LMFDB - Embedded newform 882.4.g.d.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.d.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +8.00000 q^{8} +(-7.00000 + 12.1244i) q^{10} +(17.5000 - 30.3109i) q^{11} -66.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(-29.5000 + 51.0955i) q^{17} +(68.5000 + 118.645i) q^{19} +28.0000 q^{20} -70.0000 q^{22} +(-3.50000 - 6.06218i) q^{23} +(38.0000 - 65.8179i) q^{25} +(66.0000 + 114.315i) q^{26} -106.000 q^{29} +(37.5000 - 64.9519i) q^{31} +(-16.0000 + 27.7128i) q^{32} +118.000 q^{34} +(-5.50000 - 9.52628i) q^{37} +(137.000 - 237.291i) q^{38} +(-28.0000 - 48.4974i) q^{40} -498.000 q^{41} +260.000 q^{43} +(70.0000 + 121.244i) q^{44} +(-7.00000 + 12.1244i) q^{46} +(85.5000 + 148.090i) q^{47} -152.000 q^{50} +(132.000 - 228.631i) q^{52} +(-208.500 + 361.133i) q^{53} -245.000 q^{55} +(106.000 + 183.597i) q^{58} +(8.50000 - 14.7224i) q^{59} +(25.5000 + 44.1673i) q^{61} -150.000 q^{62} +64.0000 q^{64} +(231.000 + 400.104i) q^{65} +(-219.500 + 380.185i) q^{67} +(-118.000 - 204.382i) q^{68} +784.000 q^{71} +(147.500 - 255.477i) q^{73} +(-11.0000 + 19.0526i) q^{74} -548.000 q^{76} +(247.500 + 428.683i) q^{79} +(-56.0000 + 96.9948i) q^{80} +(498.000 + 862.561i) q^{82} +932.000 q^{83} +413.000 q^{85} +(-260.000 - 450.333i) q^{86} +(140.000 - 242.487i) q^{88} +(436.500 + 756.040i) q^{89} +28.0000 q^{92} +(171.000 - 296.181i) q^{94} +(479.500 - 830.518i) q^{95} +290.000 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 4 * q^4 - 7 * q^5 + 16 * q^8	$ 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8} - 14 q^{10} + 35 q^{11} - 132 q^{13} - 16 q^{16} - 59 q^{17} + 137 q^{19} + 56 q^{20} - 140 q^{22} - 7 q^{23} + 76 q^{25} + 132 q^{26} - 212 q^{29} + 75 q^{31} - 32 q^{32} + 236 q^{34} - 11 q^{37} + 274 q^{38} - 56 q^{40} - 996 q^{41} + 520 q^{43} + 140 q^{44} - 14 q^{46} + 171 q^{47} - 304 q^{50} + 264 q^{52} - 417 q^{53} - 490 q^{55} + 212 q^{58} + 17 q^{59} + 51 q^{61} - 300 q^{62} + 128 q^{64} + 462 q^{65} - 439 q^{67} - 236 q^{68} + 1568 q^{71} + 295 q^{73} - 22 q^{74} - 1096 q^{76} + 495 q^{79} - 112 q^{80} + 996 q^{82} + 1864 q^{83} + 826 q^{85} - 520 q^{86} + 280 q^{88} + 873 q^{89} + 56 q^{92} + 342 q^{94} + 959 q^{95} + 580 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 - 7 * q^5 + 16 * q^8 - 14 * q^10 + 35 * q^11 - 132 * q^13 - 16 * q^16 - 59 * q^17 + 137 * q^19 + 56 * q^20 - 140 * q^22 - 7 * q^23 + 76 * q^25 + 132 * q^26 - 212 * q^29 + 75 * q^31 - 32 * q^32 + 236 * q^34 - 11 * q^37 + 274 * q^38 - 56 * q^40 - 996 * q^41 + 520 * q^43 + 140 * q^44 - 14 * q^46 + 171 * q^47 - 304 * q^50 + 264 * q^52 - 417 * q^53 - 490 * q^55 + 212 * q^58 + 17 * q^59 + 51 * q^61 - 300 * q^62 + 128 * q^64 + 462 * q^65 - 439 * q^67 - 236 * q^68 + 1568 * q^71 + 295 * q^73 - 22 * q^74 - 1096 * q^76 + 495 * q^79 - 112 * q^80 + 996 * q^82 + 1864 * q^83 + 826 * q^85 - 520 * q^86 + 280 * q^88 + 873 * q^89 + 56 * q^92 + 342 * q^94 + 959 * q^95 + 580 * 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
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	0.665971	−	0.745977i	$-0.268017\pi$
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	−0.979021	+	0.203760i	$0.934684\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	8.00000			0.353553
$9$		0			0
$10$	−7.00000	+	12.1244i	−0.221359	+	0.383406i
$11$	17.5000	−	30.3109i	0.479677	−	0.830825i	−0.520051	−	0.854135i	$-0.674087\pi$
$11$	17.5000	−	30.3109i	0.479677	−	0.830825i	0.999728	+	0.0233099i	$0.00742046\pi$
$12$		0			0
$13$	−66.0000			−1.40809			−0.704043	−	0.710158i	$-0.748624\pi$
$13$	−66.0000			−1.40809			−0.704043	+	0.710158i	$0.748624\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−29.5000	+	51.0955i	−0.420871	+	0.728969i	−0.996025	−	0.0890757i	$-0.971609\pi$
$17$	−29.5000	+	51.0955i	−0.420871	+	0.728969i	0.575154	+	0.818045i	$0.304942\pi$
$18$		0			0
$19$	68.5000	+	118.645i	0.827104	+	1.43259i	0.900301	+	0.435269i	$0.143347\pi$
$19$	68.5000	+	118.645i	0.827104	+	1.43259i	−0.0731965	+	0.997318i	$0.523320\pi$
$20$	28.0000			0.313050
$21$		0			0
$22$	−70.0000			−0.678366
$23$	−3.50000	−	6.06218i	−0.0317305	−	0.0549588i	0.849724	−	0.527228i	$-0.176768\pi$
$23$	−3.50000	−	6.06218i	−0.0317305	−	0.0549588i	−0.881455	+	0.472269i	$0.843435\pi$
$24$		0			0
$25$	38.0000	−	65.8179i	0.304000	−	0.526543i
$26$	66.0000	+	114.315i	0.497833	+	0.862273i
$27$		0			0
$28$		0			0
$29$	−106.000			−0.678748			−0.339374	−	0.940651i	$-0.610215\pi$
$29$	−106.000			−0.678748			−0.339374	+	0.940651i	$0.610215\pi$
$30$		0			0
$31$	37.5000	−	64.9519i	0.217264	−	0.376313i	−0.736706	−	0.676213i	$-0.763620\pi$
$31$	37.5000	−	64.9519i	0.217264	−	0.376313i	0.953971	+	0.299900i	$0.0969533\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	118.000			0.595201
$35$		0			0
$36$		0			0
$37$	−5.50000	−	9.52628i	−0.0244377	−	0.0423273i	0.853548	−	0.521014i	$-0.174446\pi$
$37$	−5.50000	−	9.52628i	−0.0244377	−	0.0423273i	−0.877986	+	0.478687i	$0.841113\pi$
$38$	137.000	−	237.291i	0.584851	−	1.01299i
$39$		0			0
$40$	−28.0000	−	48.4974i	−0.110680	−	0.191703i
$41$	−498.000			−1.89694			−0.948470	−	0.316867i	$-0.897369\pi$
$41$	−498.000			−1.89694			−0.948470	+	0.316867i	$0.897369\pi$
$42$		0			0
$43$	260.000			0.922084			0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000			0.922084			0.461042	+	0.887378i	$0.347476\pi$
$44$	70.0000	+	121.244i	0.239839	+	0.415413i
$45$		0			0
$46$	−7.00000	+	12.1244i	−0.0224368	+	0.0388617i
$47$	85.5000	+	148.090i	0.265350	+	0.459600i	0.967655	−	0.252276i	$-0.0811791\pi$
$47$	85.5000	+	148.090i	0.265350	+	0.459600i	−0.702305	+	0.711876i	$0.747846\pi$
$48$		0			0
$49$		0			0
$50$	−152.000			−0.429921
$51$		0			0
$52$	132.000	−	228.631i	0.352021	−	0.609719i
$53$	−208.500	+	361.133i	−0.540371	+	0.935951i	0.458511	+	0.888689i	$0.348383\pi$
$53$	−208.500	+	361.133i	−0.540371	+	0.935951i	−0.998883	+	0.0472619i	$0.984950\pi$
$54$		0			0
$55$	−245.000			−0.600651
$56$		0			0
$57$		0			0
$58$	106.000	+	183.597i	0.239974	+	0.415647i
$59$	8.50000	−	14.7224i	0.0187560	−	0.0324864i	−0.856495	−	0.516155i	$-0.827363\pi$
$59$	8.50000	−	14.7224i	0.0187560	−	0.0324864i	0.875251	+	0.483669i	$0.160696\pi$
$60$		0			0
$61$	25.5000	+	44.1673i	0.0535236	+	0.0927056i	0.891546	−	0.452930i	$-0.149621\pi$
$61$	25.5000	+	44.1673i	0.0535236	+	0.0927056i	−0.838022	+	0.545636i	$0.816288\pi$
$62$	−150.000			−0.307258
$63$		0			0
$64$	64.0000			0.125000
$65$	231.000	+	400.104i	0.440800	+	0.763489i
$66$		0			0
$67$	−219.500	+	380.185i	−0.400242	+	0.693239i	−0.993755	−	0.111585i	$-0.964407\pi$
$67$	−219.500	+	380.185i	−0.400242	+	0.693239i	0.593513	+	0.804824i	$0.297740\pi$
$68$	−118.000	−	204.382i	−0.210435	−	0.364485i
$69$		0			0
$70$		0			0
$71$	784.000			1.31047			0.655237	−	0.755423i	$-0.272569\pi$
$71$	784.000			1.31047			0.655237	+	0.755423i	$0.272569\pi$
$72$		0			0
$73$	147.500	−	255.477i	0.236487	−	0.409608i	−0.723217	−	0.690621i	$-0.757337\pi$
$73$	147.500	−	255.477i	0.236487	−	0.409608i	0.959704	+	0.281013i	$0.0906705\pi$
$74$	−11.0000	+	19.0526i	−0.0172801	+	0.0299299i
$75$		0			0
$76$	−548.000			−0.827104
$77$		0			0
$78$		0			0
$79$	247.500	+	428.683i	0.352480	+	0.610513i	0.986683	−	0.162653i	$-0.0520051\pi$
$79$	247.500	+	428.683i	0.352480	+	0.610513i	−0.634203	+	0.773166i	$0.718672\pi$
$80$	−56.0000	+	96.9948i	−0.0782624	+	0.135554i
$81$		0			0
$82$	498.000	+	862.561i	0.670670	+	1.16163i
$83$	932.000			1.23253			0.616267	−	0.787537i	$-0.288644\pi$
$83$	932.000			1.23253			0.616267	+	0.787537i	$0.288644\pi$
$84$		0			0
$85$	413.000			0.527013
$86$	−260.000	−	450.333i	−0.326006	−	0.564659i
$87$		0			0
$88$	140.000	−	242.487i	0.169591	−	0.293741i
$89$	436.500	+	756.040i	0.519875	+	0.900451i	0.999733	+	0.0231042i	$0.00735495\pi$
$89$	436.500	+	756.040i	0.519875	+	0.900451i	−0.479858	+	0.877346i	$0.659312\pi$
$90$		0			0
$91$		0			0
$92$	28.0000			0.0317305
$93$		0			0
$94$	171.000	−	296.181i	0.187631	−	0.324986i
$95$	479.500	−	830.518i	0.517849	−	0.896941i
$96$		0			0
$97$	290.000			0.303557			0.151779	−	0.988415i	$-0.451500\pi$
$97$	290.000			0.303557			0.151779	+	0.988415i	$0.451500\pi$
$98$		0			0
$99$		0			0
$100$	152.000	+	263.272i	0.152000	+	0.263272i
$101$	542.500	−	939.638i	0.534463	−	0.925717i	−0.464726	−	0.885454i	$-0.653847\pi$
$101$	542.500	−	939.638i	0.534463	−	0.925717i	0.999189	−	0.0402627i	$-0.0128195\pi$
$102$		0			0
$103$	776.500	+	1344.94i	0.742823	+	1.28661i	0.951205	+	0.308560i	$0.0998472\pi$
$103$	776.500	+	1344.94i	0.742823	+	1.28661i	−0.208381	+	0.978048i	$0.566819\pi$
$104$	−528.000			−0.497833
$105$		0			0
$106$	834.000			0.764200
$107$	64.5000	+	111.717i	0.0582752	+	0.100936i	0.893691	−	0.448682i	$-0.148107\pi$
$107$	64.5000	+	111.717i	0.0582752	+	0.100936i	−0.835416	+	0.549618i	$0.814773\pi$
$108$		0			0
$109$	482.500	−	835.715i	0.423992	−	0.734376i	−0.572334	−	0.820021i	$-0.693962\pi$
$109$	482.500	−	835.715i	0.423992	−	0.734376i	0.996326	+	0.0856452i	$0.0272952\pi$
$110$	245.000	+	424.352i	0.212362	+	0.367822i
$111$		0			0
$112$		0			0
$113$	50.0000			0.0416248			0.0208124	−	0.999783i	$-0.493375\pi$
$113$	50.0000			0.0416248			0.0208124	+	0.999783i	$0.493375\pi$
$114$		0			0
$115$	−24.5000	+	42.4352i	−0.0198664	+	0.0344096i
$116$	212.000	−	367.195i	0.169687	−	0.293907i
$117$		0			0
$118$	−34.0000			−0.0265250
$119$		0			0
$120$		0			0
$121$	53.0000	+	91.7987i	0.0398197	+	0.0689697i
$122$	51.0000	−	88.3346i	0.0378469	−	0.0655528i
$123$		0			0
$124$	150.000	+	259.808i	0.108632	+	0.188157i
$125$	−1407.00			−1.00677
$126$		0			0
$127$	936.000			0.653989			0.326994	−	0.945026i	$-0.393964\pi$
$127$	936.000			0.653989			0.326994	+	0.945026i	$0.393964\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	462.000	−	800.207i	0.311693	−	0.539868i
$131$	377.500	+	653.849i	0.251773	+	0.436084i	0.964014	−	0.265851i	$-0.0856529\pi$
$131$	377.500	+	653.849i	0.251773	+	0.436084i	−0.712241	+	0.701935i	$0.752320\pi$
$132$		0			0
$133$		0			0
$134$	878.000			0.566027
$135$		0			0
$136$	−236.000	+	408.764i	−0.148800	+	0.257730i
$137$	−1178.50	+	2041.22i	−0.734935	+	1.27294i	0.219817	+	0.975541i	$0.429454\pi$
$137$	−1178.50	+	2041.22i	−0.734935	+	1.27294i	−0.954752	+	0.297403i	$0.903879\pi$
$138$		0			0
$139$	−28.0000			−0.0170858			−0.00854291	−	0.999964i	$-0.502719\pi$
$139$	−28.0000			−0.0170858			−0.00854291	+	0.999964i	$0.502719\pi$
$140$		0			0
$141$		0			0
$142$	−784.000	−	1357.93i	−0.463323	−	0.802498i
$143$	−1155.00	+	2000.52i	−0.675426	+	1.16987i
$144$		0			0
$145$	371.000	+	642.591i	0.212482	+	0.368029i
$146$	−590.000			−0.334443
$147$		0			0
$148$	44.0000			0.0244377
$149$	1147.50	+	1987.53i	0.630919	+	1.09278i	0.987364	+	0.158467i	$0.0506551\pi$
$149$	1147.50	+	1987.53i	0.630919	+	1.09278i	−0.356446	+	0.934316i	$0.616012\pi$
$150$		0			0
$151$	554.500	−	960.422i	0.298838	−	0.517603i	−0.677032	−	0.735953i	$-0.736734\pi$
$151$	554.500	−	960.422i	0.298838	−	0.517603i	0.975870	+	0.218350i	$0.0700676\pi$
$152$	548.000	+	949.164i	0.292425	+	0.506496i
$153$		0			0
$154$		0			0
$155$	−525.000			−0.272058
$156$		0			0
$157$	779.500	−	1350.13i	0.396248	−	0.686321i	−0.597012	−	0.802232i	$-0.703646\pi$
$157$	779.500	−	1350.13i	0.396248	−	0.686321i	0.993260	+	0.115911i	$0.0369789\pi$
$158$	495.000	−	857.365i	0.249241	−	0.431698i
$159$		0			0
$160$	224.000			0.110680
$161$		0			0
$162$		0			0
$163$	1125.50	+	1949.42i	0.540834	+	0.936752i	0.998856	+	0.0478115i	$0.0152247\pi$
$163$	1125.50	+	1949.42i	0.540834	+	0.936752i	−0.458022	+	0.888941i	$0.651442\pi$
$164$	996.000	−	1725.12i	0.474235	−	0.821399i
$165$		0			0
$166$	−932.000	−	1614.27i	−0.435766	−	0.754770i
$167$	2788.00			1.29187			0.645934	−	0.763393i	$-0.276468\pi$
$167$	2788.00			1.29187			0.645934	+	0.763393i	$0.276468\pi$
$168$		0			0
$169$	2159.00			0.982704
$170$	−413.000	−	715.337i	−0.186327	−	0.322728i
$171$		0			0
$172$	−520.000	+	900.666i	−0.230521	+	0.399274i
$173$	−789.500	−	1367.45i	−0.346963	−	0.600957i	0.638746	−	0.769418i	$-0.279454\pi$
$173$	−789.500	−	1367.45i	−0.346963	−	0.600957i	−0.985708	+	0.168461i	$0.946120\pi$
$174$		0			0
$175$		0			0
$176$	−560.000			−0.239839
$177$		0			0
$178$	873.000	−	1512.08i	0.367607	−	0.636715i
$179$	1225.50	−	2122.63i	0.511722	−	0.886328i	−0.488186	−	0.872740i	$-0.662341\pi$
$179$	1225.50	−	2122.63i	0.511722	−	0.886328i	0.999908	−	0.0135883i	$-0.00432541\pi$
$180$		0			0
$181$	1170.00			0.480472			0.240236	−	0.970715i	$-0.422775\pi$
$181$	1170.00			0.480472			0.240236	+	0.970715i	$0.422775\pi$
$182$		0			0
$183$		0			0
$184$	−28.0000	−	48.4974i	−0.0112184	−	0.0194309i
$185$	−38.5000	+	66.6840i	−0.0153004	+	0.0265011i
$186$		0			0
$187$	1032.50	+	1788.34i	0.403764	+	0.699340i
$188$	−684.000			−0.265350
$189$		0			0
$190$	−1918.00			−0.732349
$191$	−637.500	−	1104.18i	−0.241507	−	0.418303i	0.719637	−	0.694351i	$-0.244308\pi$
$191$	−637.500	−	1104.18i	−0.241507	−	0.418303i	−0.961144	+	0.276048i	$0.910975\pi$
$192$		0			0
$193$	−17.5000	+	30.3109i	−0.00652683	+	0.0113048i	−0.869270	−	0.494337i	$-0.835411\pi$
$193$	−17.5000	+	30.3109i	−0.00652683	+	0.0113048i	0.862744	+	0.505642i	$0.168744\pi$
$194$	−290.000	−	502.295i	−0.107324	−	0.185890i
$195$		0			0
$196$		0			0
$197$	2734.00			0.988779			0.494389	−	0.869241i	$-0.335392\pi$
$197$	2734.00			0.988779			0.494389	+	0.869241i	$0.335392\pi$
$198$		0			0
$199$	1121.50	−	1942.49i	0.399503	−	0.691959i	−0.594162	−	0.804345i	$-0.702516\pi$
$199$	1121.50	−	1942.49i	0.399503	−	0.691959i	0.993665	+	0.112387i	$0.0358495\pi$
$200$	304.000	−	526.543i	0.107480	−	0.186161i
$201$		0			0
$202$	−2170.00			−0.755845
$203$		0			0
$204$		0			0
$205$	1743.00	+	3018.96i	0.593836	+	1.02855i
$206$	1553.00	−	2689.87i	0.525256	−	0.909769i
$207$		0			0
$208$	528.000	+	914.523i	0.176011	+	0.304859i
$209$	4795.00			1.58697
$210$		0			0
$211$	1172.00			0.382388			0.191194	−	0.981552i	$-0.438764\pi$
$211$	1172.00			0.382388			0.191194	+	0.981552i	$0.438764\pi$
$212$	−834.000	−	1444.53i	−0.270186	−	0.467975i
$213$		0			0
$214$	129.000	−	223.435i	0.0412068	−	0.0713723i
$215$	−910.000	−	1576.17i	−0.288658	−	0.499970i
$216$		0			0
$217$		0			0
$218$	−1930.00			−0.599615
$219$		0			0
$220$	490.000	−	848.705i	0.150163	−	0.260089i
$221$	1947.00	−	3372.30i	0.592622	−	1.02645i
$222$		0			0
$223$	−2024.00			−0.607790			−0.303895	−	0.952706i	$-0.598287\pi$
$223$	−2024.00			−0.607790			−0.303895	+	0.952706i	$0.598287\pi$
$224$		0			0
$225$		0			0
$226$	−50.0000	−	86.6025i	−0.0147166	−	0.0254899i
$227$	−1285.50	+	2226.55i	−0.375866	+	0.651019i	−0.990456	−	0.137827i	$-0.955988\pi$
$227$	−1285.50	+	2226.55i	−0.375866	+	0.651019i	0.614590	+	0.788847i	$0.289321\pi$
$228$		0			0
$229$	447.500	+	775.093i	0.129134	+	0.223666i	0.923341	−	0.383980i	$-0.125447\pi$
$229$	447.500	+	775.093i	0.129134	+	0.223666i	−0.794207	+	0.607647i	$0.792114\pi$
$230$	98.0000			0.0280953
$231$		0			0
$232$	−848.000			−0.239974
$233$	893.500	+	1547.59i	0.251224	+	0.435132i	0.963863	−	0.266398i	$-0.0858337\pi$
$233$	893.500	+	1547.59i	0.251224	+	0.435132i	−0.712639	+	0.701531i	$0.752500\pi$
$234$		0			0
$235$	598.500	−	1036.63i	0.166135	−	0.287755i
$236$	34.0000	+	58.8897i	0.00937801	+	0.0162432i
$237$		0			0
$238$		0			0
$239$	5100.00			1.38030			0.690150	−	0.723667i	$-0.257545\pi$
$239$	5100.00			1.38030			0.690150	+	0.723667i	$0.257545\pi$
$240$		0			0
$241$	−2088.50	+	3617.39i	−0.558225	+	0.966873i	0.439420	+	0.898282i	$0.355184\pi$
$241$	−2088.50	+	3617.39i	−0.558225	+	0.966873i	−0.997645	+	0.0685917i	$0.978149\pi$
$242$	106.000	−	183.597i	0.0281568	−	0.0487690i
$243$		0			0
$244$	−204.000			−0.0535236
$245$		0			0
$246$		0			0
$247$	−4521.00	−	7830.60i	−1.16463	−	2.01720i
$248$	300.000	−	519.615i	0.0768146	−	0.133047i
$249$		0			0
$250$	1407.00	+	2437.00i	0.355946	+	0.616517i
$251$	−4680.00			−1.17689			−0.588444	−	0.808538i	$-0.700259\pi$
$251$	−4680.00			−1.17689			−0.588444	+	0.808538i	$0.700259\pi$
$252$		0			0
$253$	−245.000			−0.0608815
$254$	−936.000	−	1621.20i	−0.231220	−	0.400485i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	874.500	+	1514.68i	0.212256	+	0.367638i	0.952420	−	0.304788i	$-0.0985856\pi$
$257$	874.500	+	1514.68i	0.212256	+	0.367638i	−0.740164	+	0.672426i	$0.765252\pi$
$258$		0			0
$259$		0			0
$260$	−1848.00			−0.440800
$261$		0			0
$262$	755.000	−	1307.70i	0.178031	−	0.308358i
$263$	−2236.50	+	3873.73i	−0.524367	+	0.908230i	0.475231	+	0.879861i	$0.342365\pi$
$263$	−2236.50	+	3873.73i	−0.524367	+	0.908230i	−0.999598	+	0.0283689i	$0.990969\pi$
$264$		0			0
$265$	2919.00			0.676652
$266$		0			0
$267$		0			0
$268$	−878.000	−	1520.74i	−0.200121	−	0.346619i
$269$	−987.500	+	1710.40i	−0.223825	+	0.387676i	−0.955966	−	0.293476i	$-0.905188\pi$
$269$	−987.500	+	1710.40i	−0.223825	+	0.387676i	0.732141	+	0.681153i	$0.238521\pi$
$270$		0			0
$271$	−4219.50	−	7308.39i	−0.945817	−	1.63820i	−0.754107	−	0.656751i	$-0.771930\pi$
$271$	−4219.50	−	7308.39i	−0.945817	−	1.63820i	−0.191710	−	0.981452i	$-0.561403\pi$
$272$	944.000			0.210435
$273$		0			0
$274$	4714.00			1.03935
$275$	−1330.00	−	2303.63i	−0.291644	−	0.505142i
$276$		0			0
$277$	−263.500	+	456.395i	−0.0571559	+	0.0989969i	−0.893188	−	0.449684i	$-0.851537\pi$
$277$	−263.500	+	456.395i	−0.0571559	+	0.0989969i	0.836032	+	0.548681i	$0.184870\pi$
$278$	28.0000	+	48.4974i	0.00604075	+	0.0104629i
$279$		0			0
$280$		0			0
$281$	202.000			0.0428837			0.0214418	−	0.999770i	$-0.493174\pi$
$281$	202.000			0.0428837			0.0214418	+	0.999770i	$0.493174\pi$
$282$		0			0
$283$	−3974.50	+	6884.04i	−0.834839	+	1.44598i	0.0593220	+	0.998239i	$0.481106\pi$
$283$	−3974.50	+	6884.04i	−0.834839	+	1.44598i	−0.894161	+	0.447745i	$0.852227\pi$
$284$	−1568.00	+	2715.86i	−0.327619	+	0.567452i
$285$		0			0
$286$	4620.00			0.955197
$287$		0			0
$288$		0			0
$289$	716.000	+	1240.15i	0.145736	+	0.252422i
$290$	742.000	−	1285.18i	0.150247	−	0.260236i
$291$		0			0
$292$	590.000	+	1021.91i	0.118244	+	0.204804i
$293$	318.000			0.0634053			0.0317027	−	0.999497i	$-0.489907\pi$
$293$	318.000			0.0634053			0.0317027	+	0.999497i	$0.489907\pi$
$294$		0			0
$295$	−119.000			−0.0234863
$296$	−44.0000	−	76.2102i	−0.00864003	−	0.0149650i
$297$		0			0
$298$	2295.00	−	3975.06i	0.446127	−	0.772714i
$299$	231.000	+	400.104i	0.0446792	+	0.0773866i
$300$		0			0
$301$		0			0
$302$	−2218.00			−0.422621
$303$		0			0
$304$	1096.00	−	1898.33i	0.206776	−	0.358147i
$305$	178.500	−	309.171i	0.0335111	−	0.0580429i
$306$		0			0
$307$	8132.00			1.51178			0.755892	−	0.654696i	$-0.227203\pi$
$307$	8132.00			1.51178			0.755892	+	0.654696i	$0.227203\pi$
$308$		0			0
$309$		0			0
$310$	525.000	+	909.327i	0.0961871	+	0.166601i
$311$	464.500	−	804.538i	0.0846925	−	0.146692i	−0.820568	−	0.571549i	$-0.806343\pi$
$311$	464.500	−	804.538i	0.0846925	−	0.146692i	0.905260	+	0.424858i	$0.139676\pi$
$312$		0			0
$313$	−104.500	−	180.999i	−0.0188712	−	0.0326859i	0.856436	−	0.516254i	$-0.172674\pi$
$313$	−104.500	−	180.999i	−0.0188712	−	0.0326859i	−0.875307	+	0.483568i	$0.839341\pi$
$314$	−3118.00			−0.560379
$315$		0			0
$316$	−1980.00			−0.352480
$317$	3565.50	+	6175.63i	0.631730	+	1.09419i	0.987198	+	0.159500i	$0.0509882\pi$
$317$	3565.50	+	6175.63i	0.631730	+	1.09419i	−0.355468	+	0.934689i	$0.615678\pi$
$318$		0			0
$319$	−1855.00	+	3212.95i	−0.325580	+	0.563921i
$320$	−224.000	−	387.979i	−0.0391312	−	0.0677772i
$321$		0			0
$322$		0			0
$323$	−8083.00			−1.39242
$324$		0			0
$325$	−2508.00	+	4343.98i	−0.428058	+	0.741418i
$326$	2251.00	−	3898.85i	0.382427	−	0.662384i
$327$		0			0
$328$	−3984.00			−0.670670
$329$		0			0
$330$		0			0
$331$	3285.50	+	5690.65i	0.545581	+	0.944975i	0.998570	+	0.0534583i	$0.0170244\pi$
$331$	3285.50	+	5690.65i	0.545581	+	0.944975i	−0.452989	+	0.891516i	$0.649642\pi$
$332$	−1864.00	+	3228.54i	−0.308133	+	0.533703i
$333$		0			0
$334$	−2788.00	−	4828.96i	−0.456744	−	0.791104i
$335$	3073.00			0.501182
$336$		0			0
$337$	−11466.0			−1.85339			−0.926696	−	0.375813i	$-0.877364\pi$
$337$	−11466.0			−1.85339			−0.926696	+	0.375813i	$0.877364\pi$
$338$	−2159.00	−	3739.50i	−0.347438	−	0.601781i
$339$		0			0
$340$	−826.000	+	1430.67i	−0.131753	+	0.228203i
$341$	−1312.50	−	2273.32i	−0.208434	−	0.361018i
$342$		0			0
$343$		0			0
$344$	2080.00			0.326006
$345$		0			0
$346$	−1579.00	+	2734.91i	−0.245340	+	0.424941i
$347$	−4888.50	+	8467.13i	−0.756278	+	1.30991i	0.188459	+	0.982081i	$0.439651\pi$
$347$	−4888.50	+	8467.13i	−0.756278	+	1.30991i	−0.944737	+	0.327831i	$0.893682\pi$
$348$		0			0
$349$	−11914.0			−1.82734			−0.913670	−	0.406456i	$-0.866764\pi$
$349$	−11914.0			−1.82734			−0.913670	+	0.406456i	$0.866764\pi$
$350$		0			0
$351$		0			0
$352$	560.000	+	969.948i	0.0847957	+	0.146871i
$353$	−4561.50	+	7900.75i	−0.687774	+	1.19126i	0.284783	+	0.958592i	$0.408079\pi$
$353$	−4561.50	+	7900.75i	−0.687774	+	1.19126i	−0.972556	+	0.232667i	$0.925255\pi$
$354$		0			0
$355$	−2744.00	−	4752.75i	−0.410243	−	0.710562i
$356$	−3492.00			−0.519875
$357$		0			0
$358$	−4902.00			−0.723684
$359$	4074.50	+	7057.24i	0.599008	+	1.03751i	0.992968	+	0.118385i	$0.0377716\pi$
$359$	4074.50	+	7057.24i	0.599008	+	1.03751i	−0.393960	+	0.919128i	$0.628895\pi$
$360$		0			0
$361$	−5955.00	+	10314.4i	−0.868202	+	1.50377i
$362$	−1170.00	−	2026.50i	−0.169872	−	0.294228i
$363$		0			0
$364$		0			0
$365$	−2065.00			−0.296129
$366$		0			0
$367$	4835.50	−	8375.33i	0.687769	−	1.19125i	−0.284790	−	0.958590i	$-0.591924\pi$
$367$	4835.50	−	8375.33i	0.687769	−	1.19125i	0.972558	−	0.232660i	$-0.0747429\pi$
$368$	−56.0000	+	96.9948i	−0.00793261	+	0.0137397i
$369$		0			0
$370$	154.000			0.0216381
$371$		0			0
$372$		0			0
$373$	2054.50	+	3558.50i	0.285196	+	0.493973i	0.972657	−	0.232248i	$-0.0746081\pi$
$373$	2054.50	+	3558.50i	0.285196	+	0.493973i	−0.687461	+	0.726221i	$0.741275\pi$
$374$	2065.00	−	3576.68i	0.285504	−	0.494508i
$375$		0			0
$376$	684.000	+	1184.72i	0.0938154	+	0.162493i
$377$	6996.00			0.955736
$378$		0			0
$379$	−3488.00			−0.472735			−0.236367	−	0.971664i	$-0.575957\pi$
$379$	−3488.00			−0.472735			−0.236367	+	0.971664i	$0.575957\pi$
$380$	1918.00	+	3322.07i	0.258925	+	0.448470i
$381$		0			0
$382$	−1275.00	+	2208.36i	−0.170771	+	0.295785i
$383$	−4358.50	−	7549.14i	−0.581485	−	1.00716i	−0.995304	−	0.0968028i	$-0.969138\pi$
$383$	−4358.50	−	7549.14i	−0.581485	−	1.00716i	0.413818	−	0.910360i	$-0.364195\pi$
$384$		0			0
$385$		0			0
$386$	70.0000			0.00923033
$387$		0			0
$388$	−580.000	+	1004.59i	−0.0758893	+	0.131444i
$389$	81.5000	−	141.162i	0.0106227	−	0.0183990i	−0.860665	−	0.509171i	$-0.829952\pi$
$389$	81.5000	−	141.162i	0.0106227	−	0.0183990i	0.871288	+	0.490772i	$0.163285\pi$
$390$		0			0
$391$	413.000			0.0534177
$392$		0			0
$393$		0			0
$394$	−2734.00	−	4735.43i	−0.349586	−	0.605501i
$395$	1732.50	−	3000.78i	0.220687	−	0.382242i
$396$		0			0
$397$	499.500	+	865.159i	0.0631466	+	0.109373i	0.895870	−	0.444316i	$-0.146553\pi$
$397$	499.500	+	865.159i	0.0631466	+	0.109373i	−0.832724	+	0.553689i	$0.813220\pi$
$398$	−4486.00			−0.564982
$399$		0			0
$400$	−1216.00			−0.152000
$401$	−7378.50	−	12779.9i	−0.918865	−	1.59152i	−0.801143	−	0.598474i	$-0.795774\pi$
$401$	−7378.50	−	12779.9i	−0.918865	−	1.59152i	−0.117722	−	0.993047i	$-0.537559\pi$
$402$		0			0
$403$	−2475.00	+	4286.83i	−0.305927	+	0.529881i
$404$	2170.00	+	3758.55i	0.267232	+	0.462859i
$405$		0			0
$406$		0			0
$407$	−385.000			−0.0468888
$408$		0			0
$409$	−66.5000	+	115.181i	−0.00803964	+	0.0139251i	−0.870017	−	0.493021i	$-0.835892\pi$
$409$	−66.5000	+	115.181i	−0.00803964	+	0.0139251i	0.861978	+	0.506946i	$0.169226\pi$
$410$	3486.00	−	6037.93i	0.419906	−	0.727298i
$411$		0			0
$412$	−6212.00			−0.742823
$413$		0			0
$414$		0			0
$415$	−3262.00	−	5649.95i	−0.385844	−	0.668302i
$416$	1056.00	−	1829.05i	0.124458	−	0.215568i
$417$		0			0
$418$	−4795.00	−	8305.18i	−0.561079	−	0.971818i
$419$	−6420.00			−0.748538			−0.374269	−	0.927320i	$-0.622106\pi$
$419$	−6420.00			−0.748538			−0.374269	+	0.927320i	$0.622106\pi$
$420$		0			0
$421$	10266.0			1.18844			0.594221	−	0.804302i	$-0.297460\pi$
$421$	10266.0			1.18844			0.594221	+	0.804302i	$0.297460\pi$
$422$	−1172.00	−	2029.96i	−0.135194	−	0.234164i
$423$		0			0
$424$	−1668.00	+	2889.06i	−0.191050	+	0.330908i
$425$	2242.00	+	3883.26i	0.255889	+	0.443213i
$426$		0			0
$427$		0			0
$428$	−516.000			−0.0582752
$429$		0			0
$430$	−1820.00	+	3152.33i	−0.204112	+	0.353532i
$431$	−7606.50	+	13174.8i	−0.850098	+	1.47241i	0.0310213	+	0.999519i	$0.490124\pi$
$431$	−7606.50	+	13174.8i	−0.850098	+	1.47241i	−0.881119	+	0.472894i	$0.843209\pi$
$432$		0			0
$433$	1378.00			0.152939			0.0764693	−	0.997072i	$-0.475635\pi$
$433$	1378.00			0.152939			0.0764693	+	0.997072i	$0.475635\pi$
$434$		0			0
$435$		0			0
$436$	1930.00	+	3342.86i	0.211996	+	0.367188i
$437$	479.500	−	830.518i	0.0524888	−	0.0909132i
$438$		0			0
$439$	−1381.50	−	2392.83i	−0.150195	−	0.260145i	0.781104	−	0.624401i	$-0.214657\pi$
$439$	−1381.50	−	2392.83i	−0.150195	−	0.260145i	−0.931299	+	0.364256i	$0.881323\pi$
$440$	−1960.00			−0.212362
$441$		0			0
$442$	−7788.00			−0.838094
$443$	2924.50	+	5065.38i	0.313651	+	0.543259i	0.979150	−	0.203140i	$-0.0651146\pi$
$443$	2924.50	+	5065.38i	0.313651	+	0.543259i	−0.665499	+	0.746399i	$0.731781\pi$
$444$		0			0
$445$	3055.50	−	5292.28i	0.325493	−	0.563771i
$446$	2024.00	+	3505.67i	0.214886	+	0.372194i
$447$		0			0
$448$		0			0
$449$	−4582.00			−0.481599			−0.240799	−	0.970575i	$-0.577410\pi$
$449$	−4582.00			−0.481599			−0.240799	+	0.970575i	$0.577410\pi$
$450$		0			0
$451$	−8715.00	+	15094.8i	−0.909919	+	1.57603i
$452$	−100.000	+	173.205i	−0.0104062	+	0.0180241i
$453$		0			0
$454$	5142.00			0.531555
$455$		0			0
$456$		0			0
$457$	−5775.50	−	10003.5i	−0.591174	−	1.02394i	−0.994075	−	0.108700i	$-0.965331\pi$
$457$	−5775.50	−	10003.5i	−0.591174	−	1.02394i	0.402901	−	0.915244i	$-0.368002\pi$
$458$	895.000	−	1550.19i	0.0913114	−	0.158156i
$459$		0			0
$460$	−98.0000	−	169.741i	−0.00993320	−	0.0172048i
$461$	−9494.00			−0.959175			−0.479587	−	0.877494i	$-0.659214\pi$
$461$	−9494.00			−0.959175			−0.479587	+	0.877494i	$0.659214\pi$
$462$		0			0
$463$	−10160.0			−1.01982			−0.509908	−	0.860229i	$-0.670321\pi$
$463$	−10160.0			−1.01982			−0.509908	+	0.860229i	$0.670321\pi$
$464$	848.000	+	1468.78i	0.0848436	+	0.146953i
$465$		0			0
$466$	1787.00	−	3095.17i	0.177642	−	0.307685i
$467$	653.500	+	1131.90i	0.0647545	+	0.112158i	0.896585	−	0.442872i	$-0.146040\pi$
$467$	653.500	+	1131.90i	0.0647545	+	0.112158i	−0.831831	+	0.555030i	$0.812707\pi$
$468$		0			0
$469$		0			0
$470$	−2394.00			−0.234951
$471$		0			0
$472$	68.0000	−	117.779i	0.00663126	−	0.0114857i
$473$	4550.00	−	7880.83i	0.442303	−	0.766091i
$474$		0			0
$475$	10412.0			1.00576
$476$		0			0
$477$		0			0
$478$	−5100.00	−	8833.46i	−0.488010	−	0.845257i
$479$	−9143.50	+	15837.0i	−0.872186	+	1.51067i	−0.0124559	+	0.999922i	$0.503965\pi$
$479$	−9143.50	+	15837.0i	−0.872186	+	1.51067i	−0.859730	+	0.510748i	$0.829368\pi$
$480$		0			0
$481$	363.000	+	628.734i	0.0344103	+	0.0596005i
$482$	8354.00			0.789449
$483$		0			0
$484$	−424.000			−0.0398197
$485$	−1015.00	−	1758.03i	−0.0950284	−	0.164594i
$486$		0			0
$487$	7476.50	−	12949.7i	0.695673	−	1.20494i	−0.274281	−	0.961650i	$-0.588440\pi$
$487$	7476.50	−	12949.7i	0.695673	−	1.20494i	0.969953	−	0.243291i	$-0.0782269\pi$
$488$	204.000	+	353.338i	0.0189235	+	0.0327764i
$489$		0			0
$490$		0			0
$491$	−14352.0			−1.31914			−0.659569	−	0.751644i	$-0.729261\pi$
$491$	−14352.0			−1.31914			−0.659569	+	0.751644i	$0.729261\pi$
$492$		0			0
$493$	3127.00	−	5416.12i	0.285665	−	0.494787i
$494$	−9042.00	+	15661.2i	−0.823520	+	1.42638i
$495$		0			0
$496$	−1200.00			−0.108632
$497$		0			0
$498$		0			0
$499$	2765.50	+	4789.99i	0.248098	+	0.429718i	0.962998	−	0.269509i	$-0.0868612\pi$
$499$	2765.50	+	4789.99i	0.248098	+	0.429718i	−0.714900	+	0.699226i	$0.753528\pi$
$500$	2814.00	−	4873.99i	0.251692	−	0.435943i
$501$		0			0
$502$	4680.00	+	8106.00i	0.416093	+	0.720694i
$503$	8400.00			0.744607			0.372304	−	0.928111i	$-0.378568\pi$
$503$	8400.00			0.744607			0.372304	+	0.928111i	$0.378568\pi$
$504$		0			0
$505$	−7595.00			−0.669254
$506$	245.000	+	424.352i	0.0215249	+	0.0372821i
$507$		0			0
$508$	−1872.00	+	3242.40i	−0.163497	+	0.283185i
$509$	1192.50	+	2065.47i	0.103844	+	0.179863i	0.913265	−	0.407365i	$-0.133552\pi$
$509$	1192.50	+	2065.47i	0.103844	+	0.179863i	−0.809421	+	0.587228i	$0.800219\pi$
$510$		0			0
$511$		0			0
$512$	512.000			0.0441942
$513$		0			0
$514$	1749.00	−	3029.36i	0.150088	−	0.259960i
$515$	5435.50	−	9414.56i	0.465081	−	0.805544i
$516$		0			0
$517$	5985.00			0.509130
$518$		0			0
$519$		0			0
$520$	1848.00	+	3200.83i	0.155846	+	0.269934i
$521$	4576.50	−	7926.73i	0.384837	−	0.666557i	−0.606910	−	0.794771i	$-0.707591\pi$
$521$	4576.50	−	7926.73i	0.384837	−	0.666557i	0.991747	+	0.128214i	$0.0409243\pi$
$522$		0			0
$523$	−6903.50	−	11957.2i	−0.577187	−	0.999718i	−0.995800	−	0.0915530i	$-0.970817\pi$
$523$	−6903.50	−	11957.2i	−0.577187	−	0.999718i	0.418613	−	0.908165i	$-0.362516\pi$
$524$	−3020.00			−0.251773
$525$		0			0
$526$	8946.00			0.741567
$527$	2212.50	+	3832.16i	0.182880	+	0.316758i
$528$		0			0
$529$	6059.00	−	10494.5i	0.497986	−	0.862538i
$530$	−2919.00	−	5055.86i	−0.239233	−	0.414363i
$531$		0			0
$532$		0			0
$533$	32868.0			2.67105
$534$		0			0
$535$	451.500	−	782.021i	0.0364861	−	0.0631957i
$536$	−1756.00	+	3041.48i	−0.141507	+	0.245097i
$537$		0			0
$538$	3950.00			0.316536
$539$		0			0
$540$		0			0
$541$	−4087.50	−	7079.76i	−0.324834	−	0.562629i	0.656645	−	0.754200i	$-0.271975\pi$
$541$	−4087.50	−	7079.76i	−0.324834	−	0.562629i	−0.981479	+	0.191571i	$0.938642\pi$
$542$	−8439.00	+	14616.8i	−0.668794	+	1.15838i
$543$		0			0
$544$	−944.000	−	1635.06i	−0.0744001	−	0.128865i
$545$	−6755.00			−0.530922
$546$		0			0
$547$	4656.00			0.363942			0.181971	−	0.983304i	$-0.441752\pi$
$547$	4656.00			0.363942			0.181971	+	0.983304i	$0.441752\pi$
$548$	−4714.00	−	8164.89i	−0.367467	−	0.636472i
$549$		0			0
$550$	−2660.00	+	4607.26i	−0.206223	+	0.357189i
$551$	−7261.00	−	12576.4i	−0.561396	−	0.972366i
$552$		0			0
$553$		0			0
$554$	1054.00			0.0808306
$555$		0			0
$556$	56.0000	−	96.9948i	0.00427146	−	0.00739838i
$557$	3501.50	−	6064.78i	0.266361	−	0.461352i	−0.701558	−	0.712612i	$-0.747512\pi$
$557$	3501.50	−	6064.78i	0.266361	−	0.461352i	0.967919	+	0.251261i	$0.0808452\pi$
$558$		0			0
$559$	−17160.0			−1.29837
$560$		0			0
$561$		0			0
$562$	−202.000	−	349.874i	−0.0151617	−	0.0262608i
$563$	9876.50	−	17106.6i	0.739334	−	1.28056i	−0.213462	−	0.976951i	$-0.568474\pi$
$563$	9876.50	−	17106.6i	0.739334	−	1.28056i	0.952796	−	0.303612i	$-0.0981927\pi$
$564$		0			0
$565$	−175.000	−	303.109i	−0.0130306	−	0.0225697i
$566$	15898.0			1.18064
$567$		0			0
$568$	6272.00			0.463323
$569$	−3448.50	−	5972.98i	−0.254075	−	0.440071i	0.710569	−	0.703628i	$-0.248438\pi$
$569$	−3448.50	−	5972.98i	−0.254075	−	0.440071i	−0.964644	+	0.263557i	$0.915104\pi$
$570$		0			0
$571$	−12457.5	+	21577.0i	−0.913013	+	1.58138i	−0.103227	+	0.994658i	$0.532917\pi$
$571$	−12457.5	+	21577.0i	−0.913013	+	1.58138i	−0.809785	+	0.586726i	$0.800416\pi$
$572$	−4620.00	−	8002.07i	−0.337713	−	0.584936i
$573$		0			0
$574$		0			0
$575$	−532.000			−0.0385842
$576$		0			0
$577$	63.5000	−	109.985i	0.00458152	−	0.00793543i	−0.863726	−	0.503962i	$-0.831875\pi$
$577$	63.5000	−	109.985i	0.00458152	−	0.00793543i	0.868307	+	0.496027i	$0.165208\pi$
$578$	1432.00	−	2480.30i	0.103051	−	0.178489i
$579$		0			0
$580$	−2968.00			−0.212482
$581$		0			0
$582$		0			0
$583$	7297.50	+	12639.6i	0.518407	+	0.897908i
$584$	1180.00	−	2043.82i	0.0836109	−	0.144818i
$585$		0			0
$586$	−318.000	−	550.792i	−0.0224172	−	0.0388277i
$587$	9044.00			0.635921			0.317961	−	0.948104i	$-0.397002\pi$
$587$	9044.00			0.635921			0.317961	+	0.948104i	$0.397002\pi$
$588$		0			0
$589$	10275.0			0.718801
$590$	119.000	+	206.114i	0.00830365	+	0.0143823i
$591$		0			0
$592$	−88.0000	+	152.420i	−0.00610942	+	0.0105818i
$593$	5350.50	+	9267.34i	0.370521	+	0.641760i	0.989646	−	0.143532i	$-0.0458460\pi$
$593$	5350.50	+	9267.34i	0.370521	+	0.641760i	−0.619125	+	0.785292i	$0.712513\pi$
$594$		0			0
$595$		0			0
$596$	−9180.00			−0.630919
$597$		0			0
$598$	462.000	−	800.207i	0.0315930	−	0.0547206i
$599$	10399.5	−	18012.5i	0.709369	−	1.22866i	−0.255722	−	0.966750i	$-0.582313\pi$
$599$	10399.5	−	18012.5i	0.709369	−	1.22866i	0.965091	−	0.261913i	$-0.0843533\pi$
$600$		0			0
$601$	1402.00			0.0951560			0.0475780	−	0.998868i	$-0.484850\pi$
$601$	1402.00			0.0951560			0.0475780	+	0.998868i	$0.484850\pi$
$602$		0			0
$603$		0			0
$604$	2218.00	+	3841.69i	0.149419	+	0.258801i
$605$	371.000	−	642.591i	0.0249311	−	0.0431819i
$606$		0			0
$607$	3262.50	+	5650.82i	0.218156	+	0.377858i	0.954244	−	0.299028i	$-0.0966625\pi$
$607$	3262.50	+	5650.82i	0.218156	+	0.377858i	−0.736088	+	0.676886i	$0.763329\pi$
$608$	−4384.00			−0.292425
$609$		0			0
$610$	−714.000			−0.0473918
$611$	−5643.00	−	9773.96i	−0.373636	−	0.647156i
$612$		0			0
$613$	−7525.50	+	13034.5i	−0.495844	+	0.858826i	−0.999989	−	0.00479285i	$-0.998474\pi$
$613$	−7525.50	+	13034.5i	−0.495844	+	0.858826i	0.504145	+	0.863619i	$0.331808\pi$
$614$	−8132.00	−	14085.0i	−0.534496	−	0.925775i
$615$		0			0
$616$		0			0
$617$	−11150.0			−0.727524			−0.363762	−	0.931492i	$-0.618508\pi$
$617$	−11150.0			−0.727524			−0.363762	+	0.931492i	$0.618508\pi$
$618$		0			0
$619$	1707.50	−	2957.48i	0.110873	−	0.192037i	−0.805250	−	0.592936i	$-0.797969\pi$
$619$	1707.50	−	2957.48i	0.110873	−	0.192037i	0.916122	+	0.400899i	$0.131302\pi$
$620$	1050.00	−	1818.65i	0.0680145	−	0.117805i
$621$		0			0
$622$	−1858.00			−0.119773
$623$		0			0
$624$		0			0
$625$	174.500	+	302.243i	0.0111680	+	0.0193435i
$626$	−209.000	+	361.999i	−0.0133440	+	0.0231124i
$627$		0			0
$628$	3118.00	+	5400.53i	0.198124	+	0.343160i
$629$	649.000			0.0411404
$630$		0			0
$631$	−21184.0			−1.33648			−0.668242	−	0.743944i	$-0.732953\pi$
$631$	−21184.0			−1.33648			−0.668242	+	0.743944i	$0.732953\pi$
$632$	1980.00	+	3429.46i	0.124621	+	0.215849i
$633$		0			0
$634$	7131.00	−	12351.3i	0.446701	−	0.773708i
$635$	−3276.00	−	5674.20i	−0.204731	−	0.354604i
$636$		0			0
$637$		0			0
$638$	7420.00			0.460440
$639$		0			0
$640$	−448.000	+	775.959i	−0.0276699	+	0.0479257i
$641$	−5352.50	+	9270.80i	−0.329814	+	0.571255i	−0.982475	−	0.186395i	$-0.940320\pi$
$641$	−5352.50	+	9270.80i	−0.329814	+	0.571255i	0.652660	+	0.757651i	$0.273653\pi$
$642$		0			0
$643$	−6860.00			−0.420734			−0.210367	−	0.977622i	$-0.567466\pi$
$643$	−6860.00			−0.420734			−0.210367	+	0.977622i	$0.567466\pi$
$644$		0			0
$645$		0			0
$646$	8083.00	+	14000.2i	0.492293	+	0.852677i
$647$	−7231.50	+	12525.3i	−0.439412	+	0.761084i	−0.997644	−	0.0686008i	$-0.978147\pi$
$647$	−7231.50	+	12525.3i	−0.439412	+	0.761084i	0.558232	+	0.829685i	$0.311480\pi$
$648$		0			0
$649$	−297.500	−	515.285i	−0.0179937	−	0.0311660i
$650$	10032.0			0.605365
$651$		0			0
$652$	−9004.00			−0.540834
$653$	2989.50	+	5177.97i	0.179155	+	0.310305i	0.941591	−	0.336758i	$-0.109330\pi$
$653$	2989.50	+	5177.97i	0.179155	+	0.310305i	−0.762436	+	0.647063i	$0.775997\pi$
$654$		0			0
$655$	2642.50	−	4576.94i	0.157635	−	0.273032i
$656$	3984.00	+	6900.49i	0.237117	+	0.410700i
$657$		0			0
$658$		0			0
$659$	6940.00			0.410234			0.205117	−	0.978737i	$-0.434243\pi$
$659$	6940.00			0.410234			0.205117	+	0.978737i	$0.434243\pi$
$660$		0			0
$661$	6699.50	−	11603.9i	0.394221	−	0.682812i	−0.598780	−	0.800914i	$-0.704348\pi$
$661$	6699.50	−	11603.9i	0.394221	−	0.682812i	0.993001	+	0.118102i	$0.0376810\pi$
$662$	6571.00	−	11381.3i	0.385784	−	0.668198i
$663$		0			0
$664$	7456.00			0.435766
$665$		0			0
$666$		0			0
$667$	371.000	+	642.591i	0.0215370	+	0.0373032i
$668$	−5576.00	+	9657.92i	−0.322967	+	0.559395i
$669$		0			0
$670$	−3073.00	−	5322.59i	−0.177195	−	0.306910i
$671$	1785.00			0.102696
$672$		0			0
$673$	29510.0			1.69023			0.845117	−	0.534582i	$-0.179531\pi$
$673$	29510.0			1.69023			0.845117	+	0.534582i	$0.179531\pi$
$674$	11466.0	+	19859.7i	0.655273	+	1.13497i
$675$		0			0
$676$	−4318.00	+	7479.00i	−0.245676	+	0.425523i
$677$	13000.5	+	22517.5i	0.738035	+	1.27831i	0.953379	+	0.301776i	$0.0975795\pi$
$677$	13000.5	+	22517.5i	0.738035	+	1.27831i	−0.215344	+	0.976538i	$0.569087\pi$
$678$		0			0
$679$		0			0
$680$	3304.00			0.186327
$681$		0			0
$682$	−2625.00	+	4546.63i	−0.147385	+	0.255278i
$683$	−4402.50	+	7625.35i	−0.246643	+	0.427198i	−0.962592	−	0.270954i	$-0.912661\pi$
$683$	−4402.50	+	7625.35i	−0.246643	+	0.427198i	0.715949	+	0.698152i	$0.245994\pi$
$684$		0			0
$685$	16499.0			0.920284
$686$		0			0
$687$		0			0
$688$	−2080.00	−	3602.67i	−0.115261	−	0.199637i
$689$	13761.0	−	23834.8i	0.760889	−	1.31790i
$690$		0			0
$691$	14342.5	+	24841.9i	0.789601	+	1.36763i	0.926211	+	0.377004i	$0.123046\pi$
$691$	14342.5	+	24841.9i	0.789601	+	1.36763i	−0.136610	+	0.990625i	$0.543621\pi$
$692$	6316.00			0.346963
$693$		0			0
$694$	19554.0			1.06954
$695$	98.0000	+	169.741i	0.00534871	+	0.00926423i
$696$		0			0
$697$	14691.0	−	25445.6i	0.798366	−	1.38281i
$698$	11914.0	+	20635.7i	0.646062	+	1.11901i
$699$		0			0
$700$		0			0
$701$	3146.00			0.169505			0.0847523	−	0.996402i	$-0.472990\pi$
$701$	3146.00			0.169505			0.0847523	+	0.996402i	$0.472990\pi$
$702$		0			0
$703$	753.500	−	1305.10i	0.0404250	−	0.0700182i
$704$	1120.00	−	1939.90i	0.0599596	−	0.103853i
$705$		0			0
$706$	18246.0			0.972659
$707$		0			0
$708$		0			0
$709$	−629.500	−	1090.33i	−0.0333447	−	0.0577547i	0.848871	−	0.528599i	$-0.177283\pi$
$709$	−629.500	−	1090.33i	−0.0333447	−	0.0577547i	−0.882216	+	0.470845i	$0.843949\pi$
$710$	−5488.00	+	9505.49i	−0.290086	+	0.502443i
$711$		0			0
$712$	3492.00	+	6048.32i	0.183804	+	0.318357i
$713$	−525.000			−0.0275756
$714$		0			0
$715$	16170.0			0.845767
$716$	4902.00	+	8490.51i	0.255861	+	0.443164i
$717$		0			0
$718$	8149.00	−	14114.5i	0.423563	−	0.733632i
$719$	−8212.50	−	14224.5i	−0.425973	−	0.737807i	0.570538	−	0.821271i	$-0.306735\pi$
$719$	−8212.50	−	14224.5i	−0.425973	−	0.737807i	−0.996511	+	0.0834645i	$0.973401\pi$
$720$		0			0
$721$		0			0
$722$	23820.0			1.22782
$723$		0			0
$724$	−2340.00	+	4053.00i	−0.120118	+	0.208050i
$725$	−4028.00	+	6976.70i	−0.206340	+	0.357391i
$726$		0			0
$727$	6032.00			0.307723			0.153861	−	0.988092i	$-0.450829\pi$
$727$	6032.00			0.307723			0.153861	+	0.988092i	$0.450829\pi$
$728$		0			0
$729$		0			0
$730$	2065.00	+	3576.68i	0.104697	+	0.181341i
$731$	−7670.00	+	13284.8i	−0.388078	+	0.672171i
$732$		0			0
$733$	7621.50	+	13200.8i	0.384047	+	0.665189i	0.991636	−	0.129062i	$-0.0411967\pi$
$733$	7621.50	+	13200.8i	0.384047	+	0.665189i	−0.607589	+	0.794251i	$0.707863\pi$
$734$	−19342.0			−0.972652
$735$		0			0
$736$	224.000			0.0112184
$737$	7682.50	+	13306.5i	0.383974	+	0.665062i
$738$		0			0
$739$	5026.50	−	8706.15i	0.250207	−	0.433371i	−0.713376	−	0.700782i	$-0.752835\pi$
$739$	5026.50	−	8706.15i	0.250207	−	0.433371i	0.963583	+	0.267411i	$0.0861681\pi$
$740$	−154.000	−	266.736i	−0.00765021	−	0.0132505i
$741$		0			0
$742$		0			0
$743$	−24384.0			−1.20399			−0.601993	−	0.798501i	$-0.705627\pi$
$743$	−24384.0			−1.20399			−0.601993	+	0.798501i	$0.705627\pi$
$744$		0			0
$745$	8032.50	−	13912.7i	0.395017	−	0.684190i
$746$	4109.00	−	7117.00i	0.201664	−	0.349292i
$747$		0			0
$748$	−8260.00			−0.403764
$749$		0			0
$750$		0			0
$751$	−5794.50	−	10036.4i	−0.281550	−	0.487660i	0.690216	−	0.723603i	$-0.257515\pi$
$751$	−5794.50	−	10036.4i	−0.281550	−	0.487660i	−0.971767	+	0.235943i	$0.924182\pi$
$752$	1368.00	−	2369.45i	0.0663375	−	0.114900i
$753$		0			0
$754$	−6996.00	−	12117.4i	−0.337904	−	0.585266i
$755$	−7763.00			−0.374205
$756$		0			0
$757$	14562.0			0.699161			0.349581	−	0.936906i	$-0.386324\pi$
$757$	14562.0			0.699161			0.349581	+	0.936906i	$0.386324\pi$
$758$	3488.00	+	6041.39i	0.167137	+	0.289490i
$759$		0			0
$760$	3836.00	−	6644.15i	0.183087	−	0.317116i
$761$	11382.5	+	19715.1i	0.542201	+	0.939120i	0.998777	+	0.0494360i	$0.0157424\pi$
$761$	11382.5	+	19715.1i	0.542201	+	0.939120i	−0.456576	+	0.889684i	$0.650924\pi$
$762$		0			0
$763$		0			0
$764$	5100.00			0.241507
$765$		0			0
$766$	−8717.00	+	15098.3i	−0.411172	+	0.712171i
$767$	−561.000	+	971.681i	−0.0264101	+	0.0457436i
$768$		0			0
$769$	−3766.00			−0.176600			−0.0883000	−	0.996094i	$-0.528143\pi$
$769$	−3766.00			−0.176600			−0.0883000	+	0.996094i	$0.528143\pi$
$770$		0			0
$771$		0			0
$772$	−70.0000	−	121.244i	−0.00326341	−	0.00565240i
$773$	13430.5	−	23262.3i	0.624918	−	1.08239i	−0.363639	−	0.931540i	$-0.618466\pi$
$773$	13430.5	−	23262.3i	0.624918	−	1.08239i	0.988557	−	0.150849i	$-0.0482009\pi$
$774$		0			0
$775$	−2850.00	−	4936.34i	−0.132097	−	0.228798i
$776$	2320.00			0.107324
$777$		0			0
$778$	−326.000			−0.0150227
$779$	−34113.0	−	59085.4i	−1.56897	−	2.71753i
$780$		0			0
$781$	13720.0	−	23763.7i	0.628605	−	1.08878i
$782$	−413.000	−	715.337i	−0.0188860	−	0.0327115i
$783$		0			0
$784$		0			0
$785$	−10913.0			−0.496180

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-3.50000 - 6.06218i) q^{5} +8.00000 q^{8} +(-7.00000 + 12.1244i) q^{10} +(17.5000 - 30.3109i) q^{11} -66.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(-29.5000 + 51.0955i) q^{17} +(68.5000 + 118.645i) q^{19} +28.0000 q^{20} -70.0000 q^{22} +(-3.50000 - 6.06218i) q^{23} +(38.0000 - 65.8179i) q^{25} +(66.0000 + 114.315i) q^{26} -106.000 q^{29} +(37.5000 - 64.9519i) q^{31} +(-16.0000 + 27.7128i) q^{32} +118.000 q^{34} +(-5.50000 - 9.52628i) q^{37} +(137.000 - 237.291i) q^{38} +(-28.0000 - 48.4974i) q^{40} -498.000 q^{41} +260.000 q^{43} +(70.0000 + 121.244i) q^{44} +(-7.00000 + 12.1244i) q^{46} +(85.5000 + 148.090i) q^{47} -152.000 q^{50} +(132.000 - 228.631i) q^{52} +(-208.500 + 361.133i) q^{53} -245.000 q^{55} +(106.000 + 183.597i) q^{58} +(8.50000 - 14.7224i) q^{59} +(25.5000 + 44.1673i) q^{61} -150.000 q^{62} +64.0000 q^{64} +(231.000 + 400.104i) q^{65} +(-219.500 + 380.185i) q^{67} +(-118.000 - 204.382i) q^{68} +784.000 q^{71} +(147.500 - 255.477i) q^{73} +(-11.0000 + 19.0526i) q^{74} -548.000 q^{76} +(247.500 + 428.683i) q^{79} +(-56.0000 + 96.9948i) q^{80} +(498.000 + 862.561i) q^{82} +932.000 q^{83} +413.000 q^{85} +(-260.000 - 450.333i) q^{86} +(140.000 - 242.487i) q^{88} +(436.500 + 756.040i) q^{89} +28.0000 q^{92} +(171.000 - 296.181i) q^{94} +(479.500 - 830.518i) q^{95} +290.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 4 * q^4 - 7 * q^5 + 16 * q^8	\( 2 q - 2 q^{2} - 4 q^{4} - 7 q^{5} + 16 q^{8} - 14 q^{10} + 35 q^{11} - 132 q^{13} - 16 q^{16} - 59 q^{17} + 137 q^{19} + 56 q^{20} - 140 q^{22} - 7 q^{23} + 76 q^{25} + 132 q^{26} - 212 q^{29} + 75 q^{31} - 32 q^{32} + 236 q^{34} - 11 q^{37} + 274 q^{38} - 56 q^{40} - 996 q^{41} + 520 q^{43} + 140 q^{44} - 14 q^{46} + 171 q^{47} - 304 q^{50} + 264 q^{52} - 417 q^{53} - 490 q^{55} + 212 q^{58} + 17 q^{59} + 51 q^{61} - 300 q^{62} + 128 q^{64} + 462 q^{65} - 439 q^{67} - 236 q^{68} + 1568 q^{71} + 295 q^{73} - 22 q^{74} - 1096 q^{76} + 495 q^{79} - 112 q^{80} + 996 q^{82} + 1864 q^{83} + 826 q^{85} - 520 q^{86} + 280 q^{88} + 873 q^{89} + 56 q^{92} + 342 q^{94} + 959 q^{95} + 580 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 - 7 * q^5 + 16 * q^8 - 14 * q^10 + 35 * q^11 - 132 * q^13 - 16 * q^16 - 59 * q^17 + 137 * q^19 + 56 * q^20 - 140 * q^22 - 7 * q^23 + 76 * q^25 + 132 * q^26 - 212 * q^29 + 75 * q^31 - 32 * q^32 + 236 * q^34 - 11 * q^37 + 274 * q^38 - 56 * q^40 - 996 * q^41 + 520 * q^43 + 140 * q^44 - 14 * q^46 + 171 * q^47 - 304 * q^50 + 264 * q^52 - 417 * q^53 - 490 * q^55 + 212 * q^58 + 17 * q^59 + 51 * q^61 - 300 * q^62 + 128 * q^64 + 462 * q^65 - 439 * q^67 - 236 * q^68 + 1568 * q^71 + 295 * q^73 - 22 * q^74 - 1096 * q^76 + 495 * q^79 - 112 * q^80 + 996 * q^82 + 1864 * q^83 + 826 * q^85 - 520 * q^86 + 280 * q^88 + 873 * q^89 + 56 * q^92 + 342 * q^94 + 959 * q^95 + 580 * q^97

Introduction

L-functions

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