LMFDB - Embedded newform 882.4.g.u.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(361,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.g (of order $3$, degree $2$, not 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:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	882.361
Dual form			882.4.g.u.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -8.00000 q^{8} +(-9.00000 + 15.5885i) q^{10} +(-28.5000 + 49.3634i) q^{11} +70.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(-25.5000 + 44.1673i) q^{17} +(2.50000 + 4.33013i) q^{19} -36.0000 q^{20} -114.000 q^{22} +(34.5000 + 59.7558i) q^{23} +(22.0000 - 38.1051i) q^{25} +(70.0000 + 121.244i) q^{26} -114.000 q^{29} +(11.5000 - 19.9186i) q^{31} +(16.0000 - 27.7128i) q^{32} -102.000 q^{34} +(126.500 + 219.104i) q^{37} +(-5.00000 + 8.66025i) q^{38} +(-36.0000 - 62.3538i) q^{40} -42.0000 q^{41} -124.000 q^{43} +(-114.000 - 197.454i) q^{44} +(-69.0000 + 119.512i) q^{46} +(-100.500 - 174.071i) q^{47} +88.0000 q^{50} +(-140.000 + 242.487i) q^{52} +(-196.500 + 340.348i) q^{53} -513.000 q^{55} +(-114.000 - 197.454i) q^{58} +(-109.500 + 189.660i) q^{59} +(-354.500 - 614.012i) q^{61} +46.0000 q^{62} +64.0000 q^{64} +(315.000 + 545.596i) q^{65} +(-209.500 + 362.865i) q^{67} +(-102.000 - 176.669i) q^{68} +96.0000 q^{71} +(-156.500 + 271.066i) q^{73} +(-253.000 + 438.209i) q^{74} -20.0000 q^{76} +(-230.500 - 399.238i) q^{79} +(72.0000 - 124.708i) q^{80} +(-42.0000 - 72.7461i) q^{82} -588.000 q^{83} -459.000 q^{85} +(-124.000 - 214.774i) q^{86} +(228.000 - 394.908i) q^{88} +(508.500 + 880.748i) q^{89} -276.000 q^{92} +(201.000 - 348.142i) q^{94} +(-22.5000 + 38.9711i) q^{95} +1834.00 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8	$ 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8} - 18 q^{10} - 57 q^{11} + 140 q^{13} - 16 q^{16} - 51 q^{17} + 5 q^{19} - 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} + 140 q^{26} - 228 q^{29} + 23 q^{31} + 32 q^{32} - 204 q^{34} + 253 q^{37} - 10 q^{38} - 72 q^{40} - 84 q^{41} - 248 q^{43} - 228 q^{44} - 138 q^{46} - 201 q^{47} + 176 q^{50} - 280 q^{52} - 393 q^{53} - 1026 q^{55} - 228 q^{58} - 219 q^{59} - 709 q^{61} + 92 q^{62} + 128 q^{64} + 630 q^{65} - 419 q^{67} - 204 q^{68} + 192 q^{71} - 313 q^{73} - 506 q^{74} - 40 q^{76} - 461 q^{79} + 144 q^{80} - 84 q^{82} - 1176 q^{83} - 918 q^{85} - 248 q^{86} + 456 q^{88} + 1017 q^{89} - 552 q^{92} + 402 q^{94} - 45 q^{95} + 3668 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8 - 18 * q^10 - 57 * q^11 + 140 * q^13 - 16 * q^16 - 51 * q^17 + 5 * q^19 - 72 * q^20 - 228 * q^22 + 69 * q^23 + 44 * q^25 + 140 * q^26 - 228 * q^29 + 23 * q^31 + 32 * q^32 - 204 * q^34 + 253 * q^37 - 10 * q^38 - 72 * q^40 - 84 * q^41 - 248 * q^43 - 228 * q^44 - 138 * q^46 - 201 * q^47 + 176 * q^50 - 280 * q^52 - 393 * q^53 - 1026 * q^55 - 228 * q^58 - 219 * q^59 - 709 * q^61 + 92 * q^62 + 128 * q^64 + 630 * q^65 - 419 * q^67 - 204 * q^68 + 192 * q^71 - 313 * q^73 - 506 * q^74 - 40 * q^76 - 461 * q^79 + 144 * q^80 - 84 * q^82 - 1176 * q^83 - 918 * q^85 - 248 * q^86 + 456 * q^88 + 1017 * q^89 - 552 * q^92 + 402 * q^94 - 45 * q^95 + 3668 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	4.50000	+	7.79423i	0.402492	+	0.697137i	0.994026	−	0.109143i	$-0.0348107\pi$
$5$	4.50000	+	7.79423i	0.402492	+	0.697137i	−0.591534	+	0.806280i	$0.701477\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−9.00000	+	15.5885i	−0.284605	+	0.492950i
$11$	−28.5000	+	49.3634i	−0.781188	+	1.35306i	0.150061	+	0.988677i	$0.452053\pi$
$11$	−28.5000	+	49.3634i	−0.781188	+	1.35306i	−0.931250	+	0.364381i	$0.881280\pi$
$12$		0			0
$13$	70.0000			1.49342			0.746712	−	0.665148i	$-0.231631\pi$
$13$	70.0000			1.49342			0.746712	+	0.665148i	$0.231631\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−25.5000	+	44.1673i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−25.5000	+	44.1673i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	2.50000	+	4.33013i	0.0301863	+	0.0522842i	0.880724	−	0.473630i	$-0.157057\pi$
$19$	2.50000	+	4.33013i	0.0301863	+	0.0522842i	−0.850538	+	0.525914i	$0.823723\pi$
$20$	−36.0000			−0.402492
$21$		0			0
$22$	−114.000			−1.10477
$23$	34.5000	+	59.7558i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	34.5000	+	59.7558i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	22.0000	−	38.1051i	0.176000	−	0.304841i
$26$	70.0000	+	121.244i	0.528005	+	0.914531i
$27$		0			0
$28$		0			0
$29$	−114.000			−0.729975			−0.364987	−	0.931012i	$-0.618927\pi$
$29$	−114.000			−0.729975			−0.364987	+	0.931012i	$0.618927\pi$
$30$		0			0
$31$	11.5000	−	19.9186i	0.0666278	−	0.115403i	−0.830787	−	0.556590i	$-0.812109\pi$
$31$	11.5000	−	19.9186i	0.0666278	−	0.115403i	0.897415	+	0.441188i	$0.145443\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−102.000			−0.514496
$35$		0			0
$36$		0			0
$37$	126.500	+	219.104i	0.562067	+	0.973528i	0.997316	+	0.0732182i	$0.0233270\pi$
$37$	126.500	+	219.104i	0.562067	+	0.973528i	−0.435249	+	0.900310i	$0.643340\pi$
$38$	−5.00000	+	8.66025i	−0.0213449	+	0.0369705i
$39$		0			0
$40$	−36.0000	−	62.3538i	−0.142302	−	0.246475i
$41$	−42.0000			−0.159983			−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000			−0.159983			−0.0799914	+	0.996796i	$0.525489\pi$
$42$		0			0
$43$	−124.000			−0.439763			−0.219882	−	0.975527i	$-0.570567\pi$
$43$	−124.000			−0.439763			−0.219882	+	0.975527i	$0.570567\pi$
$44$	−114.000	−	197.454i	−0.390594	−	0.676529i
$45$		0			0
$46$	−69.0000	+	119.512i	−0.221163	+	0.383065i
$47$	−100.500	−	174.071i	−0.311903	−	0.540231i	0.666871	−	0.745173i	$-0.267633\pi$
$47$	−100.500	−	174.071i	−0.311903	−	0.540231i	−0.978774	+	0.204941i	$0.934300\pi$
$48$		0			0
$49$		0			0
$50$	88.0000			0.248902
$51$		0			0
$52$	−140.000	+	242.487i	−0.373356	+	0.646671i
$53$	−196.500	+	340.348i	−0.509271	+	0.882083i	0.490672	+	0.871345i	$0.336751\pi$
$53$	−196.500	+	340.348i	−0.509271	+	0.882083i	−0.999942	+	0.0107383i	$0.996582\pi$
$54$		0			0
$55$	−513.000			−1.25769
$56$		0			0
$57$		0			0
$58$	−114.000	−	197.454i	−0.258085	−	0.447016i
$59$	−109.500	+	189.660i	−0.241622	+	0.418501i	−0.961176	−	0.275935i	$-0.911013\pi$
$59$	−109.500	+	189.660i	−0.241622	+	0.418501i	0.719555	+	0.694436i	$0.244346\pi$
$60$		0			0
$61$	−354.500	−	614.012i	−0.744083	−	1.28879i	−0.950622	−	0.310351i	$-0.899553\pi$
$61$	−354.500	−	614.012i	−0.744083	−	1.28879i	0.206539	−	0.978438i	$-0.433780\pi$
$62$	46.0000			0.0942259
$63$		0			0
$64$	64.0000			0.125000
$65$	315.000	+	545.596i	0.601091	+	1.04112i
$66$		0			0
$67$	−209.500	+	362.865i	−0.382007	+	0.661656i	−0.991349	−	0.131251i	$-0.958100\pi$
$67$	−209.500	+	362.865i	−0.382007	+	0.661656i	0.609342	+	0.792908i	$0.291434\pi$
$68$	−102.000	−	176.669i	−0.181902	−	0.315063i
$69$		0			0
$70$		0			0
$71$	96.0000			0.160466			0.0802331	−	0.996776i	$-0.474434\pi$
$71$	96.0000			0.160466			0.0802331	+	0.996776i	$0.474434\pi$
$72$		0			0
$73$	−156.500	+	271.066i	−0.250917	+	0.434601i	−0.963779	−	0.266704i	$-0.914065\pi$
$73$	−156.500	+	271.066i	−0.250917	+	0.434601i	0.712862	+	0.701305i	$0.247399\pi$
$74$	−253.000	+	438.209i	−0.397441	+	0.688388i
$75$		0			0
$76$	−20.0000			−0.0301863
$77$		0			0
$78$		0			0
$79$	−230.500	−	399.238i	−0.328269	−	0.568579i	0.653899	−	0.756582i	$-0.273132\pi$
$79$	−230.500	−	399.238i	−0.328269	−	0.568579i	−0.982169	+	0.188003i	$0.939799\pi$
$80$	72.0000	−	124.708i	0.100623	−	0.174284i
$81$		0			0
$82$	−42.0000	−	72.7461i	−0.0565625	−	0.0979691i
$83$	−588.000			−0.777607			−0.388804	−	0.921321i	$-0.627112\pi$
$83$	−588.000			−0.777607			−0.388804	+	0.921321i	$0.627112\pi$
$84$		0			0
$85$	−459.000			−0.585712
$86$	−124.000	−	214.774i	−0.155480	−	0.269299i
$87$		0			0
$88$	228.000	−	394.908i	0.276192	−	0.478378i
$89$	508.500	+	880.748i	0.605628	+	1.04898i	0.991952	+	0.126615i	$0.0404114\pi$
$89$	508.500	+	880.748i	0.605628	+	1.04898i	−0.386324	+	0.922363i	$0.626255\pi$
$90$		0			0
$91$		0			0
$92$	−276.000			−0.312772
$93$		0			0
$94$	201.000	−	348.142i	0.220549	−	0.382001i
$95$	−22.5000	+	38.9711i	−0.0242995	+	0.0420879i
$96$		0			0
$97$	1834.00			1.91974			0.959868	−	0.280451i	$-0.0904839\pi$
$97$	1834.00			1.91974			0.959868	+	0.280451i	$0.0904839\pi$
$98$		0			0
$99$		0			0
$100$	88.0000	+	152.420i	0.0880000	+	0.152420i
$101$	142.500	−	246.817i	0.140389	−	0.243161i	−0.787254	−	0.616629i	$-0.788498\pi$
$101$	142.500	−	246.817i	0.140389	−	0.243161i	0.927643	+	0.373468i	$0.121831\pi$
$102$		0			0
$103$	−249.500	−	432.147i	−0.238679	−	0.413405i	0.721656	−	0.692252i	$-0.243381\pi$
$103$	−249.500	−	432.147i	−0.238679	−	0.413405i	−0.960336	+	0.278847i	$0.910048\pi$
$104$	−560.000			−0.528005
$105$		0			0
$106$	−786.000			−0.720218
$107$	−553.500	−	958.690i	−0.500083	−	0.866169i	−1.00000		9.56665e-5i	$-0.999970\pi$
$107$	−553.500	−	958.690i	−0.500083	−	0.866169i	0.499917	−	0.866073i	$-0.333364\pi$
$108$		0			0
$109$	−461.500	+	799.341i	−0.405538	+	0.702413i	−0.994384	−	0.105832i	$-0.966249\pi$
$109$	−461.500	+	799.341i	−0.405538	+	0.702413i	0.588846	+	0.808246i	$0.299583\pi$
$110$	−513.000	−	888.542i	−0.444660	−	0.770174i
$111$		0			0
$112$		0			0
$113$	−1542.00			−1.28371			−0.641855	−	0.766826i	$-0.721835\pi$
$113$	−1542.00			−1.28371			−0.641855	+	0.766826i	$0.721835\pi$
$114$		0			0
$115$	−310.500	+	537.802i	−0.251776	+	0.436089i
$116$	228.000	−	394.908i	0.182494	−	0.316088i
$117$		0			0
$118$	−438.000			−0.341705
$119$		0			0
$120$		0			0
$121$	−959.000	−	1661.04i	−0.720511	−	1.24796i
$122$	709.000	−	1228.02i	0.526146	−	0.911312i
$123$		0			0
$124$	46.0000	+	79.6743i	0.0333139	+	0.0577013i
$125$	1521.00			1.08834
$126$		0			0
$127$	−2056.00			−1.43654			−0.718270	−	0.695765i	$-0.755066\pi$
$127$	−2056.00			−1.43654			−0.718270	+	0.695765i	$0.755066\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−630.000	+	1091.19i	−0.425036	+	0.736184i
$131$	−1024.50	−	1774.49i	−0.683290	−	1.18349i	−0.973971	−	0.226673i	$-0.927215\pi$
$131$	−1024.50	−	1774.49i	−0.683290	−	1.18349i	0.290681	−	0.956820i	$-0.406118\pi$
$132$		0			0
$133$		0			0
$134$	−838.000			−0.540240
$135$		0			0
$136$	204.000	−	353.338i	0.128624	−	0.222783i
$137$	−70.5000	+	122.110i	−0.0439651	+	0.0761498i	−0.887171	−	0.461442i	$-0.847332\pi$
$137$	−70.5000	+	122.110i	−0.0439651	+	0.0761498i	0.843205	+	0.537591i	$0.180666\pi$
$138$		0			0
$139$	−1484.00			−0.905548			−0.452774	−	0.891625i	$-0.649566\pi$
$139$	−1484.00			−0.905548			−0.452774	+	0.891625i	$0.649566\pi$
$140$		0			0
$141$		0			0
$142$	96.0000	+	166.277i	0.0567334	+	0.0982651i
$143$	−1995.00	+	3455.44i	−1.16665	+	2.02069i
$144$		0			0
$145$	−513.000	−	888.542i	−0.293809	−	0.508892i
$146$	−626.000			−0.354850
$147$		0			0
$148$	−1012.00			−0.562067
$149$	−28.5000	−	49.3634i	−0.0156699	−	0.0271410i	0.858084	−	0.513509i	$-0.171655\pi$
$149$	−28.5000	−	49.3634i	−0.0156699	−	0.0271410i	−0.873754	+	0.486368i	$0.838321\pi$
$150$		0			0
$151$	−419.500	+	726.595i	−0.226082	+	0.391586i	−0.956644	−	0.291261i	$-0.905925\pi$
$151$	−419.500	+	726.595i	−0.226082	+	0.391586i	0.730561	+	0.682847i	$0.239258\pi$
$152$	−20.0000	−	34.6410i	−0.0106725	−	0.0184852i
$153$		0			0
$154$		0			0
$155$	207.000			0.107269
$156$		0			0
$157$	−1416.50	+	2453.45i	−0.720057	+	1.24718i	0.240919	+	0.970545i	$0.422551\pi$
$157$	−1416.50	+	2453.45i	−0.720057	+	1.24718i	−0.960976	+	0.276631i	$0.910782\pi$
$158$	461.000	−	798.475i	0.232121	−	0.402046i
$159$		0			0
$160$	288.000			0.142302
$161$		0			0
$162$		0			0
$163$	1155.50	+	2001.38i	0.555250	+	0.961721i	0.997884	+	0.0650188i	$0.0207107\pi$
$163$	1155.50	+	2001.38i	0.555250	+	0.961721i	−0.442634	+	0.896702i	$0.645956\pi$
$164$	84.0000	−	145.492i	0.0399957	−	0.0692746i
$165$		0			0
$166$	−588.000	−	1018.45i	−0.274926	−	0.476185i
$167$	1260.00			0.583843			0.291921	−	0.956442i	$-0.405705\pi$
$167$	1260.00			0.583843			0.291921	+	0.956442i	$0.405705\pi$
$168$		0			0
$169$	2703.00			1.23031
$170$	−459.000	−	795.011i	−0.207081	−	0.358674i
$171$		0			0
$172$	248.000	−	429.549i	0.109941	−	0.190423i
$173$	−1633.50	−	2829.30i	−0.717877	−	1.24340i	−0.961839	−	0.273615i	$-0.911781\pi$
$173$	−1633.50	−	2829.30i	−0.717877	−	1.24340i	0.243962	−	0.969785i	$-0.421553\pi$
$174$		0			0
$175$		0			0
$176$	912.000			0.390594
$177$		0			0
$178$	−1017.00	+	1761.50i	−0.428244	+	0.741740i
$179$	643.500	−	1114.57i	0.268701	−	0.465403i	−0.699826	−	0.714314i	$-0.746739\pi$
$179$	643.500	−	1114.57i	0.268701	−	0.465403i	0.968527	+	0.248910i	$0.0800724\pi$
$180$		0			0
$181$	2674.00			1.09810			0.549052	−	0.835788i	$-0.314989\pi$
$181$	2674.00			1.09810			0.549052	+	0.835788i	$0.314989\pi$
$182$		0			0
$183$		0			0
$184$	−276.000	−	478.046i	−0.110581	−	0.191533i
$185$	−1138.50	+	1971.94i	−0.452455	+	0.783675i
$186$		0			0
$187$	−1453.50	−	2517.54i	−0.568398	−	0.984494i
$188$	804.000			0.311903
$189$		0			0
$190$	−90.0000			−0.0343647
$191$	2092.50	+	3624.32i	0.792712	+	1.37302i	0.924282	+	0.381711i	$0.124665\pi$
$191$	2092.50	+	3624.32i	0.792712	+	1.37302i	−0.131570	+	0.991307i	$0.542002\pi$
$192$		0			0
$193$	42.5000	−	73.6122i	0.0158509	−	0.0274545i	−0.857991	−	0.513664i	$-0.828288\pi$
$193$	42.5000	−	73.6122i	0.0158509	−	0.0274545i	0.873842	+	0.486210i	$0.161621\pi$
$194$	1834.00	+	3176.58i	0.678730	+	1.17559i
$195$		0			0
$196$		0			0
$197$	390.000			0.141047			0.0705237	−	0.997510i	$-0.477533\pi$
$197$	390.000			0.141047			0.0705237	+	0.997510i	$0.477533\pi$
$198$		0			0
$199$	−1416.50	+	2453.45i	−0.504588	+	0.873972i	0.495398	+	0.868666i	$0.335022\pi$
$199$	−1416.50	+	2453.45i	−0.504588	+	0.873972i	−0.999986	+	0.00530596i	$0.998311\pi$
$200$	−176.000	+	304.841i	−0.0622254	+	0.107778i
$201$		0			0
$202$	570.000			0.198540
$203$		0			0
$204$		0			0
$205$	−189.000	−	327.358i	−0.0643919	−	0.111530i
$206$	499.000	−	864.293i	0.168772	−	0.292321i
$207$		0			0
$208$	−560.000	−	969.948i	−0.186678	−	0.323336i
$209$	−285.000			−0.0943247
$210$		0			0
$211$	−124.000			−0.0404574			−0.0202287	−	0.999795i	$-0.506439\pi$
$211$	−124.000			−0.0404574			−0.0202287	+	0.999795i	$0.506439\pi$
$212$	−786.000	−	1361.39i	−0.254635	−	0.441041i
$213$		0			0
$214$	1107.00	−	1917.38i	0.353612	−	0.612474i
$215$	−558.000	−	966.484i	−0.177001	−	0.306575i
$216$		0			0
$217$		0			0
$218$	−1846.00			−0.573518
$219$		0			0
$220$	1026.00	−	1777.08i	0.314422	−	0.544595i
$221$	−1785.00	+	3091.71i	−0.543313	+	0.941045i
$222$		0			0
$223$	−56.0000			−0.0168163			−0.00840816	−	0.999965i	$-0.502676\pi$
$223$	−56.0000			−0.0168163			−0.00840816	+	0.999965i	$0.502676\pi$
$224$		0			0
$225$		0			0
$226$	−1542.00	−	2670.82i	−0.453860	−	0.786108i
$227$	1528.50	−	2647.44i	0.446917	−	0.774083i	−0.551267	−	0.834329i	$-0.685855\pi$
$227$	1528.50	−	2647.44i	0.446917	−	0.774083i	0.998184	+	0.0602465i	$0.0191887\pi$
$228$		0			0
$229$	−480.500	−	832.250i	−0.138656	−	0.240160i	0.788332	−	0.615250i	$-0.210945\pi$
$229$	−480.500	−	832.250i	−0.138656	−	0.240160i	−0.926988	+	0.375090i	$0.877612\pi$
$230$	−1242.00			−0.356065
$231$		0			0
$232$	912.000			0.258085
$233$	−1414.50	−	2449.99i	−0.397712	−	0.688858i	0.595731	−	0.803184i	$-0.296862\pi$
$233$	−1414.50	−	2449.99i	−0.397712	−	0.688858i	−0.993443	+	0.114326i	$0.963529\pi$
$234$		0			0
$235$	904.500	−	1566.64i	0.251077	−	0.434878i
$236$	−438.000	−	758.638i	−0.120811	−	0.209251i
$237$		0			0
$238$		0			0
$239$	3540.00			0.958090			0.479045	−	0.877790i	$-0.340983\pi$
$239$	3540.00			0.958090			0.479045	+	0.877790i	$0.340983\pi$
$240$		0			0
$241$	2615.50	−	4530.18i	0.699084	−	1.21085i	−0.269701	−	0.962944i	$-0.586925\pi$
$241$	2615.50	−	4530.18i	0.699084	−	1.21085i	0.968785	−	0.247904i	$-0.0797419\pi$
$242$	1918.00	−	3322.07i	0.509478	−	0.882442i
$243$		0			0
$244$	2836.00			0.744083
$245$		0			0
$246$		0			0
$247$	175.000	+	303.109i	0.0450809	+	0.0780824i
$248$	−92.0000	+	159.349i	−0.0235565	+	0.0408010i
$249$		0			0
$250$	1521.00	+	2634.45i	0.384786	+	0.666469i
$251$	5040.00			1.26742			0.633709	−	0.773571i	$-0.281532\pi$
$251$	5040.00			1.26742			0.633709	+	0.773571i	$0.281532\pi$
$252$		0			0
$253$	−3933.00			−0.977334
$254$	−2056.00	−	3561.10i	−0.507893	−	0.879697i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	718.500	+	1244.48i	0.174392	+	0.302056i	0.939951	−	0.341310i	$-0.110871\pi$
$257$	718.500	+	1244.48i	0.174392	+	0.302056i	−0.765559	+	0.643366i	$0.777537\pi$
$258$		0			0
$259$		0			0
$260$	−2520.00			−0.601091
$261$		0			0
$262$	2049.00	−	3548.97i	0.483159	−	0.836856i
$263$	−1162.50	+	2013.51i	−0.272558	+	0.472085i	−0.969516	−	0.245027i	$-0.921203\pi$
$263$	−1162.50	+	2013.51i	−0.272558	+	0.472085i	0.696958	+	0.717112i	$0.254536\pi$
$264$		0			0
$265$	−3537.00			−0.819910
$266$		0			0
$267$		0			0
$268$	−838.000	−	1451.46i	−0.191004	−	0.330828i
$269$	1192.50	−	2065.47i	0.270290	−	0.468156i	−0.698646	−	0.715467i	$-0.746214\pi$
$269$	1192.50	−	2065.47i	0.270290	−	0.468156i	0.968936	+	0.247311i	$0.0795471\pi$
$270$		0			0
$271$	−165.500	−	286.654i	−0.0370975	−	0.0642547i	0.846881	−	0.531783i	$-0.178478\pi$
$271$	−165.500	−	286.654i	−0.0370975	−	0.0642547i	−0.883978	+	0.467528i	$0.845145\pi$
$272$	816.000			0.181902
$273$		0			0
$274$	−282.000			−0.0621761
$275$	1254.00	+	2171.99i	0.274978	+	0.476276i
$276$		0			0
$277$	−2435.50	+	4218.41i	−0.528285	+	0.915017i	0.471171	+	0.882042i	$0.343831\pi$
$277$	−2435.50	+	4218.41i	−0.528285	+	0.915017i	−0.999456	+	0.0329750i	$0.989502\pi$
$278$	−1484.00	−	2570.36i	−0.320160	−	0.554533i
$279$		0			0
$280$		0			0
$281$	7026.00			1.49159			0.745794	−	0.666177i	$-0.232070\pi$
$281$	7026.00			1.49159			0.745794	+	0.666177i	$0.232070\pi$
$282$		0			0
$283$	−2676.50	+	4635.83i	−0.562196	+	0.973752i	0.435109	+	0.900378i	$0.356710\pi$
$283$	−2676.50	+	4635.83i	−0.562196	+	0.973752i	−0.997305	+	0.0733738i	$0.976623\pi$
$284$	−192.000	+	332.554i	−0.0401166	+	0.0694839i
$285$		0			0
$286$	−7980.00			−1.64989
$287$		0			0
$288$		0			0
$289$	1156.00	+	2002.25i	0.235294	+	0.407541i
$290$	1026.00	−	1777.08i	0.207754	−	0.359841i
$291$		0			0
$292$	−626.000	−	1084.26i	−0.125458	−	0.217300i
$293$	4158.00			0.829054			0.414527	−	0.910037i	$-0.363947\pi$
$293$	4158.00			0.829054			0.414527	+	0.910037i	$0.363947\pi$
$294$		0			0
$295$	−1971.00			−0.389004
$296$	−1012.00	−	1752.84i	−0.198721	−	0.344194i
$297$		0			0
$298$	57.0000	−	98.7269i	0.0110803	−	0.0191916i
$299$	2415.00	+	4182.90i	0.467101	+	0.809042i
$300$		0			0
$301$		0			0
$302$	−1678.00			−0.319729
$303$		0			0
$304$	40.0000	−	69.2820i	0.00754657	−	0.0130710i
$305$	3190.50	−	5526.11i	0.598975	−	1.03746i
$306$		0			0
$307$	9604.00			1.78544			0.892719	−	0.450615i	$-0.148795\pi$
$307$	9604.00			1.78544			0.892719	+	0.450615i	$0.148795\pi$
$308$		0			0
$309$		0			0
$310$	207.000	+	358.535i	0.0379252	+	0.0656884i
$311$	−5065.50	+	8773.70i	−0.923595	+	1.59971i	−0.129791	+	0.991541i	$0.541430\pi$
$311$	−5065.50	+	8773.70i	−0.923595	+	1.59971i	−0.793805	+	0.608173i	$0.791903\pi$
$312$		0			0
$313$	5399.50	+	9352.21i	0.975073	+	1.68888i	0.679695	+	0.733495i	$0.262112\pi$
$313$	5399.50	+	9352.21i	0.975073	+	1.68888i	0.295378	+	0.955380i	$0.404554\pi$
$314$	−5666.00			−1.01831
$315$		0			0
$316$	1844.00			0.328269
$317$	265.500	+	459.859i	0.0470409	+	0.0814772i	0.888587	−	0.458708i	$-0.151688\pi$
$317$	265.500	+	459.859i	0.0470409	+	0.0814772i	−0.841546	+	0.540185i	$0.818354\pi$
$318$		0			0
$319$	3249.00	−	5627.43i	0.570248	−	0.987698i
$320$	288.000	+	498.831i	0.0503115	+	0.0871421i
$321$		0			0
$322$		0			0
$323$	−255.000			−0.0439275
$324$		0			0
$325$	1540.00	−	2667.36i	0.262843	−	0.455257i
$326$	−2311.00	+	4002.77i	−0.392621	+	0.680040i
$327$		0			0
$328$	336.000			0.0565625
$329$		0			0
$330$		0			0
$331$	3507.50	+	6075.17i	0.582446	+	1.00883i	0.995189	+	0.0979784i	$0.0312376\pi$
$331$	3507.50	+	6075.17i	0.582446	+	1.00883i	−0.412743	+	0.910848i	$0.635429\pi$
$332$	1176.00	−	2036.89i	0.194402	−	0.336714i
$333$		0			0
$334$	1260.00	+	2182.38i	0.206420	+	0.357529i
$335$	−3771.00			−0.615020
$336$		0			0
$337$	8990.00			1.45316			0.726582	−	0.687079i	$-0.241108\pi$
$337$	8990.00			1.45316			0.726582	+	0.687079i	$0.241108\pi$
$338$	2703.00	+	4681.73i	0.434982	+	0.753410i
$339$		0			0
$340$	918.000	−	1590.02i	0.146428	−	0.253621i
$341$	655.500	+	1135.36i	0.104098	+	0.180303i
$342$		0			0
$343$		0			0
$344$	992.000			0.155480
$345$		0			0
$346$	3267.00	−	5658.61i	0.507616	−	0.879216i
$347$	−4354.50	+	7542.22i	−0.673665	+	1.16682i	0.303192	+	0.952929i	$0.401948\pi$
$347$	−4354.50	+	7542.22i	−0.673665	+	1.16682i	−0.976857	+	0.213893i	$0.931386\pi$
$348$		0			0
$349$	−6482.00			−0.994193			−0.497097	−	0.867695i	$-0.665601\pi$
$349$	−6482.00			−0.994193			−0.497097	+	0.867695i	$0.665601\pi$
$350$		0			0
$351$		0			0
$352$	912.000	+	1579.63i	0.138096	+	0.239189i
$353$	1066.50	−	1847.23i	0.160805	−	0.278522i	−0.774353	−	0.632754i	$-0.781924\pi$
$353$	1066.50	−	1847.23i	0.160805	−	0.278522i	0.935158	+	0.354232i	$0.115258\pi$
$354$		0			0
$355$	432.000	+	748.246i	0.0645864	+	0.111867i
$356$	−4068.00			−0.605628
$357$		0			0
$358$	2574.00			0.380000
$359$	1924.50	+	3333.33i	0.282928	+	0.490046i	0.972105	−	0.234548i	$-0.0753608\pi$
$359$	1924.50	+	3333.33i	0.282928	+	0.490046i	−0.689176	+	0.724594i	$0.742028\pi$
$360$		0			0
$361$	3417.00	−	5918.42i	0.498178	−	0.862869i
$362$	2674.00	+	4631.50i	0.388238	+	0.672449i
$363$		0			0
$364$		0			0
$365$	−2817.00			−0.403969
$366$		0			0
$367$	3245.50	−	5621.37i	0.461618	−	0.799545i	−0.537424	−	0.843312i	$-0.680603\pi$
$367$	3245.50	−	5621.37i	0.461618	−	0.799545i	0.999042	+	0.0437668i	$0.0139358\pi$
$368$	552.000	−	956.092i	0.0781929	−	0.135434i
$369$		0			0
$370$	−4554.00			−0.639868
$371$		0			0
$372$		0			0
$373$	−461.500	−	799.341i	−0.0640632	−	0.110961i	0.832215	−	0.554453i	$-0.187073\pi$
$373$	−461.500	−	799.341i	−0.0640632	−	0.110961i	−0.896278	+	0.443493i	$0.853739\pi$
$374$	2907.00	−	5035.07i	0.401918	−	0.696143i
$375$		0			0
$376$	804.000	+	1392.57i	0.110274	+	0.191001i
$377$	−7980.00			−1.09016
$378$		0			0
$379$	6344.00			0.859814			0.429907	−	0.902873i	$-0.358546\pi$
$379$	6344.00			0.859814			0.429907	+	0.902873i	$0.358546\pi$
$380$	−90.0000	−	155.885i	−0.0121497	−	0.0210440i
$381$		0			0
$382$	−4185.00	+	7248.63i	−0.560532	+	0.970870i
$383$	2503.50	+	4336.19i	0.334002	+	0.578509i	0.983293	−	0.182032i	$-0.0582673\pi$
$383$	2503.50	+	4336.19i	0.334002	+	0.578509i	−0.649290	+	0.760541i	$0.724934\pi$
$384$		0			0
$385$		0			0
$386$	170.000			0.0224165
$387$		0			0
$388$	−3668.00	+	6353.16i	−0.479934	+	0.831270i
$389$	6145.50	−	10644.3i	0.801001	−	1.38737i	−0.117958	−	0.993019i	$-0.537635\pi$
$389$	6145.50	−	10644.3i	0.801001	−	1.38737i	0.918958	−	0.394355i	$-0.129032\pi$
$390$		0			0
$391$	−3519.00			−0.455150
$392$		0			0
$393$		0			0
$394$	390.000	+	675.500i	0.0498678	+	0.0863736i
$395$	2074.50	−	3593.14i	0.264252	−	0.457697i
$396$		0			0
$397$	443.500	+	768.165i	0.0560671	+	0.0971110i	0.892697	−	0.450658i	$-0.148811\pi$
$397$	443.500	+	768.165i	0.0560671	+	0.0971110i	−0.836630	+	0.547769i	$0.815477\pi$
$398$	−5666.00			−0.713595
$399$		0			0
$400$	−704.000			−0.0880000
$401$	5977.50	+	10353.3i	0.744394	+	1.28933i	0.950477	+	0.310794i	$0.100595\pi$
$401$	5977.50	+	10353.3i	0.744394	+	1.28933i	−0.206083	+	0.978535i	$0.566072\pi$
$402$		0			0
$403$	805.000	−	1394.30i	0.0995035	−	0.172345i
$404$	570.000	+	987.269i	0.0701945	+	0.121580i
$405$		0			0
$406$		0			0
$407$	−14421.0			−1.75632
$408$		0			0
$409$	−1710.50	+	2962.67i	−0.206794	+	0.358178i	−0.950703	−	0.310103i	$-0.899636\pi$
$409$	−1710.50	+	2962.67i	−0.206794	+	0.358178i	0.743909	+	0.668281i	$0.232970\pi$
$410$	378.000	−	654.715i	0.0455319	−	0.0788636i
$411$		0			0
$412$	1996.00			0.238679
$413$		0			0
$414$		0			0
$415$	−2646.00	−	4583.01i	−0.312981	−	0.542099i
$416$	1120.00	−	1939.90i	0.132001	−	0.228633i
$417$		0			0
$418$	−285.000	−	493.634i	−0.0333488	−	0.0577618i
$419$	−5460.00			−0.636607			−0.318304	−	0.947989i	$-0.603113\pi$
$419$	−5460.00			−0.636607			−0.318304	+	0.947989i	$0.603113\pi$
$420$		0			0
$421$	7730.00			0.894863			0.447431	−	0.894318i	$-0.352339\pi$
$421$	7730.00			0.894863			0.447431	+	0.894318i	$0.352339\pi$
$422$	−124.000	−	214.774i	−0.0143039	−	0.0247750i
$423$		0			0
$424$	1572.00	−	2722.78i	0.180054	−	0.311863i
$425$	1122.00	+	1943.36i	0.128059	+	0.221804i
$426$		0			0
$427$		0			0
$428$	4428.00			0.500083
$429$		0			0
$430$	1116.00	−	1932.97i	0.125159	−	0.216781i
$431$	−5656.50	+	9797.35i	−0.632167	+	1.09495i	0.354941	+	0.934889i	$0.384501\pi$
$431$	−5656.50	+	9797.35i	−0.632167	+	1.09495i	−0.987108	+	0.160057i	$0.948832\pi$
$432$		0			0
$433$	−4214.00			−0.467695			−0.233847	−	0.972273i	$-0.575132\pi$
$433$	−4214.00			−0.467695			−0.233847	+	0.972273i	$0.575132\pi$
$434$		0			0
$435$		0			0
$436$	−1846.00	−	3197.37i	−0.202769	−	0.351207i
$437$	−172.500	+	298.779i	−0.0188828	+	0.0327060i
$438$		0			0
$439$	8276.50	+	14335.3i	0.899808	+	1.55851i	0.827739	+	0.561114i	$0.189627\pi$
$439$	8276.50	+	14335.3i	0.899808	+	1.55851i	0.0720696	+	0.997400i	$0.477040\pi$
$440$	4104.00			0.444660
$441$		0			0
$442$	−7140.00			−0.768360
$443$	−8197.50	−	14198.5i	−0.879176	−	1.52278i	−0.852247	−	0.523140i	$-0.824760\pi$
$443$	−8197.50	−	14198.5i	−0.879176	−	1.52278i	−0.0269294	−	0.999637i	$-0.508573\pi$
$444$		0			0
$445$	−4576.50	+	7926.73i	−0.487521	+	0.844411i
$446$	−56.0000	−	96.9948i	−0.00594546	−	0.0102978i
$447$		0			0
$448$		0			0
$449$	15090.0			1.58606			0.793030	−	0.609182i	$-0.208502\pi$
$449$	15090.0			1.58606			0.793030	+	0.609182i	$0.208502\pi$
$450$		0			0
$451$	1197.00	−	2073.26i	0.124977	−	0.216466i
$452$	3084.00	−	5341.64i	0.320927	−	0.555862i
$453$		0			0
$454$	6114.00			0.632036
$455$		0			0
$456$		0			0
$457$	7392.50	+	12804.2i	0.756688	+	1.31062i	0.944531	+	0.328423i	$0.106517\pi$
$457$	7392.50	+	12804.2i	0.756688	+	1.31062i	−0.187842	+	0.982199i	$0.560149\pi$
$458$	961.000	−	1664.50i	0.0980449	−	0.169819i
$459$		0			0
$460$	−1242.00	−	2151.21i	−0.125888	−	0.218045i
$461$	2898.00			0.292784			0.146392	−	0.989227i	$-0.453234\pi$
$461$	2898.00			0.292784			0.146392	+	0.989227i	$0.453234\pi$
$462$		0			0
$463$	464.000			0.0465743			0.0232872	−	0.999729i	$-0.492587\pi$
$463$	464.000			0.0465743			0.0232872	+	0.999729i	$0.492587\pi$
$464$	912.000	+	1579.63i	0.0912468	+	0.158044i
$465$		0			0
$466$	2829.00	−	4899.97i	0.281225	−	0.487096i
$467$	−2116.50	−	3665.89i	−0.209721	−	0.363248i	0.741905	−	0.670505i	$-0.233922\pi$
$467$	−2116.50	−	3665.89i	−0.209721	−	0.363248i	−0.951627	+	0.307256i	$0.900589\pi$
$468$		0			0
$469$		0			0
$470$	3618.00			0.355076
$471$		0			0
$472$	876.000	−	1517.28i	0.0854262	−	0.147963i
$473$	3534.00	−	6121.07i	0.343538	−	0.595025i
$474$		0			0
$475$	220.000			0.0212511
$476$		0			0
$477$		0			0
$478$	3540.00	+	6131.46i	0.338736	+	0.586708i
$479$	−1369.50	+	2372.04i	−0.130635	+	0.226266i	−0.923921	−	0.382582i	$-0.875035\pi$
$479$	−1369.50	+	2372.04i	−0.130635	+	0.226266i	0.793287	+	0.608848i	$0.208368\pi$
$480$		0			0
$481$	8855.00	+	15337.3i	0.839404	+	1.45389i
$482$	10462.0			0.988654
$483$		0			0
$484$	7672.00			0.720511
$485$	8253.00	+	14294.6i	0.772679	+	1.33832i
$486$		0			0
$487$	−8525.50	+	14766.6i	−0.793280	+	1.37400i	0.130646	+	0.991429i	$0.458295\pi$
$487$	−8525.50	+	14766.6i	−0.793280	+	1.37400i	−0.923926	+	0.382572i	$0.875038\pi$
$488$	2836.00	+	4912.10i	0.263073	+	0.455656i
$489$		0			0
$490$		0			0
$491$	4296.00			0.394859			0.197429	−	0.980317i	$-0.436741\pi$
$491$	4296.00			0.394859			0.197429	+	0.980317i	$0.436741\pi$
$492$		0			0
$493$	2907.00	−	5035.07i	0.265567	−	0.459976i
$494$	−350.000	+	606.218i	−0.0318770	+	0.0552126i
$495$		0			0
$496$	−368.000			−0.0333139
$497$		0			0
$498$		0			0
$499$	−1700.50	−	2945.35i	−0.152555	−	0.264233i	0.779611	−	0.626264i	$-0.215417\pi$
$499$	−1700.50	−	2945.35i	−0.152555	−	0.264233i	−0.932166	+	0.362031i	$0.882083\pi$
$500$	−3042.00	+	5268.90i	−0.272085	+	0.471265i
$501$		0			0
$502$	5040.00	+	8729.54i	0.448100	+	0.776132i
$503$	16800.0			1.48921			0.744607	−	0.667503i	$-0.232637\pi$
$503$	16800.0			1.48921			0.744607	+	0.667503i	$0.232637\pi$
$504$		0			0
$505$	2565.00			0.226022
$506$	−3933.00	−	6812.16i	−0.345540	−	0.598493i
$507$		0			0
$508$	4112.00	−	7122.19i	0.359135	−	0.622040i
$509$	−919.500	−	1592.62i	−0.0800710	−	0.138687i	0.823209	−	0.567738i	$-0.192181\pi$
$509$	−919.500	−	1592.62i	−0.0800710	−	0.138687i	−0.903280	+	0.429051i	$0.858848\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−1437.00	+	2488.96i	−0.123314	+	0.213586i
$515$	2245.50	−	3889.32i	0.192133	−	0.332784i
$516$		0			0
$517$	11457.0			0.974620
$518$		0			0
$519$		0			0
$520$	−2520.00	−	4364.77i	−0.212518	−	0.368092i
$521$	−151.500	+	262.406i	−0.0127396	+	0.0220656i	−0.872325	−	0.488927i	$-0.837389\pi$
$521$	−151.500	+	262.406i	−0.0127396	+	0.0220656i	0.859585	+	0.510992i	$0.170722\pi$
$522$		0			0
$523$	−10833.5	−	18764.2i	−0.905767	−	1.56883i	−0.819885	−	0.572528i	$-0.805963\pi$
$523$	−10833.5	−	18764.2i	−0.905767	−	1.56883i	−0.0858815	−	0.996305i	$-0.527371\pi$
$524$	8196.00			0.683290
$525$		0			0
$526$	−4650.00			−0.385456
$527$	586.500	+	1015.85i	0.0484788	+	0.0839678i
$528$		0			0
$529$	3703.00	−	6413.78i	0.304348	−	0.527146i
$530$	−3537.00	−	6126.26i	−0.289882	−	0.502090i
$531$		0			0
$532$		0			0
$533$	−2940.00			−0.238922
$534$		0			0
$535$	4981.50	−	8628.21i	0.402559	−	0.697253i
$536$	1676.00	−	2902.92i	0.135060	−	0.233931i
$537$		0			0
$538$	4770.00			0.382248
$539$		0			0
$540$		0			0
$541$	−2519.50	−	4363.90i	−0.200225	−	0.346800i	0.748376	−	0.663275i	$-0.230834\pi$
$541$	−2519.50	−	4363.90i	−0.200225	−	0.346800i	−0.948601	+	0.316475i	$0.897501\pi$
$542$	331.000	−	573.309i	0.0262319	−	0.0454349i
$543$		0			0
$544$	816.000	+	1413.35i	0.0643120	+	0.111392i
$545$	−8307.00			−0.652904
$546$		0			0
$547$	−2392.00			−0.186974			−0.0934868	−	0.995621i	$-0.529801\pi$
$547$	−2392.00			−0.186974			−0.0934868	+	0.995621i	$0.529801\pi$
$548$	−282.000	−	488.438i	−0.0219826	−	0.0380749i
$549$		0			0
$550$	−2508.00	+	4343.98i	−0.194439	+	0.336778i
$551$	−285.000	−	493.634i	−0.0220352	−	0.0381661i
$552$		0			0
$553$		0			0
$554$	−9742.00			−0.747108
$555$		0			0
$556$	2968.00	−	5140.73i	0.226387	−	0.392114i
$557$	−11074.5	+	19181.6i	−0.842445	+	1.45916i	0.0453775	+	0.998970i	$0.485551\pi$
$557$	−11074.5	+	19181.6i	−0.842445	+	1.45916i	−0.887822	+	0.460187i	$0.847782\pi$
$558$		0			0
$559$	−8680.00			−0.656753
$560$		0			0
$561$		0			0
$562$	7026.00	+	12169.4i	0.527356	+	0.913407i
$563$	4174.50	−	7230.45i	0.312494	−	0.541256i	−0.666408	−	0.745588i	$-0.732169\pi$
$563$	4174.50	−	7230.45i	0.312494	−	0.541256i	0.978902	+	0.204332i	$0.0655022\pi$
$564$		0			0
$565$	−6939.00	−	12018.7i	−0.516683	−	0.894921i
$566$	−10706.0			−0.795065
$567$		0			0
$568$	−768.000			−0.0567334
$569$	−7672.50	−	13289.2i	−0.565286	−	0.979105i	−0.997023	−	0.0771050i	$-0.975432\pi$
$569$	−7672.50	−	13289.2i	−0.565286	−	0.979105i	0.431737	−	0.902000i	$-0.357901\pi$
$570$		0			0
$571$	5796.50	−	10039.8i	0.424827	−	0.735821i	−0.571578	−	0.820548i	$-0.693668\pi$
$571$	5796.50	−	10039.8i	0.424827	−	0.735821i	0.996404	+	0.0847268i	$0.0270017\pi$
$572$	−7980.00	−	13821.8i	−0.583323	−	1.01034i
$573$		0			0
$574$		0			0
$575$	3036.00			0.220191
$576$		0			0
$577$	−7296.50	+	12637.9i	−0.526442	+	0.911825i	0.473083	+	0.881018i	$0.343141\pi$
$577$	−7296.50	+	12637.9i	−0.526442	+	0.911825i	−0.999525	+	0.0308071i	$0.990192\pi$
$578$	−2312.00	+	4004.50i	−0.166378	+	0.288175i
$579$		0			0
$580$	4104.00			0.293809
$581$		0			0
$582$		0			0
$583$	−11200.5	−	19399.8i	−0.795673	−	1.37815i
$584$	1252.00	−	2168.53i	0.0887125	−	0.153655i
$585$		0			0
$586$	4158.00	+	7201.87i	0.293115	+	0.507690i
$587$	−15372.0			−1.08087			−0.540435	−	0.841386i	$-0.681740\pi$
$587$	−15372.0			−1.08087			−0.540435	+	0.841386i	$0.681740\pi$
$588$		0			0
$589$	115.000			0.00804498
$590$	−1971.00	−	3413.87i	−0.137534	−	0.238215i
$591$		0			0
$592$	2024.00	−	3505.67i	0.140517	−	0.243382i
$593$	7186.50	+	12447.4i	0.497663	+	0.861978i	0.999996	−	0.00269639i	$-0.000858288\pi$
$593$	7186.50	+	12447.4i	0.497663	+	0.861978i	−0.502333	+	0.864674i	$0.667525\pi$
$594$		0			0
$595$		0			0
$596$	228.000			0.0156699
$597$		0			0
$598$	−4830.00	+	8365.81i	−0.330290	+	0.572079i
$599$	1273.50	−	2205.77i	0.0868678	−	0.150459i	−0.819318	−	0.573340i	$-0.805648\pi$
$599$	1273.50	−	2205.77i	0.0868678	−	0.150459i	0.906186	+	0.422880i	$0.138981\pi$
$600$		0			0
$601$	7042.00			0.477952			0.238976	−	0.971025i	$-0.423188\pi$
$601$	7042.00			0.477952			0.238976	+	0.971025i	$0.423188\pi$
$602$		0			0
$603$		0			0
$604$	−1678.00	−	2906.38i	−0.113041	−	0.195793i
$605$	8631.00	−	14949.3i	0.580000	−	1.00459i
$606$		0			0
$607$	−11295.5	−	19564.4i	−0.755305	−	1.30823i	−0.945223	−	0.326427i	$-0.894155\pi$
$607$	−11295.5	−	19564.4i	−0.755305	−	1.30823i	0.189917	−	0.981800i	$-0.439178\pi$
$608$	160.000			0.0106725
$609$		0			0
$610$	12762.0			0.847079
$611$	−7035.00	−	12185.0i	−0.465803	−	0.806794i
$612$		0			0
$613$	4242.50	−	7348.23i	0.279532	−	0.484163i	−0.691737	−	0.722150i	$-0.743154\pi$
$613$	4242.50	−	7348.23i	0.279532	−	0.484163i	0.971268	+	0.237987i	$0.0764874\pi$
$614$	9604.00	+	16634.6i	0.631247	+	1.09335i
$615$		0			0
$616$		0			0
$617$	18282.0			1.19288			0.596439	−	0.802658i	$-0.296582\pi$
$617$	18282.0			1.19288			0.596439	+	0.802658i	$0.296582\pi$
$618$		0			0
$619$	1145.50	−	1984.06i	0.0743805	−	0.128831i	−0.826436	−	0.563030i	$-0.809635\pi$
$619$	1145.50	−	1984.06i	0.0743805	−	0.128831i	0.900817	+	0.434200i	$0.142969\pi$
$620$	−414.000	+	717.069i	−0.0268172	+	0.0464487i
$621$		0			0
$622$	−20262.0			−1.30616
$623$		0			0
$624$		0			0
$625$	4094.50	+	7091.88i	0.262048	+	0.453880i
$626$	−10799.0	+	18704.4i	−0.689481	+	1.19422i
$627$		0			0
$628$	−5666.00	−	9813.80i	−0.360029	−	0.623588i
$629$	−12903.0			−0.817927
$630$		0			0
$631$	−6928.00			−0.437083			−0.218541	−	0.975828i	$-0.570130\pi$
$631$	−6928.00			−0.437083			−0.218541	+	0.975828i	$0.570130\pi$
$632$	1844.00	+	3193.90i	0.116061	+	0.201023i
$633$		0			0
$634$	−531.000	+	919.719i	−0.0332629	+	0.0576131i
$635$	−9252.00	−	16024.9i	−0.578196	−	1.00146i
$636$		0			0
$637$		0			0
$638$	12996.0			0.806452
$639$		0			0
$640$	−576.000	+	997.661i	−0.0355756	+	0.0616188i
$641$	12487.5	−	21629.0i	0.769464	−	1.33275i	−0.168390	−	0.985721i	$-0.553857\pi$
$641$	12487.5	−	21629.0i	0.769464	−	1.33275i	0.937854	−	0.347031i	$-0.112810\pi$
$642$		0			0
$643$	−9548.00			−0.585593			−0.292797	−	0.956175i	$-0.594586\pi$
$643$	−9548.00			−0.585593			−0.292797	+	0.956175i	$0.594586\pi$
$644$		0			0
$645$		0			0
$646$	−255.000	−	441.673i	−0.0155307	−	0.0269000i
$647$	−5065.50	+	8773.70i	−0.307798	+	0.533122i	−0.977880	−	0.209165i	$-0.932925\pi$
$647$	−5065.50	+	8773.70i	−0.307798	+	0.533122i	0.670082	+	0.742287i	$0.266259\pi$
$648$		0			0
$649$	−6241.50	−	10810.6i	−0.377504	−	0.653857i
$650$	6160.00			0.371716
$651$		0			0
$652$	−9244.00			−0.555250
$653$	8329.50	+	14427.1i	0.499171	+	0.864589i	1.00000		0.000957229i	$-0.000304695\pi$
$653$	8329.50	+	14427.1i	0.499171	+	0.864589i	−0.500829	+	0.865546i	$0.666971\pi$
$654$		0			0
$655$	9220.50	−	15970.4i	0.550038	−	0.952693i
$656$	336.000	+	581.969i	0.0199979	+	0.0346373i
$657$		0			0
$658$		0			0
$659$	−29556.0			−1.74710			−0.873550	−	0.486735i	$-0.838188\pi$
$659$	−29556.0			−1.74710			−0.873550	+	0.486735i	$0.838188\pi$
$660$		0			0
$661$	95.5000	−	165.411i	0.00561955	−	0.00973334i	−0.863202	−	0.504859i	$-0.831545\pi$
$661$	95.5000	−	165.411i	0.00561955	−	0.00973334i	0.868822	+	0.495125i	$0.164878\pi$
$662$	−7015.00	+	12150.3i	−0.411852	+	0.713348i
$663$		0			0
$664$	4704.00			0.274926
$665$		0			0
$666$		0			0
$667$	−3933.00	−	6812.16i	−0.228315	−	0.395454i
$668$	−2520.00	+	4364.77i	−0.145961	+	0.252811i
$669$		0			0
$670$	−3771.00	−	6531.56i	−0.217442	−	0.376621i
$671$	40413.0			2.32508
$672$		0			0
$673$	2606.00			0.149263			0.0746314	−	0.997211i	$-0.476222\pi$
$673$	2606.00			0.149263			0.0746314	+	0.997211i	$0.476222\pi$
$674$	8990.00	+	15571.1i	0.513771	+	0.889878i
$675$		0			0
$676$	−5406.00	+	9363.47i	−0.307579	+	0.532742i
$677$	2104.50	+	3645.10i	0.119472	+	0.206931i	0.919559	−	0.392953i	$-0.128547\pi$
$677$	2104.50	+	3645.10i	0.119472	+	0.206931i	−0.800087	+	0.599885i	$0.795213\pi$
$678$		0			0
$679$		0			0
$680$	3672.00			0.207081
$681$		0			0
$682$	−1311.00	+	2270.72i	−0.0736082	+	0.127493i
$683$	12151.5	−	21047.0i	0.680768	−	1.17912i	−0.293979	−	0.955812i	$-0.594980\pi$
$683$	12151.5	−	21047.0i	0.680768	−	1.17912i	0.974747	−	0.223312i	$-0.0716869\pi$
$684$		0			0
$685$	−1269.00			−0.0707825
$686$		0			0
$687$		0			0
$688$	992.000	+	1718.19i	0.0549704	+	0.0952116i
$689$	−13755.0	+	23824.4i	−0.760557	+	1.31732i
$690$		0			0
$691$	7520.50	+	13025.9i	0.414028	+	0.717117i	0.995326	−	0.0965734i	$-0.0307882\pi$
$691$	7520.50	+	13025.9i	0.414028	+	0.717117i	−0.581298	+	0.813691i	$0.697455\pi$
$692$	13068.0			0.717877
$693$		0			0
$694$	−17418.0			−0.952706
$695$	−6678.00	−	11566.6i	−0.364476	−	0.631291i
$696$		0			0
$697$	1071.00	−	1855.03i	0.0582023	−	0.100809i
$698$	−6482.00	−	11227.2i	−0.351500	−	0.608817i
$699$		0			0
$700$		0			0
$701$	−24726.0			−1.33222			−0.666111	−	0.745852i	$-0.732042\pi$
$701$	−24726.0			−1.33222			−0.666111	+	0.745852i	$0.732042\pi$
$702$		0			0
$703$	−632.500	+	1095.52i	−0.0339334	+	0.0587744i
$704$	−1824.00	+	3159.26i	−0.0976486	+	0.169132i
$705$		0			0
$706$	4266.00			0.227412
$707$		0			0
$708$		0			0
$709$	2478.50	+	4292.89i	0.131286	+	0.227395i	0.924173	−	0.381975i	$-0.124756\pi$
$709$	2478.50	+	4292.89i	0.131286	+	0.227395i	−0.792886	+	0.609370i	$0.791423\pi$
$710$	−864.000	+	1496.49i	−0.0456695	+	0.0791019i
$711$		0			0
$712$	−4068.00	−	7045.98i	−0.214122	−	0.370870i
$713$	1587.00			0.0833571
$714$		0			0
$715$	−35910.0			−1.87826
$716$	2574.00	+	4458.30i	0.134350	+	0.232702i
$717$		0			0
$718$	−3849.00	+	6666.66i	−0.200060	+	0.346515i
$719$	−13834.5	−	23962.1i	−0.717580	−	1.24288i	−0.961956	−	0.273204i	$-0.911917\pi$
$719$	−13834.5	−	23962.1i	−0.717580	−	1.24288i	0.244376	−	0.969680i	$-0.421417\pi$
$720$		0			0
$721$		0			0
$722$	13668.0			0.704529
$723$		0			0
$724$	−5348.00	+	9263.01i	−0.274526	+	0.475493i
$725$	−2508.00	+	4343.98i	−0.128476	+	0.222526i
$726$		0			0
$727$	13888.0			0.708497			0.354249	−	0.935151i	$-0.384737\pi$
$727$	13888.0			0.708497			0.354249	+	0.935151i	$0.384737\pi$
$728$		0			0
$729$		0			0
$730$	−2817.00	−	4879.19i	−0.142824	−	0.247379i
$731$	3162.00	−	5476.74i	0.159987	−	0.277106i
$732$		0			0
$733$	7121.50	+	12334.8i	0.358852	+	0.621550i	0.987769	−	0.155922i	$-0.0498349\pi$
$733$	7121.50	+	12334.8i	0.358852	+	0.621550i	−0.628917	+	0.777472i	$0.716502\pi$
$734$	12982.0			0.652826
$735$		0			0
$736$	2208.00			0.110581
$737$	−11941.5	−	20683.3i	−0.596840	−	1.03376i
$738$		0			0
$739$	−18479.5	+	32007.4i	−0.919864	+	1.59325i	−0.120244	+	0.992744i	$0.538368\pi$
$739$	−18479.5	+	32007.4i	−0.919864	+	1.59325i	−0.799620	+	0.600507i	$0.794966\pi$
$740$	−4554.00	−	7887.76i	−0.226228	−	0.391838i
$741$		0			0
$742$		0			0
$743$	12528.0			0.618584			0.309292	−	0.950967i	$-0.399908\pi$
$743$	12528.0			0.618584			0.309292	+	0.950967i	$0.399908\pi$
$744$		0			0
$745$	256.500	−	444.271i	0.0126140	−	0.0218481i
$746$	923.000	−	1598.68i	0.0452995	−	0.0784610i
$747$		0			0
$748$	11628.0			0.568398
$749$		0			0
$750$		0			0
$751$	8883.50	+	15386.7i	0.431643	+	0.747627i	0.997015	−	0.0772090i	$-0.0246009\pi$
$751$	8883.50	+	15386.7i	0.431643	+	0.747627i	−0.565372	+	0.824836i	$0.691268\pi$
$752$	−1608.00	+	2785.14i	−0.0779757	+	0.135058i
$753$		0			0
$754$	−7980.00	−	13821.8i	−0.385430	−	0.667585i
$755$	−7551.00			−0.363985
$756$		0			0
$757$	−28726.0			−1.37921			−0.689606	−	0.724184i	$-0.742216\pi$
$757$	−28726.0			−1.37921			−0.689606	+	0.724184i	$0.742216\pi$
$758$	6344.00	+	10988.1i	0.303990	+	0.526526i
$759$		0			0
$760$	180.000	−	311.769i	0.00859117	−	0.0148803i
$761$	13234.5	+	22922.8i	0.630421	+	1.09192i	0.987466	+	0.157834i	$0.0504510\pi$
$761$	13234.5	+	22922.8i	0.630421	+	1.09192i	−0.357045	+	0.934087i	$0.616216\pi$
$762$		0			0
$763$		0			0
$764$	−16740.0			−0.792712
$765$		0			0
$766$	−5007.00	+	8672.38i	−0.236175	+	0.409068i
$767$	−7665.00	+	13276.2i	−0.360844	+	0.625000i
$768$		0			0
$769$	−5054.00			−0.236999			−0.118499	−	0.992954i	$-0.537808\pi$
$769$	−5054.00			−0.236999			−0.118499	+	0.992954i	$0.537808\pi$
$770$		0			0
$771$		0			0
$772$	170.000	+	294.449i	0.00792543	+	0.0137273i
$773$	17782.5	−	30800.2i	0.827415	−	1.43313i	−0.0726439	−	0.997358i	$-0.523144\pi$
$773$	17782.5	−	30800.2i	0.827415	−	1.43313i	0.900059	−	0.435767i	$-0.143523\pi$
$774$		0			0
$775$	−506.000	−	876.418i	−0.0234530	−	0.0406217i
$776$	−14672.0			−0.678730
$777$		0			0
$778$	24582.0			1.13279
$779$	−105.000	−	181.865i	−0.00482929	−	0.00836457i
$780$		0			0
$781$	−2736.00	+	4738.89i	−0.125354	+	0.217120i
$782$	−3519.00	−	6095.09i	−0.160920	−	0.278721i
$783$		0			0
$784$		0			0
$785$	−25497.0			−1.15927

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(4.50000 + 7.79423i) q^{5} -8.00000 q^{8} +(-9.00000 + 15.5885i) q^{10} +(-28.5000 + 49.3634i) q^{11} +70.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(-25.5000 + 44.1673i) q^{17} +(2.50000 + 4.33013i) q^{19} -36.0000 q^{20} -114.000 q^{22} +(34.5000 + 59.7558i) q^{23} +(22.0000 - 38.1051i) q^{25} +(70.0000 + 121.244i) q^{26} -114.000 q^{29} +(11.5000 - 19.9186i) q^{31} +(16.0000 - 27.7128i) q^{32} -102.000 q^{34} +(126.500 + 219.104i) q^{37} +(-5.00000 + 8.66025i) q^{38} +(-36.0000 - 62.3538i) q^{40} -42.0000 q^{41} -124.000 q^{43} +(-114.000 - 197.454i) q^{44} +(-69.0000 + 119.512i) q^{46} +(-100.500 - 174.071i) q^{47} +88.0000 q^{50} +(-140.000 + 242.487i) q^{52} +(-196.500 + 340.348i) q^{53} -513.000 q^{55} +(-114.000 - 197.454i) q^{58} +(-109.500 + 189.660i) q^{59} +(-354.500 - 614.012i) q^{61} +46.0000 q^{62} +64.0000 q^{64} +(315.000 + 545.596i) q^{65} +(-209.500 + 362.865i) q^{67} +(-102.000 - 176.669i) q^{68} +96.0000 q^{71} +(-156.500 + 271.066i) q^{73} +(-253.000 + 438.209i) q^{74} -20.0000 q^{76} +(-230.500 - 399.238i) q^{79} +(72.0000 - 124.708i) q^{80} +(-42.0000 - 72.7461i) q^{82} -588.000 q^{83} -459.000 q^{85} +(-124.000 - 214.774i) q^{86} +(228.000 - 394.908i) q^{88} +(508.500 + 880.748i) q^{89} -276.000 q^{92} +(201.000 - 348.142i) q^{94} +(-22.5000 + 38.9711i) q^{95} +1834.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8	\( 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8} - 18 q^{10} - 57 q^{11} + 140 q^{13} - 16 q^{16} - 51 q^{17} + 5 q^{19} - 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} + 140 q^{26} - 228 q^{29} + 23 q^{31} + 32 q^{32} - 204 q^{34} + 253 q^{37} - 10 q^{38} - 72 q^{40} - 84 q^{41} - 248 q^{43} - 228 q^{44} - 138 q^{46} - 201 q^{47} + 176 q^{50} - 280 q^{52} - 393 q^{53} - 1026 q^{55} - 228 q^{58} - 219 q^{59} - 709 q^{61} + 92 q^{62} + 128 q^{64} + 630 q^{65} - 419 q^{67} - 204 q^{68} + 192 q^{71} - 313 q^{73} - 506 q^{74} - 40 q^{76} - 461 q^{79} + 144 q^{80} - 84 q^{82} - 1176 q^{83} - 918 q^{85} - 248 q^{86} + 456 q^{88} + 1017 q^{89} - 552 q^{92} + 402 q^{94} - 45 q^{95} + 3668 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 + 9 * q^5 - 16 * q^8 - 18 * q^10 - 57 * q^11 + 140 * q^13 - 16 * q^16 - 51 * q^17 + 5 * q^19 - 72 * q^20 - 228 * q^22 + 69 * q^23 + 44 * q^25 + 140 * q^26 - 228 * q^29 + 23 * q^31 + 32 * q^32 - 204 * q^34 + 253 * q^37 - 10 * q^38 - 72 * q^40 - 84 * q^41 - 248 * q^43 - 228 * q^44 - 138 * q^46 - 201 * q^47 + 176 * q^50 - 280 * q^52 - 393 * q^53 - 1026 * q^55 - 228 * q^58 - 219 * q^59 - 709 * q^61 + 92 * q^62 + 128 * q^64 + 630 * q^65 - 419 * q^67 - 204 * q^68 + 192 * q^71 - 313 * q^73 - 506 * q^74 - 40 * q^76 - 461 * q^79 + 144 * q^80 - 84 * q^82 - 1176 * q^83 - 918 * q^85 - 248 * q^86 + 456 * q^88 + 1017 * q^89 - 552 * q^92 + 402 * q^94 - 45 * q^95 + 3668 * q^97

Introduction

L-functions

Modular forms

Varieties

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