Embedded newform 429.2.bg.a.256.4

• Label: 429.2.bg.a.256.4
• 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			256.4
Character	\(\chi\)	\(=\)	429.256
Dual form			429.2.bg.a.367.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.140589 - 1.33762i) q^{2} +(0.978148 + 0.207912i) q^{3} +(0.186846 - 0.0397154i) q^{4} +(-0.768940 - 0.558668i) q^{5} +(0.140589 - 1.33762i) q^{6} +(3.61727 - 0.768874i) q^{7} +(-0.910638 - 2.80265i) q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.140589	−	1.33762i	−0.0994115	−	0.945837i	−0.924590	−	0.380964i	\(-0.875592\pi\)
							0.825178	−	0.564872i	\(-0.191075\pi\)
\(3\)	0.978148	+	0.207912i	0.564734	+	0.120038i
							\(5\)	−0.768940	−	0.558668i	−0.343881	−	0.249844i	0.402417	−	0.915457i	\(-0.368170\pi\)
−0.746297	+	0.665613i	\(0.768170\pi\)
\(7\)	3.61727	−	0.768874i	1.36720	−	0.290607i	0.534891	−	0.844921i	\(-0.320353\pi\)
							0.832307	+	0.554314i	\(0.187019\pi\)
\(11\)	2.64068	+	2.00669i	0.796196	+	0.605039i
							\(13\)	−1.40696	+	3.31971i	−0.390221	+	0.920721i
\(17\)	0.370960	−	3.52945i	0.0899710	−	0.856017i								−0.852724	−	0.522361i	\(-0.825051\pi\)
							0.942695	−	0.333655i	\(-0.108282\pi\)
\(19\)	−3.69186	+	4.10022i	−0.846970	+	0.940656i	−0.998860	−	0.0477418i	\(-0.984798\pi\)
							0.151890	+	0.988397i	\(0.451464\pi\)
\(23\)	−0.0444137	+	0.0769268i	−0.00926089	+	0.0160403i	−0.870619	−	0.491958i	\(-0.836281\pi\)
							0.861358	+	0.507999i	\(0.169615\pi\)
\(29\)	−4.92316	−	5.46772i	−0.914207	−	1.01533i	−0.999819	−	0.0190016i	\(-0.993951\pi\)
							0.0856121	−	0.996329i	\(-0.472715\pi\)
\(31\)	4.82506	−	3.50561i	0.866606	−	0.629626i	−0.0630678	−	0.998009i	\(-0.520088\pi\)
							0.929674	+	0.368383i	\(0.120088\pi\)
\(37\)	5.01419	+	5.56883i	0.824328	+	0.915509i	0.997591	−	0.0693664i	\(-0.0220977\pi\)
							−0.173263	+	0.984876i	\(0.555431\pi\)
\(41\)	−5.46233	−	1.16105i	−0.853072	−	0.181326i	−0.239433	−	0.970913i	\(-0.576962\pi\)
							−0.613639	+	0.789587i	\(0.710295\pi\)
\(43\)	1.43869	+	2.49189i	0.219399	+	0.380010i	0.954624	−	0.297813i	\(-0.0962570\pi\)
							−0.735225	+	0.677823i	\(0.762924\pi\)
\(47\)	1.12851	+	3.47319i	0.164610	+	0.506617i	0.999007	−	0.0445461i	\(-0.0141841\pi\)
							−0.834397	+	0.551163i	\(0.814184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.256.4	✓	112
11.4	even	5	inner	429.2.bg.a.334.11	yes	112
13.3	even	3	inner	429.2.bg.a.289.11	yes	112
143.81	even	15	inner	429.2.bg.a.367.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.256.4	✓	112	1.1	even	1	trivial
429.2.bg.a.289.11	yes	112	13.3	even	3	inner
429.2.bg.a.334.11	yes	112	11.4	even	5	inner
429.2.bg.a.367.4	yes	112	143.81	even	15	inner