Embedded newform 93.2.k.a.29.7

• Label: 93.2.k.a.29.7
• Level: $93$
• Weight: $2$
• Character: 93.29
• 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			29.7
Character	\(\chi\)	\(=\)	93.29
Dual form			93.2.k.a.77.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28498 - 1.76862i) q^{2} +(-0.360937 - 1.69403i) q^{3} +(-0.858812 - 2.64315i) q^{4} +3.78361i q^{5} +(-3.45988 - 1.53842i) q^{6} +(0.0623762 + 0.191974i) q^{7} +(-1.62001 - 0.526373i) q^{8} +(-2.73945 + 1.22287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{3} - 6 q^{4} + 8 q^{7} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(32\)	\(34\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28498	−	1.76862i	0.908616	−	1.25060i	−0.0590217	−	0.998257i	\(-0.518798\pi\)
							0.967637	−	0.252345i	\(-0.0812019\pi\)
\(3\)	−0.360937	−	1.69403i	−0.208387	−	0.978046i
							\(5\)			3.78361i			1.69208i	0.533118	+	0.846041i	\(0.321020\pi\)
−0.533118	+	0.846041i	\(0.678980\pi\)
\(7\)	0.0623762	+	0.191974i	0.0235760	+	0.0725594i	0.962152	−	0.272512i	\(-0.0878546\pi\)
							−0.938576	+	0.345072i	\(0.887855\pi\)
\(11\)	0.384339	+	1.18287i	0.115882	+	0.356650i	0.992130	−	0.125212i	\(-0.0399610\pi\)
							−0.876248	+	0.481861i	\(0.839961\pi\)
\(13\)	−3.67739	−	5.06149i	−1.01992	−	1.40381i	−0.912257	−	0.409618i	\(-0.865662\pi\)
							−0.107667	−	0.994187i	\(-0.534338\pi\)
\(17\)	−0.820002	+	2.52371i	−0.198880	+	0.612089i	0.801030	+	0.598625i	\(0.204286\pi\)
							−0.999909	+	0.0134641i	\(0.995714\pi\)
\(19\)	1.60582	+	1.16670i	0.368401	+	0.267659i	0.756547	−	0.653939i	\(-0.226885\pi\)
							−0.388147	+	0.921598i	\(0.626885\pi\)
\(23\)	1.02741	−	3.16205i	0.214230	−	0.659332i	−0.784977	−	0.619524i	\(-0.787325\pi\)
							0.999207	−	0.0398076i	\(-0.0126745\pi\)
\(29\)	−1.68336	−	1.22303i	−0.312592	−	0.227111i	0.420416	−	0.907331i	\(-0.361884\pi\)
							−0.733008	+	0.680220i	\(0.761884\pi\)
\(31\)	−1.84886	+	5.25183i	−0.332065	+	0.943256i
							\(37\)			4.12135i			0.677545i	0.940868	+	0.338773i	\(0.110012\pi\)
−0.940868	+	0.338773i	\(0.889988\pi\)
\(41\)	−5.19017	+	7.14366i	−0.810569	+	1.11565i	0.180667	+	0.983544i	\(0.442175\pi\)
							−0.991235	+	0.132108i	\(0.957825\pi\)
\(43\)	5.83625	−	8.03291i	0.890020	−	1.22501i	−0.0835231	−	0.996506i	\(-0.526617\pi\)
							0.973543	−	0.228502i	\(-0.0733828\pi\)
\(47\)	−2.63224	−	3.62296i	−0.383951	−	0.528463i	0.572675	−	0.819783i	\(-0.305906\pi\)
							−0.956626	+	0.291319i	\(0.905906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.k.a.29.7	yes	32
3.2	odd	2	inner	93.2.k.a.29.2	✓	32
31.15	odd	10	inner	93.2.k.a.77.2	yes	32
93.77	even	10	inner	93.2.k.a.77.7	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.k.a.29.2	✓	32	3.2	odd	2	inner
93.2.k.a.29.7	yes	32	1.1	even	1	trivial
93.2.k.a.77.2	yes	32	31.15	odd	10	inner
93.2.k.a.77.7	yes	32	93.77	even	10	inner