LMFDB - Embedded newform 108.5.k.a.65.11

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			65.11
Character	$\chi$	$=$	108.65
Dual form			108.5.k.a.5.11

$q$-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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

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	0.513367	+	0.858169i	$0.328398\pi$
$5$	−10.7879	+	29.6395i	−0.431515	+	1.18558i	−0.944883	+	0.327409i	$0.893824\pi$
$6$		0			0
$7$	15.2291	−	86.3688i	0.310799	−	1.76263i	−0.284068	−	0.958804i	$-0.591684\pi$
$7$	15.2291	−	86.3688i	0.310799	−	1.76263i	0.594867	−	0.803824i	$-0.297205\pi$
$8$		0			0
$9$	49.7738	−	63.9028i	0.614492	−	0.788923i
$10$		0			0
$11$	46.1813	+	126.882i	0.381664	+	1.04861i	0.970656	+	0.240473i	$0.0773027\pi$
$11$	46.1813	+	126.882i	0.381664	+	1.04861i	−0.588992	+	0.808139i	$0.700475\pi$
$12$		0			0
$13$	215.694	−	180.988i	1.27629	−	1.07094i	0.282549	−	0.959253i	$-0.408820\pi$
$13$	215.694	−	180.988i	1.27629	−	1.07094i	0.993744	−	0.111684i	$-0.0356245\pi$
$14$		0			0
$15$	29.8824	+	282.298i	0.132811	+	1.25466i
$16$		0			0
$17$	300.316	−	173.387i	1.03915	−	0.599956i	0.119561	−	0.992827i	$-0.461851\pi$
$17$	300.316	−	173.387i	1.03915	−	0.599956i	0.919594	+	0.392871i	$0.128518\pi$
$18$		0			0
$19$	−113.736	+	196.996i	−0.315058	+	0.545696i	−0.979450	−	0.201688i	$-0.935357\pi$
$19$	−113.736	+	196.996i	−0.315058	+	0.545696i	0.664392	+	0.747384i	$0.268690\pi$
$20$		0			0
$21$	−218.126	−	758.572i	−0.494617	−	1.72012i
$22$		0			0
$23$	−187.562	+	33.0723i	−0.354560	+	0.0625185i	−0.348091	−	0.937461i	$-0.613170\pi$
$23$	−187.562	+	33.0723i	−0.354560	+	0.0625185i	−0.00646861	+	0.999979i	$0.502059\pi$
$24$		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	−0.447234	−	0.894417i	$-0.647591\pi$
$29$	429.966	−	512.413i	0.511256	−	0.609291i	0.958490	+	0.285126i	$0.0920355\pi$
$30$		0			0
$31$	−35.8536	−	203.336i	−0.0373087	−	0.211588i	0.960454	−	0.278437i	$-0.0898164\pi$
$31$	−35.8536	−	203.336i	−0.0373087	−	0.211588i	−0.997763	+	0.0668493i	$0.978705\pi$
$32$		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	0.960337	+	0.278842i	$0.0899507\pi$
$37$	987.943	+	1711.17i	0.721653	+	1.24994i	−0.238684	+	0.971097i	$0.576716\pi$
$38$		0			0
$39$	1029.00	−	2315.79i	0.676528	−	1.52254i
$40$		0			0
$41$	−1257.73	−	1498.90i	−0.748203	−	0.891673i	0.248838	−	0.968545i	$-0.419951\pi$
$41$	−1257.73	−	1498.90i	−0.748203	−	0.891673i	−0.997041	+	0.0768719i	$0.975507\pi$
$42$		0			0
$43$	−793.064	+	288.652i	−0.428915	+	0.156112i	−0.547452	−	0.836837i	$-0.684402\pi$
$43$	−793.064	+	288.652i	−0.428915	+	0.156112i	0.118536	+	0.992950i	$0.462180\pi$
$44$		0			0
$45$	1357.09	+	2164.64i	0.670168	+	1.06896i
$46$		0			0
$47$	−1802.00	−	317.741i	−0.815754	−	0.143839i	−0.249822	−	0.968292i	$-0.580372\pi$
$47$	−1802.00	−	317.741i	−0.815754	−	0.143839i	−0.565932	+	0.824452i	$0.691483\pi$
$48$		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	$0.176944\pi$
$53$			2964.60i			1.05539i	−0.849432	+	0.527697i	$0.823056\pi$
$54$		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	0.300875	+	0.953664i	$0.402721\pi$
$59$	−1888.83	+	5189.52i	−0.542612	+	1.49081i	−0.843487	+	0.537150i	$0.819501\pi$
$60$		0			0
$61$	680.783	−	3860.91i	0.182957	−	1.03760i	−0.745595	−	0.666399i	$-0.767835\pi$
$61$	680.783	−	3860.91i	0.182957	−	1.03760i	0.928552	−	0.371202i	$-0.121054\pi$
$62$		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	−0.966497	−	0.256678i	$-0.917372\pi$
$67$	−3957.28	+	3320.55i	−0.881549	+	0.739708i	0.0849477	+	0.996385i	$0.472928\pi$
$68$		0			0
$69$	−1385.99	+	1008.55i	−0.291113	+	0.211836i
$70$		0			0
$71$	−1162.57	+	671.209i	−0.230622	+	0.133150i	−0.610859	−	0.791739i	$-0.709176\pi$
$71$	−1162.57	+	671.209i	−0.230622	+	0.133150i	0.380237	+	0.924889i	$0.375843\pi$
$72$		0			0
$73$	−1153.91	+	1998.62i	−0.216533	+	0.375047i	−0.953746	−	0.300614i	$-0.902808\pi$
$73$	−1153.91	+	1998.62i	−0.216533	+	0.375047i	0.737213	+	0.675661i	$0.236142\pi$
$74$		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	0.983224	−	0.182400i	$-0.0583867\pi$
$79$	6080.80	+	5102.39i	0.974330	+	0.817560i	−0.00889403	+	0.999960i	$0.502831\pi$
$80$		0			0
$81$	−1606.13	−	6361.37i	−0.244800	−	0.969574i
$82$		0			0
$83$	−879.339	+	1047.96i	−0.127644	+	0.152120i	−0.826081	−	0.563551i	$-0.809435\pi$
$83$	−879.339	+	1047.96i	−0.127644	+	0.152120i	0.698437	+	0.715671i	$0.253879\pi$
$84$		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	0.312717	−	0.949846i	$-0.398761\pi$
$89$	−2800.19	−	1616.69i	−0.353515	−	0.204102i	−0.666232	+	0.745744i	$0.732094\pi$
$90$		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	−0.826359	−	0.563143i	$-0.809592\pi$
$97$	−10325.5	+	3758.17i	−1.09741	+	0.399423i	−0.271046	+	0.962566i	$0.587370\pi$
$98$		0			0
$99$	10406.7	+	3364.29i	1.06180	+	0.343260i
$100$		0			0
$101$	17983.3	+	3170.94i	1.76289	+	0.310846i	0.958890	−	0.283777i	$-0.0915876\pi$
$101$	17983.3	+	3170.94i	1.76289	+	0.310846i	0.804005	+	0.594623i	$0.202699\pi$
$102$		0			0
$103$	2539.01	+	924.124i	0.239326	+	0.0871075i	0.458899	−	0.888489i	$-0.348244\pi$
$103$	2539.01	+	924.124i	0.239326	+	0.0871075i	−0.219573	+	0.975596i	$0.570466\pi$
$104$		0			0
$105$	24836.8	+	1718.25i	2.25277	+	0.155850i
$106$		0			0
$107$			6516.08i			0.569140i	0.958655	+	0.284570i	$0.0918508\pi$
$107$			6516.08i			0.569140i	−0.958655	+	0.284570i	$0.908149\pi$
$108$		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	0.389905	+	0.920855i	$0.372508\pi$
$113$	−6393.36	+	17565.6i	−0.500694	+	1.37564i	−0.890599	+	0.454790i	$0.849714\pi$
$114$		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	−0.478584	−	0.878042i	$-0.658850\pi$
$127$	8405.05	−	14558.0i	0.521114	−	0.902597i	0.999698	−	0.0245549i	$-0.00781687\pi$
$128$		0			0
$129$	−5272.33	+	5467.77i	−0.316828	+	0.328572i
$130$		0			0
$131$	−14033.3	+	2474.46i	−0.817746	+	0.144191i	−0.566848	−	0.823823i	$-0.691837\pi$
$131$	−14033.3	+	2474.46i	−0.817746	+	0.144191i	−0.250899	+	0.968013i	$0.580726\pi$
$132$		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	0.437321	−	0.899306i	$-0.355927\pi$
$137$	23405.4	−	27893.5i	1.24702	−	1.48615i	0.809703	−	0.586840i	$-0.199628\pi$
$138$		0			0
$139$	−1462.43	−	8293.86i	−0.0756913	−	0.429267i	−0.998980	−	0.0451595i	$-0.985620\pi$
$139$	−1462.43	−	8293.86i	−0.0756913	−	0.429267i	0.923288	−	0.384107i	$-0.125491\pi$
$140$		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	0.301261	−	0.953542i	$-0.402593\pi$
$149$	−15321.1	−	18259.0i	−0.690108	−	0.822439i	−0.991369	+	0.131103i	$0.958148\pi$
$150$		0			0
$151$	7165.12	−	2607.89i	0.314246	−	0.114376i	−0.180082	−	0.983652i	$-0.557636\pi$
$151$	7165.12	−	2607.89i	0.314246	−	0.114376i	0.494328	+	0.869275i	$0.335414\pi$
$152$		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	0.877366	−	0.479821i	$-0.159299\pi$
$157$	30590.5	+	11134.0i	1.24104	+	0.451703i	0.363678	+	0.931525i	$0.381521\pi$
$158$		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	−0.892592	−	0.450864i	$-0.851116\pi$
$167$	2258.54	−	6205.28i	0.0809830	−	0.222499i	0.973576	+	0.228365i	$0.0733380\pi$
$168$		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	−0.975159	−	0.221505i	$-0.928903\pi$
$173$	−11089.4	−	30467.9i	−0.370524	−	1.01801i	0.604635	−	0.796503i	$-0.293319\pi$
$174$		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	−0.789366	−	0.613923i	$-0.789591\pi$
$179$	−20902.8	+	12068.2i	−0.652377	+	0.376650i	0.136989	+	0.990573i	$0.456257\pi$
$180$		0			0
$181$	−3335.92	+	5777.98i	−0.101826	+	0.176368i	−0.912437	−	0.409217i	$-0.865802\pi$
$181$	−3335.92	+	5777.98i	−0.101826	+	0.176368i	0.810611	+	0.585585i	$0.199135\pi$
$182$		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	−0.954922	−	0.296858i	$-0.904061\pi$
$191$	−18122.1	+	21597.0i	−0.496753	+	0.592008i	0.458168	+	0.888865i	$0.348506\pi$
$192$		0			0
$193$	5929.51	+	33627.9i	0.159186	+	0.902788i	0.954859	+	0.297061i	$0.0960065\pi$
$193$	5929.51	+	33627.9i	0.159186	+	0.902788i	−0.795673	+	0.605727i	$0.792882\pi$
$194$		0			0
$195$	57538.0	+	55481.4i	1.51316	+	1.45908i
$196$		0			0
$197$	−4567.28	−	2636.92i	−0.117686	−	0.0679461i	0.440001	−	0.897997i	$-0.354978\pi$
$197$	−4567.28	−	2636.92i	−0.117686	−	0.0679461i	−0.557688	+	0.830051i	$0.688311\pi$
$198$		0			0
$199$	−24297.3	−	42084.1i	−0.613553	−	1.06270i	−0.990637	−	0.136525i	$-0.956407\pi$
$199$	−24297.3	−	42084.1i	−0.613553	−	1.06270i	0.377084	−	0.926179i	$-0.376927\pi$
$200$		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	0.179946	−	0.983677i	$-0.442408\pi$
$211$	−14002.0	−	5096.30i	−0.314503	−	0.114470i	−0.494449	+	0.869207i	$0.664630\pi$
$212$		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	−0.996504	−	0.0835505i	$-0.973374\pi$
$223$	−1567.49	+	8889.65i	−0.0315206	+	0.178762i	0.964983	+	0.262312i	$0.0844851\pi$
$224$		0			0
$225$	−29296.0	+	6272.49i	−0.578686	+	0.123901i
$226$		0			0
$227$	20863.3	+	57321.4i	0.404885	+	1.11241i	0.959844	+	0.280534i	$0.0905115\pi$
$227$	20863.3	+	57321.4i	0.404885	+	1.11241i	−0.554960	+	0.831877i	$0.687266\pi$
$228$		0			0
$229$	−45065.7	+	37814.6i	−0.859361	+	0.721089i	−0.961830	−	0.273647i	$-0.911770\pi$
$229$	−45065.7	+	37814.6i	−0.859361	+	0.721089i	0.102470	+	0.994736i	$0.467326\pi$
$230$		0			0
$231$	86175.9	−	62708.2i	1.61496	−	1.17517i
$232$		0			0
$233$	25541.4	−	14746.3i	0.470470	−	0.271626i	−0.245966	−	0.969278i	$-0.579105\pi$
$233$	25541.4	−	14746.3i	0.470470	−	0.271626i	0.716437	+	0.697652i	$0.245772\pi$
$234$		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	−0.902297	−	0.431115i	$-0.858120\pi$
$239$	−91549.5	+	16142.6i	−1.60273	+	0.282604i	−0.700432	+	0.713719i	$0.747009\pi$
$240$		0			0
$241$	75263.3	+	63153.4i	1.29583	+	1.08733i	0.990849	+	0.134976i	$0.0430957\pi$
$241$	75263.3	+	63153.4i	1.29583	+	1.08733i	0.304985	+	0.952357i	$0.401349\pi$
$242$		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	0.929196	−	0.369587i	$-0.120501\pi$
$251$	67645.6	+	39055.2i	1.07372	+	0.619914i	0.144527	+	0.989501i	$0.453834\pi$
$252$		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	0.965450	−	0.260587i	$-0.0839160\pi$
$257$	35744.0	+	42598.0i	0.541174	+	0.644946i	−0.424277	+	0.905533i	$0.639472\pi$
$258$		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	0.441805	−	0.897111i	$-0.354338\pi$
$263$	38052.4	+	6709.66i	0.550136	+	0.0970038i	0.108331	+	0.994115i	$0.465449\pi$
$264$		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	$0.0596715\pi$
$269$			26971.5i			0.372735i	−0.982480	+	0.186367i	$0.940329\pi$
$270$		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	−0.994544	−	0.104319i	$-0.966734\pi$
$277$	−1864.45	+	10573.8i	−0.0242991	+	0.137807i	0.970245	+	0.242126i	$0.0778448\pi$
$278$		0			0
$279$	−14778.3	−	7829.67i	−0.189853	−	0.100585i
$280$		0			0
$281$	−3060.57	−	8408.86i	−0.0387606	−	0.106494i	0.918803	−	0.394718i	$-0.129158\pi$
$281$	−3060.57	−	8408.86i	−0.0387606	−	0.106494i	−0.957563	+	0.288224i	$0.906935\pi$
$282$		0			0
$283$	47874.3	−	40171.3i	0.597764	−	0.501584i	−0.292962	−	0.956124i	$-0.594641\pi$
$283$	47874.3	−	40171.3i	0.597764	−	0.501584i	0.890726	+	0.454540i	$0.150196\pi$
$284$		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	−0.283329	−	0.959023i	$-0.591439\pi$
$293$	−19021.2	+	3353.96i	−0.221566	+	0.0390681i	0.0617628	+	0.998091i	$0.480328\pi$
$294$		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	−0.933678	−	0.358113i	$-0.883420\pi$
$307$	−73229.0	−	126836.i	−0.776974	−	1.34576i	0.156704	−	0.987646i	$-0.449913\pi$
$308$		0			0
$309$	24182.5	−	2559.82i	0.253270	−	0.0268098i
$310$		0			0
$311$	76390.5	+	91038.6i	0.789802	+	0.941250i	0.999332	−	0.0365526i	$-0.0116377\pi$
$311$	76390.5	+	91038.6i	0.789802	+	0.941250i	−0.209529	+	0.977802i	$0.567193\pi$
$312$		0			0
$313$	−128096.	+	46623.1i	−1.30752	+	0.475897i	−0.899437	−	0.437050i	$-0.856023\pi$
$313$	−128096.	+	46623.1i	−1.30752	+	0.475897i	−0.408078	+	0.912947i	$0.633801\pi$
$314$		0			0
$315$	207625.	−	84244.5i	2.09247	−	0.849025i
$316$		0			0
$317$	2046.11	+	360.784i	0.0203615	+	0.00359028i	0.183820	−	0.982960i	$-0.441154\pi$
$317$	2046.11	+	360.784i	0.0203615	+	0.00359028i	−0.163458	+	0.986550i	$0.552265\pi$
$318$		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	0.852939	+	0.522010i	$0.174818\pi$
$331$	−13925.1	+	78973.4i	−0.127099	+	0.720817i	−0.980039	+	0.198807i	$0.936293\pi$
$332$		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	−0.807881	−	0.589345i	$-0.799386\pi$
$337$	−41768.0	+	35047.5i	−0.367777	+	0.308601i	0.440105	+	0.897946i	$0.354941\pi$
$338$		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	−0.270923	−	0.962601i	$-0.587329\pi$
$347$	−23633.6	+	4167.25i	−0.196278	+	0.0346091i	0.0746448	+	0.997210i	$0.476218\pi$
$348$		0			0
$349$	−124185.	−	104204.i	−1.01957	−	0.855524i	−0.0300003	−	0.999550i	$-0.509551\pi$
$349$	−124185.	−	104204.i	−1.01957	−	0.855524i	−0.989574	+	0.144025i	$0.953995\pi$
$350$		0			0
$351$	−96768.2	−	181022.i	−0.785450	−	1.46932i
$352$		0			0
$353$	67927.0	−	80952.2i	0.545121	−	0.649650i	−0.421207	−	0.906965i	$-0.638393\pi$
$353$	67927.0	−	80952.2i	0.545121	−	0.649650i	0.966328	+	0.257315i	$0.0828378\pi$
$354$		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	0.503724	−	0.863864i	$-0.331963\pi$
$359$	961.228	+	554.965i	0.00745826	+	0.00430603i	−0.496266	+	0.868170i	$0.665296\pi$
$360$		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	−0.606020	−	0.795450i	$-0.707235\pi$
$367$	−75284.8	+	27401.4i	−0.558952	+	0.203442i	0.0470672	+	0.998892i	$0.485013\pi$
$368$		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	0.428409	−	0.903585i	$-0.359074\pi$
$373$	24455.5	+	8901.06i	0.175776	+	0.0639771i	−0.252633	+	0.967562i	$0.581297\pi$
$374$		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	0.402087	+	0.915601i	$0.368285\pi$
$383$	−72532.8	+	199282.i	−0.494467	+	1.35854i	−0.896554	+	0.442934i	$0.853937\pi$
$384$		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	0.998931	+	0.0462165i	$0.0147164\pi$
$389$	39859.9	+	109514.i	0.263413	+	0.723721i	−0.735519	+	0.677505i	$0.763061\pi$
$390$		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	−0.309722	−	0.950827i	$-0.600236\pi$
$397$	105374.	−	182513.i	0.668580	−	1.15801i	0.978301	−	0.207187i	$-0.0664307\pi$
$398$		0			0
$399$	174245.	+	43306.8i	1.09449	+	0.272026i
$400$		0			0
$401$	18741.2	−	3304.59i	0.116549	−	0.0205508i	−0.115069	−	0.993357i	$-0.536709\pi$
$401$	18741.2	−	3304.59i	0.116549	−	0.0205508i	0.231619	+	0.972807i	$0.425598\pi$
$402$		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	−0.991913	−	0.126918i	$-0.959492\pi$
$409$	−17268.1	−	97932.2i	−0.103228	−	0.585435i	0.888685	−	0.458518i	$-0.151619\pi$
$410$		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	0.496972	−	0.867767i	$-0.334445\pi$
$419$	−77933.2	−	92877.2i	−0.443910	−	0.529031i	−0.940882	+	0.338736i	$0.890001\pi$
$420$		0			0
$421$	203870.	−	74202.8i	1.15024	−	0.418655i	0.304641	−	0.952467i	$-0.401463\pi$
$421$	203870.	−	74202.8i	1.15024	−	0.418655i	0.845603	+	0.533812i	$0.179241\pi$
$422$		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	$-0.782150\pi$
$431$		−	234878.i		−	1.26441i	0.774802	−	0.632204i	$-0.217850\pi$
$432$		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	−0.0404229	−	0.999183i	$-0.512871\pi$
$439$	65390.8	−	370850.i	0.339303	−	1.92428i	0.379726	−	0.925099i	$-0.376018\pi$
$440$		0			0
$441$	−363077.	+	227625.i	−1.86690	+	1.17042i
$442$		0			0
$443$	31232.3	+	85809.9i	0.159146	+	0.437250i	0.993480	−	0.114010i	$-0.0363695\pi$
$443$	31232.3	+	85809.9i	0.159146	+	0.437250i	−0.834334	+	0.551260i	$0.814147\pi$
$444$		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	−0.742806	−	0.669507i	$-0.766505\pi$
$449$	−107735.	+	62201.0i	−0.534399	+	0.308535i	0.208407	+	0.978042i	$0.433172\pi$
$450$		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	0.856023	−	0.516938i	$-0.172928\pi$
$457$	103503.	+	86849.0i	0.495586	+	0.415846i	−0.360437	+	0.932783i	$0.617372\pi$
$458$		0			0
$459$	−78653.9	−	240251.i	−0.373332	−	1.14036i
$460$		0			0
$461$	−168200.	+	200453.i	−0.791450	+	0.943213i	−0.999390	−	0.0349284i	$-0.988880\pi$
$461$	−168200.	+	200453.i	−0.791450	+	0.943213i	0.207940	+	0.978142i	$0.433324\pi$
$462$		0			0
$463$	−26468.5	−	150110.i	−0.123472	−	0.700242i	−0.982204	−	0.187818i	$-0.939859\pi$
$463$	−26468.5	−	150110.i	−0.123472	−	0.700242i	0.858732	−	0.512424i	$-0.171253\pi$
$464$		0			0
$465$	56329.9	−	16197.6i	0.260515	−	0.0749107i
$466$		0			0
$467$	−310052.	−	179009.i	−1.42168	−	0.820806i	−0.425235	−	0.905083i	$-0.639809\pi$
$467$	−310052.	−	179009.i	−1.42168	−	0.820806i	−0.996442	+	0.0842767i	$0.973142\pi$
$468$		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	−0.157883	−	0.987458i	$-0.550467\pi$
$479$	−147754.	−	26053.1i	−0.643976	−	0.113550i	−0.486092	+	0.873908i	$0.661578\pi$
$480$		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	−0.559826	−	0.828610i	$-0.689132\pi$
$491$	96829.2	−	266036.i	0.401646	−	1.10351i	0.961472	−	0.274903i	$-0.0886457\pi$
$492$		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	−0.964403	−	0.264437i	$-0.914814\pi$
$499$	−216992.	+	182078.i	−0.871450	+	0.731233i	0.0929533	+	0.995670i	$0.470369\pi$
$500$		0			0
$501$	−6256.14	−	59101.5i	−0.0249248	−	0.235463i
$502$		0			0
$503$	176203.	−	101731.i	0.696429	−	0.402083i	−0.109587	−	0.993977i	$-0.534953\pi$
$503$	176203.	−	101731.i	0.696429	−	0.402083i	0.806016	+	0.591894i	$0.201620\pi$
$504$		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	−0.0677222	−	0.997704i	$-0.521573\pi$
$509$	54374.6	−	9587.70i	0.209875	−	0.0370066i	0.277597	+	0.960698i	$0.410462\pi$
$510$		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	0.542175	−	0.840265i	$-0.317601\pi$
$521$	23227.7	+	13410.5i	0.0855719	+	0.0494050i	−0.456603	+	0.889670i	$0.650934\pi$
$522$		0			0
$523$	25930.8	+	44913.4i	0.0948009	+	0.164200i	0.909525	−	0.415648i	$-0.136445\pi$
$523$	25930.8	+	44913.4i	0.0948009	+	0.164200i	−0.814725	+	0.579848i	$0.803112\pi$
$524$		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	0.618989	+	0.785399i	$0.287542\pi$
$547$	−69204.9	+	392480.i	−0.231293	+	1.31173i	−0.850282	+	0.526327i	$0.823569\pi$
$548$		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	−0.933712	−	0.358026i	$-0.883450\pi$
$557$	−338329.	+	195334.i	−1.09051	+	0.629605i	−0.156797	+	0.987631i	$0.550117\pi$
$558$		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	−0.895456	−	0.445151i	$-0.853150\pi$
$563$	−502288.	+	88566.9i	−1.58466	+	0.279418i	−0.689203	+	0.724569i	$0.742039\pi$
$564$		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	0.0782149	−	0.996937i	$-0.475078\pi$
$569$	338791.	−	403756.i	1.04642	−	1.24708i	0.968209	−	0.250143i	$-0.0804776\pi$
$570$		0			0
$571$	13409.5	+	76049.2i	0.0411284	+	0.233250i	0.998442	−	0.0558010i	$-0.0177712\pi$
$571$	13409.5	+	76049.2i	0.0411284	+	0.233250i	−0.957314	+	0.289051i	$0.906660\pi$
$572$		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	0.962842	−	0.270065i	$-0.0870452\pi$
$577$	82412.6	+	142743.i	0.247538	+	0.428748i	−0.715304	+	0.698813i	$0.753712\pi$
$578$		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	−0.0437382	−	0.999043i	$-0.513927\pi$
$587$	−146970.	−	25914.7i	−0.426532	−	0.0752090i	−0.382793	+	0.923834i	$0.625038\pi$
$588$		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	$-0.960083\pi$
$593$		−	87964.7i		−	0.250149i	0.992147	−	0.125075i	$-0.0399170\pi$
$594$		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	−0.0834914	+	0.996508i	$0.526607\pi$
$599$	−236836.	+	650702.i	−0.660077	+	1.81355i	−0.576585	+	0.817037i	$0.695615\pi$
$600$		0			0
$601$	8308.99	−	47122.6i	0.0230038	−	0.130461i	−0.971143	−	0.238497i	$-0.923345\pi$
$601$	8308.99	−	47122.6i	0.0230038	−	0.130461i	0.994147	+	0.108036i	$0.0344563\pi$
$602$		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	0.513582	−	0.858041i	$-0.328318\pi$
$607$	533429.	−	447600.i	1.44777	−	1.21482i	0.934188	−	0.356782i	$-0.116126\pi$
$608$		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	−0.339737	−	0.940521i	$-0.610338\pi$
$613$	242238.	−	419569.i	0.644646	−	1.11656i	0.984383	−	0.176040i	$-0.0563287\pi$
$614$		0			0
$615$	385553.	−	399845.i	1.01937	−	1.05716i
$616$		0			0
$617$	−15518.7	+	2736.36i	−0.0407647	+	0.00718792i	−0.193993	−	0.981003i	$-0.562144\pi$
$617$	−15518.7	+	2736.36i	−0.0407647	+	0.00718792i	0.153229	+	0.988191i	$0.451033\pi$
$618$		0			0
$619$	534132.	+	448190.i	1.39402	+	1.16972i	0.963683	+	0.267047i	$0.0860480\pi$
$619$	534132.	+	448190.i	1.39402	+	1.16972i	0.430332	+	0.902671i	$0.358396\pi$
$620$		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	0.664523	−	0.747268i	$-0.268635\pi$
$631$	−125378.	−	217160.i	−0.314892	−	0.545408i	−0.979414	+	0.201860i	$0.935302\pi$
$632$		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	0.327702	−	0.944781i	$-0.393726\pi$
$641$	128403.	+	22641.0i	0.312507	+	0.0551034i	−0.0151950	+	0.999885i	$0.504837\pi$
$642$		0			0
$643$	508983.	+	185255.i	1.23107	+	0.448072i	0.873963	−	0.485993i	$-0.161542\pi$
$643$	508983.	+	185255.i	1.23107	+	0.448072i	0.357104	+	0.934065i	$0.383764\pi$
$644$		0			0
$645$	−105184.	−	215255.i	−0.252832	−	0.517408i
$646$		0			0
$647$		−	667818.i		−	1.59533i	−0.603102	−	0.797664i	$-0.706069\pi$
$647$		−	667818.i		−	1.59533i	0.603102	−	0.797664i	$-0.293931\pi$
$648$		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	−0.712378	−	0.701796i	$-0.752382\pi$
$653$	121288.	−	333237.i	0.284441	−	0.781496i	0.996819	−	0.0796995i	$-0.0253961\pi$
$654$		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	−0.957384	−	0.288819i	$-0.906737\pi$
$659$	−177897.	−	488767.i	−0.409635	−	1.12546i	0.547749	−	0.836643i	$-0.315485\pi$
$660$		0			0
$661$	432970.	−	363305.i	0.990957	−	0.831512i	0.00525099	−	0.999986i	$-0.498329\pi$
$661$	432970.	−	363305.i	0.990957	−	0.831512i	0.985706	+	0.168475i	$0.0538841\pi$
$662$		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	0.263991	−	0.964525i	$-0.414961\pi$
$673$	−289892.	−	243249.i	−0.640039	−	0.537057i	−0.904030	+	0.427468i	$0.859406\pi$
$674$		0			0
$675$	−212109.	+	166479.i	−0.465534	+	0.365386i
$676$		0			0
$677$	−241740.	+	288095.i	−0.527438	+	0.628576i	−0.962323	−	0.271910i	$-0.912345\pi$
$677$	−241740.	+	288095.i	−0.527438	+	0.628576i	0.434884	+	0.900486i	$0.356789\pi$
$678$		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	0.438004	−	0.898973i	$-0.355686\pi$
$683$	−56691.2	−	32730.7i	−0.121527	−	0.0701639i	−0.559531	+	0.828809i	$0.689019\pi$
$684$		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	0.0554705	−	0.998460i	$-0.482334\pi$
$691$	353222.	−	128562.i	0.739761	−	0.269251i	0.684291	+	0.729209i	$0.260112\pi$
$692$		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	$-0.568341\pi$
$701$		−	960237.i		−	1.95408i	0.213055	−	0.977040i	$-0.431659\pi$
$702$		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	−0.942766	−	0.333455i	$-0.891786\pi$
$709$	28749.5	−	163047.i	0.0571924	−	0.324354i	0.999959	+	0.00910028i	$0.00289675\pi$
$710$		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	−0.246059	−	0.969255i	$-0.579136\pi$
$719$	243132.	−	140372.i	0.470310	−	0.271534i	0.716370	+	0.697721i	$0.245802\pi$
$720$		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	0.880526	−	0.473998i	$-0.157190\pi$
$727$	299480.	+	251294.i	0.566630	+	0.475459i	−0.313896	+	0.949458i	$0.601634\pi$
$728$		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	−0.985004	−	0.172532i	$-0.944805\pi$
$733$	−63621.7	−	360816.i	−0.118412	−	0.671550i	0.866592	−	0.499018i	$-0.166306\pi$
$734$		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	0.507356	−	0.861736i	$-0.330623\pi$
$739$	−269023.	−	465962.i	−0.492607	−	0.853221i	−0.999964	+	0.00851547i	$0.997289\pi$
$740$		0			0
$741$	339168.	+	466097.i	0.617701	+	0.848867i
$742$		0			0
$743$	569499.	+	678702.i	1.03161	+	1.22942i	0.972919	+	0.231147i	$0.0742479\pi$
$743$	569499.	+	678702.i	1.03161	+	1.22942i	0.0586901	+	0.998276i	$0.481308\pi$
$744$		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	0.0408271	−	0.999166i	$-0.487001\pi$
$751$	−321565.	−	117040.i	−0.570149	−	0.207517i	−0.610976	+	0.791649i	$0.709223\pi$
$752$		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	0.609855	+	0.792513i	$0.291228\pi$
$761$	−212386.	+	583525.i	−0.366738	+	1.00761i	−0.976594	+	0.215093i	$0.930995\pi$
$762$		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	−0.356783	−	0.934187i	$-0.616127\pi$
$769$	296424.	−	248729.i	0.501257	−	0.420604i	0.858040	+	0.513583i	$0.171682\pi$
$770$		0			0
$771$	457353.	+	203220.i	0.769383	+	0.341868i
$772$		0			0
$773$	−599094.	+	345887.i	−1.00262	+	0.578862i	−0.909022	−	0.416748i	$-0.863170\pi$
$773$	−599094.	+	345887.i	−1.00262	+	0.578862i	−0.0935968	+	0.995610i	$0.529836\pi$
$774$		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	−0.931324	−	0.364193i	$-0.881345\pi$
$787$	−111937.	−	634824.i	−0.180727	−	1.02495i	0.750597	−	0.660760i	$-0.229766\pi$
$788$		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	0.332157	−	0.943224i	$-0.392224\pi$
$797$	−415691.	−	495401.i	−0.654416	−	0.779903i	−0.986573	+	0.163321i	$0.947779\pi$
$798$		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	$0.0937964\pi$
$809$			380154.i			0.580849i	−0.956898	+	0.290424i	$0.906204\pi$
$810$		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	−0.887969	−	0.459903i	$-0.847884\pi$
$821$	−339289.	−	932188.i	−0.503365	−	1.38298i	0.384604	−	0.923082i	$-0.374338\pi$
$822$		0			0
$823$	−832429.	+	698491.i	−1.22899	+	1.03124i	−0.230683	+	0.973029i	$0.574096\pi$
$823$	−832429.	+	698491.i	−1.22899	+	1.03124i	−0.998304	+	0.0582138i	$0.981459\pi$
$824$		0			0
$825$	−47315.3	−	446985.i	−0.0695174	−	0.656728i
$826$		0			0
$827$	−227618.	+	131416.i	−0.332810	+	0.192148i	−0.657088	−	0.753814i	$-0.728212\pi$
$827$	−227618.	+	131416.i	−0.332810	+	0.192148i	0.324278	+	0.945962i	$0.394879\pi$
$828$		0			0
$829$	−297967.	+	516093.i	−0.433569	+	0.750964i	−0.997178	−	0.0750780i	$-0.976079\pi$
$829$	−297967.	+	516093.i	−0.433569	+	0.750964i	0.563608	+	0.826042i	$0.309413\pi$
$830$		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	−0.444672	−	0.895693i	$-0.646680\pi$
$839$	362259.	−	431724.i	0.514630	−	0.613313i	0.959302	+	0.282381i	$0.0911242\pi$
$840$		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	−0.142513	−	0.989793i	$-0.545518\pi$
$853$	279796.	−	101838.i	0.384542	−	0.139962i	0.527055	+	0.849831i	$0.323296\pi$
$854$		0			0
$855$	−580776.	+	21143.9i	−0.794469	+	0.0289236i
$856$		0			0
$857$	585082.	+	103166.i	0.796628	+	0.140467i	0.557125	−	0.830429i	$-0.311904\pi$
$857$	585082.	+	103166.i	0.796628	+	0.140467i	0.239503	+	0.970896i	$0.423016\pi$
$858$		0			0
$859$	807932.	+	294063.i	1.09494	+	0.398524i	0.825447	−	0.564480i	$-0.190923\pi$
$859$	807932.	+	294063.i	1.09494	+	0.398524i	0.269488	+	0.963004i	$0.413145\pi$
$860$		0			0
$861$	−862682.	+	1.28103e6i	−1.16371	+	1.72803i
$862$		0			0
$863$		−	395978.i		−	0.531679i	−0.964017	−	0.265839i	$-0.914351\pi$
$863$		−	395978.i		−	0.531679i	0.964017	−	0.265839i	$-0.0856491\pi$
$864$		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	−0.241065	+	0.970509i	$0.577497\pi$
$877$	−952713.	+	799421.i	−1.23869	+	1.03938i	−0.997625	+	0.0688758i	$0.978059\pi$
$878$		0			0
$879$	−140557.	+	102280.i	−0.181918	+	0.132377i
$880$		0			0
$881$	725858.	−	419074.i	0.935190	−	0.539932i	0.0467407	−	0.998907i	$-0.485117\pi$
$881$	725858.	−	419074.i	0.935190	−	0.539932i	0.888449	+	0.458975i	$0.151783\pi$
$882$		0			0
$883$	−34200.8	+	59237.6i	−0.0438647	+	0.0759759i	−0.887124	−	0.461531i	$-0.847300\pi$
$883$	−34200.8	+	59237.6i	−0.0438647	+	0.0759759i	0.843259	+	0.537507i	$0.180634\pi$
$884$		0			0
$885$	−1.52143e6	−	378137.i	−1.94252	−	0.482795i
$886$		0			0
$887$	752766.	−	132733.i	0.956781	−	0.168706i	0.326607	−	0.945160i	$-0.394095\pi$
$887$	752766.	−	132733.i	0.956781	−	0.168706i	0.630174	+	0.776454i	$0.282983\pi$
$888$		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	0.102081	−	0.994776i	$-0.467450\pi$
$907$	674332.	−	245437.i	0.819709	−	0.298349i	0.717628	+	0.696427i	$0.245228\pi$
$908$		0			0
$909$	1.09773e6	−	991352.i	1.32852	−	1.19978i
$910$		0			0
$911$	−258709.	−	45617.4i	−0.311727	−	0.0549659i	0.0155960	−	0.999878i	$-0.495035\pi$
$911$	−258709.	−	45617.4i	−0.311727	−	0.0549659i	−0.327323	+	0.944912i	$0.606147\pi$
$912$		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	−0.939870	−	0.341533i	$-0.889054\pi$
$929$	−379238.	−	1.04195e6i	−0.439421	−	1.20730i	0.500449	−	0.865766i	$-0.333168\pi$
$930$		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	−0.968810	−	0.247804i	$-0.920291\pi$
$937$	−236877.	+	410283.i	−0.269801	+	0.467309i	0.699009	+	0.715112i	$0.253625\pi$
$938$		0			0
$939$	−851588.	+	883155.i	−0.965825	+	1.00163i
$940$		0			0
$941$	−672157.	+	118519.i	−0.759087	+	0.133847i	−0.539777	−	0.841808i	$-0.681492\pi$
$941$	−672157.	+	118519.i	−0.759087	+	0.133847i	−0.219309	+	0.975655i	$0.570380\pi$
$942$		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	−0.411351	−	0.911477i	$-0.634943\pi$
$947$	500158.	−	596065.i	0.557709	−	0.664651i	0.969060	+	0.246826i	$0.0793875\pi$
$948$		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	0.860965	−	0.508664i	$-0.169860\pi$
$953$	772825.	+	446190.i	0.850932	+	0.491286i	−0.0100329	+	0.999950i	$0.503194\pi$
$954$		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	0.382743	−	0.923855i	$-0.374979\pi$
$967$	76770.3	+	27942.1i	0.0820995	+	0.0298818i	−0.300644	+	0.953736i	$0.597202\pi$
$968$		0			0
$969$	311687.	+	637852.i	0.331949	+	0.679317i
$970$		0			0
$971$		−	1.56203e6i		−	1.65673i	−0.560190	−	0.828364i	$-0.689272\pi$
$971$		−	1.56203e6i		−	1.65673i	0.560190	−	0.828364i	$-0.310728\pi$
$972$		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	0.537700	+	0.843136i	$0.319293\pi$
$977$	−397237.	+	1.09140e6i	−0.416160	+	1.14339i	−0.953860	+	0.300253i	$0.902929\pi$
$978$		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	0.998801	−	0.0489587i	$-0.0155903\pi$
$983$	195388.	+	536825.i	0.202205	+	0.555554i	−0.796596	+	0.604512i	$0.793368\pi$
$984$		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	−0.717262	−	0.696803i	$-0.754605\pi$
$991$	240431.	−	416439.i	0.244818	−	0.424037i	0.962080	+	0.272766i	$0.0879384\pi$
$992$		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	0.832161	−	0.554535i	$-0.187104\pi$
$997$	427974.	+	359113.i	0.430554	+	0.361277i	−0.401607	+	0.915812i	$0.631548\pi$
$998$		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.65.11	yes	72
3.2	odd	2		324.5.k.a.197.11		72
27.5	odd	18	inner	108.5.k.a.5.11	✓	72
27.22	even	9		324.5.k.a.125.11		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.11	✓	72	27.5	odd	18	inner
108.5.k.a.65.11	yes	72	1.1	even	1	trivial
324.5.k.a.125.11		72	27.22	even	9
324.5.k.a.197.11		72	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:	\( k \)	\(=\)	\( 5 \)
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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