LMFDB - Embedded newform 108.5.k.a.65.4

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			65.4
Character	$\chi$	$=$	108.65
Dual form			108.5.k.a.5.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.53499 + 7.09676i) q^{3} +(-14.0519 + 38.6073i) q^{5} +(-5.57324 + 31.6074i) q^{7} +(-19.7279 - 78.5609i) q^{9} +O(q^{10})$	\(q+(-5.53499 + 7.09676i) q^{3} +(-14.0519 + 38.6073i) q^{5} +(-5.57324 + 31.6074i) q^{7} +(-19.7279 - 78.5609i) q^{9} +(-4.11688 - 11.3110i) q^{11} +(91.3096 - 76.6178i) q^{13} +(-196.210 - 313.414i) q^{15} +(-175.368 + 101.249i) q^{17} +(98.3348 - 170.321i) q^{19} +(-193.462 - 214.499i) q^{21} +(-707.360 + 124.727i) q^{23} +(-814.293 - 683.273i) q^{25} +(666.721 + 294.829i) q^{27} +(523.078 - 623.381i) q^{29} +(28.4024 + 161.078i) q^{31} +(103.059 + 33.3899i) q^{33} +(-1141.96 - 659.313i) q^{35} +(-941.906 - 1631.43i) q^{37} +(38.3409 + 1072.08i) q^{39} +(1759.69 + 2097.12i) q^{41} +(-3246.83 + 1181.75i) q^{43} +(3310.24 + 342.290i) q^{45} +(89.3900 + 15.7619i) q^{47} +(1288.23 + 468.879i) q^{49} +(252.122 - 1804.96i) q^{51} +844.122i q^{53} +494.540 q^{55} +(664.444 + 1640.58i) q^{57} +(-1831.29 + 5031.42i) q^{59} +(860.325 - 4879.15i) q^{61} +(2593.06 - 185.709i) q^{63} +(1674.94 + 4601.85i) q^{65} +(561.501 - 471.156i) q^{67} +(3030.07 - 5710.32i) q^{69} +(-3351.53 + 1935.01i) q^{71} +(1199.32 - 2077.29i) q^{73} +(9356.12 - 1996.93i) q^{75} +(380.457 - 67.0849i) q^{77} +(3238.29 + 2717.25i) q^{79} +(-5782.62 + 3099.68i) q^{81} +(-6782.64 + 8083.24i) q^{83} +(-1444.69 - 8193.24i) q^{85} +(1528.75 + 7162.56i) q^{87} +(4410.52 + 2546.41i) q^{89} +(1912.80 + 3313.07i) q^{91} +(-1300.34 - 689.999i) q^{93} +(5193.84 + 6189.78i) q^{95} +(-14646.8 + 5331.01i) q^{97} +(-807.388 + 546.569i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

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
$3$	−5.53499	+	7.09676i	−0.614998	+	0.788528i
$3$	−5.53499	+	7.09676i	−0.614998	+	0.788528i
$4$		0			0
$5$	−14.0519	+	38.6073i	−0.562077	+	1.54429i	0.254509	+	0.967070i	$0.418086\pi$
$5$	−14.0519	+	38.6073i	−0.562077	+	1.54429i	−0.816586	+	0.577223i	$0.804136\pi$
$6$		0			0
$7$	−5.57324	+	31.6074i	−0.113740	+	0.645050i	0.873627	+	0.486596i	$0.161762\pi$
$7$	−5.57324	+	31.6074i	−0.113740	+	0.645050i	−0.987367	+	0.158453i	$0.949349\pi$
$8$		0			0
$9$	−19.7279	−	78.5609i	−0.243554	−	0.969887i
$10$		0			0
$11$	−4.11688	−	11.3110i	−0.0340238	−	0.0934797i	0.921518	−	0.388337i	$-0.126950\pi$
$11$	−4.11688	−	11.3110i	−0.0340238	−	0.0934797i	−0.955541	+	0.294857i	$0.904728\pi$
$12$		0			0
$13$	91.3096	−	76.6178i	0.540293	−	0.453360i	−0.331345	−	0.943510i	$-0.607502\pi$
$13$	91.3096	−	76.6178i	0.540293	−	0.453360i	0.871638	+	0.490150i	$0.163058\pi$
$14$		0			0
$15$	−196.210	−	313.414i	−0.872043	−	1.39295i
$16$		0			0
$17$	−175.368	+	101.249i	−0.606810	+	0.350342i	−0.771716	−	0.635967i	$-0.780601\pi$
$17$	−175.368	+	101.249i	−0.606810	+	0.350342i	0.164906	+	0.986309i	$0.447268\pi$
$18$		0			0
$19$	98.3348	−	170.321i	0.272395	−	0.471803i	−0.697079	−	0.716994i	$-0.745517\pi$
$19$	98.3348	−	170.321i	0.272395	−	0.471803i	0.969475	+	0.245191i	$0.0788508\pi$
$20$		0			0
$21$	−193.462	−	214.499i	−0.438690	−	0.486391i
$22$		0			0
$23$	−707.360	+	124.727i	−1.33716	+	0.235778i	−0.796081	−	0.605191i	$-0.793097\pi$
$23$	−707.360	+	124.727i	−1.33716	+	0.235778i	−0.541084	+	0.840969i	$0.681986\pi$
$24$		0			0
$25$	−814.293	−	683.273i	−1.30287	−	1.09324i
$26$		0			0
$27$	666.721	+	294.829i	0.914569	+	0.404430i
$28$		0			0
$29$	523.078	−	623.381i	0.621972	−	0.741237i	−0.359436	−	0.933170i	$-0.617031\pi$
$29$	523.078	−	623.381i	0.621972	−	0.741237i	0.981408	+	0.191932i	$0.0614754\pi$
$30$		0			0
$31$	28.4024	+	161.078i	0.0295550	+	0.167615i	0.996013	−	0.0892109i	$-0.0284345\pi$
$31$	28.4024	+	161.078i	0.0295550	+	0.167615i	−0.966458	+	0.256826i	$0.917323\pi$
$32$		0			0
$33$	103.059	+	33.3899i	0.0946360	+	0.0306611i
$34$		0			0
$35$	−1141.96	−	659.313i	−0.932216	−	0.538215i
$36$		0			0
$37$	−941.906	−	1631.43i	−0.688025	−	1.19169i	−0.972476	−	0.233004i	$-0.925145\pi$
$37$	−941.906	−	1631.43i	−0.688025	−	1.19169i	0.284451	−	0.958691i	$-0.408189\pi$
$38$		0			0
$39$	38.3409	+	1072.08i	0.0252077	+	0.704852i
$40$		0			0
$41$	1759.69	+	2097.12i	1.04681	+	1.24754i	0.968079	+	0.250646i	$0.0806429\pi$
$41$	1759.69	+	2097.12i	1.04681	+	1.24754i	0.0787328	+	0.996896i	$0.474913\pi$
$42$		0			0
$43$	−3246.83	+	1181.75i	−1.75599	+	0.639129i	−0.999883	−	0.0153148i	$-0.995125\pi$
$43$	−3246.83	+	1181.75i	−1.75599	+	0.639129i	−0.756110	+	0.654444i	$0.772903\pi$
$44$		0			0
$45$	3310.24	+	342.290i	1.63469	+	0.169032i
$46$		0			0
$47$	89.3900	+	15.7619i	0.0404663	+	0.00713530i	0.193845	−	0.981032i	$-0.437904\pi$
$47$	89.3900	+	15.7619i	0.0404663	+	0.00713530i	−0.153378	+	0.988168i	$0.549015\pi$
$48$		0			0
$49$	1288.23	+	468.879i	0.536540	+	0.195285i
$50$		0			0
$51$	252.122	−	1804.96i	0.0969326	−	0.693947i
$52$		0			0
$53$			844.122i			0.300506i	0.988648	+	0.150253i	$0.0480088\pi$
$53$			844.122i			0.300506i	−0.988648	+	0.150253i	$0.951991\pi$
$54$		0			0
$55$	494.540			0.163484
$56$		0			0
$57$	664.444	+	1640.58i	0.204507	+	0.504950i
$58$		0			0
$59$	−1831.29	+	5031.42i	−0.526081	+	1.44540i	0.337568	+	0.941301i	$0.390396\pi$
$59$	−1831.29	+	5031.42i	−0.526081	+	1.44540i	−0.863649	+	0.504094i	$0.831826\pi$
$60$		0			0
$61$	860.325	−	4879.15i	0.231208	−	1.31125i	−0.619246	−	0.785197i	$-0.712562\pi$
$61$	860.325	−	4879.15i	0.231208	−	1.31125i	0.850454	−	0.526049i	$-0.176327\pi$
$62$		0			0
$63$	2593.06	−	185.709i	0.653327	−	0.0467898i
$64$		0			0
$65$	1674.94	+	4601.85i	0.396435	+	1.08919i
$66$		0			0
$67$	561.501	−	471.156i	0.125084	−	0.104958i	−0.578100	−	0.815966i	$-0.696206\pi$
$67$	561.501	−	471.156i	0.125084	−	0.104958i	0.703184	+	0.711008i	$0.251761\pi$
$68$		0			0
$69$	3030.07	−	5710.32i	0.636436	−	1.19940i
$70$		0			0
$71$	−3351.53	+	1935.01i	−0.664855	+	0.383854i	−0.794124	−	0.607756i	$-0.792070\pi$
$71$	−3351.53	+	1935.01i	−0.664855	+	0.383854i	0.129270	+	0.991609i	$0.458737\pi$
$72$		0			0
$73$	1199.32	−	2077.29i	0.225056	−	0.389808i	−0.731280	−	0.682077i	$-0.761077\pi$
$73$	1199.32	−	2077.29i	0.225056	−	0.389808i	0.956336	+	0.292269i	$0.0944103\pi$
$74$		0			0
$75$	9356.12	−	1996.93i	1.66331	−	0.355010i
$76$		0			0
$77$	380.457	−	67.0849i	0.0641689	−	0.0113147i
$78$		0			0
$79$	3238.29	+	2717.25i	0.518874	+	0.435387i	0.864239	−	0.503082i	$-0.167801\pi$
$79$	3238.29	+	2717.25i	0.518874	+	0.435387i	−0.345365	+	0.938468i	$0.612245\pi$
$80$		0			0
$81$	−5782.62	+	3099.68i	−0.881363	+	0.472440i
$82$		0			0
$83$	−6782.64	+	8083.24i	−0.984561	+	1.17335i	0.000298097		1.00000i	$0.499905\pi$
$83$	−6782.64	+	8083.24i	−0.984561	+	1.17335i	−0.984859	+	0.173355i	$0.944539\pi$
$84$		0			0
$85$	−1444.69	−	8193.24i	−0.199957	−	1.13401i
$86$		0			0
$87$	1528.75	+	7162.56i	0.201975	+	0.946302i
$88$		0			0
$89$	4410.52	+	2546.41i	0.556813	+	0.321476i	0.751865	−	0.659317i	$-0.229154\pi$
$89$	4410.52	+	2546.41i	0.556813	+	0.321476i	−0.195052	+	0.980793i	$0.562488\pi$
$90$		0			0
$91$	1912.80	+	3313.07i	0.230987	+	0.400081i
$92$		0			0
$93$	−1300.34	−	689.999i	−0.150345	−	0.0797778i
$94$		0			0
$95$	5193.84	+	6189.78i	0.575495	+	0.685848i
$96$		0			0
$97$	−14646.8	+	5331.01i	−1.55668	+	0.566586i	−0.969973	−	0.243212i	$-0.921799\pi$
$97$	−14646.8	+	5331.01i	−1.55668	+	0.566586i	−0.586709	+	0.809798i	$0.699577\pi$
$98$		0			0
$99$	−807.388	+	546.569i	−0.0823781	+	0.0557666i
$100$		0			0
$101$	−13396.2	−	2362.10i	−1.31322	−	0.231556i	−0.527191	−	0.849747i	$-0.676755\pi$
$101$	−13396.2	−	2362.10i	−1.31322	−	0.231556i	−0.786028	+	0.618191i	$0.787866\pi$
$102$		0			0
$103$	−13783.3	−	5016.70i	−1.29921	−	0.472872i	−0.402468	−	0.915434i	$-0.631848\pi$
$103$	−13783.3	−	5016.70i	−1.29921	−	0.472872i	−0.896738	+	0.442562i	$0.854070\pi$
$104$		0			0
$105$	10999.7	−	4454.95i	0.997709	−	0.404077i
$106$		0			0
$107$		−	8097.26i		−	0.707246i	−0.935388	−	0.353623i	$-0.884950\pi$
$107$		−	8097.26i		−	0.707246i	0.935388	−	0.353623i	$-0.115050\pi$
$108$		0			0
$109$	−13401.0			−1.12793			−0.563966	−	0.825798i	$-0.690725\pi$
$109$	−13401.0			−1.12793			−0.563966	+	0.825798i	$0.690725\pi$
$110$		0			0
$111$	16791.3	+	2345.46i	1.36282	+	0.190363i
$112$		0			0
$113$	7402.08	−	20337.0i	0.579691	−	1.59269i	−0.209012	−	0.977913i	$-0.567025\pi$
$113$	7402.08	−	20337.0i	0.579691	−	1.59269i	0.788703	−	0.614775i	$-0.210753\pi$
$114$		0			0
$115$	5124.40	−	29061.9i	0.387479	−	2.19750i
$116$		0			0
$117$	−7820.51	−	5661.85i	−0.571299	−	0.413606i
$118$		0			0
$119$	−2222.85	−	6107.22i	−0.156970	−	0.431270i
$120$		0			0
$121$	11104.7	−	9317.92i	0.758464	−	0.636427i
$122$		0			0
$123$	−24622.6	+	880.579i	−1.62751	+	0.0582047i
$124$		0			0
$125$	15583.8	−	8997.30i	0.997361	−	0.575827i
$126$		0			0
$127$	−14330.6	+	24821.3i	−0.888497	+	1.53892i	−0.0468449	+	0.998902i	$0.514917\pi$
$127$	−14330.6	+	24821.3i	−0.888497	+	1.53892i	−0.841652	+	0.540020i	$0.818417\pi$
$128$		0			0
$129$	9584.57	−	29582.9i	0.575961	−	1.77771i
$130$		0			0
$131$	−7511.50	+	1324.48i	−0.437707	+	0.0771796i	−0.388160	−	0.921592i	$-0.626889\pi$
$131$	−7511.50	+	1324.48i	−0.437707	+	0.0771796i	−0.0495477	+	0.998772i	$0.515778\pi$
$132$		0			0
$133$	4835.36	+	4057.35i	0.273354	+	0.229371i
$134$		0			0
$135$	−20751.3	+	21597.4i	−1.13862	+	1.18504i
$136$		0			0
$137$	−18103.1	+	21574.5i	−0.964524	+	1.14947i	0.0241978	+	0.999707i	$0.492297\pi$
$137$	−18103.1	+	21574.5i	−0.964524	+	1.14947i	−0.988721	+	0.149767i	$0.952148\pi$
$138$		0			0
$139$	−2705.80	−	15345.3i	−0.140044	−	0.794231i	−0.971213	−	0.238211i	$-0.923439\pi$
$139$	−2705.80	−	15345.3i	−0.140044	−	0.794231i	0.831169	−	0.556020i	$-0.187672\pi$
$140$		0			0
$141$	−606.630	+	547.137i	−0.0305131	+	0.0275206i
$142$		0			0
$143$	−1242.54	−	717.380i	−0.0607628	−	0.0350814i
$144$		0			0
$145$	16716.8	+	28954.4i	0.795092	+	1.37714i
$146$		0			0
$147$	−10457.9	+	6547.04i	−0.483959	+	0.302978i
$148$		0			0
$149$	8817.28	+	10508.0i	0.397157	+	0.473313i	0.927151	−	0.374688i	$-0.122250\pi$
$149$	8817.28	+	10508.0i	0.397157	+	0.473313i	−0.529994	+	0.848001i	$0.677806\pi$
$150$		0			0
$151$	−14397.6	+	5240.31i	−0.631447	+	0.229828i	−0.637861	−	0.770152i	$-0.720181\pi$
$151$	−14397.6	+	5240.31i	−0.631447	+	0.229828i	0.00641405	+	0.999979i	$0.497958\pi$
$152$		0			0
$153$	11413.8	+	11779.6i	0.487583	+	0.503210i
$154$		0			0
$155$	−6617.89	−	1166.91i	−0.275459	−	0.0485708i
$156$		0			0
$157$	41354.0	+	15051.6i	1.67771	+	0.610638i	0.992994	−	0.118168i	$-0.0377022\pi$
$157$	41354.0	+	15051.6i	1.67771	+	0.610638i	0.684720	+	0.728806i	$0.259924\pi$
$158$		0			0
$159$	−5990.53	−	4672.20i	−0.236958	−	0.184811i
$160$		0			0
$161$		−	23053.0i		−	0.889355i
$162$		0			0
$163$	46133.7			1.73637			0.868186	−	0.496239i	$-0.165286\pi$
$163$	46133.7			1.73637			0.868186	+	0.496239i	$0.165286\pi$
$164$		0			0
$165$	−2737.27	+	3509.63i	−0.100542	+	0.128912i
$166$		0			0
$167$	−10403.4	+	28583.1i	−0.373029	+	1.02489i	0.601155	+	0.799132i	$0.294707\pi$
$167$	−10403.4	+	28583.1i	−0.373029	+	1.02489i	−0.974184	+	0.225756i	$0.927515\pi$
$168$		0			0
$169$	−2492.42	+	14135.2i	−0.0872665	+	0.494913i
$170$		0			0
$171$	−15320.5	−	4365.20i	−0.523939	−	0.149283i
$172$		0			0
$173$	−4830.95	−	13272.9i	−0.161414	−	0.443480i	0.832449	−	0.554102i	$-0.186938\pi$
$173$	−4830.95	−	13272.9i	−0.161414	−	0.443480i	−0.993863	+	0.110621i	$0.964716\pi$
$174$		0			0
$175$	26134.7	−	21929.7i	0.853379	−	0.716070i
$176$		0			0
$177$	−25570.6	−	40845.0i	−0.816196	−	1.30375i
$178$		0			0
$179$	6081.38	−	3511.09i	0.189800	−	0.109581i	−0.402089	−	0.915601i	$-0.631716\pi$
$179$	6081.38	−	3511.09i	0.189800	−	0.109581i	0.591889	+	0.806020i	$0.298382\pi$
$180$		0			0
$181$	−5455.83	+	9449.78i	−0.166534	+	0.288446i	−0.937199	−	0.348795i	$-0.886591\pi$
$181$	−5455.83	+	9449.78i	−0.166534	+	0.288446i	0.770665	+	0.637241i	$0.219924\pi$
$182$		0			0
$183$	29864.2	+	33111.5i	0.891762	+	0.988728i
$184$		0			0
$185$	76220.7	−	13439.8i	2.22705	−	0.392689i
$186$		0			0
$187$	1867.20	+	1566.77i	0.0533959	+	0.0448044i
$188$		0			0
$189$	−13034.6	+	19430.2i	−0.364900	+	0.543943i
$190$		0			0
$191$	30575.4	−	36438.4i	0.838119	−	0.998832i	−0.161809	−	0.986822i	$-0.551733\pi$
$191$	30575.4	−	36438.4i	0.838119	−	0.998832i	0.999928	−	0.0120095i	$-0.00382285\pi$
$192$		0			0
$193$	7552.68	+	42833.4i	0.202762	+	1.14992i	0.900923	+	0.433979i	$0.142891\pi$
$193$	7552.68	+	42833.4i	0.202762	+	1.14992i	−0.698161	+	0.715941i	$0.745998\pi$
$194$		0			0
$195$	−41928.9	−	13584.6i	−1.10267	−	0.357253i
$196$		0			0
$197$	−11406.4	−	6585.49i	−0.293911	−	0.169690i	0.345793	−	0.938311i	$-0.387610\pi$
$197$	−11406.4	−	6585.49i	−0.293911	−	0.169690i	−0.639704	+	0.768621i	$0.720943\pi$
$198$		0			0
$199$	17462.9	+	30246.6i	0.440971	+	0.763785i	0.997762	−	0.0668682i	$-0.0213007\pi$
$199$	17462.9	+	30246.6i	0.440971	+	0.763785i	−0.556790	+	0.830653i	$0.687967\pi$
$200$		0			0
$201$	235.774	+	6592.68i	0.00583585	+	0.163181i
$202$		0			0
$203$	16788.2	+	20007.4i	0.407392	+	0.485511i
$204$		0			0
$205$	−105691.	+	38468.4i	−2.51496	+	0.915370i
$206$		0			0
$207$	23753.3	+	53110.2i	0.554350	+	1.23947i
$208$		0			0
$209$	−2331.34	−	411.078i	−0.0533719	−	0.00941091i
$210$		0			0
$211$	51237.9	+	18649.1i	1.15087	+	0.418883i	0.845828	−	0.533456i	$-0.179107\pi$
$211$	51237.9	+	18649.1i	1.15087	+	0.418883i	0.305043	+	0.952339i	$0.401329\pi$
$212$		0			0
$213$	4818.40	−	34495.2i	0.106205	−	0.760326i
$214$		0			0
$215$		−	141957.i		−	3.07101i
$216$		0			0
$217$	−5249.55			−0.111481
$218$		0			0
$219$	8103.76	+	20009.0i	0.168966	+	0.417194i
$220$		0			0
$221$	−8255.32	+	22681.3i	−0.169024	+	0.464391i
$222$		0			0
$223$	5832.28	−	33076.5i	0.117281	−	0.665135i	−0.868314	−	0.496015i	$-0.834796\pi$
$223$	5832.28	−	33076.5i	0.117281	−	0.665135i	0.985595	−	0.169120i	$-0.0540927\pi$
$224$		0			0
$225$	−37614.2	+	77451.1i	−0.742997	+	1.52990i
$226$		0			0
$227$	2193.22	+	6025.83i	0.0425629	+	0.116941i	0.959153	−	0.282887i	$-0.0912922\pi$
$227$	2193.22	+	6025.83i	0.0425629	+	0.116941i	−0.916590	+	0.399828i	$0.869070\pi$
$228$		0			0
$229$	13000.1	−	10908.4i	0.247900	−	0.208013i	−0.510367	−	0.859956i	$-0.670491\pi$
$229$	13000.1	−	10908.4i	0.247900	−	0.208013i	0.758267	+	0.651944i	$0.226046\pi$
$230$		0			0
$231$	−1629.74	+	3071.33i	−0.0305418	+	0.0575575i
$232$		0			0
$233$	18170.9	−	10491.0i	0.334706	−	0.193243i	−0.323222	−	0.946323i	$-0.604766\pi$
$233$	18170.9	−	10491.0i	0.334706	−	0.193243i	0.657929	+	0.753080i	$0.271433\pi$
$234$		0			0
$235$	−1864.63	+	3229.63i	−0.0337642	+	0.0584812i
$236$		0			0
$237$	−37207.5	+	7941.43i	−0.662421	+	0.141385i
$238$		0			0
$239$	17008.7	−	2999.10i	0.297767	−	0.0525043i	−0.0227689	−	0.999741i	$-0.507248\pi$
$239$	17008.7	−	2999.10i	0.297767	−	0.0525043i	0.320536	+	0.947236i	$0.396137\pi$
$240$		0			0
$241$	−27227.2	−	22846.4i	−0.468780	−	0.393353i	0.377569	−	0.925981i	$-0.376760\pi$
$241$	−27227.2	−	22846.4i	−0.468780	−	0.393353i	−0.846349	+	0.532628i	$0.821205\pi$
$242$		0			0
$243$	10009.1	−	58194.5i	0.169504	−	0.985529i
$244$		0			0
$245$	−36204.3	+	43146.6i	−0.603154	+	0.718811i
$246$		0			0
$247$	−4070.71	−	23086.1i	−0.0667230	−	0.378405i
$248$		0			0
$249$	−19822.9	−	92875.4i	−0.319720	−	1.49797i
$250$		0			0
$251$	−54334.9	−	31370.3i	−0.862445	−	0.497933i	0.00238560	−	0.999997i	$-0.499241\pi$
$251$	−54334.9	−	31370.3i	−0.862445	−	0.497933i	−0.864830	+	0.502065i	$0.832574\pi$
$252$		0			0
$253$	4322.91	+	7487.50i	0.0675359	+	0.116976i
$254$		0			0
$255$	66141.7	+	35096.8i	1.01717	+	0.539744i
$256$		0			0
$257$	43512.8	+	51856.5i	0.658796	+	0.785122i	0.987212	−	0.159411i	$-0.0509594\pi$
$257$	43512.8	+	51856.5i	0.658796	+	0.785122i	−0.328417	+	0.944533i	$0.606515\pi$
$258$		0			0
$259$	56814.8	−	20678.9i	0.846958	−	0.308267i
$260$		0			0
$261$	−59292.5	−	28795.5i	−0.870400	−	0.422711i
$262$		0			0
$263$	−121068.	−	21347.5i	−1.75032	−	0.308628i	−0.795530	−	0.605914i	$-0.792808\pi$
$263$	−121068.	−	21347.5i	−1.75032	−	0.308628i	−0.954789	+	0.297285i	$0.903919\pi$
$264$		0			0
$265$	−32589.3	−	11861.5i	−0.464070	−	0.168908i
$266$		0			0
$267$	−42483.4	+	17206.0i	−0.595932	+	0.241356i
$268$		0			0
$269$		−	39540.0i		−	0.546427i	−0.961953	−	0.273213i	$-0.911913\pi$
$269$		−	39540.0i		−	0.546427i	0.961953	−	0.273213i	$-0.0880865\pi$
$270$		0			0
$271$	27986.9			0.381080			0.190540	−	0.981679i	$-0.438976\pi$
$271$	27986.9			0.381080			0.190540	+	0.981679i	$0.438976\pi$
$272$		0			0
$273$	−34099.4	−	4763.11i	−0.457532	−	0.0639095i
$274$		0			0
$275$	−4376.18	+	12023.5i	−0.0578668	+	0.158988i
$276$		0			0
$277$	−17151.2	+	97269.1i	−0.223529	+	1.26770i	0.641948	+	0.766748i	$0.278126\pi$
$277$	−17151.2	+	97269.1i	−0.223529	+	1.26770i	−0.865477	+	0.500948i	$0.832985\pi$
$278$		0			0
$279$	12094.1	−	5409.04i	0.155369	−	0.0694883i
$280$		0			0
$281$	6829.25	+	18763.2i	0.0864890	+	0.237626i	0.975396	−	0.220462i	$-0.0707565\pi$
$281$	6829.25	+	18763.2i	0.0864890	+	0.237626i	−0.888907	+	0.458088i	$0.848534\pi$
$282$		0			0
$283$	77950.3	−	65408.0i	0.973296	−	0.816692i	−0.00976888	−	0.999952i	$-0.503110\pi$
$283$	77950.3	−	65408.0i	0.973296	−	0.816692i	0.983064	+	0.183260i	$0.0586651\pi$
$284$		0			0
$285$	−72675.2	+	2599.09i	−0.894739	+	0.0319986i
$286$		0			0
$287$	−76091.7	+	43931.5i	−0.923790	+	0.533350i
$288$		0			0
$289$	−21257.9	+	36819.7i	−0.254521	+	0.440843i
$290$		0			0
$291$	43237.1	−	133452.i	0.510588	−	1.57594i
$292$		0			0
$293$	24682.6	−	4352.21i	0.287512	−	0.0506961i	−0.0280322	−	0.999607i	$-0.508924\pi$
$293$	24682.6	−	4352.21i	0.287512	−	0.0506961i	0.315544	+	0.948911i	$0.397813\pi$
$294$		0			0
$295$	−168517.	−	141402.i	−1.93642	−	1.62485i
$296$		0			0
$297$	590.016	−	8755.09i	0.00668884	−	0.0992539i
$298$		0			0
$299$	−55032.5	+	65585.1i	−0.615569	+	0.733606i
$300$		0			0
$301$	−19256.7	−	109210.i	−0.212544	−	1.20540i
$302$		0			0
$303$	90910.8	−	81995.0i	0.990216	−	0.893104i
$304$		0			0
$305$	176282.	+	101776.i	1.89499	+	1.09407i
$306$		0			0
$307$	35159.3	+	60897.6i	0.373046	+	0.646135i	0.990033	−	0.140839i	$-0.0449799\pi$
$307$	35159.3	+	60897.6i	0.373046	+	0.646135i	−0.616986	+	0.786974i	$0.711647\pi$
$308$		0			0
$309$	111893.	−	70049.2i	1.17188	−	0.733645i
$310$		0			0
$311$	−49685.8	−	59213.2i	−0.513702	−	0.612207i	0.445377	−	0.895343i	$-0.353070\pi$
$311$	−49685.8	−	59213.2i	−0.513702	−	0.612207i	−0.959080	+	0.283136i	$0.908625\pi$
$312$		0			0
$313$	143003.	−	52048.7i	1.45967	−	0.531277i	0.514397	−	0.857552i	$-0.328016\pi$
$313$	143003.	−	52048.7i	1.45967	−	0.531277i	0.945275	+	0.326275i	$0.105794\pi$
$314$		0			0
$315$	−29267.7	+	102721.i	−0.294963	+	1.03523i
$316$		0			0
$317$	12789.7	+	2255.17i	0.127275	+	0.0224419i	0.236923	−	0.971529i	$-0.423861\pi$
$317$	12789.7	+	2255.17i	0.127275	+	0.0224419i	−0.109648	+	0.993970i	$0.534972\pi$
$318$		0			0
$319$	−9204.54	−	3350.18i	−0.0904525	−	0.0329220i
$320$		0			0
$321$	57464.3	+	44818.2i	0.557684	+	0.434955i
$322$		0			0
$323$			39825.1i			0.381726i
$324$		0			0
$325$	−126704.			−1.19956
$326$		0			0
$327$	74174.1	−	95103.3i	0.693676	−	0.889406i
$328$		0			0
$329$	−996.384	+	2737.54i	−0.00920524	+	0.0252912i
$330$		0			0
$331$	−5541.63	+	31428.2i	−0.0505803	+	0.286855i	−0.999598	−	0.0283690i	$-0.990969\pi$
$331$	−5541.63	+	31428.2i	−0.0505803	+	0.286855i	0.949017	+	0.315224i	$0.102080\pi$
$332$		0			0
$333$	−109585.	+	106182.i	−0.988238	+	0.957549i
$334$		0			0
$335$	10299.9	+	28298.7i	0.0917789	+	0.252161i
$336$		0			0
$337$	−82087.2	+	68879.3i	−0.722796	+	0.606498i	−0.928157	−	0.372188i	$-0.878608\pi$
$337$	−82087.2	+	68879.3i	−0.722796	+	0.606498i	0.205361	+	0.978686i	$0.434163\pi$
$338$		0			0
$339$	103357.	+	165096.i	0.899371	+	1.43660i
$340$		0			0
$341$	1705.03	−	984.399i	0.0146630	−	0.00846569i
$342$		0			0
$343$	−60529.8	+	104841.i	−0.514494	+	0.891131i
$344$		0			0
$345$	177882.	+	197224.i	1.49449	+	1.65700i
$346$		0			0
$347$	−52650.6	+	9283.71i	−0.437264	+	0.0771015i	−0.387947	−	0.921682i	$-0.626816\pi$
$347$	−52650.6	+	9283.71i	−0.437264	+	0.0771015i	−0.0493173	+	0.998783i	$0.515705\pi$
$348$		0			0
$349$	−67043.3	−	56256.0i	−0.550433	−	0.461868i	0.324654	−	0.945833i	$-0.394752\pi$
$349$	−67043.3	−	56256.0i	−0.550433	−	0.461868i	−0.875088	+	0.483964i	$0.839196\pi$
$350$		0			0
$351$	83467.2	−	24162.0i	0.677488	−	0.196118i
$352$		0			0
$353$	−77840.5	+	92766.7i	−0.624678	+	0.744462i	−0.981867	−	0.189570i	$-0.939291\pi$
$353$	−77840.5	+	92766.7i	−0.624678	+	0.744462i	0.357189	+	0.934032i	$0.383735\pi$
$354$		0			0
$355$	−27610.0	−	156584.i	−0.219084	−	1.24249i
$356$		0			0
$357$	55644.9	+	18028.4i	0.436605	+	0.141456i
$358$		0			0
$359$	192373.	+	111067.i	1.49264	+	0.861777i	0.999964	−	0.00843513i	$-0.00268502\pi$
$359$	192373.	+	111067.i	1.49264	+	0.861777i	0.492677	+	0.870212i	$0.336018\pi$
$360$		0			0
$361$	45821.0	+	79364.4i	0.351601	+	0.608991i
$362$		0			0
$363$	4662.85	+	130382.i	0.0353865	+	0.989471i
$364$		0			0
$365$	63345.7	+	75492.5i	0.475479	+	0.566654i
$366$		0			0
$367$	29765.6	−	10833.8i	0.220995	−	0.0804356i	−0.229150	−	0.973391i	$-0.573595\pi$
$367$	29765.6	−	10833.8i	0.220995	−	0.0804356i	0.450145	+	0.892956i	$0.351372\pi$
$368$		0			0
$369$	130036.	−	179614.i	0.955019	−	1.31913i
$370$		0			0
$371$	−26680.5	−	4704.50i	−0.193841	−	0.0341795i
$372$		0			0
$373$	−136230.	−	49583.8i	−0.979167	−	0.356387i	−0.197650	−	0.980273i	$-0.563331\pi$
$373$	−136230.	−	49583.8i	−0.979167	−	0.356387i	−0.781516	+	0.623885i	$0.785553\pi$
$374$		0			0
$375$	−22404.3	+	160394.i	−0.159320	+	1.14058i
$376$		0			0
$377$		−	96997.8i		−	0.682463i
$378$		0			0
$379$	23022.1			0.160275			0.0801375	−	0.996784i	$-0.474464\pi$
$379$	23022.1			0.160275			0.0801375	+	0.996784i	$0.474464\pi$
$380$		0			0
$381$	−96831.0	−	239086.i	−0.667060	−	1.64704i
$382$		0			0
$383$	−3524.06	+	9682.28i	−0.0240240	+	0.0660055i	−0.951125	−	0.308806i	$-0.900070\pi$
$383$	−3524.06	+	9682.28i	−0.0240240	+	0.0660055i	0.927101	+	0.374812i	$0.122293\pi$
$384$		0			0
$385$	−2756.19	+	15631.1i	−0.0185946	+	0.105455i
$386$		0			0
$387$	156892.	+	231760.i	1.04756	+	1.54745i
$388$		0			0
$389$	7973.83	+	21907.9i	0.0526948	+	0.144778i	0.963248	−	0.268615i	$-0.0865659\pi$
$389$	7973.83	+	21907.9i	0.0526948	+	0.144778i	−0.910553	+	0.413393i	$0.864344\pi$
$390$		0			0
$391$	111420.	−	93492.4i	0.728802	−	0.611537i
$392$		0			0
$393$	32176.5	−	60638.2i	0.208331	−	0.392610i
$394$		0			0
$395$	−150410.	+	86839.2i	−0.964012	+	0.556573i
$396$		0			0
$397$	73141.1	−	126684.i	0.464067	−	0.803787i	−0.535092	−	0.844794i	$-0.679723\pi$
$397$	73141.1	−	126684.i	0.464067	−	0.803787i	0.999159	+	0.0410065i	$0.0130564\pi$
$398$		0			0
$399$	−55557.7	+	11858.0i	−0.348978	+	0.0744845i
$400$		0			0
$401$	8594.38	−	1515.42i	0.0534473	−	0.00942420i	−0.146861	−	0.989157i	$-0.546917\pi$
$401$	8594.38	−	1515.42i	0.0534473	−	0.00942420i	0.200308	+	0.979733i	$0.435806\pi$
$402$		0			0
$403$	14934.8	+	12531.8i	0.0919582	+	0.0771621i
$404$		0			0
$405$	−38413.4	−	266808.i	−0.234192	−	1.62663i
$406$		0			0
$407$	−14575.5	+	17370.3i	−0.0879900	+	0.104862i
$408$		0			0
$409$	36219.7	+	205412.i	0.216520	+	1.22795i	0.878249	+	0.478204i	$0.158712\pi$
$409$	36219.7	+	205412.i	0.216520	+	1.22795i	−0.661729	+	0.749743i	$0.730177\pi$
$410$		0			0
$411$	−52908.2	−	247888.i	−0.313213	−	1.46748i
$412$		0			0
$413$	−148824.	−	85923.6i	−0.872515	−	0.503747i
$414$		0			0
$415$	−216763.	−	375445.i	−1.25860	−	2.17997i
$416$		0			0
$417$	123879.	+	65733.9i	0.712401	+	0.378022i
$418$		0			0
$419$	−38386.2	−	45746.9i	−0.218649	−	0.260576i	0.645559	−	0.763710i	$-0.276624\pi$
$419$	−38386.2	−	45746.9i	−0.218649	−	0.260576i	−0.864208	+	0.503135i	$0.832180\pi$
$420$		0			0
$421$	98284.1	−	35772.5i	0.554523	−	0.201830i	−0.0495322	−	0.998773i	$-0.515773\pi$
$421$	98284.1	−	35772.5i	0.554523	−	0.201830i	0.604055	+	0.796943i	$0.293551\pi$
$422$		0			0
$423$	−525.209	−	7333.50i	−0.00293529	−	0.0409856i
$424$		0			0
$425$	211982.	+	37378.1i	1.17360	+	0.206937i
$426$		0			0
$427$	149422.	+	54385.3i	0.819521	+	0.298281i
$428$		0			0
$429$	11968.5	−	4847.31i	0.0650317	−	0.0263382i
$430$		0			0
$431$			92728.7i			0.499183i	0.968351	+	0.249591i	$0.0802963\pi$
$431$			92728.7i			0.499183i	−0.968351	+	0.249591i	$0.919704\pi$
$432$		0			0
$433$	48094.2			0.256517			0.128259	−	0.991741i	$-0.459061\pi$
$433$	48094.2			0.256517			0.128259	+	0.991741i	$0.459061\pi$
$434$		0			0
$435$	−298009.	−	41626.9i	−1.57489	−	0.219986i
$436$		0			0
$437$	−48314.5	+	132743.i	−0.252997	+	0.695103i
$438$		0			0
$439$	−4555.68	+	25836.5i	−0.0236387	+	0.134062i	−0.994343	−	0.106215i	$-0.966127\pi$
$439$	−4555.68	+	25836.5i	−0.0236387	+	0.134062i	0.970705	+	0.240276i	$0.0772381\pi$
$440$		0			0
$441$	11421.4	−	110455.i	0.0587276	−	0.567946i
$442$		0			0
$443$	−11637.5	−	31973.7i	−0.0592995	−	0.162924i	0.906505	−	0.422195i	$-0.138740\pi$
$443$	−11637.5	−	31973.7i	−0.0592995	−	0.162924i	−0.965805	+	0.259271i	$0.916518\pi$
$444$		0			0
$445$	−160287.	+	134496.i	−0.809426	+	0.679189i
$446$		0			0
$447$	−123376.	+	4412.32i	−0.617472	+	0.0220827i
$448$		0			0
$449$	−324556.	+	187382.i	−1.60989	+	0.929472i	−0.620500	+	0.784206i	$0.713070\pi$
$449$	−324556.	+	187382.i	−1.60989	+	0.929472i	−0.989393	+	0.145266i	$0.953596\pi$
$450$		0			0
$451$	16476.1	−	28537.5i	0.0810033	−	0.140302i
$452$		0			0
$453$	42501.4	−	131181.i	0.207113	−	0.639258i
$454$		0			0
$455$	−154787.	+	27293.2i	−0.747675	+	0.131835i
$456$		0			0
$457$	51282.0	+	43030.7i	0.245546	+	0.206038i	0.757252	−	0.653123i	$-0.226542\pi$
$457$	51282.0	+	43030.7i	0.245546	+	0.206038i	−0.511706	+	0.859161i	$0.670986\pi$
$458$		0			0
$459$	−146773.	+	15801.0i	−0.696658	+	0.0749998i
$460$		0			0
$461$	170802.	−	203553.i	0.803693	−	0.957804i	−0.196048	−	0.980594i	$-0.562811\pi$
$461$	170802.	−	203553.i	0.803693	−	0.957804i	0.999740	+	0.0227907i	$0.00725514\pi$
$462$		0			0
$463$	39061.0	+	221526.i	0.182214	+	1.03339i	0.929483	+	0.368865i	$0.120253\pi$
$463$	39061.0	+	221526.i	0.182214	+	1.03339i	−0.747269	+	0.664521i	$0.768635\pi$
$464$		0			0
$465$	44911.2	−	40506.7i	0.207706	−	0.187336i
$466$		0			0
$467$	187102.	+	108023.i	0.857916	+	0.495318i	0.863314	−	0.504667i	$-0.168385\pi$
$467$	187102.	+	108023.i	0.857916	+	0.495318i	−0.00539770	+	0.999985i	$0.501718\pi$
$468$		0			0
$469$	11762.6	+	20373.5i	0.0534760	+	0.0926231i
$470$		0			0
$471$	−335711.	+	210169.i	−1.51330	+	0.947384i
$472$		0			0
$473$	26733.7	+	31859.9i	0.119491	+	0.142404i
$474$		0			0
$475$	−196449.	+	71501.5i	−0.870687	+	0.316904i
$476$		0			0
$477$	66314.9	−	16652.7i	0.291457	−	0.0731895i
$478$		0			0
$479$	−23419.4	−	4129.47i	−0.102071	−	0.0179980i	0.122379	−	0.992483i	$-0.460948\pi$
$479$	−23419.4	−	4129.47i	−0.102071	−	0.0179980i	−0.224451	+	0.974485i	$0.572059\pi$
$480$		0			0
$481$	−211002.	−	76798.3i	−0.912002	−	0.331942i
$482$		0			0
$483$	163601.	+	127598.i	0.701281	+	0.546952i
$484$		0			0
$485$		−	640386.i		−	2.72244i
$486$		0			0
$487$	211780.			0.892951			0.446476	−	0.894796i	$-0.352679\pi$
$487$	211780.			0.892951			0.446476	+	0.894796i	$0.352679\pi$
$488$		0			0
$489$	−255349.	+	327399.i	−1.06787	+	1.36918i
$490$		0			0
$491$	104745.	−	287783.i	0.434479	−	1.19372i	−0.508556	−	0.861029i	$-0.669821\pi$
$491$	104745.	−	287783.i	0.434479	−	1.19372i	0.943035	−	0.332692i	$-0.107957\pi$
$492$		0			0
$493$	−28614.7	+	162282.i	−0.117732	+	0.667693i
$494$		0			0
$495$	−9756.22	−	38851.5i	−0.0398172	−	0.158561i
$496$		0			0
$497$	−42481.7	−	116718.i	−0.171985	−	0.472524i
$498$		0			0
$499$	−172810.	+	145005.i	−0.694015	+	0.582348i	−0.920064	−	0.391768i	$-0.871863\pi$
$499$	−172810.	+	145005.i	−0.694015	+	0.582348i	0.226049	+	0.974116i	$0.427419\pi$
$500$		0			0
$501$	−145265.	−	232037.i	−0.578741	−	0.924448i
$502$		0			0
$503$	−173648.	+	100256.i	−0.686333	+	0.396254i	−0.802237	−	0.597006i	$-0.796357\pi$
$503$	−173648.	+	100256.i	−0.686333	+	0.396254i	0.115904	+	0.993260i	$0.463023\pi$
$504$		0			0
$505$	279436.	−	483998.i	1.09572	−	1.89784i
$506$		0			0
$507$	−86518.6	−	95926.2i	−0.336584	−	0.373183i
$508$		0			0
$509$	−208089.	+	36691.7i	−0.803181	+	0.141622i	−0.560145	−	0.828395i	$-0.689255\pi$
$509$	−208089.	+	36691.7i	−0.803181	+	0.141622i	−0.243036	+	0.970017i	$0.578143\pi$
$510$		0			0
$511$	58973.6	+	49484.7i	0.225848	+	0.189509i
$512$		0			0
$513$	115777.	−	84564.5i	0.439936	−	0.321331i
$514$		0			0
$515$	387363.	−	461641.i	1.46051	−	1.74057i
$516$		0			0
$517$	−189.725	−	1075.98i	−0.000709812	−	0.00402555i
$518$		0			0
$519$	120934.	+	39181.4i	0.448966	+	0.145460i
$520$		0			0
$521$	162900.	+	94050.5i	0.600131	+	0.346486i	0.769093	−	0.639137i	$-0.220708\pi$
$521$	162900.	+	94050.5i	0.600131	+	0.346486i	−0.168962	+	0.985623i	$0.554041\pi$
$522$		0			0
$523$	−140142.	−	242732.i	−0.512346	−	0.887410i	−0.999898	−	0.0143156i	$-0.995443\pi$
$523$	−140142.	−	242732.i	−0.512346	−	0.887410i	0.487551	−	0.873094i	$-0.337890\pi$
$524$		0			0
$525$	10974.0	+	306852.i	0.0398149	+	1.11330i
$526$		0			0
$527$	−21289.8	−	25372.2i	−0.0766568	−	0.0913560i
$528$		0			0
$529$	221837.	−	80742.0i	0.792725	−	0.288528i
$530$		0			0
$531$	431400.	+	44608.3i	1.53000	+	0.158207i
$532$		0			0
$533$	321353.	+	56663.2i	1.13117	+	0.199456i
$534$		0			0
$535$	312614.	+	113782.i	1.09220	+	0.397527i
$536$		0			0
$537$	−8743.03	+	62591.9i	−0.0303189	+	0.217055i
$538$		0			0
$539$		−	16501.6i		−	0.0568000i
$540$		0			0
$541$	−438601.			−1.49856			−0.749281	−	0.662252i	$-0.769601\pi$
$541$	−438601.			−1.49856			−0.749281	+	0.662252i	$0.769601\pi$
$542$		0			0
$543$	−36864.8	−	91023.1i	−0.125030	−	0.308711i
$544$		0			0
$545$	188309.	−	517375.i	0.633984	−	1.74186i
$546$		0			0
$547$	−22416.7	+	127132.i	−0.0749200	+	0.424892i	0.924160	+	0.382006i	$0.124767\pi$
$547$	−22416.7	+	127132.i	−0.0749200	+	0.424892i	−0.999080	+	0.0428866i	$0.986345\pi$
$548$		0			0
$549$	−400282.	+	28667.3i	−1.32807	+	0.0951135i
$550$		0			0
$551$	−54737.9	−	150391.i	−0.180295	−	0.495358i
$552$		0			0
$553$	−103933.	+	87210.1i	−0.339862	+	0.285178i
$554$		0			0
$555$	−326502.	+	615309.i	−1.05999	+	1.99759i
$556$		0			0
$557$	−365276.	+	210892.i	−1.17737	+	0.679752i	−0.955404	−	0.295303i	$-0.904579\pi$
$557$	−365276.	+	210892.i	−1.17737	+	0.679752i	−0.221962	+	0.975055i	$0.571246\pi$
$558$		0			0
$559$	−205924.	+	356670.i	−0.658996	+	1.14141i
$560$		0			0
$561$	−21453.9	+	4579.03i	−0.0681679	+	0.0145495i
$562$		0			0
$563$	−270231.	+	47648.9i	−0.852546	+	0.150327i	−0.582811	−	0.812608i	$-0.698047\pi$
$563$	−270231.	+	47648.9i	−0.852546	+	0.150327i	−0.269735	+	0.962935i	$0.586936\pi$
$564$		0			0
$565$	681145.	+	571549.i	2.13375	+	1.79043i
$566$		0			0
$567$	−65744.9	−	200049.i	−0.204501	−	0.622258i
$568$		0			0
$569$	100787.	−	120113.i	0.311300	−	0.370993i	−0.587597	−	0.809154i	$-0.699926\pi$
$569$	100787.	−	120113.i	0.311300	−	0.370993i	0.898896	+	0.438162i	$0.144370\pi$
$570$		0			0
$571$	60411.6	+	342611.i	0.185288	+	1.05082i	0.925584	+	0.378541i	$0.123574\pi$
$571$	60411.6	+	342611.i	0.185288	+	1.05082i	−0.740296	+	0.672281i	$0.765315\pi$
$572$		0			0
$573$	89359.7	+	418672.i	0.272165	+	1.27516i
$574$		0			0
$575$	661220.	+	381756.i	1.99991	+	1.15465i
$576$		0			0
$577$	288340.	+	499420.i	0.866071	+	1.50008i	0.865980	+	0.500079i	$0.166696\pi$
$577$	288340.	+	499420.i	0.866071	+	1.50008i	9.13367e−5		1.00000i	$0.499971\pi$
$578$		0			0
$579$	−345782.	−	183483.i	−1.03144	−	0.547315i
$580$		0			0
$581$	−217689.	−	259432.i	−0.644888	−	0.768548i
$582$		0			0
$583$	9547.90	−	3475.15i	0.0280912	−	0.0102244i
$584$		0			0
$585$	328482.	−	222369.i	0.959843	−	0.649775i
$586$		0			0
$587$	318374.	+	56137.9i	0.923978	+	0.162922i	0.615344	−	0.788259i	$-0.289017\pi$
$587$	318374.	+	56137.9i	0.923978	+	0.162922i	0.308634	+	0.951181i	$0.400128\pi$
$588$		0			0
$589$	30227.8	+	11002.0i	0.0871318	+	0.0317134i
$590$		0			0
$591$	109870.	−	44497.9i	0.314560	−	0.127398i
$592$		0			0
$593$			101060.i			0.287388i	0.989622	+	0.143694i	$0.0458982\pi$
$593$			101060.i			0.287388i	−0.989622	+	0.143694i	$0.954102\pi$
$594$		0			0
$595$	267019.			0.754237
$596$		0			0
$597$	−311310.	−	43484.7i	−0.873463	−	0.122008i
$598$		0			0
$599$	−150516.	+	413539.i	−0.419497	+	1.15256i	0.532494	+	0.846434i	$0.321255\pi$
$599$	−150516.	+	413539.i	−0.419497	+	1.15256i	−0.951991	+	0.306125i	$0.900967\pi$
$600$		0			0
$601$	119609.	−	678335.i	0.331142	−	1.87800i	−0.131289	−	0.991344i	$-0.541912\pi$
$601$	119609.	−	678335.i	0.331142	−	1.87800i	0.462431	−	0.886655i	$-0.346977\pi$
$602$		0			0
$603$	−48091.6	−	34817.1i	−0.132262	−	0.0957543i
$604$		0			0
$605$	203698.	+	559656.i	0.556515	+	1.52901i
$606$		0			0
$607$	−109880.	+	92199.9i	−0.298222	+	0.250238i	−0.779604	−	0.626273i	$-0.784580\pi$
$607$	−109880.	+	92199.9i	−0.298222	+	0.250238i	0.481382	+	0.876511i	$0.340135\pi$
$608$		0			0
$609$	−234910.	+	8401.11i	−0.633384	+	0.0226518i
$610$		0			0
$611$	9369.80	−	5409.66i	0.0250985	−	0.0144906i
$612$		0			0
$613$	172287.	−	298409.i	0.458491	−	0.794130i	−0.540390	−	0.841414i	$-0.681723\pi$
$613$	172287.	−	298409.i	0.458491	−	0.794130i	0.998881	+	0.0472845i	$0.0150567\pi$
$614$		0			0
$615$	311998.	−	962987.i	0.824900	−	2.54607i
$616$		0			0
$617$	678405.	−	119621.i	1.78204	−	0.314223i	0.817065	−	0.576546i	$-0.195600\pi$
$617$	678405.	−	119621.i	1.78204	−	0.314223i	0.964980	+	0.262324i	$0.0844888\pi$
$618$		0			0
$619$	499982.	+	419535.i	1.30489	+	1.09493i	0.989278	+	0.146042i	$0.0466534\pi$
$619$	499982.	+	419535.i	1.30489	+	1.09493i	0.315610	+	0.948889i	$0.397791\pi$
$620$		0			0
$621$	−508385.	−	125393.i	−1.31828	−	0.325154i
$622$		0			0
$623$	−105066.	+	125213.i	−0.270700	+	0.322608i
$624$		0			0
$625$	13014.0	+	73805.8i	0.0333157	+	0.188943i
$626$		0			0
$627$	15821.2	−	14269.6i	0.0402444	−	0.0362976i
$628$		0			0
$629$	330361.	+	190734.i	0.835001	+	0.482088i
$630$		0			0
$631$	45688.2	+	79134.3i	0.114748	+	0.198749i	0.917679	−	0.397323i	$-0.130061\pi$
$631$	45688.2	+	79134.3i	0.114748	+	0.198749i	−0.802931	+	0.596072i	$0.796727\pi$
$632$		0			0
$633$	−415949.	+	260401.i	−1.03808	+	0.649882i
$634$		0			0
$635$	−756911.	−	902052.i	−1.87714	−	2.23709i
$636$		0			0
$637$	153553.	−	55888.6i	0.378424	−	0.137735i
$638$		0			0
$639$	218135.	+	225126.i	0.534223	+	0.551345i
$640$		0			0
$641$	−736459.	−	129858.i	−1.79239	−	0.316047i	−0.824206	−	0.566290i	$-0.808378\pi$
$641$	−736459.	−	129858.i	−1.79239	−	0.316047i	−0.968183	+	0.250243i	$0.919489\pi$
$642$		0			0
$643$	−378214.	−	137659.i	−0.914779	−	0.332952i	−0.158620	−	0.987340i	$-0.550704\pi$
$643$	−378214.	−	137659.i	−0.914779	−	0.332952i	−0.756159	+	0.654387i	$0.772927\pi$
$644$		0			0
$645$	1.00744e6	+	785732.i	2.42158	+	1.88867i
$646$		0			0
$647$			363926.i			0.869371i	0.900582	+	0.434685i	$0.143140\pi$
$647$			363926.i			0.869371i	−0.900582	+	0.434685i	$0.856860\pi$
$648$		0			0
$649$	64449.8			0.153014
$650$		0			0
$651$	29056.2	−	37254.8i	0.0685609	−	0.0879063i
$652$		0			0
$653$	−25486.2	+	70022.7i	−0.0597693	+	0.164215i	−0.965983	−	0.258605i	$-0.916737\pi$
$653$	−25486.2	+	70022.7i	−0.0597693	+	0.164215i	0.906214	+	0.422819i	$0.138960\pi$
$654$		0			0
$655$	54416.3	−	308610.i	0.126837	−	0.719330i
$656$		0			0
$657$	−186853.	−	53239.3i	−0.432883	−	0.123339i
$658$		0			0
$659$	−201632.	−	553979.i	−0.464289	−	1.27562i	−0.922230	−	0.386642i	$-0.873635\pi$
$659$	−201632.	−	553979.i	−0.464289	−	1.27562i	0.457941	−	0.888983i	$-0.348587\pi$
$660$		0			0
$661$	−101124.	+	84853.5i	−0.231448	+	0.194208i	−0.751135	−	0.660149i	$-0.770493\pi$
$661$	−101124.	+	84853.5i	−0.231448	+	0.194208i	0.519687	+	0.854357i	$0.326049\pi$
$662$		0			0
$663$	−115271.	−	184127.i	−0.262236	−	0.418880i
$664$		0			0
$665$	−224590.	+	129667.i	−0.507863	+	0.293215i
$666$		0			0
$667$	−292253.	+	506196.i	−0.656911	+	1.13780i
$668$		0			0
$669$	202454.	+	224468.i	0.452350	+	0.501536i
$670$		0			0
$671$	−58730.1	+	10355.7i	−0.130441	+	0.0230003i
$672$		0			0
$673$	338696.	+	284200.i	0.747791	+	0.627471i	0.934918	−	0.354865i	$-0.115473\pi$
$673$	338696.	+	284200.i	0.747791	+	0.627471i	−0.187127	+	0.982336i	$0.559918\pi$
$674$		0			0
$675$	−341457.	−	695629.i	−0.749426	−	1.52676i
$676$		0			0
$677$	−158293.	+	188647.i	−0.345370	+	0.411596i	−0.910568	−	0.413359i	$-0.864355\pi$
$677$	−158293.	+	188647.i	−0.345370	+	0.411596i	0.565198	+	0.824955i	$0.308800\pi$
$678$		0			0
$679$	−86869.1	−	492659.i	−0.188420	−	1.06858i
$680$		0			0
$681$	−54903.3	−	17788.1i	−0.118387	−	0.0383562i
$682$		0			0
$683$	385117.	+	222347.i	0.825565	+	0.476640i	0.852332	−	0.523002i	$-0.175188\pi$
$683$	385117.	+	222347.i	0.825565	+	0.476640i	−0.0267669	+	0.999642i	$0.508521\pi$
$684$		0			0
$685$	−578550.	−	1.00208e6i	−1.23299	−	2.13560i
$686$		0			0
$687$	5458.75	+	152636.i	0.0115659	+	0.323404i
$688$		0			0
$689$	64674.8	+	77076.4i	0.136237	+	0.162361i
$690$		0			0
$691$	−260760.	+	94909.0i	−0.546117	+	0.198770i	−0.600320	−	0.799760i	$-0.704960\pi$
$691$	−260760.	+	94909.0i	−0.546117	+	0.198770i	0.0542035	+	0.998530i	$0.482738\pi$
$692$		0			0
$693$	−12775.9	−	28565.6i	−0.0266026	−	0.0594809i
$694$		0			0
$695$	630465.	+	111168.i	1.30524	+	0.230149i
$696$		0			0
$697$	−520924.	−	189601.i	−1.07228	−	0.390279i
$698$		0			0
$699$	−26123.7	+	187021.i	−0.0534664	+	0.382769i
$700$		0			0
$701$		−	589634.i		−	1.19990i	−0.800036	−	0.599952i	$-0.795186\pi$
$701$		−	589634.i		−	1.19990i	0.800036	−	0.599952i	$-0.204814\pi$
$702$		0			0
$703$	−370489.			−0.749660
$704$		0			0
$705$	−12599.2	−	31108.7i	−0.0253492	−	0.0625899i
$706$		0			0
$707$	149320.	−	410253.i	0.298730	−	0.820755i
$708$		0			0
$709$	17459.5	−	99017.7i	0.0347327	−	0.196979i	−0.962504	−	0.271267i	$-0.912557\pi$
$709$	17459.5	−	99017.7i	0.0347327	−	0.196979i	0.997237	+	0.0742880i	$0.0236684\pi$
$710$		0			0
$711$	149585.	−	308008.i	0.295902	−	0.609289i
$712$		0			0
$713$	−40181.4	−	110397.i	−0.0790398	−	0.217160i
$714$		0			0
$715$	45156.2	−	37890.5i	0.0883294	−	0.0741172i
$716$		0			0
$717$	−72859.2	+	137307.i	−0.141725	+	0.267088i
$718$		0			0
$719$	−706697.	+	408012.i	−1.36702	+	0.789250i	−0.990547	−	0.137176i	$-0.956197\pi$
$719$	−706697.	+	408012.i	−1.36702	+	0.789250i	−0.376475	+	0.926427i	$0.622864\pi$
$720$		0			0
$721$	235383.	−	407695.i	0.452797	−	0.784268i
$722$		0			0
$723$	312837.	−	66770.7i	0.598470	−	0.127735i
$724$		0			0
$725$	−851878.	+	150209.i	−1.62069	+	0.285772i
$726$		0			0
$727$	−295988.	−	248363.i	−0.560022	−	0.469914i	0.318296	−	0.947991i	$-0.396889\pi$
$727$	−295988.	−	248363.i	−0.560022	−	0.469914i	−0.878318	+	0.478077i	$0.841334\pi$
$728$		0			0
$729$	357592.	+	393138.i	0.672873	+	0.739758i
$730$		0			0
$731$	449740.	−	535979.i	0.841640	−	1.00303i
$732$		0			0
$733$	−21566.7	−	122311.i	−0.0401399	−	0.227645i	0.958138	−	0.286307i	$-0.0924277\pi$
$733$	−21566.7	−	122311.i	−0.0401399	−	0.227645i	−0.998278	+	0.0586621i	$0.981317\pi$
$734$		0			0
$735$	−105811.	−	495749.i	−0.195864	−	0.917672i
$736$		0			0
$737$	−7640.90	−	4411.47i	−0.0140673	−	0.00812173i
$738$		0			0
$739$	42912.2	+	74326.1i	0.0785763	+	0.136098i	0.902636	−	0.430405i	$-0.141629\pi$
$739$	42912.2	+	74326.1i	0.0785763	+	0.136098i	−0.824060	+	0.566503i	$0.808296\pi$
$740$		0			0
$741$	186368.	+	98892.5i	0.339418	+	0.180106i
$742$		0			0
$743$	146820.	+	174974.i	0.265956	+	0.316953i	0.882450	−	0.470405i	$-0.155892\pi$
$743$	146820.	+	174974.i	0.265956	+	0.316953i	−0.616495	+	0.787359i	$0.711448\pi$
$744$		0			0
$745$	−529587.	+	192754.i	−0.954167	+	0.347288i
$746$		0			0
$747$	768834.	+	373385.i	1.37782	+	0.669138i
$748$		0			0
$749$	255934.	+	45128.0i	0.456209	+	0.0804419i
$750$		0			0
$751$	−518723.	−	188800.i	−0.919719	−	0.334750i	−0.161592	−	0.986858i	$-0.551663\pi$
$751$	−518723.	−	188800.i	−0.919719	−	0.334750i	−0.758127	+	0.652107i	$0.773885\pi$
$752$		0			0
$753$	523370.	−	211967.i	0.923036	−	0.373834i
$754$		0			0
$755$		−	629490.i		−	1.10432i
$756$		0			0
$757$	−291561.			−0.508788			−0.254394	−	0.967101i	$-0.581876\pi$
$757$	−291561.			−0.508788			−0.254394	+	0.967101i	$0.581876\pi$
$758$		0			0
$759$	−77064.2	−	10764.6i	−0.133773	−	0.0186858i
$760$		0			0
$761$	40839.4	−	112205.i	0.0705197	−	0.193751i	−0.899426	−	0.437073i	$-0.856015\pi$
$761$	40839.4	−	112205.i	0.0705197	−	0.193751i	0.969946	+	0.243322i	$0.0782372\pi$
$762$		0			0
$763$	74686.8	−	423570.i	0.128291	−	0.727572i
$764$		0			0
$765$	−615167.	+	275131.i	−1.05116	+	0.470129i
$766$		0			0
$767$	218283.	+	599726.i	0.371046	+	1.01944i
$768$		0			0
$769$	−625266.	+	524661.i	−1.05733	+	0.887209i	−0.993846	−	0.110774i	$-0.964667\pi$
$769$	−625266.	+	524661.i	−1.05733	+	0.887209i	−0.0634883	+	0.997983i	$0.520223\pi$
$770$		0			0
$771$	−608856.	+	21774.6i	−1.02425	+	0.0366303i
$772$		0			0
$773$	−893014.	+	515582.i	−1.49451	+	0.862856i	−0.999980	−	0.00630343i	$-0.997994\pi$
$773$	−893014.	+	515582.i	−1.49451	+	0.862856i	−0.494531	+	0.869160i	$0.664660\pi$
$774$		0			0
$775$	86932.2	−	150571.i	0.144736	−	0.250691i
$776$		0			0
$777$	−167716.	+	517658.i	−0.277800	+	0.857434i
$778$		0			0
$779$	530221.	−	93492.4i	0.873740	−	0.154064i
$780$		0			0
$781$	35684.8	+	29943.1i	0.0585034	+	0.0490902i
$782$		0			0
$783$	532538.	−	261402.i	0.868615	−	0.426369i
$784$		0			0
$785$	−1.16221e6	+	1.38506e6i	−1.88601	+	2.24766i
$786$		0			0
$787$	35062.6	+	198850.i	0.0566103	+	0.321053i	0.999942	−	0.0107993i	$-0.00343760\pi$
$787$	35062.6	+	198850.i	0.0566103	+	0.321053i	−0.943331	+	0.331852i	$0.892326\pi$
$788$		0			0
$789$	821607.	−	741031.i	1.31981	−	1.19037i
$790$		0			0
$791$	601548.	+	347304.i	0.961429	+	0.555081i
$792$		0			0
$793$	−295274.	−	511429.i	−0.469546	−	0.813278i
$794$		0			0
$795$	264560.	−	165625.i	0.418591	−	0.262054i
$796$		0			0
$797$	400366.	+	477137.i	0.630290	+	0.751150i	0.982803	−	0.184657i	$-0.0591173\pi$
$797$	400366.	+	477137.i	0.630290	+	0.751150i	−0.352513	+	0.935807i	$0.614673\pi$
$798$		0			0
$799$	−17272.0	+	6286.50i	−0.0270551	+	0.00984726i
$800$		0			0
$801$	113038.	−	396729.i	0.176182	−	0.618343i
$802$		0			0
$803$	−28433.7	−	5013.64i	−0.0440964	−	0.00777538i
$804$		0			0
$805$	890014.	+	323938.i	1.37342	+	0.499886i
$806$		0			0
$807$	280606.	+	218853.i	0.430873	+	0.336052i
$808$		0			0
$809$		−	32342.0i		−	0.0494163i	−0.999695	−	0.0247081i	$-0.992134\pi$
$809$		−	32342.0i		−	0.0494163i	0.999695	−	0.0247081i	$-0.00786564\pi$
$810$		0			0
$811$	411096.			0.625032			0.312516	−	0.949913i	$-0.398828\pi$
$811$	411096.			0.625032			0.312516	+	0.949913i	$0.398828\pi$
$812$		0			0
$813$	−154907.	+	198616.i	−0.234364	+	0.300493i
$814$		0			0
$815$	−648267.	+	1.78110e6i	−0.975975	+	2.68147i
$816$		0			0
$817$	−118000.	+	669210.i	−0.176782	+	1.00258i
$818$		0			0
$819$	222542.	−	215631.i	0.331776	−	0.321473i
$820$		0			0
$821$	−111388.	−	306037.i	−0.165255	−	0.454033i	0.829231	−	0.558906i	$-0.188779\pi$
$821$	−111388.	−	306037.i	−0.165255	−	0.454033i	−0.994486	+	0.104873i	$0.966557\pi$
$822$		0			0
$823$	540305.	−	453370.i	0.797700	−	0.669349i	−0.149939	−	0.988695i	$-0.547908\pi$
$823$	540305.	−	453370.i	0.797700	−	0.669349i	0.947638	+	0.319346i	$0.103463\pi$
$824$		0			0
$825$	−61105.4	−	97606.3i	−0.0897784	−	0.143407i
$826$		0			0
$827$	29454.2	−	17005.4i	0.0430662	−	0.0248643i	−0.478312	−	0.878190i	$-0.658751\pi$
$827$	29454.2	−	17005.4i	0.0430662	−	0.0248643i	0.521379	+	0.853326i	$0.325418\pi$
$828$		0			0
$829$	517682.	−	896652.i	0.753276	−	1.30471i	−0.192951	−	0.981208i	$-0.561806\pi$
$829$	517682.	−	896652.i	0.753276	−	1.30471i	0.946227	−	0.323504i	$-0.104861\pi$
$830$		0			0
$831$	−595363.	−	660101.i	−0.862145	−	0.955890i
$832$		0			0
$833$	−273388.	+	48205.8i	−0.393995	+	0.0694719i
$834$		0			0
$835$	−957330.	−	803295.i	−1.37306	−	1.15213i
$836$		0			0
$837$	−28554.0	+	115768.i	−0.0407583	+	0.165248i
$838$		0			0
$839$	696381.	−	829915.i	0.989289	−	1.17899i	0.00544039	−	0.999985i	$-0.498268\pi$
$839$	696381.	−	829915.i	0.989289	−	1.17899i	0.983848	−	0.179003i	$-0.0572873\pi$
$840$		0			0
$841$	7825.75	+	44382.1i	0.0110646	+	0.0627502i
$842$		0			0
$843$	−170958.	−	55388.6i	−0.240566	−	0.0779409i
$844$		0			0
$845$	−510700.	−	294853.i	−0.715240	−	0.412944i
$846$		0			0
$847$	232627.	+	402921.i	0.324259	+	0.561634i
$848$		0			0
$849$	32731.3	+	915227.i	0.0454096	+	1.26974i
$850$		0			0
$851$	869749.	+	1.03653e6i	1.20098	+	1.43127i
$852$		0			0
$853$	227061.	−	82643.5i	0.312065	−	0.113582i	−0.181240	−	0.983439i	$-0.558011\pi$
$853$	227061.	−	82643.5i	0.312065	−	0.113582i	0.493305	+	0.869857i	$0.335789\pi$
$854$		0			0
$855$	383811.	−	530144.i	0.525031	−	0.725206i
$856$		0			0
$857$	−526074.	−	92761.0i	−0.716284	−	0.126300i	−0.196384	−	0.980527i	$-0.562920\pi$
$857$	−526074.	−	92761.0i	−0.716284	−	0.126300i	−0.519900	+	0.854227i	$0.674031\pi$
$858$		0			0
$859$	932319.	+	339336.i	1.26351	+	0.459879i	0.884945	−	0.465696i	$-0.154196\pi$
$859$	932319.	+	339336.i	1.26351	+	0.459879i	0.378563	+	0.925575i	$0.376418\pi$
$860$		0			0
$861$	109395.	−	783164.i	0.147567	−	1.05644i
$862$		0			0
$863$		−	628886.i		−	0.844404i	−0.906502	−	0.422202i	$-0.861257\pi$
$863$		−	628886.i		−	0.844404i	0.906502	−	0.422202i	$-0.138743\pi$
$864$		0			0
$865$	580316.			0.775590
$866$		0			0
$867$	−143638.	−	354658.i	−0.191088	−	0.471815i
$868$		0			0
$869$	17403.3	−	47815.0i	0.0230457	−	0.0633177i
$870$		0			0
$871$	15171.5	−	86042.0i	0.0199983	−	0.113416i
$872$		0			0
$873$	707759.	+	1.04550e6i	0.928661	+	1.37181i
$874$		0			0
$875$	197529.	+	542707.i	0.257997	+	0.708842i
$876$		0			0
$877$	627237.	−	526315.i	0.815516	−	0.684300i	−0.136401	−	0.990654i	$-0.543554\pi$
$877$	627237.	−	526315.i	0.815516	−	0.684300i	0.951918	+	0.306354i	$0.0991092\pi$
$878$		0			0
$879$	−105731.	+	199256.i	−0.136844	+	0.257889i
$880$		0			0
$881$	−524079.	+	302577.i	−0.675220	+	0.389838i	−0.798051	−	0.602589i	$-0.794136\pi$
$881$	−524079.	+	302577.i	−0.675220	+	0.389838i	0.122832	+	0.992427i	$0.460802\pi$
$882$		0			0
$883$	−406102.	+	703389.i	−0.520851	+	0.902140i	0.478855	+	0.877894i	$0.341052\pi$
$883$	−406102.	+	703389.i	−0.520851	+	0.902140i	−0.999706	+	0.0242463i	$0.992281\pi$
$884$		0			0
$885$	1.93624e6	−	413262.i	2.47213	−	0.527642i
$886$		0			0
$887$	82766.8	−	14594.0i	0.105198	−	0.0185493i	−0.120801	−	0.992677i	$-0.538546\pi$
$887$	82766.8	−	14594.0i	0.105198	−	0.0185493i	0.225999	+	0.974127i	$0.427435\pi$
$888$		0			0
$889$	−704669.	−	591287.i	−0.891624	−	0.748161i
$890$		0			0
$891$	58867.0	+	52646.5i	0.0741509	+	0.0663153i
$892$		0			0
$893$	11474.7	−	13675.0i	0.0143893	−	0.0171485i
$894$		0			0
$895$	50098.6	+	284123.i	0.0625432	+	0.354700i
$896$		0			0
$897$	−160838.	−	753565.i	−0.199896	−	0.936560i
$898$		0			0
$899$	115269.	+	66550.8i	0.142625	+	0.0823444i
$900$		0			0
$901$	−85466.3	−	148032.i	−0.105280	−	0.182350i
$902$		0			0
$903$	881623.	+	467817.i	1.08120	+	0.573720i
$904$		0			0
$905$	−288166.	−	343423.i	−0.351840	−	0.419307i
$906$		0			0
$907$	60016.5	−	21844.2i	0.0729552	−	0.0265535i	−0.305285	−	0.952261i	$-0.598752\pi$
$907$	60016.5	−	21844.2i	0.0729552	−	0.0265535i	0.378240	+	0.925708i	$0.376529\pi$
$908$		0			0
$909$	78708.8	+	1.09901e6i	0.0952567	+	1.33007i
$910$		0			0
$911$	811994.	+	143176.i	0.978399	+	0.172518i	0.639908	−	0.768451i	$-0.278972\pi$
$911$	811994.	+	143176.i	0.978399	+	0.172518i	0.338491	+	0.940970i	$0.390083\pi$
$912$		0			0
$913$	119353.	+	43441.0i	0.143183	+	0.0521145i
$914$		0			0
$915$	−1.69800e6	+	687698.i	−2.02813	+	0.821401i
$916$		0			0
$917$		−	244801.i		−	0.291121i
$918$		0			0
$919$	−1.09322e6			−1.29442			−0.647211	−	0.762311i	$-0.724065\pi$
$919$	−1.09322e6			−1.29442			−0.647211	+	0.762311i	$0.724065\pi$
$920$		0			0
$921$	−626782.	−	87550.8i	−0.738919	−	0.103214i
$922$		0			0
$923$	−157771.	+	433472.i	−0.185192	+	0.508812i
$924$		0			0
$925$	−347724.	+	1.97204e6i	−0.406397	+	2.30479i
$926$		0			0
$927$	−122202.	+	1.18180e6i	−0.142206	+	1.37525i
$928$		0			0
$929$	62157.0	+	170775.i	0.0720209	+	0.197876i	0.970480	−	0.241181i	$-0.0775348\pi$
$929$	62157.0	+	170775.i	0.0720209	+	0.197876i	−0.898459	+	0.439057i	$0.855313\pi$
$930$		0			0
$931$	206538.	−	173306.i	0.238287	−	0.199947i
$932$		0			0
$933$	695232.	−	24863.6i	0.798668	−	0.0285628i
$934$		0			0
$935$	−86726.5	+	50071.5i	−0.0992038	+	0.0572753i
$936$		0			0
$937$	−288933.	+	500447.i	−0.329093	+	0.570006i	−0.982332	−	0.187146i	$-0.940076\pi$
$937$	−288933.	+	500447.i	−0.329093	+	0.570006i	0.653239	+	0.757152i	$0.273410\pi$
$938$		0			0
$939$	−422140.	+	1.30294e6i	−0.478769	+	1.47773i
$940$		0			0
$941$	1.47102e6	−	259381.i	1.66127	−	0.292927i	0.737351	−	0.675510i	$-0.236077\pi$
$941$	1.47102e6	−	259381.i	1.66127	−	0.292927i	0.923921	+	0.382583i	$0.124965\pi$
$942$		0			0
$943$	−1.50630e6	−	1.26394e6i	−1.69390	−	1.42135i
$944$		0			0
$945$	−566986.	−	776262.i	−0.634905	−	0.869250i
$946$		0			0
$947$	−9993.66	+	11910.0i	−0.0111436	+	0.0132804i	−0.771587	−	0.636123i	$-0.780537\pi$
$947$	−9993.66	+	11910.0i	−0.0111436	+	0.0132804i	0.760444	+	0.649404i	$0.224981\pi$
$948$		0			0
$949$	−49647.6	−	281566.i	−0.0551272	−	0.312642i
$950$		0			0
$951$	−86795.1	+	78283.0i	−0.0959698	+	0.0865579i
$952$		0			0
$953$	927031.	+	535222.i	1.02072	+	0.589316i	0.914314	−	0.405007i	$-0.132731\pi$
$953$	927031.	+	535222.i	1.02072	+	0.589316i	0.106411	+	0.994322i	$0.466064\pi$
$954$		0			0
$955$	977145.	+	1.69247e6i	1.07140	+	1.85572i
$956$		0			0
$957$	74722.4	−	46779.2i	0.0815881	−	0.0510774i
$958$		0			0
$959$	−581021.	−	692434.i	−0.631763	−	0.752906i
$960$		0			0
$961$	842687.	−	306713.i	0.912471	−	0.332112i
$962$		0			0
$963$	−636128.	+	159742.i	−0.685949	+	0.172253i
$964$		0			0
$965$	−1.75981e6	−	310302.i	−1.88978	−	0.333220i
$966$		0			0
$967$	−1.17361e6	−	427160.i	−1.25508	−	0.456812i	−0.372966	−	0.927845i	$-0.621659\pi$
$967$	−1.17361e6	−	427160.i	−1.25508	−	0.456812i	−0.882116	+	0.471033i	$0.843881\pi$
$968$		0			0
$969$	−282629.	−	220431.i	−0.301002	−	0.234761i
$970$		0			0
$971$		−	579172.i		−	0.614284i	−0.951664	−	0.307142i	$-0.900627\pi$
$971$		−	579172.i		−	0.614284i	0.951664	−	0.307142i	$-0.0993727\pi$
$972$		0			0
$973$	500107.			0.528247
$974$		0			0
$975$	701303.	−	899185.i	0.737728	−	0.945888i
$976$		0			0
$977$	450926.	−	1.23891e6i	0.472407	−	1.29793i	−0.443405	−	0.896321i	$-0.646230\pi$
$977$	450926.	−	1.23891e6i	0.472407	−	1.29793i	0.915812	−	0.401607i	$-0.131548\pi$
$978$		0			0
$979$	10645.0	−	60370.8i	0.0111066	−	0.0629886i
$980$		0			0
$981$	264372.	+	1.05279e6i	0.274712	+	1.09397i
$982$		0			0
$983$	−439089.	−	1.20639e6i	−0.454408	−	1.24848i	−0.929592	−	0.368589i	$-0.879841\pi$
$983$	−439089.	−	1.20639e6i	−0.454408	−	1.24848i	0.475185	−	0.879886i	$-0.342381\pi$
$984$		0			0
$985$	414530.	−	347832.i	0.427252	−	0.358507i
$986$		0			0
$987$	−13912.7	−	22223.4i	−0.0142816	−	0.0228126i
$988$		0			0
$989$	2.14928e6	−	1.24089e6i	2.19736	−	1.26865i
$990$		0			0
$991$	−656993.	+	1.13795e6i	−0.668981	+	1.15871i	0.309209	+	0.950994i	$0.399936\pi$
$991$	−656993.	+	1.13795e6i	−0.668981	+	1.15871i	−0.978190	+	0.207714i	$0.933398\pi$
$992$		0			0
$993$	−192365.	−	213282.i	−0.195087	−	0.216300i
$994$		0			0
$995$	−1.41313e6	+	249173.i	−1.42737	+	0.251683i
$996$		0			0
$997$	553254.	+	464235.i	0.556589	+	0.467033i	0.877165	−	0.480189i	$-0.159432\pi$
$997$	553254.	+	464235.i	0.556589	+	0.467033i	−0.320576	+	0.947223i	$0.603877\pi$
$998$		0			0
$999$	−146995.	−	1.36541e6i	−0.147290	−	1.36814i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.65.4	yes	72
3.2	odd	2		324.5.k.a.197.12		72
27.5	odd	18	inner	108.5.k.a.5.4	✓	72
27.22	even	9		324.5.k.a.125.12		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.4	✓	72	27.5	odd	18	inner
108.5.k.a.65.4	yes	72	1.1	even	1	trivial
324.5.k.a.125.12		72	27.22	even	9
324.5.k.a.197.12		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-5.53499 + 7.09676i) q^{3} +(-14.0519 + 38.6073i) q^{5} +(-5.57324 + 31.6074i) q^{7} +(-19.7279 - 78.5609i) q^{9} +O(q^{10})\)	\(q+(-5.53499 + 7.09676i) q^{3} +(-14.0519 + 38.6073i) q^{5} +(-5.57324 + 31.6074i) q^{7} +(-19.7279 - 78.5609i) q^{9} +(-4.11688 - 11.3110i) q^{11} +(91.3096 - 76.6178i) q^{13} +(-196.210 - 313.414i) q^{15} +(-175.368 + 101.249i) q^{17} +(98.3348 - 170.321i) q^{19} +(-193.462 - 214.499i) q^{21} +(-707.360 + 124.727i) q^{23} +(-814.293 - 683.273i) q^{25} +(666.721 + 294.829i) q^{27} +(523.078 - 623.381i) q^{29} +(28.4024 + 161.078i) q^{31} +(103.059 + 33.3899i) q^{33} +(-1141.96 - 659.313i) q^{35} +(-941.906 - 1631.43i) q^{37} +(38.3409 + 1072.08i) q^{39} +(1759.69 + 2097.12i) q^{41} +(-3246.83 + 1181.75i) q^{43} +(3310.24 + 342.290i) q^{45} +(89.3900 + 15.7619i) q^{47} +(1288.23 + 468.879i) q^{49} +(252.122 - 1804.96i) q^{51} +844.122i q^{53} +494.540 q^{55} +(664.444 + 1640.58i) q^{57} +(-1831.29 + 5031.42i) q^{59} +(860.325 - 4879.15i) q^{61} +(2593.06 - 185.709i) q^{63} +(1674.94 + 4601.85i) q^{65} +(561.501 - 471.156i) q^{67} +(3030.07 - 5710.32i) q^{69} +(-3351.53 + 1935.01i) q^{71} +(1199.32 - 2077.29i) q^{73} +(9356.12 - 1996.93i) q^{75} +(380.457 - 67.0849i) q^{77} +(3238.29 + 2717.25i) q^{79} +(-5782.62 + 3099.68i) q^{81} +(-6782.64 + 8083.24i) q^{83} +(-1444.69 - 8193.24i) q^{85} +(1528.75 + 7162.56i) q^{87} +(4410.52 + 2546.41i) q^{89} +(1912.80 + 3313.07i) q^{91} +(-1300.34 - 689.999i) q^{93} +(5193.84 + 6189.78i) q^{95} +(-14646.8 + 5331.01i) q^{97} +(-807.388 + 546.569i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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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