LMFDB - Embedded newform 288.2.i.a.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,Mod(97,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("288.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	288.i (of order $3$, degree $2$, 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:	$2.29969157821$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			97.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	288.97
Dual form			288.2.i.a.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(-1.50000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-2.50000 + 4.33013i) q^{11} +(1.00000 + 1.73205i) q^{13} +6.92820i q^{15} -3.00000 q^{17} -1.00000 q^{19} +(3.00000 - 1.73205i) q^{21} +(-3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.19615i q^{27} +(1.00000 - 1.73205i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(7.50000 - 4.33013i) q^{33} +8.00000 q^{35} -8.00000 q^{37} -3.46410i q^{39} +(-0.500000 - 0.866025i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(6.00000 - 10.3923i) q^{45} +(1.00000 - 1.73205i) q^{47} +(1.50000 + 2.59808i) q^{49} +(4.50000 + 2.59808i) q^{51} -4.00000 q^{53} +20.0000 q^{55} +(1.50000 + 0.866025i) q^{57} +(2.50000 + 4.33013i) q^{59} -6.00000 q^{63} +(4.00000 - 6.92820i) q^{65} +(-6.50000 - 11.2583i) q^{67} +10.3923i q^{69} -8.00000 q^{71} +3.00000 q^{73} +(16.5000 - 9.52628i) q^{75} +(-5.00000 - 8.66025i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(6.00000 - 10.3923i) q^{83} +(6.00000 + 10.3923i) q^{85} +(-3.00000 + 1.73205i) q^{87} -10.0000 q^{89} -4.00000 q^{91} +6.92820i q^{93} +(2.00000 + 3.46410i) q^{95} +(5.50000 - 9.52628i) q^{97} -15.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9	$ 2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{19} + 6 q^{21} - 6 q^{23} - 11 q^{25} + 2 q^{29} - 4 q^{31} + 15 q^{33} + 16 q^{35} - 16 q^{37} - q^{41} - 7 q^{43} + 12 q^{45} + 2 q^{47} + 3 q^{49} + 9 q^{51} - 8 q^{53} + 40 q^{55} + 3 q^{57} + 5 q^{59} - 12 q^{63} + 8 q^{65} - 13 q^{67} - 16 q^{71} + 6 q^{73} + 33 q^{75} - 10 q^{77} + 8 q^{79} - 9 q^{81} + 12 q^{83} + 12 q^{85} - 6 q^{87} - 20 q^{89} - 8 q^{91} + 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 6 * q^17 - 2 * q^19 + 6 * q^21 - 6 * q^23 - 11 * q^25 + 2 * q^29 - 4 * q^31 + 15 * q^33 + 16 * q^35 - 16 * q^37 - q^41 - 7 * q^43 + 12 * q^45 + 2 * q^47 + 3 * q^49 + 9 * q^51 - 8 * q^53 + 40 * q^55 + 3 * q^57 + 5 * q^59 - 12 * q^63 + 8 * q^65 - 13 * q^67 - 16 * q^71 + 6 * q^73 + 33 * q^75 - 10 * q^77 + 8 * q^79 - 9 * q^81 + 12 * q^83 + 12 * q^85 - 6 * q^87 - 20 * q^89 - 8 * q^91 + 4 * q^95 + 11 * q^97 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$		0			0
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	0.192201	+	0.981356i	$0.438437\pi$
$11$	−2.50000	+	4.33013i	−0.753778	+	1.30558i	−0.945979	+	0.324227i	$0.894896\pi$
$12$		0			0
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$			6.92820i			1.78885i
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	3.00000	−	1.73205i	0.654654	−	0.377964i
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	7.50000	−	4.33013i	1.30558	−	0.753778i
$34$		0			0
$35$	8.00000			1.35225
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$		−	3.46410i		−	0.554700i
$40$		0			0
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	0.824338	−	0.566099i	$-0.191548\pi$
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	−0.902424	+	0.430848i	$0.858214\pi$
$42$		0			0
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	0.465478	+	0.885059i	$0.345882\pi$
$43$	−3.50000	+	6.06218i	−0.533745	+	0.924473i	−0.999223	+	0.0394140i	$0.987451\pi$
$44$		0			0
$45$	6.00000	−	10.3923i	0.894427	−	1.54919i
$46$		0			0
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$	4.50000	+	2.59808i	0.630126	+	0.363803i
$52$		0			0
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	20.0000			2.69680
$56$		0			0
$57$	1.50000	+	0.866025i	0.198680	+	0.114708i
$58$		0			0
$59$	2.50000	+	4.33013i	0.325472	+	0.563735i	0.981608	−	0.190909i	$-0.0611434\pi$
$59$	2.50000	+	4.33013i	0.325472	+	0.563735i	−0.656136	+	0.754643i	$0.727810\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$	−6.00000			−0.755929
$64$		0			0
$65$	4.00000	−	6.92820i	0.496139	−	0.859338i
$66$		0			0
$67$	−6.50000	−	11.2583i	−0.794101	−	1.37542i	−0.923408	−	0.383819i	$-0.874609\pi$
$67$	−6.50000	−	11.2583i	−0.794101	−	1.37542i	0.129307	−	0.991605i	$-0.458725\pi$
$68$		0			0
$69$			10.3923i			1.25109i
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	3.00000			0.351123			0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000			0.351123			0.175562	+	0.984468i	$0.443826\pi$
$74$		0			0
$75$	16.5000	−	9.52628i	1.90526	−	1.10000i
$76$		0			0
$77$	−5.00000	−	8.66025i	−0.569803	−	0.986928i
$78$		0			0
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	−0.322396	−	0.946605i	$-0.604488\pi$
$83$	6.00000	−	10.3923i	0.658586	−	1.14070i	0.980982	−	0.194099i	$-0.0621783\pi$
$84$		0			0
$85$	6.00000	+	10.3923i	0.650791	+	1.12720i
$86$		0			0
$87$	−3.00000	+	1.73205i	−0.321634	+	0.185695i
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$			6.92820i			0.718421i
$94$		0			0
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	5.50000	−	9.52628i	0.558440	−	0.967247i	−0.439187	−	0.898396i	$-0.644733\pi$
$97$	5.50000	−	9.52628i	0.558440	−	0.967247i	0.997627	−	0.0688512i	$-0.0219334\pi$
$98$		0			0
$99$	−15.0000			−1.50756
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	0.975124	−	0.221660i	$-0.0711475\pi$
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	−0.679525	+	0.733652i	$0.737814\pi$
$104$		0			0
$105$	−12.0000	−	6.92820i	−1.17108	−	0.676123i
$106$		0			0
$107$	−5.00000			−0.483368			−0.241684	−	0.970355i	$-0.577700\pi$
$107$	−5.00000			−0.483368			−0.241684	+	0.970355i	$0.577700\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$		0			0
$111$	12.0000	+	6.92820i	1.13899	+	0.657596i
$112$		0			0
$113$	1.00000	+	1.73205i	0.0940721	+	0.162938i	0.909221	−	0.416314i	$-0.136678\pi$
$113$	1.00000	+	1.73205i	0.0940721	+	0.162938i	−0.815149	+	0.579252i	$0.803345\pi$
$114$		0			0
$115$	−12.0000	+	20.7846i	−1.11901	+	1.93817i
$116$		0			0
$117$	−3.00000	+	5.19615i	−0.277350	+	0.480384i
$118$		0			0
$119$	3.00000	−	5.19615i	0.275010	−	0.476331i
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$			1.73205i			0.156174i
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$	10.5000	−	6.06218i	0.924473	−	0.533745i
$130$		0			0
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	0.968826	+	0.247741i	$0.0796882\pi$
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	−0.269863	+	0.962899i	$0.586978\pi$
$132$		0			0
$133$	1.00000	−	1.73205i	0.0867110	−	0.150188i
$134$		0			0
$135$	−18.0000	+	10.3923i	−1.54919	+	0.894427i
$136$		0			0
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	0.922961	+	0.384893i	$0.125762\pi$
$138$		0			0
$139$	−6.50000	−	11.2583i	−0.551323	−	0.954919i	−0.998179	−	0.0603135i	$-0.980790\pi$
$139$	−6.50000	−	11.2583i	−0.551323	−	0.954919i	0.446857	−	0.894606i	$-0.352543\pi$
$140$		0			0
$141$	−3.00000	+	1.73205i	−0.252646	+	0.145865i
$142$		0			0
$143$	−10.0000			−0.836242
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		−	5.19615i		−	0.428571i
$148$		0			0
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	0.953703	+	0.300750i	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		0			0
$151$	−9.00000	+	15.5885i	−0.732410	+	1.26857i	0.223441	+	0.974717i	$0.428271\pi$
$151$	−9.00000	+	15.5885i	−0.732410	+	1.26857i	−0.955851	+	0.293853i	$0.905062\pi$
$152$		0			0
$153$	−4.50000	−	7.79423i	−0.363803	−	0.630126i
$154$		0			0
$155$	−8.00000	+	13.8564i	−0.642575	+	1.11297i
$156$		0			0
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	−0.920860	−	0.389892i	$-0.872512\pi$
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	0.122774	−	0.992435i	$-0.460821\pi$
$158$		0			0
$159$	6.00000	+	3.46410i	0.475831	+	0.274721i
$160$		0			0
$161$	12.0000			0.945732
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		0			0
$165$	−30.0000	−	17.3205i	−2.33550	−	1.34840i
$166$		0			0
$167$	−8.00000	−	13.8564i	−0.619059	−	1.07224i	−0.989658	−	0.143448i	$-0.954181\pi$
$167$	−8.00000	−	13.8564i	−0.619059	−	1.07224i	0.370599	−	0.928793i	$-0.379152\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$		0			0
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	0.289412	+	0.957205i	$0.406540\pi$
$173$	−9.00000	+	15.5885i	−0.684257	+	1.18517i	−0.973670	+	0.227964i	$0.926793\pi$
$174$		0			0
$175$	−11.0000	−	19.0526i	−0.831522	−	1.44024i
$176$		0			0
$177$		−	8.66025i		−	0.650945i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.0000	+	27.7128i	1.17634	+	2.03749i
$186$		0			0
$187$	7.50000	−	12.9904i	0.548454	−	0.949951i
$188$		0			0
$189$	9.00000	+	5.19615i	0.654654	+	0.377964i
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	0.914949	−	0.403570i	$-0.132231\pi$
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	−0.806976	+	0.590584i	$0.798898\pi$
$194$		0			0
$195$	−12.0000	+	6.92820i	−0.859338	+	0.496139i
$196$		0			0
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		0			0
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$		0			0
$201$			22.5167i			1.58820i
$202$		0			0
$203$	2.00000	+	3.46410i	0.140372	+	0.243132i
$204$		0			0
$205$	−2.00000	+	3.46410i	−0.139686	+	0.241943i
$206$		0			0
$207$	9.00000	−	15.5885i	0.625543	−	1.08347i
$208$		0			0
$209$	2.50000	−	4.33013i	0.172929	−	0.299521i
$210$		0			0
$211$	14.0000	+	24.2487i	0.963800	+	1.66935i	0.712806	+	0.701361i	$0.247424\pi$
$211$	14.0000	+	24.2487i	0.963800	+	1.66935i	0.250994	+	0.967989i	$0.419243\pi$
$212$		0			0
$213$	12.0000	+	6.92820i	0.822226	+	0.474713i
$214$		0			0
$215$	28.0000			1.90958
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$	−4.50000	−	2.59808i	−0.304082	−	0.175562i
$220$		0			0
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$		0			0
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	−0.999362	−	0.0357107i	$-0.988630\pi$
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	0.530607	+	0.847618i	$0.321964\pi$
$224$		0			0
$225$	−33.0000			−2.20000
$226$		0			0
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		0			0
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	0.536515	−	0.843891i	$-0.319740\pi$
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	−0.999088	+	0.0426906i	$0.986407\pi$
$230$		0			0
$231$			17.3205i			1.13961i
$232$		0			0
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$	−12.0000	+	6.92820i	−0.779484	+	0.450035i
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	−0.211360	−	0.977408i	$-0.567789\pi$
$241$	11.5000	−	19.9186i	0.740780	−	1.28307i	0.952141	−	0.305661i	$-0.0988773\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$		0			0
$245$	6.00000	−	10.3923i	0.383326	−	0.663940i
$246$		0			0
$247$	−1.00000	−	1.73205i	−0.0636285	−	0.110208i
$248$		0			0
$249$	−18.0000	+	10.3923i	−1.14070	+	0.658586i
$250$		0			0
$251$	21.0000			1.32551			0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000			1.32551			0.662754	+	0.748837i	$0.269387\pi$
$252$		0			0
$253$	30.0000			1.88608
$254$		0			0
$255$		−	20.7846i		−	1.30158i
$256$		0			0
$257$	2.50000	+	4.33013i	0.155946	+	0.270106i	0.933403	−	0.358830i	$-0.116824\pi$
$257$	2.50000	+	4.33013i	0.155946	+	0.270106i	−0.777457	+	0.628936i	$0.783491\pi$
$258$		0			0
$259$	8.00000	−	13.8564i	0.497096	−	0.860995i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	−0.758585	−	0.651575i	$-0.774109\pi$
$263$	3.00000	−	5.19615i	0.184988	−	0.320408i	0.943572	+	0.331166i	$0.107442\pi$
$264$		0			0
$265$	8.00000	+	13.8564i	0.491436	+	0.851192i
$266$		0			0
$267$	15.0000	+	8.66025i	0.917985	+	0.529999i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		0			0
$273$	6.00000	+	3.46410i	0.363137	+	0.209657i
$274$		0			0
$275$	−27.5000	−	47.6314i	−1.65831	−	2.87228i
$276$		0			0
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	−0.575408	−	0.817867i	$-0.695157\pi$
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	0.995997	+	0.0893846i	$0.0284900\pi$
$278$		0			0
$279$	6.00000	−	10.3923i	0.359211	−	0.622171i
$280$		0			0
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	−0.462174	−	0.886789i	$-0.652930\pi$
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	0.999069	−	0.0431402i	$-0.0137362\pi$
$282$		0			0
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	−0.896279	−	0.443491i	$-0.853740\pi$
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	0.0640654	−	0.997946i	$-0.479593\pi$
$284$		0			0
$285$		−	6.92820i		−	0.410391i
$286$		0			0
$287$	2.00000			0.118056
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−16.5000	+	9.52628i	−0.967247	+	0.558440i
$292$		0			0
$293$	−15.0000	−	25.9808i	−0.876309	−	1.51781i	−0.855361	−	0.518032i	$-0.826665\pi$
$293$	−15.0000	−	25.9808i	−0.876309	−	1.51781i	−0.0209480	−	0.999781i	$-0.506668\pi$
$294$		0			0
$295$	10.0000	−	17.3205i	0.582223	−	1.00844i
$296$		0			0
$297$	22.5000	+	12.9904i	1.30558	+	0.753778i
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$	−7.00000	−	12.1244i	−0.403473	−	0.698836i
$302$		0			0
$303$	18.0000	−	10.3923i	1.03407	−	0.597022i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	9.00000			0.513657			0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000			0.513657			0.256829	+	0.966457i	$0.417322\pi$
$308$		0			0
$309$		−	10.3923i		−	0.591198i
$310$		0			0
$311$	9.00000	+	15.5885i	0.510343	+	0.883940i	0.999928	+	0.0119847i	$0.00381495\pi$
$311$	9.00000	+	15.5885i	0.510343	+	0.883940i	−0.489585	+	0.871956i	$0.662852\pi$
$312$		0			0
$313$	−6.50000	+	11.2583i	−0.367402	+	0.636358i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−6.50000	+	11.2583i	−0.367402	+	0.636358i	0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$	12.0000	+	20.7846i	0.676123	+	1.17108i
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	5.00000	+	8.66025i	0.279946	+	0.484881i
$320$		0			0
$321$	7.50000	+	4.33013i	0.418609	+	0.241684i
$322$		0			0
$323$	3.00000			0.166924
$324$		0			0
$325$	−22.0000			−1.22034
$326$		0			0
$327$	−12.0000	−	6.92820i	−0.663602	−	0.383131i
$328$		0			0
$329$	2.00000	+	3.46410i	0.110264	+	0.190982i
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$	−12.0000	−	20.7846i	−0.657596	−	1.13899i
$334$		0			0
$335$	−26.0000	+	45.0333i	−1.42053	+	2.46043i
$336$		0			0
$337$	12.5000	+	21.6506i	0.680918	+	1.17939i	0.974701	+	0.223513i	$0.0717525\pi$
$337$	12.5000	+	21.6506i	0.680918	+	1.17939i	−0.293783	+	0.955872i	$0.594914\pi$
$338$		0			0
$339$		−	3.46410i		−	0.188144i
$340$		0			0
$341$	20.0000			1.08306
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$	36.0000	−	20.7846i	1.93817	−	1.11901i
$346$		0			0
$347$	−0.500000	−	0.866025i	−0.0268414	−	0.0464907i	0.852293	−	0.523065i	$-0.175212\pi$
$347$	−0.500000	−	0.866025i	−0.0268414	−	0.0464907i	−0.879134	+	0.476575i	$0.841878\pi$
$348$		0			0
$349$	−8.00000	+	13.8564i	−0.428230	+	0.741716i	−0.996716	−	0.0809766i	$-0.974196\pi$
$349$	−8.00000	+	13.8564i	−0.428230	+	0.741716i	0.568486	+	0.822693i	$0.307529\pi$
$350$		0			0
$351$	9.00000	−	5.19615i	0.480384	−	0.277350i
$352$		0			0
$353$	−17.5000	+	30.3109i	−0.931431	+	1.61329i	−0.150553	+	0.988602i	$0.548106\pi$
$353$	−17.5000	+	30.3109i	−0.931431	+	1.61329i	−0.780878	+	0.624684i	$0.785228\pi$
$354$		0			0
$355$	16.0000	+	27.7128i	0.849192	+	1.47084i
$356$		0			0
$357$	−9.00000	+	5.19615i	−0.476331	+	0.275010i
$358$		0			0
$359$	14.0000			0.738892			0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000			0.738892			0.369446	+	0.929252i	$0.379548\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$			24.2487i			1.27273i
$364$		0			0
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	1.50000	−	2.59808i	0.0780869	−	0.135250i
$370$		0			0
$371$	4.00000	−	6.92820i	0.207670	−	0.359694i
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$	−36.0000	−	20.7846i	−1.85903	−	1.07331i
$376$		0			0
$377$	4.00000			0.206010
$378$		0			0
$379$	−1.00000			−0.0513665			−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000			−0.0513665			−0.0256833	+	0.999670i	$0.508176\pi$
$380$		0			0
$381$	3.00000	+	1.73205i	0.153695	+	0.0887357i
$382$		0			0
$383$	10.0000	+	17.3205i	0.510976	+	0.885037i	0.999919	+	0.0127209i	$0.00404928\pi$
$383$	10.0000	+	17.3205i	0.510976	+	0.885037i	−0.488943	+	0.872316i	$0.662617\pi$
$384$		0			0
$385$	−20.0000	+	34.6410i	−1.01929	+	1.76547i
$386$		0			0
$387$	−21.0000			−1.06749
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$	9.00000	+	15.5885i	0.455150	+	0.788342i
$392$		0			0
$393$		−	27.7128i		−	1.39793i
$394$		0			0
$395$	−32.0000			−1.61009
$396$		0			0
$397$	−32.0000			−1.60603			−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000			−1.60603			−0.803017	+	0.595956i	$0.796773\pi$
$398$		0			0
$399$	−3.00000	+	1.73205i	−0.150188	+	0.0867110i
$400$		0			0
$401$	−2.50000	−	4.33013i	−0.124844	−	0.216236i	0.796828	−	0.604206i	$-0.206510\pi$
$401$	−2.50000	−	4.33013i	−0.124844	−	0.216236i	−0.921672	+	0.387970i	$0.873176\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$		0			0
$405$	36.0000			1.78885
$406$		0			0
$407$	20.0000	−	34.6410i	0.991363	−	1.71709i
$408$		0			0
$409$	−12.5000	−	21.6506i	−0.618085	−	1.07056i	−0.989835	−	0.142222i	$-0.954575\pi$
$409$	−12.5000	−	21.6506i	−0.618085	−	1.07056i	0.371750	−	0.928333i	$-0.378758\pi$
$410$		0			0
$411$	−4.50000	+	2.59808i	−0.221969	+	0.128154i
$412$		0			0
$413$	−10.0000			−0.492068
$414$		0			0
$415$	−48.0000			−2.35623
$416$		0			0
$417$			22.5167i			1.10265i
$418$		0			0
$419$	2.00000	+	3.46410i	0.0977064	+	0.169232i	0.910735	−	0.412991i	$-0.135516\pi$
$419$	2.00000	+	3.46410i	0.0977064	+	0.169232i	−0.813029	+	0.582224i	$0.802183\pi$
$420$		0			0
$421$	−4.00000	+	6.92820i	−0.194948	+	0.337660i	−0.946883	−	0.321577i	$-0.895787\pi$
$421$	−4.00000	+	6.92820i	−0.194948	+	0.337660i	0.751935	+	0.659237i	$0.229121\pi$
$422$		0			0
$423$	6.00000			0.291730
$424$		0			0
$425$	16.5000	−	28.5788i	0.800368	−	1.38628i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	15.0000	+	8.66025i	0.724207	+	0.418121i
$430$		0			0
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$		0			0
$433$	1.00000			0.0480569			0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000			0.0480569			0.0240285	+	0.999711i	$0.492351\pi$
$434$		0			0
$435$	12.0000	+	6.92820i	0.575356	+	0.332182i
$436$		0			0
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$	−4.50000	+	7.79423i	−0.214286	+	0.371154i
$442$		0			0
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	0.343715	+	0.939074i	$0.388315\pi$
$443$	−13.5000	+	23.3827i	−0.641404	+	1.11094i	−0.985119	+	0.171871i	$0.945019\pi$
$444$		0			0
$445$	20.0000	+	34.6410i	0.948091	+	1.64214i
$446$		0			0
$447$		−	31.1769i		−	1.47462i
$448$		0			0
$449$	−31.0000			−1.46298			−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000			−1.46298			−0.731490	+	0.681852i	$0.761175\pi$
$450$		0			0
$451$	5.00000			0.235441
$452$		0			0
$453$	27.0000	−	15.5885i	1.26857	−	0.732410i
$454$		0			0
$455$	8.00000	+	13.8564i	0.375046	+	0.649598i
$456$		0			0
$457$	3.50000	−	6.06218i	0.163723	−	0.283577i	−0.772478	−	0.635042i	$-0.780983\pi$
$457$	3.50000	−	6.06218i	0.163723	−	0.283577i	0.936201	+	0.351465i	$0.114316\pi$
$458$		0			0
$459$			15.5885i			0.727607i
$460$		0			0
$461$	9.00000	−	15.5885i	0.419172	−	0.726027i	−0.576685	−	0.816967i	$-0.695654\pi$
$461$	9.00000	−	15.5885i	0.419172	−	0.726027i	0.995856	+	0.0909401i	$0.0289872\pi$
$462$		0			0
$463$	−8.00000	−	13.8564i	−0.371792	−	0.643962i	0.618050	−	0.786139i	$-0.287923\pi$
$463$	−8.00000	−	13.8564i	−0.371792	−	0.643962i	−0.989841	+	0.142177i	$0.954590\pi$
$464$		0			0
$465$	24.0000	−	13.8564i	1.11297	−	0.642575i
$466$		0			0
$467$	−7.00000			−0.323921			−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000			−0.323921			−0.161961	+	0.986797i	$0.551782\pi$
$468$		0			0
$469$	26.0000			1.20057
$470$		0			0
$471$			34.6410i			1.59617i
$472$		0			0
$473$	−17.5000	−	30.3109i	−0.804651	−	1.39370i
$474$		0			0
$475$	5.50000	−	9.52628i	0.252357	−	0.437096i
$476$		0			0
$477$	−6.00000	−	10.3923i	−0.274721	−	0.475831i
$478$		0			0
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	−0.583803	−	0.811895i	$-0.698436\pi$
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	0.995023	+	0.0996406i	$0.0317693\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$		0			0
$483$	−18.0000	−	10.3923i	−0.819028	−	0.472866i
$484$		0			0
$485$	−44.0000			−1.99794
$486$		0			0
$487$	−22.0000			−0.996915			−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000			−0.996915			−0.498458	+	0.866914i	$0.666100\pi$
$488$		0			0
$489$	−30.0000	−	17.3205i	−1.35665	−	0.783260i
$490$		0			0
$491$	3.50000	+	6.06218i	0.157953	+	0.273582i	0.934130	−	0.356932i	$-0.116177\pi$
$491$	3.50000	+	6.06218i	0.157953	+	0.273582i	−0.776178	+	0.630514i	$0.782844\pi$
$492$		0			0
$493$	−3.00000	+	5.19615i	−0.135113	+	0.234023i
$494$		0			0
$495$	30.0000	+	51.9615i	1.34840	+	2.33550i
$496$		0			0
$497$	8.00000	−	13.8564i	0.358849	−	0.621545i
$498$		0			0
$499$	10.5000	+	18.1865i	0.470045	+	0.814141i	0.999413	−	0.0342508i	$-0.0109045\pi$
$499$	10.5000	+	18.1865i	0.470045	+	0.814141i	−0.529369	+	0.848392i	$0.677571\pi$
$500$		0			0
$501$			27.7128i			1.23812i
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	48.0000			2.13597
$506$		0			0
$507$	−13.5000	+	7.79423i	−0.599556	+	0.346154i
$508$		0			0
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	−0.999307	−	0.0372243i	$-0.988148\pi$
$509$	−12.0000	−	20.7846i	−0.531891	−	0.921262i	0.467416	−	0.884037i	$-0.345185\pi$
$510$		0			0
$511$	−3.00000	+	5.19615i	−0.132712	+	0.229864i
$512$		0			0
$513$			5.19615i			0.229416i
$514$		0			0
$515$	12.0000	−	20.7846i	0.528783	−	0.915879i
$516$		0			0
$517$	5.00000	+	8.66025i	0.219900	+	0.380878i
$518$		0			0
$519$	27.0000	−	15.5885i	1.18517	−	0.684257i
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$		0			0
$525$			38.1051i			1.66304i
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	−7.50000	+	12.9904i	−0.325472	+	0.563735i
$532$		0			0
$533$	1.00000	−	1.73205i	0.0433148	−	0.0750234i
$534$		0			0
$535$	10.0000	+	17.3205i	0.432338	+	0.748831i
$536$		0			0
$537$	18.0000	+	10.3923i	0.776757	+	0.448461i
$538$		0			0
$539$	−15.0000			−0.646096
$540$		0			0
$541$	−32.0000			−1.37579			−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000			−1.37579			−0.687894	+	0.725811i	$0.741464\pi$
$542$		0			0
$543$	−3.00000	−	1.73205i	−0.128742	−	0.0743294i
$544$		0			0
$545$	−16.0000	−	27.7128i	−0.685365	−	1.18709i
$546$		0			0
$547$	16.5000	−	28.5788i	0.705489	−	1.22194i	−0.261026	−	0.965332i	$-0.584061\pi$
$547$	16.5000	−	28.5788i	0.705489	−	1.22194i	0.966515	−	0.256611i	$-0.0826059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.00000	+	1.73205i	−0.0426014	+	0.0737878i
$552$		0			0
$553$	8.00000	+	13.8564i	0.340195	+	0.589234i
$554$		0			0
$555$		−	55.4256i		−	2.35269i
$556$		0			0
$557$	10.0000			0.423714			0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000			0.423714			0.211857	+	0.977301i	$0.432049\pi$
$558$		0			0
$559$	−14.0000			−0.592137
$560$		0			0
$561$	−22.5000	+	12.9904i	−0.949951	+	0.548454i
$562$		0			0
$563$	7.50000	+	12.9904i	0.316087	+	0.547479i	0.979668	−	0.200625i	$-0.0642974\pi$
$563$	7.50000	+	12.9904i	0.316087	+	0.547479i	−0.663581	+	0.748105i	$0.730964\pi$
$564$		0			0
$565$	4.00000	−	6.92820i	0.168281	−	0.291472i
$566$		0			0
$567$	−9.00000	−	15.5885i	−0.377964	−	0.654654i
$568$		0			0
$569$	5.50000	−	9.52628i	0.230572	−	0.399362i	−0.727405	−	0.686209i	$-0.759274\pi$
$569$	5.50000	−	9.52628i	0.230572	−	0.399362i	0.957977	+	0.286846i	$0.0926069\pi$
$570$		0			0
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	0.969378	−	0.245573i	$-0.0789761\pi$
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	−0.697362	+	0.716720i	$0.745643\pi$
$572$		0			0
$573$	9.00000	−	5.19615i	0.375980	−	0.217072i
$574$		0			0
$575$	66.0000			2.75239
$576$		0			0
$577$	3.00000			0.124892			0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000			0.124892			0.0624458	+	0.998048i	$0.480110\pi$
$578$		0			0
$579$		−	5.19615i		−	0.215945i
$580$		0			0
$581$	12.0000	+	20.7846i	0.497844	+	0.862291i
$582$		0			0
$583$	10.0000	−	17.3205i	0.414158	−	0.717342i
$584$		0			0
$585$	24.0000			0.992278
$586$		0			0
$587$	4.50000	−	7.79423i	0.185735	−	0.321702i	−0.758089	−	0.652151i	$-0.773867\pi$
$587$	4.50000	−	7.79423i	0.185735	−	0.321702i	0.943824	+	0.330449i	$0.107200\pi$
$588$		0			0
$589$	2.00000	+	3.46410i	0.0824086	+	0.142736i
$590$		0			0
$591$	−6.00000	−	3.46410i	−0.246807	−	0.142494i
$592$		0			0
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	−24.0000			−0.983904
$596$		0			0
$597$	3.00000	+	1.73205i	0.122782	+	0.0708881i
$598$		0			0
$599$	−20.0000	−	34.6410i	−0.817178	−	1.41539i	−0.907754	−	0.419504i	$-0.862204\pi$
$599$	−20.0000	−	34.6410i	−0.817178	−	1.41539i	0.0905757	−	0.995890i	$-0.471129\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$	19.5000	−	33.7750i	0.794101	−	1.37542i
$604$		0			0
$605$	−28.0000	+	48.4974i	−1.13836	+	1.97170i
$606$		0			0
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		−	6.92820i		−	0.280745i
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$	12.0000			0.484675			0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000			0.484675			0.242338	+	0.970192i	$0.422086\pi$
$614$		0			0
$615$	6.00000	−	3.46410i	0.241943	−	0.139686i
$616$		0			0
$617$	22.5000	+	38.9711i	0.905816	+	1.56892i	0.819818	+	0.572624i	$0.194074\pi$
$617$	22.5000	+	38.9711i	0.905816	+	1.56892i	0.0859976	+	0.996295i	$0.472592\pi$
$618$		0			0
$619$	10.5000	−	18.1865i	0.422031	−	0.730978i	−0.574107	−	0.818780i	$-0.694651\pi$
$619$	10.5000	−	18.1865i	0.422031	−	0.730978i	0.996138	+	0.0878015i	$0.0279841\pi$
$620$		0			0
$621$	−27.0000	+	15.5885i	−1.08347	+	0.625543i
$622$		0			0
$623$	10.0000	−	17.3205i	0.400642	−	0.693932i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$	−7.50000	+	4.33013i	−0.299521	+	0.172929i
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	28.0000			1.11466			0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000			1.11466			0.557331	+	0.830290i	$0.311825\pi$
$632$		0			0
$633$		−	48.4974i		−	1.92760i
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	−3.00000	+	5.19615i	−0.118864	+	0.205879i
$638$		0			0
$639$	−12.0000	−	20.7846i	−0.474713	−	0.822226i
$640$		0			0
$641$	−2.50000	+	4.33013i	−0.0987441	+	0.171030i	−0.911165	−	0.412042i	$-0.864816\pi$
$641$	−2.50000	+	4.33013i	−0.0987441	+	0.171030i	0.812421	+	0.583071i	$0.198149\pi$
$642$		0			0
$643$	−7.50000	−	12.9904i	−0.295771	−	0.512291i	0.679393	−	0.733775i	$-0.262243\pi$
$643$	−7.50000	−	12.9904i	−0.295771	−	0.512291i	−0.975164	+	0.221484i	$0.928910\pi$
$644$		0			0
$645$	−42.0000	−	24.2487i	−1.65375	−	0.954792i
$646$		0			0
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	−25.0000			−0.981336
$650$		0			0
$651$	−12.0000	−	6.92820i	−0.470317	−	0.271538i
$652$		0			0
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	0.996913	−	0.0785119i	$-0.0250169\pi$
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	−0.566450	+	0.824096i	$0.691684\pi$
$654$		0			0
$655$	32.0000	−	55.4256i	1.25034	−	2.16566i
$656$		0			0
$657$	4.50000	+	7.79423i	0.175562	+	0.304082i
$658$		0			0
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$		0			0
$663$			10.3923i			0.403604i
$664$		0			0
$665$	−8.00000			−0.310227
$666$		0			0
$667$	−12.0000			−0.464642
$668$		0			0
$669$	21.0000	−	12.1244i	0.811907	−	0.468755i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.00000	+	1.73205i	−0.0385472	+	0.0667657i	−0.884655	−	0.466246i	$-0.845606\pi$
$673$	−1.00000	+	1.73205i	−0.0385472	+	0.0667657i	0.846108	+	0.533011i	$0.178940\pi$
$674$		0			0
$675$	49.5000	+	28.5788i	1.90526	+	1.10000i
$676$		0			0
$677$	14.0000	−	24.2487i	0.538064	−	0.931954i	−0.460945	−	0.887429i	$-0.652489\pi$
$677$	14.0000	−	24.2487i	0.538064	−	0.931954i	0.999008	−	0.0445248i	$-0.0141774\pi$
$678$		0			0
$679$	11.0000	+	19.0526i	0.422141	+	0.731170i
$680$		0			0
$681$	−4.50000	+	2.59808i	−0.172440	+	0.0995585i
$682$		0			0
$683$	15.0000			0.573959			0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000			0.573959			0.286980	+	0.957937i	$0.407349\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$			24.2487i			0.925146i
$688$		0			0
$689$	−4.00000	−	6.92820i	−0.152388	−	0.263944i
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$		0			0
$693$	15.0000	−	25.9808i	0.569803	−	0.986928i
$694$		0			0
$695$	−26.0000	+	45.0333i	−0.986236	+	1.70821i
$696$		0			0
$697$	1.50000	+	2.59808i	0.0568166	+	0.0984092i
$698$		0			0
$699$	31.5000	+	18.1865i	1.19144	+	0.687878i
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	8.00000			0.301726
$704$		0			0
$705$	12.0000	+	6.92820i	0.451946	+	0.260931i
$706$		0			0
$707$	−12.0000	−	20.7846i	−0.451306	−	0.781686i
$708$		0			0
$709$	10.0000	−	17.3205i	0.375558	−	0.650485i	−0.614852	−	0.788642i	$-0.710784\pi$
$709$	10.0000	−	17.3205i	0.375558	−	0.650485i	0.990410	+	0.138157i	$0.0441178\pi$
$710$		0			0
$711$	24.0000			0.900070
$712$		0			0
$713$	−12.0000	+	20.7846i	−0.449404	+	0.778390i
$714$		0			0
$715$	20.0000	+	34.6410i	0.747958	+	1.29550i
$716$		0			0
$717$		−	10.3923i		−	0.388108i
$718$		0			0
$719$	−32.0000			−1.19340			−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000			−1.19340			−0.596699	+	0.802465i	$0.703521\pi$
$720$		0			0
$721$	−12.0000			−0.446903
$722$		0			0
$723$	−34.5000	+	19.9186i	−1.28307	+	0.740780i
$724$		0			0
$725$	11.0000	+	19.0526i	0.408530	+	0.707594i
$726$		0			0
$727$	−15.0000	+	25.9808i	−0.556319	+	0.963573i	0.441480	+	0.897271i	$0.354453\pi$
$727$	−15.0000	+	25.9808i	−0.556319	+	0.963573i	−0.997800	+	0.0663022i	$0.978880\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	10.5000	−	18.1865i	0.388357	−	0.672653i
$732$		0			0
$733$	17.0000	+	29.4449i	0.627909	+	1.08757i	0.987971	+	0.154642i	$0.0494225\pi$
$733$	17.0000	+	29.4449i	0.627909	+	1.08757i	−0.360061	+	0.932929i	$0.617244\pi$
$734$		0			0
$735$	−18.0000	+	10.3923i	−0.663940	+	0.383326i
$736$		0			0
$737$	65.0000			2.39431
$738$		0			0
$739$	−49.0000			−1.80249			−0.901247	−	0.433306i	$-0.857347\pi$
$739$	−49.0000			−1.80249			−0.901247	+	0.433306i	$0.857347\pi$
$740$		0			0
$741$			3.46410i			0.127257i
$742$		0			0
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	0.805735	−	0.592277i	$-0.201771\pi$
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	−0.915794	+	0.401648i	$0.868437\pi$
$744$		0			0
$745$	36.0000	−	62.3538i	1.31894	−	2.28447i
$746$		0			0
$747$	36.0000			1.31717
$748$		0			0
$749$	5.00000	−	8.66025i	0.182696	−	0.316439i
$750$		0			0
$751$	−26.0000	−	45.0333i	−0.948753	−	1.64329i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−26.0000	−	45.0333i	−0.948753	−	1.64329i	−0.200698	−	0.979653i	$-0.564321\pi$
$752$		0			0
$753$	−31.5000	−	18.1865i	−1.14792	−	0.662754i
$754$		0			0
$755$	72.0000			2.62035
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$	−45.0000	−	25.9808i	−1.63340	−	0.943042i
$760$		0			0
$761$	15.0000	+	25.9808i	0.543750	+	0.941802i	0.998684	+	0.0512772i	$0.0163292\pi$
$761$	15.0000	+	25.9808i	0.543750	+	0.941802i	−0.454935	+	0.890525i	$0.650337\pi$
$762$		0			0
$763$	−8.00000	+	13.8564i	−0.289619	+	0.501636i
$764$		0			0
$765$	−18.0000	+	31.1769i	−0.650791	+	1.12720i
$766$		0			0
$767$	−5.00000	+	8.66025i	−0.180540	+	0.312704i
$768$		0			0
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	0.761680	−	0.647954i	$-0.224375\pi$
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	−0.941984	+	0.335657i	$0.891042\pi$
$770$		0			0
$771$		−	8.66025i		−	0.311891i
$772$		0			0
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$	44.0000			1.58053
$776$		0			0
$777$	−24.0000	+	13.8564i	−0.860995	+	0.497096i
$778$		0			0
$779$	0.500000	+	0.866025i	0.0179144	+	0.0310286i
$780$		0			0
$781$	20.0000	−	34.6410i	0.715656	−	1.23955i
$782$		0			0
$783$	−9.00000	−	5.19615i	−0.321634	−	0.185695i
$784$		0			0
$785$	−40.0000	+	69.2820i	−1.42766	+	2.47278i
$786$		0			0
$787$	−18.0000	−	31.1769i	−0.641631	−	1.11134i	−0.985069	−	0.172162i	$-0.944925\pi$
$787$	−18.0000	−	31.1769i	−0.641631	−	1.11134i	0.343438	−	0.939175i	$-0.388408\pi$
$788$		0			0
$789$	−9.00000	+	5.19615i	−0.320408	+	0.184988i
$790$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-2.00000 - 3.46410i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-2.50000 + 4.33013i) q^{11} +(1.00000 + 1.73205i) q^{13} +6.92820i q^{15} -3.00000 q^{17} -1.00000 q^{19} +(3.00000 - 1.73205i) q^{21} +(-3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} -5.19615i q^{27} +(1.00000 - 1.73205i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(7.50000 - 4.33013i) q^{33} +8.00000 q^{35} -8.00000 q^{37} -3.46410i q^{39} +(-0.500000 - 0.866025i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(6.00000 - 10.3923i) q^{45} +(1.00000 - 1.73205i) q^{47} +(1.50000 + 2.59808i) q^{49} +(4.50000 + 2.59808i) q^{51} -4.00000 q^{53} +20.0000 q^{55} +(1.50000 + 0.866025i) q^{57} +(2.50000 + 4.33013i) q^{59} -6.00000 q^{63} +(4.00000 - 6.92820i) q^{65} +(-6.50000 - 11.2583i) q^{67} +10.3923i q^{69} -8.00000 q^{71} +3.00000 q^{73} +(16.5000 - 9.52628i) q^{75} +(-5.00000 - 8.66025i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(6.00000 - 10.3923i) q^{83} +(6.00000 + 10.3923i) q^{85} +(-3.00000 + 1.73205i) q^{87} -10.0000 q^{89} -4.00000 q^{91} +6.92820i q^{93} +(2.00000 + 3.46410i) q^{95} +(5.50000 - 9.52628i) q^{97} -15.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9	\( 2 q - 3 q^{3} - 4 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{19} + 6 q^{21} - 6 q^{23} - 11 q^{25} + 2 q^{29} - 4 q^{31} + 15 q^{33} + 16 q^{35} - 16 q^{37} - q^{41} - 7 q^{43} + 12 q^{45} + 2 q^{47} + 3 q^{49} + 9 q^{51} - 8 q^{53} + 40 q^{55} + 3 q^{57} + 5 q^{59} - 12 q^{63} + 8 q^{65} - 13 q^{67} - 16 q^{71} + 6 q^{73} + 33 q^{75} - 10 q^{77} + 8 q^{79} - 9 q^{81} + 12 q^{83} + 12 q^{85} - 6 q^{87} - 20 q^{89} - 8 q^{91} + 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 4 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 2 * q^13 - 6 * q^17 - 2 * q^19 + 6 * q^21 - 6 * q^23 - 11 * q^25 + 2 * q^29 - 4 * q^31 + 15 * q^33 + 16 * q^35 - 16 * q^37 - q^41 - 7 * q^43 + 12 * q^45 + 2 * q^47 + 3 * q^49 + 9 * q^51 - 8 * q^53 + 40 * q^55 + 3 * q^57 + 5 * q^59 - 12 * q^63 + 8 * q^65 - 13 * q^67 - 16 * q^71 + 6 * q^73 + 33 * q^75 - 10 * q^77 + 8 * q^79 - 9 * q^81 + 12 * q^83 + 12 * q^85 - 6 * q^87 - 20 * q^89 - 8 * q^91 + 4 * q^95 + 11 * q^97 - 30 * q^99

Introduction

L-functions

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