LMFDB - Embedded newform 154.2.c.a.153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(153,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("154.153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 154 = 2 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	154.c (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:	$1.22969619113$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			153.1
Root			$-1.54779 + 1.54779i$ of defining polynomial
Character	$\chi$	$=$	154.153
Dual form			154.2.c.a.153.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -3.09557i q^{3} -1.00000 q^{4} -0.646084i q^{5} -3.09557 q^{6} +(2.44949 + 1.00000i) q^{7} +1.00000i q^{8} -6.58258 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -3.09557i q^{3} -1.00000 q^{4} -0.646084i q^{5} -3.09557 q^{6} +(2.44949 + 1.00000i) q^{7} +1.00000i q^{8} -6.58258 q^{9} -0.646084 q^{10} +(2.79129 + 1.79129i) q^{11} +3.09557i q^{12} -3.09557 q^{13} +(1.00000 - 2.44949i) q^{14} -2.00000 q^{15} +1.00000 q^{16} +3.74166 q^{17} +6.58258i q^{18} -5.54506 q^{19} +0.646084i q^{20} +(3.09557 - 7.58258i) q^{21} +(1.79129 - 2.79129i) q^{22} +4.00000 q^{23} +3.09557 q^{24} +4.58258 q^{25} +3.09557i q^{26} +11.0901i q^{27} +(-2.44949 - 1.00000i) q^{28} -7.58258i q^{29} +2.00000i q^{30} -1.15732i q^{31} -1.00000i q^{32} +(5.54506 - 8.64064i) q^{33} -3.74166i q^{34} +(0.646084 - 1.58258i) q^{35} +6.58258 q^{36} -5.58258 q^{37} +5.54506i q^{38} +9.58258i q^{39} +0.646084 q^{40} +5.03383 q^{41} +(-7.58258 - 3.09557i) q^{42} +11.1652i q^{43} +(-2.79129 - 1.79129i) q^{44} +4.25290i q^{45} -4.00000i q^{46} +5.03383i q^{47} -3.09557i q^{48} +(5.00000 + 4.89898i) q^{49} -4.58258i q^{50} -11.5826i q^{51} +3.09557 q^{52} -2.41742 q^{53} +11.0901 q^{54} +(1.15732 - 1.80341i) q^{55} +(-1.00000 + 2.44949i) q^{56} +17.1652i q^{57} -7.58258 q^{58} +3.09557i q^{59} +2.00000 q^{60} -9.28672 q^{61} -1.15732 q^{62} +(-16.1240 - 6.58258i) q^{63} -1.00000 q^{64} +2.00000i q^{65} +(-8.64064 - 5.54506i) q^{66} +1.58258 q^{67} -3.74166 q^{68} -12.3823i q^{69} +(-1.58258 - 0.646084i) q^{70} +2.00000 q^{71} -6.58258i q^{72} -6.32599 q^{73} +5.58258i q^{74} -14.1857i q^{75} +5.54506 q^{76} +(5.04594 + 7.17903i) q^{77} +9.58258 q^{78} -4.00000i q^{79} -0.646084i q^{80} +14.5826 q^{81} -5.03383i q^{82} +9.15188 q^{83} +(-3.09557 + 7.58258i) q^{84} -2.41742i q^{85} +11.1652 q^{86} -23.4724 q^{87} +(-1.79129 + 2.79129i) q^{88} +9.79796i q^{89} +4.25290 q^{90} +(-7.58258 - 3.09557i) q^{91} -4.00000 q^{92} -3.58258 q^{93} +5.03383 q^{94} +3.58258i q^{95} -3.09557 q^{96} -15.9891i q^{97} +(4.89898 - 5.00000i) q^{98} +(-18.3739 - 11.7913i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 - 16 * q^9	$ 8 q - 8 q^{4} - 16 q^{9} + 4 q^{11} + 8 q^{14} - 16 q^{15} + 8 q^{16} - 4 q^{22} + 32 q^{23} + 16 q^{36} - 8 q^{37} - 24 q^{42} - 4 q^{44} + 40 q^{49} - 56 q^{53} - 8 q^{56} - 24 q^{58} + 16 q^{60} - 8 q^{64} - 24 q^{67} + 24 q^{70} + 16 q^{71} + 4 q^{77} + 40 q^{78} + 80 q^{81} + 16 q^{86} + 4 q^{88} - 24 q^{91} - 32 q^{92} + 8 q^{93} - 92 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^9 + 4 * q^11 + 8 * q^14 - 16 * q^15 + 8 * q^16 - 4 * q^22 + 32 * q^23 + 16 * q^36 - 8 * q^37 - 24 * q^42 - 4 * q^44 + 40 * q^49 - 56 * q^53 - 8 * q^56 - 24 * q^58 + 16 * q^60 - 8 * q^64 - 24 * q^67 + 24 * q^70 + 16 * q^71 + 4 * q^77 + 40 * q^78 + 80 * q^81 + 16 * q^86 + 4 * q^88 - 24 * q^91 - 32 * q^92 + 8 * q^93 - 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-1$	$-1$

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
$3$		−	3.09557i		−	1.78723i	−0.448834	−	0.893615i	$-0.648161\pi$
$3$		−	3.09557i		−	1.78723i	0.448834	−	0.893615i	$-0.351839\pi$
$4$	−1.00000			−0.500000
$5$		−	0.646084i		−	0.288937i	−0.989509	−	0.144469i	$-0.953853\pi$
$5$		−	0.646084i		−	0.288937i	0.989509	−	0.144469i	$-0.0461473\pi$
$6$	−3.09557			−1.26376
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$			1.00000i			0.353553i
$9$	−6.58258			−2.19419
$10$	−0.646084			−0.204310
$11$	2.79129	+	1.79129i	0.841605	+	0.540094i
$11$	2.79129	+	1.79129i	0.841605	+	0.540094i
$12$			3.09557i			0.893615i
$13$	−3.09557			−0.858558			−0.429279	−	0.903172i	$-0.641232\pi$
$13$	−3.09557			−0.858558			−0.429279	+	0.903172i	$0.641232\pi$
$14$	1.00000	−	2.44949i	0.267261	−	0.654654i
$15$	−2.00000			−0.516398
$16$	1.00000			0.250000
$17$	3.74166			0.907485			0.453743	−	0.891133i	$-0.350089\pi$
$17$	3.74166			0.907485			0.453743	+	0.891133i	$0.350089\pi$
$18$			6.58258i			1.55153i
$19$	−5.54506			−1.27212			−0.636062	−	0.771638i	$-0.719438\pi$
$19$	−5.54506			−1.27212			−0.636062	+	0.771638i	$0.719438\pi$
$20$			0.646084i			0.144469i
$21$	3.09557	−	7.58258i	0.675510	−	1.65465i
$22$	1.79129	−	2.79129i	0.381904	−	0.595105i
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$	3.09557			0.631881
$25$	4.58258			0.916515
$26$			3.09557i			0.607092i
$27$			11.0901i			2.13430i
$28$	−2.44949	−	1.00000i	−0.462910	−	0.188982i
$29$		−	7.58258i		−	1.40805i	−0.710176	−	0.704024i	$-0.751385\pi$
$29$		−	7.58258i		−	1.40805i	0.710176	−	0.704024i	$-0.248615\pi$
$30$			2.00000i			0.365148i
$31$		−	1.15732i		−	0.207861i	−0.994585	−	0.103931i	$-0.966858\pi$
$31$		−	1.15732i		−	0.207861i	0.994585	−	0.103931i	$-0.0331420\pi$
$32$		−	1.00000i		−	0.176777i
$33$	5.54506	−	8.64064i	0.965272	−	1.50414i
$34$		−	3.74166i		−	0.641689i
$35$	0.646084	−	1.58258i	0.109208	−	0.267504i
$36$	6.58258			1.09710
$37$	−5.58258			−0.917770			−0.458885	−	0.888496i	$-0.651751\pi$
$37$	−5.58258			−0.917770			−0.458885	+	0.888496i	$0.651751\pi$
$38$			5.54506i			0.899528i
$39$			9.58258i			1.53444i
$40$	0.646084			0.102155
$41$	5.03383			0.786151			0.393076	−	0.919506i	$-0.371411\pi$
$41$	5.03383			0.786151			0.393076	+	0.919506i	$0.371411\pi$
$42$	−7.58258	−	3.09557i	−1.17002	−	0.477657i
$43$			11.1652i			1.70267i	0.524623	+	0.851335i	$0.324206\pi$
$43$			11.1652i			1.70267i	−0.524623	+	0.851335i	$0.675794\pi$
$44$	−2.79129	−	1.79129i	−0.420802	−	0.270047i
$45$			4.25290i			0.633984i
$46$		−	4.00000i		−	0.589768i
$47$			5.03383i			0.734259i	0.930170	+	0.367129i	$0.119659\pi$
$47$			5.03383i			0.734259i	−0.930170	+	0.367129i	$0.880341\pi$
$48$		−	3.09557i		−	0.446808i
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		−	4.58258i		−	0.648074i
$51$		−	11.5826i		−	1.62189i
$52$	3.09557			0.429279
$53$	−2.41742			−0.332059			−0.166029	−	0.986121i	$-0.553095\pi$
$53$	−2.41742			−0.332059			−0.166029	+	0.986121i	$0.553095\pi$
$54$	11.0901			1.50918
$55$	1.15732	−	1.80341i	0.156053	−	0.243171i
$56$	−1.00000	+	2.44949i	−0.133631	+	0.327327i
$57$			17.1652i			2.27358i
$58$	−7.58258			−0.995641
$59$			3.09557i			0.403009i	0.979488	+	0.201505i	$0.0645831\pi$
$59$			3.09557i			0.403009i	−0.979488	+	0.201505i	$0.935417\pi$
$60$	2.00000			0.258199
$61$	−9.28672			−1.18904			−0.594521	−	0.804080i	$-0.702658\pi$
$61$	−9.28672			−1.18904			−0.594521	+	0.804080i	$0.702658\pi$
$62$	−1.15732			−0.146980
$63$	−16.1240	−	6.58258i	−2.03143	−	0.829327i
$64$	−1.00000			−0.125000
$65$			2.00000i			0.248069i
$66$	−8.64064	−	5.54506i	−1.06359	−	0.682550i
$67$	1.58258			0.193342			0.0966712	−	0.995316i	$-0.469180\pi$
$67$	1.58258			0.193342			0.0966712	+	0.995316i	$0.469180\pi$
$68$	−3.74166			−0.453743
$69$		−	12.3823i		−	1.49065i
$70$	−1.58258	−	0.646084i	−0.189154	−	0.0772218i
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		−	6.58258i		−	0.775764i
$73$	−6.32599			−0.740401			−0.370201	−	0.928952i	$-0.620711\pi$
$73$	−6.32599			−0.740401			−0.370201	+	0.928952i	$0.620711\pi$
$74$			5.58258i			0.648961i
$75$		−	14.1857i		−	1.63802i
$76$	5.54506			0.636062
$77$	5.04594	+	7.17903i	0.575039	+	0.818126i
$78$	9.58258			1.08501
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		−	0.646084i		−	0.0722344i
$81$	14.5826			1.62029
$82$		−	5.03383i		−	0.555893i
$83$	9.15188			1.00455			0.502274	−	0.864708i	$-0.332497\pi$
$83$	9.15188			1.00455			0.502274	+	0.864708i	$0.332497\pi$
$84$	−3.09557	+	7.58258i	−0.337755	+	0.827327i
$85$		−	2.41742i		−	0.262206i
$86$	11.1652			1.20397
$87$	−23.4724			−2.51651
$88$	−1.79129	+	2.79129i	−0.190952	+	0.297552i
$89$			9.79796i			1.03858i	0.854598	+	0.519291i	$0.173804\pi$
$89$			9.79796i			1.03858i	−0.854598	+	0.519291i	$0.826196\pi$
$90$	4.25290			0.448295
$91$	−7.58258	−	3.09557i	−0.794870	−	0.324504i
$92$	−4.00000			−0.417029
$93$	−3.58258			−0.371496
$94$	5.03383			0.519199
$95$			3.58258i			0.367565i
$96$	−3.09557			−0.315941
$97$		−	15.9891i		−	1.62345i	−0.584041	−	0.811724i	$-0.698529\pi$
$97$		−	15.9891i		−	1.62345i	0.584041	−	0.811724i	$-0.301471\pi$
$98$	4.89898	−	5.00000i	0.494872	−	0.505076i
$99$	−18.3739	−	11.7913i	−1.84664	−	1.18507i
$100$	−4.58258			−0.458258
$101$	−0.511238			−0.0508701			−0.0254351	−	0.999676i	$-0.508097\pi$
$101$	−0.511238			−0.0508701			−0.0254351	+	0.999676i	$0.508097\pi$
$102$	−11.5826			−1.14685
$103$			13.5396i			1.33410i	0.745014	+	0.667049i	$0.232443\pi$
$103$			13.5396i			1.33410i	−0.745014	+	0.667049i	$0.767557\pi$
$104$		−	3.09557i		−	0.303546i
$105$	−4.89898	−	2.00000i	−0.478091	−	0.195180i
$106$			2.41742i			0.234801i
$107$			8.41742i			0.813743i	0.913485	+	0.406872i	$0.133380\pi$
$107$			8.41742i			0.813743i	−0.913485	+	0.406872i	$0.866620\pi$
$108$		−	11.0901i		−	1.06715i
$109$		−	1.16515i		−	0.111601i	−0.998442	−	0.0558006i	$-0.982229\pi$
$109$		−	1.16515i		−	0.111601i	0.998442	−	0.0558006i	$-0.0177711\pi$
$110$	−1.80341	−	1.15732i	−0.171948	−	0.110346i
$111$			17.2813i			1.64027i
$112$	2.44949	+	1.00000i	0.231455	+	0.0944911i
$113$	−16.7477			−1.57549			−0.787747	−	0.615999i	$-0.788752\pi$
$113$	−16.7477			−1.57549			−0.787747	+	0.615999i	$0.788752\pi$
$114$	17.1652			1.60766
$115$		−	2.58434i		−	0.240991i
$116$			7.58258i			0.704024i
$117$	20.3768			1.88384
$118$	3.09557			0.284971
$119$	9.16515	+	3.74166i	0.840168	+	0.342997i
$120$		−	2.00000i		−	0.182574i
$121$	4.58258	+	10.0000i	0.416598	+	0.909091i
$122$			9.28672i			0.840780i
$123$		−	15.5826i		−	1.40503i
$124$			1.15732i			0.103931i
$125$		−	6.19115i		−	0.553753i
$126$	−6.58258	+	16.1240i	−0.586422	+	1.43644i
$127$			5.58258i			0.495373i	0.968840	+	0.247687i	$0.0796704\pi$
$127$			5.58258i			0.495373i	−0.968840	+	0.247687i	$0.920330\pi$
$128$			1.00000i			0.0883883i
$129$	34.5625			3.04306
$130$	2.00000			0.175412
$131$	9.42157			0.823166			0.411583	−	0.911372i	$-0.364976\pi$
$131$	9.42157			0.823166			0.411583	+	0.911372i	$0.364976\pi$
$132$	−5.54506	+	8.64064i	−0.482636	+	0.752071i
$133$	−13.5826	−	5.54506i	−1.17776	−	0.480818i
$134$		−	1.58258i		−	0.136714i
$135$	7.16515			0.616678
$136$			3.74166i			0.320844i
$137$	15.1652			1.29565			0.647823	−	0.761791i	$-0.275680\pi$
$137$	15.1652			1.29565			0.647823	+	0.761791i	$0.275680\pi$
$138$	−12.3823			−1.05405
$139$	3.23042			0.274001			0.137000	−	0.990571i	$-0.456254\pi$
$139$	3.23042			0.274001			0.137000	+	0.990571i	$0.456254\pi$
$140$	−0.646084	+	1.58258i	−0.0546040	+	0.133752i
$141$	15.5826			1.31229
$142$		−	2.00000i		−	0.167836i
$143$	−8.64064	−	5.54506i	−0.722566	−	0.463701i
$144$	−6.58258			−0.548548
$145$	−4.89898			−0.406838
$146$			6.32599i			0.523543i
$147$	15.1652	−	15.4779i	1.25080	−	1.27659i
$148$	5.58258			0.458885
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	−14.1857			−1.15826
$151$		−	3.58258i		−	0.291546i	−0.989318	−	0.145773i	$-0.953433\pi$
$151$		−	3.58258i		−	0.291546i	0.989318	−	0.145773i	$-0.0465669\pi$
$152$		−	5.54506i		−	0.449764i
$153$	−24.6297			−1.99120
$154$	7.17903	−	5.04594i	0.578503	−	0.406614i
$155$	−0.747727			−0.0600589
$156$		−	9.58258i		−	0.767220i
$157$			20.5117i			1.63701i	0.574498	+	0.818506i	$0.305197\pi$
$157$			20.5117i			1.63701i	−0.574498	+	0.818506i	$0.694803\pi$
$158$	−4.00000			−0.318223
$159$			7.48331i			0.593465i
$160$	−0.646084			−0.0510774
$161$	9.79796	+	4.00000i	0.772187	+	0.315244i
$162$		−	14.5826i		−	1.14572i
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	−5.03383			−0.393076
$165$	−5.58258	−	3.58258i	−0.434603	−	0.278903i
$166$		−	9.15188i		−	0.710323i
$167$	−6.19115			−0.479085			−0.239543	−	0.970886i	$-0.576998\pi$
$167$	−6.19115			−0.479085			−0.239543	+	0.970886i	$0.576998\pi$
$168$	7.58258	+	3.09557i	0.585008	+	0.238829i
$169$	−3.41742			−0.262879
$170$	−2.41742			−0.185408
$171$	36.5008			2.79129
$172$		−	11.1652i		−	0.851335i
$173$	−22.9612			−1.74571			−0.872853	−	0.487983i	$-0.837733\pi$
$173$	−22.9612			−1.74571			−0.872853	+	0.487983i	$0.837733\pi$
$174$			23.4724i			1.77944i
$175$	11.2250	+	4.58258i	0.848528	+	0.346410i
$176$	2.79129	+	1.79129i	0.210401	+	0.135023i
$177$	9.58258			0.720270
$178$	9.79796			0.734388
$179$	−7.16515			−0.535549			−0.267774	−	0.963482i	$-0.586288\pi$
$179$	−7.16515			−0.535549			−0.267774	+	0.963482i	$0.586288\pi$
$180$		−	4.25290i		−	0.316992i
$181$			16.6352i			1.23648i	0.785988	+	0.618242i	$0.212155\pi$
$181$			16.6352i			1.23648i	−0.785988	+	0.618242i	$0.787845\pi$
$182$	−3.09557	+	7.58258i	−0.229459	+	0.562058i
$183$			28.7477i			2.12509i
$184$			4.00000i			0.294884i
$185$			3.60681i			0.265178i
$186$			3.58258i			0.262687i
$187$	10.4440	+	6.70239i	0.763744	+	0.490127i
$188$		−	5.03383i		−	0.367129i
$189$	−11.0901	+	27.1652i	−0.806688	+	1.97597i
$190$	3.58258			0.259907
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$			3.09557i			0.223404i
$193$		−	20.3303i		−	1.46341i	−0.681623	−	0.731704i	$-0.738726\pi$
$193$		−	20.3303i		−	1.46341i	0.681623	−	0.731704i	$-0.261274\pi$
$194$	−15.9891			−1.14795
$195$	6.19115			0.443357
$196$	−5.00000	−	4.89898i	−0.357143	−	0.349927i
$197$		−	18.7477i		−	1.33572i	−0.744287	−	0.667860i	$-0.767210\pi$
$197$		−	18.7477i		−	1.33572i	0.744287	−	0.667860i	$-0.232790\pi$
$198$	−11.7913	+	18.3739i	−0.837970	+	1.30577i
$199$		−	24.6297i		−	1.74596i	−0.487759	−	0.872978i	$-0.662186\pi$
$199$		−	24.6297i		−	1.74596i	0.487759	−	0.872978i	$-0.337814\pi$
$200$			4.58258i			0.324037i
$201$		−	4.89898i		−	0.345547i
$202$			0.511238i			0.0359706i
$203$	7.58258	−	18.5734i	0.532192	−	1.30360i
$204$			11.5826i			0.810943i
$205$		−	3.25227i		−	0.227149i
$206$	13.5396			0.943350
$207$	−26.3303			−1.83008
$208$	−3.09557			−0.214639
$209$	−15.4779	−	9.93280i	−1.07063	−	0.687066i
$210$	−2.00000	+	4.89898i	−0.138013	+	0.338062i
$211$		−	10.7477i		−	0.739904i	−0.929051	−	0.369952i	$-0.879374\pi$
$211$		−	10.7477i		−	0.739904i	0.929051	−	0.369952i	$-0.120626\pi$
$212$	2.41742			0.166029
$213$		−	6.19115i		−	0.424210i
$214$	8.41742			0.575403
$215$	7.21362			0.491965
$216$	−11.0901			−0.754588
$217$	1.15732	−	2.83485i	0.0785641	−	0.192442i
$218$	−1.16515			−0.0789140
$219$			19.5826i			1.32327i
$220$	−1.15732	+	1.80341i	−0.0780266	+	0.121586i
$221$	−11.5826			−0.779128
$222$	17.2813			1.15984
$223$		−	6.32599i		−	0.423620i	−0.977311	−	0.211810i	$-0.932064\pi$
$223$		−	6.32599i		−	0.423620i	0.977311	−	0.211810i	$-0.0679358\pi$
$224$	1.00000	−	2.44949i	0.0668153	−	0.163663i
$225$	−30.1652			−2.01101
$226$			16.7477i			1.11404i
$227$	−11.7362			−0.778960			−0.389480	−	0.921035i	$-0.627345\pi$
$227$	−11.7362			−0.778960			−0.389480	+	0.921035i	$0.627345\pi$
$228$		−	17.1652i		−	1.13679i
$229$			13.0284i			0.860939i	0.902605	+	0.430470i	$0.141652\pi$
$229$			13.0284i			0.860939i	−0.902605	+	0.430470i	$0.858348\pi$
$230$	−2.58434			−0.170406
$231$	22.2232	−	15.6201i	1.46218	−	1.02773i
$232$	7.58258			0.497820
$233$		−	8.83485i		−	0.578790i	−0.957210	−	0.289395i	$-0.906546\pi$
$233$		−	8.83485i		−	0.578790i	0.957210	−	0.289395i	$-0.0934541\pi$
$234$		−	20.3768i		−	1.33208i
$235$	3.25227			0.212155
$236$		−	3.09557i		−	0.201505i
$237$	−12.3823			−0.804316
$238$	3.74166	−	9.16515i	0.242536	−	0.594089i
$239$		−	17.5826i		−	1.13732i	−0.822572	−	0.568661i	$-0.807462\pi$
$239$		−	17.5826i		−	1.13732i	0.822572	−	0.568661i	$-0.192538\pi$
$240$	−2.00000			−0.129099
$241$	−25.9219			−1.66978			−0.834889	−	0.550419i	$-0.814468\pi$
$241$	−25.9219			−1.66978			−0.834889	+	0.550419i	$0.814468\pi$
$242$	10.0000	−	4.58258i	0.642824	−	0.294579i
$243$		−	11.8711i		−	0.761529i
$244$	9.28672			0.594521
$245$	3.16515	−	3.23042i	0.202214	−	0.206384i
$246$	−15.5826			−0.993509
$247$	17.1652			1.09219
$248$	1.15732			0.0734900
$249$		−	28.3303i		−	1.79536i
$250$	−6.19115			−0.391563
$251$		−	2.07310i		−	0.130853i	−0.997857	−	0.0654264i	$-0.979159\pi$
$251$		−	2.07310i		−	0.130853i	0.997857	−	0.0654264i	$-0.0208407\pi$
$252$	16.1240	+	6.58258i	1.01571	+	0.414663i
$253$	11.1652	+	7.16515i	0.701947	+	0.450469i
$254$	5.58258			0.350282
$255$	−7.48331			−0.468623
$256$	1.00000			0.0625000
$257$		−	4.89898i		−	0.305590i	−0.988258	−	0.152795i	$-0.951173\pi$
$257$		−	4.89898i		−	0.305590i	0.988258	−	0.152795i	$-0.0488274\pi$
$258$		−	34.5625i		−	2.15177i
$259$	−13.6745	−	5.58258i	−0.849690	−	0.346884i
$260$		−	2.00000i		−	0.124035i
$261$			49.9129i			3.08953i
$262$		−	9.42157i		−	0.582066i
$263$			28.3303i			1.74692i	0.486895	+	0.873461i	$0.338130\pi$
$263$			28.3303i			1.74692i	−0.486895	+	0.873461i	$0.661870\pi$
$264$	8.64064	+	5.54506i	0.531794	+	0.341275i
$265$			1.56186i			0.0959442i
$266$	−5.54506	+	13.5826i	−0.339990	+	0.832801i
$267$	30.3303			1.85618
$268$	−1.58258			−0.0966712
$269$		−	20.2420i		−	1.23418i	−0.786894	−	0.617088i	$-0.788312\pi$
$269$		−	20.2420i		−	1.23418i	0.786894	−	0.617088i	$-0.211688\pi$
$270$		−	7.16515i		−	0.436057i
$271$	−4.89898			−0.297592			−0.148796	−	0.988868i	$-0.547540\pi$
$271$	−4.89898			−0.297592			−0.148796	+	0.988868i	$0.547540\pi$
$272$	3.74166			0.226871
$273$	−9.58258	+	23.4724i	−0.579964	+	1.42062i
$274$		−	15.1652i		−	0.916160i
$275$	12.7913	+	8.20871i	0.771344	+	0.495004i
$276$			12.3823i			0.745327i
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		−	3.23042i		−	0.193748i
$279$			7.61816i			0.456087i
$280$	1.58258	+	0.646084i	0.0945770	+	0.0386109i
$281$		−	7.16515i		−	0.427437i	−0.976895	−	0.213719i	$-0.931442\pi$
$281$		−	7.16515i		−	0.427437i	0.976895	−	0.213719i	$-0.0685575\pi$
$282$		−	15.5826i		−	0.927929i
$283$	15.6127			0.928079			0.464040	−	0.885814i	$-0.346399\pi$
$283$	15.6127			0.928079			0.464040	+	0.885814i	$0.346399\pi$
$284$	−2.00000			−0.118678
$285$	11.0901			0.656922
$286$	−5.54506	+	8.64064i	−0.327886	+	0.510932i
$287$	12.3303	+	5.03383i	0.727835	+	0.297137i
$288$			6.58258i			0.387882i
$289$	−3.00000			−0.176471
$290$			4.89898i			0.287678i
$291$	−49.4955			−2.90147
$292$	6.32599			0.370201
$293$	21.3993			1.25016			0.625081	−	0.780560i	$-0.285066\pi$
$293$	21.3993			1.25016			0.625081	+	0.780560i	$0.285066\pi$
$294$	−15.4779	−	15.1652i	−0.902688	−	0.884450i
$295$	2.00000			0.116445
$296$		−	5.58258i		−	0.324481i
$297$	−19.8656	+	30.9557i	−1.15272	+	1.79623i
$298$	−14.0000			−0.810998
$299$	−12.3823			−0.716087
$300$			14.1857i			0.819012i
$301$	−11.1652	+	27.3489i	−0.643549	+	1.57637i
$302$	−3.58258			−0.206154
$303$			1.58258i			0.0909166i
$304$	−5.54506			−0.318031
$305$			6.00000i			0.343559i
$306$			24.6297i			1.40799i
$307$	8.12940			0.463969			0.231985	−	0.972719i	$-0.425478\pi$
$307$	8.12940			0.463969			0.231985	+	0.972719i	$0.425478\pi$
$308$	−5.04594	−	7.17903i	−0.287519	−	0.409063i
$309$	41.9129			2.38434
$310$			0.747727i			0.0424680i
$311$		−	9.93280i		−	0.563238i	−0.959526	−	0.281619i	$-0.909129\pi$
$311$		−	9.93280i		−	0.563238i	0.959526	−	0.281619i	$-0.0908714\pi$
$312$	−9.58258			−0.542507
$313$		−	1.02248i		−	0.0577938i	−0.999582	−	0.0288969i	$-0.990801\pi$
$313$		−	1.02248i		−	0.0577938i	0.999582	−	0.0288969i	$-0.00919945\pi$
$314$	20.5117			1.15754
$315$	−4.25290	+	10.4174i	−0.239624	+	0.586955i
$316$			4.00000i			0.225018i
$317$	−9.16515			−0.514766			−0.257383	−	0.966309i	$-0.582860\pi$
$317$	−9.16515			−0.514766			−0.257383	+	0.966309i	$0.582860\pi$
$318$	7.48331			0.419643
$319$	13.5826	−	21.1652i	0.760478	−	1.18502i
$320$			0.646084i			0.0361172i
$321$	26.0568			1.45435
$322$	4.00000	−	9.79796i	0.222911	−	0.546019i
$323$	−20.7477			−1.15443
$324$	−14.5826			−0.810143
$325$	−14.1857			−0.786881
$326$		−	4.00000i		−	0.221540i
$327$	−3.60681			−0.199457
$328$			5.03383i			0.277946i
$329$	−5.03383	+	12.3303i	−0.277524	+	0.679792i
$330$	−3.58258	+	5.58258i	−0.197214	+	0.307311i
$331$	−14.4174			−0.792453			−0.396227	−	0.918153i	$-0.629681\pi$
$331$	−14.4174			−0.792453			−0.396227	+	0.918153i	$0.629681\pi$
$332$	−9.15188			−0.502274
$333$	36.7477			2.01376
$334$			6.19115i			0.338764i
$335$		−	1.02248i		−	0.0558639i
$336$	3.09557	−	7.58258i	0.168877	−	0.413663i
$337$			29.1652i			1.58873i	0.607443	+	0.794364i	$0.292195\pi$
$337$			29.1652i			1.58873i	−0.607443	+	0.794364i	$0.707805\pi$
$338$			3.41742i			0.185883i
$339$			51.8438i			2.81577i
$340$			2.41742i			0.131103i
$341$	2.07310	−	3.23042i	0.112264	−	0.174937i
$342$		−	36.5008i		−	1.97374i
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$	−11.1652			−0.601985
$345$	−8.00000			−0.430706
$346$			22.9612i			1.23440i
$347$		−	0.834849i		−	0.0448170i	−0.999749	−	0.0224085i	$-0.992867\pi$
$347$		−	0.834849i		−	0.0448170i	0.999749	−	0.0224085i	$-0.00713345\pi$
$348$	23.4724			1.25825
$349$	−2.07310			−0.110970			−0.0554852	−	0.998460i	$-0.517671\pi$
$349$	−2.07310			−0.110970			−0.0554852	+	0.998460i	$0.517671\pi$
$350$	4.58258	−	11.2250i	0.244949	−	0.600000i
$351$		−	34.3303i		−	1.83242i
$352$	1.79129	−	2.79129i	0.0954760	−	0.148776i
$353$		−	7.48331i		−	0.398297i	−0.979969	−	0.199148i	$-0.936182\pi$
$353$		−	7.48331i		−	0.398297i	0.979969	−	0.199148i	$-0.0638176\pi$
$354$		−	9.58258i		−	0.509308i
$355$		−	1.29217i		−	0.0685811i
$356$		−	9.79796i		−	0.519291i
$357$	11.5826	−	28.3714i	0.613015	−	1.50157i
$358$			7.16515i			0.378690i
$359$		−	0.417424i		−	0.0220308i	−0.999939	−	0.0110154i	$-0.996494\pi$
$359$		−	0.417424i		−	0.0220308i	0.999939	−	0.0110154i	$-0.00350638\pi$
$360$	−4.25290			−0.224147
$361$	11.7477			0.618301
$362$	16.6352			0.874326
$363$	30.9557	−	14.1857i	1.62475	−	0.744556i
$364$	7.58258	+	3.09557i	0.397435	+	0.162252i
$365$			4.08712i			0.213930i
$366$	28.7477			1.50267
$367$		−	1.42701i		−	0.0744895i	−0.999306	−	0.0372447i	$-0.988142\pi$
$367$		−	1.42701i		−	0.0744895i	0.999306	−	0.0372447i	$-0.0118581\pi$
$368$	4.00000			0.208514
$369$	−33.1355			−1.72497
$370$	3.60681			0.187509
$371$	−5.92146	−	2.41742i	−0.307427	−	0.125506i
$372$	3.58258			0.185748
$373$			0.417424i			0.0216134i	0.999942	+	0.0108067i	$0.00343995\pi$
$373$			0.417424i			0.0216134i	−0.999942	+	0.0108067i	$0.996560\pi$
$374$	6.70239	−	10.4440i	0.346572	−	0.540049i
$375$	−19.1652			−0.989684
$376$	−5.03383			−0.259600
$377$			23.4724i			1.20889i
$378$	27.1652	+	11.0901i	1.39722	+	0.570415i
$379$	14.3303			0.736098			0.368049	−	0.929806i	$-0.380026\pi$
$379$	14.3303			0.736098			0.368049	+	0.929806i	$0.380026\pi$
$380$		−	3.58258i		−	0.183782i
$381$	17.2813			0.885346
$382$			18.0000i			0.920960i
$383$			4.76413i			0.243436i	0.992565	+	0.121718i	$0.0388403\pi$
$383$			4.76413i			0.243436i	−0.992565	+	0.121718i	$0.961160\pi$
$384$	3.09557			0.157970
$385$	4.63825	−	3.26010i	0.236387	−	0.166150i
$386$	−20.3303			−1.03479
$387$		−	73.4955i		−	3.73598i
$388$			15.9891i			0.811724i
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		−	6.19115i		−	0.313501i
$391$	14.9666			0.756895
$392$	−4.89898	+	5.00000i	−0.247436	+	0.252538i
$393$		−	29.1652i		−	1.47119i
$394$	−18.7477			−0.944497
$395$	−2.58434			−0.130032
$396$	18.3739	+	11.7913i	0.923321	+	0.592535i
$397$		−	23.8488i		−	1.19694i	−0.801146	−	0.598469i	$-0.795776\pi$
$397$		−	23.8488i		−	1.19694i	0.801146	−	0.598469i	$-0.204224\pi$
$398$	−24.6297			−1.23458
$399$	−17.1652	+	42.0459i	−0.859332	+	2.10493i
$400$	4.58258			0.229129
$401$	−1.58258			−0.0790301			−0.0395150	−	0.999219i	$-0.512581\pi$
$401$	−1.58258			−0.0790301			−0.0395150	+	0.999219i	$0.512581\pi$
$402$	−4.89898			−0.244339
$403$			3.58258i			0.178461i
$404$	0.511238			0.0254351
$405$		−	9.42157i		−	0.468161i
$406$	−18.5734	−	7.58258i	−0.921784	−	0.376317i
$407$	−15.5826	−	10.0000i	−0.772400	−	0.495682i
$408$	11.5826			0.573423
$409$	9.93280			0.491146			0.245573	−	0.969378i	$-0.421024\pi$
$409$	9.93280			0.491146			0.245573	+	0.969378i	$0.421024\pi$
$410$	−3.25227			−0.160618
$411$		−	46.9448i		−	2.31562i
$412$		−	13.5396i		−	0.667049i
$413$	−3.09557	+	7.58258i	−0.152323	+	0.373114i
$414$			26.3303i			1.29406i
$415$		−	5.91288i		−	0.290252i
$416$			3.09557i			0.151773i
$417$		−	10.0000i		−	0.489702i
$418$	−9.93280	+	15.4779i	−0.485829	+	0.757047i
$419$		−	5.67991i		−	0.277482i	−0.990329	−	0.138741i	$-0.955694\pi$
$419$		−	5.67991i		−	0.277482i	0.990329	−	0.138741i	$-0.0443055\pi$
$420$	4.89898	+	2.00000i	0.239046	+	0.0975900i
$421$	−1.58258			−0.0771300			−0.0385650	−	0.999256i	$-0.512279\pi$
$421$	−1.58258			−0.0771300			−0.0385650	+	0.999256i	$0.512279\pi$
$422$	−10.7477			−0.523191
$423$		−	33.1355i		−	1.61110i
$424$		−	2.41742i		−	0.117401i
$425$	17.1464			0.831724
$426$	−6.19115			−0.299962
$427$	−22.7477	−	9.28672i	−1.10084	−	0.449416i
$428$		−	8.41742i		−	0.406872i
$429$	−17.1652	+	26.7477i	−0.828741	+	1.29139i
$430$		−	7.21362i		−	0.347872i
$431$			17.9129i			0.862833i	0.902153	+	0.431416i	$0.141986\pi$
$431$			17.9129i			0.862833i	−0.902153	+	0.431416i	$0.858014\pi$
$432$			11.0901i			0.533574i
$433$			28.1017i			1.35048i	0.737597	+	0.675241i	$0.235960\pi$
$433$			28.1017i			1.35048i	−0.737597	+	0.675241i	$0.764040\pi$
$434$	−2.83485	−	1.15732i	−0.136077	−	0.0555532i
$435$			15.1652i			0.727113i
$436$			1.16515i			0.0558006i
$437$	−22.1803			−1.06103
$438$	19.5826			0.935692
$439$	2.31464			0.110472			0.0552360	−	0.998473i	$-0.482409\pi$
$439$	2.31464			0.110472			0.0552360	+	0.998473i	$0.482409\pi$
$440$	1.80341	+	1.15732i	0.0859740	+	0.0551732i
$441$	−32.9129	−	32.2479i	−1.56728	−	1.53561i
$442$			11.5826i			0.550927i
$443$	−3.16515			−0.150381			−0.0751904	−	0.997169i	$-0.523956\pi$
$443$	−3.16515			−0.150381			−0.0751904	+	0.997169i	$0.523956\pi$
$444$		−	17.2813i		−	0.820133i
$445$	6.33030			0.300085
$446$	−6.32599			−0.299544
$447$	−43.3380			−2.04982
$448$	−2.44949	−	1.00000i	−0.115728	−	0.0472456i
$449$	12.8348			0.605714			0.302857	−	0.953036i	$-0.402060\pi$
$449$	12.8348			0.605714			0.302857	+	0.953036i	$0.402060\pi$
$450$			30.1652i			1.42200i
$451$	14.0509	+	9.01703i	0.661629	+	0.424595i
$452$	16.7477			0.787747
$453$	−11.0901			−0.521060
$454$			11.7362i			0.550808i
$455$	−2.00000	+	4.89898i	−0.0937614	+	0.229668i
$456$	−17.1652			−0.803832
$457$			14.8348i			0.693945i	0.937875	+	0.346972i	$0.112790\pi$
$457$			14.8348i			0.693945i	−0.937875	+	0.346972i	$0.887210\pi$
$458$	13.0284			0.608776
$459$			41.4955i			1.93684i
$460$			2.58434i			0.120495i
$461$	26.2983			1.22483			0.612417	−	0.790535i	$-0.290197\pi$
$461$	26.2983			1.22483			0.612417	+	0.790535i	$0.290197\pi$
$462$	−15.6201	−	22.2232i	−0.726712	−	1.03392i
$463$	−13.1652			−0.611836			−0.305918	−	0.952058i	$-0.598963\pi$
$463$	−13.1652			−0.611836			−0.305918	+	0.952058i	$0.598963\pi$
$464$		−	7.58258i		−	0.352012i
$465$			2.31464i			0.107339i
$466$	−8.83485			−0.409266
$467$		−	0.511238i		−	0.0236573i	−0.999930	−	0.0118286i	$-0.996235\pi$
$467$		−	0.511238i		−	0.0236573i	0.999930	−	0.0118286i	$-0.00376526\pi$
$468$	−20.3768			−0.941920
$469$	3.87650	+	1.58258i	0.179000	+	0.0730766i
$470$		−	3.25227i		−	0.150016i
$471$	63.4955			2.92572
$472$	−3.09557			−0.142485
$473$	−20.0000	+	31.1652i	−0.919601	+	1.43298i
$474$			12.3823i			0.568738i
$475$	−25.4107			−1.16592
$476$	−9.16515	−	3.74166i	−0.420084	−	0.171499i
$477$	15.9129			0.728601
$478$	−17.5826			−0.804208
$479$	−6.46084			−0.295203			−0.147602	−	0.989047i	$-0.547155\pi$
$479$	−6.46084			−0.295203			−0.147602	+	0.989047i	$0.547155\pi$
$480$			2.00000i			0.0912871i
$481$	17.2813			0.787958
$482$			25.9219i			1.18071i
$483$	12.3823	−	30.3303i	0.563414	−	1.38008i
$484$	−4.58258	−	10.0000i	−0.208299	−	0.454545i
$485$	−10.3303			−0.469075
$486$	−11.8711			−0.538482
$487$	−33.4955			−1.51782			−0.758912	−	0.651193i	$-0.774269\pi$
$487$	−33.4955			−1.51782			−0.758912	+	0.651193i	$0.774269\pi$
$488$		−	9.28672i		−	0.420390i
$489$		−	12.3823i		−	0.559947i
$490$	−3.23042	−	3.16515i	−0.145935	−	0.142987i
$491$		−	21.4955i		−	0.970076i	−0.874493	−	0.485038i	$-0.838806\pi$
$491$		−	21.4955i		−	0.970076i	0.874493	−	0.485038i	$-0.161194\pi$
$492$			15.5826i			0.702517i
$493$		−	28.3714i		−	1.27778i
$494$		−	17.1652i		−	0.772297i
$495$	−7.61816	+	11.8711i	−0.342411	+	0.533564i
$496$		−	1.15732i		−	0.0519653i
$497$	4.89898	+	2.00000i	0.219749	+	0.0897123i
$498$	−28.3303			−1.26951
$499$	27.9129			1.24955			0.624776	−	0.780804i	$-0.285190\pi$
$499$	27.9129			1.24955			0.624776	+	0.780804i	$0.285190\pi$
$500$			6.19115i			0.276877i
$501$			19.1652i			0.856236i
$502$	−2.07310			−0.0925268
$503$	−31.9782			−1.42584			−0.712919	−	0.701246i	$-0.752627\pi$
$503$	−31.9782			−1.42584			−0.712919	+	0.701246i	$0.752627\pi$
$504$	6.58258	−	16.1240i	0.293211	−	0.718218i
$505$			0.330303i			0.0146983i
$506$	7.16515	−	11.1652i	0.318530	−	0.496352i
$507$			10.5789i			0.469825i
$508$		−	5.58258i		−	0.247687i
$509$		−	17.9274i		−	0.794616i	−0.917685	−	0.397308i	$-0.869944\pi$
$509$		−	17.9274i		−	0.794616i	0.917685	−	0.397308i	$-0.130056\pi$
$510$			7.48331i			0.331367i
$511$	−15.4955	−	6.32599i	−0.685479	−	0.279845i
$512$		−	1.00000i		−	0.0441942i
$513$		−	61.4955i		−	2.71509i
$514$	−4.89898			−0.216085
$515$	8.74773			0.385471
$516$	−34.5625			−1.52153
$517$	−9.01703	+	14.0509i	−0.396569	+	0.617956i
$518$	−5.58258	+	13.6745i	−0.245284	+	0.600821i
$519$			71.0780i			3.11998i
$520$	−2.00000			−0.0877058
$521$		−	35.5850i		−	1.55901i	−0.626397	−	0.779504i	$-0.715471\pi$
$521$		−	35.5850i		−	1.55901i	0.626397	−	0.779504i	$-0.284529\pi$
$522$	49.9129			2.18463
$523$	26.7028			1.16763			0.583817	−	0.811885i	$-0.301559\pi$
$523$	26.7028			1.16763			0.583817	+	0.811885i	$0.301559\pi$
$524$	−9.42157			−0.411583
$525$	14.1857	−	34.7477i	0.619115	−	1.51652i
$526$	28.3303			1.23526
$527$		−	4.33030i		−	0.188631i
$528$	5.54506	−	8.64064i	0.241318	−	0.376035i
$529$	−7.00000			−0.304348
$530$	1.56186			0.0678428
$531$		−	20.3768i		−	0.884280i
$532$	13.5826	+	5.54506i	0.588879	+	0.240409i
$533$	−15.5826			−0.674956
$534$		−	30.3303i		−	1.31252i
$535$	5.43836			0.235121
$536$			1.58258i			0.0683569i
$537$			22.1803i			0.957149i
$538$	−20.2420			−0.872695
$539$	5.18096	+	22.6309i	0.223160	+	0.974782i
$540$	−7.16515			−0.308339
$541$		−	9.25227i		−	0.397786i	−0.980021	−	0.198893i	$-0.936265\pi$
$541$		−	9.25227i		−	0.397786i	0.980021	−	0.198893i	$-0.0637347\pi$
$542$			4.89898i			0.210429i
$543$	51.4955			2.20988
$544$		−	3.74166i		−	0.160422i
$545$	−0.752785			−0.0322458
$546$	23.4724	+	9.58258i	1.00453	+	0.410096i
$547$			2.74773i			0.117484i	0.998273	+	0.0587422i	$0.0187090\pi$
$547$			2.74773i			0.117484i	−0.998273	+	0.0587422i	$0.981291\pi$
$548$	−15.1652			−0.647823
$549$	61.1305			2.60899
$550$	8.20871	−	12.7913i	0.350021	−	0.545422i
$551$			42.0459i			1.79121i
$552$	12.3823			0.527025
$553$	4.00000	−	9.79796i	0.170097	−	0.416652i
$554$	2.00000			0.0849719
$555$	11.1652			0.473934
$556$	−3.23042			−0.137000
$557$			9.91288i			0.420022i	0.977699	+	0.210011i	$0.0673500\pi$
$557$			9.91288i			0.420022i	−0.977699	+	0.210011i	$0.932650\pi$
$558$	7.61816			0.322502
$559$		−	34.5625i		−	1.46184i
$560$	0.646084	−	1.58258i	0.0273020	−	0.0668760i
$561$	20.7477	−	32.3303i	0.875970	−	1.36499i
$562$	−7.16515			−0.302244
$563$	42.4223			1.78788			0.893942	−	0.448182i	$-0.147928\pi$
$563$	42.4223			1.78788			0.893942	+	0.448182i	$0.147928\pi$
$564$	−15.5826			−0.656145
$565$			10.8204i			0.455219i
$566$		−	15.6127i		−	0.656251i
$567$	35.7199	+	14.5826i	1.50009	+	0.612411i
$568$			2.00000i			0.0839181i
$569$		−	15.4955i		−	0.649603i	−0.945782	−	0.324802i	$-0.894702\pi$
$569$		−	15.4955i		−	0.649603i	0.945782	−	0.324802i	$-0.105298\pi$
$570$		−	11.0901i		−	0.464514i
$571$			3.58258i			0.149926i	0.997186	+	0.0749631i	$0.0238839\pi$
$571$			3.58258i			0.149926i	−0.997186	+	0.0749631i	$0.976116\pi$
$572$	8.64064	+	5.54506i	0.361283	+	0.231851i
$573$			55.7203i			2.32775i
$574$	5.03383	−	12.3303i	0.210108	−	0.514657i
$575$	18.3303			0.764426
$576$	6.58258			0.274274
$577$		−	0.269691i		−	0.0112274i	−0.999984	−	0.00561369i	$-0.998213\pi$
$577$		−	0.269691i		−	0.0112274i	0.999984	−	0.00561369i	$-0.00178690\pi$
$578$			3.00000i			0.124784i
$579$	−62.9339			−2.61545
$580$	4.89898			0.203419
$581$	22.4174	+	9.15188i	0.930031	+	0.379684i
$582$			49.4955i			2.05165i
$583$	−6.74773	−	4.33030i	−0.279462	−	0.179343i
$584$		−	6.32599i		−	0.261771i
$585$		−	13.1652i		−	0.544312i
$586$		−	21.3993i		−	0.883998i
$587$			17.0397i			0.703305i	0.936131	+	0.351652i	$0.114380\pi$
$587$			17.0397i			0.703305i	−0.936131	+	0.351652i	$0.885620\pi$
$588$	−15.1652	+	15.4779i	−0.625400	+	0.638297i
$589$			6.41742i			0.264425i
$590$		−	2.00000i		−	0.0823387i
$591$	−58.0350			−2.38724
$592$	−5.58258			−0.229442
$593$	0.134846			0.00553744			0.00276872	−	0.999996i	$-0.499119\pi$
$593$	0.134846			0.00553744			0.00276872	+	0.999996i	$0.499119\pi$
$594$	30.9557	+	19.8656i	1.27013	+	0.815096i
$595$	2.41742	−	5.92146i	0.0991047	−	0.242756i
$596$			14.0000i			0.573462i
$597$	−76.2432			−3.12043
$598$			12.3823i			0.506350i
$599$	−2.83485			−0.115829			−0.0579144	−	0.998322i	$-0.518445\pi$
$599$	−2.83485			−0.115829			−0.0579144	+	0.998322i	$0.518445\pi$
$600$	14.1857			0.579129
$601$	−1.42701			−0.0582091			−0.0291045	−	0.999576i	$-0.509266\pi$
$601$	−1.42701			−0.0582091			−0.0291045	+	0.999576i	$0.509266\pi$
$602$	27.3489	+	11.1652i	1.11466	+	0.455058i
$603$	−10.4174			−0.424230
$604$			3.58258i			0.145773i
$605$	6.46084	−	2.96073i	0.262670	−	0.120371i
$606$	1.58258			0.0642877
$607$	46.9448			1.90543			0.952716	−	0.303862i	$-0.0982761\pi$
$607$	46.9448			1.90543			0.952716	+	0.303862i	$0.0982761\pi$
$608$			5.54506i			0.224882i
$609$	−57.4955	−	23.4724i	−2.32983	−	0.951150i
$610$	6.00000			0.242933
$611$		−	15.5826i		−	0.630404i
$612$	24.6297			0.995598
$613$		−	13.9129i		−	0.561936i	−0.959717	−	0.280968i	$-0.909345\pi$
$613$		−	13.9129i		−	0.561936i	0.959717	−	0.280968i	$-0.0906555\pi$
$614$		−	8.12940i		−	0.328076i
$615$	−10.0677			−0.405967
$616$	−7.17903	+	5.04594i	−0.289251	+	0.203307i
$617$	25.9129			1.04321			0.521607	−	0.853186i	$-0.325333\pi$
$617$	25.9129			1.04321			0.521607	+	0.853186i	$0.325333\pi$
$618$		−	41.9129i		−	1.68598i
$619$			0.780929i			0.0313882i	0.999877	+	0.0156941i	$0.00499579\pi$
$619$			0.780929i			0.0313882i	−0.999877	+	0.0156941i	$0.995004\pi$
$620$	0.747727			0.0300294
$621$			44.3605i			1.78013i
$622$	−9.93280			−0.398269
$623$	−9.79796	+	24.0000i	−0.392547	+	0.961540i
$624$			9.58258i			0.383610i
$625$	18.9129			0.756515
$626$	−1.02248			−0.0408664
$627$	−30.7477	+	47.9129i	−1.22795	+	1.91346i
$628$		−	20.5117i		−	0.818506i
$629$	−20.8881			−0.832863
$630$	10.4174	+	4.25290i	0.415040	+	0.169439i
$631$	−5.16515			−0.205621			−0.102811	−	0.994701i	$-0.532784\pi$
$631$	−5.16515			−0.205621			−0.102811	+	0.994701i	$0.532784\pi$
$632$	4.00000			0.159111
$633$	−33.2704			−1.32238
$634$			9.16515i			0.363995i
$635$	3.60681			0.143132
$636$		−	7.48331i		−	0.296733i
$637$	−15.4779	−	15.1652i	−0.613255	−	0.600865i
$638$	−21.1652	−	13.5826i	−0.837936	−	0.537739i
$639$	−13.1652			−0.520805
$640$	0.646084			0.0255387
$641$	29.9129			1.18149			0.590744	−	0.806859i	$-0.298834\pi$
$641$	29.9129			1.18149			0.590744	+	0.806859i	$0.298834\pi$
$642$		−	26.0568i		−	1.02838i
$643$			16.7700i			0.661346i	0.943745	+	0.330673i	$0.107276\pi$
$643$			16.7700i			0.661346i	−0.943745	+	0.330673i	$0.892724\pi$
$644$	−9.79796	−	4.00000i	−0.386094	−	0.157622i
$645$		−	22.3303i		−	0.879255i
$646$			20.7477i			0.816308i
$647$			44.7650i			1.75990i	0.475071	+	0.879948i	$0.342423\pi$
$647$			44.7650i			1.75990i	−0.475071	+	0.879948i	$0.657577\pi$
$648$			14.5826i			0.572858i
$649$	−5.54506	+	8.64064i	−0.217663	+	0.339175i
$650$			14.1857i			0.556409i
$651$	−8.77548	−	3.58258i	−0.343938	−	0.140412i
$652$	−4.00000			−0.156652
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$			3.60681i			0.141038i
$655$		−	6.08712i		−	0.237844i
$656$	5.03383			0.196538
$657$	41.6413			1.62458
$658$	12.3303	+	5.03383i	0.480685	+	0.196239i
$659$			30.3303i			1.18150i	0.806854	+	0.590750i	$0.201168\pi$
$659$			30.3303i			1.18150i	−0.806854	+	0.590750i	$0.798832\pi$
$660$	5.58258	+	3.58258i	0.217301	+	0.139452i
$661$		−	25.4107i		−	0.988361i	−0.869359	−	0.494180i	$-0.835468\pi$
$661$		−	25.4107i		−	0.988361i	0.869359	−	0.494180i	$-0.164532\pi$
$662$			14.4174i			0.560349i
$663$			35.8547i			1.39248i
$664$			9.15188i			0.355162i
$665$	−3.58258	+	8.77548i	−0.138926	+	0.340299i
$666$		−	36.7477i		−	1.42395i
$667$		−	30.3303i		−	1.17439i
$668$	6.19115			0.239543
$669$	−19.5826			−0.757106
$670$	−1.02248			−0.0395017
$671$	−25.9219	−	16.6352i	−1.00070	−	0.642194i
$672$	−7.58258	−	3.09557i	−0.292504	−	0.119414i
$673$			18.3303i			0.706581i	0.935514	+	0.353291i	$0.114937\pi$
$673$			18.3303i			0.706581i	−0.935514	+	0.353291i	$0.885063\pi$
$674$	29.1652			1.12340
$675$			50.8213i			1.95611i
$676$	3.41742			0.131439
$677$	31.4670			1.20937			0.604687	−	0.796463i	$-0.293298\pi$
$677$	31.4670			1.20937			0.604687	+	0.796463i	$0.293298\pi$
$678$	51.8438			1.99105
$679$	15.9891	−	39.1652i	0.613606	−	1.50302i
$680$	2.41742			0.0927040
$681$			36.3303i			1.39218i
$682$	−3.23042	−	2.07310i	−0.123699	−	0.0793830i
$683$	45.4955			1.74084			0.870418	−	0.492314i	$-0.163849\pi$
$683$	45.4955			1.74084			0.870418	+	0.492314i	$0.163849\pi$
$684$	−36.5008			−1.39564
$685$		−	9.79796i		−	0.374361i
$686$	17.0000	−	7.34847i	0.649063	−	0.280566i
$687$	40.3303			1.53870
$688$			11.1652i			0.425667i
$689$	7.48331			0.285092
$690$			8.00000i			0.304555i
$691$		−	37.6581i		−	1.43258i	−0.697801	−	0.716291i	$-0.745838\pi$
$691$		−	37.6581i		−	1.43258i	0.697801	−	0.716291i	$-0.254162\pi$
$692$	22.9612			0.872853
$693$	−33.2153	−	47.2565i	−1.26175	−	1.79513i
$694$	−0.834849			−0.0316904
$695$		−	2.08712i		−	0.0791690i
$696$		−	23.4724i		−	0.889720i
$697$	18.8348			0.713421
$698$			2.07310i			0.0784679i
$699$	−27.3489			−1.03443
$700$	−11.2250	−	4.58258i	−0.424264	−	0.173205i
$701$		−	24.3303i		−	0.918943i	−0.888193	−	0.459471i	$-0.848039\pi$
$701$		−	24.3303i		−	0.918943i	0.888193	−	0.459471i	$-0.151961\pi$
$702$	−34.3303			−1.29571
$703$	30.9557			1.16752
$704$	−2.79129	−	1.79129i	−0.105201	−	0.0675117i
$705$		−	10.0677i		−	0.379170i
$706$	−7.48331			−0.281638
$707$	−1.25227	−	0.511238i	−0.0470966	−	0.0192271i
$708$	−9.58258			−0.360135
$709$	15.6697			0.588488			0.294244	−	0.955730i	$-0.404932\pi$
$709$	15.6697			0.588488			0.294244	+	0.955730i	$0.404932\pi$
$710$	−1.29217			−0.0484942
$711$			26.3303i			0.987464i
$712$	−9.79796			−0.367194
$713$		−	4.62929i		−	0.173368i
$714$	−28.3714	−	11.5826i	−1.06177	−	0.433467i
$715$	−3.58258	+	5.58258i	−0.133981	+	0.208776i
$716$	7.16515			0.267774
$717$	−54.4282			−2.03266
$718$	−0.417424			−0.0155781
$719$		−	15.8543i		−	0.591264i	−0.955302	−	0.295632i	$-0.904470\pi$
$719$		−	15.8543i		−	0.591264i	0.955302	−	0.295632i	$-0.0955302\pi$
$720$			4.25290i			0.158496i
$721$	−13.5396	+	33.1652i	−0.504242	+	1.23513i
$722$		−	11.7477i		−	0.437205i
$723$			80.2432i			2.98428i
$724$		−	16.6352i		−	0.618242i
$725$		−	34.7477i		−	1.29050i
$726$	−14.1857	−	30.9557i	−0.526481	−	1.14888i
$727$		−	52.2484i		−	1.93778i	−0.247483	−	0.968892i	$-0.579604\pi$
$727$		−	52.2484i		−	1.93778i	0.247483	−	0.968892i	$-0.420396\pi$
$728$	3.09557	−	7.58258i	0.114730	−	0.281029i
$729$	7.00000			0.259259
$730$	4.08712			0.151271
$731$			41.7762i			1.54515i
$732$		−	28.7477i		−	1.06255i
$733$	−22.9612			−0.848091			−0.424045	−	0.905641i	$-0.639390\pi$
$733$	−22.9612			−0.848091			−0.424045	+	0.905641i	$0.639390\pi$
$734$	−1.42701			−0.0526720
$735$	−10.0000	−	9.79796i	−0.368856	−	0.361403i
$736$		−	4.00000i		−	0.147442i
$737$	4.41742	+	2.83485i	0.162718	+	0.104423i
$738$			33.1355i			1.21974i
$739$		−	7.58258i		−	0.278930i	−0.990227	−	0.139465i	$-0.955462\pi$
$739$		−	7.58258i		−	0.278930i	0.990227	−	0.139465i	$-0.0445382\pi$
$740$		−	3.60681i		−	0.132589i
$741$		−	53.1360i		−	1.95200i
$742$	−2.41742	+	5.92146i	−0.0887464	+	0.217383i
$743$		−	43.9129i		−	1.61101i	−0.592591	−	0.805504i	$-0.701895\pi$
$743$		−	43.9129i		−	1.61101i	0.592591	−	0.805504i	$-0.298105\pi$
$744$		−	3.58258i		−	0.131344i
$745$	−9.04517			−0.331390
$746$	0.417424			0.0152830
$747$	−60.2429			−2.20417
$748$	−10.4440	−	6.70239i	−0.381872	−	0.245063i
$749$	−8.41742	+	20.6184i	−0.307566	+	0.753380i
$750$			19.1652i			0.699812i
$751$	13.4955			0.492456			0.246228	−	0.969212i	$-0.420809\pi$
$751$	13.4955			0.492456			0.246228	+	0.969212i	$0.420809\pi$
$752$			5.03383i			0.183565i
$753$	−6.41742			−0.233864
$754$	23.4724			0.854815
$755$	−2.31464			−0.0842385
$756$	11.0901	−	27.1652i	0.403344	−	0.987987i
$757$	−49.1652			−1.78694			−0.893469	−	0.449125i	$-0.851736\pi$
$757$	−49.1652			−1.78694			−0.893469	+	0.449125i	$0.851736\pi$
$758$		−	14.3303i		−	0.520500i
$759$	22.1803	−	34.5625i	0.805092	−	1.25454i
$760$	−3.58258			−0.129954
$761$	20.7532			0.752304			0.376152	−	0.926558i	$-0.377247\pi$
$761$	20.7532			0.752304			0.376152	+	0.926558i	$0.377247\pi$
$762$		−	17.2813i		−	0.626034i
$763$	1.16515	−	2.85403i	0.0421813	−	0.103323i
$764$	18.0000			0.651217
$765$			15.9129i			0.575331i
$766$	4.76413			0.172135
$767$		−	9.58258i		−	0.346007i
$768$		−	3.09557i		−	0.111702i
$769$	29.7984			1.07456			0.537279	−	0.843404i	$-0.319452\pi$
$769$	29.7984			1.07456			0.537279	+	0.843404i	$0.319452\pi$
$770$	−3.26010	−	4.63825i	−0.117486	−	0.167151i
$771$	−15.1652			−0.546160
$772$			20.3303i			0.731704i
$773$			51.1977i			1.84145i	0.390207	+	0.920727i	$0.372404\pi$
$773$			51.1977i			1.84145i	−0.390207	+	0.920727i	$0.627596\pi$
$774$	−73.4955			−2.64174
$775$		−	5.30352i		−	0.190508i
$776$	15.9891			0.573975
$777$	−17.2813	+	42.3303i	−0.619962	+	1.51859i
$778$		−	10.0000i		−	0.358517i
$779$	−27.9129			−1.00008
$780$	−6.19115			−0.221679
$781$	5.58258	+	3.58258i	0.199760	+

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:	\( N \)	\(=\)	\( 154 = 2 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	154.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.09557i q^{3} -1.00000 q^{4} -0.646084i q^{5} -3.09557 q^{6} +(2.44949 + 1.00000i) q^{7} +1.00000i q^{8} -6.58258 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -3.09557i q^{3} -1.00000 q^{4} -0.646084i q^{5} -3.09557 q^{6} +(2.44949 + 1.00000i) q^{7} +1.00000i q^{8} -6.58258 q^{9} -0.646084 q^{10} +(2.79129 + 1.79129i) q^{11} +3.09557i q^{12} -3.09557 q^{13} +(1.00000 - 2.44949i) q^{14} -2.00000 q^{15} +1.00000 q^{16} +3.74166 q^{17} +6.58258i q^{18} -5.54506 q^{19} +0.646084i q^{20} +(3.09557 - 7.58258i) q^{21} +(1.79129 - 2.79129i) q^{22} +4.00000 q^{23} +3.09557 q^{24} +4.58258 q^{25} +3.09557i q^{26} +11.0901i q^{27} +(-2.44949 - 1.00000i) q^{28} -7.58258i q^{29} +2.00000i q^{30} -1.15732i q^{31} -1.00000i q^{32} +(5.54506 - 8.64064i) q^{33} -3.74166i q^{34} +(0.646084 - 1.58258i) q^{35} +6.58258 q^{36} -5.58258 q^{37} +5.54506i q^{38} +9.58258i q^{39} +0.646084 q^{40} +5.03383 q^{41} +(-7.58258 - 3.09557i) q^{42} +11.1652i q^{43} +(-2.79129 - 1.79129i) q^{44} +4.25290i q^{45} -4.00000i q^{46} +5.03383i q^{47} -3.09557i q^{48} +(5.00000 + 4.89898i) q^{49} -4.58258i q^{50} -11.5826i q^{51} +3.09557 q^{52} -2.41742 q^{53} +11.0901 q^{54} +(1.15732 - 1.80341i) q^{55} +(-1.00000 + 2.44949i) q^{56} +17.1652i q^{57} -7.58258 q^{58} +3.09557i q^{59} +2.00000 q^{60} -9.28672 q^{61} -1.15732 q^{62} +(-16.1240 - 6.58258i) q^{63} -1.00000 q^{64} +2.00000i q^{65} +(-8.64064 - 5.54506i) q^{66} +1.58258 q^{67} -3.74166 q^{68} -12.3823i q^{69} +(-1.58258 - 0.646084i) q^{70} +2.00000 q^{71} -6.58258i q^{72} -6.32599 q^{73} +5.58258i q^{74} -14.1857i q^{75} +5.54506 q^{76} +(5.04594 + 7.17903i) q^{77} +9.58258 q^{78} -4.00000i q^{79} -0.646084i q^{80} +14.5826 q^{81} -5.03383i q^{82} +9.15188 q^{83} +(-3.09557 + 7.58258i) q^{84} -2.41742i q^{85} +11.1652 q^{86} -23.4724 q^{87} +(-1.79129 + 2.79129i) q^{88} +9.79796i q^{89} +4.25290 q^{90} +(-7.58258 - 3.09557i) q^{91} -4.00000 q^{92} -3.58258 q^{93} +5.03383 q^{94} +3.58258i q^{95} -3.09557 q^{96} -15.9891i q^{97} +(4.89898 - 5.00000i) q^{98} +(-18.3739 - 11.7913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 - 16 * q^9	\( 8 q - 8 q^{4} - 16 q^{9} + 4 q^{11} + 8 q^{14} - 16 q^{15} + 8 q^{16} - 4 q^{22} + 32 q^{23} + 16 q^{36} - 8 q^{37} - 24 q^{42} - 4 q^{44} + 40 q^{49} - 56 q^{53} - 8 q^{56} - 24 q^{58} + 16 q^{60} - 8 q^{64} - 24 q^{67} + 24 q^{70} + 16 q^{71} + 4 q^{77} + 40 q^{78} + 80 q^{81} + 16 q^{86} + 4 q^{88} - 24 q^{91} - 32 q^{92} + 8 q^{93} - 92 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 - 16 * q^9 + 4 * q^11 + 8 * q^14 - 16 * q^15 + 8 * q^16 - 4 * q^22 + 32 * q^23 + 16 * q^36 - 8 * q^37 - 24 * q^42 - 4 * q^44 + 40 * q^49 - 56 * q^53 - 8 * q^56 - 24 * q^58 + 16 * q^60 - 8 * q^64 - 24 * q^67 + 24 * q^70 + 16 * q^71 + 4 * q^77 + 40 * q^78 + 80 * q^81 + 16 * q^86 + 4 * q^88 - 24 * q^91 - 32 * q^92 + 8 * q^93 - 92 * q^99

Introduction

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