Embedded newform 105.4.s.a.26.2

• Label: 105.4.s.a.26.2
• Level: $105$
• Weight: $4$
• Character: 105.26
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.2
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.a.101.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.56345 - 2.63471i) q^{2} +(-4.97118 - 1.51242i) q^{3} +(9.88336 + 17.1185i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(18.7009 + 19.9994i) q^{6} +(12.9395 - 13.2503i) q^{7} -62.0037i q^{8} +(22.4252 + 15.0370i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.56345	−	2.63471i	−1.61342	−	0.931510i	−0.988569	−	0.150767i	\(-0.951826\pi\)
							−0.624853	−	0.780743i	\(-0.714841\pi\)
\(3\)	−4.97118	−	1.51242i	−0.956703	−	0.291065i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	12.9395	−	13.2503i	0.698666	−	0.715448i
							\(11\)	4.67930	−	2.70159i	0.128260	−	0.0740510i	−0.434497	−	0.900673i	\(-0.643074\pi\)
0.562757	+	0.826622i	\(0.309741\pi\)
\(13\)		−	11.1476i		−	0.237829i	−0.992905	−	0.118914i	\(-0.962059\pi\)
							0.992905	−	0.118914i	\(-0.0379414\pi\)
\(17\)	49.0554	+	84.9664i	0.699864	+	1.21220i	0.968513	+	0.248961i	\(0.0800892\pi\)
							−0.268650	+	0.963238i	\(0.586577\pi\)
\(19\)	−38.3523	−	22.1427i	−0.463085	−	0.267362i	0.250255	−	0.968180i	\(-0.419485\pi\)
							−0.713341	+	0.700818i	\(0.752819\pi\)
\(23\)	−161.199	−	93.0682i	−1.46140	−	0.843742i	−0.462327	−	0.886709i	\(-0.652986\pi\)
							−0.999076	+	0.0429674i	\(0.986319\pi\)
\(29\)		−	210.853i		−	1.35015i	−0.737747	−	0.675077i	\(-0.764110\pi\)
							0.737747	−	0.675077i	\(-0.235890\pi\)
\(31\)	143.496	−	82.8476i	0.831377	−	0.479996i	−0.0229470	−	0.999737i	\(-0.507305\pi\)
							0.854324	+	0.519741i	\(0.173972\pi\)
\(37\)	37.5633	−	65.0615i	0.166902	−	0.289082i	−0.770427	−	0.637528i	\(-0.779957\pi\)
							0.937329	+	0.348446i	\(0.113290\pi\)
\(41\)	−149.096			−0.567924			−0.283962	−	0.958836i	\(-0.591649\pi\)
							−0.283962	+	0.958836i	\(0.591649\pi\)
\(43\)	−415.711			−1.47431			−0.737155	−	0.675724i	\(-0.763831\pi\)
							−0.737155	+	0.675724i	\(0.763831\pi\)
\(47\)	276.394	−	478.728i	0.857790	−	1.48574i	−0.0162420	−	0.999868i	\(-0.505170\pi\)
							0.874032	−	0.485868i	\(-0.161496\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.26.2	✓	32
3.2	odd	2		105.4.s.b.26.15	yes	32
7.3	odd	6		105.4.s.b.101.15	yes	32
21.17	even	6	inner	105.4.s.a.101.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.2	✓	32	1.1	even	1	trivial
105.4.s.a.101.2	yes	32	21.17	even	6	inner
105.4.s.b.26.15	yes	32	3.2	odd	2
105.4.s.b.101.15	yes	32	7.3	odd	6