LMFDB - Embedded newform 246.2.d.b.163.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [246,2,Mod(163,246)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("246.163"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(246, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 246 = 2 \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	246.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.96431988972$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{57})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 196 $ x^4 + 29*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			163.4
Root			$-3.27492i$ of defining polynomial
Character	$\chi$	$=$	246.163
Dual form			246.2.d.b.163.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} -4.27492 q^{10} -4.00000i q^{11} +1.00000i q^{12} +4.27492i q^{13} +2.00000i q^{14} +4.27492i q^{15} +1.00000 q^{16} -6.27492i q^{17} +1.00000 q^{18} +6.27492i q^{19} +4.27492 q^{20} +2.00000 q^{21} +4.00000i q^{22} -1.00000i q^{24} +13.2749 q^{25} -4.27492i q^{26} -1.00000i q^{27} -2.00000i q^{28} +6.54983i q^{29} -4.27492i q^{30} -1.72508 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.27492i q^{34} -8.54983i q^{35} -1.00000 q^{36} -6.00000 q^{37} -6.27492i q^{38} -4.27492 q^{39} -4.27492 q^{40} +(1.27492 + 6.27492i) q^{41} -2.00000 q^{42} -12.5498 q^{43} -4.00000i q^{44} -4.27492 q^{45} -2.54983i q^{47} +1.00000i q^{48} +3.00000 q^{49} -13.2749 q^{50} +6.27492 q^{51} +4.27492i q^{52} -2.00000i q^{53} +1.00000i q^{54} -17.0997i q^{55} +2.00000i q^{56} -6.27492 q^{57} -6.54983i q^{58} -2.27492 q^{59} +4.27492i q^{60} -6.54983 q^{61} +1.72508 q^{62} +2.00000i q^{63} +1.00000 q^{64} +18.2749i q^{65} -4.00000 q^{66} +1.72508i q^{67} -6.27492i q^{68} +8.54983i q^{70} -3.72508i q^{71} +1.00000 q^{72} +0.274917 q^{73} +6.00000 q^{74} +13.2749i q^{75} +6.27492i q^{76} -8.00000 q^{77} +4.27492 q^{78} -6.54983i q^{79} +4.27492 q^{80} +1.00000 q^{81} +(-1.27492 - 6.27492i) q^{82} -6.27492 q^{83} +2.00000 q^{84} -26.8248i q^{85} +12.5498 q^{86} -6.54983 q^{87} +4.00000i q^{88} -14.8248i q^{89} +4.27492 q^{90} +8.54983 q^{91} -1.72508i q^{93} +2.54983i q^{94} +26.8248i q^{95} -1.00000i q^{96} +0.549834i q^{97} -3.00000 q^{98} +4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} + 4 q^{18} + 2 q^{20} + 8 q^{21} + 38 q^{25} - 22 q^{31} - 4 q^{32} + 16 q^{33} - 4 q^{36} - 24 q^{37} - 2 q^{39} - 2 q^{40} - 10 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 + 4 * q^18 + 2 * q^20 + 8 * q^21 + 38 * q^25 - 22 * q^31 - 4 * q^32 + 16 * q^33 - 4 * q^36 - 24 * q^37 - 2 * q^39 - 2 * q^40 - 10 * q^41 - 8 * q^42 - 20 * q^43 - 2 * q^45 + 12 * q^49 - 38 * q^50 + 10 * q^51 - 10 * q^57 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 - 16 * q^66 + 4 * q^72 - 14 * q^73 + 24 * q^74 - 32 * q^77 + 2 * q^78 + 2 * q^80 + 4 * q^81 + 10 * q^82 - 10 * q^83 + 8 * q^84 + 20 * q^86 + 4 * q^87 + 2 * q^90 + 4 * q^91 - 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/246\mathbb{Z}\right)^\times$.

$n$	$83$	$211$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$	4.27492			1.91180			0.955901	−	0.293691i	$-0.0948835\pi$
$5$	4.27492			1.91180			0.955901	+	0.293691i	$0.0948835\pi$
$6$		−	1.00000i		−	0.408248i
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$	−4.27492			−1.35185
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$			1.00000i			0.288675i
$13$			4.27492i			1.18565i	0.805332	+	0.592824i	$0.201987\pi$
$13$			4.27492i			1.18565i	−0.805332	+	0.592824i	$0.798013\pi$
$14$			2.00000i			0.534522i
$15$			4.27492i			1.10378i
$16$	1.00000			0.250000
$17$		−	6.27492i		−	1.52189i	−0.648816	−	0.760945i	$-0.724735\pi$
$17$		−	6.27492i		−	1.52189i	0.648816	−	0.760945i	$-0.275265\pi$
$18$	1.00000			0.235702
$19$			6.27492i			1.43956i	0.694200	+	0.719782i	$0.255758\pi$
$19$			6.27492i			1.43956i	−0.694200	+	0.719782i	$0.744242\pi$
$20$	4.27492			0.955901
$21$	2.00000			0.436436
$22$			4.00000i			0.852803i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		−	1.00000i		−	0.204124i
$25$	13.2749			2.65498
$26$		−	4.27492i		−	0.838380i
$27$		−	1.00000i		−	0.192450i
$28$		−	2.00000i		−	0.377964i
$29$			6.54983i			1.21627i	0.793832	+	0.608137i	$0.208083\pi$
$29$			6.54983i			1.21627i	−0.793832	+	0.608137i	$0.791917\pi$
$30$		−	4.27492i		−	0.780490i
$31$	−1.72508			−0.309834			−0.154917	−	0.987927i	$-0.549511\pi$
$31$	−1.72508			−0.309834			−0.154917	+	0.987927i	$0.549511\pi$
$32$	−1.00000			−0.176777
$33$	4.00000			0.696311
$34$			6.27492i			1.07614i
$35$		−	8.54983i		−	1.44519i
$36$	−1.00000			−0.166667
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		−	6.27492i		−	1.01793i
$39$	−4.27492			−0.684535
$40$	−4.27492			−0.675924
$41$	1.27492	+	6.27492i	0.199109	+	0.979977i
$41$	1.27492	+	6.27492i	0.199109	+	0.979977i
$42$	−2.00000			−0.308607
$43$	−12.5498			−1.91383			−0.956916	−	0.290365i	$-0.906223\pi$
$43$	−12.5498			−1.91383			−0.956916	+	0.290365i	$0.906223\pi$
$44$		−	4.00000i		−	0.603023i
$45$	−4.27492			−0.637267
$46$		0			0
$47$		−	2.54983i		−	0.371932i	−0.982556	−	0.185966i	$-0.940459\pi$
$47$		−	2.54983i		−	0.371932i	0.982556	−	0.185966i	$-0.0595414\pi$
$48$			1.00000i			0.144338i
$49$	3.00000			0.428571
$50$	−13.2749			−1.87736
$51$	6.27492			0.878664
$52$			4.27492i			0.592824i
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$			1.00000i			0.136083i
$55$		−	17.0997i		−	2.30572i
$56$			2.00000i			0.267261i
$57$	−6.27492			−0.831133
$58$		−	6.54983i		−	0.860035i
$59$	−2.27492			−0.296169			−0.148084	−	0.988975i	$-0.547311\pi$
$59$	−2.27492			−0.296169			−0.148084	+	0.988975i	$0.547311\pi$
$60$			4.27492i			0.551889i
$61$	−6.54983			−0.838620			−0.419310	−	0.907843i	$-0.637728\pi$
$61$	−6.54983			−0.838620			−0.419310	+	0.907843i	$0.637728\pi$
$62$	1.72508			0.219086
$63$			2.00000i			0.251976i
$64$	1.00000			0.125000
$65$			18.2749i			2.26672i
$66$	−4.00000			−0.492366
$67$			1.72508i			0.210752i	0.994432	+	0.105376i	$0.0336047\pi$
$67$			1.72508i			0.210752i	−0.994432	+	0.105376i	$0.966395\pi$
$68$		−	6.27492i		−	0.760945i
$69$		0			0
$70$			8.54983i			1.02190i
$71$		−	3.72508i		−	0.442086i	−0.975264	−	0.221043i	$-0.929054\pi$
$71$		−	3.72508i		−	0.442086i	0.975264	−	0.221043i	$-0.0709461\pi$
$72$	1.00000			0.117851
$73$	0.274917			0.0321766			0.0160883	−	0.999871i	$-0.494879\pi$
$73$	0.274917			0.0321766			0.0160883	+	0.999871i	$0.494879\pi$
$74$	6.00000			0.697486
$75$			13.2749i			1.53286i
$76$			6.27492i			0.719782i
$77$	−8.00000			−0.911685
$78$	4.27492			0.484039
$79$		−	6.54983i		−	0.736914i	−0.929645	−	0.368457i	$-0.879886\pi$
$79$		−	6.54983i		−	0.736914i	0.929645	−	0.368457i	$-0.120114\pi$
$80$	4.27492			0.477950
$81$	1.00000			0.111111
$82$	−1.27492	−	6.27492i	−0.140791	−	0.692949i
$83$	−6.27492			−0.688762			−0.344381	−	0.938830i	$-0.611911\pi$
$83$	−6.27492			−0.688762			−0.344381	+	0.938830i	$0.611911\pi$
$84$	2.00000			0.218218
$85$		−	26.8248i		−	2.90955i
$86$	12.5498			1.35328
$87$	−6.54983			−0.702216
$88$			4.00000i			0.426401i
$89$		−	14.8248i		−	1.57142i	−0.618595	−	0.785710i	$-0.712298\pi$
$89$		−	14.8248i		−	1.57142i	0.618595	−	0.785710i	$-0.287702\pi$
$90$	4.27492			0.450616
$91$	8.54983			0.896266
$92$		0			0
$93$		−	1.72508i		−	0.178883i
$94$			2.54983i			0.262995i
$95$			26.8248i			2.75216i
$96$		−	1.00000i		−	0.102062i
$97$			0.549834i			0.0558272i	0.999610	+	0.0279136i	$0.00888633\pi$
$97$			0.549834i			0.0558272i	−0.999610	+	0.0279136i	$0.991114\pi$
$98$	−3.00000			−0.303046
$99$			4.00000i			0.402015i
$100$	13.2749			1.32749
$101$		−	1.45017i		−	0.144297i	−0.997394	−	0.0721484i	$-0.977014\pi$
$101$		−	1.45017i		−	0.144297i	0.997394	−	0.0721484i	$-0.0229855\pi$
$102$	−6.27492			−0.621309
$103$	10.8248			1.06659			0.533297	−	0.845928i	$-0.320953\pi$
$103$	10.8248			1.06659			0.533297	+	0.845928i	$0.320953\pi$
$104$		−	4.27492i		−	0.419190i
$105$	8.54983			0.834378
$106$			2.00000i			0.194257i
$107$	−10.8248			−1.04647			−0.523234	−	0.852189i	$-0.675275\pi$
$107$	−10.8248			−1.04647			−0.523234	+	0.852189i	$0.675275\pi$
$108$		−	1.00000i		−	0.0962250i
$109$			11.0997i			1.06316i	0.847010	+	0.531578i	$0.178401\pi$
$109$			11.0997i			1.06316i	−0.847010	+	0.531578i	$0.821599\pi$
$110$			17.0997i			1.63039i
$111$		−	6.00000i		−	0.569495i
$112$		−	2.00000i		−	0.188982i
$113$	−10.5498			−0.992445			−0.496222	−	0.868195i	$-0.665280\pi$
$113$	−10.5498			−0.992445			−0.496222	+	0.868195i	$0.665280\pi$
$114$	6.27492			0.587700
$115$		0			0
$116$			6.54983i			0.608137i
$117$		−	4.27492i		−	0.395216i
$118$	2.27492			0.209423
$119$	−12.5498			−1.15044
$120$		−	4.27492i		−	0.390245i
$121$	−5.00000			−0.454545
$122$	6.54983			0.592994
$123$	−6.27492	+	1.27492i	−0.565790	+	0.114955i
$124$	−1.72508			−0.154917
$125$	35.3746			3.16400
$126$		−	2.00000i		−	0.178174i
$127$	−6.27492			−0.556809			−0.278404	−	0.960464i	$-0.589806\pi$
$127$	−6.27492			−0.556809			−0.278404	+	0.960464i	$0.589806\pi$
$128$	−1.00000			−0.0883883
$129$		−	12.5498i		−	1.10495i
$130$		−	18.2749i		−	1.60282i
$131$	16.0000			1.39793			0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000			1.39793			0.698963	+	0.715158i	$0.253645\pi$
$132$	4.00000			0.348155
$133$	12.5498			1.08821
$134$		−	1.72508i		−	0.149024i
$135$		−	4.27492i		−	0.367926i
$136$			6.27492i			0.538070i
$137$		−	4.54983i		−	0.388719i	−0.980930	−	0.194359i	$-0.937737\pi$
$137$		−	4.54983i		−	0.388719i	0.980930	−	0.194359i	$-0.0622628\pi$
$138$		0			0
$139$	4.54983			0.385912			0.192956	−	0.981207i	$-0.438193\pi$
$139$	4.54983			0.385912			0.192956	+	0.981207i	$0.438193\pi$
$140$		−	8.54983i		−	0.722593i
$141$	2.54983			0.214735
$142$			3.72508i			0.312602i
$143$	17.0997			1.42995
$144$	−1.00000			−0.0833333
$145$			28.0000i			2.32527i
$146$	−0.274917			−0.0227523
$147$			3.00000i			0.247436i
$148$	−6.00000			−0.493197
$149$		−	1.45017i		−	0.118802i	−0.998234	−	0.0594011i	$-0.981081\pi$
$149$		−	1.45017i		−	0.118802i	0.998234	−	0.0594011i	$-0.0189191\pi$
$150$		−	13.2749i		−	1.08389i
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		−	6.27492i		−	0.508963i
$153$			6.27492i			0.507297i
$154$	8.00000			0.644658
$155$	−7.37459			−0.592341
$156$	−4.27492			−0.342267
$157$			14.0000i			1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$			14.0000i			1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$			6.54983i			0.521077i
$159$	2.00000			0.158610
$160$	−4.27492			−0.337962
$161$		0			0
$162$	−1.00000			−0.0785674
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	1.27492	+	6.27492i	0.0995543	+	0.489989i
$165$	17.0997			1.33121
$166$	6.27492			0.487028
$167$		−	8.27492i		−	0.640332i	−0.947361	−	0.320166i	$-0.896261\pi$
$167$		−	8.27492i		−	0.640332i	0.947361	−	0.320166i	$-0.103739\pi$
$168$	−2.00000			−0.154303
$169$	−5.27492			−0.405763
$170$			26.8248i			2.05736i
$171$		−	6.27492i		−	0.479855i
$172$	−12.5498			−0.956916
$173$	−7.09967			−0.539778			−0.269889	−	0.962891i	$-0.586987\pi$
$173$	−7.09967			−0.539778			−0.269889	+	0.962891i	$0.586987\pi$
$174$	6.54983			0.496542
$175$		−	26.5498i		−	2.00698i
$176$		−	4.00000i		−	0.301511i
$177$		−	2.27492i		−	0.170993i
$178$			14.8248i			1.11116i
$179$			21.0997i			1.57706i	0.614994	+	0.788532i	$0.289158\pi$
$179$			21.0997i			1.57706i	−0.614994	+	0.788532i	$0.710842\pi$
$180$	−4.27492			−0.318634
$181$		−	3.72508i		−	0.276883i	−0.990371	−	0.138442i	$-0.955791\pi$
$181$		−	3.72508i		−	0.276883i	0.990371	−	0.138442i	$-0.0442093\pi$
$182$	−8.54983			−0.633756
$183$		−	6.54983i		−	0.484178i
$184$		0			0
$185$	−25.6495			−1.88579
$186$			1.72508i			0.126489i
$187$	−25.0997			−1.83547
$188$		−	2.54983i		−	0.185966i
$189$	−2.00000			−0.145479
$190$		−	26.8248i		−	1.94607i
$191$			10.5498i			0.763359i	0.924295	+	0.381680i	$0.124654\pi$
$191$			10.5498i			0.763359i	−0.924295	+	0.381680i	$0.875346\pi$
$192$			1.00000i			0.0721688i
$193$		−	20.0000i		−	1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$		−	20.0000i		−	1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$		−	0.549834i		−	0.0394758i
$195$	−18.2749			−1.30869
$196$	3.00000			0.214286
$197$	−15.7251			−1.12037			−0.560183	−	0.828369i	$-0.689269\pi$
$197$	−15.7251			−1.12037			−0.560183	+	0.828369i	$0.689269\pi$
$198$		−	4.00000i		−	0.284268i
$199$		−	2.00000i		−	0.141776i	−0.997484	−	0.0708881i	$-0.977417\pi$
$199$		−	2.00000i		−	0.141776i	0.997484	−	0.0708881i	$-0.0225833\pi$
$200$	−13.2749			−0.938678
$201$	−1.72508			−0.121678
$202$			1.45017i			0.102033i
$203$	13.0997			0.919417
$204$	6.27492			0.439332
$205$	5.45017	+	26.8248i	0.380656	+	1.87352i
$206$	−10.8248			−0.754196
$207$		0			0
$208$			4.27492i			0.296412i
$209$	25.0997			1.73618
$210$	−8.54983			−0.589995
$211$			1.72508i			0.118760i	0.998235	+	0.0593798i	$0.0189123\pi$
$211$			1.72508i			0.118760i	−0.998235	+	0.0593798i	$0.981088\pi$
$212$		−	2.00000i		−	0.137361i
$213$	3.72508			0.255238
$214$	10.8248			0.739965
$215$	−53.6495			−3.65887
$216$			1.00000i			0.0680414i
$217$			3.45017i			0.234212i
$218$		−	11.0997i		−	0.751764i
$219$			0.274917i			0.0185772i
$220$		−	17.0997i		−	1.15286i
$221$	26.8248			1.80443
$222$			6.00000i			0.402694i
$223$	17.7251			1.18696			0.593480	−	0.804849i	$-0.297754\pi$
$223$	17.7251			1.18696			0.593480	+	0.804849i	$0.297754\pi$
$224$			2.00000i			0.133631i
$225$	−13.2749			−0.884994
$226$	10.5498			0.701765
$227$			11.4502i			0.759974i	0.924992	+	0.379987i	$0.124072\pi$
$227$			11.4502i			0.759974i	−0.924992	+	0.379987i	$0.875928\pi$
$228$	−6.27492			−0.415567
$229$		−	11.0997i		−	0.733487i	−0.930322	−	0.366743i	$-0.880473\pi$
$229$		−	11.0997i		−	0.733487i	0.930322	−	0.366743i	$-0.119527\pi$
$230$		0			0
$231$		−	8.00000i		−	0.526361i
$232$		−	6.54983i		−	0.430018i
$233$		−	14.8248i		−	0.971202i	−0.874181	−	0.485601i	$-0.838601\pi$
$233$		−	14.8248i		−	0.971202i	0.874181	−	0.485601i	$-0.161399\pi$
$234$			4.27492i			0.279460i
$235$		−	10.9003i		−	0.711059i
$236$	−2.27492			−0.148084
$237$	6.54983			0.425457
$238$	12.5498			0.813485
$239$			7.72508i			0.499694i	0.968285	+	0.249847i	$0.0803803\pi$
$239$			7.72508i			0.499694i	−0.968285	+	0.249847i	$0.919620\pi$
$240$			4.27492i			0.275945i
$241$	−12.8248			−0.826115			−0.413057	−	0.910705i	$-0.635539\pi$
$241$	−12.8248			−0.826115			−0.413057	+	0.910705i	$0.635539\pi$
$242$	5.00000			0.321412
$243$			1.00000i			0.0641500i
$244$	−6.54983			−0.419310
$245$	12.8248			0.819343
$246$	6.27492	−	1.27492i	0.400074	−	0.0812858i
$247$	−26.8248			−1.70682
$248$	1.72508			0.109543
$249$		−	6.27492i		−	0.397657i
$250$	−35.3746			−2.23729
$251$	1.09967			0.0694105			0.0347052	−	0.999398i	$-0.488951\pi$
$251$	1.09967			0.0694105			0.0347052	+	0.999398i	$0.488951\pi$
$252$			2.00000i			0.125988i
$253$		0			0
$254$	6.27492			0.393723
$255$	26.8248			1.67983
$256$	1.00000			0.0625000
$257$		−	16.5498i		−	1.03235i	−0.856483	−	0.516175i	$-0.827355\pi$
$257$		−	16.5498i		−	1.03235i	0.856483	−	0.516175i	$-0.172645\pi$
$258$			12.5498i			0.781319i
$259$			12.0000i			0.745644i
$260$			18.2749i			1.13336i
$261$		−	6.54983i		−	0.405425i
$262$	−16.0000			−0.988483
$263$			29.9244i			1.84522i	0.385736	+	0.922609i	$0.373948\pi$
$263$			29.9244i			1.84522i	−0.385736	+	0.922609i	$0.626052\pi$
$264$	−4.00000			−0.246183
$265$		−	8.54983i		−	0.525212i
$266$	−12.5498			−0.769480
$267$	14.8248			0.907260
$268$			1.72508i			0.105376i
$269$	8.27492			0.504531			0.252265	−	0.967658i	$-0.418824\pi$
$269$	8.27492			0.504531			0.252265	+	0.967658i	$0.418824\pi$
$270$			4.27492i			0.260163i
$271$	25.0997			1.52470			0.762348	−	0.647167i	$-0.224046\pi$
$271$	25.0997			1.52470			0.762348	+	0.647167i	$0.224046\pi$
$272$		−	6.27492i		−	0.380473i
$273$			8.54983i			0.517460i
$274$			4.54983i			0.274866i
$275$		−	53.0997i		−	3.20203i
$276$		0			0
$277$	30.5498			1.83556			0.917781	−	0.397087i	$-0.129979\pi$
$277$	30.5498			1.83556			0.917781	+	0.397087i	$0.129979\pi$
$278$	−4.54983			−0.272881
$279$	1.72508			0.103278
$280$			8.54983i			0.510950i
$281$			2.27492i			0.135710i	0.997695	+	0.0678551i	$0.0216156\pi$
$281$			2.27492i			0.135710i	−0.997695	+	0.0678551i	$0.978384\pi$
$282$	−2.54983			−0.151840
$283$	21.6495			1.28693			0.643465	−	0.765476i	$-0.277496\pi$
$283$	21.6495			1.28693			0.643465	+	0.765476i	$0.277496\pi$
$284$		−	3.72508i		−	0.221043i
$285$	−26.8248			−1.58896
$286$	−17.0997			−1.01112
$287$	12.5498	−	2.54983i	0.740793	−	0.150512i
$288$	1.00000			0.0589256
$289$	−22.3746			−1.31615
$290$		−	28.0000i		−	1.64422i
$291$	−0.549834			−0.0322319
$292$	0.274917			0.0160883
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$		−	3.00000i		−	0.174964i
$295$	−9.72508			−0.566216
$296$	6.00000			0.348743
$297$	−4.00000			−0.232104
$298$			1.45017i			0.0840058i
$299$		0			0
$300$			13.2749i			0.766428i
$301$			25.0997i			1.44672i
$302$		−	2.00000i		−	0.115087i
$303$	1.45017			0.0833098
$304$			6.27492i			0.359891i
$305$	−28.0000			−1.60328
$306$		−	6.27492i		−	0.358713i
$307$	2.90033			0.165531			0.0827653	−	0.996569i	$-0.473625\pi$
$307$	2.90033			0.165531			0.0827653	+	0.996569i	$0.473625\pi$
$308$	−8.00000			−0.455842
$309$			10.8248i			0.615799i
$310$	7.37459			0.418848
$311$			28.2749i			1.60332i	0.597778	+	0.801662i	$0.296050\pi$
$311$			28.2749i			1.60332i	−0.597778	+	0.801662i	$0.703950\pi$
$312$	4.27492			0.242020
$313$			25.6495i			1.44980i	0.688856	+	0.724898i	$0.258113\pi$
$313$			25.6495i			1.44980i	−0.688856	+	0.724898i	$0.741887\pi$
$314$		−	14.0000i		−	0.790066i
$315$			8.54983i			0.481729i
$316$		−	6.54983i		−	0.368457i
$317$		−	30.5498i		−	1.71585i	−0.513775	−	0.857925i	$-0.671753\pi$
$317$		−	30.5498i		−	1.71585i	0.513775	−	0.857925i	$-0.328247\pi$
$318$	−2.00000			−0.112154
$319$	26.1993			1.46688
$320$	4.27492			0.238975
$321$		−	10.8248i		−	0.604179i
$322$		0			0
$323$	39.3746			2.19086
$324$	1.00000			0.0555556
$325$			56.7492i			3.14788i
$326$	4.00000			0.221540
$327$	−11.0997			−0.613813
$328$	−1.27492	−	6.27492i	−0.0703955	−	0.346474i
$329$	−5.09967			−0.281154
$330$	−17.0997			−0.941306
$331$			1.09967i			0.0604433i	0.999543	+	0.0302216i	$0.00962131\pi$
$331$			1.09967i			0.0604433i	−0.999543	+	0.0302216i	$0.990379\pi$
$332$	−6.27492			−0.344381
$333$	6.00000			0.328798
$334$			8.27492i			0.452783i
$335$			7.37459i			0.402917i
$336$	2.00000			0.109109
$337$	−12.8248			−0.698609			−0.349304	−	0.937009i	$-0.613582\pi$
$337$	−12.8248			−0.698609			−0.349304	+	0.937009i	$0.613582\pi$
$338$	5.27492			0.286918
$339$		−	10.5498i		−	0.572988i
$340$		−	26.8248i		−	1.45478i
$341$			6.90033i			0.373674i
$342$			6.27492i			0.339309i
$343$		−	20.0000i		−	1.07990i
$344$	12.5498			0.676642
$345$		0			0
$346$	7.09967			0.381681
$347$		−	17.0997i		−	0.917958i	−0.888447	−	0.458979i	$-0.848215\pi$
$347$		−	17.0997i		−	0.917958i	0.888447	−	0.458979i	$-0.151785\pi$
$348$	−6.54983			−0.351108
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$			26.5498i			1.41915i
$351$	4.27492			0.228178
$352$			4.00000i			0.213201i
$353$	10.5498			0.561511			0.280756	−	0.959779i	$-0.409415\pi$
$353$	10.5498			0.561511			0.280756	+	0.959779i	$0.409415\pi$
$354$			2.27492i			0.120910i
$355$		−	15.9244i		−	0.845180i
$356$		−	14.8248i		−	0.785710i
$357$		−	12.5498i		−	0.664208i
$358$		−	21.0997i		−	1.11515i
$359$	17.0997			0.902486			0.451243	−	0.892401i	$-0.350981\pi$
$359$	17.0997			0.902486			0.451243	+	0.892401i	$0.350981\pi$
$360$	4.27492			0.225308
$361$	−20.3746			−1.07235
$362$			3.72508i			0.195786i
$363$		−	5.00000i		−	0.262432i
$364$	8.54983			0.448133
$365$	1.17525			0.0615153
$366$			6.54983i			0.342365i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$	−1.27492	−	6.27492i	−0.0663695	−	0.326659i
$370$	25.6495			1.33345
$371$	−4.00000			−0.207670
$372$		−	1.72508i		−	0.0894414i
$373$	7.09967			0.367607			0.183803	−	0.982963i	$-0.441159\pi$
$373$	7.09967			0.367607			0.183803	+	0.982963i	$0.441159\pi$
$374$	25.0997			1.29787
$375$			35.3746i			1.82674i
$376$			2.54983i			0.131498i
$377$	−28.0000			−1.44207
$378$	2.00000			0.102869
$379$	28.5498			1.46651			0.733253	−	0.679956i	$-0.238001\pi$
$379$	28.5498			1.46651			0.733253	+	0.679956i	$0.238001\pi$
$380$			26.8248i			1.37608i
$381$		−	6.27492i		−	0.321474i
$382$		−	10.5498i		−	0.539776i
$383$			0.824752i			0.0421428i	0.999778	+	0.0210714i	$0.00670774\pi$
$383$			0.824752i			0.0421428i	−0.999778	+	0.0210714i	$0.993292\pi$
$384$		−	1.00000i		−	0.0510310i
$385$	−34.1993			−1.74296
$386$			20.0000i			1.01797i
$387$	12.5498			0.637944
$388$			0.549834i			0.0279136i
$389$	21.9244			1.11161			0.555806	−	0.831312i	$-0.312410\pi$
$389$	21.9244			1.11161			0.555806	+	0.831312i	$0.312410\pi$
$390$	18.2749			0.925386
$391$		0			0
$392$	−3.00000			−0.151523
$393$			16.0000i			0.807093i
$394$	15.7251			0.792218
$395$		−	28.0000i		−	1.40883i
$396$			4.00000i			0.201008i
$397$			7.17525i			0.360115i	0.983656	+	0.180058i	$0.0576284\pi$
$397$			7.17525i			0.360115i	−0.983656	+	0.180058i	$0.942372\pi$
$398$			2.00000i			0.100251i
$399$			12.5498i			0.628278i
$400$	13.2749			0.663746
$401$	−23.0997			−1.15354			−0.576771	−	0.816906i	$-0.695688\pi$
$401$	−23.0997			−1.15354			−0.576771	+	0.816906i	$0.695688\pi$
$402$	1.72508			0.0860393
$403$		−	7.37459i		−	0.367354i
$404$		−	1.45017i		−	0.0721484i
$405$	4.27492			0.212422
$406$	−13.0997			−0.650126
$407$			24.0000i			1.18964i
$408$	−6.27492			−0.310655
$409$	−8.27492			−0.409168			−0.204584	−	0.978849i	$-0.565584\pi$
$409$	−8.27492			−0.409168			−0.204584	+	0.978849i	$0.565584\pi$
$410$	−5.45017	−	26.8248i	−0.269164	−	1.32478i
$411$	4.54983			0.224427
$412$	10.8248			0.533297
$413$			4.54983i			0.223883i
$414$		0			0
$415$	−26.8248			−1.31678
$416$		−	4.27492i		−	0.209595i
$417$			4.54983i			0.222806i
$418$	−25.0997			−1.22766
$419$	−34.2749			−1.67444			−0.837220	−	0.546867i	$-0.815820\pi$
$419$	−34.2749			−1.67444			−0.837220	+	0.546867i	$0.815820\pi$
$420$	8.54983			0.417189
$421$		−	3.17525i		−	0.154752i	−0.997002	−	0.0773761i	$-0.975346\pi$
$421$		−	3.17525i		−	0.154752i	0.997002	−	0.0773761i	$-0.0246542\pi$
$422$		−	1.72508i		−	0.0839757i
$423$			2.54983i			0.123977i
$424$			2.00000i			0.0971286i
$425$		−	83.2990i		−	4.04060i
$426$	−3.72508			−0.180481
$427$			13.0997i			0.633937i
$428$	−10.8248			−0.523234
$429$			17.0997i			0.825580i
$430$	53.6495			2.58721
$431$	19.4502			0.936882			0.468441	−	0.883495i	$-0.344816\pi$
$431$	19.4502			0.936882			0.468441	+	0.883495i	$0.344816\pi$
$432$		−	1.00000i		−	0.0481125i
$433$	−16.2749			−0.782123			−0.391061	−	0.920365i	$-0.627892\pi$
$433$	−16.2749			−0.782123			−0.391061	+	0.920365i	$0.627892\pi$
$434$		−	3.45017i		−	0.165613i
$435$	−28.0000			−1.34250
$436$			11.0997i			0.531578i
$437$		0			0
$438$		−	0.274917i		−	0.0131361i
$439$		−	39.0997i		−	1.86613i	−0.359714	−	0.933063i	$-0.617126\pi$
$439$		−	39.0997i		−	1.86613i	0.359714	−	0.933063i	$-0.382874\pi$
$440$			17.0997i			0.815195i
$441$	−3.00000			−0.142857
$442$	−26.8248			−1.27592
$443$	23.3746			1.11056			0.555280	−	0.831663i	$-0.312611\pi$
$443$	23.3746			1.11056			0.555280	+	0.831663i	$0.312611\pi$
$444$		−	6.00000i		−	0.284747i
$445$		−	63.3746i		−	3.00424i
$446$	−17.7251			−0.839307
$447$	1.45017			0.0685905
$448$		−	2.00000i		−	0.0944911i
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	13.2749			0.625786
$451$	25.0997	−	5.09967i	1.18190	−	0.240134i
$452$	−10.5498			−0.496222
$453$	−2.00000			−0.0939682
$454$		−	11.4502i		−	0.537383i
$455$	36.5498			1.71348
$456$	6.27492			0.293850
$457$			29.0997i			1.36123i	0.732644	+	0.680613i	$0.238286\pi$
$457$			29.0997i			1.36123i	−0.732644	+	0.680613i	$0.761714\pi$
$458$			11.0997i			0.518653i
$459$	−6.27492			−0.292888
$460$		0			0
$461$	−27.0997			−1.26216			−0.631079	−	0.775719i	$-0.717388\pi$
$461$	−27.0997			−1.26216			−0.631079	+	0.775719i	$0.717388\pi$
$462$			8.00000i			0.372194i
$463$			26.5498i			1.23388i	0.787012	+	0.616938i	$0.211627\pi$
$463$			26.5498i			1.23388i	−0.787012	+	0.616938i	$0.788373\pi$
$464$			6.54983i			0.304068i
$465$		−	7.37459i		−	0.341988i
$466$			14.8248i			0.686743i
$467$	−21.7251			−1.00532			−0.502658	−	0.864485i	$-0.667645\pi$
$467$	−21.7251			−1.00532			−0.502658	+	0.864485i	$0.667645\pi$
$468$		−	4.27492i		−	0.197608i
$469$	3.45017			0.159314
$470$			10.9003i			0.502795i
$471$	−14.0000			−0.645086
$472$	2.27492			0.104712
$473$			50.1993i			2.30817i
$474$	−6.54983			−0.300844
$475$			83.2990i			3.82202i
$476$	−12.5498			−0.575221
$477$			2.00000i			0.0915737i
$478$		−	7.72508i		−	0.353337i
$479$		−	33.9244i		−	1.55005i	−0.631933	−	0.775023i	$-0.717738\pi$
$479$		−	33.9244i		−	1.55005i	0.631933	−	0.775023i	$-0.282262\pi$
$480$		−	4.27492i		−	0.195122i
$481$		−	25.6495i		−	1.16952i
$482$	12.8248			0.584151
$483$		0			0
$484$	−5.00000			−0.227273
$485$			2.35050i			0.106731i
$486$		−	1.00000i		−	0.0453609i
$487$	13.1752			0.597027			0.298514	−	0.954405i	$-0.403509\pi$
$487$	13.1752			0.597027			0.298514	+	0.954405i	$0.403509\pi$
$488$	6.54983			0.296497
$489$		−	4.00000i		−	0.180886i
$490$	−12.8248			−0.579363
$491$	−14.8248			−0.669032			−0.334516	−	0.942390i	$-0.608573\pi$
$491$	−14.8248			−0.669032			−0.334516	+	0.942390i	$0.608573\pi$
$492$	−6.27492	+	1.27492i	−0.282895	+	0.0574777i
$493$	41.0997			1.85104
$494$	26.8248			1.20690
$495$			17.0997i			0.768573i
$496$	−1.72508			−0.0774585
$497$	−7.45017			−0.334186
$498$			6.27492i			0.281186i
$499$			23.3746i			1.04639i	0.852213	+	0.523195i	$0.175260\pi$
$499$			23.3746i			1.04639i	−0.852213	+	0.523195i	$0.824740\pi$
$500$	35.3746			1.58200
$501$	8.27492			0.369696
$502$	−1.09967			−0.0490806
$503$		−	20.2749i		−	0.904014i	−0.892014	−	0.452007i	$-0.850708\pi$
$503$		−	20.2749i		−	0.904014i	0.892014	−	0.452007i	$-0.149292\pi$
$504$		−	2.00000i		−	0.0890871i
$505$		−	6.19934i		−	0.275867i
$506$		0			0
$507$		−	5.27492i		−	0.234267i
$508$	−6.27492			−0.278404
$509$		−	7.64950i		−	0.339058i	−0.985525	−	0.169529i	$-0.945775\pi$
$509$		−	7.64950i		−	0.339058i	0.985525	−	0.169529i	$-0.0542247\pi$
$510$	−26.8248			−1.18782
$511$		−	0.549834i		−	0.0243232i
$512$	−1.00000			−0.0441942
$513$	6.27492			0.277044
$514$			16.5498i			0.729982i
$515$	46.2749			2.03912
$516$		−	12.5498i		−	0.552476i
$517$	−10.1993			−0.448566
$518$		−	12.0000i		−	0.527250i
$519$		−	7.09967i		−	0.311641i
$520$		−	18.2749i		−	0.801408i
$521$		−	17.7251i		−	0.776550i	−0.921544	−	0.388275i	$-0.873071\pi$
$521$		−	17.7251i		−	0.776550i	0.921544	−	0.388275i	$-0.126929\pi$
$522$			6.54983i			0.286678i
$523$	−33.6495			−1.47139			−0.735695	−	0.677313i	$-0.763144\pi$
$523$	−33.6495			−1.47139			−0.735695	+	0.677313i	$0.763144\pi$
$524$	16.0000			0.698963
$525$	26.5498			1.15873
$526$		−	29.9244i		−	1.30477i
$527$			10.8248i			0.471534i
$528$	4.00000			0.174078
$529$	−23.0000			−1.00000
$530$			8.54983i			0.371381i
$531$	2.27492			0.0987230
$532$	12.5498			0.544104
$533$	−26.8248	+	5.45017i	−1.16191	+	0.236073i
$534$	−14.8248			−0.641530
$535$	−46.2749			−2.00064
$536$		−	1.72508i		−	0.0745122i
$537$	−21.0997			−0.910518
$538$	−8.27492			−0.356757
$539$		−	12.0000i		−	0.516877i
$540$		−	4.27492i		−	0.183963i
$541$	40.7492			1.75194			0.875972	−	0.482362i	$-0.160221\pi$
$541$	40.7492			1.75194			0.875972	+	0.482362i	$0.160221\pi$
$542$	−25.0997			−1.07812
$543$	3.72508			0.159859
$544$			6.27492i			0.269035i
$545$			47.4502i			2.03254i
$546$		−	8.54983i		−	0.365899i
$547$			43.3746i			1.85456i	0.374364	+	0.927282i	$0.377861\pi$
$547$			43.3746i			1.85456i	−0.374364	+	0.927282i	$0.622139\pi$
$548$		−	4.54983i		−	0.194359i
$549$	6.54983			0.279540
$550$			53.0997i			2.26418i
$551$	−41.0997			−1.75090
$552$		0			0
$553$	−13.0997			−0.557055
$554$	−30.5498			−1.29794
$555$		−	25.6495i		−	1.08876i
$556$	4.54983			0.192956
$557$		−	43.0997i		−	1.82619i	−0.407745	−	0.913096i	$-0.633685\pi$
$557$		−	43.0997i		−	1.82619i	0.407745	−	0.913096i	$-0.366315\pi$
$558$	−1.72508			−0.0730286
$559$		−	53.6495i		−	2.26913i
$560$		−	8.54983i		−	0.361296i
$561$		−	25.0997i		−	1.05971i
$562$		−	2.27492i		−	0.0959616i
$563$			4.54983i			0.191753i	0.995393	+	0.0958763i	$0.0305653\pi$
$563$			4.54983i			0.191753i	−0.995393	+	0.0958763i	$0.969435\pi$
$564$	2.54983			0.107367
$565$	−45.0997			−1.89736
$566$	−21.6495			−0.909996
$567$		−	2.00000i		−	0.0839921i
$568$			3.72508i			0.156301i
$569$	−34.5498			−1.44840			−0.724202	−	0.689588i	$-0.757792\pi$
$569$	−34.5498			−1.44840			−0.724202	+	0.689588i	$0.757792\pi$
$570$	26.8248			1.12357
$571$		−	33.0997i		−	1.38518i	−0.721332	−	0.692589i	$-0.756470\pi$
$571$		−	33.0997i		−	1.38518i	0.721332	−	0.692589i	$-0.243530\pi$
$572$	17.0997			0.714973
$573$	−10.5498			−0.440726
$574$	−12.5498	+	2.54983i	−0.523820	+	0.106428i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$			17.0997i			0.711869i	0.934511	+	0.355934i	$0.115837\pi$
$577$			17.0997i			0.711869i	−0.934511	+	0.355934i	$0.884163\pi$
$578$	22.3746			0.930660
$579$	20.0000			0.831172
$580$			28.0000i			1.16264i
$581$			12.5498i			0.520655i
$582$	0.549834			0.0227914
$583$	−8.00000			−0.331326
$584$	−0.274917			−0.0113762
$585$		−	18.2749i		−	0.755575i
$586$			2.00000i			0.0826192i
$587$		−	7.45017i		−	0.307501i	−0.988110	−	0.153751i	$-0.950865\pi$
$587$		−	7.45017i		−	0.307501i	0.988110	−	0.153751i	$-0.0491352\pi$
$588$			3.00000i			0.123718i
$589$		−	10.8248i		−	0.446026i
$590$	9.72508			0.400375
$591$		−	15.7251i		−	0.646843i
$592$	−6.00000			−0.246598
$593$			30.8248i			1.26582i	0.774225	+	0.632910i	$0.218140\pi$
$593$			30.8248i			1.26582i	−0.774225	+	0.632910i	$0.781860\pi$
$594$	4.00000			0.164122
$595$	−53.6495			−2.19942
$596$		−	1.45017i		−	0.0594011i
$597$	2.00000			0.0818546
$598$		0			0
$599$	−12.5498			−0.512772			−0.256386	−	0.966574i	$-0.582532\pi$
$599$	−12.5498			−0.512772			−0.256386	+	0.966574i	$0.582532\pi$
$600$		−	13.2749i		−	0.541946i
$601$			11.4502i			0.467062i	0.972349	+	0.233531i	$0.0750280\pi$
$601$			11.4502i			0.467062i	−0.972349	+	0.233531i	$0.924972\pi$
$602$		−	25.0997i		−	1.02299i
$603$		−	1.72508i		−	0.0702508i
$604$			2.00000i			0.0813788i
$605$	−21.3746			−0.869000
$606$	−1.45017			−0.0589089
$607$	−24.0000			−0.974130			−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000			−0.974130			−0.487065	+	0.873366i	$0.661933\pi$
$608$		−	6.27492i		−	0.254481i
$609$			13.0997i			0.530825i
$610$	28.0000			1.13369
$611$	10.9003			0.440980
$612$			6.27492i			0.253648i
$613$	6.54983			0.264545			0.132273	−	0.991213i	$-0.457773\pi$
$613$	6.54983			0.264545			0.132273	+	0.991213i	$0.457773\pi$
$614$	−2.90033			−0.117048
$615$	−26.8248	+	5.45017i	−1.08168	+	0.219772i
$616$	8.00000			0.322329
$617$	22.5498			0.907822			0.453911	−	0.891047i	$-0.350028\pi$
$617$	22.5498			0.907822			0.453911	+	0.891047i	$0.350028\pi$
$618$		−	10.8248i		−	0.435435i
$619$	−17.0997			−0.687294			−0.343647	−	0.939099i	$-0.611662\pi$
$619$	−17.0997			−0.687294			−0.343647	+	0.939099i	$0.611662\pi$
$620$	−7.37459			−0.296171
$621$		0			0
$622$		−	28.2749i		−	1.13372i
$623$	−29.6495			−1.18788
$624$	−4.27492			−0.171134
$625$	84.8488			3.39395
$626$		−	25.6495i		−	1.02516i
$627$			25.0997i			1.00238i
$628$			14.0000i			0.558661i
$629$			37.6495i			1.50118i
$630$		−	8.54983i		−	0.340634i
$631$	42.8248			1.70483			0.852413	−	0.522869i	$-0.175138\pi$
$631$	42.8248			1.70483			0.852413	+	0.522869i	$0.175138\pi$
$632$			6.54983i			0.260538i
$633$	−1.72508			−0.0685659
$634$			30.5498i			1.21329i
$635$	−26.8248			−1.06451
$636$	2.00000			0.0793052
$637$			12.8248i			0.508135i
$638$	−26.1993			−1.03724
$639$			3.72508i			0.147362i
$640$	−4.27492			−0.168981
$641$			20.5498i			0.811670i	0.913946	+	0.405835i	$0.133019\pi$
$641$			20.5498i			0.811670i	−0.913946	+	0.405835i	$0.866981\pi$
$642$			10.8248i			0.427219i
$643$		−	33.7251i		−	1.32999i	−0.746849	−	0.664994i	$-0.768434\pi$
$643$		−	33.7251i		−	1.32999i	0.746849	−	0.664994i	$-0.231566\pi$
$644$		0			0
$645$		−	53.6495i		−	2.11245i
$646$	−39.3746			−1.54917
$647$	41.0997			1.61579			0.807897	−	0.589323i	$-0.200606\pi$
$647$	41.0997			1.61579			0.807897	+	0.589323i	$0.200606\pi$
$648$	−1.00000			−0.0392837
$649$			9.09967i			0.357193i
$650$		−	56.7492i		−	2.22589i
$651$	−3.45017			−0.135223
$652$	−4.00000			−0.156652
$653$			7.64950i			0.299348i	0.988735	+	0.149674i	$0.0478224\pi$
$653$			7.64950i			0.299348i	−0.988735	+	0.149674i	$0.952178\pi$
$654$	11.0997			0.434031
$655$	68.3987			2.67256
$656$	1.27492	+	6.27492i	0.0497772	+	0.244994i
$657$	−0.274917			−0.0107255
$658$	5.09967			0.198806
$659$			12.5498i			0.488872i	0.969665	+	0.244436i	$0.0786028\pi$
$659$			12.5498i			0.488872i	−0.969665	+	0.244436i	$0.921397\pi$
$660$	17.0997			0.665604
$661$	−32.1993			−1.25241			−0.626205	−	0.779659i	$-0.715393\pi$
$661$	−32.1993			−1.25241			−0.626205	+	0.779659i	$0.715393\pi$
$662$		−	1.09967i		−	0.0427398i
$663$			26.8248i			1.04179i
$664$	6.27492			0.243514
$665$	53.6495			2.08044
$666$	−6.00000			−0.232495
$667$		0			0
$668$		−	8.27492i		−	0.320166i
$669$			17.7251i			0.685291i
$670$		−	7.37459i		−	0.284905i
$671$			26.1993i			1.01141i
$672$	−2.00000			−0.0771517
$673$		−	38.1993i		−	1.47248i	−0.676723	−	0.736238i	$-0.736600\pi$
$673$		−	38.1993i		−	1.47248i	0.676723	−	0.736238i	$-0.263400\pi$
$674$	12.8248			0.493991
$675$		−	13.2749i		−	0.510952i
$676$	−5.27492			−0.202881
$677$	30.4743			1.17122			0.585610	−	0.810593i	$-0.300855\pi$
$677$	30.4743			1.17122			0.585610	+	0.810593i	$0.300855\pi$
$678$			10.5498i			0.405164i
$679$	1.09967			0.0422014
$680$			26.8248i			1.02868i
$681$	−11.4502			−0.438771
$682$		−	6.90033i		−	0.264227i
$683$			2.90033i			0.110978i	0.998459	+	0.0554890i	$0.0176718\pi$
$683$			2.90033i			0.110978i	−0.998459	+	0.0554890i	$0.982328\pi$
$684$		−	6.27492i		−	0.239927i
$685$		−	19.4502i		−	0.743153i
$686$			20.0000i			0.763604i
$687$	11.0997			0.423479
$688$	−12.5498			−0.478458
$689$	8.54983			0.325723
$690$		0			0
$691$			13.0997i			0.498335i	0.968460	+	0.249167i	$0.0801570\pi$
$691$			13.0997i			0.498335i	−0.968460	+	0.249167i	$0.919843\pi$
$692$	−7.09967			−0.269889
$693$	8.00000			0.303895
$694$			17.0997i			0.649095i
$695$	19.4502			0.737787
$696$	6.54983			0.248271
$697$	39.3746	−	8.00000i	1.49142	−	0.303022i
$698$	6.00000			0.227103
$699$	14.8248			0.560724
$700$		−	26.5498i		−	1.00349i
$701$	−28.1993			−1.06507			−0.532537	−	0.846407i	$-0.678761\pi$
$701$	−28.1993			−1.06507			−0.532537	+	0.846407i	$0.678761\pi$
$702$	−4.27492			−0.161346
$703$		−	37.6495i		−	1.41998i
$704$		−	4.00000i		−	0.150756i
$705$	10.9003			0.410530
$706$	−10.5498			−0.397048
$707$	−2.90033			−0.109078
$708$		−	2.27492i		−	0.0854966i
$709$		−	4.27492i		−	0.160548i	−0.996773	−	0.0802739i	$-0.974420\pi$
$709$		−	4.27492i		−	0.160548i	0.996773	−	0.0802739i	$-0.0255795\pi$
$710$			15.9244i			0.597633i
$711$			6.54983i			0.245638i
$712$			14.8248i			0.555581i
$713$		0			0
$714$			12.5498i			0.469666i
$715$	73.0997			2.73377
$716$			21.0997i			0.788532i
$717$	−7.72508			−0.288499
$718$	−17.0997			−0.638154
$719$		−	12.8248i		−	0.478283i	−0.970985	−	0.239141i	$-0.923134\pi$
$719$		−	12.8248i		−	0.478283i	0.970985	−	0.239141i	$-0.0768659\pi$
$720$	−4.27492			−0.159317
$721$		−	21.6495i		−	0.806270i
$722$	20.3746			0.758264
$723$		−	12.8248i		−	0.476958i
$724$		−	3.72508i		−	0.138442i
$725$			86.9485i			3.22919i
$726$			5.00000i			0.185567i
$727$		−	1.45017i		−	0.0537837i	−0.999638	−	0.0268918i	$-0.991439\pi$
$727$		−	1.45017i		−	0.0537837i	0.999638	−	0.0268918i	$-0.00856097\pi$
$728$	−8.54983			−0.316878
$729$	−1.00000			−0.0370370
$730$	−1.17525			−0.0434979
$731$			78.7492i			2.91264i
$732$		−	6.54983i		−	0.242089i
$733$	27.6495			1.02126			0.510629	−	0.859801i	$-0.329413\pi$
$733$	27.6495			1.02126			0.510629	+	0.859801i	$0.329413\pi$
$734$	−8.00000			−0.295285
$735$			12.8248i			0.473048i
$736$		0			0
$737$	6.90033			0.254177
$738$	1.27492	+	6.27492i	0.0469304	+	0.230983i
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−25.6495			−0.942894
$741$		−	26.8248i		−	0.985432i
$742$	4.00000			0.146845
$743$	32.0000			1.17397			0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000			1.17397			0.586983	+	0.809599i	$0.300316\pi$
$744$			1.72508i			0.0632446i
$745$		−	6.19934i		−	0.227126i
$746$	−7.09967			−0.259937
$747$	6.27492			0.229587
$748$	−25.0997			−0.917735
$749$			21.6495i			0.791056i
$750$		−	35.3746i		−	1.29170i
$751$			3.09967i			0.113109i	0.998400	+	0.0565543i	$0.0180114\pi$
$751$			3.09967i			0.113109i	−0.998400	+	0.0565543i	$0.981989\pi$
$752$		−	2.54983i		−	0.0929829i
$753$			1.09967i			0.0400742i
$754$	28.0000			1.01970
$755$			8.54983i			0.311160i
$756$	−2.00000			−0.0727393
$757$		−	16.9003i		−	0.614253i	−0.951669	−	0.307126i	$-0.900633\pi$
$757$		−	16.9003i		−	0.614253i	0.951669	−	0.307126i	$-0.0993675\pi$
$758$	−28.5498			−1.03698
$759$		0			0
$760$		−	26.8248i		−	0.973036i
$761$	13.4502			0.487568			0.243784	−	0.969830i	$-0.421611\pi$
$761$	13.4502			0.487568			0.243784	+	0.969830i	$0.421611\pi$
$762$			6.27492i			0.227316i
$763$	22.1993			0.803670
$764$			10.5498i			0.381680i
$765$			26.8248i			0.969851i
$766$		−	0.824752i		−	0.0297995i
$767$		−	9.72508i		−	0.351152i
$768$			1.00000i			0.0360844i
$769$	28.8248			1.03945			0.519724	−	0.854334i	$-0.326035\pi$
$769$	28.8248			1.03945			0.519724	+	0.854334i	$0.326035\pi$
$770$	34.1993			1.23246
$771$	16.5498			0.596028
$772$		−	20.0000i		−	0.719816i
$773$		−	37.4502i		−	1.34699i	−0.739192	−	0.673494i	$-0.764793\pi$
$773$		−	37.4502i		−	1.34699i	0.739192	−	0.673494i	$-0.235207\pi$
$774$	−12.5498			−0.451094
$775$	−22.9003			−0.822604
$776$		−	0.549834i		−	0.0197379i
$777$	−12.0000			−0.430498
$778$	−21.9244			−0.786029
$779$	−39.3746	+	8.00000i	−1.41074	+	0.286630i
$780$	−18.2749			−0.654347
$781$	−14.9003			−0.533176
$782$		0			0
$783$	6.54983			0.234072
$784$	3.00000			0.107143
$785$			59.8488i			2.13610i
$786$		−	16.0000i		−	0.570701i
$787$	−33.6495			−1.19948			−0.599738	−	0.800197i	$-0.704728\pi$
$787$	−33.6495			−1.19948			−0.599738	+	0.800197i	$0.704728\pi$
$788$	−15.7251			−0.560183
$789$	−29.9244			−1.06534
$790$			28.0000i			0.996195i
$791$			21.0997i			0.750218i
$792$		−	4.00000i		−	0.142134i
$793$		−	28.0000i		−	0.994309i
$794$		−	7.17525i		−	0.254640i
$795$	8.54983			0.303231
$796$		−	2.00000i		−	0.0708881i
$797$	13.9244			0.493228			0.246614	−	0.969114i	$-0.420682\pi$
$797$	13.9244			0.493228			0.246614	+	0.969114i	$0.420682\pi$
$798$		−	12.5498i		−	0.444259i
$799$	−16.0000			−0.566039
$800$	−13.2749			−0.469339
$801$			14.8248i			0.523807i
$802$	23.0997			0.815678
$803$		−	1.09967i		−	0.0388065i
$804$	−1.72508			−0.0608390
$805$		0			0
$806$			7.37459i			0.259759i
$807$			8.27492i			0.291291i
$808$			1.45017i			0.0510166i
$809$		−	47.3746i		−	1.66560i	−0.553573	−	0.832801i	$-0.686736\pi$
$809$		−	47.3746i		−	1.66560i	0.553573	−	0.832801i	$-0.313264\pi$
$810$	−4.27492			−0.150205
$811$	−8.54983			−0.300225			−0.150113	−	0.988669i	$-0.547964\pi$
$811$	−8.54983			−0.300225			−0.150113	+	0.988669i	$0.547964\pi$
$812$	13.0997			0.459708
$813$			25.0997i			0.880284i
$814$		−	24.0000i		−	0.841200i
$815$	−17.0997			−0.598975
$816$	6.27492			0.219666
$817$		−	78.7492i		−	2.75508i
$818$	8.27492			0.289326
$819$	−8.54983			−0.298755
$820$	5.45017	+	26.8248i	0.190328	+	0.936761i
$821$	33.9244			1.18397			0.591985	−	0.805949i	$-0.298344\pi$
$821$	33.9244			1.18397			0.591985	+	0.805949i	$0.298344\pi$
$822$	−4.54983			−0.158694
$823$			21.4502i			0.747706i	0.927488	+	0.373853i	$0.121964\pi$
$823$			21.4502i			0.747706i	−0.927488	+	0.373853i	$0.878036\pi$
$824$	−10.8248			−0.377098
$825$	53.0997			1.84869
$826$		−	4.54983i		−	0.158309i
$827$		−	4.54983i		−	0.158213i	−0.996866	−	0.0791066i	$-0.974793\pi$
$827$		−	4.54983i		−	0.158213i	0.996866	−	0.0791066i	$-0.0252068\pi$
$828$		0			0
$829$	−12.9003			−0.448047			−0.224024	−	0.974584i	$-0.571919\pi$
$829$	−12.9003			−0.448047			−0.224024	+	0.974584i	$0.571919\pi$
$830$	26.8248			0.931101
$831$			30.5498i			1.05976i
$832$			4.27492i			0.148206i
$833$		−	18.8248i		−	0.652239i
$834$		−	4.54983i		−	0.157548i
$835$		−	35.3746i		−	1.22419i
$836$	25.0997			0.868090
$837$			1.72508i			0.0596276i
$838$	34.2749			1.18401
$839$		−	41.9244i		−	1.44739i	−0.690119	−	0.723696i	$-0.742442\pi$
$839$		−	41.9244i		−	1.44739i	0.690119	−	0.723696i	$-0.257558\pi$
$840$	−8.54983			−0.294997
$841$	−13.9003			−0.479322
$842$			3.17525i			0.109426i
$843$	−2.27492			−0.0783523
$844$			1.72508i			0.0593798i
$845$	−22.5498			−0.775738
$846$		−	2.54983i		−	0.0876651i
$847$			10.0000i			0.343604i
$848$		−	2.00000i		−	0.0686803i
$849$			21.6495i			0.743009i
$850$			83.2990i			2.85713i
$851$		0			0
$852$	3.72508			0.127619
$853$	9.45017			0.323568			0.161784	−	0.986826i	$-0.448275\pi$
$853$	9.45017			0.323568			0.161784	+	0.986826i	$0.448275\pi$
$854$		−	13.0997i		−	0.448261i
$855$		−	26.8248i		−	0.917387i
$856$	10.8248			0.369982
$857$	51.6495			1.76431			0.882157	−	0.470956i	$-0.156091\pi$
$857$	51.6495			1.76431			0.882157	+	0.470956i	$0.156091\pi$
$858$		−	17.0997i		−	0.583773i
$859$	−6.19934			−0.211519			−0.105759	−	0.994392i	$-0.533727\pi$
$859$	−6.19934			−0.211519			−0.105759	+	0.994392i	$0.533727\pi$
$860$	−53.6495			−1.82943
$861$	2.54983	+	12.5498i	0.0868981	+	0.427697i
$862$	−19.4502			−0.662475
$863$	11.4502			0.389768			0.194884	−	0.980826i	$-0.437567\pi$
$863$	11.4502			0.389768			0.194884	+	0.980826i	$0.437567\pi$
$864$			1.00000i			0.0340207i
$865$	−30.3505			−1.03195
$866$	16.2749			0.553044
$867$		−	22.3746i		−	0.759881i
$868$			3.45017i			0.117106i
$869$	−26.1993			−0.888752
$870$	28.0000			0.949289
$871$	−7.37459			−0.249878
$872$		−	11.0997i		−	0.375882i
$873$		−	0.549834i		−	0.0186091i
$874$		0			0
$875$		−	70.7492i		−	2.39176i
$876$			0.274917i			0.00928859i
$877$	19.6495			0.663517			0.331758	−	0.943364i	$-0.392358\pi$
$877$	19.6495			0.663517			0.331758	+	0.943364i	$0.392358\pi$
$878$			39.0997i			1.31955i
$879$	2.00000			0.0674583
$880$		−	17.0997i		−	0.576430i
$881$	−36.1993			−1.21959			−0.609793	−	0.792560i	$-0.708748\pi$
$881$	−36.1993			−1.21959			−0.609793	+	0.792560i	$0.708748\pi$
$882$	3.00000			0.101015
$883$		−	43.3746i		−	1.45967i	−0.683623	−	0.729836i	$-0.739597\pi$
$883$		−	43.3746i		−	1.45967i	0.683623	−	0.729836i	$-0.260403\pi$
$884$	26.8248			0.902214
$885$		−	9.72508i		−	0.326905i
$886$	−23.3746			−0.785285
$887$		−	25.9244i		−	0.870457i	−0.900320	−	0.435228i	$-0.856668\pi$
$887$		−	25.9244i		−	0.870457i	0.900320	−	0.435228i	$-0.143332\pi$
$888$			6.00000i			0.201347i
$889$			12.5498i			0.420908i
$890$			63.3746i			2.12432i
$891$		−	4.00000i		−	0.134005i
$892$	17.7251			0.593480
$893$	16.0000			0.535420
$894$	−1.45017			−0.0485008
$895$			90.1993i			3.01503i
$896$			2.00000i			0.0668153i
$897$		0			0
$898$	−10.0000			−0.333704
$899$		−	11.2990i		−	0.376843i
$900$	−13.2749			−0.442497
$901$	−12.5498			−0.418096
$902$	−25.0997	+	5.09967i	−0.835728	+	0.169800i
$903$	−25.0997			−0.835265
$904$	10.5498			0.350882
$905$		−	15.9244i		−	0.529346i
$906$	2.00000			0.0664455
$907$	42.7492			1.41946			0.709731	−	0.704472i	$-0.248816\pi$
$907$	42.7492			1.41946			0.709731	+	0.704472i	$0.248816\pi$
$908$			11.4502i			0.379987i
$909$			1.45017i			0.0480990i
$910$	−36.5498			−1.21162
$911$	28.5498			0.945898			0.472949	−	0.881090i	$-0.343189\pi$
$911$	28.5498			0.945898			0.472949	+	0.881090i	$0.343189\pi$
$912$	−6.27492			−0.207783
$913$			25.0997i			0.830678i
$914$		−	29.0997i		−	0.962531i
$915$		−	28.0000i		−	0.925651i
$916$		−	11.0997i		−	0.366743i
$917$		−	32.0000i		−	1.05673i
$918$	6.27492			0.207103
$919$			37.4502i			1.23537i	0.786427	+	0.617683i	$0.211929\pi$
$919$			37.4502i			1.23537i	−0.786427	+	0.617683i	$0.788071\pi$
$920$		0			0
$921$			2.90033i			0.0955692i
$922$	27.0997			0.892480
$923$	15.9244			0.524159
$924$		−	8.00000i		−	0.263181i
$925$	−79.6495			−2.61886
$926$		−	26.5498i		−	0.872482i
$927$	−10.8248			−0.355531
$928$		−	6.54983i		−	0.215009i
$929$			26.8248i			0.880092i	0.897975	+	0.440046i	$0.145038\pi$
$929$			26.8248i			0.880092i	−0.897975	+	0.440046i	$0.854962\pi$
$930$			7.37459i			0.241822i
$931$			18.8248i			0.616956i
$932$		−	14.8248i		−	0.485601i
$933$	−28.2749			−0.925679
$934$	21.7251			0.710866
$935$	−107.299			−3.50905
$936$			4.27492i			0.139730i
$937$			42.1993i			1.37859i	0.724480	+	0.689296i	$0.242080\pi$
$937$			42.1993i			1.37859i	−0.724480	+	0.689296i	$0.757920\pi$
$938$	−3.45017			−0.112652
$939$	−25.6495			−0.837040
$940$		−	10.9003i		−	0.355530i
$941$	2.62541			0.0855860			0.0427930	−	0.999084i	$-0.486374\pi$
$941$	2.62541			0.0855860			0.0427930	+	0.999084i	$0.486374\pi$
$942$	14.0000			0.456145
$943$		0			0
$944$	−2.27492			−0.0740422
$945$	−8.54983			−0.278126
$946$		−	50.1993i		−	1.63212i
$947$	−21.7251			−0.705970			−0.352985	−	0.935629i	$-0.614833\pi$
$947$	−21.7251			−0.705970			−0.352985	+	0.935629i	$0.614833\pi$
$948$	6.54983			0.212729
$949$			1.17525i			0.0381502i
$950$		−	83.2990i		−	2.70258i
$951$	30.5498			0.990646
$952$	12.5498			0.406742
$953$	−30.5498			−0.989606			−0.494803	−	0.869005i	$-0.664760\pi$
$953$	−30.5498			−0.989606			−0.494803	+	0.869005i	$0.664760\pi$
$954$		−	2.00000i		−	0.0647524i
$955$			45.0997i			1.45939i
$956$			7.72508i			0.249847i
$957$			26.1993i			0.846904i
$958$			33.9244i			1.09605i
$959$	−9.09967			−0.293844
$960$			4.27492i			0.137972i
$961$	−28.0241			−0.904003
$962$			25.6495i			0.826973i
$963$	10.8248			0.348823
$964$	−12.8248			−0.413057
$965$		−	85.4983i		−	2.75229i
$966$		0			0
$967$			17.4502i			0.561159i	0.959831	+	0.280580i	$0.0905267\pi$
$967$			17.4502i			0.561159i	−0.959831	+	0.280580i	$0.909473\pi$
$968$	5.00000			0.160706
$969$			39.3746i			1.26489i
$970$		−	2.35050i		−	0.0754699i
$971$			33.6495i			1.07986i	0.841709	+	0.539932i	$0.181550\pi$
$971$			33.6495i			1.07986i	−0.841709	+	0.539932i	$0.818450\pi$
$972$			1.00000i			0.0320750i
$973$		−	9.09967i		−	0.291722i
$974$	−13.1752			−0.422162
$975$	−56.7492			−1.81743
$976$	−6.54983			−0.209655
$977$		−	8.62541i		−	0.275951i	−0.990436	−	0.137976i	$-0.955940\pi$
$977$		−	8.62541i		−	0.275951i	0.990436	−	0.137976i	$-0.0440596\pi$
$978$			4.00000i			0.127906i
$979$	−59.2990			−1.89520
$980$	12.8248			0.409672
$981$		−	11.0997i		−	0.354385i
$982$	14.8248			0.473077
$983$	−41.0997			−1.31088			−0.655438	−	0.755249i	$-0.727516\pi$
$983$	−41.0997			−1.31088			−0.655438	+	0.755249i	$0.727516\pi$
$984$	6.27492	−	1.27492i	0.200037	−	0.0406429i
$985$	−67.2234			−2.14192
$986$	−41.0997			−1.30888
$987$		−	5.09967i		−	0.162324i
$988$	−26.8248			−0.853409
$989$		0			0
$990$		−	17.0997i		−	0.543463i
$991$		−	47.0997i		−	1.49617i	−0.663603	−	0.748085i	$-0.730973\pi$
$991$		−	47.0997i		−	1.49617i	0.663603	−	0.748085i	$-0.269027\pi$
$992$	1.72508			0.0547714
$993$	−1.09967			−0.0348969
$994$	7.45017			0.236305
$995$		−	8.54983i		−	0.271048i
$996$		−	6.27492i		−	0.198828i
$997$			9.92442i			0.314310i	0.987574	+	0.157155i	$0.0502322\pi$
$997$			9.92442i			0.314310i	−0.987574	+	0.157155i	$0.949768\pi$
$998$		−	23.3746i		−	0.739910i
$999$			6.00000i			0.189832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	246.2.d.b.163.4	yes	4
3.2	odd	2		738.2.d.h.163.1		4
4.3	odd	2		1968.2.j.c.1393.2		4
41.40	even	2	inner	246.2.d.b.163.2	✓	4
123.122	odd	2		738.2.d.h.163.2		4
164.163	odd	2		1968.2.j.c.1393.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
246.2.d.b.163.2	✓	4	41.40	even	2	inner
246.2.d.b.163.4	yes	4	1.1	even	1	trivial
738.2.d.h.163.1		4	3.2	odd	2
738.2.d.h.163.2		4	123.122	odd	2
1968.2.j.c.1393.2		4	4.3	odd	2
1968.2.j.c.1393.4		4	164.163	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 246 = 2 \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	246.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} -4.27492 q^{10} -4.00000i q^{11} +1.00000i q^{12} +4.27492i q^{13} +2.00000i q^{14} +4.27492i q^{15} +1.00000 q^{16} -6.27492i q^{17} +1.00000 q^{18} +6.27492i q^{19} +4.27492 q^{20} +2.00000 q^{21} +4.00000i q^{22} -1.00000i q^{24} +13.2749 q^{25} -4.27492i q^{26} -1.00000i q^{27} -2.00000i q^{28} +6.54983i q^{29} -4.27492i q^{30} -1.72508 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.27492i q^{34} -8.54983i q^{35} -1.00000 q^{36} -6.00000 q^{37} -6.27492i q^{38} -4.27492 q^{39} -4.27492 q^{40} +(1.27492 + 6.27492i) q^{41} -2.00000 q^{42} -12.5498 q^{43} -4.00000i q^{44} -4.27492 q^{45} -2.54983i q^{47} +1.00000i q^{48} +3.00000 q^{49} -13.2749 q^{50} +6.27492 q^{51} +4.27492i q^{52} -2.00000i q^{53} +1.00000i q^{54} -17.0997i q^{55} +2.00000i q^{56} -6.27492 q^{57} -6.54983i q^{58} -2.27492 q^{59} +4.27492i q^{60} -6.54983 q^{61} +1.72508 q^{62} +2.00000i q^{63} +1.00000 q^{64} +18.2749i q^{65} -4.00000 q^{66} +1.72508i q^{67} -6.27492i q^{68} +8.54983i q^{70} -3.72508i q^{71} +1.00000 q^{72} +0.274917 q^{73} +6.00000 q^{74} +13.2749i q^{75} +6.27492i q^{76} -8.00000 q^{77} +4.27492 q^{78} -6.54983i q^{79} +4.27492 q^{80} +1.00000 q^{81} +(-1.27492 - 6.27492i) q^{82} -6.27492 q^{83} +2.00000 q^{84} -26.8248i q^{85} +12.5498 q^{86} -6.54983 q^{87} +4.00000i q^{88} -14.8248i q^{89} +4.27492 q^{90} +8.54983 q^{91} -1.72508i q^{93} +2.54983i q^{94} +26.8248i q^{95} -1.00000i q^{96} +0.549834i q^{97} -3.00000 q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} + 4 q^{18} + 2 q^{20} + 8 q^{21} + 38 q^{25} - 22 q^{31} - 4 q^{32} + 16 q^{33} - 4 q^{36} - 24 q^{37} - 2 q^{39} - 2 q^{40} - 10 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 + 4 * q^18 + 2 * q^20 + 8 * q^21 + 38 * q^25 - 22 * q^31 - 4 * q^32 + 16 * q^33 - 4 * q^36 - 24 * q^37 - 2 * q^39 - 2 * q^40 - 10 * q^41 - 8 * q^42 - 20 * q^43 - 2 * q^45 + 12 * q^49 - 38 * q^50 + 10 * q^51 - 10 * q^57 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 - 16 * q^66 + 4 * q^72 - 14 * q^73 + 24 * q^74 - 32 * q^77 + 2 * q^78 + 2 * q^80 + 4 * q^81 + 10 * q^82 - 10 * q^83 + 8 * q^84 + 20 * q^86 + 4 * q^87 + 2 * q^90 + 4 * q^91 - 12 * q^98

Introduction

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