Embedded newform 29.6.e.a.13.6

• Label: 29.6.e.a.13.6
• Level: $29$
• Weight: $6$
• Character: 29.13
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	29.13
Dual form			29.6.e.a.9.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.242586 + 0.503734i) q^{2} +(17.2862 + 13.7853i) q^{3} +(19.7568 + 24.7742i) q^{4} +(-42.2101 - 20.3273i) q^{5} +(-11.1375 + 5.36353i) q^{6} +(-31.6252 + 39.6568i) q^{7} +(-34.7151 + 7.92348i) q^{8} +(54.7057 + 239.682i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.242586	+	0.503734i	−0.0428835	+	0.0890485i	−0.921310	−	0.388829i	\(-0.872880\pi\)
							0.878426	+	0.477878i	\(0.158594\pi\)
\(3\)	17.2862	+	13.7853i	1.10891	+	0.884325i	0.994036	−	0.109055i	\(-0.0347825\pi\)
							0.114873	+	0.993380i	\(0.463354\pi\)
\(5\)	−42.2101	−	20.3273i	−0.755076	−	0.363626i	0.0164151	−	0.999865i	\(-0.494775\pi\)
							−0.771492	+	0.636240i	\(0.780489\pi\)
\(7\)	−31.6252	+	39.6568i	−0.243943	+	0.305895i	−0.888697	−	0.458495i	\(-0.848389\pi\)
							0.644754	+	0.764391i	\(0.276960\pi\)
\(11\)	491.355	+	112.149i	1.22437	+	0.279455i	0.785368	−	0.619030i	\(-0.212474\pi\)
							0.439005	+	0.898485i	\(0.355331\pi\)
\(13\)	−6.25292	+	27.3958i	−0.0102618	+	0.0449600i	−0.979800	−	0.199982i	\(-0.935912\pi\)
							0.969538	+	0.244942i	\(0.0787688\pi\)
\(17\)		−	1871.48i		−	1.57059i	−0.619124	−	0.785293i	\(-0.712512\pi\)
							0.619124	−	0.785293i	\(-0.287488\pi\)
\(19\)	−2.66372	+	2.12425i	−0.00169280	+	0.00134996i	−0.624336	−	0.781156i	\(-0.714630\pi\)
							0.622643	+	0.782506i	\(0.286059\pi\)
\(23\)	3134.61	−	1509.55i	1.23556	−	0.595015i	0.301957	−	0.953322i	\(-0.402360\pi\)
							0.933604	+	0.358307i	\(0.116646\pi\)
\(29\)	−2034.27	+	4046.34i	−0.449173	+	0.893445i
							\(31\)	2523.58	−	5240.27i	0.471642	−	0.979376i	−0.520453	−	0.853890i	\(-0.674237\pi\)
0.992095	−	0.125485i	\(-0.0400488\pi\)
\(37\)	−6870.47	+	1568.14i	−0.825053	+	0.188313i	−0.614145	−	0.789193i	\(-0.710499\pi\)
							−0.210908	+	0.977506i	\(0.567642\pi\)
\(41\)		−	4086.90i		−	0.379695i	−0.981814	−	0.189847i	\(-0.939201\pi\)
							0.981814	−	0.189847i	\(-0.0607993\pi\)
\(43\)	63.9798	+	132.855i	0.00527681	+	0.0109574i	0.903591	−	0.428397i	\(-0.140922\pi\)
							−0.898314	+	0.439354i	\(0.855207\pi\)
\(47\)	10132.7	+	2312.73i	0.669086	+	0.152714i	0.543553	−	0.839375i	\(-0.317079\pi\)
							0.125533	+	0.992089i	\(0.459936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.13.6	yes	72
29.3	odd	28		841.6.a.l.1.34		72
29.9	even	14	inner	29.6.e.a.9.6	✓	72
29.26	odd	28		841.6.a.l.1.39		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.9.6	✓	72	29.9	even	14	inner
29.6.e.a.13.6	yes	72	1.1	even	1	trivial
841.6.a.l.1.34		72	29.3	odd	28
841.6.a.l.1.39		72	29.26	odd	28