Embedded newform 105.3.o.b.74.12

• Label: 105.3.o.b.74.12
• Level: $105$
• Weight: $3$
• Character: 105.74
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			74.12
Character	\(\chi\)	\(=\)	105.74
Dual form			105.3.o.b.44.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0859580 + 0.148884i) q^{2} +(-0.346935 + 2.97987i) q^{3} +(1.98522 - 3.43851i) q^{4} +(1.39092 - 4.80264i) q^{5} +(-0.473476 + 0.204491i) q^{6} +(6.99566 + 0.246384i) q^{7} +1.37025 q^{8} +(-8.75927 - 2.06764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 44 q^{4} + 80 q^{6} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0859580	+	0.148884i	0.0429790	+	0.0744418i	0.886715	−	0.462317i	\(-0.152982\pi\)
							−0.843736	+	0.536759i	\(0.819648\pi\)
\(3\)	−0.346935	+	2.97987i	−0.115645	+	0.993291i
							\(5\)	1.39092	−	4.80264i	0.278183	−	0.960528i
\(7\)	6.99566	+	0.246384i	0.999380	+	0.0351977i
							\(11\)	10.0440	+	5.79890i	0.913089	+	0.527172i	0.881424	−	0.472326i	\(-0.156586\pi\)
0.0316655	+	0.999499i	\(0.489919\pi\)
\(13\)			7.34521i			0.565016i	0.959265	+	0.282508i	\(0.0911665\pi\)
							−0.959265	+	0.282508i	\(0.908834\pi\)
\(17\)	−2.30611	+	3.99429i	−0.135653	+	0.234958i	−0.925847	−	0.377899i	\(-0.876647\pi\)
							0.790194	+	0.612857i	\(0.209980\pi\)
\(19\)	5.93904	+	10.2867i	0.312581	+	0.541406i	0.978920	−	0.204242i	\(-0.0654730\pi\)
							−0.666339	+	0.745649i	\(0.732140\pi\)
\(23\)	−11.8464	−	20.5186i	−0.515061	−	0.892111i	−0.999847	−	0.0174789i	\(-0.994436\pi\)
							0.484786	−	0.874633i	\(-0.338897\pi\)
\(29\)		−	32.7592i		−	1.12963i	−0.825219	−	0.564813i	\(-0.808948\pi\)
							0.825219	−	0.564813i	\(-0.191052\pi\)
\(31\)	−23.4865	+	40.6798i	−0.757629	+	1.31225i	0.186427	+	0.982469i	\(0.440309\pi\)
							−0.944057	+	0.329784i	\(0.893024\pi\)
\(37\)	−41.2822	+	23.8343i	−1.11573	+	0.644170i	−0.940309	−	0.340323i	\(-0.889464\pi\)
							−0.175426	+	0.984493i	\(0.556130\pi\)
\(41\)			70.7266i			1.72504i	0.506024	+	0.862520i	\(0.331115\pi\)
							−0.506024	+	0.862520i	\(0.668885\pi\)
\(43\)			14.1244i			0.328474i	0.986421	+	0.164237i	\(0.0525162\pi\)
							−0.986421	+	0.164237i	\(0.947484\pi\)
\(47\)	−26.1140	−	45.2307i	−0.555616	−	0.962356i	−0.997855	−	0.0654585i	\(-0.979149\pi\)
							0.442239	−	0.896897i	\(-0.354184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.o.b.74.12	yes	40
3.2	odd	2	inner	105.3.o.b.74.10	yes	40
5.4	even	2	inner	105.3.o.b.74.9	yes	40
7.2	even	3	inner	105.3.o.b.44.11	yes	40
15.14	odd	2	inner	105.3.o.b.74.11	yes	40
21.2	odd	6	inner	105.3.o.b.44.9	✓	40
35.9	even	6	inner	105.3.o.b.44.10	yes	40
105.44	odd	6	inner	105.3.o.b.44.12	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.o.b.44.9	✓	40	21.2	odd	6	inner
105.3.o.b.44.10	yes	40	35.9	even	6	inner
105.3.o.b.44.11	yes	40	7.2	even	3	inner
105.3.o.b.44.12	yes	40	105.44	odd	6	inner
105.3.o.b.74.9	yes	40	5.4	even	2	inner
105.3.o.b.74.10	yes	40	3.2	odd	2	inner
105.3.o.b.74.11	yes	40	15.14	odd	2	inner
105.3.o.b.74.12	yes	40	1.1	even	1	trivial