Embedded newform 105.2.u.a.73.4

• Label: 105.2.u.a.73.4
• Level: $105$
• Weight: $2$
• Character: 105.73
• 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			73.4
Character	\(\chi\)	\(=\)	105.73
Dual form			105.2.u.a.82.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.394487 - 0.105703i) q^{2} +(-0.258819 - 0.965926i) q^{3} +(-1.58760 - 0.916603i) q^{4} +(-2.18897 - 0.456535i) q^{5} +0.408404i q^{6} +(-0.605712 - 2.57548i) q^{7} +(1.10697 + 1.10697i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 12 q^{5} + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)	\(71\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.394487	−	0.105703i	−0.278945	−	0.0747430i	0.116634	−	0.993175i	\(-0.462789\pi\)
							−0.395579	+	0.918432i	\(0.629456\pi\)
\(3\)	−0.258819	−	0.965926i	−0.149429	−	0.557678i
							\(5\)	−2.18897	−	0.456535i	−0.978936	−	0.204169i
\(7\)	−0.605712	−	2.57548i	−0.228938	−	0.973441i
							\(11\)	−0.463738	+	0.803218i	−0.139822	+	0.242179i	−0.927429	−	0.373999i	\(-0.877986\pi\)
0.787607	+	0.616178i	\(0.211320\pi\)
\(13\)	4.08169	−	4.08169i	1.13206	−	1.13206i	0.142224	−	0.989835i	\(-0.454575\pi\)
							0.989835	−	0.142224i	\(-0.0454252\pi\)
\(17\)	−0.719705	+	0.192844i	−0.174554	+	0.0467716i	−0.345037	−	0.938589i	\(-0.612134\pi\)
							0.170483	+	0.985361i	\(0.445467\pi\)
\(19\)	−1.21966	−	2.11252i	−0.279810	−	0.484644i	0.691528	−	0.722350i	\(-0.256938\pi\)
							−0.971337	+	0.237706i	\(0.923605\pi\)
\(23\)	1.34176	−	5.00751i	0.279776	−	1.04414i	−0.672797	−	0.739827i	\(-0.734907\pi\)
							0.952573	−	0.304311i	\(-0.0984261\pi\)
\(29\)			8.08080i			1.50057i	0.661116	+	0.750284i	\(0.270083\pi\)
							−0.661116	+	0.750284i	\(0.729917\pi\)
\(31\)	−1.05279	−	0.607827i	−0.189086	−	0.109169i	0.402469	−	0.915434i	\(-0.368152\pi\)
							−0.591555	+	0.806265i	\(0.701486\pi\)
\(37\)	1.76271	+	0.472316i	0.289787	+	0.0776483i	0.400784	−	0.916172i	\(-0.368738\pi\)
							−0.110997	+	0.993821i	\(0.535404\pi\)
\(41\)		−	6.97323i		−	1.08903i	−0.838749	−	0.544517i	\(-0.816713\pi\)
							0.838749	−	0.544517i	\(-0.183287\pi\)
\(43\)	−0.781574	−	0.781574i	−0.119189	−	0.119189i	0.644997	−	0.764185i	\(-0.276859\pi\)
							−0.764185	+	0.644997i	\(0.776859\pi\)
\(47\)	2.70351	−	10.0896i	0.394347	−	1.47172i	−0.428542	−	0.903522i	\(-0.640973\pi\)
							0.822889	−	0.568202i	\(-0.192361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.u.a.73.4	yes	32
3.2	odd	2		315.2.bz.d.73.5		32
5.2	odd	4	inner	105.2.u.a.52.5	✓	32
5.3	odd	4		525.2.bc.e.157.4		32
5.4	even	2		525.2.bc.e.493.5		32
7.2	even	3		735.2.v.b.313.5		32
7.3	odd	6		735.2.m.c.538.10		32
7.4	even	3		735.2.m.c.538.9		32
7.5	odd	6	inner	105.2.u.a.103.5	yes	32
7.6	odd	2		735.2.v.b.178.4		32
15.2	even	4		315.2.bz.d.262.4		32
21.5	even	6		315.2.bz.d.208.4		32
35.2	odd	12		735.2.v.b.607.4		32
35.12	even	12	inner	105.2.u.a.82.4	yes	32
35.17	even	12		735.2.m.c.97.9		32
35.19	odd	6		525.2.bc.e.418.4		32
35.27	even	4		735.2.v.b.472.5		32
35.32	odd	12		735.2.m.c.97.10		32
35.33	even	12		525.2.bc.e.82.5		32
105.47	odd	12		315.2.bz.d.82.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.u.a.52.5	✓	32	5.2	odd	4	inner
105.2.u.a.73.4	yes	32	1.1	even	1	trivial
105.2.u.a.82.4	yes	32	35.12	even	12	inner
105.2.u.a.103.5	yes	32	7.5	odd	6	inner
315.2.bz.d.73.5		32	3.2	odd	2
315.2.bz.d.82.5		32	105.47	odd	12
315.2.bz.d.208.4		32	21.5	even	6
315.2.bz.d.262.4		32	15.2	even	4
525.2.bc.e.82.5		32	35.33	even	12
525.2.bc.e.157.4		32	5.3	odd	4
525.2.bc.e.418.4		32	35.19	odd	6
525.2.bc.e.493.5		32	5.4	even	2
735.2.m.c.97.9		32	35.17	even	12
735.2.m.c.97.10		32	35.32	odd	12
735.2.m.c.538.9		32	7.4	even	3
735.2.m.c.538.10		32	7.3	odd	6
735.2.v.b.178.4		32	7.6	odd	2
735.2.v.b.313.5		32	7.2	even	3
735.2.v.b.472.5		32	35.27	even	4
735.2.v.b.607.4		32	35.2	odd	12