Embedded newform 155.2.n.a.64.1

• Label: 155.2.n.a.64.1
• Level: $155$
• Weight: $2$
• Character: 155.64
• Analytic conductor: $1.238$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 155 = 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	155.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.23768123133\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			64.1
Character	\(\chi\)	\(=\)	155.64
Dual form			155.2.n.a.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59301 + 2.19260i) q^{2} +(0.0633434 + 0.0871848i) q^{3} +(-1.65175 - 5.08355i) q^{4} +(0.708307 - 2.12092i) q^{5} -0.292068 q^{6} +(1.26925 - 0.412403i) q^{7} +(8.62233 + 2.80156i) q^{8} +(0.923462 - 2.84212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 2 q^{4} - 10 q^{5} - 4 q^{6} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\)	\(32\)	\(96\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59301	+	2.19260i	−1.12643	+	1.55040i	−0.331754	+	0.943366i	\(0.607640\pi\)
							−0.794677	+	0.607033i	\(0.792360\pi\)
\(3\)	0.0633434	+	0.0871848i	0.0365714	+	0.0503361i	0.826911	−	0.562333i	\(-0.190096\pi\)
							−0.790340	+	0.612669i	\(0.790096\pi\)
\(5\)	0.708307	−	2.12092i	0.316764	−	0.948504i
							\(7\)	1.26925	−	0.412403i	0.479730	−	0.155874i	−0.0591622	−	0.998248i	\(-0.518843\pi\)
0.538892	+	0.842375i	\(0.318843\pi\)
\(11\)	0.645600	+	1.98695i	0.194656	+	0.599088i	0.999980	+	0.00625456i	\(0.00199090\pi\)
							−0.805325	+	0.592834i	\(0.798009\pi\)
\(13\)	−3.10754	−	4.27716i	−0.861877	−	1.18627i	−0.981119	−	0.193406i	\(-0.938046\pi\)
							0.119242	−	0.992865i	\(-0.461954\pi\)
\(17\)	0.510911	+	0.166005i	0.123914	+	0.0402621i	0.370318	−	0.928905i	\(-0.379249\pi\)
							−0.246404	+	0.969167i	\(0.579249\pi\)
\(19\)	1.33255	+	0.968155i	0.305708	+	0.222110i	0.730053	−	0.683391i	\(-0.239495\pi\)
							−0.424345	+	0.905501i	\(0.639495\pi\)
\(23\)	4.84540	+	1.57437i	1.01034	+	0.328278i	0.766988	−	0.641661i	\(-0.221754\pi\)
							0.243348	+	0.969939i	\(0.421754\pi\)
\(29\)	5.49980	+	3.99584i	1.02129	+	0.742008i	0.966546	−	0.256493i	\(-0.0825670\pi\)
							0.0547401	+	0.998501i	\(0.482567\pi\)
\(31\)	−1.02157	−	5.47324i	−0.183480	−	0.983023i
							\(37\)			9.47746i			1.55808i	0.626972	+	0.779042i	\(0.284294\pi\)
−0.626972	+	0.779042i	\(0.715706\pi\)
\(41\)	−4.83648	−	3.51391i	−0.755331	−	0.548780i	0.142144	−	0.989846i	\(-0.454600\pi\)
							−0.897475	+	0.441066i	\(0.854600\pi\)
\(43\)	5.57251	−	7.66990i	0.849799	−	1.16965i	−0.134108	−	0.990967i	\(-0.542817\pi\)
							0.983907	−	0.178682i	\(-0.0571832\pi\)
\(47\)	−0.330049	−	0.454274i	−0.0481426	−	0.0662626i	0.784267	−	0.620423i	\(-0.213039\pi\)
							−0.832410	+	0.554161i	\(0.813039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	155.2.n.a.64.1	✓	56
5.2	odd	4		775.2.k.h.126.1		56
5.3	odd	4		775.2.k.h.126.14		56
5.4	even	2	inner	155.2.n.a.64.14	yes	56
31.16	even	5	inner	155.2.n.a.109.14	yes	56
155.47	odd	20		775.2.k.h.326.1		56
155.78	odd	20		775.2.k.h.326.14		56
155.109	even	10	inner	155.2.n.a.109.1	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.n.a.64.1	✓	56	1.1	even	1	trivial
155.2.n.a.64.14	yes	56	5.4	even	2	inner
155.2.n.a.109.1	yes	56	155.109	even	10	inner
155.2.n.a.109.14	yes	56	31.16	even	5	inner
775.2.k.h.126.1		56	5.2	odd	4
775.2.k.h.126.14		56	5.3	odd	4
775.2.k.h.326.1		56	155.47	odd	20
775.2.k.h.326.14		56	155.78	odd	20