LMFDB - Embedded newform 570.2.d.b.229.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [570,2,Mod(229,570)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(570, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("570.229");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 570 = 2 \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	570.d (of order $2$, degree $1$, 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:	$4.55147291521$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			229.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	570.229
Dual form			570.2.d.b.229.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +(-1.00000 + 2.00000i) q^{10} -4.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} -4.00000i q^{22} +1.00000 q^{24} +(3.00000 + 4.00000i) q^{25} -4.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} +(-2.00000 - 1.00000i) q^{30} +1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +1.00000i q^{38} -4.00000 q^{39} +(1.00000 - 2.00000i) q^{40} -12.0000 q^{41} -6.00000i q^{43} +4.00000 q^{44} +(-2.00000 - 1.00000i) q^{45} +1.00000i q^{48} +7.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} -6.00000 q^{51} -4.00000i q^{52} +14.0000i q^{53} +1.00000 q^{54} +(-8.00000 - 4.00000i) q^{55} +1.00000i q^{57} +2.00000i q^{58} +10.0000 q^{59} +(1.00000 - 2.00000i) q^{60} -6.00000 q^{61} -1.00000 q^{64} +(-4.00000 + 8.00000i) q^{65} +4.00000 q^{66} -4.00000i q^{67} -6.00000i q^{68} +12.0000 q^{71} +1.00000i q^{72} -8.00000i q^{73} +4.00000 q^{74} +(-4.00000 + 3.00000i) q^{75} -1.00000 q^{76} -4.00000i q^{78} +8.00000 q^{79} +(2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -12.0000i q^{82} -12.0000i q^{83} +(-6.00000 + 12.0000i) q^{85} +6.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} +(1.00000 - 2.00000i) q^{90} +(2.00000 + 1.00000i) q^{95} -1.00000 q^{96} -10.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{15} + 2 q^{16} + 2 q^{19} - 4 q^{20} + 2 q^{24} + 6 q^{25} - 8 q^{26} + 4 q^{29} - 4 q^{30} - 12 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{40} - 24 q^{41} + 8 q^{44} - 4 q^{45} + 14 q^{49} - 8 q^{50} - 12 q^{51} + 2 q^{54} - 16 q^{55} + 20 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 8 q^{65} + 8 q^{66} + 24 q^{71} + 8 q^{74} - 8 q^{75} - 2 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{85} + 12 q^{86} + 16 q^{89} + 2 q^{90} + 4 q^{95} - 2 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^6 - 2 * q^9 - 2 * q^10 - 8 * q^11 - 2 * q^15 + 2 * q^16 + 2 * q^19 - 4 * q^20 + 2 * q^24 + 6 * q^25 - 8 * q^26 + 4 * q^29 - 4 * q^30 - 12 * q^34 + 2 * q^36 - 8 * q^39 + 2 * q^40 - 24 * q^41 + 8 * q^44 - 4 * q^45 + 14 * q^49 - 8 * q^50 - 12 * q^51 + 2 * q^54 - 16 * q^55 + 20 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^64 - 8 * q^65 + 8 * q^66 + 24 * q^71 + 8 * q^74 - 8 * q^75 - 2 * q^76 + 16 * q^79 + 4 * q^80 + 2 * q^81 - 12 * q^85 + 12 * q^86 + 16 * q^89 + 2 * q^90 + 4 * q^95 - 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$.

$n$	$191$	$211$	$457$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−1.00000			−0.500000
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$	−1.00000			−0.408248
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	−1.00000	+	2.00000i	−0.316228	+	0.632456i
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		−	1.00000i		−	0.288675i
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	−1.00000	+	2.00000i	−0.258199	+	0.516398i
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		−	1.00000i		−	0.235702i
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	−2.00000	−	1.00000i	−0.447214	−	0.223607i
$21$		0			0
$22$		−	4.00000i		−	0.852803i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	1.00000			0.204124
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	−4.00000			−0.784465
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$	−2.00000	−	1.00000i	−0.365148	−	0.182574i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$			1.00000i			0.176777i
$33$		−	4.00000i		−	0.696311i
$34$	−6.00000			−1.02899
$35$		0			0
$36$	1.00000			0.166667
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$			1.00000i			0.162221i
$39$	−4.00000			−0.640513
$40$	1.00000	−	2.00000i	0.158114	−	0.316228i
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	4.00000			0.603023
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$			1.00000i			0.144338i
$49$	7.00000			1.00000
$50$	−4.00000	+	3.00000i	−0.565685	+	0.424264i
$51$	−6.00000			−0.840168
$52$		−	4.00000i		−	0.554700i
$53$			14.0000i			1.92305i	0.274721	+	0.961524i	$0.411414\pi$
$53$			14.0000i			1.92305i	−0.274721	+	0.961524i	$0.588586\pi$
$54$	1.00000			0.136083
$55$	−8.00000	−	4.00000i	−1.07872	−	0.539360i
$56$		0			0
$57$			1.00000i			0.132453i
$58$			2.00000i			0.262613i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$	1.00000	−	2.00000i	0.129099	−	0.258199i
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	+	8.00000i	−0.496139	+	0.992278i
$66$	4.00000			0.492366
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$			1.00000i			0.117851i
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	4.00000			0.464991
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$	−1.00000			−0.114708
$77$		0			0
$78$		−	4.00000i		−	0.452911i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000	+	1.00000i	0.223607	+	0.111803i
$81$	1.00000			0.111111
$82$		−	12.0000i		−	1.32518i
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−6.00000	+	12.0000i	−0.650791	+	1.30158i
$86$	6.00000			0.646997
$87$			2.00000i			0.214423i
$88$			4.00000i			0.426401i
$89$	8.00000			0.847998			0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000			0.847998			0.423999	+	0.905663i	$0.360626\pi$
$90$	1.00000	−	2.00000i	0.105409	−	0.210819i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	+	1.00000i	0.205196	+	0.102598i
$96$	−1.00000			−0.102062
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$			7.00000i			0.707107i
$99$	4.00000			0.402015
$100$	−3.00000	−	4.00000i	−0.300000	−	0.400000i
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		−	6.00000i		−	0.594089i
$103$		−	2.00000i		−	0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$		−	2.00000i		−	0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	4.00000			0.392232
$105$		0			0
$106$	−14.0000			−1.35980
$107$			20.0000i			1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$			20.0000i			1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$			1.00000i			0.0962250i
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	4.00000	−	8.00000i	0.381385	−	0.762770i
$111$	4.00000			0.379663
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	−1.00000			−0.0936586
$115$		0			0
$116$	−2.00000			−0.185695
$117$		−	4.00000i		−	0.369800i
$118$			10.0000i			0.920575i
$119$		0			0
$120$	2.00000	+	1.00000i	0.182574	+	0.0912871i
$121$	5.00000			0.454545
$122$		−	6.00000i		−	0.543214i
$123$		−	12.0000i		−	1.08200i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	6.00000			0.528271
$130$	−8.00000	−	4.00000i	−0.701646	−	0.350823i
$131$	−20.0000			−1.74741			−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000			−1.74741			−0.873704	+	0.486458i	$0.838289\pi$
$132$			4.00000i			0.348155i
$133$		0			0
$134$	4.00000			0.345547
$135$	1.00000	−	2.00000i	0.0860663	−	0.172133i
$136$	6.00000			0.514496
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$			12.0000i			1.00702i
$143$		−	16.0000i		−	1.33799i
$144$	−1.00000			−0.0833333
$145$	4.00000	+	2.00000i	0.332182	+	0.166091i
$146$	8.00000			0.662085
$147$			7.00000i			0.577350i
$148$			4.00000i			0.328798i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−3.00000	−	4.00000i	−0.244949	−	0.326599i
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		−	1.00000i		−	0.0811107i
$153$		−	6.00000i		−	0.485071i
$154$		0			0
$155$		0			0
$156$	4.00000			0.320256
$157$			6.00000i			0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$			6.00000i			0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$			8.00000i			0.636446i
$159$	−14.0000			−1.11027
$160$	−1.00000	+	2.00000i	−0.0790569	+	0.158114i
$161$		0			0
$162$			1.00000i			0.0785674i
$163$			10.0000i			0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$			10.0000i			0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	12.0000			0.937043
$165$	4.00000	−	8.00000i	0.311400	−	0.622799i
$166$	12.0000			0.931381
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	−12.0000	−	6.00000i	−0.920358	−	0.460179i
$171$	−1.00000			−0.0764719
$172$			6.00000i			0.457496i
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−2.00000			−0.151620
$175$		0			0
$176$	−4.00000			−0.301511
$177$			10.0000i			0.751646i
$178$			8.00000i			0.599625i
$179$	−6.00000			−0.448461			−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000			−0.448461			−0.224231	+	0.974536i	$0.571987\pi$
$180$	2.00000	+	1.00000i	0.149071	+	0.0745356i
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	6.00000i		−	0.443533i
$184$		0			0
$185$	4.00000	−	8.00000i	0.294086	−	0.588172i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		0			0
$189$		0			0
$190$	−1.00000	+	2.00000i	−0.0725476	+	0.145095i
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			10.0000i			0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$			10.0000i			0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	10.0000			0.717958
$195$	−8.00000	−	4.00000i	−0.572892	−	0.286446i
$196$	−7.00000			−0.500000
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$			4.00000i			0.284268i
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$	4.00000	−	3.00000i	0.282843	−	0.212132i
$201$	4.00000			0.282138
$202$			12.0000i			0.844317i
$203$		0			0
$204$	6.00000			0.420084
$205$	−24.0000	−	12.0000i	−1.67623	−	0.838116i
$206$	2.00000			0.139347
$207$		0			0
$208$			4.00000i			0.277350i
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		−	14.0000i		−	0.961524i
$213$			12.0000i			0.822226i
$214$	−20.0000			−1.36717
$215$	6.00000	−	12.0000i	0.409197	−	0.818393i
$216$	−1.00000			−0.0680414
$217$		0			0
$218$			14.0000i			0.948200i
$219$	8.00000			0.540590
$220$	8.00000	+	4.00000i	0.539360	+	0.269680i
$221$	−24.0000			−1.61441
$222$			4.00000i			0.268462i
$223$		−	10.0000i		−	0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$		−	10.0000i		−	0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$		0			0
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$	−10.0000			−0.665190
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		−	1.00000i		−	0.0662266i
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		−	2.00000i		−	0.131306i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	4.00000			0.261488
$235$		0			0
$236$	−10.0000			−0.650945
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−10.0000			−0.646846			−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000			−0.646846			−0.323423	+	0.946254i	$0.604834\pi$
$240$	−1.00000	+	2.00000i	−0.0645497	+	0.129099i
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$			5.00000i			0.321412i
$243$			1.00000i			0.0641500i
$244$	6.00000			0.384111
$245$	14.0000	+	7.00000i	0.894427	+	0.447214i
$246$	12.0000			0.765092
$247$			4.00000i			0.254514i
$248$		0			0
$249$	12.0000			0.760469
$250$	−11.0000	+	2.00000i	−0.695701	+	0.126491i
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$	10.0000			0.627456
$255$	−12.0000	−	6.00000i	−0.751469	−	0.375735i
$256$	1.00000			0.0625000
$257$		−	2.00000i		−	0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$		−	2.00000i		−	0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$			6.00000i			0.373544i
$259$		0			0
$260$	4.00000	−	8.00000i	0.248069	−	0.496139i
$261$	−2.00000			−0.123797
$262$		−	20.0000i		−	1.23560i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	−4.00000			−0.246183
$265$	−14.0000	+	28.0000i	−0.860013	+	1.72003i
$266$		0			0
$267$			8.00000i			0.489592i
$268$			4.00000i			0.244339i
$269$	2.00000			0.121942			0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$	2.00000	+	1.00000i	0.121716	+	0.0608581i
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$	6.00000			0.362473
$275$	−12.0000	−	16.0000i	−0.723627	−	0.964836i
$276$		0			0
$277$		−	14.0000i		−	0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$		−	14.0000i		−	0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$			12.0000i			0.719712i
$279$		0			0
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$			2.00000i			0.118888i	0.998232	+	0.0594438i	$0.0189327\pi$
$283$			2.00000i			0.118888i	−0.998232	+	0.0594438i	$0.981067\pi$
$284$	−12.0000			−0.712069
$285$	−1.00000	+	2.00000i	−0.0592349	+	0.118470i
$286$	16.0000			0.946100
$287$		0			0
$288$		−	1.00000i		−	0.0589256i
$289$	−19.0000			−1.11765
$290$	−2.00000	+	4.00000i	−0.117444	+	0.234888i
$291$	10.0000			0.586210
$292$			8.00000i			0.468165i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−7.00000			−0.408248
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$	−4.00000			−0.232495
$297$			4.00000i			0.232104i
$298$		0			0
$299$		0			0
$300$	4.00000	−	3.00000i	0.230940	−	0.173205i
$301$		0			0
$302$			16.0000i			0.920697i
$303$			12.0000i			0.689382i
$304$	1.00000			0.0573539
$305$	−12.0000	−	6.00000i	−0.687118	−	0.343559i
$306$	6.00000			0.342997
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	2.00000			0.113776
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$			4.00000i			0.226455i
$313$		−	4.00000i		−	0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$		−	4.00000i		−	0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	−6.00000			−0.338600
$315$		0			0
$316$	−8.00000			−0.450035
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		−	14.0000i		−	0.785081i
$319$	−8.00000			−0.447914
$320$	−2.00000	−	1.00000i	−0.111803	−	0.0559017i
$321$	−20.0000			−1.11629
$322$		0			0
$323$			6.00000i			0.333849i
$324$	−1.00000			−0.0555556
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$	−10.0000			−0.553849
$327$			14.0000i			0.774202i
$328$			12.0000i			0.662589i
$329$		0			0
$330$	8.00000	+	4.00000i	0.440386	+	0.220193i
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			12.0000i			0.658586i
$333$			4.00000i			0.219199i
$334$	−12.0000			−0.656611
$335$	4.00000	−	8.00000i	0.218543	−	0.437087i
$336$		0			0
$337$			18.0000i			0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$			18.0000i			0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$		−	3.00000i		−	0.163178i
$339$	−10.0000			−0.543125
$340$	6.00000	−	12.0000i	0.325396	−	0.650791i
$341$		0			0
$342$		−	1.00000i		−	0.0540738i
$343$		0			0
$344$	−6.00000			−0.323498
$345$		0			0
$346$	−2.00000			−0.107521
$347$			32.0000i			1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$			32.0000i			1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$		−	2.00000i		−	0.107211i
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		−	4.00000i		−	0.213201i
$353$		−	22.0000i		−	1.17094i	−0.810693	−	0.585471i	$-0.800910\pi$
$353$		−	22.0000i		−	1.17094i	0.810693	−	0.585471i	$-0.199090\pi$
$354$	−10.0000			−0.531494
$355$	24.0000	+	12.0000i	1.27379	+	0.636894i
$356$	−8.00000			−0.423999
$357$		0			0
$358$		−	6.00000i		−	0.317110i
$359$	−34.0000			−1.79445			−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000			−1.79445			−0.897226	+	0.441572i	$0.854421\pi$
$360$	−1.00000	+	2.00000i	−0.0527046	+	0.105409i
$361$	1.00000			0.0526316
$362$		−	10.0000i		−	0.525588i
$363$			5.00000i			0.262432i
$364$		0			0
$365$	8.00000	−	16.0000i	0.418739	−	0.837478i
$366$	6.00000			0.313625
$367$		−	12.0000i		−	0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$		−	12.0000i		−	0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$		0			0
$369$	12.0000			0.624695
$370$	8.00000	+	4.00000i	0.415900	+	0.207950i
$371$		0			0
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	24.0000			1.24101
$375$	−11.0000	+	2.00000i	−0.568038	+	0.103280i
$376$		0			0
$377$			8.00000i			0.412021i
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$	−2.00000	−	1.00000i	−0.102598	−	0.0512989i
$381$	10.0000			0.512316
$382$			6.00000i			0.306987i
$383$		−	36.0000i		−	1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$		−	36.0000i		−	1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	1.00000			0.0510310
$385$		0			0
$386$	−10.0000			−0.508987
$387$			6.00000i			0.304997i
$388$			10.0000i			0.507673i
$389$	4.00000			0.202808			0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000			0.202808			0.101404	+	0.994845i	$0.467667\pi$
$390$	4.00000	−	8.00000i	0.202548	−	0.405096i
$391$		0			0
$392$		−	7.00000i		−	0.353553i
$393$		−	20.0000i		−	1.00887i
$394$	10.0000			0.503793
$395$	16.0000	+	8.00000i	0.805047	+	0.402524i
$396$	−4.00000			−0.201008
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$			4.00000i			0.200502i
$399$		0			0
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$	28.0000			1.39825			0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000			1.39825			0.699127	+	0.714998i	$0.253572\pi$
$402$			4.00000i			0.199502i
$403$		0			0
$404$	−12.0000			−0.597022
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$		0			0
$407$			16.0000i			0.793091i
$408$			6.00000i			0.297044i
$409$	34.0000			1.68119			0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000			1.68119			0.840596	+	0.541663i	$0.182205\pi$
$410$	12.0000	−	24.0000i	0.592638	−	1.18528i
$411$	6.00000			0.295958
$412$			2.00000i			0.0985329i
$413$		0			0
$414$		0			0
$415$	12.0000	−	24.0000i	0.589057	−	1.17811i
$416$	−4.00000			−0.196116
$417$			12.0000i			0.587643i
$418$		−	4.00000i		−	0.195646i
$419$	−8.00000			−0.390826			−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000			−0.390826			−0.195413	+	0.980721i	$0.562605\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		−	8.00000i		−	0.389434i
$423$		0			0
$424$	14.0000			0.679900
$425$	−24.0000	+	18.0000i	−1.16417	+	0.873128i
$426$	−12.0000			−0.581402
$427$		0			0
$428$		−	20.0000i		−	0.966736i
$429$	16.0000			0.772487
$430$	12.0000	+	6.00000i	0.578691	+	0.289346i
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		−	1.00000i		−	0.0481125i
$433$			18.0000i			0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$			18.0000i			0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$		0			0
$435$	−2.00000	+	4.00000i	−0.0958927	+	0.191785i
$436$	−14.0000			−0.670478
$437$		0			0
$438$			8.00000i			0.382255i
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	−4.00000	+	8.00000i	−0.190693	+	0.381385i
$441$	−7.00000			−0.333333
$442$		−	24.0000i		−	1.14156i
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−4.00000			−0.189832
$445$	16.0000	+	8.00000i	0.758473	+	0.379236i
$446$	10.0000			0.473514
$447$		0			0
$448$		0			0
$449$	32.0000			1.51017			0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000			1.51017			0.755087	+	0.655625i	$0.227595\pi$
$450$	4.00000	−	3.00000i	0.188562	−	0.141421i
$451$	48.0000			2.26023
$452$		−	10.0000i		−	0.470360i
$453$			16.0000i			0.751746i
$454$	12.0000			0.563188
$455$		0			0
$456$	1.00000			0.0468293
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$			10.0000i			0.467269i
$459$	6.00000			0.280056
$460$		0			0
$461$	24.0000			1.11779			0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000			1.11779			0.558896	+	0.829238i	$0.311225\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	2.00000			0.0928477
$465$		0			0
$466$	14.0000			0.648537
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$			4.00000i			0.184900i
$469$		0			0
$470$		0			0
$471$	−6.00000			−0.276465
$472$		−	10.0000i		−	0.460287i
$473$			24.0000i			1.10352i
$474$	−8.00000			−0.367452
$475$	3.00000	+	4.00000i	0.137649	+	0.183533i
$476$		0			0
$477$		−	14.0000i		−	0.641016i
$478$		−	10.0000i		−	0.457389i
$479$	−38.0000			−1.73626			−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000			−1.73626			−0.868132	+	0.496333i	$0.834679\pi$
$480$	−2.00000	−	1.00000i	−0.0912871	−	0.0456435i
$481$	16.0000			0.729537
$482$			26.0000i			1.18427i
$483$		0			0
$484$	−5.00000			−0.227273
$485$	10.0000	−	20.0000i	0.454077	−	0.908153i
$486$	−1.00000			−0.0453609
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$			6.00000i			0.271607i
$489$	−10.0000			−0.452216
$490$	−7.00000	+	14.0000i	−0.316228	+	0.632456i
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$			12.0000i			0.541002i
$493$			12.0000i			0.540453i
$494$	−4.00000			−0.179969
$495$	8.00000	+	4.00000i	0.359573	+	0.179787i
$496$		0			0
$497$		0			0
$498$			12.0000i			0.537733i
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	−2.00000	−	11.0000i	−0.0894427	−	0.491935i
$501$	−12.0000			−0.536120
$502$		−	24.0000i		−	1.07117i
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$		0			0
$505$	24.0000	+	12.0000i	1.06799	+	0.533993i
$506$		0			0
$507$		−	3.00000i		−	0.133235i
$508$			10.0000i			0.443678i
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$	6.00000	−	12.0000i	0.265684	−	0.531369i
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		−	1.00000i		−	0.0441511i
$514$	2.00000			0.0882162
$515$	2.00000	−	4.00000i	0.0881305	−	0.176261i
$516$	−6.00000			−0.264135
$517$		0			0
$518$		0			0
$519$	−2.00000			−0.0877903
$520$	8.00000	+	4.00000i	0.350823	+	0.175412i
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		−	2.00000i		−	0.0875376i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	20.0000			0.873704
$525$		0			0
$526$	16.0000			0.697633
$527$		0			0
$528$		−	4.00000i		−	0.174078i
$529$	23.0000			1.00000
$530$	−28.0000	−	14.0000i	−1.21624	−	0.608121i
$531$	−10.0000			−0.433963
$532$		0			0
$533$		−	48.0000i		−	2.07911i
$534$	−8.00000			−0.346194
$535$	−20.0000	+	40.0000i	−0.864675	+	1.72935i
$536$	−4.00000			−0.172774
$537$		−	6.00000i		−	0.258919i
$538$			2.00000i			0.0862261i
$539$	−28.0000			−1.20605
$540$	−1.00000	+	2.00000i	−0.0430331	+	0.0860663i
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		−	28.0000i		−	1.20270i
$543$		−	10.0000i		−	0.429141i
$544$	−6.00000			−0.257248
$545$	28.0000	+	14.0000i	1.19939	+	0.599694i
$546$		0			0
$547$			24.0000i			1.02617i	0.858339	+	0.513083i	$0.171497\pi$
$547$			24.0000i			1.02617i	−0.858339	+	0.513083i	$0.828503\pi$
$548$			6.00000i			0.256307i
$549$	6.00000			0.256074
$550$	16.0000	−	12.0000i	0.682242	−	0.511682i
$551$	2.00000			0.0852029
$552$		0			0
$553$		0			0
$554$	14.0000			0.594803
$555$	8.00000	+	4.00000i	0.339581	+	0.169791i
$556$	−12.0000			−0.508913
$557$		−	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
$559$	24.0000			1.01509
$560$		0			0
$561$	24.0000			1.01328
$562$			12.0000i			0.506189i
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$	−10.0000	+	20.0000i	−0.420703	+	0.841406i
$566$	−2.00000			−0.0840663
$567$		0			0
$568$		−	12.0000i		−	0.503509i
$569$	−32.0000			−1.34151			−0.670755	−	0.741679i	$-0.734030\pi$
$569$	−32.0000			−1.34151			−0.670755	+	0.741679i	$0.734030\pi$
$570$	−2.00000	−	1.00000i	−0.0837708	−	0.0418854i
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			16.0000i			0.668994i
$573$			6.00000i			0.250654i
$574$		0			0
$575$		0			0
$576$	1.00000			0.0416667
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		−	19.0000i		−	0.790296i
$579$	−10.0000			−0.415586
$580$	−4.00000	−	2.00000i	−0.166091	−	0.0830455i
$581$		0			0
$582$			10.0000i			0.414513i
$583$		−	56.0000i		−	2.31928i
$584$	−8.00000			−0.331042
$585$	4.00000	−	8.00000i	0.165380	−	0.330759i
$586$	14.0000			0.578335
$587$			32.0000i			1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$			32.0000i			1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$		−	7.00000i		−	0.288675i
$589$		0			0
$590$	−10.0000	+	20.0000i	−0.411693	+	0.823387i
$591$	10.0000			0.411345
$592$		−	4.00000i		−	0.164399i
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	−4.00000			−0.164122
$595$		0			0
$596$		0			0
$597$			4.00000i			0.163709i
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$	3.00000	+	4.00000i	0.122474	+	0.163299i
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$	−16.0000			−0.651031
$605$	10.0000	+	5.00000i	0.406558	+	0.203279i
$606$	−12.0000			−0.487467
$607$			30.0000i			1.21766i	0.793300	+	0.608831i	$0.208361\pi$
$607$			30.0000i			1.21766i	−0.793300	+	0.608831i	$0.791639\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	6.00000	−	12.0000i	0.242933	−	0.485866i
$611$		0			0
$612$			6.00000i			0.242536i
$613$			46.0000i			1.85792i	0.370177	+	0.928961i	$0.379297\pi$
$613$			46.0000i			1.85792i	−0.370177	+	0.928961i	$0.620703\pi$
$614$		0			0
$615$	12.0000	−	24.0000i	0.483887	−	0.967773i
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$			2.00000i			0.0804518i
$619$	28.0000			1.12542			0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000			1.12542			0.562708	+	0.826656i	$0.309760\pi$
$620$		0			0
$621$		0			0
$622$		−	18.0000i		−	0.721734i
$623$		0			0
$624$	−4.00000			−0.160128
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	4.00000			0.159872
$627$		−	4.00000i		−	0.159745i
$628$		−	6.00000i		−	0.239426i
$629$	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	$0.311825\pi$
$632$		−	8.00000i		−	0.318223i
$633$		−	8.00000i		−	0.317971i
$634$	−18.0000			−0.714871
$635$	10.0000	−	20.0000i	0.396838	−	0.793676i
$636$	14.0000			0.555136
$637$			28.0000i			1.10940i
$638$		−	8.00000i		−	0.316723i
$639$	−12.0000			−0.474713
$640$	1.00000	−	2.00000i	0.0395285	−	0.0790569i
$641$	36.0000			1.42191			0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000			1.42191			0.710957	+	0.703235i	$0.248262\pi$
$642$		−	20.0000i		−	0.789337i
$643$			26.0000i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$			26.0000i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$		0			0
$645$	12.0000	+	6.00000i	0.472500	+	0.236250i
$646$	−6.00000			−0.236067
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−40.0000			−1.57014
$650$	−12.0000	−	16.0000i	−0.470679	−	0.627572i
$651$		0			0
$652$		−	10.0000i		−	0.391630i
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	−14.0000			−0.547443
$655$	−40.0000	−	20.0000i	−1.56293	−	0.781465i
$656$	−12.0000			−0.468521
$657$			8.00000i			0.312110i
$658$		0			0
$659$	26.0000			1.01282			0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000			1.01282			0.506408	+	0.862294i	$0.330973\pi$
$660$	−4.00000	+	8.00000i	−0.155700	+	0.311400i
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$			8.00000i			0.310929i
$663$		−	24.0000i		−	0.932083i
$664$	−12.0000			−0.465690
$665$		0			0
$666$	−4.00000			−0.154997
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$	10.0000			0.386622
$670$	8.00000	+	4.00000i	0.309067	+	0.154533i
$671$	24.0000			0.926510
$672$		0			0
$673$		−	26.0000i		−	1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$		−	26.0000i		−	1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	−18.0000			−0.693334
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$	3.00000			0.115385
$677$		−	30.0000i		−	1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$		−	30.0000i		−	1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$		−	10.0000i		−	0.384048i
$679$		0			0
$680$	12.0000	+	6.00000i	0.460179	+	0.230089i
$681$	12.0000			0.459841
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	1.00000			0.0382360
$685$	6.00000	−	12.0000i	0.229248	−	0.458496i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		−	6.00000i		−	0.228748i
$689$	−56.0000			−2.13343
$690$		0			0
$691$	−4.00000			−0.152167			−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000			−0.152167			−0.0760836	+	0.997101i	$0.524242\pi$
$692$		−	2.00000i		−	0.0760286i
$693$		0			0
$694$	−32.0000			−1.21470
$695$	24.0000	+	12.0000i	0.910372	+	0.455186i
$696$	2.00000			0.0758098
$697$		−	72.0000i		−	2.72719i
$698$		−	34.0000i		−	1.28692i
$699$	14.0000			0.529529
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$			4.00000i			0.150970i
$703$		−	4.00000i		−	0.150863i
$704$	4.00000			0.150756
$705$		0			0
$706$	22.0000			0.827981
$707$		0			0
$708$		−	10.0000i		−	0.375823i
$709$	−22.0000			−0.826227			−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000			−0.826227			−0.413114	+	0.910679i	$0.635559\pi$
$710$	−12.0000	+	24.0000i	−0.450352	+	0.900704i
$711$	−8.00000			−0.300023
$712$		−	8.00000i		−	0.299813i
$713$		0			0
$714$		0			0
$715$	16.0000	−	32.0000i	0.598366	−	1.19673i
$716$	6.00000			0.224231
$717$		−	10.0000i		−	0.373457i
$718$		−	34.0000i		−	1.26887i
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$	−2.00000	−	1.00000i	−0.0745356	−	0.0372678i
$721$		0			0
$722$			1.00000i			0.0372161i
$723$			26.0000i			0.966950i
$724$	10.0000			0.371647
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$	−5.00000			−0.185567
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	16.0000	+	8.00000i	0.592187	+	0.296093i
$731$	36.0000			1.33151
$732$			6.00000i			0.221766i
$733$			26.0000i			0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$			26.0000i			0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	12.0000			0.442928
$735$	−7.00000	+	14.0000i	−0.258199	+	0.516398i
$736$		0			0
$737$			16.0000i			0.589368i
$738$			12.0000i			0.441726i
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	−4.00000	+	8.00000i	−0.147043	+	0.294086i
$741$	−4.00000			−0.146944
$742$		0			0
$743$			44.0000i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$			44.0000i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$		0			0
$745$		0			0
$746$	12.0000			0.439351
$747$			12.0000i			0.439057i
$748$			24.0000i			0.877527i
$749$		0			0
$750$	−2.00000	−	11.0000i	−0.0730297	−	0.401663i
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$		−	24.0000i		−	0.874609i
$754$	−8.00000			−0.291343
$755$	32.0000	+	16.0000i	1.16460	+	0.582300i
$756$		0			0
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$		−	12.0000i		−	0.435860i
$759$		0			0
$760$	1.00000	−	2.00000i	0.0362738	−	0.0725476i
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$			10.0000i			0.362262i
$763$		0			0
$764$	−6.00000			−0.217072
$765$	6.00000	−	12.0000i	0.216930	−	0.433861i
$766$	36.0000			1.30073
$767$			40.0000i			1.44432i
$768$			1.00000i			0.0360844i
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	2.00000			0.0720282
$772$		−	10.0000i		−	0.359908i
$773$			38.0000i			1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$			38.0000i			1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	−6.00000			−0.215666
$775$		0			0
$776$	−10.0000			−0.358979
$777$		0			0
$778$			4.00000i			0.143407i
$779$	−12.0000			−0.429945
$780$	8.00000	+	4.00000i	0.286446	+	0.143223i
$781$	−48.0000			−1.71758
$782$		0			0
$783$		−	2.00000i		−	0.0714742i
$784$	7.00000			0.250000
$785$	−6.00000	+	12.0000i	−0.214149	+	0.428298i
$786$	20.0000			0.713376
$787$			48.0000i			1.71102i	0.517790	+	0.855508i	$0.326755\pi$
$787$			48.0000i			1.71102i	−0.517790	+	0.855508i	$0.673245\pi$
$788$			10.0000i

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:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +(-1.00000 + 2.00000i) q^{10} -4.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} -4.00000i q^{22} +1.00000 q^{24} +(3.00000 + 4.00000i) q^{25} -4.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} +(-2.00000 - 1.00000i) q^{30} +1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +1.00000i q^{38} -4.00000 q^{39} +(1.00000 - 2.00000i) q^{40} -12.0000 q^{41} -6.00000i q^{43} +4.00000 q^{44} +(-2.00000 - 1.00000i) q^{45} +1.00000i q^{48} +7.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} -6.00000 q^{51} -4.00000i q^{52} +14.0000i q^{53} +1.00000 q^{54} +(-8.00000 - 4.00000i) q^{55} +1.00000i q^{57} +2.00000i q^{58} +10.0000 q^{59} +(1.00000 - 2.00000i) q^{60} -6.00000 q^{61} -1.00000 q^{64} +(-4.00000 + 8.00000i) q^{65} +4.00000 q^{66} -4.00000i q^{67} -6.00000i q^{68} +12.0000 q^{71} +1.00000i q^{72} -8.00000i q^{73} +4.00000 q^{74} +(-4.00000 + 3.00000i) q^{75} -1.00000 q^{76} -4.00000i q^{78} +8.00000 q^{79} +(2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -12.0000i q^{82} -12.0000i q^{83} +(-6.00000 + 12.0000i) q^{85} +6.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} +(1.00000 - 2.00000i) q^{90} +(2.00000 + 1.00000i) q^{95} -1.00000 q^{96} -10.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9} - 2 q^{10} - 8 q^{11} - 2 q^{15} + 2 q^{16} + 2 q^{19} - 4 q^{20} + 2 q^{24} + 6 q^{25} - 8 q^{26} + 4 q^{29} - 4 q^{30} - 12 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{40} - 24 q^{41} + 8 q^{44} - 4 q^{45} + 14 q^{49} - 8 q^{50} - 12 q^{51} + 2 q^{54} - 16 q^{55} + 20 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{64} - 8 q^{65} + 8 q^{66} + 24 q^{71} + 8 q^{74} - 8 q^{75} - 2 q^{76} + 16 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{85} + 12 q^{86} + 16 q^{89} + 2 q^{90} + 4 q^{95} - 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^6 - 2 * q^9 - 2 * q^10 - 8 * q^11 - 2 * q^15 + 2 * q^16 + 2 * q^19 - 4 * q^20 + 2 * q^24 + 6 * q^25 - 8 * q^26 + 4 * q^29 - 4 * q^30 - 12 * q^34 + 2 * q^36 - 8 * q^39 + 2 * q^40 - 24 * q^41 + 8 * q^44 - 4 * q^45 + 14 * q^49 - 8 * q^50 - 12 * q^51 + 2 * q^54 - 16 * q^55 + 20 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^64 - 8 * q^65 + 8 * q^66 + 24 * q^71 + 8 * q^74 - 8 * q^75 - 2 * q^76 + 16 * q^79 + 4 * q^80 + 2 * q^81 - 12 * q^85 + 12 * q^86 + 16 * q^89 + 2 * q^90 + 4 * q^95 - 2 * q^96 + 8 * q^99

Introduction

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