LMFDB - Embedded newform 108.6.i.a.25.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(13,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.5
Character	$\chi$	$=$	108.25
Dual form			108.6.i.a.13.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.5186 + 10.5034i) q^{3} +(0.194482 - 0.163190i) q^{5} +(83.4487 - 30.3729i) q^{7} +(22.3564 - 241.969i) q^{9} +O(q^{10})$	\(q+(-11.5186 + 10.5034i) q^{3} +(0.194482 - 0.163190i) q^{5} +(83.4487 - 30.3729i) q^{7} +(22.3564 - 241.969i) q^{9} +(422.243 + 354.304i) q^{11} +(-18.1884 - 103.151i) q^{13} +(-0.526112 + 3.92245i) q^{15} +(-108.547 - 188.010i) q^{17} +(-947.091 + 1640.41i) q^{19} +(-642.194 + 1226.35i) q^{21} +(-867.055 - 315.582i) q^{23} +(-542.639 + 3077.46i) q^{25} +(2283.99 + 3021.97i) q^{27} +(-1043.97 + 5920.66i) q^{29} +(-2498.77 - 909.479i) q^{31} +(-8585.06 + 353.909i) q^{33} +(11.2728 - 19.5250i) q^{35} +(1682.27 + 2913.77i) q^{37} +(1292.95 + 997.119i) q^{39} +(1456.87 + 8262.34i) q^{41} +(7463.76 + 6262.84i) q^{43} +(-35.1391 - 50.7071i) q^{45} +(10011.8 - 3643.99i) q^{47} +(-6833.73 + 5734.18i) q^{49} +(3225.06 + 1025.49i) q^{51} +3578.00 q^{53} +139.938 q^{55} +(-6320.74 - 28842.9i) q^{57} +(27728.4 - 23266.9i) q^{59} +(-46708.9 + 17000.6i) q^{61} +(-5483.69 - 20871.1i) q^{63} +(-20.3706 - 17.0930i) q^{65} +(4792.46 + 27179.4i) q^{67} +(13302.0 - 5471.98i) q^{69} +(10006.9 + 17332.5i) q^{71} +(-7080.10 + 12263.1i) q^{73} +(-26073.4 - 41147.6i) q^{75} +(45996.9 + 16741.5i) q^{77} +(9295.20 - 52715.7i) q^{79} +(-58049.4 - 10819.1i) q^{81} +(5269.29 - 29883.6i) q^{83} +(-51.7919 - 18.8507i) q^{85} +(-50162.1 - 79163.1i) q^{87} +(41357.3 - 71632.9i) q^{89} +(-4650.80 - 8055.41i) q^{91} +(38335.0 - 15769.7i) q^{93} +(83.5061 + 473.587i) q^{95} +(85552.1 + 71786.7i) q^{97} +(95170.6 - 94249.0i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) $ 90 * q - 87 * q^5 + 330 * q^9	$ 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) $ 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−11.5186	+	10.5034i	−0.738919	+	0.673795i
$3$	−11.5186	+	10.5034i	−0.738919	+	0.673795i
$4$		0			0
$5$	0.194482	−	0.163190i	0.00347901	−	0.00291923i	−0.641046	−	0.767502i	$-0.721499\pi$
$5$	0.194482	−	0.163190i	0.00347901	−	0.00291923i	0.644525	+	0.764583i	$0.277055\pi$
$6$		0			0
$7$	83.4487	−	30.3729i	0.643687	−	0.234283i	0.000509205	−	1.00000i	$-0.499838\pi$
$7$	83.4487	−	30.3729i	0.643687	−	0.234283i	0.643178	+	0.765717i	$0.277616\pi$
$8$		0			0
$9$	22.3564	−	241.969i	0.0920017	−	0.995759i
$10$		0			0
$11$	422.243	+	354.304i	1.05216	+	0.882866i	0.993319	−	0.115402i	$-0.0368155\pi$
$11$	422.243	+	354.304i	1.05216	+	0.882866i	0.0588397	+	0.998267i	$0.481260\pi$
$12$		0			0
$13$	−18.1884	−	103.151i	−0.0298494	−	0.169284i	0.966239	−	0.257648i	$-0.0829474\pi$
$13$	−18.1884	−	103.151i	−0.0298494	−	0.169284i	−0.996088	+	0.0883635i	$0.971836\pi$
$14$		0			0
$15$	−0.526112	+	3.92245i	−0.000603740	+	0.00450121i
$16$		0			0
$17$	−108.547	−	188.010i	−0.0910956	−	0.157782i	0.816877	−	0.576812i	$-0.195704\pi$
$17$	−108.547	−	188.010i	−0.0910956	−	0.157782i	−0.907972	+	0.419030i	$0.862370\pi$
$18$		0			0
$19$	−947.091	+	1640.41i	−0.601877	+	1.04248i	0.390660	+	0.920535i	$0.372247\pi$
$19$	−947.091	+	1640.41i	−0.601877	+	1.04248i	−0.992537	+	0.121946i	$0.961086\pi$
$20$		0			0
$21$	−642.194	+	1226.35i	−0.317774	+	0.606829i
$22$		0			0
$23$	−867.055	−	315.582i	−0.341765	−	0.124392i	0.165435	−	0.986221i	$-0.447097\pi$
$23$	−867.055	−	315.582i	−0.341765	−	0.124392i	−0.507200	+	0.861829i	$0.669319\pi$
$24$		0			0
$25$	−542.639	+	3077.46i	−0.173645	+	0.984787i
$26$		0			0
$27$	2283.99	+	3021.97i	0.602955	+	0.797775i
$28$		0			0
$29$	−1043.97	+	5920.66i	−0.230512	+	1.30730i	0.621349	+	0.783534i	$0.286585\pi$
$29$	−1043.97	+	5920.66i	−0.230512	+	1.30730i	−0.851862	+	0.523767i	$0.824526\pi$
$30$		0			0
$31$	−2498.77	−	909.479i	−0.467006	−	0.169976i	0.0977895	−	0.995207i	$-0.468823\pi$
$31$	−2498.77	−	909.479i	−0.467006	−	0.169976i	−0.564796	+	0.825231i	$0.691045\pi$
$32$		0			0
$33$	−8585.06	+	353.909i	−1.37233	+	0.0565727i
$34$		0			0
$35$	11.2728	−	19.5250i	0.00155547	−	0.00269414i
$36$		0			0
$37$	1682.27	+	2913.77i	0.202018	+	0.349906i	0.949179	−	0.314738i	$-0.101917\pi$
$37$	1682.27	+	2913.77i	0.202018	+	0.349906i	−0.747160	+	0.664644i	$0.768583\pi$
$38$		0			0
$39$	1292.95	+	997.119i	0.136119	+	0.104975i
$40$		0			0
$41$	1456.87	+	8262.34i	0.135351	+	0.767615i	0.974615	+	0.223890i	$0.0718755\pi$
$41$	1456.87	+	8262.34i	0.135351	+	0.767615i	−0.839263	+	0.543725i	$0.817013\pi$
$42$		0			0
$43$	7463.76	+	6262.84i	0.615583	+	0.516535i	0.896412	−	0.443223i	$-0.146165\pi$
$43$	7463.76	+	6262.84i	0.615583	+	0.516535i	−0.280829	+	0.959758i	$0.590609\pi$
$44$		0			0
$45$	−35.1391	−	50.7071i	−0.00258678	−	0.00373283i
$46$		0			0
$47$	10011.8	−	3643.99i	0.661099	−	0.240620i	0.0103884	−	0.999946i	$-0.496693\pi$
$47$	10011.8	−	3643.99i	0.661099	−	0.240620i	0.650711	+	0.759326i	$0.274471\pi$
$48$		0			0
$49$	−6833.73	+	5734.18i	−0.406600	+	0.341178i
$50$		0			0
$51$	3225.06	+	1025.49i	0.173625	+	0.0552085i
$52$		0			0
$53$	3578.00			0.174965			0.0874824	−	0.996166i	$-0.472118\pi$
$53$	3578.00			0.174965			0.0874824	+	0.996166i	$0.472118\pi$
$54$		0			0
$55$	139.938			0.00623776
$56$		0			0
$57$	−6320.74	−	28842.9i	−0.257680	−	1.17585i
$58$		0			0
$59$	27728.4	−	23266.9i	1.03704	−	0.870178i	0.0453664	−	0.998970i	$-0.485554\pi$
$59$	27728.4	−	23266.9i	1.03704	−	0.870178i	0.991672	+	0.128792i	$0.0411100\pi$
$60$		0			0
$61$	−46708.9	+	17000.6i	−1.60722	+	0.584980i	−0.980887	−	0.194577i	$-0.937667\pi$
$61$	−46708.9	+	17000.6i	−1.60722	+	0.584980i	−0.626331	+	0.779557i	$0.715444\pi$
$62$		0			0
$63$	−5483.69	−	20871.1i	−0.174069	−	0.662511i
$64$		0			0
$65$	−20.3706	−	17.0930i	−0.000598026	−	0.000501804i
$66$		0			0
$67$	4792.46	+	27179.4i	0.130428	+	0.739695i	0.977935	+	0.208910i	$0.0669916\pi$
$67$	4792.46	+	27179.4i	0.130428	+	0.739695i	−0.847507	+	0.530785i	$0.821897\pi$
$68$		0			0
$69$	13302.0	−	5471.98i	0.336351	−	0.138363i
$70$		0			0
$71$	10006.9	+	17332.5i	0.235589	+	0.408052i	0.959444	−	0.281900i	$-0.0909648\pi$
$71$	10006.9	+	17332.5i	0.235589	+	0.408052i	−0.723855	+	0.689952i	$0.757631\pi$
$72$		0			0
$73$	−7080.10	+	12263.1i	−0.155501	+	0.269335i	−0.933241	−	0.359250i	$-0.883032\pi$
$73$	−7080.10	+	12263.1i	−0.155501	+	0.269335i	0.777741	+	0.628585i	$0.216366\pi$
$74$		0			0
$75$	−26073.4	−	41147.6i	−0.535235	−	0.844679i
$76$		0			0
$77$	45996.9	+	16741.5i	0.884101	+	0.321786i
$78$		0			0
$79$	9295.20	−	52715.7i	0.167568	−	0.950326i	−0.778809	−	0.627261i	$-0.784176\pi$
$79$	9295.20	−	52715.7i	0.167568	−	0.950326i	0.946377	−	0.323064i	$-0.104713\pi$
$80$		0			0
$81$	−58049.4	−	10819.1i	−0.983071	−	0.183223i
$82$		0			0
$83$	5269.29	−	29883.6i	0.0839569	−	0.476143i	−0.913620	−	0.406569i	$-0.866725\pi$
$83$	5269.29	−	29883.6i	0.0839569	−	0.476143i	0.997577	−	0.0695740i	$-0.0221640\pi$
$84$		0			0
$85$	−51.7919	−	18.8507i	−0.000777525	−	0.000282996i
$86$		0			0
$87$	−50162.1	−	79163.1i	−0.710522	−	1.12131i
$88$		0			0
$89$	41357.3	−	71632.9i	0.553448	−	0.958600i	−0.444574	−	0.895742i	$-0.646645\pi$
$89$	41357.3	−	71632.9i	0.553448	−	0.958600i	0.998022	−	0.0628584i	$-0.0200216\pi$
$90$		0			0
$91$	−4650.80	−	8055.41i	−0.0588740	−	0.101973i
$92$		0			0
$93$	38335.0	−	15769.7i	0.459609	−	0.189068i
$94$		0			0
$95$	83.5061	+	473.587i	0.000949313	+	0.00538382i
$96$		0			0
$97$	85552.1	+	71786.7i	0.923212	+	0.774667i	0.974586	−	0.224013i	$-0.0719157\pi$
$97$	85552.1	+	71786.7i	0.923212	+	0.774667i	−0.0513744	+	0.998679i	$0.516360\pi$
$98$		0			0
$99$	95170.6	−	94249.0i	0.975922	−	0.966471i
$100$		0			0
$101$	88412.4	−	32179.5i	0.862402	−	0.313889i	0.127316	−	0.991862i	$-0.459364\pi$
$101$	88412.4	−	32179.5i	0.862402	−	0.313889i	0.735086	+	0.677974i	$0.237142\pi$
$102$		0			0
$103$	−81949.3	+	68763.6i	−0.761118	+	0.638654i	−0.938418	−	0.345503i	$-0.887708\pi$
$103$	−81949.3	+	68763.6i	−0.761118	+	0.638654i	0.177300	+	0.984157i	$0.443264\pi$
$104$		0			0
$105$	75.2327	+	343.303i	0.000665938	+	0.00303882i
$106$		0			0
$107$	181731.			1.53451			0.767256	−	0.641341i	$-0.221621\pi$
$107$	181731.			1.53451			0.767256	+	0.641341i	$0.221621\pi$
$108$		0			0
$109$	−168597.			−1.35920			−0.679602	−	0.733581i	$-0.737848\pi$
$109$	−168597.			−1.35920			−0.679602	+	0.733581i	$0.737848\pi$
$110$		0			0
$111$	−49982.0	−	15893.0i	−0.385040	−	0.122433i
$112$		0			0
$113$	−69295.0	+	58145.4i	−0.510512	+	0.428370i	−0.861309	−	0.508081i	$-0.830355\pi$
$113$	−69295.0	+	58145.4i	−0.510512	+	0.428370i	0.350797	+	0.936451i	$0.385911\pi$
$114$		0			0
$115$	−220.127	+	80.1196i	−0.00155213	+	0.000564930i
$116$		0			0
$117$	−25366.1	+	2094.93i	−0.171312	+	0.0141483i
$118$		0			0
$119$	−14768.5	−	12392.3i	−0.0956027	−	0.0802202i
$120$		0			0
$121$	24791.8	+	140601.i	0.153937	+	0.873022i
$122$		0			0
$123$	−103564.	−	79868.4i	−0.617228	−	0.476006i
$124$		0			0
$125$	793.363	+	1374.15i	0.00454148	+	0.00786607i
$126$		0			0
$127$	−108859.	+	188549.i	−0.598901	+	1.03733i	0.394083	+	0.919075i	$0.371062\pi$
$127$	−108859.	+	188549.i	−0.598901	+	1.03733i	−0.992984	+	0.118252i	$0.962271\pi$
$128$		0			0
$129$	−151753.	+	6255.85i	−0.802904	+	0.0330988i
$130$		0			0
$131$	190687.	+	69404.3i	0.970827	+	0.353352i	0.778267	−	0.627933i	$-0.216099\pi$
$131$	190687.	+	69404.3i	0.970827	+	0.353352i	0.192560	+	0.981285i	$0.438321\pi$
$132$		0			0
$133$	−29209.6	+	165656.i	−0.143185	+	0.812041i
$134$		0			0
$135$	937.352	+	214.995i	0.00442658	+	0.00101530i
$136$		0			0
$137$	−53656.6	+	304302.i	−0.244243	+	1.38517i	0.578001	+	0.816036i	$0.303833\pi$
$137$	−53656.6	+	304302.i	−0.244243	+	1.38517i	−0.822244	+	0.569135i	$0.807278\pi$
$138$		0			0
$139$	26011.7	+	9467.48i	0.114191	+	0.0415621i	0.398484	−	0.917175i	$-0.369537\pi$
$139$	26011.7	+	9467.48i	0.114191	+	0.0415621i	−0.284293	+	0.958738i	$0.591759\pi$
$140$		0			0
$141$	−77047.4	+	147132.i	−0.326370	+	0.623244i
$142$		0			0
$143$	28867.0	−	49999.2i	0.118049	−	0.204467i
$144$		0			0
$145$	763.160	+	1321.83i	0.00301436	+	0.00522103i
$146$		0			0
$147$	18486.5	−	137827.i	0.0705605	−	0.526068i
$148$		0			0
$149$	42788.7	+	242667.i	0.157893	+	0.895457i	0.956093	+	0.293064i	$0.0946748\pi$
$149$	42788.7	+	242667.i	0.157893	+	0.895457i	−0.798200	+	0.602393i	$0.794214\pi$
$150$		0			0
$151$	−90104.3	−	75606.5i	−0.321591	−	0.269847i	0.467672	−	0.883902i	$-0.345093\pi$
$151$	−90104.3	−	75606.5i	−0.321591	−	0.269847i	−0.789263	+	0.614055i	$0.789537\pi$
$152$		0			0
$153$	−47919.3	+	22061.9i	−0.165494	+	0.0761930i
$154$		0			0
$155$	−634.386	+	230.897i	−0.00212092	+	0.000771951i
$156$		0			0
$157$	−191266.	+	160491.i	−0.619282	+	0.519639i	−0.897578	−	0.440856i	$-0.854675\pi$
$157$	−191266.	+	160491.i	−0.619282	+	0.519639i	0.278296	+	0.960495i	$0.410231\pi$
$158$		0			0
$159$	−41213.5	+	37581.2i	−0.129285	+	0.117890i
$160$		0			0
$161$	−81939.8			−0.249132
$162$		0			0
$163$	62526.7			0.184330			0.0921651	−	0.995744i	$-0.470621\pi$
$163$	62526.7			0.184330			0.0921651	+	0.995744i	$0.470621\pi$
$164$		0			0
$165$	−1611.89	+	1469.83i	−0.00460920	+	0.00420297i
$166$		0			0
$167$	206028.	−	172878.i	0.571655	−	0.479676i	−0.310540	−	0.950560i	$-0.600510\pi$
$167$	206028.	−	172878.i	0.571655	−	0.479676i	0.882195	+	0.470885i	$0.156065\pi$
$168$		0			0
$169$	338592.	−	123237.i	0.911926	−	0.331914i
$170$		0			0
$171$	375755.	+	265841.i	0.982686	+	0.695234i
$172$		0			0
$173$	−457077.	−	383533.i	−1.16111	−	0.974288i	−0.161191	−	0.986923i	$-0.551534\pi$
$173$	−457077.	−	383533.i	−1.16111	−	0.974288i	−0.999920	+	0.0126353i	$0.995978\pi$
$174$		0			0
$175$	48188.7	+	273292.i	0.118946	+	0.674577i
$176$		0			0
$177$	−75010.5	+	559245.i	−0.179965	+	1.34174i
$178$		0			0
$179$	−89172.9	−	154452.i	−0.208018	−	0.360297i	0.743072	−	0.669211i	$-0.233368\pi$
$179$	−89172.9	−	154452.i	−0.208018	−	0.360297i	−0.951090	+	0.308914i	$0.900034\pi$
$180$		0			0
$181$	299957.	−	519541.i	0.680554	−	1.17875i	−0.294258	−	0.955726i	$-0.595073\pi$
$181$	299957.	−	519541.i	0.680554	−	1.17875i	0.974812	−	0.223028i	$-0.0715940\pi$
$182$		0			0
$183$	359456.	−	686427.i	0.793448	−	1.51519i
$184$		0			0
$185$	802.671	+	292.148i	0.00172428	+	0.000627587i
$186$		0			0
$187$	20779.2	−	117845.i	0.0434535	−	0.246437i
$188$		0			0
$189$	282382.	+	182808.i	0.575019	+	0.372255i
$190$		0			0
$191$	21965.7	−	124573.i	0.0435673	−	0.247082i	−0.955244	−	0.295818i	$-0.904408\pi$
$191$	21965.7	−	124573.i	0.0435673	−	0.247082i	0.998812	+	0.0487352i	$0.0155190\pi$
$192$		0			0
$193$	31376.9	+	11420.3i	0.0606341	+	0.0220690i	0.372159	−	0.928169i	$-0.378618\pi$
$193$	31376.9	+	11420.3i	0.0606341	+	0.0220690i	−0.311525	+	0.950238i	$0.600840\pi$
$194$		0			0
$195$	414.175	−	17.0739i	0.000780006	−	3.21548e-5i
$196$		0			0
$197$	173354.	−	300258.i	0.318250	−	0.551226i	−0.661873	−	0.749616i	$-0.730238\pi$
$197$	173354.	−	300258.i	0.318250	−	0.551226i	0.980123	+	0.198390i	$0.0635714\pi$
$198$		0			0
$199$	−456392.	−	790494.i	−0.816969	−	1.41503i	−0.907906	−	0.419175i	$-0.862320\pi$
$199$	−456392.	−	790494.i	−0.816969	−	1.41503i	0.0909370	−	0.995857i	$-0.471014\pi$
$200$		0			0
$201$	−340679.	−	262731.i	−0.594778	−	0.458693i
$202$		0			0
$203$	92709.3	+	525780.i	0.157900	+	0.895497i
$204$		0			0
$205$	1631.67	+	1369.13i	0.00271174	+	0.00227542i
$206$		0			0
$207$	−95745.5	+	202745.i	−0.155307	+	0.328871i
$208$		0			0
$209$	−981107.	+	357094.i	−1.55364	+	0.565479i
$210$		0			0
$211$	143275.	−	120222.i	0.221547	−	0.185900i	−0.525258	−	0.850943i	$-0.676031\pi$
$211$	143275.	−	120222.i	0.221547	−	0.185900i	0.746805	+	0.665043i	$0.231587\pi$
$212$		0			0
$213$	−297316.	−	94539.3i	−0.449024	−	0.142779i
$214$		0			0
$215$	2473.60			0.00364951
$216$		0			0
$217$	−236143.			−0.340428
$218$		0			0
$219$	−47251.5	−	215619.i	−0.0665742	−	0.303792i
$220$		0			0
$221$	−17419.1	+	14616.4i	−0.0239909	+	0.0201307i
$222$		0			0
$223$	1.28500e6	−	467701.i	1.73037	−	0.629805i	0.731717	−	0.681608i	$-0.238719\pi$
$223$	1.28500e6	−	467701.i	1.73037	−	0.629805i	0.998657	+	0.0518037i	$0.0164970\pi$
$224$		0			0
$225$	732520.	+	200103.i	0.964635	+	0.263510i
$226$		0			0
$227$	−23115.9	−	19396.6i	−0.0297746	−	0.0249839i	0.627779	−	0.778392i	$-0.283964\pi$
$227$	−23115.9	−	19396.6i	−0.0297746	−	0.0249839i	−0.657554	+	0.753408i	$0.728409\pi$
$228$		0			0
$229$	−40888.2	−	231889.i	−0.0515240	−	0.292207i	0.948148	−	0.317830i	$-0.102954\pi$
$229$	−40888.2	−	231889.i	−0.0515240	−	0.292207i	−0.999672	+	0.0256230i	$0.991843\pi$
$230$		0			0
$231$	−705663.	+	290286.i	−0.870097	+	0.357928i
$232$		0			0
$233$	−622944.	−	1.07897e6i	−0.751726	−	1.30203i	−0.946986	−	0.321276i	$-0.895888\pi$
$233$	−622944.	−	1.07897e6i	−0.751726	−	1.30203i	0.195260	−	0.980752i	$-0.437445\pi$
$234$		0			0
$235$	1352.45	−	2342.52i	0.00159754	−	0.00276702i
$236$		0			0
$237$	446628.	+	704843.i	0.516505	+	0.815120i
$238$		0			0
$239$	−405849.	−	147717.i	−0.459589	−	0.167277i	0.101841	−	0.994801i	$-0.467527\pi$
$239$	−405849.	−	147717.i	−0.459589	−	0.167277i	−0.561430	+	0.827524i	$0.689749\pi$
$240$		0			0
$241$	207289.	−	1.17559e6i	0.229897	−	1.30381i	−0.623201	−	0.782061i	$-0.714168\pi$
$241$	207289.	−	1.17559e6i	0.229897	−	1.30381i	0.853099	−	0.521750i	$-0.174721\pi$
$242$		0			0
$243$	782286.	−	485096.i	0.849864	−	0.527001i
$244$		0			0
$245$	−393.279	+	2230.39i	−0.000418587	+	0.00237392i
$246$		0			0
$247$	186436.	+	67857.3i	0.194441	+	0.0707708i
$248$		0			0
$249$	253185.	+	399563.i	0.258786	+	0.408401i
$250$		0			0
$251$	599430.	−	1.03824e6i	0.600557	−	1.04020i	−0.392179	−	0.919889i	$-0.628279\pi$
$251$	599430.	−	1.03824e6i	0.600557	−	1.04020i	0.992737	−	0.120307i	$-0.0383879\pi$
$252$		0			0
$253$	−254296.	−	440454.i	−0.249769	−	0.432613i
$254$		0			0
$255$	794.567	−	326.858i	0.000765209	−	0.000314781i
$256$		0			0
$257$	119504.	+	677742.i	0.112863	+	0.640077i	0.987786	+	0.155815i	$0.0498005\pi$
$257$	119504.	+	677742.i	0.112863	+	0.640077i	−0.874923	+	0.484261i	$0.839088\pi$
$258$		0			0
$259$	228883.	+	192055.i	0.212014	+	0.177901i
$260$		0			0
$261$	1.40928e6	+	384974.i	1.28055	+	0.349809i
$262$		0			0
$263$	−1.47548e6	+	537031.i	−1.31536	+	0.478752i	−0.901968	−	0.431802i	$-0.857878\pi$
$263$	−1.47548e6	+	537031.i	−1.31536	+	0.478752i	−0.413391	+	0.910554i	$0.635656\pi$
$264$		0			0
$265$	695.858	−	583.894i	0.000608704	−	0.000510763i
$266$		0			0
$267$	276012.	+	1.25950e6i	0.236947	+	1.08124i
$268$		0			0
$269$	342991.			0.289003			0.144501	−	0.989505i	$-0.453842\pi$
$269$	342991.			0.289003			0.144501	+	0.989505i	$0.453842\pi$
$270$		0			0
$271$	−1.68338e6			−1.39239			−0.696193	−	0.717855i	$-0.745124\pi$
$271$	−1.68338e6			−1.39239			−0.696193	+	0.717855i	$0.745124\pi$
$272$		0			0
$273$	138180.	+	43937.9i	0.112212	+	0.0356806i
$274$		0			0
$275$	−1.31948e6	+	1.10718e6i	−1.05214	+	0.882848i
$276$		0			0
$277$	2.23433e6	−	813231.i	1.74964	−	0.636817i	0.749944	−	0.661502i	$-0.230081\pi$
$277$	2.23433e6	−	813231.i	1.74964	−	0.636817i	0.999695	+	0.0246850i	$0.00785828\pi$
$278$		0			0
$279$	−275930.	+	584294.i	−0.212221	+	0.449387i
$280$		0			0
$281$	−1.42716e6	−	1.19753e6i	−1.07822	−	0.904733i	−0.0824470	−	0.996595i	$-0.526274\pi$
$281$	−1.42716e6	−	1.19753e6i	−1.07822	−	0.904733i	−0.995772	+	0.0918626i	$0.970718\pi$
$282$		0			0
$283$	332163.	+	1.88379e6i	0.246539	+	1.39819i	0.816891	+	0.576792i	$0.195696\pi$
$283$	332163.	+	1.88379e6i	0.246539	+	1.39819i	−0.570352	+	0.821401i	$0.693193\pi$
$284$		0			0
$285$	−5936.15	−	4577.96i	−0.00432905	−	0.00333856i
$286$		0			0
$287$	372525.	+	645232.i	0.266963	+	0.462393i
$288$		0			0
$289$	686363.	−	1.18882e6i	0.483403	−	0.837279i
$290$		0			0
$291$	−1.73945e6	+	71706.6i	−1.20414	+	0.0496394i
$292$		0			0
$293$	646916.	+	235458.i	0.440229	+	0.160230i	0.552619	−	0.833434i	$-0.313628\pi$
$293$	646916.	+	235458.i	0.440229	+	0.160230i	−0.112390	+	0.993664i	$0.535851\pi$
$294$		0			0
$295$	1595.76	−	9050.00i	0.00106761	−	0.00605471i
$296$		0			0
$297$	−106296.	+	2.08523e6i	−0.0699239	+	1.37171i
$298$		0			0
$299$	−16782.4	+	95177.8i	−0.0108562	+	0.0615684i
$300$		0			0
$301$	813061.	+	295930.i	0.517258	+	0.188266i
$302$		0			0
$303$	−680393.	+	1.29929e6i	−0.425748	+	0.813020i
$304$		0			0
$305$	−6309.72	+	10928.8i	−0.00388383	+	0.00672700i
$306$		0			0
$307$	−918696.	−	1.59123e6i	−0.556322	−	0.963577i	−0.997799	−	0.0663052i	$-0.978879\pi$
$307$	−918696.	−	1.59123e6i	−0.556322	−	0.963577i	0.441478	−	0.897272i	$-0.354454\pi$
$308$		0			0
$309$	221688.	−	1.65281e6i	0.132083	−	0.984751i
$310$		0			0
$311$	486261.	+	2.75772e6i	0.285081	+	1.61678i	0.704996	+	0.709212i	$0.250949\pi$
$311$	486261.	+	2.75772e6i	0.285081	+	1.61678i	−0.419915	+	0.907564i	$0.637940\pi$
$312$		0			0
$313$	905717.	+	759987.i	0.522555	+	0.438476i	0.865521	−	0.500872i	$-0.166987\pi$
$313$	905717.	+	759987.i	0.522555	+	0.438476i	−0.342967	+	0.939348i	$0.611432\pi$
$314$		0			0
$315$	−4472.43	−	3164.17i	−0.00253961	−	0.00179673i
$316$		0			0
$317$	−2.22573e6	+	810099.i	−1.24401	+	0.452783i	−0.878373	−	0.477975i	$-0.841371\pi$
$317$	−2.22573e6	+	810099.i	−1.24401	+	0.452783i	−0.365637	+	0.930758i	$0.619149\pi$
$318$		0			0
$319$	−2.53853e6	+	2.13008e6i	−1.39671	+	1.17198i
$320$		0			0
$321$	−2.09329e6	+	1.90880e6i	−1.13388	+	1.03395i
$322$		0			0
$323$	411217.			0.219313
$324$		0			0
$325$	327314.			0.171892
$326$		0			0
$327$	1.94201e6	−	1.77085e6i	1.00434	−	0.915825i
$328$		0			0
$329$	724792.	−	608173.i	0.369168	−	0.309768i
$330$		0			0
$331$	−2.32024e6	+	844498.i	−1.16403	+	0.423671i	−0.850534	−	0.525920i	$-0.823721\pi$
$331$	−2.32024e6	+	844498.i	−1.16403	+	0.423671i	−0.313492	+	0.949591i	$0.601499\pi$
$332$		0			0
$333$	742654.	−	341916.i	0.367008	−	0.168970i
$334$		0			0
$335$	5367.46	+	4503.83i	0.00261310	+	0.00219265i
$336$		0			0
$337$	335804.	+	1.90444e6i	0.161069	+	0.913466i	0.953026	+	0.302889i	$0.0979511\pi$
$337$	335804.	+	1.90444e6i	0.161069	+	0.913466i	−0.791957	+	0.610577i	$0.790938\pi$
$338$		0			0
$339$	187456.	−	1.39759e6i	0.0885932	−	0.660511i
$340$		0			0
$341$	−732858.	−	1.26935e6i	−0.341298	−	0.591146i
$342$		0			0
$343$	−1.14237e6	+	1.97864e6i	−0.524290	+	0.908096i
$344$		0			0
$345$	1694.02	−	3234.95i	0.000766252	−	0.00146325i
$346$		0			0
$347$	3.19377e6	+	1.16244e6i	1.42390	+	0.518257i	0.935176	−	0.354182i	$-0.115241\pi$
$347$	3.19377e6	+	1.16244e6i	1.42390	+	0.518257i	0.488723	+	0.872439i	$0.337463\pi$
$348$		0			0
$349$	596968.	−	3.38557e6i	0.262354	−	1.48788i	−0.514111	−	0.857723i	$-0.671878\pi$
$349$	596968.	−	3.38557e6i	0.262354	−	1.48788i	0.776465	−	0.630160i	$-0.217011\pi$
$350$		0			0
$351$	270178.	−	290561.i	0.117053	−	0.125884i
$352$		0			0
$353$	167889.	−	952147.i	0.0717110	−	0.406693i	−0.927730	−	0.373253i	$-0.878242\pi$
$353$	167889.	−	952147.i	0.0717110	−	0.406693i	0.999441	−	0.0334406i	$-0.0106465\pi$
$354$		0			0
$355$	4774.66	+	1737.84i	0.00201081	+	0.000731877i
$356$		0			0
$357$	300274.	−	12378.4i	0.124695	−	0.00514038i
$358$		0			0
$359$	1.37918e6	−	2.38881e6i	0.564787	−	0.978240i	−0.432282	−	0.901738i	$-0.642292\pi$
$359$	1.37918e6	−	2.38881e6i	0.564787	−	0.978240i	0.997069	−	0.0765016i	$-0.0243750\pi$
$360$		0			0
$361$	−555913.	−	962869.i	−0.224511	−	0.388865i
$362$		0			0
$363$	−1.76236e6	−	1.35913e6i	−0.701985	−	0.541370i
$364$		0			0
$365$	624.261	+	3540.36i	0.000245264	+	0.00139096i
$366$		0			0
$367$	2.25816e6	+	1.89482e6i	0.875164	+	0.734349i	0.965179	−	0.261591i	$-0.0842471\pi$
$367$	2.25816e6	+	1.89482e6i	0.875164	+	0.734349i	−0.0900153	+	0.995940i	$0.528692\pi$
$368$		0			0
$369$	2.03180e6	−	167803.i	0.776812	−	0.0641553i
$370$		0			0
$371$	298579.	−	108674.i	0.112622	−	0.0409912i
$372$		0			0
$373$	810476.	−	680070.i	0.301626	−	0.253094i	−0.479395	−	0.877599i	$-0.659144\pi$
$373$	810476.	−	680070.i	0.301626	−	0.253094i	0.781020	+	0.624505i	$0.214699\pi$
$374$		0			0
$375$	−23571.7	−	7495.21i	−0.00865589	−	0.00275236i
$376$		0			0
$377$	629713.			0.228186
$378$		0			0
$379$	−1.20424e6			−0.430640			−0.215320	−	0.976544i	$-0.569080\pi$
$379$	−1.20424e6			−0.430640			−0.215320	+	0.976544i	$0.569080\pi$
$380$		0			0
$381$	−726508.	−	3.31521e6i	−0.256406	−	1.17004i
$382$		0			0
$383$	3.59166e6	−	3.01376e6i	1.25112	−	1.04981i	0.254548	−	0.967060i	$-0.418073\pi$
$383$	3.59166e6	−	3.01376e6i	1.25112	−	1.04981i	0.996570	−	0.0827528i	$-0.0263712\pi$
$384$		0			0
$385$	11677.6	−	4250.31i	0.00401516	−	0.00146140i
$386$		0			0
$387$	1.68228e6	−	1.66599e6i	0.570979	−	0.565450i
$388$		0			0
$389$	995287.	+	835145.i	0.333483	+	0.279826i	0.794117	−	0.607764i	$-0.207933\pi$
$389$	995287.	+	835145.i	0.333483	+	0.279826i	−0.460634	+	0.887590i	$0.652378\pi$
$390$		0			0
$391$	34784.1	+	197270.i	0.0115064	+	0.0652559i
$392$		0			0
$393$	−2.92543e6	+	1.20342e6i	−0.955449	+	0.393040i
$394$		0			0
$395$	−6794.93	−	11769.2i	−0.00219125	−	0.00379536i
$396$		0			0
$397$	1.14806e6	−	1.98850e6i	0.365585	−	0.633212i	−0.623285	−	0.781995i	$-0.714202\pi$
$397$	1.14806e6	−	1.98850e6i	0.365585	−	0.633212i	0.988870	+	0.148783i	$0.0475356\pi$
$398$		0			0
$399$	−1.40350e6	−	2.21493e6i	−0.441347	−	0.696509i
$400$		0			0
$401$	−3.37532e6	−	1.22851e6i	−1.04822	−	0.381522i	−0.240231	−	0.970716i	$-0.577223\pi$
$401$	−3.37532e6	−	1.22851e6i	−1.04822	−	0.381522i	−0.807992	+	0.589194i	$0.799445\pi$
$402$		0			0
$403$	−48365.4	+	274294.i	−0.0148345	+	0.0841305i
$404$		0			0
$405$	−13055.2	+	7368.96i	−0.00395498	+	0.00223238i
$406$		0			0
$407$	−322036.	+	1.82636e6i	−0.0963648	+	0.546512i
$408$		0			0
$409$	−1.16992e6	−	425818.i	−0.345820	−	0.125868i	0.163270	−	0.986581i	$-0.447796\pi$
$409$	−1.16992e6	−	425818.i	−0.345820	−	0.125868i	−0.509090	+	0.860713i	$0.670018\pi$
$410$		0			0
$411$	−2.57816e6	−	4.06871e6i	−0.752845	−	1.18810i
$412$		0			0
$413$	1.60722e6	−	2.78378e6i	0.463660	−	0.803082i
$414$		0			0
$415$	−3851.93	−	6671.73i	−0.00109789	−	0.00190160i
$416$		0			0
$417$	−399059.	+	164160.i	−0.112382	+	0.0462302i
$418$		0			0
$419$	937638.	+	5.31761e6i	0.260916	+	1.47973i	0.780418	+	0.625258i	$0.215006\pi$
$419$	937638.	+	5.31761e6i	0.260916	+	1.47973i	−0.519502	+	0.854469i	$0.673883\pi$
$420$		0			0
$421$	1.57102e6	+	1.31824e6i	0.431992	+	0.362485i	0.832703	−	0.553720i	$-0.186792\pi$
$421$	1.57102e6	+	1.31824e6i	0.431992	+	0.362485i	−0.400711	+	0.916205i	$0.631237\pi$
$422$		0			0
$423$	−657907.	−	2.50401e6i	−0.178778	−	0.680433i
$424$		0			0
$425$	637494.	−	232029.i	0.171200	−	0.0623118i
$426$		0			0
$427$	−3.38144e6	+	2.83736e6i	−0.897495	+	0.753087i
$428$		0			0
$429$	192654.	+	879123.i	0.0505400	+	0.230625i
$430$		0			0
$431$	5.60953e6			1.45456			0.727282	−	0.686338i	$-0.240783\pi$
$431$	5.60953e6			1.45456			0.727282	+	0.686338i	$0.240783\pi$
$432$		0			0
$433$	−4.68786e6			−1.20159			−0.600793	−	0.799404i	$-0.705149\pi$
$433$	−4.68786e6			−1.20159			−0.600793	+	0.799404i	$0.705149\pi$
$434$		0			0
$435$	−22674.3	−	7209.87i	−0.00574527	−	0.00182685i
$436$		0			0
$437$	1.33886e6	−	1.12344e6i	0.335377	−	0.281414i
$438$		0			0
$439$	2.55514e6	−	929994.i	0.632780	−	0.230313i	−0.00566080	−	0.999984i	$-0.501802\pi$
$439$	2.55514e6	−	929994.i	0.632780	−	0.230313i	0.638441	+	0.769671i	$0.279580\pi$
$440$		0			0
$441$	1.23472e6	+	1.78175e6i	0.302323	+	0.436265i
$442$		0			0
$443$	12967.8	+	10881.3i	0.00313948	+	0.00263434i	0.644356	−	0.764726i	$-0.277126\pi$
$443$	12967.8	+	10881.3i	0.00313948	+	0.00263434i	−0.641217	+	0.767360i	$0.721570\pi$
$444$		0			0
$445$	−3646.52	−	20680.4i	−0.000872928	−	0.00495062i
$446$		0			0
$447$	−3.04170e6	−	2.34575e6i	−0.720024	−	0.555282i
$448$		0			0
$449$	−1.80255e6	−	3.12211e6i	−0.421961	−	0.730858i	0.574170	−	0.818736i	$-0.305325\pi$
$449$	−1.80255e6	−	3.12211e6i	−0.421961	−	0.730858i	−0.996131	+	0.0878780i	$0.971991\pi$
$450$		0			0
$451$	−2.31223e6	+	4.00489e6i	−0.535290	+	0.927149i
$452$		0			0
$453$	1.83200e6	−	75522.1i	0.419450	−	0.0172913i
$454$		0			0
$455$	−2219.06	−	807.673i	−0.000502506	−	0.000182897i
$456$		0			0
$457$	−1.16577e6	+	6.61143e6i	−0.261110	+	1.48083i	0.518778	+	0.854909i	$0.326387\pi$
$457$	−1.16577e6	+	6.61143e6i	−0.261110	+	1.48083i	−0.779888	+	0.625919i	$0.784724\pi$
$458$		0			0
$459$	320238.	−	757439.i	0.0709481	−	0.167809i
$460$		0			0
$461$	351936.	−	1.99593e6i	0.0771278	−	0.437414i	−0.921651	−	0.388019i	$-0.873160\pi$
$461$	351936.	−	1.99593e6i	0.0771278	−	0.437414i	0.998779	−	0.0493950i	$-0.0157293\pi$
$462$		0			0
$463$	−1.50784e6	−	548808.i	−0.326891	−	0.118978i	0.173362	−	0.984858i	$-0.444537\pi$
$463$	−1.50784e6	−	548808.i	−0.326891	−	0.118978i	−0.500252	+	0.865880i	$0.666759\pi$
$464$		0			0
$465$	4882.02	−	9322.83i	0.00104705	−	0.00199947i
$466$		0			0
$467$	875023.	−	1.51558e6i	0.185664	−	0.321579i	−0.758136	−	0.652096i	$-0.773890\pi$
$467$	875023.	−	1.51558e6i	0.185664	−	0.321579i	0.943800	+	0.330517i	$0.107223\pi$
$468$		0			0
$469$	1.22544e6	+	2.12252e6i	0.257253	+	0.445575i
$470$		0			0
$471$	517410.	−	3.85758e6i	0.107469	−	0.801240i
$472$		0			0
$473$	932573.	+	5.28888e6i	0.191659	+	1.08695i
$474$		0			0
$475$	−4.53437e6	−	3.80479e6i	−0.922110	−	0.773742i
$476$		0			0
$477$	79991.2	−	865766.i	0.0160970	−	0.174223i
$478$		0			0
$479$	927560.	−	337604.i	0.184715	−	0.0672309i	−0.248006	−	0.968758i	$-0.579775\pi$
$479$	927560.	−	337604.i	0.184715	−	0.0672309i	0.432722	+	0.901527i	$0.357553\pi$
$480$		0			0
$481$	269962.	−	226525.i	0.0532035	−	0.0446430i
$482$		0			0
$483$	943832.	−	860648.i	0.184089	−	0.167864i
$484$		0			0
$485$	28353.3			0.00547329
$486$		0			0
$487$	1.84095e6			0.351738			0.175869	−	0.984414i	$-0.443726\pi$
$487$	1.84095e6			0.351738			0.175869	+	0.984414i	$0.443726\pi$
$488$		0			0
$489$	−720220.	+	656744.i	−0.136205	+	0.124201i
$490$		0			0
$491$	3.74594e6	−	3.14322e6i	0.701224	−	0.588397i	−0.220897	−	0.975297i	$-0.570899\pi$
$491$	3.74594e6	−	3.14322e6i	0.701224	−	0.588397i	0.922122	+	0.386900i	$0.126454\pi$
$492$		0			0
$493$	1.22646e6	−	446396.i	0.227267	−	0.0827186i
$494$		0			0
$495$	3128.51	−	33860.7i	0.000573884	−	0.00621131i
$496$		0			0
$497$	1.36150e6	+	1.14244e6i	0.247245	+	0.207463i
$498$		0			0
$499$	1.42573e6	+	8.08573e6i	0.256323	+	1.45368i	0.792655	+	0.609671i	$0.208698\pi$
$499$	1.42573e6	+	8.08573e6i	0.256323	+	1.45368i	−0.536332	+	0.844007i	$0.680191\pi$
$500$		0			0
$501$	−557343.	+	4.15530e6i	−0.0992038	+	0.739620i
$502$		0			0
$503$	4.96453e6	+	8.59881e6i	0.874899	+	1.51537i	0.856871	+	0.515532i	$0.172405\pi$
$503$	4.96453e6	+	8.59881e6i	0.874899	+	1.51537i	0.0180281	+	0.999837i	$0.494261\pi$
$504$		0			0
$505$	11943.3	−	20686.4i	0.00208399	−	0.00360957i
$506$		0			0
$507$	−2.60569e6	+	4.97589e6i	−0.450198	+	0.859709i
$508$		0			0
$509$	−1.76798e6	−	643491.i	−0.302470	−	0.110090i	0.186327	−	0.982488i	$-0.440342\pi$
$509$	−1.76798e6	−	643491.i	−0.302470	−	0.110090i	−0.488796	+	0.872398i	$0.662564\pi$
$510$		0			0
$511$	−218360.	+	1.23838e6i	−0.0369932	+	0.209799i
$512$		0			0
$513$	−7.12041e6	+	884603.i	−1.19457	+	0.148407i
$514$		0			0
$515$	−4716.15	+	26746.6i	−0.000783555	+	0.00444376i
$516$		0			0
$517$	5.51849e6	+	2.00857e6i	0.908017	+	0.330491i
$518$		0			0
$519$	9.29329e6	−	383105.i	1.51444	−	0.0624308i
$520$		0			0
$521$	−5.69732e6	+	9.86804e6i	−0.919551	+	1.59271i	−0.119454	+	0.992840i	$0.538114\pi$
$521$	−5.69732e6	+	9.86804e6i	−0.919551	+	1.59271i	−0.800098	+	0.599870i	$0.795219\pi$
$522$		0			0
$523$	−1.57313e6	−	2.72474e6i	−0.251484	−	0.435584i	0.712450	−	0.701723i	$-0.247585\pi$
$523$	−1.57313e6	−	2.72474e6i	−0.251484	−	0.435584i	−0.963935	+	0.266139i	$0.914252\pi$
$524$		0			0
$525$	−3.42556e6	−	2.64179e6i	−0.542418	−	0.418312i
$526$		0			0
$527$	100245.	+	568515.i	0.0157230	+	0.0891693i
$528$		0			0
$529$	−4.27833e6	−	3.58995e6i	−0.664715	−	0.557762i
$530$		0			0
$531$	−5.00997e6	−	7.22959e6i	−0.771078	−	1.11270i
$532$		0			0
$533$	825773.	−	300557.i	0.125905	−	0.0458256i
$534$		0			0
$535$	35343.6	−	29656.8i	0.00533858	−	0.00447960i
$536$		0			0
$537$	2.64942e6	+	842450.i	0.396474	+	0.126069i
$538$		0			0
$539$	−4.91714e6			−0.729022
$540$		0			0
$541$	3.15282e6			0.463133			0.231566	−	0.972819i	$-0.425615\pi$
$541$	3.15282e6			0.463133			0.231566	+	0.972819i	$0.425615\pi$
$542$		0			0
$543$	2.00187e6	+	9.13495e6i	0.291364	+	1.32956i
$544$		0			0
$545$	−32789.2	+	27513.4i	−0.00472868	+	0.00396784i
$546$		0			0
$547$	−272931.	+	99338.9i	−0.0390018	+	0.0141955i	−0.361447	−	0.932392i	$-0.617717\pi$
$547$	−272931.	+	99338.9i	−0.0390018	+	0.0141955i	0.322446	+	0.946588i	$0.395495\pi$
$548$		0			0
$549$	3.06939e6	+	1.16822e7i	0.434632	+	1.65422i
$550$		0			0
$551$	−8.72358e6	−	7.31995e6i	−1.22410	−	1.02714i
$552$		0			0
$553$	−825454.	−	4.68138e6i	−0.114784	−	0.650970i
$554$		0			0
$555$	−12314.2	+	5065.65i	−0.00169697	+	0.000698076i
$556$		0			0
$557$	−1.84551e6	−	3.19652e6i	−0.252045	−	0.436555i	0.712044	−	0.702135i	$-0.247770\pi$
$557$	−1.84551e6	−	3.19652e6i	−0.252045	−	0.436555i	−0.964089	+	0.265580i	$0.914437\pi$
$558$		0			0
$559$	510266.	−	883807.i	0.0690665	−	0.119627i
$560$		0			0
$561$	998425.	+	1.57566e6i	0.133939	+	0.211376i
$562$		0			0
$563$	−7.85921e6	−	2.86052e6i	−1.04498	−	0.380341i	−0.238214	−	0.971213i	$-0.576562\pi$
$563$	−7.85921e6	−	2.86052e6i	−1.04498	−	0.380341i	−0.806766	+	0.590871i	$0.798784\pi$
$564$		0			0
$565$	−3987.90	+	22616.5i	−0.000525562	+	0.00298061i
$566$		0			0
$567$	−5.17275e6	+	860282.i	−0.675716	+	0.112379i
$568$		0			0
$569$	−1.99490e6	+	1.13136e7i	−0.258309	+	1.46494i	0.529125	+	0.848544i	$0.322520\pi$
$569$	−1.99490e6	+	1.13136e7i	−0.258309	+	1.46494i	−0.787434	+	0.616399i	$0.788591\pi$
$570$		0			0
$571$	−1.30335e6	−	474380.i	−0.167290	−	0.0608886i	0.257018	−	0.966407i	$-0.417260\pi$
$571$	−1.30335e6	−	474380.i	−0.167290	−	0.0608886i	−0.424307	+	0.905518i	$0.639482\pi$
$572$		0			0
$573$	1.05543e6	+	1.66563e6i	0.134290	+	0.211929i
$574$		0			0
$575$	1.44169e6	−	2.49708e6i	0.181845	−	0.314965i
$576$		0			0
$577$	2.93626e6	+	5.08575e6i	0.367159	+	0.635939i	0.989120	−	0.147110i	$-0.0469970\pi$
$577$	2.93626e6	+	5.08575e6i	0.367159	+	0.635939i	−0.621961	+	0.783048i	$0.713664\pi$
$578$		0			0
$579$	−481370.	+	198019.i	−0.0596736	+	0.0245477i
$580$		0			0
$581$	−467935.	−	2.65379e6i	−0.0575103	−	0.326157i
$582$		0			0
$583$	1.51079e6	+	1.26770e6i	0.184091	+	0.154470i
$584$		0			0
$585$	−4591.39	+	4546.92i	−0.000554695	+	0.000549323i
$586$		0			0
$587$	1.37156e7	−	4.99206e6i	1.64293	−	0.597977i	0.655381	−	0.755299i	$-0.272508\pi$
$587$	1.37156e7	−	4.99206e6i	1.64293	−	0.597977i	0.987547	+	0.157322i	$0.0502859\pi$
$588$		0			0
$589$	3.85848e6	−	3.23765e6i	0.458277	−	0.384540i
$590$		0			0
$591$	1.15694e6	+	5.27937e6i	0.136252	+	0.621747i
$592$		0			0
$593$	−6.94239e6			−0.810722			−0.405361	−	0.914157i	$-0.632854\pi$
$593$	−6.94239e6			−0.810722			−0.405361	+	0.914157i	$0.632854\pi$
$594$		0			0
$595$	−4894.52			−0.000566784
$596$		0			0
$597$	1.35599e7	+	4.31171e6i	1.55711	+	0.495124i
$598$		0			0
$599$	−2.73823e6	+	2.29765e6i	−0.311820	+	0.261648i	−0.785244	−	0.619187i	$-0.787462\pi$
$599$	−2.73823e6	+	2.29765e6i	−0.311820	+	0.261648i	0.473424	+	0.880835i	$0.343018\pi$
$600$		0			0
$601$	−2.34073e6	+	851955.i	−0.264341	+	0.0962123i	−0.470791	−	0.882245i	$-0.656031\pi$
$601$	−2.34073e6	+	851955.i	−0.264341	+	0.0962123i	0.206450	+	0.978457i	$0.433809\pi$
$602$		0			0
$603$	6.68372e6	−	551995.i	0.748557	−	0.0618218i
$604$		0			0
$605$	27766.3	+	23298.7i	0.00308411	+	0.00258787i
$606$		0			0
$607$	−2.50350e6	−	1.41981e7i	−0.275788	−	1.56407i	−0.736447	−	0.676496i	$-0.763498\pi$
$607$	−2.50350e6	−	1.41981e7i	−0.275788	−	1.56407i	0.460658	−	0.887578i	$-0.347613\pi$
$608$		0			0
$609$	−6.59037e6	−	5.08249e6i	−0.720057	−	0.555307i
$610$		0			0
$611$	−557980.	−	966450.i	−0.0604666	−	0.104731i
$612$		0			0
$613$	3.04813e6	−	5.27951e6i	0.327629	−	0.567469i	−0.654412	−	0.756138i	$-0.727084\pi$
$613$	3.04813e6	−	5.27951e6i	0.327629	−	0.567469i	0.982041	+	0.188669i	$0.0604172\pi$
$614$		0			0
$615$	−33175.1	+	1367.60i	−0.00353691	+	0.000145805i
$616$		0			0
$617$	−1.22405e7	−	4.45519e6i	−1.29446	−	0.471143i	−0.399268	−	0.916834i	$-0.630736\pi$
$617$	−1.22405e7	−	4.45519e6i	−1.29446	−	0.471143i	−0.895187	+	0.445691i	$0.852958\pi$
$618$		0			0
$619$	925577.	−	5.24921e6i	0.0970926	−	0.550640i	−0.896993	−	0.442044i	$-0.854254\pi$
$619$	925577.	−	5.24921e6i	0.0970926	−	0.550640i	0.994086	−	0.108596i	$-0.0346354\pi$
$620$		0			0
$621$	−1.02667e6	−	3.34100e6i	−0.106832	−	0.347654i
$622$		0			0
$623$	1.27552e6	−	7.23381e6i	0.131664	−	0.746702i
$624$		0			0
$625$	−9.17612e6	−	3.33983e6i	−0.939634	−	0.341999i
$626$		0			0
$627$	7.55028e6	−	1.44182e7i	0.766998	−	1.46468i
$628$		0			0
$629$	365212.	−	632566.i	0.0368060	−	0.0637498i
$630$		0			0
$631$	1.03718e6	+	1.79644e6i	0.103700	+	0.179614i	0.913206	−	0.407497i	$-0.133598\pi$
$631$	1.03718e6	+	1.79644e6i	0.103700	+	0.179614i	−0.809506	+	0.587111i	$0.800265\pi$
$632$		0			0
$633$	−387586.	+	2.88967e6i	−0.0384467	+	0.286642i
$634$		0			0
$635$	9598.22	+	54434.2i	0.000944619	+	0.00535720i
$636$		0			0
$637$	715783.	+	600613.i	0.0698928	+	0.0586470i
$638$		0			0
$639$	4.41765e6	−	2.03388e6i	0.427996	−	0.197048i
$640$		0			0
$641$	−9.72173e6	+	3.53842e6i	−0.934541	+	0.340145i	−0.764008	−	0.645207i	$-0.776771\pi$
$641$	−9.72173e6	+	3.53842e6i	−0.934541	+	0.340145i	−0.170533	+	0.985352i	$0.554549\pi$
$642$		0			0
$643$	1.55740e7	−	1.30681e7i	1.48550	−	1.24648i	0.585438	−	0.810717i	$-0.300923\pi$
$643$	1.55740e7	−	1.30681e7i	1.48550	−	1.24648i	0.900061	−	0.435764i	$-0.143522\pi$
$644$		0			0
$645$	−28492.5	+	25981.3i	−0.00269669	+	0.00245902i
$646$		0			0
$647$	1.27026e6			0.119297			0.0596487	−	0.998219i	$-0.481002\pi$
$647$	1.27026e6			0.119297			0.0596487	+	0.998219i	$0.481002\pi$
$648$		0			0
$649$	1.99517e7			1.85938
$650$		0			0
$651$	2.72004e6	−	2.48031e6i	0.251549	−	0.229379i
$652$		0			0
$653$	1.64318e7	−	1.37879e7i	1.50800	−	1.26537i	0.640475	−	0.767979i	$-0.278737\pi$
$653$	1.64318e7	−	1.37879e7i	1.50800	−	1.26537i	0.867529	−	0.497387i	$-0.165707\pi$
$654$		0			0
$655$	48411.3	−	17620.3i	0.00440903	−	0.00160476i
$656$		0			0
$657$	2.80901e6	+	1.98733e6i	0.253886	+	0.179620i
$658$		0			0
$659$	1.49399e7	+	1.25361e7i	1.34009	+	1.12447i	0.981602	+	0.190939i	$0.0611532\pi$
$659$	1.49399e7	+	1.25361e7i	1.34009	+	1.12447i	0.358491	+	0.933533i	$0.383291\pi$
$660$		0			0
$661$	−3.49704e6	−	1.98327e7i	−0.311312	−	1.76554i	−0.592192	−	0.805797i	$-0.701737\pi$
$661$	−3.49704e6	−	1.98327e7i	−0.311312	−	1.76554i	0.280879	−	0.959743i	$-0.409374\pi$
$662$		0			0
$663$	47122.1	−	351321.i	0.00416333	−	0.0310399i
$664$		0			0
$665$	21352.7	+	36983.9i	0.00187240	+	0.00324309i
$666$		0			0
$667$	2.77364e6	−	4.80408e6i	0.241399	−	0.418115i
$668$		0			0
$669$	−9.88892e6	+	1.88841e7i	−0.854247	+	1.63129i
$670$		0			0
$671$	−2.57459e7	−	9.37075e6i	−2.20751	−	0.803467i
$672$		0			0
$673$	−396995.	+	2.25147e6i	−0.0337868	+	0.191614i	−0.997030	−	0.0770173i	$-0.975460\pi$
$673$	−396995.	+	2.25147e6i	−0.0337868	+	0.191614i	0.963243	+	0.268632i	$0.0865715\pi$
$674$		0			0
$675$	−1.05394e7	+	5.38905e6i	−0.890339	+	0.455253i
$676$		0			0
$677$	709242.	−	4.02231e6i	0.0594734	−	0.337291i	−0.940524	−	0.339728i	$-0.889665\pi$
$677$	709242.	−	4.02231e6i	0.0594734	−	0.337291i	0.999997	+	0.00243780i	$0.000775976\pi$
$678$		0			0
$679$	9.31958e6	+	3.39205e6i	0.775750	+	0.282350i
$680$		0			0
$681$	469993.	−	19374.9i	0.0388351	−	0.00160093i
$682$		0			0
$683$	−8.77130e6	+	1.51923e7i	−0.719470	+	1.24616i	0.241741	+	0.970341i	$0.422282\pi$
$683$	−8.77130e6	+	1.51923e7i	−0.719470	+	1.24616i	−0.961210	+	0.275817i	$0.911052\pi$
$684$		0			0
$685$	39223.8	+	67937.6i	0.00319391	+	0.00553202i
$686$		0			0
$687$	2.90660e6	+	2.24157e6i	0.234960	+	0.181201i
$688$		0			0
$689$	−65077.9	−	369075.i	−0.00522259	−	0.0296188i
$690$		0			0
$691$	−3.95204e6	−	3.31616e6i	−0.314867	−	0.264205i	0.471633	−	0.881795i	$-0.343665\pi$
$691$	−3.95204e6	−	3.31616e6i	−0.314867	−	0.264205i	−0.786500	+	0.617590i	$0.788109\pi$
$692$		0			0
$693$	5.07926e6	−	1.07556e7i	0.401760	−	0.850746i
$694$		0			0
$695$	6603.82	−	2403.59i	0.000518601	−	0.000188755i
$696$		0			0
$697$	1.39526e6	−	1.17076e6i	0.108786	−	0.0912823i
$698$		0			0
$699$	1.85083e7	+	5.88520e6i	1.43276	+	0.455584i
$700$		0			0
$701$	−2.18603e7			−1.68020			−0.840099	−	0.542433i	$-0.817503\pi$
$701$	−2.18603e7			−1.68020			−0.840099	+	0.542433i	$0.817503\pi$
$702$		0			0
$703$	−6.37304e6			−0.486361
$704$		0			0
$705$	9026.07	+	41187.9i	0.000683952	+	0.00312102i
$706$		0			0
$707$	6.40052e6	−	5.37067e6i	0.481578	−	0.404092i
$708$		0			0
$709$	−1.84845e7	+	6.72782e6i	−1.38100	+	0.502642i	−0.922480	−	0.386044i	$-0.873841\pi$
$709$	−1.84845e7	+	6.72782e6i	−1.38100	+	0.502642i	−0.458517	+	0.888686i	$0.651619\pi$
$710$		0			0
$711$	−1.25478e7	−	3.42769e6i	−0.930879	−	0.254289i
$712$		0			0
$713$	1.87956e6	+	1.57714e6i	0.138462	+	0.116184i
$714$		0			0
$715$	−2545.24	−	14434.8i	−0.000186193	−	0.00105595i
$716$		0			0
$717$	6.22634e6	−	2.56131e6i	0.452309	−	0.186065i
$718$		0			0
$719$	−2.47337e6	−	4.28400e6i	−0.178430	−	0.309049i	0.762913	−	0.646501i	$-0.223768\pi$
$719$	−2.47337e6	−	4.28400e6i	−0.178430	−	0.309049i	−0.941343	+	0.337452i	$0.890435\pi$
$720$		0			0
$721$	−4.75002e6	+	8.22727e6i	−0.340296	+	0.589410i
$722$		0			0
$723$	9.96008e6	+	1.57184e7i	0.708626	+	1.11831i
$724$		0			0
$725$	−1.76541e7	−	6.42557e6i	−1.24739	−	0.454011i
$726$		0			0
$727$	1.06182e6	−	6.02185e6i	0.0745097	−	0.422566i	−0.924621	−	0.380888i	$-0.875618\pi$
$727$	1.06182e6	−	6.02185e6i	0.0745097	−	0.422566i	0.999131	−	0.0416781i	$-0.0132704\pi$
$728$		0			0
$729$	−3.91568e6	+	1.38043e7i	−0.272890	+	0.962045i
$730$		0			0
$731$	367302.	−	2.08307e6i	0.0254232	−	0.144182i
$732$		0			0
$733$	−1.05106e7	−	3.82555e6i	−0.722551	−	0.262987i	−0.0455424	−	0.998962i	$-0.514502\pi$
$733$	−1.05106e7	−	3.82555e6i	−0.722551	−	0.262987i	−0.677008	+	0.735975i	$0.736724\pi$
$734$		0			0
$735$	−18896.7	−	29821.8i	−0.00129023	−	0.00203618i
$736$		0			0
$737$	−7.60619e6	+	1.31743e7i	−0.515820	+	0.893427i
$738$		0			0
$739$	9.47611e6	+	1.64131e7i	0.638291	+	1.10555i	0.985808	+	0.167879i	$0.0536918\pi$
$739$	9.47611e6	+	1.64131e7i	0.638291	+	1.10555i	−0.347516	+	0.937674i	$0.612975\pi$
$740$		0			0
$741$	−2.86022e6	+	1.17660e6i	−0.191361	+	0.0787196i
$742$		0			0
$743$	2.85902e6	+	1.62143e7i	0.189996	+	1.07752i	0.919366	+	0.393403i	$0.128702\pi$
$743$	2.85902e6	+	1.62143e7i	0.189996	+	1.07752i	−0.729370	+	0.684119i	$0.760187\pi$
$744$		0			0
$745$	47922.5	+	40211.7i	0.00316336	+	0.00265437i
$746$		0			0
$747$	−7.11312e6	−	1.94310e6i	−0.466400	−	0.127407i
$748$		0			0
$749$	1.51653e7	−	5.51970e6i	0.987746	−	0.359510i
$750$		0			0
$751$	1.93610e6	−	1.62458e6i	0.125265	−	0.105110i	−0.578003	−	0.816034i	$-0.696168\pi$
$751$	1.93610e6	−	1.62458e6i	0.125265	−	0.105110i	0.703268	+	0.710925i	$0.251723\pi$
$752$		0			0
$753$	4.00051e6	+	1.82552e7i	0.257115	+	1.17327i
$754$		0			0
$755$	−29862.0			−0.00190656
$756$		0			0
$757$	2.77606e7			1.76072			0.880359	−	0.474308i	$-0.157302\pi$
$757$	2.77606e7			1.76072			0.880359	+	0.474308i	$0.157302\pi$
$758$		0			0
$759$	7.55541e6	+	2.40243e6i	0.476051	+	0.151373i
$760$		0			0
$761$	1.79233e7	−	1.50394e7i	1.12190	−	0.941389i	0.123204	−	0.992381i	$-0.460683\pi$
$761$	1.79233e7	−	1.50394e7i	1.12190	−	0.941389i	0.998699	+	0.0509925i	$0.0162385\pi$
$762$		0			0
$763$	−1.40692e7	+	5.12079e6i	−0.874902	+	0.318438i
$764$		0			0
$765$	−5719.18	+	12110.6i	−0.000353329	+	0.000748192i
$766$		0			0
$767$	−2.90434e6	−	2.43703e6i	−0.178262	−	0.149580i
$768$		0			0
$769$	272685.	+	1.54647e6i	0.0166282	+	0.0943031i	0.991992	−	0.126297i	$-0.0403094\pi$
$769$	272685.	+	1.54647e6i	0.0166282	+	0.0943031i	−0.975364	+	0.220601i	$0.929198\pi$
$770$		0			0
$771$	−8.49513e6	−	6.55144e6i	−0.514676	−	0.396918i
$772$		0			0
$773$	9.37520e6	+	1.62383e7i	0.564328	+	0.977445i	0.997112	+	0.0759474i	$0.0241981\pi$
$773$	9.37520e6	+	1.62383e7i	0.564328	+	0.977445i	−0.432784	+	0.901498i	$0.642469\pi$
$774$		0			0
$775$	4.15482e6	−	7.19636e6i	0.248484	−	0.430386i
$776$		0			0
$777$	−4.65365e6	+	191841.i	−0.276529	+	0.0113996i
$778$		0			0
$779$	−1.49334e7	−	5.43532e6i	−0.881689	−	0.320908i
$780$		0			0
$781$	−1.91562e6	+	1.08640e7i	−0.112378	+	0.637329i
$782$		0			0
$783$	−2.02765e7	+	1.03679e7i	−1.18192	+	0.604347i
$784$		0			0
$785$	−11007.3	+	62425.5i	−0.000637538	+	0.00361566i
$786$		0			0
$787$	1.28471e7	+	4.67596e6i	0.739380	+	0.269112i	0.684130	−	0.729360i	$-0.260182\pi$
$787$	1.28471e7	+	4.67596e6i	0.739380	+	0.269112i	0.0552501	+	0.998473i	$0.482404\pi$
$788$		0			0
$789$	1.13548e7	−	2.16834e7i	0.649363	−	1.24004i
$790$		0			0
$791$	−4.01654e6	+	6.95685e6i	−0.228250	+	0.395340i
$792$		0			0
$793$	2.60320e6	+	4.50887e6i	0.147002	+	0.254615i
$794$		0			0
$795$	−1882.43	+	14034.5i	−0.000105633	+	0.000787554i
$796$		0			0
$797$	3.07158e6	+	1.74198e7i	0.171284	+	0.971398i	0.942347	+	0.334639i	$0.108614\pi$
$797$	3.07158e6	+	1.74198e7i	0.171284	+	0.971398i	−0.771063	+	0.636759i	$0.780275\pi$
$798$		0			0
$799$	−1.77186e6	−	1.48677e6i	−0.0981888	−	0.0823902i
$800$		0			0
$801$	−1.64084e7	−	1.16087e7i	−0.903617	−	0.639294i
$802$		0			0
$803$	−7.33439e6	+	2.66950e6i	−0.401398	+	0.146097i
$804$		0			0
$805$	−15935.8	+	13371.8i	−0.000866733	+	0.000727276i
$806$		0			0
$807$	−3.95077e6	+	3.60257e6i	−0.213549	+	0.194728i
$808$		0			0
$809$	1.90929e7			1.02565			0.512827	−	0.858492i	$-0.328598\pi$
$809$	1.90929e7			1.02565			0.512827	+	0.858492i	$0.328598\pi$
$810$		0			0
$811$	−2.46287e7			−1.31489			−0.657444	−	0.753503i	$-0.728362\pi$
$811$	−2.46287e7			−1.31489			−0.657444	+	0.753503i	$0.728362\pi$
$812$		0			0
$813$	1.93902e7	−	1.76813e7i	1.02886	−	0.938182i
$814$		0			0
$815$	12160.3	−	10203.7i	0.000641286	−	0.000538103i
$816$		0			0
$817$	−1.73425e7	+	6.31215e6i	−0.908983	+	0.330843i
$818$		0			0
$819$	−2.05314e6	+	945260.i	−0.106957	+	0.0492427i
$820$		0			0
$821$	1.97857e7	+	1.66022e7i	1.02446	+	0.859622i	0.990181	−	0.139790i	$-0.0446428\pi$
$821$	1.97857e7	+	1.66022e7i	1.02446	+	0.859622i	0.0342766	+	0.999412i	$0.489087\pi$
$822$		0			0
$823$	2.69838e6	+	1.53032e7i	0.138868	+	0.787561i	0.972088	+	0.234618i	$0.0753840\pi$
$823$	2.69838e6	+	1.53032e7i	0.138868	+	0.787561i	−0.833219	+	0.552943i	$0.813505\pi$
$824$		0			0
$825$	3.56945e6	−	2.66122e7i	0.182586	−	1.36128i
$826$		0			0
$827$	1.65831e7	+	2.87227e7i	0.843144	+	1.46037i	0.887224	+	0.461340i	$0.152631\pi$
$827$	1.65831e7	+	2.87227e7i	0.843144	+	1.46037i	−0.0440800	+	0.999028i	$0.514036\pi$
$828$		0			0
$829$	−4.23971e6	+	7.34339e6i	−0.214264	+	0.371117i	−0.953045	−	0.302830i	$-0.902069\pi$
$829$	−4.23971e6	+	7.34339e6i	−0.214264	+	0.371117i	0.738780	+	0.673946i	$0.235402\pi$
$830$		0			0
$831$	−1.71947e7	+	3.28354e7i	−0.863757	+	1.64945i
$832$		0			0
$833$	1.81987e6	+	662377.i	0.0908713	+	0.0330744i
$834$		0			0
$835$	11856.8	−	67243.3i	0.000588507	−	0.00333759i
$836$		0			0
$837$	−2.95876e6	−	9.62846e6i	−0.145981	−	0.475054i
$838$		0			0
$839$	4.33782e6	−	2.46010e7i	0.212749	−	1.20656i	−0.672022	−	0.740531i	$-0.734574\pi$
$839$	4.33782e6	−	2.46010e7i	0.212749	−	1.20656i	0.884771	−	0.466027i	$-0.154315\pi$
$840$		0			0
$841$	−1.46902e7	−	5.34680e6i	−0.716206	−	0.260678i
$842$		0			0
$843$	2.90170e7	−	1.19619e6i	1.40632	−	0.0579739i
$844$		0			0
$845$	45739.1	−	79222.4i	0.00220366	−	0.00381686i
$846$		0			0
$847$	6.33930e6	+	1.09800e7i	0.303622	+	0.525888i
$848$		0			0
$849$	−2.36123e7	−	1.82098e7i	−1.12427	−	0.867034i
$850$		0			0
$851$	−539084.	−	3.05730e6i	−0.0255172	−	0.144715i
$852$		0			0
$853$	8.41131e6	+	7.05793e6i	0.395814	+	0.332127i	0.818873	−	0.573975i	$-0.194599\pi$
$853$	8.41131e6	+	7.05793e6i	0.395814	+	0.332127i	−0.423059	+	0.906102i	$0.639044\pi$
$854$		0			0
$855$	116460.	−	9618.23i	0.00544832	−	0.000449966i
$856$		0			0
$857$	2.69275e7	−	9.80079e6i	1.25240	−	0.455836i	0.371188	−	0.928558i	$-0.378950\pi$
$857$	2.69275e7	−	9.80079e6i	1.25240	−	0.455836i	0.881212	+	0.472721i	$0.156728\pi$
$858$		0			0
$859$	−1.05479e7	+	8.85076e6i	−0.487735	+	0.409259i	−0.853214	−	0.521561i	$-0.825350\pi$
$859$	−1.05479e7	+	8.85076e6i	−0.487735	+	0.409259i	0.365478	+	0.930820i	$0.380905\pi$
$860$		0			0
$861$	−1.10681e7	−	3.51939e6i	−0.508822	−	0.161793i
$862$		0			0
$863$	−9.65716e6			−0.441390			−0.220695	−	0.975343i	$-0.570833\pi$
$863$	−9.65716e6			−0.441390			−0.220695	+	0.975343i	$0.570833\pi$
$864$		0			0
$865$	−151482.			−0.00688369
$866$		0			0
$867$	4.58069e6	+	2.09027e7i	0.206958	+	0.944395i
$868$		0			0
$869$	2.26022e7	−	1.89655e7i	1.01532	−	0.851953i
$870$		0			0
$871$	2.71642e6	−	988697.i	0.121325	−	0.0441589i
$872$		0			0
$873$	1.92828e7	−	1.90961e7i	0.856318	−	0.848026i
$874$		0			0
$875$	107942.	+	90574.0i	0.00476617	+	0.00399929i
$876$		0			0
$877$	−2.97812e6	−	1.68898e7i	−0.130751	−	0.741523i	−0.977725	−	0.209890i	$-0.932690\pi$
$877$	−2.97812e6	−	1.68898e7i	−0.130751	−	0.741523i	0.846975	−	0.531633i	$-0.178422\pi$
$878$		0			0
$879$	−9.92469e6	+	4.08268e6i	−0.433256	+	0.178227i
$880$		0			0
$881$	−1.52998e7	−	2.65000e7i	−0.664119	−	1.15029i	−0.979523	−	0.201330i	$-0.935474\pi$
$881$	−1.52998e7	−	2.65000e7i	−0.664119	−	1.15029i	0.315405	−	0.948957i	$-0.397860\pi$
$882$		0			0
$883$	−1.66746e7	+	2.88813e7i	−0.719704	+	1.24656i	0.241413	+	0.970423i	$0.422389\pi$
$883$	−1.66746e7	+	2.88813e7i	−0.719704	+	1.24656i	−0.961117	+	0.276142i	$0.910944\pi$
$884$		0			0
$885$	76675.0	+	121004.i	0.00329076	+	0.00519329i
$886$		0			0
$887$	−3.77977e7	−	1.37572e7i	−1.61308	−	0.587114i	−0.631036	−	0.775754i	$-0.717370\pi$
$887$	−3.77977e7	−	1.37572e7i	−1.61308	−	0.587114i	−0.982046	+	0.188640i	$0.939592\pi$
$888$		0			0
$889$	−3.35736e6	+	1.90405e7i	−0.142477	+	0.808025i
$890$		0			0
$891$	−2.06777e7	−	2.51355e7i	−0.872586	−	1.06070i
$892$		0			0
$893$	−3.50443e6	+	1.98746e7i	−0.147058	+	0.834008i
$894$		0			0
$895$	−42547.6	−	15486.1i	−0.00177549	−	0.000646224i
$896$		0			0
$897$	−806382.	−	1.27259e6i	−0.0334626	−	0.0528088i
$898$		0			0
$899$	7.99337e6	−	1.38449e7i	0.329861	−	0.571336i
$900$		0			0
$901$	−388383.	−	672698.i	−0.0159385	−	0.0276063i
$902$		0			0
$903$	−1.24736e7	+	5.13122e6i	−0.509064	+	0.209412i
$904$		0			0
$905$	−26447.6	−	149992.i	−0.00107341	−	0.00608759i
$906$		0			0
$907$	−7.10837e6	−	5.96463e6i	−0.286914	−	0.240750i	0.487959	−	0.872867i	$-0.337742\pi$
$907$	−7.10837e6	−	5.96463e6i	−0.286914	−	0.240750i	−0.774873	+	0.632117i	$0.782186\pi$
$908$		0			0
$909$	−5.80986e6	−	2.21125e7i	−0.233215	−	0.887622i
$910$		0			0
$911$	9.45231e6	−	3.44036e6i	0.377348	−	0.137343i	−0.146381	−	0.989228i	$-0.546763\pi$
$911$	9.45231e6	−	3.44036e6i	0.377348	−	0.137343i	0.523729	+	0.851885i	$0.324540\pi$
$912$		0			0
$913$	1.28128e7	−	1.07512e7i	0.508707	−	0.426856i
$914$		0			0
$915$	−42110.1	−	192158.i	−0.00166278	−	0.00758761i
$916$		0			0
$917$	1.80206e7			0.707693
$918$		0			0
$919$	9.26150e6			0.361737			0.180868	−	0.983507i	$-0.442109\pi$
$919$	9.26150e6			0.361737			0.180868	+	0.983507i	$0.442109\pi$
$920$		0			0
$921$	2.72954e7	+	8.67928e6i	1.06033	+	0.337159i
$922$		0			0
$923$	1.60586e6	−	1.34748e6i	0.0620445	−	0.0520616i
$924$		0			0
$925$	−9.87989e6	+	3.59599e6i	−0.379663	+	0.138186i
$926$		0			0
$927$	1.48066e7	+	2.13665e7i	0.565921	+	0.816647i
$928$		0			0
$929$	2.97702e7	+	2.49802e7i	1.13173	+	0.949634i	0.999137	−	0.0415371i	$-0.0132255\pi$
$929$	2.97702e7	+	2.49802e7i	1.13173	+	0.949634i	0.132592	+	0.991171i	$0.457670\pi$
$930$		0			0
$931$	−2.93424e6	−	1.66409e7i	−0.110949	−	0.629220i
$932$		0			0
$933$	−3.45666e7	−	2.66577e7i	−1.30003	−	1.00258i
$934$		0			0
$935$	−15189.9	−	26309.7i	−0.000568232	−	0.000984207i
$936$		0			0
$937$	−6.25691e6	+	1.08373e7i	−0.232815	+	0.403247i	−0.958635	−	0.284637i	$-0.908127\pi$
$937$	−6.25691e6	+	1.08373e7i	−0.232815	+	0.403247i	0.725820	+	0.687884i	$0.241460\pi$
$938$		0			0
$939$	−1.84151e7	+	759139.i	−0.681568	+	0.0280968i
$940$		0			0
$941$	9.77924e6	+	3.55935e6i	0.360024	+	0.131038i	0.515698	−	0.856771i	$-0.327533\pi$
$941$	9.77924e6	+	3.55935e6i	0.360024	+	0.131038i	−0.155674	+	0.987808i	$0.549755\pi$
$942$		0			0
$943$	1.34426e6	−	7.62366e6i	0.0492270	−	0.279180i
$944$		0			0
$945$	84750.8	−	10529.0i	0.00308720	−	0.000383537i
$946$		0			0
$947$	2.74117e6	−	1.55459e7i	0.0993255	−	0.563303i	−0.894010	−	0.448046i	$-0.852120\pi$
$947$	2.74117e6	−	1.55459e7i	0.0993255	−	0.563303i	0.993336	−	0.115256i	$-0.0367689\pi$
$948$		0			0
$949$	1.39373e6	+	507276.i	0.0502358	+	0.0182843i
$950$		0			0
$951$	1.71285e7	−	3.27090e7i	0.614140	−	1.17278i
$952$		0			0
$953$	3.26102e6	−	5.64825e6i	0.116311	−	0.201457i	−0.801992	−	0.597335i	$-0.796226\pi$
$953$	3.26102e6	−	5.64825e6i	0.116311	−	0.201457i	0.918303	+	0.395878i	$0.129560\pi$
$954$		0			0
$955$	−16057.2	−	27811.9i	−0.000569721	−	0.000986785i
$956$		0			0
$957$	6.86720e6	−	5.11987e7i	0.242382	−	1.80709i
$958$		0			0
$959$	4.76494e6	+	2.70233e7i	0.167306	+	0.948838i
$960$		0			0
$961$	−1.65145e7	−	1.38573e7i	−0.576842	−	0.484028i
$962$		0			0
$963$	4.06286e6	−	4.39734e7i	0.141178	−	1.52800i
$964$		0			0
$965$	7965.93	−	2899.36i	0.000275371	−	0.000100227i
$966$		0			0
$967$	1.09030e7	−	9.14873e6i	0.374957	−	0.314626i	−0.435762	−	0.900062i	$-0.643521\pi$
$967$	1.09030e7	−	9.14873e6i	0.374957	−	0.314626i	0.810719	+	0.585436i	$0.199077\pi$
$968$		0			0
$969$	−4.73665e6	+	4.31919e6i	−0.162055	+	0.147772i
$970$		0			0
$971$	2.16333e7			0.736334			0.368167	−	0.929760i	$-0.379986\pi$
$971$	2.16333e7			0.736334			0.368167	+	0.929760i	$0.379986\pi$
$972$		0			0
$973$	2.45820e6			0.0832405
$974$		0			0
$975$	−3.77020e6	+	3.43791e6i	−0.127014	+	0.115820i
$976$		0			0
$977$	4.39578e7	−	3.68850e7i	1.47333	−	1.23627i	0.560352	−	0.828255i	$-0.310666\pi$
$977$	4.39578e7	−	3.68850e7i	1.47333	−	1.23627i	0.912976	−	0.408014i	$-0.133778\pi$
$978$		0			0
$979$	4.28427e7	−	1.55935e7i	1.42863	−	0.519979i
$980$		0			0
$981$	−3.76923e6	+	4.07954e7i	−0.125049	+	1.35344i
$982$		0			0
$983$	−1.74593e7	−	1.46501e7i	−0.576294	−	0.483568i	0.307434	−	0.951569i	$-0.400530\pi$
$983$	−1.74593e7	−	1.46501e7i	−0.576294	−	0.483568i	−0.883728	+	0.468001i	$0.844974\pi$
$984$		0			0
$985$	−15284.9	−	86684.7i	−0.000501962	−	0.00284677i
$986$		0			0
$987$	−1.96070e6	+	1.46181e7i	−0.0640646	+	0.477637i
$988$		0			0
$989$	−4.49505e6	−	7.78565e6i	−0.146131	−	0.253107i
$990$		0			0
$991$	2.25509e7	−	3.90593e7i	0.729423	−	1.26340i	−0.227704	−	0.973730i	$-0.573122\pi$
$991$	2.25509e7	−	3.90593e7i	0.729423	−	1.26340i	0.957127	−	0.289668i	$-0.0935448\pi$
$992$		0			0
$993$	1.78558e7	−	3.40979e7i	0.574654	−	1.09737i
$994$		0			0
$995$	−217761.	−	79258.6i	−0.00697305	−	0.00253798i
$996$		0			0
$997$	−9.87475e6	+	5.60025e7i	−0.314621	+	1.78431i	0.259713	+	0.965686i	$0.416372\pi$
$997$	−9.87475e6	+	5.60025e7i	−0.314621	+	1.78431i	−0.574334	+	0.818621i	$0.694739\pi$
$998$		0			0
$999$	−4.96305e6	+	1.17388e7i	−0.157338	+	0.372143i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.25.5	yes	90
3.2	odd	2		324.6.i.a.73.8		90
27.13	even	9	inner	108.6.i.a.13.5	✓	90
27.14	odd	18		324.6.i.a.253.8		90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.5	✓	90	27.13	even	9	inner
108.6.i.a.25.5	yes	90	1.1	even	1	trivial
324.6.i.a.73.8		90	3.2	odd	2
324.6.i.a.253.8		90	27.14	odd	18

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-11.5186 + 10.5034i) q^{3} +(0.194482 - 0.163190i) q^{5} +(83.4487 - 30.3729i) q^{7} +(22.3564 - 241.969i) q^{9} +O(q^{10})\)	\(q+(-11.5186 + 10.5034i) q^{3} +(0.194482 - 0.163190i) q^{5} +(83.4487 - 30.3729i) q^{7} +(22.3564 - 241.969i) q^{9} +(422.243 + 354.304i) q^{11} +(-18.1884 - 103.151i) q^{13} +(-0.526112 + 3.92245i) q^{15} +(-108.547 - 188.010i) q^{17} +(-947.091 + 1640.41i) q^{19} +(-642.194 + 1226.35i) q^{21} +(-867.055 - 315.582i) q^{23} +(-542.639 + 3077.46i) q^{25} +(2283.99 + 3021.97i) q^{27} +(-1043.97 + 5920.66i) q^{29} +(-2498.77 - 909.479i) q^{31} +(-8585.06 + 353.909i) q^{33} +(11.2728 - 19.5250i) q^{35} +(1682.27 + 2913.77i) q^{37} +(1292.95 + 997.119i) q^{39} +(1456.87 + 8262.34i) q^{41} +(7463.76 + 6262.84i) q^{43} +(-35.1391 - 50.7071i) q^{45} +(10011.8 - 3643.99i) q^{47} +(-6833.73 + 5734.18i) q^{49} +(3225.06 + 1025.49i) q^{51} +3578.00 q^{53} +139.938 q^{55} +(-6320.74 - 28842.9i) q^{57} +(27728.4 - 23266.9i) q^{59} +(-46708.9 + 17000.6i) q^{61} +(-5483.69 - 20871.1i) q^{63} +(-20.3706 - 17.0930i) q^{65} +(4792.46 + 27179.4i) q^{67} +(13302.0 - 5471.98i) q^{69} +(10006.9 + 17332.5i) q^{71} +(-7080.10 + 12263.1i) q^{73} +(-26073.4 - 41147.6i) q^{75} +(45996.9 + 16741.5i) q^{77} +(9295.20 - 52715.7i) q^{79} +(-58049.4 - 10819.1i) q^{81} +(5269.29 - 29883.6i) q^{83} +(-51.7919 - 18.8507i) q^{85} +(-50162.1 - 79163.1i) q^{87} +(41357.3 - 71632.9i) q^{89} +(-4650.80 - 8055.41i) q^{91} +(38335.0 - 15769.7i) q^{93} +(83.5061 + 473.587i) q^{95} +(85552.1 + 71786.7i) q^{97} +(95170.6 - 94249.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 90 * q - 87 * q^5 + 330 * q^9	\( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more