Embedded newform 49.6.g.a.11.20

• Label: 49.6.g.a.11.20
• 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			11.20
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.01874 - 6.51244i) q^{2} +(12.4965 - 1.88354i) q^{3} +(4.45950 - 59.5078i) q^{4} +(16.9650 - 43.2260i) q^{5} +(75.4433 - 94.6029i) q^{6} +(-4.29257 + 129.571i) q^{7} +(-165.210 - 207.167i) q^{8} +(-79.5893 + 24.5501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.01874	−	6.51244i	1.24075	−	1.15125i	0.258033	−	0.966136i	\(-0.416926\pi\)
							0.982717	−	0.185112i	\(-0.0592648\pi\)
\(3\)	12.4965	−	1.88354i	0.801651	−	0.120829i	0.264577	−	0.964365i	\(-0.414768\pi\)
							0.537074	+	0.843535i	\(0.319530\pi\)
\(5\)	16.9650	−	43.2260i	0.303478	−	0.773250i	−0.695047	−	0.718964i	\(-0.744616\pi\)
							0.998525	−	0.0542861i	\(-0.0172883\pi\)
\(7\)	−4.29257	+	129.571i	−0.0331110	+	0.999452i
							\(11\)	207.675	+	64.0592i	0.517490	+	0.159625i	0.542489	−	0.840063i	\(-0.317482\pi\)
−0.0249988	+	0.999687i	\(0.507958\pi\)
\(13\)	4.68983	+	20.5475i	0.00769660	+	0.0337210i	0.978630	−	0.205627i	\(-0.0659234\pi\)
							−0.970934	+	0.239348i	\(0.923066\pi\)
\(17\)	−372.558	+	254.005i	−0.312659	+	0.213167i	−0.709475	−	0.704731i	\(-0.751068\pi\)
							0.396815	+	0.917898i	\(0.370115\pi\)
\(19\)	−25.6093	−	44.3567i	−0.0162748	−	0.0281887i	0.857773	−	0.514028i	\(-0.171847\pi\)
							−0.874048	+	0.485839i	\(0.838514\pi\)
\(23\)	−1910.97	−	1302.88i	−0.753242	−	0.513552i	0.124773	−	0.992185i	\(-0.460180\pi\)
							−0.878015	+	0.478634i	\(0.841132\pi\)
\(29\)	6449.01	+	3105.68i	1.42396	+	0.685744i	0.977864	−	0.209241i	\(-0.0670992\pi\)
							0.446097	+	0.894984i	\(0.352814\pi\)
\(31\)	−971.669	+	1682.98i	−0.181599	+	0.314539i	−0.942425	−	0.334417i	\(-0.891461\pi\)
							0.760826	+	0.648956i	\(0.224794\pi\)
\(37\)	−1070.46	−	14284.4i	−0.128549	−	1.71536i	−0.572996	−	0.819558i	\(-0.694219\pi\)
							0.444447	−	0.895805i	\(-0.353400\pi\)
\(41\)	9711.13	+	12177.4i	0.902215	+	1.13134i	0.990808	+	0.135274i	\(0.0431915\pi\)
							−0.0885932	+	0.996068i	\(0.528237\pi\)
\(43\)	−517.103	+	648.427i	−0.0426487	+	0.0534798i	−0.802698	−	0.596385i	\(-0.796603\pi\)
							0.760050	+	0.649865i	\(0.225175\pi\)
\(47\)	−18068.5	+	16765.1i	−1.19310	+	1.10703i	−0.201260	+	0.979538i	\(0.564504\pi\)
							−0.991839	+	0.127497i	\(0.959306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.20	yes	264
49.9	even	21	inner	49.6.g.a.9.20	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.20	✓	264	49.9	even	21	inner
49.6.g.a.11.20	yes	264	1.1	even	1	trivial