LMFDB - Embedded newform 108.5.k.a.5.6

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			5.6
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.23486 - 8.91488i) q^{3} +(5.87553 + 16.1429i) q^{5} +(8.12577 + 46.0835i) q^{7} +(-77.9503 + 22.0172i) q^{9} +O(q^{10})$	\(q+(-1.23486 - 8.91488i) q^{3} +(5.87553 + 16.1429i) q^{5} +(8.12577 + 46.0835i) q^{7} +(-77.9503 + 22.0172i) q^{9} +(-21.9431 + 60.2882i) q^{11} +(26.5065 + 22.2416i) q^{13} +(136.657 - 72.3139i) q^{15} +(211.894 + 122.337i) q^{17} +(215.694 + 373.592i) q^{19} +(400.795 - 129.347i) q^{21} +(364.888 + 64.3396i) q^{23} +(252.707 - 212.046i) q^{25} +(292.538 + 667.729i) q^{27} +(-478.762 - 570.567i) q^{29} +(-232.834 + 1320.47i) q^{31} +(564.559 + 121.173i) q^{33} +(-696.178 + 401.939i) q^{35} +(27.6012 - 47.8066i) q^{37} +(165.550 - 263.768i) q^{39} +(-603.231 + 718.902i) q^{41} +(1586.08 + 577.286i) q^{43} +(-813.421 - 1128.98i) q^{45} +(-3718.42 + 655.657i) q^{47} +(198.539 - 72.2624i) q^{49} +(828.960 - 2040.07i) q^{51} -1222.43i q^{53} -1102.15 q^{55} +(3064.18 - 2384.22i) q^{57} +(-1312.00 - 3604.69i) q^{59} +(308.780 + 1751.18i) q^{61} +(-1648.04 - 3413.32i) q^{63} +(-203.304 + 558.573i) q^{65} +(3270.92 + 2744.63i) q^{67} +(122.995 - 3332.39i) q^{69} +(-4353.90 - 2513.72i) q^{71} +(-496.098 - 859.268i) q^{73} +(-2202.42 - 1991.00i) q^{75} +(-2956.60 - 521.328i) q^{77} +(-8677.93 + 7281.65i) q^{79} +(5591.48 - 3432.50i) q^{81} +(-2884.98 - 3438.18i) q^{83} +(-729.883 + 4139.37i) q^{85} +(-4495.33 + 4972.68i) q^{87} +(1220.28 - 704.528i) q^{89} +(-809.586 + 1402.24i) q^{91} +(12059.3 + 445.098i) q^{93} +(-4763.55 + 5676.97i) q^{95} +(5073.22 + 1846.50i) q^{97} +(383.092 - 5182.61i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.23486	−	8.91488i	−0.137206	−	0.990542i
$3$	−1.23486	−	8.91488i	−0.137206	−	0.990542i
$4$		0			0
$5$	5.87553	+	16.1429i	0.235021	+	0.645716i	0.999999	+	0.00165851i	$0.000527920\pi$
$5$	5.87553	+	16.1429i	0.235021	+	0.645716i	−0.764977	+	0.644057i	$0.777250\pi$
$6$		0			0
$7$	8.12577	+	46.0835i	0.165832	+	0.940480i	0.948202	+	0.317667i	$0.102899\pi$
$7$	8.12577	+	46.0835i	0.165832	+	0.940480i	−0.782370	+	0.622813i	$0.785990\pi$
$8$		0			0
$9$	−77.9503	+	22.0172i	−0.962349	+	0.271818i
$10$		0			0
$11$	−21.9431	+	60.2882i	−0.181348	+	0.498250i	−0.996742	−	0.0806569i	$-0.974298\pi$
$11$	−21.9431	+	60.2882i	−0.181348	+	0.498250i	0.815394	+	0.578907i	$0.196520\pi$
$12$		0			0
$13$	26.5065	+	22.2416i	0.156843	+	0.131607i	0.717832	−	0.696216i	$-0.245135\pi$
$13$	26.5065	+	22.2416i	0.156843	+	0.131607i	−0.560989	+	0.827823i	$0.689579\pi$
$14$		0			0
$15$	136.657	−	72.3139i	0.607362	−	0.321395i
$16$		0			0
$17$	211.894	+	122.337i	0.733196	+	0.423311i	0.819590	−	0.572950i	$-0.194201\pi$
$17$	211.894	+	122.337i	0.733196	+	0.423311i	−0.0863944	+	0.996261i	$0.527535\pi$
$18$		0			0
$19$	215.694	+	373.592i	0.597489	+	1.03488i	0.993190	+	0.116502i	$0.0371682\pi$
$19$	215.694	+	373.592i	0.597489	+	1.03488i	−0.395701	+	0.918379i	$0.629498\pi$
$20$		0			0
$21$	400.795	−	129.347i	0.908832	−	0.293304i
$22$		0			0
$23$	364.888	+	64.3396i	0.689770	+	0.121625i	0.507536	−	0.861630i	$-0.330556\pi$
$23$	364.888	+	64.3396i	0.689770	+	0.121625i	0.182233	+	0.983255i	$0.441667\pi$
$24$		0			0
$25$	252.707	−	212.046i	0.404331	−	0.339274i
$26$		0			0
$27$	292.538	+	667.729i	0.401287	+	0.915952i
$28$		0			0
$29$	−478.762	−	570.567i	−0.569277	−	0.678438i	0.402205	−	0.915550i	$-0.368244\pi$
$29$	−478.762	−	570.567i	−0.569277	−	0.678438i	−0.971483	+	0.237111i	$0.923799\pi$
$30$		0			0
$31$	−232.834	+	1320.47i	−0.242283	+	1.37405i	0.584436	+	0.811440i	$0.301316\pi$
$31$	−232.834	+	1320.47i	−0.242283	+	1.37405i	−0.826719	+	0.562615i	$0.809795\pi$
$32$		0			0
$33$	564.559	+	121.173i	0.518420	+	0.111270i
$34$		0			0
$35$	−696.178	+	401.939i	−0.568309	+	0.328113i
$36$		0			0
$37$	27.6012	−	47.8066i	0.0201615	−	0.0349208i	−0.855769	−	0.517359i	$-0.826915\pi$
$37$	27.6012	−	47.8066i	0.0201615	−	0.0349208i	0.875930	+	0.482438i	$0.160249\pi$
$38$		0			0
$39$	165.550	−	263.768i	0.108843	−	0.173417i
$40$		0			0
$41$	−603.231	+	718.902i	−0.358852	+	0.427664i	−0.915021	−	0.403406i	$-0.867826\pi$
$41$	−603.231	+	718.902i	−0.358852	+	0.427664i	0.556169	+	0.831069i	$0.312271\pi$
$42$		0			0
$43$	1586.08	+	577.286i	0.857805	+	0.312215i	0.733218	−	0.679993i	$-0.238017\pi$
$43$	1586.08	+	577.286i	0.857805	+	0.312215i	0.124586	+	0.992209i	$0.460240\pi$
$44$		0			0
$45$	−813.421	−	1128.98i	−0.401689	−	0.557521i
$46$		0			0
$47$	−3718.42	+	655.657i	−1.68330	+	0.296812i	−0.931814	−	0.362935i	$-0.881775\pi$
$47$	−3718.42	+	655.657i	−1.68330	+	0.296812i	−0.751488	+	0.659747i	$0.770664\pi$
$48$		0			0
$49$	198.539	−	72.2624i	0.0826903	−	0.0300968i
$50$		0			0
$51$	828.960	−	2040.07i	0.318708	−	0.784343i
$52$		0			0
$53$		−	1222.43i		−	0.435185i	−0.976040	−	0.217592i	$-0.930180\pi$
$53$		−	1222.43i		−	0.435185i	0.976040	−	0.217592i	$-0.0698203\pi$
$54$		0			0
$55$	−1102.15			−0.364348
$56$		0			0
$57$	3064.18	−	2384.22i	0.943115	−	0.733831i
$58$		0			0
$59$	−1312.00	−	3604.69i	−0.376904	−	1.03553i	−0.972633	−	0.232349i	$-0.925359\pi$
$59$	−1312.00	−	3604.69i	−0.376904	−	1.03553i	0.595729	−	0.803185i	$-0.296863\pi$
$60$		0			0
$61$	308.780	+	1751.18i	0.0829831	+	0.470620i	0.997774	+	0.0666918i	$0.0212444\pi$
$61$	308.780	+	1751.18i	0.0829831	+	0.470620i	−0.914791	+	0.403928i	$0.867644\pi$
$62$		0			0
$63$	−1648.04	−	3413.32i	−0.415227	−	0.859994i
$64$		0			0
$65$	−203.304	+	558.573i	−0.0481193	+	0.132207i
$66$		0			0
$67$	3270.92	+	2744.63i	0.728652	+	0.611411i	0.929764	−	0.368157i	$-0.120011\pi$
$67$	3270.92	+	2744.63i	0.728652	+	0.611411i	−0.201112	+	0.979568i	$0.564455\pi$
$68$		0			0
$69$	122.995	−	3332.39i	0.0258339	−	0.699934i
$70$		0			0
$71$	−4353.90	−	2513.72i	−0.863697	−	0.498656i	0.00155135	−	0.999999i	$-0.499506\pi$
$71$	−4353.90	−	2513.72i	−0.863697	−	0.498656i	−0.865249	+	0.501343i	$0.832840\pi$
$72$		0			0
$73$	−496.098	−	859.268i	−0.0930941	−	0.161244i	0.815717	−	0.578451i	$-0.196342\pi$
$73$	−496.098	−	859.268i	−0.0930941	−	0.161244i	−0.908812	+	0.417207i	$0.863009\pi$
$74$		0			0
$75$	−2202.42	−	1991.00i	−0.391542	−	0.353956i
$76$		0			0
$77$	−2956.60	−	521.328i	−0.498667	−	0.0879285i
$78$		0			0
$79$	−8677.93	+	7281.65i	−1.39047	+	1.16674i	−0.425323	+	0.905042i	$0.639840\pi$
$79$	−8677.93	+	7281.65i	−1.39047	+	1.16674i	−0.965149	+	0.261703i	$0.915716\pi$
$80$		0			0
$81$	5591.48	−	3432.50i	0.852230	−	0.523167i
$82$		0			0
$83$	−2884.98	−	3438.18i	−0.418780	−	0.499083i	0.514870	−	0.857268i	$-0.327840\pi$
$83$	−2884.98	−	3438.18i	−0.418780	−	0.499083i	−0.933651	+	0.358185i	$0.883396\pi$
$84$		0			0
$85$	−729.883	+	4139.37i	−0.101022	+	0.572923i
$86$		0			0
$87$	−4495.33	+	4972.68i	−0.593914	+	0.656980i
$88$		0			0
$89$	1220.28	−	704.528i	0.154056	−	0.0889443i	−0.420990	−	0.907065i	$-0.638317\pi$
$89$	1220.28	−	704.528i	0.154056	−	0.0889443i	0.575046	+	0.818121i	$0.304984\pi$
$90$		0			0
$91$	−809.586	+	1402.24i	−0.0977643	+	0.169333i
$92$		0			0
$93$	12059.3	+	445.098i	1.39430	+	0.0514623i
$94$		0			0
$95$	−4763.55	+	5676.97i	−0.527817	+	0.629027i
$96$		0			0
$97$	5073.22	+	1846.50i	0.539188	+	0.196248i	0.597236	−	0.802065i	$-0.296265\pi$
$97$	5073.22	+	1846.50i	0.539188	+	0.196248i	−0.0580481	+	0.998314i	$0.518488\pi$
$98$		0			0
$99$	383.092	−	5182.61i	0.0390870	−	0.528784i
$100$		0			0
$101$	−2412.93	+	425.465i	−0.236539	+	0.0417082i	−0.290661	−	0.956826i	$-0.593875\pi$
$101$	−2412.93	+	425.465i	−0.236539	+	0.0417082i	0.0541220	+	0.998534i	$0.482764\pi$
$102$		0			0
$103$	15177.4	−	5524.12i	1.43061	−	0.520701i	0.493507	−	0.869742i	$-0.335715\pi$
$103$	15177.4	−	5524.12i	1.43061	−	0.520701i	0.937107	+	0.349041i	$0.113493\pi$
$104$		0			0
$105$	4442.92	+	5710.01i	0.402986	+	0.517915i
$106$		0			0
$107$			16847.5i			1.47153i	0.677238	+	0.735764i	$0.263177\pi$
$107$			16847.5i			1.47153i	−0.677238	+	0.735764i	$0.736823\pi$
$108$		0			0
$109$	7711.13			0.649031			0.324515	−	0.945880i	$-0.394799\pi$
$109$	7711.13			0.649031			0.324515	+	0.945880i	$0.394799\pi$
$110$		0			0
$111$	−460.274	−	187.027i	−0.0373568	−	0.0151795i
$112$		0			0
$113$	−7240.01	−	19891.8i	−0.566999	−	1.55782i	−0.809165	−	0.587582i	$-0.800080\pi$
$113$	−7240.01	−	19891.8i	−0.566999	−	1.55782i	0.242166	−	0.970235i	$-0.422142\pi$
$114$		0			0
$115$	1105.28	+	6268.38i	0.0835754	+	0.473980i
$116$		0			0
$117$	−2555.89	−	1150.14i	−0.186711	−	0.0840192i
$118$		0			0
$119$	−3915.91	+	10758.9i	−0.276528	+	0.759754i
$120$		0			0
$121$	8062.49	+	6765.23i	0.550679	+	0.462074i
$122$		0			0
$123$	7153.83	+	4489.99i	0.472856	+	0.296780i
$124$		0			0
$125$	14206.2	+	8201.94i	0.909196	+	0.524924i
$126$		0			0
$127$	−2846.78	−	4930.77i	−0.176501	−	0.305709i	0.764179	−	0.645005i	$-0.223145\pi$
$127$	−2846.78	−	4930.77i	−0.176501	−	0.305709i	−0.940680	+	0.339296i	$0.889811\pi$
$128$		0			0
$129$	3187.85	−	14852.6i	0.191566	−	0.892530i
$130$		0			0
$131$	21524.9	+	3795.42i	1.25429	+	0.221165i	0.761030	−	0.648716i	$-0.224694\pi$
$131$	21524.9	+	3795.42i	1.25429	+	0.221165i	0.493260	+	0.869882i	$0.335805\pi$
$132$		0			0
$133$	−15463.8	+	12975.6i	−0.874203	+	0.733543i
$134$		0			0
$135$	−9060.26	+	8645.68i	−0.497134	+	0.474386i
$136$		0			0
$137$	−1116.26	−	1330.31i	−0.0594739	−	0.0708782i	0.735487	−	0.677538i	$-0.236953\pi$
$137$	−1116.26	−	1330.31i	−0.0594739	−	0.0708782i	−0.794961	+	0.606660i	$0.792509\pi$
$138$		0			0
$139$	2361.73	−	13394.0i	0.122236	−	0.693236i	−0.860675	−	0.509155i	$-0.829958\pi$
$139$	2361.73	−	13394.0i	0.122236	−	0.693236i	0.982911	−	0.184081i	$-0.0589310\pi$
$140$		0			0
$141$	10436.8	+	32339.6i	0.524965	+	1.62666i
$142$		0			0
$143$	−1922.54	+	1109.98i	−0.0940165	+	0.0542804i
$144$		0			0
$145$	6397.61	−	11081.0i	0.304286	−	0.527039i
$146$		0			0
$147$	−889.379	−	1680.72i	−0.0411578	−	0.0777788i
$148$		0			0
$149$	27688.4	−	32997.8i	1.24717	−	1.48632i	0.437744	−	0.899100i	$-0.355778\pi$
$149$	27688.4	−	32997.8i	1.24717	−	1.48632i	0.809427	−	0.587221i	$-0.199778\pi$
$150$		0			0
$151$	−10633.6	−	3870.31i	−0.466365	−	0.169743i	0.0981399	−	0.995173i	$-0.468711\pi$
$151$	−10633.6	−	3870.31i	−0.466365	−	0.169743i	−0.564505	+	0.825430i	$0.690933\pi$
$152$		0			0
$153$	−19210.7	−	4870.88i	−0.820653	−	0.208077i
$154$		0			0
$155$	−22684.2	+	3999.83i	−0.944190	+	0.166486i
$156$		0			0
$157$	−18511.2	+	6737.53i	−0.750992	+	0.273339i	−0.689023	−	0.724739i	$-0.741960\pi$
$157$	−18511.2	+	6737.53i	−0.750992	+	0.273339i	−0.0619689	+	0.998078i	$0.519738\pi$
$158$		0			0
$159$	−10897.9	+	1509.53i	−0.431069	+	0.0597101i
$160$		0			0
$161$			17338.1i			0.668884i
$162$		0			0
$163$	−22213.1			−0.836055			−0.418027	−	0.908434i	$-0.637278\pi$
$163$	−22213.1			−0.836055			−0.418027	+	0.908434i	$0.637278\pi$
$164$		0			0
$165$	1361.00	+	9825.57i	0.0499909	+	0.360902i
$166$		0			0
$167$	45.4798	+	124.955i	0.00163074	+	0.00448043i	0.940505	−	0.339779i	$-0.110352\pi$
$167$	45.4798	+	124.955i	0.00163074	+	0.00448043i	−0.938875	+	0.344259i	$0.888130\pi$
$168$		0			0
$169$	−4751.66	−	26948.0i	−0.166369	−	0.943524i
$170$		0			0
$171$	−25038.8	−	24372.6i	−0.856292	−	0.833509i
$172$		0			0
$173$	4304.21	−	11825.7i	0.143814	−	0.395126i	−0.846783	−	0.531939i	$-0.821464\pi$
$173$	4304.21	−	11825.7i	0.143814	−	0.395126i	0.990597	+	0.136813i	$0.0436859\pi$
$174$		0			0
$175$	11825.3	+	9922.58i	0.386131	+	0.324002i
$176$		0			0
$177$	−30515.3	+	16147.6i	−0.974027	+	0.515421i
$178$		0			0
$179$	38910.8	+	22465.2i	1.21441	+	0.701138i	0.963716	−	0.266929i	$-0.0860088\pi$
$179$	38910.8	+	22465.2i	1.21441	+	0.701138i	0.250691	+	0.968067i	$0.419342\pi$
$180$		0			0
$181$	−30521.8	−	52865.4i	−0.931652	−	1.61367i	−0.780498	−	0.625158i	$-0.785035\pi$
$181$	−30521.8	−	52865.4i	−0.931652	−	1.61367i	−0.151154	−	0.988510i	$-0.548299\pi$
$182$		0			0
$183$	15230.2	−	4915.19i	0.454784	−	0.146770i
$184$		0			0
$185$	933.908	+	164.673i	0.0272873	+	0.00481149i
$186$		0			0
$187$	−12025.1	+	10090.2i	−0.343878	+	0.288548i
$188$		0			0
$189$	−28394.2	+	18907.0i	−0.794888	+	0.529297i
$190$		0			0
$191$	−13921.7	−	16591.2i	−0.381614	−	0.454790i	0.540709	−	0.841210i	$-0.318156\pi$
$191$	−13921.7	−	16591.2i	−0.381614	−	0.454790i	−0.922323	+	0.386420i	$0.873712\pi$
$192$		0			0
$193$	2671.39	−	15150.2i	0.0717171	−	0.406728i	−0.927723	−	0.373269i	$-0.878237\pi$
$193$	2671.39	−	15150.2i	0.0717171	−	0.406728i	0.999440	−	0.0334585i	$-0.0106521\pi$
$194$		0			0
$195$	5230.67	+	1122.67i	0.137559	+	0.0295246i
$196$		0			0
$197$	42526.8	−	24552.9i	1.09580	−	0.632659i	0.160684	−	0.987006i	$-0.448630\pi$
$197$	42526.8	−	24552.9i	1.09580	−	0.632659i	0.935114	+	0.354347i	$0.115297\pi$
$198$		0			0
$199$	38457.1	−	66609.6i	0.971114	−	1.68202i	0.278910	−	0.960317i	$-0.410027\pi$
$199$	38457.1	−	66609.6i	0.971114	−	1.68202i	0.692204	−	0.721702i	$-0.256640\pi$
$200$		0			0
$201$	20428.9	−	32549.1i	0.505653	−	0.805650i
$202$		0			0
$203$	22403.4	−	26699.3i	0.543653	−	0.647901i
$204$		0			0
$205$	−15149.5	−	5513.95i	−0.360487	−	0.131207i
$206$		0			0
$207$	−29859.7	+	3018.54i	−0.696859	+	0.0704459i
$208$		0			0
$209$	−27256.2	+	4806.00i	−0.623983	+	0.110025i
$210$		0			0
$211$	80895.2	−	29443.4i	1.81701	−	0.661338i	0.821123	−	0.570751i	$-0.193348\pi$
$211$	80895.2	−	29443.4i	1.81701	−	0.661338i	0.995889	−	0.0905869i	$-0.0288743\pi$
$212$		0			0
$213$	−17033.1	+	41918.6i	−0.375435	+	0.923948i
$214$		0			0
$215$			28995.8i			0.627275i
$216$		0			0
$217$	−62743.7			−1.33245
$218$		0			0
$219$	−7047.66	+	5483.73i	−0.146946	+	0.114337i
$220$		0			0
$221$	2895.59	+	7955.58i	0.0592861	+	0.162887i
$222$		0			0
$223$	4374.88	+	24811.2i	0.0879743	+	0.498927i	0.996675	+	0.0814790i	$0.0259644\pi$
$223$	4374.88	+	24811.2i	0.0879743	+	0.498927i	−0.908701	+	0.417448i	$0.862925\pi$
$224$		0			0
$225$	−15029.9	+	22092.9i	−0.296887	+	0.436404i
$226$		0			0
$227$	−25727.7	+	70686.3i	−0.499286	+	1.37178i	0.392680	+	0.919675i	$0.371548\pi$
$227$	−25727.7	+	70686.3i	−0.499286	+	1.37178i	−0.891966	+	0.452103i	$0.850674\pi$
$228$		0			0
$229$	13639.0	+	11444.5i	0.260083	+	0.218236i	0.763500	−	0.645808i	$-0.223479\pi$
$229$	13639.0	+	11444.5i	0.260083	+	0.218236i	−0.503416	+	0.864044i	$0.667924\pi$
$230$		0			0
$231$	−996.599	+	27001.5i	−0.0186765	+	0.506015i
$232$		0			0
$233$	5151.21	+	2974.05i	0.0948850	+	0.0547819i	0.546692	−	0.837334i	$-0.315887\pi$
$233$	5151.21	+	2974.05i	0.0948850	+	0.0547819i	−0.451807	+	0.892116i	$0.649220\pi$
$234$		0			0
$235$	−32431.9	−	56173.6i	−0.587268	−	1.01718i
$236$		0			0
$237$	75631.1	+	68370.9i	1.34649	+	1.21724i
$238$		0			0
$239$	−6503.03	−	1146.66i	−0.113847	−	0.0200742i	0.116435	−	0.993198i	$-0.462853\pi$
$239$	−6503.03	−	1146.66i	−0.113847	−	0.0200742i	−0.230281	+	0.973124i	$0.573965\pi$
$240$		0			0
$241$	−66257.5	+	55596.7i	−1.14078	+	0.957226i	−0.999464	−	0.0327387i	$-0.989577\pi$
$241$	−66257.5	+	55596.7i	−1.14078	+	0.957226i	−0.141314	+	0.989965i	$0.545133\pi$
$242$		0			0
$243$	−37505.0	−	45608.8i	−0.635150	−	0.772389i
$244$		0			0
$245$	2333.05	+	2780.42i	0.0388680	+	0.0463210i
$246$		0			0
$247$	−2592.01	+	14700.0i	−0.0424857	+	0.240948i
$248$		0			0
$249$	−27088.5	+	29964.9i	−0.436904	+	0.483297i
$250$		0			0
$251$	77901.6	−	44976.5i	1.23651	−	0.713901i	0.268133	−	0.963382i	$-0.413593\pi$
$251$	77901.6	−	44976.5i	1.23651	−	0.713901i	0.968380	+	0.249481i	$0.0802599\pi$
$252$		0			0
$253$	−11885.7	+	20586.6i	−0.185688	+	0.321621i
$254$		0			0
$255$	37803.3	+	1395.28i	0.581366	+	0.0214576i
$256$		0			0
$257$	54004.6	−	64360.2i	0.817645	−	0.974432i	−0.182316	−	0.983240i	$-0.558359\pi$
$257$	54004.6	−	64360.2i	0.817645	−	0.974432i	0.999961	+	0.00880836i	$0.00280382\pi$
$258$		0			0
$259$	2427.38	+	883.493i	0.0361858	+	0.0131705i
$260$		0			0
$261$	49881.9	+	33934.8i	0.732255	+	0.498155i
$262$		0			0
$263$	−1424.17	+	251.119i	−0.0205897	+	0.00363052i	−0.183934	−	0.982939i	$-0.558883\pi$
$263$	−1424.17	+	251.119i	−0.0205897	+	0.00363052i	0.163344	+	0.986569i	$0.447772\pi$
$264$		0			0
$265$	19733.6	−	7182.45i	0.281006	−	0.102278i
$266$		0			0
$267$	−7787.65	−	10008.6i	−0.109241	−	0.140395i
$268$		0			0
$269$			124679.i			1.72301i	0.507748	+	0.861506i	$0.330478\pi$
$269$			124679.i			1.72301i	−0.507748	+	0.861506i	$0.669522\pi$
$270$		0			0
$271$	75336.2			1.02581			0.512903	−	0.858447i	$-0.328570\pi$
$271$	75336.2			1.02581			0.512903	+	0.858447i	$0.328570\pi$
$272$		0			0
$273$	13500.6	+	5485.79i	0.181145	+	0.0736061i
$274$		0			0
$275$	7238.71	+	19888.2i	0.0957184	+	0.262984i
$276$		0			0
$277$	18002.8	+	102099.i	0.234629	+	1.33064i	0.843395	+	0.537295i	$0.180554\pi$
$277$	18002.8	+	102099.i	0.234629	+	1.33064i	−0.608766	+	0.793350i	$0.708335\pi$
$278$		0			0
$279$	−10923.5	−	108057.i	−0.140332	−	1.38818i
$280$		0			0
$281$	18849.0	−	51787.1i	0.238712	−	0.655857i	−0.761260	−	0.648447i	$-0.775419\pi$
$281$	18849.0	−	51787.1i	0.238712	−	0.655857i	0.999973	−	0.00741027i	$-0.00235878\pi$
$282$		0			0
$283$	74274.8	+	62323.9i	0.927403	+	0.778183i	0.975349	−	0.220667i	$-0.0708232\pi$
$283$	74274.8	+	62323.9i	0.927403	+	0.778183i	−0.0479465	+	0.998850i	$0.515268\pi$
$284$		0			0
$285$	56491.8	+	35456.2i	0.695498	+	0.436518i
$286$		0			0
$287$	−38031.3	−	21957.4i	−0.461718	−	0.266573i
$288$		0			0
$289$	−11827.9	−	20486.5i	−0.141616	−	0.245286i
$290$		0			0
$291$	10196.6	−	47507.3i	0.120412	−	0.561015i
$292$		0			0
$293$	−3384.94	−	596.856i	−0.0394290	−	0.00695240i	0.153899	−	0.988087i	$-0.450817\pi$
$293$	−3384.94	−	596.856i	−0.0394290	−	0.00695240i	−0.193328	+	0.981134i	$0.561928\pi$
$294$		0			0
$295$	50481.5	−	42359.0i	0.580080	−	0.486745i
$296$		0			0
$297$	−46675.4	+	2984.56i	−0.529146	+	0.0338351i
$298$		0			0
$299$	8240.90	+	9821.12i	0.0921790	+	0.109855i
$300$		0			0
$301$	−13715.3	+	77783.1i	−0.151381	+	0.858523i
$302$		0			0
$303$	6772.60	+	20985.6i	0.0737683	+	0.228579i
$304$		0			0
$305$	−26454.8	+	15273.7i	−0.284384	+	0.164189i
$306$		0			0
$307$	−2091.41	+	3622.43i	−0.0221903	+	0.0384347i	−0.876907	−	0.480660i	$-0.840397\pi$
$307$	−2091.41	+	3622.43i	−0.0221903	+	0.0384347i	0.854717	+	0.519094i	$0.173731\pi$
$308$		0			0
$309$	−67988.8	−	128483.i	−0.712066	−	1.34564i
$310$		0			0
$311$	7930.44	−	9451.12i	0.0819929	−	0.0977153i	−0.723488	−	0.690337i	$-0.757462\pi$
$311$	7930.44	−	9451.12i	0.0819929	−	0.0977153i	0.805481	+	0.592621i	$0.201907\pi$
$312$		0			0
$313$	−91607.2	−	33342.3i	−0.935064	−	0.340335i	−0.170849	−	0.985297i	$-0.554651\pi$
$313$	−91607.2	−	33342.3i	−0.935064	−	0.340335i	−0.764215	+	0.644962i	$0.776873\pi$
$314$		0			0
$315$	45417.7	−	46659.1i	0.457724	−	0.470236i
$316$		0			0
$317$	−169962.	+	29968.9i	−1.69135	+	0.298230i	−0.934659	−	0.355545i	$-0.884295\pi$
$317$	−169962.	+	29968.9i	−1.69135	+	0.298230i	−0.756689	+	0.653775i	$0.773184\pi$
$318$		0			0
$319$	44904.0	−	16343.7i	0.441269	−	0.160609i
$320$		0			0
$321$	150194.	−	20804.3i	1.45761	−	0.201903i
$322$		0			0
$323$			105549.i			1.01169i
$324$		0			0
$325$	11414.6			0.108067
$326$		0			0
$327$	−9522.15	−	68743.8i	−0.0890512	−	0.642892i
$328$		0			0
$329$	−60430.0	−	166030.i	−0.558291	−	1.53389i
$330$		0			0
$331$	15558.1	+	88234.6i	0.142004	+	0.805347i	0.969724	+	0.244204i	$0.0785268\pi$
$331$	15558.1	+	88234.6i	0.142004	+	0.805347i	−0.827719	+	0.561142i	$0.810362\pi$
$332$		0			0
$333$	−1098.95	+	4334.24i	−0.00991034	+	0.0390863i
$334$		0			0
$335$	−25087.8	+	68928.2i	−0.223549	+	0.614197i
$336$		0			0
$337$	−64856.8	−	54421.3i	−0.571078	−	0.479191i	0.310925	−	0.950434i	$-0.399361\pi$
$337$	−64856.8	−	54421.3i	−0.571078	−	0.479191i	−0.882003	+	0.471243i	$0.843806\pi$
$338$		0			0
$339$	−168392.	+	89107.3i	−1.46529	+	0.775379i
$340$		0			0
$341$	−74499.4	−	43012.3i	−0.640685	−	0.369899i
$342$		0			0
$343$	61120.1	+	105863.i	0.519512	+	0.899822i
$344$		0			0
$345$	54517.0	−	17594.0i	0.458030	−	0.147818i
$346$		0			0
$347$	−169395.	−	29868.8i	−1.40683	−	0.248062i	−0.581881	−	0.813274i	$-0.697683\pi$
$347$	−169395.	−	29868.8i	−1.40683	−	0.248062i	−0.824946	+	0.565212i	$0.808794\pi$
$348$		0			0
$349$	−12779.7	+	10723.4i	−0.104923	+	0.0880405i	−0.693740	−	0.720226i	$-0.744038\pi$
$349$	−12779.7	+	10723.4i	−0.104923	+	0.0880405i	0.588817	+	0.808266i	$0.299594\pi$
$350$		0			0
$351$	−7097.20	+	24205.7i	−0.0576066	+	0.196473i
$352$		0			0
$353$	−122460.	−	145943.i	−0.982757	−	1.17120i	−0.985235	−	0.171207i	$-0.945233\pi$
$353$	−122460.	−	145943.i	−0.982757	−	1.17120i	0.00247824	−	0.999997i	$-0.499211\pi$
$354$		0			0
$355$	14997.3	−	85054.0i	0.119003	−	0.674898i
$356$		0			0
$357$	100750.	+	21624.2i	0.790510	+	0.169670i
$358$		0			0
$359$	−53628.4	+	30962.4i	−0.416108	+	0.240240i	−0.693411	−	0.720542i	$-0.743893\pi$
$359$	−53628.4	+	30962.4i	−0.416108	+	0.240240i	0.277303	+	0.960783i	$0.410560\pi$
$360$		0			0
$361$	−27887.0	+	48301.6i	−0.213987	+	0.370636i
$362$		0			0
$363$	50355.2	−	80230.2i	0.382148	−	0.608870i
$364$		0			0
$365$	10956.2	−	13057.1i	0.0822385	−	0.0980080i
$366$		0			0
$367$	−22340.3	−	8131.21i	−0.165866	−	0.0603703i	0.257753	−	0.966211i	$-0.417018\pi$
$367$	−22340.3	−	8131.21i	−0.165866	−	0.0603703i	−0.423618	+	0.905841i	$0.639240\pi$
$368$		0			0
$369$	31193.7	−	69320.1i	0.229095	−	0.509104i
$370$		0			0
$371$	56334.0	−	9933.21i	0.409282	−	0.0721675i
$372$		0			0
$373$	−61939.6	+	22544.2i	−0.445195	+	0.162038i	−0.554883	−	0.831928i	$-0.687237\pi$
$373$	−61939.6	+	22544.2i	−0.445195	+	0.162038i	0.109688	+	0.993966i	$0.465015\pi$
$374$		0			0
$375$	55576.7	−	136775.i	0.395212	−	0.972620i
$376$		0			0
$377$		−	25772.2i		−	0.181330i
$378$		0			0
$379$	−25944.4			−0.180620			−0.0903099	−	0.995914i	$-0.528786\pi$
$379$	−25944.4			−0.180620			−0.0903099	+	0.995914i	$0.528786\pi$
$380$		0			0
$381$	−40441.9	+	31467.5i	−0.278600	+	0.216777i
$382$		0			0
$383$	−7650.34	−	21019.1i	−0.0521535	−	0.143290i	0.910881	−	0.412670i	$-0.135404\pi$
$383$	−7650.34	−	21019.1i	−0.0521535	−	0.143290i	−0.963034	+	0.269379i	$0.913181\pi$
$384$		0			0
$385$	−8955.84	−	50791.1i	−0.0604206	−	0.342662i
$386$		0			0
$387$	−136346.	−	10078.5i	−0.910373	−	0.0672937i
$388$		0			0
$389$	31551.4	−	86686.7i	0.208506	−	0.572866i	−0.790721	−	0.612177i	$-0.790294\pi$
$389$	31551.4	−	86686.7i	0.208506	−	0.572866i	0.999227	+	0.0393109i	$0.0125163\pi$
$390$		0			0
$391$	69446.4	+	58272.4i	0.454251	+	0.381162i
$392$		0			0
$393$	7255.53	−	196579.i	0.0469768	−	1.27277i
$394$		0			0
$395$	−168534.	−	97303.4i	−1.08018	−	0.623640i
$396$		0			0
$397$	60496.9	+	104784.i	0.383842	+	0.664834i	0.991608	−	0.129283i	$-0.0412674\pi$
$397$	60496.9	+	104784.i	0.383842	+	0.664834i	−0.607766	+	0.794116i	$0.707934\pi$
$398$		0			0
$399$	134772.	+	121835.i	0.846552	+	0.765288i
$400$		0			0
$401$	177505.	+	31298.9i	1.10388	+	0.194644i	0.695753	−	0.718281i	$-0.255071\pi$
$401$	177505.	+	31298.9i	1.10388	+	0.194644i	0.408127	+	0.912925i	$0.366182\pi$
$402$		0			0
$403$	−35540.9	+	29822.4i	−0.218836	+	0.183625i
$404$		0			0
$405$	88263.4	+	70095.0i	0.538109	+	0.427343i
$406$		0			0
$407$	2276.52	+	2713.05i	0.0137430	+	0.0163783i
$408$		0			0
$409$	−10648.4	+	60390.1i	−0.0636558	+	0.361010i	0.936296	+	0.351211i	$0.114230\pi$
$409$	−10648.4	+	60390.1i	−0.0636558	+	0.361010i	−0.999952	+	0.00979857i	$0.996881\pi$
$410$		0			0
$411$	−10481.2	+	11594.1i	−0.0620476	+	0.0686363i
$412$		0			0
$413$	155456.	−	89752.5i	0.911396	−	0.526195i
$414$		0			0
$415$	38551.4	−	66773.1i	0.223843	−	0.387708i
$416$		0			0
$417$	−122323.	−	4514.81i	−0.703452	−	0.0259637i
$418$		0			0
$419$	−190265.	+	226748.i	−1.08375	+	1.29157i	−0.129821	+	0.991537i	$0.541440\pi$
$419$	−190265.	+	226748.i	−1.08375	+	1.29157i	−0.953930	+	0.300028i	$0.903004\pi$
$420$		0			0
$421$	103730.	+	37754.5i	0.585246	+	0.213012i	0.617637	−	0.786463i	$-0.288090\pi$
$421$	103730.	+	37754.5i	0.585246	+	0.213012i	−0.0323910	+	0.999475i	$0.510312\pi$
$422$		0			0
$423$	275416.	−	132978.i	1.53925	−	0.743188i
$424$		0			0
$425$	79488.0	−	14015.9i	0.440072	−	0.0775965i
$426$		0			0
$427$	−78191.4	+	28459.3i	−0.428848	+	0.156088i
$428$		0			0
$429$	12269.4	+	15768.6i	0.0666667	+	0.0856797i
$430$		0			0
$431$			114346.i			0.615556i	0.951458	+	0.307778i	$0.0995854\pi$
$431$			114346.i			0.615556i	−0.951458	+	0.307778i	$0.900415\pi$
$432$		0			0
$433$	68905.4			0.367517			0.183759	−	0.982971i	$-0.441174\pi$
$433$	68905.4			0.367517			0.183759	+	0.982971i	$0.441174\pi$
$434$		0			0
$435$	−106686.	−	43350.5i	−0.563804	−	0.229095i
$436$		0			0
$437$	54667.3	+	150197.i	0.286262	+	0.786500i
$438$		0			0
$439$	19037.3	+	107966.i	0.0987818	+	0.560220i	0.993523	+	0.113634i	$0.0362491\pi$
$439$	19037.3	+	107966.i	0.0987818	+	0.560220i	−0.894741	+	0.446586i	$0.852640\pi$
$440$		0			0
$441$	−13885.2	+	10004.2i	−0.0713961	+	0.0514403i
$442$		0			0
$443$	−58654.2	+	161151.i	−0.298876	+	0.821156i	0.695812	+	0.718224i	$0.255045\pi$
$443$	−58654.2	+	161151.i	−0.298876	+	0.821156i	−0.994688	+	0.102932i	$0.967178\pi$
$444$		0			0
$445$	18542.9	+	15559.3i	0.0936392	+	0.0785726i
$446$		0			0
$447$	−328363.	−	206092.i	−1.64338	−	1.03144i
$448$		0			0
$449$	−134746.	−	77795.9i	−0.668381	−	0.385890i	0.127082	−	0.991892i	$-0.459439\pi$
$449$	−134746.	−	77795.9i	−0.668381	−	0.385890i	−0.795463	+	0.606002i	$0.792772\pi$
$450$		0			0
$451$	−30104.6	−	52142.7i	−0.148006	−	0.256354i
$452$		0			0
$453$	−21372.4	+	99576.5i	−0.104149	+	0.485244i
$454$		0			0
$455$	−27393.0	−	4830.13i	−0.132317	−	0.0233311i
$456$		0			0
$457$	242462.	−	203450.i	1.16094	−	0.974147i	0.161026	−	0.986950i	$-0.448520\pi$
$457$	242462.	−	203450.i	1.16094	−	0.974147i	0.999918	+	0.0128029i	$0.00407542\pi$
$458$		0			0
$459$	−19700.8	+	177276.i	−0.0935103	+	0.841442i
$460$		0			0
$461$	180926.	+	215619.i	0.851332	+	1.01458i	0.999671	+	0.0256412i	$0.00816273\pi$
$461$	180926.	+	215619.i	0.851332	+	1.01458i	−0.148339	+	0.988936i	$0.547393\pi$
$462$		0			0
$463$	−18880.2	+	107075.i	−0.0880732	+	0.499488i	0.908578	+	0.417716i	$0.137169\pi$
$463$	−18880.2	+	107075.i	−0.0880732	+	0.499488i	−0.996651	+	0.0817723i	$0.973942\pi$
$464$		0			0
$465$	63669.7	+	197287.i	0.294461	+	0.912417i
$466$		0			0
$467$	323907.	−	187008.i	1.48520	−	0.857483i	0.485346	−	0.874322i	$-0.338693\pi$
$467$	323907.	−	187008.i	1.48520	−	0.857483i	0.999858	+	0.0168386i	$0.00536014\pi$
$468$		0			0
$469$	−99903.3	+	173038.i	−0.454186	+	0.786674i
$470$		0			0
$471$	82923.0	+	156705.i	0.373795	+	0.706386i
$472$		0			0
$473$	−69607.1	+	82954.5i	−0.311122	+	0.370781i
$474$		0			0
$475$	133726.	+	48672.3i	0.592691	+	0.215722i
$476$		0			0
$477$	26914.6	+	95289.0i	0.118291	+	0.418799i
$478$		0			0
$479$	257523.	−	45408.3i	1.12239	−	0.197909i	0.418503	−	0.908215i	$-0.362555\pi$
$479$	257523.	−	45408.3i	1.12239	−	0.197909i	0.703892	+	0.710307i	$0.251444\pi$
$480$		0			0
$481$	1794.91	−	653.293i	0.00775803	−	0.00282369i
$482$		0			0
$483$	154567.	−	21410.1i	0.662558	−	0.0917752i
$484$		0			0
$485$			92745.7i			0.394285i
$486$		0			0
$487$	−325076.			−1.37065			−0.685325	−	0.728237i	$-0.740340\pi$
$487$	−325076.			−1.37065			−0.685325	+	0.728237i	$0.740340\pi$
$488$		0			0
$489$	27430.1	+	198028.i	0.114712	+	0.828148i
$490$		0			0
$491$	16561.1	+	45501.1i	0.0686950	+	0.188738i	0.969290	−	0.245922i	$-0.0790908\pi$
$491$	16561.1	+	45501.1i	0.0686950	+	0.188738i	−0.900595	+	0.434660i	$0.856869\pi$
$492$		0			0
$493$	−31645.3	−	179470.i	−0.130202	−	0.738409i
$494$		0			0
$495$	85913.1	−	24266.4i	0.350630	−	0.0990363i
$496$		0			0
$497$	80462.5	−	221069.i	0.325747	−	0.894983i
$498$		0			0
$499$	−111893.	−	93889.6i	−0.449369	−	0.377065i	0.389833	−	0.920886i	$-0.372533\pi$
$499$	−111893.	−	93889.6i	−0.449369	−	0.377065i	−0.839202	+	0.543820i	$0.816977\pi$
$500$		0			0
$501$	1057.80	−	559.748i	0.00421431	−	0.00223006i
$502$		0			0
$503$	−404971.	−	233810.i	−1.60062	−	0.924117i	−0.991364	−	0.131141i	$-0.958136\pi$
$503$	−404971.	−	233810.i	−1.60062	−	0.924117i	−0.609253	−	0.792976i	$-0.708531\pi$
$504$		0			0
$505$	−21045.5	−	36451.9i	−0.0825233	−	0.142934i
$506$		0			0
$507$	−234371.	+	75637.4i	−0.911774	+	0.294253i
$508$		0			0
$509$	292320.	+	51543.9i	1.12830	+	0.198949i	0.706480	−	0.707733i	$-0.250282\pi$
$509$	292320.	+	51543.9i	1.12830	+	0.198949i	0.421815	+	0.906682i	$0.361393\pi$
$510$		0			0
$511$	35566.9	−	29844.2i	0.136208	−	0.114292i
$512$		0			0
$513$	−186360.	+	253315.i	−0.708137	+	0.962556i
$514$		0			0
$515$	178350.	+	212550.i	0.672450	+	0.801394i
$516$		0			0
$517$	42065.2	−	238564.i	0.157377	−	0.892531i
$518$		0			0
$519$	−110740.	−	23768.4i	−0.411121	−	0.0882401i
$520$		0			0
$521$	343568.	−	198359.i	1.26572	−	0.730763i	0.291544	−	0.956558i	$-0.405831\pi$
$521$	343568.	−	198359.i	1.26572	−	0.730763i	0.974175	+	0.225795i	$0.0724978\pi$
$522$		0			0
$523$	101598.	−	175973.i	0.371433	−	0.643342i	−0.618353	−	0.785901i	$-0.712200\pi$
$523$	101598.	−	175973.i	0.371433	−	0.643342i	0.989786	+	0.142559i	$0.0455331\pi$
$524$		0			0
$525$	73856.1	−	117674.i	0.267959	−	0.426934i
$526$		0			0
$527$	−210878.	+	251314.i	−0.759293	+	0.904890i
$528$		0			0
$529$	−133961.	−	48757.7i	−0.478703	−	0.174234i
$530$		0			0
$531$	181636.	+	252100.i	0.644189	+	0.894096i
$532$		0			0
$533$	−31979.1	+	5638.78i	−0.112567	+	0.0198486i
$534$		0			0
$535$	−271968.	+	98988.2i	−0.950189	+	0.345841i
$536$		0			0
$537$	152225.	−	374627.i	0.527883	−	1.29912i
$538$		0			0
$539$			13555.2i			0.0466584i
$540$		0			0
$541$	545062.			1.86231			0.931153	−	0.364628i	$-0.118804\pi$
$541$	545062.			1.86231			0.931153	+	0.364628i	$0.118804\pi$
$542$		0			0
$543$	−433599.	+	337380.i	−1.47058	+	1.14425i
$544$		0			0
$545$	45307.0	+	124480.i	0.152536	+	0.419089i
$546$		0			0
$547$	−77497.0	−	439507.i	−0.259006	−	1.46890i	−0.785576	−	0.618765i	$-0.787633\pi$
$547$	−77497.0	−	439507.i	−0.259006	−	1.46890i	0.526570	−	0.850132i	$-0.323478\pi$
$548$		0			0
$549$	−62625.6	−	129706.i	−0.207782	−	0.430345i
$550$		0			0
$551$	109893.	−	301929.i	0.361966	−	0.994494i
$552$		0			0
$553$	−406079.	−	340741.i	−1.32788	−	1.11423i
$554$		0			0
$555$	314.798	−	8529.03i	0.00102199	−	0.0276894i
$556$		0			0
$557$	−336771.	−	194435.i	−1.08549	−	0.626705i	−0.153115	−	0.988208i	$-0.548930\pi$
$557$	−336771.	−	194435.i	−1.08549	−	0.626705i	−0.932371	+	0.361503i	$0.882264\pi$
$558$		0			0
$559$	29201.7	+	50578.8i	0.0934511	+	0.161862i
$560$		0			0
$561$	104802.	+	94742.1i	0.333001	+	0.301035i
$562$		0			0
$563$	178666.	+	31503.7i	0.563671	+	0.0993904i	0.448221	−	0.893923i	$-0.352058\pi$
$563$	178666.	+	31503.7i	0.563671	+	0.0993904i	0.115450	+	0.993313i	$0.463169\pi$
$564$		0			0
$565$	278572.	−	233749.i	0.872650	−	0.732240i
$566$		0			0
$567$	203617.	+	229784.i	0.633355	+	0.714748i
$568$		0			0
$569$	−104051.	−	124003.i	−0.321382	−	0.383009i	0.581030	−	0.813882i	$-0.302650\pi$
$569$	−104051.	−	124003.i	−0.321382	−	0.383009i	−0.902412	+	0.430874i	$0.858205\pi$
$570$		0			0
$571$	44974.8	−	255065.i	0.137942	−	0.782308i	−0.834824	−	0.550518i	$-0.814430\pi$
$571$	44974.8	−	255065.i	0.137942	−	0.782308i	0.972766	−	0.231791i	$-0.0744585\pi$
$572$		0			0
$573$	−130717.	+	144598.i	−0.398129	+	0.440405i
$574$		0			0
$575$	105853.	−	61114.0i	0.320159	−	0.184844i
$576$		0			0
$577$	−173118.	+	299850.i	−0.519986	+	0.900642i	0.479744	+	0.877409i	$0.340730\pi$
$577$	−173118.	+	299850.i	−0.519986	+	0.900642i	−0.999730	+	0.0232338i	$0.992604\pi$
$578$		0			0
$579$	−138361.	−	5106.77i	−0.412721	−	0.0152331i
$580$		0			0
$581$	135001.	−	160888.i	0.399930	−	0.476619i
$582$		0			0
$583$	73698.3	+	26824.0i	0.216831	+	0.0789199i
$584$		0			0
$585$	3549.37	−	48017.1i	0.0103714	−	0.140309i
$586$		0			0
$587$	496369.	−	87523.2i	1.44055	−	0.254008i	0.601853	−	0.798607i	$-0.294429\pi$
$587$	496369.	−	87523.2i	1.44055	−	0.254008i	0.838697	+	0.544599i	$0.183318\pi$
$588$		0			0
$589$	−543537.	+	197831.i	−1.56674	+	0.570249i
$590$		0			0
$591$	−271400.	−	348802.i	−0.777026	−	0.998629i
$592$		0			0
$593$			637062.i			1.81164i	0.423661	+	0.905821i	$0.360745\pi$
$593$			637062.i			1.81164i	−0.423661	+	0.905821i	$0.639255\pi$
$594$		0			0
$595$	−196688.			−0.555575
$596$		0			0
$597$	−641306.	−	260587.i	−1.79935	−	0.731146i
$598$		0			0
$599$	63233.8	+	173734.i	0.176236	+	0.484206i	0.996088	−	0.0883710i	$-0.0281661\pi$
$599$	63233.8	+	173734.i	0.176236	+	0.484206i	−0.819851	+	0.572577i	$0.805944\pi$
$600$		0			0
$601$	−121142.	−	687032.i	−0.335388	−	1.90208i	−0.423373	−	0.905956i	$-0.639154\pi$
$601$	−121142.	−	687032.i	−0.335388	−	1.90208i	0.0879853	−	0.996122i	$-0.471957\pi$
$602$		0			0
$603$	−315398.	−	141928.i	−0.867410	−	0.390331i
$604$		0			0
$605$	−61839.0	+	169901.i	−0.168947	+	0.464179i
$606$		0			0
$607$	−224939.	−	188746.i	−0.610502	−	0.512272i	0.284300	−	0.958735i	$-0.408239\pi$
$607$	−224939.	−	188746.i	−0.610502	−	0.512272i	−0.894802	+	0.446463i	$0.852683\pi$
$608$		0			0
$609$	−265687.	−	166754.i	−0.716366	−	0.449616i
$610$		0			0
$611$	−113145.	−	65324.4i	−0.303077	−	0.174982i
$612$		0			0
$613$	281442.	+	487472.i	0.748976	+	1.29726i	0.948314	+	0.317333i	$0.102787\pi$
$613$	281442.	+	487472.i	0.748976	+	1.29726i	−0.199338	+	0.979931i	$0.563879\pi$
$614$		0			0
$615$	−30448.8	+	141865.i	−0.0805045	+	0.375080i
$616$		0			0
$617$	−572112.	−	100879.i	−1.50283	−	0.264990i	−0.639172	−	0.769064i	$-0.720723\pi$
$617$	−572112.	−	100879.i	−1.50283	−	0.264990i	−0.863660	+	0.504074i	$0.831834\pi$
$618$		0			0
$619$	−168647.	+	141511.i	−0.440146	+	0.369326i	−0.835764	−	0.549089i	$-0.814975\pi$
$619$	−168647.	+	141511.i	−0.440146	+	0.369326i	0.395618	+	0.918415i	$0.370530\pi$
$620$		0			0
$621$	63782.4	+	262468.i	0.165393	+	0.680603i
$622$		0			0
$623$	42382.8	+	50509.9i	0.109198	+	0.130137i
$624$		0			0
$625$	−13131.7	+	74473.7i	−0.0336172	+	0.190653i
$626$		0			0
$627$	76502.5	+	237051.i	0.194599	+	0.602985i
$628$		0			0
$629$	11697.0	−	6753.27i	0.0295647	−	0.0170692i
$630$		0			0
$631$	−143576.	+	248680.i	−0.360597	+	0.624572i	−0.988059	−	0.154075i	$-0.950760\pi$
$631$	−143576.	+	248680.i	−0.360597	+	0.624572i	0.627462	+	0.778647i	$0.284094\pi$
$632$		0			0
$633$	−362379.	−	684812.i	−0.904389	−	1.70909i
$634$		0			0
$635$	62870.6	−	74926.2i	0.155919	−	0.185817i
$636$		0			0
$637$	6869.82	+	2500.41i	0.0169304	+	0.00616215i
$638$		0			0
$639$	394733.	+	100085.i	0.966722	+	0.245113i
$640$		0			0
$641$	348199.	−	61396.9i	0.847446	−	0.149428i	0.266970	−	0.963705i	$-0.413978\pi$
$641$	348199.	−	61396.9i	0.847446	−	0.149428i	0.580476	+	0.814277i	$0.302867\pi$
$642$		0			0
$643$	604456.	−	220004.i	1.46198	−	0.532119i	0.516073	−	0.856545i	$-0.327393\pi$
$643$	604456.	−	220004.i	1.46198	−	0.532119i	0.945911	+	0.324426i	$0.105171\pi$
$644$		0			0
$645$	258494.	−	35805.7i	0.621343	−	0.0860662i
$646$		0			0
$647$		−	343334.i		−	0.820179i	−0.912045	−	0.410089i	$-0.865497\pi$
$647$		−	343334.i		−	0.820179i	0.912045	−	0.410089i	$-0.134503\pi$
$648$		0			0
$649$	246110.			0.584305
$650$		0			0
$651$	77479.5	+	559353.i	0.182821	+	1.31985i
$652$		0			0
$653$	141240.	+	388054.i	0.331231	+	0.910050i	0.987792	+	0.155777i	$0.0497883\pi$
$653$	141240.	+	388054.i	0.331231	+	0.910050i	−0.656561	+	0.754273i	$0.727990\pi$
$654$		0			0
$655$	65201.1	+	369774.i	0.151975	+	0.861894i
$656$		0			0
$657$	57589.7	+	56057.4i	0.133418	+	0.129868i
$658$		0			0
$659$	140053.	−	384794.i	0.322495	−	0.886048i	−0.667458	−	0.744648i	$-0.732617\pi$
$659$	140053.	−	384794.i	0.322495	−	0.886048i	0.989953	−	0.141400i	$-0.0451603\pi$
$660$		0			0
$661$	−223989.	−	187949.i	−0.512654	−	0.430168i	0.349408	−	0.936971i	$-0.386383\pi$
$661$	−223989.	−	187949.i	−0.512654	−	0.430168i	−0.862062	+	0.506803i	$0.830827\pi$
$662$		0			0
$663$	67347.4	−	35637.9i	0.153212	−	0.0810746i
$664$		0			0
$665$	−300322.	−	173391.i	−0.679117	−	0.392088i
$666$		0			0
$667$	−137985.	−	238996.i	−0.310155	−	0.537205i
$668$		0			0
$669$	215786.	−	69639.7i	0.482138	−	0.155598i
$670$		0			0
$671$	−112351.	−	19810.5i	−0.249535	−	0.0439998i
$672$		0			0
$673$	193341.	−	162232.i	0.426868	−	0.358185i	−0.403900	−	0.914803i	$-0.632346\pi$
$673$	193341.	−	162232.i	0.426868	−	0.358185i	0.830769	+	0.556618i	$0.187901\pi$
$674$		0			0
$675$	215516.	+	106708.i	0.473011	+	0.234201i
$676$		0			0
$677$	123728.	+	147453.i	0.269954	+	0.321719i	0.883942	−	0.467597i	$-0.154880\pi$
$677$	123728.	+	147453.i	0.269954	+	0.321719i	−0.613988	+	0.789316i	$0.710436\pi$
$678$		0			0
$679$	−43869.5	+	248796.i	−0.0951531	+	0.539640i
$680$		0			0
$681$	661930.	+	142072.i	1.42731	+	0.306348i
$682$		0			0
$683$	382344.	−	220747.i	0.819622	−	0.473209i	−0.0306644	−	0.999530i	$-0.509762\pi$
$683$	382344.	−	220747.i	0.819622	−	0.473209i	0.850286	+	0.526321i	$0.176429\pi$
$684$		0			0
$685$	14916.4	−	25836.0i	0.0317895	−	0.0550611i
$686$		0			0
$687$	85184.2	−	135723.i	0.180487	−	0.287567i
$688$		0			0
$689$	27188.9	−	32402.5i	0.0572734	−	0.0682558i
$690$		0			0
$691$	450725.	+	164051.i	0.943964	+	0.343575i	0.767730	−	0.640773i	$-0.221386\pi$
$691$	450725.	+	164051.i	0.943964	+	0.343575i	0.176234	+	0.984348i	$0.443608\pi$
$692$		0			0
$693$	241946.	−	24458.4i	0.503792	−	0.0509287i
$694$		0			0
$695$	230095.	−	40571.9i	0.476362	−	0.0839954i
$696$		0			0
$697$	−215769.	+	78533.5i	−0.444144	+	0.161655i
$698$		0			0
$699$	20152.3	−	49595.0i	0.0412450	−	0.101504i
$700$		0			0
$701$			486026.i			0.989063i	0.869160	+	0.494531i	$0.164660\pi$
$701$			486026.i			0.989063i	−0.869160	+	0.494531i	$0.835340\pi$
$702$		0			0
$703$	23813.6			0.0481852
$704$		0			0
$705$	−460733.	+	358493.i	−0.926981	+	0.721277i
$706$		0			0
$707$	−39213.8	−	107739.i	−0.0784514	−	0.215543i
$708$		0			0
$709$	−38195.6	−	216618.i	−0.0759838	−	0.430926i	−0.998940	−	0.0460216i	$-0.985346\pi$
$709$	−38195.6	−	216618.i	−0.0759838	−	0.430926i	0.922957	−	0.384904i	$-0.125765\pi$
$710$		0			0
$711$	516125.	−	758671.i	1.02098	−	1.50077i
$712$		0			0
$713$	−169917.	+	466842.i	−0.334239	+	0.918313i
$714$		0			0
$715$	−29214.3	−	24513.7i	−0.0571456	−	0.0479508i
$716$		0			0
$717$	−2192.02	+	59389.7i	−0.00426389	+	0.115524i
$718$		0			0
$719$	−757816.	−	437526.i	−1.46591	−	0.846341i	−0.466633	−	0.884451i	$-0.654533\pi$
$719$	−757816.	−	437526.i	−1.46591	−	0.846341i	−0.999274	+	0.0381097i	$0.987866\pi$
$720$		0			0
$721$	377899.	+	654540.i	0.726950	+	1.25911i
$722$		0			0
$723$	577456.	+	522024.i	1.10470	+	0.998651i
$724$		0			0
$725$	−241973.	−	42666.3i	−0.460353	−	0.0811726i
$726$		0			0
$727$	−523516.	+	439282.i	−0.990515	+	0.831141i	−0.985642	−	0.168847i	$-0.945995\pi$
$727$	−523516.	+	439282.i	−0.990515	+	0.831141i	−0.00487274	+	0.999988i	$0.501551\pi$
$728$		0			0
$729$	−360283.	+	390673.i	−0.677937	+	0.735120i
$730$		0			0
$731$	265457.	+	316359.i	0.496775	+	0.592033i
$732$		0			0
$733$	−89439.8	+	507239.i	−0.166465	+	0.944070i	0.781076	+	0.624436i	$0.214671\pi$
$733$	−89439.8	+	507239.i	−0.166465	+	0.944070i	−0.947541	+	0.319634i	$0.896440\pi$
$734$		0			0
$735$	21906.1	−	24232.3i	0.0405500	−	0.0448559i
$736$		0			0
$737$	−237243.	+	136972.i	−0.436775	+	0.252172i
$738$		0			0
$739$	134409.	−	232804.i	0.246116	−	0.426286i	−0.716329	−	0.697763i	$-0.754179\pi$
$739$	134409.	−	232804.i	0.246116	−	0.426286i	0.962445	+	0.271477i	$0.0875122\pi$
$740$		0			0
$741$	134250.	+	4955.02i	0.244499	+	0.00902421i
$742$		0			0
$743$	−367472.	+	437936.i	−0.665651	+	0.793292i	−0.988185	−	0.153266i	$-0.951021\pi$
$743$	−367472.	+	437936.i	−0.665651	+	0.793292i	0.322534	+	0.946558i	$0.395465\pi$
$744$		0			0
$745$	695364.	+	253092.i	1.25285	+	0.456001i
$746$		0			0
$747$	300584.	+	204488.i	0.538672	+	0.366460i
$748$		0			0
$749$	−776394.	+	136899.i	−1.38394	+	0.244027i
$750$		0			0
$751$	161443.	−	58760.5i	0.286246	−	0.104185i	−0.194907	−	0.980822i	$-0.562440\pi$
$751$	161443.	−	58760.5i	0.286246	−	0.104185i	0.481153	+	0.876637i	$0.340218\pi$
$752$		0			0
$753$	−497157.	−	638944.i	−0.876807	−	1.12687i
$754$		0			0
$755$		−	194397.i		−	0.341033i
$756$		0			0
$757$	87387.1			0.152495			0.0762475	−	0.997089i	$-0.475706\pi$
$757$	87387.1			0.152495			0.0762475	+	0.997089i	$0.475706\pi$
$758$		0			0
$759$	198205.	+	80538.1i	0.344057	+	0.139803i
$760$		0			0
$761$	182574.	+	501618.i	0.315260	+	0.866171i	0.991572	+	0.129555i	$0.0413550\pi$
$761$	182574.	+	501618.i	0.315260	+	0.866171i	−0.676312	+	0.736616i	$0.736423\pi$
$762$		0			0
$763$	62658.9	+	355356.i	0.107630	+	0.610400i
$764$		0			0
$765$	−34242.9	−	338735.i	−0.0585124	−	0.578811i
$766$		0			0
$767$	45397.6	−	124729.i	0.0771689	−	0.212020i
$768$		0			0
$769$	770270.	+	646334.i	1.30254	+	1.09296i	0.989701	+	0.143153i	$0.0457241\pi$
$769$	770270.	+	646334.i	1.30254	+	1.09296i	0.312838	+	0.949807i	$0.398720\pi$
$770$		0			0
$771$	−640452.	−	401969.i	−1.07740	−	0.676214i
$772$		0			0
$773$	−540306.	−	311946.i	−0.904234	−	0.522060i	−0.0256627	−	0.999671i	$-0.508170\pi$
$773$	−540306.	−	311946.i	−0.904234	−	0.522060i	−0.878572	+	0.477611i	$0.841503\pi$
$774$		0			0
$775$	221161.	+	383062.i	0.368218	+	0.637772i
$776$		0			0
$777$	4878.77	−	22730.8i	0.00808106	−	0.0376506i
$778$		0			0
$779$	−398689.	−	70299.7i	−0.656991	−	0.115845i
$780$		0			0
$781$	247086.	−	207330.i	0.405085	−	0.339907i
$782$		0			0
$783$	240928.	−	486596.i	0.392973	−	0.793680i
$784$		0			0
$785$	−217526.	−	259238.i	−0.352998	−	0.420687i
$786$		0			0
$787$	74917.1	−	424876.i	0.120957	−	0.685982i	−0.862670	−	0.505767i	$-0.831210\pi$
$787$	74917.1	−	424876.i	0.120957	−	0.685982i	0.983627	−	0.180215i	$-0.0576793\pi$
$788$		0			0
$789$	3997.34	+	12386.2i	0.00642122	+	0.0198968i
$790$		0			0
$791$	857852.	−	495281.i	1.37107	−	0.791587i
$792$		0			0
$793$	−30764.3	+	53285.4i	−0.0489217	+	0.0847348i
$794$		0			0
$795$	−88398.9	−	167054.i	−0.139866	−	0.264315i
$796$		0			0
$797$	31827.0	−	37929.9i	0.0501047	−	0.0597125i	−0.740410	−	0.672155i	$-0.765369\pi$
$797$	31827.0	−	37929.9i	0.0501047	−	0.0597125i	0.790515	+	0.612443i	$0.209813\pi$
$798$		0			0
$799$	−868119.	−	315970.i	−1.35983	−	0.494939i
$800$		0			0
$801$	−79609.2	+	81785.3i	−0.124079	+	0.127471i
$802$		0			0
$803$	62689.7	−	11053.9i	0.0972220	−	0.0171429i
$804$		0			0
$805$	−279888.	+	101871.i	−0.431909	+	0.157202i
$806$		0			0
$807$	1.11150e6	−	153961.i	1.70672	−	0.236408i
$808$		0			0
$809$			1.19267e6i			1.82231i	0.412061	+	0.911156i	$0.364809\pi$
$809$			1.19267e6i			1.82231i	−0.412061	+	0.911156i	$0.635191\pi$
$810$		0			0
$811$	−760675.			−1.15653			−0.578265	−	0.815849i	$-0.696270\pi$
$811$	−760675.			−1.15653			−0.578265	+	0.815849i	$0.696270\pi$
$812$		0			0
$813$	−93029.5	−	671613.i	−0.140747	−	1.01610i
$814$		0			0
$815$	−130514.	−	358584.i	−0.196491	−	0.539854i
$816$		0			0
$817$	126438.	+	717064.i	0.189423	+	1.07427i
$818$		0			0
$819$	32233.9	−	127130.i	0.0480557	−	0.189531i
$820$		0			0
$821$	88627.7	−	243503.i	0.131487	−	0.361258i	−0.856425	−	0.516271i	$-0.827320\pi$
$821$	88627.7	−	243503.i	0.131487	−	0.361258i	0.987912	+	0.155013i	$0.0495420\pi$
$822$		0			0
$823$	732235.	+	614418.i	1.08106	+	0.907119i	0.996009	−	0.0892572i	$-0.0284493\pi$
$823$	732235.	+	614418.i	1.08106	+	0.907119i	0.0850539	+	0.996376i	$0.472894\pi$
$824$		0			0
$825$	168362.	−	89091.3i	0.247364	−	0.130896i
$826$		0			0
$827$	−716837.	−	413866.i	−1.04812	−	0.605130i	−0.125994	−	0.992031i	$-0.540212\pi$
$827$	−716837.	−	413866.i	−1.04812	−	0.605130i	−0.922121	+	0.386901i	$0.873545\pi$
$828$		0			0
$829$	−432183.	−	748564.i	−0.628867	−	1.08923i	−0.987779	−	0.155859i	$-0.950186\pi$
$829$	−432183.	−	748564.i	−0.628867	−	1.08923i	0.358912	−	0.933371i	$-0.383148\pi$
$830$		0			0
$831$	887970.	−	286571.i	1.28587	−	0.414983i
$832$		0			0
$833$	50909.6	+	8976.73i	0.0733685	+	0.0129368i
$834$		0			0
$835$	−1749.91	+	1468.35i	−0.00250982	+	0.00210599i
$836$		0			0
$837$	−949827.	+	230817.i	−1.35579	+	0.329471i
$838$		0			0
$839$	−445949.	−	531461.i	−0.633521	−	0.755001i	0.349811	−	0.936820i	$-0.386246\pi$
$839$	−445949.	−	531461.i	−0.633521	−	0.755001i	−0.983332	+	0.181819i	$0.941801\pi$
$840$		0			0
$841$	26485.1	−	150204.i	0.0374463	−	0.212368i
$842$		0			0
$843$	−484952.	−	104087.i	−0.682407	−	0.146467i
$844$		0			0
$845$	407100.	−	235039.i	0.570148	−	0.329175i
$846$		0			0
$847$	−246252.	+	426521.i	−0.343252	+	0.594529i
$848$		0			0
$849$	463892.	−	739112.i	0.643578	−	1.02540i
$850$		0			0
$851$	13147.2	−	15668.2i	0.0181541	−	0.0216352i
$852$		0			0
$853$	542954.	+	197619.i	0.746217	+	0.271601i	0.687013	−	0.726645i	$-0.258922\pi$
$853$	542954.	+	197619.i	0.746217	+	0.271601i	0.0592040	+	0.998246i	$0.481144\pi$
$854$		0			0
$855$	246328.	−	547401.i	0.336963	−	0.748814i
$856$		0			0
$857$	−549610.	+	96911.1i	−0.748330	+	0.131951i	−0.534792	−	0.844984i	$-0.679610\pi$
$857$	−549610.	+	96911.1i	−0.748330	+	0.131951i	−0.213538	+	0.976935i	$0.568499\pi$
$858$		0			0
$859$	−268789.	+	97831.2i	−0.364271	+	0.132584i	−0.517670	−	0.855580i	$-0.673200\pi$
$859$	−268789.	+	97831.2i	−0.364271	+	0.132584i	0.153398	+	0.988164i	$0.450978\pi$
$860$		0			0
$861$	−148784.	+	366158.i	−0.200701	+	0.493927i
$862$		0			0
$863$		−	193853.i		−	0.260286i	−0.991495	−	0.130143i	$-0.958456\pi$
$863$		−	193853.i		−	0.260286i	0.991495	−	0.130143i	$-0.0415437\pi$
$864$		0			0
$865$	216191.			0.288938
$866$		0			0
$867$	−168029.	+	130742.i	−0.223536	+	0.173931i
$868$		0			0
$869$	−248577.	−	682959.i	−0.329171	−	0.904389i
$870$		0			0
$871$	25655.7	+	145501.i	0.0338180	+	0.191792i
$872$		0			0
$873$	−436114.	−	32237.0i	−0.572231	−	0.0422986i
$874$		0			0
$875$	−262538.	+	721318.i	−0.342907	+	0.942129i
$876$		0			0
$877$	−731376.	−	613697.i	−0.950915	−	0.797912i	0.0285366	−	0.999593i	$-0.490915\pi$
$877$	−731376.	−	613697.i	−0.950915	−	0.797912i	−0.979451	+	0.201681i	$0.935360\pi$
$878$		0			0
$879$	−1140.98	+	30913.4i	−0.00147673	+	0.0400100i
$880$		0			0
$881$	662276.	+	382365.i	0.853271	+	0.492636i	0.861753	−	0.507328i	$-0.169367\pi$
$881$	662276.	+	382365.i	0.853271	+	0.492636i	−0.00848230	+	0.999964i	$0.502700\pi$
$882$		0			0
$883$	−395228.	−	684556.i	−0.506905	−	0.877985i	−0.999968	−	0.00799177i	$-0.997456\pi$
$883$	−395228.	−	684556.i	−0.506905	−	0.877985i	0.493063	−	0.869994i	$-0.335877\pi$
$884$		0			0
$885$	−439963.	−	397729.i	−0.561732	−	0.507810i
$886$		0			0
$887$	−189753.	−	33458.5i	−0.241180	−	0.0425265i	0.0517515	−	0.998660i	$-0.483520\pi$
$887$	−189753.	−	33458.5i	−0.241180	−	0.0425265i	−0.292931	+	0.956133i	$0.594631\pi$
$888$		0			0
$889$	204095.	−	171256.i	0.258243	−	0.216692i
$890$		0			0
$891$	84244.5	+	412420.i	0.106117	+	0.519499i
$892$		0			0
$893$	−1.04699e6	−	1.24775e6i	−1.31292	−	1.56468i
$894$		0			0
$895$	−134031.	+	760128.i	−0.167324	+	0.948944i
$896$		0			0
$897$	77377.8	−	85594.3i	0.0961682	−	0.106380i
$898$		0			0
$899$	864886.	−	499342.i	1.07014	−	0.617844i
$900$		0			0
$901$	149549.	−	259026.i	0.184218	−	0.319076i
$902$		0			0
$903$	710363.	+	26218.8i	0.871174	+	0.0321542i
$904$		0			0
$905$	674068.	−	803323.i	0.823013	−	0.980829i
$906$		0			0
$907$	−902536.	−	328496.i	−1.09711	−	0.399315i	−0.270860	−	0.962619i	$-0.587308\pi$
$907$	−902536.	−	328496.i	−1.09711	−	0.399315i	−0.826250	+	0.563304i	$0.809530\pi$
$908$		0			0
$909$	178721.	−	86291.2i	0.216296	−	0.104433i
$910$		0			0
$911$	−1.08169e6	+	190732.i	−1.30337	+	0.229819i	−0.781875	−	0.623435i	$-0.785737\pi$
$911$	−1.08169e6	+	190732.i	−1.30337	+	0.229819i	−0.521495	+	0.853254i	$0.674626\pi$
$912$		0			0
$913$	270587.	−	98485.7i	0.324613	−	0.118149i
$914$		0			0
$915$	168831.	+	216981.i	0.201656	+	0.259167i
$916$		0			0
$917$			1.02278e6i			1.21631i
$918$		0			0
$919$	278278.			0.329495			0.164747	−	0.986336i	$-0.447319\pi$
$919$	278278.			0.329495			0.164747	+	0.986336i	$0.447319\pi$
$920$		0			0
$921$	34876.1	+	14171.5i	0.0411158	+	0.0167069i
$922$		0			0
$923$	−59497.4	−	163468.i	−0.0698385	−	0.191880i
$924$		0			0
$925$	−3162.21	−	17933.8i	−0.00369579	−	0.0209598i
$926$		0			0
$927$	−1.06146e6	+	764770.i	−1.23521	+	0.889962i
$928$		0			0
$929$	10559.2	−	29011.1i	0.0122348	−	0.0336150i	−0.933425	−	0.358773i	$-0.883195\pi$
$929$	10559.2	−	29011.1i	0.0122348	−	0.0336150i	0.945660	+	0.325158i	$0.105417\pi$
$930$		0			0
$931$	69820.4	+	58586.2i	0.0805532	+	0.0675921i
$932$		0			0
$933$	−94048.6	−	59028.1i	−0.108041	−	0.0678103i
$934$		0			0
$935$	−233539.	−	134834.i	−0.267139	−	0.154233i
$936$		0			0
$937$	249027.	+	431327.i	0.283639	+	0.491278i	0.972278	−	0.233827i	$-0.0751248\pi$
$937$	249027.	+	431327.i	0.283639	+	0.491278i	−0.688639	+	0.725104i	$0.741791\pi$
$938$		0			0
$939$	−184121.	+	857841.i	−0.208820	+	0.972916i
$940$		0			0
$941$	774493.	+	136564.i	0.874658	+	0.154226i	0.592916	−	0.805264i	$-0.297977\pi$
$941$	774493.	+	136564.i	0.874658	+	0.154226i	0.281742	+	0.959490i	$0.409088\pi$
$942$		0			0
$943$	−266366.	+	223507.i	−0.299540	+	0.251344i
$944$		0			0
$945$	−472045.	−	347276.i	−0.528591	−	0.388876i
$946$		0			0
$947$	904077.	+	1.07744e6i	1.00810	+	1.20141i	0.979422	+	0.201823i	$0.0646866\pi$
$947$	904077.	+	1.07744e6i	1.00810	+	1.20141i	0.0286819	+	0.999589i	$0.490869\pi$
$948$		0			0
$949$	5961.66	−	33810.2i	0.00661964	−	0.0375418i
$950$		0			0
$951$	477048.	+	1.47818e6i	0.527474	+	1.63443i
$952$		0			0
$953$	−1.50741e6	+	870301.i	−1.65976	+	0.958261i	−0.686930	+	0.726723i	$0.741042\pi$
$953$	−1.50741e6	+	870301.i	−1.65976	+	0.958261i	−0.972826	+	0.231537i	$0.925624\pi$
$954$		0			0
$955$	186033.	−	322218.i	0.203978	−	0.353300i
$956$		0			0
$957$	−201152.	−	380131.i	−0.219635	−	0.415059i
$958$		0			0
$959$	52235.0	−	62251.2i	0.0567968	−	0.0676878i
$960$		0			0
$961$	−821594.	−	299036.i	−0.889632	−	0.323799i
$962$		0			0
$963$	−370936.	−	1.31327e6i	−0.399987	−	1.41612i
$964$		0			0
$965$	260264.	−	45891.6i	0.279486	−	0.0492808i
$966$		0			0
$967$	986414.	−	359025.i	1.05489	−	0.383948i	0.244382	−	0.969679i	$-0.421415\pi$
$967$	986414.	−	359025.i	1.05489	−	0.383948i	0.810505	+	0.585731i	$0.199193\pi$
$968$		0			0
$969$	940958.	−	130338.i	1.00213	−	0.138811i
$970$		0			0
$971$			936946.i			0.993748i	0.867823	+	0.496874i	$0.165519\pi$
$971$			936946.i			0.993748i	−0.867823	+	0.496874i	$0.834481\pi$
$972$		0			0
$973$	636434.			0.672246
$974$		0			0
$975$	−14095.4	−	101760.i	−0.0148275	−	0.107045i
$976$		0			0
$977$	−16333.4	−	44875.6i	−0.0171115	−	0.0470134i	0.930843	−	0.365420i	$-0.119075\pi$
$977$	−16333.4	−	44875.6i	−0.0171115	−	0.0470134i	−0.947954	+	0.318406i	$0.896852\pi$
$978$		0			0
$979$	15698.0	+	89027.9i	0.0163787	+	0.0928882i
$980$		0			0
$981$	−601085.	+	169778.i	−0.624594	+	0.176418i
$982$		0			0
$983$	−14168.9	+	38928.8i	−0.0146632	+	0.0402869i	−0.946808	−	0.321798i	$-0.895713\pi$
$983$	−14168.9	+	38928.8i	−0.0146632	+	0.0402869i	0.932145	+	0.362085i	$0.117935\pi$
$984$		0			0
$985$	646222.	+	542245.i	0.666054	+	0.558885i
$986$		0			0
$987$	−1.40552e6	+	743749.i	−1.44278	+	0.763471i
$988$		0			0
$989$	541600.	+	312693.i	0.553714	+	0.319687i
$990$		0			0
$991$	−679417.	−	1.17678e6i	−0.691814	−	1.19826i	−0.971243	−	0.238090i	$-0.923479\pi$
$991$	−679417.	−	1.17678e6i	−0.691814	−	1.19826i	0.279429	−	0.960166i	$-0.409855\pi$
$992$		0			0
$993$	767389.	−	247656.i	0.778246	−	0.251160i
$994$		0			0
$995$	1.30123e6	+	229442.i	1.31434	+	0.231753i
$996$		0			0
$997$	134572.	−	112919.i	0.135383	−	0.113600i	−0.572582	−	0.819848i	$-0.694058\pi$
$997$	134572.	−	112919.i	0.135383	−	0.113600i	0.707965	+	0.706248i	$0.249614\pi$
$998$		0			0
$999$	39996.3	+	4444.83i	0.0400764	+	0.00445373i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.6	✓	72
3.2	odd	2		324.5.k.a.125.4		72
27.11	odd	18	inner	108.5.k.a.65.6	yes	72
27.16	even	9		324.5.k.a.197.4		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.6	✓	72	1.1	even	1	trivial
108.5.k.a.65.6	yes	72	27.11	odd	18	inner
324.5.k.a.125.4		72	3.2	odd	2
324.5.k.a.197.4		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)

\(f(q)\)	\(=\)	\(q+(-1.23486 - 8.91488i) q^{3} +(5.87553 + 16.1429i) q^{5} +(8.12577 + 46.0835i) q^{7} +(-77.9503 + 22.0172i) q^{9} +O(q^{10})\)	\(q+(-1.23486 - 8.91488i) q^{3} +(5.87553 + 16.1429i) q^{5} +(8.12577 + 46.0835i) q^{7} +(-77.9503 + 22.0172i) q^{9} +(-21.9431 + 60.2882i) q^{11} +(26.5065 + 22.2416i) q^{13} +(136.657 - 72.3139i) q^{15} +(211.894 + 122.337i) q^{17} +(215.694 + 373.592i) q^{19} +(400.795 - 129.347i) q^{21} +(364.888 + 64.3396i) q^{23} +(252.707 - 212.046i) q^{25} +(292.538 + 667.729i) q^{27} +(-478.762 - 570.567i) q^{29} +(-232.834 + 1320.47i) q^{31} +(564.559 + 121.173i) q^{33} +(-696.178 + 401.939i) q^{35} +(27.6012 - 47.8066i) q^{37} +(165.550 - 263.768i) q^{39} +(-603.231 + 718.902i) q^{41} +(1586.08 + 577.286i) q^{43} +(-813.421 - 1128.98i) q^{45} +(-3718.42 + 655.657i) q^{47} +(198.539 - 72.2624i) q^{49} +(828.960 - 2040.07i) q^{51} -1222.43i q^{53} -1102.15 q^{55} +(3064.18 - 2384.22i) q^{57} +(-1312.00 - 3604.69i) q^{59} +(308.780 + 1751.18i) q^{61} +(-1648.04 - 3413.32i) q^{63} +(-203.304 + 558.573i) q^{65} +(3270.92 + 2744.63i) q^{67} +(122.995 - 3332.39i) q^{69} +(-4353.90 - 2513.72i) q^{71} +(-496.098 - 859.268i) q^{73} +(-2202.42 - 1991.00i) q^{75} +(-2956.60 - 521.328i) q^{77} +(-8677.93 + 7281.65i) q^{79} +(5591.48 - 3432.50i) q^{81} +(-2884.98 - 3438.18i) q^{83} +(-729.883 + 4139.37i) q^{85} +(-4495.33 + 4972.68i) q^{87} +(1220.28 - 704.528i) q^{89} +(-809.586 + 1402.24i) q^{91} +(12059.3 + 445.098i) q^{93} +(-4763.55 + 5676.97i) q^{95} +(5073.22 + 1846.50i) q^{97} +(383.092 - 5182.61i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more