LMFDB - Embedded newform 108.6.i.a.13.14

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			13.14
Character	$\chi$	$=$	108.13
Dual form			108.6.i.a.25.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(14.3467 + 6.09679i) q^{3} +(-19.6970 - 16.5278i) q^{5} +(-125.630 - 45.7255i) q^{7} +(168.658 + 174.938i) q^{9} +O(q^{10})$	\(q+(14.3467 + 6.09679i) q^{3} +(-19.6970 - 16.5278i) q^{5} +(-125.630 - 45.7255i) q^{7} +(168.658 + 174.938i) q^{9} +(-254.551 + 213.594i) q^{11} +(-173.898 + 986.227i) q^{13} +(-181.822 - 357.208i) q^{15} +(-596.280 + 1032.79i) q^{17} +(271.733 + 470.655i) q^{19} +(-1523.60 - 1421.95i) q^{21} +(554.955 - 201.987i) q^{23} +(-427.845 - 2426.43i) q^{25} +(1353.14 + 3538.07i) q^{27} +(30.7001 + 174.109i) q^{29} +(-7839.09 + 2853.20i) q^{31} +(-4954.22 + 1512.43i) q^{33} +(1718.79 + 2977.03i) q^{35} +(2654.21 - 4597.22i) q^{37} +(-8507.70 + 13088.9i) q^{39} +(-1820.42 + 10324.1i) q^{41} +(328.079 - 275.291i) q^{43} +(-430.729 - 6233.30i) q^{45} +(5953.67 + 2166.96i) q^{47} +(817.108 + 685.635i) q^{49} +(-14851.4 + 11181.7i) q^{51} -28030.9 q^{53} +8544.12 q^{55} +(1029.00 + 8409.06i) q^{57} +(26009.3 + 21824.4i) q^{59} +(12492.6 + 4546.94i) q^{61} +(-13189.4 - 29689.4i) q^{63} +(19725.4 - 16551.6i) q^{65} +(7276.02 - 41264.3i) q^{67} +(9193.27 + 485.586i) q^{69} +(38405.8 - 66520.9i) q^{71} +(-13757.1 - 23828.1i) q^{73} +(8655.25 - 37419.9i) q^{75} +(41745.9 - 15194.3i) q^{77} +(-14989.1 - 85007.3i) q^{79} +(-2157.75 + 59009.6i) q^{81} +(20756.6 + 117717. i) q^{83} +(28814.6 - 10487.6i) q^{85} +(-621.059 + 2685.07i) q^{87} +(61600.0 + 106694. i) q^{89} +(66942.6 - 115948. i) q^{91} +(-129861. - 6859.22i) q^{93} +(2426.55 - 13761.6i) q^{95} +(12091.7 - 10146.2i) q^{97} +(-80297.9 - 8506.36i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

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
$3$	14.3467	+	6.09679i	0.920344	+	0.391109i
$3$	14.3467	+	6.09679i	0.920344	+	0.391109i
$4$		0			0
$5$	−19.6970	−	16.5278i	−0.352351	−	0.295657i	0.449382	−	0.893340i	$-0.351644\pi$
$5$	−19.6970	−	16.5278i	−0.352351	−	0.295657i	−0.801733	+	0.597682i	$0.796089\pi$
$6$		0			0
$7$	−125.630	−	45.7255i	−0.969053	−	0.352706i	−0.191478	−	0.981497i	$-0.561328\pi$
$7$	−125.630	−	45.7255i	−0.969053	−	0.352706i	−0.777575	+	0.628791i	$0.783550\pi$
$8$		0			0
$9$	168.658	+	174.938i	0.694067	+	0.719910i
$10$		0			0
$11$	−254.551	+	213.594i	−0.634298	+	0.532239i	−0.902261	−	0.431190i	$-0.858094\pi$
$11$	−254.551	+	213.594i	−0.634298	+	0.532239i	0.267963	+	0.963429i	$0.413649\pi$
$12$		0			0
$13$	−173.898	+	986.227i	−0.285389	+	1.61852i	0.418503	+	0.908215i	$0.362555\pi$
$13$	−173.898	+	986.227i	−0.285389	+	1.61852i	−0.703892	+	0.710307i	$0.748556\pi$
$14$		0			0
$15$	−181.822	−	357.208i	−0.208650	−	0.409914i
$16$		0			0
$17$	−596.280	+	1032.79i	−0.500412	+	0.866739i	0.499588	+	0.866263i	$0.333485\pi$
$17$	−596.280	+	1032.79i	−0.500412	+	0.866739i	−1.00000		0.000475970i	$0.999848\pi$
$18$		0			0
$19$	271.733	+	470.655i	0.172686	+	0.299101i	0.939358	−	0.342938i	$-0.111422\pi$
$19$	271.733	+	470.655i	0.172686	+	0.299101i	−0.766672	+	0.642039i	$0.778089\pi$
$20$		0			0
$21$	−1523.60	−	1421.95i	−0.753915	−	0.703617i
$22$		0			0
$23$	554.955	−	201.987i	0.218745	−	0.0796167i	−0.230323	−	0.973114i	$-0.573978\pi$
$23$	554.955	−	201.987i	0.218745	−	0.0796167i	0.449068	+	0.893498i	$0.351756\pi$
$24$		0			0
$25$	−427.845	−	2426.43i	−0.136910	−	0.776457i
$26$		0			0
$27$	1353.14	+	3538.07i	0.357217	+	0.934021i
$28$		0			0
$29$	30.7001	+	174.109i	0.00677868	+	0.0384438i	0.988009	−	0.154393i	$-0.0493423\pi$
$29$	30.7001	+	174.109i	0.00677868	+	0.0384438i	−0.981231	+	0.192837i	$0.938231\pi$
$30$		0			0
$31$	−7839.09	+	2853.20i	−1.46508	+	0.533246i	−0.946760	−	0.321940i	$-0.895665\pi$
$31$	−7839.09	+	2853.20i	−1.46508	+	0.533246i	−0.518321	+	0.855186i	$0.673443\pi$
$32$		0			0
$33$	−4954.22	+	1512.43i	−0.791936	+	0.241764i
$34$		0			0
$35$	1718.79	+	2977.03i	0.237166	+	0.410784i
$36$		0			0
$37$	2654.21	−	4597.22i	0.318736	−	0.552066i	−0.661489	−	0.749955i	$-0.730075\pi$
$37$	2654.21	−	4597.22i	0.318736	−	0.552066i	0.980224	+	0.197889i	$0.0634085\pi$
$38$		0			0
$39$	−8507.70	+	13088.9i	−0.895675	+	1.37798i
$40$		0			0
$41$	−1820.42	+	10324.1i	−0.169127	+	0.959165i	0.775580	+	0.631249i	$0.217457\pi$
$41$	−1820.42	+	10324.1i	−0.169127	+	0.959165i	−0.944707	+	0.327916i	$0.893654\pi$
$42$		0			0
$43$	328.079	−	275.291i	0.0270587	−	0.0227050i	−0.629158	−	0.777277i	$-0.716600\pi$
$43$	328.079	−	275.291i	0.0270587	−	0.0227050i	0.656217	+	0.754572i	$0.272156\pi$
$44$		0			0
$45$	−430.729	−	6233.30i	−0.0317083	−	0.458867i
$46$		0			0
$47$	5953.67	+	2166.96i	0.393134	+	0.143089i	0.531020	−	0.847359i	$-0.321809\pi$
$47$	5953.67	+	2166.96i	0.393134	+	0.143089i	−0.137887	+	0.990448i	$0.544031\pi$
$48$		0			0
$49$	817.108	+	685.635i	0.0486171	+	0.0407946i
$50$		0			0
$51$	−14851.4	+	11181.7i	−0.799541	+	0.601983i
$52$		0			0
$53$	−28030.9			−1.37072			−0.685359	−	0.728206i	$-0.740355\pi$
$53$	−28030.9			−1.37072			−0.685359	+	0.728206i	$0.740355\pi$
$54$		0			0
$55$	8544.12			0.380856
$56$		0			0
$57$	1029.00	+	8409.06i	0.0419495	+	0.342816i
$58$		0			0
$59$	26009.3	+	21824.4i	0.972744	+	0.816229i	0.982979	−	0.183719i	$-0.0588135\pi$
$59$	26009.3	+	21824.4i	0.972744	+	0.816229i	−0.0102349	+	0.999948i	$0.503258\pi$
$60$		0			0
$61$	12492.6	+	4546.94i	0.429862	+	0.156457i	0.547884	−	0.836554i	$-0.315433\pi$
$61$	12492.6	+	4546.94i	0.429862	+	0.156457i	−0.118023	+	0.993011i	$0.537656\pi$
$62$		0			0
$63$	−13189.4	−	29689.4i	−0.418671	−	0.942433i
$64$		0			0
$65$	19725.4	−	16551.6i	0.579085	−	0.485910i
$66$		0			0
$67$	7276.02	−	41264.3i	0.198019	−	1.12302i	−0.710035	−	0.704167i	$-0.751321\pi$
$67$	7276.02	−	41264.3i	0.198019	−	1.12302i	0.908053	−	0.418854i	$-0.137568\pi$
$68$		0			0
$69$	9193.27	+	485.586i	0.232460	+	0.0122785i
$70$		0			0
$71$	38405.8	−	66520.9i	0.904173	−	1.56607i	0.0821484	−	0.996620i	$-0.473822\pi$
$71$	38405.8	−	66520.9i	0.904173	−	1.56607i	0.822024	−	0.569453i	$-0.192845\pi$
$72$		0			0
$73$	−13757.1	−	23828.1i	−0.302149	−	0.523337i	0.674474	−	0.738299i	$-0.264371\pi$
$73$	−13757.1	−	23828.1i	−0.302149	−	0.523337i	−0.976622	+	0.214962i	$0.931037\pi$
$74$		0			0
$75$	8655.25	−	37419.9i	0.177675	−	0.768155i
$76$		0			0
$77$	41745.9	−	15194.3i	0.802393	−	0.292047i
$78$		0			0
$79$	−14989.1	−	85007.3i	−0.270214	−	1.53246i	−0.753767	−	0.657142i	$-0.771765\pi$
$79$	−14989.1	−	85007.3i	−0.270214	−	1.53246i	0.483553	−	0.875315i	$-0.339346\pi$
$80$		0			0
$81$	−2157.75	+	59009.6i	−0.0365417	+	0.999332i
$82$		0			0
$83$	20756.6	+	117717.i	0.330720	+	1.87561i	0.465975	+	0.884798i	$0.345704\pi$
$83$	20756.6	+	117717.i	0.330720	+	1.87561i	−0.135254	+	0.990811i	$0.543185\pi$
$84$		0			0
$85$	28814.6	−	10487.6i	0.432579	−	0.157446i
$86$		0			0
$87$	−621.059	+	2685.07i	−0.00879700	+	0.0380327i
$88$		0			0
$89$	61600.0	+	106694.i	0.824339	+	1.42780i	0.902423	+	0.430851i	$0.141786\pi$
$89$	61600.0	+	106694.i	0.824339	+	1.42780i	−0.0780838	+	0.996947i	$0.524880\pi$
$90$		0			0
$91$	66942.6	−	115948.i	0.847420	−	1.46777i
$92$		0			0
$93$	−129861.	−	6859.22i	−1.55694	−	0.0822370i
$94$		0			0
$95$	2426.55	−	13761.6i	0.0275854	−	0.156445i
$96$		0			0
$97$	12091.7	−	10146.2i	0.130485	−	0.109490i	−0.575210	−	0.818006i	$-0.695080\pi$
$97$	12091.7	−	10146.2i	0.130485	−	0.109490i	0.705694	+	0.708516i	$0.250635\pi$
$98$		0			0
$99$	−80297.9	−	8506.36i	−0.823410	−	0.0872280i
$100$		0			0
$101$	−541.630	−	197.137i	−0.00528323	−	0.00192294i	0.339377	−	0.940650i	$-0.389784\pi$
$101$	−541.630	−	197.137i	−0.00528323	−	0.00192294i	−0.344660	+	0.938727i	$0.612006\pi$
$102$		0			0
$103$	−38960.9	−	32692.1i	−0.361856	−	0.303633i	0.443674	−	0.896188i	$-0.353675\pi$
$103$	−38960.9	−	32692.1i	−0.361856	−	0.303633i	−0.805530	+	0.592555i	$0.798119\pi$
$104$		0			0
$105$	6508.71	+	53189.9i	0.0576132	+	0.470821i
$106$		0			0
$107$	163803.			1.38313			0.691565	−	0.722314i	$-0.256922\pi$
$107$	163803.			1.38313			0.691565	+	0.722314i	$0.256922\pi$
$108$		0			0
$109$	−13457.7			−0.108494			−0.0542468	−	0.998528i	$-0.517276\pi$
$109$	−13457.7			−0.108494			−0.0542468	+	0.998528i	$0.517276\pi$
$110$		0			0
$111$	66107.5	−	49773.0i	0.509265	−	0.383431i
$112$		0			0
$113$	−12155.2	−	10199.5i	−0.0895503	−	0.0751417i	0.596913	−	0.802306i	$-0.296394\pi$
$113$	−12155.2	−	10199.5i	−0.0895503	−	0.0751417i	−0.686464	+	0.727164i	$0.740838\pi$
$114$		0			0
$115$	−14269.4	−	5193.62i	−0.100614	−	0.0366206i
$116$		0			0
$117$	−201858.	+	135914.i	−1.36327	+	0.917908i
$118$		0			0
$119$	122135.	−	102484.i	0.790630	−	0.663418i
$120$		0			0
$121$	−8792.22	+	49863.2i	−0.0545928	+	0.309611i
$122$		0			0
$123$	−89061.1	+	137019.i	−0.530793	+	0.816615i
$124$		0			0
$125$	−71852.1	+	124452.i	−0.411306	+	0.712402i
$126$		0			0
$127$	−29007.3	−	50242.1i	−0.159587	−	0.276413i	0.775133	−	0.631798i	$-0.217683\pi$
$127$	−29007.3	−	50242.1i	−0.159587	−	0.276413i	−0.934720	+	0.355386i	$0.884350\pi$
$128$		0			0
$129$	6385.26	−	1949.30i	0.0337835	−	0.0103135i
$130$		0			0
$131$	−233975.	+	85159.8i	−1.19122	+	0.433567i	−0.860150	−	0.510040i	$-0.829630\pi$
$131$	−233975.	+	85159.8i	−1.19122	+	0.433567i	−0.331066	+	0.943608i	$0.607408\pi$
$132$		0			0
$133$	−12616.8	−	71553.4i	−0.0618472	−	0.350753i
$134$		0			0
$135$	31823.6	−	92053.7i	0.150285	−	0.434717i
$136$		0			0
$137$	37208.0	+	211017.i	0.169369	+	0.960540i	0.944444	+	0.328671i	$0.106601\pi$
$137$	37208.0	+	211017.i	0.169369	+	0.960540i	−0.775075	+	0.631869i	$0.782288\pi$
$138$		0			0
$139$	128537.	−	46783.8i	0.564277	−	0.205380i	−0.0441012	−	0.999027i	$-0.514042\pi$
$139$	128537.	−	46783.8i	0.564277	−	0.205380i	0.608379	+	0.793647i	$0.291820\pi$
$140$		0			0
$141$	72204.4	+	67387.1i	0.305855	+	0.285449i
$142$		0			0
$143$	−166386.	−	288189.i	−0.680419	−	1.17852i
$144$		0			0
$145$	2272.93	−	3936.83i	0.00897772	−	0.0155499i
$146$		0			0
$147$	7542.67	+	14818.4i	0.0287893	+	0.0565597i
$148$		0			0
$149$	−56041.7	+	317828.i	−0.206798	+	1.17281i	0.687788	+	0.725912i	$0.258582\pi$
$149$	−56041.7	+	317828.i	−0.206798	+	1.17281i	−0.894586	+	0.446896i	$0.852529\pi$
$150$		0			0
$151$	−28882.4	+	24235.2i	−0.103084	+	0.0864978i	−0.692873	−	0.721060i	$-0.743655\pi$
$151$	−28882.4	+	24235.2i	−0.103084	+	0.0864978i	0.589789	+	0.807557i	$0.299211\pi$
$152$		0			0
$153$	−281241.	+	69876.0i	−0.971294	+	0.241323i
$154$		0			0
$155$	201564.	+	73363.2i	0.673881	+	0.245272i
$156$		0			0
$157$	31328.7	+	26287.9i	0.101436	+	0.0851150i	0.692095	−	0.721806i	$-0.256688\pi$
$157$	31328.7	+	26287.9i	0.101436	+	0.0851150i	−0.590659	+	0.806921i	$0.701132\pi$
$158$		0			0
$159$	−402153.	−	170899.i	−1.26153	−	0.536100i
$160$		0			0
$161$	−78954.8			−0.240057
$162$		0			0
$163$	19318.6			0.0569516			0.0284758	−	0.999594i	$-0.490935\pi$
$163$	19318.6			0.0569516			0.0284758	+	0.999594i	$0.490935\pi$
$164$		0			0
$165$	122580.	+	52091.7i	0.350519	+	0.148956i
$166$		0			0
$167$	220900.	+	185357.i	0.612922	+	0.514302i	0.895570	−	0.444921i	$-0.146768\pi$
$167$	220900.	+	185357.i	0.612922	+	0.514302i	−0.282648	+	0.959224i	$0.591213\pi$
$168$		0			0
$169$	−593502.	−	216017.i	−1.59847	−	0.581797i
$170$		0			0
$171$	−36505.5	+	126916.i	−0.0954704	+	0.331915i
$172$		0			0
$173$	249673.	−	209500.i	0.634243	−	0.532193i	−0.268001	−	0.963419i	$-0.586363\pi$
$173$	249673.	−	209500.i	0.634243	−	0.532193i	0.902244	+	0.431225i	$0.141919\pi$
$174$		0			0
$175$	−57199.6	+	324395.i	−0.141188	+	0.800717i
$176$		0			0
$177$	240090.	+	471682.i	0.576025	+	1.13166i
$178$		0			0
$179$	48536.0	−	84066.7i	0.113222	−	0.196106i	−0.803846	−	0.594838i	$-0.797216\pi$
$179$	48536.0	−	84066.7i	0.113222	−	0.196106i	0.917068	+	0.398732i	$0.130550\pi$
$180$		0			0
$181$	401269.	+	695018.i	0.910414	+	1.57688i	0.813480	+	0.581593i	$0.197570\pi$
$181$	401269.	+	695018.i	0.910414	+	1.57688i	0.0969344	+	0.995291i	$0.469096\pi$
$182$		0			0
$183$	151507.	+	141399.i	0.334429	+	0.312117i
$184$		0			0
$185$	−128262.	+	46683.5i	−0.275529	+	0.100284i
$186$		0			0
$187$	−68813.2	−	390259.i	−0.143902	−	0.816110i
$188$		0			0
$189$	−8214.29	−	506360.i	−0.0167269	−	1.03111i
$190$		0			0
$191$	−6163.05	−	34952.4i	−0.0122240	−	0.0693255i	0.978086	−	0.208203i	$-0.0667616\pi$
$191$	−6163.05	−	34952.4i	−0.0122240	−	0.0693255i	−0.990310	+	0.138878i	$0.955650\pi$
$192$		0			0
$193$	−754460.	+	274601.i	−1.45795	+	0.530651i	−0.944800	−	0.327648i	$-0.893744\pi$
$193$	−754460.	+	274601.i	−1.45795	+	0.530651i	−0.513151	+	0.858299i	$0.671522\pi$
$194$		0			0
$195$	383907.	−	117200.i	0.723002	−	0.220719i
$196$		0			0
$197$	−326296.	−	565161.i	−0.599026	−	1.03754i	−0.992965	−	0.118407i	$-0.962221\pi$
$197$	−326296.	−	565161.i	−0.599026	−	1.03754i	0.393939	−	0.919137i	$-0.371112\pi$
$198$		0			0
$199$	428678.	−	742492.i	0.767359	−	1.32911i	−0.171631	−	0.985161i	$-0.554904\pi$
$199$	428678.	−	742492.i	0.767359	−	1.32911i	0.938990	−	0.343944i	$-0.111763\pi$
$200$		0			0
$201$	355967.	−	547649.i	0.621470	−	0.956119i
$202$		0			0
$203$	4104.37	−	23277.0i	0.00699047	−	0.0396449i
$204$		0			0
$205$	206491.	−	173267.i	0.343176	−	0.287959i
$206$		0			0
$207$	128933.	+	63016.0i	0.209141	+	0.102218i
$208$		0			0
$209$	−169699.	−	61765.3i	−0.268728	−	0.0978091i
$210$		0			0
$211$	594301.	+	498678.i	0.918969	+	0.771106i	0.973804	−	0.227389i	$-0.0730190\pi$
$211$	594301.	+	498678.i	0.918969	+	0.771106i	−0.0548354	+	0.998495i	$0.517463\pi$
$212$		0			0
$213$	956562.	−	720206.i	1.44466	−	1.08770i
$214$		0			0
$215$	−11012.1			−0.0162471
$216$		0			0
$217$	1.11529e6			1.60782
$218$		0			0
$219$	−52095.5	−	425730.i	−0.0733990	−	0.599824i
$220$		0			0
$221$	−914871.	−	767668.i	−1.26002	−	1.05729i
$222$		0			0
$223$	−397764.	−	144774.i	−0.535628	−	0.194952i	0.0600218	−	0.998197i	$-0.480883\pi$
$223$	−397764.	−	144774.i	−0.535628	−	0.194952i	−0.595649	+	0.803245i	$0.703105\pi$
$224$		0			0
$225$	352316.	−	484084.i	0.463955	−	0.637477i
$226$		0			0
$227$	253537.	−	212743.i	0.326571	−	0.274026i	−0.464730	−	0.885452i	$-0.653849\pi$
$227$	253537.	−	212743.i	0.326571	−	0.274026i	0.791301	+	0.611427i	$0.209404\pi$
$228$		0			0
$229$	−198441.	+	1.12541e6i	−0.250059	+	1.41816i	0.558385	+	0.829582i	$0.311421\pi$
$229$	−198441.	+	1.12541e6i	−0.250059	+	1.41816i	−0.808444	+	0.588573i	$0.799690\pi$
$230$		0			0
$231$	691554.	+	36527.7i	0.852700	+	0.0450394i
$232$		0			0
$233$	504053.	−	873045.i	0.608255	−	1.05353i	−0.383272	−	0.923635i	$-0.625203\pi$
$233$	504053.	−	873045.i	0.608255	−	1.05353i	0.991528	−	0.129894i	$-0.0414637\pi$
$234$		0			0
$235$	−81454.6	−	141083.i	−0.0962157	−	0.166650i
$236$		0			0
$237$	303227.	−	1.31096e6i	0.350669	−	1.51607i
$238$		0			0
$239$	−1.05980e6	+	385736.i	−1.20013	+	0.436813i	−0.863271	−	0.504741i	$-0.831588\pi$
$239$	−1.05980e6	+	385736.i	−1.20013	+	0.436813i	−0.336862	+	0.941554i	$0.609366\pi$
$240$		0			0
$241$	147706.	+	837680.i	0.163815	+	0.929043i	0.950277	+	0.311404i	$0.100799\pi$
$241$	147706.	+	837680.i	0.163815	+	0.929043i	−0.786462	+	0.617638i	$0.788090\pi$
$242$		0			0
$243$	−390726.	+	833440.i	−0.424479	+	0.905438i
$244$		0			0
$245$	−4762.58	−	27009.9i	−0.00506905	−	0.0287480i
$246$		0			0
$247$	−511427.	+	186144.i	−0.533385	+	0.194136i
$248$		0			0
$249$	−419903.	+	1.81540e6i	−0.429191	+	1.85555i
$250$		0			0
$251$	−705553.	−	1.22205e6i	−0.706879	−	1.22435i	−0.966009	−	0.258508i	$-0.916769\pi$
$251$	−705553.	−	1.22205e6i	−0.706879	−	1.22435i	0.259130	−	0.965842i	$-0.416564\pi$
$252$		0			0
$253$	−98121.3	+	169951.i	−0.0963744	+	0.166925i
$254$		0			0
$255$	477337.	+	25212.8i	0.459700	+	0.0242812i
$256$		0			0
$257$	−16452.8	+	93308.7i	−0.0155385	+	0.0881230i	−0.991591	−	0.129413i	$-0.958691\pi$
$257$	−16452.8	+	93308.7i	−0.0155385	+	0.0881230i	0.976052	+	0.217536i	$0.0698019\pi$
$258$		0			0
$259$	−543658.	+	456183.i	−0.503589	+	0.422561i
$260$		0			0
$261$	−25280.5	+	34735.5i	−0.0229712	+	0.0315626i
$262$		0			0
$263$	1.50491e6	+	547742.i	1.34159	+	0.488300i	0.910314	−	0.413919i	$-0.135840\pi$
$263$	1.50491e6	+	547742.i	1.34159	+	0.488300i	0.431279	+	0.902219i	$0.358063\pi$
$264$		0			0
$265$	552126.	+	463288.i	0.482973	+	0.405263i
$266$		0			0
$267$	233267.	+	1.90628e6i	0.200251	+	1.63647i
$268$		0			0
$269$	−1.54273e6			−1.29990			−0.649949	−	0.759978i	$-0.725210\pi$
$269$	−1.54273e6			−1.29990			−0.649949	+	0.759978i	$0.725210\pi$
$270$		0			0
$271$	−889733.			−0.735930			−0.367965	−	0.929840i	$-0.619945\pi$
$271$	−889733.			−0.735930			−0.367965	+	0.929840i	$0.619945\pi$
$272$		0			0
$273$	1.66732e6	−	1.25534e6i	1.35398	−	1.01942i
$274$		0			0
$275$	627179.	+	526265.i	0.500103	+	0.419636i
$276$		0			0
$277$	81276.5	+	29582.2i	0.0636452	+	0.0231649i	0.373647	−	0.927571i	$-0.378107\pi$
$277$	81276.5	+	29582.2i	0.0636452	+	0.0231649i	−0.310001	+	0.950736i	$0.600330\pi$
$278$		0			0
$279$	−1.82126e6	−	890142.i	−1.40075	−	0.684619i
$280$		0			0
$281$	−910157.	+	763712.i	−0.687623	+	0.576984i	−0.918223	−	0.396065i	$-0.870376\pi$
$281$	−910157.	+	763712.i	−0.687623	+	0.576984i	0.230600	+	0.973049i	$0.425931\pi$
$282$		0			0
$283$	−199055.	+	1.12890e6i	−0.147743	+	0.837893i	0.817381	+	0.576098i	$0.195425\pi$
$283$	−199055.	+	1.12890e6i	−0.147743	+	0.837893i	−0.965124	+	0.261795i	$0.915686\pi$
$284$		0			0
$285$	118715.	−	182640.i	0.0865750	−	0.133194i
$286$		0			0
$287$	700775.	−	1.21378e6i	0.502196	−	0.869830i
$288$		0			0
$289$	−1170.86	−	2027.98i	−0.000824630	−	0.00142830i
$290$		0			0
$291$	235336.	−	71843.9i	0.162913	−	0.0497344i
$292$		0			0
$293$	−2.46255e6	+	896293.i	−1.67577	+	0.609932i	−0.992721	−	0.120438i	$-0.961570\pi$
$293$	−2.46255e6	+	896293.i	−1.67577	+	0.609932i	−0.683052	+	0.730369i	$0.739348\pi$
$294$		0			0
$295$	−151597.	−	859750.i	−0.101423	−	0.575198i
$296$		0			0
$297$	−1.10015e6	−	611598.i	−0.723705	−	0.402323i
$298$		0			0
$299$	102699.	+	582437.i	0.0664339	+	0.376765i
$300$		0			0
$301$	−53804.4	+	19583.2i	−0.0342295	+	0.0124585i
$302$		0			0
$303$	−6568.73	−	6130.49i	−0.00411031	−	0.00383609i
$304$		0			0
$305$	−170916.	−	296036.i	−0.105204	−	0.182219i
$306$		0			0
$307$	270214.	−	468024.i	0.163629	−	0.283415i	−0.772538	−	0.634968i	$-0.781013\pi$
$307$	270214.	−	468024.i	0.163629	−	0.283415i	0.936168	+	0.351554i	$0.114347\pi$
$308$		0			0
$309$	−359646.	−	706562.i	−0.214278	−	0.420973i
$310$		0			0
$311$	−115604.	+	655622.i	−0.0677752	+	0.384372i	0.931985	+	0.362496i	$0.118075\pi$
$311$	−115604.	+	655622.i	−0.0677752	+	0.384372i	−0.999761	+	0.0218768i	$0.993036\pi$
$312$		0			0
$313$	−1.83121e6	+	1.53657e6i	−1.05652	+	0.886526i	−0.993764	−	0.111503i	$-0.964434\pi$
$313$	−1.83121e6	+	1.53657e6i	−1.05652	+	0.886526i	−0.0627565	+	0.998029i	$0.519989\pi$
$314$		0			0
$315$	−230909.	+	802784.i	−0.131118	+	0.455850i
$316$		0			0
$317$	1.69250e6	+	616021.i	0.945979	+	0.344308i	0.768524	−	0.639821i	$-0.220991\pi$
$317$	1.69250e6	+	616021.i	0.945979	+	0.344308i	0.177455	+	0.984129i	$0.443214\pi$
$318$		0			0
$319$	−45003.3	−	37762.3i	−0.0247610	−	0.0207769i
$320$		0			0
$321$	2.35004e6	+	998674.i	1.27296	+	0.540955i
$322$		0			0
$323$	−648115.			−0.345657
$324$		0			0
$325$	2.46741e6			1.29579
$326$		0			0
$327$	−193074.	−	82048.7i	−0.0998515	−	0.0424329i
$328$		0			0
$329$	−648873.	−	544469.i	−0.330499	−	0.277322i
$330$		0			0
$331$	−2.40321e6	−	874697.i	−1.20565	−	0.438821i	−0.340458	−	0.940260i	$-0.610582\pi$
$331$	−2.40321e6	−	874697.i	−1.20565	−	0.438821i	−0.865193	+	0.501439i	$0.832804\pi$
$332$		0			0
$333$	1.25188e6	−	311037.i	0.618662	−	0.153710i
$334$		0			0
$335$	−825323.	+	692528.i	−0.401802	+	0.337152i
$336$		0			0
$337$	−398182.	+	2.25820e6i	−0.190988	+	1.08315i	0.727029	+	0.686607i	$0.240901\pi$
$337$	−398182.	+	2.25820e6i	−0.190988	+	1.08315i	−0.918017	+	0.396541i	$0.870210\pi$
$338$		0			0
$339$	−112204.	−	220437.i	−0.0530285	−	0.104180i
$340$		0			0
$341$	1.38603e6	−	2.40067e6i	0.645484	−	1.11801i
$342$		0			0
$343$	1.05218e6	+	1.82243e6i	0.482898	+	0.836404i
$344$		0			0
$345$	−173054.	−	161509.i	−0.0782771	−	0.0730547i
$346$		0			0
$347$	−1.25171e6	+	455585.i	−0.558059	+	0.203117i	−0.605623	−	0.795751i	$-0.707076\pi$
$347$	−1.25171e6	+	455585.i	−0.558059	+	0.203117i	0.0475647	+	0.998868i	$0.484854\pi$
$348$		0			0
$349$	−504361.	−	2.86037e6i	−0.221655	−	1.25707i	−0.868977	−	0.494852i	$-0.835222\pi$
$349$	−504361.	−	2.86037e6i	−0.221655	−	1.25707i	0.647322	−	0.762217i	$-0.275889\pi$
$350$		0			0
$351$	−3.72465e6	+	719235.i	−1.61368	+	0.311604i
$352$		0			0
$353$	−311715.	−	1.76783e6i	−0.133144	−	0.755097i	−0.976134	−	0.217168i	$-0.930318\pi$
$353$	−311715.	−	1.76783e6i	−0.133144	−	0.755097i	0.842990	−	0.537929i	$-0.180793\pi$
$354$		0			0
$355$	−1.85592e6	+	675500.i	−0.781607	+	0.284482i
$356$		0			0
$357$	2.37706e6	−	725674.i	0.987121	−	0.301350i
$358$		0			0
$359$	764877.	+	1.32481e6i	0.313225	+	0.542521i	0.979058	−	0.203579i	$-0.0652574\pi$
$359$	764877.	+	1.32481e6i	0.313225	+	0.542521i	−0.665834	+	0.746100i	$0.731924\pi$
$360$		0			0
$361$	1.09037e6	−	1.88858e6i	0.440359	−	0.762724i
$362$		0			0
$363$	−430145.	+	661770.i	−0.171336	+	0.263597i
$364$		0			0
$365$	−122850.	+	696717.i	−0.0482662	+	0.273731i
$366$		0			0
$367$	1.80385e6	−	1.51361e6i	0.699092	−	0.586608i	−0.222423	−	0.974950i	$-0.571397\pi$
$367$	1.80385e6	−	1.51361e6i	0.699092	−	0.586608i	0.921515	+	0.388342i	$0.126952\pi$
$368$		0			0
$369$	−2.11311e6	+	1.42279e6i	−0.807898	+	0.543969i
$370$		0			0
$371$	3.52152e6	+	1.28173e6i	1.32830	+	0.483461i
$372$		0			0
$373$	344518.	+	289085.i	0.128215	+	0.107585i	0.704640	−	0.709565i	$-0.251108\pi$
$373$	344518.	+	289085.i	0.128215	+	0.107585i	−0.576425	+	0.817150i	$0.695553\pi$
$374$		0			0
$375$	−1.78960e6	+	1.34741e6i	−0.657170	+	0.494790i
$376$		0			0
$377$	−177050.			−0.0641567
$378$		0			0
$379$	4.85075e6			1.73465			0.867324	−	0.497744i	$-0.165838\pi$
$379$	4.85075e6			1.73465			0.867324	+	0.497744i	$0.165838\pi$
$380$		0			0
$381$	−109845.	−	897661.i	−0.0387674	−	0.316811i
$382$		0			0
$383$	3.57721e6	+	3.00164e6i	1.24609	+	1.04559i	0.997023	+	0.0771023i	$0.0245668\pi$
$383$	3.57721e6	+	3.00164e6i	1.24609	+	1.04559i	0.249062	+	0.968487i	$0.419878\pi$
$384$		0			0
$385$	−1.07340e6	−	390684.i	−0.369070	−	0.134330i
$386$		0			0
$387$	103492.	+	10963.5i	0.0351261	+	0.00372109i
$388$		0			0
$389$	3.13365e6	−	2.62944e6i	1.04997	−	0.881028i	0.0568782	−	0.998381i	$-0.481885\pi$
$389$	3.13365e6	−	2.62944e6i	1.04997	−	0.881028i	0.993090	+	0.117353i	$0.0374409\pi$
$390$		0			0
$391$	−122299.	+	693591.i	−0.0404558	+	0.229436i
$392$		0			0
$393$	−3.87598e6	−	204728.i	−1.26590	−	0.0668646i
$394$		0			0
$395$	−1.10974e6	+	1.92212e6i	−0.357872	+	0.619853i
$396$		0			0
$397$	−1.66572e6	−	2.88511e6i	−0.530428	−	0.918727i	−0.999370	−	0.0354986i	$-0.988698\pi$
$397$	−1.66572e6	−	2.88511e6i	−0.530428	−	0.918727i	0.468942	−	0.883229i	$-0.344635\pi$
$398$		0			0
$399$	255236.	−	1.10348e6i	0.0802619	−	0.347002i
$400$		0			0
$401$	−641326.	+	233424.i	−0.199167	+	0.0724910i	−0.439678	−	0.898156i	$-0.644907\pi$
$401$	−641326.	+	233424.i	−0.199167	+	0.0724910i	0.240511	+	0.970647i	$0.422685\pi$
$402$		0			0
$403$	−1.45069e6	−	8.22730e6i	−0.444952	−	2.52345i
$404$		0			0
$405$	1.01780e6	−	1.12665e6i	0.308336	−	0.341312i
$406$		0			0
$407$	306307.	+	1.73715e6i	0.0916580	+	0.519818i
$408$		0			0
$409$	4.92303e6	−	1.79184e6i	1.45521	−	0.529652i	0.511166	−	0.859482i	$-0.329214\pi$
$409$	4.92303e6	−	1.79184e6i	1.45521	−	0.529652i	0.944040	+	0.329831i	$0.106992\pi$
$410$		0			0
$411$	−752712.	+	3.25425e6i	−0.219798	+	0.950270i
$412$		0			0
$413$	−2.26961e6	−	3.93108e6i	−0.654751	−	1.13406i
$414$		0			0
$415$	1.53675e6	−	2.66172e6i	0.438008	−	0.758652i
$416$		0			0
$417$	2.12933e6	+	112470.i	0.599656	+	0.0316737i
$418$		0			0
$419$	−18724.1	+	106190.i	−0.00521034	+	0.0295493i	−0.987302	−	0.158854i	$-0.949220\pi$
$419$	−18724.1	+	106190.i	−0.00521034	+	0.0295493i	0.982092	+	0.188403i	$0.0603312\pi$
$420$		0			0
$421$	1.29702e6	−	1.08833e6i	0.356649	−	0.299264i	−0.446805	−	0.894632i	$-0.647438\pi$
$421$	1.29702e6	−	1.08833e6i	0.356649	−	0.299264i	0.803453	+	0.595368i	$0.202994\pi$
$422$		0			0
$423$	625052.	+	1.40700e6i	0.169850	+	0.382334i
$424$		0			0
$425$	2.76110e6	+	1.00496e6i	0.741498	+	0.269883i
$426$		0			0
$427$	−1.36153e6	−	1.14246e6i	−0.361375	−	0.303230i
$428$		0			0
$429$	−630070.	−	5.14899e6i	−0.165290	−	1.35076i
$430$		0			0
$431$	7.65345e6			1.98456			0.992280	−	0.124019i	$-0.0395782\pi$
$431$	7.65345e6			1.98456			0.992280	+	0.124019i	$0.0395782\pi$
$432$		0			0
$433$	−1.15426e6			−0.295857			−0.147929	−	0.988998i	$-0.547261\pi$
$433$	−1.15426e6			−0.295857			−0.147929	+	0.988998i	$0.547261\pi$
$434$		0			0
$435$	56611.2	−	42623.1i	0.0143443	−	0.0108000i
$436$		0			0
$437$	245866.	+	206306.i	0.0615878	+	0.0516783i
$438$		0			0
$439$	−170244.	−	61963.7i	−0.0421609	−	0.0153453i	0.320854	−	0.947129i	$-0.396030\pi$
$439$	−170244.	−	61963.7i	−0.0421609	−	0.0153453i	−0.363015	+	0.931783i	$0.618252\pi$
$440$		0			0
$441$	17868.3	+	258582.i	0.00437509	+	0.0633142i
$442$		0			0
$443$	5.19263e6	−	4.35713e6i	1.25712	−	1.05485i	0.261141	−	0.965301i	$-0.415901\pi$
$443$	5.19263e6	−	4.35713e6i	1.25712	−	1.05485i	0.995982	−	0.0895514i	$-0.0285433\pi$
$444$		0			0
$445$	550082.	−	3.11967e6i	0.131682	−	0.746808i
$446$		0			0
$447$	−2.74175e6	+	4.21813e6i	−0.649021	+	0.998507i
$448$		0			0
$449$	−1.47148e6	+	2.54867e6i	−0.344459	+	0.596620i	−0.985255	−	0.171091i	$-0.945271\pi$
$449$	−1.47148e6	+	2.54867e6i	−0.344459	+	0.596620i	0.640796	+	0.767711i	$0.278604\pi$
$450$		0			0
$451$	−1.74178e6	−	3.01685e6i	−0.403229	−	0.698413i
$452$		0			0
$453$	−562126.	+	171607.i	−0.128703	+	0.0392906i
$454$		0			0
$455$	−3.23493e6	+	1.17742e6i	−0.732548	+	0.266626i
$456$		0			0
$457$	1.24988e6	+	7.08842e6i	0.279948	+	1.58767i	0.722791	+	0.691067i	$0.242859\pi$
$457$	1.24988e6	+	7.08842e6i	0.279948	+	1.58767i	−0.442842	+	0.896599i	$0.646030\pi$
$458$		0			0
$459$	−4.46092e6	−	712177.i	−0.988309	−	0.157782i
$460$		0			0
$461$	939956.	+	5.33076e6i	0.205994	+	1.16825i	0.895867	+	0.444322i	$0.146555\pi$
$461$	939956.	+	5.33076e6i	0.205994	+	1.16825i	−0.689873	+	0.723930i	$0.742334\pi$
$462$		0			0
$463$	−5.81887e6	+	2.11790e6i	−1.26150	+	0.459147i	−0.884271	−	0.466974i	$-0.845344\pi$
$463$	−5.81887e6	+	2.11790e6i	−1.26150	+	0.459147i	−0.377226	+	0.926121i	$0.623122\pi$
$464$		0			0
$465$	2.44450e6	+	2.28141e6i	0.524274	+	0.489296i
$466$		0			0
$467$	−1.78253e6	−	3.08743e6i	−0.378220	−	0.655097i	0.612583	−	0.790406i	$-0.290130\pi$
$467$	−1.78253e6	−	3.08743e6i	−0.378220	−	0.655097i	−0.990803	+	0.135309i	$0.956797\pi$
$468$		0			0
$469$	−2.80092e6	+	4.85133e6i	−0.587988	+	1.01842i
$470$		0			0
$471$	289193.	+	568150.i	0.0600669	+	0.118008i
$472$		0			0
$473$	−24712.5	+	140151.i	−0.00507883	+	0.0288035i
$474$		0			0
$475$	1.02575e6	−	860708.i	0.208597	−	0.175034i
$476$		0			0
$477$	−4.72765e6	−	4.90368e6i	−0.951370	−	0.986794i
$478$		0			0
$479$	−1.63624e6	−	595542.i	−0.325842	−	0.118597i	0.173919	−	0.984760i	$-0.444357\pi$
$479$	−1.63624e6	−	595542.i	−0.325842	−	0.118597i	−0.499762	+	0.866163i	$0.666579\pi$
$480$		0			0
$481$	4.07234e6	+	3.41710e6i	0.802568	+	0.673434i
$482$		0			0
$483$	−1.13275e6	−	481371.i	−0.220935	−	0.0938885i
$484$		0			0
$485$	−405865.			−0.0783479
$486$		0			0
$487$	458780.			0.0876560			0.0438280	−	0.999039i	$-0.486045\pi$
$487$	458780.			0.0876560			0.0438280	+	0.999039i	$0.486045\pi$
$488$		0			0
$489$	277159.	+	117781.i	0.0524151	+	0.0222743i
$490$		0			0
$491$	5.43331e6	+	4.55909e6i	1.01709	+	0.853443i	0.989259	−	0.146170i	$-0.0466946\pi$
$491$	5.43331e6	+	4.55909e6i	1.01709	+	0.853443i	0.0278338	+	0.999613i	$0.491139\pi$
$492$		0			0
$493$	−198123.	−	72111.0i	−0.0367129	−	0.0133624i
$494$		0			0
$495$	1.44104e6	+	1.49469e6i	0.264340	+	0.274182i
$496$		0			0
$497$	−7.86661e6	+	6.60087e6i	−1.42855	+	1.19870i
$498$		0			0
$499$	1.30927e6	−	7.42521e6i	0.235384	−	1.33493i	−0.606420	−	0.795144i	$-0.707395\pi$
$499$	1.30927e6	−	7.42521e6i	0.235384	−	1.33493i	0.841804	−	0.539783i	$-0.181494\pi$
$500$		0			0
$501$	2.03911e6	+	4.00606e6i	0.362950	+	0.713054i
$502$		0			0
$503$	−1.42579e6	+	2.46954e6i	−0.251267	+	0.435207i	−0.963875	−	0.266355i	$-0.914181\pi$
$503$	−1.42579e6	+	2.46954e6i	−0.251267	+	0.435207i	0.712608	+	0.701563i	$0.247514\pi$
$504$		0			0
$505$	7410.26	+	12834.9i	0.00129302	+	0.00223958i
$506$		0			0
$507$	−7.19782e6	−	6.71760e6i	−1.24360	−	1.16063i
$508$		0			0
$509$	−5.66185e6	+	2.06075e6i	−0.968644	+	0.352557i	−0.777415	−	0.628988i	$-0.783469\pi$
$509$	−5.66185e6	+	2.06075e6i	−0.968644	+	0.352557i	−0.191229	+	0.981546i	$0.561247\pi$
$510$		0			0
$511$	638756.	+	3.62257e6i	0.108214	+	0.613711i
$512$		0			0
$513$	−1.29752e6	+	1.59827e6i	−0.217681	+	0.268137i
$514$		0			0
$515$	227087.	+	1.28787e6i	0.0377289	+	0.213971i
$516$		0			0
$517$	−1.97836e6	+	720066.i	−0.325522	+	0.118480i
$518$		0			0
$519$	4.85927e6	−	1.48345e6i	0.791868	−	0.241743i
$520$		0			0
$521$	4.97321e6	+	8.61386e6i	0.802681	+	1.39028i	0.917846	+	0.396937i	$0.129927\pi$
$521$	4.97321e6	+	8.61386e6i	0.802681	+	1.39028i	−0.115165	+	0.993346i	$0.536740\pi$
$522$		0			0
$523$	2.28543e6	−	3.95848e6i	0.365354	−	0.632812i	−0.623479	−	0.781840i	$-0.714281\pi$
$523$	2.28543e6	−	3.95848e6i	0.365354	−	0.632812i	0.988833	+	0.149028i	$0.0476145\pi$
$524$		0			0
$525$	−2.79840e6	+	4.30528e6i	−0.443110	+	0.681716i
$526$		0			0
$527$	1.72755e6	−	9.79742e6i	0.270959	−	1.53669i
$528$		0			0
$529$	−4.66335e6	+	3.91301e6i	−0.724534	+	0.607956i
$530$		0			0
$531$	568764.	+	8.23088e6i	0.0875378	+	1.26681i
$532$		0			0
$533$	−9.86536e6	−	3.59070e6i	−1.50416	−	0.547471i
$534$		0			0
$535$	−3.22643e6	−	2.70730e6i	−0.487347	−	0.408933i
$536$		0			0
$537$	1.20887e6	−	910171.i	0.180902	−	0.136203i
$538$		0			0
$539$	−354443.			−0.0525503
$540$		0			0
$541$	−1.89253e6			−0.278003			−0.139001	−	0.990292i	$-0.544389\pi$
$541$	−1.89253e6			−0.278003			−0.139001	+	0.990292i	$0.544389\pi$
$542$		0			0
$543$	1.51952e6	+	1.24177e7i	0.221161	+	1.80735i
$544$		0			0
$545$	265076.	+	222426.i	0.0382278	+	0.0320770i
$546$		0			0
$547$	−5.28358e6	−	1.92307e6i	−0.755023	−	0.274806i	−0.0643049	−	0.997930i	$-0.520483\pi$
$547$	−5.28358e6	−	1.92307e6i	−0.755023	−	0.274806i	−0.690718	+	0.723125i	$0.742705\pi$
$548$		0			0
$549$	1.31155e6	+	2.95231e6i	0.185718	+	0.418053i
$550$		0			0
$551$	−73603.0	+	61760.3i	−0.0103280	+	0.00866623i
$552$		0			0
$553$	−2.00393e6	+	1.13648e7i	−0.278656	+	1.58034i
$554$		0			0
$555$	−2.12476e6	−	112229.i	−0.292804	−	0.0154658i
$556$		0			0
$557$	5.81379e6	−	1.00698e7i	0.794002	−	1.37525i	−0.129469	−	0.991583i	$-0.541327\pi$
$557$	5.81379e6	−	1.00698e7i	0.794002	−	1.37525i	0.923471	−	0.383668i	$-0.125339\pi$
$558$		0			0
$559$	214447.	+	371433.i	0.0290262	+	0.0502749i
$560$		0			0
$561$	1.39208e6	−	6.01848e6i	0.186749	−	0.807384i
$562$		0			0
$563$	7.10092e6	−	2.58452e6i	0.944156	−	0.343645i	0.176350	−	0.984328i	$-0.443571\pi$
$563$	7.10092e6	−	2.58452e6i	0.944156	−	0.343645i	0.767805	+	0.640683i	$0.221349\pi$
$564$		0			0
$565$	70847.7	+	401797.i	0.00933695	+	0.0529525i
$566$		0			0
$567$	2.96932e6	−	7.31469e6i	0.387882	−	0.955517i
$568$		0			0
$569$	999047.	+	5.66588e6i	0.129362	+	0.733646i	0.978621	+	0.205671i	$0.0659376\pi$
$569$	999047.	+	5.66588e6i	0.129362	+	0.733646i	−0.849260	+	0.527975i	$0.822951\pi$
$570$		0			0
$571$	1.01235e7	−	3.68466e6i	1.29939	−	0.472941i	0.402595	−	0.915378i	$-0.368108\pi$
$571$	1.01235e7	−	3.68466e6i	1.29939	−	0.472941i	0.896799	+	0.442437i	$0.145886\pi$
$572$		0			0
$573$	124678.	−	539028.i	0.0158636	−	0.0685843i
$574$		0			0
$575$	−727542.	−	1.26014e6i	−0.0917675	−	0.158946i
$576$		0			0
$577$	−3.31856e6	+	5.74792e6i	−0.414964	+	0.718739i	−0.995425	−	0.0955494i	$-0.969539\pi$
$577$	−3.31856e6	+	5.74792e6i	−0.414964	+	0.718739i	0.580461	+	0.814288i	$0.302873\pi$
$578$		0			0
$579$	−1.24982e7	−	660153.i	−1.54936	−	0.0818367i
$580$		0			0
$581$	2.77500e6	−	1.57378e7i	0.341054	−	1.93421i
$582$		0			0
$583$	7.13531e6	−	5.98723e6i	0.869443	−	0.729550i
$584$		0			0
$585$	6.22236e6	+	659165.i	0.751736	+	0.0796351i
$586$		0			0
$587$	−5.16274e6	−	1.87908e6i	−0.618422	−	0.225087i	0.0137622	−	0.999905i	$-0.495619\pi$
$587$	−5.16274e6	−	1.87908e6i	−0.618422	−	0.225087i	−0.632184	+	0.774818i	$0.717841\pi$
$588$		0			0
$589$	−3.47301e6	−	2.91420e6i	−0.412494	−	0.346124i
$590$		0			0
$591$	−1.23562e6	−	1.00976e7i	−0.145517	−	1.18918i
$592$		0			0
$593$	−1.12082e7			−1.30888			−0.654441	−	0.756113i	$-0.727096\pi$
$593$	−1.12082e7			−1.30888			−0.654441	+	0.756113i	$0.727096\pi$
$594$		0			0
$595$	−4.09952e6			−0.474724
$596$		0			0
$597$	1.06770e7	−	8.03879e6i	1.22606	−	0.923113i
$598$		0			0
$599$	2.35762e6	+	1.97828e6i	0.268477	+	0.225279i	0.767080	−	0.641551i	$-0.221709\pi$
$599$	2.35762e6	+	1.97828e6i	0.268477	+	0.225279i	−0.498603	+	0.866831i	$0.666153\pi$
$600$		0			0
$601$	1.50944e7	+	5.49390e6i	1.70462	+	0.620432i	0.996339	−	0.0854946i	$-0.0272470\pi$
$601$	1.50944e7	+	5.49390e6i	1.70462	+	0.620432i	0.708285	+	0.705927i	$0.249469\pi$
$602$		0			0
$603$	8.44587e6	−	5.68672e6i	0.945913	−	0.636896i
$604$		0			0
$605$	997307.	−	836840.i	0.110775	−	0.0929510i
$606$		0			0
$607$	670662.	−	3.80351e6i	0.0738809	−	0.418999i	−0.925326	−	0.379173i	$-0.876208\pi$
$607$	670662.	−	3.80351e6i	0.0738809	−	0.418999i	0.999207	−	0.0398262i	$-0.0126804\pi$
$608$		0			0
$609$	200800.	−	308926.i	0.0219391	−	0.0337530i
$610$		0			0
$611$	−3.17245e6	+	5.49484e6i	−0.343789	+	0.595460i
$612$		0			0
$613$	2.38350e6	+	4.12833e6i	0.256191	+	0.443735i	0.965218	−	0.261446i	$-0.0841992\pi$
$613$	2.38350e6	+	4.12833e6i	0.256191	+	0.443735i	−0.709028	+	0.705181i	$0.750866\pi$
$614$		0			0
$615$	4.01885e6	−	1.22688e6i	0.428464	−	0.130802i
$616$		0			0
$617$	−1.28912e7	+	4.69201e6i	−1.36327	+	0.496188i	−0.917063	−	0.398743i	$-0.869446\pi$
$617$	−1.28912e7	+	4.69201e6i	−1.36327	+	0.496188i	−0.446203	+	0.894932i	$0.647224\pi$
$618$		0			0
$619$	2.59717e6	+	1.47293e7i	0.272441	+	1.54509i	0.746974	+	0.664854i	$0.231506\pi$
$619$	2.59717e6	+	1.47293e7i	0.272441	+	1.54509i	−0.474532	+	0.880238i	$0.657383\pi$
$620$		0			0
$621$	1.46557e6	+	1.69015e6i	0.152503	+	0.175872i
$622$		0			0
$623$	−2.86014e6	−	1.62207e7i	−0.295235	−	1.67436i
$624$		0			0
$625$	−3.76305e6	+	1.36964e6i	−0.385336	+	0.140251i
$626$		0			0
$627$	−2.05806e6	−	1.92075e6i	−0.209068	−	0.195120i
$628$		0			0
$629$	3.16530e6	+	5.48246e6i	0.318998	+	0.552521i
$630$		0			0
$631$	−6.29052e6	+	1.08955e7i	−0.628946	+	1.08937i	0.358818	+	0.933408i	$0.383180\pi$
$631$	−6.29052e6	+	1.08955e7i	−0.628946	+	1.08937i	−0.987764	+	0.155958i	$0.950153\pi$
$632$		0			0
$633$	5.48596e6	+	1.07777e7i	0.544181	+	1.06910i
$634$		0			0
$635$	−259032.	+	1.46904e6i	−0.0254929	+	0.144577i
$636$		0			0
$637$	−818286.	+	686623.i	−0.0799018	+	0.0670456i
$638$		0			0
$639$	1.81145e7	−	4.50065e6i	1.75499	−	0.436036i
$640$		0			0
$641$	−8.58464e6	−	3.12455e6i	−0.825234	−	0.300360i	−0.105332	−	0.994437i	$-0.533591\pi$
$641$	−8.58464e6	−	3.12455e6i	−0.825234	−	0.300360i	−0.719901	+	0.694077i	$0.755813\pi$
$642$		0			0
$643$	3.40383e6	+	2.85615e6i	0.324669	+	0.272430i	0.790523	−	0.612432i	$-0.209809\pi$
$643$	3.40383e6	+	2.85615e6i	0.324669	+	0.272430i	−0.465855	+	0.884861i	$0.654253\pi$
$644$		0			0
$645$	−157988.	−	67138.6i	−0.0149529	−	0.00635438i
$646$		0			0
$647$	−1.14683e7			−1.07705			−0.538526	−	0.842609i	$-0.681019\pi$
$647$	−1.14683e7			−1.07705			−0.538526	+	0.842609i	$0.681019\pi$
$648$		0			0
$649$	−1.12823e7			−1.05144
$650$		0			0
$651$	1.60007e7	+	6.79967e6i	1.47975	+	0.628833i
$652$		0			0
$653$	1.08730e7	+	9.12355e6i	0.997855	+	0.837300i	0.986686	−	0.162638i	$-0.0520002\pi$
$653$	1.08730e7	+	9.12355e6i	0.997855	+	0.837300i	0.0111693	+	0.999938i	$0.496445\pi$
$654$		0			0
$655$	6.01610e6	+	2.18968e6i	0.547914	+	0.199424i
$656$		0			0
$657$	1.84818e6	−	6.42545e6i	0.167044	−	0.580751i
$658$		0			0
$659$	1.04013e7	−	8.72775e6i	0.932986	−	0.782868i	−0.0433648	−	0.999059i	$-0.513808\pi$
$659$	1.04013e7	−	8.72775e6i	0.932986	−	0.782868i	0.976351	+	0.216191i	$0.0693633\pi$
$660$		0			0
$661$	1.29447e6	−	7.34133e6i	0.115236	−	0.653538i	−0.871396	−	0.490580i	$-0.836785\pi$
$661$	1.29447e6	−	7.34133e6i	0.115236	−	0.653538i	0.986633	−	0.162959i	$-0.0521038\pi$
$662$		0			0
$663$	−8.44511e6	−	1.65913e7i	−0.746142	−	1.46587i
$664$		0			0
$665$	−934104.	+	1.61791e6i	−0.0819108	+	0.141874i
$666$		0			0
$667$	52205.0	+	90421.6i	0.00454357	+	0.00786969i
$668$		0			0
$669$	−4.82396e6	−	4.50212e6i	−0.416714	−	0.388912i
$670$		0			0
$671$	−4.15121e6	+	1.51092e6i	−0.355933	+	0.129549i
$672$		0			0
$673$	278193.	+	1.57771e6i	0.0236760	+	0.134273i	0.994355	−	0.106108i	$-0.0338388\pi$
$673$	278193.	+	1.57771e6i	0.0236760	+	0.134273i	−0.970679	+	0.240381i	$0.922728\pi$
$674$		0			0
$675$	8.00594e6	−	4.79704e6i	0.676321	−	0.405241i
$676$		0			0
$677$	−2.18064e6	−	1.23670e7i	−0.182857	−	1.03703i	−0.928678	−	0.370887i	$-0.879054\pi$
$677$	−2.18064e6	−	1.23670e7i	−0.182857	−	1.03703i	0.745821	−	0.666146i	$-0.232057\pi$
$678$		0			0
$679$	−1.98302e6	+	721761.i	−0.165064	+	0.0600785i
$680$		0			0
$681$	4.93449e6	−	1.50641e6i	0.407732	−	0.124473i
$682$		0			0
$683$	91780.9	+	158969.i	0.00752836	+	0.0130395i	0.869765	−	0.493466i	$-0.164270\pi$
$683$	91780.9	+	158969.i	0.00752836	+	0.0130395i	−0.862237	+	0.506506i	$0.830937\pi$
$684$		0			0
$685$	2.75475e6	−	4.77136e6i	0.224314	−	0.388523i
$686$		0			0
$687$	−9.70840e6	+	1.49362e7i	−0.784794	+	1.20739i
$688$		0			0
$689$	4.87454e6	−	2.76449e7i	0.391188	−	2.21854i
$690$		0			0
$691$	−1.12564e7	+	9.44525e6i	−0.896819	+	0.752521i	−0.969566	−	0.244830i	$-0.921268\pi$
$691$	−1.12564e7	+	9.44525e6i	−0.896819	+	0.752521i	0.0727470	+	0.997350i	$0.476823\pi$
$692$		0			0
$693$	9.69885e6	+	4.74031e6i	0.767162	+	0.374950i
$694$		0			0
$695$	−3.30504e6	−	1.20293e6i	−0.259546	−	0.0944670i
$696$		0			0
$697$	−9.57714e6	−	8.03617e6i	−0.746713	−	0.626567i
$698$		0			0
$699$	1.25543e7	−	9.45225e6i	0.971849	−	0.731715i
$700$		0			0
$701$	1.30064e7			0.999679			0.499840	−	0.866118i	$-0.333392\pi$
$701$	1.30064e7			0.999679			0.499840	+	0.866118i	$0.333392\pi$
$702$		0			0
$703$	2.88494e6			0.220165
$704$		0			0
$705$	−308452.	−	2.52070e6i	−0.0233730	−	0.191007i
$706$		0			0
$707$	59030.7	+	49532.6i	0.00444150	+	0.00372686i
$708$		0			0
$709$	7.81176e6	+	2.84325e6i	0.583624	+	0.212422i	0.616923	−	0.787024i	$-0.288379\pi$
$709$	7.81176e6	+	2.84325e6i	0.583624	+	0.212422i	−0.0332988	+	0.999445i	$0.510601\pi$
$710$		0			0
$711$	1.23430e7	−	1.69593e7i	0.915685	−	1.25816i
$712$		0			0
$713$	−3.77404e6	+	3.16679e6i	−0.278024	+	0.233290i
$714$		0			0
$715$	−1.48581e6	+	8.42645e6i	−0.108692	+	0.616424i
$716$		0			0
$717$	−1.75564e7	−	927327.i	−1.27538	−	0.0673651i
$718$		0			0
$719$	−6.73988e6	+	1.16738e7i	−0.486217	+	0.842152i	−0.999874	−	0.0158430i	$-0.994957\pi$
$719$	−6.73988e6	+	1.16738e7i	−0.486217	+	0.842152i	0.513658	+	0.857995i	$0.328290\pi$
$720$		0			0
$721$	3.39979e6	+	5.88861e6i	0.243564	+	0.421866i
$722$		0			0
$723$	−2.98807e6	+	1.29185e7i	−0.212591	+	0.919109i
$724$		0			0
$725$	409328.	−	148983.i	0.0289219	−	0.0105267i
$726$		0			0
$727$	−2.47587e6	−	1.40414e7i	−0.173737	−	0.985310i	−0.939592	−	0.342297i	$-0.888795\pi$
$727$	−2.47587e6	−	1.40414e7i	−0.173737	−	0.985310i	0.765855	−	0.643013i	$-0.222316\pi$
$728$		0			0
$729$	−1.06870e7	+	9.57498e6i	−0.744792	+	0.667297i
$730$		0			0
$731$	88690.1	+	502987.i	0.00613878	+	0.0348147i
$732$		0			0
$733$	2.11804e7	−	7.70904e6i	1.45604	−	0.529956i	0.511771	−	0.859122i	$-0.328990\pi$
$733$	2.11804e7	−	7.70904e6i	1.45604	−	0.529956i	0.944272	+	0.329165i	$0.106767\pi$
$734$		0			0
$735$	96346.3	−	416541.i	0.00657835	−	0.0284406i
$736$		0			0
$737$	6.96169e6	+	1.20580e7i	0.472113	+	0.817724i
$738$		0			0
$739$	−6.16222e6	+	1.06733e7i	−0.415075	+	0.718931i	−0.995436	−	0.0954284i	$-0.969578\pi$
$739$	−6.16222e6	+	1.06733e7i	−0.415075	+	0.718931i	0.580362	+	0.814359i	$0.302911\pi$
$740$		0			0
$741$	−8.47219e6	−	447499.i	−0.566826	−	0.0299396i
$742$		0			0
$743$	1.39454e6	−	7.90885e6i	0.0926745	−	0.525583i	−0.902761	−	0.430143i	$-0.858463\pi$
$743$	1.39454e6	−	7.90885e6i	0.0926745	−	0.525583i	0.995435	−	0.0954399i	$-0.0304258\pi$
$744$		0			0
$745$	6.35684e6	−	5.33402e6i	0.419615	−	0.352099i
$746$		0			0
$747$	−1.70923e7	+	2.34850e7i	−1.12073	+	1.53989i
$748$		0			0
$749$	−2.05786e7	−	7.48998e6i	−1.34033	−	0.487839i
$750$		0			0
$751$	3.35271e6	+	2.81326e6i	0.216918	+	0.182016i	0.744771	−	0.667320i	$-0.232558\pi$
$751$	3.35271e6	+	2.81326e6i	0.216918	+	0.182016i	−0.527853	+	0.849336i	$0.677003\pi$
$752$		0			0
$753$	−2.67179e6	−	2.18341e7i	−0.171717	−	1.40329i
$754$		0			0
$755$	969452.			0.0618955
$756$		0			0
$757$	−2.61477e7			−1.65841			−0.829207	−	0.558941i	$-0.811208\pi$
$757$	−2.61477e7			−1.65841			−0.829207	+	0.558941i	$0.811208\pi$
$758$		0			0
$759$	−2.44388e6	+	1.84002e6i	−0.153984	+	0.115936i
$760$		0			0
$761$	1.69818e7	+	1.42495e7i	1.06298	+	0.891942i	0.994398	−	0.105704i	$-0.0337096\pi$
$761$	1.69818e7	+	1.42495e7i	1.06298	+	0.891942i	0.0685774	+	0.997646i	$0.478154\pi$
$762$		0			0
$763$	1.69069e6	+	615360.i	0.105136	+	0.0382664i
$764$		0			0
$765$	6.69451e6	+	3.27194e6i	0.413585	+	0.202140i
$766$		0			0
$767$	−2.60468e7	+	2.18558e7i	−1.59870	+	1.34146i
$768$		0			0
$769$	5.03698e6	−	2.85661e7i	0.307153	−	1.74195i	−0.306041	−	0.952018i	$-0.599005\pi$
$769$	5.03698e6	−	2.85661e7i	0.307153	−	1.74195i	0.613194	−	0.789932i	$-0.289884\pi$
$770$		0			0
$771$	−804928.	+	1.23837e6i	−0.0487665	+	0.0750263i
$772$		0			0
$773$	−6.36605e6	+	1.10263e7i	−0.383196	+	0.663715i	−0.991517	−	0.129976i	$-0.958510\pi$
$773$	−6.36605e6	+	1.10263e7i	−0.383196	+	0.663715i	0.608321	+	0.793691i	$0.291843\pi$
$774$		0			0
$775$	1.02770e7	+	1.78003e7i	0.614628	+	1.06457i
$776$		0			0
$777$	−1.05810e7	+	3.23018e6i	−0.628743	+	0.191944i
$778$		0			0
$779$	−5.35377e6	+	1.94861e6i	−0.316094	+	0.115049i
$780$		0			0
$781$	4.43219e6	+	2.51362e7i	0.260011	+	1.47459i
$782$		0			0
$783$	−574468.	+	344212.i	−0.0334859	+	0.0200642i
$784$		0			0
$785$	−182602.	−	1.03558e6i	−0.0105762	−	0.0599807i
$786$		0			0
$787$	−1.90712e7	+	6.94133e6i	−1.09759	+	0.399490i	−0.826427	−	0.563043i	$-0.809630\pi$
$787$	−1.90712e7	+	6.94133e6i	−1.09759	+	0.399490i	−0.271163	+	0.962533i	$0.587408\pi$
$788$		0			0
$789$	1.82511e7	+	1.70334e7i	1.04375	+	0.974113i
$790$		0			0
$791$	1.06068e6	+	1.83716e6i	0.0602761	+	0.104401i
$792$		0			0
$793$	−6.65676e6	+	1.15298e7i	−0.375907	+	0.651089i
$794$		0			0
$795$	5.09663e6	+	1.00129e7i	0.286000	+	0.561877i
$796$		0			0
$797$	5.83486e6	−	3.30911e7i	0.325375	−	1.84529i	−0.181650	−	0.983363i	$-0.558144\pi$
$797$	5.83486e6	−	3.30911e7i	0.325375	−	1.84529i	0.507025	−	0.861931i	$-0.330745\pi$
$798$		0			0
$799$	−5.78806e6	+	4.85676e6i	−0.320750	+	0.269141i
$800$		0			0
$801$	−8.27557e6	+	2.87711e7i	−0.455739	+	1.58444i
$802$		0			0
$803$	8.59143e6	+	3.12702e6i	0.470193	+	0.171136i
$804$		0			0
$805$	1.55517e6	+	1.30495e6i	0.0845842	+	0.0709746i
$806$		0			0
$807$	−2.21332e7	−	9.40570e6i	−1.19635	−	0.508402i
$808$		0			0
$809$	1.55120e7			0.833293			0.416647	−	0.909069i	$-0.363205\pi$
$809$	1.55120e7			0.833293			0.416647	+	0.909069i	$0.363205\pi$
$810$		0			0
$811$	−9.37585e6			−0.500563			−0.250281	−	0.968173i	$-0.580523\pi$
$811$	−9.37585e6			−0.500563			−0.250281	+	0.968173i	$0.580523\pi$
$812$		0			0
$813$	−1.27648e7	−	5.42452e6i	−0.677309	−	0.287829i
$814$		0			0
$815$	−380518.	−	319293.i	−0.0200670	−	0.0168382i
$816$		0			0
$817$	218717.	+	79606.5i	0.0114638	+	0.00417247i
$818$		0			0
$819$	3.15741e7	−	7.84477e6i	1.64483	−	0.408668i
$820$		0			0
$821$	−2.66274e7	+	2.23430e7i	−1.37870	+	1.15687i	−0.409011	+	0.912529i	$0.634126\pi$
$821$	−2.66274e7	+	2.23430e7i	−1.37870	+	1.15687i	−0.969690	+	0.244338i	$0.921429\pi$
$822$		0			0
$823$	1.98535e6	−	1.12595e7i	0.102173	−	0.579453i	−0.890139	−	0.455690i	$-0.849393\pi$
$823$	1.98535e6	−	1.12595e7i	0.102173	−	0.579453i	0.992312	−	0.123763i	$-0.0394962\pi$
$824$		0			0
$825$	5.78944e6	+	1.13740e7i	0.296143	+	0.581805i
$826$		0			0
$827$	−4.58157e6	+	7.93550e6i	−0.232943	+	0.403470i	−0.958673	−	0.284510i	$-0.908169\pi$
$827$	−4.58157e6	+	7.93550e6i	−0.232943	+	0.403470i	0.725730	+	0.687980i	$0.241502\pi$
$828$		0			0
$829$	−1.48944e7	−	2.57978e7i	−0.752723	−	1.30376i	−0.946498	−	0.322709i	$-0.895406\pi$
$829$	−1.48944e7	−	2.57978e7i	−0.752723	−	1.30376i	0.193775	−	0.981046i	$-0.437927\pi$
$830$		0			0
$831$	985697.	+	919934.i	0.0495154	+	0.0462119i
$832$		0			0
$833$	−1.19534e6	+	435068.i	−0.0596869	+	0.0217243i
$834$		0			0
$835$	−1.28753e6	−	7.30197e6i	−0.0639061	−	0.362430i
$836$		0			0
$837$	−2.07022e7	−	2.38745e7i	−1.02142	−	1.17793i
$838$		0			0
$839$	−2.63622e6	−	1.49507e7i	−0.129293	−	0.733259i	−0.978665	−	0.205464i	$-0.934130\pi$
$839$	−2.63622e6	−	1.49507e7i	−0.129293	−	0.733259i	0.849371	−	0.527796i	$-0.176981\pi$
$840$		0			0
$841$	1.92448e7	−	7.00454e6i	0.938261	−	0.341499i
$842$		0			0
$843$	−1.77140e7	+	5.40775e6i	−0.858514	+	0.262088i
$844$		0			0
$845$	8.11994e6	+	1.40642e7i	0.391211	+	0.677598i
$846$		0			0
$847$	3.38458e6	−	5.86227e6i	0.162105	−	0.280774i
$848$		0			0
$849$	−9.73844e6	+	1.49824e7i	−0.463682	+	0.713366i
$850$		0			0
$851$	544386.	−	3.08737e6i	0.0257682	−	0.146138i
$852$		0			0
$853$	−5.01684e6	+	4.20963e6i	−0.236079	+	0.198094i	−0.753150	−	0.657848i	$-0.771467\pi$
$853$	−5.01684e6	+	4.20963e6i	−0.236079	+	0.198094i	0.517071	+	0.855942i	$0.327022\pi$
$854$		0			0
$855$	2.81669e6	−	1.89652e6i	0.131772	−	0.0887241i
$856$		0			0
$857$	2.47644e7	+	9.01350e6i	1.15180	+	0.419220i	0.846160	−	0.532929i	$-0.178909\pi$
$857$	2.47644e7	+	9.01350e6i	1.15180	+	0.419220i	0.305636	+	0.952148i	$0.401131\pi$
$858$		0			0
$859$	2.95997e6	+	2.48371e6i	0.136869	+	0.114847i	0.708652	−	0.705558i	$-0.249304\pi$
$859$	2.95997e6	+	2.48371e6i	0.136869	+	0.114847i	−0.571783	+	0.820405i	$0.693748\pi$
$860$		0			0
$861$	1.74540e7	−	1.31413e7i	0.802392	−	0.604129i
$862$		0			0
$863$	1.08453e7			0.495693			0.247846	−	0.968799i	$-0.420277\pi$
$863$	1.08453e7			0.495693			0.247846	+	0.968799i	$0.420277\pi$
$864$		0			0
$865$	−8.38038e6			−0.380823
$866$		0			0
$867$	−4433.80	−	36233.4i	−0.000200322	−	0.00163705i
$868$		0			0
$869$	2.19725e7	+	1.84371e7i	0.987030	+	0.828216i
$870$		0			0
$871$	3.94307e7	+	1.43516e7i	1.76112	+	0.640996i
$872$		0			0
$873$	3.81433e6	+	404071.i	0.169388	+	0.0179441i
$874$		0			0
$875$	1.47174e7	−	1.23493e7i	0.649846	−	0.545285i
$876$		0			0
$877$	5.60681e6	−	3.17978e7i	0.246160	−	1.39604i	−0.571624	−	0.820516i	$-0.693686\pi$
$877$	5.60681e6	−	3.17978e7i	0.246160	−	1.39604i	0.817784	−	0.575525i	$-0.195202\pi$
$878$		0			0
$879$	−4.07940e7	−	2.15473e6i	−1.78084	−	0.0940634i
$880$		0			0
$881$	−985728.	+	1.70733e6i	−0.0427875	+	0.0741102i	−0.886626	−	0.462487i	$-0.846957\pi$
$881$	−985728.	+	1.70733e6i	−0.0427875	+	0.0741102i	0.843839	+	0.536597i	$0.180291\pi$
$882$		0			0
$883$	−1.29379e7	−	2.24091e7i	−0.558422	−	0.967215i	−0.997628	−	0.0688290i	$-0.978074\pi$
$883$	−1.29379e7	−	2.24091e7i	−0.558422	−	0.967215i	0.439207	−	0.898386i	$-0.355260\pi$
$884$		0			0
$885$	3.06679e6	−	1.32589e7i	0.131621	−	0.569048i
$886$		0			0
$887$	−2.05759e7	+	7.48901e6i	−0.878111	+	0.319606i	−0.741447	−	0.671011i	$-0.765860\pi$
$887$	−2.05759e7	+	7.48901e6i	−0.878111	+	0.319606i	−0.136664	+	0.990617i	$0.543638\pi$
$888$		0			0
$889$	1.34683e6	+	7.63827e6i	0.0571557	+	0.324146i
$890$		0			0
$891$	−1.20548e7	−	1.54818e7i	−0.508705	−	0.653323i
$892$		0			0
$893$	597918.	+	3.39096e6i	0.0250907	+	0.142296i
$894$		0			0
$895$	−2.34545e6	+	853673.i	−0.0978742	+	0.0356233i
$896$		0			0
$897$	−2.07759e6	+	8.98222e6i	−0.0862144	+	0.372737i
$898$		0			0
$899$	−737428.	−	1.27726e6i	−0.0304313	−	0.0527086i
$900$		0			0
$901$	1.67143e7	−	2.89500e7i	0.685924	−	1.18805i
$902$		0			0
$903$	−891312.	−	47078.9i	−0.0363756	−	0.00192135i
$904$		0			0
$905$	3.58329e6	−	2.03219e7i	0.145432	−	0.824787i
$906$		0			0
$907$	2.47389e7	−	2.07584e7i	0.998531	−	0.837867i	0.0117505	−	0.999931i	$-0.496260\pi$
$907$	2.47389e7	−	2.07584e7i	0.998531	−	0.837867i	0.986780	+	0.162064i	$0.0518152\pi$
$908$		0			0
$909$	−56863.6	−	128001.i	−0.00228257	−	0.00513810i
$910$		0			0
$911$	9.36439e6	+	3.40836e6i	0.373838	+	0.136066i	0.522104	−	0.852882i	$-0.325147\pi$
$911$	9.36439e6	+	3.40836e6i	0.373838	+	0.136066i	−0.148266	+	0.988947i	$0.547369\pi$
$912$		0			0
$913$	−3.04271e7	−	2.55314e7i	−1.20805	−	1.01367i
$914$		0			0
$915$	−647226.	−	5.28919e6i	−0.0255566	−	0.208851i
$916$		0			0
$917$	3.32882e7			1.30727
$918$		0			0
$919$	5.34548e6			0.208784			0.104392	−	0.994536i	$-0.466710\pi$
$919$	5.34548e6			0.208784			0.104392	+	0.994536i	$0.466710\pi$
$920$		0			0
$921$	6.73013e6	−	5.06719e6i	0.261442	−	0.196842i
$922$		0			0
$923$	5.89260e7	+	4.94448e7i	2.27668	+	1.91036i
$924$		0			0
$925$	−1.22904e7	−	4.47335e6i	−0.472294	−	0.171901i
$926$		0			0
$927$	−851986.	−	1.23295e7i	−0.0325637	−	0.471246i
$928$		0			0
$929$	2.90706e7	−	2.43931e7i	1.10513	−	0.927316i	0.107373	−	0.994219i	$-0.465756\pi$
$929$	2.90706e7	−	2.43931e7i	1.10513	−	0.927316i	0.997760	+	0.0669024i	$0.0213116\pi$
$930$		0			0
$931$	−100663.	+	570886.i	−0.00380622	+	0.0215861i
$932$		0			0
$933$	−5.65573e6	+	8.70123e6i	−0.212708	+	0.327247i
$934$		0			0
$935$	−5.09469e6	+	8.82426e6i	−0.190585	+	0.330103i
$936$		0			0
$937$	−7.98779e6	−	1.38353e7i	−0.297220	−	0.514800i	0.678279	−	0.734804i	$-0.262726\pi$
$937$	−7.98779e6	−	1.38353e7i	−0.297220	−	0.514800i	−0.975499	+	0.220005i	$0.929393\pi$
$938$		0			0
$939$	−3.56401e7	+	1.08803e7i	−1.31909	+	0.402694i
$940$		0			0
$941$	−3.52684e6	+	1.28367e6i	−0.129841	+	0.0472583i	−0.406123	−	0.913818i	$-0.633120\pi$
$941$	−3.52684e6	+	1.28367e6i	−0.129841	+	0.0472583i	0.276282	+	0.961077i	$0.410897\pi$
$942$		0			0
$943$	1.07509e6	+	6.09713e6i	0.0393699	+	0.223278i
$944$		0			0
$945$	−8.20719e6	+	1.01095e7i	−0.298961	+	0.368257i
$946$		0			0
$947$	−1.72770e6	−	9.79828e6i	−0.0626028	−	0.355038i	−0.999977	−	0.00671534i	$-0.997862\pi$
$947$	−1.72770e6	−	9.79828e6i	−0.0626028	−	0.355038i	0.937375	−	0.348323i	$-0.113249\pi$
$948$		0			0
$949$	2.58922e7	−	9.42401e6i	0.933263	−	0.339680i
$950$		0			0
$951$	2.05262e7	+	1.91567e7i	0.735964	+	0.686863i
$952$		0			0
$953$	−3.28674e6	−	5.69280e6i	−0.117228	−	0.203046i	0.801440	−	0.598075i	$-0.204068\pi$
$953$	−3.28674e6	−	5.69280e6i	−0.117228	−	0.203046i	−0.918668	+	0.395030i	$0.870734\pi$
$954$		0			0
$955$	−456291.	+	790318.i	−0.0161895	+	0.0280410i
$956$		0			0
$957$	−415423.	−	816142.i	−0.0146626	−	0.0288062i
$958$		0			0
$959$	4.97442e6	−	2.82113e7i	0.174661	−	0.990552i
$960$		0			0
$961$	3.13795e7	−	2.63305e7i	1.09607	−	0.919709i
$962$		0			0
$963$	2.76268e7	+	2.86554e7i	0.959985	+	0.995729i
$964$		0			0
$965$	1.93991e7	+	7.06071e6i	0.670601	+	0.244079i
$966$		0			0
$967$	−8.19735e6	−	6.87839e6i	−0.281908	−	0.236549i	0.490859	−	0.871239i	$-0.336683\pi$
$967$	−8.19735e6	−	6.87839e6i	−0.281908	−	0.236549i	−0.772766	+	0.634690i	$0.781128\pi$
$968$		0			0
$969$	−9.29834e6	−	3.95142e6i	−0.318124	−	0.135190i
$970$		0			0
$971$	−3.59515e7			−1.22368			−0.611841	−	0.790981i	$-0.709571\pi$
$971$	−3.59515e7			−1.22368			−0.611841	+	0.790981i	$0.709571\pi$
$972$		0			0
$973$	−1.82873e7			−0.619254
$974$		0			0
$975$	3.53993e7	+	1.50433e7i	1.19257	+	0.506794i
$976$		0			0
$977$	−7.82984e6	−	6.57002e6i	−0.262432	−	0.220207i	0.502072	−	0.864826i	$-0.332571\pi$
$977$	−7.82984e6	−	6.57002e6i	−0.262432	−	0.220207i	−0.764504	+	0.644620i	$0.777016\pi$
$978$		0			0
$979$	−3.84696e7	−	1.40018e7i	−1.28281	−	0.466903i
$980$		0			0
$981$	−2.26975e6	−	2.35427e6i	−0.0753019	−	0.0781057i
$982$		0			0
$983$	−1.76914e7	+	1.48448e7i	−0.583954	+	0.489995i	−0.886243	−	0.463221i	$-0.846694\pi$
$983$	−1.76914e7	+	1.48448e7i	−0.583954	+	0.489995i	0.302289	+	0.953216i	$0.402249\pi$
$984$		0			0
$985$	−2.91379e6	+	1.65249e7i	−0.0956902	+	0.542686i
$986$		0			0
$987$	−5.98971e6	−	1.17674e7i	−0.195710	−	0.384493i
$988$		0			0
$989$	126464.	−	219042.i	0.00411127	−	0.00712093i
$990$		0			0
$991$	2.56339e7	+	4.43993e7i	0.829147	+	1.43612i	0.898709	+	0.438546i	$0.144507\pi$
$991$	2.56339e7	+	4.43993e7i	0.829147	+	1.43612i	−0.0695619	+	0.997578i	$0.522160\pi$
$992$		0			0
$993$	−2.91454e7	−	2.72009e7i	−0.937987	−	0.875408i
$994$		0			0
$995$	−2.07154e7	+	7.53979e6i	−0.663339	+	0.241436i
$996$		0			0
$997$	−4.96174e6	−	2.81394e7i	−0.158087	−	0.896556i	−0.955909	−	0.293663i	$-0.905126\pi$
$997$	−4.96174e6	−	2.81394e7i	−0.158087	−	0.896556i	0.797822	−	0.602893i	$-0.205985\pi$
$998$		0			0
$999$	1.98568e7	+	3.17010e6i	0.629500	+	0.100498i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.13.14	✓	90
3.2	odd	2		324.6.i.a.253.10		90
27.2	odd	18		324.6.i.a.73.10		90
27.25	even	9	inner	108.6.i.a.25.14	yes	90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.14	✓	90	1.1	even	1	trivial
108.6.i.a.25.14	yes	90	27.25	even	9	inner
324.6.i.a.73.10		90	27.2	odd	18
324.6.i.a.253.10		90	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(14.3467 + 6.09679i) q^{3} +(-19.6970 - 16.5278i) q^{5} +(-125.630 - 45.7255i) q^{7} +(168.658 + 174.938i) q^{9} +O(q^{10})\)	\(q+(14.3467 + 6.09679i) q^{3} +(-19.6970 - 16.5278i) q^{5} +(-125.630 - 45.7255i) q^{7} +(168.658 + 174.938i) q^{9} +(-254.551 + 213.594i) q^{11} +(-173.898 + 986.227i) q^{13} +(-181.822 - 357.208i) q^{15} +(-596.280 + 1032.79i) q^{17} +(271.733 + 470.655i) q^{19} +(-1523.60 - 1421.95i) q^{21} +(554.955 - 201.987i) q^{23} +(-427.845 - 2426.43i) q^{25} +(1353.14 + 3538.07i) q^{27} +(30.7001 + 174.109i) q^{29} +(-7839.09 + 2853.20i) q^{31} +(-4954.22 + 1512.43i) q^{33} +(1718.79 + 2977.03i) q^{35} +(2654.21 - 4597.22i) q^{37} +(-8507.70 + 13088.9i) q^{39} +(-1820.42 + 10324.1i) q^{41} +(328.079 - 275.291i) q^{43} +(-430.729 - 6233.30i) q^{45} +(5953.67 + 2166.96i) q^{47} +(817.108 + 685.635i) q^{49} +(-14851.4 + 11181.7i) q^{51} -28030.9 q^{53} +8544.12 q^{55} +(1029.00 + 8409.06i) q^{57} +(26009.3 + 21824.4i) q^{59} +(12492.6 + 4546.94i) q^{61} +(-13189.4 - 29689.4i) q^{63} +(19725.4 - 16551.6i) q^{65} +(7276.02 - 41264.3i) q^{67} +(9193.27 + 485.586i) q^{69} +(38405.8 - 66520.9i) q^{71} +(-13757.1 - 23828.1i) q^{73} +(8655.25 - 37419.9i) q^{75} +(41745.9 - 15194.3i) q^{77} +(-14989.1 - 85007.3i) q^{79} +(-2157.75 + 59009.6i) q^{81} +(20756.6 + 117717. i) q^{83} +(28814.6 - 10487.6i) q^{85} +(-621.059 + 2685.07i) q^{87} +(61600.0 + 106694. i) q^{89} +(66942.6 - 115948. i) q^{91} +(-129861. - 6859.22i) q^{93} +(2426.55 - 13761.6i) q^{95} +(12091.7 - 10146.2i) q^{97} +(-80297.9 - 8506.36i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 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