Embedded newform 49.4.g.a.16.9

• Label: 49.4.g.a.16.9
• Level: $49$
• Weight: $4$
• Character: 49.16
• 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			16.9
Character	\(\chi\)	\(=\)	49.16
Dual form			49.4.g.a.46.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.701161 + 1.78653i) q^{2} +(-0.192183 - 2.56451i) q^{3} +(3.16436 - 2.93610i) q^{4} +(9.37596 - 6.39242i) q^{5} +(4.44681 - 2.14147i) q^{6} +(-18.5157 + 0.408984i) q^{7} +(21.2972 + 10.2562i) q^{8} +(20.1587 - 3.03843i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.701161	+	1.78653i	0.247898	+	0.631633i	0.999612	−	0.0278559i	\(-0.00886795\pi\)
							−0.751714	+	0.659489i	\(0.770773\pi\)
\(3\)	−0.192183	−	2.56451i	−0.0369857	−	0.493540i	−0.984681	−	0.174364i	\(-0.944213\pi\)
							0.947696	−	0.319176i	\(-0.103406\pi\)
\(5\)	9.37596	−	6.39242i	0.838611	−	0.571755i	−0.0660844	−	0.997814i	\(-0.521051\pi\)
							0.904696	+	0.426059i	\(0.140098\pi\)
\(7\)	−18.5157	+	0.408984i	−0.999756	+	0.0220831i
							\(11\)	20.8695	+	3.14557i	0.572036	+	0.0862206i	0.428689	−	0.903452i	\(-0.358975\pi\)
0.143347	+	0.989673i	\(0.454214\pi\)
\(13\)	−41.2690	+	51.7496i	−0.880458	+	1.10406i	0.113417	+	0.993547i	\(0.463820\pi\)
							−0.993875	+	0.110511i	\(0.964751\pi\)
\(17\)	−89.9801	+	27.7552i	−1.28373	+	0.395978i	−0.860224	−	0.509917i	\(-0.829676\pi\)
							−0.423504	+	0.905894i	\(0.639200\pi\)
\(19\)	52.5706	−	91.0549i	0.634764	−	1.09944i	−0.351801	−	0.936075i	\(-0.614431\pi\)
							0.986565	−	0.163369i	\(-0.0522361\pi\)
\(23\)	−105.440	−	32.5240i	−0.955904	−	0.294857i	−0.222716	−	0.974883i	\(-0.571492\pi\)
							−0.733188	+	0.680026i	\(0.761969\pi\)
\(29\)	−55.1820	+	241.768i	−0.353346	+	1.54811i	0.416052	+	0.909341i	\(0.363414\pi\)
							−0.769398	+	0.638770i	\(0.779444\pi\)
\(31\)	82.0375	+	142.093i	0.475302	+	0.823248i	0.999600	−	0.0282874i	\(-0.00900536\pi\)
							−0.524298	+	0.851535i	\(0.675672\pi\)
\(37\)	255.089	+	236.688i	1.13342	+	1.05166i	0.998147	+	0.0608465i	\(0.0193800\pi\)
							0.135268	+	0.990809i	\(0.456810\pi\)
\(41\)	−299.201	−	144.088i	−1.13969	−	0.548846i	−0.233770	−	0.972292i	\(-0.575106\pi\)
							−0.905922	+	0.423446i	\(0.860820\pi\)
\(43\)	−169.078	+	81.4237i	−0.599632	+	0.288768i	−0.708972	−	0.705237i	\(-0.750841\pi\)
							0.109340	+	0.994004i	\(0.465126\pi\)
\(47\)	−67.9876	−	173.230i	−0.211000	−	0.537620i	0.785716	−	0.618587i	\(-0.212295\pi\)
							−0.996717	+	0.0809667i	\(0.974199\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.16.9	✓	156
49.12	odd	42		2401.4.a.f.1.53		78
49.37	even	21		2401.4.a.g.1.53		78
49.46	even	21	inner	49.4.g.a.46.9	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.16.9	✓	156	1.1	even	1	trivial
49.4.g.a.46.9	yes	156	49.46	even	21	inner
2401.4.a.f.1.53		78	49.12	odd	42
2401.4.a.g.1.53		78	49.37	even	21