Embedded newform 105.3.k.d.62.12

• Label: 105.3.k.d.62.12
• Level: $105$
• Weight: $3$
• Character: 105.62
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	105.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			62.12
Character	\(\chi\)	\(=\)	105.62
Dual form			105.3.k.d.83.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88692 - 1.88692i) q^{2} +(1.39918 + 2.65373i) q^{3} -3.12092i q^{4} +(4.96950 + 0.551428i) q^{5} +(7.64751 + 2.36725i) q^{6} +(-6.68054 + 2.09055i) q^{7} +(1.65875 + 1.65875i) q^{8} +(-5.08462 + 7.42608i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88692	−	1.88692i	0.943459	−	0.943459i	−0.0550258	−	0.998485i	\(-0.517524\pi\)
							0.998485	+	0.0550258i	\(0.0175241\pi\)
\(3\)	1.39918	+	2.65373i	0.466392	+	0.884578i
							\(5\)	4.96950	+	0.551428i	0.993900	+	0.110286i
\(7\)	−6.68054	+	2.09055i	−0.954363	+	0.298651i
							\(11\)		−	17.9060i		−	1.62782i	−0.580989	−	0.813911i	\(-0.697334\pi\)
0.580989	−	0.813911i	\(-0.302666\pi\)
\(13\)	−11.1383	−	11.1383i	−0.856796	−	0.856796i	0.134164	−	0.990959i	\(-0.457165\pi\)
							−0.990959	+	0.134164i	\(0.957165\pi\)
\(17\)	−0.666845	−	0.666845i	−0.0392262	−	0.0392262i	0.687222	−	0.726448i	\(-0.258830\pi\)
							−0.726448	+	0.687222i	\(0.758830\pi\)
\(19\)	−10.8119			−0.569046			−0.284523	−	0.958669i	\(-0.591835\pi\)
							−0.284523	+	0.958669i	\(0.591835\pi\)
\(23\)	−2.20425	−	2.20425i	−0.0958368	−	0.0958368i	0.657563	−	0.753400i	\(-0.271587\pi\)
							−0.753400	+	0.657563i	\(0.771587\pi\)
\(29\)	−22.9708			−0.792098			−0.396049	−	0.918229i	\(-0.629619\pi\)
							−0.396049	+	0.918229i	\(0.629619\pi\)
\(31\)			26.1094i			0.842240i	0.907005	+	0.421120i	\(0.138363\pi\)
							−0.907005	+	0.421120i	\(0.861637\pi\)
\(37\)	41.6663	+	41.6663i	1.12612	+	1.12612i	0.990803	+	0.135314i	\(0.0432044\pi\)
							0.135314	+	0.990803i	\(0.456796\pi\)
\(41\)	6.85976			0.167311			0.0836556	−	0.996495i	\(-0.473340\pi\)
							0.0836556	+	0.996495i	\(0.473340\pi\)
\(43\)	−37.6932	+	37.6932i	−0.876587	+	0.876587i	−0.993180	−	0.116593i	\(-0.962803\pi\)
							0.116593	+	0.993180i	\(0.462803\pi\)
\(47\)	5.55606	+	5.55606i	0.118214	+	0.118214i	0.763739	−	0.645525i	\(-0.223361\pi\)
							−0.645525	+	0.763739i	\(0.723361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.k.d.62.12	yes	32
3.2	odd	2	inner	105.3.k.d.62.6	yes	32
5.3	odd	4	inner	105.3.k.d.83.5	yes	32
7.6	odd	2	inner	105.3.k.d.62.11	yes	32
15.8	even	4	inner	105.3.k.d.83.11	yes	32
21.20	even	2	inner	105.3.k.d.62.5	✓	32
35.13	even	4	inner	105.3.k.d.83.6	yes	32
105.83	odd	4	inner	105.3.k.d.83.12	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.k.d.62.5	✓	32	21.20	even	2	inner
105.3.k.d.62.6	yes	32	3.2	odd	2	inner
105.3.k.d.62.11	yes	32	7.6	odd	2	inner
105.3.k.d.62.12	yes	32	1.1	even	1	trivial
105.3.k.d.83.5	yes	32	5.3	odd	4	inner
105.3.k.d.83.6	yes	32	35.13	even	4	inner
105.3.k.d.83.11	yes	32	15.8	even	4	inner
105.3.k.d.83.12	yes	32	105.83	odd	4	inner