Embedded newform 76.3.g.c.7.2

• Label: 76.3.g.c.7.2
• Level: $76$
• Weight: $3$
• Character: 76.7
• Analytic conductor: $2.071$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.07085000914\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	76.7
Dual form			76.3.g.c.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93914 - 0.489616i) q^{2} +(-3.88623 + 2.24371i) q^{3} +(3.52055 + 1.89887i) q^{4} +(-0.133773 - 0.231701i) q^{5} +(8.63451 - 2.44812i) q^{6} -7.24937i q^{7} +(-5.89713 - 5.40590i) q^{8} +(5.56851 - 9.64495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 5 q^{2} - 11 q^{4} + 6 q^{5} - 3 q^{6} - 62 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93914	−	0.489616i	−0.969572	−	0.244808i
							\(3\)	−3.88623	+	2.24371i	−1.29541	+	0.747905i	−0.979608	−	0.200920i	\(-0.935607\pi\)
−0.315802	+	0.948825i	\(0.602273\pi\)
\(5\)	−0.133773	−	0.231701i	−0.0267546	−	0.0463403i	0.852338	−	0.522991i	\(-0.175184\pi\)
							−0.879093	+	0.476651i	\(0.841851\pi\)
\(7\)		−	7.24937i		−	1.03562i	−0.855495	−	0.517812i	\(-0.826747\pi\)
							0.855495	−	0.517812i	\(-0.173253\pi\)
\(11\)		−	11.7422i		−	1.06748i	−0.845650	−	0.533738i	\(-0.820787\pi\)
							0.845650	−	0.533738i	\(-0.179213\pi\)
\(13\)	4.17006	−	7.22276i	0.320774	−	0.555597i	−0.659874	−	0.751376i	\(-0.729390\pi\)
							0.980648	+	0.195780i	\(0.0627237\pi\)
\(17\)	−7.11880	−	12.3301i	−0.418753	−	0.725301i	0.577061	−	0.816701i	\(-0.304199\pi\)
							−0.995814	+	0.0913994i	\(0.970866\pi\)
\(19\)	11.9609	+	14.7627i	0.629522	+	0.776983i
							\(23\)	−21.4900	−	12.4073i	−0.934349	−	0.539447i	−0.0461649	−	0.998934i	\(-0.514700\pi\)
−0.888185	+	0.459487i	\(0.848033\pi\)
\(29\)	−25.7911	+	44.6716i	−0.889350	+	1.54040i	−0.0487047	+	0.998813i	\(0.515509\pi\)
							−0.840645	+	0.541586i	\(0.817824\pi\)
\(31\)		−	26.3372i		−	0.849587i	−0.905290	−	0.424793i	\(-0.860347\pi\)
							0.905290	−	0.424793i	\(-0.139653\pi\)
\(37\)	−20.1243			−0.543901			−0.271950	−	0.962311i	\(-0.587669\pi\)
							−0.271950	+	0.962311i	\(0.587669\pi\)
\(41\)	−38.5881	−	66.8366i	−0.941174	−	1.63016i	−0.763237	−	0.646119i	\(-0.776391\pi\)
							−0.177937	−	0.984042i	\(-0.556942\pi\)
\(43\)	−30.0199	+	17.3320i	−0.698138	+	0.403070i	−0.806654	−	0.591024i	\(-0.798724\pi\)
							0.108515	+	0.994095i	\(0.465390\pi\)
\(47\)	31.9968	+	18.4733i	0.680782	+	0.393050i	0.800150	−	0.599800i	\(-0.204753\pi\)
							−0.119367	+	0.992850i	\(0.538087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.3.g.c.7.2	✓	28
4.3	odd	2	inner	76.3.g.c.7.9	yes	28
19.11	even	3	inner	76.3.g.c.11.9	yes	28
76.11	odd	6	inner	76.3.g.c.11.2	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.g.c.7.2	✓	28	1.1	even	1	trivial
76.3.g.c.7.9	yes	28	4.3	odd	2	inner
76.3.g.c.11.2	yes	28	76.11	odd	6	inner
76.3.g.c.11.9	yes	28	19.11	even	3	inner