Embedded newform 49.6.g.a.11.6

• Label: 49.6.g.a.11.6
• Level: $49$
• Weight: $6$
• Character: 49.11
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.95840 + 4.60072i) q^{2} +(2.69010 - 0.405467i) q^{3} +(1.02772 - 13.7139i) q^{4} +(0.123139 - 0.313754i) q^{5} +(-11.4731 + 14.3869i) q^{6} +(-12.9481 + 128.994i) q^{7} +(-76.9560 - 96.4997i) q^{8} +(-225.132 + 69.4440i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.95840	+	4.60072i	−0.876530	+	0.813301i	−0.983490	−	0.180963i	\(-0.942079\pi\)
							0.106960	+	0.994263i	\(0.465888\pi\)
\(3\)	2.69010	−	0.405467i	0.172570	−	0.0260107i	−0.0621888	−	0.998064i	\(-0.519808\pi\)
							0.234759	+	0.972054i	\(0.424570\pi\)
\(5\)	0.123139	−	0.313754i	0.00220278	−	0.00561260i	−0.929768	−	0.368146i	\(-0.879993\pi\)
							0.931971	+	0.362533i	\(0.118088\pi\)
\(7\)	−12.9481	+	128.994i	−0.0998758	+	0.995000i
							\(11\)	101.944	+	31.4455i	0.254026	+	0.0783567i	0.419153	−	0.907916i	\(-0.362327\pi\)
−0.165127	+	0.986272i	\(0.552803\pi\)
\(13\)	−181.841	−	796.697i	−0.298424	−	1.30748i	−0.872475	−	0.488660i	\(-0.837486\pi\)
							0.574051	−	0.818820i	\(-0.305371\pi\)
\(17\)	52.3165	−	35.6688i	0.0439053	−	0.0299341i	−0.541169	−	0.840914i	\(-0.682018\pi\)
							0.585074	+	0.810980i	\(0.301066\pi\)
\(19\)	−1425.27	−	2468.64i	−0.905762	−	1.56883i	−0.819891	−	0.572519i	\(-0.805966\pi\)
							−0.0858704	−	0.996306i	\(-0.527367\pi\)
\(23\)	−3422.75	−	2333.59i	−1.34914	−	0.919825i	−0.349283	−	0.937017i	\(-0.613575\pi\)
							−0.999852	+	0.0171925i	\(0.994527\pi\)
\(29\)	−98.4334	−	47.4031i	−0.0217344	−	0.0104667i	0.422985	−	0.906137i	\(-0.360982\pi\)
							−0.444719	+	0.895670i	\(0.646697\pi\)
\(31\)	−3653.05	+	6327.27i	−0.682734	+	1.18253i	0.291410	+	0.956598i	\(0.405876\pi\)
							−0.974143	+	0.225931i	\(0.927458\pi\)
\(37\)	954.932	+	12742.7i	0.114675	+	1.53023i	0.696880	+	0.717187i	\(0.254571\pi\)
							−0.582205	+	0.813042i	\(0.697810\pi\)
\(41\)	−11145.5	−	13976.0i	−1.03547	−	1.29844i	−0.953365	−	0.301821i	\(-0.902406\pi\)
							−0.0821098	−	0.996623i	\(-0.526166\pi\)
\(43\)	−10291.4	+	12904.9i	−0.848792	+	1.06435i	0.148360	+	0.988933i	\(0.452601\pi\)
							−0.997152	+	0.0754183i	\(0.975971\pi\)
\(47\)	16793.9	−	15582.5i	1.10894	−	1.02894i	0.109550	−	0.993981i	\(-0.465059\pi\)
							0.999389	−	0.0349635i	\(-0.0111315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.6	yes	264
49.9	even	21	inner	49.6.g.a.9.6	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.6	✓	264	49.9	even	21	inner
49.6.g.a.11.6	yes	264	1.1	even	1	trivial