LMFDB - Embedded newform 108.5.k.a.5.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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, 5]))

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

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.11
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.08622 + 3.95134i) q^{3} +(-10.7879 - 29.6395i) q^{5} +(15.2291 + 86.3688i) q^{7} +(49.7738 + 63.9028i) q^{9} +O(q^{10})$	\(q+(8.08622 + 3.95134i) q^{3} +(-10.7879 - 29.6395i) q^{5} +(15.2291 + 86.3688i) q^{7} +(49.7738 + 63.9028i) q^{9} +(46.1813 - 126.882i) q^{11} +(215.694 + 180.988i) q^{13} +(29.8824 - 282.298i) q^{15} +(300.316 + 173.387i) q^{17} +(-113.736 - 196.996i) q^{19} +(-218.126 + 758.572i) q^{21} +(-187.562 - 33.0723i) q^{23} +(-283.341 + 237.752i) q^{25} +(149.980 + 713.405i) q^{27} +(429.966 + 512.413i) q^{29} +(-35.8536 + 203.336i) q^{31} +(874.786 - 843.518i) q^{33} +(2395.63 - 1383.12i) q^{35} +(987.943 - 1711.17i) q^{37} +(1029.00 + 2315.79i) q^{39} +(-1257.73 + 1498.90i) q^{41} +(-793.064 - 288.652i) q^{43} +(1357.09 - 2164.64i) q^{45} +(-1802.00 + 317.741i) q^{47} +(-4971.44 + 1809.45i) q^{49} +(1743.31 + 2588.70i) q^{51} -2964.60i q^{53} -4258.91 q^{55} +(-141.294 - 2042.36i) q^{57} +(-1888.83 - 5189.52i) q^{59} +(680.783 + 3860.91i) q^{61} +(-4761.19 + 5272.09i) q^{63} +(3037.52 - 8345.52i) q^{65} +(-3957.28 - 3320.55i) q^{67} +(-1385.99 - 1008.55i) q^{69} +(-1162.57 - 671.209i) q^{71} +(-1153.91 - 1998.62i) q^{73} +(-3230.60 + 802.933i) q^{75} +(11662.0 + 2056.32i) q^{77} +(6080.80 - 5102.39i) q^{79} +(-1606.13 + 6361.37i) q^{81} +(-879.339 - 1047.96i) q^{83} +(1899.34 - 10771.7i) q^{85} +(1452.08 + 5842.43i) q^{87} +(-2800.19 + 1616.69i) q^{89} +(-12346.9 + 21385.5i) q^{91} +(-1093.37 + 1502.55i) q^{93} +(-4611.89 + 5496.24i) q^{95} +(-10325.5 - 3758.17i) q^{97} +(10406.7 - 3364.29i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{5}{18}\right)$	$1$

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
$3$	8.08622	+	3.95134i	0.898469	+	0.439038i
$3$	8.08622	+	3.95134i	0.898469	+	0.439038i
$4$		0			0
$5$	−10.7879	−	29.6395i	−0.431515	−	1.18558i	−0.944883	−	0.327409i	$-0.893824\pi$
$5$	−10.7879	−	29.6395i	−0.431515	−	1.18558i	0.513367	−	0.858169i	$-0.328398\pi$
$6$		0			0
$7$	15.2291	+	86.3688i	0.310799	+	1.76263i	0.594867	+	0.803824i	$0.297205\pi$
$7$	15.2291	+	86.3688i	0.310799	+	1.76263i	−0.284068	+	0.958804i	$0.591684\pi$
$8$		0			0
$9$	49.7738	+	63.9028i	0.614492	+	0.788923i
$10$		0			0
$11$	46.1813	−	126.882i	0.381664	−	1.04861i	−0.588992	−	0.808139i	$-0.700475\pi$
$11$	46.1813	−	126.882i	0.381664	−	1.04861i	0.970656	−	0.240473i	$-0.0773027\pi$
$12$		0			0
$13$	215.694	+	180.988i	1.27629	+	1.07094i	0.993744	+	0.111684i	$0.0356245\pi$
$13$	215.694	+	180.988i	1.27629	+	1.07094i	0.282549	+	0.959253i	$0.408820\pi$
$14$		0			0
$15$	29.8824	−	282.298i	0.132811	−	1.25466i
$16$		0			0
$17$	300.316	+	173.387i	1.03915	+	0.599956i	0.919594	−	0.392871i	$-0.128518\pi$
$17$	300.316	+	173.387i	1.03915	+	0.599956i	0.119561	+	0.992827i	$0.461851\pi$
$18$		0			0
$19$	−113.736	−	196.996i	−0.315058	−	0.545696i	0.664392	−	0.747384i	$-0.268690\pi$
$19$	−113.736	−	196.996i	−0.315058	−	0.545696i	−0.979450	+	0.201688i	$0.935357\pi$
$20$		0			0
$21$	−218.126	+	758.572i	−0.494617	+	1.72012i
$22$		0			0
$23$	−187.562	−	33.0723i	−0.354560	−	0.0625185i	−0.00646861	−	0.999979i	$-0.502059\pi$
$23$	−187.562	−	33.0723i	−0.354560	−	0.0625185i	−0.348091	+	0.937461i	$0.613170\pi$
$24$		0			0
$25$	−283.341	+	237.752i	−0.453346	+	0.380403i
$26$		0			0
$27$	149.980	+	713.405i	0.205734	+	0.978608i
$28$		0			0
$29$	429.966	+	512.413i	0.511256	+	0.609291i	0.958490	−	0.285126i	$-0.0920355\pi$
$29$	429.966	+	512.413i	0.511256	+	0.609291i	−0.447234	+	0.894417i	$0.647591\pi$
$30$		0			0
$31$	−35.8536	+	203.336i	−0.0373087	+	0.211588i	−0.997763	−	0.0668493i	$-0.978705\pi$
$31$	−35.8536	+	203.336i	−0.0373087	+	0.211588i	0.960454	+	0.278437i	$0.0898164\pi$
$32$		0			0
$33$	874.786	−	843.518i	0.803293	−	0.774580i
$34$		0			0
$35$	2395.63	−	1383.12i	1.95562	−	1.12908i
$36$		0			0
$37$	987.943	−	1711.17i	0.721653	−	1.24994i	−0.238684	−	0.971097i	$-0.576716\pi$
$37$	987.943	−	1711.17i	0.721653	−	1.24994i	0.960337	−	0.278842i	$-0.0899507\pi$
$38$		0			0
$39$	1029.00	+	2315.79i	0.676528	+	1.52254i
$40$		0			0
$41$	−1257.73	+	1498.90i	−0.748203	+	0.891673i	−0.997041	−	0.0768719i	$-0.975507\pi$
$41$	−1257.73	+	1498.90i	−0.748203	+	0.891673i	0.248838	+	0.968545i	$0.419951\pi$
$42$		0			0
$43$	−793.064	−	288.652i	−0.428915	−	0.156112i	0.118536	−	0.992950i	$-0.462180\pi$
$43$	−793.064	−	288.652i	−0.428915	−	0.156112i	−0.547452	+	0.836837i	$0.684402\pi$
$44$		0			0
$45$	1357.09	−	2164.64i	0.670168	−	1.06896i
$46$		0			0
$47$	−1802.00	+	317.741i	−0.815754	+	0.143839i	−0.565932	−	0.824452i	$-0.691483\pi$
$47$	−1802.00	+	317.741i	−0.815754	+	0.143839i	−0.249822	+	0.968292i	$0.580372\pi$
$48$		0			0
$49$	−4971.44	+	1809.45i	−2.07057	+	0.753626i
$50$		0			0
$51$	1743.31	+	2588.70i	0.670244	+	0.995270i
$52$		0			0
$53$		−	2964.60i		−	1.05539i	−0.849432	−	0.527697i	$-0.823056\pi$
$53$		−	2964.60i		−	1.05539i	0.849432	−	0.527697i	$-0.176944\pi$
$54$		0			0
$55$	−4258.91			−1.40791
$56$		0			0
$57$	−141.294	−	2042.36i	−0.0434883	−	0.628613i
$58$		0			0
$59$	−1888.83	−	5189.52i	−0.542612	−	1.49081i	−0.843487	−	0.537150i	$-0.819501\pi$
$59$	−1888.83	−	5189.52i	−0.542612	−	1.49081i	0.300875	−	0.953664i	$-0.402721\pi$
$60$		0			0
$61$	680.783	+	3860.91i	0.182957	+	1.03760i	0.928552	+	0.371202i	$0.121054\pi$
$61$	680.783	+	3860.91i	0.182957	+	1.03760i	−0.745595	+	0.666399i	$0.767835\pi$
$62$		0			0
$63$	−4761.19	+	5272.09i	−1.19959	+	1.32832i
$64$		0			0
$65$	3037.52	−	8345.52i	0.718940	−	1.97527i
$66$		0			0
$67$	−3957.28	−	3320.55i	−0.881549	−	0.739708i	0.0849477	−	0.996385i	$-0.472928\pi$
$67$	−3957.28	−	3320.55i	−0.881549	−	0.739708i	−0.966497	+	0.256678i	$0.917372\pi$
$68$		0			0
$69$	−1385.99	−	1008.55i	−0.291113	−	0.211836i
$70$		0			0
$71$	−1162.57	−	671.209i	−0.230622	−	0.133150i	0.380237	−	0.924889i	$-0.375843\pi$
$71$	−1162.57	−	671.209i	−0.230622	−	0.133150i	−0.610859	+	0.791739i	$0.709176\pi$
$72$		0			0
$73$	−1153.91	−	1998.62i	−0.216533	−	0.375047i	0.737213	−	0.675661i	$-0.236142\pi$
$73$	−1153.91	−	1998.62i	−0.216533	−	0.375047i	−0.953746	+	0.300614i	$0.902808\pi$
$74$		0			0
$75$	−3230.60	+	802.933i	−0.574328	+	0.142744i
$76$		0			0
$77$	11662.0	+	2056.32i	1.96693	+	0.346824i
$78$		0			0
$79$	6080.80	−	5102.39i	0.974330	−	0.817560i	−0.00889403	−	0.999960i	$-0.502831\pi$
$79$	6080.80	−	5102.39i	0.974330	−	0.817560i	0.983224	+	0.182400i	$0.0583867\pi$
$80$		0			0
$81$	−1606.13	+	6361.37i	−0.244800	+	0.969574i
$82$		0			0
$83$	−879.339	−	1047.96i	−0.127644	−	0.152120i	0.698437	−	0.715671i	$-0.253879\pi$
$83$	−879.339	−	1047.96i	−0.127644	−	0.152120i	−0.826081	+	0.563551i	$0.809435\pi$
$84$		0			0
$85$	1899.34	−	10771.7i	0.262884	−	1.49089i
$86$		0			0
$87$	1452.08	+	5842.43i	0.191846	+	0.771889i
$88$		0			0
$89$	−2800.19	+	1616.69i	−0.353515	+	0.204102i	−0.666232	−	0.745744i	$-0.732094\pi$
$89$	−2800.19	+	1616.69i	−0.353515	+	0.204102i	0.312717	+	0.949846i	$0.398761\pi$
$90$		0			0
$91$	−12346.9	+	21385.5i	−1.49099	+	2.58248i
$92$		0			0
$93$	−1093.37	+	1502.55i	−0.126416	+	0.173725i
$94$		0			0
$95$	−4611.89	+	5496.24i	−0.511013	+	0.609001i
$96$		0			0
$97$	−10325.5	−	3758.17i	−1.09741	−	0.399423i	−0.271046	−	0.962566i	$-0.587370\pi$
$97$	−10325.5	−	3758.17i	−1.09741	−	0.399423i	−0.826359	+	0.563143i	$0.809592\pi$
$98$		0			0
$99$	10406.7	−	3364.29i	1.06180	−	0.343260i
$100$		0			0
$101$	17983.3	−	3170.94i	1.76289	−	0.310846i	0.804005	−	0.594623i	$-0.202699\pi$
$101$	17983.3	−	3170.94i	1.76289	−	0.310846i	0.958890	+	0.283777i	$0.0915876\pi$
$102$		0			0
$103$	2539.01	−	924.124i	0.239326	−	0.0871075i	−0.219573	−	0.975596i	$-0.570466\pi$
$103$	2539.01	−	924.124i	0.239326	−	0.0871075i	0.458899	+	0.888489i	$0.348244\pi$
$104$		0			0
$105$	24836.8	−	1718.25i	2.25277	−	0.155850i
$106$		0			0
$107$		−	6516.08i		−	0.569140i	−0.958655	−	0.284570i	$-0.908149\pi$
$107$		−	6516.08i		−	0.569140i	0.958655	−	0.284570i	$-0.0918508\pi$
$108$		0			0
$109$	−1020.46			−0.0858901			−0.0429451	−	0.999077i	$-0.513674\pi$
$109$	−1020.46			−0.0858901			−0.0429451	+	0.999077i	$0.513674\pi$
$110$		0			0
$111$	14750.1	−	9933.17i	1.19715	−	0.806199i
$112$		0			0
$113$	−6393.36	−	17565.6i	−0.500694	−	1.37564i	−0.890599	−	0.454790i	$-0.849714\pi$
$113$	−6393.36	−	17565.6i	−0.500694	−	1.37564i	0.389905	−	0.920855i	$-0.372508\pi$
$114$		0			0
$115$	1043.15	+	5916.02i	0.0788775	+	0.447336i
$116$		0			0
$117$	−829.767	+	22791.9i	−0.0606156	+	1.66498i
$118$		0			0
$119$	−10401.7	+	28578.4i	−0.734532	+	2.01811i
$120$		0			0
$121$	−2750.69	−	2308.10i	−0.187876	−	0.157647i
$122$		0			0
$123$	−16092.9	+	7150.74i	−1.06371	+	0.472651i
$124$		0			0
$125$	−6968.94	−	4023.52i	−0.446012	−	0.257505i
$126$		0			0
$127$	8405.05	+	14558.0i	0.521114	+	0.902597i	0.999698	+	0.0245549i	$0.00781687\pi$
$127$	8405.05	+	14558.0i	0.521114	+	0.902597i	−0.478584	+	0.878042i	$0.658850\pi$
$128$		0			0
$129$	−5272.33	−	5467.77i	−0.316828	−	0.328572i
$130$		0			0
$131$	−14033.3	−	2474.46i	−0.817746	−	0.144191i	−0.250899	−	0.968013i	$-0.580726\pi$
$131$	−14033.3	−	2474.46i	−0.817746	−	0.144191i	−0.566848	+	0.823823i	$0.691837\pi$
$132$		0			0
$133$	15282.2	−	12823.3i	0.863939	−	0.724931i
$134$		0			0
$135$	19527.0	−	12141.5i	1.07144	−	0.666199i
$136$		0			0
$137$	23405.4	+	27893.5i	1.24702	+	1.48615i	0.809703	+	0.586840i	$0.199628\pi$
$137$	23405.4	+	27893.5i	1.24702	+	1.48615i	0.437321	+	0.899306i	$0.355927\pi$
$138$		0			0
$139$	−1462.43	+	8293.86i	−0.0756913	+	0.429267i	0.923288	+	0.384107i	$0.125491\pi$
$139$	−1462.43	+	8293.86i	−0.0756913	+	0.429267i	−0.998980	+	0.0451595i	$0.985620\pi$
$140$		0			0
$141$	−15826.9	−	4550.99i	−0.796080	−	0.228912i
$142$		0			0
$143$	32925.2	−	19009.4i	1.61011	−	0.929599i
$144$		0			0
$145$	10549.2	−	18271.8i	0.501747	−	0.869052i
$146$		0			0
$147$	−47349.9	−	5012.19i	−2.19121	−	0.231949i
$148$		0			0
$149$	−15321.1	+	18259.0i	−0.690108	+	0.822439i	−0.991369	−	0.131103i	$-0.958148\pi$
$149$	−15321.1	+	18259.0i	−0.690108	+	0.822439i	0.301261	+	0.953542i	$0.402593\pi$
$150$		0			0
$151$	7165.12	+	2607.89i	0.314246	+	0.114376i	0.494328	−	0.869275i	$-0.335414\pi$
$151$	7165.12	+	2607.89i	0.314246	+	0.114376i	−0.180082	+	0.983652i	$0.557636\pi$
$152$		0			0
$153$	3867.93	+	27821.2i	0.165233	+	1.18848i
$154$		0			0
$155$	6413.55	−	1130.88i	0.266953	−	0.0470711i
$156$		0			0
$157$	30590.5	−	11134.0i	1.24104	−	0.451703i	0.363678	−	0.931525i	$-0.381521\pi$
$157$	30590.5	−	11134.0i	1.24104	−	0.451703i	0.877366	+	0.479821i	$0.159299\pi$
$158$		0			0
$159$	11714.2	−	23972.4i	0.463358	−	0.948239i
$160$		0			0
$161$		−	16703.2i		−	0.644388i
$162$		0			0
$163$	−11743.8			−0.442013			−0.221007	−	0.975272i	$-0.570934\pi$
$163$	−11743.8			−0.442013			−0.221007	+	0.975272i	$0.570934\pi$
$164$		0			0
$165$	−34438.5	−	16828.4i	−1.26496	−	0.618124i
$166$		0			0
$167$	2258.54	+	6205.28i	0.0809830	+	0.222499i	0.973576	−	0.228365i	$-0.0733380\pi$
$167$	2258.54	+	6205.28i	0.0809830	+	0.222499i	−0.892592	+	0.450864i	$0.851116\pi$
$168$		0			0
$169$	8807.35	+	49948.9i	0.308370	+	1.74885i
$170$		0			0
$171$	6927.54	−	17073.3i	0.236912	−	0.583882i
$172$		0			0
$173$	−11089.4	+	30467.9i	−0.370524	+	1.01801i	0.604635	+	0.796503i	$0.293319\pi$
$173$	−11089.4	+	30467.9i	−0.370524	+	1.01801i	−0.975159	+	0.221505i	$0.928903\pi$
$174$		0			0
$175$	−24849.4	−	20851.1i	−0.811408	−	0.680852i
$176$		0			0
$177$	5232.06	−	49427.0i	0.167004	−	1.57768i
$178$		0			0
$179$	−20902.8	−	12068.2i	−0.652377	−	0.376650i	0.136989	−	0.990573i	$-0.456257\pi$
$179$	−20902.8	−	12068.2i	−0.652377	−	0.376650i	−0.789366	+	0.613923i	$0.789591\pi$
$180$		0			0
$181$	−3335.92	−	5777.98i	−0.101826	−	0.176368i	0.810611	−	0.585585i	$-0.199135\pi$
$181$	−3335.92	−	5777.98i	−0.101826	−	0.176368i	−0.912437	+	0.409217i	$0.865802\pi$
$182$		0			0
$183$	−9750.82	+	33910.2i	−0.291165	+	1.01258i
$184$		0			0
$185$	−61375.9	−	10822.2i	−1.79331	−	0.316208i
$186$		0			0
$187$	35868.7	−	30097.4i	1.02573	−	0.860689i
$188$		0			0
$189$	−59331.9	+	23818.2i	−1.66098	+	0.666784i
$190$		0			0
$191$	−18122.1	−	21597.0i	−0.496753	−	0.592008i	0.458168	−	0.888865i	$-0.348506\pi$
$191$	−18122.1	−	21597.0i	−0.496753	−	0.592008i	−0.954922	+	0.296858i	$0.904061\pi$
$192$		0			0
$193$	5929.51	−	33627.9i	0.159186	−	0.902788i	−0.795673	−	0.605727i	$-0.792882\pi$
$193$	5929.51	−	33627.9i	0.159186	−	0.902788i	0.954859	−	0.297061i	$-0.0960065\pi$
$194$		0			0
$195$	57538.0	−	55481.4i	1.51316	−	1.45908i
$196$		0			0
$197$	−4567.28	+	2636.92i	−0.117686	+	0.0679461i	−0.557688	−	0.830051i	$-0.688311\pi$
$197$	−4567.28	+	2636.92i	−0.117686	+	0.0679461i	0.440001	+	0.897997i	$0.354978\pi$
$198$		0			0
$199$	−24297.3	+	42084.1i	−0.613553	+	1.06270i	0.377084	+	0.926179i	$0.376927\pi$
$199$	−24297.3	+	42084.1i	−0.613553	+	1.06270i	−0.990637	+	0.136525i	$0.956407\pi$
$200$		0			0
$201$	−18878.8	−	42487.2i	−0.467285	−	1.05164i
$202$		0			0
$203$	−37708.5	+	44939.3i	−0.915055	+	1.09052i
$204$		0			0
$205$	57994.9	+	21108.4i	1.38001	+	0.502282i
$206$		0			0
$207$	−7222.28	−	13631.9i	−0.168552	−	0.318138i
$208$		0			0
$209$	−30247.7	+	5333.49i	−0.692469	+	0.122101i
$210$		0			0
$211$	−14002.0	+	5096.30i	−0.314503	+	0.114470i	−0.494449	−	0.869207i	$-0.664630\pi$
$211$	−14002.0	+	5096.30i	−0.314503	+	0.114470i	0.179946	+	0.983677i	$0.442408\pi$
$212$		0			0
$213$	−6748.60	−	10021.2i	−0.148749	−	0.220883i
$214$		0			0
$215$			26619.9i			0.575878i
$216$		0			0
$217$	−18107.9			−0.384546
$218$		0			0
$219$	−1433.49	−	20720.8i	−0.0298887	−	0.432034i
$220$		0			0
$221$	33395.1	+	91752.2i	0.683751	+	1.87859i
$222$		0			0
$223$	−1567.49	−	8889.65i	−0.0315206	−	0.178762i	0.964983	−	0.262312i	$-0.0844851\pi$
$223$	−1567.49	−	8889.65i	−0.0315206	−	0.178762i	−0.996504	+	0.0835505i	$0.973374\pi$
$224$		0			0
$225$	−29296.0	−	6272.49i	−0.578686	−	0.123901i
$226$		0			0
$227$	20863.3	−	57321.4i	0.404885	−	1.11241i	−0.554960	−	0.831877i	$-0.687266\pi$
$227$	20863.3	−	57321.4i	0.404885	−	1.11241i	0.959844	−	0.280534i	$-0.0905115\pi$
$228$		0			0
$229$	−45065.7	−	37814.6i	−0.859361	−	0.721089i	0.102470	−	0.994736i	$-0.467326\pi$
$229$	−45065.7	−	37814.6i	−0.859361	−	0.721089i	−0.961830	+	0.273647i	$0.911770\pi$
$230$		0			0
$231$	86175.9	+	62708.2i	1.61496	+	1.17517i
$232$		0			0
$233$	25541.4	+	14746.3i	0.470470	+	0.271626i	0.716437	−	0.697652i	$-0.245772\pi$
$233$	25541.4	+	14746.3i	0.470470	+	0.271626i	−0.245966	+	0.969278i	$0.579105\pi$
$234$		0			0
$235$	28857.4	+	49982.6i	0.522543	+	0.905071i
$236$		0			0
$237$	69331.9	−	17231.8i	1.23435	−	0.306784i
$238$		0			0
$239$	−91549.5	−	16142.6i	−1.60273	−	0.282604i	−0.700432	−	0.713719i	$-0.747009\pi$
$239$	−91549.5	−	16142.6i	−1.60273	−	0.282604i	−0.902297	+	0.431115i	$0.858120\pi$
$240$		0			0
$241$	75263.3	−	63153.4i	1.29583	−	1.08733i	0.304985	−	0.952357i	$-0.401349\pi$
$241$	75263.3	−	63153.4i	1.29583	−	1.08733i	0.990849	−	0.134976i	$-0.0430957\pi$
$242$		0			0
$243$	−38123.5	+	45093.1i	−0.645624	+	0.763655i
$244$		0			0
$245$	107263.	+	127831.i	1.78696	+	2.12962i
$246$		0			0
$247$	11121.9	−	63075.6i	0.182300	−	1.03387i
$248$		0			0
$249$	−2969.70	−	11948.6i	−0.0478976	−	0.192716i
$250$		0			0
$251$	67645.6	−	39055.2i	1.07372	−	0.619914i	0.144527	−	0.989501i	$-0.453834\pi$
$251$	67645.6	−	39055.2i	1.07372	−	0.619914i	0.929196	+	0.369587i	$0.120501\pi$
$252$		0			0
$253$	−12858.2	+	22271.0i	−0.200880	+	0.347935i
$254$		0			0
$255$	57921.0	−	79597.2i	0.890750	−	1.22410i
$256$		0			0
$257$	35744.0	−	42598.0i	0.541174	−	0.644946i	−0.424277	−	0.905533i	$-0.639472\pi$
$257$	35744.0	−	42598.0i	0.541174	−	0.644946i	0.965450	+	0.260587i	$0.0839160\pi$
$258$		0			0
$259$	162837.	+	59267.8i	2.42747	+	0.883526i
$260$		0			0
$261$	−11343.6	+	52980.8i	−0.166521	+	0.777746i
$262$		0			0
$263$	38052.4	−	6709.66i	0.550136	−	0.0970038i	0.108331	−	0.994115i	$-0.465449\pi$
$263$	38052.4	−	6709.66i	0.550136	−	0.0970038i	0.441805	+	0.897111i	$0.354338\pi$
$264$		0			0
$265$	−87869.3	+	31981.8i	−1.25125	+	0.455419i
$266$		0			0
$267$	−29031.0	+	2008.41i	−0.407230	+	0.0281728i
$268$		0			0
$269$		−	26971.5i		−	0.372735i	−0.982480	−	0.186367i	$-0.940329\pi$
$269$		−	26971.5i		−	0.372735i	0.982480	−	0.186367i	$-0.0596715\pi$
$270$		0			0
$271$	15106.5			0.205696			0.102848	−	0.994697i	$-0.467204\pi$
$271$	15106.5			0.205696			0.102848	+	0.994697i	$0.467204\pi$
$272$		0			0
$273$	−184341.	+	124141.i	−2.47342	+	1.66567i
$274$		0			0
$275$	17081.3	+	46930.6i	0.225869	+	0.620570i
$276$		0			0
$277$	−1864.45	−	10573.8i	−0.0242991	−	0.137807i	0.970245	−	0.242126i	$-0.0778448\pi$
$277$	−1864.45	−	10573.8i	−0.0242991	−	0.137807i	−0.994544	+	0.104319i	$0.966734\pi$
$278$		0			0
$279$	−14778.3	+	7829.67i	−0.189853	+	0.100585i
$280$		0			0
$281$	−3060.57	+	8408.86i	−0.0387606	+	0.106494i	−0.957563	−	0.288224i	$-0.906935\pi$
$281$	−3060.57	+	8408.86i	−0.0387606	+	0.106494i	0.918803	+	0.394718i	$0.129158\pi$
$282$		0			0
$283$	47874.3	+	40171.3i	0.597764	+	0.501584i	0.890726	−	0.454540i	$-0.150196\pi$
$283$	47874.3	+	40171.3i	0.597764	+	0.501584i	−0.292962	+	0.956124i	$0.594641\pi$
$284$		0			0
$285$	−59010.3	+	26220.6i	−0.726504	+	0.322815i
$286$		0			0
$287$	−148613.	−	85801.5i	−1.80423	−	1.04167i
$288$		0			0
$289$	18365.8	+	31810.6i	0.219895	+	0.380869i
$290$		0			0
$291$	−68644.4	−	71188.9i	−0.810623	−	0.840672i
$292$		0			0
$293$	−19021.2	−	3353.96i	−0.221566	−	0.0390681i	0.0617628	−	0.998091i	$-0.480328\pi$
$293$	−19021.2	−	3353.96i	−0.221566	−	0.0390681i	−0.283329	+	0.959023i	$0.591439\pi$
$294$		0			0
$295$	−133438.	+	111968.i	−1.53333	+	1.28662i
$296$		0			0
$297$	97444.6	+	13916.1i	1.10470	+	0.157763i
$298$		0			0
$299$	−34470.3	−	41080.1i	−0.385569	−	0.459503i
$300$		0			0
$301$	12852.8	−	72891.9i	0.141862	−	0.804538i
$302$		0			0
$303$	157946.	+	45417.2i	1.72038	+	0.494692i
$304$		0			0
$305$	107091.	−	61829.1i	1.15121	−	0.664651i
$306$		0			0
$307$	−73229.0	+	126836.i	−0.776974	+	1.34576i	0.156704	+	0.987646i	$0.449913\pi$
$307$	−73229.0	+	126836.i	−0.776974	+	1.34576i	−0.933678	+	0.358113i	$0.883420\pi$
$308$		0			0
$309$	24182.5	+	2559.82i	0.253270	+	0.0268098i
$310$		0			0
$311$	76390.5	−	91038.6i	0.789802	−	0.941250i	−0.209529	−	0.977802i	$-0.567193\pi$
$311$	76390.5	−	91038.6i	0.789802	−	0.941250i	0.999332	+	0.0365526i	$0.0116377\pi$
$312$		0			0
$313$	−128096.	−	46623.1i	−1.30752	−	0.475897i	−0.408078	−	0.912947i	$-0.633801\pi$
$313$	−128096.	−	46623.1i	−1.30752	−	0.475897i	−0.899437	+	0.437050i	$0.856023\pi$
$314$		0			0
$315$	207625.	+	84244.5i	2.09247	+	0.849025i
$316$		0			0
$317$	2046.11	−	360.784i	0.0203615	−	0.00359028i	−0.163458	−	0.986550i	$-0.552265\pi$
$317$	2046.11	−	360.784i	0.0203615	−	0.00359028i	0.183820	+	0.982960i	$0.441154\pi$
$318$		0			0
$319$	84872.5	−	30891.1i	0.834037	−	0.303565i
$320$		0			0
$321$	25747.2	−	52690.4i	0.249874	−	0.511354i
$322$		0			0
$323$		−	78881.4i		−	0.756083i
$324$		0			0
$325$	−104145.			−0.985990
$326$		0			0
$327$	−8251.66	−	4032.19i	−0.0771696	−	0.0377090i
$328$		0			0
$329$	−54885.9	−	150798.i	−0.507071	−	1.39317i
$330$		0			0
$331$	−13925.1	−	78973.4i	−0.127099	−	0.720817i	−0.980039	−	0.198807i	$-0.936293\pi$
$331$	−13925.1	−	78973.4i	−0.127099	−	0.720817i	0.852939	−	0.522010i	$-0.174818\pi$
$332$		0			0
$333$	158522.	−	22039.1i	1.42956	−	0.198749i
$334$		0			0
$335$	−55728.6	+	153113.i	−0.496579	+	1.36434i
$336$		0			0
$337$	−41768.0	−	35047.5i	−0.367777	−	0.308601i	0.440105	−	0.897946i	$-0.354941\pi$
$337$	−41768.0	−	35047.5i	−0.367777	−	0.308601i	−0.807881	+	0.589345i	$0.799386\pi$
$338$		0			0
$339$	17709.6	−	167302.i	0.154102	−	1.45580i
$340$		0			0
$341$	24143.9	+	13939.5i	0.207634	+	0.119878i
$342$		0			0
$343$	−126706.	−	219461.i	−1.07698	−	1.86539i
$344$		0			0
$345$	−14941.0	+	51960.1i	−0.125529	+	0.436548i
$346$		0			0
$347$	−23633.6	−	4167.25i	−0.196278	−	0.0346091i	0.0746448	−	0.997210i	$-0.476218\pi$
$347$	−23633.6	−	4167.25i	−0.196278	−	0.0346091i	−0.270923	+	0.962601i	$0.587329\pi$
$348$		0			0
$349$	−124185.	+	104204.i	−1.01957	+	0.855524i	−0.989574	−	0.144025i	$-0.953995\pi$
$349$	−124185.	+	104204.i	−1.01957	+	0.855524i	−0.0300003	+	0.999550i	$0.509551\pi$
$350$		0			0
$351$	−96768.2	+	181022.i	−0.785450	+	1.46932i
$352$		0			0
$353$	67927.0	+	80952.2i	0.545121	+	0.649650i	0.966328	−	0.257315i	$-0.0828378\pi$
$353$	67927.0	+	80952.2i	0.545121	+	0.649650i	−0.421207	+	0.906965i	$0.638393\pi$
$354$		0			0
$355$	−7352.62	+	41698.8i	−0.0583426	+	0.330877i
$356$		0			0
$357$	−197034.	+	189991.i	−1.54598	+	1.49072i
$358$		0			0
$359$	961.228	−	554.965i	0.00745826	−	0.00430603i	−0.496266	−	0.868170i	$-0.665296\pi$
$359$	961.228	−	554.965i	0.00745826	−	0.00430603i	0.503724	+	0.863864i	$0.331963\pi$
$360$		0			0
$361$	39288.8	−	68050.3i	0.301477	−	0.522174i
$362$		0			0
$363$	−13122.6	−	29532.8i	−0.0995878	−	0.224125i
$364$		0			0
$365$	−46789.9	+	55762.1i	−0.351210	+	0.418555i
$366$		0			0
$367$	−75284.8	−	27401.4i	−0.558952	−	0.203442i	0.0470672	−	0.998892i	$-0.485013\pi$
$367$	−75284.8	−	27401.4i	−0.558952	−	0.203442i	−0.606020	+	0.795450i	$0.707235\pi$
$368$		0			0
$369$	−158386.	−	5766.23i	−1.16323	−	0.0423486i
$370$		0			0
$371$	256049.	−	45148.4i	1.86027	−	0.328016i
$372$		0			0
$373$	24455.5	−	8901.06i	0.175776	−	0.0639771i	−0.252633	−	0.967562i	$-0.581297\pi$
$373$	24455.5	−	8901.06i	0.175776	−	0.0639771i	0.428409	+	0.903585i	$0.359074\pi$
$374$		0			0
$375$	−40454.1	−	60071.7i	−0.287673	−	0.427176i
$376$		0			0
$377$			188343.i			1.32516i
$378$		0			0
$379$	−33218.9			−0.231263			−0.115632	−	0.993292i	$-0.536889\pi$
$379$	−33218.9			−0.231263			−0.115632	+	0.993292i	$0.536889\pi$
$380$		0			0
$381$	10441.6	+	150930.i	0.0719310	+	1.03974i
$382$		0			0
$383$	−72532.8	−	199282.i	−0.494467	−	1.35854i	−0.896554	−	0.442934i	$-0.853937\pi$
$383$	−72532.8	−	199282.i	−0.494467	−	1.35854i	0.402087	−	0.915601i	$-0.368285\pi$
$384$		0			0
$385$	−64859.6	−	367837.i	−0.437575	−	2.48161i
$386$		0			0
$387$	−21028.2	−	65046.3i	−0.140404	−	0.434311i
$388$		0			0
$389$	39859.9	−	109514.i	0.263413	−	0.723721i	−0.735519	−	0.677505i	$-0.763061\pi$
$389$	39859.9	−	109514.i	0.263413	−	0.723721i	0.998931	−	0.0462165i	$-0.0147164\pi$
$390$		0			0
$391$	−50593.6	−	42453.1i	−0.330934	−	0.277687i
$392$		0			0
$393$	−103699.	−	75459.5i	−0.671414	−	0.488572i
$394$		0			0
$395$	−216831.	−	125187.i	−1.38972	−	0.802355i
$396$		0			0
$397$	105374.	+	182513.i	0.668580	+	1.15801i	0.978301	+	0.207187i	$0.0664307\pi$
$397$	105374.	+	182513.i	0.668580	+	1.15801i	−0.309722	+	0.950827i	$0.600236\pi$
$398$		0			0
$399$	174245.	−	43306.8i	1.09449	−	0.272026i
$400$		0			0
$401$	18741.2	+	3304.59i	0.116549	+	0.0205508i	0.231619	−	0.972807i	$-0.425598\pi$
$401$	18741.2	+	3304.59i	0.116549	+	0.0205508i	−0.115069	+	0.993357i	$0.536709\pi$
$402$		0			0
$403$	−44534.8	+	37369.2i	−0.274214	+	0.230093i
$404$		0			0
$405$	205874.	−	21020.9i	1.25514	−	0.128156i
$406$		0			0
$407$	−171492.	−	204376.i	−1.03527	−	1.23379i
$408$		0			0
$409$	−17268.1	+	97932.2i	−0.103228	+	0.585435i	0.888685	+	0.458518i	$0.151619\pi$
$409$	−17268.1	+	97932.2i	−0.103228	+	0.585435i	−0.991913	+	0.126918i	$0.959492\pi$
$410$		0			0
$411$	79044.6	+	318035.i	0.467938	+	1.88275i
$412$		0			0
$413$	419448.	−	242168.i	2.45911	−	1.41977i
$414$		0			0
$415$	−21574.6	+	37368.4i	−0.125270	+	0.216974i
$416$		0			0
$417$	−44597.4	+	61287.4i	−0.256471	+	0.352451i
$418$		0			0
$419$	−77933.2	+	92877.2i	−0.443910	+	0.529031i	−0.940882	−	0.338736i	$-0.890001\pi$
$419$	−77933.2	+	92877.2i	−0.443910	+	0.529031i	0.496972	+	0.867767i	$0.334445\pi$
$420$		0			0
$421$	203870.	+	74202.8i	1.15024	+	0.418655i	0.845603	−	0.533812i	$-0.179241\pi$
$421$	203870.	+	74202.8i	1.15024	+	0.418655i	0.304641	+	0.952467i	$0.401463\pi$
$422$		0			0
$423$	−109997.	−	99337.6i	−0.614752	−	0.555179i
$424$		0			0
$425$	−126315.	+	22272.7i	−0.699322	+	0.123309i
$426$		0			0
$427$	−323095.	+	117597.i	−1.77204	+	0.644971i
$428$		0			0
$429$	341353.	−	23615.3i	1.85476	−	0.128315i
$430$		0			0
$431$			234878.i			1.26441i	0.774802	+	0.632204i	$0.217850\pi$
$431$			234878.i			1.26441i	−0.774802	+	0.632204i	$0.782150\pi$
$432$		0			0
$433$	−21782.0			−0.116177			−0.0580887	−	0.998311i	$-0.518501\pi$
$433$	−21782.0			−0.116177			−0.0580887	+	0.998311i	$0.518501\pi$
$434$		0			0
$435$	157502.	−	106066.i	0.832351	−	0.560530i
$436$		0			0
$437$	14817.4	+	40710.6i	0.0775908	+	0.213179i
$438$		0			0
$439$	65390.8	+	370850.i	0.339303	+	1.92428i	0.379726	+	0.925099i	$0.376018\pi$
$439$	65390.8	+	370850.i	0.339303	+	1.92428i	−0.0404229	+	0.999183i	$0.512871\pi$
$440$		0			0
$441$	−363077.	−	227625.i	−1.86690	−	1.17042i
$442$		0			0
$443$	31232.3	−	85809.9i	0.159146	−	0.437250i	−0.834334	−	0.551260i	$-0.814147\pi$
$443$	31232.3	−	85809.9i	0.159146	−	0.437250i	0.993480	+	0.114010i	$0.0363695\pi$
$444$		0			0
$445$	78125.9	+	65555.4i	0.394526	+	0.331046i
$446$		0			0
$447$	−196037.	+	87107.1i	−0.981122	+	0.435952i
$448$		0			0
$449$	−107735.	−	62201.0i	−0.534399	−	0.308535i	0.208407	−	0.978042i	$-0.433172\pi$
$449$	−107735.	−	62201.0i	−0.534399	−	0.308535i	−0.742806	+	0.669507i	$0.766505\pi$
$450$		0			0
$451$	132100.	+	228804.i	0.649458	+	1.12489i
$452$		0			0
$453$	47634.1	+	49399.8i	0.232125	+	0.240729i
$454$		0			0
$455$	767051.	+	135252.i	3.70511	+	0.653312i
$456$		0			0
$457$	103503.	−	86849.0i	0.495586	−	0.415846i	−0.360437	−	0.932783i	$-0.617372\pi$
$457$	103503.	−	86849.0i	0.495586	−	0.415846i	0.856023	+	0.516938i	$0.172928\pi$
$458$		0			0
$459$	−78653.9	+	240251.i	−0.373332	+	1.14036i
$460$		0			0
$461$	−168200.	−	200453.i	−0.791450	−	0.943213i	0.207940	−	0.978142i	$-0.433324\pi$
$461$	−168200.	−	200453.i	−0.791450	−	0.943213i	−0.999390	+	0.0349284i	$0.988880\pi$
$462$		0			0
$463$	−26468.5	+	150110.i	−0.123472	+	0.700242i	0.858732	+	0.512424i	$0.171253\pi$
$463$	−26468.5	+	150110.i	−0.123472	+	0.700242i	−0.982204	+	0.187818i	$0.939859\pi$
$464$		0			0
$465$	56329.9	+	16197.6i	0.260515	+	0.0749107i
$466$		0			0
$467$	−310052.	+	179009.i	−1.42168	+	0.820806i	−0.996442	−	0.0842767i	$-0.973142\pi$
$467$	−310052.	+	179009.i	−1.42168	+	0.820806i	−0.425235	+	0.905083i	$0.639809\pi$
$468$		0			0
$469$	226526.	−	392354.i	1.02985	−	1.78374i
$470$		0			0
$471$	291356.	+	30841.3i	1.31335	+	0.139024i
$472$		0			0
$473$	−73249.5	+	87295.3i	−0.327403	+	0.390183i
$474$		0			0
$475$	79062.2	+	28776.3i	0.350414	+	0.127540i
$476$		0			0
$477$	189446.	−	147560.i	0.832626	−	0.648531i
$478$		0			0
$479$	−147754.	+	26053.1i	−0.643976	+	0.113550i	−0.486092	−	0.873908i	$-0.661578\pi$
$479$	−147754.	+	26053.1i	−0.643976	+	0.113550i	−0.157883	+	0.987458i	$0.550467\pi$
$480$		0			0
$481$	522794.	−	190282.i	2.25965	−	0.822444i
$482$		0			0
$483$	66000.0	−	135066.i	0.282911	−	0.578963i
$484$		0			0
$485$			346585.i			1.47342i
$486$		0			0
$487$	191423.			0.807117			0.403559	−	0.914954i	$-0.367773\pi$
$487$	191423.			0.807117			0.403559	+	0.914954i	$0.367773\pi$
$488$		0			0
$489$	−94963.3	−	46403.9i	−0.397135	−	0.194060i
$490$		0			0
$491$	96829.2	+	266036.i	0.401646	+	1.10351i	0.961472	+	0.274903i	$0.0886457\pi$
$491$	96829.2	+	266036.i	0.401646	+	1.10351i	−0.559826	+	0.828610i	$0.689132\pi$
$492$		0			0
$493$	40279.5	+	228436.i	0.165726	+	0.939878i
$494$		0			0
$495$	−211982.	−	272156.i	−0.865146	−	1.11073i
$496$		0			0
$497$	40266.6	−	110631.i	0.163017	−	0.447884i
$498$		0			0
$499$	−216992.	−	182078.i	−0.871450	−	0.731233i	0.0929533	−	0.995670i	$-0.470369\pi$
$499$	−216992.	−	182078.i	−0.871450	−	0.731233i	−0.964403	+	0.264437i	$0.914814\pi$
$500$		0			0
$501$	−6256.14	+	59101.5i	−0.0249248	+	0.235463i
$502$		0			0
$503$	176203.	+	101731.i	0.696429	+	0.402083i	0.806016	−	0.591894i	$-0.201620\pi$
$503$	176203.	+	101731.i	0.696429	+	0.402083i	−0.109587	+	0.993977i	$0.534953\pi$
$504$		0			0
$505$	−287986.	−	498807.i	−1.12925	−	1.95592i
$506$		0			0
$507$	−126147.	+	438699.i	−0.490751	+	1.70667i
$508$		0			0
$509$	54374.6	+	9587.70i	0.209875	+	0.0370066i	0.277597	−	0.960698i	$-0.410462\pi$
$509$	54374.6	+	9587.70i	0.209875	+	0.0370066i	−0.0677222	+	0.997704i	$0.521573\pi$
$510$		0			0
$511$	155046.	−	130099.i	0.593769	−	0.498232i
$512$		0			0
$513$	123480.	−	110685.i	0.469204	−	0.420586i
$514$		0			0
$515$	−54781.1	−	65285.5i	−0.206546	−	0.246151i
$516$		0			0
$517$	−42903.0	+	243315.i	−0.160512	+	0.910308i
$518$		0			0
$519$	−210061.	+	202552.i	−0.779848	+	0.751973i
$520$		0			0
$521$	23227.7	−	13410.5i	0.0855719	−	0.0494050i	−0.456603	−	0.889670i	$-0.650934\pi$
$521$	23227.7	−	13410.5i	0.0855719	−	0.0494050i	0.542175	+	0.840265i	$0.317601\pi$
$522$		0			0
$523$	25930.8	−	44913.4i	0.0948009	−	0.164200i	−0.814725	−	0.579848i	$-0.803112\pi$
$523$	25930.8	−	44913.4i	0.0948009	−	0.164200i	0.909525	+	0.415648i	$0.136445\pi$
$524$		0			0
$525$	−118548.	−	266795.i	−0.430105	−	0.967963i
$526$		0			0
$527$	−46023.3	+	54848.4i	−0.165713	+	0.197489i
$528$		0			0
$529$	−228879.	−	83305.0i	−0.817888	−	0.297687i
$530$		0			0
$531$	237611.	−	379004.i	0.842707	−	1.34417i
$532$		0			0
$533$	−542568.	+	95669.3i	−1.90985	+	0.336758i
$534$		0			0
$535$	−193133.	+	70294.7i	−0.674760	+	0.245592i
$536$		0			0
$537$	−121339.	−	180181.i	−0.420777	−	0.624826i
$538$		0			0
$539$			714349.i			2.45886i
$540$		0			0
$541$	237828.			0.812585			0.406292	−	0.913743i	$-0.366821\pi$
$541$	237828.			0.812585			0.406292	+	0.913743i	$0.366821\pi$
$542$		0			0
$543$	−4144.20	−	59903.4i	−0.0140553	−	0.203166i
$544$		0			0
$545$	11008.6	+	30245.9i	0.0370629	+	0.101829i
$546$		0			0
$547$	−69204.9	−	392480.i	−0.231293	−	1.31173i	−0.850282	−	0.526327i	$-0.823569\pi$
$547$	−69204.9	−	392480.i	−0.231293	−	1.31173i	0.618989	−	0.785399i	$-0.287542\pi$
$548$		0			0
$549$	−212838.	+	235676.i	−0.706162	+	0.781937i
$550$		0			0
$551$	52041.0	−	142981.i	0.171412	−	0.470952i
$552$		0			0
$553$	533293.	+	447486.i	1.74388	+	1.46329i
$554$		0			0
$555$	−453537.	−	330028.i	−1.47240	−	1.07143i
$556$		0			0
$557$	−338329.	−	195334.i	−1.09051	−	0.629605i	−0.156797	−	0.987631i	$-0.550117\pi$
$557$	−338329.	−	195334.i	−1.09051	−	0.629605i	−0.933712	+	0.358026i	$0.883450\pi$
$558$		0			0
$559$	−118816.	−	205796.i	−0.380235	−	0.658587i
$560$		0			0
$561$	408967.	−	101645.i	1.29946	−	0.322968i
$562$		0			0
$563$	−502288.	−	88566.9i	−1.58466	−	0.279418i	−0.689203	−	0.724569i	$-0.742039\pi$
$563$	−502288.	−	88566.9i	−1.58466	−	0.279418i	−0.895456	+	0.445151i	$0.853150\pi$
$564$		0			0
$565$	−451664.	+	378991.i	−1.41488	+	1.18722i
$566$		0			0
$567$	−573884.	−	41841.3i	−1.78508	−	0.130149i
$568$		0			0
$569$	338791.	+	403756.i	1.04642	+	1.24708i	0.968209	+	0.250143i	$0.0804776\pi$
$569$	338791.	+	403756.i	1.04642	+	1.24708i	0.0782149	+	0.996937i	$0.475078\pi$
$570$		0			0
$571$	13409.5	−	76049.2i	0.0411284	−	0.233250i	−0.957314	−	0.289051i	$-0.906660\pi$
$571$	13409.5	−	76049.2i	0.0411284	−	0.233250i	0.998442	+	0.0558010i	$0.0177712\pi$
$572$		0			0
$573$	−61201.7	−	246245.i	−0.186404	−	0.749994i
$574$		0			0
$575$	61007.1	−	35222.5i	0.184521	−	0.106533i
$576$		0			0
$577$	82412.6	−	142743.i	0.247538	−	0.428748i	−0.715304	−	0.698813i	$-0.753712\pi$
$577$	82412.6	−	142743.i	0.247538	−	0.428748i	0.962842	+	0.270065i	$0.0870452\pi$
$578$		0			0
$579$	180823.	−	248493.i	0.539381	−	0.741238i
$580$		0			0
$581$	77119.1	−	91906.9i	0.228460	−	0.272268i
$582$		0			0
$583$	−376155.	−	136909.i	−1.10670	−	0.402806i
$584$		0			0
$585$	684491.	−	221282.i	2.00012	−	0.646599i
$586$		0			0
$587$	−146970.	+	25914.7i	−0.426532	+	0.0752090i	−0.382793	−	0.923834i	$-0.625038\pi$
$587$	−146970.	+	25914.7i	−0.426532	+	0.0752090i	−0.0437382	+	0.999043i	$0.513927\pi$
$588$		0			0
$589$	44134.2	−	16063.6i	0.127217	−	0.0463032i
$590$		0			0
$591$	−47351.4	+	3275.84i	−0.135568	+	0.00937880i
$592$		0			0
$593$			87964.7i			0.250149i	0.992147	+	0.125075i	$0.0399170\pi$
$593$			87964.7i			0.250149i	−0.992147	+	0.125075i	$0.960083\pi$
$594$		0			0
$595$	959262.			2.70959
$596$		0			0
$597$	−362762.	+	244295.i	−1.01782	+	0.685434i
$598$		0			0
$599$	−236836.	−	650702.i	−0.660077	−	1.81355i	−0.576585	−	0.817037i	$-0.695615\pi$
$599$	−236836.	−	650702.i	−0.660077	−	1.81355i	−0.0834914	−	0.996508i	$-0.526607\pi$
$600$		0			0
$601$	8308.99	+	47122.6i	0.0230038	+	0.130461i	0.994147	−	0.108036i	$-0.0344563\pi$
$601$	8308.99	+	47122.6i	0.0230038	+	0.130461i	−0.971143	+	0.238497i	$0.923345\pi$
$602$		0			0
$603$	15223.5	−	418157.i	0.0418678	−	1.15002i
$604$		0			0
$605$	−38736.8	+	106429.i	−0.105831	+	0.290769i
$606$		0			0
$607$	533429.	+	447600.i	1.44777	+	1.21482i	0.934188	+	0.356782i	$0.116126\pi$
$607$	533429.	+	447600.i	1.44777	+	1.21482i	0.513582	+	0.858041i	$0.328318\pi$
$608$		0			0
$609$	−482489.	+	214389.i	−1.30093	+	0.578055i
$610$		0			0
$611$	−446187.	−	257606.i	−1.19518	−	0.690040i
$612$		0			0
$613$	242238.	+	419569.i	0.644646	+	1.11656i	0.984383	+	0.176040i	$0.0563287\pi$
$613$	242238.	+	419569.i	0.644646	+	1.11656i	−0.339737	+	0.940521i	$0.610338\pi$
$614$		0			0
$615$	385553.	+	399845.i	1.01937	+	1.05716i
$616$		0			0
$617$	−15518.7	−	2736.36i	−0.0407647	−	0.00718792i	0.153229	−	0.988191i	$-0.451033\pi$
$617$	−15518.7	−	2736.36i	−0.0407647	−	0.00718792i	−0.193993	+	0.981003i	$0.562144\pi$
$618$		0			0
$619$	534132.	−	448190.i	1.39402	−	1.16972i	0.430332	−	0.902671i	$-0.358396\pi$
$619$	534132.	−	448190.i	1.39402	−	1.16972i	0.963683	−	0.267047i	$-0.0860480\pi$
$620$		0			0
$621$	−4536.73	−	138768.i	−0.0117641	−	0.359838i
$622$		0			0
$623$	−182276.	−	217228.i	−0.469628	−	0.559680i
$624$		0			0
$625$	−84217.5	+	477621.i	−0.215597	+	1.22271i
$626$		0			0
$627$	−265664.	−	76391.3i	−0.675769	−	0.194316i
$628$		0			0
$629$	593390.	−	342594.i	1.49982	−	0.865921i
$630$		0			0
$631$	−125378.	+	217160.i	−0.314892	+	0.545408i	−0.979414	−	0.201860i	$-0.935302\pi$
$631$	−125378.	+	217160.i	−0.314892	+	0.545408i	0.664523	+	0.747268i	$0.268635\pi$
$632$		0			0
$633$	−133360.	−	14116.8i	−0.332827	−	0.0352312i
$634$		0			0
$635$	340818.	−	406171.i	0.845230	−	1.00731i
$636$		0			0
$637$	−1.39980e6	−	509484.i	−3.44974	−	1.25560i
$638$		0			0
$639$	−14973.3	−	107700.i	−0.0366705	−	0.263763i
$640$		0			0
$641$	128403.	−	22641.0i	0.312507	−	0.0551034i	−0.0151950	−	0.999885i	$-0.504837\pi$
$641$	128403.	−	22641.0i	0.312507	−	0.0551034i	0.327702	+	0.944781i	$0.393726\pi$
$642$		0			0
$643$	508983.	−	185255.i	1.23107	−	0.448072i	0.357104	−	0.934065i	$-0.383764\pi$
$643$	508983.	−	185255.i	1.23107	−	0.448072i	0.873963	+	0.485993i	$0.161542\pi$
$644$		0			0
$645$	−105184.	+	215255.i	−0.252832	+	0.517408i
$646$		0			0
$647$			667818.i			1.59533i	0.603102	+	0.797664i	$0.293931\pi$
$647$			667818.i			1.59533i	−0.603102	+	0.797664i	$0.706069\pi$
$648$		0			0
$649$	−745686.			−1.77038
$650$		0			0
$651$	−146424.	−	71550.5i	−0.345503	−	0.168830i
$652$		0			0
$653$	121288.	+	333237.i	0.284441	+	0.781496i	0.996819	+	0.0796995i	$0.0253961\pi$
$653$	121288.	+	333237.i	0.284441	+	0.781496i	−0.712378	+	0.701796i	$0.752382\pi$
$654$		0			0
$655$	78048.5	+	442635.i	0.181921	+	1.03172i
$656$		0			0
$657$	70283.3	−	173217.i	0.162825	−	0.401291i
$658$		0			0
$659$	−177897.	+	488767.i	−0.409635	+	1.12546i	0.547749	+	0.836643i	$0.315485\pi$
$659$	−177897.	+	488767.i	−0.409635	+	1.12546i	−0.957384	+	0.288819i	$0.906737\pi$
$660$		0			0
$661$	432970.	+	363305.i	0.990957	+	0.831512i	0.985706	−	0.168475i	$-0.0538841\pi$
$661$	432970.	+	363305.i	0.990957	+	0.831512i	0.00525099	+	0.999986i	$0.498329\pi$
$662$		0			0
$663$	−92504.3	+	873883.i	−0.210443	+	1.98805i
$664$		0			0
$665$	−544938.	−	314620.i	−1.23227	−	0.711449i
$666$		0			0
$667$	−63698.7	−	110329.i	−0.143179	−	0.247993i
$668$		0			0
$669$	22451.0	−	78077.3i	0.0501630	−	0.174451i
$670$		0			0
$671$	521320.	+	91922.8i	1.15787	+	0.204164i
$672$		0			0
$673$	−289892.	+	243249.i	−0.640039	+	0.537057i	−0.904030	−	0.427468i	$-0.859406\pi$
$673$	−289892.	+	243249.i	−0.640039	+	0.537057i	0.263991	+	0.964525i	$0.414961\pi$
$674$		0			0
$675$	−212109.	−	166479.i	−0.465534	−	0.365386i
$676$		0			0
$677$	−241740.	−	288095.i	−0.527438	−	0.628576i	0.434884	−	0.900486i	$-0.356789\pi$
$677$	−241740.	−	288095.i	−0.527438	−	0.628576i	−0.962323	+	0.271910i	$0.912345\pi$
$678$		0			0
$679$	167340.	−	949034.i	0.362962	−	2.05846i
$680$		0			0
$681$	395202.	−	381076.i	0.852167	−	0.821707i
$682$		0			0
$683$	−56691.2	+	32730.7i	−0.121527	+	0.0701639i	−0.559531	−	0.828809i	$-0.689019\pi$
$683$	−56691.2	+	32730.7i	−0.121527	+	0.0701639i	0.438004	+	0.898973i	$0.355686\pi$
$684$		0			0
$685$	574252.	−	994634.i	1.22383	−	2.11974i
$686$		0			0
$687$	−214993.	−	483847.i	−0.455523	−	1.02517i
$688$		0			0
$689$	536559.	−	639446.i	1.13026	−	1.34699i
$690$		0			0
$691$	353222.	+	128562.i	0.739761	+	0.269251i	0.684291	−	0.729209i	$-0.260112\pi$
$691$	353222.	+	128562.i	0.739761	+	0.269251i	0.0554705	+	0.998460i	$0.482334\pi$
$692$		0			0
$693$	449056.	+	847582.i	0.935048	+	1.76488i
$694$		0			0
$695$	261602.	−	46127.5i	0.541591	−	0.0954972i
$696$		0			0
$697$	−637606.	+	232070.i	−1.31246	+	0.477697i
$698$		0			0
$699$	148265.	+	220165.i	0.303449	+	0.450602i
$700$		0			0
$701$			960237.i			1.95408i	0.213055	+	0.977040i	$0.431659\pi$
$701$			960237.i			1.95408i	−0.213055	+	0.977040i	$0.568341\pi$
$702$		0			0
$703$	−449458.			−0.909449
$704$		0			0
$705$	35849.5	+	518195.i	0.0721282	+	1.04259i
$706$		0			0
$707$	547740.	+	1.50490e6i	1.09581	+	3.01072i
$708$		0			0
$709$	28749.5	+	163047.i	0.0571924	+	0.324354i	0.999959	−	0.00910028i	$-0.00289675\pi$
$709$	28749.5	+	163047.i	0.0571924	+	0.324354i	−0.942766	+	0.333455i	$0.891786\pi$
$710$		0			0
$711$	628722.	+	134614.i	1.24371	+	0.266288i
$712$		0			0
$713$	13449.6	−	36952.4i	0.0264563	−	0.0726882i
$714$		0			0
$715$	−918620.	−	770814.i	−1.79690	−	1.50778i
$716$		0			0
$717$	−676504.	−	492276.i	−1.31593	−	0.957570i
$718$		0			0
$719$	243132.	+	140372.i	0.470310	+	0.271534i	0.716370	−	0.697721i	$-0.245802\pi$
$719$	243132.	+	140372.i	0.470310	+	0.271534i	−0.246059	+	0.969255i	$0.579136\pi$
$720$		0			0
$721$	118482.	+	205218.i	0.227920	+	0.394770i
$722$		0			0
$723$	858136.	−	213281.i	1.64165	−	0.408015i
$724$		0			0
$725$	−243654.	−	42962.8i	−0.463552	−	0.0817366i
$726$		0			0
$727$	299480.	−	251294.i	0.566630	−	0.475459i	−0.313896	−	0.949458i	$-0.601634\pi$
$727$	299480.	−	251294.i	0.566630	−	0.475459i	0.880526	+	0.473998i	$0.157190\pi$
$728$		0			0
$729$	−486453.	+	213994.i	−0.915347	+	0.402667i
$730$		0			0
$731$	−188121.	−	224194.i	−0.352049	−	0.419555i
$732$		0			0
$733$	−63621.7	+	360816.i	−0.118412	+	0.671550i	0.866592	+	0.499018i	$0.166306\pi$
$733$	−63621.7	+	360816.i	−0.118412	+	0.671550i	−0.985004	+	0.172532i	$0.944805\pi$
$734$		0			0
$735$	362246.	+	1.45750e6i	0.670547	+	2.69794i
$736$		0			0
$737$	−604070.	+	348760.i	−1.11212	+	0.642084i
$738$		0			0
$739$	−269023.	+	465962.i	−0.492607	+	0.853221i	−0.999964	−	0.00851547i	$-0.997289\pi$
$739$	−269023.	+	465962.i	−0.492607	+	0.853221i	0.507356	+	0.861736i	$0.330623\pi$
$740$		0			0
$741$	339168.	−	466097.i	0.617701	−	0.848867i
$742$		0			0
$743$	569499.	−	678702.i	1.03161	−	1.22942i	0.0586901	−	0.998276i	$-0.481308\pi$
$743$	569499.	−	678702.i	1.03161	−	1.22942i	0.972919	−	0.231147i	$-0.0742479\pi$
$744$		0			0
$745$	706468.	+	257133.i	1.27286	+	0.463282i
$746$		0			0
$747$	23199.2	−	108353.i	0.0415750	−	0.194178i
$748$		0			0
$749$	562786.	−	99234.3i	1.00318	−	0.176888i
$750$		0			0
$751$	−321565.	+	117040.i	−0.570149	+	0.207517i	−0.610976	−	0.791649i	$-0.709223\pi$
$751$	−321565.	+	117040.i	−0.570149	+	0.207517i	0.0408271	+	0.999166i	$0.487001\pi$
$752$		0			0
$753$	701317.	−	48518.2i	1.23687	−	0.0855686i
$754$		0			0
$755$		−	240504.i		−	0.421918i
$756$		0			0
$757$	360278.			0.628704			0.314352	−	0.949307i	$-0.398213\pi$
$757$	360278.			0.628704			0.314352	+	0.949307i	$0.398213\pi$
$758$		0			0
$759$	−191974.	+	129281.i	−0.333241	+	0.224415i
$760$		0			0
$761$	−212386.	−	583525.i	−0.366738	−	1.00761i	−0.976594	−	0.215093i	$-0.930995\pi$
$761$	−212386.	−	583525.i	−0.366738	−	1.00761i	0.609855	−	0.792513i	$-0.291228\pi$
$762$		0			0
$763$	−15540.7	−	88135.9i	−0.0266945	−	0.151392i
$764$		0			0
$765$	782877.	−	414775.i	1.33774	−	0.708744i
$766$		0			0
$767$	531835.	−	1.46120e6i	0.904036	−	2.48382i
$768$		0			0
$769$	296424.	+	248729.i	0.501257	+	0.420604i	0.858040	−	0.513583i	$-0.171682\pi$
$769$	296424.	+	248729.i	0.501257	+	0.420604i	−0.356783	+	0.934187i	$0.616127\pi$
$770$		0			0
$771$	457353.	−	203220.i	0.769383	−	0.341868i
$772$		0			0
$773$	−599094.	−	345887.i	−1.00262	−	0.578862i	−0.0935968	−	0.995610i	$-0.529836\pi$
$773$	−599094.	−	345887.i	−1.00262	−	0.578862i	−0.909022	+	0.416748i	$0.863170\pi$
$774$		0			0
$775$	−38184.6	−	66137.8i	−0.0635749	−	0.110115i
$776$		0			0
$777$	1.08255e6	+	1.12268e6i	1.79310	+	1.85957i
$778$		0			0
$779$	438327.	+	77288.8i	0.722309	+	0.127363i
$780$		0			0
$781$	−138853.	+	116512.i	−0.227643	+	0.191015i
$782$		0			0
$783$	−301072.	+	383592.i	−0.491074	+	0.625671i
$784$		0			0
$785$	−660013.	−	786573.i	−1.07106	−	1.27644i
$786$		0			0
$787$	−111937.	+	634824.i	−0.180727	+	1.02495i	0.750597	+	0.660760i	$0.229766\pi$
$787$	−111937.	+	634824.i	−0.180727	+	1.02495i	−0.931324	+	0.364193i	$0.881345\pi$
$788$		0			0
$789$	334212.	+	96102.0i	0.536868	+	0.154376i
$790$		0			0
$791$	1.41975e6	−	819696.i	2.26914	−	1.31009i
$792$		0			0
$793$	−551940.	+	955988.i	−0.877699	+	1.52022i
$794$		0			0
$795$	−836901.	−	88589.5i	−1.32416	−	0.140168i
$796$		0			0
$797$	−415691.	+	495401.i	−0.654416	+	0.779903i	−0.986573	−	0.163321i	$-0.947779\pi$
$797$	−415691.	+	495401.i	−0.654416	+	0.779903i	0.332157	+	0.943224i	$0.392224\pi$
$798$		0			0
$799$	−596261.	−	217021.i	−0.933992	−	0.339945i
$800$		0			0
$801$	−242687.	−	98471.1i	−0.378253	−	0.153477i
$802$		0			0
$803$	−306878.	+	54110.9i	−0.475921	+	0.0839178i
$804$		0			0
$805$	−495073.	+	180192.i	−0.763973	+	0.278063i
$806$		0			0
$807$	106573.	−	218097.i	0.163645	−	0.334891i
$808$		0			0
$809$		−	380154.i		−	0.580849i	−0.956898	−	0.290424i	$-0.906204\pi$
$809$		−	380154.i		−	0.580849i	0.956898	−	0.290424i	$-0.0937964\pi$
$810$		0			0
$811$	−854020.			−1.29845			−0.649227	−	0.760595i	$-0.724907\pi$
$811$	−854020.			−1.29845			−0.649227	+	0.760595i	$0.724907\pi$
$812$		0			0
$813$	122155.	+	59691.1i	0.184812	+	0.0903084i
$814$		0			0
$815$	126691.	+	348081.i	0.190735	+	0.524041i
$816$		0			0
$817$	33336.5	+	189061.i	0.0499431	+	0.283242i
$818$		0			0
$819$	−1.98115e6	+	275435.i	−2.95358	+	0.410631i
$820$		0			0
$821$	−339289.	+	932188.i	−0.503365	+	1.38298i	0.384604	+	0.923082i	$0.374338\pi$
$821$	−339289.	+	932188.i	−0.503365	+	1.38298i	−0.887969	+	0.459903i	$0.847884\pi$
$822$		0			0
$823$	−832429.	−	698491.i	−1.22899	−	1.03124i	−0.998304	−	0.0582138i	$-0.981459\pi$
$823$	−832429.	−	698491.i	−1.22899	−	1.03124i	−0.230683	−	0.973029i	$-0.574096\pi$
$824$		0			0
$825$	−47315.3	+	446985.i	−0.0695174	+	0.656728i
$826$		0			0
$827$	−227618.	−	131416.i	−0.332810	−	0.192148i	0.324278	−	0.945962i	$-0.394879\pi$
$827$	−227618.	−	131416.i	−0.332810	−	0.192148i	−0.657088	+	0.753814i	$0.728212\pi$
$828$		0			0
$829$	−297967.	−	516093.i	−0.433569	−	0.750964i	0.563608	−	0.826042i	$-0.309413\pi$
$829$	−297967.	−	516093.i	−0.433569	−	0.750964i	−0.997178	+	0.0750780i	$0.976079\pi$
$830$		0			0
$831$	26704.4	−	92869.1i	0.0386705	−	0.134484i
$832$		0			0
$833$	−1.80674e6	−	318576.i	−2.60378	−	0.459117i
$834$		0			0
$835$	159556.	−	133884.i	0.228845	−	0.192023i
$836$		0			0
$837$	−150438.	+	4918.26i	−0.214737	+	0.00702039i
$838$		0			0
$839$	362259.	+	431724.i	0.514630	+	0.613313i	0.959302	−	0.282381i	$-0.0911242\pi$
$839$	362259.	+	431724.i	0.514630	+	0.613313i	−0.444672	+	0.895693i	$0.646680\pi$
$840$		0			0
$841$	45121.2	−	255895.i	0.0637953	−	0.361801i
$842$		0			0
$843$	−57974.7	+	55902.5i	−0.0815800	+	0.0786640i
$844$		0			0
$845$	1.38545e6	−	799888.i	1.94033	−	1.12025i
$846$		0			0
$847$	157457.	−	272724.i	0.219481	−	0.380152i
$848$		0			0
$849$	228392.	+	514002.i	0.316858	+	0.713098i
$850$		0			0
$851$	−241893.	+	288277.i	−0.334014	+	0.398062i
$852$		0			0
$853$	279796.	+	101838.i	0.384542	+	0.139962i	0.527055	−	0.849831i	$-0.323296\pi$
$853$	279796.	+	101838.i	0.384542	+	0.139962i	−0.142513	+	0.989793i	$0.545518\pi$
$854$		0			0
$855$	−580776.	−	21143.9i	−0.794469	−	0.0289236i
$856$		0			0
$857$	585082.	−	103166.i	0.796628	−	0.140467i	0.239503	−	0.970896i	$-0.423016\pi$
$857$	585082.	−	103166.i	0.796628	−	0.140467i	0.557125	+	0.830429i	$0.311904\pi$
$858$		0			0
$859$	807932.	−	294063.i	1.09494	−	0.398524i	0.269488	−	0.963004i	$-0.413145\pi$
$859$	807932.	−	294063.i	1.09494	−	0.398524i	0.825447	+	0.564480i	$0.190923\pi$
$860$		0			0
$861$	−862682.	−	1.28103e6i	−1.16371	−	1.72803i
$862$		0			0
$863$			395978.i			0.531679i	0.964017	+	0.265839i	$0.0856491\pi$
$863$			395978.i			0.531679i	−0.964017	+	0.265839i	$0.914351\pi$
$864$		0			0
$865$	1.02268e6			1.36681
$866$		0			0
$867$	22815.8	+	329797.i	0.0303528	+	0.438741i
$868$		0			0
$869$	−366583.	−	1.00718e6i	−0.485437	−	1.33373i
$870$		0			0
$871$	−252578.	−	1.43244e6i	−0.332935	−	1.88817i
$872$		0			0
$873$	−273782.	−	846886.i	−0.359233	−	1.11121i
$874$		0			0
$875$	241375.	−	663173.i	0.315266	−	0.866186i
$876$		0			0
$877$	−952713.	−	799421.i	−1.23869	−	1.03938i	−0.997625	−	0.0688758i	$-0.978059\pi$
$877$	−952713.	−	799421.i	−1.23869	−	1.03938i	−0.241065	−	0.970509i	$-0.577497\pi$
$878$		0			0
$879$	−140557.	−	102280.i	−0.181918	−	0.132377i
$880$		0			0
$881$	725858.	+	419074.i	0.935190	+	0.539932i	0.888449	−	0.458975i	$-0.151783\pi$
$881$	725858.	+	419074.i	0.935190	+	0.539932i	0.0467407	+	0.998907i	$0.485117\pi$
$882$		0			0
$883$	−34200.8	−	59237.6i	−0.0438647	−	0.0759759i	0.843259	−	0.537507i	$-0.180634\pi$
$883$	−34200.8	−	59237.6i	−0.0438647	−	0.0759759i	−0.887124	+	0.461531i	$0.847300\pi$
$884$		0			0
$885$	−1.52143e6	+	378137.i	−1.94252	+	0.482795i
$886$		0			0
$887$	752766.	+	132733.i	0.956781	+	0.168706i	0.630174	−	0.776454i	$-0.282983\pi$
$887$	752766.	+	132733.i	0.956781	+	0.168706i	0.326607	+	0.945160i	$0.394095\pi$
$888$		0			0
$889$	−1.12935e6	+	947640.i	−1.42898	+	1.19906i
$890$		0			0
$891$	732971.	+	497566.i	0.923276	+	0.626751i
$892$		0			0
$893$	267546.	+	318849.i	0.335502	+	0.399836i
$894$		0			0
$895$	−132199.	+	749739.i	−0.165037	+	0.935974i
$896$		0			0
$897$	−116413.	−	468386.i	−0.144682	−	0.582129i
$898$		0			0
$899$	−119608.	+	69055.7i	−0.147993	+	0.0854437i
$900$		0			0
$901$	514025.	−	890317.i	0.633191	−	1.09672i
$902$		0			0
$903$	391951.	−	538634.i	0.480681	−	0.660569i
$904$		0			0
$905$	−135269.	+	161207.i	−0.165158	+	0.196828i
$906$		0			0
$907$	674332.	+	245437.i	0.819709	+	0.298349i	0.717628	−	0.696427i	$-0.245228\pi$
$907$	674332.	+	245437.i	0.819709	+	0.298349i	0.102081	+	0.994776i	$0.467450\pi$
$908$		0			0
$909$	1.09773e6	+	991352.i	1.32852	+	1.19978i
$910$		0			0
$911$	−258709.	+	45617.4i	−0.311727	+	0.0549659i	−0.327323	−	0.944912i	$-0.606147\pi$
$911$	−258709.	+	45617.4i	−0.311727	+	0.0549659i	0.0155960	+	0.999878i	$0.495035\pi$
$912$		0			0
$913$	−173576.	+	63176.4i	−0.208232	+	0.0757903i
$914$		0			0
$915$	1.11027e6	−	76810.1i	1.32613	−	0.0917437i
$916$		0			0
$917$		−	1.24973e6i		−	1.48620i
$918$		0			0
$919$	−1.20587e6			−1.42781			−0.713904	−	0.700244i	$-0.753074\pi$
$919$	−1.20587e6			−1.42781			−0.713904	+	0.700244i	$0.753074\pi$
$920$		0			0
$921$	−1.09332e6	+	736274.i	−1.28893	+	0.868001i
$922$		0			0
$923$	−129277.	−	355187.i	−0.151747	−	0.416920i
$924$		0			0
$925$	126908.	+	719730.i	0.148322	+	0.841174i
$926$		0			0
$927$	185430.	+	116253.i	0.215785	+	0.135283i
$928$		0			0
$929$	−379238.	+	1.04195e6i	−0.439421	+	1.20730i	0.500449	+	0.865766i	$0.333168\pi$
$929$	−379238.	+	1.04195e6i	−0.439421	+	1.20730i	−0.939870	+	0.341533i	$0.889054\pi$
$930$		0			0
$931$	921886.	+	773554.i	1.06360	+	0.892465i
$932$		0			0
$933$	977434.	−	434313.i	1.12286	−	0.498930i
$934$		0			0
$935$	−1.27902e6	−	738442.i	−1.46303	−	0.844682i
$936$		0			0
$937$	−236877.	−	410283.i	−0.269801	−	0.467309i	0.699009	−	0.715112i	$-0.253625\pi$
$937$	−236877.	−	410283.i	−0.269801	−	0.467309i	−0.968810	+	0.247804i	$0.920291\pi$
$938$		0			0
$939$	−851588.	−	883155.i	−0.965825	−	1.00163i
$940$		0			0
$941$	−672157.	−	118519.i	−0.759087	−	0.133847i	−0.219309	−	0.975655i	$-0.570380\pi$
$941$	−672157.	−	118519.i	−0.759087	−	0.133847i	−0.539777	+	0.841808i	$0.681492\pi$
$942$		0			0
$943$	285475.	−	239542.i	0.321029	−	0.269375i
$944$		0			0
$945$	1.34602e6	+	1.50162e6i	1.50726	+	1.68149i
$946$		0			0
$947$	500158.	+	596065.i	0.557709	+	0.664651i	0.969060	−	0.246826i	$-0.0793875\pi$
$947$	500158.	+	596065.i	0.557709	+	0.664651i	−0.411351	+	0.911477i	$0.634943\pi$
$948$		0			0
$949$	112838.	−	639934.i	0.125291	−	0.710563i
$950$		0			0
$951$	17970.9	+	5167.49i	0.0198705	+	0.00571372i
$952$		0			0
$953$	772825.	−	446190.i	0.850932	−	0.491286i	−0.0100329	−	0.999950i	$-0.503194\pi$
$953$	772825.	−	446190.i	0.850932	−	0.491286i	0.860965	+	0.508664i	$0.169860\pi$
$954$		0			0
$955$	−444626.	+	770114.i	−0.487515	+	0.844400i
$956$		0			0
$957$	808358.	+	85568.2i	0.882633	+	0.0934304i
$958$		0			0
$959$	−2.05268e6	+	2.44629e6i	−2.23195	+	2.65993i
$960$		0			0
$961$	827766.	+	301282.i	0.896315	+	0.326232i
$962$		0			0
$963$	416396.	−	324330.i	0.449007	−	0.349732i
$964$		0			0
$965$	−1.06068e6	+	187027.i	−1.13902	+	0.200839i
$966$		0			0
$967$	76770.3	−	27942.1i	0.0820995	−	0.0298818i	−0.300644	−	0.953736i	$-0.597202\pi$
$967$	76770.3	−	27942.1i	0.0820995	−	0.0298818i	0.382743	+	0.923855i	$0.374979\pi$
$968$		0			0
$969$	311687.	−	637852.i	0.331949	−	0.679317i
$970$		0			0
$971$			1.56203e6i			1.65673i	0.560190	+	0.828364i	$0.310728\pi$
$971$			1.56203e6i			1.65673i	−0.560190	+	0.828364i	$0.689272\pi$
$972$		0			0
$973$	−738603.			−0.780163
$974$		0			0
$975$	−842140.	−	411513.i	−0.885881	−	0.432887i
$976$		0			0
$977$	−397237.	−	1.09140e6i	−0.416160	−	1.14339i	−0.953860	−	0.300253i	$-0.902929\pi$
$977$	−397237.	−	1.09140e6i	−0.416160	−	1.14339i	0.537700	−	0.843136i	$-0.319293\pi$
$978$		0			0
$979$	75812.6	+	429955.i	0.0790999	+	0.448598i
$980$		0			0
$981$	−50792.2	−	65210.3i	−0.0527788	−	0.0677607i
$982$		0			0
$983$	195388.	−	536825.i	0.202205	−	0.555554i	−0.796596	−	0.604512i	$-0.793368\pi$
$983$	195388.	−	536825.i	0.202205	−	0.555554i	0.998801	+	0.0489587i	$0.0155903\pi$
$984$		0			0
$985$	127428.	+	106925.i	0.131339	+	0.110206i
$986$		0			0
$987$	152034.	−	1.43626e6i	0.156065	−	1.47434i
$988$		0			0
$989$	139203.	+	80368.7i	0.142316	+	0.0821664i
$990$		0			0
$991$	240431.	+	416439.i	0.244818	+	0.424037i	0.962080	−	0.272766i	$-0.0879384\pi$
$991$	240431.	+	416439.i	0.244818	+	0.424037i	−0.717262	+	0.696803i	$0.754605\pi$
$992$		0			0
$993$	199449.	−	693619.i	0.202271	−	0.703432i
$994$		0			0
$995$	1.50947e6	+	266160.i	1.52468	+	0.268842i
$996$		0			0
$997$	427974.	−	359113.i	0.430554	−	0.361277i	−0.401607	−	0.915812i	$-0.631548\pi$
$997$	427974.	−	359113.i	0.430554	−	0.361277i	0.832161	+	0.554535i	$0.187104\pi$
$998$		0			0
$999$	1.36893e6	+	448162.i	1.37167	+	0.449060i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.11	✓	72
3.2	odd	2		324.5.k.a.125.11		72
27.11	odd	18	inner	108.5.k.a.65.11	yes	72
27.16	even	9		324.5.k.a.197.11		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.11	✓	72	1.1	even	1	trivial
108.5.k.a.65.11	yes	72	27.11	odd	18	inner
324.5.k.a.125.11		72	3.2	odd	2
324.5.k.a.197.11		72	27.16	even	9

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(8.08622 + 3.95134i) q^{3} +(-10.7879 - 29.6395i) q^{5} +(15.2291 + 86.3688i) q^{7} +(49.7738 + 63.9028i) q^{9} +O(q^{10})\)	\(q+(8.08622 + 3.95134i) q^{3} +(-10.7879 - 29.6395i) q^{5} +(15.2291 + 86.3688i) q^{7} +(49.7738 + 63.9028i) q^{9} +(46.1813 - 126.882i) q^{11} +(215.694 + 180.988i) q^{13} +(29.8824 - 282.298i) q^{15} +(300.316 + 173.387i) q^{17} +(-113.736 - 196.996i) q^{19} +(-218.126 + 758.572i) q^{21} +(-187.562 - 33.0723i) q^{23} +(-283.341 + 237.752i) q^{25} +(149.980 + 713.405i) q^{27} +(429.966 + 512.413i) q^{29} +(-35.8536 + 203.336i) q^{31} +(874.786 - 843.518i) q^{33} +(2395.63 - 1383.12i) q^{35} +(987.943 - 1711.17i) q^{37} +(1029.00 + 2315.79i) q^{39} +(-1257.73 + 1498.90i) q^{41} +(-793.064 - 288.652i) q^{43} +(1357.09 - 2164.64i) q^{45} +(-1802.00 + 317.741i) q^{47} +(-4971.44 + 1809.45i) q^{49} +(1743.31 + 2588.70i) q^{51} -2964.60i q^{53} -4258.91 q^{55} +(-141.294 - 2042.36i) q^{57} +(-1888.83 - 5189.52i) q^{59} +(680.783 + 3860.91i) q^{61} +(-4761.19 + 5272.09i) q^{63} +(3037.52 - 8345.52i) q^{65} +(-3957.28 - 3320.55i) q^{67} +(-1385.99 - 1008.55i) q^{69} +(-1162.57 - 671.209i) q^{71} +(-1153.91 - 1998.62i) q^{73} +(-3230.60 + 802.933i) q^{75} +(11662.0 + 2056.32i) q^{77} +(6080.80 - 5102.39i) q^{79} +(-1606.13 + 6361.37i) q^{81} +(-879.339 - 1047.96i) q^{83} +(1899.34 - 10771.7i) q^{85} +(1452.08 + 5842.43i) q^{87} +(-2800.19 + 1616.69i) q^{89} +(-12346.9 + 21385.5i) q^{91} +(-1093.37 + 1502.55i) q^{93} +(-4611.89 + 5496.24i) q^{95} +(-10325.5 - 3758.17i) q^{97} +(10406.7 - 3364.29i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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