Embedded newform 429.2.bg.a.256.1

• Label: 429.2.bg.a.256.1
• Level: $429$
• Weight: $2$
• Character: 429.256
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			256.1
Character	\(\chi\)	\(=\)	429.256
Dual form			429.2.bg.a.367.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.273007 - 2.59749i) q^{2} +(0.978148 + 0.207912i) q^{3} +(-4.71612 + 1.00244i) q^{4} +(0.822674 + 0.597708i) q^{5} +(0.273007 - 2.59749i) q^{6} +(3.34488 - 0.710976i) q^{7} +(2.27719 + 7.00848i) q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.273007	−	2.59749i	−0.193045	−	1.83670i	−0.478235	−	0.878232i	\(-0.658723\pi\)
							0.285190	−	0.958471i	\(-0.407943\pi\)
\(3\)	0.978148	+	0.207912i	0.564734	+	0.120038i
							\(5\)	0.822674	+	0.597708i	0.367911	+	0.267303i	0.756344	−	0.654174i	\(-0.226984\pi\)
−0.388433	+	0.921477i	\(0.626984\pi\)
\(7\)	3.34488	−	0.710976i	1.26425	−	0.268724i	0.473459	−	0.880816i	\(-0.343005\pi\)
							0.790786	+	0.612092i	\(0.209672\pi\)
\(11\)	2.44251	−	2.24369i	0.736443	−	0.676499i
							\(13\)	−1.55895	−	3.25110i	−0.432375	−	0.901694i
\(17\)	−0.397343	+	3.78046i	−0.0963698	+	0.916897i								0.834367	+	0.551209i	\(0.185833\pi\)
							−0.930737	+	0.365689i	\(0.880834\pi\)
\(19\)	3.76063	−	4.17660i	0.862747	−	0.958178i	−0.136727	−	0.990609i	\(-0.543658\pi\)
							0.999474	+	0.0324312i	\(0.0103250\pi\)
\(23\)	−3.92245	+	6.79389i	−0.817888	+	1.41662i	0.0893480	+	0.996000i	\(0.471522\pi\)
							−0.907236	+	0.420623i	\(0.861812\pi\)
\(29\)	−4.06784	−	4.51779i	−0.755378	−	0.838933i	0.235754	−	0.971813i	\(-0.424244\pi\)
							−0.991132	+	0.132880i	\(0.957577\pi\)
\(31\)	−0.877444	+	0.637501i	−0.157594	+	0.114498i	−0.663787	−	0.747921i	\(-0.731052\pi\)
							0.506194	+	0.862420i	\(0.331052\pi\)
\(37\)	−1.58356	−	1.75872i	−0.260336	−	0.289132i	0.598780	−	0.800914i	\(-0.295652\pi\)
							−0.859116	+	0.511782i	\(0.828986\pi\)
\(41\)	4.91005	+	1.04366i	0.766822	+	0.162993i	0.574684	−	0.818375i	\(-0.305125\pi\)
							0.192137	+	0.981368i	\(0.438458\pi\)
\(43\)	5.40001	+	9.35309i	0.823493	+	1.42633i	0.903065	+	0.429503i	\(0.141311\pi\)
							−0.0795721	+	0.996829i	\(0.525355\pi\)
\(47\)	−0.225687	−	0.694594i	−0.0329199	−	0.101317i	0.933246	−	0.359237i	\(-0.116963\pi\)
							−0.966166	+	0.257920i	\(0.916963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.256.1	✓	112
11.4	even	5	inner	429.2.bg.a.334.14	yes	112
13.3	even	3	inner	429.2.bg.a.289.14	yes	112
143.81	even	15	inner	429.2.bg.a.367.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.256.1	✓	112	1.1	even	1	trivial
429.2.bg.a.289.14	yes	112	13.3	even	3	inner
429.2.bg.a.334.14	yes	112	11.4	even	5	inner
429.2.bg.a.367.1	yes	112	143.81	even	15	inner