Embedded newform 74.6.h.a.21.4

• Label: 74.6.h.a.21.4
• Level: $74$
• Weight: $6$
• Character: 74.21
• Analytic conductor: $11.868$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8684026662\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.4
Character	\(\chi\)	\(=\)	74.21
Dual form			74.6.h.a.67.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36808 + 3.75877i) q^{2} +(-10.5624 + 3.84441i) q^{3} +(-12.2567 - 10.2846i) q^{4} +(53.2500 + 9.38941i) q^{5} -44.9612i q^{6} +(-24.0440 + 136.360i) q^{7} +(55.4256 - 32.0000i) q^{8} +(-89.3632 + 74.9846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36808	+	3.75877i	−0.241845	+	0.664463i
							\(3\)	−10.5624	+	3.84441i	−0.677581	+	0.246619i	−0.657808	−	0.753185i	\(-0.728516\pi\)
−0.0197723	+	0.999805i	\(0.506294\pi\)
\(5\)	53.2500	+	9.38941i	0.952565	+	0.167963i	0.628272	−	0.777994i	\(-0.283763\pi\)
							0.324293	+	0.945957i	\(0.394874\pi\)
\(7\)	−24.0440	+	136.360i	−0.185465	+	1.05182i	0.739892	+	0.672725i	\(0.234877\pi\)
							−0.925357	+	0.379097i	\(0.876235\pi\)
\(11\)	26.5548	+	45.9943i	0.0661701	+	0.114610i	0.897212	−	0.441599i	\(-0.145589\pi\)
							−0.831042	+	0.556209i	\(0.812255\pi\)
\(13\)	−109.999	+	131.092i	−0.180522	+	0.215138i	−0.848715	−	0.528850i	\(-0.822623\pi\)
							0.668193	+	0.743988i	\(0.267068\pi\)
\(17\)	−1295.21	−	1543.57i	−1.08697	−	1.29540i	−0.952518	−	0.304484i	\(-0.901516\pi\)
							−0.134455	−	0.990920i	\(-0.542928\pi\)
\(19\)	−211.193	−	580.248i	−0.134213	−	0.368748i	0.854321	−	0.519746i	\(-0.173973\pi\)
							−0.988534	+	0.150998i	\(0.951751\pi\)
\(23\)	−373.535	−	215.661i	−0.147235	−	0.0850064i	0.424573	−	0.905394i	\(-0.360424\pi\)
							−0.571808	+	0.820388i	\(0.693758\pi\)
\(29\)	−4720.19	+	2725.20i	−1.04223	+	0.601733i	−0.920465	−	0.390826i	\(-0.872189\pi\)
							−0.121767	+	0.992559i	\(0.538856\pi\)
\(31\)			2512.29i			0.469532i	0.972052	+	0.234766i	\(0.0754323\pi\)
							−0.972052	+	0.234766i	\(0.924568\pi\)
\(37\)	−8127.46	−	1813.38i	−0.976002	−	0.217764i
							\(41\)	−9741.46	−	8174.06i	−0.905033	−	0.759413i	0.0661343	−	0.997811i	\(-0.478933\pi\)
−0.971168	+	0.238398i	\(0.923378\pi\)
\(43\)			11150.9i			0.919682i	0.888001	+	0.459841i	\(0.152094\pi\)
							−0.888001	+	0.459841i	\(0.847906\pi\)
\(47\)	−2897.83	+	5019.19i	−0.191350	+	0.331428i	−0.945698	−	0.325047i	\(-0.894620\pi\)
							0.754348	+	0.656475i	\(0.227953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.21.4	✓	96
37.30	even	18	inner	74.6.h.a.67.4	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.21.4	✓	96	1.1	even	1	trivial
74.6.h.a.67.4	yes	96	37.30	even	18	inner