LMFDB - Embedded newform 189.4.e.h.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(109,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	189.e (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.1513609911$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
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} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81 $ x^16 + 54x^14 - 12x^13 + 2361x^12 - 966x^11 + 29570x^10 - 65952x^9 + 300096x^8 - 356610x^7 + 531858x^6 + 80070x^5 + 77197x^4 - 216x^3 + 4878x^2 + 486x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{9}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-2.17666 + 3.77009i$ of defining polynomial
Character	$\chi$	$=$	189.109
Dual form			189.4.e.h.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.73089 - 4.73004i) q^{2} +(-10.9155 + 18.9063i) q^{4} +(-0.0995681 - 0.172457i) q^{5} +(-16.0061 + 9.31685i) q^{7} +75.5424 q^{8} +(-0.543819 + 0.941923i) q^{10} +(-14.2117 + 24.6153i) q^{11} +32.5809 q^{13} +(87.7801 + 50.2664i) q^{14} +(-118.974 - 206.069i) q^{16} +(57.7439 - 100.015i) q^{17} +(-10.5750 - 18.3164i) q^{19} +4.34736 q^{20} +155.242 q^{22} +(-46.8828 - 81.2034i) q^{23} +(62.4802 - 108.219i) q^{25} +(-88.9749 - 154.109i) q^{26} +(-1.43126 - 404.315i) q^{28} +231.571 q^{29} +(-140.871 + 243.996i) q^{31} +(-347.640 + 602.130i) q^{32} -630.770 q^{34} +(3.20045 + 1.83271i) q^{35} +(73.2770 + 126.919i) q^{37} +(-57.7583 + 100.040i) q^{38} +(-7.52161 - 13.0278i) q^{40} -111.001 q^{41} +392.361 q^{43} +(-310.256 - 537.380i) q^{44} +(-256.064 + 443.516i) q^{46} +(-136.584 - 236.571i) q^{47} +(169.393 - 298.253i) q^{49} -682.506 q^{50} +(-355.638 + 615.984i) q^{52} +(170.201 - 294.798i) q^{53} +5.66011 q^{55} +(-1209.14 + 703.817i) q^{56} +(-632.395 - 1095.34i) q^{58} +(348.408 - 603.461i) q^{59} +(185.001 + 320.431i) q^{61} +1538.81 q^{62} +1893.89 q^{64} +(-3.24402 - 5.61880i) q^{65} +(-43.5681 + 75.4622i) q^{67} +(1260.61 + 2183.45i) q^{68} +(-0.0713061 - 20.1432i) q^{70} -88.3772 q^{71} +(401.518 - 695.450i) q^{73} +(400.223 - 693.207i) q^{74} +461.727 q^{76} +(-1.86345 - 526.404i) q^{77} +(-182.308 - 315.766i) q^{79} +(-23.6920 + 41.0357i) q^{80} +(303.132 + 525.040i) q^{82} +921.684 q^{83} -22.9978 q^{85} +(-1071.50 - 1855.88i) q^{86} +(-1073.58 + 1859.50i) q^{88} +(105.698 + 183.074i) q^{89} +(-521.494 + 303.551i) q^{91} +2047.01 q^{92} +(-745.993 + 1292.10i) q^{94} +(-2.10586 + 3.64746i) q^{95} +845.718 q^{97} +(-1873.35 + 13.2633i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{4} - 60 q^{7} - 44 q^{10} - 168 q^{13} - 156 q^{16} - 12 q^{19} + 448 q^{22} - 408 q^{25} - 152 q^{28} - 800 q^{31} - 1896 q^{34} - 692 q^{37} - 96 q^{40} + 2912 q^{43} - 1524 q^{46} + 2020 q^{49}+ \cdots + 1168 q^{97}+O(q^{100}) $ 16 * q - 48 * q^4 - 60 * q^7 - 44 * q^10 - 168 * q^13 - 156 * q^16 - 12 * q^19 + 448 * q^22 - 408 * q^25 - 152 * q^28 - 800 * q^31 - 1896 * q^34 - 692 * q^37 - 96 * q^40 + 2912 * q^43 - 1524 * q^46 + 2020 * q^49 - 1972 * q^52 - 2560 * q^55 - 2372 * q^58 - 216 * q^61 + 9928 * q^64 - 684 * q^67 + 4316 * q^70 + 4564 * q^73 - 760 * q^76 - 556 * q^79 - 3340 * q^82 + 2592 * q^85 - 6696 * q^88 - 184 * q^91 + 492 * q^94 + 1168 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−2.73089	−	4.73004i	−0.965516	−	1.67232i	−0.708222	−	0.705990i	$-0.750502\pi$
$2$	−2.73089	−	4.73004i	−0.965516	−	1.67232i	−0.257295	−	0.966333i	$-0.582831\pi$
$3$		0			0
$3$		0			0
$4$	−10.9155	+	18.9063i	−1.36444	+	2.36328i
$5$	−0.0995681	−	0.172457i	−0.00890564	−	0.0154250i	0.861538	−	0.507693i	$-0.169501\pi$
$5$	−0.0995681	−	0.172457i	−0.00890564	−	0.0154250i	−0.870444	+	0.492268i	$0.836168\pi$
$6$		0			0
$7$	−16.0061	+	9.31685i	−0.864250	+	0.503063i
$7$	−16.0061	+	9.31685i	−0.864250	+	0.503063i
$8$	75.5424			3.33854
$9$		0			0
$10$	−0.543819	+	0.941923i	−0.0171971	+	0.0297862i
$11$	−14.2117	+	24.6153i	−0.389544	+	0.674710i	−0.992388	−	0.123149i	$-0.960701\pi$
$11$	−14.2117	+	24.6153i	−0.389544	+	0.674710i	0.602844	+	0.797859i	$0.294034\pi$
$12$		0			0
$13$	32.5809			0.695101			0.347551	−	0.937661i	$-0.387013\pi$
$13$	32.5809			0.695101			0.347551	+	0.937661i	$0.387013\pi$
$14$	87.7801	+	50.2664i	1.67573	+	0.959590i
$15$		0			0
$16$	−118.974	−	206.069i	−1.85897	−	3.21983i
$17$	57.7439	−	100.015i	0.823821	−	1.42690i	−0.0789952	−	0.996875i	$-0.525171\pi$
$17$	57.7439	−	100.015i	0.823821	−	1.42690i	0.902817	−	0.430026i	$-0.141495\pi$
$18$		0			0
$19$	−10.5750	−	18.3164i	−0.127688	−	0.221162i	0.795093	−	0.606488i	$-0.207422\pi$
$19$	−10.5750	−	18.3164i	−0.127688	−	0.221162i	−0.922780	+	0.385326i	$0.874089\pi$
$20$	4.34736			0.0486049
$21$		0			0
$22$	155.242			1.50444
$23$	−46.8828	−	81.2034i	−0.425032	−	0.736177i	0.571391	−	0.820678i	$-0.306404\pi$
$23$	−46.8828	−	81.2034i	−0.425032	−	0.736177i	−0.996423	+	0.0845003i	$0.973071\pi$
$24$		0			0
$25$	62.4802	−	108.219i	0.499841	−	0.865751i
$26$	−88.9749	−	154.109i	−0.671132	−	1.16243i
$27$		0			0
$28$	−1.43126	−	404.315i	−0.00966007	−	2.72887i
$29$	231.571			1.48282			0.741408	−	0.671055i	$-0.234159\pi$
$29$	231.571			1.48282			0.741408	+	0.671055i	$0.234159\pi$
$30$		0			0
$31$	−140.871	+	243.996i	−0.816167	+	1.41364i	0.0923206	+	0.995729i	$0.470572\pi$
$31$	−140.871	+	243.996i	−0.816167	+	1.41364i	−0.908487	+	0.417913i	$0.862762\pi$
$32$	−347.640	+	602.130i	−1.92046	+	3.32633i
$33$		0			0
$34$	−630.770			−3.18165
$35$	3.20045	+	1.83271i	0.0154564	+	0.00885098i
$36$		0			0
$37$	73.2770	+	126.919i	0.325585	+	0.563931i	0.981631	−	0.190791i	$-0.0611053\pi$
$37$	73.2770	+	126.919i	0.325585	+	0.563931i	−0.656045	+	0.754722i	$0.727772\pi$
$38$	−57.7583	+	100.040i	−0.246569	+	0.427070i
$39$		0			0
$40$	−7.52161	−	13.0278i	−0.0297318	−	0.0514970i
$41$	−111.001			−0.422816			−0.211408	−	0.977398i	$-0.567805\pi$
$41$	−111.001			−0.422816			−0.211408	+	0.977398i	$0.567805\pi$
$42$		0			0
$43$	392.361			1.39150			0.695750	−	0.718284i	$-0.255072\pi$
$43$	392.361			1.39150			0.695750	+	0.718284i	$0.255072\pi$
$44$	−310.256	−	537.380i	−1.06302	−	1.84121i
$45$		0			0
$46$	−256.064	+	443.516i	−0.820751	+	1.42158i
$47$	−136.584	−	236.571i	−0.423890	−	0.734199i	0.572426	−	0.819956i	$-0.306002\pi$
$47$	−136.584	−	236.571i	−0.423890	−	0.734199i	−0.996316	+	0.0857573i	$0.972669\pi$
$48$		0			0
$49$	169.393	−	298.253i	0.493856	−	0.869544i
$50$	−682.506			−1.93042
$51$		0			0
$52$	−355.638	+	615.984i	−0.948426	+	1.64272i
$53$	170.201	−	294.798i	0.441113	−	0.764030i	−0.556660	−	0.830741i	$-0.687917\pi$
$53$	170.201	−	294.798i	0.441113	−	0.764030i	0.997772	+	0.0667111i	$0.0212506\pi$
$54$		0			0
$55$	5.66011			0.0138765
$56$	−1209.14	+	703.817i	−2.88533	+	1.67949i
$57$		0			0
$58$	−632.395	−	1095.34i	−1.43168	−	2.47975i
$59$	348.408	−	603.461i	0.768795	−	1.33159i	−0.169421	−	0.985544i	$-0.554190\pi$
$59$	348.408	−	603.461i	0.768795	−	1.33159i	0.938217	−	0.346049i	$-0.112477\pi$
$60$		0			0
$61$	185.001	+	320.431i	0.388311	+	0.672574i	0.992222	−	0.124477i	$-0.0397254\pi$
$61$	185.001	+	320.431i	0.388311	+	0.672574i	−0.603912	+	0.797051i	$0.706392\pi$
$62$	1538.81			3.15209
$63$		0			0
$64$	1893.89			3.69900
$65$	−3.24402	−	5.61880i	−0.00619032	−	0.0107220i
$66$		0			0
$67$	−43.5681	+	75.4622i	−0.0794431	+	0.137600i	−0.903010	−	0.429620i	$-0.858648\pi$
$67$	−43.5681	+	75.4622i	−0.0794431	+	0.137600i	0.823567	+	0.567219i	$0.191981\pi$
$68$	1260.61	+	2183.45i	2.24812	+	3.89385i
$69$		0			0
$70$	−0.0713061	−	20.1432i	−0.000121753	−	0.0343939i
$71$	−88.3772			−0.147725			−0.0738623	−	0.997268i	$-0.523533\pi$
$71$	−88.3772			−0.147725			−0.0738623	+	0.997268i	$0.523533\pi$
$72$		0			0
$73$	401.518	−	695.450i	0.643755	−	1.11502i	−0.340832	−	0.940124i	$-0.610709\pi$
$73$	401.518	−	695.450i	0.643755	−	1.11502i	0.984587	−	0.174893i	$-0.0559579\pi$
$74$	400.223	−	693.207i	0.628716	−	1.08897i
$75$		0			0
$76$	461.727			0.696891
$77$	−1.86345	−	526.404i	−0.00275792	−	0.779083i
$78$		0			0
$79$	−182.308	−	315.766i	−0.259636	−	0.449702i	0.706509	−	0.707705i	$-0.250269\pi$
$79$	−182.308	−	315.766i	−0.259636	−	0.449702i	−0.966144	+	0.258002i	$0.916936\pi$
$80$	−23.6920	+	41.0357i	−0.0331106	+	0.0573492i
$81$		0			0
$82$	303.132	+	525.040i	0.408236	+	0.707085i
$83$	921.684			1.21889			0.609446	−	0.792828i	$-0.291392\pi$
$83$	921.684			1.21889			0.609446	+	0.792828i	$0.291392\pi$
$84$		0			0
$85$	−22.9978			−0.0293466
$86$	−1071.50	−	1855.88i	−1.34351	−	2.32704i
$87$		0			0
$88$	−1073.58	+	1859.50i	−1.30051	+	2.25254i
$89$	105.698	+	183.074i	0.125887	+	0.218043i	0.922079	−	0.387001i	$-0.126489\pi$
$89$	105.698	+	183.074i	0.125887	+	0.218043i	−0.796192	+	0.605044i	$0.793156\pi$
$90$		0			0
$91$	−521.494	+	303.551i	−0.600741	+	0.349679i
$92$	2047.01			2.31973
$93$		0			0
$94$	−745.993	+	1292.10i	−0.818545	+	1.41776i
$95$	−2.10586	+	3.64746i	−0.00227428	+	0.00393917i
$96$		0			0
$97$	845.718			0.885254			0.442627	−	0.896706i	$-0.354047\pi$
$97$	845.718			0.885254			0.442627	+	0.896706i	$0.354047\pi$
$98$	−1873.35	+	13.2633i	−1.93098	+	0.0136714i
$99$		0			0
$100$	1364.01	+	2362.54i	1.36401	+	2.36254i
$101$	78.4758	−	135.924i	0.0773132	−	0.133910i	−0.824777	−	0.565459i	$-0.808699\pi$
$101$	78.4758	−	135.924i	0.0773132	−	0.133910i	0.902090	+	0.431548i	$0.142033\pi$
$102$		0			0
$103$	69.6626	+	120.659i	0.0666414	+	0.115426i	0.897421	−	0.441175i	$-0.145438\pi$
$103$	69.6626	+	120.659i	0.0666414	+	0.115426i	−0.830780	+	0.556602i	$0.812105\pi$
$104$	2461.24			2.32062
$105$		0			0
$106$	−1859.21			−1.70361
$107$	207.738	+	359.813i	0.187690	+	0.325088i	0.944480	−	0.328570i	$-0.106567\pi$
$107$	207.738	+	359.813i	0.187690	+	0.325088i	−0.756790	+	0.653658i	$0.773233\pi$
$108$		0			0
$109$	422.040	−	730.995i	0.370863	−	0.642354i	−0.618835	−	0.785521i	$-0.712395\pi$
$109$	422.040	−	730.995i	0.370863	−	0.642354i	0.989699	+	0.143167i	$0.0457285\pi$
$110$	−15.4572	−	26.7726i	−0.0133980	−	0.0232061i
$111$		0			0
$112$	3824.22	+	2189.90i	3.22639	+	1.84756i
$113$	−1342.21			−1.11738			−0.558692	−	0.829375i	$-0.688697\pi$
$113$	−1342.21			−1.11738			−0.558692	+	0.829375i	$0.688697\pi$
$114$		0			0
$115$	−9.33606	+	16.1705i	−0.00757037	+	0.0131123i
$116$	−2527.72	+	4378.14i	−2.02322	+	3.50431i
$117$		0			0
$118$	−3805.86			−2.96914
$119$	7.57144	+	2138.85i	0.00583254	+	1.64763i
$120$		0			0
$121$	261.557	+	453.029i	0.196511	+	0.340368i
$122$	1010.44	−	1750.13i	0.749841	−	1.29876i
$123$		0			0
$124$	−3075.37	−	5326.69i	−2.22723	−	3.85767i
$125$	−49.7761			−0.0356169
$126$		0			0
$127$	978.750			0.683858			0.341929	−	0.939726i	$-0.388920\pi$
$127$	978.750			0.683858			0.341929	+	0.939726i	$0.388920\pi$
$128$	−2390.88	−	4141.13i	−1.65099	−	2.85959i
$129$		0			0
$130$	−17.7181	+	30.6887i	−0.0119537	+	0.0207044i
$131$	−1186.40	−	2054.91i	−0.791271	−	1.37052i	−0.925180	−	0.379528i	$-0.876086\pi$
$131$	−1186.40	−	2054.91i	−0.791271	−	1.37052i	0.133909	−	0.990994i	$-0.457247\pi$
$132$		0			0
$133$	339.916	+	194.649i	0.221612	+	0.126904i
$134$	475.919			0.306815
$135$		0			0
$136$	4362.12	−	7555.41i	2.75036	−	4.76376i
$137$	−801.337	+	1387.96i	−0.499729	+	0.865556i	−1.00000		0.000312974i	$-0.999900\pi$
$137$	−801.337	+	1387.96i	−0.499729	+	0.865556i	0.500271	+	0.865869i	$0.333234\pi$
$138$		0			0
$139$	2101.78			1.28252			0.641262	−	0.767322i	$-0.278411\pi$
$139$	2101.78			1.28252			0.641262	+	0.767322i	$0.278411\pi$
$140$	−69.5844	+	40.5037i	−0.0420068	+	0.0244513i
$141$		0			0
$142$	241.349	+	418.028i	0.142631	+	0.247043i
$143$	−463.029	+	801.990i	−0.270772	+	0.468992i
$144$		0			0
$145$	−23.0571	−	39.9360i	−0.0132054	−	0.0228724i
$146$	−4386.01			−2.48623
$147$		0			0
$148$	−3199.43			−1.77697
$149$	−262.200	−	454.144i	−0.144163	−	0.249697i	0.784897	−	0.619626i	$-0.212716\pi$
$149$	−262.200	−	454.144i	−0.144163	−	0.249697i	−0.929060	+	0.369928i	$0.879382\pi$
$150$		0			0
$151$	433.833	−	751.420i	0.233807	−	0.404965i	−0.725119	−	0.688624i	$-0.758215\pi$
$151$	433.833	−	751.420i	0.233807	−	0.404965i	0.958925	+	0.283659i	$0.0915484\pi$
$152$	−798.860	−	1383.67i	−0.426290	−	0.738356i
$153$		0			0
$154$	−2484.83	+	1446.37i	−1.30022	+	0.756829i
$155$	56.1050			0.0290739
$156$		0			0
$157$	2.96000	−	5.12687i	0.00150467	−	0.00260617i	−0.865272	−	0.501303i	$-0.832854\pi$
$157$	2.96000	−	5.12687i	0.00150467	−	0.00260617i	0.866777	+	0.498696i	$0.166188\pi$
$158$	−995.726	+	1724.65i	−0.501365	+	0.868390i
$159$		0			0
$160$	138.455			0.0684116
$161$	1506.97	+	862.953i	0.737677	+	0.422424i
$162$		0			0
$163$	416.480	+	721.364i	0.200130	+	0.346636i	0.948570	−	0.316567i	$-0.102530\pi$
$163$	416.480	+	721.364i	0.200130	+	0.346636i	−0.748440	+	0.663202i	$0.769197\pi$
$164$	1211.64	−	2098.62i	0.576909	−	0.999235i
$165$		0			0
$166$	−2517.02	−	4359.61i	−1.17686	−	2.03838i
$167$	−566.798			−0.262636			−0.131318	−	0.991340i	$-0.541921\pi$
$167$	−566.798			−0.262636			−0.131318	+	0.991340i	$0.541921\pi$
$168$		0			0
$169$	−1135.48			−0.516834
$170$	62.8045	+	108.781i	0.0283346	+	0.0490770i
$171$		0			0
$172$	−4282.83	+	7418.08i	−1.89862	+	3.28851i
$173$	2062.88	+	3573.02i	0.906579	+	1.57024i	0.818783	+	0.574103i	$0.194649\pi$
$173$	2062.88	+	3573.02i	0.906579	+	1.57024i	0.0877960	+	0.996138i	$0.472018\pi$
$174$		0			0
$175$	8.19246	+	2314.28i	0.00353881	+	0.999676i
$176$	6763.27			2.89660
$177$		0			0
$178$	577.299	−	999.912i	0.243092	−	0.421048i
$179$	−463.823	+	803.365i	−0.193675	+	0.335454i	−0.946465	−	0.322806i	$-0.895374\pi$
$179$	−463.823	+	803.365i	−0.193675	+	0.335454i	0.752791	+	0.658260i	$0.228707\pi$
$180$		0			0
$181$	1211.67			0.497585			0.248792	−	0.968557i	$-0.419966\pi$
$181$	1211.67			0.497585			0.248792	+	0.968557i	$0.419966\pi$
$182$	2859.96	+	1637.73i	1.16480	+	0.667013i
$183$		0			0
$184$	−3541.64	−	6134.30i	−1.41899	−	2.45775i
$185$	14.5921	−	25.2742i	0.00579909	−	0.0100443i
$186$		0			0
$187$	1641.28	+	2842.77i	0.641829	+	1.11168i
$188$	5963.56			2.31350
$189$		0			0
$190$	23.0035			0.00878342
$191$	913.746	+	1582.65i	0.346159	+	0.599565i	0.985564	−	0.169306i	$-0.0541525\pi$
$191$	913.746	+	1582.65i	0.346159	+	0.599565i	−0.639405	+	0.768870i	$0.720819\pi$
$192$		0			0
$193$	411.115	−	712.072i	0.153330	−	0.265576i	−0.779120	−	0.626875i	$-0.784334\pi$
$193$	411.115	−	712.072i	0.153330	−	0.265576i	0.932450	+	0.361300i	$0.117667\pi$
$194$	−2309.56	−	4000.28i	−0.854727	−	1.48043i
$195$		0			0
$196$	3789.85	+	6458.18i	1.38114	+	2.35357i
$197$	−4873.12			−1.76241			−0.881207	−	0.472731i	$-0.843268\pi$
$197$	−4873.12			−1.76241			−0.881207	+	0.472731i	$0.843268\pi$
$198$		0			0
$199$	741.113	−	1283.65i	0.264001	−	0.457262i	−0.703301	−	0.710892i	$-0.748291\pi$
$199$	741.113	−	1283.65i	0.264001	−	0.457262i	0.967301	+	0.253630i	$0.0816246\pi$
$200$	4719.90	−	8175.11i	1.66874	−	2.89034i
$201$		0			0
$202$	−857.236			−0.298589
$203$	−3706.56	+	2157.51i	−1.28152	+	0.745949i
$204$		0			0
$205$	11.0522	+	19.1429i	0.00376545	+	0.00652195i
$206$	380.482	−	659.015i	0.128687	−	0.222892i
$207$		0			0
$208$	−3876.28	−	6713.91i	−1.29217	−	2.23810i
$209$	601.153			0.198960
$210$		0			0
$211$	−4359.21			−1.42228			−0.711138	−	0.703053i	$-0.751820\pi$
$211$	−4359.21			−1.42228			−0.711138	+	0.703053i	$0.751820\pi$
$212$	3715.68	+	6435.75i	1.20375	+	2.08495i
$213$		0			0
$214$	1134.62	−	1965.22i	0.362435	−	0.627755i
$215$	−39.0666	−	67.6653i	−0.0123922	−	0.0214639i
$216$		0			0
$217$	−18.4711	−	5217.90i	−0.00577835	−	1.63232i
$218$	−4610.18			−1.43230
$219$		0			0
$220$	−61.7832	+	107.012i	−0.0189338	+	0.0327942i
$221$	1881.35	−	3258.59i	0.572639	−	0.991841i
$222$		0			0
$223$	−3312.73			−0.994784			−0.497392	−	0.867526i	$-0.665709\pi$
$223$	−3312.73			−0.994784			−0.497392	+	0.867526i	$0.665709\pi$
$224$	−45.5829	−	12876.7i	−0.0135966	−	3.84089i
$225$		0			0
$226$	3665.43	+	6348.71i	1.07885	+	1.86863i
$227$	399.161	−	691.366i	0.116710	−	0.202148i	−0.801752	−	0.597657i	$-0.796098\pi$
$227$	399.161	−	691.366i	0.116710	−	0.202148i	0.918462	+	0.395509i	$0.129432\pi$
$228$		0			0
$229$	177.457	+	307.365i	0.0512083	+	0.0886954i	0.890493	−	0.454996i	$-0.150359\pi$
$229$	177.457	+	307.365i	0.0512083	+	0.0886954i	−0.839285	+	0.543692i	$0.817026\pi$
$230$	101.983			0.0292372
$231$		0			0
$232$	17493.4			4.95043
$233$	2236.95	+	3874.52i	0.628960	+	1.08939i	0.987761	+	0.155977i	$0.0498525\pi$
$233$	2236.95	+	3874.52i	0.628960	+	1.08939i	−0.358801	+	0.933414i	$0.616814\pi$
$234$		0			0
$235$	−27.1988	+	47.1097i	−0.00755002	+	0.0130770i
$236$	7606.14	+	13174.2i	2.09795	+	3.63376i
$237$		0			0
$238$	10096.2	−	5876.79i	2.74974	−	1.60057i
$239$	−1494.65			−0.404523			−0.202262	−	0.979332i	$-0.564829\pi$
$239$	−1494.65			−0.404523			−0.202262	+	0.979332i	$0.564829\pi$
$240$		0			0
$241$	78.4424	−	135.866i	0.0209665	−	0.0363150i	−0.855352	−	0.518048i	$-0.826659\pi$
$241$	78.4424	−	135.866i	0.0209665	−	0.0363150i	0.876318	+	0.481733i	$0.159992\pi$
$242$	1428.57	−	2474.35i	0.379470	−	0.657261i
$243$		0			0
$244$	−8077.55			−2.11931
$245$	−68.3020	+	0.483578i	−0.0178108	+	0.000126101i
$246$		0			0
$247$	−344.543	−	596.765i	−0.0887559	−	0.153730i
$248$	−10641.7	+	18432.0i	−2.72480	+	4.71949i
$249$		0			0
$250$	135.933	+	235.443i	0.0343887	+	0.0595630i
$251$	3498.68			0.879819			0.439909	−	0.898042i	$-0.355011\pi$
$251$	3498.68			0.879819			0.439909	+	0.898042i	$0.355011\pi$
$252$		0			0
$253$	2665.13			0.662275
$254$	−2672.86	−	4629.53i	−0.660276	−	1.14363i
$255$		0			0
$256$	−5482.93	+	9496.72i	−1.33861	+	2.31853i
$257$	−2369.55	−	4104.18i	−0.575130	−	0.996155i	−0.996027	−	0.0890463i	$-0.971618\pi$
$257$	−2369.55	−	4104.18i	−0.575130	−	0.996155i	0.420897	−	0.907108i	$-0.361715\pi$
$258$		0			0
$259$	−2355.37	−	1348.78i	−0.565080	−	0.323587i
$260$	141.641			0.0337854
$261$		0			0
$262$	−6479.88	+	11223.5i	−1.52797	+	2.64652i
$263$	1682.22	−	2913.69i	0.394411	−	0.683141i	−0.598614	−	0.801037i	$-0.704282\pi$
$263$	1682.22	−	2913.69i	0.394411	−	0.683141i	0.993026	+	0.117897i	$0.0376152\pi$
$264$		0			0
$265$	−67.7865			−0.0157136
$266$	−7.57332	−	2139.38i	−0.00174568	−	0.493135i
$267$		0			0
$268$	−951.139	−	1647.42i	−0.216791	−	0.375494i
$269$	2603.64	−	4509.63i	0.590136	−	1.02215i	−0.404078	−	0.914725i	$-0.632408\pi$
$269$	2603.64	−	4509.63i	0.590136	−	1.02215i	0.994214	−	0.107421i	$-0.0342592\pi$
$270$		0			0
$271$	1156.00	+	2002.25i	0.259122	+	0.448813i	0.966007	−	0.258516i	$-0.0832334\pi$
$271$	1156.00	+	2002.25i	0.259122	+	0.448813i	−0.706885	+	0.707329i	$0.749900\pi$
$272$	−27480.1			−6.12583
$273$		0			0
$274$	8753.46			1.92999
$275$	1775.90	+	3075.94i	0.389420	+	0.674495i
$276$		0			0
$277$	−1303.36	+	2257.49i	−0.282713	+	0.489673i	−0.972052	−	0.234766i	$-0.924568\pi$
$277$	−1303.36	+	2257.49i	−0.282713	+	0.489673i	0.689339	+	0.724439i	$0.257901\pi$
$278$	−5739.74	−	9941.52i	−1.23830	−	2.14479i
$279$		0			0
$280$	241.770	+	138.447i	0.0516019	+	0.0295493i
$281$	−5271.62			−1.11914			−0.559570	−	0.828783i	$-0.689034\pi$
$281$	−5271.62			−1.11914			−0.559570	+	0.828783i	$0.689034\pi$
$282$		0			0
$283$	−2904.11	+	5030.06i	−0.610004	+	1.05656i	0.381235	+	0.924478i	$0.375499\pi$
$283$	−2904.11	+	5030.06i	−0.610004	+	1.05656i	−0.991239	+	0.132080i	$0.957834\pi$
$284$	964.686	−	1670.88i	0.201562	−	0.349115i
$285$		0			0
$286$	5057.93			1.04574
$287$	1776.70	−	1034.18i	0.365419	−	0.212703i
$288$		0			0
$289$	−4212.23	−	7295.79i	−0.857364	−	1.48500i
$290$	−125.933	+	218.122i	−0.0255001	+	0.0441674i
$291$		0			0
$292$	8765.58	+	15182.4i	1.75674	+	3.04275i
$293$	−1979.70			−0.394729			−0.197364	−	0.980330i	$-0.563238\pi$
$293$	−1979.70			−0.394729			−0.197364	+	0.980330i	$0.563238\pi$
$294$		0			0
$295$	−138.761			−0.0273864
$296$	5535.52	+	9587.81i	1.08698	+	1.88270i
$297$		0			0
$298$	−1432.08	+	2480.43i	−0.278383	+	0.482174i
$299$	−1527.48	−	2645.68i	−0.295441	−	0.511718i
$300$		0			0
$301$	−6280.18	+	3655.57i	−1.20260	+	0.700011i
$302$	−4739.00			−0.902977
$303$		0			0
$304$	−2516.29	+	4358.35i	−0.474735	+	0.822264i
$305$	36.8404	−	63.8094i	0.00691631	−	0.0119794i
$306$		0			0
$307$	924.005			0.171778			0.0858888	−	0.996305i	$-0.472627\pi$
$307$	924.005			0.171778			0.0858888	+	0.996305i	$0.472627\pi$
$308$	9972.69	+	5710.76i	1.84496	+	1.05650i
$309$		0			0
$310$	−153.217	−	265.379i	−0.0280714	−	0.0486210i
$311$	5054.30	−	8754.31i	0.921554	−	1.59618i	0.124542	−	0.992214i	$-0.460254\pi$
$311$	5054.30	−	8754.31i	0.921554	−	1.59618i	0.797012	−	0.603964i	$-0.206413\pi$
$312$		0			0
$313$	−3415.26	−	5915.40i	−0.616747	−	1.06824i	−0.990075	−	0.140538i	$-0.955117\pi$
$313$	−3415.26	−	5915.40i	−0.616747	−	1.06824i	0.373328	−	0.927699i	$-0.378217\pi$
$314$	−32.3337			−0.00581114
$315$		0			0
$316$	7959.96			1.41703
$317$	3811.81	+	6602.25i	0.675371	+	1.16978i	0.976360	+	0.216149i	$0.0693498\pi$
$317$	3811.81	+	6602.25i	0.675371	+	1.16978i	−0.300989	+	0.953627i	$0.597317\pi$
$318$		0			0
$319$	−3291.01	+	5700.20i	−0.577621	+	1.00047i
$320$	−188.571	−	326.614i	−0.0329419	−	0.0570571i
$321$		0			0
$322$	−33.5753	−	9484.68i	−0.00581081	−	1.64149i
$323$	−2442.56			−0.420768
$324$		0			0
$325$	2035.66	−	3525.87i	0.347440	−	0.601784i
$326$	2274.72	−	3939.94i	0.386458	−	0.669364i
$327$		0			0
$328$	−8385.30			−1.41159
$329$	4390.27	+	2514.05i	0.735695	+	0.421288i
$330$		0			0
$331$	−1241.60	−	2150.52i	−0.206177	−	0.357110i	0.744330	−	0.667812i	$-0.232769\pi$
$331$	−1241.60	−	2150.52i	−0.206177	−	0.357110i	−0.950507	+	0.310702i	$0.899436\pi$
$332$	−10060.7	+	17425.6i	−1.66311	+	2.88059i
$333$		0			0
$334$	1547.86	+	2680.98i	0.253579	+	0.439211i
$335$	17.3520			0.00282997
$336$		0			0
$337$	7895.47			1.27624			0.638121	−	0.769936i	$-0.279712\pi$
$337$	7895.47			1.27624			0.638121	+	0.769936i	$0.279712\pi$
$338$	3100.89	+	5370.89i	0.499012	+	0.864314i
$339$		0			0
$340$	251.034	−	434.803i	0.0400418	−	0.0693544i
$341$	−4004.02	−	6935.17i	−0.635865	−	1.10135i
$342$		0			0
$343$	67.4606	+	6352.09i	0.0106196	+	0.999944i
$344$	29639.9			4.64557
$345$		0			0
$346$	11267.0	−	19515.1i	1.75063	−	3.03219i
$347$	−444.712	+	770.263i	−0.0687994	+	0.119164i	−0.898373	−	0.439233i	$-0.855250\pi$
$347$	−444.712	+	770.263i	−0.0687994	+	0.119164i	0.829574	+	0.558397i	$0.188583\pi$
$348$		0			0
$349$	6962.20			1.06785			0.533923	−	0.845533i	$-0.320717\pi$
$349$	6962.20			1.06785			0.533923	+	0.845533i	$0.320717\pi$
$350$	10924.3	−	6358.81i	1.66837	−	0.971122i
$351$		0			0
$352$	−9881.09	−	17114.6i	−1.49620	−	2.59150i
$353$	5780.30	−	10011.8i	0.871542	−	1.50956i	0.0111411	−	0.999938i	$-0.496454\pi$
$353$	5780.30	−	10011.8i	0.871542	−	1.50956i	0.860401	−	0.509617i	$-0.170213\pi$
$354$		0			0
$355$	8.79955	+	15.2413i	0.00131558	+	0.00227865i
$356$	−4615.00			−0.687064
$357$		0			0
$358$	5066.60			0.747984
$359$	−4228.51	−	7323.99i	−0.621649	−	1.07673i	−0.989179	−	0.146716i	$-0.953130\pi$
$359$	−4228.51	−	7323.99i	−0.621649	−	1.07673i	0.367529	−	0.930012i	$-0.380204\pi$
$360$		0			0
$361$	3205.84	−	5552.68i	0.467392	−	0.809546i
$362$	−3308.95	−	5731.26i	−0.480426	−	0.832123i
$363$		0			0
$364$	−46.6316	−	13172.9i	−0.00671473	−	1.89684i
$365$	−159.914			−0.0229322
$366$		0			0
$367$	−3456.46	+	5986.77i	−0.491623	+	0.851517i	−0.999953	−	0.00964558i	$-0.996930\pi$
$367$	−3456.46	+	5986.77i	−0.491623	+	0.851517i	0.508330	+	0.861162i	$0.330263\pi$
$368$	−11155.7	+	19322.2i	−1.58024	+	2.73706i
$369$		0			0
$370$	−159.398			−0.0223965
$371$	22.3170	+	6304.31i	0.00312302	+	0.882220i
$372$		0			0
$373$	−211.956	−	367.118i	−0.0294226	−	0.0509615i	0.850939	−	0.525264i	$-0.176034\pi$
$373$	−211.956	−	367.118i	−0.0294226	−	0.0509615i	−0.880362	+	0.474303i	$0.842700\pi$
$374$	8964.30	−	15526.6i	1.23939	−	2.14669i
$375$		0			0
$376$	−10317.9	−	17871.1i	−1.41517	−	2.45115i
$377$	7544.79			1.03071
$378$		0			0
$379$	−9714.33			−1.31660			−0.658300	−	0.752756i	$-0.728724\pi$
$379$	−9714.33			−1.31660			−0.658300	+	0.752756i	$0.728724\pi$
$380$	−45.9732	−	79.6280i	−0.00620626	−	0.0107496i
$381$		0			0
$382$	4990.68	−	8644.12i	0.668444	−	1.15778i
$383$	1107.08	+	1917.51i	0.147699	+	0.255823i	0.930377	−	0.366605i	$-0.119480\pi$
$383$	1107.08	+	1917.51i	0.147699	+	0.255823i	−0.782677	+	0.622428i	$0.786146\pi$
$384$		0			0
$385$	−90.5966	+	52.7344i	−0.0119928	+	0.00698077i
$386$	−4490.84			−0.592171
$387$		0			0
$388$	−9231.47	+	15989.4i	−1.20788	+	2.09211i
$389$	3832.73	−	6638.48i	0.499556	−	0.865256i	−0.500444	−	0.865769i	$-0.666830\pi$
$389$	3832.73	−	6638.48i	0.499556	−	0.865256i	1.00000		0.000513038i	$0.000163305\pi$
$390$		0			0
$391$	−10828.8			−1.40060
$392$	12796.3	−	22530.8i	1.64876	−	2.90300i
$393$		0			0
$394$	13308.0	+	23050.1i	1.70164	+	2.94732i
$395$	−36.3041	+	62.8805i	−0.00462444	+	0.00800977i
$396$		0			0
$397$	−2489.10	−	4311.24i	−0.314671	−	0.545026i	0.664697	−	0.747113i	$-0.268561\pi$
$397$	−2489.10	−	4311.24i	−0.314671	−	0.545026i	−0.979367	+	0.202088i	$0.935227\pi$
$398$	−8095.60			−1.01959
$399$		0			0
$400$	−29734.0			−3.71675
$401$	2385.10	+	4131.12i	0.297023	+	0.514460i	0.975454	−	0.220205i	$-0.0706727\pi$
$401$	2385.10	+	4131.12i	0.297023	+	0.514460i	−0.678430	+	0.734665i	$0.737339\pi$
$402$		0			0
$403$	−4589.70	+	7949.60i	−0.567319	+	0.982625i
$404$	1713.21	+	2967.37i	0.210979	+	0.365426i
$405$		0			0
$406$	20327.3	+	11640.2i	2.48480	+	1.42290i
$407$	−4165.56			−0.507319
$408$		0			0
$409$	298.629	−	517.240i	0.0361033	−	0.0625327i	−0.847409	−	0.530940i	$-0.821839\pi$
$409$	298.629	−	517.240i	0.0361033	−	0.0625327i	0.883512	+	0.468408i	$0.155172\pi$
$410$	60.3646	−	104.554i	0.00727120	−	0.0125941i
$411$		0			0
$412$	−3041.62			−0.363714
$413$	45.6836	+	12905.1i	0.00544296	+	1.53758i
$414$		0			0
$415$	−91.7703	−	158.951i	−0.0108550	−	0.0188014i
$416$	−11326.4	+	19617.9i	−1.33491	+	2.31214i
$417$		0			0
$418$	−1641.68	−	2843.48i	−0.192099	−	0.332725i
$419$	9571.90			1.11603			0.558016	−	0.829830i	$-0.311563\pi$
$419$	9571.90			1.11603			0.558016	+	0.829830i	$0.311563\pi$
$420$		0			0
$421$	−5954.33			−0.689303			−0.344651	−	0.938731i	$-0.612003\pi$
$421$	−5954.33			−0.689303			−0.344651	+	0.938731i	$0.612003\pi$
$422$	11904.5	+	20619.2i	1.37323	+	2.37850i
$423$		0			0
$424$	12857.4	−	22269.7i	1.47267	−	2.55074i
$425$	−7215.70	−	12498.0i	−0.823560	−	1.42645i
$426$		0			0
$427$	−5946.56	−	3405.24i	−0.673944	−	0.385927i
$428$	−9070.29			−1.02437
$429$		0			0
$430$	−213.373	+	369.573i	−0.0239297	+	0.0414475i
$431$	1271.72	−	2202.69i	0.142127	−	0.246171i	−0.786170	−	0.618010i	$-0.787939\pi$
$431$	1271.72	−	2202.69i	0.142127	−	0.246171i	0.928297	+	0.371839i	$0.121273\pi$
$432$		0			0
$433$	9781.07			1.08556			0.542781	−	0.839874i	$-0.317371\pi$
$433$	9781.07			1.08556			0.542781	+	0.839874i	$0.317371\pi$
$434$	−24630.5	+	14336.9i	−2.72419	+	1.58570i
$435$		0			0
$436$	9213.59	+	15958.4i	1.01204	+	1.75291i
$437$	−991.570	+	1717.45i	−0.108543	+	0.188002i
$438$		0			0
$439$	8410.13	+	14566.8i	0.914336	+	1.58368i	0.807871	+	0.589360i	$0.200620\pi$
$439$	8410.13	+	14566.8i	0.914336	+	1.58368i	0.106465	+	0.994316i	$0.466047\pi$
$440$	427.579			0.0463273
$441$		0			0
$442$	−20551.1			−2.21157
$443$	−1863.37	−	3227.46i	−0.199846	−	0.346143i	0.748633	−	0.662985i	$-0.230711\pi$
$443$	−1863.37	−	3227.46i	−0.199846	−	0.346143i	−0.948478	+	0.316842i	$0.897377\pi$
$444$		0			0
$445$	21.0483	−	36.4567i	0.00224221	−	0.00388362i
$446$	9046.71	+	15669.4i	0.960480	+	1.66360i
$447$		0			0
$448$	−30313.8	+	17645.1i	−3.19686	+	1.86083i
$449$	−9287.05			−0.976131			−0.488065	−	0.872807i	$-0.662297\pi$
$449$	−9287.05			−0.976131			−0.488065	+	0.872807i	$0.662297\pi$
$450$		0			0
$451$	1577.51	−	2732.33i	0.164705	−	0.285278i
$452$	14650.9	−	25376.2i	1.52461	−	2.64070i
$453$		0			0
$454$	−4360.26			−0.450742
$455$	104.274	+	59.7113i	0.0107438	+	0.00615233i
$456$		0			0
$457$	−6009.70	−	10409.1i	−0.615147	−	1.06547i	−0.990359	−	0.138526i	$-0.955764\pi$
$457$	−6009.70	−	10409.1i	−0.615147	−	1.06547i	0.375212	−	0.926939i	$-0.377570\pi$
$458$	969.233	−	1678.76i	0.0988849	−	0.171274i
$459$		0			0
$460$	−203.816	−	353.020i	−0.0206587	−	0.0357819i
$461$	−9571.71			−0.967026			−0.483513	−	0.875337i	$-0.660639\pi$
$461$	−9571.71			−0.967026			−0.483513	+	0.875337i	$0.660639\pi$
$462$		0			0
$463$	2265.15			0.227366			0.113683	−	0.993517i	$-0.463735\pi$
$463$	2265.15			0.227366			0.113683	+	0.993517i	$0.463735\pi$
$464$	−27550.9	−	47719.5i	−2.75650	−	4.77441i
$465$		0			0
$466$	12217.8	−	21161.8i	1.21454	−	2.10365i
$467$	2660.47	+	4608.07i	0.263623	+	0.456609i	0.967202	−	0.254009i	$-0.0817492\pi$
$467$	2660.47	+	4608.07i	0.263623	+	0.456609i	−0.703579	+	0.710617i	$0.748416\pi$
$468$		0			0
$469$	−5.71269	−	1613.77i	−0.000562447	−	0.158885i
$470$	297.108			0.0291587
$471$		0			0
$472$	26319.6	−	45586.9i	2.56665	−	4.44557i
$473$	−5576.10	+	9658.10i	−0.542050	+	0.938858i
$474$		0			0
$475$	−2642.91			−0.255294
$476$	−40520.4	−	23203.6i	−3.90178	−	2.23432i
$477$		0			0
$478$	4081.74	+	7069.78i	0.390574	+	0.676494i
$479$	−1883.49	+	3262.30i	−0.179663	+	0.311186i	−0.941765	−	0.336271i	$-0.890834\pi$
$479$	−1883.49	+	3262.30i	−0.179663	+	0.311186i	0.762102	+	0.647457i	$0.224168\pi$
$480$		0			0
$481$	2387.43	+	4135.15i	0.226315	+	0.391989i
$482$	−856.871			−0.0809739
$483$		0			0
$484$	−11420.1			−1.07251
$485$	−84.2065	−	145.850i	−0.00788375	−	0.0136551i
$486$		0			0
$487$	−4428.96	+	7671.18i	−0.412105	+	0.713787i	−0.995120	−	0.0986744i	$-0.968540\pi$
$487$	−4428.96	+	7671.18i	−0.412105	+	0.713787i	0.583014	+	0.812462i	$0.301873\pi$
$488$	13975.4	+	24206.2i	1.29639	+	2.24541i
$489$		0			0
$490$	188.813	+	321.751i	0.0174075	+	0.0296637i
$491$	−17311.1			−1.59112			−0.795560	−	0.605874i	$-0.792823\pi$
$491$	−17311.1			−1.59112			−0.795560	+	0.605874i	$0.792823\pi$
$492$		0			0
$493$	13371.8	−	23160.7i	1.22157	−	2.11583i
$494$	−1881.82	+	3259.40i	−0.171391	+	0.296857i
$495$		0			0
$496$	67039.8			6.06891
$497$	1414.58	−	823.397i	0.127671	−	0.0743147i
$498$		0			0
$499$	3515.22	+	6088.55i	0.315357	+	0.546214i	0.979513	−	0.201380i	$-0.0645425\pi$
$499$	3515.22	+	6088.55i	0.315357	+	0.546214i	−0.664157	+	0.747594i	$0.731209\pi$
$500$	543.334	−	941.081i	0.0485972	−	0.0841729i
$501$		0			0
$502$	−9554.51	−	16548.9i	−0.849479	−	1.47134i
$503$	1519.74			0.134716			0.0673578	−	0.997729i	$-0.478543\pi$
$503$	1519.74			0.134716			0.0673578	+	0.997729i	$0.478543\pi$
$504$		0			0
$505$	−31.2547			−0.00275409
$506$	−7278.19	−	12606.2i	−0.639437	−	1.10754i
$507$		0			0
$508$	−10683.6	+	18504.5i	−0.933086	+	1.61615i
$509$	9727.68	+	16848.8i	0.847096	+	1.46721i	0.883788	+	0.467887i	$0.154985\pi$
$509$	9727.68	+	16848.8i	0.847096	+	1.46721i	−0.0366920	+	0.999327i	$0.511682\pi$
$510$		0			0
$511$	52.6474	+	14872.3i	0.00455770	+	1.28750i
$512$	21639.1			1.86782
$513$		0			0
$514$	−12942.0	+	22416.1i	−1.11059	+	1.92361i
$515$	13.8723	−	24.0276i	0.00118697	−	0.00205589i
$516$		0			0
$517$	7764.35			0.660495
$518$	52.4776	+	14824.4i	0.00445122	+	1.25742i
$519$		0			0
$520$	−245.061	−	424.458i	−0.0206666	−	0.0357956i
$521$	−5111.98	+	8854.21i	−0.429866	+	0.744549i	−0.996861	−	0.0791720i	$-0.974772\pi$
$521$	−5111.98	+	8854.21i	−0.429866	+	0.744549i	0.566995	+	0.823721i	$0.308106\pi$
$522$		0			0
$523$	−8303.48	−	14382.1i	−0.694237	−	1.20245i	−0.970437	−	0.241354i	$-0.922409\pi$
$523$	−8303.48	−	14382.1i	−0.694237	−	1.20245i	0.276200	−	0.961100i	$-0.410925\pi$
$524$	51801.0			4.31858
$525$		0			0
$526$	−18375.9			−1.52324
$527$	16268.9	+	28178.5i	1.34475	+	2.32918i
$528$		0			0
$529$	1687.50	−	2922.84i	0.138695	−	0.240227i
$530$	185.118	+	320.633i	0.0151717	+	0.0262781i
$531$		0			0
$532$	−7390.46	+	4301.84i	−0.602288	+	0.350580i
$533$	−3616.52			−0.293900
$534$		0			0
$535$	41.3681	−	71.6517i	0.00334299	−	0.00579023i
$536$	−3291.24	+	5700.59i	−0.265224	+	0.459381i
$537$		0			0
$538$	−28441.0			−2.27914
$539$	4934.26	+	8408.34i	0.394311	+	0.671935i
$540$		0			0
$541$	−3291.27	−	5700.65i	−0.261558	−	0.453032i	0.705098	−	0.709110i	$-0.250903\pi$
$541$	−3291.27	−	5700.65i	−0.261558	−	0.453032i	−0.966656	+	0.256078i	$0.917570\pi$
$542$	6313.83	−	10935.9i	0.500373	−	0.866672i
$543$		0			0
$544$	40148.2	+	69538.7i	3.16423	+	5.48060i
$545$	−168.087			−0.0132111
$546$		0			0
$547$	3407.73			0.266369			0.133185	−	0.991091i	$-0.457480\pi$
$547$	3407.73			0.266369			0.133185	+	0.991091i	$0.457480\pi$
$548$	−17494.1	−	30300.6i	−1.36370	−	2.36200i
$549$		0			0
$550$	9699.56	−	16800.1i	0.751983	−	1.30247i
$551$	−2448.86	−	4241.55i	−0.189337	−	0.327942i
$552$		0			0
$553$	5859.99	+	3355.66i	0.450619	+	0.258042i
$554$	14237.4			1.09185
$555$		0			0
$556$	−22942.1	+	39736.9i	−1.74993	+	3.03097i
$557$	4740.05	−	8210.02i	0.360579	−	0.624541i	−0.627477	−	0.778635i	$-0.715912\pi$
$557$	4740.05	−	8210.02i	0.360579	−	0.624541i	0.988056	+	0.154094i	$0.0492458\pi$
$558$		0			0
$559$	12783.5			0.967233
$560$	−3.10652	−	877.558i	−0.000234418	−	0.0662207i
$561$		0			0
$562$	14396.2	+	24935.0i	1.08055	+	1.87156i
$563$	10932.2	−	18935.2i	0.818363	−	1.41745i	−0.0885250	−	0.996074i	$-0.528215\pi$
$563$	10932.2	−	18935.2i	0.818363	−	1.41745i	0.906888	−	0.421372i	$-0.138451\pi$
$564$		0			0
$565$	133.641	+	231.473i	0.00995101	+	0.0172357i
$566$	31723.2			2.35588
$567$		0			0
$568$	−6676.23			−0.493184
$569$	571.695	+	990.204i	0.0421207	+	0.0729552i	0.886317	−	0.463079i	$-0.153255\pi$
$569$	571.695	+	990.204i	0.0421207	+	0.0729552i	−0.844196	+	0.536034i	$0.819922\pi$
$570$		0			0
$571$	−5235.55	+	9068.24i	−0.383715	+	0.664613i	−0.991590	−	0.129419i	$-0.958689\pi$
$571$	−5235.55	+	9068.24i	−0.383715	+	0.664613i	0.607875	+	0.794032i	$0.292022\pi$
$572$	−10108.4	−	17508.3i	−0.738907	−	1.27982i
$573$		0			0
$574$	−9743.70	−	5579.63i	−0.708526	−	0.405730i
$575$	−11717.0			−0.849795
$576$		0			0
$577$	−2379.80	+	4121.94i	−0.171703	+	0.297398i	−0.939015	−	0.343876i	$-0.888260\pi$
$577$	−2379.80	+	4121.94i	−0.171703	+	0.297398i	0.767313	+	0.641273i	$0.221594\pi$
$578$	−23006.3	+	39848.0i	−1.65560	+	2.86758i
$579$		0			0
$580$	1006.72			0.0720721
$581$	−14752.6	+	8587.19i	−1.05343	+	0.613179i
$582$		0			0
$583$	4837.70	+	8379.14i	0.343665	+	0.595246i
$584$	30331.7	−	52536.0i	2.14920	−	3.72252i
$585$		0			0
$586$	5406.36	+	9364.08i	0.381117	+	0.660114i
$587$	−6307.45			−0.443503			−0.221751	−	0.975103i	$-0.571177\pi$
$587$	−6307.45			−0.443503			−0.221751	+	0.975103i	$0.571177\pi$
$588$		0			0
$589$	5958.83			0.416858
$590$	378.942	+	656.348i	0.0264421	+	0.0457990i
$591$		0			0
$592$	17436.1	−	30200.2i	1.21051	−	2.09666i
$593$	−8865.93	−	15356.2i	−0.613963	−	1.06342i	−0.990565	−	0.137041i	$-0.956241\pi$
$593$	−8865.93	−	15356.2i	−0.613963	−	1.06342i	0.376602	−	0.926375i	$-0.377092\pi$
$594$		0			0
$595$	368.106	−	214.267i	0.0253628	−	0.0147632i
$596$	11448.2			0.786808
$597$		0			0
$598$	−8342.79	+	14450.1i	−0.570505	+	0.988144i
$599$	−9555.03	+	16549.8i	−0.651766	+	1.12889i	0.330928	+	0.943656i	$0.392638\pi$
$599$	−9555.03	+	16549.8i	−0.651766	+	1.12889i	−0.982694	+	0.185236i	$0.940695\pi$
$600$		0			0
$601$	−13118.3			−0.890357			−0.445179	−	0.895442i	$-0.646860\pi$
$601$	−13118.3			−0.890357			−0.445179	+	0.895442i	$0.646860\pi$
$602$	34441.5	+	19722.6i	2.33178	+	1.33527i
$603$		0			0
$604$	9471.04	+	16404.3i	0.638032	+	1.10510i
$605$	52.0854	−	90.2145i	0.00350012	−	0.00606238i
$606$		0			0
$607$	−8736.71	−	15132.4i	−0.584204	−	1.01187i	−0.994974	−	0.100133i	$-0.968073\pi$
$607$	−8736.71	−	15132.4i	−0.584204	−	1.01187i	0.410770	−	0.911739i	$-0.365260\pi$
$608$	14705.1			0.980876
$609$		0			0
$610$	−402.429			−0.0267112
$611$	−4450.03	−	7707.68i	−0.294647	−	0.510343i
$612$		0			0
$613$	−1547.70	+	2680.69i	−0.101975	+	0.176626i	−0.912498	−	0.409080i	$-0.865850\pi$
$613$	−1547.70	+	2680.69i	−0.101975	+	0.176626i	0.810523	+	0.585707i	$0.199183\pi$
$614$	−2523.36	−	4370.58i	−0.165854	−	0.287268i
$615$		0			0
$616$	−140.769	−	39765.9i	−0.00920740	−	2.60099i
$617$	26334.8			1.71831			0.859157	−	0.511712i	$-0.170989\pi$
$617$	26334.8			1.71831			0.859157	+	0.511712i	$0.170989\pi$
$618$		0			0
$619$	1343.02	−	2326.18i	0.0872061	−	0.151045i	−0.819123	−	0.573618i	$-0.805539\pi$
$619$	1343.02	−	2326.18i	0.0872061	−	0.151045i	0.906329	+	0.422572i	$0.138873\pi$
$620$	−612.416	+	1060.74i	−0.0396697	+	0.0687100i
$621$		0			0
$622$	−55211.0			−3.55910
$623$	−3397.49	−	1945.54i	−0.218487	−	0.125115i
$624$		0			0
$625$	−7805.07	−	13518.8i	−0.499524	−	0.865201i
$626$	−18653.4	+	32308.7i	−1.19096	+	2.06280i
$627$		0			0
$628$	64.6200	+	111.925i	0.00410608	+	0.00711194i
$629$	16925.2			1.07290
$630$		0			0
$631$	22414.2			1.41409			0.707047	−	0.707167i	$-0.250027\pi$
$631$	22414.2			1.41409			0.707047	+	0.707167i	$0.250027\pi$
$632$	−13772.0	−	23853.8i	−0.866803	−	1.50135i
$633$		0			0
$634$	20819.3	−	36060.1i	1.30416	−	2.25888i
$635$	−97.4522	−	168.792i	−0.00609019	−	0.0105485i
$636$		0			0
$637$	5518.97	−	9717.37i	0.343280	−	0.604421i
$638$	35949.6			2.23081
$639$		0			0
$640$	−476.111	+	824.648i	−0.0294062	+	0.0509329i
$641$	−11069.5	+	19173.0i	−0.682092	+	1.18142i	0.292249	+	0.956342i	$0.405596\pi$
$641$	−11069.5	+	19173.0i	−0.682092	+	1.18142i	−0.974341	+	0.225076i	$0.927737\pi$
$642$		0			0
$643$	22523.7			1.38141			0.690707	−	0.723134i	$-0.257299\pi$
$643$	22523.7			1.38141			0.690707	+	0.723134i	$0.257299\pi$
$644$	−32764.6	+	19071.6i	−2.00483	+	1.16697i
$645$		0			0
$646$	6670.38	+	11553.4i	0.406258	+	0.703659i
$647$	−5761.75	+	9979.64i	−0.350105	+	0.606399i	−0.986267	−	0.165156i	$-0.947187\pi$
$647$	−5761.75	+	9979.64i	−0.350105	+	0.606399i	0.636163	+	0.771555i	$0.280521\pi$
$648$		0			0
$649$	9902.93	+	17152.4i	0.598959	+	1.03743i
$650$	−22236.7			−1.34184
$651$		0			0
$652$	−18184.4			−1.09226
$653$	−230.968	−	400.049i	−0.0138415	−	0.0239742i	0.859022	−	0.511939i	$-0.171073\pi$
$653$	−230.968	−	400.049i	−0.0138415	−	0.0239742i	−0.872863	+	0.487965i	$0.837739\pi$
$654$		0			0
$655$	−236.256	+	409.207i	−0.0140936	+	0.0244107i
$656$	13206.2	+	22873.9i	0.786001	+	1.36139i
$657$		0			0
$658$	−97.8152	−	27631.8i	−0.00579519	−	1.63708i
$659$	−14373.3			−0.849627			−0.424813	−	0.905281i	$-0.639660\pi$
$659$	−14373.3			−0.849627			−0.424813	+	0.905281i	$0.639660\pi$
$660$		0			0
$661$	−33.9245	+	58.7589i	−0.00199623	+	0.00345758i	−0.867022	−	0.498270i	$-0.833969\pi$
$661$	−33.9245	+	58.7589i	−0.00199623	+	0.00345758i	0.865026	+	0.501728i	$0.167302\pi$
$662$	−6781.38	+	11745.7i	−0.398135	+	0.689591i
$663$		0			0
$664$	69626.3			4.06931
$665$	−0.276122	−	78.0017i	−1.61016e−5	−	0.00454853i
$666$		0			0
$667$	−10856.7	−	18804.3i	−0.630244	−	1.09162i
$668$	6186.91	−	10716.0i	0.358351	−	0.620683i
$669$		0			0
$670$	−47.3863	−	82.0755i	−0.00273238	−	0.00473262i
$671$	−10516.7			−0.605056
$672$		0			0
$673$	21970.2			1.25838			0.629190	−	0.777252i	$-0.283387\pi$
$673$	21970.2			1.25838			0.629190	+	0.777252i	$0.283387\pi$
$674$	−21561.7	−	37345.9i	−1.23223	−	2.13429i
$675$		0			0
$676$	12394.4	−	21467.8i	0.705191	−	1.22143i
$677$	1496.65	+	2592.27i	0.0849642	+	0.147162i	0.905376	−	0.424611i	$-0.139589\pi$
$677$	1496.65	+	2592.27i	0.0849642	+	0.147162i	−0.820412	+	0.571773i	$0.806256\pi$
$678$		0			0
$679$	−13536.7	+	7879.43i	−0.765081	+	0.445338i
$680$	−1737.31			−0.0979747
$681$		0			0
$682$	−21869.1	+	37878.4i	−1.22788	+	2.12674i
$683$	2730.95	−	4730.14i	0.152997	−	0.264998i	−0.779331	−	0.626612i	$-0.784441\pi$
$683$	2730.95	−	4730.14i	0.152997	−	0.264998i	0.932328	+	0.361614i	$0.117774\pi$
$684$		0			0
$685$	319.150			0.0178016
$686$	29861.4	−	17666.0i	1.66198	−	0.983221i
$687$		0			0
$688$	−46680.7	−	80853.3i	−2.58675	−	4.48038i
$689$	5545.32	−	9604.77i	0.306618	−	0.531078i
$690$		0			0
$691$	7889.49	+	13665.0i	0.434342	+	0.752303i	0.997242	−	0.0742225i	$-0.0236475\pi$
$691$	7889.49	+	13665.0i	0.434342	+	0.752303i	−0.562899	+	0.826525i	$0.690314\pi$
$692$	−90070.0			−4.94790
$693$		0			0
$694$	4857.84			0.265708
$695$	−209.270	−	362.467i	−0.0114217	−	0.0197829i
$696$		0			0
$697$	−6409.64	+	11101.8i	−0.348325	+	0.603317i
$698$	−19013.0	−	32931.5i	−1.03102	−	1.78578i
$699$		0			0
$700$	−43843.9	−	25106.8i	−2.36735	−	1.35564i
$701$	−8058.91			−0.434209			−0.217105	−	0.976148i	$-0.569661\pi$
$701$	−8058.91			−0.434209			−0.217105	+	0.976148i	$0.569661\pi$
$702$		0			0
$703$	1549.81	−	2684.34i	0.0831466	−	0.144014i
$704$	−26915.3	+	46618.7i	−1.44092	+	2.49575i
$705$		0			0
$706$	−63141.5			−3.36595
$707$	10.2898	+	2906.77i	0.000547367	+	0.154626i
$708$		0			0
$709$	−13009.5	−	22533.1i	−0.689113	−	1.19358i	−0.972125	−	0.234462i	$-0.924667\pi$
$709$	−13009.5	−	22533.1i	−0.689113	−	1.19358i	0.283013	−	0.959116i	$-0.408666\pi$
$710$	48.0612	−	83.2445i	0.00254043	−	0.00440016i
$711$		0			0
$712$	7984.68	+	13829.9i	0.420279	+	0.727944i
$713$	26417.7			1.38759
$714$		0			0
$715$	184.412			0.00964560
$716$	−10125.8	−	17538.3i	−0.528516	−	0.915416i
$717$		0			0
$718$	−23095.2	+	40002.1i	−1.20043	+	2.07920i
$719$	2331.73	+	4038.68i	0.120944	+	0.209482i	0.920140	−	0.391589i	$-0.128074\pi$
$719$	2331.73	+	4038.68i	0.120944	+	0.209482i	−0.799196	+	0.601070i	$0.794741\pi$
$720$		0			0
$721$	−2239.19	−	1282.25i	−0.115661	−	0.0662324i
$722$	−35019.2			−1.80510
$723$		0			0
$724$	−13226.1	+	22908.2i	−0.678926	+	1.17593i
$725$	14468.6	−	25060.3i	0.741172	−	1.28375i
$726$		0			0
$727$	35484.5			1.81024			0.905121	−	0.425154i	$-0.139780\pi$
$727$	35484.5			1.81024			0.905121	+	0.425154i	$0.139780\pi$
$728$	−39395.0	+	22931.0i	−2.00560	+	1.16742i
$729$		0			0
$730$	436.707	+	756.398i	0.0221414	+	0.0383501i
$731$	22656.5	−	39242.1i	1.14635	−	1.98553i
$732$		0			0
$733$	7628.55	+	13213.0i	0.384402	+	0.665804i	0.991686	−	0.128681i	$-0.0410742\pi$
$733$	7628.55	+	13213.0i	0.384402	+	0.665804i	−0.607284	+	0.794485i	$0.707741\pi$
$734$	37756.9			1.89868
$735$		0			0
$736$	65193.4			3.26503
$737$	−1238.35	−	2144.89i	−0.0618932	−	0.107202i
$738$		0			0
$739$	−12876.9	+	22303.4i	−0.640978	+	1.11021i	0.344236	+	0.938883i	$0.388138\pi$
$739$	−12876.9	+	22303.4i	−0.640978	+	1.11021i	−0.985215	+	0.171324i	$0.945195\pi$
$740$	318.561	+	551.764i	0.0158251	+	0.0274098i
$741$		0			0
$742$	29758.7	−	17322.0i	1.47234	−	0.857020i
$743$	5703.55			0.281619			0.140809	−	0.990037i	$-0.455029\pi$
$743$	5703.55			0.281619			0.140809	+	0.990037i	$0.455029\pi$
$744$		0			0
$745$	−52.2135	+	90.4364i	−0.00256772	+	0.00444743i
$746$	−1157.66	+	2005.12i	−0.0568160	+	0.0984083i
$747$		0			0
$748$	−71661.7			−3.50296
$749$	−6677.40	−	3823.75i	−0.325750	−	0.186538i
$750$		0			0
$751$	−7725.44	−	13380.9i	−0.375373	−	0.650166i	0.615010	−	0.788520i	$-0.289152\pi$
$751$	−7725.44	−	13380.9i	−0.375373	−	0.650166i	−0.990383	+	0.138354i	$0.955819\pi$
$752$	−32499.9	+	56291.4i	−1.57600	+	2.72970i
$753$		0			0
$754$	−20604.0	−	35687.2i	−0.995164	−	1.72368i
$755$	−172.784			−0.00832879
$756$		0			0
$757$	12434.7			0.597025			0.298513	−	0.954406i	$-0.403509\pi$
$757$	12434.7			0.597025			0.298513	+	0.954406i	$0.403509\pi$
$758$	26528.8	+	45949.2i	1.27120	+	2.20178i
$759$		0			0
$760$	−159.082	+	275.538i	−0.00759277	+	0.0131511i
$761$	−1159.31	−	2007.98i	−0.0552234	−	0.0956497i	0.837092	−	0.547062i	$-0.184254\pi$
$761$	−1159.31	−	2007.98i	−0.0552234	−	0.0956497i	−0.892316	+	0.451412i	$0.850920\pi$
$762$		0			0
$763$	55.3383	+	15632.5i	0.00262566	+	0.741722i
$764$	−39896.1			−1.88926
$765$		0			0
$766$	6046.61	−	10473.0i	0.285212	−	0.494003i
$767$	11351.5	−	19661.3i	0.534391	−	0.925592i
$768$		0			0
$769$	23104.3			1.08344			0.541718	−	0.840560i	$-0.317774\pi$
$769$	23104.3			1.08344			0.541718	+	0.840560i	$0.317774\pi$
$770$	496.846	+	284.514i	0.0232533	+	0.0133158i
$771$		0			0
$772$	8975.09	+	15545.3i	0.418420	+	0.724725i
$773$	−6216.80	+	10767.8i	−0.289266	+	0.501024i	−0.973635	−	0.228112i	$-0.926745\pi$
$773$	−6216.80	+	10767.8i	−0.289266	+	0.501024i	0.684369	+	0.729136i	$0.260078\pi$
$774$		0			0
$775$	17603.3	+	30489.8i	0.815908	+	1.41319i
$776$	63887.6			2.95545
$777$		0			0
$778$	−41867.1			−1.92932
$779$	1173.84	+	2033.14i	0.0539885	+	0.0935108i
$780$		0			0
$781$	1255.99	−	2175.44i	0.0575452	−	0.0996712i
$782$	29572.3	+	51220.7i	1.35230	+	2.34226i
$783$		0			0
$784$	−81614.0	+	577.827i	−3.71784	+	0.0263223i
$785$	−1.17888			−5.36002e−5
$786$		0			0
$787$	12042.7	−	20858.6i	0.545458	−	0.944762i	−0.453120	−	0.891450i	$-0.649689\pi$
$787$	12042.7	−	20858.6i	0.545458	−	0.944762i	0.998578	−	0.0533118i	$-0.0169777\pi$
$788$	53192.8	−	92132.6i	2.40471	−	4.16509i
$789$		0			0
$790$	396.570			0.0178599
$791$	21483.6	−	12505.2i	0.965699	−	0.562114i
$792$		0			0
$793$	6027.50	+	10439.9i	0.269915	+	0.467507i
$794$	−13594.9	+	23547.1i	−0.607639	+	1.05246i
$795$		0			0
$796$	16179.3	+	28023.4i	0.720427	+	1.24782i
$797$	−36596.8			−1.62651			−0.813253	−	0.581910i	$-0.802306\pi$
$797$	−36596.8			−1.62651			−0.813253	+	0.581910i	$0.802306\pi$
$798$		0			0
$799$	−31547.6			−1.39684
$800$	43441.2	+	75242.4i	1.91985	+	3.32528i
$801$		0			0
$802$	13026.9	−	22563.3i	0.573562	−	0.993438i
$803$	11412.5	+	19767.0i	0.501542	+	0.868696i
$804$		0			0
$805$	−1.22415	−	345.810i	−5.35972e−5	−	0.0151406i
$806$	50135.9			2.19102
$807$		0			0
$808$	5928.25	−	10268.0i	0.258113	−	0.447065i
$809$	10373.2	−	17966.9i	0.450805	−	0.780817i	−0.547631	−	0.836720i	$-0.684470\pi$
$809$	10373.2	−	17966.9i	0.450805	−	0.780817i	0.998436	+	0.0559027i	$0.0178037\pi$
$810$		0			0
$811$	−8538.03			−0.369680			−0.184840	−	0.982769i	$-0.559177\pi$
$811$	−8538.03			−0.369680			−0.184840	+	0.982769i	$0.559177\pi$
$812$	−331.437	−	93627.6i	−0.0143241	−	4.04641i
$813$		0			0
$814$	11375.7	+	19703.3i	0.489825	+	0.848402i
$815$	82.9362	−	143.650i	0.00356457	−	0.00617402i
$816$		0			0
$817$	−4149.21	−	7186.64i	−0.177677	−	0.307746i
$818$	−3262.09			−0.139433
$819$		0			0
$820$	−482.562			−0.0205510
$821$	−10474.9	−	18143.1i	−0.445282	−	0.771251i	0.552790	−	0.833321i	$-0.313563\pi$
$821$	−10474.9	−	18143.1i	−0.445282	−	0.771251i	−0.998072	+	0.0620697i	$0.980230\pi$
$822$		0			0
$823$	5065.25	−	8773.26i	0.214536	−	0.371588i	−0.738593	−	0.674152i	$-0.764509\pi$
$823$	5065.25	−	8773.26i	0.214536	−	0.371588i	0.953129	+	0.302564i	$0.0978426\pi$
$824$	5262.49	+	9114.89i	0.222485	+	0.385355i
$825$		0			0
$826$	60917.2	−	35458.7i	2.56608	−	1.49366i
$827$	25327.3			1.06495			0.532476	−	0.846445i	$-0.321262\pi$
$827$	25327.3			1.06495			0.532476	+	0.846445i	$0.321262\pi$
$828$		0			0
$829$	9987.94	−	17299.6i	0.418451	−	0.724778i	−0.577333	−	0.816509i	$-0.695907\pi$
$829$	9987.94	−	17299.6i	0.418451	−	0.724778i	0.995784	+	0.0917308i	$0.0292399\pi$
$830$	−501.230	+	868.155i	−0.0209614	+	0.0363062i
$831$		0			0
$832$	61704.6			2.57118
$833$	−20048.6	−	34164.2i	−0.833903	−	1.42103i
$834$		0			0
$835$	56.4350	+	97.7482i	0.00233894	+	0.00405116i
$836$	−6561.91	+	11365.6i	−0.271469	+	0.470199i
$837$		0			0
$838$	−26139.8	−	45275.5i	−1.07755	−	1.86637i
$839$	3888.98			0.160027			0.0800134	−	0.996794i	$-0.474504\pi$
$839$	3888.98			0.160027			0.0800134	+	0.996794i	$0.474504\pi$
$840$		0			0
$841$	29236.1			1.19874
$842$	16260.6	+	28164.3i	0.665533	+	1.15274i
$843$		0			0
$844$	47583.1	−	82416.4i	1.94061	−	3.36124i
$845$	113.058	+	195.822i	0.00460274	+	0.00797217i
$846$		0			0
$847$	−8407.32	−	4814.37i	−0.341061	−	0.195305i
$848$	−80998.1			−3.28006
$849$		0			0
$850$	−39410.6	+	68261.2i	−1.59032	+	2.75452i
$851$	6870.86	−	11900.7i	0.276769	−	0.479377i
$852$		0			0
$853$	−32093.5			−1.28823			−0.644116	−	0.764928i	$-0.722775\pi$
$853$	−32093.5			−1.28823			−0.644116	+	0.764928i	$0.722775\pi$
$854$	132.489	+	37426.8i	0.00530877	+	1.49967i
$855$		0			0
$856$	15693.0	+	27181.1i	0.626608	+	1.08532i
$857$	−14939.9	+	25876.6i	−0.595492	+	1.03142i	0.397985	+	0.917392i	$0.369710\pi$
$857$	−14939.9	+	25876.6i	−0.595492	+	1.03142i	−0.993477	+	0.114031i	$0.963624\pi$
$858$		0			0
$859$	−16494.2	−	28568.7i	−0.655150	−	1.13475i	−0.981856	−	0.189627i	$-0.939272\pi$
$859$	−16494.2	−	28568.7i	−0.655150	−	1.13475i	0.326707	−	0.945126i	$-0.394061\pi$
$860$	1705.73			0.0676337
$861$		0			0
$862$	−13891.7			−0.548903
$863$	−13358.4	−	23137.4i	−0.526912	−	0.912639i	−0.999508	−	0.0313596i	$-0.990016\pi$
$863$	−13358.4	−	23137.4i	−0.526912	−	0.912639i	0.472596	−	0.881279i	$-0.343317\pi$
$864$		0			0
$865$	410.795	−	711.517i	0.0161473	−	0.0279680i
$866$	−26711.1	−	46264.9i	−1.04813	−	1.81541i
$867$		0			0
$868$	98852.7	+	56607.0i	3.86553	+	2.21356i
$869$	10363.6			0.404558
$870$		0			0
$871$	−1419.49	+	2458.63i	−0.0552210	+	0.0956456i
$872$	31881.9	−	55221.1i	1.23814	−	2.14452i
$873$		0			0
$874$	10831.5			0.419199
$875$	796.723	−	463.757i	0.0307819	−	0.0179175i
$876$		0			0
$877$	8982.84	+	15558.7i	0.345871	+	0.599066i	0.985512	−	0.169608i	$-0.0542501\pi$
$877$	8982.84	+	15558.7i	0.345871	+	0.599066i	−0.639641	+	0.768674i	$0.720917\pi$
$878$	45934.3	−	79560.5i	1.76561	−	3.05813i
$879$		0			0
$880$	−673.406	−	1166.37i	−0.0257960	−	0.0446800i
$881$	15852.9			0.606239			0.303119	−	0.952953i	$-0.401972\pi$
$881$	15852.9			0.606239			0.303119	+	0.952953i	$0.401972\pi$
$882$		0			0
$883$	−22050.7			−0.840392			−0.420196	−	0.907433i	$-0.638039\pi$
$883$	−22050.7			−0.840392			−0.420196	+	0.907433i	$0.638039\pi$
$884$	41071.9	+	71138.7i	1.56267	+	2.70662i
$885$		0			0
$886$	−10177.4	+	17627.7i	−0.385908	+	0.668413i
$887$	−13886.4	−	24051.9i	−0.525658	−	0.910467i	−0.999553	−	0.0298857i	$-0.990486\pi$
$887$	−13886.4	−	24051.9i	−0.525658	−	0.910467i	0.473895	−	0.880581i	$-0.342848\pi$
$888$		0			0
$889$	−15666.0	+	9118.86i	−0.591024	+	0.344023i
$890$	−229.922			−0.00865956
$891$		0			0
$892$	36160.3	−	62631.4i	1.35733	−	2.35096i
$893$	−2888.75	+	5003.46i	−0.108251	+	0.187496i
$894$		0			0
$895$	184.728			0.00689918
$896$	76851.1	+	44008.0i	2.86542	+	1.64085i
$897$		0			0
$898$	25361.9	+	43928.1i	0.942470	+	1.63241i
$899$	−32621.6	+	56502.3i	−1.21022	+	2.09617i
$900$		0			0
$901$	−19656.2	−	34045.6i	−0.726796	−	1.25885i
$902$	−17232.1			−0.636103
$903$		0			0
$904$	−101394.			−3.73043
$905$	−120.644	−	208.961i	−0.00443131	−	0.00767525i
$906$		0			0
$907$	2948.01	−	5106.11i	0.107924	−	0.186930i	−0.807005	−	0.590545i	$-0.798913\pi$
$907$	2948.01	−	5106.11i	0.107924	−	0.186930i	0.914929	+	0.403615i	$0.132246\pi$
$908$	8714.11	+	15093.3i	0.318489	+	0.551639i
$909$		0			0
$910$	−2.32322	−	656.284i	−8.46306e−5	−	0.0239073i
$911$	42197.8			1.53466			0.767331	−	0.641251i	$-0.221584\pi$
$911$	42197.8			1.53466			0.767331	+	0.641251i	$0.221584\pi$
$912$		0			0
$913$	−13098.7	+	22687.6i	−0.474812	+	0.822398i
$914$	−32823.7	+	56852.3i	−1.18787	+	2.05745i
$915$		0			0
$916$	−7748.17			−0.279483
$917$	38135.0	+	21837.6i	1.37331	+	0.786415i
$918$		0			0
$919$	20464.3	+	35445.3i	0.734555	+	1.27229i	0.954918	+	0.296869i	$0.0959424\pi$
$919$	20464.3	+	35445.3i	0.734555	+	1.27229i	−0.220363	+	0.975418i	$0.570724\pi$
$920$	−705.269	+	1221.56i	−0.0252739	+	0.0437757i
$921$		0			0
$922$	26139.3	+	45274.6i	0.933679	+	1.61718i
$923$	−2879.41			−0.102684
$924$		0			0
$925$	18313.4			0.650964
$926$	−6185.87	−	10714.2i	−0.219525	−	0.380229i
$927$		0			0
$928$	−80503.3	+	139436.i	−2.84768	+	4.93233i
$929$	10739.5	+	18601.3i	0.379279	+	0.656930i	0.990958	−	0.134176i	$-0.0428388\pi$
$929$	10739.5	+	18601.3i	0.379279	+	0.656930i	−0.611679	+	0.791106i	$0.709505\pi$
$930$		0			0
$931$	−7254.26	+	51.3601i	−0.255369	+	0.00180801i
$932$	−97670.3			−3.43272
$933$		0			0
$934$	14530.9	−	25168.3i	0.509065	−	0.881726i
$935$	326.837	−	566.099i	0.0114318	−	0.0198004i
$936$		0			0
$937$	19560.1			0.681964			0.340982	−	0.940070i	$-0.389241\pi$
$937$	19560.1			0.681964			0.340982	+	0.940070i	$0.389241\pi$
$938$	−7617.63	+	4434.07i	−0.265164	+	0.154347i
$939$		0			0
$940$	−593.780	−	1028.46i	−0.0206031	−	0.0356857i
$941$	24301.2	−	42090.9i	0.841867	−	1.45816i	−0.0464471	−	0.998921i	$-0.514790\pi$
$941$	24301.2	−	42090.9i	0.841867	−	1.45816i	0.888314	−	0.459236i	$-0.151877\pi$
$942$		0			0
$943$	5204.05	+	9013.67i	0.179711	+	0.311268i
$944$	−165806.			−5.71666
$945$		0			0
$946$	60911.0			2.09343
$947$	14389.0	+	24922.5i	0.493748	+	0.855197i	0.999974	−	0.00720428i	$-0.00229321\pi$
$947$	14389.0	+	24922.5i	0.493748	+	0.855197i	−0.506226	+	0.862401i	$0.668960\pi$
$948$		0			0
$949$	13081.8	−	22658.4i	0.447475	−	0.775050i
$950$	7217.49	+	12501.1i	0.246491	+	0.426935i
$951$		0			0
$952$	571.965	+	161574.i	0.0194722	+	5.50068i
$953$	−26009.2			−0.884072			−0.442036	−	0.896997i	$-0.645744\pi$
$953$	−26009.2			−0.884072			−0.442036	+	0.896997i	$0.645744\pi$
$954$		0			0
$955$	181.960	−	315.164i	0.00616553	−	0.0106790i
$956$	16315.0	−	28258.3i	0.551949	−	0.956004i
$957$		0			0
$958$	20574.4			0.693872
$959$	−105.072	−	29681.8i	−0.00353801	−	0.999452i
$960$		0			0
$961$	−24793.7	−	42944.0i	−0.832256	−	1.44151i
$962$	13039.6	−	22585.3i	0.437021	−	0.756943i
$963$		0			0
$964$	1712.48	+	2966.11i	0.0572151	+	0.0990995i
$965$	−163.736			−0.00546201
$966$		0			0
$967$	−13284.5			−0.441779			−0.220889	−	0.975299i	$-0.570896\pi$
$967$	−13284.5			−0.441779			−0.220889	+	0.975299i	$0.570896\pi$
$968$	19758.6	+	34222.9i	0.656060	+	1.13633i
$969$		0			0
$970$	−459.918	+	796.601i	−0.0152238	+	0.0263684i
$971$	−22124.3	−	38320.4i	−0.731208	−	1.26649i	−0.956367	−	0.292167i	$-0.905624\pi$
$971$	−22124.3	−	38320.4i	−0.731208	−	1.26649i	0.225160	−	0.974322i	$-0.427710\pi$
$972$		0			0
$973$	−33641.4	+	19582.0i	−1.10842	+	0.645189i
$974$	48380.0			1.59158
$975$		0			0
$976$	44020.6	−	76245.9i	1.44371	−	2.50059i
$977$	−24483.7	+	42407.0i	−0.801744	+	1.38866i	0.116724	+	0.993164i	$0.462761\pi$
$977$	−24483.7	+	42407.0i	−0.801744	+	1.38866i	−0.918468	+	0.395496i	$0.870573\pi$
$978$		0			0
$979$	−6008.58			−0.196154
$980$	736.411	−	1296.61i	0.0240039	−	0.0422641i
$981$		0			0
$982$	47274.8	+	81882.4i	1.53625	+	2.66087i
$983$	−24928.2	+	43176.9i	−0.808835	+	1.40094i	0.104836	+	0.994490i	$0.466568\pi$
$983$	−24928.2	+	43176.9i	−0.808835	+	1.40094i	−0.913671	+	0.406454i	$0.866765\pi$
$984$		0			0
$985$	485.207	+	840.403i	0.0156954	+	0.0271853i
$986$	−146068.			−4.71780
$987$		0			0
$988$	15043.5			0.484410
$989$	−18395.0	−	31861.0i	−0.591432	−	1.02439i
$990$		0			0
$991$	−24324.1	+	42130.5i	−0.779696	+	1.35047i	0.152420	+	0.988316i	$0.451293\pi$
$991$	−24324.1	+	42130.5i	−0.779696	+	1.35047i	−0.932117	+	0.362158i	$0.882040\pi$
$992$	−97944.7	−	169645.i	−3.13483	−	5.42968i
$993$		0			0
$994$	−7757.77	−	4442.41i	−0.247547	−	0.141755i
$995$	−295.165			−0.00940437
$996$		0			0
$997$	−4854.58	+	8408.37i	−0.154209	+	0.267097i	−0.932771	−	0.360471i	$-0.882616\pi$
$997$	−4854.58	+	8408.37i	−0.154209	+	0.267097i	0.778562	+	0.627568i	$0.215949\pi$
$998$	19199.4	−	33254.3i	0.608964	−	1.05476i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.4.e.h.109.1	✓	16
3.2	odd	2	inner	189.4.e.h.109.8	yes	16
7.2	even	3	inner	189.4.e.h.163.1	yes	16
7.3	odd	6		1323.4.a.bn.1.8		8
7.4	even	3		1323.4.a.bo.1.8		8
21.2	odd	6	inner	189.4.e.h.163.8	yes	16
21.11	odd	6		1323.4.a.bo.1.1		8
21.17	even	6		1323.4.a.bn.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.h.109.1	✓	16	1.1	even	1	trivial
189.4.e.h.109.8	yes	16	3.2	odd	2	inner
189.4.e.h.163.1	yes	16	7.2	even	3	inner
189.4.e.h.163.8	yes	16	21.2	odd	6	inner
1323.4.a.bn.1.1		8	21.17	even	6
1323.4.a.bn.1.8		8	7.3	odd	6
1323.4.a.bo.1.1		8	21.11	odd	6
1323.4.a.bo.1.8		8	7.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	189.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.73089 - 4.73004i) q^{2} +(-10.9155 + 18.9063i) q^{4} +(-0.0995681 - 0.172457i) q^{5} +(-16.0061 + 9.31685i) q^{7} +75.5424 q^{8} +(-0.543819 + 0.941923i) q^{10} +(-14.2117 + 24.6153i) q^{11} +32.5809 q^{13} +(87.7801 + 50.2664i) q^{14} +(-118.974 - 206.069i) q^{16} +(57.7439 - 100.015i) q^{17} +(-10.5750 - 18.3164i) q^{19} +4.34736 q^{20} +155.242 q^{22} +(-46.8828 - 81.2034i) q^{23} +(62.4802 - 108.219i) q^{25} +(-88.9749 - 154.109i) q^{26} +(-1.43126 - 404.315i) q^{28} +231.571 q^{29} +(-140.871 + 243.996i) q^{31} +(-347.640 + 602.130i) q^{32} -630.770 q^{34} +(3.20045 + 1.83271i) q^{35} +(73.2770 + 126.919i) q^{37} +(-57.7583 + 100.040i) q^{38} +(-7.52161 - 13.0278i) q^{40} -111.001 q^{41} +392.361 q^{43} +(-310.256 - 537.380i) q^{44} +(-256.064 + 443.516i) q^{46} +(-136.584 - 236.571i) q^{47} +(169.393 - 298.253i) q^{49} -682.506 q^{50} +(-355.638 + 615.984i) q^{52} +(170.201 - 294.798i) q^{53} +5.66011 q^{55} +(-1209.14 + 703.817i) q^{56} +(-632.395 - 1095.34i) q^{58} +(348.408 - 603.461i) q^{59} +(185.001 + 320.431i) q^{61} +1538.81 q^{62} +1893.89 q^{64} +(-3.24402 - 5.61880i) q^{65} +(-43.5681 + 75.4622i) q^{67} +(1260.61 + 2183.45i) q^{68} +(-0.0713061 - 20.1432i) q^{70} -88.3772 q^{71} +(401.518 - 695.450i) q^{73} +(400.223 - 693.207i) q^{74} +461.727 q^{76} +(-1.86345 - 526.404i) q^{77} +(-182.308 - 315.766i) q^{79} +(-23.6920 + 41.0357i) q^{80} +(303.132 + 525.040i) q^{82} +921.684 q^{83} -22.9978 q^{85} +(-1071.50 - 1855.88i) q^{86} +(-1073.58 + 1859.50i) q^{88} +(105.698 + 183.074i) q^{89} +(-521.494 + 303.551i) q^{91} +2047.01 q^{92} +(-745.993 + 1292.10i) q^{94} +(-2.10586 + 3.64746i) q^{95} +845.718 q^{97} +(-1873.35 + 13.2633i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{4} - 60 q^{7} - 44 q^{10} - 168 q^{13} - 156 q^{16} - 12 q^{19} + 448 q^{22} - 408 q^{25} - 152 q^{28} - 800 q^{31} - 1896 q^{34} - 692 q^{37} - 96 q^{40} + 2912 q^{43} - 1524 q^{46} + 2020 q^{49}+ \cdots + 1168 q^{97}+O(q^{100}) \) 16 * q - 48 * q^4 - 60 * q^7 - 44 * q^10 - 168 * q^13 - 156 * q^16 - 12 * q^19 + 448 * q^22 - 408 * q^25 - 152 * q^28 - 800 * q^31 - 1896 * q^34 - 692 * q^37 - 96 * q^40 + 2912 * q^43 - 1524 * q^46 + 2020 * q^49 - 1972 * q^52 - 2560 * q^55 - 2372 * q^58 - 216 * q^61 + 9928 * q^64 - 684 * q^67 + 4316 * q^70 + 4564 * q^73 - 760 * q^76 - 556 * q^79 - 3340 * q^82 + 2592 * q^85 - 6696 * q^88 - 184 * q^91 + 492 * q^94 + 1168 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more