Embedded newform 21.33.h.b.2.19

• Label: 21.33.h.b.2.19
• 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			2.19
Character	\(\chi\)	\(=\)	21.2
Dual form			21.33.h.b.11.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-68807.1 - 39725.8i) q^{2} +(-3.56249e7 + 2.41638e7i) q^{3} +(1.00879e9 + 1.74728e9i) q^{4} +(-1.14635e11 - 6.61847e10i) q^{5} +(3.41117e12 - 2.47415e11i) q^{6} +(-3.31594e13 + 2.20914e12i) q^{7} +1.80942e14i q^{8} +(6.85242e14 - 1.72166e15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−68807.1	−	39725.8i	−1.04991	−	0.606167i	−0.127288	−	0.991866i	\(-0.540627\pi\)
							−0.922625	+	0.385698i	\(0.873961\pi\)
\(3\)	−3.56249e7	+	2.41638e7i	−0.827586	+	0.561339i
							\(5\)	−1.14635e11	−	6.61847e10i	−0.751274	−	0.433748i	0.0748801	−	0.997193i	\(-0.476143\pi\)
−0.826154	+	0.563444i	\(0.809476\pi\)
\(7\)	−3.31594e13	+	2.20914e12i	−0.997788	+	0.0664744i
							\(11\)	−3.22055e16	+	1.85938e16i	−0.700885	+	0.404656i	−0.807677	−	0.589625i	\(-0.799275\pi\)
0.106792	+	0.994281i	\(0.465942\pi\)
\(13\)	−2.60224e17			−0.391070			−0.195535	−	0.980697i	\(-0.562644\pi\)
							−0.195535	+	0.980697i	\(0.562644\pi\)
\(17\)	−2.15862e19	+	1.24628e19i	−0.443602	+	0.256114i	−0.705125	−	0.709083i	\(-0.749109\pi\)
							0.261522	+	0.965198i	\(0.415776\pi\)
\(19\)	−1.74419e20	+	3.02102e20i	−0.604693	+	1.04736i	0.387407	+	0.921909i	\(0.373371\pi\)
							−0.992100	+	0.125450i	\(0.959962\pi\)
\(23\)	−2.32881e21	−	1.34454e21i	−0.379742	−	0.219244i	0.297964	−	0.954577i	\(-0.403693\pi\)
							−0.677706	+	0.735333i	\(0.737026\pi\)
\(29\)			2.89529e23i			1.15698i	0.815691	+	0.578488i	\(0.196357\pi\)
							−0.815691	+	0.578488i	\(0.803643\pi\)
\(31\)	−4.80546e23	−	8.32329e23i	−0.660613	−	1.14422i	−0.980455	−	0.196745i	\(-0.936963\pi\)
							0.319841	−	0.947471i	\(-0.396370\pi\)
\(37\)	−2.66139e24	+	4.60966e24i	−0.215715	+	0.373629i	−0.953494	−	0.301414i	\(-0.902542\pi\)
							0.737779	+	0.675043i	\(0.235875\pi\)
\(41\)			4.70466e25i			0.737881i	0.929453	+	0.368941i	\(0.120279\pi\)
							−0.929453	+	0.368941i	\(0.879721\pi\)
\(43\)	1.77000e26			1.29562			0.647812	−	0.761800i	\(-0.275684\pi\)
							0.647812	+	0.761800i	\(0.275684\pi\)
\(47\)	1.22748e25	+	7.08685e24i	0.0216495	+	0.0124993i	0.510786	−	0.859708i	\(-0.329355\pi\)
							−0.489136	+	0.872207i	\(0.662688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.2.19	✓	164
3.2	odd	2	inner	21.33.h.b.2.64	yes	164
7.4	even	3	inner	21.33.h.b.11.64	yes	164
21.11	odd	6	inner	21.33.h.b.11.19	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.19	✓	164	1.1	even	1	trivial
21.33.h.b.2.64	yes	164	3.2	odd	2	inner
21.33.h.b.11.19	yes	164	21.11	odd	6	inner
21.33.h.b.11.64	yes	164	7.4	even	3	inner