Embedded newform 49.4.e.a.15.6

• Label: 49.4.e.a.15.6
• Level: $49$
• Weight: $4$
• Character: 49.15
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			15.6
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.379589 + 1.66309i) q^{2} +(-2.24247 + 2.81197i) q^{3} +(4.58597 + 2.20849i) q^{4} +(-4.66393 + 5.84838i) q^{5} +(-3.82533 - 4.79682i) q^{6} +(-14.0409 - 12.0769i) q^{7} +(-13.9224 + 17.4581i) q^{8} +(3.12958 + 13.7116i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.379589	+	1.66309i	−0.134205	+	0.587991i	0.862441	+	0.506158i	\(0.168935\pi\)
							−0.996646	+	0.0818331i	\(0.973923\pi\)
\(3\)	−2.24247	+	2.81197i	−0.431563	+	0.541163i	−0.949298	−	0.314378i	\(-0.898204\pi\)
							0.517735	+	0.855541i	\(0.326775\pi\)
\(5\)	−4.66393	+	5.84838i	−0.417154	+	0.523095i	−0.945363	−	0.326020i	\(-0.894292\pi\)
							0.528209	+	0.849115i	\(0.322864\pi\)
\(7\)	−14.0409	−	12.0769i	−0.758140	−	0.652092i
							\(11\)	−7.78469	+	34.1070i	−0.213379	+	0.934876i	0.748872	+	0.662714i	\(0.230596\pi\)
−0.962252	+	0.272162i	\(0.912261\pi\)
\(13\)	5.32468	−	23.3289i	0.113600	−	0.497714i	−0.885832	−	0.464007i	\(-0.846411\pi\)
							0.999432	−	0.0337075i	\(-0.0107315\pi\)
\(17\)	75.9808	−	36.5904i	1.08400	−	0.522028i	0.195409	−	0.980722i	\(-0.437397\pi\)
							0.888595	+	0.458693i	\(0.151682\pi\)
\(19\)	57.3716			0.692735			0.346367	−	0.938099i	\(-0.387415\pi\)
							0.346367	+	0.938099i	\(0.387415\pi\)
\(23\)	139.535	+	67.1963i	1.26500	+	0.609192i	0.941492	−	0.337034i	\(-0.109424\pi\)
							0.323507	+	0.946226i	\(0.395138\pi\)
\(29\)	98.6691	−	47.5165i	0.631807	−	0.304262i	−0.0904356	−	0.995902i	\(-0.528826\pi\)
							0.722242	+	0.691640i	\(0.243112\pi\)
\(31\)	−60.9923			−0.353372			−0.176686	−	0.984267i	\(-0.556538\pi\)
							−0.176686	+	0.984267i	\(0.556538\pi\)
\(37\)	178.296	−	85.8629i	0.792208	−	0.381507i	0.00640158	−	0.999980i	\(-0.497962\pi\)
							0.785807	+	0.618472i	\(0.212248\pi\)
\(41\)	−233.717	+	293.072i	−0.890256	+	1.11635i	0.102324	+	0.994751i	\(0.467372\pi\)
							−0.992580	+	0.121594i	\(0.961199\pi\)
\(43\)	−307.167	−	385.175i	−1.08936	−	1.36602i	−0.925155	−	0.379590i	\(-0.876065\pi\)
							−0.164206	−	0.986426i	\(-0.552506\pi\)
\(47\)	14.5738	−	63.8522i	0.0452301	−	0.198166i	−0.947265	−	0.320452i	\(-0.896165\pi\)
							0.992495	+	0.122286i	\(0.0390225\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.15.6	✓	78
49.6	odd	14		2401.4.a.c.1.25		39
49.36	even	7	inner	49.4.e.a.36.6	yes	78
49.43	even	7		2401.4.a.d.1.25		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.6	✓	78	1.1	even	1	trivial
49.4.e.a.36.6	yes	78	49.36	even	7	inner
2401.4.a.c.1.25		39	49.6	odd	14
2401.4.a.d.1.25		39	49.43	even	7