Embedded newform 74.5.g.a.51.6

• Label: 74.5.g.a.51.6
• Level: $74$
• Weight: $5$
• Character: 74.51
• 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			51.6
Character	\(\chi\)	\(=\)	74.51
Dual form			74.5.g.a.45.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.73205 + 0.732051i) q^{2} +(8.28780 - 4.78497i) q^{3} +(6.92820 - 4.00000i) q^{4} +(-17.4826 - 4.68445i) q^{5} +(-19.1399 + 19.1399i) q^{6} +(-36.2925 - 62.8605i) q^{7} +(-16.0000 + 16.0000i) q^{8} +(5.29180 - 9.16567i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.73205	+	0.732051i	−0.683013	+	0.183013i
							\(3\)	8.28780	−	4.78497i	0.920867	−	0.531663i	0.0369555	−	0.999317i	\(-0.488234\pi\)
0.883912	+	0.467654i	\(0.154901\pi\)
\(5\)	−17.4826	−	4.68445i	−0.699305	−	0.187378i	−0.108386	−	0.994109i	\(-0.534568\pi\)
							−0.590919	+	0.806731i	\(0.701235\pi\)
\(7\)	−36.2925	−	62.8605i	−0.740664	−	1.28287i	−0.952193	−	0.305496i	\(-0.901178\pi\)
							0.211529	−	0.977372i	\(-0.432156\pi\)
\(11\)			102.318i			0.845605i	0.906222	+	0.422802i	\(0.138954\pi\)
							−0.906222	+	0.422802i	\(0.861046\pi\)
\(13\)	−128.387	−	34.4011i	−0.759683	−	0.203557i	−0.141874	−	0.989885i	\(-0.545313\pi\)
							−0.617809	+	0.786328i	\(0.711980\pi\)
\(17\)	−36.3083	−	135.504i	−0.125634	−	0.468873i	0.874227	−	0.485517i	\(-0.161369\pi\)
							−0.999861	+	0.0166439i	\(0.994702\pi\)
\(19\)	−527.582	−	141.365i	−1.46145	−	0.391594i	−0.561458	−	0.827505i	\(-0.689759\pi\)
							−0.899989	+	0.435912i	\(0.856426\pi\)
\(23\)	686.570	−	686.570i	1.29786	−	1.29786i	0.368062	−	0.929801i	\(-0.380021\pi\)
							0.929801	−	0.368062i	\(-0.119979\pi\)
\(29\)	−892.638	−	892.638i	−1.06140	−	1.06140i	−0.997987	−	0.0634133i	\(-0.979801\pi\)
							−0.0634133	−	0.997987i	\(-0.520199\pi\)
\(31\)	439.389	+	439.389i	0.457220	+	0.457220i	0.897742	−	0.440522i	\(-0.145206\pi\)
							−0.440522	+	0.897742i	\(0.645206\pi\)
\(37\)	664.189	+	1197.09i	0.485164	+	0.874423i
							\(41\)	876.607	−	506.109i	0.521480	−	0.301076i	−0.216060	−	0.976380i	\(-0.569321\pi\)
0.737540	+	0.675304i	\(0.235987\pi\)
\(43\)	1235.74	−	1235.74i	0.668330	−	0.668330i	−0.288999	−	0.957329i	\(-0.593323\pi\)
							0.957329	+	0.288999i	\(0.0933226\pi\)
\(47\)	2111.90			0.956043			0.478021	−	0.878348i	\(-0.341354\pi\)
							0.478021	+	0.878348i	\(0.341354\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.a.51.6	yes	28
37.8	odd	12	inner	74.5.g.a.45.6	✓	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.a.45.6	✓	28	37.8	odd	12	inner
74.5.g.a.51.6	yes	28	1.1	even	1	trivial