Embedded newform 74.5.g.b.23.5

• Label: 74.5.g.b.23.5
• Level: $74$
• Weight: $5$
• Character: 74.23
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.5
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.b.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.732051 - 2.73205i) q^{2} +(4.65870 - 2.68970i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(6.50685 - 24.2839i) q^{5} +(-10.7588 - 10.7588i) q^{6} +(-14.6009 - 25.2895i) q^{7} +(16.0000 + 16.0000i) q^{8} +(-26.0310 + 45.0870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.732051	−	2.73205i	−0.183013	−	0.683013i
							\(3\)	4.65870	−	2.68970i	0.517634	−	0.298856i	−0.218332	−	0.975874i	\(-0.570062\pi\)
0.735966	+	0.677019i	\(0.236728\pi\)
\(5\)	6.50685	−	24.2839i	0.260274	−	0.971356i	−0.704806	−	0.709400i	\(-0.748966\pi\)
							0.965080	−	0.261956i	\(-0.0843675\pi\)
\(7\)	−14.6009	−	25.2895i	−0.297977	−	0.516112i	0.677696	−	0.735342i	\(-0.262979\pi\)
							−0.975673	+	0.219230i	\(0.929645\pi\)
\(11\)		−	77.8038i		−	0.643006i	−0.946909	−	0.321503i	\(-0.895812\pi\)
							0.946909	−	0.321503i	\(-0.104188\pi\)
\(13\)	68.0474	−	253.956i	0.402647	−	1.50270i	−0.405706	−	0.914004i	\(-0.632974\pi\)
							0.808354	−	0.588697i	\(-0.200359\pi\)
\(17\)	−441.790	+	118.377i	−1.52868	+	0.409610i	−0.922590	−	0.385783i	\(-0.873931\pi\)
							−0.606095	+	0.795393i	\(0.707265\pi\)
\(19\)	−83.4227	+	311.338i	−0.231088	+	0.862431i	0.748786	+	0.662812i	\(0.230637\pi\)
							−0.979874	+	0.199619i	\(0.936029\pi\)
\(23\)	−486.268	−	486.268i	−0.919222	−	0.919222i	0.0777510	−	0.996973i	\(-0.475226\pi\)
							−0.996973	+	0.0777510i	\(0.975226\pi\)
\(29\)	816.608	−	816.608i	0.970996	−	0.970996i	−0.0285946	−	0.999591i	\(-0.509103\pi\)
							0.999591	+	0.0285946i	\(0.00910318\pi\)
\(31\)	1062.48	−	1062.48i	1.10560	−	1.10560i	0.111877	−	0.993722i	\(-0.464314\pi\)
							0.993722	−	0.111877i	\(-0.0356862\pi\)
\(37\)	311.728	+	1333.04i	0.227705	+	0.973730i
							\(41\)	1561.19	−	901.354i	0.928727	−	0.536201i	0.0423186	−	0.999104i	\(-0.486526\pi\)
0.886409	+	0.462903i	\(0.153192\pi\)
\(43\)	1142.05	+	1142.05i	0.617656	+	0.617656i	0.944930	−	0.327274i	\(-0.106130\pi\)
							−0.327274	+	0.944930i	\(0.606130\pi\)
\(47\)	2849.88			1.29012			0.645062	−	0.764131i	\(-0.276832\pi\)
							0.645062	+	0.764131i	\(0.276832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.b.23.5	✓	28
37.29	odd	12	inner	74.5.g.b.29.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.b.23.5	✓	28	1.1	even	1	trivial
74.5.g.b.29.5	yes	28	37.29	odd	12	inner