Embedded newform 75.6.i.a.4.3

• Label: 75.6.i.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.71565 - 2.83189i) q^{2} +(5.29007 - 7.28115i) q^{3} +(42.0544 + 30.5543i) q^{4} +(-15.1679 - 53.8046i) q^{5} +(-66.7257 + 48.4791i) q^{6} -6.33250i q^{7} +(-107.635 - 148.146i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.71565	−	2.83189i	−1.54072	−	0.500611i	−0.589147	−	0.808026i	\(-0.700536\pi\)
							−0.951576	+	0.307414i	\(0.900536\pi\)
\(3\)	5.29007	−	7.28115i	0.339358	−	0.467086i
							\(5\)	−15.1679	−	53.8046i	−0.271331	−	0.962486i
\(7\)		−	6.33250i		−	0.0488461i								−0.999702	−	0.0244231i	\(-0.992225\pi\)
							0.999702	−	0.0244231i	\(-0.00777487\pi\)
\(11\)	186.031	−	572.544i	0.463557	−	1.42668i	−0.397230	−	0.917719i	\(-0.630029\pi\)
							0.860788	−	0.508964i	\(-0.169971\pi\)
\(13\)	1043.86	−	339.172i	1.71311	−	0.556623i	0.722264	−	0.691618i	\(-0.243102\pi\)
							0.990846	+	0.134994i	\(0.0431016\pi\)
\(17\)	−419.950	−	578.011i	−0.352432	−	0.485081i	0.595589	−	0.803289i	\(-0.296919\pi\)
							−0.948021	+	0.318209i	\(0.896919\pi\)
\(19\)	−1841.25	+	1337.75i	−1.17012	+	0.850140i	−0.991023	−	0.133692i	\(-0.957317\pi\)
							−0.179094	+	0.983832i	\(0.557317\pi\)
\(23\)	2371.31	+	770.487i	0.934694	+	0.303701i	0.736481	−	0.676458i	\(-0.236486\pi\)
							0.198213	+	0.980159i	\(0.436486\pi\)
\(29\)	−6378.02	−	4633.90i	−1.40829	−	1.02318i	−0.993569	−	0.113230i	\(-0.963880\pi\)
							−0.414717	−	0.909950i	\(-0.636120\pi\)
\(31\)	−3932.82	+	2857.36i	−0.735020	+	0.534024i	−0.891148	−	0.453713i	\(-0.850099\pi\)
							0.156127	+	0.987737i	\(0.450099\pi\)
\(37\)	−165.756	+	53.8575i	−0.0199052	+	0.00646759i	−0.318953	−	0.947771i	\(-0.603331\pi\)
							0.299047	+	0.954238i	\(0.403331\pi\)
\(41\)	−310.077	−	954.319i	−0.0288078	−	0.0886613i	0.935619	−	0.353012i	\(-0.114842\pi\)
							−0.964427	+	0.264350i	\(0.914842\pi\)
\(43\)		−	4256.28i		−	0.351042i	−0.984476	−	0.175521i	\(-0.943839\pi\)
							0.984476	−	0.175521i	\(-0.0561610\pi\)
\(47\)	−4518.87	+	6219.69i	−0.298391	+	0.410699i	−0.931717	−	0.363186i	\(-0.881689\pi\)
							0.633326	+	0.773885i	\(0.281689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.3	✓	96
25.19	even	10	inner	75.6.i.a.19.3	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.3	✓	96	1.1	even	1	trivial
75.6.i.a.19.3	yes	96	25.19	even	10	inner