Embedded newform 21.18.g.b.17.14

• Label: 21.18.g.b.17.14
• 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			17.14
Character	\(\chi\)	\(=\)	21.17
Dual form			21.18.g.b.5.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-279.124 + 161.152i) q^{2} +(2680.28 - 11043.4i) q^{3} +(-13596.0 + 23549.0i) q^{4} +(771218. + 1.33579e6i) q^{5} +(1.03153e6 + 3.51440e6i) q^{6} +(1.17252e6 - 1.52071e7i) q^{7} -5.10092e7i q^{8} +(-1.14772e8 - 5.91987e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 19686 q^{3} + 2752510 q^{4} - 4340518 q^{7} - 98670996 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−279.124	+	161.152i	−0.770977	+	0.445124i	−0.833223	−	0.552937i	\(-0.813507\pi\)
							0.0622459	+	0.998061i	\(0.480174\pi\)
\(3\)	2680.28	−	11043.4i	0.235857	−	0.971788i
							\(5\)	771218.	+	1.33579e6i	0.882942	+	1.52930i	0.848055	+	0.529909i	\(0.177774\pi\)
0.0348874	+	0.999391i	\(0.488893\pi\)
\(7\)	1.17252e6	−	1.52071e7i	0.0768750	−	0.997041i
							\(11\)	−2.57669e7	−	1.48765e7i	−0.0362430	−	0.0209249i	0.481769	−	0.876298i	\(-0.339994\pi\)
−0.518012	+	0.855373i	\(0.673328\pi\)
\(13\)			1.61254e7i			0.00548266i	0.999996	+	0.00274133i	\(0.000872593\pi\)
							−0.999996	+	0.00274133i	\(0.999127\pi\)
\(17\)	−1.14036e10	+	1.97516e10i	−0.396484	+	0.686731i	−0.993289	−	0.115655i	\(-0.963103\pi\)
							0.596805	+	0.802386i	\(0.296437\pi\)
\(19\)	−3.45802e10	+	1.99649e10i	−0.467112	+	0.269687i	−0.715030	−	0.699094i	\(-0.753587\pi\)
							0.247918	+	0.968781i	\(0.420254\pi\)
\(23\)	4.26008e11	−	2.45956e11i	1.13431	−	0.654893i	0.189293	−	0.981921i	\(-0.439380\pi\)
							0.945015	+	0.327028i	\(0.106047\pi\)
\(29\)		−	4.70405e12i		−	1.74618i	−0.487560	−	0.873089i	\(-0.662113\pi\)
							0.487560	−	0.873089i	\(-0.337887\pi\)
\(31\)	−3.82898e11	−	2.21066e11i	−0.0806322	−	0.0465530i	0.459142	−	0.888363i	\(-0.348157\pi\)
							−0.539774	+	0.841810i	\(0.681490\pi\)
\(37\)	−4.34808e12	−	7.53110e12i	−0.203509	−	0.352487i	0.746148	−	0.665780i	\(-0.231901\pi\)
							−0.949657	+	0.313293i	\(0.898568\pi\)
\(41\)	−6.29657e13			−1.23152			−0.615760	−	0.787934i	\(-0.711151\pi\)
							−0.615760	+	0.787934i	\(0.711151\pi\)
\(43\)	7.49751e13			0.978217			0.489109	−	0.872223i	\(-0.337322\pi\)
							0.489109	+	0.872223i	\(0.337322\pi\)
\(47\)	−8.71537e13	−	1.50955e14i	−0.533892	−	0.924729i	−0.999216	−	0.0395882i	\(-0.987395\pi\)
							0.465324	−	0.885141i	\(-0.345938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.18.g.b.17.14	yes	84
3.2	odd	2	inner	21.18.g.b.17.29	yes	84
7.5	odd	6	inner	21.18.g.b.5.29	yes	84
21.5	even	6	inner	21.18.g.b.5.14	✓	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.18.g.b.5.14	✓	84	21.5	even	6	inner
21.18.g.b.5.29	yes	84	7.5	odd	6	inner
21.18.g.b.17.14	yes	84	1.1	even	1	trivial
21.18.g.b.17.29	yes	84	3.2	odd	2	inner