Embedded newform 49.4.g.a.4.3

• Label: 49.4.g.a.4.3
• Level: $49$
• Weight: $4$
• Character: 49.4
• 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			4.3
Character	\(\chi\)	\(=\)	49.4
Dual form			49.4.g.a.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.89337 - 1.97267i) q^{2} +(-3.27221 - 3.03617i) q^{3} +(1.55746 + 3.96835i) q^{4} +(-9.83041 + 3.03228i) q^{5} +(3.47838 + 15.2398i) q^{6} +(18.2590 + 3.09998i) q^{7} +(-2.91198 + 12.7582i) q^{8} +(-0.528663 - 7.05451i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.89337	−	1.97267i	−1.02296	−	0.697444i	−0.0690758	−	0.997611i	\(-0.522005\pi\)
							−0.953886	+	0.300168i	\(0.902957\pi\)
\(3\)	−3.27221	−	3.03617i	−0.629738	−	0.584311i	0.299362	−	0.954139i	\(-0.403226\pi\)
							−0.929100	+	0.369828i	\(0.879417\pi\)
\(5\)	−9.83041	+	3.03228i	−0.879259	+	0.271215i	−0.701338	−	0.712829i	\(-0.747413\pi\)
							−0.177921	+	0.984045i	\(0.556937\pi\)
\(7\)	18.2590	+	3.09998i	0.985892	+	0.167383i
							\(11\)	−4.31229	+	57.5435i	−0.118200	+	1.57727i	0.551201	+	0.834373i	\(0.314170\pi\)
−0.669401	+	0.742901i	\(0.733449\pi\)
\(13\)	−44.8185	+	21.5835i	−0.956186	+	0.460475i	−0.845851	−	0.533419i	\(-0.820907\pi\)
							−0.110335	+	0.993894i	\(0.535192\pi\)
\(17\)	−18.3853	+	2.77114i	−0.262300	+	0.0395354i	−0.278876	−	0.960327i	\(-0.589962\pi\)
							0.0165756	+	0.999863i	\(0.494724\pi\)
\(19\)	−41.6773	−	72.1872i	−0.503233	−	0.871625i	−0.999993	−	0.00373706i	\(-0.998810\pi\)
							0.496760	−	0.867888i	\(-0.334523\pi\)
\(23\)	−163.424	−	24.6321i	−1.48157	−	0.223311i	−0.642062	−	0.766653i	\(-0.721921\pi\)
							−0.839512	+	0.543342i	\(0.817159\pi\)
\(29\)	139.183	+	174.530i	0.891228	+	1.11756i	0.992444	+	0.122700i	\(0.0391554\pi\)
							−0.101215	+	0.994865i	\(0.532273\pi\)
\(31\)	1.46638	−	2.53985i	0.00849581	−	0.0147152i	−0.861746	−	0.507340i	\(-0.830629\pi\)
							0.870242	+	0.492624i	\(0.163962\pi\)
\(37\)	−33.6111	+	85.6397i	−0.149341	+	0.380516i	−0.985894	−	0.167371i	\(-0.946472\pi\)
							0.836553	+	0.547887i	\(0.184568\pi\)
\(41\)	23.7564	−	104.083i	0.0904908	−	0.396466i	−0.909316	−	0.416105i	\(-0.863395\pi\)
							0.999807	+	0.0196395i	\(0.00625186\pi\)
\(43\)	25.9991	+	113.910i	0.0922054	+	0.403978i	0.999877	−	0.0157036i	\(-0.00499883\pi\)
							−0.907671	+	0.419682i	\(0.862142\pi\)
\(47\)	−320.678	−	218.635i	−0.995229	−	0.678535i	−0.0479178	−	0.998851i	\(-0.515259\pi\)
							−0.947311	+	0.320316i	\(0.896211\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.g.a.4.3	✓	156
49.24	odd	42		2401.4.a.f.1.18		78
49.25	even	21		2401.4.a.g.1.18		78
49.37	even	21	inner	49.4.g.a.37.3	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.g.a.4.3	✓	156	1.1	even	1	trivial
49.4.g.a.37.3	yes	156	49.37	even	21	inner
2401.4.a.f.1.18		78	49.24	odd	42
2401.4.a.g.1.18		78	49.25	even	21