Embedded newform 32.9.h.a.19.13

• Label: 32.9.h.a.19.13
• Level: $32$
• Weight: $9$
• Character: 32.19
• Analytic conductor: $13.036$
• Analytic rank: $0$
• Dimension: $124$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0361155220\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(31\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			19.13
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.10804 - 14.3344i) q^{2} +(-91.4562 - 37.8824i) q^{3} +(-154.952 + 203.779i) q^{4} +(833.599 - 345.288i) q^{5} +(107.052 + 1580.24i) q^{6} +(1943.11 + 1943.11i) q^{7} +(4022.46 + 772.669i) q^{8} +(2289.83 + 2289.83i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.10804	−	14.3344i	−0.444253	−	0.895902i
							\(3\)	−91.4562	−	37.8824i	−1.12909	−	0.467684i	−0.261618	−	0.965172i	\(-0.584256\pi\)
−0.867471	+	0.497488i	\(0.834256\pi\)
\(5\)	833.599	−	345.288i	1.33376	−	0.552461i	0.402033	−	0.915625i	\(-0.368304\pi\)
							0.931725	+	0.363164i	\(0.118304\pi\)
\(7\)	1943.11	+	1943.11i	0.809291	+	0.809291i	0.984527	−	0.175235i	\(-0.0560686\pi\)
							−0.175235	+	0.984527i	\(0.556069\pi\)
\(11\)	4420.92	−	1831.21i	0.301955	−	0.125074i	−0.226562	−	0.973997i	\(-0.572748\pi\)
							0.528517	+	0.848923i	\(0.322748\pi\)
\(13\)	14234.1	+	5895.95i	0.498375	+	0.206434i	0.617688	−	0.786423i	\(-0.288069\pi\)
							−0.119314	+	0.992857i	\(0.538069\pi\)
\(17\)		−	97200.3i		−	1.16378i	−0.813267	−	0.581891i	\(-0.802313\pi\)
							0.813267	−	0.581891i	\(-0.197687\pi\)
\(19\)	−87913.3	+	212242.i	−0.674591	+	1.62861i	0.0991266	+	0.995075i	\(0.468395\pi\)
							−0.773717	+	0.633531i	\(0.781605\pi\)
\(23\)	328139.	−	328139.i	1.17259	−	1.17259i	0.191003	−	0.981589i	\(-0.438826\pi\)
							0.981589	−	0.191003i	\(-0.0611739\pi\)
\(29\)	405188.	−	978210.i	0.572881	−	1.38306i	−0.326210	−	0.945297i	\(-0.605772\pi\)
							0.899092	−	0.437760i	\(-0.144228\pi\)
\(31\)		−	575844.i		−	0.623532i	−0.950159	−	0.311766i	\(-0.899080\pi\)
							0.950159	−	0.311766i	\(-0.100920\pi\)
\(37\)	220498.	−	91333.1i	0.117651	−	0.0487328i	−0.323081	−	0.946371i	\(-0.604719\pi\)
							0.440733	+	0.897638i	\(0.354719\pi\)
\(41\)	1.64752e6	+	1.64752e6i	0.583035	+	0.583035i	0.935736	−	0.352701i	\(-0.114737\pi\)
							−0.352701	+	0.935736i	\(0.614737\pi\)
\(43\)	−1.63171e6	+	675875.i	−0.477275	+	0.197694i	−0.608334	−	0.793681i	\(-0.708162\pi\)
							0.131060	+	0.991374i	\(0.458162\pi\)
\(47\)	3.69877e6			0.757993			0.378997	−	0.925398i	\(-0.376269\pi\)
							0.378997	+	0.925398i	\(0.376269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.13	✓	124
32.27	odd	8	inner	32.9.h.a.27.13	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.13	✓	124	1.1	even	1	trivial
32.9.h.a.27.13	yes	124	32.27	odd	8	inner