LMFDB - Embedded newform 108.5.k.a.29.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			29.5
Character	$\chi$	$=$	108.29
Dual form			108.5.k.a.41.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.24301 + 8.71602i) q^{3} +(-14.1660 - 16.8824i) q^{5} +(35.7790 - 13.0225i) q^{7} +(-70.9378 - 39.1002i) q^{9} +O(q^{10})$	\(q+(-2.24301 + 8.71602i) q^{3} +(-14.1660 - 16.8824i) q^{5} +(35.7790 - 13.0225i) q^{7} +(-70.9378 - 39.1002i) q^{9} +(-102.369 + 121.998i) q^{11} +(-49.7378 - 282.077i) q^{13} +(178.921 - 85.6037i) q^{15} +(492.251 - 284.201i) q^{17} +(135.673 - 234.992i) q^{19} +(33.2517 + 341.060i) q^{21} +(31.0578 - 85.3307i) q^{23} +(24.1910 - 137.194i) q^{25} +(499.912 - 530.593i) q^{27} +(-438.837 - 77.3788i) q^{29} +(1185.32 + 431.421i) q^{31} +(-833.724 - 1165.89i) q^{33} +(-726.696 - 419.558i) q^{35} +(-522.756 - 905.440i) q^{37} +(2570.15 + 199.185i) q^{39} +(-2177.54 + 383.960i) q^{41} +(-2656.90 - 2229.41i) q^{43} +(344.801 + 1751.49i) q^{45} +(-224.867 - 617.817i) q^{47} +(-728.720 + 611.469i) q^{49} +(1372.98 + 4927.94i) q^{51} +4302.32i q^{53} +3509.77 q^{55} +(1743.88 + 1709.61i) q^{57} +(-1523.19 - 1815.27i) q^{59} +(2634.33 - 958.816i) q^{61} +(-3047.27 - 475.177i) q^{63} +(-4057.54 + 4835.59i) q^{65} +(-672.010 - 3811.16i) q^{67} +(674.081 + 462.098i) q^{69} +(-2218.83 + 1281.04i) q^{71} +(2217.70 - 3841.17i) q^{73} +(1141.53 + 518.577i) q^{75} +(-2073.93 + 5698.06i) q^{77} +(-985.019 + 5586.32i) q^{79} +(3503.36 + 5547.36i) q^{81} +(3307.71 + 583.239i) q^{83} +(-11771.2 - 4284.37i) q^{85} +(1658.75 - 3651.35i) q^{87} +(-2394.32 - 1382.36i) q^{89} +(-5452.92 - 9444.73i) q^{91} +(-6418.95 + 9363.58i) q^{93} +(-5889.16 + 1038.42i) q^{95} +(5628.76 + 4723.09i) q^{97} +(12031.9 - 4651.65i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{18}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.24301	+	8.71602i	−0.249223	+	0.968446i
$3$	−2.24301	+	8.71602i	−0.249223	+	0.968446i
$4$		0			0
$5$	−14.1660	−	16.8824i	−0.566639	−	0.675295i	0.404298	−	0.914627i	$-0.367516\pi$
$5$	−14.1660	−	16.8824i	−0.566639	−	0.675295i	−0.970938	+	0.239333i	$0.923071\pi$
$6$		0			0
$7$	35.7790	−	13.0225i	0.730184	−	0.265765i	0.0499412	−	0.998752i	$-0.484097\pi$
$7$	35.7790	−	13.0225i	0.730184	−	0.265765i	0.680243	+	0.732987i	$0.261874\pi$
$8$		0			0
$9$	−70.9378	−	39.1002i	−0.875776	−	0.482718i
$10$		0			0
$11$	−102.369	+	121.998i	−0.846021	+	1.00825i	0.153776	+	0.988106i	$0.450856\pi$
$11$	−102.369	+	121.998i	−0.846021	+	1.00825i	−0.999797	+	0.0201426i	$0.993588\pi$
$12$		0			0
$13$	−49.7378	−	282.077i	−0.294307	−	1.66910i	−0.670008	−	0.742354i	$-0.733709\pi$
$13$	−49.7378	−	282.077i	−0.294307	−	1.66910i	0.375701	−	0.926741i	$-0.377402\pi$
$14$		0			0
$15$	178.921	−	85.6037i	0.795206	−	0.380461i
$16$		0			0
$17$	492.251	−	284.201i	1.70329	−	0.983396i	0.760909	−	0.648858i	$-0.224753\pi$
$17$	492.251	−	284.201i	1.70329	−	0.983396i	0.942382	−	0.334538i	$-0.108580\pi$
$18$		0			0
$19$	135.673	−	234.992i	0.375825	−	0.650947i	−0.614625	−	0.788819i	$-0.710693\pi$
$19$	135.673	−	234.992i	0.375825	−	0.650947i	0.990450	+	0.137872i	$0.0440262\pi$
$20$		0			0
$21$	33.2517	+	341.060i	0.0754007	+	0.773379i
$22$		0			0
$23$	31.0578	−	85.3307i	0.0587104	−	0.161306i	−0.906870	−	0.421411i	$-0.861535\pi$
$23$	31.0578	−	85.3307i	0.0587104	−	0.161306i	0.965580	+	0.260105i	$0.0837573\pi$
$24$		0			0
$25$	24.1910	−	137.194i	0.0387056	−	0.219511i
$26$		0			0
$27$	499.912	−	530.593i	0.685750	−	0.727837i
$28$		0			0
$29$	−438.837	−	77.3788i	−0.521804	−	0.0920081i	−0.0934583	−	0.995623i	$-0.529792\pi$
$29$	−438.837	−	77.3788i	−0.521804	−	0.0920081i	−0.428345	+	0.903615i	$0.640903\pi$
$30$		0			0
$31$	1185.32	+	431.421i	1.23342	+	0.448929i	0.874768	−	0.484542i	$-0.161014\pi$
$31$	1185.32	+	431.421i	1.23342	+	0.448929i	0.358654	+	0.933471i	$0.383236\pi$
$32$		0			0
$33$	−833.724	−	1165.89i	−0.765586	−	1.07060i
$34$		0			0
$35$	−726.696	−	419.558i	−0.593221	−	0.342496i
$36$		0			0
$37$	−522.756	−	905.440i	−0.381852	−	0.661388i	0.609475	−	0.792806i	$-0.291380\pi$
$37$	−522.756	−	905.440i	−0.381852	−	0.661388i	−0.991327	+	0.131418i	$0.958047\pi$
$38$		0			0
$39$	2570.15	+	199.185i	1.68978	+	0.130957i
$40$		0			0
$41$	−2177.54	+	383.960i	−1.29539	+	0.228412i	−0.778501	−	0.627643i	$-0.784020\pi$
$41$	−2177.54	+	383.960i	−1.29539	+	0.228412i	−0.516885	+	0.856055i	$0.672909\pi$
$42$		0			0
$43$	−2656.90	−	2229.41i	−1.43694	−	1.20574i	−0.941463	−	0.337116i	$-0.890549\pi$
$43$	−2656.90	−	2229.41i	−1.43694	−	1.20574i	−0.495478	−	0.868621i	$-0.665007\pi$
$44$		0			0
$45$	344.801	+	1751.49i	0.170272	+	0.864934i
$46$		0			0
$47$	−224.867	−	617.817i	−0.101796	−	0.279682i	0.878331	−	0.478053i	$-0.158657\pi$
$47$	−224.867	−	617.817i	−0.101796	−	0.279682i	−0.980127	+	0.198371i	$0.936435\pi$
$48$		0			0
$49$	−728.720	+	611.469i	−0.303507	+	0.254673i
$50$		0			0
$51$	1372.98	+	4927.94i	0.527867	+	1.89463i
$52$		0			0
$53$			4302.32i			1.53162i	0.643067	+	0.765810i	$0.277662\pi$
$53$			4302.32i			1.53162i	−0.643067	+	0.765810i	$0.722338\pi$
$54$		0			0
$55$	3509.77			1.16025
$56$		0			0
$57$	1743.88	+	1709.61i	0.536743	+	0.526197i
$58$		0			0
$59$	−1523.19	−	1815.27i	−0.437573	−	0.521479i	0.501518	−	0.865147i	$-0.332775\pi$
$59$	−1523.19	−	1815.27i	−0.437573	−	0.521479i	−0.939091	+	0.343668i	$0.888330\pi$
$60$		0			0
$61$	2634.33	−	958.816i	0.707962	−	0.257677i	0.0371553	−	0.999310i	$-0.488170\pi$
$61$	2634.33	−	958.816i	0.707962	−	0.257677i	0.670806	+	0.741633i	$0.265948\pi$
$62$		0			0
$63$	−3047.27	−	475.177i	−0.767767	−	0.119722i
$64$		0			0
$65$	−4057.54	+	4835.59i	−0.960365	+	1.14452i
$66$		0			0
$67$	−672.010	−	3811.16i	−0.149702	−	0.848999i	−0.963471	−	0.267812i	$-0.913699\pi$
$67$	−672.010	−	3811.16i	−0.149702	−	0.848999i	0.813770	−	0.581187i	$-0.197412\pi$
$68$		0			0
$69$	674.081	+	462.098i	0.141584	+	0.0970590i
$70$		0			0
$71$	−2218.83	+	1281.04i	−0.440157	+	0.254125i	−0.703664	−	0.710533i	$-0.748454\pi$
$71$	−2218.83	+	1281.04i	−0.440157	+	0.254125i	0.263507	+	0.964657i	$0.415121\pi$
$72$		0			0
$73$	2217.70	−	3841.17i	0.416157	−	0.720806i	−0.579392	−	0.815049i	$-0.696710\pi$
$73$	2217.70	−	3841.17i	0.416157	−	0.720806i	0.995549	+	0.0942434i	$0.0300432\pi$
$74$		0			0
$75$	1141.53	+	518.577i	0.202938	+	0.0921914i
$76$		0			0
$77$	−2073.93	+	5698.06i	−0.349793	+	0.961050i
$78$		0			0
$79$	−985.019	+	5586.32i	−0.157830	+	0.895100i	0.798322	+	0.602231i	$0.205721\pi$
$79$	−985.019	+	5586.32i	−0.157830	+	0.895100i	−0.956152	+	0.292870i	$0.905390\pi$
$80$		0			0
$81$	3503.36	+	5547.36i	0.533967	+	0.845505i
$82$		0			0
$83$	3307.71	+	583.239i	0.480144	+	0.0846624i	0.408482	−	0.912767i	$-0.366058\pi$
$83$	3307.71	+	583.239i	0.480144	+	0.0846624i	0.0716626	+	0.997429i	$0.477170\pi$
$84$		0			0
$85$	−11771.2	−	4284.37i	−1.62923	−	0.592993i
$86$		0			0
$87$	1658.75	−	3651.35i	0.219150	−	0.482408i
$88$		0			0
$89$	−2394.32	−	1382.36i	−0.302275	−	0.174519i	0.341189	−	0.939995i	$-0.389170\pi$
$89$	−2394.32	−	1382.36i	−0.302275	−	0.174519i	−0.643465	+	0.765476i	$0.722504\pi$
$90$		0			0
$91$	−5452.92	−	9444.73i	−0.658485	−	1.14053i
$92$		0			0
$93$	−6418.95	+	9363.58i	−0.742161	+	1.08262i
$94$		0			0
$95$	−5889.16	+	1038.42i	−0.652538	+	0.115060i
$96$		0			0
$97$	5628.76	+	4723.09i	0.598231	+	0.501976i	0.890877	−	0.454245i	$-0.150091\pi$
$97$	5628.76	+	4723.09i	0.598231	+	0.501976i	−0.292645	+	0.956221i	$0.594536\pi$
$98$		0			0
$99$	12031.9	−	4651.65i	1.22762	−	0.474610i
$100$		0			0
$101$	−3705.03	−	10179.5i	−0.363203	−	0.997891i	−0.977890	−	0.209120i	$-0.932940\pi$
$101$	−3705.03	−	10179.5i	−0.363203	−	0.997891i	0.614687	−	0.788771i	$-0.289282\pi$
$102$		0			0
$103$	12221.3	−	10254.9i	1.15198	−	0.966624i	0.152213	−	0.988348i	$-0.451360\pi$
$103$	12221.3	−	10254.9i	1.15198	−	0.966624i	0.999764	+	0.0217238i	$0.00691543\pi$
$104$		0			0
$105$	5286.86	−	5392.82i	0.479533	−	0.489145i
$106$		0			0
$107$			2421.06i			0.211465i	0.994395	+	0.105732i	$0.0337187\pi$
$107$			2421.06i			0.211465i	−0.994395	+	0.105732i	$0.966281\pi$
$108$		0			0
$109$	13754.8			1.15772			0.578859	−	0.815428i	$-0.303498\pi$
$109$	13754.8			1.15772			0.578859	+	0.815428i	$0.303498\pi$
$110$		0			0
$111$	9064.37	−	2525.44i	0.735685	−	0.204970i
$112$		0			0
$113$	−7941.63	−	9464.47i	−0.621946	−	0.741207i	0.359457	−	0.933162i	$-0.382962\pi$
$113$	−7941.63	−	9464.47i	−0.621946	−	0.741207i	−0.981404	+	0.191955i	$0.938517\pi$
$114$		0			0
$115$	−1880.55	+	684.464i	−0.142196	+	0.0517553i
$116$		0			0
$117$	−7500.97	+	21954.7i	−0.547956	+	1.60382i
$118$		0			0
$119$	13911.3	−	16578.8i	0.982364	−	1.17074i
$120$		0			0
$121$	−1861.83	−	10558.9i	−0.127165	−	0.721190i
$122$		0			0
$123$	1537.65	−	19840.7i	0.101636	−	1.31144i
$124$		0			0
$125$	−14587.5	+	8422.08i	−0.933598	+	0.539013i
$126$		0			0
$127$	−8361.83	+	14483.1i	−0.518434	+	0.897954i	0.481336	+	0.876536i	$0.340152\pi$
$127$	−8361.83	+	14483.1i	−0.518434	+	0.897954i	−0.999771	+	0.0214185i	$0.993182\pi$
$128$		0			0
$129$	25391.0	−	18157.0i	1.52581	−	1.09110i
$130$		0			0
$131$	−730.630	+	2007.39i	−0.0425750	+	0.116974i	−0.959158	−	0.282870i	$-0.908713\pi$
$131$	−730.630	+	2007.39i	−0.0425750	+	0.116974i	0.916583	+	0.399844i	$0.130936\pi$
$132$		0			0
$133$	1794.05	−	10174.6i	0.101422	−	0.575192i
$134$		0			0
$135$	−16039.4	−	923.310i	−0.880078	−	0.0506617i
$136$		0			0
$137$	12532.9	+	2209.88i	0.667743	+	0.117741i	0.497236	−	0.867615i	$-0.334348\pi$
$137$	12532.9	+	2209.88i	0.667743	+	0.117741i	0.170507	+	0.985356i	$0.445459\pi$
$138$		0			0
$139$	−11446.2	−	4166.09i	−0.592425	−	0.215625i	0.0283707	−	0.999597i	$-0.490968\pi$
$139$	−11446.2	−	4166.09i	−0.592425	−	0.215625i	−0.620796	+	0.783972i	$0.713190\pi$
$140$		0			0
$141$	5889.28	−	574.177i	0.296227	−	0.0288807i
$142$		0			0
$143$	39504.4	+	22807.9i	1.93185	+	1.11536i
$144$		0			0
$145$	4910.22	+	8504.75i	0.233542	+	0.404507i
$146$		0			0
$147$	−3695.05	−	7723.07i	−0.170996	−	0.357400i
$148$		0			0
$149$	−5612.04	+	989.554i	−0.252783	+	0.0445725i	−0.298604	−	0.954377i	$-0.596521\pi$
$149$	−5612.04	+	989.554i	−0.252783	+	0.0445725i	0.0458206	+	0.998950i	$0.485410\pi$
$150$		0			0
$151$	13097.9	+	10990.5i	0.574445	+	0.482017i	0.883118	−	0.469152i	$-0.155440\pi$
$151$	13097.9	+	10990.5i	0.574445	+	0.482017i	−0.308672	+	0.951168i	$0.599885\pi$
$152$		0			0
$153$	−46031.6	+	913.534i	−1.96640	+	0.0390249i
$154$		0			0
$155$	−9507.81	−	26122.5i	−0.395746	−	1.08730i
$156$		0			0
$157$	−24837.2	+	20840.9i	−1.00763	+	0.845505i	−0.988024	−	0.154303i	$-0.950687\pi$
$157$	−24837.2	+	20840.9i	−1.00763	+	0.845505i	−0.0196102	+	0.999808i	$0.506243\pi$
$158$		0			0
$159$	−37499.1	−	9650.13i	−1.48329	−	0.381715i
$160$		0			0
$161$		−	3457.50i		−	0.133386i
$162$		0			0
$163$	35016.5			1.31794			0.658972	−	0.752167i	$-0.270991\pi$
$163$	35016.5			1.31794			0.658972	+	0.752167i	$0.270991\pi$
$164$		0			0
$165$	−7872.43	+	30591.2i	−0.289162	+	1.12364i
$166$		0			0
$167$	13360.2	+	15922.1i	0.479050	+	0.570910i	0.950398	−	0.311038i	$-0.100677\pi$
$167$	13360.2	+	15922.1i	0.479050	+	0.570910i	−0.471347	+	0.881948i	$0.656232\pi$
$168$		0			0
$169$	−50255.1	+	18291.4i	−1.75957	+	0.640431i
$170$		0			0
$171$	−18812.6	+	11365.0i	−0.643362	+	0.388667i
$172$		0			0
$173$	11971.0	−	14266.5i	0.399979	−	0.476677i	−0.528034	−	0.849223i	$-0.677071\pi$
$173$	11971.0	−	14266.5i	0.399979	−	0.476677i	0.928014	+	0.372546i	$0.121515\pi$
$174$		0			0
$175$	−921.079	−	5223.70i	−0.0300760	−	0.170570i
$176$		0			0
$177$	19238.4	−	9204.50i	0.614077	−	0.293801i
$178$		0			0
$179$	21872.3	−	12628.0i	0.682634	−	0.394119i	−0.118213	−	0.992988i	$-0.537717\pi$
$179$	21872.3	−	12628.0i	0.682634	−	0.394119i	0.800847	+	0.598870i	$0.204383\pi$
$180$		0			0
$181$	12185.2	−	21105.4i	0.371942	−	0.644223i	−0.617922	−	0.786240i	$-0.712025\pi$
$181$	12185.2	−	21105.4i	0.371942	−	0.644223i	0.989864	+	0.142016i	$0.0453585\pi$
$182$		0			0
$183$	2448.25	+	25111.5i	0.0731060	+	0.749842i
$184$		0			0
$185$	−7880.61	+	21651.8i	−0.230259	+	0.632631i
$186$		0			0
$187$	−15719.0	+	89147.0i	−0.449513	+	2.54931i
$188$		0			0
$189$	10976.7	−	25494.2i	0.307290	−	0.713704i
$190$		0			0
$191$	11663.4	+	2056.58i	0.319712	+	0.0563739i	0.331201	−	0.943560i	$-0.392546\pi$
$191$	11663.4	+	2056.58i	0.319712	+	0.0563739i	−0.0114892	+	0.999934i	$0.503657\pi$
$192$		0			0
$193$	11024.1	+	4012.45i	0.295957	+	0.107720i	0.485731	−	0.874108i	$-0.338553\pi$
$193$	11024.1	+	4012.45i	0.295957	+	0.107720i	−0.189774	+	0.981828i	$0.560776\pi$
$194$		0			0
$195$	−33046.0	−	46211.9i	−0.869060	−	1.21530i
$196$		0			0
$197$	17211.6	+	9937.10i	0.443494	+	0.256051i	0.705079	−	0.709129i	$-0.250912\pi$
$197$	17211.6	+	9937.10i	0.443494	+	0.256051i	−0.261584	+	0.965181i	$0.584245\pi$
$198$		0			0
$199$	3015.26	+	5222.58i	0.0761410	+	0.131880i	0.901582	−	0.432608i	$-0.142407\pi$
$199$	3015.26	+	5222.58i	0.0761410	+	0.131880i	−0.825441	+	0.564489i	$0.809073\pi$
$200$		0			0
$201$	34725.4	+	2691.20i	0.859519	+	0.0666123i
$202$		0			0
$203$	−16708.8	+	2946.21i	−0.405465	+	0.0714944i
$204$		0			0
$205$	37329.2	+	31322.9i	0.888262	+	0.745340i
$206$		0			0
$207$	−5539.62	+	4838.81i	−0.129282	+	0.112927i
$208$		0			0
$209$	14780.0	+	40607.6i	0.338361	+	0.929639i
$210$		0			0
$211$	−46442.0	+	38969.5i	−1.04315	+	0.875305i	−0.992357	−	0.123404i	$-0.960619\pi$
$211$	−46442.0	+	38969.5i	−1.04315	+	0.875305i	−0.0507916	+	0.998709i	$0.516174\pi$
$212$		0			0
$213$	−6188.73	−	22212.7i	−0.136409	−	0.489602i
$214$		0			0
$215$			76436.6i			1.65358i
$216$		0			0
$217$	48027.7			1.01993
$218$		0			0
$219$	28505.4	+	27945.3i	0.594346	+	0.582667i
$220$		0			0
$221$	−104650.	−	124717.i	−2.14267	−	2.55354i
$222$		0			0
$223$	−33001.8	+	12011.7i	−0.663633	+	0.241543i	−0.651804	−	0.758387i	$-0.725988\pi$
$223$	−33001.8	+	12011.7i	−0.663633	+	0.241543i	−0.0118291	+	0.999930i	$0.503765\pi$
$224$		0			0
$225$	−7080.37	+	8786.38i	−0.139859	+	0.173558i
$226$		0			0
$227$	58153.6	−	69304.8i	1.12856	−	1.34497i	0.197415	−	0.980320i	$-0.436745\pi$
$227$	58153.6	−	69304.8i	1.12856	−	1.34497i	0.931146	−	0.364647i	$-0.118810\pi$
$228$		0			0
$229$	−8477.62	−	48079.0i	−0.161660	−	0.916820i	−0.952442	−	0.304721i	$-0.901437\pi$
$229$	−8477.62	−	48079.0i	−0.161660	−	0.916820i	0.790781	−	0.612099i	$-0.209674\pi$
$230$		0			0
$231$	−45012.6	−	30857.2i	−0.843548	−	0.578272i
$232$		0			0
$233$	79883.9	−	46121.0i	1.47146	−	0.849546i	0.471971	−	0.881614i	$-0.343543\pi$
$233$	79883.9	−	46121.0i	1.47146	−	0.849546i	0.999486	+	0.0320679i	$0.0102093\pi$
$234$		0			0
$235$	−7244.75	+	12548.3i	−0.131186	+	0.227221i
$236$		0			0
$237$	−46481.1	−	21115.6i	−0.827522	−	0.375930i
$238$		0			0
$239$	11892.0	−	32673.0i	0.208189	−	0.571996i	−0.791018	−	0.611792i	$-0.790449\pi$
$239$	11892.0	−	32673.0i	0.208189	−	0.571996i	0.999208	+	0.0397967i	$0.0126710\pi$
$240$		0			0
$241$	−3822.86	+	21680.5i	−0.0658194	+	0.373281i	0.934050	+	0.357141i	$0.116249\pi$
$241$	−3822.86	+	21680.5i	−0.0658194	+	0.373281i	−0.999870	+	0.0161391i	$0.994863\pi$
$242$		0			0
$243$	−56208.9	+	18092.5i	−0.951903	+	0.306399i
$244$		0			0
$245$	20646.1	+	3640.46i	0.343958	+	0.0606491i
$246$		0			0
$247$	−73033.9	−	26582.2i	−1.19710	−	0.435709i
$248$		0			0
$249$	−12502.7	+	27521.9i	−0.201654	+	0.443894i
$250$		0			0
$251$	18993.5	+	10965.9i	0.301480	+	0.174060i	0.643108	−	0.765776i	$-0.277645\pi$
$251$	18993.5	+	10965.9i	0.301480	+	0.174060i	−0.341628	+	0.939835i	$0.610978\pi$
$252$		0			0
$253$	7230.83	+	12524.2i	0.112966	+	0.195663i
$254$		0			0
$255$	63745.6	−	92988.2i	0.980324	−	1.43004i
$256$		0			0
$257$	−3220.03	+	567.778i	−0.0487522	+	0.00859632i	−0.197971	−	0.980208i	$-0.563435\pi$
$257$	−3220.03	+	567.778i	−0.0487522	+	0.00859632i	0.149219	+	0.988804i	$0.452324\pi$
$258$		0			0
$259$	−30494.8	−	25588.2i	−0.454596	−	0.381452i
$260$		0			0
$261$	28104.6	+	22647.7i	0.412569	+	0.332462i
$262$		0			0
$263$	−8965.93	−	24633.7i	−0.129624	−	0.356138i	0.857855	−	0.513892i	$-0.171797\pi$
$263$	−8965.93	−	24633.7i	−0.129624	−	0.356138i	−0.987478	+	0.157754i	$0.949575\pi$
$264$		0			0
$265$	72633.3	−	60946.6i	1.03429	−	0.867876i
$266$		0			0
$267$	17419.2	−	17768.3i	0.244346	−	0.249243i
$268$		0			0
$269$			107301.i			1.48286i	0.671030	+	0.741430i	$0.265852\pi$
$269$			107301.i			1.48286i	−0.671030	+	0.741430i	$0.734148\pi$
$270$		0			0
$271$	17327.7			0.235940			0.117970	−	0.993017i	$-0.462361\pi$
$271$	17327.7			0.235940			0.117970	+	0.993017i	$0.462361\pi$
$272$		0			0
$273$	94551.3	−	26343.1i	1.26865	−	0.353461i
$274$		0			0
$275$	14261.0	+	16995.6i	0.188575	+	0.224735i
$276$		0			0
$277$	95578.4	−	34787.7i	1.24566	−	0.453384i	0.366729	−	0.930328i	$-0.380478\pi$
$277$	95578.4	−	34787.7i	1.24566	−	0.453384i	0.878934	+	0.476944i	$0.158256\pi$
$278$		0			0
$279$	−67215.3	−	76950.2i	−0.863495	−	0.988556i
$280$		0			0
$281$	53939.1	−	64282.1i	0.683111	−	0.814100i	−0.307393	−	0.951583i	$-0.599457\pi$
$281$	53939.1	−	64282.1i	0.683111	−	0.814100i	0.990504	+	0.137483i	$0.0439012\pi$
$282$		0			0
$283$	11342.7	+	64327.8i	0.141626	+	0.803203i	0.970014	+	0.243048i	$0.0781474\pi$
$283$	11342.7	+	64327.8i	0.141626	+	0.803203i	−0.828388	+	0.560155i	$0.810742\pi$
$284$		0			0
$285$	4158.56	−	53659.2i	0.0511980	−	0.660624i
$286$		0			0
$287$	−72910.3	+	42094.8i	−0.885166	+	0.511051i
$288$		0			0
$289$	119780.	−	207466.i	1.43414	−	2.48399i
$290$		0			0
$291$	−53791.9	+	38466.4i	−0.635229	+	0.454251i
$292$		0			0
$293$	−58005.3	+	159368.i	−0.675667	+	1.85638i	−0.191250	+	0.981541i	$0.561254\pi$
$293$	−58005.3	+	159368.i	−0.675667	+	1.85638i	−0.484417	+	0.874837i	$0.660968\pi$
$294$		0			0
$295$	−9068.52	+	51430.1i	−0.104206	+	0.590981i
$296$		0			0
$297$	13556.2	+	115304.i	0.153682	+	1.30717i
$298$		0			0
$299$	−25614.6	−	4516.54i	−0.286513	−	0.0505200i
$300$		0			0
$301$	−124094.	−	45166.5i	−1.36967	−	0.498521i
$302$		0			0
$303$	97035.0	−	9460.45i	1.05692	−	0.103045i
$304$		0			0
$305$	−53504.9	−	30891.1i	−0.575167	−	0.332073i
$306$		0			0
$307$	25634.2	+	44399.7i	0.271984	+	0.471090i	0.969370	−	0.245606i	$-0.0789870\pi$
$307$	25634.2	+	44399.7i	0.271984	+	0.471090i	−0.697386	+	0.716696i	$0.745654\pi$
$308$		0			0
$309$	61969.5	+	129523.i	0.649024	+	1.35653i
$310$		0			0
$311$	64464.1	−	11366.8i	0.666495	−	0.117521i	0.169844	−	0.985471i	$-0.445674\pi$
$311$	64464.1	−	11366.8i	0.666495	−	0.117521i	0.496652	+	0.867950i	$0.334563\pi$
$312$		0			0
$313$	−52415.3	−	43981.7i	−0.535019	−	0.448935i	0.334811	−	0.942285i	$-0.391327\pi$
$313$	−52415.3	−	43981.7i	−0.535019	−	0.448935i	−0.869830	+	0.493351i	$0.835772\pi$
$314$		0			0
$315$	35145.4	+	58176.4i	0.354199	+	0.586308i
$316$		0			0
$317$	−43526.9	−	119589.i	−0.433150	−	1.19007i	−0.943868	−	0.330323i	$-0.892842\pi$
$317$	−43526.9	−	119589.i	−0.433150	−	1.19007i	0.510718	−	0.859749i	$-0.329380\pi$
$318$		0			0
$319$	54363.1	−	45616.1i	0.534224	−	0.448267i
$320$		0			0
$321$	−21102.0	−	5430.46i	−0.204792	−	0.0527019i
$322$		0			0
$323$		−	154233.i		−	1.47834i
$324$		0			0
$325$	−39902.5			−0.377775
$326$		0			0
$327$	−30852.2	+	119887.i	−0.288530	+	1.12119i
$328$		0			0
$329$	−16091.0	−	19176.6i	−0.148659	−	0.177165i
$330$		0			0
$331$	−48971.0	+	17824.0i	−0.446975	+	0.162685i	−0.555694	−	0.831387i	$-0.687547\pi$
$331$	−48971.0	+	17824.0i	−0.446975	+	0.162685i	0.108719	+	0.994073i	$0.465325\pi$
$332$		0			0
$333$	1680.34	+	84669.8i	0.0151534	+	0.763554i
$334$		0			0
$335$	−54821.7	+	65333.9i	−0.488498	+	0.582169i
$336$		0			0
$337$	−33649.8	−	190837.i	−0.296293	−	1.68036i	−0.661898	−	0.749594i	$-0.730249\pi$
$337$	−33649.8	−	190837.i	−0.296293	−	1.68036i	0.365605	−	0.930770i	$-0.380862\pi$
$338$		0			0
$339$	100306.	−	47990.5i	0.872822	−	0.417596i
$340$		0			0
$341$	−173972.	+	100443.i	−1.49613	+	0.863792i
$342$		0			0
$343$	−63819.4	+	110538.i	−0.542456	+	0.939561i
$344$		0			0
$345$	−1747.71	−	17926.1i	−0.0146836	−	0.150608i
$346$		0			0
$347$	−71892.7	+	197524.i	−0.597071	+	1.64044i	0.160008	+	0.987116i	$0.448848\pi$
$347$	−71892.7	+	197524.i	−0.597071	+	1.64044i	−0.757079	+	0.653324i	$0.773374\pi$
$348$		0			0
$349$	−9455.56	+	53625.2i	−0.0776312	+	0.440269i	0.921074	+	0.389389i	$0.127314\pi$
$349$	−9455.56	+	53625.2i	−0.0776312	+	0.440269i	−0.998705	+	0.0508800i	$0.983797\pi$
$350$		0			0
$351$	−174533.	−	114623.i	−1.41665	−	0.930374i
$352$		0			0
$353$	−157085.	−	27698.4i	−1.26062	−	0.222282i	−0.496891	−	0.867813i	$-0.665525\pi$
$353$	−157085.	−	27698.4i	−1.26062	−	0.222282i	−0.763734	+	0.645531i	$0.776636\pi$
$354$		0			0
$355$	53058.9	+	19311.9i	0.421019	+	0.153238i
$356$		0			0
$357$	113298.	+	158437.i	0.888967	+	1.24314i
$358$		0			0
$359$	126427.	+	72992.7i	0.980960	+	0.566358i	0.902560	−	0.430564i	$-0.141685\pi$
$359$	126427.	+	72992.7i	0.980960	+	0.566358i	0.0784002	+	0.996922i	$0.475019\pi$
$360$		0			0
$361$	28346.3	+	49097.3i	0.217512	+	0.376741i
$362$		0			0
$363$	96208.0	+	7456.07i	0.730126	+	0.0565844i
$364$		0			0
$365$	−96264.0	+	16973.9i	−0.722567	+	0.127408i
$366$		0			0
$367$	90808.8	+	76197.6i	0.674211	+	0.565730i	0.914308	−	0.405019i	$-0.132735\pi$
$367$	90808.8	+	76197.6i	0.674211	+	0.565730i	−0.240097	+	0.970749i	$0.577179\pi$
$368$		0			0
$369$	169483.	+	57905.0i	1.24473	+	0.425269i
$370$		0			0
$371$	56026.9	+	153933.i	0.407051	+	1.11836i
$372$		0			0
$373$	−82341.3	+	69092.6i	−0.591834	+	0.496608i	−0.888809	−	0.458277i	$-0.848467\pi$
$373$	−82341.3	+	69092.6i	−0.591834	+	0.496608i	0.296975	+	0.954885i	$0.404022\pi$
$374$		0			0
$375$	−40687.2	−	146035.i	−0.289331	−	1.03847i
$376$		0			0
$377$			127634.i			0.898018i
$378$		0			0
$379$	−184493.			−1.28440			−0.642200	−	0.766537i	$-0.721978\pi$
$379$	−184493.			−1.28440			−0.642200	+	0.766537i	$0.721978\pi$
$380$		0			0
$381$	−107479.	−	105368.i	−0.740415	−	0.725866i
$382$		0			0
$383$	−141634.	−	168793.i	−0.965540	−	1.15069i	−0.988541	−	0.150950i	$-0.951767\pi$
$383$	−141634.	−	168793.i	−0.965540	−	1.15069i	0.0230014	−	0.999735i	$-0.492678\pi$
$384$		0			0
$385$	125576.	−	45705.9i	0.847198	−	0.308355i
$386$		0			0
$387$	101305.	+	262035.i	0.676407	+	1.74959i
$388$		0			0
$389$	−76519.9	+	91192.9i	−0.505680	+	0.602645i	−0.957133	−	0.289649i	$-0.906461\pi$
$389$	−76519.9	+	91192.9i	−0.505680	+	0.602645i	0.451453	+	0.892295i	$0.350906\pi$
$390$		0			0
$391$	−8962.84	−	50830.8i	−0.0586263	−	0.332486i
$392$		0			0
$393$	−15857.6	−	10870.8i	−0.102672	−	0.0703842i
$394$		0			0
$395$	108264.	−	62506.3i	0.693889	−	0.400617i
$396$		0			0
$397$	−4558.39	+	7895.36i	−0.0289221	+	0.0500946i	−0.880124	−	0.474744i	$-0.842541\pi$
$397$	−4558.39	+	7895.36i	−0.0289221	+	0.0500946i	0.851202	+	0.524838i	$0.175874\pi$
$398$		0			0
$399$	84657.7	+	38458.6i	0.531766	+	0.241573i
$400$		0			0
$401$	38160.2	−	104844.i	0.237313	−	0.652013i	−0.762673	−	0.646784i	$-0.776113\pi$
$401$	38160.2	−	104844.i	0.237313	−	0.652013i	0.999986	−	0.00522863i	$-0.00166433\pi$
$402$		0			0
$403$	62738.8	−	355809.i	0.386301	−	2.19082i
$404$		0			0
$405$	44024.1	−	137729.i	0.268399	−	0.839682i
$406$		0			0
$407$	163976.	+	28913.3i	0.989898	+	0.174546i
$408$		0			0
$409$	206181.	+	75043.7i	1.23254	+	0.448609i	0.874468	−	0.485084i	$-0.161211\pi$
$409$	206181.	+	75043.7i	1.23254	+	0.448609i	0.358075	+	0.933693i	$0.383433\pi$
$410$		0			0
$411$	−47372.7	+	104280.i	−0.280443	+	0.617329i
$412$		0			0
$413$	−78137.6	−	45112.8i	−0.458100	−	0.264484i
$414$		0			0
$415$	−37010.6	−	64104.2i	−0.214897	−	0.372212i
$416$		0			0
$417$	61985.7	−	90421.1i	0.356467	−	0.519993i
$418$		0			0
$419$	163579.	−	28843.3i	0.931749	−	0.164292i	0.312885	−	0.949791i	$-0.398704\pi$
$419$	163579.	−	28843.3i	0.931749	−	0.164292i	0.618864	+	0.785499i	$0.287593\pi$
$420$		0			0
$421$	−49836.8	−	41818.0i	−0.281181	−	0.235939i	0.491279	−	0.871002i	$-0.336530\pi$
$421$	−49836.8	−	41818.0i	−0.281181	−	0.235939i	−0.772460	+	0.635063i	$0.780974\pi$
$422$		0			0
$423$	−8205.16	+	52619.0i	−0.0458571	+	0.294077i
$424$		0			0
$425$	−27082.7	−	74409.1i	−0.149939	−	0.411954i
$426$		0			0
$427$	81767.4	−	68611.0i	0.448461	−	0.376303i
$428$		0			0
$429$	−287403.	+	293163.i	−1.56162	+	1.59292i
$430$		0			0
$431$		−	175223.i		−	0.943271i	−0.881793	−	0.471636i	$-0.843664\pi$
$431$		−	175223.i		−	0.943271i	0.881793	−	0.471636i	$-0.156336\pi$
$432$		0			0
$433$	23396.7			0.124790			0.0623950	−	0.998052i	$-0.480126\pi$
$433$	23396.7			0.124790			0.0623950	+	0.998052i	$0.480126\pi$
$434$		0			0
$435$	−85141.2	+	23721.3i	−0.449947	+	0.125361i
$436$		0			0
$437$	−15838.3	−	18875.4i	−0.0829366	−	0.0988400i
$438$		0			0
$439$	−169001.	+	61511.3i	−0.876920	+	0.319173i	−0.740966	−	0.671542i	$-0.765632\pi$
$439$	−169001.	+	61511.3i	−0.876920	+	0.319173i	−0.135954	+	0.990715i	$0.543410\pi$
$440$		0			0
$441$	75602.4	−	14883.2i	0.388739	−	0.0765278i
$442$		0			0
$443$	−92863.7	+	110671.i	−0.473193	+	0.563930i	−0.948861	−	0.315696i	$-0.897762\pi$
$443$	−92863.7	+	110671.i	−0.473193	+	0.563930i	0.475667	+	0.879625i	$0.342207\pi$
$444$		0			0
$445$	10580.4	+	60004.3i	0.0534295	+	0.303014i
$446$		0			0
$447$	3962.87	−	51134.2i	0.0198333	−	0.255915i
$448$		0			0
$449$	−199330.	+	115083.i	−0.988737	+	0.570848i	−0.904896	−	0.425632i	$-0.860052\pi$
$449$	−199330.	+	115083.i	−0.988737	+	0.570848i	−0.0838404	+	0.996479i	$0.526719\pi$
$450$		0			0
$451$	176070.	−	304961.i	0.865628	−	1.49931i
$452$		0			0
$453$	−125172.	+	89510.1i	−0.609972	+	0.436190i
$454$		0			0
$455$	−82203.4	+	225852.i	−0.397070	+	1.09094i
$456$		0			0
$457$	1722.66	−	9769.66i	0.00824833	−	0.0467786i	−0.980406	−	0.196988i	$-0.936884\pi$
$457$	1722.66	−	9769.66i	0.00824833	−	0.0467786i	0.988654	+	0.150210i	$0.0479949\pi$
$458$		0			0
$459$	95286.7	−	403261.i	0.452280	−	1.91408i
$460$		0			0
$461$	226367.	+	39914.6i	1.06515	+	0.187815i	0.678641	−	0.734470i	$-0.262569\pi$
$461$	226367.	+	39914.6i	1.06515	+	0.187815i	0.386511	+	0.922285i	$0.373680\pi$
$462$		0			0
$463$	33865.3	+	12326.0i	0.157977	+	0.0574988i	0.419798	−	0.907618i	$-0.362101\pi$
$463$	33865.3	+	12326.0i	0.157977	+	0.0574988i	−0.261821	+	0.965116i	$0.584323\pi$
$464$		0			0
$465$	249010.	−	24277.3i	1.15162	−	0.112278i
$466$		0			0
$467$	−67234.9	−	38818.1i	−0.308291	−	0.177992i	0.337871	−	0.941193i	$-0.390293\pi$
$467$	−67234.9	−	38818.1i	−0.308291	−	0.177992i	−0.646161	+	0.763201i	$0.723627\pi$
$468$		0			0
$469$	−73674.7	−	127608.i	−0.334944	−	0.580140i
$470$		0			0
$471$	−125939.	−	263227.i	−0.567701	−	1.18656i
$472$		0			0
$473$	543967.	−	95916.0i	2.43136	−	0.428715i
$474$		0			0
$475$	−28957.5	−	24298.2i	−0.128343	−	0.107693i
$476$		0			0
$477$	168221.	−	305197.i	0.739340	−	1.34136i
$478$		0			0
$479$	−68424.0	−	187994.i	−0.298221	−	0.819355i	−0.994797	−	0.101872i	$-0.967517\pi$
$479$	−68424.0	−	187994.i	−0.298221	−	0.819355i	0.696577	−	0.717482i	$-0.254706\pi$
$480$		0			0
$481$	−229403.	+	192492.i	−0.991537	+	0.831999i
$482$		0			0
$483$	30135.6	+	7755.19i	0.129177	+	0.0332428i
$484$		0			0
$485$		−	161934.i		−	0.688422i
$486$		0			0
$487$	156064.			0.658027			0.329013	−	0.944325i	$-0.393284\pi$
$487$	156064.			0.658027			0.329013	+	0.944325i	$0.393284\pi$
$488$		0			0
$489$	−78542.1	+	305204.i	−0.328462	+	1.27636i
$490$		0			0
$491$	−128201.	−	152784.i	−0.531775	−	0.633745i	0.431547	−	0.902090i	$-0.357968\pi$
$491$	−128201.	−	152784.i	−0.531775	−	0.633745i	−0.963323	+	0.268345i	$0.913523\pi$
$492$		0			0
$493$	−238009.	+	86628.2i	−0.979264	+	0.356423i
$494$		0			0
$495$	−248975.	−	137232.i	−1.01612	−	0.560075i
$496$		0			0
$497$	−62705.2	+	74729.1i	−0.253858	+	0.302536i
$498$		0			0
$499$	82436.2	+	467519.i	0.331068	+	1.87758i	0.463059	+	0.886328i	$0.346752\pi$
$499$	82436.2	+	467519.i	0.331068	+	1.87758i	−0.131991	+	0.991251i	$0.542137\pi$
$500$		0			0
$501$	−168744.	+	80734.6i	−0.672286	+	0.321651i
$502$		0			0
$503$	390878.	−	225674.i	1.54492	−	0.891959i	0.546402	−	0.837523i	$-0.315997\pi$
$503$	390878.	−	225674.i	1.54492	−	0.891959i	0.998517	−	0.0544361i	$-0.0173361\pi$
$504$		0			0
$505$	−119368.	+	206752.i	−0.468065	+	0.810713i
$506$		0			0
$507$	−46705.2	−	479052.i	−0.181698	−	1.86366i
$508$		0			0
$509$	28313.2	−	77789.8i	0.109283	−	0.300253i	−0.872981	−	0.487754i	$-0.837816\pi$
$509$	28313.2	−	77789.8i	0.109283	−	0.300253i	0.982264	+	0.187501i	$0.0600387\pi$
$510$		0			0
$511$	29325.5	−	166313.i	0.112306	−	0.636921i
$512$		0			0
$513$	−56860.9	−	189462.i	−0.216062	−	0.719926i
$514$		0			0
$515$	−346254.	−	61054.0i	−1.30551	−	0.230197i
$516$		0			0
$517$	98391.8	+	35811.7i	0.368110	+	0.133981i
$518$		0			0
$519$	97495.6	+	136339.i	0.361952	+	0.506157i
$520$		0			0
$521$	−184540.	−	106544.i	−0.679852	−	0.392513i	0.119947	−	0.992780i	$-0.461727\pi$
$521$	−184540.	−	106544.i	−0.679852	−	0.392513i	−0.799799	+	0.600267i	$0.795061\pi$
$522$		0			0
$523$	182766.	+	316561.i	0.668179	+	1.15732i	0.978413	+	0.206660i	$0.0662595\pi$
$523$	182766.	+	316561.i	0.668179	+	1.15732i	−0.310233	+	0.950660i	$0.600407\pi$
$524$		0			0
$525$	47595.8	+	3688.65i	0.172683	+	0.0133829i
$526$		0			0
$527$	706085.	−	124502.i	2.54235	−	0.448285i
$528$		0			0
$529$	208054.	+	174578.i	0.743472	+	0.623847i
$530$		0			0
$531$	37074.6	+	188328.i	0.131488	+	0.667923i
$532$		0			0
$533$	216612.	+	595138.i	0.762481	+	2.09490i
$534$		0			0
$535$	40873.3	−	34296.7i	0.142801	−	0.119824i
$536$		0			0
$537$	61005.8	+	218964.i	0.211555	+	0.759317i
$538$		0			0
$539$		−	151498.i		−	0.521469i
$540$		0			0
$541$	−329733.			−1.12660			−0.563298	−	0.826254i	$-0.690468\pi$
$541$	−329733.			−1.12660			−0.563298	+	0.826254i	$0.690468\pi$
$542$		0			0
$543$	156623.	+	153546.i	0.531199	+	0.520761i
$544$		0			0
$545$	−194851.	−	232214.i	−0.656008	−	0.781800i
$546$		0			0
$547$	432146.	−	157288.i	1.44429	−	0.525680i	0.503302	−	0.864111i	$-0.332118\pi$
$547$	432146.	−	157288.i	1.44429	−	0.525680i	0.940991	+	0.338431i	$0.109896\pi$
$548$		0			0
$549$	−224363.	−	34986.2i	−0.744401	−	0.116079i
$550$		0			0
$551$	−77721.6	+	92625.0i	−0.255999	+	0.305088i
$552$		0			0
$553$	37504.8	+	212700.i	0.122641	+	0.695534i
$554$		0			0
$555$	−171041.	−	117253.i	−0.555283	−	0.380660i
$556$		0			0
$557$	390948.	−	225714.i	1.26011	−	0.727524i	0.287014	−	0.957926i	$-0.407337\pi$
$557$	390948.	−	225714.i	1.26011	−	0.727524i	0.973096	+	0.230402i	$0.0740041\pi$
$558$		0			0
$559$	−496716.	+	860338.i	−1.58959	+	2.75325i
$560$		0			0
$561$	−741748.	−	336964.i	−2.35684	−	1.07068i
$562$		0			0
$563$	−160569.	+	441159.i	−0.506576	+	1.39180i	0.378172	+	0.925735i	$0.376553\pi$
$563$	−160569.	+	441159.i	−0.506576	+	1.39180i	−0.884748	+	0.466070i	$0.845670\pi$
$564$		0			0
$565$	−47281.6	+	268147.i	−0.148114	+	0.839994i
$566$		0			0
$567$	197587.	+	152857.i	0.614600	+	0.475465i
$568$		0			0
$569$	153331.	+	27036.4i	0.473592	+	0.0835071i	0.405350	−	0.914162i	$-0.367150\pi$
$569$	153331.	+	27036.4i	0.473592	+	0.0835071i	0.0682426	+	0.997669i	$0.478261\pi$
$570$		0			0
$571$	71327.1	+	25960.9i	0.218767	+	0.0796247i	0.449079	−	0.893492i	$-0.351752\pi$
$571$	71327.1	+	25960.9i	0.218767	+	0.0796247i	−0.230312	+	0.973117i	$0.573975\pi$
$572$		0			0
$573$	−44086.3	+	97045.6i	−0.134275	+	0.295574i
$574$		0			0
$575$	−10955.5	−	6325.19i	−0.0331359	−	0.0191310i
$576$		0			0
$577$	−216466.	−	374930.i	−0.650186	−	1.12615i	−0.983078	−	0.183190i	$-0.941358\pi$
$577$	−216466.	−	374930.i	−0.650186	−	1.12615i	0.332892	−	0.942965i	$-0.391976\pi$
$578$		0			0
$579$	−59699.7	+	87086.4i	−0.178080	+	0.259772i
$580$		0			0
$581$	125942.	−	22207.0i	0.373094	−	0.0657865i
$582$		0			0
$583$	−524875.	−	440422.i	−1.54425	−	1.29578i
$584$		0			0
$585$	476906.	−	184376.i	1.39354	−	0.538756i
$586$		0			0
$587$	−42558.1	−	116928.i	−0.123511	−	0.339344i	0.862492	−	0.506071i	$-0.168903\pi$
$587$	−42558.1	−	116928.i	−0.123511	−	0.339344i	−0.986003	+	0.166726i	$0.946680\pi$
$588$		0			0
$589$	262196.	−	220008.i	0.755780	−	0.634174i
$590$		0			0
$591$	−125218.	+	127727.i	−0.358501	+	0.365686i
$592$		0			0
$593$			257653.i			0.732699i	0.930477	+	0.366349i	$0.119393\pi$
$593$			257653.i			0.732699i	−0.930477	+	0.366349i	$0.880607\pi$
$594$		0			0
$595$	−476956.			−1.34724
$596$		0			0
$597$	−52283.4	+	14566.8i	−0.146695	+	0.0408709i
$598$		0			0
$599$	194068.	+	231281.i	0.540880	+	0.644595i	0.965385	−	0.260830i	$-0.0839963\pi$
$599$	194068.	+	231281.i	0.540880	+	0.644595i	−0.424505	+	0.905426i	$0.639552\pi$
$600$		0			0
$601$	224736.	−	81797.4i	0.622192	−	0.226459i	−0.0116373	−	0.999932i	$-0.503704\pi$
$601$	224736.	−	81797.4i	0.622192	−	0.226459i	0.633829	+	0.773473i	$0.281482\pi$
$602$		0			0
$603$	−101346.	+	296631.i	−0.278722	+	0.815797i
$604$		0			0
$605$	−151885.	+	181010.i	−0.414959	+	0.494529i
$606$		0			0
$607$	87821.0	+	498057.i	0.238353	+	1.35177i	0.835436	+	0.549588i	$0.185215\pi$
$607$	87821.0	+	498057.i	0.238353	+	1.35177i	−0.597083	+	0.802180i	$0.703674\pi$
$608$		0			0
$609$	11798.7	−	152243.i	0.0318127	−	0.410489i
$610$		0			0
$611$	−163088.	+	94158.7i	−0.436856	+	0.252219i
$612$		0			0
$613$	−215234.	+	372796.i	−0.572783	+	0.992090i	0.423495	+	0.905898i	$0.360803\pi$
$613$	−215234.	+	372796.i	−0.572783	+	0.992090i	−0.996279	+	0.0861913i	$0.972530\pi$
$614$		0			0
$615$	−356741.	+	255104.i	−0.943197	+	0.674478i
$616$		0			0
$617$	−22270.7	+	61188.3i	−0.0585010	+	0.160730i	−0.965500	−	0.260402i	$-0.916145\pi$
$617$	−22270.7	+	61188.3i	−0.0585010	+	0.160730i	0.906999	+	0.421132i	$0.138367\pi$
$618$		0			0
$619$	45353.3	−	257211.i	0.118366	−	0.671287i	−0.866662	−	0.498895i	$-0.833739\pi$
$619$	45353.3	−	257211.i	0.118366	−	0.671287i	0.985028	−	0.172392i	$-0.0551497\pi$
$620$		0			0
$621$	−29749.7	−	59136.9i	−0.0771436	−	0.153347i
$622$		0			0
$623$	−103668.	−	18279.5i	−0.267098	−	0.0470965i
$624$		0			0
$625$	267012.	+	97184.5i	0.683551	+	0.248792i
$626$		0			0
$627$	−387088.	+	37739.2i	−0.984633	+	0.0959970i
$628$		0			0
$629$	−514655.	−	297136.i	−1.30081	−	0.751024i
$630$		0			0
$631$	14823.3	+	25674.8i	0.0372295	+	0.0644834i	0.884040	−	0.467412i	$-0.154813\pi$
$631$	14823.3	+	25674.8i	0.0372295	+	0.0644834i	−0.846810	+	0.531895i	$0.821480\pi$
$632$		0			0
$633$	−235489.	−	492198.i	−0.587709	−	1.22838i
$634$		0			0
$635$	362963.	−	64000.1i	0.900149	−	0.158721i
$636$		0			0
$637$	208726.	+	175142.i	0.514397	+	0.431630i
$638$		0			0
$639$	207488.	−	4117.77i	0.508149	−	0.0100846i
$640$		0			0
$641$	−19746.3	−	54252.5i	−0.0480584	−	0.132040i	0.913341	−	0.407195i	$-0.133493\pi$
$641$	−19746.3	−	54252.5i	−0.0480584	−	0.132040i	−0.961400	+	0.275155i	$0.911271\pi$
$642$		0			0
$643$	414021.	−	347405.i	1.00138	−	0.840261i	0.0142090	−	0.999899i	$-0.495477\pi$
$643$	414021.	−	347405.i	1.00138	−	0.840261i	0.987176	+	0.159638i	$0.0510326\pi$
$644$		0			0
$645$	−666222.	−	171448.i	−1.60140	−	0.412109i
$646$		0			0
$647$		−	235395.i		−	0.562326i	−0.959660	−	0.281163i	$-0.909280\pi$
$647$		−	235395.i		−	0.562326i	0.959660	−	0.281163i	$-0.0907202\pi$
$648$		0			0
$649$	377386.			0.895976
$650$		0			0
$651$	−107726.	+	418610.i	−0.254191	+	0.987752i
$652$		0			0
$653$	79303.1	+	94509.8i	0.185979	+	0.221641i	0.850976	−	0.525205i	$-0.176011\pi$
$653$	79303.1	+	94509.8i	0.185979	+	0.221641i	−0.664997	+	0.746846i	$0.731567\pi$
$654$		0			0
$655$	44239.6	−	16101.9i	0.103117	−	0.0375314i
$656$		0			0
$657$	−307510.	+	185772.i	−0.712407	+	0.430378i
$658$		0			0
$659$	222560.	−	265236.i	0.512479	−	0.610749i	−0.446306	−	0.894880i	$-0.647261\pi$
$659$	222560.	−	265236.i	0.512479	−	0.610749i	0.958785	+	0.284132i	$0.0917053\pi$
$660$		0			0
$661$	−16127.3	−	91462.5i	−0.0369113	−	0.209334i	0.960774	−	0.277332i	$-0.0894501\pi$
$661$	−16127.3	−	91462.5i	−0.0369113	−	0.209334i	−0.997685	+	0.0679976i	$0.978339\pi$
$662$		0			0
$663$	1.32177e6	−	632391.i	3.00697	−	1.43866i
$664$		0			0
$665$	−197185.	+	113845.i	−0.445894	+	0.257437i
$666$		0			0
$667$	−20232.1	+	35043.0i	−0.0454767	+	0.0787680i
$668$		0			0
$669$	−30670.7	−	314587.i	−0.0685285	−	0.702891i
$670$		0			0
$671$	−152698.	+	419535.i	−0.339148	+	0.931801i
$672$		0			0
$673$	18591.2	−	105436.i	0.0410465	−	0.232786i	−0.957382	−	0.288824i	$-0.906736\pi$
$673$	18591.2	−	105436.i	0.0410465	−	0.232786i	0.998429	+	0.0560379i	$0.0178468\pi$
$674$		0			0
$675$	−60700.9	−	81420.5i	−0.133226	−	0.178701i
$676$		0			0
$677$	410730.	+	72422.8i	0.896147	+	0.158015i	0.602707	−	0.797963i	$-0.294089\pi$
$677$	410730.	+	72422.8i	0.896147	+	0.158015i	0.293440	+	0.955978i	$0.405200\pi$
$678$		0			0
$679$	262898.	+	95687.0i	0.570227	+	0.207545i
$680$		0			0
$681$	473623.	+	662319.i	1.02126	+	1.42815i
$682$		0			0
$683$	−31295.1	−	18068.2i	−0.0670864	−	0.0387324i	0.466082	−	0.884742i	$-0.345665\pi$
$683$	−31295.1	−	18068.2i	−0.0670864	−	0.0387324i	−0.533168	+	0.846009i	$0.678999\pi$
$684$		0			0
$685$	−140232.	−	242890.i	−0.298860	−	0.517640i
$686$		0			0
$687$	438072.	+	33950.4i	0.928180	+	0.0719335i
$688$		0			0
$689$	1.21359e6	−	213988.i	2.55642	−	0.450766i
$690$		0			0
$691$	−293104.	−	245944.i	−0.613855	−	0.515086i	0.282010	−	0.959411i	$-0.408999\pi$
$691$	−293104.	−	245944.i	−0.613855	−	0.515086i	−0.895865	+	0.444326i	$0.853443\pi$
$692$		0			0
$693$	369915.	−	323117.i	0.770257	−	0.672812i
$694$		0			0
$695$	91813.8	+	252256.i	0.190081	+	0.522243i
$696$		0			0
$697$	−962777.	+	807866.i	−1.98180	+	1.66293i
$698$		0			0
$699$	222811.	+	799719.i	0.456019	+	1.63675i
$700$		0			0
$701$		−	780717.i		−	1.58876i	−0.607423	−	0.794379i	$-0.707797\pi$
$701$		−	780717.i		−	1.58876i	0.607423	−	0.794379i	$-0.292203\pi$
$702$		0			0
$703$	−283695.			−0.574038
$704$		0			0
$705$	−93121.0	−	91291.2i	−0.187357	−	0.183675i
$706$		0			0
$707$	−265125.	−	315963.i	−0.530409	−	0.632117i
$708$		0			0
$709$	341740.	−	124383.i	0.679834	−	0.247439i	0.0210578	−	0.999778i	$-0.493297\pi$
$709$	341740.	−	124383.i	0.679834	−	0.247439i	0.658776	+	0.752339i	$0.271074\pi$
$710$		0			0
$711$	288301.	−	357767.i	0.570305	−	0.707720i
$712$		0			0
$713$	73626.8	−	87745.1i	0.144830	−	0.172601i
$714$		0			0
$715$	−174568.	−	990025.i	−0.341470	−	1.93657i
$716$		0			0
$717$	258104.	+	176936.i	0.502061	+	0.344175i
$718$		0			0
$719$	−437833.	+	252783.i	−0.846936	+	0.488979i	−0.859616	−	0.510941i	$-0.829297\pi$
$719$	−437833.	+	252783.i	−0.846936	+	0.488979i	0.0126796	+	0.999920i	$0.495964\pi$
$720$		0			0
$721$	303723.	−	526063.i	0.584260	−	1.01197i
$722$		0			0
$723$	−180393.	−	81949.6i	−0.345098	−	0.156773i
$724$		0			0
$725$	−21231.8	+	58334.0i	−0.0403935	+	0.110980i
$726$		0			0
$727$	11474.5	−	65075.1i	0.0217102	−	0.123125i	−0.972027	−	0.234871i	$-0.924533\pi$
$727$	11474.5	−	65075.1i	0.0217102	−	0.123125i	0.993737	+	0.111746i	$0.0356443\pi$
$728$		0			0
$729$	−31617.8	−	530500.i	−0.0594945	−	0.998229i
$730$		0			0
$731$	−1.94147e6	−	342333.i	−3.63325	−	0.640639i
$732$		0			0
$733$	−617502.	−	224752.i	−1.14929	−	0.418308i	−0.304031	−	0.952662i	$-0.598333\pi$
$733$	−617502.	−	224752.i	−1.14929	−	0.418308i	−0.845261	+	0.534354i	$0.820555\pi$
$734$		0			0
$735$	−78039.6	+	171786.i	−0.144458	+	0.317990i
$736$		0			0
$737$	533747.	+	308159.i	0.982653	+	0.567335i
$738$		0			0
$739$	8202.93	+	14207.9i	0.0150204	+	0.0260160i	0.873438	−	0.486936i	$-0.161885\pi$
$739$	8202.93	+	14207.9i	0.0150204	+	0.0260160i	−0.858418	+	0.512952i	$0.828552\pi$
$740$		0			0
$741$	395506.	−	576941.i	0.720306	−	1.05074i
$742$		0			0
$743$	−380166.	+	67033.5i	−0.688646	+	0.121427i	−0.507011	−	0.861939i	$-0.669250\pi$
$743$	−380166.	+	67033.5i	−0.688646	+	0.121427i	−0.181634	+	0.983366i	$0.558139\pi$
$744$		0			0
$745$	96206.1	+	80726.5i	0.173336	+	0.145447i
$746$		0			0
$747$	−211837.	−	170706.i	−0.379631	−	0.305920i
$748$		0			0
$749$	31528.3	+	86623.2i	0.0562000	+	0.154408i
$750$		0			0
$751$	252123.	−	211557.i	0.447026	−	0.375100i	−0.391305	−	0.920261i	$-0.627976\pi$
$751$	252123.	−	211557.i	0.447026	−	0.375100i	0.838331	+	0.545162i	$0.183532\pi$
$752$		0			0
$753$	−138182.	+	140951.i	−0.243703	+	0.248587i
$754$		0			0
$755$		−	376815.i		−	0.661050i
$756$		0			0
$757$	1.06844e6			1.86448			0.932240	−	0.361841i	$-0.117852\pi$
$757$	1.06844e6			1.86448			0.932240	+	0.361841i	$0.117852\pi$
$758$		0			0
$759$	−125380.	+	34932.3i	−0.217642	+	0.0606377i
$760$		0			0
$761$	−655835.	−	781594.i	−1.13247	−	1.34962i	−0.928800	−	0.370580i	$-0.879159\pi$
$761$	−655835.	−	781594.i	−1.13247	−	1.34962i	−0.203666	−	0.979040i	$-0.565286\pi$
$762$		0			0
$763$	492135.	−	179122.i	0.845347	−	0.307681i
$764$		0			0
$765$	667505.	+	764181.i	1.14060	+	1.30579i
$766$		0			0
$767$	−436285.	+	519945.i	−0.741617	+	0.883825i
$768$		0			0
$769$	−24137.7	−	136892.i	−0.0408172	−	0.231486i	0.957574	−	0.288188i	$-0.0930527\pi$
$769$	−24137.7	−	136892.i	−0.0408172	−	0.231486i	−0.998391	+	0.0567019i	$0.981942\pi$
$770$		0			0
$771$	2273.79	−	29339.4i	0.00382508	−	0.0493562i
$772$		0			0
$773$	−615069.	+	355110.i	−1.02935	+	0.594298i	−0.916800	−	0.399347i	$-0.869237\pi$
$773$	−615069.	+	355110.i	−1.02935	+	0.594298i	−0.112555	+	0.993646i	$0.535903\pi$
$774$		0			0
$775$	87862.5	−	152182.i	0.146285	−	0.253373i
$776$		0			0
$777$	291427.	−	208399.i	0.482711	−	0.345186i
$778$		0			0
$779$	−205206.	+	563798.i	−0.338154	+	0.929071i
$780$		0			0
$781$	70853.7	−	401831.i	0.116161	−	0.658782i
$782$		0			0
$783$	−260436.	+	194161.i	−0.424794	+	0.316694i
$784$		0			0
$785$	703686.	+	124079.i	1.14193	+	0.201353i
$786$		0			0
$787$	1.10347e6	+	401629.i	1.78160	+	0.648448i	0.999686	+	0.0250576i	$0.00797691\pi$
$787$	1.10347e6	+	401629.i	1.78160	+	0.648448i	0.781911	+	0.623391i	$0.214245\pi$
$788$		0			0
$789$	234818.	−	22893.7i	0.377205	−	0.0367757i
$790$		0			0
$791$	−407395.	−	235210.i	−0.651122	−	0.375926i
$792$		0			0
$793$	−401486.	−	695394.i	−0.638445	−	1.10582i
$794$		0			0
$795$	368295.	+	769777.i	0.582721	+	1.21795i
$796$		0			0
$797$	−47732.3	+	8416.48i	−0.0751442	+	0.0132499i	−0.211094	−	0.977466i	$-0.567703\pi$
$797$	−47732.3	+	8416.48i	−0.0751442	+	0.0132499i	0.135950	+	0.990716i	$0.456591\pi$
$798$		0			0
$799$	−286276.	−	240214.i	−0.448426	−	0.376274i
$800$		0			0
$801$	115797.	+	191680.i	0.180482	+	0.298753i
$802$		0			0
$803$	241593.	+	663771.i	0.374673	+	1.02941i
$804$		0			0
$805$	−58370.7	+	48978.9i	−0.0900748	+	0.0755818i
$806$		0			0
$807$	−935240.	−	240678.i	−1.43607	−	0.369563i
$808$		0			0
$809$			776742.i			1.18681i	0.804906	+	0.593403i	$0.202216\pi$
$809$			776742.i			1.18681i	−0.804906	+	0.593403i	$0.797784\pi$
$810$		0			0
$811$	−630737.			−0.958974			−0.479487	−	0.877549i	$-0.659177\pi$
$811$	−630737.			−0.958974			−0.479487	+	0.877549i	$0.659177\pi$
$812$		0			0
$813$	−38866.1	+	151028.i	−0.0588017	+	0.228495i
$814$		0			0
$815$	−496043.	−	591161.i	−0.746799	−	0.890001i
$816$		0			0
$817$	−884362.	+	321881.i	−1.32491	+	0.482227i
$818$		0			0
$819$	17527.8	+	883199.i	0.0261312	+	1.31671i
$820$		0			0
$821$	304483.	−	362868.i	0.451727	−	0.538348i	−0.491332	−	0.870972i	$-0.663490\pi$
$821$	304483.	−	362868.i	0.451727	−	0.538348i	0.943059	+	0.332625i	$0.107934\pi$
$822$		0			0
$823$	37739.5	+	214031.i	0.0557181	+	0.315993i	0.999910	−	0.0134022i	$-0.00426618\pi$
$823$	37739.5	+	214031.i	0.0557181	+	0.315993i	−0.944192	+	0.329395i	$0.893155\pi$
$824$		0			0
$825$	−180122.	+	86178.0i	−0.264641	+	0.126616i
$826$		0			0
$827$	170500.	−	98438.3i	0.249295	−	0.143931i	−0.370146	−	0.928973i	$-0.620692\pi$
$827$	170500.	−	98438.3i	0.249295	−	0.143931i	0.619441	+	0.785043i	$0.287359\pi$
$828$		0			0
$829$	41377.6	−	71668.1i	0.0602083	−	0.104284i	−0.834350	−	0.551235i	$-0.814157\pi$
$829$	41377.6	−	71668.1i	0.0602083	−	0.104284i	0.894558	+	0.446951i	$0.147490\pi$
$830$		0			0
$831$	88827.1	+	911092.i	0.128630	+	1.31935i
$832$		0			0
$833$	−184933.	+	508100.i	−0.266517	+	0.732249i
$834$		0			0
$835$	79541.9	−	451105.i	0.114084	−	0.647000i
$836$		0			0
$837$	821464.	−	413250.i	1.17257	−	0.589878i
$838$		0			0
$839$	797629.	+	140644.i	1.13312	+	0.199800i	0.708596	−	0.705615i	$-0.249329\pi$
$839$	797629.	+	140644.i	1.13312	+	0.199800i	0.424528	+	0.905415i	$0.360440\pi$
$840$		0			0
$841$	−478036.	−	173991.i	−0.675879	−	0.246000i
$842$		0			0
$843$	439298.	+	614319.i	0.618165	+	0.864448i
$844$		0			0
$845$	1.02071e6	+	589310.i	1.42952	+	0.825335i
$846$		0			0
$847$	−204118.	−	353543.i	−0.284521	−	0.492805i
$848$		0			0
$849$	−586124.	−	45424.3i	−0.813156	−	0.0630191i
$850$		0			0
$851$	−93497.5	+	16486.1i	−0.129104	+	0.0227646i
$852$		0			0
$853$	362055.	+	303800.i	0.497595	+	0.417532i	0.856739	−	0.515750i	$-0.172487\pi$
$853$	362055.	+	303800.i	0.497595	+	0.417532i	−0.359144	+	0.933282i	$0.616931\pi$
$854$		0			0
$855$	458366.	+	156604.i	0.627019	+	0.214225i
$856$		0			0
$857$	−12305.5	−	33809.0i	−0.0167547	−	0.0460331i	0.931033	−	0.364934i	$-0.118908\pi$
$857$	−12305.5	−	33809.0i	−0.0167547	−	0.0460331i	−0.947788	+	0.318901i	$0.896686\pi$
$858$		0			0
$859$	1.02512e6	−	860180.i	1.38928	−	1.16574i	0.423648	−	0.905827i	$-0.360749\pi$
$859$	1.02512e6	−	860180.i	1.38928	−	1.16574i	0.965631	−	0.259916i	$-0.0836950\pi$
$860$		0			0
$861$	−203360.	−	729906.i	−0.274322	−	0.984601i
$862$		0			0
$863$			865704.i			1.16238i	0.813768	+	0.581190i	$0.197413\pi$
$863$			865704.i			1.16238i	−0.813768	+	0.581190i	$0.802587\pi$
$864$		0			0
$865$	−410432.			−0.548541
$866$		0			0
$867$	1.53961e6	+	1.50935e6i	2.04820	+	2.00795i
$868$		0			0
$869$	−580685.	−	692034.i	−0.768956	−	0.916406i
$870$		0			0
$871$	−1.04162e6	+	379117.i	−1.37300	+	0.499732i
$872$		0			0
$873$	−214618.	−	555131.i	−0.281604	−	0.728395i
$874$		0			0
$875$	−412249.	+	491299.i	−0.538447	+	0.641696i
$876$		0			0
$877$	44642.1	+	253178.i	0.0580424	+	0.329175i	0.999978	−	0.00661838i	$-0.00210671\pi$
$877$	44642.1	+	253178.i	0.0580424	+	0.329175i	−0.941936	+	0.335793i	$0.890996\pi$
$878$		0			0
$879$	−1.25895e6	−	863039.i	−1.62941	−	1.11700i
$880$		0			0
$881$	127449.	−	73582.5i	0.164204	−	0.0948032i	−0.415646	−	0.909526i	$-0.636444\pi$
$881$	127449.	−	73582.5i	0.164204	−	0.0948032i	0.579850	+	0.814723i	$0.303111\pi$
$882$		0			0
$883$	427663.	−	740733.i	0.548504	−	0.950037i	−0.449873	−	0.893092i	$-0.648531\pi$
$883$	427663.	−	740733.i	0.548504	−	0.950037i	0.998377	−	0.0569446i	$-0.0181358\pi$
$884$		0			0
$885$	−427925.	−	194399.i	−0.546363	−	0.248204i
$886$		0			0
$887$	−404835.	+	1.11227e6i	−0.514554	+	1.41372i	0.361890	+	0.932221i	$0.382132\pi$
$887$	−404835.	+	1.11227e6i	−0.514554	+	1.41372i	−0.876444	+	0.481504i	$0.840091\pi$
$888$		0			0
$889$	−110572.	+	627083.i	−0.139907	+	0.793454i
$890$		0			0
$891$	−1.03540e6	−	140473.i	−1.30423	−	0.176944i
$892$		0			0
$893$	−175690.	−	30979.0i	−0.220316	−	0.0388476i
$894$		0			0
$895$	−523032.	−	190368.i	−0.652953	−	0.237656i
$896$		0			0
$897$	96819.9	−	213126.i	0.120332	−	0.264882i
$898$		0			0
$899$	−486779.	−	281042.i	−0.602299	−	0.347738i
$900$		0			0
$901$	1.22273e6	+	2.11782e6i	1.50619	+	2.60880i
$902$		0			0
$903$	672015.	−	980295.i	0.824145	−	1.20221i
$904$		0			0
$905$	−528925.	+	93263.7i	−0.645798	+	0.113872i
$906$		0			0
$907$	237731.	+	199480.i	0.288982	+	0.242485i	0.775741	−	0.631052i	$-0.217376\pi$
$907$	237731.	+	199480.i	0.288982	+	0.242485i	−0.486758	+	0.873537i	$0.661821\pi$
$908$		0			0
$909$	−135193.	+	866978.i	−0.163616	+	1.04925i
$910$		0			0
$911$	−449233.	−	1.23426e6i	−0.541296	−	1.48720i	−0.845176	−	0.534488i	$-0.820504\pi$
$911$	−449233.	−	1.23426e6i	−0.541296	−	1.48720i	0.303880	−	0.952710i	$-0.401718\pi$
$912$		0			0
$913$	−409760.	+	343829.i	−0.491573	+	0.412478i
$914$		0			0
$915$	389259.	−	397061.i	0.464939	−	0.474258i
$916$		0			0
$917$			81337.1i			0.0967275i
$918$		0			0
$919$	665214.			0.787644			0.393822	−	0.919187i	$-0.371153\pi$
$919$	665214.			0.787644			0.393822	+	0.919187i	$0.371153\pi$
$920$		0			0
$921$	−444486.	+	123839.i	−0.524009	+	0.145995i
$922$		0			0
$923$	471712.	+	562165.i	0.553699	+	0.659873i
$924$		0			0
$925$	−136867.	+	49815.5i	−0.159961	+	0.0582212i
$926$		0			0
$927$	−1.26792e6	+	249606.i	−1.47548	+	0.290466i
$928$		0			0
$929$	−310160.	+	369634.i	−0.359380	+	0.428293i	−0.915194	−	0.403014i	$-0.867963\pi$
$929$	−310160.	+	369634.i	−0.359380	+	0.428293i	0.555813	+	0.831307i	$0.312407\pi$
$930$		0			0
$931$	44822.9	+	254203.i	0.0517131	+	0.293279i
$932$		0			0
$933$	−45520.5	+	587366.i	−0.0522931	+	0.674754i
$934$		0			0
$935$	1.72769e6	−	997481.i	1.97625	−	1.14099i
$936$		0			0
$937$	209383.	−	362662.i	0.238486	−	0.413069i	−0.721794	−	0.692108i	$-0.756682\pi$
$937$	209383.	−	362662.i	0.238486	−	0.413069i	0.960280	+	0.279038i	$0.0900157\pi$
$938$		0			0
$939$	500913.	−	358201.i	0.568108	−	0.406253i
$940$		0			0
$941$	107236.	−	294629.i	0.121105	−	0.332733i	−0.864296	−	0.502984i	$-0.832236\pi$
$941$	107236.	−	294629.i	0.121105	−	0.332733i	0.985401	+	0.170251i	$0.0544578\pi$
$942$		0			0
$943$	−34866.2	+	197736.i	−0.0392086	+	0.222363i
$944$		0			0
$945$	−585898.	+	175838.i	−0.656083	+	0.196902i
$946$		0			0
$947$	−1.14119e6	−	201222.i	−1.27250	−	0.224375i	−0.503705	−	0.863876i	$-0.668030\pi$
$947$	−1.14119e6	−	201222.i	−1.27250	−	0.224375i	−0.768791	+	0.639500i	$0.779141\pi$
$948$		0			0
$949$	−1.19381e6	−	434512.i	−1.32557	−	0.482468i
$950$		0			0
$951$	1.13997e6	−	111142.i	1.26047	−	0.122890i
$952$		0			0
$953$	−824540.	−	476049.i	−0.907875	−	0.524162i	−0.0281283	−	0.999604i	$-0.508955\pi$
$953$	−824540.	−	476049.i	−0.907875	−	0.524162i	−0.879747	+	0.475442i	$0.842288\pi$
$954$		0			0
$955$	−130504.	−	226039.i	−0.143093	−	0.247844i
$956$		0			0
$957$	275654.	+	576147.i	0.300982	+	0.629085i
$958$		0			0
$959$	477192.	−	84141.8i	0.518867	−	0.0914902i
$960$		0			0
$961$	511398.	+	429114.i	0.553749	+	0.464650i
$962$		0			0
$963$	94663.9	−	171745.i	0.102078	−	0.185196i
$964$		0			0
$965$	−88427.8	−	242953.i	−0.0949586	−	0.260896i
$966$		0			0
$967$	−465889.	+	390927.i	−0.498229	+	0.418064i	−0.856965	−	0.515375i	$-0.827653\pi$
$967$	−465889.	+	390927.i	−0.498229	+	0.418064i	0.358735	+	0.933439i	$0.383208\pi$
$968$		0			0
$969$	1.34430e6	+	345947.i	1.43169	+	0.368436i
$970$		0			0
$971$			934879.i			0.991555i	0.868449	+	0.495778i	$0.165117\pi$
$971$			934879.i			0.991555i	−0.868449	+	0.495778i	$0.834883\pi$
$972$		0			0
$973$	−463788.			−0.489885
$974$		0			0
$975$	89501.6	−	347791.i	0.0941503	−	0.365855i
$976$		0			0
$977$	−836752.	−	997202.i	−0.876612	−	1.04471i	−0.998638	−	0.0521789i	$-0.983383\pi$
$977$	−836752.	−	997202.i	−0.876612	−	1.04471i	0.122025	−	0.992527i	$-0.461061\pi$
$978$		0			0
$979$	413749.	−	150592.i	0.431689	−	0.157122i
$980$		0			0
$981$	−975739.	−	537816.i	−1.01390	−	0.558851i
$982$		0			0
$983$	1.11530e6	−	1.32916e6i	1.15421	−	1.37553i	0.239757	−	0.970833i	$-0.422932\pi$
$983$	1.11530e6	−	1.32916e6i	1.15421	−	1.37553i	0.914450	−	0.404698i	$-0.132623\pi$
$984$		0			0
$985$	−76057.0	−	431341.i	−0.0783911	−	0.444578i
$986$		0			0
$987$	203236.	−	97236.6i	0.208624	−	0.0998150i
$988$		0			0
$989$	−272754.	+	157475.i	−0.278856	+	0.160997i
$990$		0			0
$991$	51066.6	−	88450.0i	0.0519984	−	0.0900639i	−0.838855	−	0.544356i	$-0.816774\pi$
$991$	51066.6	−	88450.0i	0.0519984	−	0.0900639i	0.890853	+	0.454292i	$0.150108\pi$
$992$		0			0
$993$	−45511.9	−	466811.i	−0.0461558	−	0.473416i
$994$		0			0
$995$	45455.4	−	124888.i	0.0459134	−	0.126146i
$996$		0			0
$997$	−44930.2	+	254812.i	−0.0452010	+	0.256347i	−0.999032	−	0.0439975i	$-0.985991\pi$
$997$	−44930.2	+	254812.i	−0.0452010	+	0.256347i	0.953831	+	0.300345i	$0.0971018\pi$
$998$		0			0
$999$	−741752.	−	175269.i	−0.743238	−	0.175620i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.29.5	✓	72
3.2	odd	2		324.5.k.a.89.9		72
27.13	even	9		324.5.k.a.233.9		72
27.14	odd	18	inner	108.5.k.a.41.5	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.5	✓	72	1.1	even	1	trivial
108.5.k.a.41.5	yes	72	27.14	odd	18	inner
324.5.k.a.89.9		72	3.2	odd	2
324.5.k.a.233.9		72	27.13	even	9

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.24301 + 8.71602i) q^{3} +(-14.1660 - 16.8824i) q^{5} +(35.7790 - 13.0225i) q^{7} +(-70.9378 - 39.1002i) q^{9} +O(q^{10})\)	\(q+(-2.24301 + 8.71602i) q^{3} +(-14.1660 - 16.8824i) q^{5} +(35.7790 - 13.0225i) q^{7} +(-70.9378 - 39.1002i) q^{9} +(-102.369 + 121.998i) q^{11} +(-49.7378 - 282.077i) q^{13} +(178.921 - 85.6037i) q^{15} +(492.251 - 284.201i) q^{17} +(135.673 - 234.992i) q^{19} +(33.2517 + 341.060i) q^{21} +(31.0578 - 85.3307i) q^{23} +(24.1910 - 137.194i) q^{25} +(499.912 - 530.593i) q^{27} +(-438.837 - 77.3788i) q^{29} +(1185.32 + 431.421i) q^{31} +(-833.724 - 1165.89i) q^{33} +(-726.696 - 419.558i) q^{35} +(-522.756 - 905.440i) q^{37} +(2570.15 + 199.185i) q^{39} +(-2177.54 + 383.960i) q^{41} +(-2656.90 - 2229.41i) q^{43} +(344.801 + 1751.49i) q^{45} +(-224.867 - 617.817i) q^{47} +(-728.720 + 611.469i) q^{49} +(1372.98 + 4927.94i) q^{51} +4302.32i q^{53} +3509.77 q^{55} +(1743.88 + 1709.61i) q^{57} +(-1523.19 - 1815.27i) q^{59} +(2634.33 - 958.816i) q^{61} +(-3047.27 - 475.177i) q^{63} +(-4057.54 + 4835.59i) q^{65} +(-672.010 - 3811.16i) q^{67} +(674.081 + 462.098i) q^{69} +(-2218.83 + 1281.04i) q^{71} +(2217.70 - 3841.17i) q^{73} +(1141.53 + 518.577i) q^{75} +(-2073.93 + 5698.06i) q^{77} +(-985.019 + 5586.32i) q^{79} +(3503.36 + 5547.36i) q^{81} +(3307.71 + 583.239i) q^{83} +(-11771.2 - 4284.37i) q^{85} +(1658.75 - 3651.35i) q^{87} +(-2394.32 - 1382.36i) q^{89} +(-5452.92 - 9444.73i) q^{91} +(-6418.95 + 9363.58i) q^{93} +(-5889.16 + 1038.42i) q^{95} +(5628.76 + 4723.09i) q^{97} +(12031.9 - 4651.65i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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