Embedded newform 32.8.g.a.13.21

• Label: 32.8.g.a.13.21
• Level: $32$
• Weight: $8$
• Character: 32.13
• Analytic conductor: $9.996$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.99632081549\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(27\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.21
Character	\(\chi\)	\(=\)	32.13
Dual form			32.8.g.a.5.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.22945 + 8.70259i) q^{2} +(6.18478 - 14.9314i) q^{3} +(-23.4701 + 125.830i) q^{4} +(324.538 - 134.428i) q^{5} +(174.654 - 54.1221i) q^{6} +(441.630 - 441.630i) q^{7} +(-1264.72 + 705.431i) q^{8} +(1361.75 + 1361.75i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.22945	+	8.70259i	0.638999	+	0.769207i
							\(3\)	6.18478	−	14.9314i	0.132251	−	0.319283i	−0.843857	−	0.536569i	\(-0.819720\pi\)
0.976108	+	0.217286i	\(0.0697203\pi\)
\(5\)	324.538	−	134.428i	1.16110	−	0.480944i	0.282860	−	0.959161i	\(-0.408717\pi\)
							0.878242	+	0.478217i	\(0.158717\pi\)
\(7\)	441.630	−	441.630i	0.486649	−	0.486649i	−0.420598	−	0.907247i	\(-0.638180\pi\)
							0.907247	+	0.420598i	\(0.138180\pi\)
\(11\)	668.200	+	1613.18i	0.151367	+	0.365433i	0.981315	−	0.192408i	\(-0.0616298\pi\)
							−0.829948	+	0.557841i	\(0.811630\pi\)
\(13\)	2810.31	+	1164.07i	0.354774	+	0.146952i	0.552950	−	0.833214i	\(-0.313502\pi\)
							−0.198176	+	0.980166i	\(0.563502\pi\)
\(17\)			14405.6i			0.711149i	0.934648	+	0.355575i	\(0.115715\pi\)
							−0.934648	+	0.355575i	\(0.884285\pi\)
\(19\)	7868.56	+	3259.26i	0.263183	+	0.109014i	0.510373	−	0.859953i	\(-0.329507\pi\)
							−0.247190	+	0.968967i	\(0.579507\pi\)
\(23\)	−22703.4	−	22703.4i	−0.389084	−	0.389084i	0.485276	−	0.874361i	\(-0.338719\pi\)
							−0.874361	+	0.485276i	\(0.838719\pi\)
\(29\)	87780.8	−	211922.i	0.668354	−	1.61355i	−0.116009	−	0.993248i	\(-0.537010\pi\)
							0.784363	−	0.620302i	\(-0.212990\pi\)
\(31\)	−147902.			−0.891679			−0.445840	−	0.895113i	\(-0.647095\pi\)
							−0.445840	+	0.895113i	\(0.647095\pi\)
\(37\)	−226436.	+	93792.7i	−0.734918	+	0.304413i	−0.718571	−	0.695453i	\(-0.755204\pi\)
							−0.0163467	+	0.999866i	\(0.505204\pi\)
\(41\)	−208298.	−	208298.i	−0.471999	−	0.471999i	0.430562	−	0.902561i	\(-0.358315\pi\)
							−0.902561	+	0.430562i	\(0.858315\pi\)
\(43\)	−315056.	−	760613.i	−0.604294	−	1.45890i	−0.869121	−	0.494600i	\(-0.835315\pi\)
							0.264827	−	0.964296i	\(-0.414685\pi\)
\(47\)		−	1.10861e6i		−	1.55753i	−0.627318	−	0.778764i	\(-0.715847\pi\)
							0.627318	−	0.778764i	\(-0.284153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.13.21	yes	108
32.5	even	8	inner	32.8.g.a.5.21	✓	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.5.21	✓	108	32.5	even	8	inner
32.8.g.a.13.21	yes	108	1.1	even	1	trivial