Embedded newform 95.3.h.a.69.2

• Label: 95.3.h.a.69.2
• Level: $95$
• Weight: $3$
• Character: 95.69
• Analytic conductor: $2.589$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	95.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.58856251142\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			69.2
Character	\(\chi\)	\(=\)	95.69
Dual form			95.3.h.a.84.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71056 - 2.96278i) q^{2} +(-0.826102 - 1.43085i) q^{3} +(-3.85204 + 6.67193i) q^{4} +(-0.955782 + 4.90780i) q^{5} +(-2.82620 + 4.89512i) q^{6} +10.6356i q^{7} +12.6721 q^{8} +(3.13511 - 5.43017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 32 q^{4} - 2 q^{5} + 20 q^{6} - 44 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71056	−	2.96278i	−0.855281	−	1.48139i	−0.876384	−	0.481613i	\(-0.840051\pi\)
							0.0211035	−	0.999777i	\(-0.493282\pi\)
\(3\)	−0.826102	−	1.43085i	−0.275367	−	0.476950i	0.694860	−	0.719145i	\(-0.255466\pi\)
							−0.970228	+	0.242194i	\(0.922133\pi\)
\(5\)	−0.955782	+	4.90780i	−0.191156	+	0.981560i
							\(7\)			10.6356i			1.51937i	0.650289	+	0.759687i	\(0.274648\pi\)
−0.650289	+	0.759687i	\(0.725352\pi\)
\(11\)	−20.6808			−1.88007			−0.940035	−	0.341078i	\(-0.889208\pi\)
							−0.940035	+	0.341078i	\(0.889208\pi\)
\(13\)	−0.159956	+	0.277052i	−0.0123043	+	0.0213117i	−0.872112	−	0.489306i	\(-0.837250\pi\)
							0.859808	+	0.510618i	\(0.170583\pi\)
\(17\)	−8.34411	+	4.81747i	−0.490830	+	0.283381i	−0.724919	−	0.688834i	\(-0.758123\pi\)
							0.234089	+	0.972215i	\(0.424789\pi\)
\(19\)	8.58254	+	16.9511i	0.451713	+	0.892163i
							\(23\)	22.2695	+	12.8573i	0.968238	+	0.559012i	0.898699	−	0.438567i	\(-0.144514\pi\)
0.0695394	+	0.997579i	\(0.477847\pi\)
\(29\)	−25.4279	−	14.6808i	−0.876824	−	0.506234i	−0.00721376	−	0.999974i	\(-0.502296\pi\)
							−0.869610	+	0.493740i	\(0.835630\pi\)
\(31\)			28.9056i			0.932440i	0.884669	+	0.466220i	\(0.154384\pi\)
							−0.884669	+	0.466220i	\(0.845616\pi\)
\(37\)	−32.9041			−0.889300			−0.444650	−	0.895704i	\(-0.646672\pi\)
							−0.444650	+	0.895704i	\(0.646672\pi\)
\(41\)	−3.71221	+	2.14325i	−0.0905418	+	0.0522743i	−0.544587	−	0.838704i	\(-0.683314\pi\)
							0.454045	+	0.890979i	\(0.349980\pi\)
\(43\)	13.8937	−	8.02156i	0.323110	−	0.186548i	−0.329668	−	0.944097i	\(-0.606937\pi\)
							0.652778	+	0.757549i	\(0.273603\pi\)
\(47\)	29.8528	+	17.2355i	0.635166	+	0.366714i	0.782750	−	0.622336i	\(-0.213816\pi\)
							−0.147584	+	0.989050i	\(0.547150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.3.h.a.69.2	✓	36
5.4	even	2	inner	95.3.h.a.69.17	yes	36
19.8	odd	6	inner	95.3.h.a.84.17	yes	36
95.84	odd	6	inner	95.3.h.a.84.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.h.a.69.2	✓	36	1.1	even	1	trivial
95.3.h.a.69.17	yes	36	5.4	even	2	inner
95.3.h.a.84.2	yes	36	95.84	odd	6	inner
95.3.h.a.84.17	yes	36	19.8	odd	6	inner