Embedded newform 74.5.g.b.45.1

• Label: 74.5.g.b.45.1
• Level: $74$
• Weight: $5$
• Character: 74.45
• 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			45.1
Character	\(\chi\)	\(=\)	74.45
Dual form			74.5.g.b.51.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.73205 + 0.732051i) q^{2} +(-15.3929 - 8.88709i) q^{3} +(6.92820 + 4.00000i) q^{4} +(-1.55696 + 0.417185i) q^{5} +(-35.5484 - 35.5484i) q^{6} +(-4.44075 + 7.69161i) q^{7} +(16.0000 + 16.0000i) q^{8} +(117.461 + 203.448i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{2} - 102 q^{5} + 40 q^{6} - 48 q^{7} + 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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
							\(3\)	−15.3929	−	8.88709i	−1.71032	−	0.987455i	−0.934112	−	0.356981i	\(-0.883806\pi\)
−0.776210	−	0.630474i	\(-0.782861\pi\)
\(5\)	−1.55696	+	0.417185i	−0.0622783	+	0.0166874i	−0.289824	−	0.957080i	\(-0.593597\pi\)
							0.227545	+	0.973767i	\(0.426930\pi\)
\(7\)	−4.44075	+	7.69161i	−0.0906276	+	0.156972i	−0.907775	−	0.419457i	\(-0.862221\pi\)
							0.817148	+	0.576428i	\(0.195554\pi\)
\(11\)			118.207i			0.976918i	0.872587	+	0.488459i	\(0.162441\pi\)
							−0.872587	+	0.488459i	\(0.837559\pi\)
\(13\)	−175.029	+	46.8990i	−1.03568	+	0.277509i	−0.736321	−	0.676633i	\(-0.763439\pi\)
							−0.299356	+	0.954141i	\(0.596772\pi\)
\(17\)	−93.1886	+	347.785i	−0.322452	+	1.20341i	0.594397	+	0.804172i	\(0.297391\pi\)
							−0.916849	+	0.399235i	\(0.869276\pi\)
\(19\)	476.474	−	127.671i	1.31987	−	0.353659i	0.470942	−	0.882164i	\(-0.343914\pi\)
							0.848930	+	0.528506i	\(0.177247\pi\)
\(23\)	−194.181	−	194.181i	−0.367071	−	0.367071i	0.499337	−	0.866408i	\(-0.333577\pi\)
							−0.866408	+	0.499337i	\(0.833577\pi\)
\(29\)	−74.3734	+	74.3734i	−0.0884344	+	0.0884344i	−0.749940	−	0.661506i	\(-0.769918\pi\)
							0.661506	+	0.749940i	\(0.269918\pi\)
\(31\)	−352.754	+	352.754i	−0.367069	+	0.367069i	−0.866407	−	0.499338i	\(-0.833577\pi\)
							0.499338	+	0.866407i	\(0.333577\pi\)
\(37\)	−1342.85	+	266.318i	−0.980896	+	0.194534i
							\(41\)	41.3831	+	23.8926i	0.0246182	+	0.0142133i	0.512259	−	0.858831i	\(-0.328809\pi\)
−0.487640	+	0.873045i	\(0.662142\pi\)
\(43\)	311.825	+	311.825i	0.168645	+	0.168645i	0.786384	−	0.617738i	\(-0.211951\pi\)
							−0.617738	+	0.786384i	\(0.711951\pi\)
\(47\)	−3387.10			−1.53332			−0.766658	−	0.642055i	\(-0.778082\pi\)
							−0.766658	+	0.642055i	\(0.778082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.b.45.1	✓	28
37.14	odd	12	inner	74.5.g.b.51.1	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.b.45.1	✓	28	1.1	even	1	trivial
74.5.g.b.51.1	yes	28	37.14	odd	12	inner