Embedded newform 21.33.h.b.2.7

• Label: 21.33.h.b.2.7
• Level: $21$
• Weight: $33$
• Character: 21.2
• Analytic conductor: $136.220$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(136.219975799\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-100095. - 57789.7i) q^{2} +(-3.45400e7 - 2.56906e7i) q^{3} +(4.53182e9 + 7.84935e9i) q^{4} +(-4.47113e9 - 2.58141e9i) q^{5} +(1.97262e12 + 4.56755e12i) q^{6} +(-8.18737e12 - 3.22086e13i) q^{7} -5.51161e14i q^{8} +(5.33007e14 + 1.77471e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	164 * q + 43046720 * q^3 + 176093659134 * q^4 + 3915256954876 * q^6 - 41868332499126 * q^7 + 1232720974221986 * q^9

Character values

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

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−100095.	−	57789.7i	−1.52732	−	0.881801i	−0.999473	−	0.0324650i	\(-0.989664\pi\)
							−0.527852	−	0.849336i	\(-0.677002\pi\)
\(3\)	−3.45400e7	−	2.56906e7i	−0.802385	−	0.596807i
							\(5\)	−4.47113e9	−	2.58141e9i	−0.0293020	−	0.0169175i	0.485277	−	0.874360i	\(-0.338719\pi\)
−0.514579	+	0.857443i	\(0.672052\pi\)
\(7\)	−8.18737e12	−	3.22086e13i	−0.246363	−	0.969178i
							\(11\)	−5.49555e16	+	3.17286e16i	−1.19599	+	0.690506i	−0.959659	−	0.281166i	\(-0.909279\pi\)
−0.236333	+	0.971672i	\(0.575946\pi\)
\(13\)	8.95079e17			1.34514			0.672570	−	0.740033i	\(-0.265190\pi\)
							0.672570	+	0.740033i	\(0.265190\pi\)
\(17\)	1.11558e19	−	6.44081e18i	0.229255	−	0.132360i	−0.380973	−	0.924586i	\(-0.624411\pi\)
							0.610228	+	0.792226i	\(0.291078\pi\)
\(19\)	−1.47932e20	+	2.56226e20i	−0.512867	+	0.888312i	0.487022	+	0.873390i	\(0.338083\pi\)
							−0.999889	+	0.0149219i	\(0.995250\pi\)
\(23\)	8.13695e21	+	4.69787e21i	1.32683	+	0.766047i	0.984809	−	0.173644i	\(-0.0555541\pi\)
							0.342024	+	0.939691i	\(0.388887\pi\)
\(29\)		−	3.48157e23i		−	1.39126i	−0.718402	−	0.695628i	\(-0.755126\pi\)
							0.718402	−	0.695628i	\(-0.244874\pi\)
\(31\)	−5.48453e23	−	9.49949e23i	−0.753967	−	1.30591i	−0.945886	−	0.324499i	\(-0.894804\pi\)
							0.191919	−	0.981411i	\(-0.438529\pi\)
\(37\)	4.86908e24	−	8.43349e24i	0.394656	−	0.683565i	−0.598401	−	0.801197i	\(-0.704197\pi\)
							0.993057	+	0.117632i	\(0.0375303\pi\)
\(41\)			4.39129e25i			0.688732i	0.938835	+	0.344366i	\(0.111906\pi\)
							−0.938835	+	0.344366i	\(0.888094\pi\)
\(43\)	1.75298e26			1.28316			0.641580	−	0.767056i	\(-0.278279\pi\)
							0.641580	+	0.767056i	\(0.278279\pi\)
\(47\)	3.59706e26	+	2.07676e26i	0.634427	+	0.366287i	0.782465	−	0.622695i	\(-0.213962\pi\)
							−0.148038	+	0.988982i	\(0.547296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.7	✓	164
3.2	odd	2	inner	21.33.h.b.2.76	yes	164
7.4	even	3	inner	21.33.h.b.11.76	yes	164
21.11	odd	6	inner	21.33.h.b.11.7	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.7	✓	164	1.1	even	1	trivial
21.33.h.b.2.76	yes	164	3.2	odd	2	inner
21.33.h.b.11.7	yes	164	21.11	odd	6	inner
21.33.h.b.11.76	yes	164	7.4	even	3	inner