Embedded newform 105.4.x.a.2.13

• Label: 105.4.x.a.2.13
• Level: $105$
• Weight: $4$
• Character: 105.2
• Analytic conductor: $6.195$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.19520055060\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	105.2
Dual form			105.4.x.a.53.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.718246 - 2.68053i) q^{2} +(1.46722 - 4.98470i) q^{3} +(0.258841 - 0.149442i) q^{4} +(-10.9729 - 2.14346i) q^{5} +(-14.4155 - 0.352673i) q^{6} +(-5.48584 + 17.6891i) q^{7} +(-16.2848 - 16.2848i) q^{8} +(-22.6946 - 14.6273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 2 q^{3} + 4 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.718246	−	2.68053i	−0.253938	−	0.947710i	−0.968678	−	0.248320i	\(-0.920122\pi\)
							0.714740	−	0.699390i	\(-0.246545\pi\)
\(3\)	1.46722	−	4.98470i	0.282366	−	0.959307i
							\(5\)	−10.9729	−	2.14346i	−0.981450	−	0.191716i
\(7\)	−5.48584	+	17.6891i	−0.296207	+	0.955124i
							\(11\)	−3.15128	+	1.81939i	−0.0863768	+	0.0498697i	−0.542566	−	0.840013i	\(-0.682547\pi\)
0.456189	+	0.889883i	\(0.349214\pi\)
\(13\)	27.2333	−	27.2333i	0.581012	−	0.581012i	−0.354169	−	0.935181i	\(-0.615236\pi\)
							0.935181	+	0.354169i	\(0.115236\pi\)
\(17\)	22.0651	+	5.91233i	0.314799	+	0.0843500i	0.412759	−	0.910840i	\(-0.364565\pi\)
							−0.0979604	+	0.995190i	\(0.531232\pi\)
\(19\)	−94.3831	−	54.4921i	−1.13963	−	0.657966i	−0.193291	−	0.981141i	\(-0.561916\pi\)
							−0.946339	+	0.323176i	\(0.895250\pi\)
\(23\)	97.4178	−	26.1030i	0.883175	−	0.236646i	0.211398	−	0.977400i	\(-0.432198\pi\)
							0.671776	+	0.740754i	\(0.265532\pi\)
\(29\)	−255.874			−1.63844			−0.819219	−	0.573481i	\(-0.805593\pi\)
							−0.819219	+	0.573481i	\(0.805593\pi\)
\(31\)	−45.3048	−	78.4702i	−0.262483	−	0.454634i	0.704418	−	0.709785i	\(-0.251208\pi\)
							−0.966901	+	0.255151i	\(0.917875\pi\)
\(37\)	354.132	−	94.8892i	1.57348	−	0.421613i	0.636582	−	0.771209i	\(-0.280348\pi\)
							0.936901	+	0.349596i	\(0.113681\pi\)
\(41\)		−	273.534i		−	1.04192i	−0.853580	−	0.520961i	\(-0.825574\pi\)
							0.853580	−	0.520961i	\(-0.174426\pi\)
\(43\)	111.978	−	111.978i	0.397129	−	0.397129i	−0.480090	−	0.877219i	\(-0.659396\pi\)
							0.877219	+	0.480090i	\(0.159396\pi\)
\(47\)	−144.517	−	539.345i	−0.448511	−	1.67386i	−0.706497	−	0.707716i	\(-0.749726\pi\)
							0.257987	−	0.966148i	\(-0.416941\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.x.a.2.13	✓	176
3.2	odd	2	inner	105.4.x.a.2.32	yes	176
5.3	odd	4	inner	105.4.x.a.23.13	yes	176
7.4	even	3	inner	105.4.x.a.32.32	yes	176
15.8	even	4	inner	105.4.x.a.23.32	yes	176
21.11	odd	6	inner	105.4.x.a.32.13	yes	176
35.18	odd	12	inner	105.4.x.a.53.32	yes	176
105.53	even	12	inner	105.4.x.a.53.13	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.x.a.2.13	✓	176	1.1	even	1	trivial
105.4.x.a.2.32	yes	176	3.2	odd	2	inner
105.4.x.a.23.13	yes	176	5.3	odd	4	inner
105.4.x.a.23.32	yes	176	15.8	even	4	inner
105.4.x.a.32.13	yes	176	21.11	odd	6	inner
105.4.x.a.32.32	yes	176	7.4	even	3	inner
105.4.x.a.53.13	yes	176	105.53	even	12	inner
105.4.x.a.53.32	yes	176	35.18	odd	12	inner