Embedded newform 29.6.d.a.7.3

• Label: 29.6.d.a.7.3
• Level: $29$
• Weight: $6$
• Character: 29.7
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73099 - 7.58398i) q^{2} +(-20.8543 - 10.0429i) q^{3} +(-25.6894 + 12.3713i) q^{4} +(-4.14155 - 18.1453i) q^{5} +(-40.0664 + 175.543i) q^{6} +(89.9242 + 43.3052i) q^{7} +(-16.9122 - 21.2072i) q^{8} +(182.533 + 228.889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73099	−	7.58398i	−0.305999	−	1.34067i	−0.860909	−	0.508760i	\(-0.830104\pi\)
							0.554909	−	0.831911i	\(-0.312753\pi\)
\(3\)	−20.8543	−	10.0429i	−1.33780	−	0.644252i	−0.378229	−	0.925712i	\(-0.623467\pi\)
							−0.959573	+	0.281460i	\(0.909181\pi\)
\(5\)	−4.14155	−	18.1453i	−0.0740863	−	0.324593i	0.924281	−	0.381713i	\(-0.124666\pi\)
							−0.998367	+	0.0571193i	\(0.981808\pi\)
\(7\)	89.9242	+	43.3052i	0.693636	+	0.334038i	0.747268	−	0.664523i	\(-0.231365\pi\)
							−0.0536315	+	0.998561i	\(0.517080\pi\)
\(11\)	−167.194	+	209.655i	−0.416619	+	0.522424i	−0.945214	−	0.326450i	\(-0.894147\pi\)
							0.528595	+	0.848874i	\(0.322719\pi\)
\(13\)	122.130	−	153.146i	0.200430	−	0.251331i	−0.671451	−	0.741049i	\(-0.734329\pi\)
							0.871881	+	0.489718i	\(0.162900\pi\)
\(17\)	−2303.29			−1.93297			−0.966486	−	0.256719i	\(-0.917359\pi\)
							−0.966486	+	0.256719i	\(0.917359\pi\)
\(19\)	−313.607	+	151.025i	−0.199298	+	0.0959766i	−0.530871	−	0.847453i	\(-0.678135\pi\)
							0.331573	+	0.943430i	\(0.392421\pi\)
\(23\)	680.951	−	2983.44i	0.268409	−	1.17598i	−0.643456	−	0.765483i	\(-0.722500\pi\)
							0.911864	−	0.410492i	\(-0.134643\pi\)
\(29\)	−4336.80	+	1305.12i	−0.957578	+	0.288175i
							\(31\)	51.7407	+	226.691i	0.00967003	+	0.0423672i	0.979533	−	0.201285i	\(-0.0645117\pi\)
−0.969863	+	0.243652i	\(0.921655\pi\)
\(37\)	−2979.13	−	3735.71i	−0.357755	−	0.448610i	0.570087	−	0.821584i	\(-0.306909\pi\)
							−0.927842	+	0.372974i	\(0.878338\pi\)
\(41\)	−12007.1			−1.11552			−0.557760	−	0.830002i	\(-0.688339\pi\)
							−0.557760	+	0.830002i	\(0.688339\pi\)
\(43\)	−3255.87	+	14264.9i	−0.268532	+	1.17651i	0.643190	+	0.765707i	\(0.277611\pi\)
							−0.911722	+	0.410808i	\(0.865247\pi\)
\(47\)	18547.0	−	23257.2i	1.22470	−	1.53572i	0.465216	−	0.885197i	\(-0.345977\pi\)
							0.759483	−	0.650527i	\(-0.225452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.3	✓	66
29.5	even	14		841.6.a.h.1.6		33
29.24	even	7		841.6.a.i.1.28		33
29.25	even	7	inner	29.6.d.a.25.3	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.3	✓	66	1.1	even	1	trivial
29.6.d.a.25.3	yes	66	29.25	even	7	inner
841.6.a.h.1.6		33	29.5	even	14
841.6.a.i.1.28		33	29.24	even	7