Embedded newform 74.6.h.a.3.15

• Label: 74.6.h.a.3.15
• Level: $74$
• Weight: $6$
• Character: 74.3
• 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			3.15
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(12.3631 - 10.3739i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(33.5955 - 92.3028i) q^{5} -64.5555i q^{6} +(200.133 + 72.8426i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(3.03242 - 17.1977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.57115	−	3.06418i	0.454519	−	0.541675i
							\(3\)	12.3631	−	10.3739i	0.793092	−	0.665484i	−0.153416	−	0.988162i	\(-0.549028\pi\)
0.946509	+	0.322678i	\(0.104583\pi\)
\(5\)	33.5955	−	92.3028i	0.600974	−	1.65116i	−0.148332	−	0.988938i	\(-0.547390\pi\)
							0.749306	−	0.662224i	\(-0.230387\pi\)
\(7\)	200.133	+	72.8426i	1.54374	+	0.561876i	0.966939	−	0.255008i	\(-0.0820780\pi\)
							0.576803	+	0.816884i	\(0.304300\pi\)
\(11\)	−140.949	+	244.130i	−0.351220	+	0.608331i	−0.986464	−	0.163981i	\(-0.947567\pi\)
							0.635243	+	0.772312i	\(0.280900\pi\)
\(13\)	−298.433	+	52.6217i	−0.489765	+	0.0863588i	−0.413077	−	0.910696i	\(-0.635546\pi\)
							−0.0766885	+	0.997055i	\(0.524435\pi\)
\(17\)	−456.139	−	80.4295i	−0.382802	−	0.0674984i	−0.0210642	−	0.999778i	\(-0.506705\pi\)
							−0.361738	+	0.932280i	\(0.617817\pi\)
\(19\)	964.259	+	1149.16i	0.612788	+	0.730292i	0.979812	−	0.199919i	\(-0.0640680\pi\)
							−0.367025	+	0.930211i	\(0.619624\pi\)
\(23\)	986.234	−	569.402i	0.388741	−	0.224440i	−0.292873	−	0.956151i	\(-0.594611\pi\)
							0.681614	+	0.731711i	\(0.261278\pi\)
\(29\)	−6594.31	−	3807.23i	−1.45604	−	0.840647i	−0.457230	−	0.889348i	\(-0.651159\pi\)
							−0.998813	+	0.0487010i	\(0.984492\pi\)
\(31\)			6349.86i			1.18675i	0.804925	+	0.593376i	\(0.202205\pi\)
							−0.804925	+	0.593376i	\(0.797795\pi\)
\(37\)	2710.68	−	7873.77i	0.325517	−	0.945536i
							\(41\)	1793.40	+	10170.9i	0.166617	+	0.944931i	0.947382	+	0.320106i	\(0.103718\pi\)
−0.780765	+	0.624825i	\(0.785170\pi\)
\(43\)			19803.2i			1.63330i	0.577135	+	0.816649i	\(0.304171\pi\)
							−0.577135	+	0.816649i	\(0.695829\pi\)
\(47\)	−8345.19	−	14454.3i	−0.551051	−	0.954448i	−0.998199	−	0.0599882i	\(-0.980894\pi\)
							0.447148	−	0.894460i	\(-0.352440\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.15	✓	96
37.25	even	18	inner	74.6.h.a.25.15	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.15	✓	96	1.1	even	1	trivial
74.6.h.a.25.15	yes	96	37.25	even	18	inner