Embedded newform 150.6.g.c.31.3

• Label: 150.6.g.c.31.3
• Level: $150$
• Weight: $6$
• Character: 150.31
• Analytic conductor: $24.058$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0575729719\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	150.31
Dual form			150.6.g.c.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23607 - 3.80423i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-12.9443 + 9.40456i) q^{4} +(-5.54795 - 55.6257i) q^{5} +(29.1246 + 21.1603i) q^{6} -223.096 q^{7} +(51.7771 + 37.6183i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} - 54 q^{3} - 96 q^{4} + 80 q^{5} + 216 q^{6} - 454 q^{7} + 384 q^{8} - 486 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23607	−	3.80423i	−0.218508	−	0.672499i
							\(3\)	−7.28115	+	5.29007i	−0.467086	+	0.339358i
\(5\)	−5.54795	−	55.6257i	−0.0992448	−	0.995063i
							\(7\)	−223.096			−1.72087			−0.860433	−	0.509563i	\(-0.829807\pi\)
−0.860433	+	0.509563i	\(0.829807\pi\)
\(11\)	−72.9553	−	224.533i	−0.181792	−	0.559499i	0.818086	−	0.575096i	\(-0.195035\pi\)
							−0.999878	+	0.0155968i	\(0.995035\pi\)
\(13\)	31.4259	−	96.7191i	0.0515739	−	0.158728i	−0.921952	−	0.387303i	\(-0.873407\pi\)
							0.973526	+	0.228575i	\(0.0734067\pi\)
\(17\)	−174.146	−	126.524i	−0.146147	−	0.106182i	0.512309	−	0.858801i	\(-0.328790\pi\)
							−0.658456	+	0.752619i	\(0.728790\pi\)
\(19\)	−1645.08	−	1195.22i	−1.04545	−	0.759564i	−0.0741087	−	0.997250i	\(-0.523611\pi\)
							−0.971342	+	0.237686i	\(0.923611\pi\)
\(23\)	1223.60	+	3765.84i	0.482302	+	1.48437i	0.835851	+	0.548956i	\(0.184975\pi\)
							−0.353549	+	0.935416i	\(0.615025\pi\)
\(29\)	3244.64	−	2357.37i	0.716426	−	0.520514i	−0.168815	−	0.985648i	\(-0.553994\pi\)
							0.885240	+	0.465134i	\(0.153994\pi\)
\(31\)	3888.23	+	2824.96i	0.726687	+	0.527969i	0.888514	−	0.458850i	\(-0.151739\pi\)
							−0.161827	+	0.986819i	\(0.551739\pi\)
\(37\)	−3192.15	+	9824.43i	−0.383335	+	1.17979i	0.554345	+	0.832287i	\(0.312969\pi\)
							−0.937681	+	0.347498i	\(0.887031\pi\)
\(41\)	−729.374	+	2244.78i	−0.0677627	+	0.208552i	−0.979204	−	0.202878i	\(-0.934971\pi\)
							0.911441	+	0.411430i	\(0.134971\pi\)
\(43\)	19448.8			1.60406			0.802030	−	0.597284i	\(-0.203753\pi\)
							0.802030	+	0.597284i	\(0.203753\pi\)
\(47\)	−5055.53	+	3673.06i	−0.333827	+	0.242540i	−0.742053	−	0.670341i	\(-0.766148\pi\)
							0.408225	+	0.912881i	\(0.366148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.c.31.3	✓	24
25.21	even	5	inner	150.6.g.c.121.3	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.c.31.3	✓	24	1.1	even	1	trivial
150.6.g.c.121.3	yes	24	25.21	even	5	inner