Embedded newform 105.3.v.a.37.12

• Label: 105.3.v.a.37.12
• Level: $105$
• Weight: $3$
• Character: 105.37
• Analytic conductor: $2.861$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.86104277578\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.12
Character	\(\chi\)	\(=\)	105.37
Dual form			105.3.v.a.88.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13226 - 0.571337i) q^{2} +(1.67303 + 0.448288i) q^{3} +(0.756006 - 0.436480i) q^{4} +(2.78563 + 4.15214i) q^{5} +3.82346 q^{6} +(2.73668 - 6.44287i) q^{7} +(-4.88107 + 4.88107i) q^{8} +(2.59808 + 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13226	−	0.571337i	1.06613	−	0.285669i	0.317228	−	0.948349i	\(-0.397248\pi\)
							0.748902	+	0.662681i	\(0.230581\pi\)
\(3\)	1.67303	+	0.448288i	0.557678	+	0.149429i
							\(5\)	2.78563	+	4.15214i	0.557125	+	0.830429i
\(7\)	2.73668	−	6.44287i	0.390955	−	0.920410i
							\(11\)	−4.69282	−	8.12820i	−0.426620	−	0.738927i	0.569950	−	0.821679i	\(-0.306962\pi\)
−0.996570	+	0.0827519i	\(0.973629\pi\)
\(13\)	−0.405222	+	0.405222i	−0.0311709	+	0.0311709i	−0.722520	−	0.691350i	\(-0.757016\pi\)
							0.691350	+	0.722520i	\(0.257016\pi\)
\(17\)	2.82961	−	10.5602i	0.166448	−	0.621191i	−0.831404	−	0.555669i	\(-0.812462\pi\)
							0.997851	−	0.0655217i	\(-0.0208712\pi\)
\(19\)	−23.8500	−	13.7698i	−1.25526	−	0.724725i	−0.283111	−	0.959087i	\(-0.591367\pi\)
							−0.972149	+	0.234362i	\(0.924700\pi\)
\(23\)	1.02354	+	3.81990i	0.0445017	+	0.166083i	0.984601	−	0.174819i	\(-0.0559342\pi\)
							−0.940099	+	0.340902i	\(0.889268\pi\)
\(29\)			34.6601i			1.19518i	0.801803	+	0.597588i	\(0.203874\pi\)
							−0.801803	+	0.597588i	\(0.796126\pi\)
\(31\)	−11.8284	−	20.4874i	−0.381561	−	0.660883i	0.609724	−	0.792613i	\(-0.291280\pi\)
							−0.991286	+	0.131730i	\(0.957947\pi\)
\(37\)	25.6130	−	6.86297i	0.692242	−	0.185486i	0.104489	−	0.994526i	\(-0.466679\pi\)
							0.587753	+	0.809040i	\(0.300013\pi\)
\(41\)	54.6731			1.33349			0.666745	−	0.745286i	\(-0.267687\pi\)
							0.666745	+	0.745286i	\(0.267687\pi\)
\(43\)	−47.3942	+	47.3942i	−1.10219	+	1.10219i	−0.108044	+	0.994146i	\(0.534459\pi\)
							−0.994146	+	0.108044i	\(0.965541\pi\)
\(47\)	−16.9830	+	4.55058i	−0.361340	+	0.0968208i	−0.434921	−	0.900468i	\(-0.643224\pi\)
							0.0735811	+	0.997289i	\(0.476557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.3.v.a.37.12	✓	64
3.2	odd	2		315.3.ca.b.37.5		64
5.3	odd	4	inner	105.3.v.a.58.5	yes	64
7.4	even	3	inner	105.3.v.a.67.5	yes	64
15.8	even	4		315.3.ca.b.163.12		64
21.11	odd	6		315.3.ca.b.172.12		64
35.18	odd	12	inner	105.3.v.a.88.12	yes	64
105.53	even	12		315.3.ca.b.298.5		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.v.a.37.12	✓	64	1.1	even	1	trivial
105.3.v.a.58.5	yes	64	5.3	odd	4	inner
105.3.v.a.67.5	yes	64	7.4	even	3	inner
105.3.v.a.88.12	yes	64	35.18	odd	12	inner
315.3.ca.b.37.5		64	3.2	odd	2
315.3.ca.b.163.12		64	15.8	even	4
315.3.ca.b.172.12		64	21.11	odd	6
315.3.ca.b.298.5		64	105.53	even	12