Embedded newform 49.4.e.a.15.13

• Label: 49.4.e.a.15.13
• 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			15.13
Character	\(\chi\)	\(=\)	49.15
Dual form			49.4.e.a.36.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.21752 - 5.33429i) q^{2} +(-5.18210 + 6.49815i) q^{3} +(-19.7645 - 9.51810i) q^{4} +(-3.70379 + 4.64440i) q^{5} +(28.3537 + 35.5544i) q^{6} +(-12.4428 - 13.7178i) q^{7} +(-47.5447 + 59.6192i) q^{8} +(-9.36372 - 41.0251i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 5 q^{2} - 5 q^{3} - 53 q^{4} - 23 q^{5} + 19 q^{6} - 31 q^{8} - 174 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.21752	−	5.33429i	0.430457	−	1.88596i	−0.0323131	−	0.999478i	\(-0.510287\pi\)
							0.462770	−	0.886478i	\(-0.346855\pi\)
\(3\)	−5.18210	+	6.49815i	−0.997296	+	1.25057i	−0.0293090	+	0.999570i	\(0.509331\pi\)
							−0.967987	+	0.250999i	\(0.919241\pi\)
\(5\)	−3.70379	+	4.64440i	−0.331277	+	0.415408i	−0.919375	−	0.393381i	\(-0.871305\pi\)
							0.588099	+	0.808789i	\(0.299877\pi\)
\(7\)	−12.4428	−	13.7178i	−0.671846	−	0.740691i
							\(11\)	6.97871	−	30.5757i	0.191287	−	0.838084i	−0.784634	−	0.619960i	\(-0.787149\pi\)
0.975921	−	0.218124i	\(-0.0699939\pi\)
\(13\)	3.78378	−	16.5778i	0.0807255	−	0.353681i	−0.918393	−	0.395670i	\(-0.870512\pi\)
							0.999118	+	0.0419889i	\(0.0133694\pi\)
\(17\)	−50.3378	+	24.2414i	−0.718159	+	0.345847i	−0.757015	−	0.653398i	\(-0.773343\pi\)
							0.0388558	+	0.999245i	\(0.487629\pi\)
\(19\)	−16.2635			−0.196374			−0.0981871	−	0.995168i	\(-0.531304\pi\)
							−0.0981871	+	0.995168i	\(0.531304\pi\)
\(23\)	−75.6471	−	36.4297i	−0.685805	−	0.330266i	0.0583285	−	0.998297i	\(-0.481423\pi\)
							−0.744133	+	0.668031i	\(0.767137\pi\)
\(29\)	90.4478	−	43.5574i	0.579163	−	0.278910i	−0.121280	−	0.992618i	\(-0.538700\pi\)
							0.700443	+	0.713708i	\(0.252986\pi\)
\(31\)	−153.655			−0.890234			−0.445117	−	0.895472i	\(-0.646838\pi\)
							−0.445117	+	0.895472i	\(0.646838\pi\)
\(37\)	88.5755	−	42.6557i	0.393560	−	0.189529i	−0.226628	−	0.973981i	\(-0.572770\pi\)
							0.620189	+	0.784453i	\(0.287056\pi\)
\(41\)	205.766	−	258.022i	0.783785	−	0.982835i	−0.216194	−	0.976350i	\(-0.569364\pi\)
							0.999979	−	0.00648484i	\(-0.00206420\pi\)
\(43\)	−212.506	−	266.474i	−0.753647	−	0.945043i	0.246060	−	0.969255i	\(-0.420864\pi\)
							−0.999707	+	0.0242113i	\(0.992293\pi\)
\(47\)	−17.7790	+	77.8949i	−0.0551773	+	0.241748i	−0.994994	−	0.0999302i	\(-0.968138\pi\)
							0.939817	+	0.341678i	\(0.110995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.e.a.15.13	✓	78
49.6	odd	14		2401.4.a.c.1.2		39
49.36	even	7	inner	49.4.e.a.36.13	yes	78
49.43	even	7		2401.4.a.d.1.2		39

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.13	✓	78	1.1	even	1	trivial
49.4.e.a.36.13	yes	78	49.36	even	7	inner
2401.4.a.c.1.2		39	49.6	odd	14
2401.4.a.d.1.2		39	49.43	even	7