Embedded newform 74.3.g.b.23.2

• Label: 74.3.g.b.23.2
• Level: $74$
• Weight: $3$
• Character: 74.23
• Analytic conductor: $2.016$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01635395627\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 82x^{10} + 2505x^{8} + 34456x^{6} + 196096x^{4} + 262464x^{2} + 69696 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.2
Root			\(-0.594165i\) of defining polynomial
Character	\(\chi\)	\(=\)	74.23
Dual form			74.3.g.b.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-0.514562 + 0.297082i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-0.866025 + 3.23205i) q^{5} +(-0.594165 - 0.594165i) q^{6} +(5.01184 + 8.68076i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-4.32348 + 7.48849i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 4 q^{6} - 8 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	+	1.36603i	0.183013	+	0.683013i
							\(3\)	−0.514562	+	0.297082i	−0.171521	+	0.0990275i	−0.583303	−	0.812255i	\(-0.698240\pi\)
0.411782	+	0.911282i	\(0.364907\pi\)
\(5\)	−0.866025	+	3.23205i	−0.173205	+	0.646410i	0.823645	+	0.567105i	\(0.191937\pi\)
							−0.996850	+	0.0793049i	\(0.974730\pi\)
\(7\)	5.01184	+	8.68076i	0.715977	+	1.24011i	0.962581	+	0.270992i	\(0.0873518\pi\)
							−0.246604	+	0.969116i	\(0.579315\pi\)
\(11\)		−	8.99456i		−	0.817687i	−0.912605	−	0.408843i	\(-0.865932\pi\)
							0.912605	−	0.408843i	\(-0.134068\pi\)
\(13\)	3.55840	−	13.2801i	0.273723	−	1.02155i	−0.682969	−	0.730447i	\(-0.739312\pi\)
							0.956692	−	0.291101i	\(-0.0940215\pi\)
\(17\)	25.4974	−	6.83202i	1.49985	−	0.401884i	0.586800	−	0.809732i	\(-0.300387\pi\)
							0.913050	+	0.407848i	\(0.133721\pi\)
\(19\)	−3.45143	+	12.8809i	−0.181654	+	0.677943i	0.813668	+	0.581330i	\(0.197468\pi\)
							−0.995322	+	0.0966129i	\(0.969199\pi\)
\(23\)	−9.01648	−	9.01648i	−0.392021	−	0.392021i	0.483386	−	0.875407i	\(-0.339407\pi\)
							−0.875407	+	0.483386i	\(0.839407\pi\)
\(29\)	38.3405	−	38.3405i	1.32209	−	1.32209i	0.410004	−	0.912084i	\(-0.365527\pi\)
							0.912084	−	0.410004i	\(-0.134473\pi\)
\(31\)	−15.4088	+	15.4088i	−0.497058	+	0.497058i	−0.910521	−	0.413463i	\(-0.864319\pi\)
							0.413463	+	0.910521i	\(0.364319\pi\)
\(37\)	−36.9507	+	1.90954i	−0.998667	+	0.0516093i
							\(41\)	12.7230	−	7.34562i	0.310317	−	0.179162i	−0.336751	−	0.941594i	\(-0.609328\pi\)
0.647068	+	0.762432i	\(0.275995\pi\)
\(43\)	19.9072	+	19.9072i	0.462958	+	0.462958i	0.899624	−	0.436666i	\(-0.143841\pi\)
							−0.436666	+	0.899624i	\(0.643841\pi\)
\(47\)	−62.2361			−1.32417			−0.662086	−	0.749428i	\(-0.730329\pi\)
							−0.662086	+	0.749428i	\(0.730329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.3.g.b.23.2	✓	12
37.29	odd	12	inner	74.3.g.b.29.2	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.3.g.b.23.2	✓	12	1.1	even	1	trivial
74.3.g.b.29.2	yes	12	37.29	odd	12	inner