Embedded newform 74.5.i.b.5.3

• Label: 74.5.i.b.5.3
• Level: $74$
• Weight: $5$
• Character: 74.5
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	74.5
Dual form			74.5.i.b.15.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19534 - 2.56343i) q^{2} +(-0.0952242 + 0.261626i) q^{3} +(-5.14230 - 6.12836i) q^{4} +(-15.0442 + 21.4853i) q^{5} +(0.556834 + 0.556834i) q^{6} +(8.44350 + 47.8855i) q^{7} +(-21.8564 + 5.85641i) q^{8} +(61.9902 + 52.0160i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 114 q^{5} - 576 q^{8} - 210 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{23}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19534	−	2.56343i	0.298836	−	0.640856i
							\(3\)	−0.0952242	+	0.261626i	−0.0105805	+	0.0290696i	−0.944870	−	0.327444i	\(-0.893813\pi\)
0.934290	+	0.356514i	\(0.116035\pi\)
\(5\)	−15.0442	+	21.4853i	−0.601766	+	0.859411i	−0.998281	−	0.0586174i	\(-0.981331\pi\)
							0.396514	+	0.918028i	\(0.370220\pi\)
\(7\)	8.44350	+	47.8855i	0.172316	+	0.977255i	0.941196	+	0.337861i	\(0.109703\pi\)
							−0.768880	+	0.639394i	\(0.779185\pi\)
\(11\)	33.4805	+	19.3300i	0.276698	+	0.159752i	0.631928	−	0.775027i	\(-0.282264\pi\)
							−0.355229	+	0.934779i	\(0.615597\pi\)
\(13\)	−41.9889	−	3.67355i	−0.248455	−	0.0217370i	−0.0377521	−	0.999287i	\(-0.512020\pi\)
							−0.210703	+	0.977550i	\(0.567575\pi\)
\(17\)	17.2858	+	197.577i	0.0598123	+	0.683658i	0.965618	+	0.259965i	\(0.0837110\pi\)
							−0.905806	+	0.423693i	\(0.860733\pi\)
\(19\)	−33.3024	−	71.4172i	−0.0922503	−	0.197831i	0.854769	−	0.519009i	\(-0.173699\pi\)
							−0.947019	+	0.321178i	\(0.895921\pi\)
\(23\)	−99.2028	+	370.230i	−0.187529	+	0.699868i	0.806546	+	0.591171i	\(0.201334\pi\)
							−0.994075	+	0.108696i	\(0.965332\pi\)
\(29\)	24.7140	+	92.2339i	0.0293864	+	0.109672i	0.979061	−	0.203566i	\(-0.0652531\pi\)
							−0.949675	+	0.313237i	\(0.898586\pi\)
\(31\)	291.001	−	291.001i	0.302811	−	0.302811i	−0.539302	−	0.842113i	\(-0.681312\pi\)
							0.842113	+	0.539302i	\(0.181312\pi\)
\(37\)	−1058.51	+	868.169i	−0.773200	+	0.634163i
							\(41\)	−901.615	−	1074.50i	−0.536356	−	0.639205i	0.428010	−	0.903774i	\(-0.359215\pi\)
−0.964367	+	0.264569i	\(0.914770\pi\)
\(43\)	−1116.82	−	1116.82i	−0.604012	−	0.604012i	0.337363	−	0.941375i	\(-0.390465\pi\)
							−0.941375	+	0.337363i	\(0.890465\pi\)
\(47\)	367.690	+	636.858i	0.166451	+	0.288301i	0.937170	−	0.348874i	\(-0.113436\pi\)
							−0.770719	+	0.637176i	\(0.780103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.i.b.5.3	✓	72
37.15	odd	36	inner	74.5.i.b.15.3	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.i.b.5.3	✓	72	1.1	even	1	trivial
74.5.i.b.15.3	yes	72	37.15	odd	36	inner