Embedded newform 21.33.h.b.2.12

• Label: 21.33.h.b.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-85963.2 - 49630.9i) q^{2} +(3.66132e7 - 2.26384e7i) q^{3} +(2.77897e9 + 4.81331e9i) q^{4} +(-1.43172e11 - 8.26606e10i) q^{5} +(-4.27095e12 + 1.28924e11i) q^{6} +(-1.23935e13 + 3.08355e13i) q^{7} -1.25364e14i q^{8} +(8.28028e14 - 1.65773e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−85963.2	−	49630.9i	−1.31169	−	0.757307i	−0.329318	−	0.944219i	\(-0.606819\pi\)
							−0.982377	+	0.186912i	\(0.940152\pi\)
\(3\)	3.66132e7	−	2.26384e7i	0.850545	−	0.525903i
							\(5\)	−1.43172e11	−	8.26606e10i	−0.938294	−	0.541724i	−0.0488690	−	0.998805i	\(-0.515562\pi\)
−0.889425	+	0.457081i	\(0.848895\pi\)
\(7\)	−1.23935e13	+	3.08355e13i	−0.372930	+	0.927860i
							\(11\)	−6.66573e15	+	3.84846e15i	−0.145066	+	0.0837537i	−0.570776	−	0.821106i	\(-0.693358\pi\)
0.425710	+	0.904859i	\(0.360024\pi\)
\(13\)	1.10624e18			1.66248			0.831241	−	0.555912i	\(-0.187631\pi\)
							0.831241	+	0.555912i	\(0.187631\pi\)
\(17\)	5.46560e19	−	3.15556e19i	1.12319	−	0.648477i	0.180980	−	0.983487i	\(-0.442073\pi\)
							0.942214	+	0.335010i	\(0.108740\pi\)
\(19\)	−1.84634e20	+	3.19796e20i	−0.640110	+	1.10870i	0.345298	+	0.938493i	\(0.387778\pi\)
							−0.985408	+	0.170210i	\(0.945555\pi\)
\(23\)	−8.03818e21	−	4.64084e21i	−1.31073	−	0.756748i	−0.328510	−	0.944500i	\(-0.606547\pi\)
							−0.982216	+	0.187752i	\(0.939880\pi\)
\(29\)		−	3.97555e22i		−	0.158865i	−0.996840	−	0.0794326i	\(-0.974689\pi\)
							0.996840	−	0.0794326i	\(-0.0253109\pi\)
\(31\)	1.02820e23	+	1.78089e23i	0.141348	+	0.244822i	0.928004	−	0.372569i	\(-0.121523\pi\)
							−0.786657	+	0.617391i	\(0.788190\pi\)
\(37\)	−7.13533e24	+	1.23588e25i	−0.578344	+	1.00172i	0.417325	+	0.908757i	\(0.362968\pi\)
							−0.995669	+	0.0929645i	\(0.970366\pi\)
\(41\)		−	5.56503e25i		−	0.872822i	−0.899747	−	0.436411i	\(-0.856249\pi\)
							0.899747	−	0.436411i	\(-0.143751\pi\)
\(43\)	1.52729e26			1.11796			0.558980	−	0.829181i	\(-0.311193\pi\)
							0.558980	+	0.829181i	\(0.311193\pi\)
\(47\)	2.94828e26	+	1.70219e26i	0.519999	+	0.300222i	0.736934	−	0.675964i	\(-0.236273\pi\)
							−0.216935	+	0.976186i	\(0.569606\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.12	✓	164
3.2	odd	2	inner	21.33.h.b.2.71	yes	164
7.4	even	3	inner	21.33.h.b.11.71	yes	164
21.11	odd	6	inner	21.33.h.b.11.12	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.12	✓	164	1.1	even	1	trivial
21.33.h.b.2.71	yes	164	3.2	odd	2	inner
21.33.h.b.11.12	yes	164	21.11	odd	6	inner
21.33.h.b.11.71	yes	164	7.4	even	3	inner