Embedded newform 76.7.l.a.23.6

• Label: 76.7.l.a.23.6
• Level: $76$
• Weight: $7$
• Character: 76.23
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(58\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.6
Character	\(\chi\)	\(=\)	76.23
Dual form			76.7.l.a.43.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.71608 + 2.11236i) q^{2} +(26.4348 - 4.66117i) q^{3} +(55.0759 - 32.5982i) q^{4} +(6.31656 - 5.30022i) q^{5} +(-194.127 + 91.8057i) q^{6} +(-142.890 - 82.4974i) q^{7} +(-356.111 + 367.871i) q^{8} +(-7.96412 + 2.89870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 192 q^{4} - 12 q^{5} - 492 q^{6} - 3 q^{8} + 1368 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.71608	+	2.11236i	−0.964510	+	0.264045i
							\(3\)	26.4348	−	4.66117i	0.979066	−	0.172636i	0.338858	−	0.940837i	\(-0.389959\pi\)
0.640208	+	0.768202i	\(0.278848\pi\)
\(5\)	6.31656	−	5.30022i	0.0505325	−	0.0424018i	−0.617172	−	0.786829i	\(-0.711722\pi\)
							0.667704	+	0.744427i	\(0.267277\pi\)
\(7\)	−142.890	−	82.4974i	−0.416588	−	0.240517i	0.277029	−	0.960862i	\(-0.410650\pi\)
							−0.693616	+	0.720345i	\(0.743984\pi\)
\(11\)	−1593.34	+	919.915i	−1.19710	+	0.691146i	−0.959907	−	0.280317i	\(-0.909560\pi\)
							−0.237192	+	0.971463i	\(0.576227\pi\)
\(13\)	−195.286	+	1107.52i	−0.0888875	+	0.504106i	0.907562	+	0.419917i	\(0.137941\pi\)
							−0.996450	+	0.0841886i	\(0.973170\pi\)
\(17\)	1286.92	+	468.400i	0.261941	+	0.0953389i	0.469653	−	0.882851i	\(-0.344379\pi\)
							−0.207711	+	0.978190i	\(0.566601\pi\)
\(19\)	6200.52	+	2932.47i	0.903998	+	0.427537i
							\(23\)	−587.608	+	700.284i	−0.0482952	+	0.0575560i	−0.789651	−	0.613556i	\(-0.789739\pi\)
0.741356	+	0.671112i	\(0.234183\pi\)
\(29\)	−32574.3	+	11856.1i	−1.33562	+	0.486125i	−0.908429	−	0.418039i	\(-0.862718\pi\)
							−0.427187	+	0.904163i	\(0.640495\pi\)
\(31\)	10678.3	+	6165.09i	0.358439	+	0.206945i	0.668396	−	0.743806i	\(-0.266981\pi\)
							−0.309957	+	0.950751i	\(0.600315\pi\)
\(37\)	−30468.8			−0.601521			−0.300761	−	0.953700i	\(-0.597240\pi\)
							−0.300761	+	0.953700i	\(0.597240\pi\)
\(41\)	−3866.56	−	21928.3i	−0.0561013	−	0.318166i	0.943823	−	0.330451i	\(-0.107201\pi\)
							−0.999925	+	0.0122845i	\(0.996090\pi\)
\(43\)	29628.1	+	35309.4i	0.372648	+	0.444104i	0.919479	−	0.393138i	\(-0.128611\pi\)
							−0.546832	+	0.837243i	\(0.684166\pi\)
\(47\)	51557.7	+	141654.i	0.496593	+	1.36438i	0.894548	+	0.446972i	\(0.147498\pi\)
							−0.397955	+	0.917405i	\(0.630280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.l.a.23.6	✓	348
4.3	odd	2	inner	76.7.l.a.23.18	yes	348
19.5	even	9	inner	76.7.l.a.43.18	yes	348
76.43	odd	18	inner	76.7.l.a.43.6	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.l.a.23.6	✓	348	1.1	even	1	trivial
76.7.l.a.23.18	yes	348	4.3	odd	2	inner
76.7.l.a.43.6	yes	348	76.43	odd	18	inner
76.7.l.a.43.18	yes	348	19.5	even	9	inner