Embedded newform 429.2.bg.a.256.2

• Label: 429.2.bg.a.256.2
• 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			256.2
Character	\(\chi\)	\(=\)	429.256
Dual form			429.2.bg.a.367.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.258546 - 2.45990i) q^{2} +(0.978148 + 0.207912i) q^{3} +(-4.02797 + 0.856172i) q^{4} +(-0.872789 - 0.634118i) q^{5} +(0.258546 - 2.45990i) q^{6} +(-3.11026 + 0.661107i) q^{7} +(1.61884 + 4.98227i) q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} + 20 q^{4} + 6 q^{5} + 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.258546	−	2.45990i	−0.182820	−	1.73941i	−0.573784	−	0.819007i	\(-0.694525\pi\)
							0.390964	−	0.920406i	\(-0.372142\pi\)
\(3\)	0.978148	+	0.207912i	0.564734	+	0.120038i
							\(5\)	−0.872789	−	0.634118i	−0.390323	−	0.283586i	0.375265	−	0.926918i	\(-0.377552\pi\)
−0.765588	+	0.643331i	\(0.777552\pi\)
\(7\)	−3.11026	+	0.661107i	−1.17557	+	0.249875i	−0.753971	−	0.656908i	\(-0.771864\pi\)
							−0.421598	+	0.906783i	\(0.638531\pi\)
\(11\)	−2.99246	−	1.43009i	−0.902262	−	0.431189i
							\(13\)	−2.31690	+	2.76261i	−0.642591	+	0.766209i
\(17\)	0.547666	−	5.21070i	0.132829	−	1.26378i								−0.701563	−	0.712608i	\(-0.747514\pi\)
							0.834392	−	0.551172i	\(-0.185819\pi\)
\(19\)	−2.58415	+	2.86999i	−0.592845	+	0.658421i	−0.962669	−	0.270681i	\(-0.912751\pi\)
							0.369824	+	0.929102i	\(0.379418\pi\)
\(23\)	−1.30426	+	2.25905i	−0.271957	+	0.471044i	−0.969363	−	0.245633i	\(-0.921004\pi\)
							0.697406	+	0.716677i	\(0.254338\pi\)
\(29\)	3.42717	+	3.80626i	0.636410	+	0.706805i	0.971941	−	0.235224i	\(-0.0755825\pi\)
							−0.335531	+	0.942029i	\(0.608916\pi\)
\(31\)	3.91380	−	2.84354i	0.702939	−	0.510715i	−0.177949	−	0.984040i	\(-0.556946\pi\)
							0.880888	+	0.473324i	\(0.156946\pi\)
\(37\)	−6.19878	−	6.88445i	−1.01907	−	1.13180i	−0.991225	−	0.132184i	\(-0.957801\pi\)
							−0.0278486	−	0.999612i	\(-0.508866\pi\)
\(41\)	−10.4560	−	2.22250i	−1.63296	−	0.347096i	−0.701989	−	0.712188i	\(-0.747704\pi\)
							−0.930972	+	0.365092i	\(0.881038\pi\)
\(43\)	−1.23783	−	2.14399i	−0.188768	−	0.326956i	0.756072	−	0.654489i	\(-0.227116\pi\)
							−0.944840	+	0.327533i	\(0.893783\pi\)
\(47\)	1.00226	+	3.08463i	0.146194	+	0.449939i	0.997163	−	0.0752775i	\(-0.0239843\pi\)
							−0.850968	+	0.525217i	\(0.823984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.a.256.2	✓	112
11.4	even	5	inner	429.2.bg.a.334.13	yes	112
13.3	even	3	inner	429.2.bg.a.289.13	yes	112
143.81	even	15	inner	429.2.bg.a.367.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.a.256.2	✓	112	1.1	even	1	trivial
429.2.bg.a.289.13	yes	112	13.3	even	3	inner
429.2.bg.a.334.13	yes	112	11.4	even	5	inner
429.2.bg.a.367.2	yes	112	143.81	even	15	inner