LMFDB - Embedded newform 108.5.d.b.55.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(55,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("108.55");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.d (of order $2$, degree $1$, 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38x^{14} + 1016x^{12} + 13512x^{10} + 130640x^{8} + 569472x^{6} + 1783808x^{4} + 352256x^{2} + 65536 $ x^16 + 38x^14 + 1016x^12 + 13512x^10 + 130640x^8 + 569472x^6 + 1783808x^4 + 352256*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.14
Root			$1.12787 + 1.95353i$ of defining polynomial
Character	$\chi$	$=$	108.55
Dual form			108.5.d.b.55.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.49052 + 1.95353i) q^{2} +(8.36746 + 13.6377i) q^{4} +47.1174 q^{5} -67.4003i q^{7} +(2.56528 + 63.9486i) q^{8} +O(q^{10})$	\(q+(3.49052 + 1.95353i) q^{2} +(8.36746 + 13.6377i) q^{4} +47.1174 q^{5} -67.4003i q^{7} +(2.56528 + 63.9486i) q^{8} +(164.464 + 92.0451i) q^{10} -69.3981i q^{11} +3.28838 q^{13} +(131.668 - 235.262i) q^{14} +(-115.971 + 228.225i) q^{16} -116.780 q^{17} +513.525i q^{19} +(394.253 + 642.571i) q^{20} +(135.571 - 242.236i) q^{22} -134.254i q^{23} +1595.05 q^{25} +(11.4782 + 6.42394i) q^{26} +(919.182 - 563.970i) q^{28} -772.602 q^{29} +1069.97i q^{31} +(-850.643 + 570.072i) q^{32} +(-407.623 - 228.133i) q^{34} -3175.73i q^{35} -1034.09 q^{37} +(-1003.19 + 1792.47i) q^{38} +(120.869 + 3013.09i) q^{40} -2272.62 q^{41} -2523.25i q^{43} +(946.428 - 580.686i) q^{44} +(262.270 - 468.618i) q^{46} +694.871i q^{47} -2141.80 q^{49} +(5567.55 + 3115.97i) q^{50} +(27.5154 + 44.8458i) q^{52} +1166.03 q^{53} -3269.86i q^{55} +(4310.15 - 172.901i) q^{56} +(-2696.78 - 1509.30i) q^{58} -5539.90i q^{59} -3204.11 q^{61} +(-2090.21 + 3734.75i) q^{62} +(-4082.84 + 328.092i) q^{64} +154.940 q^{65} +5366.14i q^{67} +(-977.152 - 1592.60i) q^{68} +(6203.87 - 11084.9i) q^{70} +166.225i q^{71} +5085.51 q^{73} +(-3609.51 - 2020.12i) q^{74} +(-7003.28 + 4296.90i) q^{76} -4677.46 q^{77} -724.474i q^{79} +(-5464.26 + 10753.4i) q^{80} +(-7932.63 - 4439.62i) q^{82} -8153.13i q^{83} -5502.37 q^{85} +(4929.24 - 8807.46i) q^{86} +(4437.91 - 178.026i) q^{88} +6963.40 q^{89} -221.638i q^{91} +(1830.92 - 1123.37i) q^{92} +(-1357.45 + 2425.46i) q^{94} +24196.0i q^{95} -278.895 q^{97} +(-7476.00 - 4184.07i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 28 q^{4}+O(q^{10}) $ 16 * q + 28 * q^4	$ 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) $ 16 * q + 28 * q^4 + 176 * q^10 + 176 * q^13 + 88 * q^16 + 384 * q^22 + 2736 * q^25 + 1812 * q^28 + 1520 * q^34 + 80 * q^37 - 688 * q^40 - 1824 * q^46 - 7904 * q^49 - 5236 * q^52 - 11584 * q^58 - 1648 * q^61 + 5056 * q^64 + 26688 * q^70 + 80 * q^73 - 8388 * q^76 - 38464 * q^82 - 16832 * q^85 - 29520 * q^88 - 4512 * q^94 + 14864 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	3.49052	+	1.95353i	0.872630	+	0.488382i
$2$	3.49052	+	1.95353i	0.872630	+	0.488382i
$3$		0			0
$3$		0			0
$4$	8.36746	+	13.6377i	0.522967	+	0.852353i
$5$	47.1174			1.88470			0.942348	−	0.334634i	$-0.108613\pi$
$5$	47.1174			1.88470			0.942348	+	0.334634i	$0.108613\pi$
$6$		0			0
$7$		−	67.4003i		−	1.37552i	−0.725940	−	0.687758i	$-0.758595\pi$
$7$		−	67.4003i		−	1.37552i	0.725940	−	0.687758i	$-0.241405\pi$
$8$	2.56528	+	63.9486i	0.0400825	+	0.999196i
$9$		0			0
$10$	164.464	+	92.0451i	1.64464	+	0.920451i
$11$		−	69.3981i		−	0.573538i	−0.958000	−	0.286769i	$-0.907419\pi$
$11$		−	69.3981i		−	0.573538i	0.958000	−	0.286769i	$-0.0925813\pi$
$12$		0			0
$13$	3.28838			0.0194579			0.00972893	−	0.999953i	$-0.496903\pi$
$13$	3.28838			0.0194579			0.00972893	+	0.999953i	$0.496903\pi$
$14$	131.668	−	235.262i	0.671777	−	1.20032i
$15$		0			0
$16$	−115.971	+	228.225i	−0.453012	+	0.891504i
$17$	−116.780			−0.404083			−0.202041	−	0.979377i	$-0.564758\pi$
$17$	−116.780			−0.404083			−0.202041	+	0.979377i	$0.564758\pi$
$18$		0			0
$19$			513.525i			1.42251i	0.702935	+	0.711254i	$0.251872\pi$
$19$			513.525i			1.42251i	−0.702935	+	0.711254i	$0.748128\pi$
$20$	394.253	+	642.571i	0.985633	+	1.60643i
$21$		0			0
$22$	135.571	−	242.236i	0.280106	−	0.500487i
$23$		−	134.254i		−	0.253789i	−0.991916	−	0.126895i	$-0.959499\pi$
$23$		−	134.254i		−	0.253789i	0.991916	−	0.126895i	$-0.0405010\pi$
$24$		0			0
$25$	1595.05			2.55208
$26$	11.4782	+	6.42394i	0.0169795	+	0.00950287i
$27$		0			0
$28$	919.182	−	563.970i	1.17243	−	0.719349i
$29$	−772.602			−0.918671			−0.459335	−	0.888263i	$-0.651912\pi$
$29$	−772.602			−0.918671			−0.459335	+	0.888263i	$0.651912\pi$
$30$		0			0
$31$			1069.97i			1.11339i	0.830717	+	0.556695i	$0.187931\pi$
$31$			1069.97i			1.11339i	−0.830717	+	0.556695i	$0.812069\pi$
$32$	−850.643	+	570.072i	−0.830706	+	0.556711i
$33$		0			0
$34$	−407.623	−	228.133i	−0.352615	−	0.197347i
$35$		−	3175.73i		−	2.59243i
$36$		0			0
$37$	−1034.09			−0.755362			−0.377681	−	0.925936i	$-0.623278\pi$
$37$	−1034.09			−0.755362			−0.377681	+	0.925936i	$0.623278\pi$
$38$	−1003.19	+	1792.47i	−0.694727	+	1.24132i
$39$		0			0
$40$	120.869	+	3013.09i	0.0755434	+	1.88318i
$41$	−2272.62			−1.35195			−0.675973	−	0.736927i	$-0.736276\pi$
$41$	−2272.62			−1.35195			−0.675973	+	0.736927i	$0.736276\pi$
$42$		0			0
$43$		−	2523.25i		−	1.36466i	−0.731046	−	0.682328i	$-0.760967\pi$
$43$		−	2523.25i		−	1.36466i	0.731046	−	0.682328i	$-0.239033\pi$
$44$	946.428	−	580.686i	0.488857	−	0.299941i
$45$		0			0
$46$	262.270	−	468.618i	0.123946	−	0.221464i
$47$			694.871i			0.314564i	0.987554	+	0.157282i	$0.0502731\pi$
$47$			694.871i			0.314564i	−0.987554	+	0.157282i	$0.949727\pi$
$48$		0			0
$49$	−2141.80			−0.892045
$50$	5567.55	+	3115.97i	2.22702	+	1.24639i
$51$		0			0
$52$	27.5154	+	44.8458i	0.0101758	+	0.0165850i
$53$	1166.03			0.415106			0.207553	−	0.978224i	$-0.433450\pi$
$53$	1166.03			0.415106			0.207553	+	0.978224i	$0.433450\pi$
$54$		0			0
$55$		−	3269.86i		−	1.08095i
$56$	4310.15	−	172.901i	1.37441	−	0.0551342i
$57$		0			0
$58$	−2696.78	−	1509.30i	−0.801660	−	0.448662i
$59$		−	5539.90i		−	1.59147i	−0.605647	−	0.795733i	$-0.707086\pi$
$59$		−	5539.90i		−	1.59147i	0.605647	−	0.795733i	$-0.292914\pi$
$60$		0			0
$61$	−3204.11			−0.861089			−0.430544	−	0.902569i	$-0.641678\pi$
$61$	−3204.11			−0.861089			−0.430544	+	0.902569i	$0.641678\pi$
$62$	−2090.21	+	3734.75i	−0.543760	+	0.971578i
$63$		0			0
$64$	−4082.84	+	328.092i	−0.996787	+	0.0801006i
$65$	154.940			0.0366722
$66$		0			0
$67$			5366.14i			1.19540i	0.801721	+	0.597699i	$0.203918\pi$
$67$			5366.14i			1.19540i	−0.801721	+	0.597699i	$0.796082\pi$
$68$	−977.152	−	1592.60i	−0.211322	−	0.344421i
$69$		0			0
$70$	6203.87	−	11084.9i	1.26610	−	2.26223i
$71$			166.225i			0.0329746i	0.999864	+	0.0164873i	$0.00524832\pi$
$71$			166.225i			0.0329746i	−0.999864	+	0.0164873i	$0.994752\pi$
$72$		0			0
$73$	5085.51			0.954309			0.477154	−	0.878819i	$-0.341668\pi$
$73$	5085.51			0.954309			0.477154	+	0.878819i	$0.341668\pi$
$74$	−3609.51	−	2020.12i	−0.659151	−	0.368905i
$75$		0			0
$76$	−7003.28	+	4296.90i	−1.21248	+	0.743924i
$77$	−4677.46			−0.788911
$78$		0			0
$79$		−	724.474i		−	0.116083i	−0.998314	−	0.0580415i	$-0.981514\pi$
$79$		−	724.474i		−	0.116083i	0.998314	−	0.0580415i	$-0.0184856\pi$
$80$	−5464.26	+	10753.4i	−0.853790	+	1.68021i
$81$		0			0
$82$	−7932.63	−	4439.62i	−1.17975	−	0.660265i
$83$		−	8153.13i		−	1.18350i	−0.806121	−	0.591750i	$-0.798437\pi$
$83$		−	8153.13i		−	1.18350i	0.806121	−	0.591750i	$-0.201563\pi$
$84$		0			0
$85$	−5502.37			−0.761573
$86$	4929.24	−	8807.46i	0.666473	−	1.19084i
$87$		0			0
$88$	4437.91	−	178.026i	0.573077	−	0.0229889i
$89$	6963.40			0.879107			0.439553	−	0.898217i	$-0.355137\pi$
$89$	6963.40			0.879107			0.439553	+	0.898217i	$0.355137\pi$
$90$		0			0
$91$		−	221.638i		−	0.0267646i
$92$	1830.92	−	1123.37i	0.216318	−	0.132723i
$93$		0			0
$94$	−1357.45	+	2425.46i	−0.153627	+	0.274498i
$95$			24196.0i			2.68099i
$96$		0			0
$97$	−278.895			−0.0296413			−0.0148207	−	0.999890i	$-0.504718\pi$
$97$	−278.895			−0.0296413			−0.0148207	+	0.999890i	$0.504718\pi$
$98$	−7476.00	−	4184.07i	−0.778426	−	0.435659i
$99$		0			0
$100$	13346.5	+	21752.7i	1.33465	+	2.17527i
$101$	−9308.10			−0.912469			−0.456235	−	0.889860i	$-0.650802\pi$
$101$	−9308.10			−0.912469			−0.456235	+	0.889860i	$0.650802\pi$
$102$		0			0
$103$		−	3271.88i		−	0.308406i	−0.988039	−	0.154203i	$-0.950719\pi$
$103$		−	3271.88i		−	0.308406i	0.988039	−	0.154203i	$-0.0492810\pi$
$104$	8.43562	+	210.287i	0.000779921	+	0.0194422i
$105$		0			0
$106$	4070.06	+	2277.88i	0.362234	+	0.202730i
$107$		−	1845.07i		−	0.161156i	−0.996748	−	0.0805778i	$-0.974323\pi$
$107$		−	1845.07i		−	0.161156i	0.996748	−	0.0805778i	$-0.0256765\pi$
$108$		0			0
$109$	6021.20			0.506793			0.253396	−	0.967363i	$-0.418452\pi$
$109$	6021.20			0.506793			0.253396	+	0.967363i	$0.418452\pi$
$110$	6387.76	−	11413.5i	0.527914	−	0.943266i
$111$		0			0
$112$	15382.4	+	7816.49i	1.22628	+	0.623126i
$113$	−8841.22			−0.692398			−0.346199	−	0.938161i	$-0.612528\pi$
$113$	−8841.22			−0.692398			−0.346199	+	0.938161i	$0.612528\pi$
$114$		0			0
$115$		−	6325.72i		−	0.478315i
$116$	−6464.72	−	10536.5i	−0.480434	−	0.783032i
$117$		0			0
$118$	10822.3	−	19337.1i	0.777243	−	1.38876i
$119$			7871.00i			0.555822i
$120$		0			0
$121$	9824.90			0.671054
$122$	−11184.0	−	6259.32i	−0.751412	−	0.420540i
$123$		0			0
$124$	−14591.9	+	8952.92i	−0.949002	+	0.582266i
$125$	45706.2			2.92520
$126$		0			0
$127$			9165.05i			0.568234i	0.958790	+	0.284117i	$0.0917004\pi$
$127$			9165.05i			0.568234i	−0.958790	+	0.284117i	$0.908300\pi$
$128$	−14892.2	−	6830.72i	−0.908946	−	0.416914i
$129$		0			0
$130$	540.821	+	302.679i	0.0320012	+	0.0179100i
$131$			16239.9i			0.946325i	0.880975	+	0.473162i	$0.156888\pi$
$131$			16239.9i			0.946325i	−0.880975	+	0.473162i	$0.843112\pi$
$132$		0			0
$133$	34611.7			1.95668
$134$	−10482.9	+	18730.6i	−0.583810	+	1.04314i
$135$		0			0
$136$	−299.573	−	7467.91i	−0.0161967	−	0.403758i
$137$	30410.5			1.62025			0.810125	−	0.586257i	$-0.199399\pi$
$137$	30410.5			1.62025			0.810125	+	0.586257i	$0.199399\pi$
$138$		0			0
$139$			32689.0i			1.69189i	0.533269	+	0.845946i	$0.320963\pi$
$139$			32689.0i			1.69189i	−0.533269	+	0.845946i	$0.679037\pi$
$140$	43309.5	−	26572.8i	2.20967	−	1.35575i
$141$		0			0
$142$	−324.725	+	580.212i	−0.0161042	+	0.0287747i
$143$		−	228.207i		−	0.0111598i
$144$		0			0
$145$	−36403.0			−1.73141
$146$	17751.1	+	9934.68i	0.832759	+	0.466067i
$147$		0			0
$148$	−8652.71	−	14102.6i	−0.395029	−	0.643835i
$149$	−13388.8			−0.603073			−0.301536	−	0.953455i	$-0.597499\pi$
$149$	−13388.8			−0.603073			−0.301536	+	0.953455i	$0.597499\pi$
$150$		0			0
$151$			9106.99i			0.399412i	0.979856	+	0.199706i	$0.0639987\pi$
$151$			9106.99i			0.399412i	−0.979856	+	0.199706i	$0.936001\pi$
$152$	−32839.2	+	1317.34i	−1.42136	+	0.0570177i
$153$		0			0
$154$	−16326.8	−	9137.54i	−0.688428	−	0.385290i
$155$			50414.1i			2.09840i
$156$		0			0
$157$	4555.18			0.184802			0.0924009	−	0.995722i	$-0.470546\pi$
$157$	4555.18			0.184802			0.0924009	+	0.995722i	$0.470546\pi$
$158$	1415.28	−	2528.79i	0.0566928	−	0.101298i
$159$		0			0
$160$	−40080.1	+	26860.3i	−1.56563	+	1.04923i
$161$	−9048.79			−0.349091
$162$		0			0
$163$			466.424i			0.0175552i	0.999961	+	0.00877760i	$0.00279403\pi$
$163$			466.424i			0.0175552i	−0.999961	+	0.00877760i	$0.997206\pi$
$164$	−19016.1	−	30993.2i	−0.707022	−	1.15233i
$165$		0			0
$166$	15927.4	−	28458.7i	0.578000	−	1.03276i
$167$		−	30165.9i		−	1.08164i	−0.841138	−	0.540821i	$-0.818114\pi$
$167$		−	30165.9i		−	1.08164i	0.841138	−	0.540821i	$-0.181886\pi$
$168$		0			0
$169$	−28550.2			−0.999621
$170$	−19206.1	−	10749.0i	−0.664572	−	0.371938i
$171$		0			0
$172$	34411.2	−	21113.2i	1.16317	−	0.713670i
$173$	−51017.1			−1.70460			−0.852302	−	0.523051i	$-0.824794\pi$
$173$	−51017.1			−1.70460			−0.852302	+	0.523051i	$0.824794\pi$
$174$		0			0
$175$		−	107507.i		−	3.51043i
$176$	15838.4	+	8048.18i	0.511312	+	0.259820i
$177$		0			0
$178$	24305.9	+	13603.2i	0.767135	+	0.429340i
$179$		−	26293.5i		−	0.820622i	−0.911946	−	0.410311i	$-0.865420\pi$
$179$		−	26293.5i		−	0.820622i	0.911946	−	0.410311i	$-0.134580\pi$
$180$		0			0
$181$	3132.04			0.0956026			0.0478013	−	0.998857i	$-0.484779\pi$
$181$	3132.04			0.0956026			0.0478013	+	0.998857i	$0.484779\pi$
$182$	432.975	−	773.631i	0.0130713	−	0.0233556i
$183$		0			0
$184$	8585.38	−	344.401i	0.253585	−	0.0101725i
$185$	−48723.6			−1.42363
$186$		0			0
$187$			8104.31i			0.231757i
$188$	−9476.41	+	5814.31i	−0.268119	+	0.164506i
$189$		0			0
$190$	−47267.5	+	84456.5i	−1.30935	+	2.33952i
$191$		−	17160.1i		−	0.470384i	−0.971949	−	0.235192i	$-0.924428\pi$
$191$		−	17160.1i		−	0.470384i	0.971949	−	0.235192i	$-0.0755718\pi$
$192$		0			0
$193$	−16841.7			−0.452139			−0.226070	−	0.974111i	$-0.572588\pi$
$193$	−16841.7			−0.452139			−0.226070	+	0.974111i	$0.572588\pi$
$194$	−973.490	−	544.829i	−0.0258659	−	0.0144763i
$195$		0			0
$196$	−17921.4	−	29209.1i	−0.466510	−	0.760338i
$197$	67293.3			1.73396			0.866981	−	0.498342i	$-0.166058\pi$
$197$	67293.3			1.73396			0.866981	+	0.498342i	$0.166058\pi$
$198$		0			0
$199$		−	13276.3i		−	0.335253i	−0.985851	−	0.167626i	$-0.946390\pi$
$199$		−	13276.3i		−	0.335253i	0.985851	−	0.167626i	$-0.0536102\pi$
$200$	4091.75	+	102001.i	0.102294	+	2.55003i
$201$		0			0
$202$	−32490.1	−	18183.6i	−0.796248	−	0.445633i
$203$			52073.6i			1.26365i
$204$		0			0
$205$	−107080.			−2.54801
$206$	6391.71	−	11420.6i	0.150620	−	0.269124i
$207$		0			0
$208$	−381.357	+	750.491i	−0.00881465	+	0.0173468i
$209$	35637.7			0.815863
$210$		0			0
$211$			17830.2i			0.400489i	0.979746	+	0.200245i	$0.0641737\pi$
$211$			17830.2i			0.400489i	−0.979746	+	0.200245i	$0.935826\pi$
$212$	9756.73	+	15901.9i	0.217086	+	0.353817i
$213$		0			0
$214$	3604.39	−	6440.26i	0.0787055	−	0.140629i
$215$		−	118889.i		−	2.57196i
$216$		0			0
$217$	72116.2			1.53149
$218$	21017.1	+	11762.6i	0.442242	+	0.247508i
$219$		0			0
$220$	44593.2	−	27360.4i	0.921347	−	0.565298i
$221$	−384.017			−0.00786259
$222$		0			0
$223$			68248.8i			1.37241i	0.727406	+	0.686207i	$0.240726\pi$
$223$			68248.8i			1.37241i	−0.727406	+	0.686207i	$0.759274\pi$
$224$	38423.0	+	57333.6i	0.765765	+	1.14265i
$225$		0			0
$226$	−30860.5	−	17271.6i	−0.604207	−	0.338154i
$227$		−	56672.7i		−	1.09982i	−0.835223	−	0.549911i	$-0.814662\pi$
$227$		−	56672.7i		−	1.09982i	0.835223	−	0.549911i	$-0.185338\pi$
$228$		0			0
$229$	65653.5			1.25195			0.625975	−	0.779843i	$-0.284701\pi$
$229$	65653.5			1.25195			0.625975	+	0.779843i	$0.284701\pi$
$230$	12357.5	−	22080.1i	0.233601	−	0.417392i
$231$		0			0
$232$	−1981.94	−	49406.8i	−0.0368226	−	0.917932i
$233$	−55652.7			−1.02512			−0.512559	−	0.858652i	$-0.671303\pi$
$233$	−55652.7			−1.02512			−0.512559	+	0.858652i	$0.671303\pi$
$234$		0			0
$235$			32740.5i			0.592857i
$236$	75551.2	−	46354.9i	1.35649	−	0.832284i
$237$		0			0
$238$	−15376.2	+	27473.9i	−0.271454	+	0.485027i
$239$		−	13031.8i		−	0.228144i	−0.993472	−	0.114072i	$-0.963611\pi$
$239$		−	13031.8i		−	0.228144i	0.993472	−	0.114072i	$-0.0363895\pi$
$240$		0			0
$241$	57972.1			0.998124			0.499062	−	0.866566i	$-0.333678\pi$
$241$	57972.1			0.998124			0.499062	+	0.866566i	$0.333678\pi$
$242$	34294.0	+	19193.2i	0.585582	+	0.327730i
$243$		0			0
$244$	−26810.3	−	43696.5i	−0.450321	−	0.733952i
$245$	−100916.			−1.68123
$246$		0			0
$247$			1688.67i			0.0276790i
$248$	−68422.9	+	2744.77i	−1.11250	+	0.0446275i
$249$		0			0
$250$	159538.	+	89288.3i	2.55262	+	1.42861i
$251$			81292.8i			1.29034i	0.764038	+	0.645171i	$0.223214\pi$
$251$			81292.8i			1.29034i	−0.764038	+	0.645171i	$0.776786\pi$
$252$		0			0
$253$	−9317.01			−0.145558
$254$	−17904.2	+	31990.8i	−0.277515	+	0.495858i
$255$		0			0
$256$	−38637.4	−	52935.0i	−0.589560	−	0.807724i
$257$	18818.7			0.284920			0.142460	−	0.989801i	$-0.454499\pi$
$257$	18818.7			0.284920			0.142460	+	0.989801i	$0.454499\pi$
$258$		0			0
$259$			69698.0i			1.03901i
$260$	1296.45	+	2113.02i	0.0191783	+	0.0312576i
$261$		0			0
$262$	−31725.0	+	56685.6i	−0.462168	+	0.825791i
$263$		−	96169.6i		−	1.39036i	−0.718837	−	0.695179i	$-0.755325\pi$
$263$		−	96169.6i		−	1.39036i	0.718837	−	0.695179i	$-0.244675\pi$
$264$		0			0
$265$	54940.4			0.782348
$266$	120813.	+	67615.0i	1.70746	+	0.955608i
$267$		0			0
$268$	−73181.5	+	44901.0i	−1.01890	+	0.625153i
$269$	−20853.4			−0.288186			−0.144093	−	0.989564i	$-0.546026\pi$
$269$	−20853.4			−0.288186			−0.144093	+	0.989564i	$0.546026\pi$
$270$		0			0
$271$		−	101517.i		−	1.38230i	−0.722712	−	0.691149i	$-0.757105\pi$
$271$		−	101517.i		−	1.38230i	0.722712	−	0.691149i	$-0.242895\pi$
$272$	13543.1	−	26652.1i	0.183054	−	0.360242i
$273$		0			0
$274$	106148.	+	59407.7i	1.41388	+	0.791301i
$275$		−	110693.i		−	1.46372i
$276$		0			0
$277$	53726.5			0.700212			0.350106	−	0.936710i	$-0.386146\pi$
$277$	53726.5			0.700212			0.350106	+	0.936710i	$0.386146\pi$
$278$	−63858.9	+	114102.i	−0.826289	+	1.47640i
$279$		0			0
$280$	203083.	−	8146.64i	2.59035	−	0.103911i
$281$	99500.6			1.26012			0.630061	−	0.776545i	$-0.283030\pi$
$281$	99500.6			1.26012			0.630061	+	0.776545i	$0.283030\pi$
$282$		0			0
$283$			69719.8i			0.870529i	0.900303	+	0.435265i	$0.143345\pi$
$283$			69719.8i			0.870529i	−0.900303	+	0.435265i	$0.856655\pi$
$284$	−2266.92	+	1390.88i	−0.0281060	+	0.0172446i
$285$		0			0
$286$	445.809	−	796.563i	0.00545026	−	0.00973840i
$287$			153175.i			1.85962i
$288$		0			0
$289$	−69883.5			−0.836717
$290$	−127065.	−	71114.2i	−1.51088	−	0.845592i
$291$		0			0
$292$	42552.8	+	69354.4i	0.499072	+	0.813408i
$293$	12440.0			0.144905			0.0724526	−	0.997372i	$-0.476917\pi$
$293$	12440.0			0.144905			0.0724526	+	0.997372i	$0.476917\pi$
$294$		0			0
$295$		−	261025.i		−	2.99943i
$296$	−2652.73	−	66128.6i	−0.0302768	−	0.754755i
$297$		0			0
$298$	−46733.9	−	26155.4i	−0.526259	−	0.294530i
$299$		−	441.480i		−	0.00493819i
$300$		0			0
$301$	−170068.			−1.87711
$302$	−17790.8	+	31788.1i	−0.195065	+	0.348539i
$303$		0			0
$304$	−117199.	−	59554.1i	−1.26817	−	0.644413i
$305$	−150969.			−1.62289
$306$		0			0
$307$			151986.i			1.61260i	0.591504	+	0.806302i	$0.298534\pi$
$307$			151986.i			1.61260i	−0.591504	+	0.806302i	$0.701466\pi$
$308$	−39138.4	−	63789.5i	−0.412574	−	0.672431i
$309$		0			0
$310$	−98485.4	+	175972.i	−1.02482	+	1.83113i
$311$		−	50086.5i		−	0.517845i	−0.965898	−	0.258923i	$-0.916633\pi$
$311$		−	50086.5i		−	0.517845i	0.965898	−	0.258923i	$-0.0833675\pi$
$312$		0			0
$313$	61969.9			0.632546			0.316273	−	0.948668i	$-0.397568\pi$
$313$	61969.9			0.632546			0.316273	+	0.948668i	$0.397568\pi$
$314$	15900.0	+	8898.67i	0.161264	+	0.0902539i
$315$		0			0
$316$	9880.12	−	6062.01i	0.0989437	−	0.0607075i
$317$	80.9272			0.000805334			0.000402667	−	1.00000i	$-0.499872\pi$
$317$	80.9272			0.000805334			0.000402667		1.00000i	$0.499872\pi$
$318$		0			0
$319$			53617.1i			0.526893i
$320$	−192373.	+	15458.9i	−1.87864	+	0.150965i
$321$		0			0
$322$	−31585.0	−	17677.1i	−0.304627	−	0.170490i
$323$		−	59969.4i		−	0.574811i
$324$		0			0
$325$	5245.13			0.0496580
$326$	−911.172	+	1628.06i	−0.00857364	+	0.0153192i
$327$		0			0
$328$	−5829.91	−	145331.i	−0.0541894	−	1.35086i
$329$	46834.5			0.432687
$330$		0			0
$331$			31123.5i			0.284074i	0.989861	+	0.142037i	$0.0453652\pi$
$331$			31123.5i			0.284074i	−0.989861	+	0.142037i	$0.954635\pi$
$332$	111190.	−	68221.0i	1.00876	−	0.618931i
$333$		0			0
$334$	58929.9	−	105295.i	0.528254	−	0.943873i
$335$			252839.i			2.25296i
$336$		0			0
$337$	5863.61			0.0516304			0.0258152	−	0.999667i	$-0.491782\pi$
$337$	5863.61			0.0516304			0.0258152	+	0.999667i	$0.491782\pi$
$338$	−99655.0	−	55773.6i	−0.872300	−	0.488197i
$339$		0			0
$340$	−46040.9	−	75039.4i	−0.398277	−	0.649129i
$341$	74253.8			0.638572
$342$		0			0
$343$		−	17470.1i		−	0.148493i
$344$	161358.	−	6472.85i	1.36356	−	0.0546989i
$345$		0			0
$346$	−178076.	−	99663.2i	−1.48749	−	0.832497i
$347$			138155.i			1.14738i	0.819073	+	0.573689i	$0.194488\pi$
$347$			138155.i			1.14738i	−0.819073	+	0.573689i	$0.805512\pi$
$348$		0			0
$349$	180947.			1.48559			0.742797	−	0.669517i	$-0.233499\pi$
$349$	180947.			1.48559			0.742797	+	0.669517i	$0.233499\pi$
$350$	210018.	−	375255.i	1.71443	−	3.06330i
$351$		0			0
$352$	39561.9	+	59033.1i	0.319295	+	0.476442i
$353$	98784.9			0.792759			0.396380	−	0.918087i	$-0.370266\pi$
$353$	98784.9			0.792759			0.396380	+	0.918087i	$0.370266\pi$
$354$		0			0
$355$			7832.10i			0.0621472i
$356$	58266.0	+	94964.5i	0.459743	+	0.749309i
$357$		0			0
$358$	51365.2	−	91778.2i	0.400777	−	0.716099i
$359$			151427.i			1.17493i	0.809248	+	0.587467i	$0.199875\pi$
$359$			151427.i			1.17493i	−0.809248	+	0.587467i	$0.800125\pi$
$360$		0			0
$361$	−133387.			−1.02353
$362$	10932.4	+	6118.52i	0.0834257	+	0.0466906i
$363$		0			0
$364$	3022.62	−	1854.55i	0.0228129	−	0.0139970i
$365$	239616.			1.79858
$366$		0			0
$367$		−	101304.i		−	0.752132i	−0.926593	−	0.376066i	$-0.877277\pi$
$367$		−	101304.i		−	0.752132i	0.926593	−	0.376066i	$-0.122723\pi$
$368$	30640.2	+	15569.6i	0.226254	+	0.114970i
$369$		0			0
$370$	−170071.	−	95182.9i	−1.24230	−	0.695273i
$371$		−	78590.9i		−	0.570985i
$372$		0			0
$373$	−181796.			−1.30667			−0.653336	−	0.757068i	$-0.726631\pi$
$373$	−181796.			−1.30667			−0.653336	+	0.757068i	$0.726631\pi$
$374$	−15832.0	+	28288.3i	−0.113186	+	0.202238i
$375$		0			0
$376$	−44436.0	+	1782.54i	−0.314311	+	0.0126085i
$377$	−2540.61			−0.0178754
$378$		0			0
$379$			2324.09i			0.0161798i	0.999967	+	0.00808991i	$0.00257513\pi$
$379$			2324.09i			0.0161798i	−0.999967	+	0.00808991i	$0.997425\pi$
$380$	−329976.	+	202459.i	−2.28515	+	1.40207i
$381$		0			0
$382$	33522.7	−	59897.6i	0.229727	−	0.410471i
$383$			129732.i			0.884402i	0.896916	+	0.442201i	$0.145802\pi$
$383$			129732.i			0.884402i	−0.896916	+	0.442201i	$0.854198\pi$
$384$		0			0
$385$	−220390.			−1.48686
$386$	−58786.4	−	32900.8i	−0.394550	−	0.220817i
$387$		0			0
$388$	−2333.65	−	3803.48i	−0.0155014	−	0.0252649i
$389$	53603.2			0.354235			0.177118	−	0.984190i	$-0.443323\pi$
$389$	53603.2			0.354235			0.177118	+	0.984190i	$0.443323\pi$
$390$		0			0
$391$			15678.2i			0.102552i
$392$	−5494.32	−	136965.i	−0.0357554	−	0.891329i
$393$		0			0
$394$	234889.	+	131459.i	1.51311	+	0.846835i
$395$		−	34135.3i		−	0.218781i
$396$		0			0
$397$	156598.			0.993584			0.496792	−	0.867870i	$-0.334511\pi$
$397$	156598.			0.993584			0.496792	+	0.867870i	$0.334511\pi$
$398$	25935.7	−	46341.3i	0.163731	−	0.292551i
$399$		0			0
$400$	−184980.	+	364030.i	−1.15612	+	2.27519i
$401$	207347.			1.28946			0.644731	−	0.764410i	$-0.276969\pi$
$401$	207347.			1.28946			0.644731	+	0.764410i	$0.276969\pi$
$402$		0			0
$403$			3518.46i			0.0216642i
$404$	−77885.2	−	126941.i	−0.477191	−	0.777746i
$405$		0			0
$406$	−101727.	+	181764.i	−0.617142	+	1.10270i
$407$			71763.9i			0.433229i
$408$		0			0
$409$	35322.3			0.211156			0.105578	−	0.994411i	$-0.466331\pi$
$409$	35322.3			0.211156			0.105578	+	0.994411i	$0.466331\pi$
$410$	−373765.	−	209184.i	−2.22347	−	1.24440i
$411$		0			0
$412$	44620.8	−	27377.3i	0.262871	−	0.161286i
$413$	−373391.			−2.18909
$414$		0			0
$415$		−	384154.i		−	2.23054i
$416$	−2797.24	+	1874.61i	−0.0161638	+	0.0108324i
$417$		0			0
$418$	124394.	+	69619.2i	0.711946	+	0.398452i
$419$		−	272874.i		−	1.55430i	−0.629318	−	0.777148i	$-0.716666\pi$
$419$		−	272874.i		−	1.55430i	0.629318	−	0.777148i	$-0.283334\pi$
$420$		0			0
$421$	−223490.			−1.26094			−0.630468	−	0.776215i	$-0.717137\pi$
$421$	−223490.			−1.26094			−0.630468	+	0.776215i	$0.717137\pi$
$422$	−34831.8	+	62236.6i	−0.195592	+	0.349479i
$423$		0			0
$424$	2991.20	+	74566.1i	0.0166385	+	0.414772i
$425$	−186270.			−1.03125
$426$		0			0
$427$			215958.i			1.18444i
$428$	25162.4	−	15438.6i	0.137361	−	0.0842790i
$429$		0			0
$430$	232253.	−	414984.i	1.25610	−	2.24437i
$431$			158123.i			0.851216i	0.904908	+	0.425608i	$0.139940\pi$
$431$			158123.i			0.851216i	−0.904908	+	0.425608i	$0.860060\pi$
$432$		0			0
$433$	135143.			0.720803			0.360401	−	0.932797i	$-0.382640\pi$
$433$	135143.			0.720803			0.360401	+	0.932797i	$0.382640\pi$
$434$	251723.	+	140881.i	1.33642	+	0.747950i
$435$		0			0
$436$	50382.2	+	82115.1i	0.265036	+	0.431966i
$437$	68943.0			0.361017
$438$		0			0
$439$		−	75873.6i		−	0.393697i	−0.980434	−	0.196848i	$-0.936929\pi$
$439$		−	75873.6i		−	0.393697i	0.980434	−	0.196848i	$-0.0630707\pi$
$440$	209103.	−	8388.11i	1.08008	−	0.0433270i
$441$		0			0
$442$	−1340.42	−	750.187i	−0.00686113	−	0.00383994i
$443$			99747.7i			0.508271i	0.967169	+	0.254135i	$0.0817909\pi$
$443$			99747.7i			0.508271i	−0.967169	+	0.254135i	$0.918209\pi$
$444$		0			0
$445$	328097.			1.65685
$446$	−133326.	+	238224.i	−0.670262	+	1.19761i
$447$		0			0
$448$	22113.5	+	275185.i	0.110180	+	1.37110i
$449$	−156931.			−0.778424			−0.389212	−	0.921148i	$-0.627253\pi$
$449$	−156931.			−0.778424			−0.389212	+	0.921148i	$0.627253\pi$
$450$		0			0
$451$			157716.i			0.775392i
$452$	−73978.6	−	120574.i	−0.362101	−	0.590167i
$453$		0			0
$454$	110712.	−	197817.i	0.537133	−	0.959737i
$455$		−	10443.0i		−	0.0504432i
$456$		0			0
$457$	109830.			0.525882			0.262941	−	0.964812i	$-0.415308\pi$
$457$	109830.			0.525882			0.262941	+	0.964812i	$0.415308\pi$
$458$	229165.	+	128256.i	1.09249	+	0.611430i
$459$		0			0
$460$	86268.0	−	52930.2i	0.407694	−	0.250143i
$461$	172802.			0.813104			0.406552	−	0.913628i	$-0.366731\pi$
$461$	172802.			0.813104			0.406552	+	0.913628i	$0.366731\pi$
$462$		0			0
$463$		−	159146.i		−	0.742395i	−0.928554	−	0.371198i	$-0.878947\pi$
$463$		−	159146.i		−	0.742395i	0.928554	−	0.371198i	$-0.121053\pi$
$464$	89599.5	−	176327.i	0.416169	−	0.818999i
$465$		0			0
$466$	−194257.	−	108719.i	−0.894549	−	0.500649i
$467$			153806.i			0.705246i	0.935765	+	0.352623i	$0.114710\pi$
$467$			153806.i			0.705246i	−0.935765	+	0.352623i	$0.885290\pi$
$468$		0			0
$469$	361679.			1.64429
$470$	−63959.5	+	114281.i	−0.289540	+	0.517345i
$471$		0			0
$472$	354268.	−	14211.4i	1.59019	−	0.0637900i
$473$	−175109.			−0.782683
$474$		0			0
$475$			819098.i			3.63035i
$476$	−107342.	+	65860.3i	−0.473757	+	0.290677i
$477$		0			0
$478$	25458.0	−	45487.8i	0.111421	−	0.199085i
$479$		−	246496.i		−	1.07433i	−0.843477	−	0.537166i	$-0.819495\pi$
$479$		−	246496.i		−	1.07433i	0.843477	−	0.537166i	$-0.180505\pi$
$480$		0			0
$481$	−3400.48			−0.0146977
$482$	202353.	+	113250.i	0.870993	+	0.487466i
$483$		0			0
$484$	82209.5	+	133989.i	0.350939	+	0.571975i
$485$	−13140.8			−0.0558649
$486$		0			0
$487$			341810.i			1.44121i	0.693347	+	0.720603i	$0.256135\pi$
$487$			341810.i			1.44121i	−0.693347	+	0.720603i	$0.743865\pi$
$488$	−8219.45	−	204898.i	−0.0345146	−	0.860397i
$489$		0			0
$490$	−352250.	−	197142.i	−1.46710	−	0.821084i
$491$		−	302270.i		−	1.25381i	−0.779095	−	0.626906i	$-0.784321\pi$
$491$		−	302270.i		−	1.25381i	0.779095	−	0.626906i	$-0.215679\pi$
$492$		0			0
$493$	90224.4			0.371219
$494$	−3298.85	+	5894.32i	−0.0135179	+	0.0241535i
$495$		0			0
$496$	−244194.	−	124085.i	−0.992592	−	0.504379i
$497$	11203.6			0.0453572
$498$		0			0
$499$		−	193507.i		−	0.777132i	−0.921421	−	0.388566i	$-0.872971\pi$
$499$		−	193507.i		−	0.777132i	0.921421	−	0.388566i	$-0.127029\pi$
$500$	382445.	+	623325.i	1.52978	+	2.49330i
$501$		0			0
$502$	−158808.	+	283754.i	−0.630179	+	1.12599i
$503$		−	452328.i		−	1.78779i	−0.448273	−	0.893897i	$-0.647961\pi$
$503$		−	452328.i		−	1.78779i	0.448273	−	0.893897i	$-0.352039\pi$
$504$		0			0
$505$	−438573.			−1.71973
$506$	−32521.2	−	18201.0i	−0.127018	−	0.0710878i
$507$		0			0
$508$	−124990.	+	76688.2i	−0.484336	+	0.297167i
$509$	−232364.			−0.896876			−0.448438	−	0.893814i	$-0.648020\pi$
$509$	−232364.			−0.896876			−0.448438	+	0.893814i	$0.648020\pi$
$510$		0			0
$511$		−	342765.i		−	1.31267i
$512$	−31454.7	−	260250.i	−0.119990	−	0.992775i
$513$		0			0
$514$	65687.0	+	36762.8i	0.248630	+	0.139150i
$515$		−	154163.i		−	0.581252i
$516$		0			0
$517$	48222.8			0.180414
$518$	−136157.	+	243282.i	−0.507435	+	0.906673i
$519$		0			0
$520$	397.465	+	9908.18i	0.00146991	+	0.0366427i
$521$	425181.			1.56638			0.783192	−	0.621780i	$-0.213590\pi$
$521$	425181.			1.56638			0.783192	+	0.621780i	$0.213590\pi$
$522$		0			0
$523$		−	91819.6i		−	0.335685i	−0.985814	−	0.167842i	$-0.946320\pi$
$523$		−	91819.6i		−	0.335685i	0.985814	−	0.167842i	$-0.0536800\pi$
$524$	−221474.	+	135887.i	−0.806603	+	0.494896i
$525$		0			0
$526$	187870.	−	335682.i	0.679025	−	1.21327i
$527$		−	124951.i		−	0.449902i
$528$		0			0
$529$	261817.			0.935591
$530$	191771.	+	107328.i	0.682701	+	0.382085i
$531$		0			0
$532$	289613.	+	472023.i	1.02328	+	1.66778i
$533$	−7473.24			−0.0263060
$534$		0			0
$535$		−	86934.9i		−	0.303729i
$536$	−343157.	+	13765.7i	−1.19444	+	0.0479146i
$537$		0			0
$538$	−72789.3	−	40737.7i	−0.251480	−	0.140745i
$539$			148637.i			0.511622i
$540$		0			0
$541$	−419310.			−1.43265			−0.716326	−	0.697766i	$-0.754178\pi$
$541$	−419310.			−1.43265			−0.716326	+	0.697766i	$0.754178\pi$
$542$	198317.	−	354349.i	0.675090	−	1.20624i
$543$		0			0
$544$	99338.1	−	66572.9i	0.335674	−	0.224957i
$545$	283703.			0.955150
$546$		0			0
$547$		−	372884.i		−	1.24623i	−0.782129	−	0.623116i	$-0.785866\pi$
$547$		−	372884.i		−	1.24623i	0.782129	−	0.623116i	$-0.214134\pi$
$548$	254459.	+	414727.i	0.847337	+	1.38103i
$549$		0			0
$550$	216243.	−	386378.i	0.714852	−	1.27728i
$551$		−	396751.i		−	1.30682i
$552$		0			0
$553$	−48829.8			−0.159674
$554$	187534.	+	104956.i	0.611026	+	0.341971i
$555$		0			0
$556$	−445802.	+	273524.i	−1.44209	+	0.884803i
$557$	−249244.			−0.803369			−0.401685	−	0.915778i	$-0.631575\pi$
$557$	−249244.			−0.803369			−0.401685	+	0.915778i	$0.631575\pi$
$558$		0			0
$559$		−	8297.40i		−	0.0265533i
$560$	724781.	+	368293.i	2.31116	+	1.17440i
$561$		0			0
$562$	347309.	+	194377.i	1.09962	+	0.615421i
$563$			469821.i			1.48223i	0.671377	+	0.741116i	$0.265703\pi$
$563$			469821.i			1.48223i	−0.671377	+	0.741116i	$0.734297\pi$
$564$		0			0
$565$	−416575.			−1.30496
$566$	−136200.	+	243358.i	−0.425151	+	0.759650i
$567$		0			0
$568$	−10629.9	+	426.415i	−0.0329482	+	0.00132171i
$569$	−501734.			−1.54971			−0.774853	−	0.632142i	$-0.782176\pi$
$569$	−501734.			−1.54971			−0.774853	+	0.632142i	$0.782176\pi$
$570$		0			0
$571$		−	514877.i		−	1.57918i	−0.613636	−	0.789589i	$-0.710294\pi$
$571$		−	514877.i		−	1.57918i	0.613636	−	0.789589i	$-0.289706\pi$
$572$	3112.21	−	1909.52i	0.00951212	−	0.00583622i
$573$		0			0
$574$	−299232.	+	534661.i	−0.908206	+	1.62276i
$575$		−	214143.i		−	0.647690i
$576$		0			0
$577$	55168.5			0.165707			0.0828533	−	0.996562i	$-0.473597\pi$
$577$	55168.5			0.165707			0.0828533	+	0.996562i	$0.473597\pi$
$578$	−243930.	−	136519.i	−0.730145	−	0.408637i
$579$		0			0
$580$	−304601.	−	496451.i	−0.905472	−	1.47578i
$581$	−549524.			−1.62792
$582$		0			0
$583$		−	80920.5i		−	0.238079i
$584$	13045.8	+	325211.i	0.0382511	+	0.953542i
$585$		0			0
$586$	43421.9	+	24301.8i	0.126449	+	0.0707690i
$587$			221207.i			0.641980i	0.947083	+	0.320990i	$0.104016\pi$
$587$			221207.i			0.641980i	−0.947083	+	0.320990i	$0.895984\pi$
$588$		0			0
$589$	−549456.			−1.58381
$590$	509920.	−	911115.i	1.46487	−	2.61739i
$591$		0			0
$592$	119925.	−	236005.i	0.342188	−	0.673408i
$593$	−167379.			−0.475982			−0.237991	−	0.971267i	$-0.576489\pi$
$593$	−167379.			−0.475982			−0.237991	+	0.971267i	$0.576489\pi$
$594$		0			0
$595$			370861.i			1.04756i
$596$	−112030.	−	182592.i	−0.315387	−	0.514031i
$597$		0			0
$598$	862.442	−	1540.99i	0.00241172	−	0.00430922i
$599$		−	285002.i		−	0.794319i	−0.917750	−	0.397160i	$-0.869996\pi$
$599$		−	285002.i		−	0.794319i	0.917750	−	0.397160i	$-0.130004\pi$
$600$		0			0
$601$	30586.9			0.0846810			0.0423405	−	0.999103i	$-0.486519\pi$
$601$	30586.9			0.0846810			0.0423405	+	0.999103i	$0.486519\pi$
$602$	−593625.	−	332232.i	−1.63802	−	0.916745i
$603$		0			0
$604$	−124198.	+	76202.4i	−0.340440	+	0.208879i
$605$	462924.			1.26473
$606$		0			0
$607$		−	165312.i		−	0.448670i	−0.974512	−	0.224335i	$-0.927979\pi$
$607$		−	165312.i		−	0.448670i	0.974512	−	0.224335i	$-0.0720209\pi$
$608$	−292746.	−	436827.i	−0.791925	−	1.18169i
$609$		0			0
$610$	−526962.	−	294923.i	−1.41618	−	0.792590i
$611$			2285.00i			0.00612074i
$612$		0			0
$613$	−64752.4			−0.172320			−0.0861599	−	0.996281i	$-0.527460\pi$
$613$	−64752.4			−0.172320			−0.0861599	+	0.996281i	$0.527460\pi$
$614$	−296909.	+	530511.i	−0.787566	+	1.40721i
$615$		0			0
$616$	−11999.0	−	299117.i	−0.0316216	−	0.788277i
$617$	−321272.			−0.843923			−0.421961	−	0.906614i	$-0.638658\pi$
$617$	−321272.			−0.843923			−0.421961	+	0.906614i	$0.638658\pi$
$618$		0			0
$619$			409948.i			1.06991i	0.844880	+	0.534956i	$0.179672\pi$
$619$			409948.i			1.06991i	−0.844880	+	0.534956i	$0.820328\pi$
$620$	−687530.	+	421838.i	−1.78858	+	1.09739i
$621$		0			0
$622$	97845.4	−	174828.i	0.252906	−	0.451887i
$623$		−	469335.i		−	1.20923i
$624$		0			0
$625$	1.15665e6			2.96103
$626$	216307.	+	121060.i	0.551979	+	0.308924i
$627$		0			0
$628$	38115.3	+	62122.0i	0.0966452	+	0.157516i
$629$	120761.			0.305229
$630$		0			0
$631$		−	453263.i		−	1.13839i	−0.822202	−	0.569195i	$-0.807255\pi$
$631$		−	453263.i		−	1.13839i	0.822202	−	0.569195i	$-0.192745\pi$
$632$	46329.1	−	1858.48i	0.115990	−	0.00465290i
$633$		0			0
$634$	282.478	+	158.093i	0.000702758	+	0.000393310i
$635$			431833.i			1.07095i
$636$		0			0
$637$	−7043.05			−0.0173573
$638$	−104743.	+	187152.i	−0.257325	+	0.459783i
$639$		0			0
$640$	−701680.	−	321846.i	−1.71309	−	0.785757i
$641$	−764363.			−1.86030			−0.930152	−	0.367175i	$-0.880325\pi$
$641$	−764363.			−1.86030			−0.930152	+	0.367175i	$0.880325\pi$
$642$		0			0
$643$			674655.i			1.63177i	0.578211	+	0.815887i	$0.303751\pi$
$643$			674655.i			1.63177i	−0.578211	+	0.815887i	$0.696249\pi$
$644$	−75715.4	−	123404.i	−0.182563	−	0.297549i
$645$		0			0
$646$	117152.	−	209324.i	0.280727	−	0.501597i
$647$			372446.i			0.889723i	0.895599	+	0.444861i	$0.146747\pi$
$647$			372446.i			0.889723i	−0.895599	+	0.444861i	$0.853253\pi$
$648$		0			0
$649$	−384458.			−0.912767
$650$	18308.2	+	10246.5i	0.0433331	+	0.0242521i
$651$		0			0
$652$	−6360.93	+	3902.79i	−0.0149632	+	0.00918078i
$653$	620680.			1.45560			0.727799	−	0.685791i	$-0.240543\pi$
$653$	620680.			1.45560			0.727799	+	0.685791i	$0.240543\pi$
$654$		0			0
$655$			765181.i			1.78353i
$656$	263558.	−	518669.i	0.612447	−	1.20527i
$657$		0			0
$658$	163477.	+	91492.5i	0.377576	+	0.211317i
$659$			416907.i			0.959993i	0.877271	+	0.479996i	$0.159362\pi$
$659$			416907.i			0.959993i	−0.877271	+	0.479996i	$0.840638\pi$
$660$		0			0
$661$	−724690.			−1.65863			−0.829315	−	0.558782i	$-0.811269\pi$
$661$	−724690.			−1.65863			−0.829315	+	0.558782i	$0.811269\pi$
$662$	−60800.5	+	108637.i	−0.138737	+	0.247892i
$663$		0			0
$664$	521381.	−	20915.1i	1.18255	−	0.0474377i
$665$	1.63082e6			3.68775
$666$		0			0
$667$			103725.i			0.233149i
$668$	411392.	−	252412.i	0.921941	−	0.565662i
$669$		0			0
$670$	−493927.	+	882538.i	−1.10030	+	1.96600i
$671$			222359.i			0.493867i
$672$		0			0
$673$	−239394.			−0.528546			−0.264273	−	0.964448i	$-0.585132\pi$
$673$	−239394.			−0.528546			−0.264273	+	0.964448i	$0.585132\pi$
$674$	20467.0	+	11454.7i	0.0450542	+	0.0252153i
$675$		0			0
$676$	−238893.	−	389357.i	−0.522769	−	0.852031i
$677$	731049.			1.59503			0.797515	−	0.603299i	$-0.206147\pi$
$677$	731049.			1.59503			0.797515	+	0.603299i	$0.206147\pi$
$678$		0			0
$679$			18797.6i			0.0407721i
$680$	−14115.1	−	351868.i	−0.0305258	−	0.760961i
$681$		0			0
$682$	259184.	+	145057.i	0.557237	+	0.311867i
$683$		−	804575.i		−	1.72475i	−0.506273	−	0.862373i	$-0.668977\pi$
$683$		−	804575.i		−	1.72475i	0.506273	−	0.862373i	$-0.331023\pi$
$684$		0			0
$685$	1.43286e6			3.05368
$686$	34128.3	−	60979.7i	0.0725214	−	0.129580i
$687$		0			0
$688$	575869.	+	292624.i	1.21660	+	0.618206i
$689$	3834.36			0.00807707
$690$		0			0
$691$		−	734097.i		−	1.53744i	−0.639587	−	0.768719i	$-0.720895\pi$
$691$		−	734097.i		−	1.53744i	0.639587	−	0.768719i	$-0.279105\pi$
$692$	−426883.	−	695753.i	−0.891450	−	1.45292i
$693$		0			0
$694$	−269889.	+	482232.i	−0.560359	+	1.00124i
$695$			1.54022e6i			3.18870i
$696$		0			0
$697$	265396.			0.546298
$698$	631598.	+	353484.i	1.29637	+	0.725537i
$699$		0			0
$700$	1.46614e6	−	899560.i	2.99212	−	1.83584i
$701$	−158522.			−0.322592			−0.161296	−	0.986906i	$-0.551567\pi$
$701$	−158522.			−0.322592			−0.161296	+	0.986906i	$0.551567\pi$
$702$		0			0
$703$		−	531031.i		−	1.07451i
$704$	22769.0	+	283341.i	0.0459408	+	0.571695i
$705$		0			0
$706$	344811.	+	192979.i	0.691785	+	0.387169i
$707$			627369.i			1.25512i
$708$		0			0
$709$	365334.			0.726770			0.363385	−	0.931639i	$-0.381621\pi$
$709$	365334.			0.726770			0.363385	+	0.931639i	$0.381621\pi$
$710$	−15300.2	+	27338.1i	−0.0303516	+	0.0542315i
$711$		0			0
$712$	17863.1	+	445300.i	0.0352368	+	0.878400i
$713$	143648.			0.282566
$714$		0			0
$715$		−	10752.5i		−	0.0210329i
$716$	358582.	−	220010.i	0.699460	−	0.429158i
$717$		0			0
$718$	−295816.	+	528558.i	−0.573817	+	1.02528i
$719$		−	801198.i		−	1.54982i	−0.632070	−	0.774911i	$-0.717795\pi$
$719$		−	801198.i		−	1.54982i	0.632070	−	0.774911i	$-0.282205\pi$
$720$		0			0
$721$	−220526.			−0.424218
$722$	−465590.	−	260575.i	−0.893160	−	0.499872i
$723$		0			0
$724$	26207.2	+	42713.6i	0.0499970	+	0.0814872i
$725$	−1.23234e6			−2.34452
$726$		0			0
$727$			207794.i			0.393155i	0.980488	+	0.196577i	$0.0629827\pi$
$727$			207794.i			0.393155i	−0.980488	+	0.196577i	$0.937017\pi$
$728$	14173.4	−	568.563i	0.0267431	−	0.00107279i
$729$		0			0
$730$	836385.	+	468097.i	1.56950	+	0.878395i
$731$			294665.i			0.551434i
$732$		0			0
$733$	177976.			0.331248			0.165624	−	0.986189i	$-0.447036\pi$
$733$	177976.			0.331248			0.165624	+	0.986189i	$0.447036\pi$
$734$	197900.	−	353603.i	0.367328	−	0.656333i
$735$		0			0
$736$	76534.7	+	114203.i	0.141287	+	0.210824i
$737$	372400.			0.685606
$738$		0			0
$739$		−	587913.i		−	1.07653i	−0.842777	−	0.538263i	$-0.819081\pi$
$739$		−	587913.i		−	1.07653i	0.842777	−	0.538263i	$-0.180919\pi$
$740$	−407693.	−	664476.i	−0.744509	−	1.21343i
$741$		0			0
$742$	153530.	−	274323.i	0.278859	−	0.498259i
$743$			675642.i			1.22388i	0.790904	+	0.611940i	$0.209611\pi$
$743$			675642.i			1.22388i	−0.790904	+	0.611940i	$0.790389\pi$
$744$		0			0
$745$	−630846.			−1.13661
$746$	−634562.	−	355143.i	−1.14024	−	0.638155i
$747$		0			0
$748$	−110524.	+	67812.5i	−0.197539	+	0.121201i
$749$	−124358.			−0.221672
$750$		0			0
$751$			267589.i			0.474447i	0.971455	+	0.237224i	$0.0762374\pi$
$751$			267589.i			0.474447i	−0.971455	+	0.237224i	$0.923763\pi$
$752$	−158587.	−	80584.9i	−0.280435	−	0.142501i
$753$		0			0
$754$	−8868.04	−	4963.15i	−0.0155986	−	0.00873000i
$755$			429098.i			0.752770i
$756$		0			0
$757$	400491.			0.698878			0.349439	−	0.936959i	$-0.386372\pi$
$757$	400491.			0.698878			0.349439	+	0.936959i	$0.386372\pi$
$758$	−4540.17	+	8112.27i	−0.00790193	+	0.0141190i
$759$		0			0
$760$	−1.54730e6	+	62069.5i	−2.67884	+	0.107461i
$761$	−755370.			−1.30434			−0.652170	−	0.758073i	$-0.726141\pi$
$761$	−755370.			−1.30434			−0.652170	+	0.758073i	$0.726141\pi$
$762$		0			0
$763$		−	405831.i		−	0.697101i
$764$	234023.	−	143586.i	0.400933	−	0.245995i
$765$		0			0
$766$	−253435.	+	452832.i	−0.431926	+	0.771756i
$767$		−	18217.3i		−	0.0309665i
$768$		0			0
$769$	−482211.			−0.815427			−0.407713	−	0.913110i	$-0.633674\pi$
$769$	−482211.			−0.815427			−0.407713	+	0.913110i	$0.633674\pi$
$770$	−769274.	−	430537.i	−1.29748	−	0.726155i
$771$		0			0
$772$	−140923.	−	229682.i	−0.236454	−	0.385382i
$773$	1.06175e6			1.77691			0.888454	−	0.458965i	$-0.151780\pi$
$773$	1.06175e6			1.77691			0.888454	+	0.458965i	$0.151780\pi$
$774$		0			0
$775$			1.70665e6i			2.84146i
$776$	−715.445	−	17835.0i	−0.00118810	−	0.0296175i
$777$		0			0
$778$	187103.	+	104715.i	0.309116	+	0.173002i
$779$		−	1.16705e6i		−	1.92315i
$780$		0			0
$781$	11535.7			0.0189122
$782$	−30627.8	+	54725.2i	−0.0500844	+	0.0894898i
$783$		0			0
$784$	248387.	−	488813.i	0.404107	−	0.795262i
$785$	214628.			0.348295
$786$		0			0
$787$			313747.i			0.506559i	0.967393	+	0.253280i	$0.0815093\pi$
$787$			313747.i			0.506559i	−0.967393	+	0.253280i	$0.918491\pi$
$788$	563074.	+	917723.i	0.906804	+	1.47795i
$789$		0			0
$790$	66684.3	−	119150.i	0.106849	−	0.190915i
$791$			595901.i			0.952404i
$792$		0			0
$793$	−10536.3			−0.0167549
$794$	546608.	+	305918.i	0.867031	+	0.485248i
$795$		0			0
$796$	181058.	−	111089.i	0.285754	−	0.175326i
$797$	756561.			1.19104			0.595522	−	0.803339i	$-0.296945\pi$
$797$	756561.			1.19104			0.595522	+	0.803339i	$0.296945\pi$
$798$		0			0
$799$		−	81147.0i		−	0.127110i
$800$	−1.35682e6	+	909293.i	−2.12003	+	1.42077i
$801$		0			0
$802$	723748.	+	405058.i	1.12522	+	0.629750i
$803$		−	352925.i		−	0.547333i
$804$		0			0
$805$	−426356.			−0.657931
$806$	−6873.41	+	12281.3i	−0.0105804	+	0.0189048i
$807$		0			0
$808$	−23877.9	−	595240.i	−0.0365741	−	0.911736i
$809$	−259808.			−0.396968			−0.198484	−	0.980104i	$-0.563602\pi$
$809$	−259808.			−0.396968			−0.198484	+	0.980104i	$0.563602\pi$
$810$		0			0
$811$			108740.i			0.165329i	0.996577	+	0.0826644i	$0.0263430\pi$
$811$			108740.i			0.165329i	−0.996577	+	0.0826644i	$0.973657\pi$
$812$	−710162.	+	435724.i	−1.07707	+	0.660845i
$813$		0			0
$814$	−140193.	+	250493.i	−0.211581	+	0.378049i
$815$			21976.7i			0.0330862i
$816$		0			0
$817$	1.29575e6			1.94123
$818$	123293.	+	69003.1i	0.184261	+	0.103125i
$819$		0			0
$820$	−895988.	−	1.46032e6i	−1.33252	−	2.17180i
$821$	−468489.			−0.695045			−0.347523	−	0.937672i	$-0.612977\pi$
$821$	−468489.			−0.695045			−0.347523	+	0.937672i	$0.612977\pi$
$822$		0			0
$823$			487197.i			0.719291i	0.933089	+	0.359646i	$0.117102\pi$
$823$			487197.i			0.719291i	−0.933089	+	0.359646i	$0.882898\pi$
$824$	209232.	−	8393.30i	0.308158	−	0.0123617i
$825$		0			0
$826$	−1.30333e6	−	729429.i	−1.91026	−	1.06911i
$827$			161820.i			0.236603i	0.992978	+	0.118301i	$0.0377449\pi$
$827$			161820.i			0.236603i	−0.992978	+	0.118301i	$0.962255\pi$
$828$		0			0
$829$	−77603.0			−0.112920			−0.0564598	−	0.998405i	$-0.517981\pi$
$829$	−77603.0			−0.112920			−0.0564598	+	0.998405i	$0.517981\pi$
$830$	750456.	−	1.34090e6i	1.08935	−	1.94643i
$831$		0			0
$832$	−13425.9	+	1078.89i	−0.0193953	+	0.00155859i
$833$	250119.			0.360460
$834$		0			0
$835$		−	1.42134e6i		−	2.03857i
$836$	298197.	+	486014.i	0.426669	+	0.695403i
$837$		0			0
$838$	533066.	−	952471.i	0.759089	−	1.35632i
$839$		−	33592.5i		−	0.0477220i	−0.999715	−	0.0238610i	$-0.992404\pi$
$839$		−	33592.5i		−	0.0477220i	0.999715	−	0.0238610i	$-0.00759591\pi$
$840$		0			0
$841$	−110367.			−0.156044
$842$	−780095.	−	436593.i	−1.10033	−	0.615818i
$843$		0			0
$844$	−243162.	+	149193.i	−0.341358	+	0.209443i
$845$	−1.34521e6			−1.88398
$846$		0			0
$847$		−	662201.i		−	0.923045i
$848$	−135226.	+	266118.i	−0.188048	+	0.370069i
$849$		0			0
$850$	−650178.	−	363883.i	−0.899901	−	0.503644i
$851$			138831.i			0.191703i
$852$		0			0
$853$	713456.			0.980549			0.490274	−	0.871568i	$-0.336897\pi$
$853$	713456.			0.980549			0.490274	+	0.871568i	$0.336897\pi$
$854$	−421880.	+	753806.i	−0.578460	+	1.03358i
$855$		0			0
$856$	117990.	−	4733.13i	0.161026	−	0.00645952i
$857$	−345801.			−0.470830			−0.235415	−	0.971895i	$-0.575645\pi$
$857$	−345801.			−0.470830			−0.235415	+	0.971895i	$0.575645\pi$
$858$		0			0
$859$			942252.i			1.27697i	0.769634	+	0.638485i	$0.220439\pi$
$859$			942252.i			1.27697i	−0.769634	+	0.638485i	$0.779561\pi$
$860$	1.62137e6	−	994799.i	2.19222	−	1.34505i
$861$		0			0
$862$	−308897.	+	551931.i	−0.415719	+	0.742797i
$863$			110905.i			0.148912i	0.997224	+	0.0744561i	$0.0237221\pi$
$863$			110905.i			0.148912i	−0.997224	+	0.0744561i	$0.976278\pi$
$864$		0			0
$865$	−2.40379e6			−3.21266
$866$	471718.	+	264005.i	0.628994	+	0.352027i
$867$		0			0
$868$	603430.	+	983495.i	0.800916	+	1.30537i
$869$	−50277.1			−0.0665780
$870$		0			0
$871$			17645.9i			0.0232599i
$872$	15446.1	+	385047.i	0.0203135	+	0.506385i
$873$		0			0
$874$	240647.	+	134682.i	0.315034	+	0.176314i
$875$		−	3.08061e6i		−	4.02366i
$876$		0			0
$877$	−573424.			−0.745550			−0.372775	−	0.927922i	$-0.621594\pi$
$877$	−573424.			−0.745550			−0.372775	+	0.927922i	$0.621594\pi$
$878$	148221.	−	264838.i	0.192274	−	0.343552i
$879$		0			0
$880$	746264.	+	379209.i	0.963668	+	0.489681i
$881$	−446113.			−0.574768			−0.287384	−	0.957815i	$-0.592786\pi$
$881$	−446113.			−0.574768			−0.287384	+	0.957815i	$0.592786\pi$
$882$		0			0
$883$		−	1.28083e6i		−	1.64274i	−0.570393	−	0.821372i	$-0.693209\pi$
$883$		−	1.28083e6i		−	1.64274i	0.570393	−	0.821372i	$-0.306791\pi$
$884$	−3213.25	−	5237.09i	−0.00411187	−	0.00670170i
$885$		0			0
$886$	−194860.	+	348171.i	−0.248230	+	0.443532i
$887$			901749.i			1.14614i	0.819506	+	0.573071i	$0.194248\pi$
$887$			901749.i			1.14614i	−0.819506	+	0.573071i	$0.805752\pi$
$888$		0			0
$889$	617727.			0.781615
$890$	1.14523e6	+	640947.i	1.44582	+	0.809175i
$891$		0			0
$892$	−930753.	+	571069.i	−1.16978	+	0.717727i
$893$	−356834.			−0.447469
$894$		0			0
$895$		−	1.23888e6i		−	1.54662i
$896$	−460393.	+	1.00374e6i	−0.573472	+	1.25027i
$897$		0			0
$898$	−547771.	−	306569.i	−0.679276	−	0.380168i
$899$		−	826660.i		−	1.02284i
$900$		0			0
$901$	−136169.			−0.167737
$902$	−308102.	+	550510.i	−0.378688	+	0.676631i
$903$		0			0
$904$	−22680.2	−	565384.i	−0.0277530	−	0.691841i
$905$	147573.			0.180182
$906$		0			0
$907$			181413.i			0.220523i	0.993903	+	0.110261i	$0.0351688\pi$
$907$			181413.i			0.220523i	−0.993903	+	0.110261i	$0.964831\pi$
$908$	772883.	−	474207.i	0.937436	−	0.575170i
$909$		0			0
$910$	20400.7	−	36451.5i	0.0246355	−	0.0440182i
$911$			47687.4i			0.0574602i	0.999587	+	0.0287301i	$0.00914634\pi$
$911$			47687.4i			0.0574602i	−0.999587	+	0.0287301i	$0.990854\pi$
$912$		0			0
$913$	−565812.			−0.678783
$914$	383364.	+	214556.i	0.458900	+	0.256831i
$915$		0			0
$916$	549354.	+	895360.i	0.654728	+	1.06710i
$917$	1.09457e6			1.30169
$918$		0			0
$919$			1.10248e6i			1.30539i	0.757622	+	0.652694i	$0.226361\pi$
$919$			1.10248e6i			1.30539i	−0.757622	+	0.652694i	$0.773639\pi$
$920$	404521.	−	16227.3i	0.477931	−	0.0191721i
$921$		0			0
$922$	603168.	+	337573.i	0.709539	+	0.397105i
$923$			546.611i			0.000641616i
$924$		0			0
$925$	−1.64942e6			−1.92774
$926$	310897.	−	555504.i	0.362572	−	0.647836i
$927$		0			0
$928$	657209.	−	440439.i	0.763146	−	0.511434i
$929$	924812.			1.07157			0.535787	−	0.844353i	$-0.320015\pi$
$929$	924812.			1.07157			0.535787	+	0.844353i	$0.320015\pi$
$930$		0			0
$931$		−	1.09987e6i		−	1.26894i
$932$	−465672.	−	758972.i	−0.536103	−	0.873763i
$933$		0			0
$934$	−300465.	+	536865.i	−0.344429	+	0.615419i
$935$			381854.i			0.436791i
$936$		0			0
$937$	−35819.1			−0.0407977			−0.0203988	−	0.999792i	$-0.506494\pi$
$937$	−35819.1			−0.0407977			−0.0203988	+	0.999792i	$0.506494\pi$
$938$	1.26245e6	+	706550.i	1.43486	+	0.803041i
$939$		0			0
$940$	−446504.	+	273955.i	−0.505323	+	0.310044i
$941$	−434924.			−0.491172			−0.245586	−	0.969375i	$-0.578980\pi$
$941$	−434924.			−0.491172			−0.245586	+	0.969375i	$0.578980\pi$
$942$		0			0
$943$			305109.i			0.343109i
$944$	1.26434e6	+	642468.i	1.41880	+	0.720954i
$945$		0			0
$946$	−611221.	−	342080.i	−0.682993	−	0.382248i
$947$			1.40078e6i			1.56196i	0.624553	+	0.780982i	$0.285281\pi$
$947$			1.40078e6i			1.56196i	−0.624553	+	0.780982i	$0.714719\pi$
$948$		0			0
$949$	16723.1			0.0185688
$950$	−1.60013e6	+	2.85908e6i	−1.77300	+	3.16795i
$951$		0			0
$952$	−503339.	+	20191.3i	−0.555376	+	0.0222788i
$953$	−320073.			−0.352422			−0.176211	−	0.984352i	$-0.556384\pi$
$953$	−320073.			−0.352422			−0.176211	+	0.984352i	$0.556384\pi$
$954$		0			0
$955$		−	808538.i		−	0.886530i
$956$	177723.	−	109043.i	0.194459	−	0.119312i
$957$		0			0
$958$	481536.	−	860398.i	0.524684	−	0.937494i
$959$		−	2.04968e6i		−	2.22868i
$960$		0			0
$961$	−221311.			−0.239638
$962$	−11869.4	−	6642.93i	−0.0128257	−	0.00717810i
$963$		0			0
$964$	485079.	+	790603.i	0.521986	+	0.850754i
$965$	−793539.			−0.852145
$966$		0			0
$967$			1.06890e6i			1.14310i	0.820568	+	0.571549i	$0.193657\pi$
$967$			1.06890e6i			1.14310i	−0.820568	+	0.571549i	$0.806343\pi$
$968$	25203.6	+	628288.i	0.0268975	+	0.670514i
$969$		0			0
$970$	−45868.3	−	25670.9i	−0.0487494	−	0.0272834i
$971$		−	1.25688e6i		−	1.33308i	−0.745471	−	0.666538i	$-0.767775\pi$
$971$		−	1.25688e6i		−	1.33308i	0.745471	−	0.666538i	$-0.232225\pi$
$972$		0			0
$973$	2.20325e6			2.32722
$974$	−667734.	+	1.19309e6i	−0.703859	+	1.25764i
$975$		0			0
$976$	371584.	−	731259.i	0.390084	−	0.767664i
$977$	748167.			0.783807			0.391903	−	0.920006i	$-0.371817\pi$
$977$	748167.			0.783807			0.391903	+	0.920006i	$0.371817\pi$
$978$		0			0
$979$		−	483247.i		−	0.504201i
$980$	−844412.	−	1.37626e6i	−0.879229	−	1.43301i
$981$		0			0
$982$	590493.	−	1.05508e6i	0.612339	−	1.09411i
$983$		−	1.53724e6i		−	1.59087i	−0.606038	−	0.795436i	$-0.707242\pi$
$983$		−	1.53724e6i		−	1.59087i	0.606038	−	0.795436i	$-0.292758\pi$
$984$		0			0
$985$	3.17069e6			3.26799
$986$	314930.	+	176256.i	0.323937	+	0.181297i
$987$		0			0
$988$	−23029.4	+	14129.8i	−0.0235922	+	0.0144752i
$989$	−338758.			−0.346335
$990$		0			0
$991$			1.36196e6i			1.38682i	0.720546	+	0.693408i	$0.243891\pi$
$991$			1.36196e6i			1.38682i	−0.720546	+	0.693408i	$0.756109\pi$
$992$	−609959.	−	910161.i	−0.619836	−	0.924901i
$993$		0			0
$994$	39106.5	+	21886.6i	0.0395800	+	0.0221516i
$995$		−	625547.i		−	0.631849i
$996$		0			0
$997$	2016.78			0.00202893			0.00101447	−	0.999999i	$-0.499677\pi$
$997$	2016.78			0.00202893			0.00101447	+	0.999999i	$0.499677\pi$
$998$	378020.	−	675439.i	0.379537	−	0.678148i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.d.b.55.14	yes	16
3.2	odd	2	inner	108.5.d.b.55.3	✓	16
4.3	odd	2	inner	108.5.d.b.55.13	yes	16
12.11	even	2	inner	108.5.d.b.55.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.d.b.55.3	✓	16	3.2	odd	2	inner
108.5.d.b.55.4	yes	16	12.11	even	2	inner
108.5.d.b.55.13	yes	16	4.3	odd	2	inner
108.5.d.b.55.14	yes	16	1.1	even	1	trivial

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(3.49052 + 1.95353i) q^{2} +(8.36746 + 13.6377i) q^{4} +47.1174 q^{5} -67.4003i q^{7} +(2.56528 + 63.9486i) q^{8} +O(q^{10})\)	\(q+(3.49052 + 1.95353i) q^{2} +(8.36746 + 13.6377i) q^{4} +47.1174 q^{5} -67.4003i q^{7} +(2.56528 + 63.9486i) q^{8} +(164.464 + 92.0451i) q^{10} -69.3981i q^{11} +3.28838 q^{13} +(131.668 - 235.262i) q^{14} +(-115.971 + 228.225i) q^{16} -116.780 q^{17} +513.525i q^{19} +(394.253 + 642.571i) q^{20} +(135.571 - 242.236i) q^{22} -134.254i q^{23} +1595.05 q^{25} +(11.4782 + 6.42394i) q^{26} +(919.182 - 563.970i) q^{28} -772.602 q^{29} +1069.97i q^{31} +(-850.643 + 570.072i) q^{32} +(-407.623 - 228.133i) q^{34} -3175.73i q^{35} -1034.09 q^{37} +(-1003.19 + 1792.47i) q^{38} +(120.869 + 3013.09i) q^{40} -2272.62 q^{41} -2523.25i q^{43} +(946.428 - 580.686i) q^{44} +(262.270 - 468.618i) q^{46} +694.871i q^{47} -2141.80 q^{49} +(5567.55 + 3115.97i) q^{50} +(27.5154 + 44.8458i) q^{52} +1166.03 q^{53} -3269.86i q^{55} +(4310.15 - 172.901i) q^{56} +(-2696.78 - 1509.30i) q^{58} -5539.90i q^{59} -3204.11 q^{61} +(-2090.21 + 3734.75i) q^{62} +(-4082.84 + 328.092i) q^{64} +154.940 q^{65} +5366.14i q^{67} +(-977.152 - 1592.60i) q^{68} +(6203.87 - 11084.9i) q^{70} +166.225i q^{71} +5085.51 q^{73} +(-3609.51 - 2020.12i) q^{74} +(-7003.28 + 4296.90i) q^{76} -4677.46 q^{77} -724.474i q^{79} +(-5464.26 + 10753.4i) q^{80} +(-7932.63 - 4439.62i) q^{82} -8153.13i q^{83} -5502.37 q^{85} +(4929.24 - 8807.46i) q^{86} +(4437.91 - 178.026i) q^{88} +6963.40 q^{89} -221.638i q^{91} +(1830.92 - 1123.37i) q^{92} +(-1357.45 + 2425.46i) q^{94} +24196.0i q^{95} -278.895 q^{97} +(-7476.00 - 4184.07i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 28 q^{4}+O(q^{10}) \) 16 * q + 28 * q^4	\( 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) \) 16 * q + 28 * q^4 + 176 * q^10 + 176 * q^13 + 88 * q^16 + 384 * q^22 + 2736 * q^25 + 1812 * q^28 + 1520 * q^34 + 80 * q^37 - 688 * q^40 - 1824 * q^46 - 7904 * q^49 - 5236 * q^52 - 11584 * q^58 - 1648 * q^61 + 5056 * q^64 + 26688 * q^70 + 80 * q^73 - 8388 * q^76 - 38464 * q^82 - 16832 * q^85 - 29520 * q^88 - 4512 * q^94 + 14864 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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