Embedded newform 21.33.h.b.2.9

• Label: 21.33.h.b.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-95143.0 - 54930.9i) q^{2} +(4.10114e7 - 1.30799e7i) q^{3} +(3.88731e9 + 6.73302e9i) q^{4} +(-1.11850e11 - 6.45768e10i) q^{5} +(-4.62044e12 - 1.00833e12i) q^{6} +(2.02925e13 - 2.63181e13i) q^{7} -3.82281e14i q^{8} +(1.51085e15 - 1.07285e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−95143.0	−	54930.9i	−1.45177	−	0.838178i	−0.453186	−	0.891416i	\(-0.649712\pi\)
							−0.998582	+	0.0532379i	\(0.983046\pi\)
\(3\)	4.10114e7	−	1.30799e7i	0.952719	−	0.303854i
							\(5\)	−1.11850e11	−	6.45768e10i	−0.733022	−	0.423210i	0.0865047	−	0.996251i	\(-0.472430\pi\)
−0.819527	+	0.573041i	\(0.805764\pi\)
\(7\)	2.02925e13	−	2.63181e13i	0.610613	−	0.791929i
							\(11\)	1.13384e16	−	6.54622e15i	0.246756	−	0.142465i	−0.371522	−	0.928424i	\(-0.621164\pi\)
0.618278	+	0.785959i	\(0.287831\pi\)
\(13\)	−3.93803e17			−0.591813			−0.295907	−	0.955217i	\(-0.595622\pi\)
							−0.295907	+	0.955217i	\(0.595622\pi\)
\(17\)	2.41230e18	−	1.39274e18i	0.0495734	−	0.0286212i	−0.475008	−	0.879981i	\(-0.657555\pi\)
							0.524582	+	0.851360i	\(0.324222\pi\)
\(19\)	2.34816e20	−	4.06713e20i	0.814085	−	1.41004i	−0.0958973	−	0.995391i	\(-0.530572\pi\)
							0.909983	−	0.414646i	\(-0.136095\pi\)
\(23\)	3.57428e21	+	2.06361e21i	0.582832	+	0.336498i	0.762258	−	0.647273i	\(-0.224091\pi\)
							−0.179426	+	0.983771i	\(0.557424\pi\)
\(29\)			9.20826e22i			0.367968i	0.982929	+	0.183984i	\(0.0588994\pi\)
							−0.982929	+	0.183984i	\(0.941101\pi\)
\(31\)	−4.13435e23	−	7.16091e23i	−0.568356	−	0.984421i	−0.996729	−	0.0808190i	\(-0.974246\pi\)
							0.428373	−	0.903602i	\(-0.359087\pi\)
\(37\)	4.88390e24	−	8.45916e24i	0.395857	−	0.685645i	−0.597353	−	0.801979i	\(-0.703781\pi\)
							0.993210	+	0.116333i	\(0.0371141\pi\)
\(41\)			9.10766e25i			1.42845i	0.699916	+	0.714225i	\(0.253221\pi\)
							−0.699916	+	0.714225i	\(0.746779\pi\)
\(43\)	5.26326e25			0.385265			0.192632	−	0.981271i	\(-0.438298\pi\)
							0.192632	+	0.981271i	\(0.438298\pi\)
\(47\)	6.86237e26	+	3.96199e26i	1.21034	+	0.698792i	0.962834	−	0.270094i	\(-0.0870548\pi\)
							0.247509	+	0.968886i	\(0.420388\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.9	✓	164
3.2	odd	2	inner	21.33.h.b.2.74	yes	164
7.4	even	3	inner	21.33.h.b.11.74	yes	164
21.11	odd	6	inner	21.33.h.b.11.9	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.9	✓	164	1.1	even	1	trivial
21.33.h.b.2.74	yes	164	3.2	odd	2	inner
21.33.h.b.11.9	yes	164	21.11	odd	6	inner
21.33.h.b.11.74	yes	164	7.4	even	3	inner