Embedded newform 75.6.i.a.19.18

• Label: 75.6.i.a.19.18
• Level: $75$
• Weight: $6$
• Character: 75.19
• 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			19.18
Character	\(\chi\)	\(=\)	75.19
Dual form			75.6.i.a.4.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.18267 - 1.68395i) q^{2} +(5.29007 + 7.28115i) q^{3} +(-1.86417 + 1.35440i) q^{4} +(-41.4891 + 37.4654i) q^{5} +(39.6778 + 28.8276i) q^{6} -121.054i q^{7} +(-109.879 + 151.235i) q^{8} +(-25.0304 + 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.18267	−	1.68395i	0.916175	−	0.297683i	0.187278	−	0.982307i	\(-0.440033\pi\)
							0.728897	+	0.684623i	\(0.240033\pi\)
\(3\)	5.29007	+	7.28115i	0.339358	+	0.467086i
							\(5\)	−41.4891	+	37.4654i	−0.742180	+	0.670201i
\(7\)		−	121.054i		−	0.933758i								−0.884321	−	0.466879i	\(-0.845378\pi\)
							0.884321	−	0.466879i	\(-0.154622\pi\)
\(11\)	212.022	+	652.537i	0.528323	+	1.62601i	0.757650	+	0.652661i	\(0.226348\pi\)
							−0.229327	+	0.973349i	\(0.573652\pi\)
\(13\)	−953.707	−	309.878i	−1.56515	−	0.508549i	−0.606974	−	0.794722i	\(-0.707617\pi\)
							−0.958178	+	0.286173i	\(0.907617\pi\)
\(17\)	−690.177	+	949.947i	−0.579213	+	0.797218i	−0.993609	−	0.112879i	\(-0.963993\pi\)
							0.414396	+	0.910097i	\(0.363993\pi\)
\(19\)	464.056	+	337.156i	0.294908	+	0.214263i	0.725394	−	0.688334i	\(-0.241658\pi\)
							−0.430486	+	0.902597i	\(0.641658\pi\)
\(23\)	1217.89	−	395.718i	0.480054	−	0.155979i	−0.0589867	−	0.998259i	\(-0.518787\pi\)
							0.539041	+	0.842280i	\(0.318787\pi\)
\(29\)	1841.56	−	1337.97i	0.406621	−	0.295428i	−0.365611	−	0.930768i	\(-0.619140\pi\)
							0.772233	+	0.635340i	\(0.219140\pi\)
\(31\)	4922.19	+	3576.18i	0.919929	+	0.668367i	0.943506	−	0.331355i	\(-0.107506\pi\)
							−0.0235777	+	0.999722i	\(0.507506\pi\)
\(37\)	627.273	+	203.813i	0.0753273	+	0.0244753i	0.346438	−	0.938073i	\(-0.387391\pi\)
							−0.271111	+	0.962548i	\(0.587391\pi\)
\(41\)	1731.94	−	5330.36i	0.160906	−	0.495218i	−0.837805	−	0.545969i	\(-0.816162\pi\)
							0.998711	+	0.0507511i	\(0.0161615\pi\)
\(43\)			10057.6i			0.829512i	0.909933	+	0.414756i	\(0.136133\pi\)
							−0.909933	+	0.414756i	\(0.863867\pi\)
\(47\)	12527.2	+	17242.2i	0.827196	+	1.13854i	0.988438	+	0.151623i	\(0.0484499\pi\)
							−0.161242	+	0.986915i	\(0.551550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.19.18	yes	96
25.4	even	10	inner	75.6.i.a.4.18	✓	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.18	✓	96	25.4	even	10	inner
75.6.i.a.19.18	yes	96	1.1	even	1	trivial