LMFDB - Embedded newform 108.6.b.b.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(107,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("108.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 30x^{14} + 619x^{12} + 5604x^{10} + 40971x^{8} - 4866x^{6} + 568069x^{4} - 7909632x^{2} + 20340100 $ x^16 + 30x^14 + 619x^12 + 5604x^10 + 40971x^8 - 4866x^6 + 568069x^4 - 7909632*x^2 + 20340100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{32}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			107.4
Root			$1.73205 + 0.353400i$ of defining polynomial
Character	$\chi$	$=$	108.107
Dual form			108.6.b.b.107.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.08799 + 2.47231i) q^{2} +(19.7753 - 25.1582i) q^{4} +33.1536i q^{5} -59.4073i q^{7} +(-38.4178 + 176.896i) q^{8} +O(q^{10})$	\(q+(-5.08799 + 2.47231i) q^{2} +(19.7753 - 25.1582i) q^{4} +33.1536i q^{5} -59.4073i q^{7} +(-38.4178 + 176.896i) q^{8} +(-81.9660 - 168.685i) q^{10} +179.135 q^{11} -143.222 q^{13} +(146.873 + 302.264i) q^{14} +(-241.872 - 995.025i) q^{16} -93.8221i q^{17} -1617.39i q^{19} +(834.085 + 655.624i) q^{20} +(-911.438 + 442.878i) q^{22} +2414.48 q^{23} +2025.84 q^{25} +(728.711 - 354.089i) q^{26} +(-1494.58 - 1174.80i) q^{28} +7910.14i q^{29} +4726.80i q^{31} +(3690.65 + 4464.70i) q^{32} +(231.958 + 477.366i) q^{34} +1969.56 q^{35} +9387.10 q^{37} +(3998.68 + 8229.25i) q^{38} +(-5864.73 - 1273.69i) q^{40} +11838.5i q^{41} +19531.8i q^{43} +(3542.46 - 4506.72i) q^{44} +(-12284.8 + 5969.34i) q^{46} +7373.21 q^{47} +13277.8 q^{49} +(-10307.5 + 5008.51i) q^{50} +(-2832.26 + 3603.20i) q^{52} -26895.4i q^{53} +5938.97i q^{55} +(10508.9 + 2282.30i) q^{56} +(-19556.3 - 40246.7i) q^{58} +6144.64 q^{59} +55252.3 q^{61} +(-11686.1 - 24049.9i) q^{62} +(-29816.1 - 13591.9i) q^{64} -4748.31i q^{65} -38888.6i q^{67} +(-2360.40 - 1855.36i) q^{68} +(-10021.1 + 4869.38i) q^{70} -45277.6 q^{71} +21000.4 q^{73} +(-47761.5 + 23207.8i) q^{74} +(-40690.5 - 31984.4i) q^{76} -10641.9i q^{77} +76460.5i q^{79} +(32988.6 - 8018.91i) q^{80} +(-29268.4 - 60234.0i) q^{82} -58515.8 q^{83} +3110.54 q^{85} +(-48288.6 - 99377.5i) q^{86} +(-6881.99 + 31688.2i) q^{88} -18558.5i q^{89} +8508.41i q^{91} +(47747.1 - 60744.0i) q^{92} +(-37514.9 + 18228.9i) q^{94} +53622.2 q^{95} +137248. q^{97} +(-67557.2 + 32826.8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 94 q^{4}+O(q^{10}) $ 16 * q + 94 * q^4	$ 16 q + 94 q^{4} + 1454 q^{10} + 896 q^{13} + 178 q^{16} + 30 q^{22} + 9888 q^{25} + 11454 q^{28} - 6172 q^{34} - 71008 q^{37} - 16618 q^{40} + 35304 q^{46} - 49376 q^{49} + 14876 q^{52} - 10492 q^{58} + 77888 q^{61} + 89206 q^{64} + 229398 q^{70} - 38032 q^{73} + 48960 q^{76} - 224488 q^{82} - 371264 q^{85} + 249102 q^{88} + 68772 q^{94} - 976 q^{97}+O(q^{100}) $ 16 * q + 94 * q^4 + 1454 * q^10 + 896 * q^13 + 178 * q^16 + 30 * q^22 + 9888 * q^25 + 11454 * q^28 - 6172 * q^34 - 71008 * q^37 - 16618 * q^40 + 35304 * q^46 - 49376 * q^49 + 14876 * q^52 - 10492 * q^58 + 77888 * q^61 + 89206 * q^64 + 229398 * q^70 - 38032 * q^73 + 48960 * q^76 - 224488 * q^82 - 371264 * q^85 + 249102 * q^88 + 68772 * q^94 - 976 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−5.08799	+	2.47231i	−0.899439	+	0.437047i
$2$	−5.08799	+	2.47231i	−0.899439	+	0.437047i
$3$		0			0
$3$		0			0
$4$	19.7753	−	25.1582i	0.617979	−	0.786194i
$5$			33.1536i			0.593069i	0.955022	+	0.296535i	$0.0958310\pi$
$5$			33.1536i			0.593069i	−0.955022	+	0.296535i	$0.904169\pi$
$6$		0			0
$7$		−	59.4073i		−	0.458242i	−0.973398	−	0.229121i	$-0.926415\pi$
$7$		−	59.4073i		−	0.458242i	0.973398	−	0.229121i	$-0.0735851\pi$
$8$	−38.4178	+	176.896i	−0.212231	+	0.977220i
$9$		0			0
$10$	−81.9660	−	168.685i	−0.259199	−	0.533430i
$11$	179.135			0.446374			0.223187	−	0.974776i	$-0.428354\pi$
$11$	179.135			0.446374			0.223187	+	0.974776i	$0.428354\pi$
$12$		0			0
$13$	−143.222			−0.235045			−0.117522	−	0.993070i	$-0.537495\pi$
$13$	−143.222			−0.235045			−0.117522	+	0.993070i	$0.537495\pi$
$14$	146.873	+	302.264i	0.200273	+	0.412160i
$15$		0			0
$16$	−241.872	−	995.025i	−0.236203	−	0.971704i
$17$		−	93.8221i		−	0.0787377i	−0.999225	−	0.0393689i	$-0.987465\pi$
$17$		−	93.8221i		−	0.0787377i	0.999225	−	0.0393689i	$-0.0125347\pi$
$18$		0			0
$19$		−	1617.39i		−	1.02785i	−0.857835	−	0.513925i	$-0.828191\pi$
$19$		−	1617.39i		−	1.02785i	0.857835	−	0.513925i	$-0.171809\pi$
$20$	834.085	+	655.624i	0.466268	+	0.366505i
$21$		0			0
$22$	−911.438	+	442.878i	−0.401486	+	0.195087i
$23$	2414.48			0.951708			0.475854	−	0.879524i	$-0.342139\pi$
$23$	2414.48			0.951708			0.475854	+	0.879524i	$0.342139\pi$
$24$		0			0
$25$	2025.84			0.648269
$26$	728.711	−	354.089i	0.211408	−	0.102726i
$27$		0			0
$28$	−1494.58	−	1174.80i	−0.360267	−	0.283184i
$29$			7910.14i			1.74658i	0.487198	+	0.873291i	$0.338019\pi$
$29$			7910.14i			1.74658i	−0.487198	+	0.873291i	$0.661981\pi$
$30$		0			0
$31$			4726.80i			0.883412i	0.897160	+	0.441706i	$0.145627\pi$
$31$			4726.80i			0.883412i	−0.897160	+	0.441706i	$0.854373\pi$
$32$	3690.65	+	4464.70i	0.637130	+	0.770756i
$33$		0			0
$34$	231.958	+	477.366i	0.0344121	+	0.0708198i
$35$	1969.56			0.271769
$36$		0			0
$37$	9387.10			1.12727			0.563634	−	0.826025i	$-0.309403\pi$
$37$	9387.10			1.12727			0.563634	+	0.826025i	$0.309403\pi$
$38$	3998.68	+	8229.25i	0.449219	+	0.924488i
$39$		0			0
$40$	−5864.73	−	1273.69i	−0.579559	−	0.125867i
$41$			11838.5i			1.09986i	0.835212	+	0.549928i	$0.185345\pi$
$41$			11838.5i			1.09986i	−0.835212	+	0.549928i	$0.814655\pi$
$42$		0			0
$43$			19531.8i			1.61091i	0.592659	+	0.805454i	$0.298078\pi$
$43$			19531.8i			1.61091i	−0.592659	+	0.805454i	$0.701922\pi$
$44$	3542.46	−	4506.72i	0.275850	−	0.350937i
$45$		0			0
$46$	−12284.8	+	5969.34i	−0.856003	+	0.415941i
$47$	7373.21			0.486869			0.243435	−	0.969917i	$-0.421726\pi$
$47$	7373.21			0.486869			0.243435	+	0.969917i	$0.421726\pi$
$48$		0			0
$49$	13277.8			0.790015
$50$	−10307.5	+	5008.51i	−0.583078	+	0.283324i
$51$		0			0
$52$	−2832.26	+	3603.20i	−0.145253	+	0.184791i
$53$		−	26895.4i		−	1.31519i	−0.753372	−	0.657594i	$-0.771574\pi$
$53$		−	26895.4i		−	1.31519i	0.753372	−	0.657594i	$-0.228426\pi$
$54$		0			0
$55$			5938.97i			0.264731i
$56$	10508.9	+	2282.30i	0.447803	+	0.0972529i
$57$		0			0
$58$	−19556.3	−	40246.7i	−0.763339	−	1.57094i
$59$	6144.64			0.229809			0.114904	−	0.993377i	$-0.463344\pi$
$59$	6144.64			0.229809			0.114904	+	0.993377i	$0.463344\pi$
$60$		0			0
$61$	55252.3			1.90119			0.950596	−	0.310430i	$-0.100473\pi$
$61$	55252.3			1.90119			0.950596	+	0.310430i	$0.100473\pi$
$62$	−11686.1	−	24049.9i	−0.386093	−	0.794575i
$63$		0			0
$64$	−29816.1	−	13591.9i	−0.909916	−	0.414792i
$65$		−	4748.31i		−	0.139398i
$66$		0			0
$67$		−	38888.6i		−	1.05836i	−0.848509	−	0.529182i	$-0.822499\pi$
$67$		−	38888.6i		−	1.05836i	0.848509	−	0.529182i	$-0.177501\pi$
$68$	−2360.40	−	1855.36i	−0.0619032	−	0.0486583i
$69$		0			0
$70$	−10021.1	+	4869.38i	−0.244440	+	0.118776i
$71$	−45277.6			−1.06595			−0.532976	−	0.846131i	$-0.678926\pi$
$71$	−45277.6			−1.06595			−0.532976	+	0.846131i	$0.678926\pi$
$72$		0			0
$73$	21000.4			0.461233			0.230617	−	0.973045i	$-0.425926\pi$
$73$	21000.4			0.461233			0.230617	+	0.973045i	$0.425926\pi$
$74$	−47761.5	+	23207.8i	−1.01391	+	0.492669i
$75$		0			0
$76$	−40690.5	−	31984.4i	−0.808090	−	0.635190i
$77$		−	10641.9i		−	0.204547i
$78$		0			0
$79$			76460.5i			1.37838i	0.724580	+	0.689191i	$0.242034\pi$
$79$			76460.5i			1.37838i	−0.724580	+	0.689191i	$0.757966\pi$
$80$	32988.6	−	8018.91i	0.576288	−	0.140085i
$81$		0			0
$82$	−29268.4	−	60234.0i	−0.480689	−	0.989253i
$83$	−58515.8			−0.932347			−0.466174	−	0.884693i	$-0.654368\pi$
$83$	−58515.8			−0.932347			−0.466174	+	0.884693i	$0.654368\pi$
$84$		0			0
$85$	3110.54			0.0466969
$86$	−48288.6	−	99377.5i	−0.704042	−	1.44891i
$87$		0			0
$88$	−6881.99	+	31688.2i	−0.0947343	+	0.436206i
$89$		−	18558.5i		−	0.248352i	−0.992260	−	0.124176i	$-0.960371\pi$
$89$		−	18558.5i		−	0.248352i	0.992260	−	0.124176i	$-0.0396287\pi$
$90$		0			0
$91$			8508.41i			0.107707i
$92$	47747.1	−	60744.0i	0.588136	−	0.748227i
$93$		0			0
$94$	−37514.9	+	18228.9i	−0.437909	+	0.212785i
$95$	53622.2			0.609586
$96$		0			0
$97$	137248.			1.48107			0.740536	−	0.672017i	$-0.234572\pi$
$97$	137248.			1.48107			0.740536	+	0.672017i	$0.234572\pi$
$98$	−67557.2	+	32826.8i	−0.710570	+	0.345274i
$99$		0			0
$100$	40061.7	−	50966.5i	0.400617	−	0.509665i
$101$		−	97319.1i		−	0.949280i	−0.880180	−	0.474640i	$-0.842578\pi$
$101$		−	97319.1i		−	0.949280i	0.880180	−	0.474640i	$-0.157422\pi$
$102$		0			0
$103$			38694.4i			0.359381i	0.983723	+	0.179690i	$0.0575096\pi$
$103$			38694.4i			0.359381i	−0.983723	+	0.179690i	$0.942490\pi$
$104$	5502.27	−	25335.3i	0.0498837	−	0.229690i
$105$		0			0
$106$	66493.8	+	136844.i	0.574799	+	1.18293i
$107$	−42882.4			−0.362093			−0.181046	−	0.983475i	$-0.557948\pi$
$107$	−42882.4			−0.362093			−0.181046	+	0.983475i	$0.557948\pi$
$108$		0			0
$109$	−41832.7			−0.337248			−0.168624	−	0.985680i	$-0.553932\pi$
$109$	−41832.7			−0.337248			−0.168624	+	0.985680i	$0.553932\pi$
$110$	−14683.0	−	30217.5i	−0.115700	−	0.238109i
$111$		0			0
$112$	−59111.7	+	14368.9i	−0.445275	+	0.108238i
$113$			30797.2i			0.226890i	0.993544	+	0.113445i	$0.0361885\pi$
$113$			30797.2i			0.226890i	−0.993544	+	0.113445i	$0.963811\pi$
$114$		0			0
$115$			80048.6i			0.564429i
$116$	199005.	+	156426.i	1.37315	+	1.07935i
$117$		0			0
$118$	−31263.9	+	15191.5i	−0.206699	+	0.100437i
$119$	−5573.72			−0.0360809
$120$		0			0
$121$	−128962.			−0.800750
$122$	−281123.	+	136601.i	−1.71001	+	0.830911i
$123$		0			0
$124$	118918.	+	93474.2i	0.694534	+	0.545931i
$125$			170769.i			0.977538i
$126$		0			0
$127$		−	77161.9i		−	0.424515i	−0.977214	−	0.212258i	$-0.931918\pi$
$127$		−	77161.9i		−	0.424515i	0.977214	−	0.212258i	$-0.0680816\pi$
$128$	185308.	−	4559.32i	0.999697	−	0.0245966i
$129$		0			0
$130$	11739.3	+	24159.4i	0.0609234	+	0.125380i
$131$	309317.			1.57480			0.787401	−	0.616441i	$-0.211426\pi$
$131$	309317.			1.57480			0.787401	+	0.616441i	$0.211426\pi$
$132$		0			0
$133$	−96084.5			−0.471004
$134$	96144.6	+	197865.i	0.462555	+	0.951933i
$135$		0			0
$136$	16596.7	+	3604.44i	0.0769441	+	0.0167106i
$137$			69310.1i			0.315497i	0.987479	+	0.157749i	$0.0504236\pi$
$137$			69310.1i			0.315497i	−0.987479	+	0.157749i	$0.949576\pi$
$138$		0			0
$139$		−	327514.i		−	1.43778i	−0.695122	−	0.718891i	$-0.744650\pi$
$139$		−	327514.i		−	1.43778i	0.695122	−	0.718891i	$-0.255350\pi$
$140$	38948.8	−	49550.7i	0.167948	−	0.213663i
$141$		0			0
$142$	230372.	−	111940.i	0.958758	−	0.465871i
$143$	−25656.0			−0.104918
$144$		0			0
$145$	−262250.			−1.03584
$146$	−106850.	+	51919.6i	−0.414851	+	0.201581i
$147$		0			0
$148$	185633.	−	236163.i	0.696628	−	0.886251i
$149$			317624.i			1.17205i	0.810292	+	0.586026i	$0.199308\pi$
$149$			317624.i			1.17205i	−0.810292	+	0.586026i	$0.800692\pi$
$150$		0			0
$151$		−	32871.2i		−	0.117320i	−0.998278	−	0.0586602i	$-0.981317\pi$
$151$		−	32871.2i		−	0.117320i	0.998278	−	0.0586602i	$-0.0186829\pi$
$152$	286109.	+	62136.5i	1.00444	+	0.218141i
$153$		0			0
$154$	26310.2	+	54146.1i	0.0893968	+	0.183978i
$155$	−156711.			−0.523925
$156$		0			0
$157$	−292389.			−0.946700			−0.473350	−	0.880874i	$-0.656955\pi$
$157$	−292389.			−0.946700			−0.473350	+	0.880874i	$0.656955\pi$
$158$	−189034.	−	389031.i	−0.602418	−	1.23977i
$159$		0			0
$160$	−148021.	+	122358.i	−0.457112	+	0.377862i
$161$		−	143438.i		−	0.436112i
$162$		0			0
$163$		−	324860.i		−	0.957695i	−0.877898	−	0.478848i	$-0.841055\pi$
$163$		−	324860.i		−	0.957695i	0.877898	−	0.478848i	$-0.158945\pi$
$164$	297835.	+	234110.i	0.864701	+	0.679689i
$165$		0			0
$166$	297728.	−	144669.i	0.838589	−	0.407480i
$167$	563689.			1.56404			0.782021	−	0.623251i	$-0.214189\pi$
$167$	563689.			1.56404			0.782021	+	0.623251i	$0.214189\pi$
$168$		0			0
$169$	−350781.			−0.944754
$170$	−15826.4	+	7690.23i	−0.0420010	+	0.0204088i
$171$		0			0
$172$	491384.	+	386247.i	1.26649	+	0.995508i
$173$		−	126447.i		−	0.321214i	−0.987018	−	0.160607i	$-0.948655\pi$
$173$		−	126447.i		−	0.321214i	0.987018	−	0.160607i	$-0.0513452\pi$
$174$		0			0
$175$		−	120350.i		−	0.297064i
$176$	−43327.7	−	178244.i	−0.105435	−	0.433744i
$177$		0			0
$178$	45882.4	+	94425.4i	0.108541	+	0.223377i
$179$	−719928.			−1.67941			−0.839705	−	0.543043i	$-0.817272\pi$
$179$	−719928.			−1.67941			−0.839705	+	0.543043i	$0.817272\pi$
$180$		0			0
$181$	−51023.8			−0.115765			−0.0578823	−	0.998323i	$-0.518435\pi$
$181$	−51023.8			−0.115765			−0.0578823	+	0.998323i	$0.518435\pi$
$182$	−21035.5	−	43290.7i	−0.0470732	−	0.0968761i
$183$		0			0
$184$	−92759.1	+	427111.i	−0.201982	+	0.930028i
$185$			311216.i			0.668548i
$186$		0			0
$187$		−	16806.8i		−	0.0351465i
$188$	145808.	−	185497.i	0.300875	−	0.382774i
$189$		0			0
$190$	−272829.	+	132571.i	−0.548285	+	0.266418i
$191$	−94246.4			−0.186931			−0.0934655	−	0.995623i	$-0.529794\pi$
$191$	−94246.4			−0.186931			−0.0934655	+	0.995623i	$0.529794\pi$
$192$		0			0
$193$	−17335.4			−0.0334997			−0.0167498	−	0.999860i	$-0.505332\pi$
$193$	−17335.4			−0.0334997			−0.0167498	+	0.999860i	$0.505332\pi$
$194$	−698316.	+	339319.i	−1.33213	+	0.647298i
$195$		0			0
$196$	262573.	−	334045.i	0.488213	−	0.621105i
$197$		−	680902.i		−	1.25003i	−0.780614	−	0.625013i	$-0.785094\pi$
$197$		−	680902.i		−	1.25003i	0.780614	−	0.625013i	$-0.214906\pi$
$198$		0			0
$199$			320286.i			0.573331i	0.958031	+	0.286665i	$0.0925468\pi$
$199$			320286.i			0.573331i	−0.958031	+	0.286665i	$0.907453\pi$
$200$	−77828.4	+	358362.i	−0.137582	+	0.633501i
$201$		0			0
$202$	240603.	+	495159.i	0.414880	+	0.853819i
$203$	469920.			0.800357
$204$		0			0
$205$	−392488.			−0.652291
$206$	−95664.5	−	196877.i	−0.157066	−	0.323241i
$207$		0			0
$208$	34641.3	+	142509.i	0.0555182	+	0.228394i
$209$		−	289731.i		−	0.458806i
$210$		0			0
$211$			1.09747e6i			1.69702i	0.529178	+	0.848511i	$0.322500\pi$
$211$			1.09747e6i			1.69702i	−0.529178	+	0.848511i	$0.677500\pi$
$212$	−676640.	−	531865.i	−1.03399	−	0.812760i
$213$		0			0
$214$	218186.	−	106019.i	0.325680	−	0.158252i
$215$	−647548.			−0.955380
$216$		0			0
$217$	280807.			0.404816
$218$	212845.	−	103424.i	0.303334	−	0.147393i
$219$		0			0
$220$	149414.	+	117445.i	0.208130	+	0.163598i
$221$			13437.4i			0.0185069i
$222$		0			0
$223$		−	1.14571e6i		−	1.54281i	−0.636344	−	0.771406i	$-0.719554\pi$
$223$		−	1.14571e6i		−	1.54281i	0.636344	−	0.771406i	$-0.280446\pi$
$224$	265235.	−	219252.i	0.353193	−	0.291960i
$225$		0			0
$226$	−76140.3	−	156696.i	−0.0991615	−	0.204073i
$227$	−827252.			−1.06555			−0.532774	−	0.846257i	$-0.678851\pi$
$227$	−827252.			−1.06555			−0.532774	+	0.846257i	$0.678851\pi$
$228$		0			0
$229$	800776.			1.00907			0.504536	−	0.863390i	$-0.331663\pi$
$229$	800776.			1.00907			0.504536	+	0.863390i	$0.331663\pi$
$230$	−197905.	−	407287.i	−0.246682	−	0.507669i
$231$		0			0
$232$	−1.39927e6	−	303891.i	−1.70680	−	0.370678i
$233$		−	1.65496e6i		−	1.99709i	−0.0539626	−	0.998543i	$-0.517185\pi$
$233$		−	1.65496e6i		−	1.99709i	0.0539626	−	0.998543i	$-0.482815\pi$
$234$		0			0
$235$			244449.i			0.288747i
$236$	121512.	−	154588.i	0.142017	−	0.180674i
$237$		0			0
$238$	28359.0	−	13780.0i	0.0324526	−	0.0157691i
$239$	924466.			1.04688			0.523439	−	0.852063i	$-0.324649\pi$
$239$	924466.			1.04688			0.523439	+	0.852063i	$0.324649\pi$
$240$		0			0
$241$	−108394.			−0.120216			−0.0601082	−	0.998192i	$-0.519145\pi$
$241$	−108394.			−0.120216			−0.0601082	+	0.998192i	$0.519145\pi$
$242$	656156.	−	318833.i	0.720225	−	0.349966i
$243$		0			0
$244$	1.09263e6	−	1.39005e6i	1.17490	−	1.49471i
$245$			440206.i			0.468533i
$246$		0			0
$247$			231645.i			0.241591i
$248$	−836151.	−	181594.i	−0.863288	−	0.187487i
$249$		0			0
$250$	−422194.	−	868871.i	−0.427230	−	0.879235i
$251$	−665770.			−0.667022			−0.333511	−	0.942746i	$-0.608233\pi$
$251$	−665770.			−0.667022			−0.333511	+	0.942746i	$0.608233\pi$
$252$		0			0
$253$	432518.			0.424818
$254$	190768.	+	392599.i	0.185533	+	0.381826i
$255$		0			0
$256$	−931572.	+	481336.i	−0.888417	+	0.459038i
$257$			1.17372e6i			1.10849i	0.832354	+	0.554244i	$0.186993\pi$
$257$			1.17372e6i			1.10849i	−0.832354	+	0.554244i	$0.813007\pi$
$258$		0			0
$259$		−	557662.i		−	0.516561i
$260$	−119459.	−	93899.5i	−0.109594	−	0.0861450i
$261$		0			0
$262$	−1.57381e6	+	764729.i	−1.41644	+	0.688263i
$263$	307275.			0.273929			0.136965	−	0.990576i	$-0.456265\pi$
$263$	307275.			0.273929			0.136965	+	0.990576i	$0.456265\pi$
$264$		0			0
$265$	891678.			0.779998
$266$	488877.	−	237551.i	0.423639	−	0.205851i
$267$		0			0
$268$	−978366.	−	769034.i	−0.832079	−	0.654047i
$269$			1.40037e6i			1.17994i	0.807424	+	0.589972i	$0.200861\pi$
$269$			1.40037e6i			1.17994i	−0.807424	+	0.589972i	$0.799139\pi$
$270$		0			0
$271$			637013.i			0.526896i	0.964674	+	0.263448i	$0.0848598\pi$
$271$			637013.i			0.526896i	−0.964674	+	0.263448i	$0.915140\pi$
$272$	−93355.3	+	22692.9i	−0.0765098	+	0.0185981i
$273$		0			0
$274$	−171356.	−	352650.i	−0.137887	−	0.283770i
$275$	362899.			0.289370
$276$		0			0
$277$	820888.			0.642813			0.321406	−	0.946941i	$-0.395844\pi$
$277$	820888.			0.642813			0.321406	+	0.946941i	$0.395844\pi$
$278$	809718.	+	1.66639e6i	0.628379	+	1.29320i
$279$		0			0
$280$	−75666.4	+	348407.i	−0.0576777	+	0.265578i
$281$			446685.i			0.337470i	0.985661	+	0.168735i	$0.0539683\pi$
$281$			446685.i			0.337470i	−0.985661	+	0.168735i	$0.946032\pi$
$282$		0			0
$283$		−	270685.i		−	0.200908i	−0.994942	−	0.100454i	$-0.967970\pi$
$283$		−	270685.i		−	0.200908i	0.994942	−	0.100454i	$-0.0320296\pi$
$284$	−895380.	+	1.13910e6i	−0.658736	+	0.838045i
$285$		0			0
$286$	130538.	−	63429.8i	0.0943672	−	0.0458541i
$287$	703291.			0.504000
$288$		0			0
$289$	1.41105e6			0.993800
$290$	1.33432e6	−	648363.i	0.931679	−	0.452713i
$291$		0			0
$292$	415290.	−	528333.i	0.285033	−	0.362619i
$293$		−	384363.i		−	0.261561i	−0.991411	−	0.130781i	$-0.958252\pi$
$293$		−	384363.i		−	0.261561i	0.991411	−	0.130781i	$-0.0417483\pi$
$294$		0			0
$295$			203717.i			0.136293i
$296$	−360632.	+	1.66054e6i	−0.239241	+	1.10159i
$297$		0			0
$298$	−785265.	−	1.61607e6i	−0.512242	−	1.05419i
$299$	−345806.			−0.223694
$300$		0			0
$301$	1.16033e6			0.738185
$302$	81268.0	+	167249.i	0.0512746	+	0.105523i
$303$		0			0
$304$	−1.60934e6	+	391200.i	−0.998766	+	0.242781i
$305$			1.83181e6i			1.12754i
$306$		0			0
$307$			2.20012e6i			1.33229i	0.745820	+	0.666147i	$0.232058\pi$
$307$			2.20012e6i			1.33229i	−0.745820	+	0.666147i	$0.767942\pi$
$308$	−267732.	−	210448.i	−0.160814	−	0.126406i
$309$		0			0
$310$	797342.	−	387437.i	0.471238	−	0.228980i
$311$	2.43025e6			1.42479			0.712395	−	0.701779i	$-0.247611\pi$
$311$	2.43025e6			1.42479			0.712395	+	0.701779i	$0.247611\pi$
$312$		0			0
$313$	406568.			0.234570			0.117285	−	0.993098i	$-0.462581\pi$
$313$	406568.			0.234570			0.117285	+	0.993098i	$0.462581\pi$
$314$	1.48767e6	−	722878.i	0.851498	−	0.413753i
$315$		0			0
$316$	1.92361e6	+	1.51203e6i	1.08368	+	0.851812i
$317$		−	2.49960e6i		−	1.39709i	−0.715568	−	0.698543i	$-0.753832\pi$
$317$		−	2.49960e6i		−	1.39709i	0.715568	−	0.698543i	$-0.246168\pi$
$318$		0			0
$319$			1.41698e6i			0.779630i
$320$	450620.	−	988512.i	0.246000	−	0.539644i
$321$		0			0
$322$	354623.	+	729809.i	0.190602	+	0.392256i
$323$	−151747.			−0.0809306
$324$		0			0
$325$	−290144.			−0.152372
$326$	803155.	+	1.65289e6i	0.418558	+	0.861388i
$327$		0			0
$328$	−2.09417e6	−	454809.i	−1.07480	−	0.233423i
$329$		−	438023.i		−	0.223104i
$330$		0			0
$331$			2.55857e6i			1.28359i	0.766875	+	0.641797i	$0.221811\pi$
$331$			2.55857e6i			1.28359i	−0.766875	+	0.641797i	$0.778189\pi$
$332$	−1.15717e6	+	1.47215e6i	−0.576171	+	0.733006i
$333$		0			0
$334$	−2.86805e6	+	1.39362e6i	−1.40676	+	0.683561i
$335$	1.28929e6			0.627683
$336$		0			0
$337$	320043.			0.153509			0.0767544	−	0.997050i	$-0.475544\pi$
$337$	320043.			0.153509			0.0767544	+	0.997050i	$0.475544\pi$
$338$	1.78477e6	−	867239.i	0.849748	−	0.412902i
$339$		0			0
$340$	61512.0	−	78255.6i	0.0288578	−	0.0367129i
$341$			846737.i			0.394332i
$342$		0			0
$343$		−	1.78725e6i		−	0.820259i
$344$	−3.45509e6	−	750369.i	−1.57421	−	0.341884i
$345$		0			0
$346$	312617.	+	643363.i	0.140386	+	0.288912i
$347$	−3.56951e6			−1.59142			−0.795710	−	0.605678i	$-0.792902\pi$
$347$	−3.56951e6			−1.59142			−0.795710	+	0.605678i	$0.792902\pi$
$348$		0			0
$349$	−319652.			−0.140480			−0.0702400	−	0.997530i	$-0.522377\pi$
$349$	−319652.			−0.140480			−0.0702400	+	0.997530i	$0.522377\pi$
$350$	297542.	+	612338.i	0.129831	+	0.267191i
$351$		0			0
$352$	661126.	+	799784.i	0.284399	+	0.344046i
$353$			1.78942e6i			0.764320i	0.924096	+	0.382160i	$0.124820\pi$
$353$			1.78942e6i			0.764320i	−0.924096	+	0.382160i	$0.875180\pi$
$354$		0			0
$355$		−	1.50111e6i		−	0.632183i
$356$	−466898.	−	367000.i	−0.195253	−	0.153476i
$357$		0			0
$358$	3.66299e6	−	1.77989e6i	1.51053	−	0.733981i
$359$	1.91480e6			0.784130			0.392065	−	0.919937i	$-0.371761\pi$
$359$	1.91480e6			0.784130			0.392065	+	0.919937i	$0.371761\pi$
$360$		0			0
$361$	−139839.			−0.0564754
$362$	259609.	−	126147.i	0.104123	−	0.0505946i
$363$		0			0
$364$	214056.	+	168257.i	0.0846788	+	0.0665609i
$365$			696239.i			0.273543i
$366$		0			0
$367$		−	882321.i		−	0.341949i	−0.985275	−	0.170974i	$-0.945308\pi$
$367$		−	882321.i		−	0.341949i	0.985275	−	0.170974i	$-0.0546916\pi$
$368$	−583994.	−	2.40247e6i	−0.224796	−	0.924778i
$369$		0			0
$370$	−769423.	−	1.58346e6i	−0.292187	−	0.601318i
$371$	−1.59778e6			−0.602674
$372$		0			0
$373$	−2.86790e6			−1.06731			−0.533657	−	0.845701i	$-0.679183\pi$
$373$	−2.86790e6			−1.06731			−0.533657	+	0.845701i	$0.679183\pi$
$374$	41551.8	+	85513.1i	0.0153607	+	0.0316121i
$375$		0			0
$376$	−283263.	+	1.30429e6i	−0.103329	+	0.475778i
$377$		−	1.13290e6i		−	0.410525i
$378$		0			0
$379$		−	4.53351e6i		−	1.62120i	−0.585602	−	0.810599i	$-0.699142\pi$
$379$		−	4.53351e6i		−	1.62120i	0.585602	−	0.810599i	$-0.300858\pi$
$380$	1.06040e6	−	1.34904e6i	0.376712	−	0.479253i
$381$		0			0
$382$	479525.	−	233007.i	0.168133	−	0.0816977i
$383$	−1.75464e6			−0.611210			−0.305605	−	0.952158i	$-0.598859\pi$
$383$	−1.75464e6			−0.611210			−0.305605	+	0.952158i	$0.598859\pi$
$384$		0			0
$385$	352818.			0.121311
$386$	88202.4	−	42858.5i	0.0301309	−	0.0146409i
$387$		0			0
$388$	2.71412e6	−	3.45291e6i	0.915272	−	1.16441i
$389$		−	1.84834e6i		−	0.619311i	−0.950849	−	0.309656i	$-0.899786\pi$
$389$		−	1.84834e6i		−	0.619311i	0.950849	−	0.309656i	$-0.100214\pi$
$390$		0			0
$391$		−	226531.i		−	0.0749353i
$392$	−510104.	+	2.34878e6i	−0.167665	+	0.772018i
$393$		0			0
$394$	1.68340e6	+	3.46443e6i	0.546321	+	1.12432i
$395$	−2.53494e6			−0.817476
$396$		0			0
$397$	−1.30700e6			−0.416198			−0.208099	−	0.978108i	$-0.566728\pi$
$397$	−1.30700e6			−0.416198			−0.208099	+	0.978108i	$0.566728\pi$
$398$	−791847.	−	1.62961e6i	−0.250573	−	0.515676i
$399$		0			0
$400$	−489993.	−	2.01576e6i	−0.153123	−	0.629925i
$401$		−	3.45140e6i		−	1.07185i	−0.844265	−	0.535925i	$-0.819963\pi$
$401$		−	3.45140e6i		−	1.07185i	0.844265	−	0.535925i	$-0.180037\pi$
$402$		0			0
$403$		−	676981.i		−	0.207641i
$404$	−2.44837e6	−	1.92452e6i	−0.746319	−	0.586636i
$405$		0			0
$406$	−2.39095e6	+	1.16179e6i	−0.719872	+	0.349794i
$407$	1.68156e6			0.503183
$408$		0			0
$409$	−5.83836e6			−1.72577			−0.862884	−	0.505402i	$-0.831344\pi$
$409$	−5.83836e6			−1.72577			−0.862884	+	0.505402i	$0.831344\pi$
$410$	1.99697e6	−	970352.i	0.586696	−	0.285082i
$411$		0			0
$412$	973481.	+	765194.i	0.282543	+	0.222090i
$413$		−	365036.i		−	0.105308i
$414$		0			0
$415$		−	1.94001e6i		−	0.552947i
$416$	−528582.	−	639442.i	−0.149754	−	0.181162i
$417$		0			0
$418$	716305.	+	1.47415e6i	0.200520	+	0.412668i
$419$	−5.29746e6			−1.47412			−0.737060	−	0.675827i	$-0.763786\pi$
$419$	−5.29746e6			−1.47412			−0.737060	+	0.675827i	$0.763786\pi$
$420$		0			0
$421$	−631619.			−0.173680			−0.0868400	−	0.996222i	$-0.527677\pi$
$421$	−631619.			−0.173680			−0.0868400	+	0.996222i	$0.527677\pi$
$422$	−2.71329e6	−	5.58393e6i	−0.741679	−	1.52637i
$423$		0			0
$424$	4.75768e6	+	1.03326e6i	1.28523	+	0.279123i
$425$		−	190069.i		−	0.0510432i
$426$		0			0
$427$		−	3.28239e6i		−	0.871205i
$428$	−848015.	+	1.07885e6i	−0.223766	+	0.284675i
$429$		0			0
$430$	3.29472e6	−	1.60094e6i	0.859305	−	0.417546i
$431$	6.01340e6			1.55929			0.779645	−	0.626222i	$-0.215400\pi$
$431$	6.01340e6			1.55929			0.779645	+	0.626222i	$0.215400\pi$
$432$		0			0
$433$	−5.60717e6			−1.43722			−0.718611	−	0.695412i	$-0.755222\pi$
$433$	−5.60717e6			−1.43722			−0.718611	+	0.695412i	$0.755222\pi$
$434$	−1.42874e6	+	694242.i	−0.364107	+	0.176924i
$435$		0			0
$436$	−827256.	+	1.05244e6i	−0.208413	+	0.265143i
$437$		−	3.90514e6i		−	0.978213i
$438$		0			0
$439$			7.08822e6i			1.75540i	0.479211	+	0.877700i	$0.340923\pi$
$439$			7.08822e6i			1.75540i	−0.479211	+	0.877700i	$0.659077\pi$
$440$	−1.05058e6	−	228163.i	−0.258700	−	0.0561840i
$441$		0			0
$442$	−33221.4	−	68369.2i	−0.00808838	−	0.0166458i
$443$	405533.			0.0981786			0.0490893	−	0.998794i	$-0.484368\pi$
$443$	405533.			0.0981786			0.0490893	+	0.998794i	$0.484368\pi$
$444$		0			0
$445$	615280.			0.147290
$446$	2.83255e6	+	5.82937e6i	0.674281	+	1.38766i
$447$		0			0
$448$	−807458.	+	1.77130e6i	−0.190075	+	0.416962i
$449$		−	5.81570e6i		−	1.36140i	−0.732561	−	0.680701i	$-0.761675\pi$
$449$		−	5.81570e6i		−	1.36140i	0.732561	−	0.680701i	$-0.238325\pi$
$450$		0			0
$451$			2.12069e6i			0.490947i
$452$	774802.	+	609025.i	0.178379	+	0.140213i
$453$		0			0
$454$	4.20905e6	−	2.04523e6i	0.958395	−	0.465695i
$455$	−282084.			−0.0638779
$456$		0			0
$457$	1.51155e6			0.338558			0.169279	−	0.985568i	$-0.445856\pi$
$457$	1.51155e6			0.338558			0.169279	+	0.985568i	$0.445856\pi$
$458$	−4.07434e6	+	1.97977e6i	−0.907599	+	0.441012i
$459$		0			0
$460$	2.01388e6	+	1.58299e6i	0.443751	+	0.348805i
$461$			8.52192e6i			1.86761i	0.357787	+	0.933803i	$0.383531\pi$
$461$			8.52192e6i			1.86761i	−0.357787	+	0.933803i	$0.616469\pi$
$462$		0			0
$463$		−	5.84253e6i		−	1.26663i	−0.773896	−	0.633313i	$-0.781695\pi$
$463$		−	5.84253e6i		−	1.26663i	0.773896	−	0.633313i	$-0.218305\pi$
$464$	7.87079e6	−	1.91324e6i	1.69716	−	0.412548i
$465$		0			0
$466$	4.09157e6	+	8.42041e6i	0.872821	+	1.79626i
$467$	−1.60988e6			−0.341586			−0.170793	−	0.985307i	$-0.554633\pi$
$467$	−1.60988e6			−0.341586			−0.170793	+	0.985307i	$0.554633\pi$
$468$		0			0
$469$	−2.31026e6			−0.484986
$470$	−604353.	−	1.24375e6i	−0.126196	−	0.259710i
$471$		0			0
$472$	−236064.	+	1.08696e6i	−0.0487724	+	0.224574i
$473$			3.49883e6i			0.719067i
$474$		0			0
$475$		−	3.27656e6i		−	0.666323i
$476$	−110222.	+	140225.i	−0.0222973	+	0.0283666i
$477$		0			0
$478$	−4.70367e6	+	2.28557e6i	−0.941602	+	0.457535i
$479$	4.03682e6			0.803898			0.401949	−	0.915662i	$-0.368333\pi$
$479$	4.03682e6			0.803898			0.401949	+	0.915662i	$0.368333\pi$
$480$		0			0
$481$	−1.34444e6			−0.264958
$482$	551509.	−	267984.i	0.108127	−	0.0525402i
$483$		0			0
$484$	−2.55026e6	+	3.24444e6i	−0.494847	+	0.629545i
$485$			4.55026e6i			0.878378i
$486$		0			0
$487$			6.28957e6i			1.20171i	0.799359	+	0.600854i	$0.205173\pi$
$487$			6.28957e6i			1.20171i	−0.799359	+	0.600854i	$0.794827\pi$
$488$	−2.12268e6	+	9.77390e6i	−0.403491	+	1.85788i
$489$		0			0
$490$	−1.08833e6	−	2.23976e6i	−0.204771	−	0.421417i
$491$	−5.26683e6			−0.985928			−0.492964	−	0.870050i	$-0.664087\pi$
$491$	−5.26683e6			−0.985928			−0.492964	+	0.870050i	$0.664087\pi$
$492$		0			0
$493$	742146.			0.137522
$494$	−572698.	−	1.17861e6i	−0.105587	−	0.217296i
$495$		0			0
$496$	4.70329e6	−	1.14328e6i	0.858415	−	0.208664i
$497$			2.68982e6i			0.488463i
$498$		0			0
$499$			2.24696e6i			0.403966i	0.979389	+	0.201983i	$0.0647385\pi$
$499$			2.24696e6i			0.403966i	−0.979389	+	0.201983i	$0.935261\pi$
$500$	4.29624e6	+	3.37701e6i	0.768535	+	0.604098i
$501$		0			0
$502$	3.38744e6	−	1.64599e6i	0.599945	−	0.291520i
$503$	7.96470e6			1.40362			0.701809	−	0.712365i	$-0.252376\pi$
$503$	7.96470e6			1.40362			0.701809	+	0.712365i	$0.252376\pi$
$504$		0			0
$505$	3.22648e6			0.562989
$506$	−2.20065e6	+	1.06932e6i	−0.382098	+	0.185665i
$507$		0			0
$508$	−1.94125e6	−	1.52590e6i	−0.333752	−	0.262342i
$509$		−	2.49761e6i		−	0.427297i	−0.976911	−	0.213648i	$-0.931465\pi$
$509$		−	2.49761e6i		−	0.427297i	0.976911	−	0.213648i	$-0.0685347\pi$
$510$		0			0
$511$		−	1.24758e6i		−	0.211356i
$512$	3.54982e6	−	4.75217e6i	0.598455	−	0.801157i
$513$		0			0
$514$	−2.90180e6	−	5.97187e6i	−0.484462	−	0.997017i
$515$	−1.28286e6			−0.213138
$516$		0			0
$517$	1.32080e6			0.217326
$518$	1.37871e6	+	2.83738e6i	0.225761	+	0.464615i
$519$		0			0
$520$	839956.	+	182420.i	0.136222	+	0.0295845i
$521$			9.44582e6i			1.52456i	0.647246	+	0.762281i	$0.275921\pi$
$521$			9.44582e6i			1.52456i	−0.647246	+	0.762281i	$0.724079\pi$
$522$		0			0
$523$		−	3.14550e6i		−	0.502846i	−0.967877	−	0.251423i	$-0.919101\pi$
$523$		−	3.14550e6i		−	0.502846i	0.967877	−	0.251423i	$-0.0808986\pi$
$524$	6.11686e6	−	7.78188e6i	0.973196	−	1.23810i
$525$		0			0
$526$	−1.56341e6	+	759680.i	−0.246382	+	0.119720i
$527$	443479.			0.0695579
$528$		0			0
$529$	−606637.			−0.0942519
$530$	−4.53685e6	+	2.20451e6i	−0.701560	+	0.340896i
$531$		0			0
$532$	−1.90010e6	+	2.41731e6i	−0.291071	+	0.370300i
$533$		−	1.69553e6i		−	0.258515i
$534$		0			0
$535$		−	1.42171e6i		−	0.214746i
$536$	6.87922e6	+	1.49401e6i	1.03425	+	0.224617i
$537$		0			0
$538$	−3.46215e6	−	7.12506e6i	−0.515691	−	1.06129i
$539$	2.37852e6			0.352642
$540$		0			0
$541$	−1.17569e6			−0.172703			−0.0863515	−	0.996265i	$-0.527521\pi$
$541$	−1.17569e6			−0.172703			−0.0863515	+	0.996265i	$0.527521\pi$
$542$	−1.57490e6	−	3.24112e6i	−0.230279	−	0.473911i
$543$		0			0
$544$	418887.	−	346265.i	0.0606876	−	0.0501662i
$545$		−	1.38690e6i		−	0.200012i
$546$		0			0
$547$			1.69619e6i			0.242386i	0.992629	+	0.121193i	$0.0386719\pi$
$547$			1.69619e6i			0.242386i	−0.992629	+	0.121193i	$0.961328\pi$
$548$	1.74372e6	+	1.37063e6i	0.248042	+	0.194971i
$549$		0			0
$550$	−1.84643e6	+	897200.i	−0.260271	+	0.126469i
$551$	1.27938e7			1.79523
$552$		0			0
$553$	4.54231e6			0.631632
$554$	−4.17667e6	+	2.02949e6i	−0.578171	+	0.280939i
$555$		0			0
$556$	−8.23968e6	−	6.47671e6i	−1.13038	−	0.888520i
$557$		−	713498.i		−	0.0974440i	−0.998812	−	0.0487220i	$-0.984485\pi$
$557$		−	713498.i		−	0.0974440i	0.998812	−	0.0487220i	$-0.0155148\pi$
$558$		0			0
$559$		−	2.79737e6i		−	0.378635i
$560$	−476382.	−	1.95977e6i	−0.0641926	−	0.264079i
$561$		0			0
$562$	−1.10435e6	−	2.27273e6i	−0.147491	−	0.303534i
$563$	−1.28328e7			−1.70627			−0.853137	−	0.521687i	$-0.825303\pi$
$563$	−1.28328e7			−1.70627			−0.853137	+	0.521687i	$0.825303\pi$
$564$		0			0
$565$	−1.02104e6			−0.134561
$566$	669217.	+	1.37724e6i	0.0878064	+	0.180705i
$567$		0			0
$568$	1.73947e6	−	8.00941e6i	0.226227	−	1.04167i
$569$		−	7.48951e6i		−	0.969779i	−0.874575	−	0.484890i	$-0.838860\pi$
$569$		−	7.48951e6i		−	0.969779i	0.874575	−	0.484890i	$-0.161140\pi$
$570$		0			0
$571$			8.20920e6i			1.05368i	0.849963	+	0.526842i	$0.176624\pi$
$571$			8.20920e6i			1.05368i	−0.849963	+	0.526842i	$0.823376\pi$
$572$	−507357.	+	645460.i	−0.0648371	+	0.0824859i
$573$		0			0
$574$	−3.57834e6	+	1.73876e6i	−0.453317	+	0.220272i
$575$	4.89135e6			0.616962
$576$		0			0
$577$	1.12522e7			1.40701			0.703505	−	0.710691i	$-0.251617\pi$
$577$	1.12522e7			1.40701			0.703505	+	0.710691i	$0.251617\pi$
$578$	−7.17943e6	+	3.48857e6i	−0.893862	+	0.434338i
$579$		0			0
$580$	−5.18607e6	+	6.59773e6i	−0.640131	+	0.814375i
$581$			3.47626e6i			0.427240i
$582$		0			0
$583$		−	4.81791e6i		−	0.587066i
$584$	−806791.	+	3.71488e6i	−0.0978879	+	0.450726i
$585$		0			0
$586$	950266.	+	1.95564e6i	0.114315	+	0.235258i
$587$	−8.39201e6			−1.00524			−0.502621	−	0.864507i	$-0.667631\pi$
$587$	−8.39201e6			−1.00524			−0.502621	+	0.864507i	$0.667631\pi$
$588$		0			0
$589$	7.64507e6			0.908015
$590$	−503652.	−	1.03651e6i	−0.0595663	−	0.122587i
$591$		0			0
$592$	−2.27047e6	−	9.34039e6i	−0.266264	−	1.09537i
$593$			5.97649e6i			0.697926i	0.937137	+	0.348963i	$0.113466\pi$
$593$			5.97649e6i			0.697926i	−0.937137	+	0.348963i	$0.886534\pi$
$594$		0			0
$595$		−	184789.i		−	0.0213985i
$596$	7.99084e6	+	6.28112e6i	0.921461	+	0.724305i
$597$		0			0
$598$	1.75946e6	−	854940.i	0.201199	−	0.0977648i
$599$	−2.33722e6			−0.266153			−0.133077	−	0.991106i	$-0.542486\pi$
$599$	−2.33722e6			−0.266153			−0.133077	+	0.991106i	$0.542486\pi$
$600$		0			0
$601$	5.95099e6			0.672052			0.336026	−	0.941853i	$-0.390917\pi$
$601$	5.95099e6			0.672052			0.336026	+	0.941853i	$0.390917\pi$
$602$	−5.90375e6	+	2.86870e6i	−0.663952	+	0.322622i
$603$		0			0
$604$	−826982.	−	650040.i	−0.0922367	−	0.0725016i
$605$		−	4.27554e6i		−	0.474900i
$606$		0			0
$607$			1.38582e7i			1.52663i	0.646026	+	0.763316i	$0.276430\pi$
$607$			1.38582e7i			1.52663i	−0.646026	+	0.763316i	$0.723570\pi$
$608$	7.22114e6	−	5.96921e6i	0.792222	−	0.654874i
$609$		0			0
$610$	−4.52881e6	−	9.32025e6i	−0.492788	−	1.01415i
$611$	−1.05600e6			−0.114436
$612$		0			0
$613$	8.69155e6			0.934214			0.467107	−	0.884201i	$-0.345296\pi$
$613$	8.69155e6			0.934214			0.467107	+	0.884201i	$0.345296\pi$
$614$	−5.43938e6	−	1.11942e7i	−0.582276	−	1.19832i
$615$		0			0
$616$	1.88251e6	+	408840.i	0.199888	+	0.0434112i
$617$		−	1.37669e7i		−	1.45587i	−0.685646	−	0.727936i	$-0.740480\pi$
$617$		−	1.37669e7i		−	1.45587i	0.685646	−	0.727936i	$-0.259520\pi$
$618$		0			0
$619$		−	5.73601e6i		−	0.601704i	−0.953671	−	0.300852i	$-0.902729\pi$
$619$		−	5.73601e6i		−	0.601704i	0.953671	−	0.300852i	$-0.0972710\pi$
$620$	−3.09900e6	+	3.94256e6i	−0.323775	+	0.411907i
$621$		0			0
$622$	−1.23651e7	+	6.00834e6i	−1.28151	+	0.622700i
$623$	−1.10251e6			−0.113805
$624$		0			0
$625$	669150.			0.0685209
$626$	−2.06862e6	+	1.00516e6i	−0.210981	+	0.102518i
$627$		0			0
$628$	−5.78210e6	+	7.35599e6i	−0.585041	+	0.744290i
$629$		−	880717.i		−	0.0887585i
$630$		0			0
$631$			3.53427e6i			0.353368i	0.984268	+	0.176684i	$0.0565370\pi$
$631$			3.53427e6i			0.353368i	−0.984268	+	0.176684i	$0.943463\pi$
$632$	−1.35255e7	−	2.93745e6i	−1.34698	−	0.292535i
$633$		0			0
$634$	6.17980e6	+	1.27180e7i	0.610592	+	1.25659i
$635$	2.55819e6			0.251767
$636$		0			0
$637$	−1.90167e6			−0.185689
$638$	−3.50323e6	−	7.20961e6i	−0.340735	−	0.701229i
$639$		0			0
$640$	151158.	+	6.14362e6i	0.0145875	+	0.592890i
$641$			717761.i			0.0689978i	0.999405	+	0.0344989i	$0.0109835\pi$
$641$			717761.i			0.0689978i	−0.999405	+	0.0344989i	$0.989016\pi$
$642$		0			0
$643$			7.19240e6i			0.686035i	0.939329	+	0.343017i	$0.111449\pi$
$643$			7.19240e6i			0.686035i	−0.939329	+	0.343017i	$0.888551\pi$
$644$	−3.60863e6	−	2.83653e6i	−0.342869	−	0.269508i
$645$		0			0
$646$	772086.	−	375165.i	0.0727921	−	0.0353705i
$647$	−1.35541e6			−0.127294			−0.0636472	−	0.997972i	$-0.520273\pi$
$647$	−1.35541e6			−0.127294			−0.0636472	+	0.997972i	$0.520273\pi$
$648$		0			0
$649$	1.10072e6			0.102581
$650$	1.47625e6	−	717327.i	0.137049	−	0.0665938i
$651$		0			0
$652$	−8.17290e6	−	6.42422e6i	−0.752934	−	0.591836i
$653$		−	7.90801e6i		−	0.725745i	−0.931839	−	0.362873i	$-0.881796\pi$
$653$		−	7.90801e6i		−	0.725745i	0.931839	−	0.362873i	$-0.118204\pi$
$654$		0			0
$655$			1.02550e7i			0.933967i
$656$	1.17796e7	−	2.86339e6i	1.06873	−	0.259789i
$657$		0			0
$658$	1.08293e6	+	2.22866e6i	0.0975068	+	0.200668i
$659$	−1.60268e7			−1.43758			−0.718792	−	0.695226i	$-0.755304\pi$
$659$	−1.60268e7			−1.43758			−0.718792	+	0.695226i	$0.755304\pi$
$660$		0			0
$661$	1.22510e7			1.09061			0.545303	−	0.838239i	$-0.316415\pi$
$661$	1.22510e7			1.09061			0.545303	+	0.838239i	$0.316415\pi$
$662$	−6.32558e6	−	1.30180e7i	−0.560991	−	1.15451i
$663$		0			0
$664$	2.24805e6	−	1.03512e7i	0.197873	−	0.911108i
$665$		−	3.18555e6i		−	0.279338i
$666$		0			0
$667$			1.90989e7i			1.66224i
$668$	1.11472e7	−	1.41814e7i	0.966546	−	1.22964i
$669$		0			0
$670$	−6.55992e6	+	3.18754e6i	−0.564562	+	0.274327i
$671$	9.89764e6			0.848643
$672$		0			0
$673$	−8.62568e6			−0.734101			−0.367051	−	0.930201i	$-0.619632\pi$
$673$	−8.62568e6			−0.734101			−0.367051	+	0.930201i	$0.619632\pi$
$674$	−1.62838e6	+	791246.i	−0.138072	+	0.0670906i
$675$		0			0
$676$	−6.93681e6	+	8.82501e6i	−0.583839	+	0.742760i
$677$		−	3.13206e6i		−	0.262638i	−0.991340	−	0.131319i	$-0.958079\pi$
$677$		−	3.13206e6i		−	0.262638i	0.991340	−	0.131319i	$-0.0419213\pi$
$678$		0			0
$679$		−	8.15352e6i		−	0.678689i
$680$	−119500.	+	550241.i	−0.00991052	+	0.0456332i
$681$		0			0
$682$	−2.09340e6	−	4.30819e6i	−0.172342	−	0.354678i
$683$	−1.42490e7			−1.16878			−0.584389	−	0.811473i	$-0.698666\pi$
$683$	−1.42490e7			−1.16878			−0.584389	+	0.811473i	$0.698666\pi$
$684$		0			0
$685$	−2.29788e6			−0.187112
$686$	4.41865e6	+	9.09354e6i	0.358492	+	0.737773i
$687$		0			0
$688$	1.94346e7	−	4.72418e6i	1.56532	−	0.380501i
$689$			3.85200e6i			0.309128i
$690$		0			0
$691$		−	6.95060e6i		−	0.553767i	−0.960904	−	0.276883i	$-0.910698\pi$
$691$		−	6.95060e6i		−	0.553767i	0.960904	−	0.276883i	$-0.0893016\pi$
$692$	−3.18119e6	−	2.50054e6i	−0.252537	−	0.198504i
$693$		0			0
$694$	1.81616e7	−	8.82494e6i	1.43138	−	0.695526i
$695$	1.08583e7			0.852705
$696$		0			0
$697$	1.11071e6			0.0866002
$698$	1.62639e6	−	790281.i	0.126353	−	0.0613964i
$699$		0			0
$700$	−3.02778e6	−	2.37996e6i	−0.233550	−	0.183579i
$701$		−	4.21552e6i		−	0.324008i	−0.986790	−	0.162004i	$-0.948204\pi$
$701$		−	4.21552e6i		−	0.324008i	0.986790	−	0.162004i	$-0.0517958\pi$
$702$		0			0
$703$		−	1.51826e7i		−	1.15866i
$704$	−5.34112e6	−	2.43479e6i	−0.406163	−	0.185152i
$705$		0			0
$706$	−4.42400e6	−	9.10455e6i	−0.334044	−	0.687459i
$707$	−5.78146e6			−0.435000
$708$		0			0
$709$	−5.08494e6			−0.379901			−0.189950	−	0.981794i	$-0.560833\pi$
$709$	−5.08494e6			−0.379901			−0.189950	+	0.981794i	$0.560833\pi$
$710$	3.71122e6	+	7.63766e6i	0.276294	+	0.568610i
$711$		0			0
$712$	3.28291e6	+	712977.i	0.242694	+	0.0527079i
$713$			1.14128e7i			0.840750i
$714$		0			0
$715$		−	850590.i		−	0.0622236i
$716$	−1.42368e7	+	1.81121e7i	−1.03784	+	1.32034i
$717$		0			0
$718$	−9.74251e6	+	4.73399e6i	−0.705277	+	0.342702i
$719$	−2.52087e6			−0.181856			−0.0909281	−	0.995857i	$-0.528983\pi$
$719$	−2.52087e6			−0.181856			−0.0909281	+	0.995857i	$0.528983\pi$
$720$		0			0
$721$	2.29873e6			0.164683
$722$	711498.	−	345725.i	0.0507962	−	0.0246824i
$723$		0			0
$724$	−1.00901e6	+	1.28367e6i	−0.0715402	+	0.0910135i
$725$			1.60247e7i			1.13225i
$726$		0			0
$727$		−	2.28089e7i		−	1.60054i	−0.599637	−	0.800272i	$-0.704688\pi$
$727$		−	2.28089e7i		−	1.60054i	0.599637	−	0.800272i	$-0.295312\pi$
$728$	−1.50510e6	−	326875.i	−0.105254	−	0.0228588i
$729$		0			0
$730$	−1.72132e6	−	3.54246e6i	−0.119551	−	0.246036i
$731$	1.83251e6			0.126839
$732$		0			0
$733$	−4.13592e6			−0.284323			−0.142161	−	0.989843i	$-0.545405\pi$
$733$	−4.13592e6			−0.284323			−0.142161	+	0.989843i	$0.545405\pi$
$734$	2.18137e6	+	4.48924e6i	0.149448	+	0.307562i
$735$		0			0
$736$	8.91100e6	+	1.07799e7i	0.606362	+	0.733535i
$737$		−	6.96631e6i		−	0.472426i
$738$		0			0
$739$			2.06353e6i			0.138995i	0.997582	+	0.0694974i	$0.0221396\pi$
$739$			2.06353e6i			0.138995i	−0.997582	+	0.0694974i	$0.977860\pi$
$740$	7.82964e6	+	6.15440e6i	0.525608	+	0.413149i
$741$		0			0
$742$	8.12950e6	−	3.95021e6i	0.542068	−	0.263397i
$743$	2.14217e7			1.42358			0.711789	−	0.702393i	$-0.247885\pi$
$743$	2.14217e7			1.42358			0.711789	+	0.702393i	$0.247885\pi$
$744$		0			0
$745$	−1.05304e7			−0.695109
$746$	1.45919e7	−	7.09035e6i	0.959984	−	0.466467i
$747$		0			0
$748$	−422830.	−	332361.i	−0.0276320	−	0.0217198i
$749$			2.54753e6i			0.165926i
$750$		0			0
$751$		−	1.18689e7i		−	0.767910i	−0.923352	−	0.383955i	$-0.874562\pi$
$751$		−	1.18689e7i		−	0.767910i	0.923352	−	0.383955i	$-0.125438\pi$
$752$	−1.78337e6	−	7.33653e6i	−0.115000	−	0.473093i
$753$		0			0
$754$	2.80089e6	+	5.76421e6i	0.179419	+	0.369242i
$755$	1.08980e6			0.0695792
$756$		0			0
$757$	−1.89960e7			−1.20482			−0.602412	−	0.798186i	$-0.705793\pi$
$757$	−1.89960e7			−1.20482			−0.602412	+	0.798186i	$0.705793\pi$
$758$	1.12082e7	+	2.30664e7i	0.708540	+	1.45817i
$759$		0			0
$760$	−2.06005e6	+	9.48553e6i	−0.129373	+	0.595700i
$761$			2.59190e7i			1.62240i	0.584771	+	0.811198i	$0.301184\pi$
$761$			2.59190e7i			1.62240i	−0.584771	+	0.811198i	$0.698816\pi$
$762$		0			0
$763$			2.48517e6i			0.154541i
$764$	−1.86375e6	+	2.37107e6i	−0.115520	+	0.146964i
$765$		0			0
$766$	8.92758e6	−	4.33801e6i	0.549746	−	0.267128i
$767$	−880046.			−0.0540153
$768$		0			0
$769$	−2.19627e7			−1.33927			−0.669637	−	0.742689i	$-0.733550\pi$
$769$	−2.19627e7			−1.33927			−0.669637	+	0.742689i	$0.733550\pi$
$770$	−1.79514e6	+	872277.i	−0.109112	+	0.0530185i
$771$		0			0
$772$	−342814.	+	436128.i	−0.0207021	+	0.0263373i
$773$		−	1.80022e7i		−	1.08362i	−0.840501	−	0.541810i	$-0.817739\pi$
$773$		−	1.80022e7i		−	1.08362i	0.840501	−	0.541810i	$-0.182261\pi$
$774$		0			0
$775$			9.57575e6i			0.572688i
$776$	−5.27276e6	+	2.42785e7i	−0.314329	+	1.44733i
$777$		0			0
$778$	4.56969e6	+	9.40436e6i	0.270668	+	0.557032i
$779$	1.91474e7			1.13049
$780$		0			0
$781$	−8.11081e6			−0.475813
$782$	560057.	+	1.15259e6i	0.0327503	+	0.0673997i
$783$		0			0
$784$	−3.21152e6	−	1.32117e7i	−0.186604	−	0.767660i
$785$		−	9.69375e6i		−	0.561459i
$786$		0			0
$787$		−	2.05610e7i		−	1.18333i	−0.806182	−	0.591667i	$-0.798470\pi$
$787$		−	2.05610e7i		−	1.18333i	0.806182	−	0.591667i	$-0.201530\pi$
$788$	−1.71303e7	−	1.34651e7i	−0.982764	−	0.772491i
$789$		0			0
$790$	1.28978e7	−	6.26717e6i	0.735270	−	0.357276i
$791$	1.82958e6			0.103970
$792$		0			0
$793$	−7.91333e6			−0.446865
$794$	6.65002e6	−	3.23132e6i	0.374345	−	0.181898i
$795$		0			0
$796$	8.05782e6	+	6.33376e6i	0.450749	+	0.354307i
$797$			6.25656e6i			0.348891i	0.984667	+	0.174446i	$0.0558133\pi$
$797$			6.25656e6i			0.348891i	−0.984667	+	0.174446i	$0.944187\pi$
$798$		0			0
$799$		−	691771.i		−	0.0383350i
$800$	7.47667e6	+	9.04476e6i	0.413032	+	0.499657i
$801$		0			0
$802$	8.53294e6	+	1.75607e7i	0.468449	+	0.964064i
$803$	3.76191e6			0.205883
$804$		0			0
$805$	4.75547e6			0.258645
$806$	1.67371e6	+	3.44447e6i	0.0907491	+	0.186761i
$807$		0			0
$808$	1.72153e7	+	3.73879e6i	0.927655	+	0.201466i
$809$			774632.i			0.0416125i	0.999784	+	0.0208063i	$0.00662332\pi$
$809$			774632.i			0.0416125i	−0.999784	+	0.0208063i	$0.993377\pi$
$810$		0			0
$811$		−	2.83962e7i		−	1.51603i	−0.652238	−	0.758014i	$-0.726170\pi$
$811$		−	2.83962e7i		−	1.51603i	0.652238	−	0.758014i	$-0.273830\pi$
$812$	9.29283e6	−	1.18223e7i	0.494604	−	0.629236i
$813$		0			0
$814$	−8.55576e6	+	4.15734e6i	−0.452582	+	0.219915i
$815$	1.07703e7			0.567980
$816$		0			0
$817$	3.15904e7			1.65577
$818$	2.97055e7	−	1.44342e7i	1.55222	−	0.754242i
$819$		0			0
$820$	−7.76158e6	+	9.87429e6i	−0.403103	+	0.512827i
$821$			2.55262e7i			1.32169i	0.750524	+	0.660843i	$0.229801\pi$
$821$			2.55262e7i			1.32169i	−0.750524	+	0.660843i	$0.770199\pi$
$822$		0			0
$823$			2.87596e7i			1.48007i	0.672566	+	0.740037i	$0.265192\pi$
$823$			2.87596e7i			1.48007i	−0.672566	+	0.740037i	$0.734808\pi$
$824$	−6.84486e6	−	1.48655e6i	−0.351194	−	0.0762716i
$825$		0			0
$826$	902484.	+	1.85730e6i	0.0460245	+	0.0947180i
$827$	−2.19785e7			−1.11747			−0.558733	−	0.829348i	$-0.688712\pi$
$827$	−2.19785e7			−1.11747			−0.558733	+	0.829348i	$0.688712\pi$
$828$		0			0
$829$	−1.87554e7			−0.947852			−0.473926	−	0.880565i	$-0.657164\pi$
$829$	−1.87554e7			−0.947852			−0.473926	+	0.880565i	$0.657164\pi$
$830$	4.79630e6	+	9.87074e6i	0.241664	+	0.497341i
$831$		0			0
$832$	4.27032e6	+	1.94666e6i	0.213871	+	0.0974946i
$833$		−	1.24575e6i		−	0.0622040i
$834$		0			0
$835$			1.86883e7i			0.927586i
$836$	−7.28911e6	−	5.72952e6i	−0.360710	−	0.283533i
$837$		0			0
$838$	2.69534e7	−	1.30970e7i	1.32588	−	0.644260i
$839$	−2.04395e7			−1.00246			−0.501228	−	0.865315i	$-0.667118\pi$
$839$	−2.04395e7			−1.00246			−0.501228	+	0.865315i	$0.667118\pi$
$840$		0			0
$841$	−4.20592e7			−2.05055
$842$	3.21367e6	−	1.56156e6i	0.156215	−	0.0759064i
$843$		0			0
$844$	2.76104e7	+	2.17029e7i	1.33419	+	1.04872i
$845$		−	1.16296e7i		−	0.560305i
$846$		0			0
$847$			7.66126e6i			0.366937i
$848$	−2.67616e7	+	6.50523e6i	−1.27797	+	0.310651i
$849$		0			0
$850$	469909.	+	967068.i	0.0223083	+	0.0459102i
$851$	2.26649e7			1.07283
$852$		0			0
$853$	−1.27007e7			−0.597663			−0.298831	−	0.954306i	$-0.596597\pi$
$853$	−1.27007e7			−0.597663			−0.298831	+	0.954306i	$0.596597\pi$
$854$	8.11509e6	+	1.67008e7i	0.380758	+	0.783596i
$855$		0			0
$856$	1.64745e6	−	7.58572e6i	0.0768472	−	0.353844i
$857$		−	1.16731e7i		−	0.542917i	−0.962450	−	0.271459i	$-0.912494\pi$
$857$		−	1.16731e7i		−	0.542917i	0.962450	−	0.271459i	$-0.0875060\pi$
$858$		0			0
$859$			27093.7i			0.00125281i	1.00000		0.000626405i	$0.000199391\pi$
$859$			27093.7i			0.00125281i	−1.00000		0.000626405i	$0.999801\pi$
$860$	−1.28055e7	+	1.62912e7i	−0.590405	+	0.751114i
$861$		0			0
$862$	−3.05961e7	+	1.48670e7i	−1.40249	+	0.681483i
$863$	−9.03157e6			−0.412797			−0.206398	−	0.978468i	$-0.566174\pi$
$863$	−9.03157e6			−0.412797			−0.206398	+	0.978468i	$0.566174\pi$
$864$		0			0
$865$	4.19218e6			0.190502
$866$	2.85293e7	−	1.38627e7i	1.29269	−	0.628134i
$867$		0			0
$868$	5.55305e6	−	7.06459e6i	0.250168	−	0.318264i
$869$			1.36968e7i			0.615274i
$870$		0			0
$871$			5.56969e6i			0.248763i
$872$	1.60712e6	−	7.40003e6i	0.0715744	−	0.329566i
$873$		0			0
$874$	9.65473e6	+	1.98693e7i	0.427525	+	0.879842i
$875$	1.01449e7			0.447948
$876$		0			0
$877$	2.19962e7			0.965713			0.482856	−	0.875700i	$-0.339599\pi$
$877$	2.19962e7			0.965713			0.482856	+	0.875700i	$0.339599\pi$
$878$	−1.75243e7	−	3.60648e7i	−0.767192	−	1.57887i
$879$		0			0
$880$	5.90943e6	−	1.43647e6i	0.257240	−	0.0625301i
$881$			3.05186e7i			1.32472i	0.749184	+	0.662362i	$0.230446\pi$
$881$			3.05186e7i			1.32472i	−0.749184	+	0.662362i	$0.769554\pi$
$882$		0			0
$883$			1.64489e6i			0.0709962i	0.999370	+	0.0354981i	$0.0113018\pi$
$883$			1.64489e6i			0.0709962i	−0.999370	+	0.0354981i	$0.988698\pi$
$884$	338060.	+	265729.i	0.0145500	+	0.0114369i
$885$		0			0
$886$	−2.06335e6	+	1.00260e6i	−0.0883056	+	0.0429087i
$887$	9.31338e6			0.397464			0.198732	−	0.980054i	$-0.436318\pi$
$887$	9.31338e6			0.397464			0.198732	+	0.980054i	$0.436318\pi$
$888$		0			0
$889$	−4.58398e6			−0.194531
$890$	−3.13054e6	+	1.52116e6i	−0.132478	+	0.0643726i
$891$		0			0
$892$	−2.88240e7	−	2.26568e7i	−1.21295	−	0.953426i
$893$		−	1.19253e7i		−	0.500428i
$894$		0			0
$895$		−	2.38682e7i		−	0.996006i
$896$	−270857.	−	1.10086e7i	−0.0112712	−	0.458103i
$897$		0			0
$898$	1.43782e7	+	2.95903e7i	0.594997	+	1.22450i
$899$	−3.73897e7			−1.54295
$900$		0			0
$901$	−2.52338e6			−0.103555
$902$	−5.24300e6	−	1.07900e7i	−0.214567	−	0.441577i
$903$		0			0
$904$	−5.44789e6	−	1.18316e6i	−0.221721	−	0.0481530i
$905$		−	1.69162e6i		−	0.0686565i
$906$		0			0
$907$			1.85183e7i			0.747452i	0.927539	+	0.373726i	$0.121920\pi$
$907$			1.85183e7i			0.747452i	−0.927539	+	0.373726i	$0.878080\pi$
$908$	−1.63592e7	+	2.08122e7i	−0.658487	+	0.837728i
$909$		0			0
$910$	1.43524e6	−	697401.i	0.0574542	−	0.0279177i
$911$	−1.72695e7			−0.689419			−0.344710	−	0.938709i	$-0.612023\pi$
$911$	−1.72695e7			−0.689419			−0.344710	+	0.938709i	$0.612023\pi$
$912$		0			0
$913$	−1.04822e7			−0.416176
$914$	−7.69077e6	+	3.73703e6i	−0.304512	+	0.147966i
$915$		0			0
$916$	1.58356e7	−	2.01461e7i	0.623586	−	0.793327i
$917$		−	1.83757e7i		−	0.721640i
$918$		0			0
$919$		−	3.54182e7i		−	1.38337i	−0.722200	−	0.691684i	$-0.756869\pi$
$919$		−	3.54182e7i		−	1.38337i	0.722200	−	0.691684i	$-0.243131\pi$
$920$	−1.41603e7	−	3.07530e6i	−0.551571	−	0.119789i
$921$		0			0
$922$	−2.10689e7	−	4.33595e7i	−0.816232	−	1.67980i
$923$	6.48473e6			0.250546
$924$		0			0
$925$	1.90168e7			0.730772
$926$	1.44446e7	+	2.97267e7i	0.553575	+	1.13925i
$927$		0			0
$928$	−3.53164e7	+	2.91936e7i	−1.34619	+	1.11280i
$929$		−	4.08533e7i		−	1.55306i	−0.630080	−	0.776530i	$-0.716978\pi$
$929$		−	4.08533e7i		−	1.55306i	0.630080	−	0.776530i	$-0.283022\pi$
$930$		0			0
$931$		−	2.14753e7i		−	0.812016i
$932$	−4.16357e7	−	3.27273e7i	−1.57010	−	1.23416i
$933$		0			0
$934$	8.19105e6	−	3.98012e6i	0.307236	−	0.149289i
$935$	557207.			0.0208443
$936$		0			0
$937$	3.49763e7			1.30144			0.650721	−	0.759317i	$-0.274467\pi$
$937$	3.49763e7			1.30144			0.650721	+	0.759317i	$0.274467\pi$
$938$	1.17546e7	−	5.71169e6i	0.436215	−	0.211962i
$939$		0			0
$940$	6.14989e6	+	4.83405e6i	0.227011	+	0.178440i
$941$		−	1.01395e7i		−	0.373285i	−0.982428	−	0.186642i	$-0.940239\pi$
$941$		−	1.01395e7i		−	0.373285i	0.982428	−	0.186642i	$-0.0597606\pi$
$942$		0			0
$943$			2.85837e7i			1.04674i
$944$	−1.48621e6	−	6.11407e6i	−0.0542814	−	0.223306i
$945$		0			0
$946$	−8.65019e6	−	1.78020e7i	−0.314266	−	0.646757i
$947$	−3.64042e7			−1.31910			−0.659549	−	0.751662i	$-0.729253\pi$
$947$	−3.64042e7			−1.31910			−0.659549	+	0.751662i	$0.729253\pi$
$948$		0			0
$949$	−3.00772e6			−0.108410
$950$	8.10069e6	+	1.66711e7i	0.291215	+	0.599317i
$951$		0			0
$952$	214130.	−	985966.i	0.00765747	−	0.0352590i
$953$		−	3.96453e7i		−	1.41403i	−0.707198	−	0.707016i	$-0.750041\pi$
$953$		−	3.96453e7i		−	1.41403i	0.707198	−	0.707016i	$-0.249959\pi$
$954$		0			0
$955$		−	3.12461e6i		−	0.110863i
$956$	1.82816e7	−	2.32579e7i	0.646949	−	0.823049i
$957$		0			0
$958$	−2.05393e7	+	9.98029e6i	−0.723057	+	0.351341i
$959$	4.11753e6			0.144574
$960$		0			0
$961$	6.28647e6			0.219583
$962$	6.84048e6	−	3.32387e6i	0.238314	−	0.115799i
$963$		0			0
$964$	−2.14353e6	+	2.72700e6i	−0.0742912	+	0.0945134i
$965$		−	574731.i		−	0.0198676i
$966$		0			0
$967$		−	1.14182e7i		−	0.392675i	−0.980536	−	0.196337i	$-0.937095\pi$
$967$		−	1.14182e7i		−	0.392675i	0.980536	−	0.196337i	$-0.0629048\pi$
$968$	4.95443e6	−	2.28127e7i	0.169944	−	0.782509i
$969$		0			0
$970$	−1.12497e7	−	2.31517e7i	−0.383893	−	0.790047i
$971$	4.35838e7			1.48346			0.741732	−	0.670697i	$-0.234005\pi$
$971$	4.35838e7			1.48346			0.741732	+	0.670697i	$0.234005\pi$
$972$		0			0
$973$	−1.94567e7			−0.658852
$974$	−1.55498e7	−	3.20013e7i	−0.525203	−	1.08086i
$975$		0			0
$976$	−1.33640e7	−	5.49774e7i	−0.449067	−	1.84740i
$977$		−	4.35775e7i		−	1.46058i	−0.683136	−	0.730291i	$-0.739384\pi$
$977$		−	4.35775e7i		−	1.46058i	0.683136	−	0.730291i	$-0.260616\pi$
$978$		0			0
$979$		−	3.32448e6i		−	0.110858i
$980$	1.10748e7	+	8.70522e6i	0.368358	+	0.289544i
$981$		0			0
$982$	2.67976e7	−	1.30212e7i	0.886782	−	0.430897i
$983$	7.18955e6			0.237311			0.118656	−	0.992935i	$-0.462142\pi$
$983$	7.18955e6			0.237311			0.118656	+	0.992935i	$0.462142\pi$
$984$		0			0
$985$	2.25744e7			0.741352
$986$	−3.77603e6	+	1.83482e6i	−0.123693	+	0.0601036i
$987$		0			0
$988$	5.82777e6	+	4.58086e6i	0.189937	+	0.149298i
$989$			4.71590e7i			1.53311i
$990$		0			0
$991$			3.64691e7i			1.17962i	0.807543	+	0.589808i	$0.200797\pi$
$991$			3.64691e7i			1.17962i	−0.807543	+	0.589808i	$0.799203\pi$
$992$	−2.11037e7	+	1.74450e7i	−0.680895	+	0.562849i
$993$		0			0
$994$	−6.65007e6	−	1.36858e7i	−0.213481	−	0.439343i
$995$	−1.06186e7			−0.340025
$996$		0			0
$997$	3.18842e7			1.01587			0.507934	−	0.861396i	$-0.330409\pi$
$997$	3.18842e7			1.01587			0.507934	+	0.861396i	$0.330409\pi$
$998$	−5.55519e6	−	1.14325e7i	−0.176552	−	0.363342i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.b.b.107.4	yes	16
3.2	odd	2	inner	108.6.b.b.107.13	yes	16
4.3	odd	2	inner	108.6.b.b.107.14	yes	16
12.11	even	2	inner	108.6.b.b.107.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.b.b.107.3	✓	16	12.11	even	2	inner
108.6.b.b.107.4	yes	16	1.1	even	1	trivial
108.6.b.b.107.13	yes	16	3.2	odd	2	inner
108.6.b.b.107.14	yes	16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.08799 + 2.47231i) q^{2} +(19.7753 - 25.1582i) q^{4} +33.1536i q^{5} -59.4073i q^{7} +(-38.4178 + 176.896i) q^{8} +O(q^{10})\)	\(q+(-5.08799 + 2.47231i) q^{2} +(19.7753 - 25.1582i) q^{4} +33.1536i q^{5} -59.4073i q^{7} +(-38.4178 + 176.896i) q^{8} +(-81.9660 - 168.685i) q^{10} +179.135 q^{11} -143.222 q^{13} +(146.873 + 302.264i) q^{14} +(-241.872 - 995.025i) q^{16} -93.8221i q^{17} -1617.39i q^{19} +(834.085 + 655.624i) q^{20} +(-911.438 + 442.878i) q^{22} +2414.48 q^{23} +2025.84 q^{25} +(728.711 - 354.089i) q^{26} +(-1494.58 - 1174.80i) q^{28} +7910.14i q^{29} +4726.80i q^{31} +(3690.65 + 4464.70i) q^{32} +(231.958 + 477.366i) q^{34} +1969.56 q^{35} +9387.10 q^{37} +(3998.68 + 8229.25i) q^{38} +(-5864.73 - 1273.69i) q^{40} +11838.5i q^{41} +19531.8i q^{43} +(3542.46 - 4506.72i) q^{44} +(-12284.8 + 5969.34i) q^{46} +7373.21 q^{47} +13277.8 q^{49} +(-10307.5 + 5008.51i) q^{50} +(-2832.26 + 3603.20i) q^{52} -26895.4i q^{53} +5938.97i q^{55} +(10508.9 + 2282.30i) q^{56} +(-19556.3 - 40246.7i) q^{58} +6144.64 q^{59} +55252.3 q^{61} +(-11686.1 - 24049.9i) q^{62} +(-29816.1 - 13591.9i) q^{64} -4748.31i q^{65} -38888.6i q^{67} +(-2360.40 - 1855.36i) q^{68} +(-10021.1 + 4869.38i) q^{70} -45277.6 q^{71} +21000.4 q^{73} +(-47761.5 + 23207.8i) q^{74} +(-40690.5 - 31984.4i) q^{76} -10641.9i q^{77} +76460.5i q^{79} +(32988.6 - 8018.91i) q^{80} +(-29268.4 - 60234.0i) q^{82} -58515.8 q^{83} +3110.54 q^{85} +(-48288.6 - 99377.5i) q^{86} +(-6881.99 + 31688.2i) q^{88} -18558.5i q^{89} +8508.41i q^{91} +(47747.1 - 60744.0i) q^{92} +(-37514.9 + 18228.9i) q^{94} +53622.2 q^{95} +137248. q^{97} +(-67557.2 + 32826.8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 94 q^{4}+O(q^{10}) \) 16 * q + 94 * q^4	\( 16 q + 94 q^{4} + 1454 q^{10} + 896 q^{13} + 178 q^{16} + 30 q^{22} + 9888 q^{25} + 11454 q^{28} - 6172 q^{34} - 71008 q^{37} - 16618 q^{40} + 35304 q^{46} - 49376 q^{49} + 14876 q^{52} - 10492 q^{58} + 77888 q^{61} + 89206 q^{64} + 229398 q^{70} - 38032 q^{73} + 48960 q^{76} - 224488 q^{82} - 371264 q^{85} + 249102 q^{88} + 68772 q^{94} - 976 q^{97}+O(q^{100}) \) 16 * q + 94 * q^4 + 1454 * q^10 + 896 * q^13 + 178 * q^16 + 30 * q^22 + 9888 * q^25 + 11454 * q^28 - 6172 * q^34 - 71008 * q^37 - 16618 * q^40 + 35304 * q^46 - 49376 * q^49 + 14876 * q^52 - 10492 * q^58 + 77888 * q^61 + 89206 * q^64 + 229398 * q^70 - 38032 * q^73 + 48960 * q^76 - 224488 * q^82 - 371264 * q^85 + 249102 * q^88 + 68772 * q^94 - 976 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more