Embedded newform 93.2.k.a.77.3

• Label: 93.2.k.a.77.3
• Level: $93$
• Weight: $2$
• Character: 93.77
• Analytic conductor: $0.743$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 93 = 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	93.k (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.742608738798\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			77.3
Character	\(\chi\)	\(=\)	93.77
Dual form			93.2.k.a.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.708516 - 0.975189i) q^{2} +(1.63179 - 0.580750i) q^{3} +(0.169036 - 0.520239i) q^{4} +0.0993369i q^{5} +(-1.72249 - 1.17983i) q^{6} +(-0.720350 + 2.21701i) q^{7} +(-2.91990 + 0.948734i) q^{8} +(2.32546 - 1.89532i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{3} - 6 q^{4} + 8 q^{7} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.708516	−	0.975189i	−0.500997	−	0.689563i	0.481372	−	0.876516i	\(-0.340139\pi\)
							−0.982369	+	0.186954i	\(0.940139\pi\)
\(3\)	1.63179	−	0.580750i	0.942113	−	0.335296i
							\(5\)			0.0993369i			0.0444248i	0.999753	+	0.0222124i	\(0.00707101\pi\)
−0.999753	+	0.0222124i	\(0.992929\pi\)
\(7\)	−0.720350	+	2.21701i	−0.272267	+	0.837951i	0.717663	+	0.696391i	\(0.245212\pi\)
							−0.989930	+	0.141560i	\(0.954788\pi\)
\(11\)	0.770563	−	2.37155i	0.232334	−	0.715049i	−0.765130	−	0.643875i	\(-0.777326\pi\)
							0.997464	−	0.0711737i	\(-0.0226745\pi\)
\(13\)	−1.72393	+	2.37279i	−0.478132	+	0.658093i	−0.978145	−	0.207926i	\(-0.933329\pi\)
							0.500012	+	0.866018i	\(0.333329\pi\)
\(17\)	0.143373	+	0.441255i	0.0347729	+	0.107020i	0.966936	−	0.255017i	\(-0.0820812\pi\)
							−0.932164	+	0.362038i	\(0.882081\pi\)
\(19\)	−1.94105	+	1.41026i	−0.445308	+	0.323535i	−0.787740	−	0.616007i	\(-0.788749\pi\)
							0.342433	+	0.939542i	\(0.388749\pi\)
\(23\)	2.36442	+	7.27694i	0.493016	+	1.51735i	0.820027	+	0.572326i	\(0.193959\pi\)
							−0.327011	+	0.945021i	\(0.606041\pi\)
\(29\)	−2.51354	+	1.82620i	−0.466753	+	0.339116i	−0.796175	−	0.605067i	\(-0.793146\pi\)
							0.329421	+	0.944183i	\(0.393146\pi\)
\(31\)	−5.55636	−	0.356243i	−0.997951	−	0.0639832i
							\(37\)		−	4.25854i		−	0.700100i	−0.936731	−	0.350050i	\(-0.886165\pi\)
0.936731	−	0.350050i	\(-0.113835\pi\)
\(41\)	−6.25806	−	8.61349i	−0.977345	−	1.34520i	−0.938247	−	0.345965i	\(-0.887552\pi\)
							−0.0390980	−	0.999235i	\(-0.512448\pi\)
\(43\)	−1.56179	−	2.14961i	−0.238170	−	0.327813i	0.673154	−	0.739502i	\(-0.264939\pi\)
							−0.911324	+	0.411689i	\(0.864939\pi\)
\(47\)	6.38164	−	8.78358i	0.930858	−	1.28122i	−0.0286654	−	0.999589i	\(-0.509126\pi\)
							0.959524	−	0.281628i	\(-0.0908743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.k.a.77.3	yes	32
3.2	odd	2	inner	93.2.k.a.77.6	yes	32
31.29	odd	10	inner	93.2.k.a.29.6	yes	32
93.29	even	10	inner	93.2.k.a.29.3	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.k.a.29.3	✓	32	93.29	even	10	inner
93.2.k.a.29.6	yes	32	31.29	odd	10	inner
93.2.k.a.77.3	yes	32	1.1	even	1	trivial
93.2.k.a.77.6	yes	32	3.2	odd	2	inner