Embedded newform 105.3.o.b.44.12

• Label: 105.3.o.b.44.12
• Level: $105$
• Weight: $3$
• Character: 105.44
• 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			44.12
Character	\(\chi\)	\(=\)	105.44
Dual form			105.3.o.b.74.12

$q$-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 \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.o.b.44.9	✓	40	5.4	even	2	inner
105.3.o.b.44.10	yes	40	3.2	odd	2	inner
105.3.o.b.44.11	yes	40	15.14	odd	2	inner
105.3.o.b.44.12	yes	40	1.1	even	1	trivial
105.3.o.b.74.9	yes	40	21.11	odd	6	inner
105.3.o.b.74.10	yes	40	35.4	even	6	inner
105.3.o.b.74.11	yes	40	7.4	even	3	inner
105.3.o.b.74.12	yes	40	105.74	odd	6	inner