Embedded newform 49.4.e.a.43.9

• Label: 49.4.e.a.43.9
• Level: $49$
• Weight: $4$
• Character: 49.43
• 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			43.9
Character	\(\chi\)	\(=\)	49.43
Dual form			49.4.e.a.8.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.777062 - 0.974405i) q^{2} +(-0.189548 - 0.0912813i) q^{3} +(1.43453 + 6.28508i) q^{4} +(7.66912 + 3.69325i) q^{5} +(-0.236235 + 0.113765i) q^{6} +(18.3768 - 2.30053i) q^{7} +(16.2220 + 7.81211i) q^{8} +(-16.8066 - 21.0748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.777062	−	0.974405i	0.274733	−	0.344504i	−0.625254	−	0.780421i	\(-0.715005\pi\)
							0.899987	+	0.435917i	\(0.143576\pi\)
\(3\)	−0.189548	−	0.0912813i	−0.0364784	−	0.0175671i	0.415556	−	0.909568i	\(-0.363587\pi\)
							−0.452034	+	0.892001i	\(0.649301\pi\)
\(5\)	7.66912	+	3.69325i	0.685947	+	0.330335i	0.744190	−	0.667968i	\(-0.232836\pi\)
							−0.0582431	+	0.998302i	\(0.518550\pi\)
\(7\)	18.3768	−	2.30053i	0.992255	−	0.124217i
							\(11\)	2.58443	−	3.24077i	0.0708394	−	0.0888299i	−0.745149	−	0.666898i	\(-0.767622\pi\)
0.815989	+	0.578068i	\(0.196193\pi\)
\(13\)	−11.7770	+	14.7678i	−0.251257	+	0.315066i	−0.891425	−	0.453169i	\(-0.850293\pi\)
							0.640168	+	0.768235i	\(0.278865\pi\)
\(17\)	−4.87109	+	21.3416i	−0.0694948	+	0.304477i	−0.997716	−	0.0675529i	\(-0.978481\pi\)
							0.928221	+	0.372030i	\(0.121338\pi\)
\(19\)	−46.4772			−0.561190			−0.280595	−	0.959826i	\(-0.590532\pi\)
							−0.280595	+	0.959826i	\(0.590532\pi\)
\(23\)	−18.3535	−	80.4119i	−0.166390	−	0.729002i	−0.987420	−	0.158117i	\(-0.949458\pi\)
							0.821030	−	0.570884i	\(-0.193400\pi\)
\(29\)	44.6635	−	195.683i	0.285993	−	1.25302i	−0.603978	−	0.797001i	\(-0.706419\pi\)
							0.889971	−	0.456017i	\(-0.150724\pi\)
\(31\)	−276.703			−1.60314			−0.801570	−	0.597901i	\(-0.796002\pi\)
							−0.801570	+	0.597901i	\(0.796002\pi\)
\(37\)	48.3784	−	211.960i	0.214956	−	0.941782i	−0.746188	−	0.665735i	\(-0.768118\pi\)
							0.961144	−	0.276047i	\(-0.0890247\pi\)
\(41\)	386.663	+	186.207i	1.47284	+	0.709284i	0.986390	−	0.164423i	\(-0.0525762\pi\)
							0.486453	+	0.873707i	\(0.338291\pi\)
\(43\)	188.263	−	90.6627i	0.667670	−	0.321533i	−0.0691650	−	0.997605i	\(-0.522034\pi\)
							0.736835	+	0.676072i	\(0.236319\pi\)
\(47\)	−258.855	+	324.593i	−0.803358	+	1.00738i	0.196282	+	0.980547i	\(0.437113\pi\)
							−0.999640	+	0.0268314i	\(0.991458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.43.9	yes	78
49.8	even	7	inner	49.4.e.a.8.9	✓	78
49.20	odd	14		2401.4.a.c.1.24		39
49.29	even	7		2401.4.a.d.1.24		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.9	✓	78	49.8	even	7	inner
49.4.e.a.43.9	yes	78	1.1	even	1	trivial
2401.4.a.c.1.24		39	49.20	odd	14
2401.4.a.d.1.24		39	49.29	even	7