Embedded newform 49.4.g.a.2.6

• Label: 49.4.g.a.2.6
• Level: $49$
• Weight: $4$
• Character: 49.2
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.6
Character	\(\chi\)	\(=\)	49.2
Dual form			49.4.g.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.42883 - 0.440736i) q^{2} +(-1.06746 + 2.71984i) q^{3} +(-4.76260 - 3.24708i) q^{4} +(14.5837 - 2.19813i) q^{5} +(2.72395 - 3.41573i) q^{6} +(18.2694 + 3.03821i) q^{7} +(12.8321 + 16.0909i) q^{8} +(13.5343 + 12.5580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.42883	−	0.440736i	−0.505168	−	0.155824i	0.0316874	−	0.999498i	\(-0.489912\pi\)
							−0.536856	+	0.843674i	\(0.680388\pi\)
\(3\)	−1.06746	+	2.71984i	−0.205433	+	0.523434i	−0.996042	−	0.0888874i	\(-0.971669\pi\)
							0.790609	+	0.612321i	\(0.209764\pi\)
\(5\)	14.5837	−	2.19813i	1.30440	−	0.196607i	0.540174	−	0.841553i	\(-0.318358\pi\)
							0.764228	+	0.644946i	\(0.223120\pi\)
\(7\)	18.2694	+	3.03821i	0.986452	+	0.164048i
							\(11\)	14.3407	−	13.3063i	0.393081	−	0.364726i	−0.458729	−	0.888576i	\(-0.651695\pi\)
0.851811	+	0.523850i	\(0.175505\pi\)
\(13\)	−8.78116	−	38.4728i	−0.187343	−	0.820802i	−0.978010	−	0.208557i	\(-0.933123\pi\)
							0.790668	−	0.612246i	\(-0.209734\pi\)
\(17\)	−0.467335	−	6.23615i	−0.00666737	−	0.0889699i	0.992877	−	0.119141i	\(-0.0380139\pi\)
							−0.999545	+	0.0301707i	\(0.990395\pi\)
\(19\)	−9.64247	+	16.7012i	−0.116428	+	0.201659i	−0.918350	−	0.395770i	\(-0.870478\pi\)
							0.801922	+	0.597429i	\(0.203811\pi\)
\(23\)	−1.26949	+	16.9402i	−0.0115090	+	0.153577i	0.988488	+	0.151297i	\(0.0483449\pi\)
							−0.999997	+	0.00228027i	\(0.999274\pi\)
\(29\)	−153.637	−	73.9875i	−0.983779	−	0.473763i	−0.128376	−	0.991726i	\(-0.540976\pi\)
							−0.855403	+	0.517962i	\(0.826691\pi\)
\(31\)	130.010	+	225.183i	0.753240	+	1.30465i	0.946245	+	0.323452i	\(0.104843\pi\)
							−0.193005	+	0.981198i	\(0.561823\pi\)
\(37\)	324.125	−	220.985i	1.44016	−	0.981884i	0.443845	−	0.896103i	\(-0.353614\pi\)
							0.996314	−	0.0857808i	\(-0.0273385\pi\)
\(41\)	−101.263	−	126.979i	−0.385721	−	0.483679i	0.550627	−	0.834751i	\(-0.314389\pi\)
							−0.936348	+	0.351072i	\(0.885817\pi\)
\(43\)	−232.984	+	292.153i	−0.826274	+	1.03611i	0.172420	+	0.985023i	\(0.444841\pi\)
							−0.998694	+	0.0510910i	\(0.983730\pi\)
\(47\)	−238.782	−	73.6545i	−0.741062	−	0.228587i	−0.0988373	−	0.995104i	\(-0.531512\pi\)
							−0.642225	+	0.766516i	\(0.721989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.2.6	✓	156
49.5	odd	42		2401.4.a.f.1.30		78
49.25	even	21	inner	49.4.g.a.25.6	yes	156
49.44	even	21		2401.4.a.g.1.30		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.2.6	✓	156	1.1	even	1	trivial
49.4.g.a.25.6	yes	156	49.25	even	21	inner
2401.4.a.f.1.30		78	49.5	odd	42
2401.4.a.g.1.30		78	49.44	even	21