Embedded newform 187.2.j.a.135.4

• Label: 187.2.j.a.135.4
• Level: $187$
• Weight: $2$
• Character: 187.135
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			135.4
Character	\(\chi\)	\(=\)	187.135
Dual form			187.2.j.a.169.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68340 + 1.22306i) q^{2} +(0.00997681 - 0.00324166i) q^{3} +(0.719917 - 2.21568i) q^{4} +(-0.572549 + 0.788046i) q^{5} +(-0.0128302 + 0.0176592i) q^{6} +(3.51500 + 1.14209i) q^{7} +(0.211997 + 0.652459i) q^{8} +(-2.42696 + 1.76329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68340	+	1.22306i	−1.19034	+	0.864834i	−0.993300	−	0.115562i	\(-0.963133\pi\)
							−0.197041	+	0.980395i	\(0.563133\pi\)
\(3\)	0.00997681	−	0.00324166i	0.00576012	−	0.00187157i	−0.306136	−	0.951988i	\(-0.599036\pi\)
							0.311896	+	0.950116i	\(0.399036\pi\)
\(5\)	−0.572549	+	0.788046i	−0.256052	+	0.352425i	−0.917619	−	0.397461i	\(-0.869892\pi\)
							0.661567	+	0.749886i	\(0.269892\pi\)
\(7\)	3.51500	+	1.14209i	1.32855	+	0.431671i	0.885422	−	0.464789i	\(-0.153870\pi\)
							0.443125	+	0.896460i	\(0.353870\pi\)
\(11\)	1.16291	+	3.10606i	0.350631	+	0.936514i
							\(13\)	−2.42207	+	1.75974i	−0.671761	+	0.488063i	−0.870614	−	0.491966i	\(-0.836278\pi\)
0.198853	+	0.980029i	\(0.436278\pi\)
\(17\)	−4.10089	−	0.427445i	−0.994612	−	0.103671i
							\(19\)	−0.957240	−	2.94608i	−0.219606	−	0.675877i	−0.998794	−	0.0490878i	\(-0.984369\pi\)
0.779189	−	0.626790i	\(-0.215631\pi\)
\(23\)			6.57541i			1.37107i	0.728041	+	0.685534i	\(0.240431\pi\)
							−0.728041	+	0.685534i	\(0.759569\pi\)
\(29\)	−4.03317	−	1.31046i	−0.748941	−	0.243346i	−0.0904156	−	0.995904i	\(-0.528820\pi\)
							−0.658526	+	0.752558i	\(0.728820\pi\)
\(31\)	−3.16431	−	4.35530i	−0.568327	−	0.782235i	0.424028	−	0.905649i	\(-0.360616\pi\)
							−0.992355	+	0.123414i	\(0.960616\pi\)
\(37\)	11.1662	+	3.62812i	1.83571	+	0.596459i	0.998792	+	0.0491333i	\(0.0156459\pi\)
							0.836920	+	0.547326i	\(0.184354\pi\)
\(41\)	11.2375	−	3.65128i	1.75500	−	0.570233i	0.758336	−	0.651864i	\(-0.226013\pi\)
							0.996663	+	0.0816309i	\(0.0260129\pi\)
\(43\)	4.38252			0.668327			0.334164	−	0.942515i	\(-0.391546\pi\)
							0.334164	+	0.942515i	\(0.391546\pi\)
\(47\)	2.21635	+	6.82124i	0.323288	+	0.994980i	0.972207	+	0.234122i	\(0.0752215\pi\)
							−0.648919	+	0.760858i	\(0.724779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.135.4	yes	64
11.4	even	5	inner	187.2.j.a.169.3	yes	64
17.16	even	2	inner	187.2.j.a.135.3	✓	64
187.169	even	10	inner	187.2.j.a.169.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.135.3	✓	64	17.16	even	2	inner
187.2.j.a.135.4	yes	64	1.1	even	1	trivial
187.2.j.a.169.3	yes	64	11.4	even	5	inner
187.2.j.a.169.4	yes	64	187.169	even	10	inner