LMFDB - Embedded newform 168.5.z.a.145.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,5,Mod(73,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("168.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	168.z (of order $6$, degree $2$, 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:	$17.3661537981$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 $ x^16 - 6x^15 + 99x^14 - 1810x^13 + 14212x^12 - 199882x^11 + 1800935x^10 - 9073270x^9 + 62016647x^8 - 13497916x^7 - 872612906x^6 + 8208467240x^5 + 11144877252x^4 - 185824445600x^3 + 3750935164160x^2 - 13970712492800*x + 41390114348800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.5
Root			$4.88974 + 3.43978i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.a.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{3} +(1.51540 + 0.874918i) q^{5} +(-2.69843 - 48.9256i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 2.59808i) q^{3} +(1.51540 + 0.874918i) q^{5} +(-2.69843 - 48.9256i) q^{7} +(13.5000 - 23.3827i) q^{9} +(63.6775 + 110.293i) q^{11} +156.315i q^{13} -9.09241 q^{15} +(-245.977 + 142.015i) q^{17} +(-378.760 - 218.677i) q^{19} +(139.255 + 213.155i) q^{21} +(249.038 - 431.347i) q^{23} +(-310.969 - 538.614i) q^{25} +140.296i q^{27} -1437.62 q^{29} +(-633.503 + 365.753i) q^{31} +(-573.097 - 330.878i) q^{33} +(38.7167 - 76.5029i) q^{35} +(-1268.39 + 2196.91i) q^{37} +(-406.119 - 703.419i) q^{39} +52.0993i q^{41} +67.7783 q^{43} +(40.9159 - 23.6228i) q^{45} +(-2212.18 - 1277.20i) q^{47} +(-2386.44 + 264.045i) q^{49} +(737.931 - 1278.13i) q^{51} +(791.895 + 1371.60i) q^{53} +222.850i q^{55} +2272.56 q^{57} +(-22.1078 + 12.7639i) q^{59} +(2220.90 + 1282.24i) q^{61} +(-1180.44 - 597.400i) q^{63} +(-136.763 + 236.880i) q^{65} +(-2430.05 - 4208.97i) q^{67} +2588.08i q^{69} +8622.80 q^{71} +(-7453.63 + 4303.36i) q^{73} +(2798.72 + 1615.84i) q^{75} +(5224.31 - 3413.08i) q^{77} +(-1649.04 + 2856.22i) q^{79} +(-364.500 - 631.333i) q^{81} -2435.04i q^{83} -497.006 q^{85} +(6469.30 - 3735.05i) q^{87} +(-31.5162 - 18.1959i) q^{89} +(7647.82 - 421.806i) q^{91} +(1900.51 - 3291.78i) q^{93} +(-382.650 - 662.768i) q^{95} +9888.12i q^{97} +3438.58 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.50000	+	2.59808i	−0.500000	+	0.288675i
$3$	−4.50000	+	2.59808i	−0.500000	+	0.288675i
$4$		0			0
$5$	1.51540	+	0.874918i	0.0606161	+	0.0349967i	0.530002	−	0.847997i	$-0.322191\pi$
$5$	1.51540	+	0.874918i	0.0606161	+	0.0349967i	−0.469386	+	0.882993i	$0.655525\pi$
$6$		0			0
$7$	−2.69843	−	48.9256i	−0.0550701	−	0.998482i
$7$	−2.69843	−	48.9256i	−0.0550701	−	0.998482i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	63.6775	+	110.293i	0.526260	+	0.911509i	0.999532	+	0.0305926i	$0.00973946\pi$
$11$	63.6775	+	110.293i	0.526260	+	0.911509i	−0.473272	+	0.880916i	$0.656927\pi$
$12$		0			0
$13$			156.315i			0.924942i	0.886634	+	0.462471i	$0.153037\pi$
$13$			156.315i			0.924942i	−0.886634	+	0.462471i	$0.846963\pi$
$14$		0			0
$15$	−9.09241			−0.0404107
$16$		0			0
$17$	−245.977	+	142.015i	−0.851132	+	0.491401i	−0.861033	−	0.508550i	$-0.830182\pi$
$17$	−245.977	+	142.015i	−0.851132	+	0.491401i	0.00990080	+	0.999951i	$0.496848\pi$
$18$		0			0
$19$	−378.760	−	218.677i	−1.04920	−	0.605755i	−0.126773	−	0.991932i	$-0.540462\pi$
$19$	−378.760	−	218.677i	−1.04920	−	0.605755i	−0.922425	+	0.386177i	$0.873795\pi$
$20$		0			0
$21$	139.255	+	213.155i	0.315772	+	0.483344i
$22$		0			0
$23$	249.038	−	431.347i	0.470771	−	0.815400i	−0.528670	−	0.848828i	$-0.677309\pi$
$23$	249.038	−	431.347i	0.470771	−	0.815400i	0.999441	+	0.0334276i	$0.0106423\pi$
$24$		0			0
$25$	−310.969	−	538.614i	−0.497550	−	0.861783i
$26$		0			0
$27$			140.296i			0.192450i
$28$		0			0
$29$	−1437.62			−1.70942			−0.854711	−	0.519105i	$-0.826265\pi$
$29$	−1437.62			−1.70942			−0.854711	+	0.519105i	$0.826265\pi$
$30$		0			0
$31$	−633.503	+	365.753i	−0.659212	+	0.380596i	−0.791977	−	0.610551i	$-0.790948\pi$
$31$	−633.503	+	365.753i	−0.659212	+	0.380596i	0.132764	+	0.991148i	$0.457615\pi$
$32$		0			0
$33$	−573.097	−	330.878i	−0.526260	−	0.303836i
$34$		0			0
$35$	38.7167	−	76.5029i	0.0316055	−	0.0624514i
$36$		0			0
$37$	−1268.39	+	2196.91i	−0.926505	+	1.60475i	−0.137382	+	0.990518i	$0.543869\pi$
$37$	−1268.39	+	2196.91i	−0.926505	+	1.60475i	−0.789123	+	0.614235i	$0.789465\pi$
$38$		0			0
$39$	−406.119	−	703.419i	−0.267008	−	0.462471i
$40$		0			0
$41$			52.0993i			0.0309931i	0.999880	+	0.0154965i	$0.00493290\pi$
$41$			52.0993i			0.0309931i	−0.999880	+	0.0154965i	$0.995067\pi$
$42$		0			0
$43$	67.7783			0.0366567			0.0183284	−	0.999832i	$-0.494166\pi$
$43$	67.7783			0.0366567			0.0183284	+	0.999832i	$0.494166\pi$
$44$		0			0
$45$	40.9159	−	23.6228i	0.0202054	−	0.0116656i
$46$		0			0
$47$	−2212.18	−	1277.20i	−1.00144	−	0.578181i	−0.0927650	−	0.995688i	$-0.529571\pi$
$47$	−2212.18	−	1277.20i	−1.00144	−	0.578181i	−0.908674	+	0.417507i	$0.862904\pi$
$48$		0			0
$49$	−2386.44	+	264.045i	−0.993935	+	0.109973i
$50$		0			0
$51$	737.931	−	1278.13i	0.283711	−	0.491401i
$52$		0			0
$53$	791.895	+	1371.60i	0.281914	+	0.488289i	0.971856	−	0.235576i	$-0.0756975\pi$
$53$	791.895	+	1371.60i	0.281914	+	0.488289i	−0.689942	+	0.723864i	$0.742364\pi$
$54$		0			0
$55$			222.850i			0.0736695i
$56$		0			0
$57$	2272.56			0.699465
$58$		0			0
$59$	−22.1078	+	12.7639i	−0.00635099	+	0.00366675i	−0.503172	−	0.864186i	$-0.667834\pi$
$59$	−22.1078	+	12.7639i	−0.00635099	+	0.00366675i	0.496821	+	0.867853i	$0.334500\pi$
$60$		0			0
$61$	2220.90	+	1282.24i	0.596856	+	0.344595i	0.767804	−	0.640685i	$-0.221350\pi$
$61$	2220.90	+	1282.24i	0.596856	+	0.344595i	−0.170948	+	0.985280i	$0.554683\pi$
$62$		0			0
$63$	−1180.44	−	597.400i	−0.297415	−	0.150516i
$64$		0			0
$65$	−136.763	+	236.880i	−0.0323699	+	0.0560664i
$66$		0			0
$67$	−2430.05	−	4208.97i	−0.541335	−	0.937619i	−0.998828	−	0.0484058i	$-0.984586\pi$
$67$	−2430.05	−	4208.97i	−0.541335	−	0.937619i	0.457493	−	0.889213i	$-0.348747\pi$
$68$		0			0
$69$			2588.08i			0.543600i
$70$		0			0
$71$	8622.80			1.71053			0.855267	−	0.518188i	$-0.173393\pi$
$71$	8622.80			1.71053			0.855267	+	0.518188i	$0.173393\pi$
$72$		0			0
$73$	−7453.63	+	4303.36i	−1.39869	+	0.807535i	−0.994256	−	0.107031i	$-0.965866\pi$
$73$	−7453.63	+	4303.36i	−1.39869	+	0.807535i	−0.404436	+	0.914566i	$0.632532\pi$
$74$		0			0
$75$	2798.72	+	1615.84i	0.497550	+	0.287261i
$76$		0			0
$77$	5224.31	−	3413.08i	0.881145	−	0.575658i
$78$		0			0
$79$	−1649.04	+	2856.22i	−0.264227	+	0.457654i	−0.967361	−	0.253403i	$-0.918450\pi$
$79$	−1649.04	+	2856.22i	−0.264227	+	0.457654i	0.703134	+	0.711057i	$0.251783\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	2435.04i		−	0.353468i	−0.984259	−	0.176734i	$-0.943447\pi$
$83$		−	2435.04i		−	0.353468i	0.984259	−	0.176734i	$-0.0565533\pi$
$84$		0			0
$85$	−497.006			−0.0687897
$86$		0			0
$87$	6469.30	−	3735.05i	0.854711	−	0.493467i
$88$		0			0
$89$	−31.5162	−	18.1959i	−0.00397882	−	0.00229717i	0.498009	−	0.867172i	$-0.334065\pi$
$89$	−31.5162	−	18.1959i	−0.00397882	−	0.00229717i	−0.501988	+	0.864875i	$0.667398\pi$
$90$		0			0
$91$	7647.82	−	421.806i	0.923539	−	0.0509366i
$92$		0			0
$93$	1900.51	−	3291.78i	0.219737	−	0.380596i
$94$		0			0
$95$	−382.650	−	662.768i	−0.0423988	−	0.0734370i
$96$		0			0
$97$			9888.12i			1.05092i	0.850818	+	0.525461i	$0.176107\pi$
$97$			9888.12i			1.05092i	−0.850818	+	0.525461i	$0.823893\pi$
$98$		0			0
$99$	3438.58			0.350840
$100$		0			0
$101$	−1707.34	+	985.734i	−0.167370	+	0.0966311i	−0.581345	−	0.813657i	$-0.697473\pi$
$101$	−1707.34	+	985.734i	−0.167370	+	0.0966311i	0.413975	+	0.910288i	$0.364140\pi$
$102$		0			0
$103$	−8369.47	−	4832.11i	−0.788902	−	0.455473i	0.0506735	−	0.998715i	$-0.483863\pi$
$103$	−8369.47	−	4832.11i	−0.788902	−	0.455473i	−0.839576	+	0.543242i	$0.817197\pi$
$104$		0			0
$105$	24.5353	+	444.852i	0.00222542	+	0.0403494i
$106$		0			0
$107$	6918.90	−	11983.9i	0.604323	−	1.04672i	−0.387835	−	0.921729i	$-0.626777\pi$
$107$	6918.90	−	11983.9i	0.604323	−	1.04672i	0.992158	−	0.124990i	$-0.0398898\pi$
$108$		0			0
$109$	2383.46	+	4128.28i	0.200611	+	0.347469i	0.948726	−	0.316101i	$-0.102374\pi$
$109$	2383.46	+	4128.28i	0.200611	+	0.347469i	−0.748114	+	0.663570i	$0.769041\pi$
$110$		0			0
$111$		−	13181.4i		−	1.06984i
$112$		0			0
$113$	16507.6			1.29279			0.646395	−	0.763003i	$-0.276276\pi$
$113$	16507.6			1.29279			0.646395	+	0.763003i	$0.276276\pi$
$114$		0			0
$115$	754.786	−	435.776i	0.0570726	−	0.0329509i
$116$		0			0
$117$	3655.07	+	2110.26i	0.267008	+	0.154157i
$118$		0			0
$119$	7611.92	+	11651.4i	0.537527	+	0.822779i
$120$		0			0
$121$	−789.137	+	1366.82i	−0.0538991	+	0.0933560i
$122$		0			0
$123$	−135.358	−	234.447i	−0.00894693	−	0.0154965i
$124$		0			0
$125$		−	2181.94i		−	0.139644i
$126$		0			0
$127$	−3118.96			−0.193376			−0.0966878	−	0.995315i	$-0.530825\pi$
$127$	−3118.96			−0.193376			−0.0966878	+	0.995315i	$0.530825\pi$
$128$		0			0
$129$	−305.002	+	176.093i	−0.0183284	+	0.0105819i
$130$		0			0
$131$	19341.8	+	11167.0i	1.12708	+	0.650718i	0.943198	−	0.332231i	$-0.107802\pi$
$131$	19341.8	+	11167.0i	1.12708	+	0.650718i	0.183878	+	0.982949i	$0.441135\pi$
$132$		0			0
$133$	−9676.87	+	19121.2i	−0.547056	+	1.08096i
$134$		0			0
$135$	−122.748	+	212.605i	−0.00673512	+	0.0116656i
$136$		0			0
$137$	−12715.5	−	22024.0i	−0.677476	−	1.17342i	−0.975739	−	0.218938i	$-0.929741\pi$
$137$	−12715.5	−	22024.0i	−0.677476	−	1.17342i	0.298263	−	0.954484i	$-0.403593\pi$
$138$		0			0
$139$			20794.0i			1.07624i	0.842869	+	0.538119i	$0.180865\pi$
$139$			20794.0i			1.07624i	−0.842869	+	0.538119i	$0.819135\pi$
$140$		0			0
$141$	13273.1			0.667626
$142$		0			0
$143$	−17240.4	+	9953.76i	−0.843093	+	0.486760i
$144$		0			0
$145$	−2178.58	−	1257.80i	−0.103618	−	0.0598241i
$146$		0			0
$147$	10053.0	−	7388.35i	0.465221	−	0.341911i
$148$		0			0
$149$	13064.3	−	22628.0i	0.588456	−	1.01924i	−0.405979	−	0.913882i	$-0.633069\pi$
$149$	13064.3	−	22628.0i	0.588456	−	1.01924i	0.994435	−	0.105353i	$-0.0335973\pi$
$150$		0			0
$151$	−10089.0	−	17474.7i	−0.442482	−	0.766401i	0.555391	−	0.831589i	$-0.312568\pi$
$151$	−10089.0	−	17474.7i	−0.442482	−	0.766401i	−0.997873	+	0.0651884i	$0.979235\pi$
$152$		0			0
$153$			7668.81i			0.327601i
$154$		0			0
$155$	−1280.02			−0.0532785
$156$		0			0
$157$	−24960.2	+	14410.8i	−1.01263	+	0.584640i	−0.911960	−	0.410280i	$-0.865431\pi$
$157$	−24960.2	+	14410.8i	−1.01263	+	0.584640i	−0.100667	+	0.994920i	$0.532098\pi$
$158$		0			0
$159$	−7127.06	−	4114.81i	−0.281914	−	0.162763i
$160$		0			0
$161$	−21775.9	−	11020.4i	−0.840088	−	0.425153i
$162$		0			0
$163$	−3090.07	+	5352.16i	−0.116304	+	0.201444i	−0.918300	−	0.395885i	$-0.870438\pi$
$163$	−3090.07	+	5352.16i	−0.116304	+	0.201444i	0.801996	+	0.597329i	$0.203771\pi$
$164$		0			0
$165$	−578.982	−	1002.83i	−0.0212665	−	0.0368347i
$166$		0			0
$167$			11075.1i			0.397113i	0.980089	+	0.198556i	$0.0636253\pi$
$167$			11075.1i			0.397113i	−0.980089	+	0.198556i	$0.936375\pi$
$168$		0			0
$169$	4126.55			0.144482
$170$		0			0
$171$	−10226.5	+	5904.29i	−0.349733	+	0.201918i
$172$		0			0
$173$	−452.346	−	261.162i	−0.0151140	−	0.00872605i	0.492424	−	0.870355i	$-0.336111\pi$
$173$	−452.346	−	261.162i	−0.0151140	−	0.00872605i	−0.507538	+	0.861629i	$0.669444\pi$
$174$		0			0
$175$	−25512.9	+	16667.8i	−0.833075	+	0.544254i
$176$		0			0
$177$	66.3234	−	114.876i	0.00211700	−	0.00366675i
$178$		0			0
$179$	−11486.7	−	19895.6i	−0.358501	−	0.620941i	0.629210	−	0.777235i	$-0.283379\pi$
$179$	−11486.7	−	19895.6i	−0.358501	−	0.620941i	−0.987711	+	0.156294i	$0.950045\pi$
$180$		0			0
$181$		−	55628.9i		−	1.69802i	−0.528374	−	0.849012i	$-0.677198\pi$
$181$		−	55628.9i		−	1.69802i	0.528374	−	0.849012i	$-0.322802\pi$
$182$		0			0
$183$	−13325.4			−0.397904
$184$		0			0
$185$	−3844.23	+	2219.47i	−0.112322	+	0.0648492i
$186$		0			0
$187$	−31326.4	−	18086.3i	−0.895833	−	0.517209i
$188$		0			0
$189$	6864.08	−	378.580i	0.192158	−	0.0105982i
$190$		0			0
$191$	−35490.1	+	61470.7i	−0.972838	+	1.68501i	−0.285946	+	0.958246i	$0.592308\pi$
$191$	−35490.1	+	61470.7i	−0.972838	+	1.68501i	−0.686892	+	0.726759i	$0.741026\pi$
$192$		0			0
$193$	−7468.63	−	12936.0i	−0.200506	−	0.347286i	0.748186	−	0.663489i	$-0.230925\pi$
$193$	−7468.63	−	12936.0i	−0.200506	−	0.347286i	−0.948691	+	0.316203i	$0.897592\pi$
$194$		0			0
$195$		−	1421.28i		−	0.0373776i
$196$		0			0
$197$	347.397			0.00895146			0.00447573	−	0.999990i	$-0.498575\pi$
$197$	347.397			0.00895146			0.00447573	+	0.999990i	$0.498575\pi$
$198$		0			0
$199$	−57523.7	+	33211.3i	−1.45258	+	0.838648i	−0.998627	−	0.0523775i	$-0.983320\pi$
$199$	−57523.7	+	33211.3i	−1.45258	+	0.838648i	−0.453953	+	0.891025i	$0.649987\pi$
$200$		0			0
$201$	21870.5	+	12626.9i	0.541335	+	0.312540i
$202$		0			0
$203$	3879.33	+	70336.6i	0.0941379	+	1.70683i
$204$		0			0
$205$	−45.5826	+	78.9514i	−0.00108466	+	0.00187868i
$206$		0			0
$207$	−6724.03	−	11646.4i	−0.156924	−	0.271800i
$208$		0			0
$209$		−	55699.3i		−	1.27514i
$210$		0			0
$211$	49350.2			1.10847			0.554235	−	0.832360i	$-0.313011\pi$
$211$	49350.2			1.10847			0.554235	+	0.832360i	$0.313011\pi$
$212$		0			0
$213$	−38802.6	+	22402.7i	−0.855267	+	0.493789i
$214$		0			0
$215$	102.711	+	59.3004i	0.00222199	+	0.00128286i
$216$		0			0
$217$	19604.2	+	30007.6i	0.416322	+	0.637253i
$218$		0			0
$219$	22360.9	−	38730.2i	0.466231	−	0.807535i
$220$		0			0
$221$	−22199.1	−	38450.0i	−0.454518	−	0.787248i
$222$		0			0
$223$		−	42635.1i		−	0.857349i	−0.903459	−	0.428674i	$-0.858981\pi$
$223$		−	42635.1i		−	0.857349i	0.903459	−	0.428674i	$-0.141019\pi$
$224$		0			0
$225$	−16792.3			−0.331700
$226$		0			0
$227$	9232.35	−	5330.30i	0.179168	−	0.103443i	−0.407734	−	0.913101i	$-0.633681\pi$
$227$	9232.35	−	5330.30i	0.179168	−	0.103443i	0.586902	+	0.809658i	$0.300347\pi$
$228$		0			0
$229$	62336.6	+	35990.1i	1.18870	+	0.686296i	0.958011	−	0.286731i	$-0.0925686\pi$
$229$	62336.6	+	35990.1i	1.18870	+	0.686296i	0.230689	+	0.973028i	$0.425902\pi$
$230$		0			0
$231$	−14641.9	+	28932.0i	−0.274394	+	0.542194i
$232$		0			0
$233$	36639.3	−	63461.1i	0.674894	−	1.16895i	−0.301606	−	0.953433i	$-0.597523\pi$
$233$	36639.3	−	63461.1i	0.674894	−	1.16895i	0.976500	−	0.215518i	$-0.0691438\pi$
$234$		0			0
$235$	−2234.89	−	3870.95i	−0.0404689	−	0.0700941i
$236$		0			0
$237$		−	17137.3i		−	0.305103i
$238$		0			0
$239$	−44926.0			−0.786507			−0.393253	−	0.919430i	$-0.628650\pi$
$239$	−44926.0			−0.786507			−0.393253	+	0.919430i	$0.628650\pi$
$240$		0			0
$241$	1838.22	−	1061.30i	0.0316493	−	0.0182727i	−0.484092	−	0.875017i	$-0.660850\pi$
$241$	1838.22	−	1061.30i	0.0316493	−	0.0182727i	0.515741	+	0.856744i	$0.327517\pi$
$242$		0			0
$243$	3280.50	+	1894.00i	0.0555556	+	0.0320750i
$244$		0			0
$245$	−3847.43	−	1687.80i	−0.0640971	−	0.0281183i
$246$		0			0
$247$	34182.6	−	59206.0i	0.560288	−	0.970447i
$248$		0			0
$249$	6326.43	+	10957.7i	0.102037	+	0.176734i
$250$		0			0
$251$			108614.i			1.72401i	0.506904	+	0.862003i	$0.330790\pi$
$251$			108614.i			1.72401i	−0.506904	+	0.862003i	$0.669210\pi$
$252$		0			0
$253$	63432.4			0.990993
$254$		0			0
$255$	2236.53	−	1291.26i	0.0343948	−	0.0198579i
$256$		0			0
$257$	38942.0	+	22483.2i	0.589593	+	0.340402i	0.764937	−	0.644106i	$-0.222770\pi$
$257$	38942.0	+	22483.2i	0.589593	+	0.340402i	−0.175344	+	0.984507i	$0.556104\pi$
$258$		0			0
$259$	110908.	+	56128.3i	1.65334	+	0.836725i
$260$		0			0
$261$	−19407.9	+	33615.5i	−0.284904	+	0.493467i
$262$		0			0
$263$	−4983.42	−	8631.54i	−0.0720470	−	0.124789i	0.827751	−	0.561095i	$-0.189620\pi$
$263$	−4983.42	−	8631.54i	−0.0720470	−	0.124789i	−0.899798	+	0.436306i	$0.856286\pi$
$264$		0			0
$265$			2771.37i			0.0394642i
$266$		0			0
$267$	189.097			0.00265255
$268$		0			0
$269$	−71691.1	+	41390.9i	−0.990743	+	0.572005i	−0.905496	−	0.424354i	$-0.860501\pi$
$269$	−71691.1	+	41390.9i	−0.990743	+	0.572005i	−0.0852464	+	0.996360i	$0.527168\pi$
$270$		0			0
$271$	39010.9	+	22522.9i	0.531186	+	0.306681i	0.741499	−	0.670953i	$-0.234115\pi$
$271$	39010.9	+	22522.9i	0.531186	+	0.306681i	−0.210313	+	0.977634i	$0.567448\pi$
$272$		0			0
$273$	−33319.3	+	21767.8i	−0.447065	+	0.292071i
$274$		0			0
$275$	39603.4	−	68595.2i	0.523682	−	0.907043i
$276$		0			0
$277$	−37203.6	−	64438.5i	−0.484870	−	0.839820i	0.514979	−	0.857203i	$-0.327800\pi$
$277$	−37203.6	−	64438.5i	−0.484870	−	0.839820i	−0.999849	+	0.0173833i	$0.994466\pi$
$278$		0			0
$279$			19750.7i			0.253731i
$280$		0			0
$281$	100781.			1.27634			0.638168	−	0.769897i	$-0.279692\pi$
$281$	100781.			1.27634			0.638168	+	0.769897i	$0.279692\pi$
$282$		0			0
$283$	115969.	−	66954.9i	1.44800	−	0.836006i	0.449642	−	0.893209i	$-0.351552\pi$
$283$	115969.	−	66954.9i	1.44800	−	0.836006i	0.998363	+	0.0572033i	$0.0182183\pi$
$284$		0			0
$285$	3443.85	+	1988.31i	0.0423988	+	0.0244790i
$286$		0			0
$287$	2548.99	−	140.587i	0.0309460	−	0.00170679i
$288$		0			0
$289$	−1424.02	+	2466.47i	−0.0170498	+	0.0295312i
$290$		0			0
$291$	−25690.1	−	44496.5i	−0.303375	−	0.525461i
$292$		0			0
$293$		−	55274.5i		−	0.643857i	−0.946764	−	0.321929i	$-0.895669\pi$
$293$		−	55274.5i		−	0.643857i	0.946764	−	0.321929i	$-0.104331\pi$
$294$		0			0
$295$	−44.6696			−0.000513296
$296$		0			0
$297$	−15473.6	+	8933.70i	−0.175420	+	0.101279i
$298$		0			0
$299$	67426.0	+	38928.4i	0.754198	+	0.435436i
$300$		0			0
$301$	−182.895	−	3316.10i	−0.00201869	−	0.0366011i
$302$		0			0
$303$	5122.03	−	8871.61i	0.0557900	−	0.0966311i
$304$		0			0
$305$	2243.71	+	3886.21i	0.0241194	+	0.0417760i
$306$		0			0
$307$		−	127850.i		−	1.35651i	−0.734825	−	0.678256i	$-0.762736\pi$
$307$		−	127850.i		−	1.35651i	0.734825	−	0.678256i	$-0.237264\pi$
$308$		0			0
$309$	50216.8			0.525935
$310$		0			0
$311$	−138969.	+	80233.7i	−1.43680	+	0.829538i	−0.997626	−	0.0688639i	$-0.978063\pi$
$311$	−138969.	+	80233.7i	−1.43680	+	0.829538i	−0.439175	+	0.898401i	$0.644729\pi$
$312$		0			0
$313$	−5146.01	−	2971.05i	−0.0525269	−	0.0303264i	0.473507	−	0.880790i	$-0.342988\pi$
$313$	−5146.01	−	2971.05i	−0.0525269	−	0.0303264i	−0.526033	+	0.850464i	$0.676321\pi$
$314$		0			0
$315$	−1266.17	−	1938.09i	−0.0127606	−	0.0195323i
$316$		0			0
$317$	−23436.9	+	40593.9i	−0.233229	+	0.403964i	−0.958756	−	0.284229i	$-0.908262\pi$
$317$	−23436.9	+	40593.9i	−0.233229	+	0.403964i	0.725528	+	0.688193i	$0.241596\pi$
$318$		0			0
$319$	−91544.2	−	158559.i	−0.899600	−	1.55815i
$320$		0			0
$321$			71903.3i			0.697812i
$322$		0			0
$323$	124222.			1.19067
$324$		0			0
$325$	84193.6	−	48609.2i	0.797099	−	0.460205i
$326$		0			0
$327$	−21451.2	−	12384.8i	−0.200611	−	0.115823i
$328$		0			0
$329$	−56518.5	+	111679.i	−0.522154	+	1.03176i
$330$		0			0
$331$	30665.3	−	53113.9i	0.279892	−	0.484788i	−0.691465	−	0.722410i	$-0.743035\pi$
$331$	30665.3	−	53113.9i	0.279892	−	0.484788i	0.971358	+	0.237622i	$0.0763678\pi$
$332$		0			0
$333$	34246.4	+	59316.5i	0.308835	+	0.534918i
$334$		0			0
$335$		−	8504.38i		−	0.0757797i
$336$		0			0
$337$	170476.			1.50108			0.750540	−	0.660825i	$-0.229793\pi$
$337$	170476.			1.50108			0.750540	+	0.660825i	$0.229793\pi$
$338$		0			0
$339$	−74284.3	+	42888.1i	−0.646395	+	0.373196i
$340$		0			0
$341$	−80679.7	−	46580.5i	−0.693834	−	0.400585i
$342$		0			0
$343$	19358.2	+	116045.i	0.164542	+	0.986370i
$344$		0			0
$345$	−2264.36	+	3921.98i	−0.0190242	+	0.0329509i
$346$		0			0
$347$	3585.68	+	6210.58i	0.0297792	+	0.0515790i	0.880531	−	0.473989i	$-0.157186\pi$
$347$	3585.68	+	6210.58i	0.0297792	+	0.0515790i	−0.850752	+	0.525568i	$0.823853\pi$
$348$		0			0
$349$			67214.9i			0.551842i	0.961180	+	0.275921i	$0.0889828\pi$
$349$			67214.9i			0.551842i	−0.961180	+	0.275921i	$0.911017\pi$
$350$		0			0
$351$	−21930.4			−0.178005
$352$		0			0
$353$	−146558.	+	84615.3i	−1.17614	+	0.679047i	−0.955119	−	0.296222i	$-0.904273\pi$
$353$	−146558.	+	84615.3i	−1.17614	+	0.679047i	−0.221024	+	0.975268i	$0.570940\pi$
$354$		0			0
$355$	13067.0	+	7544.24i	0.103686	+	0.0598631i
$356$		0			0
$357$	−64524.8	−	32654.8i	−0.506279	−	0.256219i
$358$		0			0
$359$	63573.5	−	110112.i	0.493273	−	0.854373i	−0.506697	−	0.862124i	$-0.669134\pi$
$359$	63573.5	−	110112.i	0.493273	−	0.854373i	0.999970	+	0.00775078i	$0.00246718\pi$
$360$		0			0
$361$	30479.1	+	52791.4i	0.233877	+	0.405088i
$362$		0			0
$363$		−	8200.95i		−	0.0622373i
$364$		0			0
$365$	−15060.3			−0.113044
$366$		0			0
$367$	26136.3	−	15089.8i	0.194050	−	0.112035i	−0.399827	−	0.916590i	$-0.630930\pi$
$367$	26136.3	−	15089.8i	0.194050	−	0.112035i	0.593877	+	0.804556i	$0.297597\pi$
$368$		0			0
$369$	1218.22	+	703.341i	0.00894693	+	0.00516551i
$370$		0			0
$371$	64969.7	−	42445.2i	0.472023	−	0.308376i
$372$		0			0
$373$	−79937.4	+	138456.i	−0.574556	+	0.995160i	0.421534	+	0.906813i	$0.361492\pi$
$373$	−79937.4	+	138456.i	−0.574556	+	0.995160i	−0.996090	+	0.0883474i	$0.971841\pi$
$374$		0			0
$375$	5668.84	+	9818.71i	0.0403117	+	0.0698220i
$376$		0			0
$377$		−	224722.i		−	1.58112i
$378$		0			0
$379$	−2456.29			−0.0171002			−0.00855011	−	0.999963i	$-0.502722\pi$
$379$	−2456.29			−0.0171002			−0.00855011	+	0.999963i	$0.502722\pi$
$380$		0			0
$381$	14035.3	−	8103.28i	0.0966878	−	0.0558227i
$382$		0			0
$383$	−15732.9	−	9083.37i	−0.107253	−	0.0619226i	0.445414	−	0.895325i	$-0.353057\pi$
$383$	−15732.9	−	9083.37i	−0.107253	−	0.0619226i	−0.552667	+	0.833402i	$0.686390\pi$
$384$		0			0
$385$	10903.1	−	601.346i	0.0735577	−	0.00405698i
$386$		0			0
$387$	915.007	−	1584.84i	0.00610945	−	0.0105819i
$388$		0			0
$389$	8769.74	+	15189.6i	0.0579546	+	0.100380i	0.893547	−	0.448969i	$-0.148209\pi$
$389$	8769.74	+	15189.6i	0.0579546	+	0.100380i	−0.835593	+	0.549350i	$0.814875\pi$
$390$		0			0
$391$			141469.i			0.925350i
$392$		0			0
$393$	−116051.			−0.751384
$394$		0			0
$395$	−4997.92	+	2885.55i	−0.0320328	+	0.0184941i
$396$		0			0
$397$	−134013.	−	77372.4i	−0.850287	−	0.490914i	0.0104604	−	0.999945i	$-0.496670\pi$
$397$	−134013.	−	77372.4i	−0.850287	−	0.490914i	−0.860748	+	0.509032i	$0.830004\pi$
$398$		0			0
$399$	−6132.36	−	111187.i	−0.0385196	−	0.698404i
$400$		0			0
$401$	−69083.4	+	119656.i	−0.429621	+	0.744125i	−0.996839	−	0.0794423i	$-0.974686\pi$
$401$	−69083.4	+	119656.i	−0.429621	+	0.744125i	0.567219	+	0.823567i	$0.308019\pi$
$402$		0			0
$403$	−57172.8	−	99026.2i	−0.352030	−	0.609733i
$404$		0			0
$405$		−	1275.63i		−	0.00777705i
$406$		0			0
$407$	−323070.			−1.95033
$408$		0			0
$409$	−277220.	+	160053.i	−1.65721	+	0.956790i	−0.683215	+	0.730218i	$0.739419\pi$
$409$	−277220.	+	160053.i	−1.65721	+	0.956790i	−0.973994	+	0.226572i	$0.927248\pi$
$410$		0			0
$411$	114440.	+	66071.9i	0.677476	+	0.391141i
$412$		0			0
$413$	684.141	+	1047.20i	0.00401093	+	0.00613943i
$414$		0			0
$415$	2130.46	−	3690.07i	0.0123702	−	0.0214259i
$416$		0			0
$417$	−54024.3	−	93572.9i	−0.310683	−	0.538119i
$418$		0			0
$419$			194626.i			1.10859i	0.832319	+	0.554297i	$0.187013\pi$
$419$			194626.i			1.10859i	−0.832319	+	0.554297i	$0.812987\pi$
$420$		0			0
$421$	183907.			1.03761			0.518806	−	0.854892i	$-0.326377\pi$
$421$	183907.			1.03761			0.518806	+	0.854892i	$0.326377\pi$
$422$		0			0
$423$	−59728.8	+	34484.4i	−0.333813	+	0.192727i
$424$		0			0
$425$	152983.	+	88324.5i	0.846962	+	0.488994i
$426$		0			0
$427$	56741.3	−	112119.i	0.311203	−	0.614927i
$428$		0			0
$429$	51721.2	−	89583.8i	0.281031	−	0.486760i
$430$		0			0
$431$	−103627.	−	179487.i	−0.557849	−	0.966223i	−0.997676	−	0.0681403i	$-0.978293\pi$
$431$	−103627.	−	179487.i	−0.557849	−	0.966223i	0.439827	−	0.898083i	$-0.355040\pi$
$432$		0			0
$433$		−	267809.i		−	1.42840i	−0.699943	−	0.714199i	$-0.746791\pi$
$433$		−	267809.i		−	1.42840i	0.699943	−	0.714199i	$-0.253209\pi$
$434$		0			0
$435$	13071.5			0.0690789
$436$		0			0
$437$	−188652.	+	108918.i	−0.987865	+	0.570344i
$438$		0			0
$439$	−235335.	−	135871.i	−1.22112	−	0.705013i	−0.255961	−	0.966687i	$-0.582392\pi$
$439$	−235335.	−	135871.i	−1.22112	−	0.705013i	−0.965156	+	0.261674i	$0.915725\pi$
$440$		0			0
$441$	−26042.8	+	59365.9i	−0.133909	+	0.305253i
$442$		0			0
$443$	61590.6	−	106678.i	0.313839	−	0.543585i	−0.665351	−	0.746531i	$-0.731718\pi$
$443$	61590.6	−	106678.i	0.313839	−	0.543585i	0.979190	+	0.202946i	$0.0650515\pi$
$444$		0			0
$445$	−31.8398	−	55.1482i	−0.000160787	−	0.000278491i
$446$		0			0
$447$			135768.i			0.679490i
$448$		0			0
$449$	137080.			0.679956			0.339978	−	0.940433i	$-0.389580\pi$
$449$	137080.			0.679956			0.339978	+	0.940433i	$0.389580\pi$
$450$		0			0
$451$	−5746.17	+	3317.55i	−0.0282505	+	0.0163104i
$452$		0			0
$453$	90801.2	+	52424.1i	0.442482	+	0.255467i
$454$		0			0
$455$	11958.6	+	6052.01i	0.0577639	+	0.0292332i
$456$		0			0
$457$	−135713.	+	235062.i	−0.649815	+	1.12551i	0.333351	+	0.942803i	$0.391820\pi$
$457$	−135713.	+	235062.i	−0.649815	+	1.12551i	−0.983167	+	0.182710i	$0.941513\pi$
$458$		0			0
$459$	−19924.1	−	34509.6i	−0.0945702	−	0.163800i
$460$		0			0
$461$			275292.i			1.29536i	0.761911	+	0.647682i	$0.224261\pi$
$461$			275292.i			1.29536i	−0.761911	+	0.647682i	$0.775739\pi$
$462$		0			0
$463$	−176341.			−0.822606			−0.411303	−	0.911499i	$-0.634926\pi$
$463$	−176341.			−0.822606			−0.411303	+	0.911499i	$0.634926\pi$
$464$		0			0
$465$	5760.07	−	3325.58i	0.0266392	−	0.0153802i
$466$		0			0
$467$	−222929.	−	128708.i	−1.02219	−	0.590163i	−0.107454	−	0.994210i	$-0.534270\pi$
$467$	−222929.	−	128708.i	−1.02219	−	0.590163i	−0.914738	+	0.404047i	$0.867603\pi$
$468$		0			0
$469$	−199369.	+	130249.i	−0.906385	+	0.592148i
$470$		0			0
$471$	74880.7	−	129697.i	0.337542	−	0.584640i
$472$		0			0
$473$	4315.95	+	7475.44i	0.0192910	+	0.0334129i
$474$		0			0
$475$			272008.i			1.20557i
$476$		0			0
$477$	42762.3			0.187942
$478$		0			0
$479$	248731.	−	143605.i	1.08407	−	0.625889i	0.152080	−	0.988368i	$-0.451403\pi$
$479$	248731.	−	143605.i	1.08407	−	0.625889i	0.931992	+	0.362479i	$0.118069\pi$
$480$		0			0
$481$	−343410.	−	198268.i	−1.48430	−	0.856963i
$482$		0			0
$483$	126623.	−	6983.76i	0.542775	−	0.0299361i
$484$		0			0
$485$	−8651.29	+	14984.5i	−0.0367788	+	0.0637028i
$486$		0			0
$487$	103569.	+	179386.i	0.436688	+	0.756365i	0.997432	−	0.0716241i	$-0.0228182\pi$
$487$	103569.	+	179386.i	0.436688	+	0.756365i	−0.560744	+	0.827989i	$0.689485\pi$
$488$		0			0
$489$		−	32113.0i		−	0.134296i
$490$		0			0
$491$	246870.			1.02401			0.512007	−	0.858981i	$-0.328902\pi$
$491$	246870.			1.02401			0.512007	+	0.858981i	$0.328902\pi$
$492$		0			0
$493$	353622.	−	204164.i	1.45494	−	0.840011i
$494$		0			0
$495$	5210.84	+	3008.48i	0.0212665	+	0.0122782i
$496$		0			0
$497$	−23268.1	−	421876.i	−0.0941992	−	1.70794i
$498$		0			0
$499$	−52583.9	+	91078.0i	−0.211179	+	0.365774i	−0.952084	−	0.305837i	$-0.901064\pi$
$499$	−52583.9	+	91078.0i	−0.211179	+	0.365774i	0.740904	+	0.671610i	$0.234397\pi$
$500$		0			0
$501$	−28773.9	−	49837.9i	−0.114637	−	0.198556i
$502$		0			0
$503$			95424.4i			0.377158i	0.982058	+	0.188579i	$0.0603882\pi$
$503$			95424.4i			0.377158i	−0.982058	+	0.188579i	$0.939612\pi$
$504$		0			0
$505$	−3449.75			−0.0135271
$506$		0			0
$507$	−18569.5	+	10721.1i	−0.0722410	+	0.0417084i
$508$		0			0
$509$	231972.	+	133929.i	0.895364	+	0.516939i	0.875693	−	0.482868i	$-0.160405\pi$
$509$	231972.	+	133929.i	0.895364	+	0.516939i	0.0196710	+	0.999807i	$0.493738\pi$
$510$		0			0
$511$	230658.	+	353061.i	0.883336	+	1.35210i
$512$		0			0
$513$	30679.6	−	53138.6i	0.116578	−	0.201918i
$514$		0			0
$515$	−8455.40	−	14645.2i	−0.0318801	−	0.0552180i
$516$		0			0
$517$		−	325316.i		−	1.21709i
$518$		0			0
$519$	2714.07			0.0100760
$520$		0			0
$521$	405893.	−	234342.i	1.49533	−	0.863327i	0.495341	−	0.868698i	$-0.335043\pi$
$521$	405893.	−	234342.i	1.49533	−	0.863327i	0.999986	+	0.00537116i	$0.00170970\pi$
$522$		0			0
$523$	45127.2	+	26054.2i	0.164981	+	0.0952520i	0.580217	−	0.814462i	$-0.302968\pi$
$523$	45127.2	+	26054.2i	0.164981	+	0.0952520i	−0.415236	+	0.909714i	$0.636301\pi$
$524$		0			0
$525$	71504.0	−	141289.i	0.259425	−	0.512615i
$526$		0			0
$527$	103885.	−	179934.i	0.374051	−	0.647875i
$528$		0			0
$529$	15880.6	+	27505.9i	0.0567485	+	0.0982913i
$530$		0			0
$531$			689.253i			0.00244450i
$532$		0			0
$533$	−8143.92			−0.0286668
$534$		0			0
$535$	20969.8	−	12106.9i	0.0732634	−	0.0422987i
$536$		0			0
$537$	103380.	+	59686.8i	0.358501	+	0.206980i
$538$		0			0
$539$	−181084.	−	246393.i	−0.623309	−	0.848106i
$540$		0			0
$541$	−201109.	+	348332.i	−0.687128	+	1.19014i	0.285635	+	0.958339i	$0.407796\pi$
$541$	−201109.	+	348332.i	−0.687128	+	1.19014i	−0.972763	+	0.231802i	$0.925538\pi$
$542$		0			0
$543$	144528.	+	250330.i	0.490177	+	0.849012i
$544$		0			0
$545$			8341.34i			0.0280829i
$546$		0			0
$547$	−321233.			−1.07361			−0.536804	−	0.843707i	$-0.680369\pi$
$547$	−321233.			−1.07361			−0.536804	+	0.843707i	$0.680369\pi$
$548$		0			0
$549$	59964.3	−	34620.4i	0.198952	−	0.114865i
$550$		0			0
$551$	544515.	+	314376.i	1.79352	+	1.03549i
$552$		0			0
$553$	144192.	+	72973.0i	0.471511	+	0.238623i
$554$		0			0
$555$	11532.7	−	19975.2i	0.0374407	−	0.0648492i
$556$		0			0
$557$	254429.	+	440684.i	0.820081	+	1.42042i	0.905621	+	0.424087i	$0.139405\pi$
$557$	254429.	+	440684.i	0.820081	+	1.42042i	−0.0855405	+	0.996335i	$0.527262\pi$
$558$		0			0
$559$			10594.8i			0.0339053i
$560$		0			0
$561$	187958.			0.597222
$562$		0			0
$563$	−17207.7	+	9934.85i	−0.0542882	+	0.0313433i	−0.526898	−	0.849928i	$-0.676645\pi$
$563$	−17207.7	+	9934.85i	−0.0542882	+	0.0313433i	0.472610	+	0.881272i	$0.343312\pi$
$564$		0			0
$565$	25015.7	+	14442.8i	0.0783638	+	0.0452434i
$566$		0			0
$567$	−29904.8	+	19537.0i	−0.0930196	+	0.0607704i
$568$		0			0
$569$	67594.1	−	117076.i	0.208778	−	0.361614i	−0.742552	−	0.669788i	$-0.766385\pi$
$569$	67594.1	−	117076.i	0.208778	−	0.361614i	0.951330	+	0.308175i	$0.0997182\pi$
$570$		0			0
$571$	228508.	+	395788.i	0.700857	+	1.21392i	0.968166	+	0.250309i	$0.0805323\pi$
$571$	228508.	+	395788.i	0.700857	+	1.21392i	−0.267309	+	0.963611i	$0.586134\pi$
$572$		0			0
$573$		−	368824.i		−	1.12334i
$574$		0			0
$575$	−309773.			−0.936930
$576$		0			0
$577$	252798.	−	145953.i	0.759314	−	0.438390i	−0.0697351	−	0.997566i	$-0.522215\pi$
$577$	252798.	−	145953.i	0.759314	−	0.438390i	0.829050	+	0.559175i	$0.188882\pi$
$578$		0			0
$579$	67217.7	+	38808.1i	0.200506	+	0.115762i
$580$		0			0
$581$	−119136.	+	6570.80i	−0.352932	+	0.0194655i
$582$		0			0
$583$	−100852.	+	174680.i	−0.296720	+	0.513933i
$584$		0			0
$585$	3692.60	+	6395.77i	0.0107900	+	0.0186888i
$586$		0			0
$587$			7684.37i			0.0223014i	0.999938	+	0.0111507i	$0.00354945\pi$
$587$			7684.37i			0.0223014i	−0.999938	+	0.0111507i	$0.996451\pi$
$588$		0			0
$589$	319928.			0.922192
$590$		0			0
$591$	−1563.29	+	902.564i	−0.00447573	+	0.00258406i
$592$		0			0
$593$	−128962.	−	74456.1i	−0.366734	−	0.211734i	0.305297	−	0.952257i	$-0.401244\pi$
$593$	−128962.	−	74456.1i	−0.366734	−	0.211734i	−0.672031	+	0.740523i	$0.734578\pi$
$594$		0			0
$595$	1341.14	+	24316.3i	0.00378825	+	0.0686853i
$596$		0			0
$597$	172571.	−	298902.i	0.484194	−	0.838648i
$598$		0			0
$599$	−21716.0	−	37613.1i	−0.0605237	−	0.104830i	0.834176	−	0.551498i	$-0.185944\pi$
$599$	−21716.0	−	37613.1i	−0.0605237	−	0.104830i	−0.894700	+	0.446668i	$0.852610\pi$
$600$		0			0
$601$		−	109502.i		−	0.303161i	−0.988445	−	0.151581i	$-0.951564\pi$
$601$		−	109502.i		−	0.303161i	0.988445	−	0.151581i	$-0.0484363\pi$
$602$		0			0
$603$	−131223.			−0.360890
$604$		0			0
$605$	−2391.72	+	1380.86i	−0.00653430	+	0.00377258i
$606$		0			0
$607$	−471628.	−	272295.i	−1.28004	−	0.739029i	−0.303181	−	0.952933i	$-0.598049\pi$
$607$	−471628.	−	272295.i	−1.28004	−	0.739029i	−0.976855	+	0.213904i	$0.931382\pi$
$608$		0			0
$609$	−200197.	−	306436.i	−0.539787	−	0.826238i
$610$		0			0
$611$	199646.	−	345797.i	0.534784	−	0.926273i
$612$		0			0
$613$	−150466.	−	260615.i	−0.400422	−	0.693552i	0.593354	−	0.804941i	$-0.297803\pi$
$613$	−150466.	−	260615.i	−0.400422	−	0.693552i	−0.993777	+	0.111389i	$0.964470\pi$
$614$		0			0
$615$		−	473.709i		−	0.00125245i
$616$		0			0
$617$	383479.			1.00733			0.503665	−	0.863899i	$-0.331985\pi$
$617$	383479.			1.00733			0.503665	+	0.863899i	$0.331985\pi$
$618$		0			0
$619$	−150892.	+	87117.5i	−0.393808	+	0.227365i	−0.683809	−	0.729661i	$-0.739678\pi$
$619$	−150892.	+	87117.5i	−0.393808	+	0.227365i	0.290001	+	0.957026i	$0.406344\pi$
$620$		0			0
$621$	60516.3	+	34939.1i	0.156924	+	0.0906000i
$622$		0			0
$623$	−805.202	+	1591.05i	−0.00207457	+	0.00409929i
$624$		0			0
$625$	−192447.	+	333327.i	−0.492663	+	0.853318i
$626$		0			0
$627$	144711.	+	250647.i	0.368101	+	0.637569i
$628$		0			0
$629$		−	720518.i		−	1.82114i
$630$		0			0
$631$	−361662.			−0.908332			−0.454166	−	0.890917i	$-0.650063\pi$
$631$	−361662.			−0.908332			−0.454166	+	0.890917i	$0.650063\pi$
$632$		0			0
$633$	−222076.	+	128216.i	−0.554235	+	0.319988i
$634$		0			0
$635$	−4726.47	−	2728.83i	−0.0117217	−	0.00676751i
$636$		0			0
$637$	−41274.3	−	373036.i	−0.101719	−	0.919332i
$638$		0			0
$639$	116408.	−	201624.i	0.285089	−	0.493789i
$640$		0			0
$641$	340871.	+	590405.i	0.829609	+	1.43693i	0.898345	+	0.439290i	$0.144770\pi$
$641$	340871.	+	590405.i	0.829609	+	1.43693i	−0.0687361	+	0.997635i	$0.521897\pi$
$642$		0			0
$643$			519321.i			1.25607i	0.778185	+	0.628035i	$0.216141\pi$
$643$			519321.i			1.25607i	−0.778185	+	0.628035i	$0.783859\pi$
$644$		0			0
$645$	−616.268			−0.00148132
$646$		0			0
$647$	467422.	−	269866.i	1.11661	−	0.644673i	0.176074	−	0.984377i	$-0.443660\pi$
$647$	467422.	−	269866.i	1.11661	−	0.644673i	0.940533	+	0.339704i	$0.110327\pi$
$648$		0			0
$649$	−2815.54	−	1625.55i	−0.00668455	−	0.00385932i
$650$		0			0
$651$	−166181.	−	84101.0i	−0.392120	−	0.198445i
$652$		0			0
$653$	−285028.	+	493682.i	−0.668437	+	1.15777i	0.309904	+	0.950768i	$0.399703\pi$
$653$	−285028.	+	493682.i	−0.668437	+	1.15777i	−0.978341	+	0.206999i	$0.933630\pi$
$654$		0			0
$655$	19540.4	+	33844.9i	0.0455460	+	0.0788879i
$656$		0			0
$657$			232381.i			0.538357i
$658$		0			0
$659$	−710514.			−1.63607			−0.818035	−	0.575168i	$-0.804937\pi$
$659$	−710514.			−1.63607			−0.818035	+	0.575168i	$0.804937\pi$
$660$		0			0
$661$	477550.	−	275713.i	1.09299	−	0.631037i	0.158618	−	0.987340i	$-0.449296\pi$
$661$	477550.	−	275713.i	1.09299	−	0.631037i	0.934371	+	0.356303i	$0.115963\pi$
$662$		0			0
$663$	199792.	+	115350.i	0.454518	+	0.262416i
$664$		0			0
$665$	−31393.8	+	20509.8i	−0.0709906	+	0.0463787i
$666$		0			0
$667$	−358023.	+	620114.i	−0.804747	+	1.39386i
$668$		0			0
$669$	110769.	+	191858.i	0.247495	+	0.428674i
$670$		0			0
$671$			326599.i			0.725386i
$672$		0			0
$673$	−654979.			−1.44610			−0.723048	−	0.690798i	$-0.757259\pi$
$673$	−654979.			−1.44610			−0.723048	+	0.690798i	$0.757259\pi$
$674$		0			0
$675$	75565.5	−	43627.7i	0.165850	−	0.0957536i
$676$		0			0
$677$	−416606.	−	240527.i	−0.908967	−	0.524792i	−0.0288681	−	0.999583i	$-0.509190\pi$
$677$	−416606.	−	240527.i	−0.908967	−	0.524792i	−0.880099	+	0.474791i	$0.842524\pi$
$678$		0			0
$679$	483783.	−	26682.4i	1.04933	−	0.0578743i
$680$		0			0
$681$	−27697.1	+	47972.7i	−0.0597227	+	0.103443i
$682$		0			0
$683$	−326682.	−	565830.i	−0.700300	−	1.21295i	−0.968361	−	0.249553i	$-0.919716\pi$
$683$	−326682.	−	565830.i	−0.700300	−	1.21295i	0.268062	−	0.963402i	$-0.413617\pi$
$684$		0			0
$685$		−	44500.2i		−	0.0948377i
$686$		0			0
$687$	−374020.			−0.792467
$688$		0			0
$689$	−214402.	+	123785.i	−0.451639	+	0.260754i
$690$		0			0
$691$	−545296.	−	314827.i	−1.14203	−	0.659349i	−0.195094	−	0.980784i	$-0.562501\pi$
$691$	−545296.	−	314827.i	−1.14203	−	0.659349i	−0.946931	+	0.321436i	$0.895835\pi$
$692$		0			0
$693$	−9278.79	−	168235.i	−0.0193208	−	0.350308i
$694$		0			0
$695$	−18193.0	+	31511.2i	−0.0376648	+	0.0652373i
$696$		0			0
$697$	−7398.88	−	12815.2i	−0.0152300	−	0.0263792i
$698$		0			0
$699$			380767.i			0.779300i
$700$		0			0
$701$	−779759.			−1.58681			−0.793404	−	0.608696i	$-0.791693\pi$
$701$	−779759.			−1.58681			−0.793404	+	0.608696i	$0.791693\pi$
$702$		0			0
$703$	960828.	−	554734.i	1.94417	−	1.12247i
$704$		0			0
$705$	20114.0	+	11612.8i	0.0404689	+	0.0233647i
$706$		0			0
$707$	52834.8	+	80872.9i	0.105702	+	0.161795i
$708$		0			0
$709$	−26713.1	+	46268.5i	−0.0531413	+	0.0920435i	−0.891372	−	0.453272i	$-0.850257\pi$
$709$	−26713.1	+	46268.5i	−0.0531413	+	0.0920435i	0.838231	+	0.545315i	$0.183590\pi$
$710$		0			0
$711$	44524.1	+	77118.0i	0.0880756	+	0.152551i
$712$		0			0
$713$			364346.i			0.716696i
$714$		0			0
$715$	−34834.9			−0.0681400
$716$		0			0
$717$	202167.	−	116721.i	0.393253	−	0.227045i
$718$		0			0
$719$	231366.	+	133579.i	0.447550	+	0.258393i	0.706795	−	0.707419i	$-0.250140\pi$
$719$	231366.	+	133579.i	0.447550	+	0.258393i	−0.259245	+	0.965812i	$0.583474\pi$
$720$		0			0
$721$	−213830.	+	422521.i	−0.411337	+	0.812788i
$722$		0			0
$723$	−5514.66	+	9551.68i	−0.0105498	+	0.0182727i
$724$		0			0
$725$	447056.	+	774324.i	0.850523	+	1.47315i
$726$		0			0
$727$		−	628775.i		−	1.18967i	−0.803848	−	0.594835i	$-0.797218\pi$
$727$		−	628775.i		−	1.18967i	0.803848	−	0.594835i	$-0.202782\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	−16671.9	+	9625.53i	−0.0311997	+	0.0180132i
$732$		0			0
$733$	−343139.	−	198112.i	−0.638650	−	0.368725i	0.145444	−	0.989366i	$-0.453539\pi$
$733$	−343139.	−	198112.i	−0.638650	−	0.368725i	−0.784094	+	0.620642i	$0.786872\pi$
$734$		0			0
$735$	21698.5	−	2400.81i	0.0401656	−	0.00444409i
$736$		0			0
$737$	309479.	−	536033.i	0.569765	−	0.986863i
$738$		0			0
$739$	−194841.	−	337475.i	−0.356773	−	0.617948i	0.630647	−	0.776070i	$-0.282790\pi$
$739$	−194841.	−	337475.i	−0.356773	−	0.617948i	−0.987420	+	0.158121i	$0.949456\pi$
$740$		0			0
$741$			355236.i			0.646965i
$742$		0			0
$743$	−104210.			−0.188769			−0.0943844	−	0.995536i	$-0.530088\pi$
$743$	−104210.			−0.188769			−0.0943844	+	0.995536i	$0.530088\pi$
$744$		0			0
$745$	39595.4	−	22860.4i	0.0713398	−	0.0411880i
$746$		0			0
$747$	−56937.8	−	32873.1i	−0.102037	−	0.0589114i
$748$		0			0
$749$	−604989.	−	306174.i	−1.07841	−	0.545763i
$750$		0			0
$751$	−287909.	+	498673.i	−0.510477	+	0.884171i	0.489450	+	0.872031i	$0.337198\pi$
$751$	−287909.	+	498673.i	−0.510477	+	0.884171i	−0.999926	+	0.0121398i	$0.996136\pi$
$752$		0			0
$753$	−282188.	−	488763.i	−0.497677	−	0.862003i
$754$		0			0
$755$		−	35308.3i		−	0.0619416i
$756$		0			0
$757$	−722889.			−1.26148			−0.630739	−	0.775995i	$-0.717248\pi$
$757$	−722889.			−1.26148			−0.630739	+	0.775995i	$0.717248\pi$
$758$		0			0
$759$	−285446.	+	164802.i	−0.495496	+	0.286075i
$760$		0			0
$761$	876394.	+	505987.i	1.51332	+	0.873715i	0.999878	+	0.0155892i	$0.00496239\pi$
$761$	876394.	+	505987.i	1.51332	+	0.873715i	0.513440	+	0.858126i	$0.328371\pi$
$762$		0			0
$763$	195547.	−	127752.i	0.335894	−	0.219442i
$764$		0			0
$765$	−6709.58	+	11621.3i	−0.0114649	+	0.0198579i
$766$		0			0
$767$	−1995.20	−	3455.79i	−0.00339153	−	0.00587430i
$768$		0			0
$769$		−	627672.i		−	1.06140i	−0.847559	−	0.530701i	$-0.821929\pi$
$769$		−	627672.i		−	1.06140i	0.847559	−	0.530701i	$-0.178071\pi$
$770$		0			0
$771$	−233652.			−0.393062
$772$		0			0
$773$	−916460.	+	529118.i	−1.53375	+	0.885511i	−0.534565	+	0.845127i	$0.679525\pi$
$773$	−916460.	+	529118.i	−1.53375	+	0.885511i	−0.999184	+	0.0403832i	$0.987142\pi$
$774$		0			0
$775$	394000.	+	227476.i	0.655983	+	0.378732i
$776$		0			0
$777$	−644911.	+	35569.2i	−1.06821	+	0.0589159i
$778$		0			0
$779$	11392.9	−	19733.2i	0.0187742	−	0.0325179i
$780$		0			0
$781$	549078.	+	951031.i	0.900185	+	1.55917i
$782$		0			0
$783$		−	201693.i		−	0.328978i
$784$		0			0
$785$	−50433.0			−0.0818419
$786$		0			0
$787$	584854.	−	337665.i	0.944273	−	0.545176i	0.0529760	−	0.998596i	$-0.483129\pi$
$787$	584854.	−	337665.i	0.944273	−	0.545176i	0.891297	+	0.453419i	$0.149796\pi$
$788$		0			0
$789$	44850.8	+	25894.6i	0.0720470	+	0.0415964i
$790$		0			0
$791$	−44544.7	−	807646.i	−0.0711940	−	1.29083i
$792$		0			0
$793$	−200433.	+	347161.i	−0.318730	+	0.552057i
$794$		0			0
$795$	−7200.24	−	12471.2i	−0.0113923	−	0.0197321i
$796$		0			0
$797$			693604.i			1.09193i	0.837808	+	0.545965i	$0.183837\pi$
$797$			693604.i			1.09193i	−0.837808	+	0.545965i	$0.816163\pi$
$798$		0			0
$799$	725527.			1.13647
$800$		0			0
$801$	−850.938	+	491.289i	−0.00132627	+	0.000765724i
$802$		0			0
$803$	−949256.	−	548053.i	−1.47215	−	0.849947i
$804$		0			0
$805$	−23357.3	−	35752.5i	−0.0360439	−	0.0551714i
$806$		0			0
$807$	215073.	−	372518.i	0.330248	−	0.572005i
$808$		0			0
$809$	−237348.	−	411099.i	−0.362651	−	0.628130i	0.625745	−	0.780027i	$-0.284795\pi$
$809$	−237348.	−	411099.i	−0.362651	−	0.628130i	−0.988396	+	0.151898i	$0.951462\pi$
$810$		0			0
$811$		−	665829.i		−	1.01233i	−0.862438	−	0.506163i	$-0.831063\pi$
$811$		−	665829.i		−	1.01233i	0.862438	−	0.506163i	$-0.168937\pi$
$812$		0			0
$813$	−234065.			−0.354124
$814$		0			0
$815$	−9365.40	+	5407.12i	−0.0140997	+	0.00814049i
$816$		0			0
$817$	−25671.7	−	14821.6i	−0.0384602	−	0.0222050i
$818$		0			0
$819$	93382.6	−	184521.i	0.139219	−	0.275092i
$820$		0			0
$821$	−156268.	+	270663.i	−0.231837	+	0.401553i	−0.958349	−	0.285601i	$-0.907807\pi$
$821$	−156268.	+	270663.i	−0.231837	+	0.401553i	0.726512	+	0.687154i	$0.241140\pi$
$822$		0			0
$823$	308276.	+	533951.i	0.455136	+	0.788318i	0.998696	−	0.0510522i	$-0.0162575\pi$
$823$	308276.	+	533951.i	0.455136	+	0.788318i	−0.543560	+	0.839370i	$0.682924\pi$
$824$		0			0
$825$			411571.i			0.604696i
$826$		0			0
$827$	−797964.			−1.16673			−0.583367	−	0.812208i	$-0.698265\pi$
$827$	−797964.			−1.16673			−0.583367	+	0.812208i	$0.698265\pi$
$828$		0			0
$829$	935480.	−	540099.i	1.36121	−	0.785895i	0.371425	−	0.928463i	$-0.378869\pi$
$829$	935480.	−	540099.i	1.36121	−	0.785895i	0.989785	+	0.142568i	$0.0455359\pi$
$830$		0			0
$831$	334832.	+	193316.i	0.484870	+	0.279940i
$832$		0			0
$833$	549510.	−	403859.i	0.791928	−	0.582022i
$834$		0			0
$835$	−9689.79	+	16783.2i	−0.0138976	+	0.0240714i
$836$		0			0
$837$	−51313.7	−	88878.0i	−0.0732458	−	0.126865i
$838$		0			0
$839$			254929.i			0.362156i	0.983469	+	0.181078i	$0.0579587\pi$
$839$			254929.i			0.362156i	−0.983469	+	0.181078i	$0.942041\pi$
$840$		0			0
$841$	1.35948e6			1.92212
$842$		0			0
$843$	−453514.	+	261836.i	−0.638168	+	0.368447i
$844$		0			0
$845$	6253.38	+	3610.39i	0.00875793	+	0.00505639i
$846$		0			0
$847$	69002.2	+	34920.7i	0.0961825	+	0.0486762i
$848$		0			0
$849$	−347908.	+	602594.i	−0.482668	+	0.836006i
$850$		0			0
$851$	631752.	+	1.09423e6i	0.872344	+	1.51094i
$852$		0			0
$853$			98835.9i			0.135837i	0.997691	+	0.0679183i	$0.0216357\pi$
$853$			98835.9i			0.135837i	−0.997691	+	0.0679183i	$0.978364\pi$
$854$		0			0
$855$	−20663.1			−0.0282659
$856$		0			0
$857$	−171269.	+	98882.3i	−0.233194	+	0.134635i	−0.612045	−	0.790823i	$-0.709653\pi$
$857$	−171269.	+	98882.3i	−0.233194	+	0.134635i	0.378851	+	0.925458i	$0.376319\pi$
$858$		0			0
$859$	−1.22789e6	−	708923.i	−1.66408	−	0.960755i	−0.970738	−	0.240143i	$-0.922806\pi$
$859$	−1.22789e6	−	708923.i	−1.66408	−	0.960755i	−0.693339	−	0.720612i	$-0.743861\pi$
$860$		0			0
$861$	−11105.2	+	7255.12i	−0.0149803	+	0.00978674i
$862$		0			0
$863$	388485.	−	672876.i	0.521618	−	0.903469i	−0.478066	−	0.878324i	$-0.658662\pi$
$863$	388485.	−	672876.i	0.521618	−	0.903469i	0.999684	−	0.0251450i	$-0.00800474\pi$
$864$		0			0
$865$	−456.990	−	791.530i	−0.000610766	−	0.00105788i
$866$		0			0
$867$		−	14798.8i		−	0.0196874i
$868$		0			0
$869$	−420027.			−0.556208
$870$		0			0
$871$	657926.	−	379854.i	0.867243	−	0.500703i
$872$		0			0
$873$	231211.	+	133490.i	0.303375	+	0.175154i
$874$		0			0
$875$	−106753.	+	5887.81i	−0.139432	+	0.00769020i
$876$		0			0
$877$	−84199.4	+	145838.i	−0.109474	+	0.189614i	−0.915557	−	0.402188i	$-0.868250\pi$
$877$	−84199.4	+	145838.i	−0.109474	+	0.189614i	0.806083	+	0.591802i	$0.201583\pi$
$878$		0			0
$879$	143607.	+	248735.i	0.185866	+	0.321929i
$880$		0			0
$881$		−	903011.i		−	1.16343i	−0.813392	−	0.581716i	$-0.802382\pi$
$881$		−	903011.i		−	1.16343i	0.813392	−	0.581716i	$-0.197618\pi$
$882$		0			0
$883$	1.04089e6			1.33500			0.667502	−	0.744608i	$-0.267364\pi$
$883$	1.04089e6			1.33500			0.667502	+	0.744608i	$0.267364\pi$
$884$		0			0
$885$	201.013	−	116.055i	0.000256648	−	0.000148176i
$886$		0			0
$887$	197462.	+	114005.i	0.250978	+	0.144902i	0.620212	−	0.784434i	$-0.287047\pi$
$887$	197462.	+	114005.i	0.250978	+	0.144902i	−0.369234	+	0.929337i	$0.620380\pi$
$888$		0			0
$889$	8416.29	+	152597.i	0.0106492	+	0.193082i
$890$		0			0
$891$	46420.9	−	80403.3i	0.0584733	−	0.101279i
$892$		0			0
$893$	558590.	+	967507.i	0.700472	+	1.21325i
$894$		0			0
$895$		−	40199.7i		−	0.0501854i
$896$		0			0
$897$	−404556.			−0.502799
$898$		0			0
$899$	910739.	−	525815.i	1.12687	−	0.650600i
$900$		0			0
$901$	−389576.	−	224922.i	−0.479891	−	0.277065i
$902$		0			0
$903$	9438.50	+	14447.3i	0.0115752	+	0.0177178i
$904$		0			0
$905$	48670.8	−	84300.2i	0.0594252	−	0.102928i
$906$		0			0
$907$	−521769.	−	903730.i	−0.634254	−	1.09856i	−0.986673	−	0.162718i	$-0.947974\pi$
$907$	−521769.	−	903730.i	−0.634254	−	1.09856i	0.352418	−	0.935843i	$-0.385360\pi$
$908$		0			0
$909$			53229.7i			0.0644208i
$910$		0			0
$911$	1.18075e6			1.42273			0.711363	−	0.702825i	$-0.248078\pi$
$911$	1.18075e6			1.42273			0.711363	+	0.702825i	$0.248078\pi$
$912$		0			0
$913$	268567.	−	155057.i	0.322189	−	0.186016i
$914$		0			0
$915$	−20193.3	−	11658.6i	−0.0241194	−	0.0139253i
$916$		0			0
$917$	494159.	−	976441.i	0.587662	−	1.16120i
$918$		0			0
$919$	574040.	−	994267.i	0.679691	−	1.17726i	−0.295383	−	0.955379i	$-0.595447\pi$
$919$	574040.	−	994267.i	0.679691	−	1.17726i	0.975074	−	0.221880i	$-0.0712194\pi$
$920$		0			0
$921$	332164.	+	575325.i	0.391591	+	0.678256i
$922$		0			0
$923$			1.34788e6i			1.58214i
$924$		0			0
$925$	1.57771e6			1.84393
$926$		0			0
$927$	−225976.	+	130467.i	−0.262967	+	0.151824i
$928$		0			0
$929$	65085.0	+	37576.8i	0.0754136	+	0.0435400i	0.537233	−	0.843434i	$-0.319470\pi$
$929$	65085.0	+	37576.8i	0.0754136	+	0.0435400i	−0.461819	+	0.886974i	$0.652803\pi$
$930$		0			0
$931$	961629.	+	421850.i	1.10945	+	0.486697i
$932$		0			0
$933$	416907.	−	722103.i	0.478934	−	0.829538i
$934$		0			0
$935$	−31648.0	−	54816.0i	−0.0362013	−	0.0627024i
$936$		0			0
$937$			484162.i			0.551457i	0.961236	+	0.275728i	$0.0889191\pi$
$937$			484162.i			0.551457i	−0.961236	+	0.275728i	$0.911081\pi$
$938$		0			0
$939$	30876.1			0.0350180
$940$		0			0
$941$	−839101.	+	484455.i	−0.947622	+	0.547110i	−0.892341	−	0.451361i	$-0.850939\pi$
$941$	−839101.	+	484455.i	−0.947622	+	0.547110i	−0.0552805	+	0.998471i	$0.517605\pi$
$942$		0			0
$943$	22472.9	+	12974.7i	0.0252717	+	0.0145906i
$944$		0			0
$945$	10733.1	+	5431.80i	0.0120188	+	0.00608248i
$946$		0			0
$947$	−696362.	+	1.20613e6i	−0.776488	+	1.34492i	0.157466	+	0.987524i	$0.449667\pi$
$947$	−696362.	+	1.20613e6i	−0.776488	+	1.34492i	−0.933954	+	0.357392i	$0.883666\pi$
$948$		0			0
$949$	−672680.	−	1.16512e6i	−0.746923	−	1.29371i
$950$		0			0
$951$		−	243564.i		−	0.269309i
$952$		0			0
$953$	491345.			0.541004			0.270502	−	0.962719i	$-0.412810\pi$
$953$	491345.			0.541004			0.270502	+	0.962719i	$0.412810\pi$
$954$		0			0
$955$	−107564.	+	62101.9i	−0.117939	+	0.0680923i
$956$		0			0
$957$	823898.	+	475678.i	0.899600	+	0.519384i
$958$		0			0
$959$	−1.04322e6	+	681546.i	−1.13433	+	0.741068i
$960$		0			0
$961$	−194210.	+	336381.i	−0.210293	+	0.364238i
$962$		0			0
$963$	−186810.	−	323565.i	−0.201441	−	0.348906i
$964$		0			0
$965$		−	26137.8i		−	0.0280681i
$966$		0			0
$967$	−508743.			−0.544058			−0.272029	−	0.962289i	$-0.587695\pi$
$967$	−508743.			−0.544058			−0.272029	+	0.962289i	$0.587695\pi$
$968$		0			0
$969$	−558998.	+	322738.i	−0.595337	+	0.343718i
$970$		0			0
$971$	1.38817e6	+	801461.i	1.47233	+	0.850048i	0.999516	−	0.0311152i	$-0.00990589\pi$
$971$	1.38817e6	+	801461.i	1.47233	+	0.850048i	0.472811	+	0.881164i	$0.343239\pi$
$972$		0			0
$973$	1.01736e6	−	56111.2i	1.07460	−	0.0592685i
$974$		0			0
$975$	−252581.	+	437483.i	−0.265700	+	0.460205i
$976$		0			0
$977$	−563129.	−	975368.i	−0.589955	−	1.02183i	−0.994238	−	0.107198i	$-0.965812\pi$
$977$	−563129.	−	975368.i	−0.589955	−	1.02183i	0.404283	−	0.914634i	$-0.367521\pi$
$978$		0			0
$979$		−	4634.68i		−	0.00483564i
$980$		0			0
$981$	128707.			0.133741
$982$		0			0
$983$	714115.	−	412295.i	0.739029	−	0.426678i	−0.0826873	−	0.996576i	$-0.526350\pi$
$983$	714115.	−	412295.i	0.739029	−	0.426678i	0.821716	+	0.569897i	$0.193017\pi$
$984$		0			0
$985$	526.446	+	303.944i	0.000542602	+	0.000313272i
$986$		0			0
$987$	−35816.5	−	649393.i	−0.0367662	−	0.666613i
$988$		0			0
$989$	16879.4	−	29235.9i	0.0172569	−	0.0298899i
$990$		0			0
$991$	491464.	+	851240.i	0.500431	+	0.866772i	1.00000		0.000497836i	$0.000158466\pi$
$991$	491464.	+	851240.i	0.500431	+	0.866772i	−0.499569	+	0.866274i	$0.666508\pi$
$992$		0			0
$993$			318683.i			0.323192i
$994$		0			0
$995$	−116229.			−0.117400
$996$		0			0
$997$	961660.	−	555215.i	0.967456	−	0.558561i	0.0689966	−	0.997617i	$-0.478020\pi$
$997$	961660.	−	555215.i	0.967456	−	0.558561i	0.898460	+	0.439056i	$0.144687\pi$
$998$		0			0
$999$	−308218.	−	177949.i	−0.308835	−	0.178306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.a.145.5	yes	16
3.2	odd	2		504.5.by.a.145.4		16
4.3	odd	2		336.5.bh.i.145.5		16
7.2	even	3		1176.5.f.b.97.4		16
7.3	odd	6	inner	168.5.z.a.73.5	✓	16
7.5	odd	6		1176.5.f.b.97.13		16
21.17	even	6		504.5.by.a.73.4		16
28.3	even	6		336.5.bh.i.241.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.a.73.5	✓	16	7.3	odd	6	inner
168.5.z.a.145.5	yes	16	1.1	even	1	trivial
336.5.bh.i.145.5		16	4.3	odd	2
336.5.bh.i.241.5		16	28.3	even	6
504.5.by.a.73.4		16	21.17	even	6
504.5.by.a.145.4		16	3.2	odd	2
1176.5.f.b.97.4		16	7.2	even	3
1176.5.f.b.97.13		16	7.5	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 2.59808i) q^{3} +(1.51540 + 0.874918i) q^{5} +(-2.69843 - 48.9256i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 2.59808i) q^{3} +(1.51540 + 0.874918i) q^{5} +(-2.69843 - 48.9256i) q^{7} +(13.5000 - 23.3827i) q^{9} +(63.6775 + 110.293i) q^{11} +156.315i q^{13} -9.09241 q^{15} +(-245.977 + 142.015i) q^{17} +(-378.760 - 218.677i) q^{19} +(139.255 + 213.155i) q^{21} +(249.038 - 431.347i) q^{23} +(-310.969 - 538.614i) q^{25} +140.296i q^{27} -1437.62 q^{29} +(-633.503 + 365.753i) q^{31} +(-573.097 - 330.878i) q^{33} +(38.7167 - 76.5029i) q^{35} +(-1268.39 + 2196.91i) q^{37} +(-406.119 - 703.419i) q^{39} +52.0993i q^{41} +67.7783 q^{43} +(40.9159 - 23.6228i) q^{45} +(-2212.18 - 1277.20i) q^{47} +(-2386.44 + 264.045i) q^{49} +(737.931 - 1278.13i) q^{51} +(791.895 + 1371.60i) q^{53} +222.850i q^{55} +2272.56 q^{57} +(-22.1078 + 12.7639i) q^{59} +(2220.90 + 1282.24i) q^{61} +(-1180.44 - 597.400i) q^{63} +(-136.763 + 236.880i) q^{65} +(-2430.05 - 4208.97i) q^{67} +2588.08i q^{69} +8622.80 q^{71} +(-7453.63 + 4303.36i) q^{73} +(2798.72 + 1615.84i) q^{75} +(5224.31 - 3413.08i) q^{77} +(-1649.04 + 2856.22i) q^{79} +(-364.500 - 631.333i) q^{81} -2435.04i q^{83} -497.006 q^{85} +(6469.30 - 3735.05i) q^{87} +(-31.5162 - 18.1959i) q^{89} +(7647.82 - 421.806i) q^{91} +(1900.51 - 3291.78i) q^{93} +(-382.650 - 662.768i) q^{95} +9888.12i q^{97} +3438.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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