Embedded newform 91.3.z.a.44.1

• Label: 91.3.z.a.44.1
• Level: $91$
• Weight: $3$
• Character: 91.44
• Analytic conductor: $2.480$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	91.z (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47957040568\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			44.1
Character	\(\chi\)	\(=\)	91.44
Dual form			91.3.z.a.60.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909975 - 3.39607i) q^{2} +(-1.39738 - 2.42033i) q^{3} +(-7.24117 + 4.18069i) q^{4} +(0.577586 + 2.15558i) q^{5} +(-6.94805 + 6.94805i) q^{6} +(-6.99920 + 0.105956i) q^{7} +(10.8428 + 10.8428i) q^{8} +(0.594658 - 1.02998i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 8 q^{6} - 2 q^{7} - 24 q^{8} - 64 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.909975	−	3.39607i	−0.454988	−	1.69804i	−0.688124	−	0.725593i	\(-0.741565\pi\)
							0.233136	−	0.972444i	\(-0.425101\pi\)
\(3\)	−1.39738	−	2.42033i	−0.465793	−	0.806778i	0.533444	−	0.845836i	\(-0.320898\pi\)
							−0.999237	+	0.0390579i	\(0.987564\pi\)
\(5\)	0.577586	+	2.15558i	0.115517	+	0.431116i	0.999325	−	0.0367341i	\(-0.0116955\pi\)
							−0.883808	+	0.467850i	\(0.845029\pi\)
\(7\)	−6.99920	+	0.105956i	−0.999885	+	0.0151366i
							\(11\)	−0.241581	+	0.901593i	−0.0219619	+	0.0819630i	−0.976037	−	0.217604i	\(-0.930176\pi\)
0.954075	+	0.299568i	\(0.0968424\pi\)
\(13\)	−4.64242	−	12.1428i	−0.357109	−	0.934063i
							\(17\)	−7.77275	+	4.48760i	−0.457221	+	0.263977i	−0.710875	−	0.703318i	\(-0.751701\pi\)
0.253654	+	0.967295i	\(0.418367\pi\)
\(19\)	0.919537	+	3.43176i	0.0483967	+	0.180619i	0.985893	−	0.167376i	\(-0.0535293\pi\)
							−0.937496	+	0.347995i	\(0.886863\pi\)
\(23\)	−20.3168	−	11.7299i	−0.883340	−	0.509997i	−0.0115820	−	0.999933i	\(-0.503687\pi\)
							−0.871758	+	0.489936i	\(0.837020\pi\)
\(29\)	−22.3316			−0.770054			−0.385027	−	0.922905i	\(-0.625808\pi\)
							−0.385027	+	0.922905i	\(0.625808\pi\)
\(31\)	−45.5100	−	12.1944i	−1.46806	−	0.393367i	−0.565798	−	0.824544i	\(-0.691432\pi\)
							−0.902267	+	0.431177i	\(0.858098\pi\)
\(37\)	70.6728	−	18.9367i	1.91008	−	0.511803i	0.916286	−	0.400525i	\(-0.131172\pi\)
							0.993789	−	0.111278i	\(-0.0354943\pi\)
\(41\)	1.71364	−	1.71364i	0.0417962	−	0.0417962i	−0.685900	−	0.727696i	\(-0.740591\pi\)
							0.727696	+	0.685900i	\(0.240591\pi\)
\(43\)		−	65.0534i		−	1.51287i	−0.654069	−	0.756434i	\(-0.726940\pi\)
							0.654069	−	0.756434i	\(-0.273060\pi\)
\(47\)	43.2988	−	11.6019i	0.921252	−	0.246849i	0.233131	−	0.972445i	\(-0.425103\pi\)
							0.688120	+	0.725597i	\(0.258436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.3.z.a.44.1	yes	64
7.4	even	3	inner	91.3.z.a.18.16	✓	64
13.8	odd	4	inner	91.3.z.a.86.16	yes	64
91.60	odd	12	inner	91.3.z.a.60.1	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.3.z.a.18.16	✓	64	7.4	even	3	inner
91.3.z.a.44.1	yes	64	1.1	even	1	trivial
91.3.z.a.60.1	yes	64	91.60	odd	12	inner
91.3.z.a.86.16	yes	64	13.8	odd	4	inner