Embedded newform 309.3.h.b.56.4

• Label: 309.3.h.b.56.4
• Level: $309$
• Weight: $3$
• Character: 309.56
• Analytic conductor: $8.420$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	309.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.41964016873\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			56.4
Character	\(\chi\)	\(=\)	309.56
Dual form			309.3.h.b.149.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.03097 - 1.74993i) q^{2} +(2.46539 + 1.70935i) q^{3} +(4.12451 + 7.14386i) q^{4} +(3.52313 - 2.03408i) q^{5} +(-4.48128 - 9.49523i) q^{6} +(4.15103 + 7.18980i) q^{7} -14.8710i q^{8} +(3.15628 + 8.42840i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 4 q^{3} + 130 q^{4} + 7 q^{6} + 16 q^{7} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/309\mathbb{Z}\right)^\times\).

\(n\)	\(104\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.03097	−	1.74993i	−1.51548	−	0.874965i	−0.999835	−	0.0181649i	\(-0.994218\pi\)
							−0.515649	−	0.856800i	\(-0.672449\pi\)
\(3\)	2.46539	+	1.70935i	0.821796	+	0.569782i
							\(5\)	3.52313	−	2.03408i	0.704626	−	0.406816i	−0.104442	−	0.994531i	\(-0.533306\pi\)
0.809068	+	0.587715i	\(0.199972\pi\)
\(7\)	4.15103	+	7.18980i	0.593004	+	1.02711i	0.993825	+	0.110957i	\(0.0353917\pi\)
							−0.400821	+	0.916157i	\(0.631275\pi\)
\(11\)	14.1765	−	8.18481i	1.28877	−	0.744074i	0.310338	−	0.950626i	\(-0.399558\pi\)
							0.978435	+	0.206553i	\(0.0662245\pi\)
\(13\)	0.931874			0.0716826			0.0358413	−	0.999357i	\(-0.488589\pi\)
							0.0358413	+	0.999357i	\(0.488589\pi\)
\(17\)	−27.0721	+	15.6301i	−1.59248	+	0.919416i	−0.599594	+	0.800304i	\(0.704671\pi\)
							−0.992881	+	0.119112i	\(0.961995\pi\)
\(19\)	10.0761	−	17.4523i	0.530320	−	0.918541i	−0.469054	−	0.883169i	\(-0.655405\pi\)
							0.999374	−	0.0353715i	\(-0.0112615\pi\)
\(23\)			23.2388i			1.01038i	0.863008	+	0.505191i	\(0.168578\pi\)
							−0.863008	+	0.505191i	\(0.831422\pi\)
\(29\)	12.6415	+	7.29857i	0.435913	+	0.251675i	0.701863	−	0.712312i	\(-0.252352\pi\)
							−0.265949	+	0.963987i	\(0.585685\pi\)
\(31\)	7.73350			0.249468			0.124734	−	0.992190i	\(-0.460192\pi\)
							0.124734	+	0.992190i	\(0.460192\pi\)
\(37\)	−34.6254			−0.935821			−0.467911	−	0.883776i	\(-0.654993\pi\)
							−0.467911	+	0.883776i	\(0.654993\pi\)
\(41\)	−18.5133	−	10.6887i	−0.451545	−	0.260699i	0.256938	−	0.966428i	\(-0.417286\pi\)
							−0.708482	+	0.705729i	\(0.750620\pi\)
\(43\)	20.0998	−	34.8138i	0.467436	−	0.809624i	−0.531871	−	0.846825i	\(-0.678511\pi\)
							0.999308	+	0.0372016i	\(0.0118444\pi\)
\(47\)	77.9836	−	45.0239i	1.65923	−	0.957954i	0.686153	−	0.727458i	\(-0.259298\pi\)
							0.973073	−	0.230497i	\(-0.0740351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.3.h.b.56.4	✓	132
3.2	odd	2	inner	309.3.h.b.56.63	yes	132
103.46	even	3	inner	309.3.h.b.149.63	yes	132
309.149	odd	6	inner	309.3.h.b.149.4	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.3.h.b.56.4	✓	132	1.1	even	1	trivial
309.3.h.b.56.63	yes	132	3.2	odd	2	inner
309.3.h.b.149.4	yes	132	309.149	odd	6	inner
309.3.h.b.149.63	yes	132	103.46	even	3	inner