Embedded newform 155.2.n.a.109.1

• Label: 155.2.n.a.109.1
• Level: $155$
• Weight: $2$
• Character: 155.109
• Analytic conductor: $1.238$
• Analytic rank: $0$
• Dimension: $56$
• 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			109.1
Character	\(\chi\)	\(=\)	155.109
Dual form			155.2.n.a.64.1

$q$-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 \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.n.a.64.1	✓	56	155.64	even	10	inner
155.2.n.a.64.14	yes	56	31.2	even	5	inner
155.2.n.a.109.1	yes	56	1.1	even	1	trivial
155.2.n.a.109.14	yes	56	5.4	even	2	inner
775.2.k.h.126.1		56	155.33	odd	20
775.2.k.h.126.14		56	155.2	odd	20
775.2.k.h.326.1		56	5.3	odd	4
775.2.k.h.326.14		56	5.2	odd	4