Embedded newform 93.2.k.a.77.8

• Label: 93.2.k.a.77.8
• 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			77.8
Character	\(\chi\)	\(=\)	93.77
Dual form			93.2.k.a.29.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.43914 + 1.98081i) q^{2} +(-1.57528 - 0.720074i) q^{3} +(-1.23444 + 3.79921i) q^{4} +2.68893i q^{5} +(-0.840716 - 4.15661i) q^{6} +(1.40279 - 4.31734i) q^{7} +(-4.64488 + 1.50921i) q^{8} +(1.96299 + 2.26863i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{3} - 6 q^{4} + 8 q^{7} - q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.43914	+	1.98081i	1.01763	+	1.40064i	0.913855	+	0.406042i	\(0.133091\pi\)
							0.103772	+	0.994601i	\(0.466909\pi\)
\(3\)	−1.57528	−	0.720074i	−0.909486	−	0.415735i
							\(5\)			2.68893i			1.20253i	0.799051	+	0.601263i	\(0.205336\pi\)
−0.799051	+	0.601263i	\(0.794664\pi\)
\(7\)	1.40279	−	4.31734i	0.530204	−	1.63180i	−0.223586	−	0.974684i	\(-0.571776\pi\)
							0.753790	−	0.657116i	\(-0.228224\pi\)
\(11\)	0.328658	−	1.01150i	0.0990940	−	0.304980i	−0.889205	−	0.457509i	\(-0.848742\pi\)
							0.988299	+	0.152529i	\(0.0487417\pi\)
\(13\)	0.720736	−	0.992008i	0.199896	−	0.275133i	−0.697287	−	0.716792i	\(-0.745610\pi\)
							0.897183	+	0.441659i	\(0.145610\pi\)
\(17\)	−1.20033	−	3.69424i	−0.291123	−	0.895984i	−0.984496	−	0.175405i	\(-0.943876\pi\)
							0.693373	−	0.720578i	\(-0.256124\pi\)
\(19\)	−2.60903	+	1.89557i	−0.598552	+	0.434873i	−0.845365	−	0.534190i	\(-0.820617\pi\)
							0.246813	+	0.969063i	\(0.420617\pi\)
\(23\)	−0.773101	−	2.37936i	−0.161203	−	0.496131i	0.837534	−	0.546386i	\(-0.183997\pi\)
							−0.998736	+	0.0502547i	\(0.983997\pi\)
\(29\)	−2.50790	+	1.82210i	−0.465706	+	0.338355i	−0.795766	−	0.605605i	\(-0.792931\pi\)
							0.330059	+	0.943960i	\(0.392931\pi\)
\(31\)	−4.53596	+	3.22879i	−0.814682	+	0.579908i
							\(37\)			4.63604i			0.762161i	0.924542	+	0.381081i	\(0.124448\pi\)
−0.924542	+	0.381081i	\(0.875552\pi\)
\(41\)	−4.50228	−	6.19685i	−0.703137	−	0.967785i	−0.999918	−	0.0128397i	\(-0.995913\pi\)
							0.296780	−	0.954946i	\(-0.404087\pi\)
\(43\)	1.32879	+	1.82893i	0.202639	+	0.278909i	0.898227	−	0.439532i	\(-0.144856\pi\)
							−0.695588	+	0.718441i	\(0.744856\pi\)
\(47\)	0.902574	−	1.24229i	0.131654	−	0.181206i	−0.738101	−	0.674691i	\(-0.764277\pi\)
							0.869755	+	0.493485i	\(0.164277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	93.2.k.a.77.8	yes	32
3.2	odd	2	inner	93.2.k.a.77.1	yes	32
31.29	odd	10	inner	93.2.k.a.29.1	✓	32
93.29	even	10	inner	93.2.k.a.29.8	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.k.a.29.1	✓	32	31.29	odd	10	inner
93.2.k.a.29.8	yes	32	93.29	even	10	inner
93.2.k.a.77.1	yes	32	3.2	odd	2	inner
93.2.k.a.77.8	yes	32	1.1	even	1	trivial