Embedded newform 29.6.d.a.7.2

• Label: 29.6.d.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	29.7
Dual form			29.6.d.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89325 - 8.29487i) q^{2} +(14.6426 + 7.05150i) q^{3} +(-36.3895 + 17.5242i) q^{4} +(-19.7959 - 86.7313i) q^{5} +(30.7692 - 134.809i) q^{6} +(23.9398 + 11.5288i) q^{7} +(44.5031 + 55.8051i) q^{8} +(13.1736 + 16.5192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89325	−	8.29487i	−0.334682	−	1.46634i	−0.809950	−	0.586499i	\(-0.800506\pi\)
							0.475268	−	0.879841i	\(-0.342351\pi\)
\(3\)	14.6426	+	7.05150i	0.939322	+	0.452354i	0.839930	−	0.542695i	\(-0.182596\pi\)
							0.0993920	+	0.995048i	\(0.468310\pi\)
\(5\)	−19.7959	−	86.7313i	−0.354119	−	1.55150i	−0.767567	−	0.640969i	\(-0.778533\pi\)
							0.413448	−	0.910528i	\(-0.364324\pi\)
\(7\)	23.9398	+	11.5288i	0.184661	+	0.0889282i	0.523930	−	0.851761i	\(-0.324465\pi\)
							−0.339269	+	0.940689i	\(0.610180\pi\)
\(11\)	−69.7923	+	87.5168i	−0.173911	+	0.218077i	−0.861146	−	0.508358i	\(-0.830253\pi\)
							0.687235	+	0.726435i	\(0.258824\pi\)
\(13\)	−33.6503	+	42.1961i	−0.0552243	+	0.0692491i	−0.808677	−	0.588254i	\(-0.799816\pi\)
							0.753452	+	0.657503i	\(0.228387\pi\)
\(17\)	1908.24			1.60144			0.800718	−	0.599041i	\(-0.204451\pi\)
							0.800718	+	0.599041i	\(0.204451\pi\)
\(19\)	1110.97	−	535.015i	0.706023	−	0.340003i	−0.0461811	−	0.998933i	\(-0.514705\pi\)
							0.752204	+	0.658930i	\(0.228991\pi\)
\(23\)	755.078	−	3308.21i	0.297627	−	1.30399i	−0.576023	−	0.817433i	\(-0.695396\pi\)
							0.873650	−	0.486555i	\(-0.161747\pi\)
\(29\)	3016.57	+	3378.09i	0.666067	+	0.745892i
							\(31\)	1812.67	+	7941.84i	0.338778	+	1.48428i	0.801615	+	0.597841i	\(0.203975\pi\)
−0.462837	+	0.886444i	\(0.653168\pi\)
\(37\)	8058.71	+	10105.3i	0.967746	+	1.21352i	0.976931	+	0.213557i	\(0.0685050\pi\)
							−0.00918457	+	0.999958i	\(0.502924\pi\)
\(41\)	−1885.90			−0.175210			−0.0876052	−	0.996155i	\(-0.527921\pi\)
							−0.0876052	+	0.996155i	\(0.527921\pi\)
\(43\)	4234.14	−	18551.0i	0.349216	−	1.53002i	−0.429749	−	0.902949i	\(-0.641398\pi\)
							0.778965	−	0.627068i	\(-0.215745\pi\)
\(47\)	−5268.22	+	6606.13i	−0.347872	+	0.436217i	−0.924728	−	0.380628i	\(-0.875708\pi\)
							0.576857	+	0.816845i	\(0.304279\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.7.2	✓	66
29.5	even	14		841.6.a.h.1.5		33
29.24	even	7		841.6.a.i.1.29		33
29.25	even	7	inner	29.6.d.a.25.2	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.7.2	✓	66	1.1	even	1	trivial
29.6.d.a.25.2	yes	66	29.25	even	7	inner
841.6.a.h.1.5		33	29.5	even	14
841.6.a.i.1.29		33	29.24	even	7