Embedded newform 429.2.bn.b.49.4

• Label: 429.2.bn.b.49.4
• Level: $429$
• Weight: $2$
• Character: 429.49
• 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.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			49.4
Character	\(\chi\)	\(=\)	429.49
Dual form			429.2.bn.b.394.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.977278 - 0.879945i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(-0.0282879 - 0.269141i) q^{4} +(1.70562 + 0.554188i) q^{5} +(0.977278 - 0.879945i) q^{6} +(2.04084 - 0.214501i) q^{7} +(-1.75513 + 2.41573i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} - 8 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.977278	−	0.879945i	−0.691040	−	0.622215i	0.246888	−	0.969044i	\(-0.420592\pi\)
							−0.937929	+	0.346829i	\(0.887259\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	1.70562	+	0.554188i	0.762775	+	0.247841i	0.664469	−	0.747316i	\(-0.268658\pi\)
0.0983058	+	0.995156i	\(0.468658\pi\)
\(7\)	2.04084	−	0.214501i	0.771367	−	0.0810739i	0.289335	−	0.957228i	\(-0.406566\pi\)
							0.482031	+	0.876154i	\(0.339899\pi\)
\(11\)	3.26701	+	0.571551i	0.985039	+	0.172329i
							\(13\)	−3.27004	−	1.51884i	−0.906945	−	0.421250i
\(17\)	3.68280	+	4.09017i	0.893211	+	0.992011i								0.999997	−	0.00227079i	\(-0.000722815\pi\)
							−0.106787	+	0.994282i	\(0.534056\pi\)
\(19\)	1.56718	+	3.51994i	0.359536	+	0.807531i	0.999243	+	0.0389117i	\(0.0123891\pi\)
							−0.639707	+	0.768619i	\(0.720944\pi\)
\(23\)	−1.71865	−	2.97679i	−0.358364	−	0.620704i	0.629324	−	0.777143i	\(-0.283332\pi\)
							−0.987688	+	0.156439i	\(0.949999\pi\)
\(29\)	7.01413	+	3.12289i	1.30249	+	0.579906i	0.936485	−	0.350707i	\(-0.114059\pi\)
							0.366005	+	0.930613i	\(0.380725\pi\)
\(31\)	1.90464	−	0.618855i	0.342083	−	0.111150i	−0.132937	−	0.991125i	\(-0.542441\pi\)
							0.475020	+	0.879975i	\(0.342441\pi\)
\(37\)	2.41128	−	5.41582i	0.396412	−	0.890356i	−0.599530	−	0.800353i	\(-0.704646\pi\)
							0.995941	−	0.0900031i	\(-0.0286877\pi\)
\(41\)	8.21500	+	0.863432i	1.28297	+	0.134845i	0.721361	−	0.692559i	\(-0.243517\pi\)
							0.561607	+	0.827404i	\(0.310183\pi\)
\(43\)	3.86747	−	6.69866i	0.589784	−	1.02154i	−0.404476	−	0.914549i	\(-0.632546\pi\)
							0.994260	−	0.106988i	\(-0.0341205\pi\)
\(47\)	0.729060	−	1.00347i	0.106344	−	0.146370i	−0.752528	−	0.658561i	\(-0.771166\pi\)
							0.858872	+	0.512190i	\(0.171166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.b.49.4	✓	112
11.9	even	5	inner	429.2.bn.b.361.11	yes	112
13.4	even	6	inner	429.2.bn.b.82.11	yes	112
143.108	even	30	inner	429.2.bn.b.394.4	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.b.49.4	✓	112	1.1	even	1	trivial
429.2.bn.b.82.11	yes	112	13.4	even	6	inner
429.2.bn.b.361.11	yes	112	11.9	even	5	inner
429.2.bn.b.394.4	yes	112	143.108	even	30	inner