Embedded newform 21.18.g.b.17.33

• Label: 21.18.g.b.17.33
• Level: $21$
• Weight: $18$
• Character: 21.17
• Analytic conductor: $38.477$
• Analytic rank: $0$
• Dimension: $84$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(38.4766383424\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.33
Character	\(\chi\)	\(=\)	21.17
Dual form			21.18.g.b.5.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(361.472 - 208.696i) q^{2} +(-9069.23 - 6847.58i) q^{3} +(21571.9 - 37363.5i) q^{4} +(-59813.0 - 103599. i) q^{5} +(-4.70733e6 - 582496. i) q^{6} +(1.32833e7 - 7.49561e6i) q^{7} +3.67005e7i q^{8} +(3.53616e7 + 1.24204e8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 19686 q^{3} + 2752510 q^{4} - 4340518 q^{7} - 98670996 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	361.472	−	208.696i	0.998434	−	0.576446i	0.0906494	−	0.995883i	\(-0.471106\pi\)
							0.907785	+	0.419437i	\(0.137772\pi\)
\(3\)	−9069.23	−	6847.58i	−0.798067	−	0.602568i
							\(5\)	−59813.0	−	103599.i	−0.0684779	−	0.118607i	0.829754	−	0.558130i	\(-0.188481\pi\)
−0.898231	+	0.439523i	\(0.855148\pi\)
\(7\)	1.32833e7	−	7.49561e6i	0.870910	−	0.491443i
							\(11\)	2.99200e8	+	1.72743e8i	0.420846	+	0.242976i	0.695439	−	0.718585i	\(-0.255210\pi\)
−0.274593	+	0.961561i	\(0.588543\pi\)
\(13\)			1.30777e9i			0.444646i	0.974973	+	0.222323i	\(0.0713640\pi\)
							−0.974973	+	0.222323i	\(0.928636\pi\)
\(17\)	3.29745e9	−	5.71134e9i	0.114647	−	0.198574i	−0.802992	−	0.595990i	\(-0.796760\pi\)
							0.917638	+	0.397416i	\(0.130093\pi\)
\(19\)	−2.85681e10	+	1.64938e10i	−0.385901	+	0.222800i	−0.680382	−	0.732857i	\(-0.738186\pi\)
							0.294482	+	0.955657i	\(0.404853\pi\)
\(23\)	6.04453e11	−	3.48981e11i	1.60944	−	0.929212i	0.619949	−	0.784642i	\(-0.287153\pi\)
							0.989495	−	0.144570i	\(-0.0461799\pi\)
\(29\)		−	1.44033e12i		−	0.534662i	−0.963605	−	0.267331i	\(-0.913858\pi\)
							0.963605	−	0.267331i	\(-0.0861417\pi\)
\(31\)	3.72693e11	+	2.15174e11i	0.0784832	+	0.0453123i	0.538728	−	0.842480i	\(-0.318905\pi\)
							−0.460245	+	0.887792i	\(0.652238\pi\)
\(37\)	2.71809e12	+	4.70786e12i	0.127218	+	0.220348i	0.922598	−	0.385763i	\(-0.126062\pi\)
							−0.795380	+	0.606111i	\(0.792729\pi\)
\(41\)	−6.87272e13			−1.34421			−0.672103	−	0.740458i	\(-0.734609\pi\)
							−0.672103	+	0.740458i	\(0.734609\pi\)
\(43\)	6.41495e13			0.836973			0.418486	−	0.908223i	\(-0.362561\pi\)
							0.418486	+	0.908223i	\(0.362561\pi\)
\(47\)	8.81633e12	+	1.52703e13i	0.0540078	+	0.0935442i	0.891765	−	0.452498i	\(-0.149467\pi\)
							−0.837758	+	0.546042i	\(0.816134\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.18.g.b.17.33	yes	84
3.2	odd	2	inner	21.18.g.b.17.10	yes	84
7.5	odd	6	inner	21.18.g.b.5.10	✓	84
21.5	even	6	inner	21.18.g.b.5.33	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.18.g.b.5.10	✓	84	7.5	odd	6	inner
21.18.g.b.5.33	yes	84	21.5	even	6	inner
21.18.g.b.17.10	yes	84	3.2	odd	2	inner
21.18.g.b.17.33	yes	84	1.1	even	1	trivial