Embedded newform 74.4.h.a.21.5

• Label: 74.4.h.a.21.5
• Level: $74$
• Weight: $4$
• Character: 74.21
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.5
Character	\(\chi\)	\(=\)	74.21
Dual form			74.4.h.a.67.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.684040 + 1.87939i) q^{2} +(8.12302 - 2.95654i) q^{3} +(-3.06418 - 2.57115i) q^{4} +(5.91804 + 1.04351i) q^{5} +17.2887i q^{6} +(0.991911 - 5.62541i) q^{7} +(6.92820 - 4.00000i) q^{8} +(36.5591 - 30.6767i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 12 q^{3} + 18 q^{5} + 150 q^{7} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.684040	+	1.87939i	−0.241845	+	0.664463i
							\(3\)	8.12302	−	2.95654i	1.56328	−	0.568986i	0.591791	−	0.806091i	\(-0.298421\pi\)
0.971484	+	0.237105i	\(0.0761987\pi\)
\(5\)	5.91804	+	1.04351i	0.529325	+	0.0933343i	0.431922	−	0.901911i	\(-0.357835\pi\)
							0.0974027	+	0.995245i	\(0.468947\pi\)
\(7\)	0.991911	−	5.62541i	0.0535582	−	0.303743i	−0.946248	−	0.323443i	\(-0.895160\pi\)
							0.999806	+	0.0196992i	\(0.00627085\pi\)
\(11\)	−5.57191	−	9.65083i	−0.152727	−	0.264530i	0.779502	−	0.626399i	\(-0.215472\pi\)
							−0.932229	+	0.361869i	\(0.882139\pi\)
\(13\)	−10.4325	+	12.4329i	−0.222572	+	0.265252i	−0.865763	−	0.500455i	\(-0.833166\pi\)
							0.643190	+	0.765707i	\(0.277611\pi\)
\(17\)	14.7681	+	17.5999i	0.210693	+	0.251094i	0.861033	−	0.508549i	\(-0.169818\pi\)
							−0.650340	+	0.759643i	\(0.725373\pi\)
\(19\)	54.8812	+	150.785i	0.662663	+	1.82065i	0.564416	+	0.825490i	\(0.309101\pi\)
							0.0982471	+	0.995162i	\(0.468676\pi\)
\(23\)	−69.8082	−	40.3038i	−0.632870	−	0.365388i	0.148993	−	0.988838i	\(-0.452397\pi\)
							−0.781863	+	0.623451i	\(0.785730\pi\)
\(29\)	−114.307	+	65.9949i	−0.731938	+	0.422584i	−0.819131	−	0.573607i	\(-0.805544\pi\)
							0.0871930	+	0.996191i	\(0.472210\pi\)
\(31\)		−	166.351i		−	0.963792i	−0.876228	−	0.481896i	\(-0.839948\pi\)
							0.876228	−	0.481896i	\(-0.160052\pi\)
\(37\)	−174.082	−	142.648i	−0.773484	−	0.633816i
							\(41\)	−276.545	−	232.049i	−1.05339	−	0.883902i	−0.0599468	−	0.998202i	\(-0.519093\pi\)
−0.993446	+	0.114300i	\(0.963538\pi\)
\(43\)			367.642i			1.30383i	0.758290	+	0.651917i	\(0.226035\pi\)
							−0.758290	+	0.651917i	\(0.773965\pi\)
\(47\)	71.4140	−	123.693i	0.221634	−	0.383882i	−0.733670	−	0.679506i	\(-0.762194\pi\)
							0.955304	+	0.295624i	\(0.0955276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.h.a.21.5	✓	60
37.30	even	18	inner	74.4.h.a.67.5	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.h.a.21.5	✓	60	1.1	even	1	trivial
74.4.h.a.67.5	yes	60	37.30	even	18	inner