Embedded newform 105.2.u.a.82.1

• Label: 105.2.u.a.82.1
• Level: $105$
• Weight: $2$
• Character: 105.82
• Analytic conductor: $0.838$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.838429221223\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			82.1
Character	\(\chi\)	\(=\)	105.82
Dual form			105.2.u.a.73.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.42777 + 0.650518i) q^{2} +(-0.258819 + 0.965926i) q^{3} +(3.73883 - 2.15861i) q^{4} +(0.780598 - 2.09539i) q^{5} -2.51341i q^{6} +(1.61838 + 2.09305i) q^{7} +(-4.11829 + 4.11829i) q^{8} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.42777	+	0.650518i	−1.71669	+	0.459986i	−0.977049	−	0.213015i	\(-0.931672\pi\)
							−0.739642	+	0.673001i	\(0.765005\pi\)
\(3\)	−0.258819	+	0.965926i	−0.149429	+	0.557678i
							\(5\)	0.780598	−	2.09539i	0.349094	−	0.937088i
\(7\)	1.61838	+	2.09305i	0.611691	+	0.791097i
							\(11\)	2.73807	+	4.74248i	0.825561	+	1.42991i	0.901490	+	0.432800i	\(0.142474\pi\)
−0.0759295	+	0.997113i	\(0.524192\pi\)
\(13\)	−0.579674	−	0.579674i	−0.160773	−	0.160773i	0.622136	−	0.782909i	\(-0.286265\pi\)
							−0.782909	+	0.622136i	\(0.786265\pi\)
\(17\)	4.58934	+	1.22971i	1.11308	+	0.298248i	0.768079	−	0.640355i	\(-0.221213\pi\)
							0.344999	+	0.938603i	\(0.387879\pi\)
\(19\)	0.220281	−	0.381538i	0.0505359	−	0.0875308i	−0.839651	−	0.543127i	\(-0.817240\pi\)
							0.890187	+	0.455596i	\(0.150574\pi\)
\(23\)	−0.457316	−	1.70673i	−0.0953570	−	0.355877i	0.901716	−	0.432328i	\(-0.142308\pi\)
							−0.997073	+	0.0764510i	\(0.975641\pi\)
\(29\)			0.853158i			0.158427i	0.996858	+	0.0792137i	\(0.0252410\pi\)
							−0.996858	+	0.0792137i	\(0.974759\pi\)
\(31\)	2.32463	−	1.34213i	0.417516	−	0.241053i	−0.276498	−	0.961014i	\(-0.589174\pi\)
							0.694014	+	0.719962i	\(0.255841\pi\)
\(37\)	−0.249579	+	0.0668745i	−0.0410306	+	0.0109941i	−0.279276	−	0.960211i	\(-0.590094\pi\)
							0.238245	+	0.971205i	\(0.423428\pi\)
\(41\)			0.321873i			0.0502681i	0.999684	+	0.0251341i	\(0.00800127\pi\)
							−0.999684	+	0.0251341i	\(0.991999\pi\)
\(43\)	−0.631635	+	0.631635i	−0.0963234	+	0.0963234i	−0.753626	−	0.657303i	\(-0.771697\pi\)
							0.657303	+	0.753626i	\(0.271697\pi\)
\(47\)	−2.12084	−	7.91508i	−0.309356	−	1.15453i	−0.929130	−	0.369752i	\(-0.879443\pi\)
							0.619774	−	0.784780i	\(-0.287224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.u.a.82.1	yes	32
3.2	odd	2		315.2.bz.d.82.8		32
5.2	odd	4		525.2.bc.e.418.1		32
5.3	odd	4	inner	105.2.u.a.103.8	yes	32
5.4	even	2		525.2.bc.e.82.8		32
7.2	even	3		735.2.m.c.97.15		32
7.3	odd	6	inner	105.2.u.a.52.8	✓	32
7.4	even	3		735.2.v.b.472.8		32
7.5	odd	6		735.2.m.c.97.16		32
7.6	odd	2		735.2.v.b.607.1		32
15.8	even	4		315.2.bz.d.208.1		32
21.17	even	6		315.2.bz.d.262.1		32
35.3	even	12	inner	105.2.u.a.73.1	yes	32
35.13	even	4		735.2.v.b.313.8		32
35.17	even	12		525.2.bc.e.493.8		32
35.18	odd	12		735.2.v.b.178.1		32
35.23	odd	12		735.2.m.c.538.16		32
35.24	odd	6		525.2.bc.e.157.1		32
35.33	even	12		735.2.m.c.538.15		32
105.38	odd	12		315.2.bz.d.73.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.8	✓	32	7.3	odd	6	inner
105.2.u.a.73.1	yes	32	35.3	even	12	inner
105.2.u.a.82.1	yes	32	1.1	even	1	trivial
105.2.u.a.103.8	yes	32	5.3	odd	4	inner
315.2.bz.d.73.8		32	105.38	odd	12
315.2.bz.d.82.8		32	3.2	odd	2
315.2.bz.d.208.1		32	15.8	even	4
315.2.bz.d.262.1		32	21.17	even	6
525.2.bc.e.82.8		32	5.4	even	2
525.2.bc.e.157.1		32	35.24	odd	6
525.2.bc.e.418.1		32	5.2	odd	4
525.2.bc.e.493.8		32	35.17	even	12
735.2.m.c.97.15		32	7.2	even	3
735.2.m.c.97.16		32	7.5	odd	6
735.2.m.c.538.15		32	35.33	even	12
735.2.m.c.538.16		32	35.23	odd	12
735.2.v.b.178.1		32	35.18	odd	12
735.2.v.b.313.8		32	35.13	even	4
735.2.v.b.472.8		32	7.4	even	3
735.2.v.b.607.1		32	7.6	odd	2