Embedded newform 105.4.s.a.26.1

• Label: 105.4.s.a.26.1
• 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			26.1
Character	\(\chi\)	\(=\)	105.26
Dual form			105.4.s.a.101.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.57212 - 2.63971i) q^{2} +(4.83065 - 1.91438i) q^{3} +(9.93617 + 17.2100i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-27.1397 - 3.99877i) q^{6} +(-17.5731 + 5.84690i) q^{7} -62.6791i q^{8} +(19.6703 - 18.4954i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{3} + 64 q^{4} - 80 q^{5} - 28 q^{6} + 46 q^{7} + 100 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.57212	−	2.63971i	−1.61649	−	0.933279i	−0.987819	−	0.155606i	\(-0.950267\pi\)
							−0.628668	−	0.777674i	\(-0.716400\pi\)
\(3\)	4.83065	−	1.91438i	0.929659	−	0.368422i
							\(5\)	−2.50000	+	4.33013i	−0.223607	+	0.387298i
\(7\)	−17.5731	+	5.84690i	−0.948858	+	0.315703i
							\(11\)	19.7811	−	11.4206i	0.542202	−	0.313041i	−0.203769	−	0.979019i	\(-0.565319\pi\)
0.745971	+	0.665979i	\(0.231986\pi\)
\(13\)		−	81.4876i		−	1.73851i	−0.494367	−	0.869253i	\(-0.664600\pi\)
							0.494367	−	0.869253i	\(-0.335400\pi\)
\(17\)	−61.9810	−	107.354i	−0.884271	−	1.53160i	−0.846547	−	0.532314i	\(-0.821323\pi\)
							−0.0377235	−	0.999288i	\(-0.512011\pi\)
\(19\)	−40.3436	−	23.2924i	−0.487130	−	0.281245i	0.236253	−	0.971692i	\(-0.424081\pi\)
							−0.723383	+	0.690447i	\(0.757414\pi\)
\(23\)	98.1751	+	56.6814i	0.890040	+	0.513865i	0.873956	−	0.486006i	\(-0.161547\pi\)
							0.0160847	+	0.999871i	\(0.494880\pi\)
\(29\)		−	80.2506i		−	0.513868i	−0.966429	−	0.256934i	\(-0.917288\pi\)
							0.966429	−	0.256934i	\(-0.0827123\pi\)
\(31\)	−15.7036	+	9.06650i	−0.0909825	+	0.0525288i	−0.544801	−	0.838565i	\(-0.683395\pi\)
							0.453818	+	0.891094i	\(0.350061\pi\)
\(37\)	−49.5305	+	85.7894i	−0.220075	+	0.381181i	−0.954830	−	0.297151i	\(-0.903963\pi\)
							0.734756	+	0.678332i	\(0.237297\pi\)
\(41\)	−142.889			−0.544282			−0.272141	−	0.962257i	\(-0.587732\pi\)
							−0.272141	+	0.962257i	\(0.587732\pi\)
\(43\)	184.906			0.655766			0.327883	−	0.944718i	\(-0.393665\pi\)
							0.327883	+	0.944718i	\(0.393665\pi\)
\(47\)	60.6901	−	105.118i	0.188352	−	0.326236i	−0.756349	−	0.654169i	\(-0.773019\pi\)
							0.944701	+	0.327933i	\(0.106352\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.s.a.26.1	✓	32
3.2	odd	2		105.4.s.b.26.16	yes	32
7.3	odd	6		105.4.s.b.101.16	yes	32
21.17	even	6	inner	105.4.s.a.101.1	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.s.a.26.1	✓	32	1.1	even	1	trivial
105.4.s.a.101.1	yes	32	21.17	even	6	inner
105.4.s.b.26.16	yes	32	3.2	odd	2
105.4.s.b.101.16	yes	32	7.3	odd	6