Embedded newform 75.6.i.a.4.10

• Label: 75.6.i.a.4.10
• Level: $75$
• Weight: $6$
• Character: 75.4
• Analytic conductor: $12.029$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.i (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.0287864860\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.10
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.43009 - 0.789586i) q^{2} +(-5.29007 + 7.28115i) q^{3} +(-20.6066 - 14.9716i) q^{4} +(-22.3194 + 51.2528i) q^{5} +(18.6045 - 13.5169i) q^{6} +116.776i q^{7} +(86.3149 + 118.802i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.43009	−	0.789586i	−0.429584	−	0.139580i	0.0862409	−	0.996274i	\(-0.472515\pi\)
							−0.515825	+	0.856694i	\(0.672515\pi\)
\(3\)	−5.29007	+	7.28115i	−0.339358	+	0.467086i
							\(5\)	−22.3194	+	51.2528i	−0.399261	+	0.916837i
\(7\)			116.776i			0.900755i								0.892838	+	0.450378i	\(0.148711\pi\)
							−0.892838	+	0.450378i	\(0.851289\pi\)
\(11\)	127.876	−	393.561i	0.318644	−	0.980686i	−0.655584	−	0.755122i	\(-0.727578\pi\)
							0.974228	−	0.225564i	\(-0.0724224\pi\)
\(13\)	−771.004	+	250.514i	−1.26531	+	0.411125i	−0.863385	−	0.504545i	\(-0.831660\pi\)
							−0.401929	+	0.915671i	\(0.631660\pi\)
\(17\)	−714.881	−	983.949i	−0.599945	−	0.825754i	0.395758	−	0.918355i	\(-0.370482\pi\)
							−0.995703	+	0.0926012i	\(0.970482\pi\)
\(19\)	2506.78	−	1821.28i	1.59306	−	1.15743i	0.693663	−	0.720300i	\(-0.255996\pi\)
							0.899400	−	0.437127i	\(-0.144004\pi\)
\(23\)	867.388	+	281.831i	0.341896	+	0.111089i	0.474932	−	0.880022i	\(-0.342473\pi\)
							−0.133036	+	0.991111i	\(0.542473\pi\)
\(29\)	−5684.48	−	4130.01i	−1.25515	−	0.911920i	−0.256641	−	0.966507i	\(-0.582616\pi\)
							−0.998509	+	0.0545873i	\(0.982616\pi\)
\(31\)	1534.69	−	1115.02i	0.286825	−	0.208391i	−0.435064	−	0.900400i	\(-0.643274\pi\)
							0.721889	+	0.692009i	\(0.243274\pi\)
\(37\)	12194.6	−	3962.27i	1.46441	−	0.475817i	0.534999	−	0.844852i	\(-0.320312\pi\)
							0.929415	+	0.369035i	\(0.120312\pi\)
\(41\)	489.616	+	1506.88i	0.0454879	+	0.139997i	0.971221	−	0.238180i	\(-0.0765509\pi\)
							−0.925733	+	0.378178i	\(0.876551\pi\)
\(43\)			12604.6i			1.03958i	0.854293	+	0.519792i	\(0.173990\pi\)
							−0.854293	+	0.519792i	\(0.826010\pi\)
\(47\)	−7118.31	+	9797.52i	−0.470038	+	0.646951i	−0.976552	−	0.215280i	\(-0.930934\pi\)
							0.506515	+	0.862231i	\(0.330934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.10	✓	96
25.19	even	10	inner	75.6.i.a.19.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.10	✓	96	1.1	even	1	trivial
75.6.i.a.19.10	yes	96	25.19	even	10	inner