Embedded newform 76.7.l.a.23.10

• Label: 76.7.l.a.23.10
• 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			23.10
Character	\(\chi\)	\(=\)	76.23
Dual form			76.7.l.a.43.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.18899 - 3.50976i) q^{2} +(-51.9029 + 9.15188i) q^{3} +(39.3632 + 50.4632i) q^{4} +(-46.4533 + 38.9790i) q^{5} +(405.250 + 116.374i) q^{6} +(-69.3277 - 40.0264i) q^{7} +(-105.868 - 500.935i) q^{8} +(1925.12 - 700.686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 192 q^{4} - 12 q^{5} - 492 q^{6} - 3 q^{8} + 1368 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.18899	−	3.50976i	−0.898624	−	0.438720i
							\(3\)	−51.9029	+	9.15188i	−1.92233	+	0.338959i	−0.998986	−	0.0450157i	\(-0.985666\pi\)
−0.923344	+	0.383974i	\(0.874555\pi\)
\(5\)	−46.4533	+	38.9790i	−0.371627	+	0.311832i	−0.809405	−	0.587251i	\(-0.800210\pi\)
							0.437778	+	0.899083i	\(0.355766\pi\)
\(7\)	−69.3277	−	40.0264i	−0.202122	−	0.116695i	0.395523	−	0.918456i	\(-0.370563\pi\)
							−0.597645	+	0.801761i	\(0.703897\pi\)
\(11\)	−272.825	+	157.516i	−0.204978	+	0.118344i	−0.598975	−	0.800768i	\(-0.704425\pi\)
							0.393998	+	0.919112i	\(0.371092\pi\)
\(13\)	384.519	−	2180.72i	0.175020	−	0.992588i	−0.763101	−	0.646280i	\(-0.776324\pi\)
							0.938121	−	0.346308i	\(-0.112565\pi\)
\(17\)	3932.62	+	1431.36i	0.800452	+	0.291341i	0.709674	−	0.704530i	\(-0.248842\pi\)
							0.0907782	+	0.995871i	\(0.471065\pi\)
\(19\)	4579.19	−	5106.55i	0.667618	−	0.744504i
							\(23\)	−7587.19	+	9042.06i	−0.623587	+	0.743162i	−0.981683	−	0.190523i	\(-0.938982\pi\)
0.358096	+	0.933685i	\(0.383426\pi\)
\(29\)	30141.3	−	10970.5i	1.23585	−	0.449814i	0.360256	−	0.932854i	\(-0.382689\pi\)
							0.875599	+	0.483039i	\(0.160467\pi\)
\(31\)	−4707.49	−	2717.87i	−0.158017	−	0.0912312i	0.418906	−	0.908029i	\(-0.362414\pi\)
							−0.576923	+	0.816798i	\(0.695747\pi\)
\(37\)	−25203.8			−0.497578			−0.248789	−	0.968558i	\(-0.580033\pi\)
							−0.248789	+	0.968558i	\(0.580033\pi\)
\(41\)	3288.58	+	18650.5i	0.0477152	+	0.270607i	0.999326	−	0.0366982i	\(-0.0116840\pi\)
							−0.951611	+	0.307305i	\(0.900573\pi\)
\(43\)	−89124.9	−	106215.i	−1.12097	−	1.33592i	−0.935524	−	0.353264i	\(-0.885072\pi\)
							−0.185445	−	0.982655i	\(-0.559373\pi\)
\(47\)	51416.0	+	141264.i	0.495228	+	1.36063i	0.895839	+	0.444379i	\(0.146576\pi\)
							−0.400611	+	0.916248i	\(0.631202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.l.a.23.10	yes	348
4.3	odd	2	inner	76.7.l.a.23.4	✓	348
19.5	even	9	inner	76.7.l.a.43.4	yes	348
76.43	odd	18	inner	76.7.l.a.43.10	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.l.a.23.4	✓	348	4.3	odd	2	inner
76.7.l.a.23.10	yes	348	1.1	even	1	trivial
76.7.l.a.43.4	yes	348	19.5	even	9	inner
76.7.l.a.43.10	yes	348	76.43	odd	18	inner