LMFDB - Embedded newform 48.16.c.d.47.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,16,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	48.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:	$68.4928824480$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 10 x^{19} + 8885809 x^{18} - 79971996 x^{17} + 21106062365235 x^{16} - 168846686224596 x^{15} + \cdots + 85\!\cdots\!61 $ x^20 - 10x^19 + 8885809x^18 - 79971996x^17 + 21106062365235x^16 - 168846686224596x^15 + 6104796603193796014x^14 - 42730621411692080224x^13 + 612830101834948128969517x^12 - 3676425120613837434937014x^11 + 23515317858743883397971890679x^10 - 117542889748506812960062582524x^9 + 442824580967063105118845295045457x^8 - 1770593106968691891463936968183484x^7 + 4516272979574514115212279965662378678x^6 - 13542622356488850024523824403808515248x^5 + 25368182055011473290428060306984553604575x^4 - 50713795138219024400871126679601580299346x^3 + 73555006160055781174118405609684271136003209x^2 - 73529651519295334152034903037741123700144732*x + 85774090629870699032700202010530622843520083061
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{194}\cdot 3^{63}\cdot 5^{6}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.9
Root			$0.500000 + 363.024i$ of defining polynomial
Character	$\chi$	$=$	48.47
Dual form			48.16.c.d.47.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-660.169 - 3730.02i) q^{3} -233484. i q^{5} -674260. i q^{7} +(-1.34773e7 + 4.92490e6i) q^{9} +O(q^{10})$	\(q+(-660.169 - 3730.02i) q^{3} -233484. i q^{5} -674260. i q^{7} +(-1.34773e7 + 4.92490e6i) q^{9} +1.99491e7 q^{11} -6.08051e7 q^{13} +(-8.70900e8 + 1.54139e8i) q^{15} +3.15809e9i q^{17} +1.41564e9i q^{19} +(-2.51500e9 + 4.45126e8i) q^{21} -1.57568e10 q^{23} -2.39970e10 q^{25} +(2.72673e10 + 4.70192e10i) q^{27} -1.11059e11i q^{29} +2.32822e11i q^{31} +(-1.31698e10 - 7.44108e10i) q^{33} -1.57429e11 q^{35} -3.75428e11 q^{37} +(4.01417e10 + 2.26804e11i) q^{39} +3.06884e11i q^{41} -1.01153e12i q^{43} +(1.14988e12 + 3.14672e12i) q^{45} -4.21635e12 q^{47} +4.29294e12 q^{49} +(1.17798e13 - 2.08488e12i) q^{51} -9.07188e12i q^{53} -4.65779e12i q^{55} +(5.28035e12 - 9.34559e11i) q^{57} -5.24088e12 q^{59} +2.39988e13 q^{61} +(3.32066e12 + 9.08717e12i) q^{63} +1.41970e13i q^{65} +8.59195e13i q^{67} +(1.04022e13 + 5.87733e13i) q^{69} -1.22093e14 q^{71} +4.98083e13 q^{73} +(1.58421e13 + 8.95094e13i) q^{75} -1.34509e13i q^{77} +1.48059e14i q^{79} +(1.57382e14 - 1.32748e14i) q^{81} +3.90447e14 q^{83} +7.37363e14 q^{85} +(-4.14252e14 + 7.33176e13i) q^{87} +3.10097e14i q^{89} +4.09984e13i q^{91} +(8.68433e14 - 1.53702e14i) q^{93} +3.30528e14 q^{95} +6.87836e14 q^{97} +(-2.68860e14 + 9.82474e13i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2271972 q^{9}+O(q^{10}) $ 20 * q + 2271972 * q^9	$ 20 q + 2271972 q^{9} + 318771128 q^{13} + 13145285784 q^{21} - 57334310012 q^{25} + 628079136192 q^{33} - 1811120039336 q^{37} + 7518335948928 q^{45} - 8329580497444 q^{49} - 36365149089912 q^{57} + 46120845287032 q^{61} - 117111587094144 q^{69} + 83221863805064 q^{73} + 73507522500468 q^{81} - 12\!\cdots\!52 q^{85}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100}) $ 20 * q + 2271972 * q^9 + 318771128 * q^13 + 13145285784 * q^21 - 57334310012 * q^25 + 628079136192 * q^33 - 1811120039336 * q^37 + 7518335948928 * q^45 - 8329580497444 * q^49 - 36365149089912 * q^57 + 46120845287032 * q^61 - 117111587094144 * q^69 + 83221863805064 * q^73 + 73507522500468 * q^81 - 1297131567063552 * q^85 + 1162806897994488 * q^93 - 1219036801794712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−660.169	−	3730.02i	−0.174279	−	0.984696i
$3$	−660.169	−	3730.02i	−0.174279	−	0.984696i
$4$		0			0
$5$		−	233484.i		−	1.33654i	−0.743920	−	0.668269i	$-0.767036\pi$
$5$		−	233484.i		−	1.33654i	0.743920	−	0.668269i	$-0.232964\pi$
$6$		0			0
$7$		−	674260.i		−	0.309451i	−0.987958	−	0.154725i	$-0.950551\pi$
$7$		−	674260.i		−	0.309451i	0.987958	−	0.154725i	$-0.0494493\pi$
$8$		0			0
$9$	−1.34773e7	+	4.92490e6i	−0.939253	+	0.343224i
$10$		0			0
$11$	1.99491e7			0.308659			0.154329	−	0.988019i	$-0.450678\pi$
$11$	1.99491e7			0.308659			0.154329	+	0.988019i	$0.450678\pi$
$12$		0			0
$13$	−6.08051e7			−0.268760			−0.134380	−	0.990930i	$-0.542904\pi$
$13$	−6.08051e7			−0.268760			−0.134380	+	0.990930i	$0.542904\pi$
$14$		0			0
$15$	−8.70900e8	+	1.54139e8i	−1.31608	+	0.232931i
$16$		0			0
$17$			3.15809e9i			1.86663i	0.359058	+	0.933315i	$0.383098\pi$
$17$			3.15809e9i			1.86663i	−0.359058	+	0.933315i	$0.616902\pi$
$18$		0			0
$19$			1.41564e9i			0.363328i	0.983361	+	0.181664i	$0.0581484\pi$
$19$			1.41564e9i			0.363328i	−0.983361	+	0.181664i	$0.941852\pi$
$20$		0			0
$21$	−2.51500e9	+	4.45126e8i	−0.304715	+	0.0539309i
$22$		0			0
$23$	−1.57568e10			−0.964961			−0.482481	−	0.875907i	$-0.660264\pi$
$23$	−1.57568e10			−0.964961			−0.482481	+	0.875907i	$0.660264\pi$
$24$		0			0
$25$	−2.39970e10			−0.786334
$26$		0			0
$27$	2.72673e10	+	4.70192e10i	0.501664	+	0.865062i
$28$		0			0
$29$		−	1.11059e11i		−	1.19555i	−0.801664	−	0.597775i	$-0.796051\pi$
$29$		−	1.11059e11i		−	1.19555i	0.801664	−	0.597775i	$-0.203949\pi$
$30$		0			0
$31$			2.32822e11i			1.51989i	0.649988	+	0.759944i	$0.274774\pi$
$31$			2.32822e11i			1.51989i	−0.649988	+	0.759944i	$0.725226\pi$
$32$		0			0
$33$	−1.31698e10	−	7.44108e10i	−0.0537929	−	0.303935i
$34$		0			0
$35$	−1.57429e11			−0.413593
$36$		0			0
$37$	−3.75428e11			−0.650151			−0.325075	−	0.945688i	$-0.605390\pi$
$37$	−3.75428e11			−0.650151			−0.325075	+	0.945688i	$0.605390\pi$
$38$		0			0
$39$	4.01417e10	+	2.26804e11i	0.0468393	+	0.264647i
$40$		0			0
$41$			3.06884e11i			0.246091i	0.992401	+	0.123045i	$0.0392661\pi$
$41$			3.06884e11i			0.246091i	−0.992401	+	0.123045i	$0.960734\pi$
$42$		0			0
$43$		−	1.01153e12i		−	0.567498i	−0.958899	−	0.283749i	$-0.908422\pi$
$43$		−	1.01153e12i		−	0.567498i	0.958899	−	0.283749i	$-0.0915783\pi$
$44$		0			0
$45$	1.14988e12	+	3.14672e12i	0.458733	+	1.25535i
$46$		0			0
$47$	−4.21635e12			−1.21396			−0.606978	−	0.794719i	$-0.707618\pi$
$47$	−4.21635e12			−1.21396			−0.606978	+	0.794719i	$0.707618\pi$
$48$		0			0
$49$	4.29294e12			0.904240
$50$		0			0
$51$	1.17798e13	−	2.08488e12i	1.83806	−	0.325315i
$52$		0			0
$53$		−	9.07188e12i		−	1.06079i	−0.847751	−	0.530394i	$-0.822044\pi$
$53$		−	9.07188e12i		−	1.06079i	0.847751	−	0.530394i	$-0.177956\pi$
$54$		0			0
$55$		−	4.65779e12i		−	0.412534i
$56$		0			0
$57$	5.28035e12	−	9.34559e11i	0.357768	−	0.0633206i
$58$		0			0
$59$	−5.24088e12			−0.274166			−0.137083	−	0.990560i	$-0.543773\pi$
$59$	−5.24088e12			−0.274166			−0.137083	+	0.990560i	$0.543773\pi$
$60$		0			0
$61$	2.39988e13			0.977723			0.488862	−	0.872361i	$-0.337412\pi$
$61$	2.39988e13			0.977723			0.488862	+	0.872361i	$0.337412\pi$
$62$		0			0
$63$	3.32066e12	+	9.08717e12i	0.106211	+	0.290653i
$64$		0			0
$65$			1.41970e13i			0.359208i
$66$		0			0
$67$			8.59195e13i			1.73193i	0.500102	+	0.865966i	$0.333296\pi$
$67$			8.59195e13i			1.73193i	−0.500102	+	0.865966i	$0.666704\pi$
$68$		0			0
$69$	1.04022e13	+	5.87733e13i	0.168173	+	0.950194i
$70$		0			0
$71$	−1.22093e14			−1.59314			−0.796572	−	0.604544i	$-0.793355\pi$
$71$	−1.22093e14			−1.59314			−0.796572	+	0.604544i	$0.793355\pi$
$72$		0			0
$73$	4.98083e13			0.527692			0.263846	−	0.964565i	$-0.415009\pi$
$73$	4.98083e13			0.527692			0.263846	+	0.964565i	$0.415009\pi$
$74$		0			0
$75$	1.58421e13	+	8.95094e13i	0.137042	+	0.774300i
$76$		0			0
$77$		−	1.34509e13i		−	0.0955148i
$78$		0			0
$79$			1.48059e14i			0.867422i	0.901052	+	0.433711i	$0.142796\pi$
$79$			1.48059e14i			0.867422i	−0.901052	+	0.433711i	$0.857204\pi$
$80$		0			0
$81$	1.57382e14	−	1.32748e14i	0.764394	−	0.644749i
$82$		0			0
$83$	3.90447e14			1.57934			0.789672	−	0.613530i	$-0.210251\pi$
$83$	3.90447e14			1.57934			0.789672	+	0.613530i	$0.210251\pi$
$84$		0			0
$85$	7.37363e14			2.49482
$86$		0			0
$87$	−4.14252e14	+	7.33176e13i	−1.17725	+	0.208360i
$88$		0			0
$89$			3.10097e14i			0.743143i	0.928404	+	0.371571i	$0.121181\pi$
$89$			3.10097e14i			0.743143i	−0.928404	+	0.371571i	$0.878819\pi$
$90$		0			0
$91$			4.09984e13i			0.0831681i
$92$		0			0
$93$	8.68433e14	−	1.53702e14i	1.49663	−	0.264885i
$94$		0			0
$95$	3.30528e14			0.485602
$96$		0			0
$97$	6.87836e14			0.864365			0.432182	−	0.901786i	$-0.357744\pi$
$97$	6.87836e14			0.864365			0.432182	+	0.901786i	$0.357744\pi$
$98$		0			0
$99$	−2.68860e14	+	9.82474e13i	−0.289909	+	0.105939i
$100$		0			0
$101$			1.57452e15i			1.46130i	0.682753	+	0.730649i	$0.260782\pi$
$101$			1.57452e15i			1.46130i	−0.682753	+	0.730649i	$0.739218\pi$
$102$		0			0
$103$			9.78318e14i			0.783792i	0.920010	+	0.391896i	$0.128181\pi$
$103$			9.78318e14i			0.783792i	−0.920010	+	0.391896i	$0.871819\pi$
$104$		0			0
$105$	1.03930e14	+	5.87212e14i	0.0720807	+	0.407263i
$106$		0			0
$107$	−1.57547e15			−0.948487			−0.474244	−	0.880394i	$-0.657278\pi$
$107$	−1.57547e15			−0.948487			−0.474244	+	0.880394i	$0.657278\pi$
$108$		0			0
$109$	2.81784e15			1.47644			0.738222	−	0.674558i	$-0.235666\pi$
$109$	2.81784e15			1.47644			0.738222	+	0.674558i	$0.235666\pi$
$110$		0			0
$111$	2.47846e14	+	1.40036e15i	0.113308	+	0.640201i
$112$		0			0
$113$		−	3.29733e15i		−	1.31848i	−0.751931	−	0.659241i	$-0.770878\pi$
$113$		−	3.29733e15i		−	1.31848i	0.751931	−	0.659241i	$-0.229122\pi$
$114$		0			0
$115$			3.67896e15i			1.28971i
$116$		0			0
$117$	8.19486e14	−	2.99459e14i	0.252434	−	0.0922450i
$118$		0			0
$119$	2.12938e15			0.577630
$120$		0			0
$121$	−3.77928e15			−0.904730
$122$		0			0
$123$	1.14468e15	−	2.02595e14i	0.242325	−	0.0428885i
$124$		0			0
$125$		−	1.52245e15i		−	0.285573i
$126$		0			0
$127$		−	1.51152e15i		−	0.251701i	−0.992049	−	0.125851i	$-0.959834\pi$
$127$		−	1.51152e15i		−	0.251701i	0.992049	−	0.125851i	$-0.0401660\pi$
$128$		0			0
$129$	−3.77303e15	+	6.67780e14i	−0.558814	+	0.0989033i
$130$		0			0
$131$	−1.11587e16			−1.47258			−0.736292	−	0.676664i	$-0.763425\pi$
$131$	−1.11587e16			−1.47258			−0.736292	+	0.676664i	$0.763425\pi$
$132$		0			0
$133$	9.54506e14			0.112432
$134$		0			0
$135$	1.09782e16	−	6.36646e15i	1.15619	−	0.670493i
$136$		0			0
$137$			6.37722e15i			0.601488i	0.953705	+	0.300744i	$0.0972349\pi$
$137$			6.37722e15i			0.601488i	−0.953705	+	0.300744i	$0.902765\pi$
$138$		0			0
$139$		−	1.45074e16i		−	1.22738i	−0.789547	−	0.613690i	$-0.789684\pi$
$139$		−	1.45074e16i		−	1.22738i	0.789547	−	0.613690i	$-0.210316\pi$
$140$		0			0
$141$	2.78351e15	+	1.57271e16i	0.211567	+	1.19538i
$142$		0			0
$143$	−1.21301e15			−0.0829552
$144$		0			0
$145$	−2.59304e16			−1.59790
$146$		0			0
$147$	−2.83406e15	−	1.60128e16i	−0.157590	−	0.890402i
$148$		0			0
$149$		−	1.83936e16i		−	0.924208i	−0.886826	−	0.462104i	$-0.847095\pi$
$149$		−	1.83936e16i		−	0.924208i	0.886826	−	0.462104i	$-0.152905\pi$
$150$		0			0
$151$		−	1.85994e16i		−	0.845612i	−0.906220	−	0.422806i	$-0.861045\pi$
$151$		−	1.85994e16i		−	0.845612i	0.906220	−	0.422806i	$-0.138955\pi$
$152$		0			0
$153$	−1.55533e16	−	4.25625e16i	−0.640673	−	1.75324i
$154$		0			0
$155$	5.43602e16			2.03139
$156$		0			0
$157$	−3.69034e16			−1.25262			−0.626310	−	0.779574i	$-0.715436\pi$
$157$	−3.69034e16			−1.25262			−0.626310	+	0.779574i	$0.715436\pi$
$158$		0			0
$159$	−3.38383e16	+	5.98898e15i	−1.04455	+	0.184873i
$160$		0			0
$161$			1.06242e16i			0.298608i
$162$		0			0
$163$			3.09611e16i			0.793249i	0.917981	+	0.396624i	$0.129818\pi$
$163$			3.09611e16i			0.793249i	−0.917981	+	0.396624i	$0.870182\pi$
$164$		0			0
$165$	−1.73737e16	+	3.07493e15i	−0.406221	+	0.0718962i
$166$		0			0
$167$	−2.54317e16			−0.543253			−0.271626	−	0.962403i	$-0.587561\pi$
$167$	−2.54317e16			−0.543253			−0.271626	+	0.962403i	$0.587561\pi$
$168$		0			0
$169$	−4.74886e16			−0.927768
$170$		0			0
$171$	−6.97186e15	−	1.90789e16i	−0.124703	−	0.341257i
$172$		0			0
$173$			6.45522e13i			0.00105819i	1.00000		0.000529097i	$0.000168417\pi$
$173$			6.45522e13i			0.00105819i	−1.00000		0.000529097i	$0.999832\pi$
$174$		0			0
$175$			1.61802e16i			0.243332i
$176$		0			0
$177$	3.45987e15	+	1.95486e16i	0.0477815	+	0.269970i
$178$		0			0
$179$	−1.13379e16			−0.143925			−0.0719623	−	0.997407i	$-0.522926\pi$
$179$	−1.13379e16			−0.143925			−0.0719623	+	0.997407i	$0.522926\pi$
$180$		0			0
$181$	−4.68399e16			−0.547050			−0.273525	−	0.961865i	$-0.588190\pi$
$181$	−4.68399e16			−0.547050			−0.273525	+	0.961865i	$0.588190\pi$
$182$		0			0
$183$	−1.58433e16	−	8.95162e16i	−0.170397	−	0.962760i
$184$		0			0
$185$			8.76563e16i			0.868951i
$186$		0			0
$187$			6.30013e16i			0.576152i
$188$		0			0
$189$	3.17032e16	−	1.83852e16i	0.267694	−	0.155241i
$190$		0			0
$191$	5.78711e16			0.451556			0.225778	−	0.974179i	$-0.427508\pi$
$191$	5.78711e16			0.451556			0.225778	+	0.974179i	$0.427508\pi$
$192$		0			0
$193$	2.27639e17			1.64274			0.821368	−	0.570399i	$-0.193211\pi$
$193$	2.27639e17			1.64274			0.821368	+	0.570399i	$0.193211\pi$
$194$		0			0
$195$	5.29551e16	−	9.37242e15i	0.353711	−	0.0626025i
$196$		0			0
$197$		−	1.44969e17i		−	0.896972i	−0.893790	−	0.448486i	$-0.851963\pi$
$197$		−	1.44969e17i		−	0.896972i	0.893790	−	0.448486i	$-0.148037\pi$
$198$		0			0
$199$			3.50807e16i			0.201220i	0.994926	+	0.100610i	$0.0320794\pi$
$199$			3.50807e16i			0.201220i	−0.994926	+	0.100610i	$0.967921\pi$
$200$		0			0
$201$	3.20482e17	−	5.67214e16i	1.70543	−	0.301840i
$202$		0			0
$203$	−7.48825e16			−0.369964
$204$		0			0
$205$	7.16524e16			0.328910
$206$		0			0
$207$	2.12359e17	−	7.76007e16i	0.906343	−	0.331198i
$208$		0			0
$209$			2.82407e16i			0.112145i
$210$		0			0
$211$			3.61413e17i			1.33624i	0.744053	+	0.668121i	$0.232901\pi$
$211$			3.61413e17i			1.33624i	−0.744053	+	0.668121i	$0.767099\pi$
$212$		0			0
$213$	8.06024e16	+	4.55412e17i	0.277652	+	1.56876i
$214$		0			0
$215$	−2.36175e17			−0.758483
$216$		0			0
$217$	1.56983e17			0.470331
$218$		0			0
$219$	−3.28819e16	−	1.85786e17i	−0.0919657	−	0.519616i
$220$		0			0
$221$		−	1.92028e17i		−	0.501676i
$222$		0			0
$223$			6.88674e17i			1.68162i	0.541333	+	0.840808i	$0.317920\pi$
$223$			6.88674e17i			1.68162i	−0.541333	+	0.840808i	$0.682080\pi$
$224$		0			0
$225$	3.23414e17	−	1.18183e17i	0.738567	−	0.269889i
$226$		0			0
$227$	−6.73830e17			−1.43998			−0.719990	−	0.693985i	$-0.755853\pi$
$227$	−6.73830e17			−1.43998			−0.719990	+	0.693985i	$0.755853\pi$
$228$		0			0
$229$	−2.84722e17			−0.569710			−0.284855	−	0.958571i	$-0.591946\pi$
$229$	−2.84722e17			−0.569710			−0.284855	+	0.958571i	$0.591946\pi$
$230$		0			0
$231$	−5.01722e16	+	8.87987e15i	−0.0940531	+	0.0166463i
$232$		0			0
$233$			9.22753e17i			1.62150i	0.585395	+	0.810748i	$0.300939\pi$
$233$			9.22753e17i			1.62150i	−0.585395	+	0.810748i	$0.699061\pi$
$234$		0			0
$235$			9.84449e17i			1.62250i
$236$		0			0
$237$	5.52262e17	−	9.77437e16i	0.854147	−	0.151174i
$238$		0			0
$239$	1.07511e18			1.56124			0.780620	−	0.625006i	$-0.214903\pi$
$239$	1.07511e18			1.56124			0.780620	+	0.625006i	$0.214903\pi$
$240$		0			0
$241$	−2.39636e17			−0.326907			−0.163453	−	0.986551i	$-0.552263\pi$
$241$	−2.39636e17			−0.326907			−0.163453	+	0.986551i	$0.552263\pi$
$242$		0			0
$243$	−5.99053e17	−	4.99402e17i	−0.768100	−	0.640329i
$244$		0			0
$245$		−	1.00233e18i		−	1.20855i
$246$		0			0
$247$		−	8.60778e16i		−	0.0976482i
$248$		0			0
$249$	−2.57761e17	−	1.45638e18i	−0.275247	−	1.55517i
$250$		0			0
$251$	1.33508e18			1.34263			0.671314	−	0.741173i	$-0.265730\pi$
$251$	1.33508e18			1.34263			0.671314	+	0.741173i	$0.265730\pi$
$252$		0			0
$253$	−3.14335e17			−0.297844
$254$		0			0
$255$	−4.86785e17	−	2.75038e18i	−0.434796	−	2.45664i
$256$		0			0
$257$			2.84718e17i			0.239837i	0.992784	+	0.119918i	$0.0382633\pi$
$257$			2.84718e17i			0.239837i	−0.992784	+	0.119918i	$0.961737\pi$
$258$		0			0
$259$			2.53136e17i			0.201190i
$260$		0			0
$261$	5.46953e17	+	1.49677e18i	0.410342	+	1.12293i
$262$		0			0
$263$	−6.91491e17			−0.489912			−0.244956	−	0.969534i	$-0.578774\pi$
$263$	−6.91491e17			−0.489912			−0.244956	+	0.969534i	$0.578774\pi$
$264$		0			0
$265$	−2.11814e18			−1.41778
$266$		0			0
$267$	1.15667e18	−	2.04716e17i	0.731770	−	0.129514i
$268$		0			0
$269$			1.32968e18i			0.795437i	0.917508	+	0.397718i	$0.130198\pi$
$269$			1.32968e18i			0.795437i	−0.917508	+	0.397718i	$0.869802\pi$
$270$		0			0
$271$		−	1.14254e18i		−	0.646549i	−0.946305	−	0.323274i	$-0.895216\pi$
$271$		−	1.14254e18i		−	0.646549i	0.946305	−	0.323274i	$-0.104784\pi$
$272$		0			0
$273$	1.52925e17	−	2.70659e16i	0.0818953	−	0.0144945i
$274$		0			0
$275$	−4.78720e17			−0.242709
$276$		0			0
$277$	−2.39679e18			−1.15089			−0.575443	−	0.817842i	$-0.695170\pi$
$277$	−2.39679e18			−1.15089			−0.575443	+	0.817842i	$0.695170\pi$
$278$		0			0
$279$	−1.14663e18	−	3.13781e18i	−0.521663	−	1.42756i
$280$		0			0
$281$		−	5.48051e17i		−	0.236332i	−0.992994	−	0.118166i	$-0.962298\pi$
$281$		−	5.48051e17i		−	0.236332i	0.992994	−	0.118166i	$-0.0377016\pi$
$282$		0			0
$283$			3.91086e18i			1.59909i	0.600604	+	0.799547i	$0.294927\pi$
$283$			3.91086e18i			1.59909i	−0.600604	+	0.799547i	$0.705073\pi$
$284$		0			0
$285$	−2.18204e17	−	1.23288e18i	−0.0846304	−	0.478171i
$286$		0			0
$287$	2.06919e17			0.0761530
$288$		0			0
$289$	−7.11114e18			−2.48431
$290$		0			0
$291$	−4.54088e17	−	2.56565e18i	−0.150641	−	0.851137i
$292$		0			0
$293$		−	2.82414e18i		−	0.889978i	−0.895536	−	0.444989i	$-0.853208\pi$
$293$		−	2.82414e18i		−	0.889978i	0.895536	−	0.444989i	$-0.146792\pi$
$294$		0			0
$295$			1.22366e18i			0.366434i
$296$		0			0
$297$	5.43958e17	+	9.37993e17i	0.154843	+	0.267009i
$298$		0			0
$299$	9.58095e17			0.259343
$300$		0			0
$301$	−6.82033e17			−0.175613
$302$		0			0
$303$	5.87301e18	−	1.03945e18i	1.43894	−	0.254674i
$304$		0			0
$305$		−	5.60333e18i		−	1.30676i
$306$		0			0
$307$			6.34266e18i			1.40843i	0.709989	+	0.704213i	$0.248700\pi$
$307$			6.34266e18i			1.40843i	−0.709989	+	0.704213i	$0.751300\pi$
$308$		0			0
$309$	3.64915e18	−	6.45855e17i	0.771797	−	0.136599i
$310$		0			0
$311$	−7.91698e18			−1.59535			−0.797676	−	0.603086i	$-0.793938\pi$
$311$	−7.91698e18			−1.59535			−0.797676	+	0.603086i	$0.793938\pi$
$312$		0			0
$313$	1.58201e18			0.303827			0.151913	−	0.988394i	$-0.451457\pi$
$313$	1.58201e18			0.303827			0.151913	+	0.988394i	$0.451457\pi$
$314$		0			0
$315$	2.12171e18	−	7.75319e17i	0.388469	−	0.141955i
$316$		0			0
$317$			6.87775e18i			1.20089i	0.799668	+	0.600443i	$0.205009\pi$
$317$			6.87775e18i			1.20089i	−0.799668	+	0.600443i	$0.794991\pi$
$318$		0			0
$319$		−	2.21553e18i		−	0.369017i
$320$		0			0
$321$	1.04008e18	+	5.87654e18i	0.165302	+	0.933972i
$322$		0			0
$323$	−4.47071e18			−0.678200
$324$		0			0
$325$	1.45914e18			0.211335
$326$		0			0
$327$	−1.86025e18	−	1.05106e19i	−0.257314	−	1.45385i
$328$		0			0
$329$			2.84292e18i			0.375660i
$330$		0			0
$331$			1.06554e19i			1.34543i	0.739901	+	0.672716i	$0.234872\pi$
$331$			1.06554e19i			1.34543i	−0.739901	+	0.672716i	$0.765128\pi$
$332$		0			0
$333$	5.05974e18	−	1.84895e18i	0.610656	−	0.223148i
$334$		0			0
$335$	2.00608e19			2.31479
$336$		0			0
$337$	9.07898e18			1.00187			0.500937	−	0.865484i	$-0.332989\pi$
$337$	9.07898e18			1.00187			0.500937	+	0.865484i	$0.332989\pi$
$338$		0			0
$339$	−1.22991e19	+	2.17680e18i	−1.29831	+	0.229784i
$340$		0			0
$341$			4.64460e18i			0.469127i
$342$		0			0
$343$		−	6.09564e18i		−	0.589269i
$344$		0			0
$345$	1.37226e19	−	2.42874e18i	1.26997	−	0.224769i
$346$		0			0
$347$	−7.54972e18			−0.669052			−0.334526	−	0.942386i	$-0.608576\pi$
$347$	−7.54972e18			−0.669052			−0.334526	+	0.942386i	$0.608576\pi$
$348$		0			0
$349$	1.08687e19			0.922544			0.461272	−	0.887259i	$-0.347393\pi$
$349$	1.08687e19			0.922544			0.461272	+	0.887259i	$0.347393\pi$
$350$		0			0
$351$	−1.65799e18	−	2.85901e18i	−0.134827	−	0.232494i
$352$		0			0
$353$			2.68337e18i			0.209108i	0.994519	+	0.104554i	$0.0333415\pi$
$353$			2.68337e18i			0.209108i	−0.994519	+	0.104554i	$0.966658\pi$
$354$		0			0
$355$			2.85068e19i			2.12930i
$356$		0			0
$357$	−1.40575e18	−	7.94262e18i	−0.100669	−	0.568791i
$358$		0			0
$359$	1.47485e19			1.01284			0.506420	−	0.862287i	$-0.330969\pi$
$359$	1.47485e19			1.01284			0.506420	+	0.862287i	$0.330969\pi$
$360$		0			0
$361$	1.31771e19			0.867992
$362$		0			0
$363$	2.49497e18	+	1.40968e19i	0.157676	+	0.890884i
$364$		0			0
$365$		−	1.16294e19i		−	0.705280i
$366$		0			0
$367$		−	1.50154e18i		−	0.0874058i	−0.999045	−	0.0437029i	$-0.986085\pi$
$367$		−	1.50154e18i		−	0.0874058i	0.999045	−	0.0437029i	$-0.0139155\pi$
$368$		0			0
$369$	−1.51137e18	−	4.13595e18i	−0.0844643	−	0.231142i
$370$		0			0
$371$	−6.11680e18			−0.328262
$372$		0			0
$373$	−2.12055e19			−1.09303			−0.546515	−	0.837449i	$-0.684046\pi$
$373$	−2.12055e19			−1.09303			−0.546515	+	0.837449i	$0.684046\pi$
$374$		0			0
$375$	−5.67876e18	+	1.00507e18i	−0.281202	+	0.0497694i
$376$		0			0
$377$			6.75294e18i			0.321316i
$378$		0			0
$379$			1.63029e19i			0.745538i	0.927924	+	0.372769i	$0.121592\pi$
$379$			1.63029e19i			0.745538i	−0.927924	+	0.372769i	$0.878408\pi$
$380$		0			0
$381$	−5.63799e18	+	9.97857e17i	−0.247849	+	0.0438663i
$382$		0			0
$383$	−3.83078e19			−1.61918			−0.809591	−	0.586994i	$-0.800311\pi$
$383$	−3.83078e19			−1.61918			−0.809591	+	0.586994i	$0.800311\pi$
$384$		0			0
$385$	−3.14056e18			−0.127659
$386$		0			0
$387$	4.98167e18	+	1.36326e19i	0.194779	+	0.533025i
$388$		0			0
$389$			2.75023e19i			1.03454i	0.855823	+	0.517269i	$0.173051\pi$
$389$			2.75023e19i			1.03454i	−0.855823	+	0.517269i	$0.826949\pi$
$390$		0			0
$391$		−	4.97615e19i		−	1.80123i
$392$		0			0
$393$	7.36665e18	+	4.16223e19i	0.256641	+	1.45005i
$394$		0			0
$395$	3.45692e19			1.15934
$396$		0			0
$397$	−8.07740e18			−0.260821			−0.130411	−	0.991460i	$-0.541630\pi$
$397$	−8.07740e18			−0.260821			−0.130411	+	0.991460i	$0.541630\pi$
$398$		0			0
$399$	−6.30135e17	−	3.56033e18i	−0.0195946	−	0.110712i
$400$		0			0
$401$		−	3.96351e19i		−	1.18713i	−0.804787	−	0.593564i	$-0.797720\pi$
$401$		−	3.96351e19i		−	1.18713i	0.804787	−	0.593564i	$-0.202280\pi$
$402$		0			0
$403$		−	1.41568e19i		−	0.408485i
$404$		0			0
$405$	−3.09945e19	−	3.67461e19i	−0.861732	−	1.02164i
$406$		0			0
$407$	−7.48947e18			−0.200675
$408$		0			0
$409$	2.14027e19			0.552768			0.276384	−	0.961047i	$-0.410864\pi$
$409$	2.14027e19			0.552768			0.276384	+	0.961047i	$0.410864\pi$
$410$		0			0
$411$	2.37872e19	−	4.21004e18i	0.592283	−	0.104827i
$412$		0			0
$413$			3.53371e18i			0.0848410i
$414$		0			0
$415$		−	9.11630e19i		−	2.11085i
$416$		0			0
$417$	−5.41130e19	+	9.57735e18i	−1.20860	+	0.213907i
$418$		0			0
$419$	−2.09808e19			−0.452081			−0.226041	−	0.974118i	$-0.572578\pi$
$419$	−2.09808e19			−0.452081			−0.226041	+	0.974118i	$0.572578\pi$
$420$		0			0
$421$	−2.10248e19			−0.437136			−0.218568	−	0.975822i	$-0.570138\pi$
$421$	−2.10248e19			−0.437136			−0.218568	+	0.975822i	$0.570138\pi$
$422$		0			0
$423$	5.68249e19	−	2.07651e19i	1.14021	−	0.416659i
$424$		0			0
$425$		−	7.57848e19i		−	1.46779i
$426$		0			0
$427$		−	1.61814e19i		−	0.302557i
$428$		0			0
$429$	8.00791e17	+	4.52455e18i	0.0144574	+	0.0816857i
$430$		0			0
$431$	3.00534e19			0.523979			0.261989	−	0.965071i	$-0.415621\pi$
$431$	3.00534e19			0.523979			0.261989	+	0.965071i	$0.415621\pi$
$432$		0			0
$433$	−8.42776e19			−1.41923			−0.709615	−	0.704590i	$-0.751131\pi$
$433$	−8.42776e19			−1.41923			−0.709615	+	0.704590i	$0.751131\pi$
$434$		0			0
$435$	1.71185e19	+	9.67211e19i	0.278481	+	1.57345i
$436$		0			0
$437$		−	2.23059e19i		−	0.350598i
$438$		0			0
$439$		−	6.62509e19i		−	1.00625i	−0.864212	−	0.503127i	$-0.832183\pi$
$439$		−	6.62509e19i		−	1.00625i	0.864212	−	0.503127i	$-0.167817\pi$
$440$		0			0
$441$	−5.78570e19	+	2.11423e19i	−0.849311	+	0.310357i
$442$		0			0
$443$	−6.17664e19			−0.876444			−0.438222	−	0.898867i	$-0.644392\pi$
$443$	−6.17664e19			−0.876444			−0.438222	+	0.898867i	$0.644392\pi$
$444$		0			0
$445$	7.24025e19			0.993239
$446$		0			0
$447$	−6.86086e19	+	1.21429e19i	−0.910064	+	0.161070i
$448$		0			0
$449$			1.35538e20i			1.73865i	0.494241	+	0.869325i	$0.335446\pi$
$449$			1.35538e20i			1.73865i	−0.494241	+	0.869325i	$0.664554\pi$
$450$		0			0
$451$			6.12207e18i			0.0759581i
$452$		0			0
$453$	−6.93761e19	+	1.22787e19i	−0.832671	+	0.147373i
$454$		0			0
$455$	9.57246e18			0.111157
$456$		0			0
$457$	−1.22417e19			−0.137553			−0.0687765	−	0.997632i	$-0.521910\pi$
$457$	−1.22417e19			−0.137553			−0.0687765	+	0.997632i	$0.521910\pi$
$458$		0			0
$459$	−1.48491e20	+	8.61126e19i	−1.61475	+	0.936422i
$460$		0			0
$461$			6.28302e19i			0.661319i	0.943750	+	0.330660i	$0.107271\pi$
$461$			6.28302e19i			0.661319i	−0.943750	+	0.330660i	$0.892729\pi$
$462$		0			0
$463$		−	5.58526e19i		−	0.569097i	−0.958662	−	0.284548i	$-0.908156\pi$
$463$		−	5.58526e19i		−	0.569097i	0.958662	−	0.284548i	$-0.0918436\pi$
$464$		0			0
$465$	−3.58869e19	−	2.02765e20i	−0.354029	−	2.00030i
$466$		0			0
$467$	1.26782e20			1.21111			0.605553	−	0.795805i	$-0.292952\pi$
$467$	1.26782e20			1.21111			0.605553	+	0.795805i	$0.292952\pi$
$468$		0			0
$469$	5.79321e19			0.535948
$470$		0			0
$471$	2.43625e19	+	1.37651e20i	0.218306	+	1.23345i
$472$		0			0
$473$		−	2.01791e19i		−	0.175163i
$474$		0			0
$475$		−	3.39710e19i		−	0.285697i
$476$		0			0
$477$	4.46781e19	+	1.22264e20i	0.364088	+	0.996348i
$478$		0			0
$479$	1.41200e20			1.11511			0.557556	−	0.830139i	$-0.311739\pi$
$479$	1.41200e20			1.11511			0.557556	+	0.830139i	$0.311739\pi$
$480$		0			0
$481$	2.28279e19			0.174735
$482$		0			0
$483$	3.96285e19	−	7.01376e18i	0.294038	−	0.0520412i
$484$		0			0
$485$		−	1.60598e20i		−	1.15526i
$486$		0			0
$487$			3.60932e19i			0.251744i	0.992047	+	0.125872i	$0.0401728\pi$
$487$			3.60932e19i			0.251744i	−0.992047	+	0.125872i	$0.959827\pi$
$488$		0			0
$489$	1.15486e20	−	2.04396e19i	0.781109	−	0.138247i
$490$		0			0
$491$	−2.60429e20			−1.70836			−0.854179	−	0.519979i	$-0.825940\pi$
$491$	−2.60429e20			−1.70836			−0.854179	+	0.519979i	$0.825940\pi$
$492$		0			0
$493$	3.50734e20			2.23165
$494$		0			0
$495$	2.29392e19	+	6.27743e19i	0.141592	+	0.387474i
$496$		0			0
$497$			8.23227e19i			0.493000i
$498$		0			0
$499$		−	1.54438e20i		−	0.897431i	−0.893675	−	0.448715i	$-0.851882\pi$
$499$		−	1.54438e20i		−	0.897431i	0.893675	−	0.448715i	$-0.148118\pi$
$500$		0			0
$501$	1.67892e19	+	9.48608e19i	0.0946777	+	0.534939i
$502$		0			0
$503$	8.90819e19			0.487562			0.243781	−	0.969830i	$-0.421612\pi$
$503$	8.90819e19			0.487562			0.243781	+	0.969830i	$0.421612\pi$
$504$		0			0
$505$	3.67625e20			1.95308
$506$		0			0
$507$	3.13505e19	+	1.77134e20i	0.161691	+	0.913570i
$508$		0			0
$509$		−	3.52183e19i		−	0.176354i	−0.996105	−	0.0881769i	$-0.971896\pi$
$509$		−	3.52183e19i		−	0.176354i	0.996105	−	0.0881769i	$-0.0281041\pi$
$510$		0			0
$511$		−	3.35837e19i		−	0.163295i
$512$		0			0
$513$	−6.65621e19	+	3.86005e19i	−0.314302	+	0.182269i
$514$		0			0
$515$	2.28421e20			1.04757
$516$		0			0
$517$	−8.41126e19			−0.374698
$518$		0			0
$519$	2.40781e17	−	4.26154e16i	0.00104200	−	0.000184421i
$520$		0			0
$521$			4.03731e20i			1.69750i	0.528796	+	0.848749i	$0.322644\pi$
$521$			4.03731e20i			1.69750i	−0.528796	+	0.848749i	$0.677356\pi$
$522$		0			0
$523$		−	8.24552e19i		−	0.336865i	−0.985713	−	0.168432i	$-0.946130\pi$
$523$		−	8.24552e19i		−	0.336865i	0.985713	−	0.168432i	$-0.0538705\pi$
$524$		0			0
$525$	6.03526e19	−	1.06817e19i	0.239608	−	0.0424077i
$526$		0			0
$527$	−7.35275e20			−2.83707
$528$		0			0
$529$	−1.83577e19			−0.0688495
$530$		0			0
$531$	7.06327e19	−	2.58108e19i	0.257512	−	0.0941005i
$532$		0			0
$533$		−	1.86601e19i		−	0.0661393i
$534$		0			0
$535$			3.67846e20i			1.26769i
$536$		0			0
$537$	7.48493e18	+	4.22906e19i	0.0250831	+	0.141722i
$538$		0			0
$539$	8.56403e19			0.279102
$540$		0			0
$541$	−1.46914e18			−0.00465675			−0.00232838	−	0.999997i	$-0.500741\pi$
$541$	−1.46914e18			−0.00465675			−0.00232838	+	0.999997i	$0.500741\pi$
$542$		0			0
$543$	3.09223e19	+	1.74714e20i	0.0953395	+	0.538678i
$544$		0			0
$545$		−	6.57918e20i		−	1.97332i
$546$		0			0
$547$			2.66460e20i			0.777547i	0.921333	+	0.388773i	$0.127101\pi$
$547$			2.66460e20i			0.777547i	−0.921333	+	0.388773i	$0.872899\pi$
$548$		0			0
$549$	−3.23438e20	+	1.18192e20i	−0.918330	+	0.335579i
$550$		0			0
$551$	1.57219e20			0.434378
$552$		0			0
$553$	9.98299e19			0.268425
$554$		0			0
$555$	3.26960e20	−	5.78680e19i	0.855653	−	0.151440i
$556$		0			0
$557$		−	1.49699e20i		−	0.381333i	−0.981655	−	0.190666i	$-0.938935\pi$
$557$		−	1.49699e20i		−	0.381333i	0.981655	−	0.190666i	$-0.0610648\pi$
$558$		0			0
$559$			6.15061e19i			0.152521i
$560$		0			0
$561$	2.34996e20	−	4.15915e19i	0.567335	−	0.100411i
$562$		0			0
$563$	−2.49912e20			−0.587456			−0.293728	−	0.955889i	$-0.594896\pi$
$563$	−2.49912e20			−0.587456			−0.293728	+	0.955889i	$0.594896\pi$
$564$		0			0
$565$	−7.69873e20			−1.76220
$566$		0			0
$567$	−8.95067e19	−	1.06116e20i	−0.199518	−	0.236542i
$568$		0			0
$569$			1.43611e20i			0.311777i	0.987775	+	0.155889i	$0.0498241\pi$
$569$			1.43611e20i			0.311777i	−0.987775	+	0.155889i	$0.950176\pi$
$570$		0			0
$571$		−	5.22654e20i		−	1.10521i	−0.833444	−	0.552603i	$-0.813635\pi$
$571$		−	5.22654e20i		−	1.10521i	0.833444	−	0.552603i	$-0.186365\pi$
$572$		0			0
$573$	−3.82047e19	−	2.15861e20i	−0.0786969	−	0.444645i
$574$		0			0
$575$	3.78117e20			0.758782
$576$		0			0
$577$	−5.91872e20			−1.15720			−0.578601	−	0.815611i	$-0.696401\pi$
$577$	−5.91872e20			−1.15720			−0.578601	+	0.815611i	$0.696401\pi$
$578$		0			0
$579$	−1.50281e20	−	8.49101e20i	−0.286295	−	1.61760i
$580$		0			0
$581$		−	2.63263e20i		−	0.488729i
$582$		0			0
$583$		−	1.80976e20i		−	0.327422i
$584$		0			0
$585$	−6.99187e19	−	1.91337e20i	−0.123289	−	0.337387i
$586$		0			0
$587$	5.64727e20			0.970627			0.485314	−	0.874340i	$-0.338705\pi$
$587$	5.64727e20			0.970627			0.485314	+	0.874340i	$0.338705\pi$
$588$		0			0
$589$	−3.29591e20			−0.552219
$590$		0			0
$591$	−5.40739e20	+	9.57042e19i	−0.883245	+	0.156324i
$592$		0			0
$593$			1.01714e21i			1.61984i	0.586541	+	0.809919i	$0.300489\pi$
$593$			1.01714e21i			1.61984i	−0.586541	+	0.809919i	$0.699511\pi$
$594$		0			0
$595$		−	4.97174e20i		−	0.772025i
$596$		0			0
$597$	1.30852e20	−	2.31592e19i	0.198140	−	0.0350684i
$598$		0			0
$599$	1.30711e20			0.193024			0.0965122	−	0.995332i	$-0.469231\pi$
$599$	1.30711e20			0.193024			0.0965122	+	0.995332i	$0.469231\pi$
$600$		0			0
$601$	−3.46190e20			−0.498604			−0.249302	−	0.968426i	$-0.580201\pi$
$601$	−3.46190e20			−0.498604			−0.249302	+	0.968426i	$0.580201\pi$
$602$		0			0
$603$	−4.23145e20	−	1.15796e21i	−0.594442	−	1.62672i
$604$		0			0
$605$			8.82400e20i			1.20921i
$606$		0			0
$607$			1.03144e21i			1.37888i	0.724342	+	0.689440i	$0.242143\pi$
$607$			1.03144e21i			1.37888i	−0.724342	+	0.689440i	$0.757857\pi$
$608$		0			0
$609$	4.94351e19	+	2.79313e20i	0.0644772	+	0.364303i
$610$		0			0
$611$	2.56376e20			0.326263
$612$		0			0
$613$	−7.60952e19			−0.0944939			−0.0472469	−	0.998883i	$-0.515045\pi$
$613$	−7.60952e19			−0.0944939			−0.0472469	+	0.998883i	$0.515045\pi$
$614$		0			0
$615$	−4.73027e19	−	2.67265e20i	−0.0573221	−	0.323876i
$616$		0			0
$617$		−	6.93023e20i		−	0.819613i	−0.912173	−	0.409806i	$-0.865596\pi$
$617$		−	6.93023e20i		−	0.819613i	0.912173	−	0.409806i	$-0.134404\pi$
$618$		0			0
$619$		−	8.36199e20i		−	0.965228i	−0.875833	−	0.482614i	$-0.839687\pi$
$619$		−	8.36199e20i		−	0.965228i	0.875833	−	0.482614i	$-0.160313\pi$
$620$		0			0
$621$	−4.29645e20	−	7.40874e20i	−0.484087	−	0.834752i
$622$		0			0
$623$	2.09086e20			0.229966
$624$		0			0
$625$	−1.08780e21			−1.16801
$626$		0			0
$627$	1.05338e20	−	1.86436e19i	0.110428	−	0.0195445i
$628$		0			0
$629$		−	1.18564e21i		−	1.21359i
$630$		0			0
$631$			1.82218e21i			1.82125i	0.413231	+	0.910626i	$0.364400\pi$
$631$			1.82218e21i			1.82125i	−0.413231	+	0.910626i	$0.635600\pi$
$632$		0			0
$633$	1.34808e21	−	2.38594e20i	1.31579	−	0.232879i
$634$		0			0
$635$	−3.52914e20			−0.336408
$636$		0			0
$637$	−2.61032e20			−0.243024
$638$		0			0
$639$	1.64549e21	−	6.01298e20i	1.49637	−	0.546806i
$640$		0			0
$641$		−	1.29181e21i		−	1.14753i	−0.819020	−	0.573765i	$-0.805482\pi$
$641$		−	1.29181e21i		−	1.14753i	0.819020	−	0.573765i	$-0.194518\pi$
$642$		0			0
$643$			7.29523e20i			0.633077i	0.948580	+	0.316538i	$0.102521\pi$
$643$			7.29523e20i			0.633077i	−0.948580	+	0.316538i	$0.897479\pi$
$644$		0			0
$645$	1.55916e20	+	8.80940e20i	0.132188	+	0.746876i
$646$		0			0
$647$	−7.41481e20			−0.614211			−0.307106	−	0.951675i	$-0.599361\pi$
$647$	−7.41481e20			−0.614211			−0.307106	+	0.951675i	$0.599361\pi$
$648$		0			0
$649$	−1.04551e20			−0.0846238
$650$		0			0
$651$	−1.03635e20	−	5.85549e20i	−0.0819690	−	0.463133i
$652$		0			0
$653$			1.77635e21i			1.37303i	0.727116	+	0.686515i	$0.240860\pi$
$653$			1.77635e21i			1.37303i	−0.727116	+	0.686515i	$0.759140\pi$
$654$		0			0
$655$			2.60538e21i			1.96816i
$656$		0			0
$657$	−6.71279e20	+	2.45301e20i	−0.495636	+	0.181117i
$658$		0			0
$659$	−1.76695e21			−1.27522			−0.637608	−	0.770361i	$-0.720076\pi$
$659$	−1.76695e21			−1.27522			−0.637608	+	0.770361i	$0.720076\pi$
$660$		0			0
$661$	1.56633e21			1.10503			0.552514	−	0.833504i	$-0.313669\pi$
$661$	1.56633e21			1.10503			0.552514	+	0.833504i	$0.313669\pi$
$662$		0			0
$663$	−7.16270e20	+	1.26771e20i	−0.493998	+	0.0874317i
$664$		0			0
$665$		−	2.22861e20i		−	0.150270i
$666$		0			0
$667$			1.74993e21i			1.15366i
$668$		0			0
$669$	2.56877e21	−	4.54641e20i	1.65588	−	0.293071i
$670$		0			0
$671$	4.78755e20			0.301783
$672$		0			0
$673$	3.66276e20			0.225785			0.112892	−	0.993607i	$-0.463988\pi$
$673$	3.66276e20			0.225785			0.112892	+	0.993607i	$0.463988\pi$
$674$		0			0
$675$	−6.54333e20	−	1.12832e21i	−0.394476	−	0.680228i
$676$		0			0
$677$			7.07554e20i			0.417200i	0.978001	+	0.208600i	$0.0668907\pi$
$677$			7.07554e20i			0.417200i	−0.978001	+	0.208600i	$0.933109\pi$
$678$		0			0
$679$		−	4.63780e20i		−	0.267478i
$680$		0			0
$681$	4.44842e20	+	2.51340e21i	0.250959	+	1.41794i
$682$		0			0
$683$	1.58706e21			0.875869			0.437934	−	0.899007i	$-0.355710\pi$
$683$	1.58706e21			0.875869			0.437934	+	0.899007i	$0.355710\pi$
$684$		0			0
$685$	1.48898e21			0.803911
$686$		0			0
$687$	1.87964e20	+	1.06202e21i	0.0992888	+	0.560992i
$688$		0			0
$689$			5.51617e20i			0.285097i
$690$		0			0
$691$		−	5.17696e20i		−	0.261812i	−0.991395	−	0.130906i	$-0.958211\pi$
$691$		−	5.17696e20i		−	0.261812i	0.991395	−	0.130906i	$-0.0417886\pi$
$692$		0			0
$693$	6.62442e19	+	1.81281e20i	0.0327830	+	0.0897126i
$694$		0			0
$695$	−3.38724e21			−1.64044
$696$		0			0
$697$	−9.69168e20			−0.459360
$698$		0			0
$699$	3.44189e21	−	6.09173e20i	1.59668	−	0.282593i
$700$		0			0
$701$			1.19828e20i			0.0544095i	0.999630	+	0.0272048i	$0.00866061\pi$
$701$			1.19828e20i			0.0544095i	−0.999630	+	0.0272048i	$0.991339\pi$
$702$		0			0
$703$		−	5.31469e20i		−	0.236218i
$704$		0			0
$705$	3.67202e21	−	6.49903e20i	1.59767	−	0.282768i
$706$		0			0
$707$	1.06164e21			0.452200
$708$		0			0
$709$	3.33867e21			1.39228			0.696141	−	0.717905i	$-0.254899\pi$
$709$	3.33867e21			1.39228			0.696141	+	0.717905i	$0.254899\pi$
$710$		0			0
$711$	−7.29173e20	−	1.99542e21i	−0.297720	−	0.814729i
$712$		0			0
$713$		−	3.66854e21i		−	1.46663i
$714$		0			0
$715$			2.83218e20i			0.110873i
$716$		0			0
$717$	−7.09757e20	−	4.01020e21i	−0.272092	−	1.53735i
$718$		0			0
$719$	6.60481e20			0.247967			0.123983	−	0.992284i	$-0.460433\pi$
$719$	6.60481e20			0.247967			0.123983	+	0.992284i	$0.460433\pi$
$720$		0			0
$721$	6.59640e20			0.242545
$722$		0			0
$723$	1.58200e20	+	8.93848e20i	0.0569731	+	0.321904i
$724$		0			0
$725$			2.66508e21i			0.940102i
$726$		0			0
$727$			3.40066e21i			1.17505i	0.809207	+	0.587524i	$0.199897\pi$
$727$			3.40066e21i			1.17505i	−0.809207	+	0.587524i	$0.800103\pi$
$728$		0			0
$729$	−1.46731e21	+	2.56417e21i	−0.496666	+	0.867942i
$730$		0			0
$731$	3.19450e21			1.05931
$732$		0			0
$733$	−1.06235e21			−0.345135			−0.172568	−	0.984998i	$-0.555206\pi$
$733$	−1.06235e21			−0.345135			−0.172568	+	0.984998i	$0.555206\pi$
$734$		0			0
$735$	−3.73872e21	+	6.61708e20i	−1.19006	+	0.210626i
$736$		0			0
$737$			1.71402e21i			0.534576i
$738$		0			0
$739$		−	1.26887e21i		−	0.387779i	−0.981023	−	0.193890i	$-0.937890\pi$
$739$		−	1.26887e21i		−	0.387779i	0.981023	−	0.193890i	$-0.0621104\pi$
$740$		0			0
$741$	−3.21072e20	+	5.68259e19i	−0.0961538	+	0.0170181i
$742$		0			0
$743$	3.98374e21			1.16916			0.584581	−	0.811336i	$-0.301259\pi$
$743$	3.98374e21			1.16916			0.584581	+	0.811336i	$0.301259\pi$
$744$		0			0
$745$	−4.29460e21			−1.23524
$746$		0			0
$747$	−5.26216e21	+	1.92291e21i	−1.48340	+	0.542069i
$748$		0			0
$749$			1.06228e21i			0.293510i
$750$		0			0
$751$		−	1.62217e21i		−	0.439336i	−0.975575	−	0.219668i	$-0.929503\pi$
$751$		−	1.62217e21i		−	0.439336i	0.975575	−	0.219668i	$-0.0704973\pi$
$752$		0			0
$753$	−8.81381e20	−	4.97990e21i	−0.233992	−	1.32208i
$754$		0			0
$755$	−4.34265e21			−1.13019
$756$		0			0
$757$	−4.96207e21			−1.26603			−0.633015	−	0.774140i	$-0.718183\pi$
$757$	−4.96207e21			−1.26603			−0.633015	+	0.774140i	$0.718183\pi$
$758$		0			0
$759$	2.07514e20	+	1.17248e21i	0.0519081	+	0.293286i
$760$		0			0
$761$			8.21368e20i			0.201443i	0.994915	+	0.100722i	$0.0321152\pi$
$761$			8.21368e20i			0.201443i	−0.994915	+	0.100722i	$0.967885\pi$
$762$		0			0
$763$		−	1.89995e21i		−	0.456887i
$764$		0			0
$765$	−9.93764e21	+	3.63144e21i	−2.34327	+	0.856284i
$766$		0			0
$767$	3.18672e20			0.0736849
$768$		0			0
$769$	2.71911e21			0.616566			0.308283	−	0.951295i	$-0.400246\pi$
$769$	2.71911e21			0.616566			0.308283	+	0.951295i	$0.400246\pi$
$770$		0			0
$771$	1.06200e21	−	1.87962e20i	0.236167	−	0.0417986i
$772$		0			0
$773$		−	5.76469e21i		−	1.25727i	−0.777699	−	0.628636i	$-0.783613\pi$
$773$		−	5.76469e21i		−	1.25727i	0.777699	−	0.628636i	$-0.216387\pi$
$774$		0			0
$775$		−	5.58704e21i		−	1.19514i
$776$		0			0
$777$	9.44204e20	−	1.67113e20i	0.198111	−	0.0350632i
$778$		0			0
$779$	−4.34436e20			−0.0894117
$780$		0			0
$781$	−2.43566e21			−0.491738
$782$		0			0
$783$	5.22190e21	−	3.02827e21i	1.03423	−	0.599765i
$784$		0			0
$785$			8.61634e21i			1.67417i
$786$		0			0
$787$			1.95348e21i			0.372391i	0.982513	+	0.186196i	$0.0596158\pi$
$787$			1.95348e21i			0.372391i	−0.982513	+	0.186196i	$0.940384\pi$
$788$		0			0
$789$	4.56501e20	+	2.57928e21i	0.0853816	+	0.482415i
$790$		0			0
$791$	−2.22326e21			−0.408006
$792$		0			0
$793$	−1.45925e21			−0.262773
$794$		0			0
$795$	1.39833e21	+	7.90070e21i	0.247090	+	1.39609i
$796$		0			0
$797$		−	9.60790e21i		−	1.66606i	−0.553227	−	0.833031i	$-0.686604\pi$
$797$		−	9.60790e21i		−	1.66606i	0.553227	−	0.833031i	$-0.313396\pi$
$798$		0			0
$799$		−	1.33156e22i		−	2.26601i
$800$		0			0
$801$	−1.52719e21	−	4.17926e21i	−0.255065	−	0.697999i
$802$		0			0
$803$	9.93632e20			0.162877
$804$		0			0
$805$	2.48057e21			0.399101
$806$		0			0
$807$	4.95975e21	−	8.77815e20i	0.783264	−	0.138628i
$808$		0			0
$809$		−	6.95217e21i		−	1.07772i	−0.842395	−	0.538861i	$-0.818855\pi$
$809$		−	6.95217e21i		−	1.07772i	0.842395	−	0.538861i	$-0.181145\pi$
$810$		0			0
$811$		−	4.95049e21i		−	0.753341i	−0.926347	−	0.376671i	$-0.877069\pi$
$811$		−	4.95049e21i		−	0.753341i	0.926347	−	0.376671i	$-0.122931\pi$
$812$		0			0
$813$	−4.26170e21	+	7.54269e20i	−0.636654	+	0.112680i
$814$		0			0
$815$	7.22891e21			1.06021
$816$		0			0
$817$	1.43196e21			0.206188
$818$		0			0
$819$	−2.01913e20	−	5.52546e20i	−0.0285453	−	0.0781159i
$820$		0			0
$821$		−	4.90883e21i		−	0.681404i	−0.940171	−	0.340702i	$-0.889335\pi$
$821$		−	4.90883e21i		−	0.681404i	0.940171	−	0.340702i	$-0.110665\pi$
$822$		0			0
$823$		−	1.19170e22i		−	1.62430i	−0.583448	−	0.812151i	$-0.698297\pi$
$823$		−	1.19170e22i		−	1.62430i	0.583448	−	0.812151i	$-0.301703\pi$
$824$		0			0
$825$	3.16036e20	+	1.78564e21i	0.0422992	+	0.238995i
$826$		0			0
$827$	−1.34832e22			−1.77215			−0.886075	−	0.463541i	$-0.846579\pi$
$827$	−1.34832e22			−1.77215			−0.886075	+	0.463541i	$0.846579\pi$
$828$		0			0
$829$	3.23135e21			0.417086			0.208543	−	0.978013i	$-0.433128\pi$
$829$	3.23135e21			0.417086			0.208543	+	0.978013i	$0.433128\pi$
$830$		0			0
$831$	1.58229e21	+	8.94009e21i	0.200576	+	1.13327i
$832$		0			0
$833$			1.35575e22i			1.68788i
$834$		0			0
$835$			5.93788e21i			0.726078i
$836$		0			0
$837$	−1.09471e22	+	6.34842e21i	−1.31480	+	0.762474i
$838$		0			0
$839$	2.00134e21			0.236105			0.118053	−	0.993007i	$-0.462335\pi$
$839$	2.00134e21			0.236105			0.118053	+	0.993007i	$0.462335\pi$
$840$		0			0
$841$	−3.70487e21			−0.429342
$842$		0			0
$843$	−2.04424e21	+	3.61806e20i	−0.232716	+	0.0411879i
$844$		0			0
$845$			1.10878e22i			1.24000i
$846$		0			0
$847$			2.54822e21i			0.279969i
$848$		0			0
$849$	1.45876e22	−	2.58183e21i	1.57462	−	0.278689i
$850$		0			0
$851$	5.91556e21			0.627370
$852$		0			0
$853$	1.80434e22			1.88019			0.940093	−	0.340918i	$-0.110738\pi$
$853$	1.80434e22			1.88019			0.940093	+	0.340918i	$0.110738\pi$
$854$		0			0
$855$	−4.45461e21	+	1.62781e21i	−0.456104	+	0.166671i
$856$		0			0
$857$		−	9.02494e21i		−	0.908005i	−0.891001	−	0.454002i	$-0.849996\pi$
$857$		−	9.02494e21i		−	0.908005i	0.891001	−	0.454002i	$-0.150004\pi$
$858$		0			0
$859$			2.00161e21i			0.197894i	0.995093	+	0.0989468i	$0.0315474\pi$
$859$			2.00161e21i			0.197894i	−0.995093	+	0.0989468i	$0.968453\pi$
$860$		0			0
$861$	−1.36602e20	−	7.71815e20i	−0.0132719	−	0.0749876i
$862$		0			0
$863$	−1.87787e22			−1.79301			−0.896506	−	0.443031i	$-0.853903\pi$
$863$	−1.87787e22			−1.79301			−0.896506	+	0.443031i	$0.853903\pi$
$864$		0			0
$865$	1.50719e19			0.00141432
$866$		0			0
$867$	4.69456e21	+	2.65247e22i	0.432964	+	2.44629i
$868$		0			0
$869$			2.95364e21i			0.267737i
$870$		0			0
$871$		−	5.22434e21i		−	0.465474i
$872$		0			0
$873$	−9.27015e21	+	3.38752e21i	−0.811858	+	0.296671i
$874$		0			0
$875$	−1.02652e21			−0.0883707
$876$		0			0
$877$	9.78746e21			0.828272			0.414136	−	0.910215i	$-0.364084\pi$
$877$	9.78746e21			0.828272			0.414136	+	0.910215i	$0.364084\pi$
$878$		0			0
$879$	−1.05341e22	+	1.86441e21i	−0.876358	+	0.155105i
$880$		0			0
$881$		−	9.05471e21i		−	0.740551i	−0.928922	−	0.370276i	$-0.879263\pi$
$881$		−	9.05471e21i		−	0.740551i	0.928922	−	0.370276i	$-0.120737\pi$
$882$		0			0
$883$			1.31593e22i			1.05810i	0.848589	+	0.529052i	$0.177452\pi$
$883$			1.31593e22i			1.05810i	−0.848589	+	0.529052i	$0.822548\pi$
$884$		0			0
$885$	4.56428e21	−	8.07822e20i	0.360826	−	0.0638618i
$886$		0			0
$887$	1.51131e22			1.17470			0.587349	−	0.809333i	$-0.300171\pi$
$887$	1.51131e22			1.17470			0.587349	+	0.809333i	$0.300171\pi$
$888$		0			0
$889$	−1.01915e21			−0.0778891
$890$		0			0
$891$	3.13963e21	−	2.64821e21i	0.235937	−	0.199008i
$892$		0			0
$893$		−	5.96882e21i		−	0.441065i
$894$		0			0
$895$			2.64721e21i			0.192361i
$896$		0			0
$897$	−6.32505e20	−	3.57372e21i	−0.0451981	−	0.255374i
$898$		0			0
$899$	2.58570e22			1.81710
$900$		0			0
$901$	2.86499e22			1.98010
$902$		0			0
$903$	4.50257e20	+	2.54400e21i	0.0306057	+	0.172925i
$904$		0			0
$905$			1.09364e22i			0.731153i
$906$		0			0
$907$			1.17651e22i			0.773642i	0.922155	+	0.386821i	$0.126427\pi$
$907$			1.17651e22i			0.773642i	−0.922155	+	0.386821i	$0.873573\pi$
$908$		0			0
$909$	−7.75437e21	−	2.12203e22i	−0.501553	−	1.37253i
$910$		0			0
$911$	−1.36959e22			−0.871372			−0.435686	−	0.900099i	$-0.643494\pi$
$911$	−1.36959e22			−0.871372			−0.435686	+	0.900099i	$0.643494\pi$
$912$		0			0
$913$	7.78908e21			0.487478
$914$		0			0
$915$	−2.09006e22	+	3.69915e21i	−1.28677	+	0.227742i
$916$		0			0
$917$			7.52388e21i			0.455693i
$918$		0			0
$919$		−	2.78557e22i		−	1.65977i	−0.557937	−	0.829883i	$-0.688407\pi$
$919$		−	2.78557e22i		−	1.65977i	0.557937	−	0.829883i	$-0.311593\pi$
$920$		0			0
$921$	2.36583e22	−	4.18723e21i	1.38687	−	0.245460i
$922$		0			0
$923$	7.42390e21			0.428173
$924$		0			0
$925$	9.00916e21			0.511236
$926$		0			0
$927$	−4.81811e21	−	1.31850e22i	−0.269017	−	0.736179i
$928$		0			0
$929$			2.05970e22i			1.13158i	0.824549	+	0.565791i	$0.191429\pi$
$929$			2.05970e22i			1.13158i	−0.824549	+	0.565791i	$0.808571\pi$
$930$		0			0
$931$			6.07723e21i			0.328536i
$932$		0			0
$933$	5.22655e21	+	2.95305e22i	0.278037	+	1.57094i
$934$		0			0
$935$	1.47098e22			0.770049
$936$		0			0
$937$	−1.97549e22			−1.01772			−0.508860	−	0.860849i	$-0.669933\pi$
$937$	−1.97549e22			−1.01772			−0.508860	+	0.860849i	$0.669933\pi$
$938$		0			0
$939$	−1.04439e21	−	5.90092e21i	−0.0529507	−	0.299177i
$940$		0			0
$941$			2.70307e22i			1.34876i	0.738385	+	0.674380i	$0.235589\pi$
$941$			2.70307e22i			1.34876i	−0.738385	+	0.674380i	$0.764411\pi$
$942$		0			0
$943$		−	4.83552e21i		−	0.237468i
$944$		0			0
$945$	−4.29264e21	−	7.40217e21i	−0.207485	−	0.357784i
$946$		0			0
$947$	−3.69694e22			−1.75881			−0.879403	−	0.476078i	$-0.842058\pi$
$947$	−3.69694e22			−1.75881			−0.879403	+	0.476078i	$0.842058\pi$
$948$		0			0
$949$	−3.02860e21			−0.141822
$950$		0			0
$951$	2.56542e22	−	4.54048e21i	1.18251	−	0.209290i
$952$		0			0
$953$		−	2.08028e22i		−	0.943896i	−0.881626	−	0.471948i	$-0.843551\pi$
$953$		−	2.08028e22i		−	0.943896i	0.881626	−	0.471948i	$-0.156449\pi$
$954$		0			0
$955$		−	1.35120e22i		−	0.603522i
$956$		0			0
$957$	−8.26397e21	+	1.46262e21i	−0.363370	+	0.0643121i
$958$		0			0
$959$	4.29990e21			0.186131
$960$		0			0
$961$	−3.07410e22			−1.31006
$962$		0			0
$963$	2.12330e22	−	7.75902e21i	0.890870	−	0.325544i
$964$		0			0
$965$		−	5.31501e22i		−	2.19558i
$966$		0			0
$967$			7.67088e21i			0.311994i	0.987758	+	0.155997i	$0.0498590\pi$
$967$			7.67088e21i			0.311994i	−0.987758	+	0.155997i	$0.950141\pi$
$968$		0			0
$969$	2.95143e21	+	1.66759e22i	0.118196	+	0.667821i
$970$		0			0
$971$	−2.34132e22			−0.923244			−0.461622	−	0.887077i	$-0.652732\pi$
$971$	−2.34132e22			−0.923244			−0.461622	+	0.887077i	$0.652732\pi$
$972$		0			0
$973$	−9.78177e21			−0.379814
$974$		0			0
$975$	−9.63280e20	−	5.44263e21i	−0.0368314	−	0.208101i
$976$		0			0
$977$		−	2.67718e22i		−	1.00802i	−0.863698	−	0.504010i	$-0.831858\pi$
$977$		−	2.67718e22i		−	1.00802i	0.863698	−	0.504010i	$-0.168142\pi$
$978$		0			0
$979$			6.18616e21i			0.229378i
$980$		0			0
$981$	−3.79767e22	+	1.38775e22i	−1.38675	+	0.506752i
$982$		0			0
$983$	1.75557e21			0.0631345			0.0315673	−	0.999502i	$-0.489950\pi$
$983$	1.75557e21			0.0631345			0.0315673	+	0.999502i	$0.489950\pi$
$984$		0			0
$985$	−3.38479e22			−1.19884
$986$		0			0
$987$	1.06041e22	−	1.87681e21i	0.369911	−	0.0654698i
$988$		0			0
$989$			1.59385e22i			0.547614i
$990$		0			0
$991$			5.14963e22i			1.74270i	0.490659	+	0.871352i	$0.336756\pi$
$991$			5.14963e22i			1.74270i	−0.490659	+	0.871352i	$0.663244\pi$
$992$		0			0
$993$	3.97451e22	−	7.03440e21i	1.32484	−	0.234481i
$994$		0			0
$995$	8.19078e21			0.268938
$996$		0			0
$997$	−2.52853e22			−0.817812			−0.408906	−	0.912576i	$-0.634090\pi$
$997$	−2.52853e22			−0.817812			−0.408906	+	0.912576i	$0.634090\pi$
$998$		0			0
$999$	−1.02369e22	−	1.76524e22i	−0.326157	−	0.562421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.c.d.47.9	✓	20
3.2	odd	2	inner	48.16.c.d.47.11	yes	20
4.3	odd	2	inner	48.16.c.d.47.12	yes	20
12.11	even	2	inner	48.16.c.d.47.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.16.c.d.47.9	✓	20	1.1	even	1	trivial
48.16.c.d.47.10	yes	20	12.11	even	2	inner
48.16.c.d.47.11	yes	20	3.2	odd	2	inner
48.16.c.d.47.12	yes	20	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	48.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-660.169 - 3730.02i) q^{3} -233484. i q^{5} -674260. i q^{7} +(-1.34773e7 + 4.92490e6i) q^{9} +O(q^{10})\)	\(q+(-660.169 - 3730.02i) q^{3} -233484. i q^{5} -674260. i q^{7} +(-1.34773e7 + 4.92490e6i) q^{9} +1.99491e7 q^{11} -6.08051e7 q^{13} +(-8.70900e8 + 1.54139e8i) q^{15} +3.15809e9i q^{17} +1.41564e9i q^{19} +(-2.51500e9 + 4.45126e8i) q^{21} -1.57568e10 q^{23} -2.39970e10 q^{25} +(2.72673e10 + 4.70192e10i) q^{27} -1.11059e11i q^{29} +2.32822e11i q^{31} +(-1.31698e10 - 7.44108e10i) q^{33} -1.57429e11 q^{35} -3.75428e11 q^{37} +(4.01417e10 + 2.26804e11i) q^{39} +3.06884e11i q^{41} -1.01153e12i q^{43} +(1.14988e12 + 3.14672e12i) q^{45} -4.21635e12 q^{47} +4.29294e12 q^{49} +(1.17798e13 - 2.08488e12i) q^{51} -9.07188e12i q^{53} -4.65779e12i q^{55} +(5.28035e12 - 9.34559e11i) q^{57} -5.24088e12 q^{59} +2.39988e13 q^{61} +(3.32066e12 + 9.08717e12i) q^{63} +1.41970e13i q^{65} +8.59195e13i q^{67} +(1.04022e13 + 5.87733e13i) q^{69} -1.22093e14 q^{71} +4.98083e13 q^{73} +(1.58421e13 + 8.95094e13i) q^{75} -1.34509e13i q^{77} +1.48059e14i q^{79} +(1.57382e14 - 1.32748e14i) q^{81} +3.90447e14 q^{83} +7.37363e14 q^{85} +(-4.14252e14 + 7.33176e13i) q^{87} +3.10097e14i q^{89} +4.09984e13i q^{91} +(8.68433e14 - 1.53702e14i) q^{93} +3.30528e14 q^{95} +6.87836e14 q^{97} +(-2.68860e14 + 9.82474e13i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2271972 q^{9}+O(q^{10}) \) 20 * q + 2271972 * q^9	\( 20 q + 2271972 q^{9} + 318771128 q^{13} + 13145285784 q^{21} - 57334310012 q^{25} + 628079136192 q^{33} - 1811120039336 q^{37} + 7518335948928 q^{45} - 8329580497444 q^{49} - 36365149089912 q^{57} + 46120845287032 q^{61} - 117111587094144 q^{69} + 83221863805064 q^{73} + 73507522500468 q^{81} - 12\!\cdots\!52 q^{85}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100}) \) 20 * q + 2271972 * q^9 + 318771128 * q^13 + 13145285784 * q^21 - 57334310012 * q^25 + 628079136192 * q^33 - 1811120039336 * q^37 + 7518335948928 * q^45 - 8329580497444 * q^49 - 36365149089912 * q^57 + 46120845287032 * q^61 - 117111587094144 * q^69 + 83221863805064 * q^73 + 73507522500468 * q^81 - 1297131567063552 * q^85 + 1162806897994488 * q^93 - 1219036801794712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more