Embedded newform 49.4.g.a.11.9

• Label: 49.4.g.a.11.9
• Level: $49$
• Weight: $4$
• Character: 49.11
• 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			11.9
Character	\(\chi\)	\(=\)	49.11
Dual form			49.4.g.a.9.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.69240 - 1.57032i) q^{2} +(2.54191 - 0.383132i) q^{3} +(-0.199521 + 2.66243i) q^{4} +(4.20113 - 10.7043i) q^{5} +(3.70029 - 4.64002i) q^{6} +(16.1098 - 9.13640i) q^{7} +(15.3588 + 19.2593i) q^{8} +(-19.4859 + 6.01061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.69240	−	1.57032i	0.598353	−	0.555191i	−0.321771	−	0.946818i	\(-0.604278\pi\)
							0.920124	+	0.391627i	\(0.128088\pi\)
\(3\)	2.54191	−	0.383132i	0.489192	−	0.0737338i	0.100185	−	0.994969i	\(-0.468056\pi\)
							0.389006	+	0.921235i	\(0.372818\pi\)
\(5\)	4.20113	−	10.7043i	0.375761	−	0.957423i	−0.610469	−	0.792040i	\(-0.709019\pi\)
							0.986229	−	0.165383i	\(-0.0528859\pi\)
\(7\)	16.1098	−	9.13640i	0.869848	−	0.493319i
							\(11\)	−24.7623	−	7.63817i	−0.678739	−	0.209363i	−0.0638291	−	0.997961i	\(-0.520331\pi\)
−0.614909	+	0.788598i	\(0.710807\pi\)
\(13\)	1.02753	+	4.50188i	0.0219219	+	0.0960459i	0.984705	−	0.174228i	\(-0.0557429\pi\)
							−0.962784	+	0.270274i	\(0.912886\pi\)
\(17\)	−75.7284	+	51.6307i	−1.08040	+	0.736605i	−0.966443	−	0.256880i	\(-0.917305\pi\)
							−0.113958	+	0.993486i	\(0.536353\pi\)
\(19\)	66.9393	+	115.942i	0.808259	+	1.39995i	0.914069	+	0.405559i	\(0.132923\pi\)
							−0.105809	+	0.994386i	\(0.533743\pi\)
\(23\)	−161.923	−	110.397i	−1.46797	−	1.00084i	−0.992649	−	0.121032i	\(-0.961379\pi\)
							−0.475321	−	0.879812i	\(-0.657668\pi\)
\(29\)	106.973	+	51.5156i	0.684980	+	0.329869i	0.743803	−	0.668399i	\(-0.233020\pi\)
							−0.0588224	+	0.998268i	\(0.518735\pi\)
\(31\)	62.4955	−	108.245i	0.362081	−	0.627143i	−0.626222	−	0.779645i	\(-0.715400\pi\)
							0.988303	+	0.152502i	\(0.0487329\pi\)
\(37\)	−19.2050	−	256.273i	−0.0853319	−	1.13867i	−0.861366	−	0.507984i	\(-0.830391\pi\)
							0.776034	−	0.630691i	\(-0.217228\pi\)
\(41\)	50.4371	+	63.2461i	0.192121	+	0.240912i	0.868557	−	0.495590i	\(-0.165048\pi\)
							−0.676436	+	0.736501i	\(0.736477\pi\)
\(43\)	−61.5611	+	77.1952i	−0.218325	+	0.273771i	−0.878918	−	0.476974i	\(-0.841734\pi\)
							0.660592	+	0.750745i	\(0.270305\pi\)
\(47\)	370.808	−	344.060i	1.15081	−	1.06779i	0.153991	−	0.988072i	\(-0.450787\pi\)
							0.996817	−	0.0797214i	\(-0.0254031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.11.9	yes	156
49.3	odd	42		2401.4.a.f.1.27		78
49.9	even	21	inner	49.4.g.a.9.9	✓	156
49.46	even	21		2401.4.a.g.1.27		78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.9.9	✓	156	49.9	even	21	inner
49.4.g.a.11.9	yes	156	1.1	even	1	trivial
2401.4.a.f.1.27		78	49.3	odd	42
2401.4.a.g.1.27		78	49.46	even	21