Embedded newform 29.8.e.a.13.4

• Label: 29.8.e.a.13.4
• Level: $29$
• Weight: $8$
• Character: 29.13
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	29.13
Dual form			29.8.e.a.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.01313 + 12.4864i) q^{2} +(66.1997 + 52.7925i) q^{3} +(-39.9455 - 50.0901i) q^{4} +(-439.672 - 211.735i) q^{5} +(-1057.26 + 509.148i) q^{6} +(122.568 - 153.696i) q^{7} +(-863.814 + 197.160i) q^{8} +(1108.70 + 4857.54i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.01313	+	12.4864i	−0.531491	+	1.10365i	0.446458	+	0.894805i	\(0.352685\pi\)
							−0.977948	+	0.208847i	\(0.933029\pi\)
\(3\)	66.1997	+	52.7925i	1.41557	+	1.12888i	0.972638	+	0.232327i	\(0.0746338\pi\)
							0.442934	+	0.896554i	\(0.353938\pi\)
\(5\)	−439.672	−	211.735i	−1.57302	−	0.757525i	−0.574862	−	0.818250i	\(-0.694944\pi\)
							−0.998155	+	0.0607252i	\(0.980659\pi\)
\(7\)	122.568	−	153.696i	0.135063	−	0.169363i	−0.709701	−	0.704503i	\(-0.751170\pi\)
							0.844763	+	0.535140i	\(0.179741\pi\)
\(11\)	−4294.49	−	980.190i	−0.972831	−	0.222042i	−0.293569	−	0.955938i	\(-0.594843\pi\)
							−0.679262	+	0.733895i	\(0.737700\pi\)
\(13\)	−777.164	+	3404.98i	−0.0981095	+	0.429846i	−0.999998	−	0.00212373i	\(-0.999324\pi\)
							0.901888	+	0.431969i	\(0.142181\pi\)
\(17\)			31431.5i			1.55165i	0.630948	+	0.775825i	\(0.282666\pi\)
							−0.630948	+	0.775825i	\(0.717334\pi\)
\(19\)	4249.36	−	3388.75i	0.142130	−	0.113345i	−0.549844	−	0.835267i	\(-0.685313\pi\)
							0.691974	+	0.721923i	\(0.256741\pi\)
\(23\)	63833.8	−	30740.7i	1.09396	−	0.526825i	0.202209	−	0.979342i	\(-0.435188\pi\)
							0.891756	+	0.452517i	\(0.149474\pi\)
\(29\)	−113361.	+	66326.6i	−0.863117	+	0.505004i
							\(31\)	−73226.1	+	152056.i	−0.441469	+	0.916719i	0.554927	+	0.831899i	\(0.312746\pi\)
−0.996396	+	0.0848206i	\(0.972968\pi\)
\(37\)	271807.	−	62038.2i	0.882176	−	0.201351i	0.242644	−	0.970115i	\(-0.421985\pi\)
							0.639532	+	0.768765i	\(0.279128\pi\)
\(41\)			329211.i			0.745986i	0.927834	+	0.372993i	\(0.121669\pi\)
							−0.927834	+	0.372993i	\(0.878331\pi\)
\(43\)	−31614.4	−	65648.0i	−0.0606381	−	0.125916i	0.868447	−	0.495781i	\(-0.165118\pi\)
							−0.929086	+	0.369865i	\(0.879404\pi\)
\(47\)	−387785.	−	88509.4i	−0.544814	−	0.124350i	−0.0587490	−	0.998273i	\(-0.518711\pi\)
							−0.486065	+	0.873922i	\(0.661568\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.13.4	yes	96
29.9	even	14	inner	29.8.e.a.9.4	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.9.4	✓	96	29.9	even	14	inner
29.8.e.a.13.4	yes	96	1.1	even	1	trivial