Embedded newform 9.15.d.a.5.6

• Label: 9.15.d.a.5.6
• Level: $9$
• Weight: $15$
• Character: 9.5
• Analytic conductor: $11.190$
• Analytic rank: $0$
• Dimension: $26$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	9.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1896071337\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Relative dimension:	\(13\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	9.5
Dual form			9.15.d.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-40.5755 + 23.4263i) q^{2} +(-594.691 + 2104.59i) q^{3} +(-7094.42 + 12287.9i) q^{4} +(-113204. - 65358.5i) q^{5} +(-25172.9 - 99326.3i) q^{6} +(742609. + 1.28624e6i) q^{7} -1.43242e6i q^{8} +(-4.07565e6 - 2.50316e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 3 q^{2} - 2199 q^{3} + 98303 q^{4} - 107994 q^{5} + 24957 q^{6} - 146330 q^{7} - 5316525 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−40.5755	+	23.4263i	−0.316996	+	0.183018i	−0.650053	−	0.759889i	\(-0.725253\pi\)
							0.333057	+	0.942907i	\(0.391920\pi\)
\(3\)	−594.691	+	2104.59i	−0.271921	+	0.962320i
							\(5\)	−113204.	−	65358.5i	−1.44901	−	0.836589i	−0.450591	−	0.892730i	\(-0.648787\pi\)
−0.998423	+	0.0561414i	\(0.982120\pi\)
\(7\)	742609.	+	1.28624e6i	0.901725	+	1.56183i	0.825254	+	0.564762i	\(0.191032\pi\)
							0.0764707	+	0.997072i	\(0.475635\pi\)
\(11\)	8.68401e6	−	5.01371e6i	0.445627	−	0.257283i	−0.260355	−	0.965513i	\(-0.583839\pi\)
							0.705982	+	0.708230i	\(0.250506\pi\)
\(13\)	2.29172e7	−	3.96937e7i	0.365222	−	0.632584i	−0.623589	−	0.781752i	\(-0.714326\pi\)
							0.988812	+	0.149168i	\(0.0476596\pi\)
\(17\)		−	1.65730e7i		−	0.0403886i	−0.999796	−	0.0201943i	\(-0.993572\pi\)
							0.999796	−	0.0201943i	\(-0.00642848\pi\)
\(19\)	−5.21001e8			−0.582858			−0.291429	−	0.956592i	\(-0.594131\pi\)
							−0.291429	+	0.956592i	\(0.594131\pi\)
\(23\)	−4.97079e8	−	2.86989e8i	−0.145993	−	0.0842889i	0.425224	−	0.905088i	\(-0.360195\pi\)
							−0.571217	+	0.820799i	\(0.693529\pi\)
\(29\)	−1.62092e10	+	9.35841e9i	−0.939673	+	0.542520i	−0.889858	−	0.456238i	\(-0.849196\pi\)
							−0.0498149	+	0.998758i	\(0.515863\pi\)
\(31\)	5.39381e9	−	9.34235e9i	0.196049	−	0.339566i	−0.751195	−	0.660080i	\(-0.770522\pi\)
							0.947244	+	0.320514i	\(0.103856\pi\)
\(37\)	−4.41250e10			−0.464807			−0.232404	−	0.972619i	\(-0.574659\pi\)
							−0.232404	+	0.972619i	\(0.574659\pi\)
\(41\)	−1.90859e11	−	1.10192e11i	−0.979999	−	0.565802i	−0.0777289	−	0.996975i	\(-0.524767\pi\)
							−0.902270	+	0.431172i	\(0.858100\pi\)
\(43\)	−1.30820e11	−	2.26588e11i	−0.481279	−	0.833599i	0.518491	−	0.855083i	\(-0.326494\pi\)
							−0.999769	+	0.0214844i	\(0.993161\pi\)
\(47\)	1.53768e11	−	8.87781e10i	0.303516	−	0.175235i	−0.340505	−	0.940243i	\(-0.610598\pi\)
							0.644021	+	0.765008i	\(0.277265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.15.d.a.5.6	yes	26
3.2	odd	2		27.15.d.a.17.8		26
9.2	odd	6	inner	9.15.d.a.2.6	✓	26
9.4	even	3		81.15.b.a.80.11		26
9.5	odd	6		81.15.b.a.80.16		26
9.7	even	3		27.15.d.a.8.8		26

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.15.d.a.2.6	✓	26	9.2	odd	6	inner
9.15.d.a.5.6	yes	26	1.1	even	1	trivial
27.15.d.a.8.8		26	9.7	even	3
27.15.d.a.17.8		26	3.2	odd	2
81.15.b.a.80.11		26	9.4	even	3
81.15.b.a.80.16		26	9.5	odd	6