Embedded newform 29.6.d.a.16.7

• Label: 29.6.d.a.16.7
• Level: $29$
• Weight: $6$
• Character: 29.16
• 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			16.7
Character	\(\chi\)	\(=\)	29.16
Dual form			29.6.d.a.20.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.69866 + 0.818031i) q^{2} +(5.27812 - 6.61856i) q^{3} +(-17.7354 - 22.2395i) q^{4} +(-69.8948 - 33.6596i) q^{5} +(14.3799 - 6.92500i) q^{6} +(10.7481 - 13.4777i) q^{7} +(-25.3589 - 111.105i) q^{8} +(38.1259 + 167.040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.69866	+	0.818031i	0.300283	+	0.144609i	0.577958	−	0.816066i	\(-0.303850\pi\)
							−0.277675	+	0.960675i	\(0.589564\pi\)
\(3\)	5.27812	−	6.61856i	0.338592	−	0.424581i	−0.583162	−	0.812356i	\(-0.698185\pi\)
							0.921754	+	0.387775i	\(0.126756\pi\)
\(5\)	−69.8948	−	33.6596i	−1.25032	−	0.602121i	−0.312720	−	0.949845i	\(-0.601240\pi\)
							−0.937597	+	0.347725i	\(0.886954\pi\)
\(7\)	10.7481	−	13.4777i	0.0829060	−	0.103961i	−0.738649	−	0.674091i	\(-0.764536\pi\)
							0.821555	+	0.570130i	\(0.193107\pi\)
\(11\)	117.034	−	512.758i	0.291628	−	1.27770i	−0.590632	−	0.806941i	\(-0.701121\pi\)
							0.882260	−	0.470763i	\(-0.156021\pi\)
\(13\)	61.6907	−	270.284i	0.101242	−	0.443570i	−0.898745	−	0.438472i	\(-0.855520\pi\)
							0.999987	−	0.00509864i	\(-0.00162295\pi\)
\(17\)	1485.03			1.24627			0.623135	−	0.782114i	\(-0.285859\pi\)
							0.623135	+	0.782114i	\(0.285859\pi\)
\(19\)	−231.628	−	290.453i	−0.147200	−	0.184583i	0.702765	−	0.711422i	\(-0.251948\pi\)
							−0.849965	+	0.526839i	\(0.823377\pi\)
\(23\)	−1786.39	+	860.281i	−0.704137	+	0.339094i	−0.751454	−	0.659786i	\(-0.770647\pi\)
							0.0473172	+	0.998880i	\(0.484933\pi\)
\(29\)	3807.75	−	2451.98i	0.840762	−	0.541405i
							\(31\)	−3776.10	−	1818.47i	−0.705731	−	0.339862i	0.0463572	−	0.998925i	\(-0.485239\pi\)
−0.752088	+	0.659063i	\(0.770953\pi\)
\(37\)	−2986.26	−	13083.7i	−0.358611	−	1.57118i	−0.756657	−	0.653812i	\(-0.773169\pi\)
							0.398046	−	0.917366i	\(-0.369689\pi\)
\(41\)	−2200.85			−0.204471			−0.102235	−	0.994760i	\(-0.532600\pi\)
							−0.102235	+	0.994760i	\(0.532600\pi\)
\(43\)	3897.78	−	1877.07i	0.321475	−	0.154814i	−0.266186	−	0.963922i	\(-0.585763\pi\)
							0.587660	+	0.809108i	\(0.300049\pi\)
\(47\)	4690.03	−	20548.4i	0.309693	−	1.35685i	−0.545313	−	0.838233i	\(-0.683589\pi\)
							0.855005	−	0.518619i	\(-0.173554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.16.7	✓	66
29.7	even	7		841.6.a.i.1.14		33
29.20	even	7	inner	29.6.d.a.20.7	yes	66
29.22	even	14		841.6.a.h.1.20		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.7	✓	66	1.1	even	1	trivial
29.6.d.a.20.7	yes	66	29.20	even	7	inner
841.6.a.h.1.20		33	29.22	even	14
841.6.a.i.1.14		33	29.7	even	7