LMFDB - Embedded newform 54.4.c.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,4,Mod(19,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, 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("54.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	54.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:	$3.18610314031$
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 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			19.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	54.19
Dual form			54.4.c.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.50000 - 7.79423i) q^{5} +(15.5000 - 26.8468i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.50000 - 7.79423i) q^{5} +(15.5000 - 26.8468i) q^{7} -8.00000 q^{8} -18.0000 q^{10} +(-7.50000 + 12.9904i) q^{11} +(18.5000 + 32.0429i) q^{13} +(-31.0000 - 53.6936i) q^{14} +(-8.00000 + 13.8564i) q^{16} +42.0000 q^{17} -28.0000 q^{19} +(-18.0000 + 31.1769i) q^{20} +(15.0000 + 25.9808i) q^{22} +(97.5000 + 168.875i) q^{23} +(22.0000 - 38.1051i) q^{25} +74.0000 q^{26} -124.000 q^{28} +(55.5000 - 96.1288i) q^{29} +(102.500 + 177.535i) q^{31} +(16.0000 + 27.7128i) q^{32} +(42.0000 - 72.7461i) q^{34} -279.000 q^{35} -166.000 q^{37} +(-28.0000 + 48.4974i) q^{38} +(36.0000 + 62.3538i) q^{40} +(-130.500 - 226.033i) q^{41} +(21.5000 - 37.2391i) q^{43} +60.0000 q^{44} +390.000 q^{46} +(88.5000 - 153.286i) q^{47} +(-309.000 - 535.204i) q^{49} +(-44.0000 - 76.2102i) q^{50} +(74.0000 - 128.172i) q^{52} -114.000 q^{53} +135.000 q^{55} +(-124.000 + 214.774i) q^{56} +(-111.000 - 192.258i) q^{58} +(79.5000 + 137.698i) q^{59} +(-95.5000 + 165.411i) q^{61} +410.000 q^{62} +64.0000 q^{64} +(166.500 - 288.386i) q^{65} +(210.500 + 364.597i) q^{67} +(-84.0000 - 145.492i) q^{68} +(-279.000 + 483.242i) q^{70} -156.000 q^{71} +182.000 q^{73} +(-166.000 + 287.520i) q^{74} +(56.0000 + 96.9948i) q^{76} +(232.500 + 402.702i) q^{77} +(-566.500 + 981.207i) q^{79} +144.000 q^{80} -522.000 q^{82} +(-541.500 + 937.906i) q^{83} +(-189.000 - 327.358i) q^{85} +(-43.0000 - 74.4782i) q^{86} +(60.0000 - 103.923i) q^{88} +1050.00 q^{89} +1147.00 q^{91} +(390.000 - 675.500i) q^{92} +(-177.000 - 306.573i) q^{94} +(126.000 + 218.238i) q^{95} +(450.500 - 780.289i) q^{97} -1236.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8	$ 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8} - 36 q^{10} - 15 q^{11} + 37 q^{13} - 62 q^{14} - 16 q^{16} + 84 q^{17} - 56 q^{19} - 36 q^{20} + 30 q^{22} + 195 q^{23} + 44 q^{25} + 148 q^{26} - 248 q^{28} + 111 q^{29} + 205 q^{31} + 32 q^{32} + 84 q^{34} - 558 q^{35} - 332 q^{37} - 56 q^{38} + 72 q^{40} - 261 q^{41} + 43 q^{43} + 120 q^{44} + 780 q^{46} + 177 q^{47} - 618 q^{49} - 88 q^{50} + 148 q^{52} - 228 q^{53} + 270 q^{55} - 248 q^{56} - 222 q^{58} + 159 q^{59} - 191 q^{61} + 820 q^{62} + 128 q^{64} + 333 q^{65} + 421 q^{67} - 168 q^{68} - 558 q^{70} - 312 q^{71} + 364 q^{73} - 332 q^{74} + 112 q^{76} + 465 q^{77} - 1133 q^{79} + 288 q^{80} - 1044 q^{82} - 1083 q^{83} - 378 q^{85} - 86 q^{86} + 120 q^{88} + 2100 q^{89} + 2294 q^{91} + 780 q^{92} - 354 q^{94} + 252 q^{95} + 901 q^{97} - 2472 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8 - 36 * q^10 - 15 * q^11 + 37 * q^13 - 62 * q^14 - 16 * q^16 + 84 * q^17 - 56 * q^19 - 36 * q^20 + 30 * q^22 + 195 * q^23 + 44 * q^25 + 148 * q^26 - 248 * q^28 + 111 * q^29 + 205 * q^31 + 32 * q^32 + 84 * q^34 - 558 * q^35 - 332 * q^37 - 56 * q^38 + 72 * q^40 - 261 * q^41 + 43 * q^43 + 120 * q^44 + 780 * q^46 + 177 * q^47 - 618 * q^49 - 88 * q^50 + 148 * q^52 - 228 * q^53 + 270 * q^55 - 248 * q^56 - 222 * q^58 + 159 * q^59 - 191 * q^61 + 820 * q^62 + 128 * q^64 + 333 * q^65 + 421 * q^67 - 168 * q^68 - 558 * q^70 - 312 * q^71 + 364 * q^73 - 332 * q^74 + 112 * q^76 + 465 * q^77 - 1133 * q^79 + 288 * q^80 - 1044 * q^82 - 1083 * q^83 - 378 * q^85 - 86 * q^86 + 120 * q^88 + 2100 * q^89 + 2294 * q^91 + 780 * q^92 - 354 * q^94 + 252 * q^95 + 901 * q^97 - 2472 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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
$5$	−4.50000	−	7.79423i	−0.402492	−	0.697137i	0.591534	−	0.806280i	$-0.298523\pi$
$5$	−4.50000	−	7.79423i	−0.402492	−	0.697137i	−0.994026	+	0.109143i	$0.965189\pi$
$6$		0			0
$7$	15.5000	−	26.8468i	0.836921	−	1.44959i	−0.0555351	−	0.998457i	$-0.517686\pi$
$7$	15.5000	−	26.8468i	0.836921	−	1.44959i	0.892456	−	0.451134i	$-0.148980\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−18.0000			−0.569210
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	−0.950316	−	0.311287i	$-0.899240\pi$
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	0.744740	+	0.667355i	$0.232573\pi$
$12$		0			0
$13$	18.5000	+	32.0429i	0.394691	+	0.683624i	0.993062	−	0.117595i	$-0.0375185\pi$
$13$	18.5000	+	32.0429i	0.394691	+	0.683624i	−0.598371	+	0.801219i	$0.704185\pi$
$14$	−31.0000	−	53.6936i	−0.591793	−	1.02502i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	42.0000			0.599206			0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000			0.599206			0.299603	+	0.954064i	$0.403146\pi$
$18$		0			0
$19$	−28.0000			−0.338086			−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000			−0.338086			−0.169043	+	0.985609i	$0.554068\pi$
$20$	−18.0000	+	31.1769i	−0.201246	+	0.348569i
$21$		0			0
$22$	15.0000	+	25.9808i	0.145364	+	0.251778i
$23$	97.5000	+	168.875i	0.883920	+	1.53099i	0.846947	+	0.531678i	$0.178438\pi$
$23$	97.5000	+	168.875i	0.883920	+	1.53099i	0.0369731	+	0.999316i	$0.488228\pi$
$24$		0			0
$25$	22.0000	−	38.1051i	0.176000	−	0.304841i
$26$	74.0000			0.558177
$27$		0			0
$28$	−124.000			−0.836921
$29$	55.5000	−	96.1288i	0.355382	−	0.615540i	−0.631801	−	0.775131i	$-0.717684\pi$
$29$	55.5000	−	96.1288i	0.355382	−	0.615540i	0.987183	+	0.159590i	$0.0510173\pi$
$30$		0			0
$31$	102.500	+	177.535i	0.593856	+	1.02859i	0.993707	+	0.112009i	$0.0357285\pi$
$31$	102.500	+	177.535i	0.593856	+	1.02859i	−0.399851	+	0.916580i	$0.630938\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	42.0000	−	72.7461i	0.211851	−	0.366937i
$35$	−279.000			−1.34742
$36$		0			0
$37$	−166.000			−0.737574			−0.368787	−	0.929514i	$-0.620227\pi$
$37$	−166.000			−0.737574			−0.368787	+	0.929514i	$0.620227\pi$
$38$	−28.0000	+	48.4974i	−0.119532	+	0.207035i
$39$		0			0
$40$	36.0000	+	62.3538i	0.142302	+	0.246475i
$41$	−130.500	−	226.033i	−0.497090	−	0.860985i	0.502905	−	0.864342i	$-0.332265\pi$
$41$	−130.500	−	226.033i	−0.497090	−	0.860985i	−0.999994	+	0.00335732i	$0.998931\pi$
$42$		0			0
$43$	21.5000	−	37.2391i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	21.5000	−	37.2391i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	60.0000			0.205576
$45$		0			0
$46$	390.000			1.25005
$47$	88.5000	−	153.286i	0.274661	−	0.475726i	−0.695389	−	0.718634i	$-0.744768\pi$
$47$	88.5000	−	153.286i	0.274661	−	0.475726i	0.970049	+	0.242907i	$0.0781011\pi$
$48$		0			0
$49$	−309.000	−	535.204i	−0.900875	−	1.56036i
$50$	−44.0000	−	76.2102i	−0.124451	−	0.215555i
$51$		0			0
$52$	74.0000	−	128.172i	0.197345	−	0.341812i
$53$	−114.000			−0.295455			−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000			−0.295455			−0.147727	+	0.989028i	$0.547196\pi$
$54$		0			0
$55$	135.000			0.330971
$56$	−124.000	+	214.774i	−0.295896	+	0.512508i
$57$		0			0
$58$	−111.000	−	192.258i	−0.251293	−	0.435253i
$59$	79.5000	+	137.698i	0.175424	+	0.303843i	0.940308	−	0.340325i	$-0.110537\pi$
$59$	79.5000	+	137.698i	0.175424	+	0.303843i	−0.764884	+	0.644168i	$0.777204\pi$
$60$		0			0
$61$	−95.5000	+	165.411i	−0.200451	+	0.347192i	−0.948674	−	0.316256i	$-0.897574\pi$
$61$	−95.5000	+	165.411i	−0.200451	+	0.347192i	0.748223	+	0.663448i	$0.230907\pi$
$62$	410.000			0.839840
$63$		0			0
$64$	64.0000			0.125000
$65$	166.500	−	288.386i	0.317720	−	0.550307i
$66$		0			0
$67$	210.500	+	364.597i	0.383831	+	0.664815i	0.991606	−	0.129294i	$-0.0412712\pi$
$67$	210.500	+	364.597i	0.383831	+	0.664815i	−0.607775	+	0.794109i	$0.707938\pi$
$68$	−84.0000	−	145.492i	−0.149801	−	0.259464i
$69$		0			0
$70$	−279.000	+	483.242i	−0.476384	+	0.825121i
$71$	−156.000			−0.260758			−0.130379	−	0.991464i	$-0.541619\pi$
$71$	−156.000			−0.260758			−0.130379	+	0.991464i	$0.541619\pi$
$72$		0			0
$73$	182.000			0.291801			0.145901	−	0.989299i	$-0.453392\pi$
$73$	182.000			0.291801			0.145901	+	0.989299i	$0.453392\pi$
$74$	−166.000	+	287.520i	−0.260772	+	0.451670i
$75$		0			0
$76$	56.0000	+	96.9948i	0.0845216	+	0.146396i
$77$	232.500	+	402.702i	0.344102	+	0.596002i
$78$		0			0
$79$	−566.500	+	981.207i	−0.806788	+	1.39740i	0.108290	+	0.994119i	$0.465463\pi$
$79$	−566.500	+	981.207i	−0.806788	+	1.39740i	−0.915078	+	0.403278i	$0.867871\pi$
$80$	144.000			0.201246
$81$		0			0
$82$	−522.000			−0.702991
$83$	−541.500	+	937.906i	−0.716113	+	1.24034i	0.246416	+	0.969164i	$0.420747\pi$
$83$	−541.500	+	937.906i	−0.716113	+	1.24034i	−0.962529	+	0.271179i	$0.912586\pi$
$84$		0			0
$85$	−189.000	−	327.358i	−0.241176	−	0.417728i
$86$	−43.0000	−	74.4782i	−0.0539164	−	0.0933859i
$87$		0			0
$88$	60.0000	−	103.923i	0.0726821	−	0.125889i
$89$	1050.00			1.25056			0.625280	−	0.780401i	$-0.284985\pi$
$89$	1050.00			1.25056			0.625280	+	0.780401i	$0.284985\pi$
$90$		0			0
$91$	1147.00			1.32130
$92$	390.000	−	675.500i	0.441960	−	0.765497i
$93$		0			0
$94$	−177.000	−	306.573i	−0.194214	−	0.336389i
$95$	126.000	+	218.238i	0.136077	+	0.235693i
$96$		0			0
$97$	450.500	−	780.289i	0.471560	−	0.816766i	−0.527910	−	0.849300i	$-0.677024\pi$
$97$	450.500	−	780.289i	0.471560	−	0.816766i	0.999471	+	0.0325338i	$0.0103576\pi$
$98$	−1236.00			−1.27403
$99$		0			0
$100$	−176.000			−0.176000
$101$	193.500	−	335.152i	0.190633	−	0.330187i	−0.754827	−	0.655924i	$-0.772279\pi$
$101$	193.500	−	335.152i	0.190633	−	0.330187i	0.945460	+	0.325737i	$0.105613\pi$
$102$		0			0
$103$	−275.500	−	477.180i	−0.263552	−	0.456485i	0.703631	−	0.710565i	$-0.251561\pi$
$103$	−275.500	−	477.180i	−0.263552	−	0.456485i	−0.967183	+	0.254080i	$0.918227\pi$
$104$	−148.000	−	256.344i	−0.139544	−	0.241698i
$105$		0			0
$106$	−114.000	+	197.454i	−0.104459	+	0.180928i
$107$	12.0000			0.0108419			0.00542095	−	0.999985i	$-0.498274\pi$
$107$	12.0000			0.0108419			0.00542095	+	0.999985i	$0.498274\pi$
$108$		0			0
$109$	−502.000			−0.441127			−0.220564	−	0.975373i	$-0.570790\pi$
$109$	−502.000			−0.441127			−0.220564	+	0.975373i	$0.570790\pi$
$110$	135.000	−	233.827i	0.117016	−	0.202677i
$111$		0			0
$112$	248.000	+	429.549i	0.209230	+	0.362398i
$113$	−700.500	−	1213.30i	−0.583164	−	1.01007i	−0.995102	−	0.0988572i	$-0.968481\pi$
$113$	−700.500	−	1213.30i	−0.583164	−	1.01007i	0.411938	−	0.911212i	$-0.364852\pi$
$114$		0			0
$115$	877.500	−	1519.87i	0.711542	−	1.23243i
$116$	−444.000			−0.355382
$117$		0			0
$118$	318.000			0.248087
$119$	651.000	−	1127.57i	0.501488	−	0.868603i
$120$		0			0
$121$	553.000	+	957.824i	0.415477	+	0.719627i
$122$	191.000	+	330.822i	0.141740	+	0.245502i
$123$		0			0
$124$	410.000	−	710.141i	0.296928	−	0.514295i
$125$	−1521.00			−1.08834
$126$		0			0
$127$	−880.000			−0.614861			−0.307431	−	0.951571i	$-0.599469\pi$
$127$	−880.000			−0.614861			−0.307431	+	0.951571i	$0.599469\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	−333.000	−	576.773i	−0.224662	−	0.389126i
$131$	751.500	+	1301.64i	0.501213	+	0.868126i	0.999999	+	0.00140084i	$0.000445901\pi$
$131$	751.500	+	1301.64i	0.501213	+	0.868126i	−0.498786	+	0.866725i	$0.666221\pi$
$132$		0			0
$133$	−434.000	+	751.710i	−0.282952	+	0.490087i
$134$	842.000			0.542819
$135$		0			0
$136$	−336.000			−0.211851
$137$	−1330.50	+	2304.49i	−0.829725	+	1.43713i	0.0685295	+	0.997649i	$0.478169\pi$
$137$	−1330.50	+	2304.49i	−0.829725	+	1.43713i	−0.898254	+	0.439476i	$0.855164\pi$
$138$		0			0
$139$	60.5000	+	104.789i	0.0369176	+	0.0639431i	0.883894	−	0.467688i	$-0.154913\pi$
$139$	60.5000	+	104.789i	0.0369176	+	0.0639431i	−0.846976	+	0.531631i	$0.821579\pi$
$140$	558.000	+	966.484i	0.336854	+	0.583449i
$141$		0			0
$142$	−156.000	+	270.200i	−0.0921918	+	0.159681i
$143$	−555.000			−0.324555
$144$		0			0
$145$	−999.000			−0.572155
$146$	182.000	−	315.233i	0.103167	−	0.178691i
$147$		0			0
$148$	332.000	+	575.041i	0.184393	+	0.319379i
$149$	−1414.50	−	2449.99i	−0.777721	−	1.34705i	−0.933253	−	0.359221i	$-0.883043\pi$
$149$	−1414.50	−	2449.99i	−0.777721	−	1.34705i	0.155532	−	0.987831i	$-0.450291\pi$
$150$		0			0
$151$	−230.500	+	399.238i	−0.124224	+	0.215162i	−0.921429	−	0.388546i	$-0.872977\pi$
$151$	−230.500	+	399.238i	−0.124224	+	0.215162i	0.797205	+	0.603708i	$0.206311\pi$
$152$	224.000			0.119532
$153$		0			0
$154$	930.000			0.486633
$155$	922.500	−	1597.82i	0.478045	−	0.827998i
$156$		0			0
$157$	1488.50	+	2578.16i	0.756658	+	1.31057i	0.944546	+	0.328379i	$0.106502\pi$
$157$	1488.50	+	2578.16i	0.756658	+	1.31057i	−0.187889	+	0.982190i	$0.560164\pi$
$158$	1133.00	+	1962.41i	0.570485	+	0.988109i
$159$		0			0
$160$	144.000	−	249.415i	0.0711512	−	0.123238i
$161$	6045.00			2.95909
$162$		0			0
$163$	−3316.00			−1.59343			−0.796715	−	0.604355i	$-0.793431\pi$
$163$	−3316.00			−1.59343			−0.796715	+	0.604355i	$0.793431\pi$
$164$	−522.000	+	904.131i	−0.248545	+	0.430492i
$165$		0			0
$166$	1083.00	+	1875.81i	0.506368	+	0.877055i
$167$	−340.500	−	589.763i	−0.157777	−	0.273277i	0.776290	−	0.630376i	$-0.217099\pi$
$167$	−340.500	−	589.763i	−0.157777	−	0.273277i	−0.934067	+	0.357099i	$0.883766\pi$
$168$		0			0
$169$	414.000	−	717.069i	0.188439	−	0.326386i
$170$	−756.000			−0.341074
$171$		0			0
$172$	−172.000			−0.0762493
$173$	−1990.50	+	3447.65i	−0.874768	+	1.51514i	−0.0177589	+	0.999842i	$0.505653\pi$
$173$	−1990.50	+	3447.65i	−0.874768	+	1.51514i	−0.857009	+	0.515301i	$0.827680\pi$
$174$		0			0
$175$	−682.000	−	1181.26i	−0.294596	−	0.510256i
$176$	−120.000	−	207.846i	−0.0513940	−	0.0890170i
$177$		0			0
$178$	1050.00	−	1818.65i	0.442139	−	0.765808i
$179$	−2004.00			−0.836793			−0.418397	−	0.908264i	$-0.637408\pi$
$179$	−2004.00			−0.836793			−0.418397	+	0.908264i	$0.637408\pi$
$180$		0			0
$181$	1274.00			0.523181			0.261590	−	0.965179i	$-0.415753\pi$
$181$	1274.00			0.523181			0.261590	+	0.965179i	$0.415753\pi$
$182$	1147.00	−	1986.66i	0.467150	−	0.809128i
$183$		0			0
$184$	−780.000	−	1351.00i	−0.312513	−	0.541288i
$185$	747.000	+	1293.84i	0.296868	+	0.514190i
$186$		0			0
$187$	−315.000	+	545.596i	−0.123182	+	0.213358i
$188$	−708.000			−0.274661
$189$		0			0
$190$	504.000			0.192442
$191$	580.500	−	1005.46i	0.219914	−	0.380902i	−0.734868	−	0.678210i	$-0.762756\pi$
$191$	580.500	−	1005.46i	0.219914	−	0.380902i	0.954781	+	0.297309i	$0.0960891\pi$
$192$		0			0
$193$	−1805.50	−	3127.22i	−0.673382	−	1.16633i	−0.976939	−	0.213519i	$-0.931508\pi$
$193$	−1805.50	−	3127.22i	−0.673382	−	1.16633i	0.303557	−	0.952813i	$-0.401826\pi$
$194$	−901.000	−	1560.58i	−0.333443	−	0.577541i
$195$		0			0
$196$	−1236.00	+	2140.81i	−0.450437	+	0.780180i
$197$	−2046.00			−0.739957			−0.369978	−	0.929040i	$-0.620635\pi$
$197$	−2046.00			−0.739957			−0.369978	+	0.929040i	$0.620635\pi$
$198$		0			0
$199$	2996.00			1.06724			0.533620	−	0.845724i	$-0.320831\pi$
$199$	2996.00			1.06724			0.533620	+	0.845724i	$0.320831\pi$
$200$	−176.000	+	304.841i	−0.0622254	+	0.107778i
$201$		0			0
$202$	−387.000	−	670.304i	−0.134798	−	0.233477i
$203$	−1720.50	−	2979.99i	−0.594854	−	1.03032i
$204$		0			0
$205$	−1174.50	+	2034.29i	−0.400149	+	0.693079i
$206$	−1102.00			−0.372718
$207$		0			0
$208$	−592.000			−0.197345
$209$	210.000	−	363.731i	0.0695024	−	0.120382i
$210$		0			0
$211$	−377.500	−	653.849i	−0.123167	−	0.213331i	0.797848	−	0.602858i	$-0.205972\pi$
$211$	−377.500	−	653.849i	−0.123167	−	0.213331i	−0.921015	+	0.389528i	$0.872638\pi$
$212$	228.000	+	394.908i	0.0738637	+	0.127936i
$213$		0			0
$214$	12.0000	−	20.7846i	0.00383319	−	0.00663928i
$215$	−387.000			−0.122759
$216$		0			0
$217$	6355.00			1.98804
$218$	−502.000	+	869.490i	−0.155962	+	0.270134i
$219$		0			0
$220$	−270.000	−	467.654i	−0.0827427	−	0.143315i
$221$	777.000	+	1345.80i	0.236501	+	0.409631i
$222$		0			0
$223$	1731.50	−	2999.05i	0.519954	−	0.900587i	−0.479777	−	0.877391i	$-0.659282\pi$
$223$	1731.50	−	2999.05i	0.519954	−	0.900587i	0.999731	−	0.0231966i	$-0.00738438\pi$
$224$	992.000			0.295896
$225$		0			0
$226$	−2802.00			−0.824718
$227$	3112.50	−	5391.01i	0.910061	−	1.57627i	0.0960856	−	0.995373i	$-0.469368\pi$
$227$	3112.50	−	5391.01i	0.910061	−	1.57627i	0.813976	−	0.580899i	$-0.197299\pi$
$228$		0			0
$229$	732.500	+	1268.73i	0.211375	+	0.366113i	0.952145	−	0.305646i	$-0.0988724\pi$
$229$	732.500	+	1268.73i	0.211375	+	0.366113i	−0.740770	+	0.671759i	$0.765539\pi$
$230$	−1755.00	−	3039.75i	−0.503136	−	0.871457i
$231$		0			0
$232$	−444.000	+	769.031i	−0.125647	+	0.217626i
$233$	−2634.00			−0.740597			−0.370298	−	0.928913i	$-0.620745\pi$
$233$	−2634.00			−0.740597			−0.370298	+	0.928913i	$0.620745\pi$
$234$		0			0
$235$	−1593.00			−0.442195
$236$	318.000	−	550.792i	0.0877120	−	0.151922i
$237$		0			0
$238$	−1302.00	−	2255.13i	−0.354606	−	0.614195i
$239$	3457.50	+	5988.57i	0.935762	+	1.62079i	0.773270	+	0.634077i	$0.218620\pi$
$239$	3457.50	+	5988.57i	0.935762	+	1.62079i	0.162492	+	0.986710i	$0.448047\pi$
$240$		0			0
$241$	744.500	−	1289.51i	0.198994	−	0.344667i	−0.749209	−	0.662334i	$-0.769566\pi$
$241$	744.500	−	1289.51i	0.198994	−	0.344667i	0.948202	+	0.317667i	$0.102899\pi$
$242$	2212.00			0.587573
$243$		0			0
$244$	764.000			0.200451
$245$	−2781.00	+	4816.83i	−0.725190	+	1.25607i
$246$		0			0
$247$	−518.000	−	897.202i	−0.133439	−	0.231124i
$248$	−820.000	−	1420.28i	−0.209960	−	0.363661i
$249$		0			0
$250$	−1521.00	+	2634.45i	−0.384786	+	0.666469i
$251$	4620.00			1.16180			0.580900	−	0.813975i	$-0.302701\pi$
$251$	4620.00			1.16180			0.580900	+	0.813975i	$0.302701\pi$
$252$		0			0
$253$	−2925.00			−0.726850
$254$	−880.000	+	1524.20i	−0.217386	+	0.376524i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1675.50	+	2902.05i	0.406672	+	0.704377i	0.994515	−	0.104598i	$-0.0333557\pi$
$257$	1675.50	+	2902.05i	0.406672	+	0.704377i	−0.587842	+	0.808976i	$0.700022\pi$
$258$		0			0
$259$	−2573.00	+	4456.57i	−0.617291	+	1.06918i
$260$	−1332.00			−0.317720
$261$		0			0
$262$	3006.00			0.708822
$263$	−301.500	+	522.213i	−0.0706893	+	0.122437i	−0.899204	−	0.437530i	$-0.855853\pi$
$263$	−301.500	+	522.213i	−0.0706893	+	0.122437i	0.828514	+	0.559968i	$0.189187\pi$
$264$		0			0
$265$	513.000	+	888.542i	0.118918	+	0.205972i
$266$	868.000	+	1503.42i	0.200077	+	0.346544i
$267$		0			0
$268$	842.000	−	1458.39i	0.191915	−	0.332407i
$269$	1470.00			0.333188			0.166594	−	0.986026i	$-0.446723\pi$
$269$	1470.00			0.333188			0.166594	+	0.986026i	$0.446723\pi$
$270$		0			0
$271$	2072.00			0.464447			0.232223	−	0.972662i	$-0.425400\pi$
$271$	2072.00			0.464447			0.232223	+	0.972662i	$0.425400\pi$
$272$	−336.000	+	581.969i	−0.0749007	+	0.129732i
$273$		0			0
$274$	2661.00	+	4608.99i	0.586704	+	1.01620i
$275$	330.000	+	571.577i	0.0723627	+	0.125336i
$276$		0			0
$277$	−3569.50	+	6182.56i	−0.774262	+	1.34106i	0.160947	+	0.986963i	$0.448545\pi$
$277$	−3569.50	+	6182.56i	−0.774262	+	1.34106i	−0.935209	+	0.354097i	$0.884788\pi$
$278$	242.000			0.0522093
$279$		0			0
$280$	2232.00			0.476384
$281$	4213.50	−	7298.00i	0.894507	−	1.54933i	0.0600924	−	0.998193i	$-0.480860\pi$
$281$	4213.50	−	7298.00i	0.894507	−	1.54933i	0.834414	−	0.551138i	$-0.185806\pi$
$282$		0			0
$283$	228.500	+	395.774i	0.0479962	+	0.0831318i	0.889025	−	0.457858i	$-0.151383\pi$
$283$	228.500	+	395.774i	0.0479962	+	0.0831318i	−0.841029	+	0.540990i	$0.818050\pi$
$284$	312.000	+	540.400i	0.0651894	+	0.112911i
$285$		0			0
$286$	−555.000	+	961.288i	−0.114748	+	0.198749i
$287$	−8091.00			−1.66410
$288$		0			0
$289$	−3149.00			−0.640953
$290$	−999.000	+	1730.32i	−0.202287	+	0.350372i
$291$		0			0
$292$	−364.000	−	630.466i	−0.0729503	−	0.126354i
$293$	−2944.50	−	5100.02i	−0.587097	−	1.01688i	−0.994610	−	0.103683i	$-0.966937\pi$
$293$	−2944.50	−	5100.02i	−0.587097	−	1.01688i	0.407513	−	0.913199i	$-0.366396\pi$
$294$		0			0
$295$	715.500	−	1239.28i	0.141214	−	0.244589i
$296$	1328.00			0.260772
$297$		0			0
$298$	−5658.00			−1.09986
$299$	−3607.50	+	6248.37i	−0.697750	+	1.20854i
$300$		0			0
$301$	−666.500	−	1154.41i	−0.127629	−	0.221060i
$302$	461.000	+	798.475i	0.0878396	+	0.152143i
$303$		0			0
$304$	224.000	−	387.979i	0.0422608	−	0.0731978i
$305$	1719.00			0.322720
$306$		0			0
$307$	−1204.00			−0.223830			−0.111915	−	0.993718i	$-0.535698\pi$
$307$	−1204.00			−0.223830			−0.111915	+	0.993718i	$0.535698\pi$
$308$	930.000	−	1610.81i	0.172051	−	0.298001i
$309$		0			0
$310$	−1845.00	−	3195.63i	−0.338029	−	0.585483i
$311$	−1642.50	−	2844.89i	−0.299478	−	0.518711i	0.676539	−	0.736407i	$-0.263479\pi$
$311$	−1642.50	−	2844.89i	−0.299478	−	0.518711i	−0.976017	+	0.217696i	$0.930146\pi$
$312$		0			0
$313$	5028.50	−	8709.62i	0.908075	−	1.57283i	0.0913406	−	0.995820i	$-0.470885\pi$
$313$	5028.50	−	8709.62i	0.908075	−	1.57283i	0.816735	−	0.577013i	$-0.195782\pi$
$314$	5954.00			1.07008
$315$		0			0
$316$	4532.00			0.806788
$317$	1147.50	−	1987.53i	0.203312	−	0.352147i	−0.746281	−	0.665631i	$-0.768163\pi$
$317$	1147.50	−	1987.53i	0.203312	−	0.352147i	0.949594	+	0.313483i	$0.101496\pi$
$318$		0			0
$319$	832.500	+	1441.93i	0.146116	+	0.253081i
$320$	−288.000	−	498.831i	−0.0503115	−	0.0871421i
$321$		0			0
$322$	6045.00	−	10470.2i	1.04619	−	1.81206i
$323$	−1176.00			−0.202583
$324$		0			0
$325$	1628.00			0.277862
$326$	−3316.00	+	5743.48i	−0.563363	+	0.975773i
$327$		0			0
$328$	1044.00	+	1808.26i	0.175748	+	0.304404i
$329$	−2743.50	−	4751.88i	−0.459739	−	0.796291i
$330$		0			0
$331$	3339.50	−	5784.18i	0.554548	−	0.960506i	−0.443390	−	0.896329i	$-0.646224\pi$
$331$	3339.50	−	5784.18i	0.554548	−	0.960506i	0.997939	−	0.0641773i	$-0.0204423\pi$
$332$	4332.00			0.716113
$333$		0			0
$334$	−1362.00			−0.223130
$335$	1894.50	−	3281.37i	0.308978	−	0.535165i
$336$		0			0
$337$	−1091.50	−	1890.53i	−0.176433	−	0.305590i	0.764224	−	0.644951i	$-0.223122\pi$
$337$	−1091.50	−	1890.53i	−0.176433	−	0.305590i	−0.940656	+	0.339361i	$0.889789\pi$
$338$	−828.000	−	1434.14i	−0.133246	−	0.230789i
$339$		0			0
$340$	−756.000	+	1309.43i	−0.120588	+	0.208864i
$341$	−3075.00			−0.488330
$342$		0			0
$343$	−8525.00			−1.34200
$344$	−172.000	+	297.913i	−0.0269582	+	0.0466930i
$345$		0			0
$346$	3981.00	+	6895.29i	0.618555	+	1.07137i
$347$	1945.50	+	3369.70i	0.300980	+	0.521312i	0.976358	−	0.216159i	$-0.0693531\pi$
$347$	1945.50	+	3369.70i	0.300980	+	0.521312i	−0.675379	+	0.737471i	$0.736020\pi$
$348$		0			0
$349$	−1397.50	+	2420.54i	−0.214345	+	0.371257i	−0.953070	−	0.302751i	$-0.902095\pi$
$349$	−1397.50	+	2420.54i	−0.214345	+	0.371257i	0.738725	+	0.674007i	$0.235428\pi$
$350$	−2728.00			−0.416622
$351$		0			0
$352$	−480.000			−0.0726821
$353$	2377.50	−	4117.95i	0.358475	−	0.620896i	−0.629232	−	0.777218i	$-0.716630\pi$
$353$	2377.50	−	4117.95i	0.358475	−	0.620896i	0.987706	+	0.156322i	$0.0499636\pi$
$354$		0			0
$355$	702.000	+	1215.90i	0.104953	+	0.181784i
$356$	−2100.00	−	3637.31i	−0.312640	−	0.541508i
$357$		0			0
$358$	−2004.00	+	3471.03i	−0.295851	+	0.512429i
$359$	−4608.00			−0.677440			−0.338720	−	0.940887i	$-0.609994\pi$
$359$	−4608.00			−0.677440			−0.338720	+	0.940887i	$0.609994\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$	1274.00	−	2206.63i	0.184972	−	0.320381i
$363$		0			0
$364$	−2294.00	−	3973.32i	−0.330325	−	0.572140i
$365$	−819.000	−	1418.55i	−0.117448	−	0.203425i
$366$		0			0
$367$	−1922.50	+	3329.87i	−0.273443	+	0.473618i	−0.969741	−	0.244135i	$-0.921496\pi$
$367$	−1922.50	+	3329.87i	−0.273443	+	0.473618i	0.696298	+	0.717753i	$0.254829\pi$
$368$	−3120.00			−0.441960
$369$		0			0
$370$	2988.00			0.419834
$371$	−1767.00	+	3060.53i	−0.247272	+	0.428288i
$372$		0			0
$373$	4158.50	+	7202.73i	0.577263	+	0.999848i	0.995792	+	0.0916449i	$0.0292125\pi$
$373$	4158.50	+	7202.73i	0.577263	+	0.999848i	−0.418529	+	0.908203i	$0.637454\pi$
$374$	630.000	+	1091.19i	0.0871030	+	0.150867i
$375$		0			0
$376$	−708.000	+	1226.29i	−0.0971072	+	0.168195i
$377$	4107.00			0.561064
$378$		0			0
$379$	12560.0			1.70228			0.851140	−	0.524939i	$-0.175912\pi$
$379$	12560.0			1.70228			0.851140	+	0.524939i	$0.175912\pi$
$380$	504.000	−	872.954i	0.0680386	−	0.117846i
$381$		0			0
$382$	−1161.00	−	2010.91i	−0.155502	−	0.269338i
$383$	6043.50	+	10467.6i	0.806288	+	1.39653i	0.915418	+	0.402505i	$0.131860\pi$
$383$	6043.50	+	10467.6i	0.806288	+	1.39653i	−0.109130	+	0.994028i	$0.534806\pi$
$384$		0			0
$385$	2092.50	−	3624.32i	0.276997	−	0.479772i
$386$	−7222.00			−0.952306
$387$		0			0
$388$	−3604.00			−0.471560
$389$	−4270.50	+	7396.72i	−0.556614	+	0.964084i	0.441162	+	0.897428i	$0.354567\pi$
$389$	−4270.50	+	7396.72i	−0.556614	+	0.964084i	−0.997776	+	0.0666565i	$0.978767\pi$
$390$		0			0
$391$	4095.00	+	7092.75i	0.529650	+	0.917380i
$392$	2472.00	+	4281.63i	0.318507	+	0.551671i
$393$		0			0
$394$	−2046.00	+	3543.78i	−0.261614	+	0.453129i
$395$	10197.0			1.29890
$396$		0			0
$397$	−13174.0			−1.66545			−0.832726	−	0.553686i	$-0.813221\pi$
$397$	−13174.0			−1.66545			−0.832726	+	0.553686i	$0.813221\pi$
$398$	2996.00	−	5189.22i	0.377326	−	0.653549i
$399$		0			0
$400$	352.000	+	609.682i	0.0440000	+	0.0762102i
$401$	4801.50	+	8316.44i	0.597944	+	1.03567i	0.993124	+	0.117066i	$0.0373488\pi$
$401$	4801.50	+	8316.44i	0.597944	+	1.03567i	−0.395180	+	0.918604i	$0.629318\pi$
$402$		0			0
$403$	−3792.50	+	6568.80i	−0.468779	+	0.811949i
$404$	−1548.00			−0.190633
$405$		0			0
$406$	−6882.00			−0.841251
$407$	1245.00	−	2156.40i	0.151627	−	0.262626i
$408$		0			0
$409$	−5735.50	−	9934.18i	−0.693404	−	1.20101i	−0.970716	−	0.240231i	$-0.922777\pi$
$409$	−5735.50	−	9934.18i	−0.693404	−	1.20101i	0.277312	−	0.960780i	$-0.410557\pi$
$410$	2349.00	+	4068.59i	0.282948	+	0.490081i
$411$		0			0
$412$	−1102.00	+	1908.72i	−0.131776	+	0.228242i
$413$	4929.00			0.587264
$414$		0			0
$415$	9747.00			1.15292
$416$	−592.000	+	1025.37i	−0.0697721	+	0.120849i
$417$		0			0
$418$	−420.000	−	727.461i	−0.0491456	−	0.0851227i
$419$	−2986.50	−	5172.77i	−0.348210	−	0.603118i	0.637721	−	0.770267i	$-0.279877\pi$
$419$	−2986.50	−	5172.77i	−0.348210	−	0.603118i	−0.985932	+	0.167149i	$0.946544\pi$
$420$		0			0
$421$	4452.50	−	7711.96i	0.515443	−	0.892774i	−0.484396	−	0.874849i	$-0.660961\pi$
$421$	4452.50	−	7711.96i	0.515443	−	0.892774i	0.999839	−	0.0179250i	$-0.00570601\pi$
$422$	−1510.00			−0.174184
$423$		0			0
$424$	912.000			0.104459
$425$	924.000	−	1600.41i	0.105460	−	0.182662i
$426$		0			0
$427$	2960.50	+	5127.74i	0.335524	+	0.581144i
$428$	−24.0000	−	41.5692i	−0.00271048	−	0.00469468i
$429$		0			0
$430$	−387.000	+	670.304i	−0.0434019	+	0.0751742i
$431$	−1416.00			−0.158251			−0.0791257	−	0.996865i	$-0.525213\pi$
$431$	−1416.00			−0.158251			−0.0791257	+	0.996865i	$0.525213\pi$
$432$		0			0
$433$	10766.0			1.19488			0.597438	−	0.801915i	$-0.296186\pi$
$433$	10766.0			1.19488			0.597438	+	0.801915i	$0.296186\pi$
$434$	6355.00	−	11007.2i	0.702880	−	1.21742i
$435$		0			0
$436$	1004.00	+	1738.98i	0.110282	+	0.191014i
$437$	−2730.00	−	4728.50i	−0.298841	−	0.517608i
$438$		0			0
$439$	−2174.50	+	3766.34i	−0.236408	+	0.409471i	−0.959681	−	0.281091i	$-0.909304\pi$
$439$	−2174.50	+	3766.34i	−0.236408	+	0.409471i	0.723273	+	0.690562i	$0.242637\pi$
$440$	−1080.00			−0.117016
$441$		0			0
$442$	3108.00			0.334463
$443$	−7273.50	+	12598.1i	−0.780078	+	1.35113i	0.151818	+	0.988408i	$0.451487\pi$
$443$	−7273.50	+	12598.1i	−0.780078	+	1.35113i	−0.931896	+	0.362726i	$0.881846\pi$
$444$		0			0
$445$	−4725.00	−	8183.94i	−0.503340	−	0.871811i
$446$	−3463.00	−	5998.09i	−0.367663	−	0.636811i
$447$		0			0
$448$	992.000	−	1718.19i	0.104615	−	0.181199i
$449$	3330.00			0.350005			0.175003	−	0.984568i	$-0.444007\pi$
$449$	3330.00			0.350005			0.175003	+	0.984568i	$0.444007\pi$
$450$		0			0
$451$	3915.00			0.408759
$452$	−2802.00	+	4853.21i	−0.291582	+	0.505035i
$453$		0			0
$454$	−6225.00	−	10782.0i	−0.643510	−	1.11459i
$455$	−5161.50	−	8939.98i	−0.531813	−	0.921127i
$456$		0			0
$457$	−4073.50	+	7055.51i	−0.416959	+	0.722194i	−0.995632	−	0.0933655i	$-0.970237\pi$
$457$	−4073.50	+	7055.51i	−0.416959	+	0.722194i	0.578673	+	0.815560i	$0.303571\pi$
$458$	2930.00			0.298930
$459$		0			0
$460$	−7020.00			−0.711542
$461$	4015.50	−	6955.05i	0.405684	−	0.702666i	−0.588717	−	0.808340i	$-0.700367\pi$
$461$	4015.50	−	6955.05i	0.405684	−	0.702666i	0.994401	+	0.105674i	$0.0336999\pi$
$462$		0			0
$463$	−2141.50	−	3709.19i	−0.214955	−	0.372312i	0.738304	−	0.674468i	$-0.235627\pi$
$463$	−2141.50	−	3709.19i	−0.214955	−	0.372312i	−0.953259	+	0.302156i	$0.902294\pi$
$464$	888.000	+	1538.06i	0.0888456	+	0.153885i
$465$		0			0
$466$	−2634.00	+	4562.22i	−0.261841	+	0.453521i
$467$	−5460.00			−0.541025			−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00			−0.541025			−0.270512	+	0.962716i	$0.587193\pi$
$468$		0			0
$469$	13051.0			1.28494
$470$	−1593.00	+	2759.16i	−0.156340	+	0.270788i
$471$		0			0
$472$	−636.000	−	1101.58i	−0.0620218	−	0.107425i
$473$	322.500	+	558.586i	0.0313500	+	0.0542999i
$474$		0			0
$475$	−616.000	+	1066.94i	−0.0595032	+	0.103063i
$476$	−5208.00			−0.501488
$477$		0			0
$478$	13830.0			1.32337
$479$	214.500	−	371.525i	0.0204609	−	0.0354393i	−0.855614	−	0.517615i	$-0.826820\pi$
$479$	214.500	−	371.525i	0.0204609	−	0.0354393i	0.876075	+	0.482176i	$0.160153\pi$
$480$		0			0
$481$	−3071.00	−	5319.13i	−0.291113	−	0.504223i
$482$	−1489.00	−	2579.02i	−0.140710	−	0.243716i
$483$		0			0
$484$	2212.00	−	3831.30i	0.207739	−	0.359814i
$485$	−8109.00			−0.759197
$486$		0			0
$487$	−11296.0			−1.05107			−0.525535	−	0.850772i	$-0.676135\pi$
$487$	−11296.0			−1.05107			−0.525535	+	0.850772i	$0.676135\pi$
$488$	764.000	−	1323.29i	0.0708702	−	0.122751i
$489$		0			0
$490$	5562.00	+	9633.67i	0.512787	+	0.888173i
$491$	−7336.50	−	12707.2i	−0.674321	−	1.16796i	−0.976667	−	0.214760i	$-0.931103\pi$
$491$	−7336.50	−	12707.2i	−0.674321	−	1.16796i	0.302346	−	0.953198i	$-0.402230\pi$
$492$		0			0
$493$	2331.00	−	4037.41i	0.212947	−	0.368835i
$494$	−2072.00			−0.188712
$495$		0			0
$496$	−3280.00			−0.296928
$497$	−2418.00	+	4188.10i	−0.218234	+	0.377992i
$498$		0			0
$499$	−6719.50	−	11638.5i	−0.602818	−	1.04411i	−0.992392	−	0.123116i	$-0.960711\pi$
$499$	−6719.50	−	11638.5i	−0.602818	−	1.04411i	0.389574	−	0.920995i	$-0.372622\pi$
$500$	3042.00	+	5268.90i	0.272085	+	0.471265i
$501$		0			0
$502$	4620.00	−	8002.07i	0.410758	−	0.711454i
$503$	17388.0			1.54134			0.770669	−	0.637236i	$-0.219922\pi$
$503$	17388.0			1.54134			0.770669	+	0.637236i	$0.219922\pi$
$504$		0			0
$505$	−3483.00			−0.306914
$506$	−2925.00	+	5066.25i	−0.256980	+	0.445103i
$507$		0			0
$508$	1760.00	+	3048.41i	0.153715	+	0.266243i
$509$	−1894.50	−	3281.37i	−0.164975	−	0.285745i	0.771671	−	0.636021i	$-0.219421\pi$
$509$	−1894.50	−	3281.37i	−0.164975	−	0.285745i	−0.936646	+	0.350276i	$0.886088\pi$
$510$		0			0
$511$	2821.00	−	4886.12i	0.244215	−	0.422992i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	6702.00			0.575122
$515$	−2479.50	+	4294.62i	−0.212155	+	0.367463i
$516$		0			0
$517$	1327.50	+	2299.30i	0.112927	+	0.195596i
$518$	5146.00	+	8913.13i	0.436491	+	0.756024i
$519$		0			0
$520$	−1332.00	+	2307.09i	−0.112331	+	0.194563i
$521$	−9786.00			−0.822903			−0.411451	−	0.911432i	$-0.634978\pi$
$521$	−9786.00			−0.822903			−0.411451	+	0.911432i	$0.634978\pi$
$522$		0			0
$523$	−8008.00			−0.669532			−0.334766	−	0.942301i	$-0.608657\pi$
$523$	−8008.00			−0.669532			−0.334766	+	0.942301i	$0.608657\pi$
$524$	3006.00	−	5206.54i	0.250606	−	0.434063i
$525$		0			0
$526$	603.000	+	1044.43i	0.0499849	+	0.0865764i
$527$	4305.00	+	7456.48i	0.355842	+	0.616336i
$528$		0			0
$529$	−12929.0	+	22393.7i	−1.06263	+	1.84053i
$530$	2052.00			0.168176
$531$		0			0
$532$	3472.00			0.282952
$533$	4828.50	−	8363.21i	0.392393	−	0.679645i
$534$		0			0
$535$	−54.0000	−	93.5307i	−0.00436378	−	0.00755829i
$536$	−1684.00	−	2916.77i	−0.135705	−	0.235047i
$537$		0			0
$538$	1470.00	−	2546.11i	0.117800	−	0.204035i
$539$	9270.00			0.740793
$540$		0			0
$541$	−2938.00			−0.233483			−0.116742	−	0.993162i	$-0.537245\pi$
$541$	−2938.00			−0.233483			−0.116742	+	0.993162i	$0.537245\pi$
$542$	2072.00	−	3588.81i	0.164207	−	0.284414i
$543$		0			0
$544$	672.000	+	1163.94i	0.0529628	+	0.0917343i
$545$	2259.00	+	3912.70i	0.177550	+	0.307526i
$546$		0			0
$547$	5187.50	−	8985.01i	0.405487	−	0.702324i	−0.588891	−	0.808213i	$-0.700435\pi$
$547$	5187.50	−	8985.01i	0.405487	−	0.702324i	0.994378	+	0.105888i	$0.0337686\pi$
$548$	10644.0			0.829725
$549$		0			0
$550$	1320.00			0.102336
$551$	−1554.00	+	2691.61i	−0.120150	+	0.208106i
$552$		0			0
$553$	17561.5	+	30417.4i	1.35044	+	2.33902i
$554$	7139.00	+	12365.1i	0.547486	+	0.948273i
$555$		0			0
$556$	242.000	−	419.156i	0.0184588	−	0.0319716i
$557$	−3306.00			−0.251490			−0.125745	−	0.992063i	$-0.540132\pi$
$557$	−3306.00			−0.251490			−0.125745	+	0.992063i	$0.540132\pi$
$558$		0			0
$559$	1591.00			0.120379
$560$	2232.00	−	3865.94i	0.168427	−	0.291724i
$561$		0			0
$562$	−8427.00	−	14596.0i	−0.632512	−	1.09554i
$563$	−10546.5	−	18267.1i	−0.789488	−	1.36743i	−0.926281	−	0.376834i	$-0.877013\pi$
$563$	−10546.5	−	18267.1i	−0.789488	−	1.36743i	0.136792	−	0.990600i	$-0.456321\pi$
$564$		0			0
$565$	−6304.50	+	10919.7i	−0.469438	+	0.813090i
$566$	914.000			0.0678768
$567$		0			0
$568$	1248.00			0.0921918
$569$	643.500	−	1114.57i	0.0474111	−	0.0821185i	−0.841346	−	0.540497i	$-0.818236\pi$
$569$	643.500	−	1114.57i	0.0474111	−	0.0821185i	0.888757	+	0.458379i	$0.151570\pi$
$570$		0			0
$571$	−7517.50	−	13020.7i	−0.550959	−	0.954289i	−0.998206	−	0.0598783i	$-0.980929\pi$
$571$	−7517.50	−	13020.7i	−0.550959	−	0.954289i	0.447247	−	0.894411i	$-0.352405\pi$
$572$	1110.00	+	1922.58i	0.0811389	+	0.140537i
$573$		0			0
$574$	−8091.00	+	14014.0i	−0.588348	+	1.01905i
$575$	8580.00			0.622280
$576$		0			0
$577$	1190.00			0.0858585			0.0429292	−	0.999078i	$-0.486331\pi$
$577$	1190.00			0.0858585			0.0429292	+	0.999078i	$0.486331\pi$
$578$	−3149.00	+	5454.23i	−0.226611	+	0.392502i
$579$		0			0
$580$	1998.00	+	3460.64i	0.143039	+	0.247750i
$581$	16786.5	+	29075.1i	1.19866	+	2.07614i
$582$		0			0
$583$	855.000	−	1480.90i	0.0607384	−	0.105202i
$584$	−1456.00			−0.103167
$585$		0			0
$586$	−11778.0			−0.830281
$587$	−8941.50	+	15487.1i	−0.628714	+	1.08896i	0.359096	+	0.933301i	$0.383085\pi$
$587$	−8941.50	+	15487.1i	−0.628714	+	1.08896i	−0.987810	+	0.155664i	$0.950248\pi$
$588$		0			0
$589$	−2870.00	−	4970.99i	−0.200775	−	0.347752i
$590$	−1431.00	−	2478.56i	−0.0998531	−	0.172951i
$591$		0			0
$592$	1328.00	−	2300.16i	0.0921967	−	0.159689i
$593$	−20118.0			−1.39317			−0.696583	−	0.717476i	$-0.745297\pi$
$593$	−20118.0			−1.39317			−0.696583	+	0.717476i	$0.745297\pi$
$594$		0			0
$595$	−11718.0			−0.807380
$596$	−5658.00	+	9799.94i	−0.388860	+	0.673526i
$597$		0			0
$598$	7215.00	+	12496.7i	0.493383	+	0.854565i
$599$	−532.500	−	922.317i	−0.0363228	−	0.0629129i	0.847293	−	0.531127i	$-0.178231\pi$
$599$	−532.500	−	922.317i	−0.0363228	−	0.0629129i	−0.883615	+	0.468214i	$0.844898\pi$
$600$		0			0
$601$	10362.5	−	17948.4i	0.703320	−	1.21819i	−0.263975	−	0.964530i	$-0.585034\pi$
$601$	10362.5	−	17948.4i	0.703320	−	1.21819i	0.967294	−	0.253656i	$-0.0816331\pi$
$602$	−2666.00			−0.180495
$603$		0			0
$604$	1844.00			0.124224
$605$	4977.00	−	8620.42i	0.334453	−	0.579289i
$606$		0			0
$607$	7872.50	+	13635.6i	0.526417	+	0.911780i	0.999526	+	0.0307768i	$0.00979812\pi$
$607$	7872.50	+	13635.6i	0.526417	+	0.911780i	−0.473110	+	0.881004i	$0.656869\pi$
$608$	−448.000	−	775.959i	−0.0298829	−	0.0517587i
$609$		0			0
$610$	1719.00	−	2977.40i	0.114099	−	0.197625i
$611$	6549.00			0.433624
$612$		0			0
$613$	5042.00			0.332210			0.166105	−	0.986108i	$-0.446881\pi$
$613$	5042.00			0.332210			0.166105	+	0.986108i	$0.446881\pi$
$614$	−1204.00	+	2085.39i	−0.0791360	+	0.137068i
$615$		0			0
$616$	−1860.00	−	3221.61i	−0.121658	−	0.210718i
$617$	−5026.50	−	8706.15i	−0.327973	−	0.568066i	0.654137	−	0.756376i	$-0.273032\pi$
$617$	−5026.50	−	8706.15i	−0.327973	−	0.568066i	−0.982110	+	0.188311i	$0.939699\pi$
$618$		0			0
$619$	2991.50	−	5181.43i	0.194246	−	0.336445i	−0.752407	−	0.658699i	$-0.771107\pi$
$619$	2991.50	−	5181.43i	0.194246	−	0.336445i	0.946653	+	0.322254i	$0.104441\pi$
$620$	−7380.00			−0.478045
$621$		0			0
$622$	−6570.00			−0.423526
$623$	16275.0	−	28189.1i	1.04662	−	1.81280i
$624$		0			0
$625$	4094.50	+	7091.88i	0.262048	+	0.453880i
$626$	−10057.0	−	17419.2i	−0.642106	−	1.11216i
$627$		0			0
$628$	5954.00	−	10312.6i	0.378329	−	0.655285i
$629$	−6972.00			−0.441958
$630$		0			0
$631$	−19696.0			−1.24261			−0.621304	−	0.783570i	$-0.713397\pi$
$631$	−19696.0			−1.24261			−0.621304	+	0.783570i	$0.713397\pi$
$632$	4532.00	−	7849.65i	0.285243	−	0.494055i
$633$		0			0
$634$	−2295.00	−	3975.06i	−0.143764	−	0.249006i
$635$	3960.00	+	6858.92i	0.247477	+	0.428642i
$636$		0			0
$637$	11433.0	−	19802.5i	0.711133	−	1.23172i
$638$	3330.00			0.206639
$639$		0			0
$640$	−1152.00			−0.0711512
$641$	−5488.50	+	9506.36i	−0.338195	+	0.585770i	−0.984093	−	0.177653i	$-0.943150\pi$
$641$	−5488.50	+	9506.36i	−0.338195	+	0.585770i	0.645899	+	0.763423i	$0.276483\pi$
$642$		0			0
$643$	7914.50	+	13708.3i	0.485408	+	0.840752i	0.999859	−	0.0167681i	$-0.00533770\pi$
$643$	7914.50	+	13708.3i	0.485408	+	0.840752i	−0.514451	+	0.857520i	$0.672004\pi$
$644$	−12090.0	−	20940.5i	−0.739771	−	1.28132i
$645$		0			0
$646$	−1176.00	+	2036.89i	−0.0716240	+	0.124056i
$647$	−28224.0			−1.71499			−0.857496	−	0.514490i	$-0.827981\pi$
$647$	−28224.0			−1.71499			−0.857496	+	0.514490i	$0.827981\pi$
$648$		0			0
$649$	−2385.00			−0.144252
$650$	1628.00	−	2819.78i	0.0982391	−	0.170155i
$651$		0			0
$652$	6632.00	+	11487.0i	0.398358	+	0.689976i
$653$	14083.5	+	24393.3i	0.843997	+	1.46185i	0.886490	+	0.462749i	$0.153137\pi$
$653$	14083.5	+	24393.3i	0.843997	+	1.46185i	−0.0424927	+	0.999097i	$0.513530\pi$
$654$		0			0
$655$	6763.50	−	11714.7i	0.403468	−	0.698828i
$656$	4176.00			0.248545
$657$		0			0
$658$	−10974.0			−0.650169
$659$	5368.50	−	9298.51i	0.317340	−	0.549649i	−0.662592	−	0.748980i	$-0.730544\pi$
$659$	5368.50	−	9298.51i	0.317340	−	0.549649i	0.979932	+	0.199331i	$0.0638770\pi$
$660$		0			0
$661$	−5063.50	−	8770.24i	−0.297954	−	0.516071i	0.677714	−	0.735326i	$-0.262971\pi$
$661$	−5063.50	−	8770.24i	−0.297954	−	0.516071i	−0.975668	+	0.219255i	$0.929637\pi$
$662$	−6679.00	−	11568.4i	−0.392125	−	0.679180i
$663$		0			0
$664$	4332.00	−	7503.24i	0.253184	−	0.438528i
$665$	7812.00			0.455543
$666$		0			0
$667$	21645.0			1.25652
$668$	−1362.00	+	2359.05i	−0.0788883	+	0.136638i
$669$		0			0
$670$	−3789.00	−	6562.74i	−0.218480	−	0.378419i
$671$	−1432.50	−	2481.16i	−0.0824159	−	0.142748i
$672$		0			0
$673$	−125.500	+	217.372i	−0.00718822	+	0.0124504i	−0.869597	−	0.493762i	$-0.835621\pi$
$673$	−125.500	+	217.372i	−0.00718822	+	0.0124504i	0.862409	+	0.506212i	$0.168955\pi$
$674$	−4366.00			−0.249513
$675$		0			0
$676$	−3312.00			−0.188439
$677$	4225.50	−	7318.78i	0.239881	−	0.415485i	−0.720799	−	0.693144i	$-0.756225\pi$
$677$	4225.50	−	7318.78i	0.239881	−	0.415485i	0.960680	+	0.277659i	$0.0895584\pi$
$678$		0			0
$679$	−13965.5	−	24189.0i	−0.789318	−	1.36714i
$680$	1512.00	+	2618.86i	0.0852685	+	0.147689i
$681$		0			0
$682$	−3075.00	+	5326.06i	−0.172651	+	0.299040i
$683$	25884.0			1.45011			0.725054	−	0.688692i	$-0.241815\pi$
$683$	25884.0			1.45011			0.725054	+	0.688692i	$0.241815\pi$
$684$		0			0
$685$	23949.0			1.33583
$686$	−8525.00	+	14765.7i	−0.474469	+	0.821805i
$687$		0			0
$688$	344.000	+	595.825i	0.0190623	+	0.0330169i
$689$	−2109.00	−	3652.90i	−0.116613	−	0.201980i
$690$		0			0
$691$	−3182.50	+	5512.25i	−0.175207	+	0.303467i	−0.940233	−	0.340532i	$-0.889393\pi$
$691$	−3182.50	+	5512.25i	−0.175207	+	0.303467i	0.765026	+	0.643999i	$0.222726\pi$
$692$	15924.0			0.874768
$693$		0			0
$694$	7782.00			0.425649
$695$	544.500	−	943.102i	0.0297181	−	0.0514732i
$696$		0			0
$697$	−5481.00	−	9493.37i	−0.297859	−	0.515907i
$698$	2795.00	+	4841.08i	0.151565	+	0.262518i
$699$		0			0
$700$	−2728.00	+	4725.03i	−0.147298	+	0.255128i
$701$	−1122.00			−0.0604527			−0.0302264	−	0.999543i	$-0.509623\pi$
$701$	−1122.00			−0.0604527			−0.0302264	+	0.999543i	$0.509623\pi$
$702$		0			0
$703$	4648.00			0.249364
$704$	−480.000	+	831.384i	−0.0256970	+	0.0445085i
$705$		0			0
$706$	−4755.00	−	8235.90i	−0.253480	−	0.439040i
$707$	−5998.50	−	10389.7i	−0.319090	−	0.552681i
$708$		0			0
$709$	−2141.50	+	3709.19i	−0.113435	+	0.196476i	−0.917153	−	0.398535i	$-0.869519\pi$
$709$	−2141.50	+	3709.19i	−0.113435	+	0.196476i	0.803718	+	0.595011i	$0.202852\pi$
$710$	2808.00			0.148426
$711$		0			0
$712$	−8400.00			−0.442139
$713$	−19987.5	+	34619.4i	−1.04984	+	1.81838i
$714$		0			0
$715$	2497.50	+	4325.80i	0.130631	+	0.226260i
$716$	4008.00	+	6942.06i	0.209198	+	0.362342i
$717$		0			0
$718$	−4608.00	+	7981.29i	−0.239511	+	0.414846i
$719$	−4032.00			−0.209135			−0.104568	−	0.994518i	$-0.533346\pi$
$719$	−4032.00			−0.209135			−0.104568	+	0.994518i	$0.533346\pi$
$720$		0			0
$721$	−17081.0			−0.882288
$722$	−6075.00	+	10522.2i	−0.313141	+	0.542377i
$723$		0			0
$724$	−2548.00	−	4413.27i	−0.130795	−	0.226544i
$725$	−2442.00	−	4229.67i	−0.125095	−	0.216670i
$726$		0			0
$727$	−12002.5	+	20788.9i	−0.612308	+	1.06055i	0.378542	+	0.925584i	$0.376426\pi$
$727$	−12002.5	+	20788.9i	−0.612308	+	1.06055i	−0.990850	+	0.134965i	$0.956908\pi$
$728$	−9176.00			−0.467150
$729$		0			0
$730$	−3276.00			−0.166096
$731$	903.000	−	1564.04i	0.0456890	−	0.0791357i
$732$		0			0
$733$	18750.5	+	32476.8i	0.944837	+	1.63651i	0.756077	+	0.654482i	$0.227113\pi$
$733$	18750.5	+	32476.8i	0.944837	+	1.63651i	0.188760	+	0.982023i	$0.439553\pi$
$734$	3845.00	+	6659.74i	0.193354	+	0.334898i
$735$		0			0
$736$	−3120.00	+	5404.00i	−0.156256	+	0.270644i
$737$	−6315.00			−0.315626
$738$		0			0
$739$	−880.000			−0.0438042			−0.0219021	−	0.999760i	$-0.506972\pi$
$739$	−880.000			−0.0438042			−0.0219021	+	0.999760i	$0.506972\pi$
$740$	2988.00	−	5175.37i	0.148434	−	0.257095i
$741$		0			0
$742$	3534.00	+	6121.07i	0.174848	+	0.302846i
$743$	811.500	+	1405.56i	0.0400687	+	0.0694010i	0.885364	−	0.464898i	$-0.153909\pi$
$743$	811.500	+	1405.56i	0.0400687	+	0.0694010i	−0.845296	+	0.534299i	$0.820576\pi$
$744$		0			0
$745$	−12730.5	+	22049.9i	−0.626053	+	1.08436i
$746$	16634.0			0.816373
$747$		0			0
$748$	2520.00			0.123182
$749$	186.000	−	322.161i	0.00907382	−	0.0157163i
$750$		0			0
$751$	3444.50	+	5966.05i	0.167366	+	0.289886i	0.937493	−	0.348005i	$-0.113141\pi$
$751$	3444.50	+	5966.05i	0.167366	+	0.289886i	−0.770127	+	0.637890i	$0.779807\pi$
$752$	1416.00	+	2452.58i	0.0686652	+	0.118932i
$753$		0			0
$754$	4107.00	−	7113.53i	0.198366	−	0.343580i
$755$	4149.00			0.199997
$756$		0			0
$757$	−12850.0			−0.616963			−0.308482	−	0.951230i	$-0.599821\pi$
$757$	−12850.0			−0.616963			−0.308482	+	0.951230i	$0.599821\pi$
$758$	12560.0	−	21754.6i	0.601847	−	1.04243i
$759$		0			0
$760$	−1008.00	−	1745.91i	−0.0481105	−	0.0833299i
$761$	2305.50	+	3993.24i	0.109822	+	0.190217i	0.915698	−	0.401867i	$-0.131639\pi$
$761$	2305.50	+	3993.24i	0.109822	+	0.190217i	−0.805876	+	0.592084i	$0.798305\pi$
$762$		0			0
$763$	−7781.00	+	13477.1i	−0.369189	+	0.639454i
$764$	−4644.00			−0.219914
$765$		0			0
$766$	24174.0			1.14026
$767$	−2941.50	+	5094.83i	−0.138476	+	0.239848i
$768$		0			0
$769$	1152.50	+	1996.19i	0.0540445	+	0.0936078i	0.891782	−	0.452465i	$-0.149455\pi$
$769$	1152.50	+	1996.19i	0.0540445	+	0.0936078i	−0.837737	+	0.546073i	$0.816122\pi$
$770$	−4185.00	−	7248.63i	−0.195866	−	0.339250i
$771$		0			0
$772$	−7222.00	+	12508.9i	−0.336691	+	0.583166i
$773$	34902.0			1.62398			0.811991	−	0.583670i	$-0.198384\pi$
$773$	34902.0			1.62398			0.811991	+	0.583670i	$0.198384\pi$
$774$		0			0
$775$	9020.00			0.418075
$776$	−3604.00	+	6242.31i	−0.166722	+	0.288771i
$777$		0			0
$778$	8541.00	+	14793.4i	0.393586	+	0.681710i
$779$	3654.00	+	6328.91i	0.168059	+	0.291087i

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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.50000 - 7.79423i) q^{5} +(15.5000 - 26.8468i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.50000 - 7.79423i) q^{5} +(15.5000 - 26.8468i) q^{7} -8.00000 q^{8} -18.0000 q^{10} +(-7.50000 + 12.9904i) q^{11} +(18.5000 + 32.0429i) q^{13} +(-31.0000 - 53.6936i) q^{14} +(-8.00000 + 13.8564i) q^{16} +42.0000 q^{17} -28.0000 q^{19} +(-18.0000 + 31.1769i) q^{20} +(15.0000 + 25.9808i) q^{22} +(97.5000 + 168.875i) q^{23} +(22.0000 - 38.1051i) q^{25} +74.0000 q^{26} -124.000 q^{28} +(55.5000 - 96.1288i) q^{29} +(102.500 + 177.535i) q^{31} +(16.0000 + 27.7128i) q^{32} +(42.0000 - 72.7461i) q^{34} -279.000 q^{35} -166.000 q^{37} +(-28.0000 + 48.4974i) q^{38} +(36.0000 + 62.3538i) q^{40} +(-130.500 - 226.033i) q^{41} +(21.5000 - 37.2391i) q^{43} +60.0000 q^{44} +390.000 q^{46} +(88.5000 - 153.286i) q^{47} +(-309.000 - 535.204i) q^{49} +(-44.0000 - 76.2102i) q^{50} +(74.0000 - 128.172i) q^{52} -114.000 q^{53} +135.000 q^{55} +(-124.000 + 214.774i) q^{56} +(-111.000 - 192.258i) q^{58} +(79.5000 + 137.698i) q^{59} +(-95.5000 + 165.411i) q^{61} +410.000 q^{62} +64.0000 q^{64} +(166.500 - 288.386i) q^{65} +(210.500 + 364.597i) q^{67} +(-84.0000 - 145.492i) q^{68} +(-279.000 + 483.242i) q^{70} -156.000 q^{71} +182.000 q^{73} +(-166.000 + 287.520i) q^{74} +(56.0000 + 96.9948i) q^{76} +(232.500 + 402.702i) q^{77} +(-566.500 + 981.207i) q^{79} +144.000 q^{80} -522.000 q^{82} +(-541.500 + 937.906i) q^{83} +(-189.000 - 327.358i) q^{85} +(-43.0000 - 74.4782i) q^{86} +(60.0000 - 103.923i) q^{88} +1050.00 q^{89} +1147.00 q^{91} +(390.000 - 675.500i) q^{92} +(-177.000 - 306.573i) q^{94} +(126.000 + 218.238i) q^{95} +(450.500 - 780.289i) q^{97} -1236.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8	\( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} + 31 q^{7} - 16 q^{8} - 36 q^{10} - 15 q^{11} + 37 q^{13} - 62 q^{14} - 16 q^{16} + 84 q^{17} - 56 q^{19} - 36 q^{20} + 30 q^{22} + 195 q^{23} + 44 q^{25} + 148 q^{26} - 248 q^{28} + 111 q^{29} + 205 q^{31} + 32 q^{32} + 84 q^{34} - 558 q^{35} - 332 q^{37} - 56 q^{38} + 72 q^{40} - 261 q^{41} + 43 q^{43} + 120 q^{44} + 780 q^{46} + 177 q^{47} - 618 q^{49} - 88 q^{50} + 148 q^{52} - 228 q^{53} + 270 q^{55} - 248 q^{56} - 222 q^{58} + 159 q^{59} - 191 q^{61} + 820 q^{62} + 128 q^{64} + 333 q^{65} + 421 q^{67} - 168 q^{68} - 558 q^{70} - 312 q^{71} + 364 q^{73} - 332 q^{74} + 112 q^{76} + 465 q^{77} - 1133 q^{79} + 288 q^{80} - 1044 q^{82} - 1083 q^{83} - 378 q^{85} - 86 q^{86} + 120 q^{88} + 2100 q^{89} + 2294 q^{91} + 780 q^{92} - 354 q^{94} + 252 q^{95} + 901 q^{97} - 2472 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 - 9 * q^5 + 31 * q^7 - 16 * q^8 - 36 * q^10 - 15 * q^11 + 37 * q^13 - 62 * q^14 - 16 * q^16 + 84 * q^17 - 56 * q^19 - 36 * q^20 + 30 * q^22 + 195 * q^23 + 44 * q^25 + 148 * q^26 - 248 * q^28 + 111 * q^29 + 205 * q^31 + 32 * q^32 + 84 * q^34 - 558 * q^35 - 332 * q^37 - 56 * q^38 + 72 * q^40 - 261 * q^41 + 43 * q^43 + 120 * q^44 + 780 * q^46 + 177 * q^47 - 618 * q^49 - 88 * q^50 + 148 * q^52 - 228 * q^53 + 270 * q^55 - 248 * q^56 - 222 * q^58 + 159 * q^59 - 191 * q^61 + 820 * q^62 + 128 * q^64 + 333 * q^65 + 421 * q^67 - 168 * q^68 - 558 * q^70 - 312 * q^71 + 364 * q^73 - 332 * q^74 + 112 * q^76 + 465 * q^77 - 1133 * q^79 + 288 * q^80 - 1044 * q^82 - 1083 * q^83 - 378 * q^85 - 86 * q^86 + 120 * q^88 + 2100 * q^89 + 2294 * q^91 + 780 * q^92 - 354 * q^94 + 252 * q^95 + 901 * q^97 - 2472 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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