Embedded newform 49.4.e.a.43.10

• Label: 49.4.e.a.43.10
• 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			43.10
Character	\(\chi\)	\(=\)	49.43
Dual form			49.4.e.a.8.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88794 - 2.36740i) q^{2} +(-5.83533 - 2.81015i) q^{3} +(-0.260101 - 1.13958i) q^{4} +(-12.6471 - 6.09051i) q^{5} +(-17.6695 + 8.50917i) q^{6} +(-3.94107 - 18.0961i) q^{7} +(18.6363 + 8.97477i) q^{8} +(9.31994 + 11.6868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88794	−	2.36740i	0.667487	−	0.837002i	−0.326649	−	0.945146i	\(-0.605919\pi\)
							0.994135	+	0.108144i	\(0.0344908\pi\)
\(3\)	−5.83533	−	2.81015i	−1.12301	−	0.540813i	−0.222189	−	0.975004i	\(-0.571320\pi\)
							−0.900821	+	0.434191i	\(0.857034\pi\)
\(5\)	−12.6471	−	6.09051i	−1.13119	−	0.544751i	−0.227857	−	0.973695i	\(-0.573172\pi\)
							−0.903331	+	0.428943i	\(0.858886\pi\)
\(7\)	−3.94107	−	18.0961i	−0.212798	−	0.977096i
							\(11\)	1.11842	−	1.40245i	0.0306560	−	0.0384415i	−0.766267	−	0.642522i	\(-0.777888\pi\)
0.796923	+	0.604081i	\(0.206460\pi\)
\(13\)	19.6402	−	24.6280i	0.419017	−	0.525430i	−0.526862	−	0.849951i	\(-0.676632\pi\)
							0.945879	+	0.324521i	\(0.105203\pi\)
\(17\)	3.96139	−	17.3560i	0.0565163	−	0.247614i	−0.938777	−	0.344526i	\(-0.888039\pi\)
							0.995293	+	0.0969123i	\(0.0308966\pi\)
\(19\)	34.8025			0.420224			0.210112	−	0.977677i	\(-0.432617\pi\)
							0.210112	+	0.977677i	\(0.432617\pi\)
\(23\)	−44.3595	−	194.352i	−0.402156	−	1.76196i	−0.618639	−	0.785675i	\(-0.712316\pi\)
							0.216483	−	0.976286i	\(-0.430541\pi\)
\(29\)	−61.3317	+	268.712i	−0.392725	+	1.72064i	0.262260	+	0.964997i	\(0.415532\pi\)
							−0.654984	+	0.755642i	\(0.727325\pi\)
\(31\)	116.667			0.675936			0.337968	−	0.941158i	\(-0.390260\pi\)
							0.337968	+	0.941158i	\(0.390260\pi\)
\(37\)	7.15720	−	31.3578i	0.0318010	−	0.139329i	−0.956535	−	0.291617i	\(-0.905807\pi\)
							0.988336	+	0.152287i	\(0.0486640\pi\)
\(41\)	−294.218	−	141.688i	−1.12071	−	0.539706i	−0.220600	−	0.975364i	\(-0.570801\pi\)
							−0.900112	+	0.435658i	\(0.856516\pi\)
\(43\)	405.907	−	195.474i	1.43954	−	0.693245i	0.458795	−	0.888542i	\(-0.348281\pi\)
							0.980744	+	0.195297i	\(0.0625671\pi\)
\(47\)	−227.386	+	285.133i	−0.705694	+	0.884912i	−0.997434	−	0.0715852i	\(-0.977194\pi\)
							0.291740	+	0.956498i	\(0.405766\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.43.10	yes	78
49.8	even	7	inner	49.4.e.a.8.10	✓	78
49.20	odd	14		2401.4.a.c.1.28		39
49.29	even	7		2401.4.a.d.1.28		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.8.10	✓	78	49.8	even	7	inner
49.4.e.a.43.10	yes	78	1.1	even	1	trivial
2401.4.a.c.1.28		39	49.20	odd	14
2401.4.a.d.1.28		39	49.29	even	7