LMFDB - Embedded newform 162.4.c.d.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,4,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.c (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:	$9.55830942093$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.4.c.d.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(10.5000 + 18.1865i) q^{5} +(-4.00000 + 6.92820i) q^{7} +8.00000 q^{8} -42.0000 q^{10} +(18.0000 - 31.1769i) q^{11} +(24.5000 + 42.4352i) q^{13} +(-8.00000 - 13.8564i) q^{14} +(-8.00000 + 13.8564i) q^{16} -21.0000 q^{17} -112.000 q^{19} +(42.0000 - 72.7461i) q^{20} +(36.0000 + 62.3538i) q^{22} +(90.0000 + 155.885i) q^{23} +(-158.000 + 273.664i) q^{25} -98.0000 q^{26} +32.0000 q^{28} +(-67.5000 + 116.913i) q^{29} +(-154.000 - 266.736i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(21.0000 - 36.3731i) q^{34} -168.000 q^{35} -1.00000 q^{37} +(112.000 - 193.990i) q^{38} +(84.0000 + 145.492i) q^{40} +(-21.0000 - 36.3731i) q^{41} +(-10.0000 + 17.3205i) q^{43} -144.000 q^{44} -360.000 q^{46} +(42.0000 - 72.7461i) q^{47} +(139.500 + 241.621i) q^{49} +(-316.000 - 547.328i) q^{50} +(98.0000 - 169.741i) q^{52} +174.000 q^{53} +756.000 q^{55} +(-32.0000 + 55.4256i) q^{56} +(-135.000 - 233.827i) q^{58} +(252.000 + 436.477i) q^{59} +(192.500 - 333.420i) q^{61} +616.000 q^{62} +64.0000 q^{64} +(-514.500 + 891.140i) q^{65} +(-136.000 - 235.559i) q^{67} +(42.0000 + 72.7461i) q^{68} +(168.000 - 290.985i) q^{70} +888.000 q^{71} +371.000 q^{73} +(1.00000 - 1.73205i) q^{74} +(224.000 + 387.979i) q^{76} +(144.000 + 249.415i) q^{77} +(326.000 - 564.649i) q^{79} -336.000 q^{80} +84.0000 q^{82} +(42.0000 - 72.7461i) q^{83} +(-220.500 - 381.917i) q^{85} +(-20.0000 - 34.6410i) q^{86} +(144.000 - 249.415i) q^{88} -21.0000 q^{89} -392.000 q^{91} +(360.000 - 623.538i) q^{92} +(84.0000 + 145.492i) q^{94} +(-1176.00 - 2036.89i) q^{95} +(623.000 - 1079.07i) q^{97} -558.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26}+ \cdots - 1116 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 + 21 * q^5 - 8 * q^7 + 16 * q^8 - 84 * q^10 + 36 * q^11 + 49 * q^13 - 16 * q^14 - 16 * q^16 - 42 * q^17 - 224 * q^19 + 84 * q^20 + 72 * q^22 + 180 * q^23 - 316 * q^25 - 196 * q^26 + 64 * q^28 - 135 * q^29 - 308 * q^31 - 32 * q^32 + 42 * q^34 - 336 * q^35 - 2 * q^37 + 224 * q^38 + 168 * q^40 - 42 * q^41 - 20 * q^43 - 288 * q^44 - 720 * q^46 + 84 * q^47 + 279 * q^49 - 632 * q^50 + 196 * q^52 + 348 * q^53 + 1512 * q^55 - 64 * q^56 - 270 * q^58 + 504 * q^59 + 385 * q^61 + 1232 * q^62 + 128 * q^64 - 1029 * q^65 - 272 * q^67 + 84 * q^68 + 336 * q^70 + 1776 * q^71 + 742 * q^73 + 2 * q^74 + 448 * q^76 + 288 * q^77 + 652 * q^79 - 672 * q^80 + 168 * q^82 + 84 * q^83 - 441 * q^85 - 40 * q^86 + 288 * q^88 - 42 * q^89 - 784 * q^91 + 720 * q^92 + 168 * q^94 - 2352 * q^95 + 1246 * q^97 - 1116 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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
$5$	10.5000	+	18.1865i	0.939149	+	1.62665i	0.767064	+	0.641571i	$0.221717\pi$
$5$	10.5000	+	18.1865i	0.939149	+	1.62665i	0.172085	+	0.985082i	$0.444950\pi$
$6$		0			0
$7$	−4.00000	+	6.92820i	−0.215980	+	0.374088i	−0.953575	−	0.301155i	$-0.902628\pi$
$7$	−4.00000	+	6.92820i	−0.215980	+	0.374088i	0.737595	+	0.675243i	$0.235961\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	−42.0000			−1.32816
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	−0.506589	−	0.862188i	$-0.669094\pi$
$11$	18.0000	−	31.1769i	0.493382	−	0.854563i	0.999971	+	0.00762479i	$0.00242707\pi$
$12$		0			0
$13$	24.5000	+	42.4352i	0.522698	+	0.905340i	0.999651	+	0.0264111i	$0.00840789\pi$
$13$	24.5000	+	42.4352i	0.522698	+	0.905340i	−0.476953	+	0.878929i	$0.658259\pi$
$14$	−8.00000	−	13.8564i	−0.152721	−	0.264520i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−21.0000			−0.299603			−0.149801	−	0.988716i	$-0.547863\pi$
$17$	−21.0000			−0.299603			−0.149801	+	0.988716i	$0.547863\pi$
$18$		0			0
$19$	−112.000			−1.35235			−0.676173	−	0.736743i	$-0.736363\pi$
$19$	−112.000			−1.35235			−0.676173	+	0.736743i	$0.736363\pi$
$20$	42.0000	−	72.7461i	0.469574	−	0.813327i
$21$		0			0
$22$	36.0000	+	62.3538i	0.348874	+	0.604267i
$23$	90.0000	+	155.885i	0.815926	+	1.41323i	0.908661	+	0.417534i	$0.137106\pi$
$23$	90.0000	+	155.885i	0.815926	+	1.41323i	−0.0927351	+	0.995691i	$0.529561\pi$
$24$		0			0
$25$	−158.000	+	273.664i	−1.26400	+	2.18931i
$26$	−98.0000			−0.739207
$27$		0			0
$28$	32.0000			0.215980
$29$	−67.5000	+	116.913i	−0.432222	+	0.748630i	−0.997064	−	0.0765685i	$-0.975604\pi$
$29$	−67.5000	+	116.913i	−0.432222	+	0.748630i	0.564842	+	0.825199i	$0.308937\pi$
$30$		0			0
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.837192	−	0.546908i	$-0.815805\pi$
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.0550403	−	0.998484i	$-0.517529\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	21.0000	−	36.3731i	0.105926	−	0.183469i
$35$	−168.000			−0.811348
$36$		0			0
$37$	−1.00000			−0.00444322			−0.00222161	−	0.999998i	$-0.500707\pi$
$37$	−1.00000			−0.00444322			−0.00222161	+	0.999998i	$0.500707\pi$
$38$	112.000	−	193.990i	0.478126	−	0.828139i
$39$		0			0
$40$	84.0000	+	145.492i	0.332039	+	0.575109i
$41$	−21.0000	−	36.3731i	−0.0799914	−	0.138549i	0.823255	−	0.567672i	$-0.192156\pi$
$41$	−21.0000	−	36.3731i	−0.0799914	−	0.138549i	−0.903246	+	0.429123i	$0.858823\pi$
$42$		0			0
$43$	−10.0000	+	17.3205i	−0.0354648	+	0.0614268i	−0.883213	−	0.468972i	$-0.844624\pi$
$43$	−10.0000	+	17.3205i	−0.0354648	+	0.0614268i	0.847748	+	0.530399i	$0.177958\pi$
$44$	−144.000			−0.493382
$45$		0			0
$46$	−360.000			−1.15389
$47$	42.0000	−	72.7461i	0.130347	−	0.225768i	−0.793463	−	0.608618i	$-0.791724\pi$
$47$	42.0000	−	72.7461i	0.130347	−	0.225768i	0.923811	+	0.382850i	$0.125057\pi$
$48$		0			0
$49$	139.500	+	241.621i	0.406706	+	0.704435i
$50$	−316.000	−	547.328i	−0.893783	−	1.54808i
$51$		0			0
$52$	98.0000	−	169.741i	0.261349	−	0.452670i
$53$	174.000			0.450957			0.225479	−	0.974248i	$-0.427605\pi$
$53$	174.000			0.450957			0.225479	+	0.974248i	$0.427605\pi$
$54$		0			0
$55$	756.000			1.85344
$56$	−32.0000	+	55.4256i	−0.0763604	+	0.132260i
$57$		0			0
$58$	−135.000	−	233.827i	−0.305627	−	0.529362i
$59$	252.000	+	436.477i	0.556061	+	0.963126i	0.997820	+	0.0659923i	$0.0210213\pi$
$59$	252.000	+	436.477i	0.556061	+	0.963126i	−0.441759	+	0.897134i	$0.645645\pi$
$60$		0			0
$61$	192.500	−	333.420i	0.404051	−	0.699837i	−0.590160	−	0.807287i	$-0.700935\pi$
$61$	192.500	−	333.420i	0.404051	−	0.699837i	0.994210	+	0.107450i	$0.0342686\pi$
$62$	616.000			1.26181
$63$		0			0
$64$	64.0000			0.125000
$65$	−514.500	+	891.140i	−0.981783	+	1.70050i
$66$		0			0
$67$	−136.000	−	235.559i	−0.247986	−	0.429524i	0.714981	−	0.699144i	$-0.246435\pi$
$67$	−136.000	−	235.559i	−0.247986	−	0.429524i	−0.962967	+	0.269620i	$0.913102\pi$
$68$	42.0000	+	72.7461i	0.0749007	+	0.129732i
$69$		0			0
$70$	168.000	−	290.985i	0.286855	−	0.496847i
$71$	888.000			1.48431			0.742156	−	0.670227i	$-0.233803\pi$
$71$	888.000			1.48431			0.742156	+	0.670227i	$0.233803\pi$
$72$		0			0
$73$	371.000			0.594826			0.297413	−	0.954749i	$-0.403876\pi$
$73$	371.000			0.594826			0.297413	+	0.954749i	$0.403876\pi$
$74$	1.00000	−	1.73205i	0.00157091	−	0.00272090i
$75$		0			0
$76$	224.000	+	387.979i	0.338086	+	0.585583i
$77$	144.000	+	249.415i	0.213121	+	0.369137i
$78$		0			0
$79$	326.000	−	564.649i	0.464277	−	0.804151i	−0.534892	−	0.844921i	$-0.679648\pi$
$79$	326.000	−	564.649i	0.464277	−	0.804151i	0.999169	+	0.0407696i	$0.0129810\pi$
$80$	−336.000			−0.469574
$81$		0			0
$82$	84.0000			0.113125
$83$	42.0000	−	72.7461i	0.0555434	−	0.0962039i	−0.836917	−	0.547330i	$-0.815644\pi$
$83$	42.0000	−	72.7461i	0.0555434	−	0.0962039i	0.892460	+	0.451126i	$0.148978\pi$
$84$		0			0
$85$	−220.500	−	381.917i	−0.281372	−	0.487350i
$86$	−20.0000	−	34.6410i	−0.0250774	−	0.0434353i
$87$		0			0
$88$	144.000	−	249.415i	0.174437	−	0.302134i
$89$	−21.0000			−0.0250112			−0.0125056	−	0.999922i	$-0.503981\pi$
$89$	−21.0000			−0.0250112			−0.0125056	+	0.999922i	$0.503981\pi$
$90$		0			0
$91$	−392.000			−0.451569
$92$	360.000	−	623.538i	0.407963	−	0.706613i
$93$		0			0
$94$	84.0000	+	145.492i	0.0921696	+	0.159642i
$95$	−1176.00	−	2036.89i	−1.27005	−	2.19980i
$96$		0			0
$97$	623.000	−	1079.07i	0.652124	−	1.12951i	−0.330482	−	0.943812i	$-0.607211\pi$
$97$	623.000	−	1079.07i	0.652124	−	1.12951i	0.982606	−	0.185700i	$-0.0594554\pi$
$98$	−558.000			−0.575168
$99$		0			0
$100$	1264.00			1.26400
$101$	273.000	−	472.850i	0.268956	−	0.465845i	−0.699637	−	0.714499i	$-0.746655\pi$
$101$	273.000	−	472.850i	0.268956	−	0.465845i	0.968592	+	0.248654i	$0.0799882\pi$
$102$		0			0
$103$	98.0000	+	169.741i	0.0937498	+	0.162379i	0.909086	−	0.416608i	$-0.136781\pi$
$103$	98.0000	+	169.741i	0.0937498	+	0.162379i	−0.815336	+	0.578988i	$0.803448\pi$
$104$	196.000	+	339.482i	0.184802	+	0.320086i
$105$		0			0
$106$	−174.000	+	301.377i	−0.159437	+	0.276154i
$107$	300.000			0.271048			0.135524	−	0.990774i	$-0.456728\pi$
$107$	300.000			0.271048			0.135524	+	0.990774i	$0.456728\pi$
$108$		0			0
$109$	−1069.00			−0.939373			−0.469686	−	0.882833i	$-0.655633\pi$
$109$	−1069.00			−0.939373			−0.469686	+	0.882833i	$0.655633\pi$
$110$	−756.000	+	1309.43i	−0.655289	+	1.13499i
$111$		0			0
$112$	−64.0000	−	110.851i	−0.0539949	−	0.0935220i
$113$	448.500	+	776.825i	0.373375	+	0.646704i	0.990082	−	0.140488i	$-0.0448673\pi$
$113$	448.500	+	776.825i	0.373375	+	0.646704i	−0.616708	+	0.787192i	$0.711534\pi$
$114$		0			0
$115$	−1890.00	+	3273.58i	−1.53255	+	2.65446i
$116$	540.000			0.432222
$117$		0			0
$118$	−1008.00			−0.786389
$119$	84.0000	−	145.492i	0.0647081	−	0.112078i
$120$		0			0
$121$	17.5000	+	30.3109i	0.0131480	+	0.0227730i
$122$	385.000	+	666.840i	0.285707	+	0.494859i
$123$		0			0
$124$	−616.000	+	1066.94i	−0.446116	+	0.772696i
$125$	−4011.00			−2.87004
$126$		0			0
$127$	1532.00			1.07042			0.535209	−	0.844720i	$-0.320233\pi$
$127$	1532.00			1.07042			0.535209	+	0.844720i	$0.320233\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−1029.00	−	1782.28i	−0.694225	−	1.20243i
$131$	420.000	+	727.461i	0.280119	+	0.485180i	0.971414	−	0.237393i	$-0.0762928\pi$
$131$	420.000	+	727.461i	0.280119	+	0.485180i	−0.691295	+	0.722573i	$0.742959\pi$
$132$		0			0
$133$	448.000	−	775.959i	0.292079	−	0.505896i
$134$	544.000			0.350705
$135$		0			0
$136$	−168.000			−0.105926
$137$	364.500	−	631.333i	0.227309	−	0.393711i	−0.729701	−	0.683767i	$-0.760341\pi$
$137$	364.500	−	631.333i	0.227309	−	0.393711i	0.957010	+	0.290056i	$0.0936739\pi$
$138$		0			0
$139$	1022.00	+	1770.16i	0.623632	+	1.08016i	0.988804	+	0.149223i	$0.0476771\pi$
$139$	1022.00	+	1770.16i	0.623632	+	1.08016i	−0.365171	+	0.930940i	$0.618990\pi$
$140$	336.000	+	581.969i	0.202837	+	0.351324i
$141$		0			0
$142$	−888.000	+	1538.06i	−0.524784	+	0.908952i
$143$	1764.00			1.03156
$144$		0			0
$145$	−2835.00			−1.62368
$146$	−371.000	+	642.591i	−0.210303	+	0.364255i
$147$		0			0
$148$	2.00000	+	3.46410i	0.00111080	+	0.00192397i
$149$	−643.500	−	1114.57i	−0.353809	−	0.612816i	0.633104	−	0.774067i	$-0.281780\pi$
$149$	−643.500	−	1114.57i	−0.353809	−	0.612816i	−0.986913	+	0.161251i	$0.948447\pi$
$150$		0			0
$151$	368.000	−	637.395i	0.198327	−	0.343513i	−0.749659	−	0.661824i	$-0.769782\pi$
$151$	368.000	−	637.395i	0.198327	−	0.343513i	0.947986	+	0.318311i	$0.103116\pi$
$152$	−896.000			−0.478126
$153$		0			0
$154$	−576.000			−0.301399
$155$	3234.00	−	5601.45i	1.67588	−	2.90271i
$156$		0			0
$157$	1074.50	+	1861.09i	0.546207	+	0.946058i	0.998530	+	0.0542035i	$0.0172620\pi$
$157$	1074.50	+	1861.09i	0.546207	+	0.946058i	−0.452323	+	0.891854i	$0.649405\pi$
$158$	652.000	+	1129.30i	0.328293	+	0.568621i
$159$		0			0
$160$	336.000	−	581.969i	0.166020	−	0.287554i
$161$	−1440.00			−0.704894
$162$		0			0
$163$	−3088.00			−1.48387			−0.741935	−	0.670472i	$-0.766092\pi$
$163$	−3088.00			−1.48387			−0.741935	+	0.670472i	$0.766092\pi$
$164$	−84.0000	+	145.492i	−0.0399957	+	0.0692746i
$165$		0			0
$166$	84.0000	+	145.492i	0.0392751	+	0.0680264i
$167$	84.0000	+	145.492i	0.0389228	+	0.0674163i	0.884831	−	0.465913i	$-0.154274\pi$
$167$	84.0000	+	145.492i	0.0389228	+	0.0674163i	−0.845908	+	0.533329i	$0.820941\pi$
$168$		0			0
$169$	−102.000	+	176.669i	−0.0464269	+	0.0804138i
$170$	882.000			0.397919
$171$		0			0
$172$	80.0000			0.0354648
$173$	−1501.50	+	2600.67i	−0.659867	+	1.14292i	0.320783	+	0.947153i	$0.396054\pi$
$173$	−1501.50	+	2600.67i	−0.659867	+	1.14292i	−0.980650	+	0.195770i	$0.937279\pi$
$174$		0			0
$175$	−1264.00	−	2189.31i	−0.545997	−	0.945694i
$176$	288.000	+	498.831i	0.123346	+	0.213641i
$177$		0			0
$178$	21.0000	−	36.3731i	0.00884279	−	0.0153162i
$179$	1164.00			0.486042			0.243021	−	0.970021i	$-0.421862\pi$
$179$	1164.00			0.486042			0.243021	+	0.970021i	$0.421862\pi$
$180$		0			0
$181$	−1666.00			−0.684159			−0.342080	−	0.939671i	$-0.611131\pi$
$181$	−1666.00			−0.684159			−0.342080	+	0.939671i	$0.611131\pi$
$182$	392.000	−	678.964i	0.159654	−	0.276528i
$183$		0			0
$184$	720.000	+	1247.08i	0.288473	+	0.499651i
$185$	−10.5000	−	18.1865i	−0.00417284	−	0.00722757i
$186$		0			0
$187$	−378.000	+	654.715i	−0.147819	+	0.256030i
$188$	−336.000			−0.130347
$189$		0			0
$190$	4704.00			1.79613
$191$	−1032.00	+	1787.48i	−0.390958	+	0.677158i	−0.992576	−	0.121625i	$-0.961189\pi$
$191$	−1032.00	+	1787.48i	−0.390958	+	0.677158i	0.601619	+	0.798784i	$0.294523\pi$
$192$		0			0
$193$	282.500	+	489.304i	0.105362	+	0.182492i	0.913886	−	0.405971i	$-0.133067\pi$
$193$	282.500	+	489.304i	0.105362	+	0.182492i	−0.808524	+	0.588463i	$0.799733\pi$
$194$	1246.00	+	2158.14i	0.461122	+	0.798686i
$195$		0			0
$196$	558.000	−	966.484i	0.203353	−	0.352217i
$197$	4731.00			1.71101			0.855507	−	0.517791i	$-0.173246\pi$
$197$	4731.00			1.71101			0.855507	+	0.517791i	$0.173246\pi$
$198$		0			0
$199$	4676.00			1.66569			0.832846	−	0.553504i	$-0.186710\pi$
$199$	4676.00			1.66569			0.832846	+	0.553504i	$0.186710\pi$
$200$	−1264.00	+	2189.31i	−0.446891	+	0.774039i
$201$		0			0
$202$	546.000	+	945.700i	0.190180	+	0.329402i
$203$	−540.000	−	935.307i	−0.186702	−	0.323378i
$204$		0			0
$205$	441.000	−	763.834i	0.150248	−	0.260237i
$206$	−392.000			−0.132582
$207$		0			0
$208$	−784.000			−0.261349
$209$	−2016.00	+	3491.81i	−0.667223	+	1.15566i
$210$		0			0
$211$	−1690.00	−	2927.17i	−0.551395	−	0.955045i	−0.998174	−	0.0604002i	$-0.980762\pi$
$211$	−1690.00	−	2927.17i	−0.551395	−	0.955045i	0.446779	−	0.894644i	$-0.352571\pi$
$212$	−348.000	−	602.754i	−0.112739	−	0.195270i
$213$		0			0
$214$	−300.000	+	519.615i	−0.0958298	+	0.165982i
$215$	−420.000			−0.133227
$216$		0			0
$217$	2464.00			0.770817
$218$	1069.00	−	1851.56i	0.332118	−	0.575246i
$219$		0			0
$220$	−1512.00	−	2618.86i	−0.463359	−	0.802562i
$221$	−514.500	−	891.140i	−0.156602	−	0.271242i
$222$		0			0
$223$	2618.00	−	4534.51i	0.786163	−	1.36167i	−0.142140	−	0.989847i	$-0.545398\pi$
$223$	2618.00	−	4534.51i	0.786163	−	1.36167i	0.928302	−	0.371827i	$-0.121268\pi$
$224$	256.000			0.0763604
$225$		0			0
$226$	−1794.00			−0.528031
$227$	1932.00	−	3346.32i	0.564896	−	0.978428i	−0.432164	−	0.901795i	$-0.642250\pi$
$227$	1932.00	−	3346.32i	0.564896	−	0.978428i	0.997059	−	0.0766330i	$-0.0244170\pi$
$228$		0			0
$229$	1956.50	+	3388.76i	0.564581	+	0.977884i	0.997088	+	0.0762532i	$0.0242957\pi$
$229$	1956.50	+	3388.76i	0.564581	+	0.977884i	−0.432507	+	0.901631i	$0.642371\pi$
$230$	−3780.00	−	6547.15i	−1.08368	−	1.87698i
$231$		0			0
$232$	−540.000	+	935.307i	−0.152814	+	0.264681i
$233$	−6333.00			−1.78064			−0.890319	−	0.455337i	$-0.849519\pi$
$233$	−6333.00			−1.78064			−0.890319	+	0.455337i	$0.849519\pi$
$234$		0			0
$235$	1764.00			0.489662
$236$	1008.00	−	1745.91i	0.278031	−	0.481563i
$237$		0			0
$238$	168.000	+	290.985i	0.0457556	+	0.0792509i
$239$	−1914.00	−	3315.15i	−0.518018	−	0.897234i	−0.999781	−	0.0209322i	$-0.993337\pi$
$239$	−1914.00	−	3315.15i	−0.518018	−	0.897234i	0.481763	−	0.876302i	$-0.339997\pi$
$240$		0			0
$241$	738.500	−	1279.12i	0.197390	−	0.341889i	−0.750291	−	0.661107i	$-0.770087\pi$
$241$	738.500	−	1279.12i	0.197390	−	0.341889i	0.947681	+	0.319218i	$0.103420\pi$
$242$	−70.0000			−0.0185941
$243$		0			0
$244$	−1540.00			−0.404051
$245$	−2929.50	+	5074.04i	−0.763914	+	1.32314i
$246$		0			0
$247$	−2744.00	−	4752.75i	−0.706869	−	1.22433i
$248$	−1232.00	−	2133.89i	−0.315452	−	0.546379i
$249$		0			0
$250$	4011.00	−	6947.26i	1.01471	−	1.75753i
$251$	3612.00			0.908316			0.454158	−	0.890921i	$-0.349940\pi$
$251$	3612.00			0.908316			0.454158	+	0.890921i	$0.349940\pi$
$252$		0			0
$253$	6480.00			1.61025
$254$	−1532.00	+	2653.50i	−0.378450	+	0.655494i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−199.500	−	345.544i	−0.0484221	−	0.0838695i	0.840798	−	0.541348i	$-0.182086\pi$
$257$	−199.500	−	345.544i	−0.0484221	−	0.0838695i	−0.889221	+	0.457479i	$0.848753\pi$
$258$		0			0
$259$	4.00000	−	6.92820i	0.000959644	−	0.00166215i
$260$	4116.00			0.981783
$261$		0			0
$262$	−1680.00			−0.396148
$263$	1614.00	−	2795.53i	0.378416	−	0.655436i	−0.612416	−	0.790536i	$-0.709802\pi$
$263$	1614.00	−	2795.53i	0.378416	−	0.655436i	0.990832	+	0.135100i	$0.0431354\pi$
$264$		0			0
$265$	1827.00	+	3164.46i	0.423516	+	0.733551i
$266$	896.000	+	1551.92i	0.206531	+	0.357722i
$267$		0			0
$268$	−544.000	+	942.236i	−0.123993	+	0.214762i
$269$	147.000			0.0333188			0.0166594	−	0.999861i	$-0.494697\pi$
$269$	147.000			0.0333188			0.0166594	+	0.999861i	$0.494697\pi$
$270$		0			0
$271$	3332.00			0.746880			0.373440	−	0.927654i	$-0.378178\pi$
$271$	3332.00			0.746880			0.373440	+	0.927654i	$0.378178\pi$
$272$	168.000	−	290.985i	0.0374504	−	0.0648659i
$273$		0			0
$274$	729.000	+	1262.67i	0.160732	+	0.278396i
$275$	5688.00	+	9851.90i	1.24727	+	2.16034i
$276$		0			0
$277$	−1207.00	+	2090.59i	−0.261811	+	0.453470i	−0.966723	−	0.255825i	$-0.917653\pi$
$277$	−1207.00	+	2090.59i	−0.261811	+	0.453470i	0.704912	+	0.709294i	$0.250986\pi$
$278$	−4088.00			−0.881949
$279$		0			0
$280$	−1344.00			−0.286855
$281$	−1777.50	+	3078.72i	−0.377355	+	0.653598i	−0.990677	−	0.136235i	$-0.956500\pi$
$281$	−1777.50	+	3078.72i	−0.377355	+	0.653598i	0.613321	+	0.789833i	$0.289833\pi$
$282$		0			0
$283$	−2674.00	−	4631.50i	−0.561671	−	0.972842i	−0.997351	−	0.0727406i	$-0.976825\pi$
$283$	−2674.00	−	4631.50i	−0.561671	−	0.972842i	0.435680	−	0.900102i	$-0.356508\pi$
$284$	−1776.00	−	3076.12i	−0.371078	−	0.642726i
$285$		0			0
$286$	−1764.00	+	3055.34i	−0.364712	+	0.631699i
$287$	336.000			0.0691061
$288$		0			0
$289$	−4472.00			−0.910238
$290$	2835.00	−	4910.36i	0.574058	−	0.994298i
$291$		0			0
$292$	−742.000	−	1285.18i	−0.148706	−	0.257567i
$293$	3244.50	+	5619.64i	0.646914	+	1.12049i	0.983856	+	0.178962i	$0.0572739\pi$
$293$	3244.50	+	5619.64i	0.646914	+	1.12049i	−0.336942	+	0.941525i	$0.609393\pi$
$294$		0			0
$295$	−5292.00	+	9166.01i	−1.04445	+	1.80904i
$296$	−8.00000			−0.00157091
$297$		0			0
$298$	2574.00			0.500362
$299$	−4410.00	+	7638.34i	−0.852966	+	1.47738i
$300$		0			0
$301$	−80.0000	−	138.564i	−0.0153193	−	0.0265339i
$302$	736.000	+	1274.79i	0.140239	+	0.242900i
$303$		0			0
$304$	896.000	−	1551.92i	0.169043	−	0.292791i
$305$	8085.00			1.51785
$306$		0			0
$307$	−1204.00			−0.223830			−0.111915	−	0.993718i	$-0.535698\pi$
$307$	−1204.00			−0.223830			−0.111915	+	0.993718i	$0.535698\pi$
$308$	576.000	−	997.661i	0.106561	−	0.184568i
$309$		0			0
$310$	6468.00	+	11202.9i	1.18502	+	2.05252i
$311$	−1596.00	−	2764.35i	−0.291000	−	0.504026i	0.683047	−	0.730375i	$-0.260654\pi$
$311$	−1596.00	−	2764.35i	−0.291000	−	0.504026i	−0.974046	+	0.226349i	$0.927321\pi$
$312$		0			0
$313$	1620.50	−	2806.79i	0.292639	−	0.506866i	−0.681794	−	0.731544i	$-0.738800\pi$
$313$	1620.50	−	2806.79i	0.292639	−	0.506866i	0.974433	+	0.224678i	$0.0721332\pi$
$314$	−4298.00			−0.772453
$315$		0			0
$316$	−2608.00			−0.464277
$317$	2662.50	−	4611.59i	0.471738	−	0.817074i	−0.527739	−	0.849406i	$-0.676960\pi$
$317$	2662.50	−	4611.59i	0.471738	−	0.817074i	0.999477	+	0.0323325i	$0.0102935\pi$
$318$		0			0
$319$	2430.00	+	4208.88i	0.426501	+	0.738722i
$320$	672.000	+	1163.94i	0.117394	+	0.203332i
$321$		0			0
$322$	1440.00	−	2494.15i	0.249218	−	0.431658i
$323$	2352.00			0.405167
$324$		0			0
$325$	−15484.0			−2.64276
$326$	3088.00	−	5348.57i	0.524627	−	0.908681i
$327$		0			0
$328$	−168.000	−	290.985i	−0.0282812	−	0.0489846i
$329$	336.000	+	581.969i	0.0563048	+	0.0975228i
$330$		0			0
$331$	−484.000	+	838.313i	−0.0803717	+	0.139208i	−0.903410	−	0.428779i	$-0.858944\pi$
$331$	−484.000	+	838.313i	−0.0803717	+	0.139208i	0.823038	+	0.567986i	$0.192277\pi$
$332$	−336.000			−0.0555434
$333$		0			0
$334$	−336.000			−0.0550452
$335$	2856.00	−	4946.74i	0.465791	−	0.806773i
$336$		0			0
$337$	−4945.00	−	8564.99i	−0.799321	−	1.38447i	−0.920059	−	0.391781i	$-0.871859\pi$
$337$	−4945.00	−	8564.99i	−0.799321	−	1.38447i	0.120737	−	0.992685i	$-0.461474\pi$
$338$	−204.000	−	353.338i	−0.0328288	−	0.0568612i
$339$		0			0
$340$	−882.000	+	1527.67i	−0.140686	+	0.243675i
$341$	−11088.0			−1.76085
$342$		0			0
$343$	−4976.00			−0.783320
$344$	−80.0000	+	138.564i	−0.0125387	+	0.0217177i
$345$		0			0
$346$	−3003.00	−	5201.35i	−0.466596	−	0.808168i
$347$	−780.000	−	1351.00i	−0.120670	−	0.209007i	0.799362	−	0.600850i	$-0.205171\pi$
$347$	−780.000	−	1351.00i	−0.120670	−	0.209007i	−0.920032	+	0.391843i	$0.871838\pi$
$348$		0			0
$349$	−1435.00	+	2485.49i	−0.220097	+	0.381219i	−0.954837	−	0.297130i	$-0.903971\pi$
$349$	−1435.00	+	2485.49i	−0.220097	+	0.381219i	0.734740	+	0.678348i	$0.237304\pi$
$350$	5056.00			0.772156
$351$		0			0
$352$	−1152.00			−0.174437
$353$	3591.00	−	6219.79i	0.541444	−	0.937808i	−0.457378	−	0.889273i	$-0.651211\pi$
$353$	3591.00	−	6219.79i	0.541444	−	0.937808i	0.998821	−	0.0485356i	$-0.0154554\pi$
$354$		0			0
$355$	9324.00	+	16149.6i	1.39399	+	2.41446i
$356$	42.0000	+	72.7461i	0.00625280	+	0.0108302i
$357$		0			0
$358$	−1164.00	+	2016.11i	−0.171842	+	0.297638i
$359$	−8100.00			−1.19081			−0.595406	−	0.803425i	$-0.703009\pi$
$359$	−8100.00			−1.19081			−0.595406	+	0.803425i	$0.703009\pi$
$360$		0			0
$361$	5685.00			0.828838
$362$	1666.00	−	2885.60i	0.241887	−	0.418960i
$363$		0			0
$364$	784.000	+	1357.93i	0.112892	+	0.195535i
$365$	3895.50	+	6747.20i	0.558630	+	0.967575i
$366$		0			0
$367$	−5572.00	+	9650.99i	−0.792523	+	1.37269i	0.131877	+	0.991266i	$0.457900\pi$
$367$	−5572.00	+	9650.99i	−0.792523	+	1.37269i	−0.924400	+	0.381424i	$0.875434\pi$
$368$	−2880.00			−0.407963
$369$		0			0
$370$	42.0000			0.00590129
$371$	−696.000	+	1205.51i	−0.0973976	+	0.168698i
$372$		0			0
$373$	−6919.00	−	11984.1i	−0.960462	−	1.66357i	−0.721343	−	0.692578i	$-0.756475\pi$
$373$	−6919.00	−	11984.1i	−0.960462	−	1.66357i	−0.239119	−	0.970990i	$-0.576859\pi$
$374$	−756.000	−	1309.43i	−0.104524	−	0.181040i
$375$		0			0
$376$	336.000	−	581.969i	0.0460848	−	0.0798212i
$377$	−6615.00			−0.903687
$378$		0			0
$379$	1196.00			0.162096			0.0810480	−	0.996710i	$-0.474173\pi$
$379$	1196.00			0.162096			0.0810480	+	0.996710i	$0.474173\pi$
$380$	−4704.00	+	8147.57i	−0.635027	+	1.09990i
$381$		0			0
$382$	−2064.00	−	3574.95i	−0.276449	−	0.478823i
$383$	1932.00	+	3346.32i	0.257756	+	0.446447i	0.965640	−	0.259882i	$-0.0836836\pi$
$383$	1932.00	+	3346.32i	0.257756	+	0.446447i	−0.707884	+	0.706328i	$0.750350\pi$
$384$		0			0
$385$	−3024.00	+	5237.72i	−0.400305	+	0.693348i
$386$	−1130.00			−0.149004
$387$		0			0
$388$	−4984.00			−0.652124
$389$	−2535.00	+	4390.75i	−0.330410	+	0.572287i	−0.982592	−	0.185775i	$-0.940520\pi$
$389$	−2535.00	+	4390.75i	−0.330410	+	0.572287i	0.652182	+	0.758062i	$0.273854\pi$
$390$		0			0
$391$	−1890.00	−	3273.58i	−0.244454	−	0.423406i
$392$	1116.00	+	1932.97i	0.143792	+	0.249055i
$393$		0			0
$394$	−4731.00	+	8194.33i	−0.604935	+	1.04778i
$395$	13692.0			1.74410
$396$		0			0
$397$	15239.0			1.92651			0.963254	−	0.268593i	$-0.0865587\pi$
$397$	15239.0			1.92651			0.963254	+	0.268593i	$0.0865587\pi$
$398$	−4676.00	+	8099.07i	−0.588911	+	1.02002i
$399$		0			0
$400$	−2528.00	−	4378.62i	−0.316000	−	0.547328i
$401$	−853.500	−	1478.31i	−0.106289	−	0.184097i	0.807975	−	0.589216i	$-0.200563\pi$
$401$	−853.500	−	1478.31i	−0.106289	−	0.184097i	−0.914264	+	0.405119i	$0.867230\pi$
$402$		0			0
$403$	7546.00	−	13070.1i	0.932737	−	1.61555i
$404$	−2184.00			−0.268956
$405$		0			0
$406$	2160.00			0.264037
$407$	−18.0000	+	31.1769i	−0.00219220	+	0.00379701i
$408$		0			0
$409$	6660.50	+	11536.3i	0.805234	+	1.39471i	0.916133	+	0.400874i	$0.131293\pi$
$409$	6660.50	+	11536.3i	0.805234	+	1.39471i	−0.110900	+	0.993832i	$0.535373\pi$
$410$	882.000	+	1527.67i	0.106241	+	0.184015i
$411$		0			0
$412$	392.000	−	678.964i	0.0468749	−	0.0811897i
$413$	−4032.00			−0.480392
$414$		0			0
$415$	1764.00			0.208654
$416$	784.000	−	1357.93i	0.0924009	−	0.160043i
$417$		0			0
$418$	−4032.00	−	6983.63i	−0.471798	−	0.817178i
$419$	−6972.00	−	12075.9i	−0.812899	−	1.40798i	−0.910827	−	0.412788i	$-0.864555\pi$
$419$	−6972.00	−	12075.9i	−0.812899	−	1.40798i	0.0979285	−	0.995193i	$-0.468778\pi$
$420$		0			0
$421$	5418.50	−	9385.12i	0.627272	−	1.08647i	−0.360825	−	0.932634i	$-0.617505\pi$
$421$	5418.50	−	9385.12i	0.627272	−	1.08647i	0.988097	−	0.153833i	$-0.0491619\pi$
$422$	6760.00			0.779791
$423$		0			0
$424$	1392.00			0.159437
$425$	3318.00	−	5746.94i	0.378698	−	0.655924i
$426$		0			0
$427$	1540.00	+	2667.36i	0.174534	+	0.302301i
$428$	−600.000	−	1039.23i	−0.0677619	−	0.117367i
$429$		0			0
$430$	420.000	−	727.461i	0.0471028	−	0.0815844i
$431$	−12612.0			−1.40951			−0.704755	−	0.709451i	$-0.748943\pi$
$431$	−12612.0			−1.40951			−0.704755	+	0.709451i	$0.748943\pi$
$432$		0			0
$433$	−9709.00			−1.07756			−0.538781	−	0.842446i	$-0.681115\pi$
$433$	−9709.00			−1.07756			−0.538781	+	0.842446i	$0.681115\pi$
$434$	−2464.00	+	4267.77i	−0.272525	+	0.472027i
$435$		0			0
$436$	2138.00	+	3703.12i	0.234843	+	0.406760i
$437$	−10080.0	−	17459.1i	−1.10341	−	1.91117i
$438$		0			0
$439$	−5194.00	+	8996.27i	−0.564684	+	0.978061i	0.432395	+	0.901684i	$0.357668\pi$
$439$	−5194.00	+	8996.27i	−0.564684	+	0.978061i	−0.997079	+	0.0763766i	$0.975665\pi$
$440$	6048.00			0.655289
$441$		0			0
$442$	2058.00			0.221469
$443$	−1254.00	+	2171.99i	−0.134491	+	0.232945i	−0.925403	−	0.378985i	$-0.876273\pi$
$443$	−1254.00	+	2171.99i	−0.134491	+	0.232945i	0.790912	+	0.611930i	$0.209606\pi$
$444$		0			0
$445$	−220.500	−	381.917i	−0.0234892	−	0.0406845i
$446$	5236.00	+	9069.02i	0.555901	+	0.962849i
$447$		0			0
$448$	−256.000	+	443.405i	−0.0269975	+	0.0467610i
$449$	13698.0			1.43975			0.719876	−	0.694103i	$-0.244199\pi$
$449$	13698.0			1.43975			0.719876	+	0.694103i	$0.244199\pi$
$450$		0			0
$451$	−1512.00			−0.157865
$452$	1794.00	−	3107.30i	0.186687	−	0.323352i
$453$		0			0
$454$	3864.00	+	6692.64i	0.399442	+	0.691853i
$455$	−4116.00	−	7129.12i	−0.424090	−	0.734546i
$456$		0			0
$457$	4872.50	−	8439.42i	0.498744	−	0.863850i	−0.501255	−	0.865300i	$-0.667128\pi$
$457$	4872.50	−	8439.42i	0.498744	−	0.863850i	0.999999	+	0.00144988i	$0.000461511\pi$
$458$	−7826.00			−0.798439
$459$		0			0
$460$	15120.0			1.53255
$461$	8757.00	−	15167.6i	0.884716	−	1.53237i	0.0386775	−	0.999252i	$-0.487685\pi$
$461$	8757.00	−	15167.6i	0.884716	−	1.53237i	0.846039	−	0.533122i	$-0.178981\pi$
$462$		0			0
$463$	−2320.00	−	4018.36i	−0.232872	−	0.403345i	0.725780	−	0.687926i	$-0.241479\pi$
$463$	−2320.00	−	4018.36i	−0.232872	−	0.403345i	−0.958652	+	0.284581i	$0.908146\pi$
$464$	−1080.00	−	1870.61i	−0.108055	−	0.187158i
$465$		0			0
$466$	6333.00	−	10969.1i	0.629551	−	1.09041i
$467$	4368.00			0.432820			0.216410	−	0.976303i	$-0.430565\pi$
$467$	4368.00			0.432820			0.216410	+	0.976303i	$0.430565\pi$
$468$		0			0
$469$	2176.00			0.214240
$470$	−1764.00	+	3055.34i	−0.173122	+	0.299856i
$471$		0			0
$472$	2016.00	+	3491.81i	0.196597	+	0.340516i
$473$	360.000	+	623.538i	0.0349954	+	0.0606138i
$474$		0			0
$475$	17696.0	−	30650.4i	1.70936	−	2.96071i
$476$	−672.000			−0.0647081
$477$		0			0
$478$	7656.00			0.732588
$479$	−9408.00	+	16295.1i	−0.897416	+	1.55437i	−0.0666313	+	0.997778i	$0.521225\pi$
$479$	−9408.00	+	16295.1i	−0.897416	+	1.55437i	−0.830785	+	0.556593i	$0.812108\pi$
$480$		0			0
$481$	−24.5000	−	42.4352i	−0.00232246	−	0.00402262i
$482$	1477.00	+	2558.24i	0.139576	+	0.241752i
$483$		0			0
$484$	70.0000	−	121.244i	0.00657400	−	0.0113865i
$485$	26166.0			2.44977
$486$		0			0
$487$	−13756.0			−1.27997			−0.639983	−	0.768389i	$-0.721059\pi$
$487$	−13756.0			−1.27997			−0.639983	+	0.768389i	$0.721059\pi$
$488$	1540.00	−	2667.36i	0.142854	−	0.247430i
$489$		0			0
$490$	−5859.00	−	10148.1i	−0.540169	−	0.935600i
$491$	3870.00	+	6703.04i	0.355704	+	0.616097i	0.987238	−	0.159251i	$-0.0509078\pi$
$491$	3870.00	+	6703.04i	0.355704	+	0.616097i	−0.631534	+	0.775348i	$0.717574\pi$
$492$		0			0
$493$	1417.50	−	2455.18i	0.129495	−	0.224292i
$494$	10976.0			0.999663
$495$		0			0
$496$	4928.00			0.446116
$497$	−3552.00	+	6152.24i	−0.320581	+	0.555263i
$498$		0			0
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	0.807272	−	0.590180i	$-0.200943\pi$
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	−0.914747	+	0.404028i	$0.867610\pi$
$500$	8022.00	+	13894.5i	0.717509	+	1.24276i
$501$		0			0
$502$	−3612.00	+	6256.17i	−0.321138	+	0.556228i
$503$	−12096.0			−1.07223			−0.536117	−	0.844144i	$-0.680110\pi$
$503$	−12096.0			−1.07223			−0.536117	+	0.844144i	$0.680110\pi$
$504$		0			0
$505$	11466.0			1.01036
$506$	−6480.00	+	11223.7i	−0.569311	+	0.986075i
$507$		0			0
$508$	−3064.00	−	5307.00i	−0.267604	−	0.463504i
$509$	861.000	+	1491.30i	0.0749767	+	0.129864i	0.901076	−	0.433661i	$-0.142778\pi$
$509$	861.000	+	1491.30i	0.0749767	+	0.129864i	−0.826099	+	0.563524i	$0.809445\pi$
$510$		0			0
$511$	−1484.00	+	2570.36i	−0.128470	+	0.222517i
$512$	512.000			0.0441942
$513$		0			0
$514$	798.000			0.0684791
$515$	−2058.00	+	3564.56i	−0.176090	+	0.304997i
$516$		0			0
$517$	−1512.00	−	2618.86i	−0.128622	−	0.222780i
$518$	8.00000	+	13.8564i	0.000678571	+	0.00117532i
$519$		0			0
$520$	−4116.00	+	7129.12i	−0.347113	+	0.601217i
$521$	−2982.00			−0.250756			−0.125378	−	0.992109i	$-0.540014\pi$
$521$	−2982.00			−0.250756			−0.125378	+	0.992109i	$0.540014\pi$
$522$		0			0
$523$	812.000			0.0678896			0.0339448	−	0.999424i	$-0.489193\pi$
$523$	812.000			0.0678896			0.0339448	+	0.999424i	$0.489193\pi$
$524$	1680.00	−	2909.85i	0.140059	−	0.242590i
$525$		0			0
$526$	3228.00	+	5591.06i	0.267581	+	0.463464i
$527$	3234.00	+	5601.45i	0.267315	+	0.463004i
$528$		0			0
$529$	−10116.5	+	17522.3i	−0.831470	+	1.44015i
$530$	−7308.00			−0.598942
$531$		0			0
$532$	−3584.00			−0.292079
$533$	1029.00	−	1782.28i	0.0836228	−	0.144839i
$534$		0			0
$535$	3150.00	+	5455.96i	0.254554	+	0.440900i
$536$	−1088.00	−	1884.47i	−0.0876762	−	0.151860i
$537$		0			0
$538$	−147.000	+	254.611i	−0.0117800	+	0.0204035i
$539$	10044.0			0.802645
$540$		0			0
$541$	7055.00			0.560662			0.280331	−	0.959903i	$-0.409556\pi$
$541$	7055.00			0.560662			0.280331	+	0.959903i	$0.409556\pi$
$542$	−3332.00	+	5771.19i	−0.264062	+	0.457369i
$543$		0			0
$544$	336.000	+	581.969i	0.0264814	+	0.0458671i
$545$	−11224.5	−	19441.4i	−0.882211	−	1.52803i
$546$		0			0
$547$	7298.00	−	12640.5i	0.570457	−	0.988060i	−0.426062	−	0.904694i	$-0.640099\pi$
$547$	7298.00	−	12640.5i	0.570457	−	0.988060i	0.996519	−	0.0833664i	$-0.0265672\pi$
$548$	−2916.00			−0.227309
$549$		0			0
$550$	−22752.0			−1.76391
$551$	7560.00	−	13094.3i	0.584513	−	1.01241i
$552$		0			0
$553$	2608.00	+	4517.19i	0.200549	+	0.347361i
$554$	−2414.00	−	4181.17i	−0.185128	−	0.320651i
$555$		0			0
$556$	4088.00	−	7080.62i	0.311816	−	0.540082i
$557$	7755.00			0.589928			0.294964	−	0.955508i	$-0.404692\pi$
$557$	7755.00			0.589928			0.294964	+	0.955508i	$0.404692\pi$
$558$		0			0
$559$	−980.000			−0.0741495
$560$	1344.00	−	2327.88i	0.101419	−	0.175662i
$561$		0			0
$562$	−3555.00	−	6157.44i	−0.266830	−	0.462164i
$563$	8022.00	+	13894.5i	0.600510	+	1.04011i	0.992744	+	0.120248i	$0.0383690\pi$
$563$	8022.00	+	13894.5i	0.600510	+	1.04011i	−0.392234	+	0.919865i	$0.628298\pi$
$564$		0			0
$565$	−9418.50	+	16313.3i	−0.701308	+	1.21470i
$566$	10696.0			0.794322
$567$		0			0
$568$	7104.00			0.524784
$569$	8512.50	−	14744.1i	0.627175	−	1.08630i	−0.360941	−	0.932589i	$-0.617544\pi$
$569$	8512.50	−	14744.1i	0.627175	−	1.08630i	0.988116	−	0.153710i	$-0.0491222\pi$
$570$		0			0
$571$	−1660.00	−	2875.20i	−0.121662	−	0.210724i	0.798761	−	0.601648i	$-0.205489\pi$
$571$	−1660.00	−	2875.20i	−0.121662	−	0.210724i	−0.920423	+	0.390924i	$0.872156\pi$
$572$	−3528.00	−	6110.68i	−0.257890	−	0.446679i
$573$		0			0
$574$	−336.000	+	581.969i	−0.0244327	+	0.0423187i
$575$	−56880.0			−4.12532
$576$		0			0
$577$	1127.00			0.0813130			0.0406565	−	0.999173i	$-0.487055\pi$
$577$	1127.00			0.0813130			0.0406565	+	0.999173i	$0.487055\pi$
$578$	4472.00	−	7745.73i	0.321818	−	0.557405i
$579$		0			0
$580$	5670.00	+	9820.73i	0.405921	+	0.703075i
$581$	336.000	+	581.969i	0.0239925	+	0.0415562i
$582$		0			0
$583$	3132.00	−	5424.78i	0.222494	−	0.385371i
$584$	2968.00			0.210303
$585$		0			0
$586$	−12978.0			−0.914874
$587$	−42.0000	+	72.7461i	−0.00295320	+	0.00511508i	−0.867498	−	0.497440i	$-0.834273\pi$
$587$	−42.0000	+	72.7461i	−0.00295320	+	0.00511508i	0.864545	+	0.502555i	$0.167607\pi$
$588$		0			0
$589$	17248.0	+	29874.4i	1.20661	+	2.08990i
$590$	−10584.0	−	18332.0i	−0.738536	−	1.27918i
$591$		0			0
$592$	8.00000	−	13.8564i	0.000555402	−	0.000961984i
$593$	1743.00			0.120702			0.0603511	−	0.998177i	$-0.480778\pi$
$593$	1743.00			0.120702			0.0603511	+	0.998177i	$0.480778\pi$
$594$		0			0
$595$	3528.00			0.243082
$596$	−2574.00	+	4458.30i	−0.176905	+	0.306408i
$597$		0			0
$598$	−8820.00	−	15276.7i	−0.603138	−	1.04467i
$599$	−8046.00	−	13936.1i	−0.548832	−	0.950606i	−0.998355	−	0.0573367i	$-0.981739\pi$
$599$	−8046.00	−	13936.1i	−0.548832	−	0.950606i	0.449522	−	0.893269i	$-0.351594\pi$
$600$		0			0
$601$	−10517.5	+	18216.8i	−0.713840	+	1.23641i	0.249565	+	0.968358i	$0.419712\pi$
$601$	−10517.5	+	18216.8i	−0.713840	+	1.23641i	−0.963405	+	0.268049i	$0.913621\pi$
$602$	320.000			0.0216648
$603$		0			0
$604$	−2944.00			−0.198327
$605$	−367.500	+	636.529i	−0.0246959	+	0.0427745i
$606$		0			0
$607$	−3388.00	−	5868.19i	−0.226548	−	0.392393i	0.730235	−	0.683196i	$-0.239411\pi$
$607$	−3388.00	−	5868.19i	−0.226548	−	0.392393i	−0.956783	+	0.290804i	$0.906077\pi$
$608$	1792.00	+	3103.84i	0.119532	+	0.207035i
$609$		0			0
$610$	−8085.00	+	14003.6i	−0.536643	+	0.929493i
$611$	4116.00			0.272530
$612$		0			0
$613$	−23794.0			−1.56775			−0.783875	−	0.620919i	$-0.786760\pi$
$613$	−23794.0			−1.56775			−0.783875	+	0.620919i	$0.786760\pi$
$614$	1204.00	−	2085.39i	0.0791360	−	0.137068i
$615$		0			0
$616$	1152.00	+	1995.32i	0.0753497	+	0.130509i
$617$	10810.5	+	18724.3i	0.705372	+	1.22174i	0.966557	+	0.256451i	$0.0825532\pi$
$617$	10810.5	+	18724.3i	0.705372	+	1.22174i	−0.261186	+	0.965289i	$0.584113\pi$
$618$		0			0
$619$	−11116.0	+	19253.5i	−0.721793	+	1.25018i	0.238488	+	0.971146i	$0.423348\pi$
$619$	−11116.0	+	19253.5i	−0.721793	+	1.25018i	−0.960281	+	0.279036i	$0.909985\pi$
$620$	−25872.0			−1.67588
$621$		0			0
$622$	6384.00			0.411535
$623$	84.0000	−	145.492i	0.00540191	−	0.00935638i
$624$		0			0
$625$	−22365.5	−	38738.2i	−1.43139	−	2.47924i
$626$	3241.00	+	5613.58i	0.206927	+	0.358408i
$627$		0			0
$628$	4298.00	−	7444.35i	0.273103	−	0.473029i
$629$	21.0000			0.00133120
$630$		0			0
$631$	−9280.00			−0.585469			−0.292735	−	0.956194i	$-0.594565\pi$
$631$	−9280.00			−0.585469			−0.292735	+	0.956194i	$0.594565\pi$
$632$	2608.00	−	4517.19i	0.164147	−	0.284310i
$633$		0			0
$634$	5325.00	+	9223.17i	0.333569	+	0.577759i
$635$	16086.0	+	27861.8i	1.00528	+	1.74120i
$636$		0			0
$637$	−6835.50	+	11839.4i	−0.425169	+	0.736414i
$638$	−9720.00			−0.603164
$639$		0			0
$640$	−2688.00			−0.166020
$641$	−9589.50	+	16609.5i	−0.590893	+	1.02346i	0.403219	+	0.915103i	$0.367891\pi$
$641$	−9589.50	+	16609.5i	−0.590893	+	1.02346i	−0.994112	+	0.108354i	$0.965442\pi$
$642$		0			0
$643$	1610.00	+	2788.60i	0.0987437	+	0.171029i	0.911165	−	0.412042i	$-0.135184\pi$
$643$	1610.00	+	2788.60i	0.0987437	+	0.171029i	−0.812421	+	0.583071i	$0.801851\pi$
$644$	2880.00	+	4988.31i	0.176223	+	0.305228i
$645$		0			0
$646$	−2352.00	+	4073.78i	−0.143248	+	0.248113i
$647$	−14112.0			−0.857496			−0.428748	−	0.903424i	$-0.641045\pi$
$647$	−14112.0			−0.857496			−0.428748	+	0.903424i	$0.641045\pi$
$648$		0			0
$649$	18144.0			1.09740
$650$	15484.0	−	26819.1i	0.934358	−	1.61835i
$651$		0			0
$652$	6176.00	+	10697.1i	0.370968	+	0.642535i
$653$	11421.0	+	19781.8i	0.684438	+	1.18548i	0.973613	+	0.228206i	$0.0732859\pi$
$653$	11421.0	+	19781.8i	0.684438	+	1.18548i	−0.289175	+	0.957276i	$0.593381\pi$
$654$		0			0
$655$	−8820.00	+	15276.7i	−0.526146	+	0.911312i
$656$	672.000			0.0399957
$657$		0			0
$658$	−1344.00			−0.0796270
$659$	−10860.0	+	18810.1i	−0.641951	+	1.11189i	0.343046	+	0.939319i	$0.388541\pi$
$659$	−10860.0	+	18810.1i	−0.641951	+	1.11189i	−0.984997	+	0.172573i	$0.944792\pi$
$660$		0			0
$661$	−13163.5	−	22799.9i	−0.774585	−	1.34162i	−0.935027	−	0.354576i	$-0.884625\pi$
$661$	−13163.5	−	22799.9i	−0.774585	−	1.34162i	0.160442	−	0.987045i	$-0.448708\pi$
$662$	−968.000	−	1676.63i	−0.0568314	−	0.0984349i
$663$		0			0
$664$	336.000	−	581.969i	0.0196375	−	0.0340132i
$665$	18816.0			1.09722
$666$		0			0
$667$	−24300.0			−1.41064
$668$	336.000	−	581.969i	0.0194614	−	0.0337082i
$669$		0			0
$670$	5712.00	+	9893.47i	0.329364	+	0.570475i
$671$	−6930.00	−	12003.1i	−0.398703	−	0.690574i
$672$		0			0
$673$	9870.50	−	17096.2i	0.565349	−	0.979213i	−0.431668	−	0.902032i	$-0.642075\pi$
$673$	9870.50	−	17096.2i	0.565349	−	0.979213i	0.997017	−	0.0771806i	$-0.0245918\pi$
$674$	19780.0			1.13041
$675$		0			0
$676$	816.000			0.0464269
$677$	6321.00	−	10948.3i	0.358842	−	0.621532i	−0.628926	−	0.777465i	$-0.716505\pi$
$677$	6321.00	−	10948.3i	0.358842	−	0.621532i	0.987768	+	0.155933i	$0.0498385\pi$
$678$		0			0
$679$	4984.00	+	8632.54i	0.281691	+	0.487904i
$680$	−1764.00	−	3055.34i	−0.0994799	−	0.172304i
$681$		0			0
$682$	11088.0	−	19205.0i	0.622553	−	1.07829i
$683$	26172.0			1.46624			0.733121	−	0.680098i	$-0.238063\pi$
$683$	26172.0			1.46624			0.733121	+	0.680098i	$0.238063\pi$
$684$		0			0
$685$	15309.0			0.853908
$686$	4976.00	−	8618.68i	0.276945	−	0.479684i
$687$		0			0
$688$	−160.000	−	277.128i	−0.00886620	−	0.0153567i
$689$	4263.00	+	7383.73i	0.235715	+	0.408270i
$690$		0			0
$691$	4760.00	−	8244.56i	0.262053	−	0.453890i	−0.704734	−	0.709472i	$-0.748934\pi$
$691$	4760.00	−	8244.56i	0.262053	−	0.453890i	0.966787	+	0.255582i	$0.0822670\pi$
$692$	12012.0			0.659867
$693$		0			0
$694$	3120.00			0.170654
$695$	−21462.0	+	37173.3i	−1.17137	+	2.02887i
$696$		0			0
$697$	441.000	+	763.834i	0.0239657	+	0.0415097i
$698$	−2870.00	−	4970.99i	−0.155632	−	0.269562i
$699$		0			0
$700$	−5056.00	+	8757.25i	−0.272998	+	0.472847i
$701$	−16773.0			−0.903720			−0.451860	−	0.892089i	$-0.649239\pi$
$701$	−16773.0			−0.903720			−0.451860	+	0.892089i	$0.649239\pi$
$702$		0			0
$703$	112.000			0.00600876
$704$	1152.00	−	1995.32i	0.0616728	−	0.106820i
$705$		0			0
$706$	7182.00	+	12439.6i	0.382859	+	0.663130i
$707$	2184.00	+	3782.80i	0.116178	+	0.201226i
$708$		0			0
$709$	−6383.50	+	11056.5i	−0.338135	+	0.585666i	−0.984082	−	0.177716i	$-0.943129\pi$
$709$	−6383.50	+	11056.5i	−0.338135	+	0.585666i	0.645947	+	0.763382i	$0.276463\pi$
$710$	−37296.0			−1.97140
$711$		0			0
$712$	−168.000			−0.00884279
$713$	27720.0	−	48012.4i	1.45599	−	2.52185i
$714$		0			0
$715$	18522.0	+	32081.0i	0.968788	+	1.67799i
$716$	−2328.00	−	4032.21i	−0.121510	−	0.210462i
$717$		0			0
$718$	8100.00	−	14029.6i	0.421016	−	0.729221i
$719$	24948.0			1.29402			0.647012	−	0.762480i	$-0.276018\pi$
$719$	24948.0			1.29402			0.647012	+	0.762480i	$0.276018\pi$
$720$		0			0
$721$	−1568.00			−0.0809922
$722$	−5685.00	+	9846.71i	−0.293038	+	0.507558i
$723$		0			0
$724$	3332.00	+	5771.19i	0.171040	+	0.296250i
$725$	−21330.0	−	36944.6i	−1.09266	−	1.89254i
$726$		0			0
$727$	−28.0000	+	48.4974i	−0.00142842	+	0.00247410i	−0.866739	−	0.498762i	$-0.833788\pi$
$727$	−28.0000	+	48.4974i	−0.00142842	+	0.00247410i	0.865310	+	0.501237i	$0.167121\pi$
$728$	−3136.00			−0.159654
$729$		0			0
$730$	−15582.0			−0.790021
$731$	210.000	−	363.731i	0.0106253	−	0.0184036i
$732$		0			0
$733$	−595.000	−	1030.57i	−0.0299820	−	0.0519304i	0.850645	−	0.525740i	$-0.176212\pi$
$733$	−595.000	−	1030.57i	−0.0299820	−	0.0519304i	−0.880627	+	0.473810i	$0.842878\pi$
$734$	−11144.0	−	19302.0i	−0.560399	−	0.970639i
$735$		0			0
$736$	2880.00	−	4988.31i	0.144237	−	0.249825i
$737$	−9792.00			−0.489407
$738$		0			0
$739$	−26692.0			−1.32866			−0.664331	−	0.747439i	$-0.731283\pi$
$739$	−26692.0			−1.32866			−0.664331	+	0.747439i	$0.731283\pi$
$740$	−42.0000	+	72.7461i	−0.00208642	+	0.00361379i
$741$		0			0
$742$	−1392.00	−	2411.01i	−0.0688705	−	0.119287i
$743$	9426.00	+	16326.3i	0.465419	+	0.806130i	0.999220	−	0.0394804i	$-0.0125703\pi$
$743$	9426.00	+	16326.3i	0.465419	+	0.806130i	−0.533801	+	0.845610i	$0.679237\pi$
$744$		0			0
$745$	13513.5	−	23406.1i	0.664559	−	1.15105i
$746$	27676.0			1.35830
$747$		0			0
$748$	3024.00			0.147819
$749$	−1200.00	+	2078.46i	−0.0585408	+	0.101396i
$750$		0			0
$751$	−11308.0	−	19586.0i	−0.549447	−	0.951670i	−0.998312	−	0.0580709i	$-0.981505\pi$
$751$	−11308.0	−	19586.0i	−0.549447	−	0.951670i	0.448865	−	0.893599i	$-0.351828\pi$
$752$	672.000	+	1163.94i	0.0325869	+	0.0564421i
$753$		0			0
$754$	6615.00	−	11457.5i	0.319501	−	0.553393i
$755$	15456.0			0.745035
$756$		0			0
$757$	9326.00			0.447766			0.223883	−	0.974616i	$-0.428127\pi$
$757$	9326.00			0.447766			0.223883	+	0.974616i	$0.428127\pi$
$758$	−1196.00	+	2071.53i	−0.0573096	+	0.0992631i
$759$		0			0
$760$	−9408.00	−	16295.1i	−0.449032	−	0.777746i
$761$	10384.5	+	17986.5i	0.494662	+	0.856780i	0.999981	−	0.00615279i	$-0.00195851\pi$
$761$	10384.5	+	17986.5i	0.494662	+	0.856780i	−0.505319	+	0.862933i	$0.668625\pi$
$762$		0			0
$763$	4276.00	−	7406.25i	0.202885	−	0.351408i
$764$	8256.00			0.390958
$765$		0			0
$766$	−7728.00			−0.364522
$767$	−12348.0	+	21387.4i	−0.581304	+	1.00685i
$768$		0			0
$769$	150.500	+	260.674i	0.00705744	+	0.0122238i	0.869533	−	0.493876i	$-0.164420\pi$
$769$	150.500	+	260.674i	0.00705744	+	0.0122238i	−0.862475	+	0.506099i	$0.831087\pi$
$770$	−6048.00	−	10475.4i	−0.283058	−	0.490271i
$771$		0			0
$772$	1130.00	−	1957.22i	0.0526808	−	0.0912458i
$773$	17955.0			0.835442			0.417721	−	0.908575i	$-0.362829\pi$
$773$	17955.0			0.835442			0.417721	+	0.908575i	$0.362829\pi$
$774$		0			0
$775$	97328.0			4.51113
$776$	4984.00	−	8632.54i	0.230561	−	0.399343i
$777$		0			0
$778$	−5070.00	−	8781.50i	−0.233635	−	0.404668i
$779$	2352.00	+	4073.78i	0.108176	+	0.187366i
$780$		0			0
$781$	15984.0	−	27685.1i	0.732334	−	1.26844i

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(10.5000 + 18.1865i) q^{5} +(-4.00000 + 6.92820i) q^{7} +8.00000 q^{8} -42.0000 q^{10} +(18.0000 - 31.1769i) q^{11} +(24.5000 + 42.4352i) q^{13} +(-8.00000 - 13.8564i) q^{14} +(-8.00000 + 13.8564i) q^{16} -21.0000 q^{17} -112.000 q^{19} +(42.0000 - 72.7461i) q^{20} +(36.0000 + 62.3538i) q^{22} +(90.0000 + 155.885i) q^{23} +(-158.000 + 273.664i) q^{25} -98.0000 q^{26} +32.0000 q^{28} +(-67.5000 + 116.913i) q^{29} +(-154.000 - 266.736i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(21.0000 - 36.3731i) q^{34} -168.000 q^{35} -1.00000 q^{37} +(112.000 - 193.990i) q^{38} +(84.0000 + 145.492i) q^{40} +(-21.0000 - 36.3731i) q^{41} +(-10.0000 + 17.3205i) q^{43} -144.000 q^{44} -360.000 q^{46} +(42.0000 - 72.7461i) q^{47} +(139.500 + 241.621i) q^{49} +(-316.000 - 547.328i) q^{50} +(98.0000 - 169.741i) q^{52} +174.000 q^{53} +756.000 q^{55} +(-32.0000 + 55.4256i) q^{56} +(-135.000 - 233.827i) q^{58} +(252.000 + 436.477i) q^{59} +(192.500 - 333.420i) q^{61} +616.000 q^{62} +64.0000 q^{64} +(-514.500 + 891.140i) q^{65} +(-136.000 - 235.559i) q^{67} +(42.0000 + 72.7461i) q^{68} +(168.000 - 290.985i) q^{70} +888.000 q^{71} +371.000 q^{73} +(1.00000 - 1.73205i) q^{74} +(224.000 + 387.979i) q^{76} +(144.000 + 249.415i) q^{77} +(326.000 - 564.649i) q^{79} -336.000 q^{80} +84.0000 q^{82} +(42.0000 - 72.7461i) q^{83} +(-220.500 - 381.917i) q^{85} +(-20.0000 - 34.6410i) q^{86} +(144.000 - 249.415i) q^{88} -21.0000 q^{89} -392.000 q^{91} +(360.000 - 623.538i) q^{92} +(84.0000 + 145.492i) q^{94} +(-1176.00 - 2036.89i) q^{95} +(623.000 - 1079.07i) q^{97} -558.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26}+ \cdots - 1116 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 + 21 * q^5 - 8 * q^7 + 16 * q^8 - 84 * q^10 + 36 * q^11 + 49 * q^13 - 16 * q^14 - 16 * q^16 - 42 * q^17 - 224 * q^19 + 84 * q^20 + 72 * q^22 + 180 * q^23 - 316 * q^25 - 196 * q^26 + 64 * q^28 - 135 * q^29 - 308 * q^31 - 32 * q^32 + 42 * q^34 - 336 * q^35 - 2 * q^37 + 224 * q^38 + 168 * q^40 - 42 * q^41 - 20 * q^43 - 288 * q^44 - 720 * q^46 + 84 * q^47 + 279 * q^49 - 632 * q^50 + 196 * q^52 + 348 * q^53 + 1512 * q^55 - 64 * q^56 - 270 * q^58 + 504 * q^59 + 385 * q^61 + 1232 * q^62 + 128 * q^64 - 1029 * q^65 - 272 * q^67 + 84 * q^68 + 336 * q^70 + 1776 * q^71 + 742 * q^73 + 2 * q^74 + 448 * q^76 + 288 * q^77 + 652 * q^79 - 672 * q^80 + 168 * q^82 + 84 * q^83 - 441 * q^85 - 40 * q^86 + 288 * q^88 - 42 * q^89 - 784 * q^91 + 720 * q^92 + 168 * q^94 - 2352 * q^95 + 1246 * q^97 - 1116 * q^98

Introduction

L-functions

Modular forms

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