Embedded newform 29.15.c.a.12.6

• Label: 29.15.c.a.12.6
• Level: $29$
• Weight: $15$
• Character: 29.12
• Analytic conductor: $36.055$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.0554007641\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.6
Character	\(\chi\)	\(=\)	29.12
Dual form			29.15.c.a.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-126.672 - 126.672i) q^{2} +(1067.39 + 1067.39i) q^{3} +15707.8i q^{4} -7103.16i q^{5} -270419. i q^{6} -354580. q^{7} +(-85651.6 + 85651.6i) q^{8} -2.50431e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 312 q^{2} - 2 q^{3} - 4 q^{7} - 689310 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−126.672	−	126.672i	−0.989629	−	0.989629i	0.0103180	−	0.999947i	\(-0.496716\pi\)
							−0.999947	+	0.0103180i	\(0.996716\pi\)
\(3\)	1067.39	+	1067.39i	0.488063	+	0.488063i	0.907695	−	0.419631i	\(-0.137841\pi\)
							−0.419631	+	0.907695i	\(0.637841\pi\)
\(5\)		−	7103.16i		−	0.0909204i	−0.998966	−	0.0454602i	\(-0.985525\pi\)
							0.998966	−	0.0454602i	\(-0.0144754\pi\)
\(7\)	−354580.			−0.430554			−0.215277	−	0.976553i	\(-0.569066\pi\)
							−0.215277	+	0.976553i	\(0.569066\pi\)
\(11\)	2.45019e7	+	2.45019e7i	1.25734	+	1.25734i	0.952360	+	0.304975i	\(0.0986482\pi\)
							0.304975	+	0.952360i	\(0.401352\pi\)
\(13\)		−	2.32501e7i		−	0.370529i	−0.982689	−	0.185264i	\(-0.940686\pi\)
							0.982689	−	0.185264i	\(-0.0593142\pi\)
\(17\)	−2.48694e7	−	2.48694e7i	−0.0606071	−	0.0606071i	0.676154	−	0.736761i	\(-0.263646\pi\)
							−0.736761	+	0.676154i	\(0.763646\pi\)
\(19\)	−8.49811e8	−	8.49811e8i	−0.950708	−	0.950708i	0.0481330	−	0.998841i	\(-0.484673\pi\)
							−0.998841	+	0.0481330i	\(0.984673\pi\)
\(23\)	−4.32714e9			−1.27088			−0.635442	−	0.772149i	\(-0.719182\pi\)
							−0.635442	+	0.772149i	\(0.719182\pi\)
\(29\)	−7.41993e9	−	1.55725e10i	−0.430144	−	0.902760i
							\(31\)	7.24206e9	+	7.24206e9i	0.263227	+	0.263227i	0.826364	−	0.563137i	\(-0.190406\pi\)
−0.563137	+	0.826364i	\(0.690406\pi\)
\(37\)	−1.20993e11	+	1.20993e11i	−1.27453	+	1.27453i	−0.330842	+	0.943686i	\(0.607333\pi\)
							−0.943686	+	0.330842i	\(0.892667\pi\)
\(41\)	−4.37614e10	+	4.37614e10i	−0.224701	+	0.224701i	−0.810475	−	0.585774i	\(-0.800791\pi\)
							0.585774	+	0.810475i	\(0.300791\pi\)
\(43\)	5.22081e10	+	5.22081e10i	0.192070	+	0.192070i	0.796590	−	0.604520i	\(-0.206635\pi\)
							−0.604520	+	0.796590i	\(0.706635\pi\)
\(47\)	−5.43683e11	+	5.43683e11i	−1.07315	+	1.07315i	−0.0760463	+	0.997104i	\(0.524230\pi\)
							−0.997104	+	0.0760463i	\(0.975770\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.15.c.a.12.6	✓	68
29.17	odd	4	inner	29.15.c.a.17.6	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.15.c.a.12.6	✓	68	1.1	even	1	trivial
29.15.c.a.17.6	yes	68	29.17	odd	4	inner