LMFDB - Embedded newform 229.2.b.a.228.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [229,2,Mod(228,229)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(229, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("229.228");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	229.b (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:	$1.82857420629$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
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			228.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	229.228
Dual form			229.2.b.a.228.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} +1.00000 q^{3} -3.00000 q^{4} +3.00000 q^{5} -2.23607i q^{6} +2.23607i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-2.23607i q^{2} +1.00000 q^{3} -3.00000 q^{4} +3.00000 q^{5} -2.23607i q^{6} +2.23607i q^{8} -2.00000 q^{9} -6.70820i q^{10} +3.00000 q^{11} -3.00000 q^{12} +3.00000 q^{15} -1.00000 q^{16} -3.00000 q^{17} +4.47214i q^{18} -1.00000 q^{19} -9.00000 q^{20} -6.70820i q^{22} +4.47214i q^{23} +2.23607i q^{24} +4.00000 q^{25} -5.00000 q^{27} +4.47214i q^{29} -6.70820i q^{30} +6.70820i q^{32} +3.00000 q^{33} +6.70820i q^{34} +6.00000 q^{36} +2.00000 q^{37} +2.23607i q^{38} +6.70820i q^{40} +4.47214i q^{41} -1.00000 q^{43} -9.00000 q^{44} -6.00000 q^{45} +10.0000 q^{46} -8.94427i q^{47} -1.00000 q^{48} +7.00000 q^{49} -8.94427i q^{50} -3.00000 q^{51} +6.00000 q^{53} +11.1803i q^{54} +9.00000 q^{55} -1.00000 q^{57} +10.0000 q^{58} -8.94427i q^{59} -9.00000 q^{60} +5.00000 q^{61} +13.0000 q^{64} -6.70820i q^{66} -13.4164i q^{67} +9.00000 q^{68} +4.47214i q^{69} -15.0000 q^{71} -4.47214i q^{72} +13.4164i q^{73} -4.47214i q^{74} +4.00000 q^{75} +3.00000 q^{76} +13.4164i q^{79} -3.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -9.00000 q^{83} -9.00000 q^{85} +2.23607i q^{86} +4.47214i q^{87} +6.70820i q^{88} +17.8885i q^{89} +13.4164i q^{90} -13.4164i q^{92} -20.0000 q^{94} -3.00000 q^{95} +6.70820i q^{96} -7.00000 q^{97} -15.6525i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 6 * q^4 + 6 * q^5 - 4 * q^9	$ 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9} + 6 q^{11} - 6 q^{12} + 6 q^{15} - 2 q^{16} - 6 q^{17} - 2 q^{19} - 18 q^{20} + 8 q^{25} - 10 q^{27} + 6 q^{33} + 12 q^{36} + 4 q^{37} - 2 q^{43} - 18 q^{44} - 12 q^{45} + 20 q^{46} - 2 q^{48} + 14 q^{49} - 6 q^{51} + 12 q^{53} + 18 q^{55} - 2 q^{57} + 20 q^{58} - 18 q^{60} + 10 q^{61} + 26 q^{64} + 18 q^{68} - 30 q^{71} + 8 q^{75} + 6 q^{76} - 6 q^{80} + 2 q^{81} + 20 q^{82} - 18 q^{83} - 18 q^{85} - 40 q^{94} - 6 q^{95} - 14 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 6 * q^4 + 6 * q^5 - 4 * q^9 + 6 * q^11 - 6 * q^12 + 6 * q^15 - 2 * q^16 - 6 * q^17 - 2 * q^19 - 18 * q^20 + 8 * q^25 - 10 * q^27 + 6 * q^33 + 12 * q^36 + 4 * q^37 - 2 * q^43 - 18 * q^44 - 12 * q^45 + 20 * q^46 - 2 * q^48 + 14 * q^49 - 6 * q^51 + 12 * q^53 + 18 * q^55 - 2 * q^57 + 20 * q^58 - 18 * q^60 + 10 * q^61 + 26 * q^64 + 18 * q^68 - 30 * q^71 + 8 * q^75 + 6 * q^76 - 6 * q^80 + 2 * q^81 + 20 * q^82 - 18 * q^83 - 18 * q^85 - 40 * q^94 - 6 * q^95 - 14 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/229\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$		−	2.23607i		−	1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$		−	2.23607i		−	1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	1.00000			0.577350			0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000			0.577350			0.288675	+	0.957427i	$0.406785\pi$
$4$	−3.00000			−1.50000
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		−	2.23607i		−	0.912871i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			2.23607i			0.790569i
$9$	−2.00000			−0.666667
$10$		−	6.70820i		−	2.12132i
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$	−3.00000			−0.866025
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	3.00000			0.774597
$16$	−1.00000			−0.250000
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$			4.47214i			1.05409i
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$	−9.00000			−2.01246
$21$		0			0
$22$		−	6.70820i		−	1.43019i
$23$			4.47214i			0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$			4.47214i			0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$			2.23607i			0.456435i
$25$	4.00000			0.800000
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$			4.47214i			0.830455i	0.909718	+	0.415227i	$0.136298\pi$
$29$			4.47214i			0.830455i	−0.909718	+	0.415227i	$0.863702\pi$
$30$		−	6.70820i		−	1.22474i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			6.70820i			1.18585i
$33$	3.00000			0.522233
$34$			6.70820i			1.15045i
$35$		0			0
$36$	6.00000			1.00000
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$			2.23607i			0.362738i
$39$		0			0
$40$			6.70820i			1.06066i
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−9.00000			−1.35680
$45$	−6.00000			−0.894427
$46$	10.0000			1.47442
$47$		−	8.94427i		−	1.30466i	−0.757937	−	0.652328i	$-0.773792\pi$
$47$		−	8.94427i		−	1.30466i	0.757937	−	0.652328i	$-0.226208\pi$
$48$	−1.00000			−0.144338
$49$	7.00000			1.00000
$50$		−	8.94427i		−	1.26491i
$51$	−3.00000			−0.420084
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$			11.1803i			1.52145i
$55$	9.00000			1.21356
$56$		0			0
$57$	−1.00000			−0.132453
$58$	10.0000			1.31306
$59$		−	8.94427i		−	1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$		−	8.94427i		−	1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$	−9.00000			−1.16190
$61$	5.00000			0.640184			0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000			0.640184			0.320092	+	0.947386i	$0.396286\pi$
$62$		0			0
$63$		0			0
$64$	13.0000			1.62500
$65$		0			0
$66$		−	6.70820i		−	0.825723i
$67$		−	13.4164i		−	1.63908i	−0.573025	−	0.819538i	$-0.694230\pi$
$67$		−	13.4164i		−	1.63908i	0.573025	−	0.819538i	$-0.305770\pi$
$68$	9.00000			1.09141
$69$			4.47214i			0.538382i
$70$		0			0
$71$	−15.0000			−1.78017			−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000			−1.78017			−0.890086	+	0.455792i	$0.849356\pi$
$72$		−	4.47214i		−	0.527046i
$73$			13.4164i			1.57027i	0.619324	+	0.785136i	$0.287407\pi$
$73$			13.4164i			1.57027i	−0.619324	+	0.785136i	$0.712593\pi$
$74$		−	4.47214i		−	0.519875i
$75$	4.00000			0.461880
$76$	3.00000			0.344124
$77$		0			0
$78$		0			0
$79$			13.4164i			1.50946i	0.656033	+	0.754732i	$0.272233\pi$
$79$			13.4164i			1.50946i	−0.656033	+	0.754732i	$0.727767\pi$
$80$	−3.00000			−0.335410
$81$	1.00000			0.111111
$82$	10.0000			1.10432
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$		0			0
$85$	−9.00000			−0.976187
$86$			2.23607i			0.241121i
$87$			4.47214i			0.479463i
$88$			6.70820i			0.715097i
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$			13.4164i			1.41421i
$91$		0			0
$92$		−	13.4164i		−	1.39876i
$93$		0			0
$94$	−20.0000			−2.06284
$95$	−3.00000			−0.307794
$96$			6.70820i			0.684653i
$97$	−7.00000			−0.710742			−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000			−0.710742			−0.355371	+	0.934725i	$0.615646\pi$
$98$		−	15.6525i		−	1.58114i
$99$	−6.00000			−0.603023
$100$	−12.0000			−1.20000
$101$			4.47214i			0.444994i	0.974933	+	0.222497i	$0.0714208\pi$
$101$			4.47214i			0.444994i	−0.974933	+	0.222497i	$0.928579\pi$
$102$			6.70820i			0.664211i
$103$	11.0000			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$104$		0			0
$105$		0			0
$106$		−	13.4164i		−	1.30312i
$107$		−	8.94427i		−	0.864675i	−0.901712	−	0.432338i	$-0.857689\pi$
$107$		−	8.94427i		−	0.864675i	0.901712	−	0.432338i	$-0.142311\pi$
$108$	15.0000			1.44338
$109$		−	13.4164i		−	1.28506i	−0.766261	−	0.642529i	$-0.777885\pi$
$109$		−	13.4164i		−	1.28506i	0.766261	−	0.642529i	$-0.222115\pi$
$110$		−	20.1246i		−	1.91881i
$111$	2.00000			0.189832
$112$		0			0
$113$		−	8.94427i		−	0.841406i	−0.907198	−	0.420703i	$-0.861783\pi$
$113$		−	8.94427i		−	0.841406i	0.907198	−	0.420703i	$-0.138217\pi$
$114$			2.23607i			0.209427i
$115$			13.4164i			1.25109i
$116$		−	13.4164i		−	1.24568i
$117$		0			0
$118$	−20.0000			−1.84115
$119$		0			0
$120$			6.70820i			0.612372i
$121$	−2.00000			−0.181818
$122$		−	11.1803i		−	1.01222i
$123$			4.47214i			0.403239i
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	15.6525i		−	1.38350i
$129$	−1.00000			−0.0880451
$130$		0			0
$131$			4.47214i			0.390732i	0.980730	+	0.195366i	$0.0625895\pi$
$131$			4.47214i			0.390732i	−0.980730	+	0.195366i	$0.937410\pi$
$132$	−9.00000			−0.783349
$133$		0			0
$134$	−30.0000			−2.59161
$135$	−15.0000			−1.29099
$136$		−	6.70820i		−	0.575224i
$137$		−	8.94427i		−	0.764161i	−0.924129	−	0.382080i	$-0.875208\pi$
$137$		−	8.94427i		−	0.764161i	0.924129	−	0.382080i	$-0.124792\pi$
$138$	10.0000			0.851257
$139$		−	13.4164i		−	1.13796i	−0.822350	−	0.568982i	$-0.807337\pi$
$139$		−	13.4164i		−	1.13796i	0.822350	−	0.568982i	$-0.192663\pi$
$140$		0			0
$141$		−	8.94427i		−	0.753244i
$142$			33.5410i			2.81470i
$143$		0			0
$144$	2.00000			0.166667
$145$			13.4164i			1.11417i
$146$	30.0000			2.48282
$147$	7.00000			0.577350
$148$	−6.00000			−0.493197
$149$	−21.0000			−1.72039			−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000			−1.72039			−0.860194	+	0.509968i	$0.829657\pi$
$150$		−	8.94427i		−	0.730297i
$151$	−7.00000			−0.569652			−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000			−0.569652			−0.284826	+	0.958579i	$0.591936\pi$
$152$		−	2.23607i		−	0.181369i
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	13.4164i		−	1.07075i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$		−	13.4164i		−	1.07075i	0.844616	−	0.535373i	$-0.179829\pi$
$158$	30.0000			2.38667
$159$	6.00000			0.475831
$160$			20.1246i			1.59099i
$161$		0			0
$162$		−	2.23607i		−	0.175682i
$163$		−	13.4164i		−	1.05085i	−0.850839	−	0.525427i	$-0.823906\pi$
$163$		−	13.4164i		−	1.05085i	0.850839	−	0.525427i	$-0.176094\pi$
$164$		−	13.4164i		−	1.04765i
$165$	9.00000			0.700649
$166$			20.1246i			1.56197i
$167$	3.00000			0.232147			0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000			0.232147			0.116073	+	0.993241i	$0.462969\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$			20.1246i			1.54349i
$171$	2.00000			0.152944
$172$	3.00000			0.228748
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$	10.0000			0.758098
$175$		0			0
$176$	−3.00000			−0.226134
$177$		−	8.94427i		−	0.672293i
$178$	40.0000			2.99813
$179$			17.8885i			1.33705i	0.743689	+	0.668526i	$0.233075\pi$
$179$			17.8885i			1.33705i	−0.743689	+	0.668526i	$0.766925\pi$
$180$	18.0000			1.34164
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	5.00000			0.369611
$184$	−10.0000			−0.737210
$185$	6.00000			0.441129
$186$		0			0
$187$	−9.00000			−0.658145
$188$			26.8328i			1.95698i
$189$		0			0
$190$			6.70820i			0.486664i
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$	13.0000			0.938194
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$			15.6525i			1.12378i
$195$		0			0
$196$	−21.0000			−1.50000
$197$			17.8885i			1.27451i	0.770655	+	0.637253i	$0.219929\pi$
$197$			17.8885i			1.27451i	−0.770655	+	0.637253i	$0.780071\pi$
$198$			13.4164i			0.953463i
$199$		−	13.4164i		−	0.951064i	−0.879698	−	0.475532i	$-0.842256\pi$
$199$		−	13.4164i		−	0.951064i	0.879698	−	0.475532i	$-0.157744\pi$
$200$			8.94427i			0.632456i
$201$		−	13.4164i		−	0.946320i
$202$	10.0000			0.703598
$203$		0			0
$204$	9.00000			0.630126
$205$			13.4164i			0.937043i
$206$		−	24.5967i		−	1.71374i
$207$		−	8.94427i		−	0.621670i
$208$		0			0
$209$	−3.00000			−0.207514
$210$		0			0
$211$		−	13.4164i		−	0.923624i	−0.886978	−	0.461812i	$-0.847200\pi$
$211$		−	13.4164i		−	0.923624i	0.886978	−	0.461812i	$-0.152800\pi$
$212$	−18.0000			−1.23625
$213$	−15.0000			−1.02778
$214$	−20.0000			−1.36717
$215$	−3.00000			−0.204598
$216$		−	11.1803i		−	0.760726i
$217$		0			0
$218$	−30.0000			−2.03186
$219$			13.4164i			0.906597i
$220$	−27.0000			−1.82034
$221$		0			0
$222$		−	4.47214i		−	0.300150i
$223$		−	13.4164i		−	0.898429i	−0.893424	−	0.449215i	$-0.851704\pi$
$223$		−	13.4164i		−	0.898429i	0.893424	−	0.449215i	$-0.148296\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$	−20.0000			−1.33038
$227$			4.47214i			0.296826i	0.988925	+	0.148413i	$0.0474165\pi$
$227$			4.47214i			0.296826i	−0.988925	+	0.148413i	$0.952583\pi$
$228$	3.00000			0.198680
$229$	7.00000	−	13.4164i	0.462573	−	0.886581i
$229$	7.00000	−	13.4164i	0.462573	−	0.886581i
$230$	30.0000			1.97814
$231$		0			0
$232$	−10.0000			−0.656532
$233$	9.00000			0.589610			0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000			0.589610			0.294805	+	0.955557i	$0.404745\pi$
$234$		0			0
$235$		−	26.8328i		−	1.75038i
$236$			26.8328i			1.74667i
$237$			13.4164i			0.871489i
$238$		0			0
$239$		−	22.3607i		−	1.44639i	−0.690643	−	0.723196i	$-0.742672\pi$
$239$		−	22.3607i		−	1.44639i	0.690643	−	0.723196i	$-0.257328\pi$
$240$	−3.00000			−0.193649
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$			4.47214i			0.287480i
$243$	16.0000			1.02640
$244$	−15.0000			−0.960277
$245$	21.0000			1.34164
$246$	10.0000			0.637577
$247$		0			0
$248$		0			0
$249$	−9.00000			−0.570352
$250$			6.70820i			0.424264i
$251$			17.8885i			1.12911i	0.825394	+	0.564557i	$0.190953\pi$
$251$			17.8885i			1.12911i	−0.825394	+	0.564557i	$0.809047\pi$
$252$		0			0
$253$			13.4164i			0.843482i
$254$		0			0
$255$	−9.00000			−0.563602
$256$	−9.00000			−0.562500
$257$			17.8885i			1.11586i	0.829889	+	0.557928i	$0.188404\pi$
$257$			17.8885i			1.11586i	−0.829889	+	0.557928i	$0.811596\pi$
$258$			2.23607i			0.139212i
$259$		0			0
$260$		0			0
$261$		−	8.94427i		−	0.553637i
$262$	10.0000			0.617802
$263$			4.47214i			0.275764i	0.990449	+	0.137882i	$0.0440294\pi$
$263$			4.47214i			0.275764i	−0.990449	+	0.137882i	$0.955971\pi$
$264$			6.70820i			0.412861i
$265$	18.0000			1.10573
$266$		0			0
$267$			17.8885i			1.09476i
$268$			40.2492i			2.45861i
$269$		−	22.3607i		−	1.36335i	−0.731653	−	0.681677i	$-0.761251\pi$
$269$		−	22.3607i		−	1.36335i	0.731653	−	0.681677i	$-0.238749\pi$
$270$			33.5410i			2.04124i
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$	3.00000			0.181902
$273$		0			0
$274$	−20.0000			−1.20824
$275$	12.0000			0.723627
$276$		−	13.4164i		−	0.807573i
$277$	17.0000			1.02143			0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000			1.02143			0.510716	+	0.859750i	$0.329381\pi$
$278$	−30.0000			−1.79928
$279$		0			0
$280$		0			0
$281$			31.3050i			1.86750i	0.357930	+	0.933748i	$0.383483\pi$
$281$			31.3050i			1.86750i	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−20.0000			−1.19098
$283$		−	13.4164i		−	0.797523i	−0.917055	−	0.398761i	$-0.869440\pi$
$283$		−	13.4164i		−	0.797523i	0.917055	−	0.398761i	$-0.130560\pi$
$284$	45.0000			2.67026
$285$	−3.00000			−0.177705
$286$		0			0
$287$		0			0
$288$		−	13.4164i		−	0.790569i
$289$	−8.00000			−0.470588
$290$	30.0000			1.76166
$291$	−7.00000			−0.410347
$292$		−	40.2492i		−	2.35541i
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		−	15.6525i		−	0.912871i
$295$		−	26.8328i		−	1.56227i
$296$			4.47214i			0.259938i
$297$	−15.0000			−0.870388
$298$			46.9574i			2.72017i
$299$		0			0
$300$	−12.0000			−0.692820
$301$		0			0
$302$			15.6525i			0.900699i
$303$			4.47214i			0.256917i
$304$	1.00000			0.0573539
$305$	15.0000			0.858898
$306$		−	13.4164i		−	0.766965i
$307$	17.0000			0.970241			0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000			0.970241			0.485121	+	0.874447i	$0.338776\pi$
$308$		0			0
$309$	11.0000			0.625768
$310$		0			0
$311$	15.0000			0.850572			0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000			0.850572			0.425286	+	0.905059i	$0.360174\pi$
$312$		0			0
$313$			13.4164i			0.758340i	0.925327	+	0.379170i	$0.123790\pi$
$313$			13.4164i			0.758340i	−0.925327	+	0.379170i	$0.876210\pi$
$314$	−30.0000			−1.69300
$315$		0			0
$316$		−	40.2492i		−	2.26420i
$317$		−	8.94427i		−	0.502360i	−0.967940	−	0.251180i	$-0.919181\pi$
$317$		−	8.94427i		−	0.502360i	0.967940	−	0.251180i	$-0.0808187\pi$
$318$		−	13.4164i		−	0.752355i
$319$			13.4164i			0.751175i
$320$	39.0000			2.18017
$321$		−	8.94427i		−	0.499221i
$322$		0			0
$323$	3.00000			0.166924
$324$	−3.00000			−0.166667
$325$		0			0
$326$	−30.0000			−1.66155
$327$		−	13.4164i		−	0.741929i
$328$	−10.0000			−0.552158
$329$		0			0
$330$		−	20.1246i		−	1.10782i
$331$			13.4164i			0.737432i	0.929542	+	0.368716i	$0.120203\pi$
$331$			13.4164i			0.737432i	−0.929542	+	0.368716i	$0.879797\pi$
$332$	27.0000			1.48182
$333$	−4.00000			−0.219199
$334$		−	6.70820i		−	0.367057i
$335$		−	40.2492i		−	2.19905i
$336$		0			0
$337$	−13.0000			−0.708155			−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000			−0.708155			−0.354078	+	0.935216i	$0.615205\pi$
$338$		−	29.0689i		−	1.58114i
$339$		−	8.94427i		−	0.485786i
$340$	27.0000			1.46428
$341$		0			0
$342$		−	4.47214i		−	0.241825i
$343$		0			0
$344$		−	2.23607i		−	0.120561i
$345$			13.4164i			0.722315i
$346$		−	13.4164i		−	0.721271i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		−	13.4164i		−	0.719195i
$349$		−	13.4164i		−	0.718164i	−0.933306	−	0.359082i	$-0.883090\pi$
$349$		−	13.4164i		−	0.718164i	0.933306	−	0.359082i	$-0.116910\pi$
$350$		0			0
$351$		0			0
$352$			20.1246i			1.07265i
$353$		−	8.94427i		−	0.476056i	−0.971258	−	0.238028i	$-0.923499\pi$
$353$		−	8.94427i		−	0.476056i	0.971258	−	0.238028i	$-0.0765009\pi$
$354$	−20.0000			−1.06299
$355$	−45.0000			−2.38835
$356$		−	53.6656i		−	2.84427i
$357$		0			0
$358$	40.0000			2.11407
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		−	13.4164i		−	0.707107i
$361$	−18.0000			−0.947368
$362$			15.6525i			0.822676i
$363$	−2.00000			−0.104973
$364$		0			0
$365$			40.2492i			2.10674i
$366$		−	11.1803i		−	0.584406i
$367$	17.0000			0.887393			0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000			0.887393			0.443696	+	0.896177i	$0.353667\pi$
$368$		−	4.47214i		−	0.233126i
$369$		−	8.94427i		−	0.465620i
$370$		−	13.4164i		−	0.697486i
$371$		0			0
$372$		0			0
$373$	−19.0000			−0.983783			−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000			−0.983783			−0.491891	+	0.870657i	$0.663694\pi$
$374$			20.1246i			1.04062i
$375$	−3.00000			−0.154919
$376$	20.0000			1.03142
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	9.00000			0.461690
$381$		0			0
$382$	−20.0000			−1.02329
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		−	15.6525i		−	0.798762i
$385$		0			0
$386$		−	31.3050i		−	1.59338i
$387$	2.00000			0.101666
$388$	21.0000			1.06611
$389$			31.3050i			1.58722i	0.608424	+	0.793612i	$0.291802\pi$
$389$			31.3050i			1.58722i	−0.608424	+	0.793612i	$0.708198\pi$
$390$		0			0
$391$		−	13.4164i		−	0.678497i
$392$			15.6525i			0.790569i
$393$			4.47214i			0.225589i
$394$	40.0000			2.01517
$395$			40.2492i			2.02516i
$396$	18.0000			0.904534
$397$	23.0000			1.15434			0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000			1.15434			0.577168	+	0.816625i	$0.304158\pi$
$398$	−30.0000			−1.50376
$399$		0			0
$400$	−4.00000			−0.200000
$401$	27.0000			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$402$	−30.0000			−1.49626
$403$		0			0
$404$		−	13.4164i		−	0.667491i
$405$	3.00000			0.149071
$406$		0			0
$407$	6.00000			0.297409
$408$		−	6.70820i		−	0.332106i
$409$	35.0000			1.73064			0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000			1.73064			0.865319	+	0.501221i	$0.167116\pi$
$410$	30.0000			1.48159
$411$		−	8.94427i		−	0.441188i
$412$	−33.0000			−1.62579
$413$		0			0
$414$	−20.0000			−0.982946
$415$	−27.0000			−1.32538
$416$		0			0
$417$		−	13.4164i		−	0.657004i
$418$			6.70820i			0.328109i
$419$			4.47214i			0.218478i	0.994016	+	0.109239i	$0.0348414\pi$
$419$			4.47214i			0.218478i	−0.994016	+	0.109239i	$0.965159\pi$
$420$		0			0
$421$	17.0000			0.828529			0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000			0.828529			0.414265	+	0.910156i	$0.364039\pi$
$422$	−30.0000			−1.46038
$423$			17.8885i			0.869771i
$424$			13.4164i			0.651558i
$425$	−12.0000			−0.582086
$426$			33.5410i			1.62507i
$427$		0			0
$428$			26.8328i			1.29701i
$429$		0			0
$430$			6.70820i			0.323498i
$431$	15.0000			0.722525			0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000			0.722525			0.361262	+	0.932464i	$0.382346\pi$
$432$	5.00000			0.240563
$433$	−19.0000			−0.913082			−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000			−0.913082			−0.456541	+	0.889702i	$0.650912\pi$
$434$		0			0
$435$			13.4164i			0.643268i
$436$			40.2492i			1.92759i
$437$		−	4.47214i		−	0.213931i
$438$	30.0000			1.43346
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$			20.1246i			0.959403i
$441$	−14.0000			−0.666667
$442$		0			0
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$	−6.00000			−0.284747
$445$			53.6656i			2.54399i
$446$	−30.0000			−1.42054
$447$	−21.0000			−0.993266
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$			17.8885i			0.843274i
$451$			13.4164i			0.631754i
$452$			26.8328i			1.26211i
$453$	−7.00000			−0.328889
$454$	10.0000			0.469323
$455$		0			0
$456$		−	2.23607i		−	0.104713i
$457$	23.0000			1.07589			0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000			1.07589			0.537947	+	0.842978i	$0.319200\pi$
$458$	−30.0000	−	15.6525i	−1.40181	−	0.731392i
$459$	15.0000			0.700140
$460$		−	40.2492i		−	1.87663i
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	−1.00000			−0.0464739			−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000			−0.0464739			−0.0232370	+	0.999730i	$0.507397\pi$
$464$		−	4.47214i		−	0.207614i
$465$		0			0
$466$		−	20.1246i		−	0.932255i
$467$	27.0000			1.24941			0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000			1.24941			0.624705	+	0.780860i	$0.285219\pi$
$468$		0			0
$469$		0			0
$470$	−60.0000			−2.76759
$471$		−	13.4164i		−	0.618195i
$472$	20.0000			0.920575
$473$	−3.00000			−0.137940
$474$	30.0000			1.37795
$475$	−4.00000			−0.183533
$476$		0			0
$477$	−12.0000			−0.549442
$478$	−50.0000			−2.28695
$479$		−	8.94427i		−	0.408674i	−0.978901	−	0.204337i	$-0.934496\pi$
$479$		−	8.94427i		−	0.408674i	0.978901	−	0.204337i	$-0.0655039\pi$
$480$			20.1246i			0.918559i
$481$		0			0
$482$			22.3607i			1.01850i
$483$		0			0
$484$	6.00000			0.272727
$485$	−21.0000			−0.953561
$486$		−	35.7771i		−	1.62288i
$487$			13.4164i			0.607955i	0.952679	+	0.303978i	$0.0983148\pi$
$487$			13.4164i			0.607955i	−0.952679	+	0.303978i	$0.901685\pi$
$488$			11.1803i			0.506110i
$489$		−	13.4164i		−	0.606711i
$490$		−	46.9574i		−	2.12132i
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		−	13.4164i		−	0.604858i
$493$		−	13.4164i		−	0.604245i
$494$		0			0
$495$	−18.0000			−0.809040
$496$		0			0
$497$		0			0
$498$			20.1246i			0.901805i
$499$			26.8328i			1.20120i	0.799549	+	0.600601i	$0.205072\pi$
$499$			26.8328i			1.20120i	−0.799549	+	0.600601i	$0.794928\pi$
$500$	9.00000			0.402492
$501$	3.00000			0.134030
$502$	40.0000			1.78529
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$			13.4164i			0.597022i
$506$	30.0000			1.33366
$507$	13.0000			0.577350
$508$		0			0
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$			20.1246i			0.891133i
$511$		0			0
$512$		−	11.1803i		−	0.494106i
$513$	5.00000			0.220755
$514$	40.0000			1.76432
$515$	33.0000			1.45415
$516$	3.00000			0.132068
$517$		−	26.8328i		−	1.18011i
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$			31.3050i			1.37149i	0.727840	+	0.685747i	$0.240525\pi$
$521$			31.3050i			1.37149i	−0.727840	+	0.685747i	$0.759475\pi$
$522$	−20.0000			−0.875376
$523$			26.8328i			1.17332i	0.809834	+	0.586659i	$0.199557\pi$
$523$			26.8328i			1.17332i	−0.809834	+	0.586659i	$0.800443\pi$
$524$		−	13.4164i		−	0.586098i
$525$		0			0
$526$	10.0000			0.436021
$527$		0			0
$528$	−3.00000			−0.130558
$529$	3.00000			0.130435
$530$		−	40.2492i		−	1.74831i
$531$			17.8885i			0.776297i
$532$		0			0
$533$		0			0
$534$	40.0000			1.73097
$535$		−	26.8328i		−	1.16008i
$536$	30.0000			1.29580
$537$			17.8885i			0.771948i
$538$	−50.0000			−2.15565
$539$	21.0000			0.904534
$540$	45.0000			1.93649
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$			55.9017i			2.40118i
$543$	−7.00000			−0.300399
$544$		−	20.1246i		−	0.862836i
$545$		−	40.2492i		−	1.72409i
$546$		0			0
$547$		−	13.4164i		−	0.573644i	−0.957984	−	0.286822i	$-0.907401\pi$
$547$		−	13.4164i		−	0.573644i	0.957984	−	0.286822i	$-0.0925989\pi$
$548$			26.8328i			1.14624i
$549$	−10.0000			−0.426790
$550$		−	26.8328i		−	1.14416i
$551$		−	4.47214i		−	0.190519i
$552$	−10.0000			−0.425628
$553$		0			0
$554$		−	38.0132i		−	1.61502i
$555$	6.00000			0.254686
$556$			40.2492i			1.70695i
$557$	−3.00000			−0.127114			−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000			−0.127114			−0.0635570	+	0.997978i	$0.520244\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−9.00000			−0.379980
$562$	70.0000			2.95277
$563$			44.7214i			1.88478i	0.334515	+	0.942390i	$0.391427\pi$
$563$			44.7214i			1.88478i	−0.334515	+	0.942390i	$0.608573\pi$
$564$			26.8328i			1.12987i
$565$		−	26.8328i		−	1.12887i
$566$	−30.0000			−1.26099
$567$		0			0
$568$		−	33.5410i		−	1.40735i
$569$	45.0000			1.88650			0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000			1.88650			0.943249	+	0.332086i	$0.107752\pi$
$570$			6.70820i			0.280976i
$571$			26.8328i			1.12292i	0.827504	+	0.561459i	$0.189760\pi$
$571$			26.8328i			1.12292i	−0.827504	+	0.561459i	$0.810240\pi$
$572$		0			0
$573$		−	8.94427i		−	0.373652i
$574$		0			0
$575$			17.8885i			0.746004i
$576$	−26.0000			−1.08333
$577$		−	40.2492i		−	1.67560i	−0.545979	−	0.837799i	$-0.683842\pi$
$577$		−	40.2492i		−	1.67560i	0.545979	−	0.837799i	$-0.316158\pi$
$578$			17.8885i			0.744065i
$579$	14.0000			0.581820
$580$		−	40.2492i		−	1.67126i
$581$		0			0
$582$			15.6525i			0.648816i
$583$	18.0000			0.745484
$584$	−30.0000			−1.24141
$585$		0			0
$586$		−	20.1246i		−	0.831340i
$587$	−27.0000			−1.11441			−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000			−1.11441			−0.557205	+	0.830375i	$0.688126\pi$
$588$	−21.0000			−0.866025
$589$		0			0
$590$	−60.0000			−2.47016
$591$			17.8885i			0.735836i
$592$	−2.00000			−0.0821995
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$			33.5410i			1.37620i
$595$		0			0
$596$	63.0000			2.58058
$597$		−	13.4164i		−	0.549097i
$598$		0			0
$599$		−	8.94427i		−	0.365453i	−0.983164	−	0.182727i	$-0.941508\pi$
$599$		−	8.94427i		−	0.365453i	0.983164	−	0.182727i	$-0.0584923\pi$
$600$			8.94427i			0.365148i
$601$			40.2492i			1.64180i	0.571072	+	0.820900i	$0.306528\pi$
$601$			40.2492i			1.64180i	−0.571072	+	0.820900i	$0.693472\pi$
$602$		0			0
$603$			26.8328i			1.09272i
$604$	21.0000			0.854478
$605$	−6.00000			−0.243935
$606$	10.0000			0.406222
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		−	6.70820i		−	0.272054i
$609$		0			0
$610$		−	33.5410i		−	1.35804i
$611$		0			0
$612$	−18.0000			−0.727607
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	38.0132i		−	1.53409i
$615$			13.4164i			0.541002i
$616$		0			0
$617$	42.0000			1.69086			0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000			1.69086			0.845428	+	0.534089i	$0.179345\pi$
$618$		−	24.5967i		−	0.989426i
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$		−	22.3607i		−	0.897303i
$622$		−	33.5410i		−	1.34487i
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$	30.0000			1.19904
$627$	−3.00000			−0.119808
$628$			40.2492i			1.60612i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$	−30.0000			−1.19334
$633$		−	13.4164i		−	0.533254i
$634$	−20.0000			−0.794301
$635$		0			0
$636$	−18.0000			−0.713746
$637$		0			0
$638$	30.0000			1.18771
$639$	30.0000			1.18678
$640$		−	46.9574i		−	1.85616i
$641$	−45.0000			−1.77739			−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000			−1.77739			−0.888697	+	0.458496i	$0.848388\pi$
$642$	−20.0000			−0.789337
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$		0			0
$645$	−3.00000			−0.118125
$646$		−	6.70820i		−	0.263931i
$647$			17.8885i			0.703271i	0.936137	+	0.351636i	$0.114374\pi$
$647$			17.8885i			0.703271i	−0.936137	+	0.351636i	$0.885626\pi$
$648$			2.23607i			0.0878410i
$649$		−	26.8328i		−	1.05328i
$650$		0			0
$651$		0			0
$652$			40.2492i			1.57628i
$653$		−	35.7771i		−	1.40007i	−0.714111	−	0.700033i	$-0.753169\pi$
$653$		−	35.7771i		−	1.40007i	0.714111	−	0.700033i	$-0.246831\pi$
$654$	−30.0000			−1.17309
$655$			13.4164i			0.524222i
$656$		−	4.47214i		−	0.174608i
$657$		−	26.8328i		−	1.04685i
$658$		0			0
$659$			31.3050i			1.21947i	0.792606	+	0.609734i	$0.208724\pi$
$659$			31.3050i			1.21947i	−0.792606	+	0.609734i	$0.791276\pi$
$660$	−27.0000			−1.05097
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$	30.0000			1.16598
$663$		0			0
$664$		−	20.1246i		−	0.780986i
$665$		0			0
$666$			8.94427i			0.346583i
$667$	−20.0000			−0.774403
$668$	−9.00000			−0.348220
$669$		−	13.4164i		−	0.518708i
$670$	−90.0000			−3.47700
$671$	15.0000			0.579069
$672$		0			0
$673$	−19.0000			−0.732396			−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000			−0.732396			−0.366198	+	0.930537i	$0.619341\pi$
$674$			29.0689i			1.11969i
$675$	−20.0000			−0.769800
$676$	−39.0000			−1.50000
$677$		−	8.94427i		−	0.343756i	−0.985118	−	0.171878i	$-0.945016\pi$
$677$		−	8.94427i		−	0.343756i	0.985118	−	0.171878i	$-0.0549835\pi$
$678$	−20.0000			−0.768095
$679$		0			0
$680$		−	20.1246i		−	0.771744i
$681$			4.47214i			0.171373i
$682$		0			0
$683$	9.00000			0.344375			0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000			0.344375			0.172188	+	0.985064i	$0.444916\pi$
$684$	−6.00000			−0.229416
$685$		−	26.8328i		−	1.02523i
$686$		0			0
$687$	7.00000	−	13.4164i	0.267067	−	0.511868i
$688$	1.00000			0.0381246
$689$		0			0
$690$	30.0000			1.14208
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$			26.8328i			1.01856i
$695$		−	40.2492i		−	1.52674i
$696$	−10.0000			−0.379049
$697$		−	13.4164i		−	0.508183i
$698$	−30.0000			−1.13552
$699$	9.00000			0.340411
$700$		0			0
$701$	33.0000			1.24639			0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000			1.24639			0.623196	+	0.782065i	$0.285834\pi$
$702$		0			0
$703$	−2.00000			−0.0754314
$704$	39.0000			1.46987
$705$		−	26.8328i		−	1.01058i
$706$	−20.0000			−0.752710
$707$		0			0
$708$			26.8328i			1.00844i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$			100.623i			3.77632i
$711$		−	26.8328i		−	1.00631i
$712$	−40.0000			−1.49906
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	53.6656i		−	2.00558i
$717$		−	22.3607i		−	0.835075i
$718$			53.6656i			2.00278i
$719$			4.47214i			0.166783i	0.996517	+	0.0833913i	$0.0265751\pi$
$719$			4.47214i			0.166783i	−0.996517	+	0.0833913i	$0.973425\pi$
$720$	6.00000			0.223607
$721$		0			0
$722$			40.2492i			1.49792i
$723$	−10.0000			−0.371904
$724$	21.0000			0.780459
$725$			17.8885i			0.664364i
$726$			4.47214i			0.165977i
$727$		−	26.8328i		−	0.995174i	−0.867414	−	0.497587i	$-0.834220\pi$
$727$		−	26.8328i		−	0.995174i	0.867414	−	0.497587i	$-0.165780\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$	90.0000			3.33105
$731$	3.00000			0.110959
$732$	−15.0000			−0.554416
$733$	11.0000			0.406294			0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000			0.406294			0.203147	+	0.979148i	$0.434883\pi$
$734$		−	38.0132i		−	1.40309i
$735$	21.0000			0.774597
$736$	−30.0000			−1.10581
$737$		−	40.2492i		−	1.48260i
$738$	−20.0000			−0.736210
$739$		−	13.4164i		−	0.493531i	−0.969075	−	0.246765i	$-0.920632\pi$
$739$		−	13.4164i		−	0.493531i	0.969075	−	0.246765i	$-0.0793677\pi$
$740$	−18.0000			−0.661693
$741$		0			0
$742$		0			0
$743$	−39.0000			−1.43077			−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000			−1.43077			−0.715386	+	0.698730i	$0.753749\pi$
$744$		0			0
$745$	−63.0000			−2.30814
$746$			42.4853i			1.55550i
$747$	18.0000			0.658586
$748$	27.0000			0.987218
$749$		0			0
$750$			6.70820i			0.244949i
$751$	5.00000			0.182453			0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000			0.182453			0.0912263	+	0.995830i	$0.470921\pi$
$752$			8.94427i			0.326164i
$753$			17.8885i			0.651895i
$754$		0			0
$755$	−21.0000			−0.764268
$756$		0			0
$757$	−13.0000			−0.472493			−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000			−0.472493			−0.236247	+	0.971693i	$0.575917\pi$
$758$		0			0
$759$			13.4164i			0.486985i
$760$		−	6.70820i		−	0.243332i
$761$			44.7214i			1.62115i	0.585636	+	0.810574i	$0.300845\pi$
$761$			44.7214i			1.62115i	−0.585636	+	0.810574i	$0.699155\pi$
$762$		0			0
$763$		0			0
$764$			26.8328i			0.970777i
$765$	18.0000			0.650791
$766$			53.6656i			1.93902i
$767$		0			0
$768$	−9.00000			−0.324760
$769$	5.00000			0.180305			0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000			0.180305			0.0901523	+	0.995928i	$0.471265\pi$
$770$		0			0
$771$			17.8885i			0.644240i
$772$	−42.0000			−1.51161
$773$		−	35.7771i		−	1.28681i	−0.765525	−	0.643406i	$-0.777521\pi$
$773$		−	35.7771i		−	1.28681i	0.765525	−	0.643406i	$-0.222479\pi$
$774$		−	4.47214i		−	0.160748i
$775$		0			0
$776$		−	15.6525i		−	0.561891i
$777$		0			0
$778$	70.0000			2.50962
$779$		−	4.47214i		−	0.160231i
$780$		0			0
$781$	−45.0000			−1.61023
$782$	−30.0000			−1.07280
$783$		−	22.3607i		−	0.799106i
$784$	−7.00000			−0.250000
$785$		−	40.2492i		−	1.43656i
$786$	10.0000			0.356688
$787$	23.0000			0.819861			0.409931	−	0.912117i	$-0.365553\pi$
$787$	23.0000			0.819861			0.409931	+	0.912117i	$0.365553\pi$
$788$		−	53.6656i		−	1.91176i
$789$			4.47214i			0.159212i
$790$	90.0000			3.20206
$791$		0			0
$792$		−	13.4164i		−	0.476731i
$793$		0			0
$794$		−	51.4296i		−	1.82517i
$795$	18.0000			0.638394
$796$			40.2492i			1.42660i
$797$			44.7214i			1.58411i	0.610449	+	0.792056i	$0.290989\pi$
$797$			44.7214i			1.58411i	−0.610449	+	0.792056i	$0.709011\pi$
$798$		0			0
$799$			26.8328i			0.949277i
$800$			26.8328i			0.948683i
$801$		−	35.7771i		−	1.26412i
$802$		−	60.3738i		−	2.13187i
$803$			40.2492i			1.42036i
$804$			40.2492i			1.41948i
$805$		0			0
$806$		0			0
$807$		−	22.3607i		−	0.787133i
$808$	−10.0000			−0.351799
$809$			4.47214i			0.157232i	0.996905	+	0.0786160i	$0.0250501\pi$
$809$			4.47214i			0.157232i	−0.996905	+	0.0786160i	$0.974950\pi$
$810$		−	6.70820i		−	0.235702i
$811$			13.4164i			0.471114i	0.971861	+	0.235557i	$0.0756914\pi$
$811$			13.4164i			0.471114i	−0.971861	+	0.235557i	$0.924309\pi$
$812$		0			0
$813$	−25.0000			−0.876788
$814$		−	13.4164i		−	0.470245i
$815$		−	40.2492i		−	1.40987i
$816$	3.00000			0.105021
$817$	1.00000			0.0349856
$818$		−	78.2624i		−	2.73638i
$819$		0			0
$820$		−	40.2492i		−	1.40556i
$821$	3.00000			0.104701			0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000			0.104701			0.0523504	+	0.998629i	$0.483329\pi$
$822$	−20.0000			−0.697580
$823$			26.8328i			0.935333i	0.883905	+	0.467667i	$0.154905\pi$
$823$			26.8328i			0.935333i	−0.883905	+	0.467667i	$0.845095\pi$
$824$			24.5967i			0.856868i
$825$	12.0000			0.417786
$826$		0			0
$827$			17.8885i			0.622046i	0.950402	+	0.311023i	$0.100672\pi$
$827$			17.8885i			0.622046i	−0.950402	+	0.311023i	$0.899328\pi$
$828$			26.8328i			0.932505i
$829$			26.8328i			0.931942i	0.884800	+	0.465971i	$0.154295\pi$
$829$			26.8328i			0.931942i	−0.884800	+	0.465971i	$0.845705\pi$
$830$			60.3738i			2.09561i
$831$	17.0000			0.589723
$832$		0			0
$833$	−21.0000			−0.727607
$834$	−30.0000			−1.03882
$835$	9.00000			0.311458
$836$	9.00000			0.311272
$837$		0			0
$838$	10.0000			0.345444
$839$		−	22.3607i		−	0.771976i	−0.922504	−	0.385988i	$-0.873861\pi$
$839$		−	22.3607i		−	0.771976i	0.922504	−	0.385988i	$-0.126139\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		−	38.0132i		−	1.31002i
$843$			31.3050i			1.07820i
$844$			40.2492i			1.38544i
$845$	39.0000			1.34164
$846$	40.0000			1.37523
$847$		0			0
$848$	−6.00000			−0.206041
$849$		−	13.4164i		−	0.460450i
$850$			26.8328i			0.920358i
$851$			8.94427i			0.306606i
$852$	45.0000			1.54167
$853$		−	26.8328i		−	0.918738i	−0.888246	−	0.459369i	$-0.848076\pi$
$853$		−	26.8328i		−	0.918738i	0.888246	−	0.459369i	$-0.151924\pi$
$854$		0			0
$855$	6.00000			0.205196
$856$	20.0000			0.683586
$857$			4.47214i			0.152765i	0.997079	+	0.0763826i	$0.0243370\pi$
$857$			4.47214i			0.152765i	−0.997079	+	0.0763826i	$0.975663\pi$
$858$		0			0
$859$	5.00000			0.170598			0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000			0.170598			0.0852989	+	0.996355i	$0.472815\pi$
$860$	9.00000			0.306897
$861$		0			0
$862$		−	33.5410i		−	1.14241i
$863$	9.00000			0.306364			0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000			0.306364			0.153182	+	0.988198i	$0.451048\pi$
$864$		−	33.5410i		−	1.14109i
$865$	18.0000			0.612018
$866$			42.4853i			1.44371i
$867$	−8.00000			−0.271694
$868$		0			0
$869$			40.2492i			1.36536i
$870$	30.0000			1.01710
$871$		0			0
$872$	30.0000			1.01593
$873$	14.0000			0.473828
$874$	−10.0000			−0.338255
$875$		0			0
$876$		−	40.2492i		−	1.35990i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		−	44.7214i		−	1.50927i
$879$	9.00000			0.303562
$880$	−9.00000			−0.303390
$881$			4.47214i			0.150670i	0.997158	+	0.0753350i	$0.0240026\pi$
$881$			4.47214i			0.150670i	−0.997158	+	0.0753350i	$0.975997\pi$
$882$			31.3050i			1.05409i
$883$	29.0000			0.975928			0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000			0.975928			0.487964	+	0.872864i	$0.337740\pi$
$884$		0			0
$885$		−	26.8328i		−	0.901975i
$886$		−	80.4984i		−	2.70440i
$887$		−	35.7771i		−	1.20128i	−0.799521	−	0.600639i	$-0.794913\pi$
$887$		−	35.7771i		−	1.20128i	0.799521	−	0.600639i	$-0.205087\pi$
$888$			4.47214i			0.150075i
$889$		0			0
$890$	120.000			4.02241
$891$	3.00000			0.100504
$892$			40.2492i			1.34764i
$893$			8.94427i			0.299309i
$894$			46.9574i			1.57049i
$895$			53.6656i			1.79384i
$896$		0			0
$897$		0			0
$898$			67.0820i			2.23856i
$899$		0			0
$900$	24.0000			0.800000
$901$	−18.0000			−0.599667
$902$	30.0000			0.998891
$903$		0			0
$904$	20.0000			0.665190
$905$	−21.0000			−0.698064
$906$			15.6525i			0.520019i
$907$	−52.0000			−1.72663			−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000			−1.72663			−0.863316	+	0.504664i	$0.831616\pi$
$908$		−	13.4164i		−	0.445239i
$909$		−	8.94427i		−	0.296663i
$910$		0			0
$911$	−15.0000			−0.496972			−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000			−0.496972			−0.248486	+	0.968635i	$0.579933\pi$
$912$	1.00000			0.0331133
$913$	−27.0000			−0.893570
$914$		−	51.4296i		−	1.70114i
$915$	15.0000			0.495885
$916$	−21.0000	+	40.2492i	−0.693860	+	1.32987i
$917$		0			0
$918$		−	33.5410i		−	1.10702i
$919$	−25.0000			−0.824674			−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000			−0.824674			−0.412337	+	0.911031i	$0.635287\pi$
$920$	−30.0000			−0.989071
$921$	17.0000			0.560169
$922$			40.2492i			1.32554i
$923$		0			0
$924$		0			0
$925$	8.00000			0.263038
$926$			2.23607i			0.0734818i
$927$	−22.0000			−0.722575
$928$	−30.0000			−0.984798
$929$			4.47214i			0.146726i	0.997305	+	0.0733630i	$0.0233732\pi$
$929$			4.47214i			0.146726i	−0.997305	+	0.0733630i	$0.976627\pi$
$930$		0			0
$931$	−7.00000			−0.229416
$932$	−27.0000			−0.884414
$933$	15.0000			0.491078
$934$		−	60.3738i		−	1.97549i
$935$	−27.0000			−0.882994
$936$		0			0
$937$		−	40.2492i		−	1.31488i	−0.753505	−	0.657442i	$-0.771638\pi$
$937$		−	40.2492i		−	1.31488i	0.753505	−	0.657442i	$-0.228362\pi$
$938$		0			0
$939$			13.4164i			0.437828i
$940$			80.4984i			2.62557i
$941$	−45.0000			−1.46696			−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000			−1.46696			−0.733479	+	0.679712i	$0.762105\pi$
$942$	−30.0000			−0.977453
$943$	−20.0000			−0.651290
$944$			8.94427i			0.291111i
$945$		0			0
$946$			6.70820i			0.218103i
$947$			4.47214i			0.145325i	0.997357	+	0.0726624i	$0.0231496\pi$
$947$			4.47214i			0.145325i	−0.997357	+	0.0726624i	$0.976850\pi$
$948$		−	40.2492i		−	1.30723i
$949$		0			0
$950$			8.94427i			0.290191i
$951$		−	8.94427i		−	0.290038i
$952$		0			0
$953$	−39.0000			−1.26333			−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000			−1.26333			−0.631667	+	0.775240i	$0.717629\pi$
$954$			26.8328i			0.868744i
$955$		−	26.8328i		−	0.868290i
$956$			67.0820i			2.16959i
$957$			13.4164i			0.433691i
$958$	−20.0000			−0.646171
$959$		0			0
$960$	39.0000			1.25872
$961$	31.0000			1.00000
$962$		0			0
$963$			17.8885i			0.576450i
$964$	30.0000			0.966235
$965$	42.0000			1.35203
$966$		0			0
$967$	−28.0000			−0.900419			−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000			−0.900419			−0.450210	+	0.892923i	$0.648651\pi$
$968$		−	4.47214i		−	0.143740i
$969$	3.00000			0.0963739
$970$			46.9574i			1.50771i
$971$	33.0000			1.05902			0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000			1.05902			0.529510	+	0.848304i	$0.322376\pi$
$972$	−48.0000			−1.53960
$973$		0			0
$974$	30.0000			0.961262
$975$		0			0
$976$	−5.00000			−0.160046
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$	−30.0000			−0.959294
$979$			53.6656i			1.71516i
$980$	−63.0000			−2.01246
$981$			26.8328i			0.856706i
$982$			33.5410i			1.07034i
$983$		−	22.3607i		−	0.713195i	−0.934258	−	0.356597i	$-0.883937\pi$
$983$		−	22.3607i		−	0.713195i	0.934258	−	0.356597i	$-0.116063\pi$
$984$	−10.0000			−0.318788
$985$			53.6656i			1.70993i
$986$	−30.0000			−0.955395
$987$		0			0
$988$		0			0
$989$		−	4.47214i		−	0.142206i
$990$			40.2492i			1.27920i
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		0			0
$993$			13.4164i			0.425757i
$994$		0			0
$995$		−	40.2492i		−	1.27599i
$996$	27.0000			0.855528
$997$	−37.0000			−1.17180			−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000			−1.17180			−0.585901	+	0.810383i	$0.699259\pi$
$998$	60.0000			1.89927
$999$	−10.0000			−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	229.2.b.a.228.1	✓	2
3.2	odd	2		2061.2.c.a.1144.2		2
229.228	even	2	inner	229.2.b.a.228.2	yes	2
687.686	odd	2		2061.2.c.a.1144.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
229.2.b.a.228.1	✓	2	1.1	even	1	trivial
229.2.b.a.228.2	yes	2	229.228	even	2	inner
2061.2.c.a.1144.1		2	687.686	odd	2
2061.2.c.a.1144.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	229.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} +1.00000 q^{3} -3.00000 q^{4} +3.00000 q^{5} -2.23607i q^{6} +2.23607i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-2.23607i q^{2} +1.00000 q^{3} -3.00000 q^{4} +3.00000 q^{5} -2.23607i q^{6} +2.23607i q^{8} -2.00000 q^{9} -6.70820i q^{10} +3.00000 q^{11} -3.00000 q^{12} +3.00000 q^{15} -1.00000 q^{16} -3.00000 q^{17} +4.47214i q^{18} -1.00000 q^{19} -9.00000 q^{20} -6.70820i q^{22} +4.47214i q^{23} +2.23607i q^{24} +4.00000 q^{25} -5.00000 q^{27} +4.47214i q^{29} -6.70820i q^{30} +6.70820i q^{32} +3.00000 q^{33} +6.70820i q^{34} +6.00000 q^{36} +2.00000 q^{37} +2.23607i q^{38} +6.70820i q^{40} +4.47214i q^{41} -1.00000 q^{43} -9.00000 q^{44} -6.00000 q^{45} +10.0000 q^{46} -8.94427i q^{47} -1.00000 q^{48} +7.00000 q^{49} -8.94427i q^{50} -3.00000 q^{51} +6.00000 q^{53} +11.1803i q^{54} +9.00000 q^{55} -1.00000 q^{57} +10.0000 q^{58} -8.94427i q^{59} -9.00000 q^{60} +5.00000 q^{61} +13.0000 q^{64} -6.70820i q^{66} -13.4164i q^{67} +9.00000 q^{68} +4.47214i q^{69} -15.0000 q^{71} -4.47214i q^{72} +13.4164i q^{73} -4.47214i q^{74} +4.00000 q^{75} +3.00000 q^{76} +13.4164i q^{79} -3.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -9.00000 q^{83} -9.00000 q^{85} +2.23607i q^{86} +4.47214i q^{87} +6.70820i q^{88} +17.8885i q^{89} +13.4164i q^{90} -13.4164i q^{92} -20.0000 q^{94} -3.00000 q^{95} +6.70820i q^{96} -7.00000 q^{97} -15.6525i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 6 * q^4 + 6 * q^5 - 4 * q^9	\( 2 q + 2 q^{3} - 6 q^{4} + 6 q^{5} - 4 q^{9} + 6 q^{11} - 6 q^{12} + 6 q^{15} - 2 q^{16} - 6 q^{17} - 2 q^{19} - 18 q^{20} + 8 q^{25} - 10 q^{27} + 6 q^{33} + 12 q^{36} + 4 q^{37} - 2 q^{43} - 18 q^{44} - 12 q^{45} + 20 q^{46} - 2 q^{48} + 14 q^{49} - 6 q^{51} + 12 q^{53} + 18 q^{55} - 2 q^{57} + 20 q^{58} - 18 q^{60} + 10 q^{61} + 26 q^{64} + 18 q^{68} - 30 q^{71} + 8 q^{75} + 6 q^{76} - 6 q^{80} + 2 q^{81} + 20 q^{82} - 18 q^{83} - 18 q^{85} - 40 q^{94} - 6 q^{95} - 14 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 6 * q^4 + 6 * q^5 - 4 * q^9 + 6 * q^11 - 6 * q^12 + 6 * q^15 - 2 * q^16 - 6 * q^17 - 2 * q^19 - 18 * q^20 + 8 * q^25 - 10 * q^27 + 6 * q^33 + 12 * q^36 + 4 * q^37 - 2 * q^43 - 18 * q^44 - 12 * q^45 + 20 * q^46 - 2 * q^48 + 14 * q^49 - 6 * q^51 + 12 * q^53 + 18 * q^55 - 2 * q^57 + 20 * q^58 - 18 * q^60 + 10 * q^61 + 26 * q^64 + 18 * q^68 - 30 * q^71 + 8 * q^75 + 6 * q^76 - 6 * q^80 + 2 * q^81 + 20 * q^82 - 18 * q^83 - 18 * q^85 - 40 * q^94 - 6 * q^95 - 14 * q^97 - 12 * q^99

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