LMFDB - Embedded newform 882.4.g.l.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 42)
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.l.667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(7.50000 + 12.9904i) q^{5} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(7.50000 + 12.9904i) q^{5} +8.00000 q^{8} +(15.0000 - 25.9808i) q^{10} +(-4.50000 + 7.79423i) q^{11} +88.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(42.0000 - 72.7461i) q^{17} +(52.0000 + 90.0666i) q^{19} -60.0000 q^{20} +18.0000 q^{22} +(-42.0000 - 72.7461i) q^{23} +(-50.0000 + 86.6025i) q^{25} +(-88.0000 - 152.420i) q^{26} -51.0000 q^{29} +(92.5000 - 160.215i) q^{31} +(-16.0000 + 27.7128i) q^{32} -168.000 q^{34} +(-22.0000 - 38.1051i) q^{37} +(104.000 - 180.133i) q^{38} +(60.0000 + 103.923i) q^{40} -168.000 q^{41} +326.000 q^{43} +(-18.0000 - 31.1769i) q^{44} +(-84.0000 + 145.492i) q^{46} +(69.0000 + 119.512i) q^{47} +200.000 q^{50} +(-176.000 + 304.841i) q^{52} +(319.500 - 553.390i) q^{53} -135.000 q^{55} +(51.0000 + 88.3346i) q^{58} +(-79.5000 + 137.698i) q^{59} +(361.000 + 625.270i) q^{61} -370.000 q^{62} +64.0000 q^{64} +(660.000 + 1143.15i) q^{65} +(83.0000 - 143.760i) q^{67} +(168.000 + 290.985i) q^{68} -1086.00 q^{71} +(109.000 - 188.794i) q^{73} +(-44.0000 + 76.2102i) q^{74} -416.000 q^{76} +(291.500 + 504.893i) q^{79} +(120.000 - 207.846i) q^{80} +(168.000 + 290.985i) q^{82} -597.000 q^{83} +1260.00 q^{85} +(-326.000 - 564.649i) q^{86} +(-36.0000 + 62.3538i) q^{88} +(519.000 + 898.934i) q^{89} +336.000 q^{92} +(138.000 - 239.023i) q^{94} +(-780.000 + 1351.00i) q^{95} +169.000 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8	$ 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8} + 30 q^{10} - 9 q^{11} + 176 q^{13} - 16 q^{16} + 84 q^{17} + 104 q^{19} - 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} - 176 q^{26} - 102 q^{29} + 185 q^{31} - 32 q^{32} - 336 q^{34} - 44 q^{37} + 208 q^{38} + 120 q^{40} - 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} + 138 q^{47} + 400 q^{50} - 352 q^{52} + 639 q^{53} - 270 q^{55} + 102 q^{58} - 159 q^{59} + 722 q^{61} - 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} + 336 q^{68} - 2172 q^{71} + 218 q^{73} - 88 q^{74} - 832 q^{76} + 583 q^{79} + 240 q^{80} + 336 q^{82} - 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} + 1038 q^{89} + 672 q^{92} + 276 q^{94} - 1560 q^{95} + 338 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8 + 30 * q^10 - 9 * q^11 + 176 * q^13 - 16 * q^16 + 84 * q^17 + 104 * q^19 - 120 * q^20 + 36 * q^22 - 84 * q^23 - 100 * q^25 - 176 * q^26 - 102 * q^29 + 185 * q^31 - 32 * q^32 - 336 * q^34 - 44 * q^37 + 208 * q^38 + 120 * q^40 - 336 * q^41 + 652 * q^43 - 36 * q^44 - 168 * q^46 + 138 * q^47 + 400 * q^50 - 352 * q^52 + 639 * q^53 - 270 * q^55 + 102 * q^58 - 159 * q^59 + 722 * q^61 - 740 * q^62 + 128 * q^64 + 1320 * q^65 + 166 * q^67 + 336 * q^68 - 2172 * q^71 + 218 * q^73 - 88 * q^74 - 832 * q^76 + 583 * q^79 + 240 * q^80 + 336 * q^82 - 1194 * q^83 + 2520 * q^85 - 652 * q^86 - 72 * q^88 + 1038 * q^89 + 672 * q^92 + 276 * q^94 - 1560 * q^95 + 338 * 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
$5$	7.50000	+	12.9904i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	7.50000	+	12.9904i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	8.00000			0.353553
$9$		0			0
$10$	15.0000	−	25.9808i	0.474342	−	0.821584i
$11$	−4.50000	+	7.79423i	−0.123346	+	0.213641i	−0.921085	−	0.389362i	$-0.872696\pi$
$11$	−4.50000	+	7.79423i	−0.123346	+	0.213641i	0.797739	+	0.603002i	$0.206029\pi$
$12$		0			0
$13$	88.0000			1.87745			0.938723	−	0.344671i	$-0.112010\pi$
$13$	88.0000			1.87745			0.938723	+	0.344671i	$0.112010\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	42.0000	−	72.7461i	0.599206	−	1.03785i	−0.393733	−	0.919225i	$-0.628817\pi$
$17$	42.0000	−	72.7461i	0.599206	−	1.03785i	0.992939	−	0.118630i	$-0.0378502\pi$
$18$		0			0
$19$	52.0000	+	90.0666i	0.627875	+	1.08751i	0.987977	+	0.154598i	$0.0494083\pi$
$19$	52.0000	+	90.0666i	0.627875	+	1.08751i	−0.360103	+	0.932913i	$0.617258\pi$
$20$	−60.0000			−0.670820
$21$		0			0
$22$	18.0000			0.174437
$23$	−42.0000	−	72.7461i	−0.380765	−	0.659505i	0.610406	−	0.792088i	$-0.291006\pi$
$23$	−42.0000	−	72.7461i	−0.380765	−	0.659505i	−0.991172	+	0.132583i	$0.957673\pi$
$24$		0			0
$25$	−50.0000	+	86.6025i	−0.400000	+	0.692820i
$26$	−88.0000	−	152.420i	−0.663778	−	1.14970i
$27$		0			0
$28$		0			0
$29$	−51.0000			−0.326568			−0.163284	−	0.986579i	$-0.552209\pi$
$29$	−51.0000			−0.326568			−0.163284	+	0.986579i	$0.552209\pi$
$30$		0			0
$31$	92.5000	−	160.215i	0.535919	−	0.928239i	−0.463199	−	0.886254i	$-0.653299\pi$
$31$	92.5000	−	160.215i	0.535919	−	0.928239i	0.999118	−	0.0419848i	$-0.0133681\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−168.000			−0.847405
$35$		0			0
$36$		0			0
$37$	−22.0000	−	38.1051i	−0.0977507	−	0.169309i	0.813003	−	0.582260i	$-0.197831\pi$
$37$	−22.0000	−	38.1051i	−0.0977507	−	0.169309i	−0.910753	+	0.412951i	$0.864498\pi$
$38$	104.000	−	180.133i	0.443974	−	0.768986i
$39$		0			0
$40$	60.0000	+	103.923i	0.237171	+	0.410792i
$41$	−168.000			−0.639932			−0.319966	−	0.947429i	$-0.603671\pi$
$41$	−168.000			−0.639932			−0.319966	+	0.947429i	$0.603671\pi$
$42$		0			0
$43$	326.000			1.15615			0.578076	−	0.815983i	$-0.303804\pi$
$43$	326.000			1.15615			0.578076	+	0.815983i	$0.303804\pi$
$44$	−18.0000	−	31.1769i	−0.0616728	−	0.106820i
$45$		0			0
$46$	−84.0000	+	145.492i	−0.269242	+	0.466341i
$47$	69.0000	+	119.512i	0.214142	+	0.370905i	0.953007	−	0.302949i	$-0.0979711\pi$
$47$	69.0000	+	119.512i	0.214142	+	0.370905i	−0.738865	+	0.673854i	$0.764638\pi$
$48$		0			0
$49$		0			0
$50$	200.000			0.565685
$51$		0			0
$52$	−176.000	+	304.841i	−0.469362	+	0.812958i
$53$	319.500	−	553.390i	0.828051	−	1.43423i	−0.0715141	−	0.997440i	$-0.522783\pi$
$53$	319.500	−	553.390i	0.828051	−	1.43423i	0.899565	−	0.436787i	$-0.143884\pi$
$54$		0			0
$55$	−135.000			−0.330971
$56$		0			0
$57$		0			0
$58$	51.0000	+	88.3346i	0.115459	+	0.199981i
$59$	−79.5000	+	137.698i	−0.175424	+	0.303843i	−0.940308	−	0.340325i	$-0.889463\pi$
$59$	−79.5000	+	137.698i	−0.175424	+	0.303843i	0.764884	+	0.644168i	$0.222796\pi$
$60$		0			0
$61$	361.000	+	625.270i	0.757726	+	1.31242i	0.944007	+	0.329924i	$0.107023\pi$
$61$	361.000	+	625.270i	0.757726	+	1.31242i	−0.186281	+	0.982497i	$0.559643\pi$
$62$	−370.000			−0.757904
$63$		0			0
$64$	64.0000			0.125000
$65$	660.000	+	1143.15i	1.25943	+	2.18140i
$66$		0			0
$67$	83.0000	−	143.760i	0.151344	−	0.262136i	−0.780378	−	0.625309i	$-0.784973\pi$
$67$	83.0000	−	143.760i	0.151344	−	0.262136i	0.931722	+	0.363173i	$0.118306\pi$
$68$	168.000	+	290.985i	0.299603	+	0.518927i
$69$		0			0
$70$		0			0
$71$	−1086.00			−1.81527			−0.907637	−	0.419755i	$-0.862116\pi$
$71$	−1086.00			−1.81527			−0.907637	+	0.419755i	$0.862116\pi$
$72$		0			0
$73$	109.000	−	188.794i	0.174760	−	0.302693i	−0.765318	−	0.643652i	$-0.777418\pi$
$73$	109.000	−	188.794i	0.174760	−	0.302693i	0.940078	+	0.340959i	$0.110752\pi$
$74$	−44.0000	+	76.2102i	−0.0691202	+	0.119720i
$75$		0			0
$76$	−416.000			−0.627875
$77$		0			0
$78$		0			0
$79$	291.500	+	504.893i	0.415143	+	0.719049i	0.995443	−	0.0953535i	$-0.0303981\pi$
$79$	291.500	+	504.893i	0.415143	+	0.719049i	−0.580300	+	0.814403i	$0.697065\pi$
$80$	120.000	−	207.846i	0.167705	−	0.290474i
$81$		0			0
$82$	168.000	+	290.985i	0.226250	+	0.391876i
$83$	−597.000			−0.789509			−0.394755	−	0.918787i	$-0.629170\pi$
$83$	−597.000			−0.789509			−0.394755	+	0.918787i	$0.629170\pi$
$84$		0			0
$85$	1260.00			1.60784
$86$	−326.000	−	564.649i	−0.408761	−	0.707996i
$87$		0			0
$88$	−36.0000	+	62.3538i	−0.0436092	+	0.0755334i
$89$	519.000	+	898.934i	0.618134	+	1.07064i	0.989826	+	0.142283i	$0.0454443\pi$
$89$	519.000	+	898.934i	0.618134	+	1.07064i	−0.371692	+	0.928356i	$0.621222\pi$
$90$		0			0
$91$		0			0
$92$	336.000			0.380765
$93$		0			0
$94$	138.000	−	239.023i	0.151421	−	0.262270i
$95$	−780.000	+	1351.00i	−0.842382	+	1.45905i
$96$		0			0
$97$	169.000			0.176901			0.0884503	−	0.996081i	$-0.471809\pi$
$97$	169.000			0.176901			0.0884503	+	0.996081i	$0.471809\pi$
$98$		0			0
$99$		0			0
$100$	−200.000	−	346.410i	−0.200000	−	0.346410i
$101$	−321.000	+	555.988i	−0.316244	+	0.547752i	−0.979701	−	0.200463i	$-0.935755\pi$
$101$	−321.000	+	555.988i	−0.316244	+	0.547752i	0.663457	+	0.748215i	$0.269089\pi$
$102$		0			0
$103$	232.000	+	401.836i	0.221938	+	0.384408i	0.955396	−	0.295326i	$-0.0954283\pi$
$103$	232.000	+	401.836i	0.221938	+	0.384408i	−0.733458	+	0.679735i	$0.762095\pi$
$104$	704.000			0.663778
$105$		0			0
$106$	−1278.00			−1.17104
$107$	196.500	+	340.348i	0.177536	+	0.307502i	0.941036	−	0.338306i	$-0.109854\pi$
$107$	196.500	+	340.348i	0.177536	+	0.307502i	−0.763500	+	0.645808i	$0.776521\pi$
$108$		0			0
$109$	−7.00000	+	12.1244i	−0.00615118	+	0.0106542i	−0.869085	−	0.494663i	$-0.835291\pi$
$109$	−7.00000	+	12.1244i	−0.00615118	+	0.0106542i	0.862933	+	0.505318i	$0.168625\pi$
$110$	135.000	+	233.827i	0.117016	+	0.202677i
$111$		0			0
$112$		0			0
$113$	2184.00			1.81817			0.909086	−	0.416608i	$-0.136781\pi$
$113$	2184.00			1.81817			0.909086	+	0.416608i	$0.136781\pi$
$114$		0			0
$115$	630.000	−	1091.19i	0.510850	−	0.884819i
$116$	102.000	−	176.669i	0.0816419	−	0.141408i
$117$		0			0
$118$	318.000			0.248087
$119$		0			0
$120$		0			0
$121$	625.000	+	1082.53i	0.469572	+	0.813322i
$122$	722.000	−	1250.54i	0.535794	−	0.928022i
$123$		0			0
$124$	370.000	+	640.859i	0.267960	+	0.464120i
$125$	375.000			0.268328
$126$		0			0
$127$	−373.000			−0.260617			−0.130309	−	0.991473i	$-0.541597\pi$
$127$	−373.000			−0.260617			−0.130309	+	0.991473i	$0.541597\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	1320.00	−	2286.31i	0.890551	−	1.54248i
$131$	586.500	+	1015.85i	0.391166	+	0.677519i	0.992604	−	0.121400i	$-0.0387385\pi$
$131$	586.500	+	1015.85i	0.391166	+	0.677519i	−0.601438	+	0.798920i	$0.705405\pi$
$132$		0			0
$133$		0			0
$134$	−332.000			−0.214033
$135$		0			0
$136$	336.000	−	581.969i	0.211851	−	0.366937i
$137$	15.0000	−	25.9808i	0.00935428	−	0.0162021i	−0.861310	−	0.508079i	$-0.830356\pi$
$137$	15.0000	−	25.9808i	0.00935428	−	0.0162021i	0.870665	+	0.491877i	$0.163689\pi$
$138$		0			0
$139$	82.0000			0.0500370			0.0250185	−	0.999687i	$-0.492036\pi$
$139$	82.0000			0.0500370			0.0250185	+	0.999687i	$0.492036\pi$
$140$		0			0
$141$		0			0
$142$	1086.00	+	1881.01i	0.641796	+	1.11162i
$143$	−396.000	+	685.892i	−0.231575	+	0.401099i
$144$		0			0
$145$	−382.500	−	662.509i	−0.219068	−	0.379437i
$146$	−436.000			−0.247148
$147$		0			0
$148$	176.000			0.0977507
$149$	−717.000	−	1241.88i	−0.394221	−	0.682811i	0.598780	−	0.800913i	$-0.295652\pi$
$149$	−717.000	−	1241.88i	−0.394221	−	0.682811i	−0.993001	+	0.118102i	$0.962319\pi$
$150$		0			0
$151$	1335.50	−	2313.15i	0.719745	−	1.24663i	−0.241356	−	0.970437i	$-0.577592\pi$
$151$	1335.50	−	2313.15i	0.719745	−	1.24663i	0.961101	−	0.276198i	$-0.0890745\pi$
$152$	416.000	+	720.533i	0.221987	+	0.384493i
$153$		0			0
$154$		0			0
$155$	2775.00			1.43802
$156$		0			0
$157$	1126.00	−	1950.29i	0.572386	−	0.991401i	−0.423934	−	0.905693i	$-0.639351\pi$
$157$	1126.00	−	1950.29i	0.572386	−	0.991401i	0.996320	−	0.0857085i	$-0.0273154\pi$
$158$	583.000	−	1009.79i	0.293551	−	0.508444i
$159$		0			0
$160$	−480.000			−0.237171
$161$		0			0
$162$		0			0
$163$	−838.000	−	1451.46i	−0.402682	−	0.697466i	0.591366	−	0.806403i	$-0.298589\pi$
$163$	−838.000	−	1451.46i	−0.402682	−	0.697466i	−0.994049	+	0.108937i	$0.965255\pi$
$164$	336.000	−	581.969i	0.159983	−	0.277098i
$165$		0			0
$166$	597.000	+	1034.03i	0.279134	+	0.483474i
$167$	3030.00			1.40400			0.702001	−	0.712176i	$-0.252290\pi$
$167$	3030.00			1.40400			0.702001	+	0.712176i	$0.252290\pi$
$168$		0			0
$169$	5547.00			2.52481
$170$	−1260.00	−	2182.38i	−0.568456	−	0.984595i
$171$		0			0
$172$	−652.000	+	1129.30i	−0.289038	+	0.500628i
$173$	−1719.00	−	2977.40i	−0.755452	−	1.30848i	−0.945149	−	0.326638i	$-0.894084\pi$
$173$	−1719.00	−	2977.40i	−0.755452	−	1.30848i	0.189698	−	0.981843i	$-0.439249\pi$
$174$		0			0
$175$		0			0
$176$	144.000			0.0616728
$177$		0			0
$178$	1038.00	−	1797.87i	0.437086	−	0.757056i
$179$	−606.000	+	1049.62i	−0.253042	+	0.438282i	−0.964362	−	0.264587i	$-0.914765\pi$
$179$	−606.000	+	1049.62i	−0.253042	+	0.438282i	0.711320	+	0.702869i	$0.248098\pi$
$180$		0			0
$181$	−3032.00			−1.24512			−0.622560	−	0.782572i	$-0.713907\pi$
$181$	−3032.00			−1.24512			−0.622560	+	0.782572i	$0.713907\pi$
$182$		0			0
$183$		0			0
$184$	−336.000	−	581.969i	−0.134621	−	0.233170i
$185$	330.000	−	571.577i	0.131146	−	0.227152i
$186$		0			0
$187$	378.000	+	654.715i	0.147819	+	0.256030i
$188$	−552.000			−0.214142
$189$		0			0
$190$	3120.00			1.19131
$191$	1260.00	+	2182.38i	0.477332	+	0.826763i	0.999662	−	0.0259799i	$-0.00827060\pi$
$191$	1260.00	+	2182.38i	0.477332	+	0.826763i	−0.522331	+	0.852743i	$0.674937\pi$
$192$		0			0
$193$	−182.500	+	316.099i	−0.0680655	+	0.117893i	−0.898050	−	0.439894i	$-0.855016\pi$
$193$	−182.500	+	316.099i	−0.0680655	+	0.117893i	0.829984	+	0.557787i	$0.188349\pi$
$194$	−169.000	−	292.717i	−0.0625438	−	0.108329i
$195$		0			0
$196$		0			0
$197$	1590.00			0.575040			0.287520	−	0.957775i	$-0.407169\pi$
$197$	1590.00			0.575040			0.287520	+	0.957775i	$0.407169\pi$
$198$		0			0
$199$	−2690.00	+	4659.22i	−0.958236	+	1.65971i	−0.231455	+	0.972846i	$0.574348\pi$
$199$	−2690.00	+	4659.22i	−0.958236	+	1.65971i	−0.726782	+	0.686868i	$0.758985\pi$
$200$	−400.000	+	692.820i	−0.141421	+	0.244949i
$201$		0			0
$202$	1284.00			0.447237
$203$		0			0
$204$		0			0
$205$	−1260.00	−	2182.38i	−0.429279	−	0.743533i
$206$	464.000	−	803.672i	0.156934	−	0.271818i
$207$		0			0
$208$	−704.000	−	1219.36i	−0.234681	−	0.406479i
$209$	−936.000			−0.309782
$210$		0			0
$211$	−5362.00			−1.74946			−0.874728	−	0.484614i	$-0.838960\pi$
$211$	−5362.00			−1.74946			−0.874728	+	0.484614i	$0.838960\pi$
$212$	1278.00	+	2213.56i	0.414025	+	0.717113i
$213$		0			0
$214$	393.000	−	680.696i	0.125537	−	0.217437i
$215$	2445.00	+	4234.86i	0.775570	+	1.34333i
$216$		0			0
$217$		0			0
$218$	28.0000			0.00869908
$219$		0			0
$220$	270.000	−	467.654i	0.0827427	−	0.143315i
$221$	3696.00	−	6401.66i	1.12498	−	1.94852i
$222$		0			0
$223$	1573.00			0.472358			0.236179	−	0.971710i	$-0.424105\pi$
$223$	1573.00			0.472358			0.236179	+	0.971710i	$0.424105\pi$
$224$		0			0
$225$		0			0
$226$	−2184.00	−	3782.80i	−0.642821	−	1.11340i
$227$	−460.500	+	797.609i	−0.134645	+	0.233212i	−0.925462	−	0.378841i	$-0.876323\pi$
$227$	−460.500	+	797.609i	−0.134645	+	0.233212i	0.790817	+	0.612053i	$0.209656\pi$
$228$		0			0
$229$	2026.00	+	3509.13i	0.584637	+	1.01262i	0.994921	+	0.100663i	$0.0320963\pi$
$229$	2026.00	+	3509.13i	0.584637	+	1.01262i	−0.410284	+	0.911958i	$0.634570\pi$
$230$	−2520.00			−0.722452
$231$		0			0
$232$	−408.000			−0.115459
$233$	−234.000	−	405.300i	−0.0657933	−	0.113957i	0.831252	−	0.555895i	$-0.187624\pi$
$233$	−234.000	−	405.300i	−0.0657933	−	0.113957i	−0.897046	+	0.441938i	$0.854291\pi$
$234$		0			0
$235$	−1035.00	+	1792.67i	−0.287302	+	0.497622i
$236$	−318.000	−	550.792i	−0.0877120	−	0.151922i
$237$		0			0
$238$		0			0
$239$	−4932.00			−1.33483			−0.667415	−	0.744686i	$-0.732599\pi$
$239$	−4932.00			−1.33483			−0.667415	+	0.744686i	$0.732599\pi$
$240$		0			0
$241$	−768.500	+	1331.08i	−0.205408	+	0.355778i	−0.950263	−	0.311449i	$-0.899186\pi$
$241$	−768.500	+	1331.08i	−0.205408	+	0.355778i	0.744854	+	0.667227i	$0.232519\pi$
$242$	1250.00	−	2165.06i	0.332037	−	0.575106i
$243$		0			0
$244$	−2888.00			−0.757726
$245$		0			0
$246$		0			0
$247$	4576.00	+	7925.86i	1.17880	+	2.04174i
$248$	740.000	−	1281.72i	0.189476	−	0.328182i
$249$		0			0
$250$	−375.000	−	649.519i	−0.0948683	−	0.164317i
$251$	5319.00			1.33758			0.668789	−	0.743452i	$-0.266813\pi$
$251$	5319.00			1.33758			0.668789	+	0.743452i	$0.266813\pi$
$252$		0			0
$253$	756.000			0.187863
$254$	373.000	+	646.055i	0.0921421	+	0.159595i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	−2673.00	−	4629.77i	−0.648783	−	1.12372i	−0.983414	−	0.181375i	$-0.941945\pi$
$257$	−2673.00	−	4629.77i	−0.648783	−	1.12372i	0.334631	−	0.942349i	$-0.391388\pi$
$258$		0			0
$259$		0			0
$260$	−5280.00			−1.25943
$261$		0			0
$262$	1173.00	−	2031.70i	0.276596	−	0.479079i
$263$	387.000	−	670.304i	0.0907355	−	0.157159i	−0.817085	−	0.576517i	$-0.804412\pi$
$263$	387.000	−	670.304i	0.0907355	−	0.157159i	0.907821	+	0.419358i	$0.137745\pi$
$264$		0			0
$265$	9585.00			2.22189
$266$		0			0
$267$		0			0
$268$	332.000	+	575.041i	0.0756721	+	0.131068i
$269$	−1207.50	+	2091.45i	−0.273690	+	0.474045i	−0.969804	−	0.243887i	$-0.921578\pi$
$269$	−1207.50	+	2091.45i	−0.273690	+	0.474045i	0.696114	+	0.717931i	$0.254911\pi$
$270$		0			0
$271$	−237.500	−	411.362i	−0.0532365	−	0.0922084i	0.838179	−	0.545395i	$-0.183620\pi$
$271$	−237.500	−	411.362i	−0.0532365	−	0.0922084i	−0.891416	+	0.453187i	$0.850287\pi$
$272$	−1344.00			−0.299603
$273$		0			0
$274$	−60.0000			−0.0132290
$275$	−450.000	−	779.423i	−0.0986764	−	0.170913i
$276$		0			0
$277$	−1864.00	+	3228.54i	−0.404321	+	0.700304i	−0.994242	−	0.107156i	$-0.965825\pi$
$277$	−1864.00	+	3228.54i	−0.404321	+	0.700304i	0.589921	+	0.807461i	$0.299159\pi$
$278$	−82.0000	−	142.028i	−0.0176908	−	0.0306413i
$279$		0			0
$280$		0			0
$281$	−1602.00			−0.340097			−0.170049	−	0.985436i	$-0.554392\pi$
$281$	−1602.00			−0.340097			−0.170049	+	0.985436i	$0.554392\pi$
$282$		0			0
$283$	343.000	−	594.093i	0.0720468	−	0.124789i	−0.827751	−	0.561095i	$-0.810380\pi$
$283$	343.000	−	594.093i	0.0720468	−	0.124789i	0.899798	+	0.436306i	$0.143714\pi$
$284$	2172.00	−	3762.01i	0.453819	−	0.786037i
$285$		0			0
$286$	1584.00			0.327496
$287$		0			0
$288$		0			0
$289$	−1071.50	−	1855.89i	−0.218095	−	0.377751i
$290$	−765.000	+	1325.02i	−0.154905	+	0.268303i
$291$		0			0
$292$	436.000	+	755.174i	0.0873800	+	0.151347i
$293$	−1101.00			−0.219526			−0.109763	−	0.993958i	$-0.535009\pi$
$293$	−1101.00			−0.219526			−0.109763	+	0.993958i	$0.535009\pi$
$294$		0			0
$295$	−2385.00			−0.470712
$296$	−176.000	−	304.841i	−0.0345601	−	0.0598599i
$297$		0			0
$298$	−1434.00	+	2483.76i	−0.278756	+	0.482820i
$299$	−3696.00	−	6401.66i	−0.714867	−	1.23819i
$300$		0			0
$301$		0			0
$302$	−5342.00			−1.01787
$303$		0			0
$304$	832.000	−	1441.07i	0.156969	−	0.271878i
$305$	−5415.00	+	9379.06i	−1.01660	+	1.76080i
$306$		0			0
$307$	−2780.00			−0.516818			−0.258409	−	0.966036i	$-0.583198\pi$
$307$	−2780.00			−0.516818			−0.258409	+	0.966036i	$0.583198\pi$
$308$		0			0
$309$		0			0
$310$	−2775.00	−	4806.44i	−0.508417	−	0.880605i
$311$	−2148.00	+	3720.45i	−0.391646	+	0.678351i	−0.992667	−	0.120883i	$-0.961428\pi$
$311$	−2148.00	+	3720.45i	−0.391646	+	0.678351i	0.601021	+	0.799233i	$0.294761\pi$
$312$		0			0
$313$	2744.50	+	4753.61i	0.495618	+	0.858435i	0.999987	−	0.00505298i	$-0.00160842\pi$
$313$	2744.50	+	4753.61i	0.495618	+	0.858435i	−0.504370	+	0.863488i	$0.668275\pi$
$314$	−4504.00			−0.809476
$315$		0			0
$316$	−2332.00			−0.415143
$317$	2245.50	+	3889.32i	0.397854	+	0.689104i	0.993461	−	0.114172i	$-0.0364216\pi$
$317$	2245.50	+	3889.32i	0.397854	+	0.689104i	−0.595607	+	0.803276i	$0.703088\pi$
$318$		0			0
$319$	229.500	−	397.506i	0.0402807	−	0.0697682i
$320$	480.000	+	831.384i	0.0838525	+	0.145237i
$321$		0			0
$322$		0			0
$323$	8736.00			1.50490
$324$		0			0
$325$	−4400.00	+	7621.02i	−0.750979	+	1.30073i
$326$	−1676.00	+	2902.92i	−0.284739	+	0.493183i
$327$		0			0
$328$	−1344.00			−0.226250
$329$		0			0
$330$		0			0
$331$	1982.00	+	3432.92i	0.329126	+	0.570062i	0.982339	−	0.187112i	$-0.0599127\pi$
$331$	1982.00	+	3432.92i	0.329126	+	0.570062i	−0.653213	+	0.757174i	$0.726579\pi$
$332$	1194.00	−	2068.07i	0.197377	−	0.341868i
$333$		0			0
$334$	−3030.00	−	5248.11i	−0.496390	−	0.859773i
$335$	2490.00			0.406099
$336$		0			0
$337$	161.000			0.0260244			0.0130122	−	0.999915i	$-0.495858\pi$
$337$	161.000			0.0260244			0.0130122	+	0.999915i	$0.495858\pi$
$338$	−5547.00	−	9607.69i	−0.892654	−	1.54612i
$339$		0			0
$340$	−2520.00	+	4364.77i	−0.401959	+	0.696214i
$341$	832.500	+	1441.93i	0.132206	+	0.228988i
$342$		0			0
$343$		0			0
$344$	2608.00			0.408761
$345$		0			0
$346$	−3438.00	+	5954.79i	−0.534185	+	0.925236i
$347$	−2958.00	+	5123.41i	−0.457619	+	0.792619i	−0.998835	−	0.0482646i	$-0.984631\pi$
$347$	−2958.00	+	5123.41i	−0.457619	+	0.792619i	0.541216	+	0.840884i	$0.317964\pi$
$348$		0			0
$349$	142.000			0.0217796			0.0108898	−	0.999941i	$-0.496534\pi$
$349$	142.000			0.0217796			0.0108898	+	0.999941i	$0.496534\pi$
$350$		0			0
$351$		0			0
$352$	−144.000	−	249.415i	−0.0218046	−	0.0377667i
$353$	2220.00	−	3845.15i	0.334727	−	0.579764i	−0.648705	−	0.761040i	$-0.724689\pi$
$353$	2220.00	−	3845.15i	0.334727	−	0.579764i	0.983432	+	0.181275i	$0.0580225\pi$
$354$		0			0
$355$	−8145.00	−	14107.6i	−1.21772	−	2.10916i
$356$	−4152.00			−0.618134
$357$		0			0
$358$	2424.00			0.357856
$359$	1143.00	+	1979.73i	0.168037	+	0.291048i	0.937730	−	0.347366i	$-0.112924\pi$
$359$	1143.00	+	1979.73i	0.168037	+	0.291048i	−0.769693	+	0.638415i	$0.779591\pi$
$360$		0			0
$361$	−1978.50	+	3426.86i	−0.288453	+	0.499615i
$362$	3032.00	+	5251.58i	0.440217	+	0.762477i
$363$		0			0
$364$		0			0
$365$	3270.00			0.468930
$366$		0			0
$367$	−1434.50	+	2484.63i	−0.204033	+	0.353396i	−0.949824	−	0.312784i	$-0.898738\pi$
$367$	−1434.50	+	2484.63i	−0.204033	+	0.353396i	0.745791	+	0.666180i	$0.232072\pi$
$368$	−672.000	+	1163.94i	−0.0951914	+	0.164876i
$369$		0			0
$370$	−1320.00			−0.185469
$371$		0			0
$372$		0			0
$373$	1532.00	+	2653.50i	0.212665	+	0.368346i	0.952548	−	0.304390i	$-0.0984524\pi$
$373$	1532.00	+	2653.50i	0.212665	+	0.368346i	−0.739883	+	0.672736i	$0.765119\pi$
$374$	756.000	−	1309.43i	0.104524	−	0.181040i
$375$		0			0
$376$	552.000	+	956.092i	0.0757107	+	0.131135i
$377$	−4488.00			−0.613113
$378$		0			0
$379$	−6040.00			−0.818612			−0.409306	−	0.912397i	$-0.634229\pi$
$379$	−6040.00			−0.818612			−0.409306	+	0.912397i	$0.634229\pi$
$380$	−3120.00	−	5404.00i	−0.421191	−	0.729524i
$381$		0			0
$382$	2520.00	−	4364.77i	0.337525	−	0.584610i
$383$	−921.000	−	1595.22i	−0.122874	−	0.212825i	0.798026	−	0.602623i	$-0.205878\pi$
$383$	−921.000	−	1595.22i	−0.122874	−	0.212825i	−0.920900	+	0.389799i	$0.872545\pi$
$384$		0			0
$385$		0			0
$386$	730.000			0.0962591
$387$		0			0
$388$	−338.000	+	585.433i	−0.0442251	+	0.0766002i
$389$	3915.00	−	6780.98i	0.510279	−	0.883828i	−0.489650	−	0.871919i	$-0.662876\pi$
$389$	3915.00	−	6780.98i	0.510279	−	0.883828i	0.999929	−	0.0119097i	$-0.00379105\pi$
$390$		0			0
$391$	−7056.00			−0.912627
$392$		0			0
$393$		0			0
$394$	−1590.00	−	2753.96i	−0.203307	−	0.352138i
$395$	−4372.50	+	7573.39i	−0.556973	+	0.964706i
$396$		0			0
$397$	−7382.00	−	12786.0i	−0.933229	−	1.61640i	−0.777762	−	0.628559i	$-0.783645\pi$
$397$	−7382.00	−	12786.0i	−0.933229	−	1.61640i	−0.155467	−	0.987841i	$-0.549688\pi$
$398$	10760.0			1.35515
$399$		0			0
$400$	1600.00			0.200000
$401$	3132.00	+	5424.78i	0.390036	+	0.675563i	0.992454	−	0.122618i	$-0.0391291\pi$
$401$	3132.00	+	5424.78i	0.390036	+	0.675563i	−0.602417	+	0.798181i	$0.705796\pi$
$402$		0			0
$403$	8140.00	−	14098.9i	1.00616	−	1.74272i
$404$	−1284.00	−	2223.95i	−0.158122	−	0.273876i
$405$		0			0
$406$		0			0
$407$	396.000			0.0482285
$408$		0			0
$409$	2375.50	−	4114.49i	0.287191	−	0.497429i	−0.685948	−	0.727651i	$-0.740612\pi$
$409$	2375.50	−	4114.49i	0.287191	−	0.497429i	0.973138	+	0.230222i	$0.0739454\pi$
$410$	−2520.00	+	4364.77i	−0.303546	+	0.525757i
$411$		0			0
$412$	−1856.00			−0.221938
$413$		0			0
$414$		0			0
$415$	−4477.50	−	7755.26i	−0.529619	−	0.917327i
$416$	−1408.00	+	2438.73i	−0.165944	+	0.287424i
$417$		0			0
$418$	936.000	+	1621.20i	0.109525	+	0.189702i
$419$	−4704.00			−0.548462			−0.274231	−	0.961664i	$-0.588423\pi$
$419$	−4704.00			−0.548462			−0.274231	+	0.961664i	$0.588423\pi$
$420$		0			0
$421$	−4474.00			−0.517932			−0.258966	−	0.965886i	$-0.583382\pi$
$421$	−4474.00			−0.517932			−0.258966	+	0.965886i	$0.583382\pi$
$422$	5362.00	+	9287.26i	0.618526	+	1.07132i
$423$		0			0
$424$	2556.00	−	4427.12i	0.292760	−	0.507076i
$425$	4200.00	+	7274.61i	0.479365	+	0.830284i
$426$		0			0
$427$		0			0
$428$	−1572.00			−0.177536
$429$		0			0
$430$	4890.00	−	8469.73i	0.548411	−	0.949876i
$431$	−6402.00	+	11088.6i	−0.715484	+	1.23925i	0.247289	+	0.968942i	$0.420460\pi$
$431$	−6402.00	+	11088.6i	−0.715484	+	1.23925i	−0.962773	+	0.270312i	$0.912873\pi$
$432$		0			0
$433$	5074.00			0.563143			0.281571	−	0.959540i	$-0.409144\pi$
$433$	5074.00			0.563143			0.281571	+	0.959540i	$0.409144\pi$
$434$		0			0
$435$		0			0
$436$	−28.0000	−	48.4974i	−0.00307559	−	0.00532708i
$437$	4368.00	−	7565.60i	0.478146	−	0.828173i
$438$		0			0
$439$	−633.500	−	1097.25i	−0.0688731	−	0.119292i	0.829532	−	0.558459i	$-0.188607\pi$
$439$	−633.500	−	1097.25i	−0.0688731	−	0.119292i	−0.898406	+	0.439167i	$0.855274\pi$
$440$	−1080.00			−0.117016
$441$		0			0
$442$	−14784.0			−1.59096
$443$	−3466.50	−	6004.15i	−0.371780	−	0.643941i	0.618060	−	0.786131i	$-0.287919\pi$
$443$	−3466.50	−	6004.15i	−0.371780	−	0.643941i	−0.989839	+	0.142190i	$0.954586\pi$
$444$		0			0
$445$	−7785.00	+	13484.0i	−0.829313	+	1.43641i
$446$	−1573.00	−	2724.52i	−0.167004	−	0.289259i
$447$		0			0
$448$		0			0
$449$	−11688.0			−1.22849			−0.614244	−	0.789116i	$-0.710539\pi$
$449$	−11688.0			−1.22849			−0.614244	+	0.789116i	$0.710539\pi$
$450$		0			0
$451$	756.000	−	1309.43i	0.0789327	−	0.136715i
$452$	−4368.00	+	7565.60i	−0.454543	+	0.787292i
$453$		0			0
$454$	1842.00			0.190417
$455$		0			0
$456$		0			0
$457$	−275.500	−	477.180i	−0.0281999	−	0.0488436i	0.851581	−	0.524223i	$-0.175644\pi$
$457$	−275.500	−	477.180i	−0.0281999	−	0.0488436i	−0.879781	+	0.475379i	$0.842311\pi$
$458$	4052.00	−	7018.27i	0.413401	−	0.716031i
$459$		0			0
$460$	2520.00	+	4364.77i	0.255425	+	0.442409i
$461$	13386.0			1.35238			0.676191	−	0.736726i	$-0.263629\pi$
$461$	13386.0			1.35238			0.676191	+	0.736726i	$0.263629\pi$
$462$		0			0
$463$	−6376.00			−0.639995			−0.319998	−	0.947418i	$-0.603682\pi$
$463$	−6376.00			−0.639995			−0.319998	+	0.947418i	$0.603682\pi$
$464$	408.000	+	706.677i	0.0408210	+	0.0707040i
$465$		0			0
$466$	−468.000	+	810.600i	−0.0465229	+	0.0805801i
$467$	−2850.00	−	4936.34i	−0.282403	−	0.489137i	0.689573	−	0.724216i	$-0.257798\pi$
$467$	−2850.00	−	4936.34i	−0.282403	−	0.489137i	−0.971976	+	0.235080i	$0.924465\pi$
$468$		0			0
$469$		0			0
$470$	4140.00			0.406306
$471$		0			0
$472$	−636.000	+	1101.58i	−0.0620218	+	0.107425i
$473$	−1467.00	+	2540.92i	−0.142606	+	0.247001i
$474$		0			0
$475$	−10400.0			−1.00460
$476$		0			0
$477$		0			0
$478$	4932.00	+	8542.47i	0.471934	+	0.817414i
$479$	−9897.00	+	17142.1i	−0.944062	+	1.63516i	−0.186442	+	0.982466i	$0.559696\pi$
$479$	−9897.00	+	17142.1i	−0.944062	+	1.63516i	−0.757619	+	0.652697i	$0.773638\pi$
$480$		0			0
$481$	−1936.00	−	3353.25i	−0.183522	−	0.317869i
$482$	3074.00			0.290491
$483$		0			0
$484$	−5000.00			−0.469572
$485$	1267.50	+	2195.37i	0.118668	+	0.205540i
$486$		0			0
$487$	−7967.50	+	13800.1i	−0.741359	+	1.28407i	0.210517	+	0.977590i	$0.432485\pi$
$487$	−7967.50	+	13800.1i	−0.741359	+	1.28407i	−0.951877	+	0.306482i	$0.900848\pi$
$488$	2888.00	+	5002.16i	0.267897	+	0.464011i
$489$		0			0
$490$		0			0
$491$	−9963.00			−0.915731			−0.457865	−	0.889021i	$-0.651386\pi$
$491$	−9963.00			−0.915731			−0.457865	+	0.889021i	$0.651386\pi$
$492$		0			0
$493$	−2142.00	+	3710.05i	−0.195681	+	0.338930i
$494$	9152.00	−	15851.7i	0.833538	−	1.44373i
$495$		0			0
$496$	−2960.00			−0.267960
$497$		0			0
$498$		0			0
$499$	−9571.00	−	16577.5i	−0.858631	−	1.48719i	−0.873235	−	0.487299i	$-0.837982\pi$
$499$	−9571.00	−	16577.5i	−0.858631	−	1.48719i	0.0146043	−	0.999893i	$-0.495351\pi$
$500$	−750.000	+	1299.04i	−0.0670820	+	0.116190i
$501$		0			0
$502$	−5319.00	−	9212.78i	−0.472906	−	0.819096i
$503$	−12192.0			−1.08074			−0.540372	−	0.841426i	$-0.681717\pi$
$503$	−12192.0			−1.08074			−0.540372	+	0.841426i	$0.681717\pi$
$504$		0			0
$505$	−9630.00			−0.848573
$506$	−756.000	−	1309.43i	−0.0664196	−	0.115042i
$507$		0			0
$508$	746.000	−	1292.11i	0.0651543	−	0.112851i
$509$	9904.50	+	17155.1i	0.862494	+	1.49388i	0.869515	+	0.493907i	$0.164432\pi$
$509$	9904.50	+	17155.1i	0.862494	+	1.49388i	−0.00702091	+	0.999975i	$0.502235\pi$
$510$		0			0
$511$		0			0
$512$	512.000			0.0441942
$513$		0			0
$514$	−5346.00	+	9259.54i	−0.458759	+	0.794593i
$515$	−3480.00	+	6027.54i	−0.297761	+	0.515738i
$516$		0			0
$517$	−1242.00			−0.105654
$518$		0			0
$519$		0			0
$520$	5280.00	+	9145.23i	0.445276	+	0.771240i
$521$	897.000	−	1553.65i	0.0754286	−	0.130646i	−0.825844	−	0.563899i	$-0.809301\pi$
$521$	897.000	−	1553.65i	0.0754286	−	0.130646i	0.901273	+	0.433253i	$0.142634\pi$
$522$		0			0
$523$	−3224.00	−	5584.13i	−0.269552	−	0.466878i	0.699194	−	0.714932i	$-0.253542\pi$
$523$	−3224.00	−	5584.13i	−0.269552	−	0.466878i	−0.968746	+	0.248054i	$0.920209\pi$
$524$	−4692.00			−0.391166
$525$		0			0
$526$	−1548.00			−0.128319
$527$	−7770.00	−	13458.0i	−0.642251	−	1.11241i
$528$		0			0
$529$	2555.50	−	4426.26i	0.210035	−	0.363792i
$530$	−9585.00	−	16601.7i	−0.785558	−	1.36063i
$531$		0			0
$532$		0			0
$533$	−14784.0			−1.20144
$534$		0			0
$535$	−2947.50	+	5105.22i	−0.238190	+	0.412557i
$536$	664.000	−	1150.08i	0.0535083	−	0.0926790i
$537$		0			0
$538$	4830.00			0.387056
$539$		0			0
$540$		0			0
$541$	−3631.00	−	6289.08i	−0.288556	−	0.499794i	0.684909	−	0.728628i	$-0.259842\pi$
$541$	−3631.00	−	6289.08i	−0.288556	−	0.499794i	−0.973465	+	0.228835i	$0.926509\pi$
$542$	−475.000	+	822.724i	−0.0376439	+	0.0652012i
$543$		0			0
$544$	1344.00	+	2327.88i	0.105926	+	0.183469i
$545$	−210.000			−0.0165053
$546$		0			0
$547$	14204.0			1.11027			0.555136	−	0.831759i	$-0.312666\pi$
$547$	14204.0			1.11027			0.555136	+	0.831759i	$0.312666\pi$
$548$	60.0000	+	103.923i	0.00467714	+	0.00810104i
$549$		0			0
$550$	−900.000	+	1558.85i	−0.0697748	+	0.120853i
$551$	−2652.00	−	4593.40i	−0.205044	−	0.355146i
$552$		0			0
$553$		0			0
$554$	7456.00			0.571796
$555$		0			0
$556$	−164.000	+	284.056i	−0.0125093	+	0.0216667i
$557$	7912.50	−	13704.9i	0.601909	−	1.04254i	−0.390623	−	0.920551i	$-0.627740\pi$
$557$	7912.50	−	13704.9i	0.601909	−	1.04254i	0.992532	−	0.121986i	$-0.0389264\pi$
$558$		0			0
$559$	28688.0			2.17061
$560$		0			0
$561$		0			0
$562$	1602.00	+	2774.75i	0.120243	+	0.208266i
$563$	−529.500	+	917.121i	−0.0396372	+	0.0686537i	−0.885163	−	0.465280i	$-0.845954\pi$
$563$	−529.500	+	917.121i	−0.0396372	+	0.0686537i	0.845526	+	0.533934i	$0.179287\pi$
$564$		0			0
$565$	16380.0	+	28371.0i	1.21967	+	2.11252i
$566$	−1372.00			−0.101890
$567$		0			0
$568$	−8688.00			−0.641796
$569$	1980.00	+	3429.46i	0.145880	+	0.252672i	0.929701	−	0.368315i	$-0.120065\pi$
$569$	1980.00	+	3429.46i	0.145880	+	0.252672i	−0.783821	+	0.620987i	$0.786732\pi$
$570$		0			0
$571$	1265.00	−	2191.04i	0.0927121	−	0.160582i	−0.815939	−	0.578138i	$-0.803780\pi$
$571$	1265.00	−	2191.04i	0.0927121	−	0.160582i	0.908651	+	0.417555i	$0.137113\pi$
$572$	−1584.00	−	2743.57i	−0.115787	−	0.200550i
$573$		0			0
$574$		0			0
$575$	8400.00			0.609225
$576$		0			0
$577$	5915.50	−	10245.9i	0.426803	−	0.739245i	−0.569784	−	0.821795i	$-0.692973\pi$
$577$	5915.50	−	10245.9i	0.426803	−	0.739245i	0.996587	+	0.0825498i	$0.0263063\pi$
$578$	−2143.00	+	3711.78i	−0.154216	+	0.267111i
$579$		0			0
$580$	3060.00			0.219068
$581$		0			0
$582$		0			0
$583$	2875.50	+	4980.51i	0.204273	+	0.353811i
$584$	872.000	−	1510.35i	0.0617870	−	0.107018i
$585$		0			0
$586$	1101.00	+	1906.99i	0.0776141	+	0.134432i
$587$	4809.00			0.338141			0.169070	−	0.985604i	$-0.445923\pi$
$587$	4809.00			0.338141			0.169070	+	0.985604i	$0.445923\pi$
$588$		0			0
$589$	19240.0			1.34596
$590$	2385.00	+	4130.94i	0.166422	+	0.288251i
$591$		0			0
$592$	−352.000	+	609.682i	−0.0244377	+	0.0423273i
$593$	−10902.0	−	18882.8i	−0.754960	−	1.30763i	−0.945394	−	0.325930i	$-0.894323\pi$
$593$	−10902.0	−	18882.8i	−0.754960	−	1.30763i	0.190434	−	0.981700i	$-0.439011\pi$
$594$		0			0
$595$		0			0
$596$	5736.00			0.394221
$597$		0			0
$598$	−7392.00	+	12803.3i	−0.505487	+	0.875530i
$599$	7083.00	−	12268.1i	0.483144	−	0.836831i	−0.516668	−	0.856186i	$-0.672828\pi$
$599$	7083.00	−	12268.1i	0.483144	−	0.836831i	0.999813	+	0.0193549i	$0.00616125\pi$
$600$		0			0
$601$	−5891.00			−0.399832			−0.199916	−	0.979813i	$-0.564067\pi$
$601$	−5891.00			−0.399832			−0.199916	+	0.979813i	$0.564067\pi$
$602$		0			0
$603$		0			0
$604$	5342.00	+	9252.62i	0.359872	+	0.623317i
$605$	−9375.00	+	16238.0i	−0.629997	+	1.09119i
$606$		0			0
$607$	−1368.50	−	2370.31i	−0.0915086	−	0.158497i	0.816638	−	0.577151i	$-0.195836\pi$
$607$	−1368.50	−	2370.31i	−0.0915086	−	0.158497i	−0.908146	+	0.418653i	$0.862502\pi$
$608$	−3328.00			−0.221987
$609$		0			0
$610$	21660.0			1.43768
$611$	6072.00	+	10517.0i	0.402041	+	0.696355i
$612$		0			0
$613$	13094.0	−	22679.5i	0.862743	−	1.49432i	−0.00652719	−	0.999979i	$-0.502078\pi$
$613$	13094.0	−	22679.5i	0.862743	−	1.49432i	0.869271	−	0.494337i	$-0.164589\pi$
$614$	2780.00	+	4815.10i	0.182723	+	0.316485i
$615$		0			0
$616$		0			0
$617$	2358.00			0.153857			0.0769283	−	0.997037i	$-0.475489\pi$
$617$	2358.00			0.153857			0.0769283	+	0.997037i	$0.475489\pi$
$618$		0			0
$619$	6883.00	−	11921.7i	0.446932	−	0.774110i	−0.551252	−	0.834339i	$-0.685850\pi$
$619$	6883.00	−	11921.7i	0.446932	−	0.774110i	0.998185	+	0.0602291i	$0.0191831\pi$
$620$	−5550.00	+	9612.88i	−0.359505	+	0.622682i
$621$		0			0
$622$	8592.00			0.553871
$623$		0			0
$624$		0			0
$625$	9062.50	+	15696.7i	0.580000	+	1.00459i
$626$	5489.00	−	9507.23i	0.350455	−	0.607005i
$627$		0			0
$628$	4504.00	+	7801.16i	0.286193	+	0.495701i
$629$	−3696.00			−0.234291
$630$		0			0
$631$	21287.0			1.34298			0.671491	−	0.741012i	$-0.265654\pi$
$631$	21287.0			1.34298			0.671491	+	0.741012i	$0.265654\pi$
$632$	2332.00	+	4039.14i	0.146775	+	0.254222i
$633$		0			0
$634$	4491.00	−	7778.64i	0.281326	−	0.487270i
$635$	−2797.50	−	4845.41i	−0.174827	−	0.302810i
$636$		0			0
$637$		0			0
$638$	−918.000			−0.0569655
$639$		0			0
$640$	960.000	−	1662.77i	0.0592927	−	0.102698i
$641$	10713.0	−	18555.5i	0.660122	−	1.14336i	−0.320462	−	0.947262i	$-0.603838\pi$
$641$	10713.0	−	18555.5i	0.660122	−	1.14336i	0.980583	−	0.196103i	$-0.0628287\pi$
$642$		0			0
$643$	−9962.00			−0.610984			−0.305492	−	0.952195i	$-0.598821\pi$
$643$	−9962.00			−0.610984			−0.305492	+	0.952195i	$0.598821\pi$
$644$		0			0
$645$		0			0
$646$	−8736.00	−	15131.2i	−0.532064	−	0.921562i
$647$	9087.00	−	15739.1i	0.552159	−	0.956367i	−0.445960	−	0.895053i	$-0.647137\pi$
$647$	9087.00	−	15739.1i	0.552159	−	0.956367i	0.998119	−	0.0613142i	$-0.0195292\pi$
$648$		0			0
$649$	−715.500	−	1239.28i	−0.0432755	−	0.0749555i
$650$	17600.0			1.06204
$651$		0			0
$652$	6704.00			0.402682
$653$	−9583.50	−	16599.1i	−0.574321	−	0.994752i	−0.996115	−	0.0880610i	$-0.971933\pi$
$653$	−9583.50	−	16599.1i	−0.574321	−	0.994752i	0.421795	−	0.906691i	$-0.361400\pi$
$654$		0			0
$655$	−8797.50	+	15237.7i	−0.524804	+	0.908988i
$656$	1344.00	+	2327.88i	0.0799914	+	0.138549i
$657$		0			0
$658$		0			0
$659$	−13080.0			−0.773178			−0.386589	−	0.922252i	$-0.626347\pi$
$659$	−13080.0			−0.773178			−0.386589	+	0.922252i	$0.626347\pi$
$660$		0			0
$661$	−7595.00	+	13154.9i	−0.446916	+	0.774081i	−0.998183	−	0.0602477i	$-0.980811\pi$
$661$	−7595.00	+	13154.9i	−0.446916	+	0.774081i	0.551268	+	0.834328i	$0.314144\pi$
$662$	3964.00	−	6865.85i	0.232727	−	0.403095i
$663$		0			0
$664$	−4776.00			−0.279134
$665$		0			0
$666$		0			0
$667$	2142.00	+	3710.05i	0.124346	+	0.215373i
$668$	−6060.00	+	10496.2i	−0.351001	+	0.607951i
$669$		0			0
$670$	−2490.00	−	4312.81i	−0.143578	−	0.248684i
$671$	−6498.00			−0.373849
$672$		0			0
$673$	4397.00			0.251845			0.125923	−	0.992040i	$-0.459811\pi$
$673$	4397.00			0.251845			0.125923	+	0.992040i	$0.459811\pi$
$674$	−161.000	−	278.860i	−0.00920102	−	0.0159366i
$675$		0			0
$676$	−11094.0	+	19215.4i	−0.631202	+	1.09327i
$677$	−2014.50	−	3489.22i	−0.114363	−	0.198082i	0.803162	−	0.595761i	$-0.203149\pi$
$677$	−2014.50	−	3489.22i	−0.114363	−	0.198082i	−0.917525	+	0.397679i	$0.869816\pi$
$678$		0			0
$679$		0			0
$680$	10080.0			0.568456
$681$		0			0
$682$	1665.00	−	2883.86i	0.0934841	−	0.161919i
$683$	7510.50	−	13008.6i	0.420763	−	0.728783i	−0.575251	−	0.817977i	$-0.695096\pi$
$683$	7510.50	−	13008.6i	0.420763	−	0.728783i	0.996014	+	0.0891936i	$0.0284290\pi$
$684$		0			0
$685$	450.000			0.0251002
$686$		0			0
$687$		0			0
$688$	−2608.00	−	4517.19i	−0.144519	−	0.250314i
$689$	28116.0	−	48698.3i	1.55462	−	2.69268i
$690$		0			0
$691$	−6992.00	−	12110.5i	−0.384932	−	0.666722i	0.606828	−	0.794834i	$-0.292442\pi$
$691$	−6992.00	−	12110.5i	−0.384932	−	0.666722i	−0.991760	+	0.128111i	$0.959109\pi$
$692$	13752.0			0.755452
$693$		0			0
$694$	11832.0			0.647171
$695$	615.000	+	1065.21i	0.0335659	+	0.0581378i
$696$		0			0
$697$	−7056.00	+	12221.4i	−0.383451	+	0.664156i
$698$	−142.000	−	245.951i	−0.00770026	−	0.0133372i
$699$		0			0
$700$		0			0
$701$	31053.0			1.67312			0.836559	−	0.547877i	$-0.184564\pi$
$701$	31053.0			1.67312			0.836559	+	0.547877i	$0.184564\pi$
$702$		0			0
$703$	2288.00	−	3962.93i	0.122750	−	0.212610i
$704$	−288.000	+	498.831i	−0.0154182	+	0.0267051i
$705$		0			0
$706$	−8880.00			−0.473376
$707$		0			0
$708$		0			0
$709$	−7543.00	−	13064.9i	−0.399553	−	0.692047i	0.594117	−	0.804378i	$-0.297501\pi$
$709$	−7543.00	−	13064.9i	−0.399553	−	0.692047i	−0.993671	+	0.112332i	$0.964168\pi$
$710$	−16290.0	+	28215.1i	−0.861060	+	1.49140i
$711$		0			0
$712$	4152.00	+	7191.47i	0.218543	+	0.378528i
$713$	−15540.0			−0.816238
$714$		0			0
$715$	−11880.0			−0.621380
$716$	−2424.00	−	4198.49i	−0.126521	−	0.219141i
$717$		0			0
$718$	2286.00	−	3959.47i	0.118820	−	0.205802i
$719$	3189.00	+	5523.51i	0.165410	+	0.286498i	0.936801	−	0.349863i	$-0.113772\pi$
$719$	3189.00	+	5523.51i	0.165410	+	0.286498i	−0.771391	+	0.636362i	$0.780439\pi$
$720$		0			0
$721$		0			0
$722$	7914.00			0.407934
$723$		0			0
$724$	6064.00	−	10503.2i	0.311280	−	0.539153i
$725$	2550.00	−	4416.73i	0.130627	−	0.226253i
$726$		0			0
$727$	7363.00			0.375624			0.187812	−	0.982205i	$-0.439860\pi$
$727$	7363.00			0.375624			0.187812	+	0.982205i	$0.439860\pi$
$728$		0			0
$729$		0			0
$730$	−3270.00	−	5663.81i	−0.165792	−	0.287160i
$731$	13692.0	−	23715.2i	0.692773	−	1.19992i
$732$		0			0
$733$	16405.0	+	28414.3i	0.826647	+	1.43180i	0.900654	+	0.434537i	$0.143088\pi$
$733$	16405.0	+	28414.3i	0.826647	+	1.43180i	−0.0740064	+	0.997258i	$0.523579\pi$
$734$	5738.00			0.288547
$735$		0			0
$736$	2688.00			0.134621
$737$	747.000	+	1293.84i	0.0373353	+	0.0646666i
$738$		0			0
$739$	12017.0	−	20814.1i	0.598177	−	1.03607i	−0.394914	−	0.918718i	$-0.629225\pi$
$739$	12017.0	−	20814.1i	0.598177	−	1.03607i	0.993090	−	0.117354i	$-0.0374412\pi$
$740$	1320.00	+	2286.31i	0.0655732	+	0.113576i
$741$		0			0
$742$		0			0
$743$	8022.00			0.396095			0.198048	−	0.980192i	$-0.436540\pi$
$743$	8022.00			0.396095			0.198048	+	0.980192i	$0.436540\pi$
$744$		0			0
$745$	10755.0	−	18628.2i	0.528903	−	0.916087i
$746$	3064.00	−	5307.00i	0.150377	−	0.260460i
$747$		0			0
$748$	−3024.00			−0.147819
$749$		0			0
$750$		0			0
$751$	−14759.5	−	25564.2i	−0.717153	−	1.24215i	−0.962123	−	0.272615i	$-0.912112\pi$
$751$	−14759.5	−	25564.2i	−0.717153	−	1.24215i	0.244970	−	0.969531i	$-0.421222\pi$
$752$	1104.00	−	1912.18i	0.0535356	−	0.0927263i
$753$		0			0
$754$	4488.00	+	7773.44i	0.216768	+	0.375454i
$755$	40065.0			1.93128
$756$		0			0
$757$	−3742.00			−0.179664			−0.0898318	−	0.995957i	$-0.528633\pi$
$757$	−3742.00			−0.179664			−0.0898318	+	0.995957i	$0.528633\pi$
$758$	6040.00	+	10461.6i	0.289423	+	0.501295i
$759$		0			0
$760$	−6240.00	+	10808.0i	−0.297827	+	0.515852i
$761$	5448.00	+	9436.21i	0.259514	+	0.449491i	0.966112	−	0.258124i	$-0.0831044\pi$
$761$	5448.00	+	9436.21i	0.259514	+	0.449491i	−0.706598	+	0.707615i	$0.749771\pi$
$762$		0			0
$763$		0			0
$764$	−10080.0			−0.477332
$765$		0			0
$766$	−1842.00	+	3190.44i	−0.0868853	+	0.150490i
$767$	−6996.00	+	12117.4i	−0.329349	+	0.570450i
$768$		0			0
$769$	−17285.0			−0.810550			−0.405275	−	0.914195i	$-0.632824\pi$
$769$	−17285.0			−0.810550			−0.405275	+	0.914195i	$0.632824\pi$
$770$		0			0
$771$		0			0
$772$	−730.000	−	1264.40i	−0.0340327	−	0.0589464i
$773$	−5913.00	+	10241.6i	−0.275130	+	0.476540i	−0.970168	−	0.242434i	$-0.922054\pi$
$773$	−5913.00	+	10241.6i	−0.275130	+	0.476540i	0.695038	+	0.718973i	$0.255388\pi$
$774$		0			0
$775$	9250.00	+	16021.5i	0.428735	+	0.742591i
$776$	1352.00			0.0625438
$777$		0			0
$778$	−15660.0			−0.721643
$779$	−8736.00	−	15131.2i	−0.401797	−	0.695932i
$780$		0			0
$781$	4887.00	−	8464.53i	0.223906	−	0.387817i
$782$	7056.00	+	12221.4i	0.322662	+	0.558868i
$783$		0			0
$784$		0			0

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(7.50000 + 12.9904i) q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(7.50000 + 12.9904i) q^{5} +8.00000 q^{8} +(15.0000 - 25.9808i) q^{10} +(-4.50000 + 7.79423i) q^{11} +88.0000 q^{13} +(-8.00000 - 13.8564i) q^{16} +(42.0000 - 72.7461i) q^{17} +(52.0000 + 90.0666i) q^{19} -60.0000 q^{20} +18.0000 q^{22} +(-42.0000 - 72.7461i) q^{23} +(-50.0000 + 86.6025i) q^{25} +(-88.0000 - 152.420i) q^{26} -51.0000 q^{29} +(92.5000 - 160.215i) q^{31} +(-16.0000 + 27.7128i) q^{32} -168.000 q^{34} +(-22.0000 - 38.1051i) q^{37} +(104.000 - 180.133i) q^{38} +(60.0000 + 103.923i) q^{40} -168.000 q^{41} +326.000 q^{43} +(-18.0000 - 31.1769i) q^{44} +(-84.0000 + 145.492i) q^{46} +(69.0000 + 119.512i) q^{47} +200.000 q^{50} +(-176.000 + 304.841i) q^{52} +(319.500 - 553.390i) q^{53} -135.000 q^{55} +(51.0000 + 88.3346i) q^{58} +(-79.5000 + 137.698i) q^{59} +(361.000 + 625.270i) q^{61} -370.000 q^{62} +64.0000 q^{64} +(660.000 + 1143.15i) q^{65} +(83.0000 - 143.760i) q^{67} +(168.000 + 290.985i) q^{68} -1086.00 q^{71} +(109.000 - 188.794i) q^{73} +(-44.0000 + 76.2102i) q^{74} -416.000 q^{76} +(291.500 + 504.893i) q^{79} +(120.000 - 207.846i) q^{80} +(168.000 + 290.985i) q^{82} -597.000 q^{83} +1260.00 q^{85} +(-326.000 - 564.649i) q^{86} +(-36.0000 + 62.3538i) q^{88} +(519.000 + 898.934i) q^{89} +336.000 q^{92} +(138.000 - 239.023i) q^{94} +(-780.000 + 1351.00i) q^{95} +169.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8	\( 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8} + 30 q^{10} - 9 q^{11} + 176 q^{13} - 16 q^{16} + 84 q^{17} + 104 q^{19} - 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} - 176 q^{26} - 102 q^{29} + 185 q^{31} - 32 q^{32} - 336 q^{34} - 44 q^{37} + 208 q^{38} + 120 q^{40} - 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} + 138 q^{47} + 400 q^{50} - 352 q^{52} + 639 q^{53} - 270 q^{55} + 102 q^{58} - 159 q^{59} + 722 q^{61} - 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} + 336 q^{68} - 2172 q^{71} + 218 q^{73} - 88 q^{74} - 832 q^{76} + 583 q^{79} + 240 q^{80} + 336 q^{82} - 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} + 1038 q^{89} + 672 q^{92} + 276 q^{94} - 1560 q^{95} + 338 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8 + 30 * q^10 - 9 * q^11 + 176 * q^13 - 16 * q^16 + 84 * q^17 + 104 * q^19 - 120 * q^20 + 36 * q^22 - 84 * q^23 - 100 * q^25 - 176 * q^26 - 102 * q^29 + 185 * q^31 - 32 * q^32 - 336 * q^34 - 44 * q^37 + 208 * q^38 + 120 * q^40 - 336 * q^41 + 652 * q^43 - 36 * q^44 - 168 * q^46 + 138 * q^47 + 400 * q^50 - 352 * q^52 + 639 * q^53 - 270 * q^55 + 102 * q^58 - 159 * q^59 + 722 * q^61 - 740 * q^62 + 128 * q^64 + 1320 * q^65 + 166 * q^67 + 336 * q^68 - 2172 * q^71 + 218 * q^73 - 88 * q^74 - 832 * q^76 + 583 * q^79 + 240 * q^80 + 336 * q^82 - 1194 * q^83 + 2520 * q^85 - 652 * q^86 - 72 * q^88 + 1038 * q^89 + 672 * q^92 + 276 * q^94 - 1560 * q^95 + 338 * q^97

Introduction

L-functions

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