LMFDB - Embedded newform 112.6.p.b.47.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 $ x^14 - 3x^13 + 691x^12 - 8602x^11 + 416261x^10 - 3521447x^9 + 66162087x^8 - 31847546x^7 + 3819865057x^6 + 14320412589x^5 + 280505627703x^4 + 1315235310354x^3 + 6574080917124x^2 + 11537944725888*x + 17213603549184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{6}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.7
Root			$7.14120 - 12.3689i$ of defining polynomial
Character	$\chi$	$=$	112.47
Dual form			112.6.p.b.31.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(11.5627 + 20.0272i) q^{3} +(-75.7872 - 43.7557i) q^{5} +(90.8476 - 92.4863i) q^{7} +(-145.892 + 252.692i) q^{9} +(350.920 - 202.604i) q^{11} -640.275i q^{13} -2023.74i q^{15} +(1286.81 - 742.939i) q^{17} +(-699.973 + 1212.39i) q^{19} +(2902.68 + 750.029i) q^{21} +(1039.79 + 600.322i) q^{23} +(2266.63 + 3925.92i) q^{25} -1128.15 q^{27} -1968.20 q^{29} +(-406.957 - 704.870i) q^{31} +(8115.18 + 4685.30i) q^{33} +(-10931.9 + 3034.17i) q^{35} +(6956.39 - 12048.8i) q^{37} +(12822.9 - 7403.31i) q^{39} +17041.2i q^{41} -19253.7i q^{43} +(22113.5 - 12767.2i) q^{45} +(11325.4 - 19616.1i) q^{47} +(-300.442 - 16804.3i) q^{49} +(29757.9 + 17180.8i) q^{51} +(8009.36 + 13872.6i) q^{53} -35460.4 q^{55} -32374.3 q^{57} +(-15061.0 - 26086.5i) q^{59} +(-37809.1 - 21829.1i) q^{61} +(10116.7 + 36449.5i) q^{63} +(-28015.7 + 48524.7i) q^{65} +(9185.38 - 5303.18i) q^{67} +27765.4i q^{69} +57405.1i q^{71} +(56304.1 - 32507.2i) q^{73} +(-52416.7 + 90788.5i) q^{75} +(13142.2 - 50861.4i) q^{77} +(15991.3 + 9232.58i) q^{79} +(22407.3 + 38810.6i) q^{81} -17093.5 q^{83} -130031. q^{85} +(-22757.7 - 39417.4i) q^{87} +(-40872.2 - 23597.6i) q^{89} +(-59216.7 - 58167.5i) q^{91} +(9411.04 - 16300.4i) q^{93} +(106098. - 61255.7i) q^{95} -52847.4i q^{97} +118233. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) $ 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	11.5627	+	20.0272i	0.741748	+	1.28474i	0.951699	+	0.307033i	$0.0993361\pi$
$3$	11.5627	+	20.0272i	0.741748	+	1.28474i	−0.209951	+	0.977712i	$0.567331\pi$
$4$		0			0
$5$	−75.7872	−	43.7557i	−1.35572	−	0.782727i	−0.366678	−	0.930348i	$-0.619505\pi$
$5$	−75.7872	−	43.7557i	−1.35572	−	0.782727i	−0.989044	+	0.147621i	$0.952838\pi$
$6$		0			0
$7$	90.8476	−	92.4863i	0.700758	−	0.713399i
$7$	90.8476	−	92.4863i	0.700758	−	0.713399i
$8$		0			0
$9$	−145.892	+	252.692i	−0.600379	+	1.03989i
$10$		0			0
$11$	350.920	−	202.604i	0.874434	−	0.504855i	0.00561475	−	0.999984i	$-0.498213\pi$
$11$	350.920	−	202.604i	0.874434	−	0.504855i	0.868819	+	0.495130i	$0.164879\pi$
$12$		0			0
$13$		−	640.275i		−	1.05077i	−0.850864	−	0.525386i	$-0.823921\pi$
$13$		−	640.275i		−	1.05077i	0.850864	−	0.525386i	$-0.176079\pi$
$14$		0			0
$15$		−	2023.74i		−	2.32234i
$16$		0			0
$17$	1286.81	−	742.939i	1.07992	−	0.623492i	0.149045	−	0.988830i	$-0.452380\pi$
$17$	1286.81	−	742.939i	1.07992	−	0.623492i	0.930875	+	0.365339i	$0.119047\pi$
$18$		0			0
$19$	−699.973	+	1212.39i	−0.444833	+	0.770474i	−0.998041	−	0.0625698i	$-0.980070\pi$
$19$	−699.973	+	1212.39i	−0.444833	+	0.770474i	0.553207	+	0.833044i	$0.313404\pi$
$20$		0			0
$21$	2902.68	+	750.029i	1.43632	+	0.371133i
$22$		0			0
$23$	1039.79	+	600.322i	0.409850	+	0.236627i	0.690725	−	0.723117i	$-0.257291\pi$
$23$	1039.79	+	600.322i	0.409850	+	0.236627i	−0.280875	+	0.959744i	$0.590625\pi$
$24$		0			0
$25$	2266.63	+	3925.92i	0.725322	+	1.25629i
$26$		0			0
$27$	−1128.15			−0.297823
$28$		0			0
$29$	−1968.20			−0.434584			−0.217292	−	0.976107i	$-0.569722\pi$
$29$	−1968.20			−0.434584			−0.217292	+	0.976107i	$0.569722\pi$
$30$		0			0
$31$	−406.957	−	704.870i	−0.0760579	−	0.131736i	0.825488	−	0.564420i	$-0.190900\pi$
$31$	−406.957	−	704.870i	−0.0760579	−	0.131736i	−0.901546	+	0.432684i	$0.857567\pi$
$32$		0			0
$33$	8115.18	+	4685.30i	1.29722	+	0.748949i
$34$		0			0
$35$	−10931.9	+	3034.17i	−1.50843	+	0.418669i
$36$		0			0
$37$	6956.39	−	12048.8i	0.835372	−	1.44691i	−0.0583560	−	0.998296i	$-0.518586\pi$
$37$	6956.39	−	12048.8i	0.835372	−	1.44691i	0.893728	−	0.448610i	$-0.148081\pi$
$38$		0			0
$39$	12822.9	−	7403.31i	1.34997	−	0.779408i
$40$		0			0
$41$			17041.2i			1.58322i	0.611026	+	0.791611i	$0.290757\pi$
$41$			17041.2i			1.58322i	−0.611026	+	0.791611i	$0.709243\pi$
$42$		0			0
$43$		−	19253.7i		−	1.58797i	−0.607937	−	0.793986i	$-0.708003\pi$
$43$		−	19253.7i		−	1.58797i	0.607937	−	0.793986i	$-0.291997\pi$
$44$		0			0
$45$	22113.5	−	12767.2i	1.62789	−	0.939865i
$46$		0			0
$47$	11325.4	−	19616.1i	0.747837	−	1.29529i	−0.201020	−	0.979587i	$-0.564426\pi$
$47$	11325.4	−	19616.1i	0.747837	−	1.29529i	0.948857	−	0.315705i	$-0.102241\pi$
$48$		0			0
$49$	−300.442	−	16804.3i	−0.0178760	−	0.999840i
$50$		0			0
$51$	29757.9	+	17180.8i	1.60206	+	0.924947i
$52$		0			0
$53$	8009.36	+	13872.6i	0.391659	+	0.678374i	0.992669	−	0.120868i	$-0.0385679\pi$
$53$	8009.36	+	13872.6i	0.391659	+	0.678374i	−0.601009	+	0.799242i	$0.705235\pi$
$54$		0			0
$55$	−35460.4			−1.58065
$56$		0			0
$57$	−32374.3			−1.31982
$58$		0			0
$59$	−15061.0	−	26086.5i	−0.563281	−	0.975631i	−0.997207	−	0.0746831i	$-0.976205\pi$
$59$	−15061.0	−	26086.5i	−0.563281	−	0.975631i	0.433926	−	0.900948i	$-0.357128\pi$
$60$		0			0
$61$	−37809.1	−	21829.1i	−1.30098	−	0.751124i	−0.320411	−	0.947278i	$-0.603821\pi$
$61$	−37809.1	−	21829.1i	−1.30098	−	0.751124i	−0.980573	+	0.196155i	$0.937155\pi$
$62$		0			0
$63$	10116.7	+	36449.5i	0.321134	+	1.15702i
$64$		0			0
$65$	−28015.7	+	48524.7i	−0.822467	+	1.42455i
$66$		0			0
$67$	9185.38	−	5303.18i	0.249983	−	0.144328i	−0.369774	−	0.929122i	$-0.620565\pi$
$67$	9185.38	−	5303.18i	0.249983	−	0.144328i	0.619756	+	0.784794i	$0.287231\pi$
$68$		0			0
$69$			27765.4i			0.702071i
$70$		0			0
$71$			57405.1i			1.35146i	0.737148	+	0.675732i	$0.236172\pi$
$71$			57405.1i			1.35146i	−0.737148	+	0.675732i	$0.763828\pi$
$72$		0			0
$73$	56304.1	−	32507.2i	1.23661	−	0.713957i	0.268210	−	0.963360i	$-0.413568\pi$
$73$	56304.1	−	32507.2i	1.23661	−	0.713957i	0.968400	+	0.249403i	$0.0802345\pi$
$74$		0			0
$75$	−52416.7	+	90788.5i	−1.07601	+	1.86371i
$76$		0			0
$77$	13142.2	−	50861.4i	0.252604	−	0.977601i
$78$		0			0
$79$	15991.3	+	9232.58i	0.288281	+	0.166439i	0.637166	−	0.770726i	$-0.280106\pi$
$79$	15991.3	+	9232.58i	0.288281	+	0.166439i	−0.348885	+	0.937165i	$0.613440\pi$
$80$		0			0
$81$	22407.3	+	38810.6i	0.379469	+	0.657260i
$82$		0			0
$83$	−17093.5			−0.272355			−0.136178	−	0.990684i	$-0.543482\pi$
$83$	−17093.5			−0.272355			−0.136178	+	0.990684i	$0.543482\pi$
$84$		0			0
$85$	−130031.			−1.95209
$86$		0			0
$87$	−22757.7	−	39417.4i	−0.322351	−	0.558329i
$88$		0			0
$89$	−40872.2	−	23597.6i	−0.546956	−	0.315785i	0.200937	−	0.979604i	$-0.435601\pi$
$89$	−40872.2	−	23597.6i	−0.546956	−	0.315785i	−0.747893	+	0.663819i	$0.768935\pi$
$90$		0			0
$91$	−59216.7	−	58167.5i	−0.749620	−	0.736337i
$92$		0			0
$93$	9411.04	−	16300.4i	0.112832	−	0.195430i
$94$		0			0
$95$	106098.	−	61255.7i	1.20614	−	0.696366i
$96$		0			0
$97$		−	52847.4i		−	0.570288i	−0.958485	−	0.285144i	$-0.907959\pi$
$97$		−	52847.4i		−	0.570288i	0.958485	−	0.285144i	$-0.0920414\pi$
$98$		0			0
$99$			118233.i			1.21242i
$100$		0			0
$101$	−155978.	+	90053.7i	−1.52145	+	0.878412i	−0.521775	+	0.853083i	$0.674730\pi$
$101$	−155978.	+	90053.7i	−1.52145	+	0.878412i	−0.999679	+	0.0253292i	$0.991937\pi$
$102$		0			0
$103$	−12339.7	+	21372.9i	−0.114607	+	0.198505i	−0.917623	−	0.397453i	$-0.869894\pi$
$103$	−12339.7	+	21372.9i	−0.114607	+	0.198505i	0.803016	+	0.595958i	$0.203227\pi$
$104$		0			0
$105$	−187168.	−	183852.i	−1.65676	−	1.62740i
$106$		0			0
$107$	12330.8	+	7119.17i	0.104119	+	0.0601132i	0.551155	−	0.834403i	$-0.314187\pi$
$107$	12330.8	+	7119.17i	0.104119	+	0.0601132i	−0.447036	+	0.894516i	$0.647520\pi$
$108$		0			0
$109$	90398.4	+	156575.i	0.728777	+	1.26228i	0.957401	+	0.288763i	$0.0932440\pi$
$109$	90398.4	+	156575.i	0.728777	+	1.26228i	−0.228624	+	0.973515i	$0.573423\pi$
$110$		0			0
$111$	321739.			2.47854
$112$		0			0
$113$	−46481.6			−0.342440			−0.171220	−	0.985233i	$-0.554771\pi$
$113$	−46481.6			−0.342440			−0.171220	+	0.985233i	$0.554771\pi$
$114$		0			0
$115$	−52535.1	−	90993.4i	−0.370429	−	0.641602i
$116$		0			0
$117$	161793.	+	93411.1i	1.09268	+	0.630861i
$118$		0			0
$119$	48191.7	−	186506.i	0.311964	−	1.20733i
$120$		0			0
$121$	1571.28	−	2721.54i	0.00975641	−	0.0168986i
$122$		0			0
$123$	−341288.	+	197043.i	−2.03404	+	1.17435i
$124$		0			0
$125$		−	123239.i		−	0.705462i
$126$		0			0
$127$		−	129698.i		−	0.713551i	−0.934190	−	0.356775i	$-0.883876\pi$
$127$		−	129698.i		−	0.713551i	0.934190	−	0.356775i	$-0.116124\pi$
$128$		0			0
$129$	385597.	−	222625.i	2.04014	−	1.17787i
$130$		0			0
$131$	−79543.3	+	137773.i	−0.404973	+	0.701433i	−0.994318	−	0.106448i	$-0.966052\pi$
$131$	−79543.3	+	137773.i	−0.404973	+	0.701433i	0.589346	+	0.807881i	$0.299386\pi$
$132$		0			0
$133$	48538.6	+	174880.i	0.237935	+	0.857259i
$134$		0			0
$135$	85499.5	+	49363.2i	0.403765	+	0.233114i
$136$		0			0
$137$	65112.8	+	112779.i	0.296391	+	0.513364i	0.975308	−	0.220851i	$-0.0708835\pi$
$137$	65112.8	+	112779.i	0.296391	+	0.513364i	−0.678916	+	0.734216i	$0.737550\pi$
$138$		0			0
$139$	−317422.			−1.39348			−0.696739	−	0.717325i	$-0.745366\pi$
$139$	−317422.			−1.39348			−0.696739	+	0.717325i	$0.745366\pi$
$140$		0			0
$141$	523807.			2.21883
$142$		0			0
$143$	−129722.	−	224686.i	−0.530487	−	0.918831i
$144$		0			0
$145$	149164.	+	86119.9i	0.589175	+	0.340160i
$146$		0			0
$147$	333069.	−	200320.i	1.27128	−	0.764595i
$148$		0			0
$149$	9923.11	−	17187.3i	0.0366169	−	0.0634224i	−0.847136	−	0.531376i	$-0.821675\pi$
$149$	9923.11	−	17187.3i	0.0366169	−	0.0634224i	0.883753	+	0.467953i	$0.155009\pi$
$150$		0			0
$151$	181950.	−	105049.i	0.649396	−	0.374929i	−0.138829	−	0.990316i	$-0.544334\pi$
$151$	181950.	−	105049.i	0.649396	−	0.374929i	0.788225	+	0.615387i	$0.211000\pi$
$152$		0			0
$153$			433556.i			1.49733i
$154$		0			0
$155$			71226.8i			0.238130i
$156$		0			0
$157$	−342669.	+	197840.i	−1.10950	+	0.640568i	−0.938699	−	0.344739i	$-0.887968\pi$
$157$	−342669.	+	197840.i	−1.10950	+	0.640568i	−0.170797	+	0.985306i	$0.554634\pi$
$158$		0			0
$159$	−185220.	+	320810.i	−0.581024	+	1.00636i
$160$		0			0
$161$	149984.	−	41628.4i	0.456016	−	0.126568i
$162$		0			0
$163$	190266.	+	109850.i	0.560907	+	0.323840i	0.753510	−	0.657437i	$-0.228359\pi$
$163$	190266.	+	109850.i	0.560907	+	0.323840i	−0.192602	+	0.981277i	$0.561693\pi$
$164$		0			0
$165$	−410018.	−	710171.i	−1.17245	−	2.03073i
$166$		0			0
$167$	28110.7			0.0779974			0.0389987	−	0.999239i	$-0.487583\pi$
$167$	28110.7			0.0779974			0.0389987	+	0.999239i	$0.487583\pi$
$168$		0			0
$169$	−38659.7			−0.104122
$170$		0			0
$171$	−204241.	−	353756.i	−0.534137	−	0.925152i
$172$		0			0
$173$	372652.	+	215150.i	0.946646	+	0.546546i	0.892037	−	0.451961i	$-0.149276\pi$
$173$	372652.	+	215150.i	0.946646	+	0.546546i	0.0546086	+	0.998508i	$0.482609\pi$
$174$		0			0
$175$	569012.	+	147028.i	1.40451	+	0.362915i
$176$		0			0
$177$	348293.	−	603261.i	0.835625	−	1.44734i
$178$		0			0
$179$	36003.6	−	20786.7i	0.0839871	−	0.0484900i	−0.457418	−	0.889252i	$-0.651226\pi$
$179$	36003.6	−	20786.7i	0.0839871	−	0.0484900i	0.541405	+	0.840762i	$0.317892\pi$
$180$		0			0
$181$		−	273268.i		−	0.620001i	−0.950736	−	0.310000i	$-0.899671\pi$
$181$		−	273268.i		−	0.620001i	0.950736	−	0.310000i	$-0.100329\pi$
$182$		0			0
$183$		−	1.00961e6i		−	2.22858i
$184$		0			0
$185$	−1.05441e6	+	608764.i	−2.26506	+	1.30774i
$186$		0			0
$187$	301045.	−	521425.i	0.629545	−	1.09040i
$188$		0			0
$189$	−102490.	+	104339.i	−0.208702	+	0.212467i
$190$		0			0
$191$	430294.	+	248431.i	0.853458	+	0.492744i	0.861816	−	0.507221i	$-0.169327\pi$
$191$	430294.	+	248431.i	0.853458	+	0.492744i	−0.00835809	+	0.999965i	$0.502660\pi$
$192$		0			0
$193$	114986.	+	199162.i	0.222204	+	0.384868i	0.955477	−	0.295066i	$-0.0953416\pi$
$193$	114986.	+	199162.i	0.222204	+	0.384868i	−0.733273	+	0.679934i	$0.762008\pi$
$194$		0			0
$195$	−1.29575e6			−2.44025
$196$		0			0
$197$	67861.0			0.124582			0.0622909	−	0.998058i	$-0.480159\pi$
$197$	67861.0			0.124582			0.0622909	+	0.998058i	$0.480159\pi$
$198$		0			0
$199$	331890.	+	574851.i	0.594103	+	1.02902i	0.993673	+	0.112314i	$0.0358262\pi$
$199$	331890.	+	574851.i	0.594103	+	1.02902i	−0.399570	+	0.916703i	$0.630840\pi$
$200$		0			0
$201$	212416.	+	122638.i	0.370848	+	0.214109i
$202$		0			0
$203$	−178806.	+	182031.i	−0.304538	+	0.310031i
$204$		0			0
$205$	745652.	−	1.29151e6i	1.23923	−	2.14641i
$206$		0			0
$207$	−303394.	+	175164.i	−0.492131	+	0.284132i
$208$		0			0
$209$			567269.i			0.898305i
$210$		0			0
$211$		−	129670.i		−	0.200509i	−0.994962	−	0.100254i	$-0.968034\pi$
$211$		−	129670.i		−	0.200509i	0.994962	−	0.100254i	$-0.0319656\pi$
$212$		0			0
$213$	−1.14966e6	+	663757.i	−1.73628	+	1.00244i
$214$		0			0
$215$	−842459.	+	1.45918e6i	−1.24295	+	2.15285i
$216$		0			0
$217$	−102162.	−	26397.8i	−0.147279	−	0.0380556i
$218$		0			0
$219$	1.30205e6	+	751741.i	1.83450	+	1.05915i
$220$		0			0
$221$	−475686.	−	823912.i	−0.655148	−	1.13475i
$222$		0			0
$223$	−1.16536e6			−1.56927			−0.784637	−	0.619956i	$-0.787151\pi$
$223$	−1.16536e6			−1.56927			−0.784637	+	0.619956i	$0.787151\pi$
$224$		0			0
$225$	−1.32273e6			−1.74187
$226$		0			0
$227$	545332.	+	944543.i	0.702420	+	1.21663i	0.967615	+	0.252432i	$0.0812303\pi$
$227$	545332.	+	944543.i	0.702420	+	1.21663i	−0.265195	+	0.964195i	$0.585436\pi$
$228$		0			0
$229$	−37293.9	−	21531.7i	−0.0469947	−	0.0271324i	0.476319	−	0.879273i	$-0.341971\pi$
$229$	−37293.9	−	21531.7i	−0.0469947	−	0.0271324i	−0.523313	+	0.852140i	$0.675304\pi$
$230$		0			0
$231$	1.17057e6	−	324895.i	1.44334	−	0.400602i
$232$		0			0
$233$	−400725.	+	694075.i	−0.483567	+	0.837562i	−0.999822	−	0.0188729i	$-0.993992\pi$
$233$	−400725.	+	694075.i	−0.483567	+	0.837562i	0.516255	+	0.856435i	$0.327326\pi$
$234$		0			0
$235$	−1.71663e6	+	991099.i	−2.02772	+	1.17070i
$236$		0			0
$237$			427014.i			0.493823i
$238$		0			0
$239$		−	841735.i		−	0.953193i	−0.879122	−	0.476596i	$-0.841870\pi$
$239$		−	841735.i		−	0.953193i	0.879122	−	0.476596i	$-0.158130\pi$
$240$		0			0
$241$	715405.	−	413040.i	0.793432	−	0.458088i	−0.0477375	−	0.998860i	$-0.515201\pi$
$241$	715405.	−	413040.i	0.793432	−	0.458088i	0.841169	+	0.540772i	$0.181868\pi$
$242$		0			0
$243$	−655248.	+	1.13492e6i	−0.711852	+	1.23296i
$244$		0			0
$245$	−712516.	+	1.28670e6i	−0.758367	+	1.36950i
$246$		0			0
$247$	776263.	+	448176.i	0.809592	+	0.467418i
$248$		0			0
$249$	−197647.	−	342335.i	−0.202019	−	0.349907i
$250$		0			0
$251$	1.51586e6			1.51871			0.759357	−	0.650674i	$-0.225513\pi$
$251$	1.51586e6			1.51871			0.759357	+	0.650674i	$0.225513\pi$
$252$		0			0
$253$	486511.			0.477849
$254$		0			0
$255$	−1.50351e6	−	2.60416e6i	−1.44796	−	2.50794i
$256$		0			0
$257$	−363693.	−	209978.i	−0.343480	−	0.198309i	0.318330	−	0.947980i	$-0.396878\pi$
$257$	−363693.	−	209978.i	−0.343480	−	0.198309i	−0.661810	+	0.749672i	$0.730211\pi$
$258$		0			0
$259$	−482380.	−	1.73798e6i	−0.446828	−	1.60988i
$260$		0			0
$261$	287144.	−	497348.i	0.260915	−	0.451918i
$262$		0			0
$263$	1.08817e6	−	628258.i	0.970083	−	0.560078i	0.0708218	−	0.997489i	$-0.477438\pi$
$263$	1.08817e6	−	628258.i	0.970083	−	0.560078i	0.899262	+	0.437411i	$0.144105\pi$
$264$		0			0
$265$		−	1.40182e6i		−	1.22625i
$266$		0			0
$267$		−	1.09141e6i		−	0.936932i
$268$		0			0
$269$	−236769.	+	136699.i	−0.199500	+	0.115182i	−0.596422	−	0.802671i	$-0.703412\pi$
$269$	−236769.	+	136699.i	−0.199500	+	0.115182i	0.396922	+	0.917852i	$0.370078\pi$
$270$		0			0
$271$	−386345.	+	669169.i	−0.319560	+	0.553494i	−0.980396	−	0.197036i	$-0.936868\pi$
$271$	−386345.	+	669169.i	−0.319560	+	0.553494i	0.660837	+	0.750530i	$0.270202\pi$
$272$		0			0
$273$	480225.	−	1.85852e6i	0.389976	−	1.50925i
$274$		0			0
$275$	1.59081e6	+	918457.i	1.26849	+	0.732364i
$276$		0			0
$277$	−15930.9	−	27593.2i	−0.0124750	−	0.0216074i	0.859720	−	0.510765i	$-0.170638\pi$
$277$	−15930.9	−	27593.2i	−0.0124750	−	0.0216074i	−0.872196	+	0.489157i	$0.837304\pi$
$278$		0			0
$279$	237487.			0.182654
$280$		0			0
$281$	510770.			0.385887			0.192943	−	0.981210i	$-0.438197\pi$
$281$	510770.			0.385887			0.192943	+	0.981210i	$0.438197\pi$
$282$		0			0
$283$	−973793.	−	1.68666e6i	−0.722771	−	1.25188i	−0.959885	−	0.280394i	$-0.909535\pi$
$283$	−973793.	−	1.68666e6i	−0.722771	−	1.25188i	0.237114	−	0.971482i	$-0.423798\pi$
$284$		0			0
$285$	2.45356e6	+	1.41656e6i	1.78930	+	1.03306i
$286$		0			0
$287$	1.57608e6	+	1.54816e6i	1.12947	+	1.10946i
$288$		0			0
$289$	393988.	−	682407.i	0.277484	−	0.480617i
$290$		0			0
$291$	1.05838e6	−	611059.i	0.732675	−	0.423010i
$292$		0			0
$293$			1.59231e6i			1.08357i	0.840516	+	0.541787i	$0.182252\pi$
$293$			1.59231e6i			1.08357i	−0.840516	+	0.541787i	$0.817748\pi$
$294$		0			0
$295$			2.63603e6i			1.76358i
$296$		0			0
$297$	−395892.	+	228568.i	−0.260427	+	0.150357i
$298$		0			0
$299$	384372.	−	665751.i	0.248641	−	0.430659i
$300$		0			0
$301$	−1.78070e6	−	1.74915e6i	−1.13286	−	1.11278i
$302$		0			0
$303$	−3.60705e6	−	2.08253e6i	−2.25707	−	1.30312i
$304$		0			0
$305$	1.91030e6	+	3.30873e6i	1.17585	+	2.03663i
$306$		0			0
$307$	148691.			0.0900404			0.0450202	−	0.998986i	$-0.485665\pi$
$307$	148691.			0.0900404			0.0450202	+	0.998986i	$0.485665\pi$
$308$		0			0
$309$	−570720.			−0.340037
$310$		0			0
$311$	−364641.	−	631576.i	−0.213779	−	0.370275i	0.739115	−	0.673579i	$-0.235244\pi$
$311$	−364641.	−	631576.i	−0.213779	−	0.370275i	−0.952894	+	0.303303i	$0.901910\pi$
$312$		0			0
$313$	1.57730e6	+	910655.i	0.910026	+	0.525404i	0.880439	−	0.474159i	$-0.157248\pi$
$313$	1.57730e6	+	910655.i	0.910026	+	0.525404i	0.0295862	+	0.999562i	$0.490581\pi$
$314$		0			0
$315$	828163.	−	3.20507e6i	0.470261	−	1.81996i
$316$		0			0
$317$	−790103.	+	1.36850e6i	−0.441607	+	0.764885i	−0.997809	−	0.0661618i	$-0.978925\pi$
$317$	−790103.	+	1.36850e6i	−0.441607	+	0.764885i	0.556202	+	0.831047i	$0.312258\pi$
$318$		0			0
$319$	−690680.	+	398764.i	−0.380015	+	0.219402i
$320$		0			0
$321$			329268.i			0.178355i
$322$		0			0
$323$			2.08015e6i			1.10940i
$324$		0			0
$325$	2.51367e6	−	1.45127e6i	1.32008	−	0.762148i
$326$		0			0
$327$	−2.09050e6	+	3.62085e6i	−1.08114	+	1.87258i
$328$		0			0
$329$	−785339.	−	2.82951e6i	−0.400007	−	1.44119i
$330$		0			0
$331$	2.71356e6	+	1.56667e6i	1.36135	+	0.785974i	0.989803	−	0.142442i	$-0.0454954\pi$
$331$	2.71356e6	+	1.56667e6i	1.36135	+	0.785974i	0.371543	+	0.928416i	$0.378829\pi$
$332$		0			0
$333$	2.02976e6	+	3.51566e6i	1.00308	+	1.73738i
$334$		0			0
$335$	−928178.			−0.451876
$336$		0			0
$337$	3.16854e6			1.51979			0.759897	−	0.650043i	$-0.225249\pi$
$337$	3.16854e6			1.51979			0.759897	+	0.650043i	$0.225249\pi$
$338$		0			0
$339$	−537452.	−	930895.i	−0.254004	−	0.439948i
$340$		0			0
$341$	−285619.	−	164902.i	−0.133015	−	0.0767964i
$342$		0			0
$343$	−1.58146e6	−	1.49884e6i	−0.725812	−	0.687893i
$344$		0			0
$345$	1.21489e6	−	2.10426e6i	0.549529	−	0.951813i
$346$		0			0
$347$	−237589.	+	137172.i	−0.105926	+	0.0611565i	−0.552027	−	0.833826i	$-0.686146\pi$
$347$	−237589.	+	137172.i	−0.105926	+	0.0611565i	0.446101	+	0.894983i	$0.352812\pi$
$348$		0			0
$349$		−	528446.i		−	0.232240i	−0.993235	−	0.116120i	$-0.962954\pi$
$349$		−	528446.i		−	0.232240i	0.993235	−	0.116120i	$-0.0370457\pi$
$350$		0			0
$351$			722328.i			0.312944i
$352$		0			0
$353$	−135096.	+	77998.0i	−0.0577042	+	0.0333155i	−0.528575	−	0.848887i	$-0.677273\pi$
$353$	−135096.	+	77998.0i	−0.0577042	+	0.0333155i	0.470870	+	0.882202i	$0.343940\pi$
$354$		0			0
$355$	2.51180e6	−	4.35057e6i	1.05783	−	1.83221i
$356$		0			0
$357$	4.29242e6	−	1.19137e6i	1.78251	−	0.494740i
$358$		0			0
$359$	−258854.	−	149449.i	−0.106003	−	0.0612009i	0.446061	−	0.895002i	$-0.352826\pi$
$359$	−258854.	−	149449.i	−0.106003	−	0.0612009i	−0.552064	+	0.833802i	$0.686160\pi$
$360$		0			0
$361$	258125.	+	447086.i	0.104247	+	0.180561i
$362$		0			0
$363$	72672.9			0.0289472
$364$		0			0
$365$	−5.68950e6			−2.23533
$366$		0			0
$367$	−1.36459e6	−	2.36354e6i	−0.528855	−	0.916004i	−0.999434	−	0.0336463i	$-0.989288\pi$
$367$	−1.36459e6	−	2.36354e6i	−0.528855	−	0.916004i	0.470578	−	0.882358i	$-0.344045\pi$
$368$		0			0
$369$	−4.30619e6	−	2.48618e6i	−1.64637	−	0.950533i
$370$		0			0
$371$	2.01066e6	+	519537.i	0.758409	+	0.195967i
$372$		0			0
$373$	−976118.	+	1.69069e6i	−0.363270	+	0.629203i	−0.988497	−	0.151240i	$-0.951673\pi$
$373$	−976118.	+	1.69069e6i	−0.363270	+	0.629203i	0.625227	+	0.780443i	$0.285007\pi$
$374$		0			0
$375$	2.46813e6	−	1.42498e6i	0.906338	−	0.523274i
$376$		0			0
$377$			1.26019e6i			0.456648i
$378$		0			0
$379$		−	223755.i		−	0.0800157i	−0.999199	−	0.0400079i	$-0.987262\pi$
$379$		−	223755.i		−	0.0800157i	0.999199	−	0.0400079i	$-0.0127383\pi$
$380$		0			0
$381$	2.59749e6	−	1.49966e6i	0.916730	−	0.529274i
$382$		0			0
$383$	−2.04849e6	+	3.54809e6i	−0.713572	+	1.23594i	0.249936	+	0.968262i	$0.419590\pi$
$383$	−2.04849e6	+	3.54809e6i	−0.713572	+	1.23594i	−0.963508	+	0.267680i	$0.913743\pi$
$384$		0			0
$385$	−3.22149e6	+	3.27960e6i	−1.10766	+	1.12764i
$386$		0			0
$387$	4.86526e6	+	2.80896e6i	1.65131	+	0.953384i
$388$		0			0
$389$	−688529.	−	1.19257e6i	−0.230700	−	0.399584i	0.727314	−	0.686305i	$-0.240768\pi$
$389$	−688529.	−	1.19257e6i	−0.230700	−	0.399584i	−0.958014	+	0.286720i	$0.907435\pi$
$390$		0			0
$391$	1.78401e6			0.590141
$392$		0			0
$393$	−3.67894e6			−1.20155
$394$		0			0
$395$	−807957.	−	1.39942e6i	−0.260553	−	0.451290i
$396$		0			0
$397$	−3.87505e6	−	2.23726e6i	−1.23396	−	0.712427i	−0.266106	−	0.963944i	$-0.585737\pi$
$397$	−3.87505e6	−	2.23726e6i	−1.23396	−	0.712427i	−0.967853	+	0.251517i	$0.919071\pi$
$398$		0			0
$399$	−2.94113e6	+	2.99418e6i	−0.924872	+	0.941555i
$400$		0			0
$401$	−2.98261e6	+	5.16604e6i	−0.926266	+	1.60434i	−0.136755	+	0.990605i	$0.543667\pi$
$401$	−2.98261e6	+	5.16604e6i	−0.926266	+	1.60434i	−0.789512	+	0.613735i	$0.789666\pi$
$402$		0			0
$403$	−451311.	+	260565.i	−0.138425	+	0.0799195i
$404$		0			0
$405$		−	3.92179e6i		−	1.18808i
$406$		0			0
$407$		−	5.63757e6i		−	1.68696i
$408$		0			0
$409$	−1.05768e6	+	610654.i	−0.312642	+	0.180504i	−0.648108	−	0.761548i	$-0.724440\pi$
$409$	−1.05768e6	+	610654.i	−0.312642	+	0.180504i	0.335466	+	0.942052i	$0.391106\pi$
$410$		0			0
$411$	−1.50576e6	+	2.60805e6i	−0.439695	+	0.761573i
$412$		0			0
$413$	−3.78090e6	−	976954.i	−1.09074	−	0.281838i
$414$		0			0
$415$	1.29547e6	+	747939.i	0.369238	+	0.213180i
$416$		0			0
$417$	−3.67026e6	−	6.35707e6i	−1.03361	−	1.79026i
$418$		0			0
$419$	−3.12252e6			−0.868901			−0.434450	−	0.900696i	$-0.643057\pi$
$419$	−3.12252e6			−0.868901			−0.434450	+	0.900696i	$0.643057\pi$
$420$		0			0
$421$	−937555.			−0.257805			−0.128903	−	0.991657i	$-0.541145\pi$
$421$	−937555.			−0.257805			−0.128903	+	0.991657i	$0.541145\pi$
$422$		0			0
$423$	3.30456e6	+	5.72366e6i	0.897971	+	1.55533i
$424$		0			0
$425$	5.83344e6	+	3.36794e6i	1.56658	+	0.904464i
$426$		0			0
$427$	−5.45376e6	+	1.51371e6i	−1.44753	+	0.401765i
$428$		0			0
$429$	2.99988e6	−	5.19595e6i	0.786975	−	1.36308i
$430$		0			0
$431$	235746.	−	136108.i	0.0611296	−	0.0352932i	−0.469124	−	0.883132i	$-0.655430\pi$
$431$	235746.	−	136108.i	0.0611296	−	0.0352932i	0.530253	+	0.847839i	$0.322097\pi$
$432$		0			0
$433$			6.33614e6i			1.62407i	0.583608	+	0.812035i	$0.301640\pi$
$433$			6.33614e6i			1.62407i	−0.583608	+	0.812035i	$0.698360\pi$
$434$		0			0
$435$			3.98311e6i			1.00925i
$436$		0			0
$437$	−1.45565e6	+	840419.i	−0.364630	+	0.210519i
$438$		0			0
$439$	59115.3	−	102391.i	0.0146399	−	0.0253571i	−0.858613	−	0.512625i	$-0.828673\pi$
$439$	59115.3	−	102391.i	0.0146399	−	0.0253571i	0.873253	+	0.487268i	$0.162006\pi$
$440$		0			0
$441$	4.29016e6	+	2.37570e6i	1.05045	+	0.581694i
$442$		0			0
$443$	−3.87402e6	−	2.23667e6i	−0.937891	−	0.541492i	−0.0485926	−	0.998819i	$-0.515474\pi$
$443$	−3.87402e6	−	2.23667e6i	−0.937891	−	0.541492i	−0.889299	+	0.457327i	$0.848807\pi$
$444$		0			0
$445$	2.06506e6	+	3.57678e6i	0.494347	+	0.856234i
$446$		0			0
$447$	458952.			0.108642
$448$		0			0
$449$	1.27906e6			0.299416			0.149708	−	0.988730i	$-0.452167\pi$
$449$	1.27906e6			0.299416			0.149708	+	0.988730i	$0.452167\pi$
$450$		0			0
$451$	3.45262e6	+	5.98012e6i	0.799297	+	1.38442i
$452$		0			0
$453$	4.20767e6	+	2.42930e6i	0.963376	+	0.556206i
$454$		0			0
$455$	1.94271e6	+	6.99942e6i	0.439925	+	1.58502i
$456$		0			0
$457$	2.26066e6	−	3.91558e6i	0.506343	−	0.877012i	−0.493630	−	0.869672i	$-0.664330\pi$
$457$	2.26066e6	−	3.91558e6i	0.506343	−	0.877012i	0.999973	−	0.00734012i	$-0.00233645\pi$
$458$		0			0
$459$	−1.45172e6	+	838148.i	−0.321625	+	0.185690i
$460$		0			0
$461$		−	1.58045e6i		−	0.346360i	−0.984890	−	0.173180i	$-0.944596\pi$
$461$		−	1.58045e6i		−	0.346360i	0.984890	−	0.173180i	$-0.0554043\pi$
$462$		0			0
$463$		−	7.54852e6i		−	1.63648i	−0.574880	−	0.818238i	$-0.694951\pi$
$463$		−	7.54852e6i		−	1.63648i	0.574880	−	0.818238i	$-0.305049\pi$
$464$		0			0
$465$	−1.42647e6	+	823574.i	−0.305936	+	0.176632i
$466$		0			0
$467$	3.27541e6	−	5.67317e6i	0.694982	−	1.20374i	−0.275205	−	0.961385i	$-0.588746\pi$
$467$	3.27541e6	−	5.67317e6i	0.694982	−	1.20374i	0.970187	−	0.242358i	$-0.0779208\pi$
$468$		0			0
$469$	343997.	−	1.33130e6i	0.0722143	−	0.279476i
$470$		0			0
$471$	−7.92436e6	−	4.57513e6i	−1.64593	−	0.950279i
$472$		0			0
$473$	−3.90087e6	−	6.75651e6i	−0.801695	−	1.38858i
$474$		0			0
$475$	−6.34632e6			−1.29059
$476$		0			0
$477$	−4.67401e6			−0.940576
$478$		0			0
$479$	2.09770e6	+	3.63332e6i	0.417739	+	0.723545i	0.995712	−	0.0925112i	$-0.0294894\pi$
$479$	2.09770e6	+	3.63332e6i	0.417739	+	0.723545i	−0.577973	+	0.816056i	$0.696156\pi$
$480$		0			0
$481$	−7.71457e6	−	4.45401e6i	−1.52037	−	0.877785i
$482$		0			0
$483$	2.56792e6	+	2.52242e6i	0.500857	+	0.491982i
$484$		0			0
$485$	−2.31238e6	+	4.00516e6i	−0.446380	+	0.773152i
$486$		0			0
$487$	−2.42013e6	+	1.39726e6i	−0.462399	+	0.266966i	−0.713052	−	0.701111i	$-0.752688\pi$
$487$	−2.42013e6	+	1.39726e6i	−0.462399	+	0.266966i	0.250653	+	0.968077i	$0.419355\pi$
$488$		0			0
$489$			5.08064e6i			0.960830i
$490$		0			0
$491$		−	3.03606e6i		−	0.568338i	−0.958774	−	0.284169i	$-0.908282\pi$
$491$		−	3.03606e6i		−	0.568338i	0.958774	−	0.284169i	$-0.0917177\pi$
$492$		0			0
$493$	−2.53269e6	+	1.46225e6i	−0.469315	+	0.270959i
$494$		0			0
$495$	5.17339e6	−	8.96057e6i	0.948990	−	1.64370i
$496$		0			0
$497$	5.30918e6	+	5.21511e6i	0.964132	+	0.947049i
$498$		0			0
$499$	3.90725e6	+	2.25585e6i	0.702458	+	0.405564i	0.808262	−	0.588823i	$-0.200408\pi$
$499$	3.90725e6	+	2.25585e6i	0.702458	+	0.405564i	−0.105804	+	0.994387i	$0.533742\pi$
$500$		0			0
$501$	325035.	+	562977.i	0.0578544	+	0.100207i
$502$		0			0
$503$	5.19848e6			0.916129			0.458065	−	0.888919i	$-0.348543\pi$
$503$	5.19848e6			0.916129			0.458065	+	0.888919i	$0.348543\pi$
$504$		0			0
$505$	1.57615e7			2.75023
$506$		0			0
$507$	−447011.	−	774245.i	−0.0772321	−	0.133770i
$508$		0			0
$509$	−3.58777e6	−	2.07140e6i	−0.613805	−	0.354380i	0.160648	−	0.987012i	$-0.448642\pi$
$509$	−3.58777e6	−	2.07140e6i	−0.613805	−	0.354380i	−0.774453	+	0.632631i	$0.781975\pi$
$510$		0			0
$511$	2.10862e6	−	8.16055e6i	0.357228	−	1.38251i
$512$		0			0
$513$	789676.	−	1.36776e6i	0.132482	−	0.229465i
$514$		0			0
$515$	1.87038e6	−	1.07986e6i	0.310750	−	0.179412i
$516$		0			0
$517$		−	9.17825e6i		−	1.51020i
$518$		0			0
$519$			9.95088e6i			1.62160i
$520$		0			0
$521$	−4.24991e6	+	2.45369e6i	−0.685940	+	0.396027i	−0.802089	−	0.597204i	$-0.796278\pi$
$521$	−4.24991e6	+	2.45369e6i	−0.685940	+	0.396027i	0.116150	+	0.993232i	$0.462945\pi$
$522$		0			0
$523$	4.49822e6	−	7.79114e6i	0.719095	−	1.24551i	−0.242263	−	0.970211i	$-0.577890\pi$
$523$	4.49822e6	−	7.79114e6i	0.719095	−	1.24551i	0.961359	−	0.275299i	$-0.0887769\pi$
$524$		0			0
$525$	3.63476e6	+	1.30957e7i	0.575542	+	2.07363i
$526$		0			0
$527$	−1.04735e6	−	604688.i	−0.164273	−	0.0948429i
$528$		0			0
$529$	−2.49740e6	−	4.32562e6i	−0.388015	−	0.672062i
$530$		0			0
$531$	8.78915e6			1.35273
$532$		0			0
$533$	1.09111e7			1.66360
$534$		0			0
$535$	−623010.	−	1.07908e6i	−0.0941045	−	0.162994i
$536$		0			0
$537$	832597.	+	480700.i	0.124595	+	0.0719347i
$538$		0			0
$539$	−3.51005e6	−	5.83611e6i	−0.520405	−	0.865269i
$540$		0			0
$541$	−557536.	+	965681.i	−0.0818993	+	0.141854i	−0.904066	−	0.427393i	$-0.859432\pi$
$541$	−557536.	+	965681.i	−0.0818993	+	0.141854i	0.822166	+	0.569247i	$0.192765\pi$
$542$		0			0
$543$	5.47279e6	−	3.15971e6i	0.796543	−	0.459884i
$544$		0			0
$545$		−	1.58218e7i		−	2.28173i
$546$		0			0
$547$			164832.i			0.0235545i	0.999931	+	0.0117772i	$0.00374889\pi$
$547$			164832.i			0.0235545i	−0.999931	+	0.0117772i	$0.996251\pi$
$548$		0			0
$549$	1.10321e7	−	6.36939e6i	1.56217	−	0.901918i
$550$		0			0
$551$	1.37768e6	−	2.38622e6i	0.193317	−	0.334835i
$552$		0			0
$553$	2.30666e6	−	640219.i	0.320753	−	0.0890258i
$554$		0			0
$555$	−2.43837e7	−	1.40779e7i	−3.36021	−	1.94002i
$556$		0			0
$557$	4.56783e6	+	7.91171e6i	0.623838	+	1.08052i	0.988764	+	0.149482i	$0.0477607\pi$
$557$	4.56783e6	+	7.91171e6i	0.623838	+	1.08052i	−0.364927	+	0.931036i	$0.618906\pi$
$558$		0			0
$559$	−1.23277e7			−1.66860
$560$		0			0
$561$	1.39236e7			1.86786
$562$		0			0
$563$	3.84849e6	+	6.66579e6i	0.511705	+	0.886300i	0.999908	+	0.0135693i	$0.00431937\pi$
$563$	3.84849e6	+	6.66579e6i	0.511705	+	0.886300i	−0.488203	+	0.872730i	$0.662347\pi$
$564$		0			0
$565$	3.52271e6	+	2.03384e6i	0.464254	+	0.268037i
$566$		0			0
$567$	5.62509e6	+	1.45348e6i	0.734805	+	0.189867i
$568$		0			0
$569$	−2.96198e6	+	5.13030e6i	−0.383532	+	0.664297i	−0.991564	−	0.129615i	$-0.958626\pi$
$569$	−2.96198e6	+	5.13030e6i	−0.383532	+	0.664297i	0.608032	+	0.793912i	$0.291959\pi$
$570$		0			0
$571$	−7.44621e6	+	4.29907e6i	−0.955752	+	0.551804i	−0.894863	−	0.446341i	$-0.852727\pi$
$571$	−7.44621e6	+	4.29907e6i	−0.955752	+	0.551804i	−0.0608891	+	0.998145i	$0.519394\pi$
$572$		0			0
$573$			1.14901e7i			1.46197i
$574$		0			0
$575$			5.44283e6i			0.686524i
$576$		0			0
$577$	−5.98064e6	+	3.45293e6i	−0.747839	+	0.431765i	−0.824913	−	0.565260i	$-0.808776\pi$
$577$	−5.98064e6	+	3.45293e6i	−0.747839	+	0.431765i	0.0770732	+	0.997025i	$0.475442\pi$
$578$		0			0
$579$	−2.65910e6	+	4.60569e6i	−0.329638	+	0.570950i
$580$		0			0
$581$	−1.55290e6	+	1.58091e6i	−0.190855	+	0.194298i
$582$		0			0
$583$	5.62130e6	+	3.24546e6i	0.684960	+	0.395462i
$584$		0			0
$585$	−8.17455e6	−	1.41587e7i	−0.987584	−	1.71055i
$586$		0			0
$587$	9.16568e6			1.09792			0.548958	−	0.835850i	$-0.315025\pi$
$587$	9.16568e6			1.09792			0.548958	+	0.835850i	$0.315025\pi$
$588$		0			0
$589$	1.13944e6			0.135332
$590$		0			0
$591$	784657.	+	1.35907e6i	0.0924083	+	0.160056i
$592$		0			0
$593$	3.29964e6	+	1.90505e6i	0.385328	+	0.222469i	0.680134	−	0.733088i	$-0.261922\pi$
$593$	3.29964e6	+	1.90505e6i	0.385328	+	0.222469i	−0.294806	+	0.955557i	$0.595255\pi$
$594$		0			0
$595$	−1.18130e7	+	1.20261e7i	−1.36795	+	1.39262i
$596$		0			0
$597$	−7.67510e6	+	1.32937e7i	−0.881349	+	1.52654i
$598$		0			0
$599$	−3.17064e6	+	1.83057e6i	−0.361061	+	0.208459i	−0.669546	−	0.742771i	$-0.733511\pi$
$599$	−3.17064e6	+	1.83057e6i	−0.361061	+	0.208459i	0.308485	+	0.951229i	$0.400178\pi$
$600$		0			0
$601$			1.35098e6i			0.152568i	0.997086	+	0.0762840i	$0.0243056\pi$
$601$			1.35098e6i			0.152568i	−0.997086	+	0.0762840i	$0.975694\pi$
$602$		0			0
$603$			3.09477e6i			0.346605i
$604$		0			0
$605$	−238166.	+	137505.i	−0.0264540	+	0.0152732i
$606$		0			0
$607$	6.08662e6	−	1.05423e7i	0.670509	−	1.16136i	−0.307251	−	0.951629i	$-0.599409\pi$
$607$	6.08662e6	−	1.05423e7i	0.670509	−	1.16136i	0.977760	−	0.209727i	$-0.0672576\pi$
$608$		0			0
$609$	−5.71305e6	−	1.47620e6i	−0.624201	−	0.161288i
$610$		0			0
$611$	−1.25597e7	−	7.25135e6i	−1.36106	−	0.785806i
$612$		0			0
$613$	1.38851e6	+	2.40497e6i	0.149245	+	0.258499i	0.930948	−	0.365151i	$-0.118983\pi$
$613$	1.38851e6	+	2.40497e6i	0.149245	+	0.258499i	−0.781704	+	0.623650i	$0.785649\pi$
$614$		0			0
$615$	3.44870e7			3.67678
$616$		0			0
$617$	−3.36595e6			−0.355955			−0.177978	−	0.984035i	$-0.556955\pi$
$617$	−3.36595e6			−0.355955			−0.177978	+	0.984035i	$0.556955\pi$
$618$		0			0
$619$	8.38142e6	+	1.45171e7i	0.879207	+	1.52283i	0.852212	+	0.523196i	$0.175260\pi$
$619$	8.38142e6	+	1.45171e7i	0.879207	+	1.52283i	0.0269947	+	0.999636i	$0.491406\pi$
$620$		0			0
$621$	−1.17304e6	−	677255.i	−0.122063	−	0.0704731i
$622$		0			0
$623$	−5.89559e6	+	1.63634e6i	−0.608565	+	0.168909i
$624$		0			0
$625$	1.69080e6	−	2.92856e6i	0.173138	−	0.299884i
$626$		0			0
$627$	−1.13608e7	+	6.55917e6i	−1.15409	+	0.666315i
$628$		0			0
$629$		−	2.06727e7i		−	2.08339i
$630$		0			0
$631$			7.07244e6i			0.707124i	0.935411	+	0.353562i	$0.115030\pi$
$631$			7.07244e6i			0.707124i	−0.935411	+	0.353562i	$0.884970\pi$
$632$		0			0
$633$	2.59692e6	−	1.49933e6i	0.257602	−	0.148727i
$634$		0			0
$635$	−5.67504e6	+	9.82946e6i	−0.558515	+	0.967376i
$636$		0			0
$637$	−1.07594e7	+	192366.i	−1.05060	+	0.0187836i
$638$		0			0
$639$	−1.45058e7	−	8.37494e6i	−1.40537	−	0.811390i
$640$		0			0
$641$	8.79135e6	+	1.52271e7i	0.845105	+	1.46376i	0.885530	+	0.464582i	$0.153796\pi$
$641$	8.79135e6	+	1.52271e7i	0.845105	+	1.46376i	−0.0404249	+	0.999183i	$0.512871\pi$
$642$		0			0
$643$	1.07973e7			1.02988			0.514940	−	0.857226i	$-0.327814\pi$
$643$	1.07973e7			1.02988			0.514940	+	0.857226i	$0.327814\pi$
$644$		0			0
$645$	−3.89644e7			−3.68781
$646$		0			0
$647$	−1.76538e6	−	3.05773e6i	−0.165797	−	0.287169i	0.771141	−	0.636665i	$-0.219686\pi$
$647$	−1.76538e6	−	3.05773e6i	−0.165797	−	0.287169i	−0.936938	+	0.349495i	$0.886353\pi$
$648$		0			0
$649$	−1.05705e7	−	6.10286e6i	−0.985104	−	0.568750i
$650$		0			0
$651$	−652594.	−	2.35124e6i	−0.0603519	−	0.217443i
$652$		0			0
$653$	7.06624e6	−	1.22391e7i	0.648493	−	1.12322i	−0.334990	−	0.942222i	$-0.608733\pi$
$653$	7.06624e6	−	1.22391e7i	0.648493	−	1.12322i	0.983483	−	0.181001i	$-0.0579338\pi$
$654$		0			0
$655$	1.20567e7	−	6.96096e6i	1.09806	−	0.633966i
$656$		0			0
$657$			1.89702e7i			1.71458i
$658$		0			0
$659$		−	8.40793e6i		−	0.754181i	−0.926176	−	0.377090i	$-0.876925\pi$
$659$		−	8.40793e6i		−	0.754181i	0.926176	−	0.377090i	$-0.123075\pi$
$660$		0			0
$661$	5.66468e6	−	3.27051e6i	0.504280	−	0.291146i	−0.226199	−	0.974081i	$-0.572630\pi$
$661$	5.66468e6	−	3.27051e6i	0.504280	−	0.291146i	0.730479	+	0.682935i	$0.239297\pi$
$662$		0			0
$663$	1.10004e7	−	1.90533e7i	0.971909	−	1.68339i
$664$		0			0
$665$	3.97343e6	−	1.53775e7i	0.348426	−	1.34844i
$666$		0			0
$667$	−2.04651e6	−	1.18155e6i	−0.178114	−	0.102834i
$668$		0			0
$669$	−1.34747e7	−	2.33389e7i	−1.16401	−	2.01612i
$670$		0			0
$671$	−1.76907e7			−1.51683
$672$		0			0
$673$	−1.09027e6			−0.0927890			−0.0463945	−	0.998923i	$-0.514773\pi$
$673$	−1.09027e6			−0.0927890			−0.0463945	+	0.998923i	$0.514773\pi$
$674$		0			0
$675$	−2.55711e6	−	4.42904e6i	−0.216018	−	0.374153i
$676$		0			0
$677$	−1.40358e6	−	810359.i	−0.117697	−	0.0679525i	0.439996	−	0.898000i	$-0.354980\pi$
$677$	−1.40358e6	−	810359.i	−0.117697	−	0.0679525i	−0.557693	+	0.830047i	$0.688313\pi$
$678$		0			0
$679$	−4.88766e6	−	4.80106e6i	−0.406843	−	0.399634i
$680$		0			0
$681$	−1.26110e7	+	2.18429e7i	−1.04204	+	1.80486i
$682$		0			0
$683$	1.83458e7	−	1.05919e7i	1.50482	−	0.868807i	0.504834	−	0.863216i	$-0.331554\pi$
$683$	1.83458e7	−	1.05919e7i	1.50482	−	0.868807i	0.999984	−	0.00559114i	$-0.00177972\pi$
$684$		0			0
$685$		−	1.13962e7i		−	0.927973i
$686$		0			0
$687$		−	995856.i		−	0.0805017i
$688$		0			0
$689$	8.88230e6	−	5.12820e6i	0.712816	−	0.411545i
$690$		0			0
$691$	−4.73880e6	+	8.20784e6i	−0.377549	+	0.653934i	−0.990705	−	0.136028i	$-0.956566\pi$
$691$	−4.73880e6	+	8.20784e6i	−0.377549	+	0.653934i	0.613156	+	0.789962i	$0.289900\pi$
$692$		0			0
$693$	1.09350e7	+	1.07412e7i	0.864936	+	0.849611i
$694$		0			0
$695$	2.40565e7	+	1.38890e7i	1.88917	+	1.09071i
$696$		0			0
$697$	1.26606e7	+	2.19288e7i	0.987126	+	1.70975i
$698$		0			0
$699$	−1.85338e7			−1.43474
$700$		0			0
$701$	1.44960e7			1.11417			0.557087	−	0.830454i	$-0.311919\pi$
$701$	1.44960e7			1.11417			0.557087	+	0.830454i	$0.311919\pi$
$702$		0			0
$703$	9.73857e6	+	1.68677e7i	0.743202	+	1.28726i
$704$		0			0
$705$	−3.96978e7	−	2.29196e7i	−3.00811	−	1.73673i
$706$		0			0
$707$	−5.84145e6	+	2.26070e7i	−0.439513	+	1.70096i
$708$		0			0
$709$	−1.16308e7	+	2.01451e7i	−0.868945	+	1.50506i	−0.00586960	+	0.999983i	$0.501868\pi$
$709$	−1.16308e7	+	2.01451e7i	−0.868945	+	1.50506i	−0.863076	+	0.505075i	$0.831465\pi$
$710$		0			0
$711$	−4.66601e6	+	2.69392e6i	−0.346156	+	0.199853i
$712$		0			0
$713$		−	977221.i		−	0.0719895i
$714$		0			0
$715$			2.27044e7i			1.66091i
$716$		0			0
$717$	1.68576e7	−	9.73273e6i	1.22461	−	0.707028i
$718$		0			0
$719$	−1.89565e6	+	3.28337e6i	−0.136753	+	0.236863i	−0.926266	−	0.376871i	$-0.877000\pi$
$719$	−1.89565e6	+	3.28337e6i	−0.136753	+	0.236863i	0.789513	+	0.613734i	$0.210333\pi$
$720$		0			0
$721$	855675.	+	3.08293e6i	0.0613015	+	0.220864i
$722$		0			0
$723$	1.65440e7	+	9.55170e6i	1.17705	+	0.679571i
$724$		0			0
$725$	−4.46117e6	−	7.72698e6i	−0.315213	−	0.545965i
$726$		0			0
$727$	−1.04927e7			−0.736294			−0.368147	−	0.929768i	$-0.620008\pi$
$727$	−1.04927e7			−0.736294			−0.368147	+	0.929768i	$0.620008\pi$
$728$		0			0
$729$	−1.94158e7			−1.35312
$730$		0			0
$731$	−1.43043e7	−	2.47758e7i	−0.990087	−	1.71488i
$732$		0			0
$733$	7.94841e6	+	4.58902e6i	0.546412	+	0.315471i	0.747674	−	0.664066i	$-0.231171\pi$
$733$	7.94841e6	+	4.58902e6i	0.546412	+	0.315471i	−0.201262	+	0.979538i	$0.564504\pi$
$734$		0			0
$735$	−3.40075e7	+	608016.i	−2.32197	+	0.0415142i
$736$		0			0
$737$	2.14889e6	−	3.72199e6i	0.145729	−	0.252410i
$738$		0			0
$739$	1.98212e7	−	1.14437e7i	1.33511	−	0.770827i	0.349034	−	0.937110i	$-0.386510\pi$
$739$	1.98212e7	−	1.14437e7i	1.33511	−	0.770827i	0.986078	+	0.166283i	$0.0531764\pi$
$740$		0			0
$741$			2.07285e7i			1.38683i
$742$		0			0
$743$			8.13890e6i			0.540871i	0.962738	+	0.270435i	$0.0871676\pi$
$743$			8.13890e6i			0.540871i	−0.962738	+	0.270435i	$0.912832\pi$
$744$		0			0
$745$	−1.50409e6	+	868386.i	−0.0992848	+	0.0573221i
$746$		0			0
$747$	2.49381e6	−	4.31940e6i	0.163516	−	0.283219i
$748$		0			0
$749$	1.77865e6	−	493668.i	0.115847	−	0.0321537i
$750$		0			0
$751$	6.93489e6	+	4.00386e6i	0.448683	+	0.259048i	0.707274	−	0.706940i	$-0.249925\pi$
$751$	6.93489e6	+	4.00386e6i	0.448683	+	0.259048i	−0.258591	+	0.965987i	$0.583258\pi$
$752$		0			0
$753$	1.75275e7	+	3.03585e7i	1.12650	+	1.95116i
$754$		0			0
$755$	−1.83860e7			−1.17387
$756$		0			0
$757$	−1.42534e7			−0.904020			−0.452010	−	0.892013i	$-0.649293\pi$
$757$	−1.42534e7			−0.904020			−0.452010	+	0.892013i	$0.649293\pi$
$758$		0			0
$759$	5.62538e6	+	9.74344e6i	0.354444	+	0.613914i
$760$		0			0
$761$	−6.94692e6	−	4.01081e6i	−0.434841	−	0.251056i	0.266566	−	0.963817i	$-0.414111\pi$
$761$	−6.94692e6	−	4.01081e6i	−0.434841	−	0.251056i	−0.701407	+	0.712761i	$0.747444\pi$
$762$		0			0
$763$	2.26935e7	+	5.86380e6i	1.41120	+	0.364643i
$764$		0			0
$765$	1.89705e7	−	3.28579e7i	1.17200	−	2.02996i
$766$		0			0
$767$	−1.67025e7	+	9.64322e6i	−1.02517	+	0.591880i
$768$		0			0
$769$			6.02804e6i			0.367587i	0.982965	+	0.183794i	$0.0588378\pi$
$769$			6.02804e6i			0.367587i	−0.982965	+	0.183794i	$0.941162\pi$
$770$		0			0
$771$		−	9.71166e6i		−	0.588380i
$772$		0			0
$773$	5.18009e6	−	2.99073e6i	0.311809	−	0.180023i	−0.335927	−	0.941888i	$-0.609049\pi$
$773$	5.18009e6	−	2.99073e6i	0.311809	−	0.180023i	0.647736	+	0.761865i	$0.275716\pi$
$774$		0			0
$775$	1.84484e6	−	3.19536e6i	0.110333	−	0.191102i
$776$		0			0
$777$	2.92292e7	−	2.97564e7i	1.73686	−	1.76819i
$778$		0			0
$779$	−2.06606e7	−	1.19284e7i	−1.21983	−	0.704270i
$780$		0			0
$781$	1.16305e7	+	2.01446e7i	0.682293	+	1.18177i
$782$		0			0
$783$	2.22043e6			0.129429
$784$		0			0
$785$	3.46265e7			2.00556
$786$		0			0
$787$	−5.44997e6	−	9.43963e6i	−0.313659	−	0.543273i	0.665493	−	0.746404i	$-0.268221\pi$
$787$	−5.44997e6	−	9.43963e6i	−0.313659	−	0.543273i	−0.979151	+	0.203132i	$0.934888\pi$
$788$		0			0
$789$	2.51645e7	+	1.45287e7i	1.43911	+	0.830873i
$790$		0			0
$791$	−4.22274e6	+	4.29891e6i	−0.239968	+	0.244296i
$792$		0			0
$793$	−1.39766e7	+	2.42083e7i	−0.789260	+	1.36704i
$794$		0			0
$795$	2.80746e7	−	1.62089e7i	1.57542	−	0.909567i
$796$		0			0
$797$		−	9.04729e6i		−	0.504514i	−0.967660	−	0.252257i	$-0.918827\pi$
$797$		−	9.04729e6i		−	0.504514i	0.967660	−	0.252257i	$-0.0811728\pi$
$798$		0			0
$799$		−	3.36562e7i		−	1.86508i
$800$		0			0
$801$	1.19258e7	−	6.88539e6i	0.656762	−	0.379182i
$802$		0			0
$803$	1.31722e7	−	2.28149e7i	0.720889	−	1.24862i
$804$		0			0
$805$	−1.31883e7	−	3.40775e6i	−0.717299	−	0.185344i
$806$		0			0
$807$	−5.47537e6	−	3.16121e6i	−0.295958	−	0.170871i
$808$		0			0
$809$	−1.33927e7	−	2.31969e7i	−0.719445	−	1.24612i	−0.961220	−	0.275783i	$-0.911063\pi$
$809$	−1.33927e7	−	2.31969e7i	−0.719445	−	1.24612i	0.241775	−	0.970332i	$-0.422270\pi$
$810$		0			0
$811$	1.53751e7			0.820853			0.410427	−	0.911894i	$-0.365380\pi$
$811$	1.53751e7			0.820853			0.410427	+	0.911894i	$0.365380\pi$
$812$		0			0
$813$	−1.78688e7			−0.948130
$814$		0			0
$815$	−9.61312e6	−	1.66504e7i	−0.506956	−	0.878074i
$816$		0			0
$817$	2.33429e7	+	1.34771e7i	1.22349	+	0.706382i
$818$		0			0
$819$	2.33377e7	−	6.47745e6i	1.21576	−	0.337438i
$820$		0			0
$821$	1.10075e7	−	1.90655e7i	0.569940	−	0.987166i	−0.426631	−	0.904426i	$-0.640300\pi$
$821$	1.10075e7	−	1.90655e7i	0.569940	−	0.987166i	0.996571	−	0.0827399i	$-0.0263671\pi$
$822$		0			0
$823$	−3.81561e6	+	2.20294e6i	−0.196365	+	0.113371i	−0.594959	−	0.803756i	$-0.702832\pi$
$823$	−3.81561e6	+	2.20294e6i	−0.196365	+	0.113371i	0.398594	+	0.917128i	$0.369498\pi$
$824$		0			0
$825$			4.24794e7i			2.17292i
$826$		0			0
$827$		−	1.96164e7i		−	0.997369i	−0.866783	−	0.498685i	$-0.833817\pi$
$827$		−	1.96164e7i		−	0.997369i	0.866783	−	0.498685i	$-0.166183\pi$
$828$		0			0
$829$	4.99433e6	−	2.88348e6i	0.252401	−	0.145724i	−0.368462	−	0.929643i	$-0.620116\pi$
$829$	4.99433e6	−	2.88348e6i	0.252401	−	0.145724i	0.620863	+	0.783919i	$0.286782\pi$
$830$		0			0
$831$	368409.	−	638104.i	0.0185067	−	0.0320545i
$832$		0			0
$833$	−1.28712e7	−	2.14007e7i	−0.642697	−	1.06860i
$834$		0			0
$835$	−2.13043e6	−	1.23000e6i	−0.105743	−	0.0610506i
$836$		0			0
$837$	459110.	+	795201.i	0.0226518	+	0.0392341i
$838$		0			0
$839$	1.04959e6			0.0514774			0.0257387	−	0.999669i	$-0.491806\pi$
$839$	1.04959e6			0.0514774			0.0257387	+	0.999669i	$0.491806\pi$
$840$		0			0
$841$	−1.66374e7			−0.811137
$842$		0			0
$843$	5.90588e6	+	1.02293e7i	0.286231	+	0.495766i
$844$		0			0
$845$	2.92991e6	+	1.69158e6i	0.141160	+	0.0814989i
$846$		0			0
$847$	−108958.	−	392567.i	−0.00521856	−	0.0188020i
$848$		0			0
$849$	2.25194e7	−	3.90047e7i	1.07223	−	1.85715i
$850$		0			0
$851$	1.44664e7	−	8.35215e6i	0.684755	−	0.395343i
$852$		0			0
$853$		−	2.19938e7i		−	1.03497i	−0.855692	−	0.517485i	$-0.826868\pi$
$853$		−	2.19938e7i		−	1.03497i	0.855692	−	0.517485i	$-0.173132\pi$
$854$		0			0
$855$			3.57469e7i			1.67233i
$856$		0			0
$857$	1.41287e7	−	8.15723e6i	0.657130	−	0.379394i	−0.134052	−	0.990974i	$-0.542799\pi$
$857$	1.41287e7	−	8.15723e6i	0.657130	−	0.379394i	0.791183	+	0.611580i	$0.209466\pi$
$858$		0			0
$859$	−1.57232e7	+	2.72333e7i	−0.727038	+	1.25927i	0.231092	+	0.972932i	$0.425770\pi$
$859$	−1.57232e7	+	2.72333e7i	−0.727038	+	1.25927i	−0.958130	+	0.286335i	$0.907563\pi$
$860$		0			0
$861$	−1.27814e7	+	4.94653e7i	−0.587586	+	2.27401i
$862$		0			0
$863$	1.68377e7	+	9.72127e6i	0.769585	+	0.444320i	0.832727	−	0.553684i	$-0.186779\pi$
$863$	1.68377e7	+	9.72127e6i	0.769585	+	0.444320i	−0.0631415	+	0.998005i	$0.520112\pi$
$864$		0			0
$865$	−1.88281e7	−	3.26113e7i	−0.855593	−	1.48193i
$866$		0			0
$867$	1.82222e7			0.823293
$868$		0			0
$869$	7.48223e6			0.336110
$870$		0			0
$871$	−3.39550e6	−	5.88117e6i	−0.151655	−	0.262675i
$872$		0			0
$873$	1.33541e7	+	7.71002e6i	0.593035	+	0.342389i
$874$		0			0
$875$	−1.13979e7	−	1.11960e7i	−0.503276	−	0.494358i
$876$		0			0
$877$	−2.07888e7	+	3.60072e7i	−0.912704	+	1.58085i	−0.102476	+	0.994735i	$0.532677\pi$
$877$	−2.07888e7	+	3.60072e7i	−0.912704	+	1.58085i	−0.810228	+	0.586115i	$0.800657\pi$
$878$		0			0
$879$	−3.18895e7	+	1.84114e7i	−1.39212	+	0.803739i
$880$		0			0
$881$			7.22354e6i			0.313553i	0.987634	+	0.156776i	$0.0501102\pi$
$881$			7.22354e6i			0.313553i	−0.987634	+	0.156776i	$0.949890\pi$
$882$		0			0
$883$		−	2.73652e7i		−	1.18113i	−0.806992	−	0.590563i	$-0.798906\pi$
$883$		−	2.73652e7i		−	1.18113i	0.806992	−	0.590563i	$-0.201094\pi$
$884$		0			0
$885$	−5.27923e7	+	3.04796e7i	−2.26575	+	1.30813i
$886$		0			0
$887$	1.05430e6	−	1.82610e6i	0.0449939	−	0.0779317i	−0.842651	−	0.538460i	$-0.819006\pi$
$887$	1.05430e6	−	1.82610e6i	0.0449939	−	0.0779317i	0.887645	+	0.460528i	$0.152340\pi$
$888$		0			0
$889$	−1.19953e7	−	1.17828e7i	−0.509046	−	0.500026i
$890$		0			0
$891$	1.57263e7	+	9.07961e6i	0.663642	+	0.383154i
$892$		0			0
$893$	1.58549e7	+	2.74615e7i	0.665326	+	1.15238i
$894$		0			0
$895$	−3.63814e6			−0.151818
$896$		0			0
$897$	1.77775e7			0.737716
$898$		0			0
$899$	800971.	+	1.38732e6i	0.0330535	+	0.0572504i
$900$		0			0
$901$	2.06130e7	+	1.19009e7i	0.845921	+	0.488393i
$902$		0			0
$903$	1.44408e7	−	5.58873e7i	0.589349	−	2.28084i
$904$		0			0
$905$	−1.19570e7	+	2.07102e7i	−0.485291	+	0.840549i
$906$		0			0
$907$	−3.21121e7	+	1.85399e7i	−1.29614	+	0.748324i	−0.979734	−	0.200301i	$-0.935808\pi$
$907$	−3.21121e7	+	1.85399e7i	−1.29614	+	0.748324i	−0.316401	+	0.948625i	$0.602475\pi$
$908$		0			0
$909$		−	5.25525e7i		−	2.10952i
$910$		0			0
$911$		−	8.94944e6i		−	0.357273i	−0.983915	−	0.178636i	$-0.942831\pi$
$911$		−	8.94944e6i		−	0.357273i	0.983915	−	0.178636i	$-0.0571686\pi$
$912$		0			0
$913$	−5.99846e6	+	3.46321e6i	−0.238157	+	0.137500i
$914$		0			0
$915$	−4.41764e7	+	7.65158e7i	−1.74437	+	3.02133i
$916$		0			0
$917$	5.51581e6	+	1.98730e7i	0.216614	+	0.780442i
$918$		0			0
$919$	−2.38735e7	−	1.37834e7i	−0.932455	−	0.538353i	−0.0448676	−	0.998993i	$-0.514287\pi$
$919$	−2.38735e7	−	1.37834e7i	−0.932455	−	0.538353i	−0.887587	+	0.460640i	$0.847620\pi$
$920$		0			0
$921$	1.71926e6	+	2.97785e6i	0.0667873	+	0.115679i
$922$		0			0
$923$	3.67550e7			1.42008
$924$		0			0
$925$	6.30703e7			2.42365
$926$		0			0
$927$	−3.60052e6	−	6.23628e6i	−0.137615	−	0.238356i
$928$		0			0
$929$	−2.99496e7	−	1.72914e7i	−1.13855	−	0.657341i	−0.192477	−	0.981301i	$-0.561652\pi$
$929$	−2.99496e7	−	1.72914e7i	−1.13855	−	0.657341i	−0.946071	+	0.323960i	$0.894986\pi$
$930$		0			0
$931$	2.05837e7	+	1.13983e7i	0.778303	+	0.430989i
$932$		0			0
$933$	8.43246e6	−	1.46055e7i	0.317140	−	0.549302i
$934$		0			0
$935$	−4.56307e7	+	2.63449e7i	−1.70698	+	0.985524i
$936$		0			0
$937$		−	2.64955e7i		−	0.985877i	−0.870064	−	0.492938i	$-0.835923\pi$
$937$		−	2.64955e7i		−	0.985877i	0.870064	−	0.492938i	$-0.164077\pi$
$938$		0			0
$939$			4.21185e7i			1.55887i
$940$		0			0
$941$	−2.08986e7	+	1.20658e7i	−0.769386	+	0.444205i	−0.832655	−	0.553791i	$-0.813181\pi$
$941$	−2.08986e7	+	1.20658e7i	−0.769386	+	0.444205i	0.0632697	+	0.997996i	$0.479847\pi$
$942$		0			0
$943$	−1.02302e7	+	1.77193e7i	−0.374633	+	0.648884i
$944$		0			0
$945$	1.23328e7	−	3.42301e6i	0.449245	−	0.124689i
$946$		0			0
$947$	2.82110e7	+	1.62876e7i	1.02222	+	0.590178i	0.914746	−	0.404030i	$-0.132391\pi$
$947$	2.82110e7	+	1.62876e7i	1.02222	+	0.590178i	0.107472	+	0.994208i	$0.465724\pi$
$948$		0			0
$949$	−2.08135e7	−	3.60501e7i	−0.750206	−	1.29939i
$950$		0			0
$951$	−3.65429e7			−1.31024
$952$		0			0
$953$	2.20095e7			0.785016			0.392508	−	0.919749i	$-0.371608\pi$
$953$	2.20095e7			0.785016			0.392508	+	0.919749i	$0.371608\pi$
$954$		0			0
$955$	−2.17405e7	−	3.76557e7i	−0.771368	−	1.33605i
$956$		0			0
$957$	−1.59723e7	−	9.22159e6i	−0.563750	−	0.325481i
$958$		0			0
$959$	1.63458e7	+	4.22362e6i	0.573932	+	0.148299i
$960$		0			0
$961$	1.39833e7	−	2.42199e7i	0.488430	−	0.845986i
$962$		0			0
$963$	−3.59792e6	+	2.07726e6i	−0.125022	+	0.0721814i
$964$		0			0
$965$		−	2.01252e7i		−	0.695699i
$966$		0			0
$967$			4.35658e7i			1.49823i	0.662439	+	0.749116i	$0.269522\pi$
$967$			4.35658e7i			1.49823i	−0.662439	+	0.749116i	$0.730478\pi$
$968$		0			0
$969$	−4.16595e7	+	2.40521e7i	−1.42530	+	0.822895i
$970$		0			0
$971$	1.75209e7	−	3.03471e7i	0.596360	−	1.03293i	−0.396994	−	0.917821i	$-0.629947\pi$
$971$	1.75209e7	−	3.03471e7i	0.596360	−	1.03293i	0.993353	−	0.115104i	$-0.0367201\pi$
$972$		0			0
$973$	−2.88370e7	+	2.93572e7i	−0.976491	+	0.994105i
$974$		0			0
$975$	5.81296e7	+	3.35612e7i	1.95833	+	1.13064i
$976$		0			0
$977$	−4.14260e6	−	7.17519e6i	−0.138847	−	0.240490i	0.788214	−	0.615402i	$-0.211006\pi$
$977$	−4.14260e6	−	7.17519e6i	−0.138847	−	0.240490i	−0.927060	+	0.374912i	$0.877673\pi$
$978$		0			0
$979$	−1.91238e7			−0.637703
$980$		0			0
$981$	−5.27536e7			−1.75017
$982$		0			0
$983$	3.83932e6	+	6.64989e6i	0.126727	+	0.219498i	0.922407	−	0.386220i	$-0.126219\pi$
$983$	3.83932e6	+	6.64989e6i	0.126727	+	0.219498i	−0.795680	+	0.605718i	$0.792886\pi$
$984$		0			0
$985$	−5.14299e6	−	2.96931e6i	−0.168898	−	0.0975135i
$986$		0			0
$987$	4.75866e7	−	4.84450e7i	1.55486	−	1.58291i
$988$		0			0
$989$	1.15584e7	−	2.00198e7i	0.375757	−	0.650831i
$990$		0			0
$991$	3.93535e7	−	2.27207e7i	1.27291	−	0.734917i	0.297379	−	0.954760i	$-0.403888\pi$
$991$	3.93535e7	−	2.27207e7i	1.27291	−	0.734917i	0.975535	+	0.219842i	$0.0705543\pi$
$992$		0			0
$993$			7.24598e7i			2.33198i
$994$		0			0
$995$		−	5.80884e7i		−	1.86008i
$996$		0			0
$997$	−4.58960e7	+	2.64981e7i	−1.46230	+	0.844261i	−0.999118	−	0.0420007i	$-0.986627\pi$
$997$	−4.58960e7	+	2.64981e7i	−1.46230	+	0.844261i	−0.463185	+	0.886262i	$0.653293\pi$
$998$		0			0
$999$	−7.84787e6	+	1.35929e7i	−0.248793	+	0.430922i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.p.b.47.7	yes	14
4.3	odd	2		112.6.p.c.47.1	yes	14
7.2	even	3		784.6.f.d.783.2		14
7.3	odd	6		112.6.p.c.31.1	yes	14
7.5	odd	6		784.6.f.c.783.13		14
28.3	even	6	inner	112.6.p.b.31.7	✓	14
28.19	even	6		784.6.f.d.783.1		14
28.23	odd	6		784.6.f.c.783.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.b.31.7	✓	14	28.3	even	6	inner
112.6.p.b.47.7	yes	14	1.1	even	1	trivial
112.6.p.c.31.1	yes	14	7.3	odd	6
112.6.p.c.47.1	yes	14	4.3	odd	2
784.6.f.c.783.13		14	7.5	odd	6
784.6.f.c.783.14		14	28.23	odd	6
784.6.f.d.783.1		14	28.19	even	6
784.6.f.d.783.2		14	7.2	even	3

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(11.5627 + 20.0272i) q^{3} +(-75.7872 - 43.7557i) q^{5} +(90.8476 - 92.4863i) q^{7} +(-145.892 + 252.692i) q^{9} +(350.920 - 202.604i) q^{11} -640.275i q^{13} -2023.74i q^{15} +(1286.81 - 742.939i) q^{17} +(-699.973 + 1212.39i) q^{19} +(2902.68 + 750.029i) q^{21} +(1039.79 + 600.322i) q^{23} +(2266.63 + 3925.92i) q^{25} -1128.15 q^{27} -1968.20 q^{29} +(-406.957 - 704.870i) q^{31} +(8115.18 + 4685.30i) q^{33} +(-10931.9 + 3034.17i) q^{35} +(6956.39 - 12048.8i) q^{37} +(12822.9 - 7403.31i) q^{39} +17041.2i q^{41} -19253.7i q^{43} +(22113.5 - 12767.2i) q^{45} +(11325.4 - 19616.1i) q^{47} +(-300.442 - 16804.3i) q^{49} +(29757.9 + 17180.8i) q^{51} +(8009.36 + 13872.6i) q^{53} -35460.4 q^{55} -32374.3 q^{57} +(-15061.0 - 26086.5i) q^{59} +(-37809.1 - 21829.1i) q^{61} +(10116.7 + 36449.5i) q^{63} +(-28015.7 + 48524.7i) q^{65} +(9185.38 - 5303.18i) q^{67} +27765.4i q^{69} +57405.1i q^{71} +(56304.1 - 32507.2i) q^{73} +(-52416.7 + 90788.5i) q^{75} +(13142.2 - 50861.4i) q^{77} +(15991.3 + 9232.58i) q^{79} +(22407.3 + 38810.6i) q^{81} -17093.5 q^{83} -130031. q^{85} +(-22757.7 - 39417.4i) q^{87} +(-40872.2 - 23597.6i) q^{89} +(-59216.7 - 58167.5i) q^{91} +(9411.04 - 16300.4i) q^{93} +(106098. - 61255.7i) q^{95} -52847.4i q^{97} +118233. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) \) 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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