LMFDB - Embedded newform 225.2.k.a.124.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(49,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.k (of order $6$, 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:	$1.79663404548$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			124.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	225.124
Dual form			225.2.k.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 1.50000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.73205i q^{6} +(2.59808 + 1.50000i) q^{7} +3.00000i q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 1.50000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.73205i q^{6} +(2.59808 + 1.50000i) q^{7} +3.00000i q^{8} +(-1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} +(0.866025 - 1.50000i) q^{12} +(1.73205 - 1.00000i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(0.500000 - 0.866025i) q^{16} +4.00000i q^{17} +(2.59808 - 1.50000i) q^{18} +8.00000 q^{19} +5.19615i q^{21} +(-1.73205 + 1.00000i) q^{22} +(-2.59808 + 1.50000i) q^{23} +(-4.50000 + 2.59808i) q^{24} -2.00000 q^{26} -5.19615 q^{27} -3.00000i q^{28} +(-0.500000 + 0.866025i) q^{29} +(4.33013 - 2.50000i) q^{32} +3.46410 q^{33} +(2.00000 - 3.46410i) q^{34} +3.00000 q^{36} -4.00000i q^{37} +(-6.92820 - 4.00000i) q^{38} +(3.00000 + 1.73205i) q^{39} +(-2.50000 - 4.33013i) q^{41} +(2.59808 - 4.50000i) q^{42} +(-6.92820 - 4.00000i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(-6.06218 - 3.50000i) q^{47} +1.73205 q^{48} +(1.00000 + 1.73205i) q^{49} +(-6.00000 + 3.46410i) q^{51} +(-1.73205 - 1.00000i) q^{52} +2.00000i q^{53} +(4.50000 + 2.59808i) q^{54} +(-4.50000 + 7.79423i) q^{56} +(6.92820 + 12.0000i) q^{57} +(0.866025 - 0.500000i) q^{58} +(-7.00000 - 12.1244i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(-7.79423 + 4.50000i) q^{63} -7.00000 q^{64} +(-3.00000 - 1.73205i) q^{66} +(-2.59808 + 1.50000i) q^{67} +(3.46410 - 2.00000i) q^{68} +(-4.50000 - 2.59808i) q^{69} +2.00000 q^{71} +(-7.79423 - 4.50000i) q^{72} -4.00000i q^{73} +(-2.00000 + 3.46410i) q^{74} +(-4.00000 - 6.92820i) q^{76} +(5.19615 - 3.00000i) q^{77} +(-1.73205 - 3.00000i) q^{78} +(-3.00000 + 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +5.00000i q^{82} +(7.79423 + 4.50000i) q^{83} +(4.50000 - 2.59808i) q^{84} +(4.00000 + 6.92820i) q^{86} -1.73205 q^{87} +(5.19615 + 3.00000i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(2.59808 + 1.50000i) q^{92} +(3.50000 + 6.06218i) q^{94} +(7.50000 + 4.33013i) q^{96} +(-1.73205 - 1.00000i) q^{97} -2.00000i q^{98} +(3.00000 + 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^4 - 6 * q^9	$ 4 q - 2 q^{4} - 6 q^{9} + 4 q^{11} - 6 q^{14} + 2 q^{16} + 32 q^{19} - 18 q^{24} - 8 q^{26} - 2 q^{29} + 8 q^{34} + 12 q^{36} + 12 q^{39} - 10 q^{41} - 8 q^{44} + 12 q^{46} + 4 q^{49} - 24 q^{51} + 18 q^{54} - 18 q^{56} - 28 q^{59} - 14 q^{61} - 28 q^{64} - 12 q^{66} - 18 q^{69} + 8 q^{71} - 8 q^{74} - 16 q^{76} - 12 q^{79} - 18 q^{81} + 18 q^{84} + 16 q^{86} + 60 q^{89} + 24 q^{91} + 14 q^{94} + 30 q^{96} + 12 q^{99}+O(q^{100}) $ 4 * q - 2 * q^4 - 6 * q^9 + 4 * q^11 - 6 * q^14 + 2 * q^16 + 32 * q^19 - 18 * q^24 - 8 * q^26 - 2 * q^29 + 8 * q^34 + 12 * q^36 + 12 * q^39 - 10 * q^41 - 8 * q^44 + 12 * q^46 + 4 * q^49 - 24 * q^51 + 18 * q^54 - 18 * q^56 - 28 * q^59 - 14 * q^61 - 28 * q^64 - 12 * q^66 - 18 * q^69 + 8 * q^71 - 8 * q^74 - 16 * q^76 - 12 * q^79 - 18 * q^81 + 18 * q^84 + 16 * q^86 + 60 * q^89 + 24 * q^91 + 14 * q^94 + 30 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i	0.161521	−	0.986869i	$-0.448360\pi$
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i	−0.773893	+	0.633316i	$0.781693\pi$
$3$	0.866025	+	1.50000i	0.500000	+	0.866025i
$3$	0.866025	+	1.50000i	0.500000	+	0.866025i
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0
$5$		0			0
$6$		−	1.73205i		−	0.707107i
$7$	2.59808	+	1.50000i	0.981981	+	0.566947i	0.902867	−	0.429919i	$-0.141458\pi$
$7$	2.59808	+	1.50000i	0.981981	+	0.566947i	0.0791130	+	0.996866i	$0.474791\pi$
$8$			3.00000i			1.06066i
$9$	−1.50000	+	2.59808i	−0.500000	+	0.866025i
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$	0.866025	−	1.50000i	0.250000	−	0.433013i
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	0.720577	+	0.693375i	$0.243877\pi$
$14$	−1.50000	−	2.59808i	−0.400892	−	0.694365i
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	2.59808	−	1.50000i	0.612372	−	0.353553i
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$			5.19615i			1.13389i
$22$	−1.73205	+	1.00000i	−0.369274	+	0.213201i
$23$	−2.59808	+	1.50000i	−0.541736	+	0.312772i	−0.745782	−	0.666190i	$-0.767924\pi$
$23$	−2.59808	+	1.50000i	−0.541736	+	0.312772i	0.204046	+	0.978961i	$0.434591\pi$
$24$	−4.50000	+	2.59808i	−0.918559	+	0.530330i
$25$		0			0
$26$	−2.00000			−0.392232
$27$	−5.19615			−1.00000
$28$		−	3.00000i		−	0.566947i
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	−0.908708	−	0.417432i	$-0.862930\pi$
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	0.815861	+	0.578249i	$0.196264\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	4.33013	−	2.50000i	0.765466	−	0.441942i
$33$	3.46410			0.603023
$34$	2.00000	−	3.46410i	0.342997	−	0.594089i
$35$		0			0
$36$	3.00000			0.500000
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	−6.92820	−	4.00000i	−1.12390	−	0.648886i
$39$	3.00000	+	1.73205i	0.480384	+	0.277350i
$40$		0			0
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	0.602072	−	0.798441i	$-0.294342\pi$
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	−0.992507	+	0.122189i	$0.961009\pi$
$42$	2.59808	−	4.50000i	0.400892	−	0.694365i
$43$	−6.92820	−	4.00000i	−1.05654	−	0.609994i	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	−6.92820	−	4.00000i	−1.05654	−	0.609994i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	3.00000			0.442326
$47$	−6.06218	−	3.50000i	−0.884260	−	0.510527i	−0.0121990	−	0.999926i	$-0.503883\pi$
$47$	−6.06218	−	3.50000i	−0.884260	−	0.510527i	−0.872060	+	0.489398i	$0.837217\pi$
$48$	1.73205			0.250000
$49$	1.00000	+	1.73205i	0.142857	+	0.247436i
$50$		0			0
$51$	−6.00000	+	3.46410i	−0.840168	+	0.485071i
$52$	−1.73205	−	1.00000i	−0.240192	−	0.138675i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	4.50000	+	2.59808i	0.612372	+	0.353553i
$55$		0			0
$56$	−4.50000	+	7.79423i	−0.601338	+	1.04155i
$57$	6.92820	+	12.0000i	0.917663	+	1.58944i
$58$	0.866025	−	0.500000i	0.113715	−	0.0656532i
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.812198	−	0.583382i	$-0.801729\pi$
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.0991242	−	0.995075i	$-0.531604\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$	−7.79423	+	4.50000i	−0.981981	+	0.566947i
$64$	−7.00000			−0.875000
$65$		0			0
$66$	−3.00000	−	1.73205i	−0.369274	−	0.213201i
$67$	−2.59808	+	1.50000i	−0.317406	+	0.183254i	−0.650236	−	0.759733i	$-0.725330\pi$
$67$	−2.59808	+	1.50000i	−0.317406	+	0.183254i	0.332830	+	0.942987i	$0.391996\pi$
$68$	3.46410	−	2.00000i	0.420084	−	0.242536i
$69$	−4.50000	−	2.59808i	−0.541736	−	0.312772i
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$	−7.79423	−	4.50000i	−0.918559	−	0.530330i
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$	−4.00000	−	6.92820i	−0.458831	−	0.794719i
$77$	5.19615	−	3.00000i	0.592157	−	0.341882i
$78$	−1.73205	−	3.00000i	−0.196116	−	0.339683i
$79$	−3.00000	+	5.19615i	−0.337526	+	0.584613i	−0.983967	−	0.178352i	$-0.942924\pi$
$79$	−3.00000	+	5.19615i	−0.337526	+	0.584613i	0.646440	+	0.762964i	$0.276257\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$			5.00000i			0.552158i
$83$	7.79423	+	4.50000i	0.855528	+	0.493939i	0.862512	−	0.506036i	$-0.168890\pi$
$83$	7.79423	+	4.50000i	0.855528	+	0.493939i	−0.00698436	+	0.999976i	$0.502223\pi$
$84$	4.50000	−	2.59808i	0.490990	−	0.283473i
$85$		0			0
$86$	4.00000	+	6.92820i	0.431331	+	0.747087i
$87$	−1.73205			−0.185695
$88$	5.19615	+	3.00000i	0.553912	+	0.319801i
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	2.59808	+	1.50000i	0.270868	+	0.156386i
$93$		0			0
$94$	3.50000	+	6.06218i	0.360997	+	0.625266i
$95$		0			0
$96$	7.50000	+	4.33013i	0.765466	+	0.441942i
$97$	−1.73205	−	1.00000i	−0.175863	−	0.101535i	0.409484	−	0.912317i	$-0.365709\pi$
$97$	−1.73205	−	1.00000i	−0.175863	−	0.101535i	−0.585348	+	0.810782i	$0.699042\pi$
$98$		−	2.00000i		−	0.202031i
$99$	3.00000	+	5.19615i	0.301511	+	0.522233i
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$	6.92820			0.685994
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	−0.800855	−	0.598858i	$-0.795621\pi$
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	0.118199	+	0.992990i	$0.462288\pi$
$104$	3.00000	+	5.19615i	0.294174	+	0.509525i
$105$		0			0
$106$	1.00000	−	1.73205i	0.0971286	−	0.168232i
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	2.59808	+	4.50000i	0.250000	+	0.433013i
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$	6.00000	−	3.46410i	0.569495	−	0.328798i
$112$	2.59808	−	1.50000i	0.245495	−	0.141737i
$113$	6.92820	−	4.00000i	0.651751	−	0.376288i	−0.137376	−	0.990519i	$-0.543867\pi$
$113$	6.92820	−	4.00000i	0.651751	−	0.376288i	0.789127	+	0.614231i	$0.210534\pi$
$114$		−	13.8564i		−	1.29777i
$115$		0			0
$116$	1.00000			0.0928477
$117$			6.00000i			0.554700i
$118$			14.0000i			1.28880i
$119$	−6.00000	+	10.3923i	−0.550019	+	0.952661i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	6.06218	−	3.50000i	0.548844	−	0.316875i
$123$	4.33013	−	7.50000i	0.390434	−	0.676252i
$124$		0			0
$125$		0			0
$126$	9.00000			0.801784
$127$		−	5.00000i		−	0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$		−	5.00000i		−	0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	−2.59808	−	1.50000i	−0.229640	−	0.132583i
$129$		−	13.8564i		−	1.21999i
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$	−1.73205	−	3.00000i	−0.150756	−	0.261116i
$133$	20.7846	+	12.0000i	1.80225	+	1.04053i
$134$	3.00000			0.259161
$135$		0			0
$136$	−12.0000			−1.02899
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.0146279	−	0.999893i	$-0.504656\pi$
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.873247	+	0.487278i	$0.837990\pi$
$138$	2.59808	+	4.50000i	0.221163	+	0.383065i
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		−	12.1244i		−	1.02105i
$142$	−1.73205	−	1.00000i	−0.145350	−	0.0839181i
$143$		−	4.00000i		−	0.334497i
$144$	1.50000	+	2.59808i	0.125000	+	0.216506i
$145$		0			0
$146$	−2.00000	+	3.46410i	−0.165521	+	0.286691i
$147$	−1.73205	+	3.00000i	−0.142857	+	0.247436i
$148$	−3.46410	+	2.00000i	−0.284747	+	0.164399i
$149$	8.50000	+	14.7224i	0.696347	+	1.20611i	0.969724	+	0.244202i	$0.0785259\pi$
$149$	8.50000	+	14.7224i	0.696347	+	1.20611i	−0.273377	+	0.961907i	$0.588141\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	0.903842	+	0.427865i	$0.140734\pi$
$152$			24.0000i			1.94666i
$153$	−10.3923	−	6.00000i	−0.840168	−	0.485071i
$154$	−6.00000			−0.483494
$155$		0			0
$156$		−	3.46410i		−	0.277350i
$157$	12.1244	−	7.00000i	0.967629	−	0.558661i	0.0691164	−	0.997609i	$-0.477982\pi$
$157$	12.1244	−	7.00000i	0.967629	−	0.558661i	0.898513	+	0.438948i	$0.144649\pi$
$158$	5.19615	−	3.00000i	0.413384	−	0.238667i
$159$	−3.00000	+	1.73205i	−0.237915	+	0.137361i
$160$		0			0
$161$	−9.00000			−0.709299
$162$			9.00000i			0.707107i
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.50000	+	4.33013i	−0.195217	+	0.338126i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	−7.79423	+	4.50000i	−0.603136	+	0.348220i	−0.770274	−	0.637713i	$-0.779881\pi$
$167$	−7.79423	+	4.50000i	−0.603136	+	0.348220i	0.167139	+	0.985933i	$0.446547\pi$
$168$	−15.5885			−1.20268
$169$	−4.50000	+	7.79423i	−0.346154	+	0.599556i
$170$		0			0
$171$	−12.0000	+	20.7846i	−0.917663	+	1.58944i
$172$			8.00000i			0.609994i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	1.50000	+	0.866025i	0.113715	+	0.0656532i
$175$		0			0
$176$	−1.00000	−	1.73205i	−0.0753778	−	0.130558i
$177$	12.1244	−	21.0000i	0.911322	−	1.57846i
$178$	−12.9904	−	7.50000i	−0.973670	−	0.562149i
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	−5.19615	−	3.00000i	−0.385164	−	0.222375i
$183$	−12.1244			−0.896258
$184$	−4.50000	−	7.79423i	−0.331744	−	0.574598i
$185$		0			0
$186$		0			0
$187$	6.92820	+	4.00000i	0.506640	+	0.292509i
$188$			7.00000i			0.510527i
$189$	−13.5000	−	7.79423i	−0.981981	−	0.566947i
$190$		0			0
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	−0.973674	−	0.227946i	$-0.926799\pi$
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	0.684244	+	0.729253i	$0.260132\pi$
$192$	−6.06218	−	10.5000i	−0.437500	−	0.757772i
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	−0.154805	−	0.987945i	$-0.549475\pi$
$193$	8.66025	−	5.00000i	0.623379	−	0.359908i	0.778183	+	0.628037i	$0.216141\pi$
$194$	1.00000	+	1.73205i	0.0717958	+	0.124354i
$195$		0			0
$196$	1.00000	−	1.73205i	0.0714286	−	0.123718i
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		−	6.00000i		−	0.426401i
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	−4.50000	−	2.59808i	−0.317406	−	0.183254i
$202$	−15.5885	+	9.00000i	−1.09680	+	0.633238i
$203$	−2.59808	+	1.50000i	−0.182349	+	0.105279i
$204$	6.00000	+	3.46410i	0.420084	+	0.242536i
$205$		0			0
$206$	8.00000			0.557386
$207$		−	9.00000i		−	0.625543i
$208$		−	2.00000i		−	0.138675i
$209$	8.00000	−	13.8564i	0.553372	−	0.958468i
$210$		0			0
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	0.944237	+	0.329266i	$0.106801\pi$
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	−0.186966	+	0.982366i	$0.559865\pi$
$212$	1.73205	−	1.00000i	0.118958	−	0.0686803i
$213$	1.73205	+	3.00000i	0.118678	+	0.205557i
$214$	1.50000	−	2.59808i	0.102538	−	0.177601i
$215$		0			0
$216$		−	15.5885i		−	1.06066i
$217$		0			0
$218$	4.33013	+	2.50000i	0.293273	+	0.169321i
$219$	6.00000	−	3.46410i	0.405442	−	0.234082i
$220$		0			0
$221$	4.00000	+	6.92820i	0.269069	+	0.466041i
$222$	−6.92820			−0.464991
$223$	−16.4545	−	9.50000i	−1.10187	−	0.636167i	−0.165161	−	0.986267i	$-0.552814\pi$
$223$	−16.4545	−	9.50000i	−1.10187	−	0.636167i	−0.936713	+	0.350100i	$0.886148\pi$
$224$	15.0000			1.00223
$225$		0			0
$226$	−8.00000			−0.532152
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	0.610535	−	0.791989i	$-0.290954\pi$
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	−0.380615	+	0.924734i	$0.624288\pi$
$228$	6.92820	−	12.0000i	0.458831	−	0.794719i
$229$	7.50000	+	12.9904i	0.495614	+	0.858429i	0.999987	−	0.00505719i	$-0.00160976\pi$
$229$	7.50000	+	12.9904i	0.495614	+	0.858429i	−0.504373	+	0.863486i	$0.668276\pi$
$230$		0			0
$231$	9.00000	+	5.19615i	0.592157	+	0.341882i
$232$	−2.59808	−	1.50000i	−0.170572	−	0.0984798i
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	3.00000	−	5.19615i	0.196116	−	0.339683i
$235$		0			0
$236$	−7.00000	+	12.1244i	−0.455661	+	0.789228i
$237$	−10.3923			−0.675053
$238$	10.3923	−	6.00000i	0.673633	−	0.388922i
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	0.707166	−	0.707048i	$-0.249973\pi$
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	−0.965904	+	0.258900i	$0.916640\pi$
$240$		0			0
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	−0.632709	−	0.774389i	$-0.718057\pi$
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	0.986996	+	0.160748i	$0.0513906\pi$
$242$		−	7.00000i		−	0.449977i
$243$	7.79423	−	13.5000i	0.500000	−	0.866025i
$244$	7.00000			0.448129
$245$		0			0
$246$	−7.50000	+	4.33013i	−0.478183	+	0.276079i
$247$	13.8564	−	8.00000i	0.881662	−	0.509028i
$248$		0			0
$249$			15.5885i			0.987878i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	7.79423	+	4.50000i	0.490990	+	0.283473i
$253$			6.00000i			0.377217i
$254$	−2.50000	+	4.33013i	−0.156864	+	0.271696i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	−5.19615	+	3.00000i	−0.324127	+	0.187135i	−0.653231	−	0.757159i	$-0.726587\pi$
$257$	−5.19615	+	3.00000i	−0.324127	+	0.187135i	0.329104	+	0.944294i	$0.393253\pi$
$258$	−6.92820	+	12.0000i	−0.431331	+	0.747087i
$259$	6.00000	−	10.3923i	0.372822	−	0.645746i
$260$		0			0
$261$	−1.50000	−	2.59808i	−0.0928477	−	0.160817i
$262$		−	6.00000i		−	0.370681i
$263$	−13.8564	−	8.00000i	−0.854423	−	0.493301i	0.00771799	−	0.999970i	$-0.497543\pi$
$263$	−13.8564	−	8.00000i	−0.854423	−	0.493301i	−0.862141	+	0.506669i	$0.830877\pi$
$264$			10.3923i			0.639602i
$265$		0			0
$266$	−12.0000	−	20.7846i	−0.735767	−	1.27439i
$267$	12.9904	+	22.5000i	0.794998	+	1.37698i
$268$	2.59808	+	1.50000i	0.158703	+	0.0916271i
$269$	−25.0000			−1.52428			−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000			−1.52428			−0.762138	+	0.647414i	$0.775850\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	3.46410	+	2.00000i	0.210042	+	0.121268i
$273$	5.19615	+	9.00000i	0.314485	+	0.544705i
$274$	6.00000	+	10.3923i	0.362473	+	0.627822i
$275$		0			0
$276$			5.19615i			0.312772i
$277$	−10.3923	−	6.00000i	−0.624413	−	0.360505i	0.154172	−	0.988044i	$-0.450729\pi$
$277$	−10.3923	−	6.00000i	−0.624413	−	0.360505i	−0.778585	+	0.627539i	$0.784062\pi$
$278$			16.0000i			0.959616i
$279$		0			0
$280$		0			0
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	−0.550804	−	0.834634i	$-0.685679\pi$
$281$	7.50000	−	12.9904i	0.447412	−	0.774941i	0.998217	+	0.0596933i	$0.0190123\pi$
$282$	−6.06218	+	10.5000i	−0.360997	+	0.625266i
$283$	−18.1865	+	10.5000i	−1.08108	+	0.624160i	−0.931187	−	0.364542i	$-0.881225\pi$
$283$	−18.1865	+	10.5000i	−1.08108	+	0.624160i	−0.149890	+	0.988703i	$0.547892\pi$
$284$	−1.00000	−	1.73205i	−0.0593391	−	0.102778i
$285$		0			0
$286$	−2.00000	+	3.46410i	−0.118262	+	0.204837i
$287$		−	15.0000i		−	0.885422i
$288$			15.0000i			0.883883i
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	3.46410i		−	0.203069i
$292$	−3.46410	+	2.00000i	−0.202721	+	0.117041i
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	−0.771839	−	0.635818i	$-0.780663\pi$
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	0.164714	+	0.986341i	$0.447330\pi$
$294$	3.00000	−	1.73205i	0.174964	−	0.101015i
$295$		0			0
$296$	12.0000			0.697486
$297$	−5.19615	+	9.00000i	−0.301511	+	0.522233i
$298$		−	17.0000i		−	0.984784i
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−12.0000	−	20.7846i	−0.691669	−	1.19800i
$302$	−1.73205	+	1.00000i	−0.0996683	+	0.0575435i
$303$	31.1769			1.79107
$304$	4.00000	−	6.92820i	0.229416	−	0.397360i
$305$		0			0
$306$	6.00000	+	10.3923i	0.342997	+	0.594089i
$307$			7.00000i			0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$			7.00000i			0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	−5.19615	−	3.00000i	−0.296078	−	0.170941i
$309$	−12.0000	−	6.92820i	−0.682656	−	0.394132i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$	−5.19615	+	9.00000i	−0.294174	+	0.509525i
$313$	12.1244	+	7.00000i	0.685309	+	0.395663i	0.801852	−	0.597522i	$-0.203848\pi$
$313$	12.1244	+	7.00000i	0.685309	+	0.395663i	−0.116543	+	0.993186i	$0.537181\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	6.00000			0.337526
$317$	29.4449	+	17.0000i	1.65379	+	0.954815i	0.975494	+	0.220024i	$0.0706137\pi$
$317$	29.4449	+	17.0000i	1.65379	+	0.954815i	0.678294	+	0.734791i	$0.262720\pi$
$318$	3.46410			0.194257
$319$	1.00000	+	1.73205i	0.0559893	+	0.0969762i
$320$		0			0
$321$	−4.50000	+	2.59808i	−0.251166	+	0.145010i
$322$	7.79423	+	4.50000i	0.434355	+	0.250775i
$323$			32.0000i			1.78053i
$324$	−4.50000	+	7.79423i	−0.250000	+	0.433013i
$325$		0			0
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$	−4.33013	−	7.50000i	−0.239457	−	0.414751i
$328$	12.9904	−	7.50000i	0.717274	−	0.414118i
$329$	−10.5000	−	18.1865i	−0.578884	−	1.00266i
$330$		0			0
$331$	3.00000	−	5.19615i	0.164895	−	0.285606i	−0.771723	−	0.635959i	$-0.780605\pi$
$331$	3.00000	−	5.19615i	0.164895	−	0.285606i	0.936618	+	0.350352i	$0.113938\pi$
$332$		−	9.00000i		−	0.493939i
$333$	10.3923	+	6.00000i	0.569495	+	0.328798i
$334$	9.00000			0.492458
$335$		0			0
$336$	4.50000	+	2.59808i	0.245495	+	0.141737i
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	−0.676688	−	0.736270i	$-0.736585\pi$
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	0.299285	+	0.954164i	$0.403252\pi$
$338$	7.79423	−	4.50000i	0.423950	−	0.244768i
$339$	12.0000	+	6.92820i	0.651751	+	0.376288i
$340$		0			0
$341$		0			0
$342$	20.7846	−	12.0000i	1.12390	−	0.648886i
$343$		−	15.0000i		−	0.809924i
$344$	12.0000	−	20.7846i	0.646997	−	1.12063i
$345$		0			0
$346$		0			0
$347$	3.46410	−	2.00000i	0.185963	−	0.107366i	−0.404128	−	0.914702i	$-0.632425\pi$
$347$	3.46410	−	2.00000i	0.185963	−	0.107366i	0.590091	+	0.807337i	$0.299092\pi$
$348$	0.866025	+	1.50000i	0.0464238	+	0.0804084i
$349$	−2.50000	+	4.33013i	−0.133822	+	0.231786i	−0.925147	−	0.379610i	$-0.876058\pi$
$349$	−2.50000	+	4.33013i	−0.133822	+	0.231786i	0.791325	+	0.611396i	$0.209392\pi$
$350$		0			0
$351$	−9.00000	+	5.19615i	−0.480384	+	0.277350i
$352$		−	10.0000i		−	0.533002i
$353$	20.7846	+	12.0000i	1.10625	+	0.638696i	0.937856	−	0.347024i	$-0.112808\pi$
$353$	20.7846	+	12.0000i	1.10625	+	0.638696i	0.168397	+	0.985719i	$0.446141\pi$
$354$	−21.0000	+	12.1244i	−1.11614	+	0.644402i
$355$		0			0
$356$	−7.50000	−	12.9904i	−0.397499	−	0.688489i
$357$	−20.7846			−1.10004
$358$	−1.73205	−	1.00000i	−0.0915417	−	0.0528516i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$	6.06218	+	3.50000i	0.318621	+	0.183956i
$363$	−6.06218	+	10.5000i	−0.318182	+	0.551107i
$364$	−3.00000	−	5.19615i	−0.157243	−	0.272352i
$365$		0			0
$366$	10.5000	+	6.06218i	0.548844	+	0.316875i
$367$	−20.7846	−	12.0000i	−1.08495	−	0.626395i	−0.152721	−	0.988269i	$-0.548804\pi$
$367$	−20.7846	−	12.0000i	−1.08495	−	0.626395i	−0.932227	+	0.361874i	$0.882137\pi$
$368$			3.00000i			0.156386i
$369$	15.0000			0.780869
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	−8.66025	+	5.00000i	−0.448411	+	0.258890i	−0.707159	−	0.707055i	$-0.750023\pi$
$373$	−8.66025	+	5.00000i	−0.448411	+	0.258890i	0.258748	+	0.965945i	$0.416690\pi$
$374$	−4.00000	−	6.92820i	−0.206835	−	0.358249i
$375$		0			0
$376$	10.5000	−	18.1865i	0.541496	−	0.937899i
$377$			2.00000i			0.103005i
$378$	7.79423	+	13.5000i	0.400892	+	0.694365i
$379$	26.0000			1.33553			0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000			1.33553			0.667765	+	0.744372i	$0.267251\pi$
$380$		0			0
$381$	7.50000	−	4.33013i	0.384237	−	0.221839i
$382$	6.92820	−	4.00000i	0.354478	−	0.204658i
$383$	−31.1769	+	18.0000i	−1.59307	+	0.919757i	−0.600289	+	0.799783i	$0.704948\pi$
$383$	−31.1769	+	18.0000i	−1.59307	+	0.919757i	−0.992777	+	0.119974i	$0.961719\pi$
$384$		−	5.19615i		−	0.265165i
$385$		0			0
$386$	−10.0000			−0.508987
$387$	20.7846	−	12.0000i	1.05654	−	0.609994i
$388$			2.00000i			0.101535i
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	−0.0561516	−	0.998422i	$-0.517883\pi$
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	0.892735	−	0.450582i	$-0.148784\pi$
$390$		0			0
$391$	−6.00000	−	10.3923i	−0.303433	−	0.525561i
$392$	−5.19615	+	3.00000i	−0.262445	+	0.151523i
$393$	−5.19615	+	9.00000i	−0.262111	+	0.453990i
$394$	−6.00000	+	10.3923i	−0.302276	+	0.523557i
$395$		0			0
$396$	3.00000	−	5.19615i	0.150756	−	0.261116i
$397$			34.0000i			1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$			34.0000i			1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	3.46410	+	2.00000i	0.173640	+	0.100251i
$399$			41.5692i			2.08106i
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$	2.59808	+	4.50000i	0.129580	+	0.224440i
$403$		0			0
$404$	−18.0000			−0.895533
$405$		0			0
$406$	3.00000			0.148888
$407$	−6.92820	−	4.00000i	−0.343418	−	0.198273i
$408$	−10.3923	−	18.0000i	−0.514496	−	0.891133i
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$		−	20.7846i		−	1.02523i
$412$	6.92820	+	4.00000i	0.341328	+	0.197066i
$413$		−	42.0000i		−	2.06668i
$414$	−4.50000	+	7.79423i	−0.221163	+	0.383065i
$415$		0			0
$416$	5.00000	−	8.66025i	0.245145	−	0.424604i
$417$	13.8564	−	24.0000i	0.678551	−	1.17529i
$418$	−13.8564	+	8.00000i	−0.677739	+	0.391293i
$419$	13.0000	+	22.5167i	0.635092	+	1.10001i	0.986496	+	0.163787i	$0.0523710\pi$
$419$	13.0000	+	22.5167i	0.635092	+	1.10001i	−0.351404	+	0.936224i	$0.614296\pi$
$420$		0			0
$421$	−17.0000	+	29.4449i	−0.828529	+	1.43505i	0.0706626	+	0.997500i	$0.477489\pi$
$421$	−17.0000	+	29.4449i	−0.828529	+	1.43505i	−0.899192	+	0.437555i	$0.855845\pi$
$422$		−	22.0000i		−	1.07094i
$423$	18.1865	−	10.5000i	0.884260	−	0.510527i
$424$	−6.00000			−0.291386
$425$		0			0
$426$		−	3.46410i		−	0.167836i
$427$	−18.1865	+	10.5000i	−0.880108	+	0.508131i
$428$	2.59808	−	1.50000i	0.125583	−	0.0725052i
$429$	6.00000	−	3.46410i	0.289683	−	0.167248i
$430$		0			0
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$	−2.59808	+	4.50000i	−0.125000	+	0.216506i
$433$		−	28.0000i		−	1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$		−	28.0000i		−	1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$		0			0
$435$		0			0
$436$	2.50000	+	4.33013i	0.119728	+	0.207375i
$437$	−20.7846	+	12.0000i	−0.994263	+	0.574038i
$438$	−6.92820			−0.331042
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		−	8.00000i		−	0.380521i
$443$	12.9904	+	7.50000i	0.617192	+	0.356336i	0.775775	−	0.631010i	$-0.217359\pi$
$443$	12.9904	+	7.50000i	0.617192	+	0.356336i	−0.158583	+	0.987346i	$0.550693\pi$
$444$	−6.00000	−	3.46410i	−0.284747	−	0.164399i
$445$		0			0
$446$	9.50000	+	16.4545i	0.449838	+	0.779142i
$447$	−14.7224	+	25.5000i	−0.696347	+	1.20611i
$448$	−18.1865	−	10.5000i	−0.859233	−	0.496078i
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$	−6.92820	−	4.00000i	−0.325875	−	0.188144i
$453$	3.46410			0.162758
$454$	−2.00000	−	3.46410i	−0.0938647	−	0.162578i
$455$		0			0
$456$	−36.0000	+	20.7846i	−1.68585	+	0.973329i
$457$	17.3205	+	10.0000i	0.810219	+	0.467780i	0.847032	−	0.531542i	$-0.178387\pi$
$457$	17.3205	+	10.0000i	0.810219	+	0.467780i	−0.0368128	+	0.999322i	$0.511721\pi$
$458$		−	15.0000i		−	0.700904i
$459$		−	20.7846i		−	0.970143i
$460$		0			0
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	−0.951584	−	0.307388i	$-0.900545\pi$
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	0.741998	+	0.670402i	$0.233878\pi$
$462$	−5.19615	−	9.00000i	−0.241747	−	0.418718i
$463$	31.1769	−	18.0000i	1.44891	−	0.836531i	0.450497	−	0.892778i	$-0.351247\pi$
$463$	31.1769	−	18.0000i	1.44891	−	0.836531i	0.998417	+	0.0562469i	$0.0179134\pi$
$464$	0.500000	+	0.866025i	0.0232119	+	0.0402042i
$465$		0			0
$466$	−12.0000	+	20.7846i	−0.555889	+	0.962828i
$467$			20.0000i			0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$			20.0000i			0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	5.19615	−	3.00000i	0.240192	−	0.138675i
$469$	−9.00000			−0.415581
$470$		0			0
$471$	21.0000	+	12.1244i	0.967629	+	0.558661i
$472$	36.3731	−	21.0000i	1.67421	−	0.966603i
$473$	−13.8564	+	8.00000i	−0.637118	+	0.367840i
$474$	9.00000	+	5.19615i	0.413384	+	0.238667i
$475$		0			0
$476$	12.0000			0.550019
$477$	−5.19615	−	3.00000i	−0.237915	−	0.137361i
$478$			8.00000i			0.365911i
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	−0.995023	−	0.0996406i	$-0.968231\pi$
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	0.583803	+	0.811895i	$0.301564\pi$
$480$		0			0
$481$	−4.00000	−	6.92820i	−0.182384	−	0.315899i
$482$	−9.52628	+	5.50000i	−0.433910	+	0.250518i
$483$	−7.79423	−	13.5000i	−0.354650	−	0.614271i
$484$	3.50000	−	6.06218i	0.159091	−	0.275554i
$485$		0			0
$486$	−13.5000	+	7.79423i	−0.612372	+	0.353553i
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	−18.1865	−	10.5000i	−0.823266	−	0.475313i
$489$	−6.00000	+	3.46410i	−0.271329	+	0.156652i
$490$		0			0
$491$	−10.0000	−	17.3205i	−0.451294	−	0.781664i	0.547173	−	0.837020i	$-0.315704\pi$
$491$	−10.0000	−	17.3205i	−0.451294	−	0.781664i	−0.998467	+	0.0553560i	$0.982371\pi$
$492$	−8.66025			−0.390434
$493$	−3.46410	−	2.00000i	−0.156015	−	0.0900755i
$494$	−16.0000			−0.719874
$495$		0			0
$496$		0			0
$497$	5.19615	+	3.00000i	0.233079	+	0.134568i
$498$	7.79423	−	13.5000i	0.349268	−	0.604949i
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	0.246214	−	0.969216i	$-0.420813\pi$
$500$		0			0
$501$	−13.5000	−	7.79423i	−0.603136	−	0.348220i
$502$		0			0
$503$			7.00000i			0.312115i	0.987748	+	0.156057i	$0.0498784\pi$
$503$			7.00000i			0.312115i	−0.987748	+	0.156057i	$0.950122\pi$
$504$	−13.5000	−	23.3827i	−0.601338	−	1.04155i
$505$		0			0
$506$	3.00000	−	5.19615i	0.133366	−	0.230997i
$507$	−15.5885			−0.692308
$508$	−4.33013	+	2.50000i	−0.192118	+	0.110920i
$509$	21.5000	+	37.2391i	0.952971	+	1.65059i	0.738945	+	0.673766i	$0.235324\pi$
$509$	21.5000	+	37.2391i	0.952971	+	1.65059i	0.214026	+	0.976828i	$0.431342\pi$
$510$		0			0
$511$	6.00000	−	10.3923i	0.265424	−	0.459728i
$512$		−	11.0000i		−	0.486136i
$513$	−41.5692			−1.83533
$514$	6.00000			0.264649
$515$		0			0
$516$	−12.0000	+	6.92820i	−0.528271	+	0.304997i
$517$	−12.1244	+	7.00000i	−0.533229	+	0.307860i
$518$	−10.3923	+	6.00000i	−0.456612	+	0.263625i
$519$		0			0
$520$		0			0
$521$	11.0000			0.481919			0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000			0.481919			0.240959	+	0.970535i	$0.422538\pi$
$522$			3.00000i			0.131306i
$523$			29.0000i			1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$			29.0000i			1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	3.00000	−	5.19615i	0.131056	−	0.226995i
$525$		0			0
$526$	8.00000	+	13.8564i	0.348817	+	0.604168i
$527$		0			0
$528$	1.73205	−	3.00000i	0.0753778	−	0.130558i
$529$	−7.00000	+	12.1244i	−0.304348	+	0.527146i
$530$		0			0
$531$	42.0000			1.82264
$532$		−	24.0000i		−	1.04053i
$533$	−8.66025	−	5.00000i	−0.375117	−	0.216574i
$534$		−	25.9808i		−	1.12430i
$535$		0			0
$536$	−4.50000	−	7.79423i	−0.194370	−	0.336659i
$537$	1.73205	+	3.00000i	0.0747435	+	0.129460i
$538$	21.6506	+	12.5000i	0.933425	+	0.538913i
$539$	4.00000			0.172292
$540$		0			0
$541$	−39.0000			−1.67674			−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000			−1.67674			−0.838370	+	0.545101i	$0.816491\pi$
$542$	6.92820	+	4.00000i	0.297592	+	0.171815i
$543$	−6.06218	−	10.5000i	−0.260153	−	0.450598i
$544$	10.0000	+	17.3205i	0.428746	+	0.742611i
$545$		0			0
$546$		−	10.3923i		−	0.444750i
$547$	25.1147	+	14.5000i	1.07383	+	0.619975i	0.929225	−	0.369514i	$-0.120476\pi$
$547$	25.1147	+	14.5000i	1.07383	+	0.619975i	0.144604	+	0.989490i	$0.453809\pi$
$548$			12.0000i			0.512615i
$549$	−10.5000	−	18.1865i	−0.448129	−	0.776182i
$550$		0			0
$551$	−4.00000	+	6.92820i	−0.170406	+	0.295151i
$552$	7.79423	−	13.5000i	0.331744	−	0.574598i
$553$	−15.5885	+	9.00000i	−0.662889	+	0.382719i
$554$	6.00000	+	10.3923i	0.254916	+	0.441527i
$555$		0			0
$556$	−8.00000	+	13.8564i	−0.339276	+	0.587643i
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$			13.8564i			0.585018i
$562$	−12.9904	+	7.50000i	−0.547966	+	0.316368i
$563$	18.1865	−	10.5000i	0.766471	−	0.442522i	−0.0651433	−	0.997876i	$-0.520750\pi$
$563$	18.1865	−	10.5000i	0.766471	−	0.442522i	0.831614	+	0.555354i	$0.187417\pi$
$564$	−10.5000	+	6.06218i	−0.442130	+	0.255264i
$565$		0			0
$566$	21.0000			0.882696
$567$		−	27.0000i		−	1.13389i
$568$			6.00000i			0.251754i
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$	−3.46410	+	2.00000i	−0.144841	+	0.0836242i
$573$	−13.8564			−0.578860
$574$	−7.50000	+	12.9904i	−0.313044	+	0.542208i
$575$		0			0
$576$	10.5000	−	18.1865i	0.437500	−	0.757772i
$577$		−	10.0000i		−	0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$		−	10.0000i		−	0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	−0.866025	−	0.500000i	−0.0360219	−	0.0207973i
$579$	15.0000	+	8.66025i	0.623379	+	0.359908i
$580$		0			0
$581$	13.5000	+	23.3827i	0.560074	+	0.970077i
$582$	−1.73205	+	3.00000i	−0.0717958	+	0.124354i
$583$	3.46410	+	2.00000i	0.143468	+	0.0828315i
$584$	12.0000			0.496564
$585$		0			0
$586$	12.0000			0.495715
$587$	28.5788	+	16.5000i	1.17957	+	0.681028i	0.955916	−	0.293640i	$-0.0948666\pi$
$587$	28.5788	+	16.5000i	1.17957	+	0.681028i	0.223659	+	0.974668i	$0.428200\pi$
$588$	3.46410			0.142857
$589$		0			0
$590$		0			0
$591$	18.0000	−	10.3923i	0.740421	−	0.427482i
$592$	−3.46410	−	2.00000i	−0.142374	−	0.0821995i
$593$		−	20.0000i		−	0.821302i	−0.911793	−	0.410651i	$-0.865302\pi$
$593$		−	20.0000i		−	0.821302i	0.911793	−	0.410651i	$-0.134698\pi$
$594$	9.00000	−	5.19615i	0.369274	−	0.213201i
$595$		0			0
$596$	8.50000	−	14.7224i	0.348174	−	0.603054i
$597$	−3.46410	−	6.00000i	−0.141776	−	0.245564i
$598$	5.19615	−	3.00000i	0.212486	−	0.122679i
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	0.745613	−	0.666379i	$-0.232157\pi$
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	−0.949908	+	0.312531i	$0.898823\pi$
$600$		0			0
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	0.844909	+	0.534910i	$0.179654\pi$
$602$			24.0000i			0.978167i
$603$		−	9.00000i		−	0.366508i
$604$	−2.00000			−0.0813788
$605$		0			0
$606$	−27.0000	−	15.5885i	−1.09680	−	0.633238i
$607$	−35.5070	+	20.5000i	−1.44119	+	0.832069i	−0.997929	−	0.0643251i	$-0.979511\pi$
$607$	−35.5070	+	20.5000i	−1.44119	+	0.832069i	−0.443257	+	0.896394i	$0.646177\pi$
$608$	34.6410	−	20.0000i	1.40488	−	0.811107i
$609$	−4.50000	−	2.59808i	−0.182349	−	0.105279i
$610$		0			0
$611$	−14.0000			−0.566379
$612$			12.0000i			0.485071i
$613$			44.0000i			1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$			44.0000i			1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$	3.50000	−	6.06218i	0.141249	−	0.244650i
$615$		0			0
$616$	9.00000	+	15.5885i	0.362620	+	0.628077i
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.972125	−	0.234464i	$-0.924666\pi$
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.283011	+	0.959117i	$0.591333\pi$
$618$	6.92820	+	12.0000i	0.278693	+	0.482711i
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	−0.903416	−	0.428765i	$-0.858949\pi$
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	0.823029	+	0.567999i	$0.192282\pi$
$620$		0			0
$621$	13.5000	−	7.79423i	0.541736	−	0.312772i
$622$		0			0
$623$	38.9711	+	22.5000i	1.56135	+	0.901443i
$624$	3.00000	−	1.73205i	0.120096	−	0.0693375i
$625$		0			0
$626$	−7.00000	−	12.1244i	−0.279776	−	0.484587i
$627$	27.7128			1.10674
$628$	−12.1244	−	7.00000i	−0.483814	−	0.279330i
$629$	16.0000			0.637962
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−15.5885	−	9.00000i	−0.620076	−	0.358001i
$633$	−19.0526	+	33.0000i	−0.757271	+	1.31163i
$634$	−17.0000	−	29.4449i	−0.675156	−	1.16940i
$635$		0			0
$636$	3.00000	+	1.73205i	0.118958	+	0.0686803i
$637$	3.46410	+	2.00000i	0.137253	+	0.0792429i
$638$		−	2.00000i		−	0.0791808i
$639$	−3.00000	+	5.19615i	−0.118678	+	0.205557i
$640$		0			0
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	0.330997	+	0.943632i	$0.392615\pi$
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	−0.982708	+	0.185164i	$0.940718\pi$
$642$	5.19615			0.205076
$643$	7.79423	−	4.50000i	0.307374	−	0.177463i	−0.338377	−	0.941011i	$-0.609878\pi$
$643$	7.79423	−	4.50000i	0.307374	−	0.177463i	0.645751	+	0.763548i	$0.276544\pi$
$644$	4.50000	+	7.79423i	0.177325	+	0.307136i
$645$		0			0
$646$	16.0000	−	27.7128i	0.629512	−	1.09035i
$647$		−	17.0000i		−	0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$		−	17.0000i		−	0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$	23.3827	−	13.5000i	0.918559	−	0.530330i
$649$	−28.0000			−1.09910
$650$		0			0
$651$		0			0
$652$	3.46410	−	2.00000i	0.135665	−	0.0783260i
$653$	3.46410	−	2.00000i	0.135561	−	0.0782660i	−0.430686	−	0.902502i	$-0.641728\pi$
$653$	3.46410	−	2.00000i	0.135561	−	0.0782660i	0.566247	+	0.824236i	$0.308395\pi$
$654$			8.66025i			0.338643i
$655$		0			0
$656$	−5.00000			−0.195217
$657$	10.3923	+	6.00000i	0.405442	+	0.234082i
$658$			21.0000i			0.818665i
$659$	−4.00000	+	6.92820i	−0.155818	+	0.269884i	−0.933357	−	0.358951i	$-0.883135\pi$
$659$	−4.00000	+	6.92820i	−0.155818	+	0.269884i	0.777539	+	0.628835i	$0.216468\pi$
$660$		0			0
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	0.969442	−	0.245319i	$-0.0788928\pi$
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	−0.697174	+	0.716902i	$0.745559\pi$
$662$	−5.19615	+	3.00000i	−0.201954	+	0.116598i
$663$	−6.92820	+	12.0000i	−0.269069	+	0.466041i
$664$	−13.5000	+	23.3827i	−0.523902	+	0.907424i
$665$		0			0
$666$	−6.00000	−	10.3923i	−0.232495	−	0.402694i
$667$		−	3.00000i		−	0.116160i
$668$	7.79423	+	4.50000i	0.301568	+	0.174110i
$669$		−	32.9090i		−	1.27233i
$670$		0			0
$671$	7.00000	+	12.1244i	0.270232	+	0.468056i
$672$	12.9904	+	22.5000i	0.501115	+	0.867956i
$673$	5.19615	+	3.00000i	0.200297	+	0.115642i	0.596794	−	0.802395i	$-0.296441\pi$
$673$	5.19615	+	3.00000i	0.200297	+	0.115642i	−0.396497	+	0.918036i	$0.629774\pi$
$674$	8.00000			0.308148
$675$		0			0
$676$	9.00000			0.346154
$677$	36.3731	+	21.0000i	1.39793	+	0.807096i	0.994176	−	0.107772i	$-0.0343715\pi$
$677$	36.3731	+	21.0000i	1.39793	+	0.807096i	0.403755	+	0.914867i	$0.367705\pi$
$678$	−6.92820	−	12.0000i	−0.266076	−	0.460857i
$679$	−3.00000	−	5.19615i	−0.115129	−	0.199410i
$680$		0			0
$681$			6.92820i			0.265489i
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	24.0000			0.917663
$685$		0			0
$686$	−7.50000	+	12.9904i	−0.286351	+	0.495975i
$687$	−12.9904	+	22.5000i	−0.495614	+	0.858429i
$688$	−6.92820	+	4.00000i	−0.264135	+	0.152499i
$689$	2.00000	+	3.46410i	0.0761939	+	0.131972i
$690$		0			0
$691$	7.00000	−	12.1244i	0.266293	−	0.461232i	−0.701609	−	0.712562i	$-0.747535\pi$
$691$	7.00000	−	12.1244i	0.266293	−	0.461232i	0.967901	+	0.251330i	$0.0808679\pi$
$692$		0			0
$693$			18.0000i			0.683763i
$694$	−4.00000			−0.151838
$695$		0			0
$696$		−	5.19615i		−	0.196960i
$697$	17.3205	−	10.0000i	0.656061	−	0.378777i
$698$	4.33013	−	2.50000i	0.163898	−	0.0946264i
$699$	36.0000	−	20.7846i	1.36165	−	0.786146i
$700$		0			0
$701$	23.0000			0.868698			0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000			0.868698			0.434349	+	0.900745i	$0.356978\pi$
$702$	10.3923			0.392232
$703$		−	32.0000i		−	1.20690i
$704$	−7.00000	+	12.1244i	−0.263822	+	0.456954i
$705$		0			0
$706$	−12.0000	−	20.7846i	−0.451626	−	0.782239i
$707$	46.7654	−	27.0000i	1.75879	−	1.01544i
$708$	−24.2487			−0.911322
$709$	−20.5000	+	35.5070i	−0.769894	+	1.33349i	0.167727	+	0.985834i	$0.446357\pi$
$709$	−20.5000	+	35.5070i	−0.769894	+	1.33349i	−0.937620	+	0.347661i	$0.886976\pi$
$710$		0			0
$711$	−9.00000	−	15.5885i	−0.337526	−	0.584613i
$712$			45.0000i			1.68645i
$713$		0			0
$714$	18.0000	+	10.3923i	0.673633	+	0.388922i
$715$		0			0
$716$	−1.00000	−	1.73205i	−0.0373718	−	0.0647298i
$717$	6.92820	−	12.0000i	0.258738	−	0.448148i
$718$	20.7846	+	12.0000i	0.775675	+	0.447836i
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	−38.9711	−	22.5000i	−1.45036	−	0.837363i
$723$	19.0526			0.708572
$724$	3.50000	+	6.06218i	0.130076	+	0.225299i
$725$		0			0
$726$	10.5000	−	6.06218i	0.389692	−	0.224989i
$727$	−19.9186	−	11.5000i	−0.738739	−	0.426511i	0.0828714	−	0.996560i	$-0.473591\pi$
$727$	−19.9186	−	11.5000i	−0.738739	−	0.426511i	−0.821611	+	0.570049i	$0.806924\pi$
$728$			18.0000i			0.667124i
$729$	27.0000			1.00000
$730$		0			0
$731$	16.0000	−	27.7128i	0.591781	−	1.02500i
$732$	6.06218	+	10.5000i	0.224065	+	0.388091i
$733$	29.4449	−	17.0000i	1.08757	−	0.627909i	0.154642	−	0.987971i	$-0.450578\pi$
$733$	29.4449	−	17.0000i	1.08757	−	0.627909i	0.932929	+	0.360061i	$0.117244\pi$
$734$	12.0000	+	20.7846i	0.442928	+	0.767174i
$735$		0			0
$736$	−7.50000	+	12.9904i	−0.276454	+	0.478832i
$737$			6.00000i			0.221013i
$738$	−12.9904	−	7.50000i	−0.478183	−	0.276079i
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$		0			0
$741$	24.0000	+	13.8564i	0.881662	+	0.509028i
$742$	5.19615	−	3.00000i	0.190757	−	0.110133i
$743$	25.1147	−	14.5000i	0.921370	−	0.531953i	0.0372984	−	0.999304i	$-0.488125\pi$
$743$	25.1147	−	14.5000i	0.921370	−	0.531953i	0.884072	+	0.467351i	$0.154791\pi$
$744$		0			0
$745$		0			0
$746$	10.0000			0.366126
$747$	−23.3827	+	13.5000i	−0.855528	+	0.493939i
$748$		−	8.00000i		−	0.292509i
$749$	−4.50000	+	7.79423i	−0.164426	+	0.284795i
$750$		0			0
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	0.760263	−	0.649616i	$-0.225070\pi$
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	−0.942715	+	0.333599i	$0.891737\pi$
$752$	−6.06218	+	3.50000i	−0.221065	+	0.127632i
$753$		0			0
$754$	1.00000	−	1.73205i	0.0364179	−	0.0630776i
$755$		0			0
$756$			15.5885i			0.566947i
$757$		−	26.0000i		−	0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$		−	26.0000i		−	0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	−22.5167	−	13.0000i	−0.817842	−	0.472181i
$759$	−9.00000	+	5.19615i	−0.326679	+	0.188608i
$760$		0			0
$761$	−7.50000	−	12.9904i	−0.271875	−	0.470901i	0.697467	−	0.716617i	$-0.254310\pi$
$761$	−7.50000	−	12.9904i	−0.271875	−	0.470901i	−0.969342	+	0.245716i	$0.920977\pi$
$762$	−8.66025			−0.313728
$763$	−12.9904	−	7.50000i	−0.470283	−	0.271518i
$764$	8.00000			0.289430
$765$		0			0
$766$	36.0000			1.30073
$767$	−24.2487	−	14.0000i	−0.875570	−	0.505511i
$768$	−14.7224	+	25.5000i	−0.531250	+	0.920152i
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	0.907575	−	0.419890i	$-0.137931\pi$
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	−0.817423	+	0.576038i	$0.804598\pi$
$770$		0			0
$771$	−9.00000	−	5.19615i	−0.324127	−	0.187135i
$772$	−8.66025	−	5.00000i	−0.311689	−	0.179954i
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	−24.0000			−0.862662
$775$		0			0
$776$	3.00000	−	5.19615i	0.107694	−	0.186531i
$777$	20.7846			0.745644
$778$	−28.5788	+	16.5000i	−1.02460	+	0.591554i
$779$	−20.0000	−	34.6410i

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 1.50000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.73205i q^{6} +(2.59808 + 1.50000i) q^{7} +3.00000i q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 1.50000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.73205i q^{6} +(2.59808 + 1.50000i) q^{7} +3.00000i q^{8} +(-1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{11} +(0.866025 - 1.50000i) q^{12} +(1.73205 - 1.00000i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(0.500000 - 0.866025i) q^{16} +4.00000i q^{17} +(2.59808 - 1.50000i) q^{18} +8.00000 q^{19} +5.19615i q^{21} +(-1.73205 + 1.00000i) q^{22} +(-2.59808 + 1.50000i) q^{23} +(-4.50000 + 2.59808i) q^{24} -2.00000 q^{26} -5.19615 q^{27} -3.00000i q^{28} +(-0.500000 + 0.866025i) q^{29} +(4.33013 - 2.50000i) q^{32} +3.46410 q^{33} +(2.00000 - 3.46410i) q^{34} +3.00000 q^{36} -4.00000i q^{37} +(-6.92820 - 4.00000i) q^{38} +(3.00000 + 1.73205i) q^{39} +(-2.50000 - 4.33013i) q^{41} +(2.59808 - 4.50000i) q^{42} +(-6.92820 - 4.00000i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(-6.06218 - 3.50000i) q^{47} +1.73205 q^{48} +(1.00000 + 1.73205i) q^{49} +(-6.00000 + 3.46410i) q^{51} +(-1.73205 - 1.00000i) q^{52} +2.00000i q^{53} +(4.50000 + 2.59808i) q^{54} +(-4.50000 + 7.79423i) q^{56} +(6.92820 + 12.0000i) q^{57} +(0.866025 - 0.500000i) q^{58} +(-7.00000 - 12.1244i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(-7.79423 + 4.50000i) q^{63} -7.00000 q^{64} +(-3.00000 - 1.73205i) q^{66} +(-2.59808 + 1.50000i) q^{67} +(3.46410 - 2.00000i) q^{68} +(-4.50000 - 2.59808i) q^{69} +2.00000 q^{71} +(-7.79423 - 4.50000i) q^{72} -4.00000i q^{73} +(-2.00000 + 3.46410i) q^{74} +(-4.00000 - 6.92820i) q^{76} +(5.19615 - 3.00000i) q^{77} +(-1.73205 - 3.00000i) q^{78} +(-3.00000 + 5.19615i) q^{79} +(-4.50000 - 7.79423i) q^{81} +5.00000i q^{82} +(7.79423 + 4.50000i) q^{83} +(4.50000 - 2.59808i) q^{84} +(4.00000 + 6.92820i) q^{86} -1.73205 q^{87} +(5.19615 + 3.00000i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(2.59808 + 1.50000i) q^{92} +(3.50000 + 6.06218i) q^{94} +(7.50000 + 4.33013i) q^{96} +(-1.73205 - 1.00000i) q^{97} -2.00000i q^{98} +(3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^4 - 6 * q^9	\( 4 q - 2 q^{4} - 6 q^{9} + 4 q^{11} - 6 q^{14} + 2 q^{16} + 32 q^{19} - 18 q^{24} - 8 q^{26} - 2 q^{29} + 8 q^{34} + 12 q^{36} + 12 q^{39} - 10 q^{41} - 8 q^{44} + 12 q^{46} + 4 q^{49} - 24 q^{51} + 18 q^{54} - 18 q^{56} - 28 q^{59} - 14 q^{61} - 28 q^{64} - 12 q^{66} - 18 q^{69} + 8 q^{71} - 8 q^{74} - 16 q^{76} - 12 q^{79} - 18 q^{81} + 18 q^{84} + 16 q^{86} + 60 q^{89} + 24 q^{91} + 14 q^{94} + 30 q^{96} + 12 q^{99}+O(q^{100}) \) 4 * q - 2 * q^4 - 6 * q^9 + 4 * q^11 - 6 * q^14 + 2 * q^16 + 32 * q^19 - 18 * q^24 - 8 * q^26 - 2 * q^29 + 8 * q^34 + 12 * q^36 + 12 * q^39 - 10 * q^41 - 8 * q^44 + 12 * q^46 + 4 * q^49 - 24 * q^51 + 18 * q^54 - 18 * q^56 - 28 * q^59 - 14 * q^61 - 28 * q^64 - 12 * q^66 - 18 * q^69 + 8 * q^71 - 8 * q^74 - 16 * q^76 - 12 * q^79 - 18 * q^81 + 18 * q^84 + 16 * q^86 + 60 * q^89 + 24 * q^91 + 14 * q^94 + 30 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more