	0.697572	+	0.716515i	$0.254264\pi$
$348$		0			0
$349$	−3.25801e6			−1.43182			−0.715910	−	0.698193i	$-0.753988\pi$
$349$	−3.25801e6			−1.43182			−0.715910	+	0.698193i	$0.753988\pi$
$350$		0			0
$351$	−719313.			−0.311638
$352$		0			0
$353$	317724.	−	550315.i	0.135711	−	0.235058i	−0.790158	−	0.612903i	$-0.790002\pi$
$353$	317724.	−	550315.i	0.135711	−	0.235058i	0.925869	+	0.377845i	$0.123335\pi$
$354$		0			0
$355$	−643485.	−	1.11455e6i	−0.270999	−	0.469384i
$356$		0			0
$357$	566559.	−	311424.i	0.235274	−	0.129325i
$358$		0			0
$359$	471991.	+	817512.i	0.193285	+	0.334779i	0.946337	−	0.323182i	$-0.104753\pi$
$359$	471991.	+	817512.i	0.193285	+	0.334779i	−0.753052	+	0.657961i	$0.771419\pi$
$360$		0			0
$361$	1.19904e6	−	2.07679e6i	0.484244	−	0.838736i
$362$		0			0
$363$	282846.			0.112663
$364$		0			0
$365$	−1.15711e6			−0.454615
$366$		0			0
$367$	2.00724e6	−	3.47664e6i	0.777918	−	1.34739i	−0.155222	−	0.987880i	$-0.549609\pi$
$367$	2.00724e6	−	3.47664e6i	0.777918	−	1.34739i	0.933140	−	0.359514i	$-0.117057\pi$
$368$		0			0
$369$	−284704.	−	493122.i	−0.108850	−	0.188533i
$370$		0			0
$371$	−942235.	−	570716.i	−0.355406	−	0.215271i
$372$		0			0
$373$	−83508.2	−	144641.i	−0.0310783	−	0.0538292i	0.850068	−	0.526673i	$-0.176561\pi$
$373$	−83508.2	−	144641.i	−0.0310783	−	0.0538292i	−0.881146	+	0.472844i	$0.843227\pi$
$374$		0			0
$375$	−212769.	+	368527.i	−0.0781322	+	0.135329i
$376$		0			0
$377$	3.46561e6			1.25582
$378$		0			0
$379$	2.51875e6			0.900716			0.450358	−	0.892848i	$-0.351296\pi$
$379$	2.51875e6			0.900716			0.450358	+	0.892848i	$0.351296\pi$
$380$		0			0
$381$	−379405.	+	657148.i	−0.133903	+	0.231927i
$382$		0			0
$383$	572049.	+	990818.i	0.199267	+	0.345141i	0.948291	−	0.317402i	$-0.102810\pi$
$383$	572049.	+	990818.i	0.199267	+	0.345141i	−0.749024	+	0.662543i	$0.769477\pi$
$384$		0			0
$385$	−26410.7	+	1.25719e6i	−0.00908089	+	0.432263i
$386$		0			0
$387$	−2.05249e6	−	3.55502e6i	−0.696633	−	1.20660i
$388$		0			0
$389$	−1.21736e6	+	2.10853e6i	−0.407892	+	0.706490i	−0.994653	−	0.103270i	$-0.967069\pi$
$389$	−1.21736e6	+	2.10853e6i	−0.407892	+	0.706490i	0.586761	+	0.809760i	$0.300403\pi$
$390$		0			0
$391$	8.13044e6			2.68950
$392$		0			0
$393$	−132630.			−0.0433173
$394$		0			0
$395$	−1.15971e6	+	2.00868e6i	−0.373987	+	0.647764i
$396$		0			0
$397$	1.37560e6	+	2.38261e6i	0.438042	+	0.758711i	0.997538	−	0.0701215i	$-0.0223387\pi$
$397$	1.37560e6	+	2.38261e6i	0.438042	+	0.758711i	−0.559496	+	0.828833i	$0.689005\pi$
$398$		0			0
$399$	−1732.44	+	82466.5i	−0.000544786	+	0.0259326i
$400$		0			0
$401$	55226.2	+	95654.6i	0.0171508	+	0.0297061i	0.874473	−	0.485073i	$-0.161207\pi$
$401$	55226.2	+	95654.6i	0.0171508	+	0.0297061i	−0.857323	+	0.514779i	$0.827874\pi$
$402$		0			0
$403$	2.12993e6	−	3.68915e6i	0.653285	−	1.13152i
$404$		0			0
$405$	2.79294e6			0.846105
$406$		0			0
$407$	−566497.			−0.169516
$408$		0			0
$409$	85347.5	−	147826.i	0.0252280	−	0.0436962i	−0.853136	−	0.521689i	$-0.825302\pi$
$409$	85347.5	−	147826.i	0.0252280	−	0.0436962i	0.878364	+	0.477993i	$0.158635\pi$
$410$		0			0
$411$	373196.	+	646395.i	0.108976	+	0.188753i
$412$		0			0
$413$	990776.	+	600117.i	0.285825	+	0.173126i
$414$		0			0
$415$	−2.45435e6	−	4.25106e6i	−0.699546	−	1.21165i
$416$		0			0
$417$	−188001.	+	325627.i	−0.0529444	+	0.0917024i
$418$		0			0
$419$	−2852.47			−0.000793754			−0.000396877	−	1.00000i	$-0.500126\pi$
$419$	−2852.47			−0.000793754			−0.000396877		1.00000i	$0.500126\pi$
$420$		0			0
$421$	−730555.			−0.200885			−0.100443	−	0.994943i	$-0.532026\pi$
$421$	−730555.			−0.200885			−0.100443	+	0.994943i	$0.532026\pi$
$422$		0			0
$423$	−918044.	+	1.59010e6i	−0.249467	+	0.432089i
$424$		0			0
$425$	627951.	+	1.08764e6i	0.168637	+	0.292088i
$426$		0			0
$427$	2.54368e6	−	1.39820e6i	0.675138	−	0.371107i
$428$		0			0
$429$	143640.	+	248792.i	0.0376819	+	0.0652671i
$430$		0			0
$431$	1.84705e6	−	3.19918e6i	0.478945	−	0.829557i	−0.520764	−	0.853701i	$-0.674353\pi$
$431$	1.84705e6	−	3.19918e6i	0.478945	−	0.829557i	0.999708	+	0.0241442i	$0.00768610\pi$
$432$		0			0
$433$	−674656.			−0.172927			−0.0864635	−	0.996255i	$-0.527557\pi$
$433$	−674656.			−0.172927			−0.0864635	+	0.996255i	$0.527557\pi$
$434$		0			0
$435$	607077.			0.153823
$436$		0			0
$437$	−518660.	+	898346.i	−0.129921	+	0.225030i
$438$		0			0
$439$	1.97813e6	+	3.42622e6i	0.489884	+	0.848505i	0.999932	−	0.0116414i	$-0.00370567\pi$
$439$	1.97813e6	+	3.42622e6i	0.489884	+	0.848505i	−0.510048	+	0.860146i	$0.670372\pi$
$440$		0			0
$441$	−1.85132e6	−	3.54229e6i	−0.453299	−	0.867337i
$442$		0			0
$443$	3.64706e6	+	6.31689e6i	0.882945	+	1.52931i	0.848051	+	0.529914i	$0.177776\pi$
$443$	3.64706e6	+	6.31689e6i	0.882945	+	1.52931i	0.0348935	+	0.999391i	$0.488891\pi$
$444$		0			0
$445$	1.49339e6	−	2.58663e6i	0.357499	−	0.619206i
$446$		0			0
$447$	−268750.			−0.0636179
$448$		0			0
$449$	5.49637e6			1.28665			0.643325	−	0.765593i	$-0.277554\pi$
$449$	5.49637e6			1.28665			0.643325	+	0.765593i	$0.277554\pi$
$450$		0			0
$451$	−229892.	+	398185.i	−0.0532210	+	0.0921815i
$452$		0			0
$453$	−472980.	−	819226.i	−0.108292	−	0.187568i
$454$		0			0
$455$	3.76907e6	−	2.07176e6i	0.853504	−	0.469150i
$456$		0			0
$457$	1.86877e6	+	3.23681e6i	0.418568	+	0.724982i	0.995796	−	0.0916018i	$-0.0291987\pi$
$457$	1.86877e6	+	3.23681e6i	0.418568	+	0.724982i	−0.577227	+	0.816583i	$0.695865\pi$
$458$		0			0
$459$	−1.19888e6	+	2.07652e6i	−0.265609	+	0.460048i
$460$		0			0
$461$	414493.			0.0908374			0.0454187	−	0.998968i	$-0.485538\pi$
$461$	414493.			0.0908374			0.0454187	+	0.998968i	$0.485538\pi$
$462$		0			0
$463$	−6.90895e6			−1.49782			−0.748910	−	0.662672i	$-0.769423\pi$
$463$	−6.90895e6			−1.49782			−0.748910	+	0.662672i	$0.769423\pi$
$464$		0			0
$465$	373104.	−	646235.i	0.0800198	−	0.138598i
$466$		0			0
$467$	−2.65352e6	−	4.59604e6i	−0.563029	−	0.975194i	−0.997230	−	0.0743786i	$-0.976303\pi$
$467$	−2.65352e6	−	4.59604e6i	−0.563029	−	0.975194i	0.434201	−	0.900816i	$-0.357031\pi$
$468$		0			0
$469$	−5.84318e6	−	3.53924e6i	−1.22664	−	0.742982i
$470$		0			0
$471$	−516065.	−	893851.i	−0.107189	−	0.185658i
$472$		0			0
$473$	−1.65734e6	+	2.87060e6i	−0.340612	+	0.589957i
$474$		0			0
$475$	−160234.			−0.0325852
$476$		0			0
$477$	2.02075e6			0.406646
$478$		0			0
$479$	−4.51875e6	+	7.82671e6i	−0.899870	+	1.55862i	−0.0722119	+	0.997389i	$0.523006\pi$
$479$	−4.51875e6	+	7.82671e6i	−0.899870	+	1.55862i	−0.827659	+	0.561232i	$0.810328\pi$
$480$		0			0
$481$	968804.	+	1.67802e6i	0.190930	+	0.330700i
$482$		0			0
$483$	−23032.2	+	1.09636e6i	−0.00449229	+	0.213839i
$484$		0			0
$485$	1.85607e6	+	3.21480e6i	0.358293	+	0.620582i
$486$		0			0
$487$	−2.22641e6	+	3.85625e6i	−0.425385	+	0.736789i	−0.996456	−	0.0841119i	$-0.973195\pi$
$487$	−2.22641e6	+	3.85625e6i	−0.425385	+	0.736789i	0.571071	+	0.820901i	$0.306528\pi$
$488$		0			0
$489$	−984758.			−0.186233
$490$		0			0
$491$	−2.27049e6			−0.425027			−0.212513	−	0.977158i	$-0.568165\pi$
$491$	−2.27049e6			−0.425027			−0.212513	+	0.977158i	$0.568165\pi$
$492$		0			0
$493$	5.77612e6	−	1.00045e7i	1.07033	−	1.85387i
$494$		0			0
$495$	−1.15333e6	−	1.99763e6i	−0.211563	−	0.366439i
$496$		0			0
$497$	69376.5	−	3.30242e6i	0.0125986	−	0.599709i
$498$		0			0
$499$	3.73551e6	+	6.47010e6i	0.671582	+	1.16321i	0.977456	+	0.211141i	$0.0677181\pi$
$499$	3.73551e6	+	6.47010e6i	0.671582	+	1.16321i	−0.305874	+	0.952072i	$0.598949\pi$
$500$		0			0
$501$	−390731.	+	676765.i	−0.0695477	+	0.120460i
$502$		0			0
$503$	6.64380e6			1.17084			0.585418	−	0.810731i	$-0.300930\pi$
$503$	6.64380e6			1.17084			0.585418	+	0.810731i	$0.300930\pi$
$504$		0			0
$505$	−9.03438e6			−1.57641
$506$		0			0
$507$	68437.1	−	118537.i	0.0118242	−	0.0204801i
$508$		0			0
$509$	−4.33107e6	−	7.50163e6i	−0.740970	−	1.28340i	−0.952054	−	0.305929i	$-0.901033\pi$
$509$	−4.33107e6	−	7.50163e6i	−0.740970	−	1.28340i	0.211085	−	0.977468i	$-0.432300\pi$
$510$		0			0
$511$	−2.54021e6	−	1.53862e6i	−0.430346	−	0.260663i
$512$		0			0
$513$	−152958.	−	264932.i	−0.0256614	−	0.0444468i
$514$		0			0
$515$	220025.	−	381095.i	0.0365556	−	0.0633162i
$516$		0			0
$517$	1.48260e6			0.243948
$518$		0			0
$519$	1.01020e6			0.164623
$520$		0			0
$521$	1.24435e6	−	2.15528e6i	0.200839	−	0.347864i	−0.747960	−	0.663744i	$-0.768966\pi$
$521$	1.24435e6	−	2.15528e6i	0.200839	−	0.347864i	0.948799	+	0.315880i	$0.102300\pi$
$522$		0			0
$523$	−4.18199e6	−	7.24342e6i	−0.668542	−	1.15795i	−0.978312	−	0.207137i	$-0.933585\pi$
$523$	−4.18199e6	−	7.24342e6i	−0.668542	−	1.15795i	0.309770	−	0.950812i	$-0.399748\pi$
$524$		0			0
$525$	−148444.	+	81596.0i	−0.0235052	+	0.0129202i
$526$		0			0
$527$	−7.09990e6	−	1.22974e7i	−1.11359	−	1.92880i
$528$		0			0
$529$	−3.67724e6	+	6.36916e6i	−0.571324	+	0.989562i
$530$		0			0
$531$	−2.12485e6			−0.327033
$532$		0			0
$533$	1.57262e6			0.239775
$534$		0			0
$535$	−4.36210e6	+	7.55538e6i	−0.658888	+	1.14123i
$536$		0			0
$537$	108535.	+	187989.i	0.0162419	+	0.0281317i
$538$		0			0
$539$	−1.72966e6	+	2.72478e6i	−0.256442	+	0.403980i
$540$		0			0
$541$	2.56279e6	+	4.43888e6i	0.376461	+	0.652049i	0.990545	−	0.137192i	$-0.0438076\pi$
$541$	2.56279e6	+	4.43888e6i	0.376461	+	0.652049i	−0.614084	+	0.789241i	$0.710474\pi$
$542$		0			0
$543$	−89473.5	+	154973.i	−0.0130225	+	0.0225557i
$544$		0			0
$545$	−8.40031e6			−1.21145
$546$		0			0
$547$	−8.27719e6			−1.18281			−0.591404	−	0.806375i	$-0.701426\pi$
$547$	−8.27719e6			−1.18281			−0.591404	+	0.806375i	$0.701426\pi$
$548$		0			0
$549$	−2.66225e6	+	4.61116e6i	−0.376980	+	0.652949i
$550$		0			0
$551$	736945.	+	1.27643e6i	0.103408	+	0.179109i
$552$		0			0
$553$	−5.21685e6	+	2.86758e6i	−0.725430	+	0.398751i
$554$		0			0
$555$	169707.	+	293942.i	0.0233867	+	0.0405069i
$556$		0			0
$557$	−5.72960e6	+	9.92396e6i	−0.782503	+	1.35534i	0.147976	+	0.988991i	$0.452724\pi$
$557$	−5.72960e6	+	9.92396e6i	−0.782503	+	1.35534i	−0.930479	+	0.366345i	$0.880609\pi$
$558$		0			0
$559$	1.13373e7			1.53455
$560$		0			0
$561$	957621.			0.128465
$562$		0			0
$563$	4.61535e6	−	7.99402e6i	0.613668	−	1.06290i	−0.376948	−	0.926234i	$-0.623027\pi$
$563$	4.61535e6	−	7.99402e6i	0.613668	−	1.06290i	0.990617	−	0.136670i	$-0.0436401\pi$
$564$		0			0
$565$	−747681.	−	1.29502e6i	−0.0985361	−	0.170670i
$566$		0			0
$567$	6.13134e6	+	3.71378e6i	0.800937	+	0.485131i
$568$		0			0
$569$	4.14874e6	+	7.18582e6i	0.537199	+	0.930456i	0.999053	+	0.0435000i	$0.0138509\pi$
$569$	4.14874e6	+	7.18582e6i	0.537199	+	0.930456i	−0.461855	+	0.886956i	$0.652816\pi$
$570$		0			0
$571$	−5.03677e6	+	8.72394e6i	−0.646490	+	1.11975i	0.337465	+	0.941338i	$0.390430\pi$
$571$	−5.03677e6	+	8.72394e6i	−0.646490	+	1.11975i	−0.983955	+	0.178416i	$0.942903\pi$
$572$		0			0
$573$	671399.			0.0854268
$574$		0			0
$575$	−2.13025e6			−0.268696
$576$		0			0
$577$	−3.21257e6	+	5.56434e6i	−0.401711	+	0.695784i	−0.993933	−	0.109991i	$-0.964918\pi$
$577$	−3.21257e6	+	5.56434e6i	−0.401711	+	0.695784i	0.592222	+	0.805775i	$0.298251\pi$
$578$		0			0
$579$	−996134.	−	1.72535e6i	−0.123487	−	0.213886i
$580$		0			0
$581$	264612.	−	1.25959e7i	0.0325214	−	1.54806i
$582$		0			0
$583$	−815855.	−	1.41310e6i	−0.0994126	−	0.172188i
$584$		0			0
$585$	−3.94477e6	+	6.83254e6i	−0.476576	+	0.825454i
$586$		0			0
$587$	3.97124e6			0.475698			0.237849	−	0.971302i	$-0.423558\pi$
$587$	3.97124e6			0.475698			0.237849	+	0.971302i	$0.423558\pi$
$588$		0			0
$589$	1.81168e6			0.215176
$590$		0			0
$591$	385032.	−	666895.i	0.0453449	−	0.0785396i
$592$		0			0
$593$	−2.44661e6	−	4.23765e6i	−0.285712	−	0.494867i	0.687070	−	0.726591i	$-0.258897\pi$
$593$	−2.44661e6	−	4.23765e6i	−0.285712	−	0.494867i	−0.972781	+	0.231724i	$0.925563\pi$
$594$		0			0
$595$	301117.	−	1.43336e7i	0.0348693	−	1.65982i
$596$		0			0
$597$	402077.	+	696419.i	0.0461715	+	0.0799714i
$598$		0			0
$599$	−1.98993e6	+	3.44666e6i	−0.226606	+	0.392493i	−0.956800	−	0.290747i	$-0.906096\pi$
$599$	−1.98993e6	+	3.44666e6i	−0.226606	+	0.392493i	0.730194	+	0.683240i	$0.239430\pi$
$600$		0			0
$601$	6.77069e6			0.764622			0.382311	−	0.924034i	$-0.375128\pi$
$601$	6.77069e6			0.764622			0.382311	+	0.924034i	$0.375128\pi$
$602$		0			0
$603$	1.25315e7			1.40349
$604$		0			0
$605$	3.13614e6	−	5.43195e6i	0.348343	−	0.603347i
$606$		0			0
$607$	−7.61245e6	−	1.31851e7i	−0.838596	−	1.45249i	−0.891069	−	0.453868i	$-0.850044\pi$
$607$	−7.61245e6	−	1.31851e7i	−0.838596	−	1.45249i	0.0524729	−	0.998622i	$-0.483290\pi$
$608$		0			0
$609$	1.33272e6	+	807233.i	0.145611	+	0.0881973i
$610$		0			0
$611$	−2.53549e6	−	4.39160e6i	−0.274764	−	0.475905i
$612$		0			0
$613$	5.80178e6	−	1.00490e7i	0.623606	−	1.08012i	−0.365203	−	0.930928i	$-0.619000\pi$
$613$	5.80178e6	−	1.00490e7i	0.623606	−	1.08012i	0.988809	−	0.149189i	$-0.0476662\pi$
$614$		0			0
$615$	275478.			0.0293697
$616$		0			0
$617$	−4.51891e6			−0.477882			−0.238941	−	0.971034i	$-0.576800\pi$
$617$	−4.51891e6			−0.477882			−0.238941	+	0.971034i	$0.576800\pi$
$618$		0			0
$619$	−3.43539e6	+	5.95026e6i	−0.360370	+	0.624180i	−0.988022	−	0.154315i	$-0.950683\pi$
$619$	−3.43539e6	+	5.95026e6i	−0.360370	+	0.624180i	0.627651	+	0.778494i	$0.284016\pi$
$620$		0			0
$621$	−2.03353e6	−	3.52218e6i	−0.211603	−	0.366507i
$622$		0			0
$623$	6.71791e6	−	3.69267e6i	0.693448	−	0.381171i
$624$		0			0
$625$	3.82198e6	+	6.61986e6i	0.391370	+	0.677873i
$626$		0			0
$627$	−61088.9	+	105809.i	−0.00620574	+	0.0107487i
$628$		0			0
$629$	6.45882e6			0.650918
$630$		0			0
$631$	−3.17504e6			−0.317450			−0.158725	−	0.987323i	$-0.550738\pi$
$631$	−3.17504e6			−0.317450			−0.158725	+	0.987323i	$0.550738\pi$
$632$		0			0
$633$	−400747.	+	694114.i	−0.0397522	+	0.0688528i
$634$		0			0
$635$	8.41353e6	+	1.45727e7i	0.828026	+	1.43418i
$636$		0			0
$637$	1.10291e7	+	463598.i	1.07694	+	0.0452682i
$638$		0			0
$639$	3.02960e6	+	5.24743e6i	0.293517	+	0.508387i
$640$		0			0
$641$	−1.85599e6	+	3.21466e6i	−0.178414	+	0.309023i	−0.941338	−	0.337466i	$-0.890430\pi$
$641$	−1.85599e6	+	3.21466e6i	−0.178414	+	0.309023i	0.762923	+	0.646489i	$0.223763\pi$
$642$		0			0
$643$	2.23112e6			0.212811			0.106406	−	0.994323i	$-0.466066\pi$
$643$	2.23112e6			0.212811			0.106406	+	0.994323i	$0.466066\pi$
$644$		0			0
$645$	1.98598e6			0.187964
$646$		0			0
$647$	−3.53325e6	+	6.11976e6i	−0.331828	+	0.574743i	−0.982870	−	0.184298i	$-0.940999\pi$
$647$	−3.53325e6	+	6.11976e6i	−0.331828	+	0.574743i	0.651042	+	0.759042i	$0.274332\pi$
$648$		0			0
$649$	857885.	+	1.48590e6i	0.0799498	+	0.138477i
$650$		0			0
$651$	1.67838e6	−	922563.i	0.155216	−	0.0853186i
$652$		0			0
$653$	−9.19619e6	−	1.59283e7i	−0.843966	−	1.46179i	−0.886516	−	0.462697i	$-0.846882\pi$
$653$	−9.19619e6	−	1.59283e7i	−0.843966	−	1.46179i	0.0425508	−	0.999094i	$-0.486452\pi$
$654$		0			0
$655$	−1.47058e6	+	2.54712e6i	−0.133932	+	0.231977i
$656$		0			0
$657$	5.44782e6			0.492390
$658$		0			0
$659$	−4.02375e6			−0.360926			−0.180463	−	0.983582i	$-0.557760\pi$
$659$	−4.02375e6			−0.360926			−0.180463	+	0.983582i	$0.557760\pi$
$660$		0			0
$661$	1.26461e6	−	2.19037e6i	0.112578	−	0.194991i	−0.804231	−	0.594317i	$-0.797423\pi$
$661$	1.26461e6	−	2.19037e6i	0.112578	−	0.194991i	0.916809	+	0.399326i	$0.130756\pi$
$662$		0			0
$663$	−1.63769e6	−	2.83656e6i	−0.144693	−	0.250616i
$664$		0			0
$665$	1.56453e6	+	947644.i	0.137192	+	0.0830981i
$666$		0			0
$667$	9.79743e6	+	1.69696e7i	0.852702	+	1.47692i
$668$		0			0
$669$	−495224.	+	857754.i	−0.0427796	+	0.0740964i
$670$		0			0
$671$	4.29942e6			0.368641
$672$		0			0
$673$	1.31377e7			1.11811			0.559053	−	0.829132i	$-0.311165\pi$
$673$	1.31377e7			1.11811			0.559053	+	0.829132i	$0.311165\pi$
$674$		0			0
$675$	314117.	−	544067.i	0.0265358	−	0.0459614i
$676$		0			0
$677$	−1.00407e6	−	1.73911e6i	−0.0841965	−	0.145833i	0.820852	−	0.571141i	$-0.193499\pi$
$677$	−1.00407e6	−	1.73911e6i	−0.0841965	−	0.145833i	−0.905049	+	0.425308i	$0.860166\pi$
$678$		0			0
$679$	−200109.	+	9.52547e6i	−0.0166568	+	0.792888i
$680$		0			0
$681$	−1.21925e6	−	2.11180e6i	−0.100745	−	0.174496i
$682$		0			0
$683$	1.67913e6	−	2.90833e6i	0.137731	−	0.238557i	−0.788906	−	0.614513i	$-0.789352\pi$
$683$	1.67913e6	−	2.90833e6i	0.137731	−	0.238557i	0.926637	+	0.375956i	$0.122686\pi$
$684$		0			0
$685$	1.65517e7			1.34777
$686$		0			0
$687$	−300362.			−0.0242802
$688$		0			0
$689$	−2.79049e6	+	4.83328e6i	−0.223941	+	0.387877i
$690$		0			0
$691$	−898758.	−	1.55669e6i	−0.0716057	−	0.124025i	0.827999	−	0.560729i	$-0.189479\pi$
$691$	−898758.	−	1.55669e6i	−0.0716057	−	0.124025i	−0.899605	+	0.436704i	$0.856146\pi$
$692$		0			0
$693$	124345.	−	5.91898e6i	0.00983545	−	0.468181i
$694$		0			0
$695$	4.16904e6	+	7.22099e6i	0.327396	+	0.567067i
$696$		0			0
$697$	2.62108e6	−	4.53984e6i	0.204361	−	0.353963i
$698$		0			0
$699$	−142732.			−0.0110492
$700$		0			0
$701$	−1.74200e7			−1.33891			−0.669457	−	0.742851i	$-0.733473\pi$
$701$	−1.74200e7			−1.33891			−0.669457	+	0.742851i	$0.733473\pi$
$702$		0			0
$703$	−412023.	+	713645.i	−0.0314437	+	0.0544621i
$704$		0			0
$705$	−444147.	−	769286.i	−0.0336554	−	0.0582928i
$706$		0			0
$707$	−1.98332e7	−	1.20131e7i	−1.49226	−	0.903868i
$708$		0			0
$709$	−86766.3	−	150284.i	−0.00648240	−	0.0112278i	0.862766	−	0.505603i	$-0.168730\pi$
$709$	−86766.3	−	150284.i	−0.00648240	−	0.0112278i	−0.869248	+	0.494376i	$0.835397\pi$
$710$		0			0
$711$	5.46004e6	−	9.45708e6i	0.405063	−	0.701589i
$712$		0			0
$713$	2.40856e7			1.77433
$714$		0			0
$715$	6.37063e6			0.466034
$716$		0			0
$717$	1.47300e6	−	2.55130e6i	0.107005	−	0.185338i
$718$		0			0
$719$	−7.20941e6	−	1.24871e7i	−0.520089	−	0.900820i	−0.999727	−	0.0233538i	$-0.992566\pi$
$719$	−7.20941e6	−	1.24871e7i	−0.520089	−	0.900820i	0.479639	−	0.877466i	$-0.340768\pi$
$720$		0			0
$721$	989764.	−	544049.i	0.0709077	−	0.0389762i
$722$		0			0
$723$	132432.	+	229379.i	0.00942211	+	0.0163196i
$724$		0			0
$725$	−1.51340e6	+	2.62128e6i	−0.106932	+	0.185212i
$726$		0			0
$727$	1.47752e7			1.03681			0.518403	−	0.855136i	$-0.326527\pi$
$727$	1.47752e7			1.03681			0.518403	+	0.855136i	$0.326527\pi$
$728$		0			0
$729$	−1.25433e7			−0.874165
$730$		0			0
$731$	1.88959e7	−	3.27286e7i	1.30790	−	2.26535i
$732$		0			0
$733$	4.36723e6	+	7.56427e6i	0.300225	+	0.520004i	0.976187	−	0.216932i	$-0.0696050\pi$
$733$	4.36723e6	+	7.56427e6i	0.300225	+	0.520004i	−0.675962	+	0.736936i	$0.736272\pi$
$734$		0			0
$735$	1.93198e6	+	81209.3i	0.131912	+	0.00554482i
$736$		0			0
$737$	−5.05945e6	−	8.76322e6i	−0.343111	−	0.594286i
$738$		0			0
$739$	−1.28465e7	+	2.22508e7i	−0.865314	+	1.49877i	0.00142018	+	0.999999i	$0.499548\pi$
$739$	−1.28465e7	+	2.22508e7i	−0.865314	+	1.49877i	−0.866735	+	0.498770i	$0.833785\pi$
$740$		0			0
$741$	417889.			0.0279586
$742$		0			0
$743$	3.71107e6			0.246619			0.123310	−	0.992368i	$-0.460649\pi$
$743$	3.71107e6			0.246619			0.123310	+	0.992368i	$0.460649\pi$
$744$		0			0
$745$	−2.97985e6	+	5.16124e6i	−0.196699	+	0.340693i
$746$		0			0
$747$	1.15554e7	+	2.00145e7i	0.757673	+	1.31233i
$748$		0			0
$749$	−1.96226e7	+	1.07860e7i	−1.27806	+	0.702518i
$750$		0			0
$751$	−438014.	−	758663.i	−0.0283392	−	0.0490850i	0.851508	−	0.524342i	$-0.175689\pi$
$751$	−438014.	−	758663.i	−0.0283392	−	0.0490850i	−0.879847	+	0.475257i	$0.842355\pi$
$752$		0			0
$753$	−411109.	+	712062.i	−0.0264222	+	0.0457646i
$754$		0			0
$755$	−2.09773e7			−1.33931
$756$		0			0
$757$	1.30192e7			0.825743			0.412871	−	0.910789i	$-0.364526\pi$
$757$	1.30192e7			0.825743			0.412871	+	0.910789i	$0.364526\pi$
$758$		0			0
$759$	−812156.	+	1.40670e6i	−0.0511723	+	0.0886330i
$760$		0			0
$761$	−3.90135e6	−	6.75734e6i	−0.244204	−	0.422974i	0.717703	−	0.696349i	$-0.245193\pi$
$761$	−3.90135e6	−	6.75734e6i	−0.244204	−	0.422974i	−0.961908	+	0.273375i	$0.911860\pi$
$762$		0			0
$763$	−1.84412e7	−	1.11699e7i	−1.14677	−	0.694606i
$764$		0			0
$765$	1.31495e7	+	2.27756e7i	0.812372	+	1.40707i
$766$		0			0
$767$	2.93425e6	−	5.08227e6i	0.180098	−	0.311939i
$768$		0			0
$769$	−1.99560e7			−1.21691			−0.608454	−	0.793589i	$-0.708210\pi$
$769$	−1.99560e7			−1.21691			−0.608454	+	0.793589i	$0.708210\pi$
$770$		0			0
$771$	4.40374e6			0.266800
$772$		0			0
$773$	−1.37454e7	+	2.38078e7i	−0.827390	+	1.43308i	0.0726897	+	0.997355i	$0.476842\pi$
$773$	−1.37454e7	+	2.38078e7i	−0.827390	+	1.43308i	−0.900079	+	0.435726i	$0.856492\pi$
$774$		0			0
$775$	1.86024e6	+	3.22204e6i	0.111254	+	0.192698i
$776$		0			0
$777$	−18296.8	+	870951.i	−0.00108723	+	0.0517537i
$778$		0			0
$779$	334409.	+	579214.i	0.0197440	+	0.0341976i
$780$		0			0
$781$	2.44634e6	−	4.23718e6i	0.143512	−	0.248570i
$782$		0			0
$783$	−5.77873e6			−0.336844
$784$		0			0
$785$	−2.28881e7			−1.32567
$786$		0			0
$787$	1.42384e7	−	2.46616e7i	0.819451	−	1.41933i	−0.0866365	−	0.996240i	$-0.527612\pi$
$787$	1.42384e7	−	2.46616e7i	0.819451	−	1.41933i	0.906087	−	0.423091i	$-0.139055\pi$
$788$		0			0
$789$	−244620.	−	423694.i	−0.0139894	−	0.0242303i
$790$		0			0
$791$	80610.3	−	3.83716e6i	0.00458088	−	0.218056i
$792$		0			0
$793$	−7.35273e6	−	1.27353e7i	−0.415208	−	0.719161i
$794$		0			0
$795$	−488816.	+	846655.i	−0.0274301	+	0.0475104i
$796$		0			0
$797$	−1.32245e7			−0.737450			−0.368725	−	0.929538i	$-0.620206\pi$
$797$	−1.32245e7			−0.737450			−0.368725	+	0.929538i	$0.620206\pi$
$798$		0			0
$799$	−1.69036e7			−0.936726
$800$		0			0
$801$	−7.03107e6	+	1.21782e7i	−0.387204	+	0.670658i
$802$		0			0
$803$	−2.19950e6	−	3.80964e6i	−0.120375	−	0.208495i
$804$		0			0
$805$	2.07999e7	+	1.25986e7i	1.13128	+	0.685224i
$806$		0			0
$807$	2.16702e6	+	3.75338e6i	0.117133	+	0.202880i
$808$		0			0
$809$	5.67095e6	−	9.82237e6i	0.304638	−	0.527649i	−0.672543	−	0.740058i	$-0.734798\pi$
$809$	5.67095e6	−	9.82237e6i	0.304638	−	0.527649i	0.977181	+	0.212410i	$0.0681311\pi$
$810$		0			0
$811$	−6.47233e6			−0.345548			−0.172774	−	0.984961i	$-0.555273\pi$
$811$	−6.47233e6			−0.345548			−0.172774	+	0.984961i	$0.555273\pi$
$812$		0			0
$813$	−2.92677e6			−0.155296
$814$		0			0
$815$	−1.09188e7	+	1.89119e7i	−0.575813	+	0.997337i
$816$		0			0
$817$	2.41083e6	+	4.17568e6i	0.126360	+	0.218863i
$818$		0			0
$819$	−1.77452e7	+	9.75411e6i	−0.924424	+	0.508133i
$820$		0			0
$821$	−9.17380e6	−	1.58895e7i	−0.474998	−	0.822720i	0.524592	−	0.851354i	$-0.324218\pi$
$821$	−9.17380e6	−	1.58895e7i	−0.474998	−	0.822720i	−0.999590	+	0.0286334i	$0.990884\pi$
$822$		0			0
$823$	5.02115e6	−	8.69689e6i	0.258407	−	0.447574i	−0.707408	−	0.706805i	$-0.750136\pi$
$823$	5.02115e6	−	8.69689e6i	0.258407	−	0.447574i	0.965815	+	0.259231i	$0.0834692\pi$
$824$		0			0
$825$	−250906.			−0.0128344
$826$		0			0
$827$	334853.			0.0170251			0.00851256	−	0.999964i	$-0.497290\pi$
$827$	334853.			0.0170251			0.00851256	+	0.999964i	$0.497290\pi$
$828$		0			0
$829$	−1.56447e7	+	2.70974e7i	−0.790643	+	1.36943i	0.134926	+	0.990856i	$0.456920\pi$
$829$	−1.56447e7	+	2.70974e7i	−0.790643	+	1.36943i	−0.925569	+	0.378578i	$0.876413\pi$
$830$		0			0
$831$	1.62485e6	+	2.81432e6i	0.0816224	+	0.141374i
$832$		0			0
$833$	1.97204e7	−	3.10661e7i	0.984701	−	1.55122i
$834$		0			0
$835$	8.66469e6	+	1.50077e7i	0.430068	+	0.744899i
$836$		0			0
$837$	−3.55156e6	+	6.15148e6i	−0.175229	+	0.303505i
$838$		0			0
$839$	−2.42772e7			−1.19068			−0.595339	−	0.803475i	$-0.702982\pi$
$839$	−2.42772e7			−1.19068			−0.595339	+	0.803475i	$0.702982\pi$
$840$		0			0
$841$	7.33046e6			0.357389
$842$		0			0
$843$	561057.	−	971779.i	0.0271918	−	0.0470976i
$844$		0			0
$845$	−1.51764e6	−	2.62862e6i	−0.0731182	−	0.126644i
$846$		0			0
$847$	1.41076e7	−	7.75463e6i	0.675688	−	0.371409i
$848$		0			0
$849$	−2.29253e6	−	3.97078e6i	−0.109156	−	0.189063i
$850$		0			0
$851$	−5.47771e6	+	9.48767e6i	−0.259284	+	0.449092i
$852$		0			0
$853$	−3.17995e7			−1.49640			−0.748200	−	0.663474i	$-0.769081\pi$
$853$	−3.17995e7			−1.49640			−0.748200	+	0.663474i	$0.769081\pi$
$854$		0			0
$855$	−3.35535e6			−0.156972
$856$		0			0
$857$	1.71424e7	−	2.96915e7i	0.797296	−	1.38096i	−0.124075	−	0.992273i	$-0.539596\pi$
$857$	1.71424e7	−	2.96915e7i	0.797296	−	1.38096i	0.921371	−	0.388685i	$-0.127070\pi$
$858$		0			0
$859$	−3.81780e6	−	6.61263e6i	−0.176535	−	0.305767i	0.764157	−	0.645031i	$-0.223155\pi$
$859$	−3.81780e6	−	6.61263e6i	−0.176535	−	0.305767i	−0.940691	+	0.339264i	$0.889822\pi$
$860$		0			0
$861$	604757.	+	366304.i	0.0278018	+	0.0168397i
$862$		0			0
$863$	−1.80224e7	−	3.12156e7i	−0.823729	−	1.42674i	−0.902887	−	0.429879i	$-0.858556\pi$
$863$	−1.80224e7	−	3.12156e7i	−0.823729	−	1.42674i	0.0791571	−	0.996862i	$-0.474777\pi$
$864$		0			0
$865$	1.12009e7	−	1.94006e7i	0.508996	−	0.881607i
$866$		0			0
$867$	−7.68402e6			−0.347169
$868$		0			0
$869$	−8.81773e6			−0.396102
$870$		0			0
$871$	−1.73050e7	+	2.99731e7i	−0.772905	+	1.33871i
$872$		0			0
$873$	−8.73857e6	−	1.51356e7i	−0.388065	−	0.672149i
$874$		0			0
$875$	−508695.	+	2.42146e7i	−0.0224614	+	1.06920i
$876$		0			0
$877$	−1.62877e7	−	2.82112e7i	−0.715092	−	1.23858i	−0.962924	−	0.269773i	$-0.913051\pi$
$877$	−1.62877e7	−	2.82112e7i	−0.715092	−	1.23858i	0.247832	−	0.968803i	$-0.420282\pi$
$878$		0			0
$879$	−66075.8	+	114447.i	−0.00288450	+	0.00499610i
$880$		0			0
$881$	3.83192e7			1.66332			0.831661	−	0.555284i	$-0.187390\pi$
$881$	3.83192e7			1.66332			0.831661	+	0.555284i	$0.187390\pi$
$882$		0			0
$883$	3.50607e7			1.51328			0.756640	−	0.653832i	$-0.226840\pi$
$883$	3.50607e7			1.51328			0.756640	+	0.653832i	$0.226840\pi$
$884$		0			0
$885$	513998.	−	890272.i	0.0220599	−	0.0382089i
$886$		0			0
$887$	9.80677e6	+	1.69858e7i	0.418521	+	0.724899i	0.995791	−	0.0916541i	$-0.0292154\pi$
$887$	9.80677e6	+	1.69858e7i	0.418521	+	0.724899i	−0.577270	+	0.816553i	$0.695882\pi$
$888$		0			0
$889$	−907093.	+	4.31789e7i	−0.0384944	+	1.83239i
$890$		0			0
$891$	5.30896e6	+	9.19539e6i	0.224035	+	0.388039i
$892$		0			0
$893$	1.07832e6	−	1.86771e6i	0.0452501	−	0.0783755i
$894$		0			0
$895$	4.81368e6			0.200872
$896$		0			0
$897$	5.55568e6			0.230545
$898$		0			0
$899$	1.71112e7	−	2.96375e7i	0.706125	−	1.22304i
$900$		0			0
$901$	9.30182e6	+	1.61112e7i	0.381730	+	0.661175i
$902$		0			0
$903$	4.35983e6	+	2.64077e6i	0.177930	+	0.107773i
$904$		0			0
$905$	1.98413e6	+	3.43662e6i	0.0805284	+	0.139479i
$906$		0			0
$907$	−1.20621e7	+	2.08922e7i	−0.486860	+	0.843267i	−0.999886	−	0.0151065i	$-0.995191\pi$
$907$	−1.20621e7	+	2.08922e7i	−0.486860	+	0.843267i	0.513026	+	0.858373i	$0.328525\pi$
$908$		0			0
$909$	4.25349e7			1.70740
$910$		0			0
$911$	346019.			0.0138135			0.00690676	−	0.999976i	$-0.497801\pi$
$911$	346019.			0.0138135			0.00690676	+	0.999976i	$0.497801\pi$
$912$		0			0
$913$	9.33070e6	−	1.61612e7i	0.370457	−	0.641650i
$914$		0			0
$915$	−1.28799e6	−	2.23087e6i	−0.0508581	−	0.0880888i
$916$		0			0
$917$	−6.61527e6	+	3.63625e6i	−0.259791	+	0.142801i
$918$		0			0
$919$	−3.27320e6	−	5.66935e6i	−0.127845	−	0.221434i	0.794996	−	0.606614i	$-0.207473\pi$
$919$	−3.27320e6	−	5.66935e6i	−0.127845	−	0.221434i	−0.922841	+	0.385180i	$0.874139\pi$
$920$		0			0
$921$	−3.50818e6	+	6.07635e6i	−0.136280	+	0.236044i
$922$		0			0
$923$	−1.67346e7			−0.646562
$924$		0			0
$925$	−1.69227e6			−0.0650303
$926$		0			0
$927$	−1.03590e6	+	1.79424e6i	−0.0395931	+	0.0685773i
$928$		0			0
$929$	9.24280e6	+	1.60090e7i	0.351370	+	0.608590i	0.986490	−	0.163823i	$-0.0523827\pi$
$929$	9.24280e6	+	1.60090e7i	0.351370	+	0.608590i	−0.635120	+	0.772414i	$0.719049\pi$
$930$		0			0
$931$	2.17453e6	+	4.16073e6i	0.0822227	+	0.157324i
$932$		0			0
$933$	639887.	+	1.10832e6i	0.0240657	+	0.0416831i
$934$		0			0
$935$	1.06179e7	−	1.83908e7i	0.397201	−	0.687972i
$936$		0			0
$937$	−4.28083e7			−1.59286			−0.796432	−	0.604729i	$-0.793282\pi$
$937$	−4.28083e7			−1.59286			−0.796432	+	0.604729i	$0.793282\pi$
$938$		0			0
$939$	−6.76789e6			−0.250490
$940$		0			0
$941$	9.20301e6	−	1.59401e7i	0.338810	−	0.586836i	−0.645399	−	0.763845i	$-0.723309\pi$
$941$	9.20301e6	−	1.59401e7i	0.338810	−	0.586836i	0.984209	+	0.177010i	$0.0566424\pi$
$942$		0			0
$943$	4.44586e6	+	7.70045e6i	0.162808	+	0.281992i
$944$		0			0
$945$	−6.28474e6	+	3.45457e6i	−0.228933	+	0.125839i
$946$		0			0
$947$	8.59421e6	+	1.48856e7i	0.311409	+	0.539376i	0.978668	−	0.205450i	$-0.0658658\pi$
$947$	8.59421e6	+	1.48856e7i	0.311409	+	0.539376i	−0.667259	+	0.744826i	$0.732532\pi$
$948$		0			0
$949$	−7.52301e6	+	1.30302e7i	−0.271160	+	0.469664i
$950$		0			0
$951$	1.70944e6			0.0612918
$952$		0			0
$953$	3.40220e7			1.21347			0.606733	−	0.794905i	$-0.292480\pi$
$953$	3.40220e7			1.21347			0.606733	+	0.794905i	$0.292480\pi$
$954$		0			0
$955$	7.44434e6	−	1.28940e7i	0.264130	−	0.457487i
$956$		0			0
$957$	1.15396e6	+	1.99872e6i	0.0407297	+	0.0705460i
$958$		0			0
$959$	3.63360e7	+	2.20089e7i	1.27582	+	0.772772i
$960$		0			0
$961$	−6.71821e6	−	1.16363e7i	−0.234663	−	0.406449i
$962$		0			0
$963$	2.05373e7	−	3.55716e7i	0.713637	−	1.23606i
$964$		0			0
$965$	−4.41798e7			−1.52723
$966$		0			0
$967$	1.22029e6			0.0419658			0.0209829	−	0.999780i	$-0.493320\pi$
$967$	1.22029e6			0.0419658			0.0209829	+	0.999780i	$0.493320\pi$
$968$		0			0
$969$	696494.	−	1.20636e6i	0.0238291	−	0.0412732i
$970$		0			0
$971$	−9.34534e6	−	1.61866e7i	−0.318088	−	0.550944i	0.662001	−	0.749503i	$-0.269707\pi$
$971$	−9.34534e6	−	1.61866e7i	−0.318088	−	0.550944i	−0.980089	+	0.198558i	$0.936374\pi$
$972$		0			0
$973$	−449479.	+	2.13958e7i	−0.0152205	+	0.724514i
$974$		0			0
$975$	429091.	+	743207.i	0.0144556	+	0.0250379i
$976$		0			0
$977$	1.37254e7	−	2.37730e7i	0.460031	−	0.796797i	−0.538931	−	0.842350i	$-0.681172\pi$
$977$	1.37254e7	−	2.37730e7i	0.460031	−	0.796797i	0.998962	+	0.0455527i	$0.0145049\pi$
$978$		0			0
$979$	1.13549e7			0.378639
$980$		0			0
$981$	3.95496e7			1.31211
$982$		0			0
$983$	5.58131e6	−	9.66711e6i	0.184227	−	0.319090i	−0.759089	−	0.650987i	$-0.774355\pi$
$983$	5.58131e6	−	9.66711e6i	0.184227	−	0.319090i	0.943316	+	0.331897i	$0.107689\pi$
$984$		0			0
$985$	−8.53832e6	−	1.47888e7i	−0.280403	−	0.485672i
$986$		0			0
$987$	47885.1	−	2.27940e6i	0.00156462	−	0.0744779i
$988$		0			0
$989$	3.20511e7	+	5.55142e7i	1.04196	+	1.80473i
$990$		0			0
$991$	2.24404e7	−	3.88679e7i	0.725850	−	1.25721i	−0.232773	−	0.972531i	$-0.574780\pi$
$991$	2.24404e7	−	3.88679e7i	0.725850	−	1.25721i	0.958623	−	0.284678i	$-0.0918867\pi$
$992$		0			0
$993$	6.11985e6			0.196955
$994$		0			0
$995$	1.78326e7			0.571028
$996$		0			0
$997$	1.03154e6	−	1.78668e6i	0.0328661	−	0.0569257i	−0.849125	−	0.528193i	$-0.822870\pi$
$997$	1.03154e6	−	1.78668e6i	0.0328661	−	0.0569257i	0.881991	+	0.471267i	$0.156203\pi$
$998$		0			0
$999$	−1.61543e6	−	2.79801e6i	−0.0512125	−	0.0887026i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.i.f.81.3		10
4.3	odd	2		56.6.i.b.25.3	yes	10
7.2	even	3	inner	112.6.i.f.65.3		10
7.3	odd	6		784.6.a.bl.1.3		5
7.4	even	3		784.6.a.bk.1.3		5
12.11	even	2		504.6.s.b.361.5		10
28.3	even	6		392.6.a.j.1.3		5
28.11	odd	6		392.6.a.k.1.3		5
28.19	even	6		392.6.i.o.177.3		10
28.23	odd	6		56.6.i.b.9.3	✓	10
28.27	even	2		392.6.i.o.361.3		10
84.23	even	6		504.6.s.b.289.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.3	✓	10	28.23	odd	6
56.6.i.b.25.3	yes	10	4.3	odd	2
112.6.i.f.65.3		10	7.2	even	3	inner
112.6.i.f.81.3		10	1.1	even	1	trivial
392.6.a.j.1.3		5	28.3	even	6
392.6.a.k.1.3		5	28.11	odd	6
392.6.i.o.177.3		10	28.19	even	6
392.6.i.o.361.3		10	28.27	even	2
504.6.s.b.289.5		10	84.23	even	6
504.6.s.b.361.5		10	12.11	even	2
784.6.a.bk.1.3		5	7.4	even	3
784.6.a.bl.1.3		5	7.3	odd	6

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.13889 - 1.97261i) q^{3} +(-25.2555 - 43.7438i) q^{5} +(2.72289 - 129.613i) q^{7} +(118.906 + 205.951i) q^{9} +(96.0139 - 166.301i) q^{11} -656.798 q^{13} -115.053 q^{15} +(-1094.68 + 1896.05i) q^{17} +(-139.665 - 241.907i) q^{19} +(-252.575 - 152.986i) q^{21} +(-1856.80 - 3216.07i) q^{23} +(286.818 - 496.783i) q^{25} +1095.18 q^{27} -5276.51 q^{29} +(-3242.90 + 5616.87i) q^{31} +(-218.698 - 378.796i) q^{33} +(-5738.55 + 3154.34i) q^{35} +(-1475.04 - 2554.84i) q^{37} +(-748.019 + 1295.61i) q^{39} -2394.37 q^{41} -17261.5 q^{43} +(6006.06 - 10402.8i) q^{45} +(3860.38 + 6686.38i) q^{47} +(-16792.2 - 705.845i) q^{49} +(2493.44 + 4318.77i) q^{51} +(4248.63 - 7358.84i) q^{53} -9699.52 q^{55} -636.251 q^{57} +(-4467.51 + 7737.95i) q^{59} +(11194.8 + 19390.0i) q^{61} +(27017.7 - 14851.0i) q^{63} +(16587.8 + 28730.9i) q^{65} +(26347.5 - 45635.2i) q^{67} -8458.73 q^{69} +25479.0 q^{71} +(11454.1 - 19839.0i) q^{73} +(-653.306 - 1131.56i) q^{75} +(-21293.4 - 12897.5i) q^{77} +(-22959.5 - 39767.1i) q^{79} +(-27646.8 + 47885.7i) q^{81} +97180.7 q^{83} +110587. q^{85} +(-6009.35 + 10408.5i) q^{87} +(29565.7 + 51209.3i) q^{89} +(-1788.39 + 85129.8i) q^{91} +(7386.59 + 12793.9i) q^{93} +(-7054.63 + 12219.0i) q^{95} -73491.5 q^{97} +45666.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{3} + 81 q^{5} - 116 q^{7} - 390 q^{9} + 361 q^{11} - 684 q^{13} + 2098 q^{15} + 1809 q^{17} - 1277 q^{19} - 5253 q^{21} + 911 q^{23} - 3940 q^{25} - 9502 q^{27} + 10884 q^{29} - 2187 q^{31}+ \cdots - 499596 q^{99}+O(q^{100}) \) 10 * q + 5 * q^3 + 81 * q^5 - 116 * q^7 - 390 * q^9 + 361 * q^11 - 684 * q^13 + 2098 * q^15 + 1809 * q^17 - 1277 * q^19 - 5253 * q^21 + 911 * q^23 - 3940 * q^25 - 9502 * q^27 + 10884 * q^29 - 2187 * q^31 - 5553 * q^33 + 16845 * q^35 + 8181 * q^37 - 3422 * q^39 - 33156 * q^41 - 12664 * q^43 + 41310 * q^45 - 16101 * q^47 - 10070 * q^49 - 67865 * q^51 - 16047 * q^53 + 91258 * q^55 + 44694 * q^57 - 71027 * q^59 + 31093 * q^61 + 127978 * q^63 + 64370 * q^65 + 47981 * q^67 - 274498 * q^69 - 45024 * q^71 + 123333 * q^73 + 45460 * q^75 - 245825 * q^77 - 212481 * q^79 - 52917 * q^81 + 174920 * q^83 + 444282 * q^85 - 318070 * q^87 + 129045 * q^89 + 526488 * q^91 + 252835 * q^93 + 300417 * q^95 - 656548 * q^97 - 499596 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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