Embedded newform 74.4.f.a.53.1

• Label: 74.4.f.a.53.1
• Level: $74$
• Weight: $4$
• Character: 74.53
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			53.1
Character	\(\chi\)	\(=\)	74.53
Dual form			74.4.f.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.87939 + 0.684040i) q^{2} +(-4.40012 + 1.60151i) q^{3} +(3.06418 + 2.57115i) q^{4} +(-0.0856491 + 0.485740i) q^{5} -9.36501 q^{6} +(-4.85295 + 27.5224i) q^{7} +(4.00000 + 6.92820i) q^{8} +(-3.88702 + 3.26159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.87939	+	0.684040i	0.664463	+	0.241845i
							\(3\)	−4.40012	+	1.60151i	−0.846803	+	0.308211i	−0.728736	−	0.684795i	\(-0.759892\pi\)
−0.118067	+	0.993006i	\(0.537670\pi\)
\(5\)	−0.0856491	+	0.485740i	−0.00766069	+	0.0434459i	−0.988398	−	0.151883i	\(-0.951466\pi\)
							0.980738	+	0.195329i	\(0.0625774\pi\)
\(7\)	−4.85295	+	27.5224i	−0.262034	+	1.48607i	0.515314	+	0.857001i	\(0.327675\pi\)
							−0.777349	+	0.629070i	\(0.783436\pi\)
\(11\)	14.3294	+	24.8193i	0.392772	+	0.680301i	0.992814	−	0.119667i	\(-0.0381828\pi\)
							−0.600042	+	0.799968i	\(0.704850\pi\)
\(13\)	−32.8237	−	27.5424i	−0.700282	−	0.587607i	0.221572	−	0.975144i	\(-0.428881\pi\)
							−0.921854	+	0.387538i	\(0.873326\pi\)
\(17\)	−34.3208	+	28.7986i	−0.489648	+	0.410864i	−0.853900	−	0.520437i	\(-0.825769\pi\)
							0.364252	+	0.931300i	\(0.381325\pi\)
\(19\)	146.256	−	53.2327i	1.76597	−	0.642759i	0.765966	−	0.642881i	\(-0.222261\pi\)
							1.00000		0.000121544i	\(3.86885e-5\pi\)
\(23\)	17.3086	−	29.9795i	0.156917	−	0.271789i	−0.776838	−	0.629700i	\(-0.783178\pi\)
							0.933756	+	0.357911i	\(0.116511\pi\)
\(29\)	−4.83096	−	8.36746i	−0.0309340	−	0.0535792i	0.850144	−	0.526550i	\(-0.176515\pi\)
							−0.881078	+	0.472971i	\(0.843182\pi\)
\(31\)	9.17583			0.0531622			0.0265811	−	0.999647i	\(-0.491538\pi\)
							0.0265811	+	0.999647i	\(0.491538\pi\)
\(37\)	−74.8199	+	212.262i	−0.332441	+	0.943124i
							\(41\)	−7.15149	−	6.00082i	−0.0272409	−	0.0228578i	0.629066	−	0.777352i	\(-0.283438\pi\)
−0.656306	+	0.754494i	\(0.727882\pi\)
\(43\)	−87.8577			−0.311585			−0.155793	−	0.987790i	\(-0.549793\pi\)
							−0.155793	+	0.987790i	\(0.549793\pi\)
\(47\)	125.517	−	217.402i	0.389543	−	0.674709i	−0.602845	−	0.797858i	\(-0.705966\pi\)
							0.992388	+	0.123150i	\(0.0392996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.53.1	yes	24
37.7	even	9	inner	74.4.f.a.7.1	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.7.1	✓	24	37.7	even	9	inner
74.4.f.a.53.1	yes	24	1.1	even	1	trivial