Embedded newform 29.11.f.a.2.14

• Label: 29.11.f.a.2.14
• Level: $29$
• Weight: $11$
• Character: 29.2
• Analytic conductor: $18.425$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.4253603275\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(24\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			2.14
Character	\(\chi\)	\(=\)	29.2
Dual form			29.11.f.a.15.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.3156 + 1.38763i) q^{2} +(-317.080 - 199.234i) q^{3} +(-848.579 - 193.683i) q^{4} +(-3318.67 + 2646.55i) q^{5} +(-3628.55 - 2893.67i) q^{6} +(-7010.74 - 30716.1i) q^{7} +(-22160.7 - 7754.37i) q^{8} +(35224.8 + 73145.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 22 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 76784 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.3156	+	1.38763i	0.384861	+	0.0433634i	0.302276	−	0.953221i	\(-0.402254\pi\)
							0.0825855	+	0.996584i	\(0.473682\pi\)
\(3\)	−317.080	−	199.234i	−1.30485	−	0.819895i	−0.313309	−	0.949651i	\(-0.601438\pi\)
							−0.991546	+	0.129757i	\(0.958580\pi\)
\(5\)	−3318.67	+	2646.55i	−1.06198	+	0.846897i	−0.988624	−	0.150409i	\(-0.951941\pi\)
							−0.0733518	+	0.997306i	\(0.523370\pi\)
\(7\)	−7010.74	−	30716.1i	−0.417132	−	1.82758i	−0.548385	−	0.836226i	\(-0.684757\pi\)
							0.131253	−	0.991349i	\(-0.458100\pi\)
\(11\)	−223241.	+	78115.3i	−1.38615	+	0.485035i	−0.917284	−	0.398235i	\(-0.869623\pi\)
							−0.468866	+	0.883269i	\(0.655337\pi\)
\(13\)	38910.3	−	80798.0i	0.104797	−	0.217612i	−0.841976	−	0.539514i	\(-0.818608\pi\)
							0.946773	+	0.321902i	\(0.104322\pi\)
\(17\)	419705.	−	419705.i	0.295597	−	0.295597i	−0.543690	−	0.839286i	\(-0.682973\pi\)
							0.839286	+	0.543690i	\(0.182973\pi\)
\(19\)	−224534.	−	357344.i	−0.0906806	−	0.144317i	0.798304	−	0.602255i	\(-0.205731\pi\)
							−0.888984	+	0.457938i	\(0.848588\pi\)
\(23\)	5.76300e6	−	7.22657e6i	0.895384	−	1.12278i	−0.0964623	−	0.995337i	\(-0.530753\pi\)
							0.991846	−	0.127439i	\(-0.0406758\pi\)
\(29\)	−1.98990e7	−	4.97344e6i	−0.970158	−	0.242475i
							\(31\)	−4.06063e7	−	4.57524e6i	−1.41836	−	0.159810i	−0.630749	−	0.775987i	\(-0.717252\pi\)
−0.787608	+	0.616177i	\(0.788681\pi\)
\(37\)	−7.49963e7	−	2.62423e7i	−1.08151	−	0.378437i	−0.270052	−	0.962846i	\(-0.587041\pi\)
							−0.811459	+	0.584409i	\(0.801326\pi\)
\(41\)	−5.18551e7	−	5.18551e7i	−0.447581	−	0.447581i	0.446968	−	0.894550i	\(-0.352504\pi\)
							−0.894550	+	0.446968i	\(0.852504\pi\)
\(43\)	8.41849e6	+	7.47161e7i	0.0572653	+	0.508244i	0.989302	+	0.145880i	\(0.0466015\pi\)
							−0.932037	+	0.362363i	\(0.881970\pi\)
\(47\)	−3.64946e7	−	1.04296e8i	−0.159125	−	0.454754i	0.836524	−	0.547930i	\(-0.184584\pi\)
							−0.995650	+	0.0931756i	\(0.970298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.f.a.2.14	✓	288
29.15	odd	28	inner	29.11.f.a.15.14	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.f.a.2.14	✓	288	1.1	even	1	trivial
29.11.f.a.15.14	yes	288	29.15	odd	28	inner