Embedded newform 18.12.c.b.7.2

• Label: 18.12.c.b.7.2
• Level: $18$
• Weight: $12$
• Character: 18.7
• Analytic conductor: $13.830$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.8301772501\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 6*x^11 + 142057*x^10 - 710230*x^9 + 7490763955*x^8 - 29958794506*x^7 + 178413594963643*x^6 - 535135932093058*x^5 + 1856017130395912051*x^4 - 3711142402522175502*x^3 + 7014772917042471764113*x^2 - 7012917435025539772518*x + 1996943090208352847168412
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{14}\cdot 3^{25}\cdot 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.2
Root			\(0.500000 - 193.579i\) of defining polynomial
Character	\(\chi\)	\(=\)	18.7
Dual form			18.12.c.b.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 - 27.7128i) q^{2} +(-328.215 - 263.481i) q^{3} +(-512.000 - 886.810i) q^{4} +(4650.37 + 8054.68i) q^{5} +(-12553.2 + 4880.06i) q^{6} +(-18776.9 + 32522.5i) q^{7} -32768.0 q^{8} +(38302.8 + 172957. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 192 q^{2} - 6144 q^{4} + 4104 q^{5} - 22080 q^{6} - 61554 q^{7} - 393216 q^{8} + 233022 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	16.0000	−	27.7128i	0.353553	−	0.612372i
							\(3\)	−328.215	−	263.481i	−0.779814	−	0.626011i
\(5\)	4650.37	+	8054.68i	0.665507	+	1.15269i								0.979148	+	0.203150i	\(0.0651180\pi\)
							−0.313640	+	0.949542i	\(0.601549\pi\)
\(7\)	−18776.9	+	32522.5i	−0.422264	+	0.731383i	−0.996161	−	0.0875446i	\(-0.972098\pi\)
							0.573896	+	0.818928i	\(0.305431\pi\)
\(11\)	481426.	−	833854.i	0.901300	−	1.56110i	0.0754919	−	0.997146i	\(-0.475947\pi\)
							0.825808	−	0.563951i	\(-0.190719\pi\)
\(13\)	863297.	+	1.49527e6i	0.644870	+	1.11695i	0.984332	+	0.176327i	\(0.0564217\pi\)
							−0.339462	+	0.940620i	\(0.610245\pi\)
\(17\)	−1.55638e6			−0.265855			−0.132928	−	0.991126i	\(-0.542438\pi\)
							−0.132928	+	0.991126i	\(0.542438\pi\)
\(19\)	7.71268e6			0.714596			0.357298	−	0.933990i	\(-0.383698\pi\)
							0.357298	+	0.933990i	\(0.383698\pi\)
\(23\)	2.96272e7	+	5.13157e7i	0.959813	+	1.66245i	0.722947	+	0.690903i	\(0.242787\pi\)
							0.236866	+	0.971542i	\(0.423880\pi\)
\(29\)	−2.23618e7	+	3.87318e7i	−0.202450	+	0.350654i	−0.949317	−	0.314319i	\(-0.898224\pi\)
							0.746867	+	0.664973i	\(0.231557\pi\)
\(31\)	2.36158e7	+	4.09037e7i	0.148154	+	0.256610i	0.930545	−	0.366177i	\(-0.119334\pi\)
							−0.782391	+	0.622787i	\(0.786000\pi\)
\(37\)	−3.98988e8			−0.945910			−0.472955	−	0.881087i	\(-0.656813\pi\)
							−0.472955	+	0.881087i	\(0.656813\pi\)
\(41\)	5.26939e8	+	9.12685e8i	0.710312	+	1.23030i	0.964740	+	0.263204i	\(0.0847794\pi\)
							−0.254428	+	0.967092i	\(0.581887\pi\)
\(43\)	1.75614e8	−	3.04173e8i	0.182173	−	0.315532i	−0.760448	−	0.649399i	\(-0.775020\pi\)
							0.942620	+	0.333867i	\(0.108354\pi\)
\(47\)	−8.72375e8	+	1.51100e9i	−0.554836	+	0.961005i	0.443080	+	0.896482i	\(0.353886\pi\)
							−0.997916	+	0.0645227i	\(0.979448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.12.c.b.7.2	✓	12
3.2	odd	2		54.12.c.b.19.2		12
9.2	odd	6		162.12.a.i.1.5		6
9.4	even	3	inner	18.12.c.b.13.2	yes	12
9.5	odd	6		54.12.c.b.37.2		12
9.7	even	3		162.12.a.h.1.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.12.c.b.7.2	✓	12	1.1	even	1	trivial
18.12.c.b.13.2	yes	12	9.4	even	3	inner
54.12.c.b.19.2		12	3.2	odd	2
54.12.c.b.37.2		12	9.5	odd	6
162.12.a.h.1.2		6	9.7	even	3
162.12.a.i.1.5		6	9.2	odd	6