LMFDB - Embedded newform 168.5.z.a.145.4

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			145.4
Root			$-0.530737 - 7.91948i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.a.73.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{3} +(-2.25900 - 1.30423i) q^{5} +(16.7036 + 46.0651i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 2.59808i) q^{3} +(-2.25900 - 1.30423i) q^{5} +(16.7036 + 46.0651i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-63.9217 - 110.716i) q^{11} +246.652i q^{13} +13.5540 q^{15} +(299.463 - 172.895i) q^{17} +(-386.590 - 223.198i) q^{19} +(-194.847 - 163.895i) q^{21} +(-321.366 + 556.622i) q^{23} +(-309.098 - 535.373i) q^{25} +140.296i q^{27} -1541.46 q^{29} +(-1003.96 + 579.638i) q^{31} +(575.296 + 332.147i) q^{33} +(22.3461 - 125.846i) q^{35} +(701.134 - 1214.40i) q^{37} +(-640.821 - 1109.93i) q^{39} -2653.41i q^{41} -2738.20 q^{43} +(-60.9930 + 35.2143i) q^{45} +(-581.445 - 335.697i) q^{47} +(-1842.98 + 1538.91i) q^{49} +(-898.390 + 1556.06i) q^{51} +(1998.13 + 3460.86i) q^{53} +333.476i q^{55} +2319.54 q^{57} +(-928.546 + 536.096i) q^{59} +(-3982.94 - 2299.55i) q^{61} +(1302.62 + 231.303i) q^{63} +(321.692 - 557.187i) q^{65} +(1840.42 + 3187.69i) q^{67} -3339.73i q^{69} +1202.53 q^{71} +(3619.63 - 2089.79i) q^{73} +(2781.88 + 1606.12i) q^{75} +(4032.40 - 4793.91i) q^{77} +(-3594.11 + 6225.19i) q^{79} +(-364.500 - 631.333i) q^{81} -664.275i q^{83} -901.983 q^{85} +(6936.56 - 4004.83i) q^{87} +(10203.9 + 5891.25i) q^{89} +(-11362.0 + 4119.98i) q^{91} +(3011.89 - 5216.74i) q^{93} +(582.205 + 1008.41i) q^{95} +3922.36i q^{97} -3451.77 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.25900	−	1.30423i	−0.0903600	−	0.0521693i	0.454139	−	0.890931i	$-0.349947\pi$
$5$	−2.25900	−	1.30423i	−0.0903600	−	0.0521693i	−0.544499	+	0.838761i	$0.683280\pi$
$6$		0			0
$7$	16.7036	+	46.0651i	0.340890	+	0.940103i
$7$	16.7036	+	46.0651i	0.340890	+	0.940103i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	−63.9217	−	110.716i	−0.528279	−	0.915006i	−0.999456	−	0.0329674i	$-0.989504\pi$
$11$	−63.9217	−	110.716i	−0.528279	−	0.915006i	0.471178	−	0.882038i	$-0.343829\pi$
$12$		0			0
$13$			246.652i			1.45948i	0.683725	+	0.729740i	$0.260359\pi$
$13$			246.652i			1.45948i	−0.683725	+	0.729740i	$0.739641\pi$
$14$		0			0
$15$	13.5540			0.0602400
$16$		0			0
$17$	299.463	−	172.895i	1.03621	−	0.598253i	0.117449	−	0.993079i	$-0.462528\pi$
$17$	299.463	−	172.895i	1.03621	−	0.598253i	0.918756	+	0.394826i	$0.129195\pi$
$18$		0			0
$19$	−386.590	−	223.198i	−1.07089	−	0.618277i	−0.142463	−	0.989800i	$-0.545502\pi$
$19$	−386.590	−	223.198i	−1.07089	−	0.618277i	−0.928424	+	0.371523i	$0.878836\pi$
$20$		0			0
$21$	−194.847	−	163.895i	−0.441830	−	0.371645i
$22$		0			0
$23$	−321.366	+	556.622i	−0.607497	+	1.05222i	0.384154	+	0.923269i	$0.374493\pi$
$23$	−321.366	+	556.622i	−0.607497	+	1.05222i	−0.991651	+	0.128947i	$0.958840\pi$
$24$		0			0
$25$	−309.098	−	535.373i	−0.494557	−	0.856597i
$26$		0			0
$27$			140.296i			0.192450i
$28$		0			0
$29$	−1541.46			−1.83289			−0.916444	−	0.400164i	$-0.868953\pi$
$29$	−1541.46			−1.83289			−0.916444	+	0.400164i	$0.868953\pi$
$30$		0			0
$31$	−1003.96	+	579.638i	−1.04471	+	0.603161i	−0.921163	−	0.389178i	$-0.872759\pi$
$31$	−1003.96	+	579.638i	−1.04471	+	0.603161i	−0.123544	+	0.992339i	$0.539426\pi$
$32$		0			0
$33$	575.296	+	332.147i	0.528279	+	0.305002i
$34$		0			0
$35$	22.3461	−	125.846i	0.0182417	−	0.102732i
$36$		0			0
$37$	701.134	−	1214.40i	0.512151	−	0.887071i	−0.487750	−	0.872983i	$-0.662182\pi$
$37$	701.134	−	1214.40i	0.512151	−	0.887071i	0.999901	−	0.0140878i	$-0.00448445\pi$
$38$		0			0
$39$	−640.821	−	1109.93i	−0.421315	−	0.729740i
$40$		0			0
$41$		−	2653.41i		−	1.57847i	−0.614089	−	0.789236i	$-0.710477\pi$
$41$		−	2653.41i		−	1.57847i	0.614089	−	0.789236i	$-0.289523\pi$
$42$		0			0
$43$	−2738.20			−1.48091			−0.740453	−	0.672108i	$-0.765389\pi$
$43$	−2738.20			−1.48091			−0.740453	+	0.672108i	$0.765389\pi$
$44$		0			0
$45$	−60.9930	+	35.2143i	−0.0301200	+	0.0173898i
$46$		0			0
$47$	−581.445	−	335.697i	−0.263216	−	0.151968i	0.362585	−	0.931951i	$-0.381894\pi$
$47$	−581.445	−	335.697i	−0.263216	−	0.151968i	−0.625801	+	0.779983i	$0.715228\pi$
$48$		0			0
$49$	−1842.98	+	1538.91i	−0.767588	+	0.640944i
$50$		0			0
$51$	−898.390	+	1556.06i	−0.345402	+	0.598253i
$52$		0			0
$53$	1998.13	+	3460.86i	0.711331	+	1.23206i	0.964358	+	0.264602i	$0.0852406\pi$
$53$	1998.13	+	3460.86i	0.711331	+	1.23206i	−0.253027	+	0.967459i	$0.581426\pi$
$54$		0			0
$55$			333.476i			0.110240i
$56$		0			0
$57$	2319.54			0.713925
$58$		0			0
$59$	−928.546	+	536.096i	−0.266747	+	0.154006i	−0.627408	−	0.778690i	$-0.715884\pi$
$59$	−928.546	+	536.096i	−0.266747	+	0.154006i	0.360662	+	0.932697i	$0.382551\pi$
$60$		0			0
$61$	−3982.94	−	2299.55i	−1.07040	−	0.617993i	−0.142106	−	0.989851i	$-0.545387\pi$
$61$	−3982.94	−	2299.55i	−1.07040	−	0.617993i	−0.928289	+	0.371858i	$0.878721\pi$
$62$		0			0
$63$	1302.62	+	231.303i	0.328199	+	0.0582773i
$64$		0			0
$65$	321.692	−	557.187i	0.0761401	−	0.131878i
$66$		0			0
$67$	1840.42	+	3187.69i	0.409983	+	0.710112i	0.994887	−	0.100990i	$-0.0322012\pi$
$67$	1840.42	+	3187.69i	0.409983	+	0.710112i	−0.584904	+	0.811103i	$0.698868\pi$
$68$		0			0
$69$		−	3339.73i		−	0.701477i
$70$		0			0
$71$	1202.53			0.238551			0.119275	−	0.992861i	$-0.461943\pi$
$71$	1202.53			0.238551			0.119275	+	0.992861i	$0.461943\pi$
$72$		0			0
$73$	3619.63	−	2089.79i	0.679232	−	0.392155i	−0.120334	−	0.992734i	$-0.538396\pi$
$73$	3619.63	−	2089.79i	0.679232	−	0.392155i	0.799566	+	0.600579i	$0.205063\pi$
$74$		0			0
$75$	2781.88	+	1606.12i	0.494557	+	0.285532i
$76$		0			0
$77$	4032.40	−	4793.91i	0.680115	−	0.808553i
$78$		0			0
$79$	−3594.11	+	6225.19i	−0.575887	+	0.997466i	0.420057	+	0.907498i	$0.362010\pi$
$79$	−3594.11	+	6225.19i	−0.575887	+	0.997466i	−0.995945	+	0.0899685i	$0.971323\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	664.275i		−	0.0964255i	−0.998837	−	0.0482127i	$-0.984647\pi$
$83$		−	664.275i		−	0.0964255i	0.998837	−	0.0482127i	$-0.0153525\pi$
$84$		0			0
$85$	−901.983			−0.124842
$86$		0			0
$87$	6936.56	−	4004.83i	0.916444	−	0.529109i
$88$		0			0
$89$	10203.9	+	5891.25i	1.28821	+	0.743750i	0.978335	−	0.207026i	$-0.0663785\pi$
$89$	10203.9	+	5891.25i	1.28821	+	0.743750i	0.309878	+	0.950776i	$0.399712\pi$
$90$		0			0
$91$	−11362.0	+	4119.98i	−1.37206	+	0.497522i
$92$		0			0
$93$	3011.89	−	5216.74i	0.348235	−	0.603161i
$94$		0			0
$95$	582.205	+	1008.41i	0.0645102	+	0.111735i
$96$		0			0
$97$			3922.36i			0.416873i	0.978036	+	0.208437i	$0.0668375\pi$
$97$			3922.36i			0.416873i	−0.978036	+	0.208437i	$0.933162\pi$
$98$		0			0
$99$	−3451.77			−0.352186
$100$		0			0
$101$	8429.27	−	4866.64i	0.826318	−	0.477075i	−0.0262720	−	0.999655i	$-0.508364\pi$
$101$	8429.27	−	4866.64i	0.826318	−	0.477075i	0.852590	+	0.522580i	$0.175030\pi$
$102$		0			0
$103$	4825.94	+	2786.26i	0.454891	+	0.262632i	0.709894	−	0.704309i	$-0.248743\pi$
$103$	4825.94	+	2786.26i	0.454891	+	0.262632i	−0.255002	+	0.966940i	$0.582076\pi$
$104$		0			0
$105$	226.401	+	624.365i	0.0205352	+	0.0566318i
$106$		0			0
$107$	6374.53	−	11041.0i	0.556776	−	0.964364i	−0.440987	−	0.897514i	$-0.645371\pi$
$107$	6374.53	−	11041.0i	0.556776	−	0.964364i	0.997763	−	0.0668509i	$-0.0212952\pi$
$108$		0			0
$109$	276.408	+	478.753i	0.0232647	+	0.0402957i	0.877423	−	0.479717i	$-0.159261\pi$
$109$	276.408	+	478.753i	0.0232647	+	0.0402957i	−0.854159	+	0.520013i	$0.825927\pi$
$110$		0			0
$111$			7286.40i			0.591381i
$112$		0			0
$113$	−1493.40			−0.116955			−0.0584774	−	0.998289i	$-0.518625\pi$
$113$	−1493.40			−0.116955			−0.0584774	+	0.998289i	$0.518625\pi$
$114$		0			0
$115$	1451.93	−	838.273i	0.109787	−	0.0633855i
$116$		0			0
$117$	5767.39	+	3329.80i	0.421315	+	0.243247i
$118$		0			0
$119$	12966.5	+	10906.8i	0.915652	+	0.770201i
$120$		0			0
$121$	−851.477	+	1474.80i	−0.0581570	+	0.100731i
$122$		0			0
$123$	6893.77	+	11940.4i	0.455666	+	0.789236i
$124$		0			0
$125$			3242.84i			0.207542i
$126$		0			0
$127$	−27734.6			−1.71955			−0.859774	−	0.510675i	$-0.829396\pi$
$127$	−27734.6			−1.71955			−0.859774	+	0.510675i	$0.829396\pi$
$128$		0			0
$129$	12321.9	−	7114.04i	0.740453	−	0.427501i
$130$		0			0
$131$	−12340.6	−	7124.84i	−0.719107	−	0.415177i	0.0953170	−	0.995447i	$-0.469614\pi$
$131$	−12340.6	−	7124.84i	−0.719107	−	0.415177i	−0.814424	+	0.580270i	$0.802947\pi$
$132$		0			0
$133$	3824.17	−	21536.5i	0.216189	−	1.21751i
$134$		0			0
$135$	182.979	−	316.929i	0.0100400	−	0.0173898i
$136$		0			0
$137$	9799.17	+	16972.7i	0.522093	+	0.904292i	0.999670	+	0.0257018i	$0.00818204\pi$
$137$	9799.17	+	16972.7i	0.522093	+	0.904292i	−0.477576	+	0.878590i	$0.658485\pi$
$138$		0			0
$139$		−	7335.90i		−	0.379686i	−0.981815	−	0.189843i	$-0.939202\pi$
$139$		−	7335.90i		−	0.379686i	0.981815	−	0.189843i	$-0.0607978\pi$
$140$		0			0
$141$	3488.67			0.175478
$142$		0			0
$143$	27308.2	−	15766.4i	1.33543	−	0.771012i
$144$		0			0
$145$	3482.15	+	2010.42i	0.165620	+	0.0956205i
$146$		0			0
$147$	4295.20	−	11713.3i	0.198769	−	0.542055i
$148$		0			0
$149$	−10653.0	+	18451.5i	−0.479842	+	0.831112i	−0.999733	−	0.0231216i	$-0.992640\pi$
$149$	−10653.0	+	18451.5i	−0.479842	+	0.831112i	0.519890	+	0.854233i	$0.325973\pi$
$150$		0			0
$151$	8836.04	+	15304.5i	0.387529	+	0.671219i	0.992116	−	0.125319i	$-0.0399955\pi$
$151$	8836.04	+	15304.5i	0.387529	+	0.671219i	−0.604588	+	0.796538i	$0.706662\pi$
$152$		0			0
$153$		−	9336.34i		−	0.398836i
$154$		0			0
$155$	3023.93			0.125866
$156$		0			0
$157$	−40515.0	+	23391.4i	−1.64368	+	0.948978i	−0.664169	+	0.747583i	$0.731214\pi$
$157$	−40515.0	+	23391.4i	−1.64368	+	0.948978i	−0.979510	+	0.201396i	$0.935452\pi$
$158$		0			0
$159$	−17983.2	−	10382.6i	−0.711331	−	0.410687i
$160$		0			0
$161$	−31008.8	−	5506.13i	−1.19628	−	0.212420i
$162$		0			0
$163$	−20495.1	+	35498.6i	−0.771393	+	1.33609i	0.165407	+	0.986225i	$0.447106\pi$
$163$	−20495.1	+	35498.6i	−0.771393	+	1.33609i	−0.936800	+	0.349866i	$0.886227\pi$
$164$		0			0
$165$	−866.395	−	1500.64i	−0.0318235	−	0.0551199i
$166$		0			0
$167$		−	10562.3i		−	0.378726i	−0.981907	−	0.189363i	$-0.939358\pi$
$167$		−	10562.3i		−	0.378726i	0.981907	−	0.189363i	$-0.0606423\pi$
$168$		0			0
$169$	−32276.2			−1.13008
$170$		0			0
$171$	−10437.9	+	6026.34i	−0.356962	+	0.206092i
$172$		0			0
$173$	−12088.5	−	6979.29i	−0.403906	−	0.233195i	0.284262	−	0.958747i	$-0.408251\pi$
$173$	−12088.5	−	6979.29i	−0.403906	−	0.233195i	−0.688168	+	0.725552i	$0.741585\pi$
$174$		0			0
$175$	19498.9	−	23181.3i	0.636700	−	0.756940i
$176$		0			0
$177$	2785.64	−	4824.87i	0.0889156	−	0.154006i
$178$		0			0
$179$	−15107.6	−	26167.1i	−0.471507	−	0.816674i	0.527961	−	0.849268i	$-0.322957\pi$
$179$	−15107.6	−	26167.1i	−0.471507	−	0.816674i	−0.999469	+	0.0325939i	$0.989623\pi$
$180$		0			0
$181$		−	3891.36i		−	0.118780i	−0.998235	−	0.0593902i	$-0.981084\pi$
$181$		−	3891.36i		−	0.118780i	0.998235	−	0.0593902i	$-0.0189156\pi$
$182$		0			0
$183$	23897.6			0.713597
$184$		0			0
$185$	−3167.72	+	1828.89i	−0.0925559	+	0.0534371i
$186$		0			0
$187$	−38284.4	−	22103.5i	−1.09481	−	0.632089i
$188$		0			0
$189$	−6462.75	+	2343.45i	−0.180923	+	0.0656044i
$190$		0			0
$191$	6866.00	−	11892.3i	0.188208	−	0.325985i	−0.756445	−	0.654057i	$-0.773066\pi$
$191$	6866.00	−	11892.3i	0.188208	−	0.325985i	0.944653	+	0.328072i	$0.106399\pi$
$192$		0			0
$193$	4394.06	+	7610.73i	0.117964	+	0.204320i	0.918961	−	0.394349i	$-0.129030\pi$
$193$	4394.06	+	7610.73i	0.117964	+	0.204320i	−0.800996	+	0.598669i	$0.795696\pi$
$194$		0			0
$195$			3343.12i			0.0879190i
$196$		0			0
$197$	−48671.4			−1.25413			−0.627063	−	0.778968i	$-0.715743\pi$
$197$	−48671.4			−1.25413			−0.627063	+	0.778968i	$0.715743\pi$
$198$		0			0
$199$	51616.9	−	29801.0i	1.30342	−	0.752533i	0.322435	−	0.946592i	$-0.395499\pi$
$199$	51616.9	−	29801.0i	1.30342	−	0.752533i	0.980990	+	0.194059i	$0.0621653\pi$
$200$		0			0
$201$	−16563.7	−	9563.08i	−0.409983	−	0.236704i
$202$		0			0
$203$	−25747.9	−	71007.4i	−0.624814	−	1.72310i
$204$		0			0
$205$	−3460.67	+	5994.06i	−0.0823479	+	0.142631i
$206$		0			0
$207$	8676.88	+	15028.8i	0.202499	+	0.350739i
$208$		0			0
$209$			57068.8i			1.30649i
$210$		0			0
$211$	−24415.4			−0.548401			−0.274201	−	0.961673i	$-0.588413\pi$
$211$	−24415.4			−0.548401			−0.274201	+	0.961673i	$0.588413\pi$
$212$		0			0
$213$	−5411.40	+	3124.27i	−0.119275	+	0.0688636i
$214$		0			0
$215$	6185.58	+	3571.25i	0.133815	+	0.0772579i
$216$		0			0
$217$	−43470.9	−	36565.5i	−0.923164	−	0.776520i
$218$		0			0
$219$	−10858.9	+	18808.1i	−0.226411	+	0.392155i
$220$		0			0
$221$	42645.0	+	73863.2i	0.873138	+	1.51232i
$222$		0			0
$223$			74422.4i			1.49656i	0.663384	+	0.748280i	$0.269120\pi$
$223$			74422.4i			1.49656i	−0.663384	+	0.748280i	$0.730880\pi$
$224$		0			0
$225$	−16691.3			−0.329704
$226$		0			0
$227$	64392.0	−	37176.7i	1.24963	−	0.721472i	0.278591	−	0.960410i	$-0.410133\pi$
$227$	64392.0	−	37176.7i	1.24963	−	0.721472i	0.971035	+	0.238938i	$0.0767993\pi$
$228$		0			0
$229$	2661.08	+	1536.37i	0.0507442	+	0.0292972i	0.525157	−	0.851005i	$-0.324006\pi$
$229$	2661.08	+	1536.37i	0.0507442	+	0.0292972i	−0.474413	+	0.880302i	$0.657340\pi$
$230$		0			0
$231$	−5690.85	+	32049.1i	−0.106648	+	0.600609i
$232$		0			0
$233$	−35530.5	+	61540.7i	−0.654470	+	1.13358i	0.327556	+	0.944832i	$0.393775\pi$
$233$	−35530.5	+	61540.7i	−0.654470	+	1.13358i	−0.982026	+	0.188744i	$0.939558\pi$
$234$		0			0
$235$	875.656	+	1516.68i	0.0158561	+	0.0274636i
$236$		0			0
$237$		−	37351.1i		−	0.664977i
$238$		0			0
$239$	−26590.5			−0.465513			−0.232756	−	0.972535i	$-0.574774\pi$
$239$	−26590.5			−0.465513			−0.232756	+	0.972535i	$0.574774\pi$
$240$		0			0
$241$	86179.6	−	49755.8i	1.48378	−	0.856662i	0.483953	−	0.875094i	$-0.339201\pi$
$241$	86179.6	−	49755.8i	1.48378	−	0.856662i	0.999830	+	0.0184317i	$0.00586734\pi$
$242$		0			0
$243$	3280.50	+	1894.00i	0.0555556	+	0.0320750i
$244$		0			0
$245$	6170.38	−	1072.72i	0.102797	−	0.0178711i
$246$		0			0
$247$	55052.2	−	95353.2i	0.902362	−	1.56294i
$248$		0			0
$249$	1725.84	+	2989.24i	0.0278356	+	0.0482127i
$250$		0			0
$251$			10354.2i			0.164350i	0.996618	+	0.0821750i	$0.0261866\pi$
$251$			10354.2i			0.164350i	−0.996618	+	0.0821750i	$0.973813\pi$
$252$		0			0
$253$	82169.1			1.28371
$254$		0			0
$255$	4058.92	−	2343.42i	0.0624210	−	0.0360388i
$256$		0			0
$257$	−49442.8	−	28545.8i	−0.748577	−	0.432191i	0.0766026	−	0.997062i	$-0.475593\pi$
$257$	−49442.8	−	28545.8i	−0.748577	−	0.432191i	−0.825179	+	0.564871i	$0.808926\pi$
$258$		0			0
$259$	67652.9	+	12012.9i	1.00853	+	0.179081i
$260$		0			0
$261$	−20809.7	+	36043.4i	−0.305481	+	0.529109i
$262$		0			0
$263$	7941.41	+	13754.9i	0.114812	+	0.198860i	0.917704	−	0.397264i	$-0.130040\pi$
$263$	7941.41	+	13754.9i	0.114812	+	0.198860i	−0.802893	+	0.596124i	$0.796707\pi$
$264$		0			0
$265$		−	10424.1i		−	0.148439i
$266$		0			0
$267$	−61223.6			−0.858809
$268$		0			0
$269$	26116.4	−	15078.3i	0.360918	−	0.208376i	−0.308565	−	0.951203i	$-0.599849\pi$
$269$	26116.4	−	15078.3i	0.360918	−	0.208376i	0.669483	+	0.742827i	$0.266516\pi$
$270$		0			0
$271$	75529.9	+	43607.2i	1.02844	+	0.593772i	0.916538	−	0.399947i	$-0.130972\pi$
$271$	75529.9	+	43607.2i	1.02844	+	0.593772i	0.111904	+	0.993719i	$0.464305\pi$
$272$		0			0
$273$	40425.1	−	48059.4i	0.542408	−	0.644841i
$274$		0			0
$275$	−39516.2	+	68444.0i	−0.522528	+	0.905045i
$276$		0			0
$277$	57379.4	+	99384.0i	0.747818	+	1.29526i	0.948866	+	0.315678i	$0.102232\pi$
$277$	57379.4	+	99384.0i	0.747818	+	1.29526i	−0.201048	+	0.979581i	$0.564435\pi$
$278$		0			0
$279$			31300.5i			0.402108i
$280$		0			0
$281$	54629.8			0.691858			0.345929	−	0.938261i	$-0.387564\pi$
$281$	54629.8			0.691858			0.345929	+	0.938261i	$0.387564\pi$
$282$		0			0
$283$	−34501.8	+	19919.6i	−0.430793	+	0.248719i	−0.699684	−	0.714452i	$-0.746676\pi$
$283$	−34501.8	+	19919.6i	−0.430793	+	0.248719i	0.268891	+	0.963171i	$0.413343\pi$
$284$		0			0
$285$	−5239.84	−	3025.22i	−0.0645102	−	0.0372450i
$286$		0			0
$287$	122230.	−	44321.6i	1.48393	−	0.538086i
$288$		0			0
$289$	18025.0	−	31220.2i	0.215814	−	0.373801i
$290$		0			0
$291$	−10190.6	−	17650.6i	−0.120341	−	0.208437i
$292$		0			0
$293$			35360.2i			0.411889i	0.978564	+	0.205944i	$0.0660266\pi$
$293$			35360.2i			0.411889i	−0.978564	+	0.205944i	$0.933973\pi$
$294$		0			0
$295$	2796.78			0.0321376
$296$		0			0
$297$	15533.0	−	8967.97i	0.176093	−	0.101667i
$298$		0			0
$299$	−137292.	−	79265.6i	−1.53569	−	0.886629i
$300$		0			0
$301$	−45737.8	−	126135.i	−0.504827	−	1.39220i
$302$		0			0
$303$	−25287.8	+	43799.8i	−0.275439	+	0.477075i
$304$		0			0
$305$	5998.31	+	10389.4i	0.0644806	+	0.111684i
$306$		0			0
$307$			11532.7i			0.122364i	0.998127	+	0.0611819i	$0.0194870\pi$
$307$			11532.7i			0.122364i	−0.998127	+	0.0611819i	$0.980513\pi$
$308$		0			0
$309$	−28955.6			−0.303261
$310$		0			0
$311$	79818.6	−	46083.3i	0.825246	−	0.476456i	−0.0269760	−	0.999636i	$-0.508588\pi$
$311$	79818.6	−	46083.3i	0.825246	−	0.476456i	0.852222	+	0.523180i	$0.175254\pi$
$312$		0			0
$313$	42768.0	+	24692.1i	0.436546	+	0.252040i	0.702131	−	0.712047i	$-0.252232\pi$
$313$	42768.0	+	24692.1i	0.436546	+	0.252040i	−0.265585	+	0.964087i	$0.585565\pi$
$314$		0			0
$315$	−2640.95	−	2221.44i	−0.0266158	−	0.0223879i
$316$		0			0
$317$	−7152.19	+	12388.0i	−0.0711738	+	0.123277i	−0.899416	−	0.437094i	$-0.856008\pi$
$317$	−7152.19	+	12388.0i	−0.0711738	+	0.123277i	0.828242	+	0.560370i	$0.189341\pi$
$318$		0			0
$319$	98532.7	+	170664.i	0.968276	+	1.67710i
$320$		0			0
$321$			66246.0i			0.642910i
$322$		0			0
$323$	−154359.			−1.47954
$324$		0			0
$325$	132051.	−	76239.6i	1.25019	−	0.721795i
$326$		0			0
$327$	−2487.68	−	1436.26i	−0.0232647	−	0.0134319i
$328$		0			0
$329$	5751.68	−	32391.6i	0.0531377	−	0.299255i
$330$		0			0
$331$	27903.4	−	48330.0i	0.254683	−	0.441125i	−0.710126	−	0.704075i	$-0.751362\pi$
$331$	27903.4	−	48330.0i	0.254683	−	0.441125i	0.964809	+	0.262950i	$0.0846954\pi$
$332$		0			0
$333$	−18930.6	−	32788.8i	−0.170717	−	0.295690i
$334$		0			0
$335$		−	9601.33i		−	0.0855543i
$336$		0			0
$337$	74489.7			0.655899			0.327949	−	0.944695i	$-0.393642\pi$
$337$	74489.7			0.655899			0.327949	+	0.944695i	$0.393642\pi$
$338$		0			0
$339$	6720.28	−	3879.95i	0.0584774	−	0.0337619i
$340$		0			0
$341$	128350.	+	74103.0i	1.10379	+	0.637275i
$342$		0			0
$343$	−101674.	−	59191.5i	−0.864217	−	0.503120i
$344$		0			0
$345$	−4355.79	+	7544.45i	−0.0365956	+	0.0633855i
$346$		0			0
$347$	59004.1	+	102198.i	0.490030	+	0.848758i	0.999934	−	0.0114739i	$-0.00365232\pi$
$347$	59004.1	+	102198.i	0.490030	+	0.848758i	−0.509904	+	0.860231i	$0.670319\pi$
$348$		0			0
$349$		−	111351.i		−	0.914204i	−0.889414	−	0.457102i	$-0.848887\pi$
$349$		−	111351.i		−	0.914204i	0.889414	−	0.457102i	$-0.151113\pi$
$350$		0			0
$351$	−34604.3			−0.280877
$352$		0			0
$353$	−191585.	+	110612.i	−1.53749	+	0.887671i	−0.538506	+	0.842622i	$0.681011\pi$
$353$	−191585.	+	110612.i	−1.53749	+	0.887671i	−0.998985	+	0.0450489i	$0.985656\pi$
$354$		0			0
$355$	−2716.52	−	1568.39i	−0.0215554	−	0.0124450i
$356$		0			0
$357$	−86686.2	−	15392.6i	−0.680164	−	0.120774i
$358$		0			0
$359$	8955.34	−	15511.1i	0.0694854	−	0.120352i	−0.829190	−	0.558968i	$-0.811198\pi$
$359$	8955.34	−	15511.1i	0.0694854	−	0.120352i	0.898675	+	0.438615i	$0.144531\pi$
$360$		0			0
$361$	34474.1	+	59711.0i	0.264533	+	0.458184i
$362$		0			0
$363$		−	8848.81i		−	0.0671540i
$364$		0			0
$365$	−10902.3			−0.0818338
$366$		0			0
$367$	83920.0	−	48451.2i	0.623065	−	0.359727i	−0.154996	−	0.987915i	$-0.549537\pi$
$367$	83920.0	−	48451.2i	0.623065	−	0.359727i	0.778061	+	0.628188i	$0.216203\pi$
$368$		0			0
$369$	−62043.9	−	35821.1i	−0.455666	−	0.263079i
$370$		0			0
$371$	−126049.	+	149853.i	−0.915779	+	1.08872i
$372$		0			0
$373$	−12557.6	+	21750.4i	−0.0902587	+	0.156333i	−0.907620	−	0.419793i	$-0.862103\pi$
$373$	−12557.6	+	21750.4i	−0.0902587	+	0.156333i	0.817361	+	0.576126i	$0.195436\pi$
$374$		0			0
$375$	−8425.14	−	14592.8i	−0.0599121	−	0.103771i
$376$		0			0
$377$		−	380204.i		−	2.67506i
$378$		0			0
$379$	−137071.			−0.954262			−0.477131	−	0.878832i	$-0.658323\pi$
$379$	−137071.			−0.954262			−0.477131	+	0.878832i	$0.658323\pi$
$380$		0			0
$381$	124806.	−	72056.6i	0.859774	−	0.496391i
$382$		0			0
$383$	73436.9	+	42398.8i	0.500630	+	0.289039i	0.728974	−	0.684542i	$-0.239998\pi$
$383$	73436.9	+	42398.8i	0.500630	+	0.289039i	−0.228344	+	0.973581i	$0.573331\pi$
$384$		0			0
$385$	−15361.6	+	5570.25i	−0.103637	+	0.0375797i
$386$		0			0
$387$	−36965.6	+	64026.4i	−0.246818	+	0.427501i
$388$		0			0
$389$	−39637.3	−	68653.9i	−0.261942	−	0.453697i	0.704816	−	0.709390i	$-0.251030\pi$
$389$	−39637.3	−	68653.9i	−0.261942	−	0.453697i	−0.966758	+	0.255693i	$0.917696\pi$
$390$		0			0
$391$			222251.i			1.45375i
$392$		0			0
$393$	74043.6			0.479405
$394$		0			0
$395$	16238.2	−	9375.13i	0.104074	−	0.0600873i
$396$		0			0
$397$	26367.3	+	15223.2i	0.167296	+	0.0965881i	0.581310	−	0.813682i	$-0.302540\pi$
$397$	26367.3	+	15223.2i	0.167296	+	0.0965881i	−0.414014	+	0.910270i	$0.635874\pi$
$398$		0			0
$399$	38744.7	+	106850.i	0.243370	+	0.671163i
$400$		0			0
$401$	−119352.	+	206724.i	−0.742234	+	1.28559i	0.209241	+	0.977864i	$0.432901\pi$
$401$	−119352.	+	206724.i	−0.742234	+	1.28559i	−0.951476	+	0.307724i	$0.900433\pi$
$402$		0			0
$403$	−142969.	−	247629.i	−0.880302	−	1.52473i
$404$		0			0
$405$			1901.57i			0.0115932i
$406$		0			0
$407$	−179271.			−1.08223
$408$		0			0
$409$	−14222.4	+	8211.29i	−0.0850208	+	0.0490868i	−0.541908	−	0.840438i	$-0.682298\pi$
$409$	−14222.4	+	8211.29i	−0.0850208	+	0.0490868i	0.456887	+	0.889525i	$0.348964\pi$
$410$		0			0
$411$	−88192.5	−	50918.0i	−0.522093	−	0.301431i
$412$		0			0
$413$	−40205.4	−	33818.8i	−0.235713	−	0.198270i
$414$		0			0
$415$	−866.370	+	1500.60i	−0.00503045	+	0.00871300i
$416$		0			0
$417$	19059.2	+	33011.6i	0.109606	+	0.189843i
$418$		0			0
$419$			290326.i			1.65370i	0.562419	+	0.826852i	$0.309871\pi$
$419$			290326.i			1.65370i	−0.562419	+	0.826852i	$0.690129\pi$
$420$		0			0
$421$	−302213.			−1.70510			−0.852548	−	0.522648i	$-0.824944\pi$
$421$	−302213.			−1.70510			−0.852548	+	0.522648i	$0.824944\pi$
$422$		0			0
$423$	−15699.0	+	9063.83i	−0.0877388	+	0.0506560i
$424$		0			0
$425$	−185127.	−	106883.i	−1.02492	−	0.591740i
$426$		0			0
$427$	39399.4	−	221885.i	0.216090	−	1.21695i
$428$		0			0
$429$	−81924.7	+	141898.i	−0.445144	+	0.771012i
$430$		0			0
$431$	−43308.1	−	75011.8i	−0.233139	−	0.403808i	0.725591	−	0.688126i	$-0.241566\pi$
$431$	−43308.1	−	75011.8i	−0.233139	−	0.403808i	−0.958730	+	0.284318i	$0.908233\pi$
$432$		0			0
$433$		−	110391.i		−	0.588784i	−0.955685	−	0.294392i	$-0.904883\pi$
$433$		−	110391.i		−	0.588784i	0.955685	−	0.294392i	$-0.0951171\pi$
$434$		0			0
$435$	−20892.9			−0.110413
$436$		0			0
$437$	248474.	−	143456.i	1.30112	−	0.751203i
$438$		0			0
$439$	39285.4	+	22681.4i	0.203846	+	0.117691i	0.598448	−	0.801161i	$-0.295784\pi$
$439$	39285.4	+	22681.4i	0.203846	+	0.117691i	−0.394602	+	0.918852i	$0.629118\pi$
$440$		0			0
$441$	11103.6	+	63869.0i	0.0570934	+	0.328407i
$442$		0			0
$443$	167902.	−	290814.i	0.855554	−	1.48186i	−0.0205752	−	0.999788i	$-0.506550\pi$
$443$	167902.	−	290814.i	0.855554	−	1.48186i	0.876130	−	0.482075i	$-0.160117\pi$
$444$		0			0
$445$	−15367.1	−	26616.6i	−0.0776019	−	0.134410i
$446$		0			0
$447$		−	110709.i		−	0.554074i
$448$		0			0
$449$	161409.			0.800637			0.400318	−	0.916376i	$-0.368900\pi$
$449$	161409.			0.800637			0.400318	+	0.916376i	$0.368900\pi$
$450$		0			0
$451$	−293774.	+	169611.i	−1.44431	+	0.833874i
$452$		0			0
$453$	−79524.3	−	45913.4i	−0.387529	−	0.223740i
$454$		0			0
$455$	31040.3	+	5511.71i	0.149935	+	0.0266234i
$456$		0			0
$457$	69058.4	−	119613.i	0.330662	−	0.572724i	−0.651980	−	0.758236i	$-0.726061\pi$
$457$	69058.4	−	119613.i	0.330662	−	0.572724i	0.982642	+	0.185513i	$0.0593946\pi$
$458$		0			0
$459$	24256.5	+	42013.5i	0.115134	+	0.199418i
$460$		0			0
$461$		−	132545.i		−	0.623680i	−0.950135	−	0.311840i	$-0.899055\pi$
$461$		−	132545.i		−	0.623680i	0.950135	−	0.311840i	$-0.100945\pi$
$462$		0			0
$463$	−208868.			−0.974337			−0.487169	−	0.873308i	$-0.661970\pi$
$463$	−208868.			−0.974337			−0.487169	+	0.873308i	$0.661970\pi$
$464$		0			0
$465$	−13607.7	+	7856.41i	−0.0629331	+	0.0363344i
$466$		0			0
$467$	−225360.	−	130112.i	−1.03334	−	0.596599i	−0.115401	−	0.993319i	$-0.536815\pi$
$467$	−225360.	−	130112.i	−1.03334	−	0.596599i	−0.917940	+	0.396720i	$0.870149\pi$
$468$		0			0
$469$	−116100.	+	138025.i	−0.527819	+	0.627497i
$470$		0			0
$471$	121545.	−	210522.i	0.547893	−	0.948978i
$472$		0			0
$473$	175030.	+	303161.i	0.782332	+	1.35504i
$474$		0			0
$475$			275960.i			1.22309i
$476$		0			0
$477$	107899.			0.474221
$478$		0			0
$479$	−6852.09	+	3956.05i	−0.0298643	+	0.0172421i	−0.514858	−	0.857276i	$-0.672155\pi$
$479$	−6852.09	+	3956.05i	−0.0298643	+	0.0172421i	0.484994	+	0.874518i	$0.338822\pi$
$480$		0			0
$481$	299534.	+	172936.i	1.29466	+	0.747474i
$482$		0			0
$483$	153845.	−	55785.7i	0.659461	−	0.239127i
$484$		0			0
$485$	5115.67	−	8860.61i	0.0217480	−	0.0376687i
$486$		0			0
$487$	8538.02	+	14788.3i	0.0359997	+	0.0623533i	0.883464	−	0.468499i	$-0.155205\pi$
$487$	8538.02	+	14788.3i	0.0359997	+	0.0623533i	−0.847464	+	0.530853i	$0.821872\pi$
$488$		0			0
$489$		−	212992.i		−	0.890728i
$490$		0			0
$491$	−51807.0			−0.214895			−0.107447	−	0.994211i	$-0.534268\pi$
$491$	−51807.0			−0.214895			−0.107447	+	0.994211i	$0.534268\pi$
$492$		0			0
$493$	−461610.	+	266511.i	−1.89925	+	1.09653i
$494$		0			0
$495$	7797.55	+	4501.92i	0.0318235	+	0.0183733i
$496$		0			0
$497$	20086.7	+	55394.8i	0.0813196	+	0.224262i
$498$		0			0
$499$	119066.	−	206228.i	0.478174	−	0.828222i	−0.521513	−	0.853243i	$-0.674632\pi$
$499$	119066.	−	206228.i	0.478174	−	0.828222i	0.999687	+	0.0250217i	$0.00796548\pi$
$500$		0			0
$501$	27441.6	+	47530.3i	0.109329	+	0.189363i
$502$		0			0
$503$		−	31001.0i		−	0.122529i	−0.998122	−	0.0612646i	$-0.980487\pi$
$503$		−	31001.0i		−	0.122529i	0.998122	−	0.0612646i	$-0.0195134\pi$
$504$		0			0
$505$	−25389.0			−0.0995548
$506$		0			0
$507$	145243.	−	83856.0i	0.565040	−	0.326226i
$508$		0			0
$509$	−155334.	−	89682.3i	−0.599559	−	0.346155i	0.169309	−	0.985563i	$-0.445846\pi$
$509$	−155334.	−	89682.3i	−0.599559	−	0.346155i	−0.768868	+	0.639408i	$0.779180\pi$
$510$		0			0
$511$	156727.	+	131831.i	0.600210	+	0.504866i
$512$		0			0
$513$	31313.8	−	54237.1i	0.118987	−	0.206092i
$514$		0			0
$515$	−7267.86	−	12588.3i	−0.0274026	−	0.0474628i
$516$		0			0
$517$			85833.4i			0.321126i
$518$		0			0
$519$	72531.0			0.269270
$520$		0			0
$521$	175302.	−	101211.i	0.645820	−	0.372864i	−0.141033	−	0.990005i	$-0.545042\pi$
$521$	175302.	−	101211.i	0.645820	−	0.372864i	0.786853	+	0.617141i	$0.211709\pi$
$522$		0			0
$523$	384649.	+	222077.i	1.40625	+	0.811897i	0.995024	−	0.0996379i	$-0.0317684\pi$
$523$	384649.	+	222077.i	1.40625	+	0.811897i	0.411223	+	0.911535i	$0.365102\pi$
$524$		0			0
$525$	−27518.5	+	154976.i	−0.0998403	+	0.562270i
$526$		0			0
$527$	−200433.	+	347161.i	−0.721687	+	1.25000i
$528$		0			0
$529$	−66631.6	−	115409.i	−0.238105	−	0.412411i
$530$		0			0
$531$			28949.2i			0.102671i
$532$		0			0
$533$	654470.			2.30375
$534$		0			0
$535$	−28800.1	+	16627.8i	−0.100621	+	0.0580933i
$536$		0			0
$537$	135968.	+	78501.2i	0.471507	+	0.272225i
$538$		0			0
$539$	288187.	+	105677.i	0.991968	+	0.363750i
$540$		0			0
$541$	−78577.7	+	136101.i	−0.268475	+	0.465013i	−0.968468	−	0.249137i	$-0.919853\pi$
$541$	−78577.7	+	136101.i	−0.268475	+	0.465013i	0.699993	+	0.714150i	$0.253186\pi$
$542$		0			0
$543$	10110.1	+	17511.1i	0.0342889	+	0.0593902i
$544$		0			0
$545$		−	1442.00i		−	0.00485482i
$546$		0			0
$547$	−47511.9			−0.158792			−0.0793958	−	0.996843i	$-0.525299\pi$
$547$	−47511.9			−0.158792			−0.0793958	+	0.996843i	$0.525299\pi$
$548$		0			0
$549$	−107539.	+	62087.9i	−0.356798	+	0.205998i
$550$		0			0
$551$	595913.	+	344050.i	1.96282	+	1.13323i
$552$		0			0
$553$	−346798.	−	61579.8i	−1.13404	−	0.201367i
$554$		0			0
$555$	9503.17	−	16460.0i	0.0308520	−	0.0534371i
$556$		0			0
$557$	−145931.	−	252761.i	−0.470369	−	0.814703i	0.529057	−	0.848586i	$-0.322546\pi$
$557$	−145931.	−	252761.i	−0.470369	−	0.814703i	−0.999426	+	0.0338836i	$0.989212\pi$
$558$		0			0
$559$		−	675382.i		−	2.16135i
$560$		0			0
$561$	229707.			0.729874
$562$		0			0
$563$	322932.	−	186445.i	1.01881	−	0.588212i	0.105054	−	0.994467i	$-0.466499\pi$
$563$	322932.	−	186445.i	1.01881	−	0.588212i	0.913760	+	0.406254i	$0.133165\pi$
$564$		0			0
$565$	3373.58	+	1947.74i	0.0105680	+	0.00610145i
$566$		0			0
$567$	22993.9	−	27336.3i	0.0715231	−	0.0850301i
$568$		0			0
$569$	−116590.	+	201940.i	−0.360112	+	0.623733i	−0.987979	−	0.154588i	$-0.950595\pi$
$569$	−116590.	+	201940.i	−0.360112	+	0.623733i	0.627867	+	0.778321i	$0.283928\pi$
$570$		0			0
$571$	−24085.7	−	41717.6i	−0.0738732	−	0.127952i	0.826723	−	0.562610i	$-0.190203\pi$
$571$	−24085.7	−	41717.6i	−0.0738732	−	0.127952i	−0.900596	+	0.434658i	$0.856869\pi$
$572$		0			0
$573$			71353.6i			0.217323i
$574$		0			0
$575$	397334.			1.20177
$576$		0			0
$577$	−345896.	+	199703.i	−1.03895	+	0.599837i	−0.919535	−	0.393008i	$-0.871434\pi$
$577$	−345896.	+	199703.i	−1.03895	+	0.599837i	−0.119413	+	0.992845i	$0.538101\pi$
$578$		0			0
$579$	−39546.5	−	22832.2i	−0.117964	−	0.0681068i
$580$		0			0
$581$	30599.9	−	11095.8i	0.0906499	−	0.0328705i
$582$		0			0
$583$	255448.	−	442448.i	0.751562	−	1.30174i
$584$		0			0
$585$	−8685.68	−	15044.0i	−0.0253800	−	0.0439595i
$586$		0			0
$587$			246043.i			0.714061i	0.934093	+	0.357031i	$0.116211\pi$
$587$			246043.i			0.714061i	−0.934093	+	0.357031i	$0.883789\pi$
$588$		0			0
$589$	517496.			1.49168
$590$		0			0
$591$	219021.	−	126452.i	0.627063	−	0.362035i
$592$		0			0
$593$	−45883.4	−	26490.8i	−0.130481	−	0.0753331i	0.433339	−	0.901231i	$-0.357335\pi$
$593$	−45883.4	−	26490.8i	−0.130481	−	0.0753331i	−0.563820	+	0.825898i	$0.690669\pi$
$594$		0			0
$595$	−15066.4	−	41549.9i	−0.0425574	−	0.117364i
$596$		0			0
$597$	−154851.	+	268209.i	−0.434475	+	0.752533i
$598$		0			0
$599$	−327585.	−	567394.i	−0.912999	−	1.58136i	−0.809805	−	0.586699i	$-0.800427\pi$
$599$	−327585.	−	567394.i	−0.912999	−	1.58136i	−0.103193	−	0.994661i	$-0.532906\pi$
$600$		0			0
$601$			448466.i			1.24160i	0.783970	+	0.620798i	$0.213191\pi$
$601$			448466.i			1.24160i	−0.783970	+	0.620798i	$0.786809\pi$
$602$		0			0
$603$	99382.4			0.273322
$604$		0			0
$605$	3846.97	−	2221.05i	0.0105101	−	0.00606803i
$606$		0			0
$607$	−418865.	−	241832.i	−1.13683	−	0.656351i	−0.191189	−	0.981553i	$-0.561234\pi$
$607$	−418865.	−	241832.i	−1.13683	−	0.656351i	−0.945645	+	0.325202i	$0.894568\pi$
$608$		0			0
$609$	300348.	+	252638.i	0.809824	+	0.681183i
$610$		0			0
$611$	82800.4	−	143415.i	0.221794	−	0.384159i
$612$		0			0
$613$	237873.	+	412008.i	0.633029	+	1.09644i	0.986929	+	0.161155i	$0.0515219\pi$
$613$	237873.	+	412008.i	0.633029	+	1.09644i	−0.353900	+	0.935283i	$0.615145\pi$
$614$		0			0
$615$		−	35964.3i		−	0.0950872i
$616$		0			0
$617$	−491207.			−1.29031			−0.645156	−	0.764051i	$-0.723208\pi$
$617$	−491207.			−1.29031			−0.645156	+	0.764051i	$0.723208\pi$
$618$		0			0
$619$	−268368.	+	154942.i	−0.700405	+	0.404379i	−0.807498	−	0.589870i	$-0.799179\pi$
$619$	−268368.	+	154942.i	−0.700405	+	0.404379i	0.107093	+	0.994249i	$0.465846\pi$
$620$		0			0
$621$	−78091.9	−	45086.4i	−0.202499	−	0.116913i
$622$		0			0
$623$	−100938.	+	568450.i	−0.260062	+	1.46459i
$624$		0			0
$625$	−188957.	+	327283.i	−0.483729	+	0.837844i
$626$		0			0
$627$	−148269.	−	256810.i	−0.377151	−	0.653245i
$628$		0			0
$629$		−	484891.i		−	1.22558i
$630$		0			0
$631$	−262697.			−0.659775			−0.329888	−	0.944020i	$-0.607011\pi$
$631$	−262697.			−0.659775			−0.329888	+	0.944020i	$0.607011\pi$
$632$		0			0
$633$	109869.	−	63433.0i	0.274201	−	0.158310i
$634$		0			0
$635$	62652.4	+	36172.4i	0.155378	+	0.0897077i
$636$		0			0
$637$	−379574.	−	454574.i	−0.935445	−	1.12028i
$638$		0			0
$639$	16234.2	−	28118.5i	0.0397584	−	0.0688636i
$640$		0			0
$641$	−143571.	−	248672.i	−0.349423	−	0.605218i	0.636724	−	0.771091i	$-0.280289\pi$
$641$	−143571.	−	248672.i	−0.349423	−	0.605218i	−0.986147	+	0.165874i	$0.946956\pi$
$642$		0			0
$643$			637473.i			1.54184i	0.636930	+	0.770922i	$0.280204\pi$
$643$			637473.i			1.54184i	−0.636930	+	0.770922i	$0.719796\pi$
$644$		0			0
$645$	−37113.5			−0.0892098
$646$		0			0
$647$	−155574.	+	89820.5i	−0.371645	+	0.214569i	−0.674177	−	0.738570i	$-0.735501\pi$
$647$	−155574.	+	89820.5i	−0.371645	+	0.214569i	0.302532	+	0.953139i	$0.402168\pi$
$648$		0			0
$649$	118709.	+	68536.4i	0.281833	+	0.162717i
$650$		0			0
$651$	290619.	+	51604.3i	0.685744	+	0.121765i
$652$		0			0
$653$	178913.	−	309887.i	0.419582	−	0.726737i	−0.576316	−	0.817227i	$-0.695510\pi$
$653$	178913.	−	309887.i	0.419582	−	0.726737i	0.995897	+	0.0904903i	$0.0288434\pi$
$654$		0			0
$655$	18584.9	+	32190.0i	0.0433190	+	0.0750307i
$656$		0			0
$657$		−	112849.i		−	0.261437i
$658$		0			0
$659$	633593.			1.45895			0.729473	−	0.684009i	$-0.239765\pi$
$659$	633593.			1.45895			0.729473	+	0.684009i	$0.239765\pi$
$660$		0			0
$661$	−694573.	+	401012.i	−1.58970	+	0.917813i	−0.596342	+	0.802730i	$0.703380\pi$
$661$	−694573.	+	401012.i	−1.58970	+	0.917813i	−0.993356	+	0.115083i	$0.963287\pi$
$662$		0			0
$663$	−383805.	−	221590.i	−0.873138	−	0.504107i
$664$		0			0
$665$	−36727.4	+	43663.4i	−0.0830515	+	0.0987356i
$666$		0			0
$667$	495372.	−	858010.i	1.11347	−	1.92859i
$668$		0			0
$669$	−193355.	−	334901.i	−0.432019	−	0.748280i
$670$		0			0
$671$			587966.i			1.30589i
$672$		0			0
$673$	784704.			1.73251			0.866255	−	0.499602i	$-0.166520\pi$
$673$	784704.			1.73251			0.866255	+	0.499602i	$0.166520\pi$
$674$		0			0
$675$	75110.8	−	43365.2i	0.164852	−	0.0951775i
$676$		0			0
$677$	−12073.1	−	6970.40i	−0.0263415	−	0.0152083i	0.486771	−	0.873529i	$-0.338174\pi$
$677$	−12073.1	−	6970.40i	−0.0263415	−	0.0152083i	−0.513113	+	0.858321i	$0.671508\pi$
$678$		0			0
$679$	−180684.	+	65517.6i	−0.391904	+	0.142108i
$680$		0			0
$681$	−193176.	+	334591.i	−0.416542	+	0.721472i
$682$		0			0
$683$	−357496.	−	619202.i	−0.766356	−	1.32737i	−0.939527	−	0.342475i	$-0.888735\pi$
$683$	−357496.	−	619202.i	−0.766356	−	1.32737i	0.173171	−	0.984892i	$-0.444599\pi$
$684$		0			0
$685$		−	51121.6i		−	0.108949i
$686$		0			0
$687$	−15966.5			−0.0338295
$688$		0			0
$689$	−853628.	+	492842.i	−1.79817	+	1.03817i
$690$		0			0
$691$	172559.	+	99626.8i	0.361394	+	0.208651i	0.669692	−	0.742639i	$-0.266426\pi$
$691$	172559.	+	99626.8i	0.361394	+	0.208651i	−0.308298	+	0.951290i	$0.599759\pi$
$692$		0			0
$693$	−57657.1	−	159006.i	−0.120057	−	0.331091i
$694$		0			0
$695$	−9567.73	+	16571.8i	−0.0198079	+	0.0343084i
$696$		0			0
$697$	−458762.	−	794600.i	−0.944327	−	1.63562i
$698$		0			0
$699$		−	369244.i		−	0.755717i
$700$		0			0
$701$	413576.			0.841626			0.420813	−	0.907147i	$-0.361745\pi$
$701$	413576.			0.841626			0.420813	+	0.907147i	$0.361745\pi$
$702$		0			0
$703$	−542103.	+	312984.i	−1.09691	+	0.633302i
$704$		0			0
$705$	−7880.90	−	4550.04i	−0.0158561	−	0.00915455i
$706$		0			0
$707$	364982.	+	307004.i	0.730184	+	0.614194i
$708$		0			0
$709$	175157.	−	303381.i	0.348446	−	0.603527i	−0.637527	−	0.770428i	$-0.720043\pi$
$709$	175157.	−	303381.i	0.348446	−	0.603527i	0.985974	+	0.166901i	$0.0533760\pi$
$710$		0			0
$711$	97041.0	+	168080.i	0.191962	+	0.332489i
$712$		0			0
$713$		−	745104.i		−	1.46568i
$714$		0			0
$715$	−82252.4			−0.160893
$716$		0			0
$717$	119657.	−	69084.3i	0.232756	−	0.134382i
$718$		0			0
$719$	−747596.	−	431625.i	−1.44614	−	0.834927i	−0.447887	−	0.894090i	$-0.647823\pi$
$719$	−747596.	−	431625.i	−1.44614	−	0.834927i	−0.998248	+	0.0591635i	$0.981157\pi$
$720$		0			0
$721$	−47738.4	+	268848.i	−0.0918327	+	0.517173i
$722$		0			0
$723$	−258539.	+	447802.i	−0.494594	+	0.856662i
$724$		0			0
$725$	476462.	+	825256.i	0.906467	+	1.57005i
$726$		0			0
$727$			423146.i			0.800611i	0.916382	+	0.400305i	$0.131096\pi$
$727$			423146.i			0.800611i	−0.916382	+	0.400305i	$0.868904\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	−819989.	+	473421.i	−1.53452	+	0.885957i
$732$		0			0
$733$	180508.	+	104216.i	0.335960	+	0.193967i	0.658484	−	0.752595i	$-0.271198\pi$
$733$	180508.	+	104216.i	0.335960	+	0.193967i	−0.322524	+	0.946561i	$0.604531\pi$
$734$		0			0
$735$	−24979.7	+	20858.3i	−0.0462395	+	0.0386105i
$736$		0			0
$737$	235285.	−	407526.i	0.433171	−	0.750274i
$738$		0			0
$739$	37349.7	+	64691.6i	0.0683909	+	0.118457i	0.898193	−	0.439601i	$-0.144880\pi$
$739$	37349.7	+	64691.6i	0.0683909	+	0.118457i	−0.829802	+	0.558058i	$0.811547\pi$
$740$		0			0
$741$			572119.i			1.04196i
$742$		0			0
$743$	−1.01033e6			−1.83015			−0.915073	−	0.403289i	$-0.867867\pi$
$743$	−1.01033e6			−1.83015			−0.915073	+	0.403289i	$0.867867\pi$
$744$		0			0
$745$	48130.2	−	27788.0i	0.0867171	−	0.0500661i
$746$		0			0
$747$	−15532.5	−	8967.71i	−0.0278356	−	0.0160709i
$748$		0			0
$749$	615082.	+	109218.i	1.09640	+	0.194684i
$750$		0			0
$751$	276869.	−	479551.i	0.490902	−	0.850267i	−0.509043	−	0.860741i	$-0.670001\pi$
$751$	276869.	−	479551.i	0.490902	−	0.850267i	0.999945	+	0.0104742i	$0.00333410\pi$
$752$		0			0
$753$	−26901.0	−	46594.0i	−0.0474437	−	0.0821750i
$754$		0			0
$755$		−	46097.0i		−	0.0808684i
$756$		0			0
$757$	−799409.			−1.39501			−0.697505	−	0.716580i	$-0.745706\pi$
$757$	−799409.			−1.39501			−0.697505	+	0.716580i	$0.745706\pi$
$758$		0			0
$759$	−369761.	+	213482.i	−0.641856	+	0.370576i
$760$		0			0
$761$	−245061.	−	141486.i	−0.423160	−	0.244311i	0.273268	−	0.961938i	$-0.411895\pi$
$761$	−245061.	−	141486.i	−0.423160	−	0.244311i	−0.696428	+	0.717626i	$0.745229\pi$
$762$		0			0
$763$	−17436.8	+	20729.7i	−0.0299514	+	0.0356077i
$764$		0			0
$765$	−12176.8	+	21090.8i	−0.0208070	+	0.0360388i
$766$		0			0
$767$	−132229.	−	229028.i	−0.224769	−	0.389311i
$768$		0			0
$769$		−	179710.i		−	0.303892i	−0.988389	−	0.151946i	$-0.951446\pi$
$769$		−	179710.i		−	0.303892i	0.988389	−	0.151946i	$-0.0485540\pi$
$770$		0			0
$771$	296657.			0.499051
$772$		0			0
$773$	407704.	−	235388.i	0.682317	−	0.393936i	−0.118410	−	0.992965i	$-0.537780\pi$
$773$	407704.	−	235388.i	0.682317	−	0.393936i	0.800728	+	0.599029i	$0.204446\pi$
$774$		0			0
$775$	620646.	+	358330.i	1.03333	+	0.596595i
$776$		0			0
$777$	−335649.	+	121709.i	−0.555959	+	0.201596i
$778$		0			0
$779$	−592236.	+	1.02578e6i	−0.975933	+	1.69037i
$780$		0			0
$781$	−76868.1	−	133139.i	−0.126021	−	0.218275i
$782$		0			0
$783$		−	216261.i		−	0.352739i
$784$		0			0
$785$	122031.			0.198030
$786$		0			0
$787$	315696.	−	182267.i	0.509705	−	0.294279i	−0.223007	−	0.974817i	$-0.571587\pi$
$787$	315696.	−	182267.i	0.509705	−	0.294279i	0.732713	+	0.680538i	$0.238254\pi$
$788$		0			0
$789$	−71472.7	−	41264.8i	−0.114812	−	0.0662866i
$790$		0			0
$791$	−24945.1	−	68793.3i	−0.0398687	−	0.109950i
$792$		0			0
$793$	567189.	−	982401.i	0.901948	−	1.56222i
$794$		0			0
$795$	27082.6	+	46908.5i	0.0428506	+	0.0742193i
$796$		0			0
$797$		−	652893.i		−	1.02784i	−0.857838	−	0.513920i	$-0.828193\pi$
$797$		−	652893.i		−	1.02784i	0.857838	−	0.513920i	$-0.171807\pi$
$798$		0			0
$799$	−232162.			−0.363661
$800$		0			0
$801$	275506.	−	159064.i	0.429404	−	0.247917i
$802$		0			0
$803$	−462746.	−	267166.i	−0.717648	−	0.414334i
$804$		0			0
$805$	62867.6	+	52881.1i	0.0970141	+	0.0816034i
$806$		0			0
$807$	−78349.2	+	135705.i	−0.120306	+	0.208376i
$808$		0			0
$809$	−618638.	−	1.07151e6i	−0.945235	−	1.63719i	−0.755280	−	0.655402i	$-0.772499\pi$
$809$	−618638.	−	1.07151e6i	−0.945235	−	1.63719i	−0.189955	−	0.981793i	$-0.560834\pi$
$810$		0			0
$811$		−	36848.4i		−	0.0560244i	−0.999608	−	0.0280122i	$-0.991082\pi$
$811$		−	36848.4i		−	0.0560244i	0.999608	−	0.0280122i	$-0.00891773\pi$
$812$		0			0
$813$	−453179.			−0.685628
$814$		0			0
$815$	92597.0	−	53460.9i	0.139406	−	0.0804861i
$816$		0			0
$817$	1.05856e6	+	611160.i	1.58588	+	0.915610i
$818$		0			0
$819$	−57051.2	+	321295.i	−0.0850545	+	0.479000i
$820$		0			0
$821$	−505623.	+	875765.i	−0.750137	+	1.29928i	0.197618	+	0.980279i	$0.436679\pi$
$821$	−505623.	+	875765.i	−0.750137	+	1.29928i	−0.947756	+	0.318997i	$0.896654\pi$
$822$		0			0
$823$	−485882.	−	841572.i	−0.717349	−	1.24249i	−0.962046	−	0.272886i	$-0.912022\pi$
$823$	−485882.	−	841572.i	−0.717349	−	1.24249i	0.244697	−	0.969600i	$-0.421312\pi$
$824$		0			0
$825$		−	410664.i		−	0.603363i
$826$		0			0
$827$	979935.			1.43280			0.716401	−	0.697689i	$-0.245788\pi$
$827$	979935.			1.43280			0.716401	+	0.697689i	$0.245788\pi$
$828$		0			0
$829$	229141.	−	132295.i	0.333422	−	0.192501i	−0.323938	−	0.946078i	$-0.605007\pi$
$829$	229141.	−	132295.i	0.333422	−	0.192501i	0.657359	+	0.753577i	$0.271673\pi$
$830$		0			0
$831$	−516414.	−	298152.i	−0.747818	−	0.431753i
$832$		0			0
$833$	−285835.	+	779488.i	−0.411931	+	1.12336i
$834$		0			0
$835$	−13775.7	+	23860.2i	−0.0197579	+	0.0342217i
$836$		0			0
$837$	−81321.0	−	140852.i	−0.116078	−	0.201054i
$838$		0			0
$839$			95391.1i			0.135514i	0.997702	+	0.0677569i	$0.0215842\pi$
$839$			95391.1i			0.135514i	−0.997702	+	0.0677569i	$0.978416\pi$
$840$		0			0
$841$	1.66881e6			2.35948
$842$		0			0
$843$	−245834.	+	141932.i	−0.345929	+	0.199722i
$844$		0			0
$845$	72911.9	+	42095.7i	0.102114	+	0.0589555i
$846$		0			0
$847$	−82159.6	−	14588.8i	−0.114523	−	0.0203354i
$848$		0			0
$849$	103505.	−	179277.i	0.143598	−	0.248719i
$850$		0			0
$851$	450641.	+	780534.i	0.622260	+	1.07779i
$852$		0			0
$853$		−	147477.i		−	0.202687i	−0.994852	−	0.101343i	$-0.967686\pi$
$853$		−	147477.i		−	0.202687i	0.994852	−	0.101343i	$-0.0323141\pi$
$854$		0			0
$855$	31439.0			0.0430068
$856$		0			0
$857$	−708826.	+	409241.i	−0.965113	+	0.557208i	−0.897743	−	0.440520i	$-0.854794\pi$
$857$	−708826.	+	409241.i	−0.965113	+	0.557208i	−0.0673700	+	0.997728i	$0.521461\pi$
$858$		0			0
$859$	666760.	+	384954.i	0.903615	+	0.521702i	0.878371	−	0.477979i	$-0.158630\pi$
$859$	666760.	+	384954.i	0.903615	+	0.521702i	0.0252438	+	0.999681i	$0.491964\pi$
$860$		0			0
$861$	−434882.	+	517009.i	−0.586632	+	0.697416i
$862$		0			0
$863$	161515.	−	279751.i	0.216865	−	0.375622i	−0.736983	−	0.675912i	$-0.763750\pi$
$863$	161515.	−	279751.i	0.216865	−	0.375622i	0.953848	+	0.300290i	$0.0970834\pi$
$864$		0			0
$865$	18205.3	+	31532.4i	0.0243313	+	0.0421430i
$866$		0			0
$867$			187321.i			0.249201i
$868$		0			0
$869$	918968.			1.21692
$870$		0			0
$871$	−786251.	+	453942.i	−1.03639	+	0.598362i
$872$		0			0
$873$	91715.3	+	52951.9i	0.120341	+	0.0694789i
$874$		0			0
$875$	−149381.	+	54167.1i	−0.195110	+	0.0707489i
$876$		0			0
$877$	409507.	−	709287.i	0.532430	−	0.922195i	−0.466854	−	0.884335i	$-0.654612\pi$
$877$	409507.	−	709287.i	0.532430	−	0.922195i	0.999283	−	0.0378603i	$-0.0120542\pi$
$878$		0			0
$879$	−91868.6	−	159121.i	−0.118902	−	0.205944i
$880$		0			0
$881$			524257.i			0.675449i	0.941245	+	0.337725i	$0.109657\pi$
$881$			524257.i			0.675449i	−0.941245	+	0.337725i	$0.890343\pi$
$882$		0			0
$883$	813096.			1.04285			0.521423	−	0.853298i	$-0.325401\pi$
$883$	813096.			1.04285			0.521423	+	0.853298i	$0.325401\pi$
$884$		0			0
$885$	−12585.5	+	7266.24i	−0.0160688	+	0.00927734i
$886$		0			0
$887$	−103528.	−	59772.0i	−0.131586	−	0.0759714i	0.432762	−	0.901508i	$-0.357539\pi$
$887$	−103528.	−	59772.0i	−0.131586	−	0.0759714i	−0.564348	+	0.825537i	$0.690872\pi$
$888$		0			0
$889$	−463268.	−	1.27760e6i	−0.586177	−	1.61655i
$890$		0			0
$891$	−46598.9	+	80711.7i	−0.0586976	+	0.101667i
$892$		0			0
$893$	149854.	+	259555.i	0.187917	+	0.325481i
$894$		0			0
$895$			78815.2i			0.0983929i
$896$		0			0
$897$	823752.			1.02379
$898$		0			0
$899$	1.54757e6	−	893488.i	1.91483	−	1.10553i
$900$		0			0
$901$	1.19673e6	+	690934.i	1.47417	+	0.851112i
$902$		0			0
$903$	533529.	+	448778.i	0.654308	+	0.550371i
$904$		0			0
$905$	−5075.25	+	8790.59i	−0.00619669	+	0.0107330i
$906$		0			0
$907$	398252.	+	689793.i	0.484110	+	0.838502i	0.999833	−	0.0182526i	$-0.00581030\pi$
$907$	398252.	+	689793.i	0.484110	+	0.838502i	−0.515724	+	0.856755i	$0.672477\pi$
$908$		0			0
$909$		−	262799.i		−	0.318050i
$910$		0			0
$911$	−913425.			−1.10062			−0.550308	−	0.834961i	$-0.685490\pi$
$911$	−913425.			−1.10062			−0.550308	+	0.834961i	$0.685490\pi$
$912$		0			0
$913$	−73545.7	+	42461.6i	−0.0882299	+	0.0509395i
$914$		0			0
$915$	−53984.8	−	31168.1i	−0.0644806	−	0.0372279i
$916$		0			0
$917$	122074.	−	687481.i	0.145172	−	0.817564i
$918$		0			0
$919$	−82282.9	+	142518.i	−0.0974268	+	0.168748i	−0.910619	−	0.413247i	$-0.864395\pi$
$919$	−82282.9	+	142518.i	−0.0974268	+	0.168748i	0.813192	+	0.581995i	$0.197728\pi$
$920$		0			0
$921$	−29962.7	−	51897.0i	−0.0353234	−	0.0611819i
$922$		0			0
$923$			296607.i			0.348160i
$924$		0			0
$925$	−866877.			−1.01315
$926$		0			0
$927$	130300.	−	75229.0i	0.151630	−	0.0875439i
$928$		0			0
$929$	−82212.6	−	47465.5i	−0.0952592	−	0.0549979i	0.451614	−	0.892214i	$-0.350849\pi$
$929$	−82212.6	−	47465.5i	−0.0952592	−	0.0549979i	−0.546873	+	0.837216i	$0.684182\pi$
$930$		0			0
$931$	1.05596e6	−	183577.i	1.21828	−	0.211797i
$932$		0			0
$933$	−239456.	+	414750.i	−0.275082	+	0.476456i
$934$		0			0
$935$	57656.3	+	99863.7i	0.0659514	+	0.114231i
$936$		0			0
$937$			771016.i			0.878181i	0.898443	+	0.439091i	$0.144699\pi$
$937$			771016.i			0.878181i	−0.898443	+	0.439091i	$0.855301\pi$
$938$		0			0
$939$	−256608.			−0.291031
$940$		0			0
$941$	−229465.	+	132482.i	−0.259141	+	0.149615i	−0.623943	−	0.781470i	$-0.714470\pi$
$941$	−229465.	+	132482.i	−0.259141	+	0.149615i	0.364801	+	0.931085i	$0.381137\pi$
$942$		0			0
$943$	1.47695e6	+	852717.i	1.66089	+	0.958918i
$944$		0			0
$945$	17655.8	+	3135.07i	0.0197707	+	0.00351062i
$946$		0			0
$947$	20816.0	−	36054.4i	0.0232112	−	0.0402030i	−0.854186	−	0.519967i	$-0.825944\pi$
$947$	20816.0	−	36054.4i	0.0232112	−	0.0402030i	0.877398	+	0.479764i	$0.159278\pi$
$948$		0			0
$949$	515452.	+	892788.i	0.572342	+	0.991325i
$950$		0			0
$951$		−	74327.7i		−	0.0821845i
$952$		0			0
$953$	−688355.			−0.757926			−0.378963	−	0.925412i	$-0.623719\pi$
$953$	−688355.			−0.757926			−0.378963	+	0.925412i	$0.623719\pi$
$954$		0			0
$955$	−31020.6	+	17909.7i	−0.0340129	+	0.0196373i
$956$		0			0
$957$	−886794.	−	511991.i	−0.968276	−	0.559034i
$958$		0			0
$959$	−618165.	+	734904.i	−0.672151	+	0.799086i
$960$		0			0
$961$	210200.	−	364078.i	0.227608	−	0.394228i
$962$		0			0
$963$	−172112.	−	298107.i	−0.185592	−	0.321455i
$964$		0			0
$965$		−	22923.5i		−	0.0246165i
$966$		0			0
$967$	−292240.			−0.312527			−0.156263	−	0.987715i	$-0.549945\pi$
$967$	−292240.			−0.312527			−0.156263	+	0.987715i	$0.549945\pi$
$968$		0			0
$969$	694617.	−	401038.i	0.739772	−	0.427108i
$970$		0			0
$971$	480873.	+	277632.i	0.510026	+	0.294464i	0.732844	−	0.680396i	$-0.238192\pi$
$971$	480873.	+	277632.i	0.510026	+	0.294464i	−0.222818	+	0.974860i	$0.571526\pi$
$972$		0			0
$973$	337929.	−	122536.i	0.356944	−	0.129431i
$974$		0			0
$975$	−396153.	+	686157.i	−0.416729	+	0.721795i
$976$		0			0
$977$	451279.	+	781639.i	0.472777	+	0.818874i	0.999515	−	0.0311542i	$-0.00991830\pi$
$977$	451279.	+	781639.i	0.472777	+	0.818874i	−0.526738	+	0.850028i	$0.676585\pi$
$978$		0			0
$979$		−	1.50631e6i		−	1.57163i
$980$		0			0
$981$	14926.1			0.0155098
$982$		0			0
$983$	1.60311e6	−	925558.i	1.65904	−	0.957848i	0.685884	−	0.727711i	$-0.259416\pi$
$983$	1.60311e6	−	925558.i	1.65904	−	0.957848i	0.973158	−	0.230137i	$-0.0739175\pi$
$984$		0			0
$985$	109949.	+	63478.9i	0.113323	+	0.0654270i
$986$		0			0
$987$	58273.4	+	160706.i	0.0598186	+	0.164967i
$988$		0			0
$989$	879963.	−	1.52414e6i	0.899646	−	1.55823i
$990$		0			0
$991$	79030.3	+	136885.i	0.0804723	+	0.139382i	0.903453	−	0.428687i	$-0.141024\pi$
$991$	79030.3	+	136885.i	0.0804723	+	0.139382i	−0.822981	+	0.568069i	$0.807690\pi$
$992$		0			0
$993$			289980.i			0.294083i
$994$		0			0
$995$	−155470.			−0.157037
$996$		0			0
$997$	475172.	−	274340.i	0.478035	−	0.275994i	−0.241562	−	0.970385i	$-0.577660\pi$
$997$	475172.	−	274340.i	0.478035	−	0.275994i	0.719597	+	0.694391i	$0.244326\pi$
$998$		0			0
$999$	170376.	+	98366.4i	0.170717	+	0.0985635i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.a.145.4	yes	16
3.2	odd	2		504.5.by.a.145.5		16
4.3	odd	2		336.5.bh.i.145.4		16
7.2	even	3		1176.5.f.b.97.5		16
7.3	odd	6	inner	168.5.z.a.73.4	✓	16
7.5	odd	6		1176.5.f.b.97.12		16
21.17	even	6		504.5.by.a.73.5		16
28.3	even	6		336.5.bh.i.241.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.a.73.4	✓	16	7.3	odd	6	inner
168.5.z.a.145.4	yes	16	1.1	even	1	trivial
336.5.bh.i.145.4		16	4.3	odd	2
336.5.bh.i.241.4		16	28.3	even	6
504.5.by.a.73.5		16	21.17	even	6
504.5.by.a.145.5		16	3.2	odd	2
1176.5.f.b.97.5		16	7.2	even	3
1176.5.f.b.97.12		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)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-2.25900 - 1.30423i) q^{5} +(16.7036 + 46.0651i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-2.25900 - 1.30423i) q^{5} +(16.7036 + 46.0651i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-63.9217 - 110.716i) q^{11} +246.652i q^{13} +13.5540 q^{15} +(299.463 - 172.895i) q^{17} +(-386.590 - 223.198i) q^{19} +(-194.847 - 163.895i) q^{21} +(-321.366 + 556.622i) q^{23} +(-309.098 - 535.373i) q^{25} +140.296i q^{27} -1541.46 q^{29} +(-1003.96 + 579.638i) q^{31} +(575.296 + 332.147i) q^{33} +(22.3461 - 125.846i) q^{35} +(701.134 - 1214.40i) q^{37} +(-640.821 - 1109.93i) q^{39} -2653.41i q^{41} -2738.20 q^{43} +(-60.9930 + 35.2143i) q^{45} +(-581.445 - 335.697i) q^{47} +(-1842.98 + 1538.91i) q^{49} +(-898.390 + 1556.06i) q^{51} +(1998.13 + 3460.86i) q^{53} +333.476i q^{55} +2319.54 q^{57} +(-928.546 + 536.096i) q^{59} +(-3982.94 - 2299.55i) q^{61} +(1302.62 + 231.303i) q^{63} +(321.692 - 557.187i) q^{65} +(1840.42 + 3187.69i) q^{67} -3339.73i q^{69} +1202.53 q^{71} +(3619.63 - 2089.79i) q^{73} +(2781.88 + 1606.12i) q^{75} +(4032.40 - 4793.91i) q^{77} +(-3594.11 + 6225.19i) q^{79} +(-364.500 - 631.333i) q^{81} -664.275i q^{83} -901.983 q^{85} +(6936.56 - 4004.83i) q^{87} +(10203.9 + 5891.25i) q^{89} +(-11362.0 + 4119.98i) q^{91} +(3011.89 - 5216.74i) q^{93} +(582.205 + 1008.41i) q^{95} +3922.36i q^{97} -3451.77 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) 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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