Embedded newform 21.7.h.a.2.4

• Label: 21.7.h.a.2.4
• Level: $21$
• Weight: $7$
• Character: 21.2
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83113575602\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	21.2
Dual form			21.7.h.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.42751 - 4.28827i) q^{2} +(19.2952 + 18.8863i) q^{3} +(4.77856 + 8.27670i) q^{4} +(-88.3745 - 51.0231i) q^{5} +(-62.3258 - 223.022i) q^{6} +(-127.732 + 318.329i) q^{7} +466.932i q^{8} +(15.6130 + 728.833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.42751	−	4.28827i	−0.928438	−	0.536034i	−0.0421209	−	0.999113i	\(-0.513411\pi\)
							−0.886317	+	0.463079i	\(0.846745\pi\)
\(3\)	19.2952	+	18.8863i	0.714639	+	0.699494i
							\(5\)	−88.3745	−	51.0231i	−0.706996	−	0.408185i	0.102952	−	0.994686i	\(-0.467171\pi\)
−0.809948	+	0.586502i	\(0.800505\pi\)
\(7\)	−127.732	+	318.329i	−0.372398	+	0.928073i
							\(11\)	−1435.62	+	828.858i	−1.07861	+	0.622733i	−0.930520	−	0.366242i	\(-0.880644\pi\)
−0.148085	+	0.988975i	\(0.547311\pi\)
\(13\)	−278.963			−0.126974			−0.0634872	−	0.997983i	\(-0.520222\pi\)
							−0.0634872	+	0.997983i	\(0.520222\pi\)
\(17\)	−4951.96	+	2859.02i	−1.00793	+	0.581929i	−0.910585	−	0.413322i	\(-0.864369\pi\)
							−0.0973456	+	0.995251i	\(0.531035\pi\)
\(19\)	2933.29	−	5080.60i	0.427655	−	0.740721i	−0.569009	−	0.822331i	\(-0.692673\pi\)
							0.996664	+	0.0816105i	\(0.0260063\pi\)
\(23\)	11371.0	+	6565.05i	0.934577	+	0.539578i	0.888256	−	0.459348i	\(-0.151917\pi\)
							0.0463208	+	0.998927i	\(0.485250\pi\)
\(29\)		−	38808.9i		−	1.59124i	−0.605793	−	0.795622i	\(-0.707144\pi\)
							0.605793	−	0.795622i	\(-0.292856\pi\)
\(31\)	6289.55	+	10893.8i	0.211123	+	0.365675i	0.952066	−	0.305892i	\(-0.0989547\pi\)
							−0.740944	+	0.671567i	\(0.765621\pi\)
\(37\)	−38133.2	+	66048.7i	−0.752832	+	1.30394i	0.193613	+	0.981078i	\(0.437979\pi\)
							−0.946445	+	0.322866i	\(0.895354\pi\)
\(41\)			57478.9i			0.833983i	0.908910	+	0.416991i	\(0.136916\pi\)
							−0.908910	+	0.416991i	\(0.863084\pi\)
\(43\)	−2510.49			−0.0315757			−0.0157879	−	0.999875i	\(-0.505026\pi\)
							−0.0157879	+	0.999875i	\(0.505026\pi\)
\(47\)	41606.0	+	24021.2i	0.400739	+	0.231367i	0.686803	−	0.726844i	\(-0.259014\pi\)
							−0.286064	+	0.958211i	\(0.592347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.7.h.a.2.4	✓	28
3.2	odd	2	inner	21.7.h.a.2.11	yes	28
7.2	even	3		147.7.b.e.50.11		14
7.4	even	3	inner	21.7.h.a.11.11	yes	28
7.5	odd	6		147.7.b.d.50.11		14
21.2	odd	6		147.7.b.e.50.4		14
21.5	even	6		147.7.b.d.50.4		14
21.11	odd	6	inner	21.7.h.a.11.4	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.h.a.2.4	✓	28	1.1	even	1	trivial
21.7.h.a.2.11	yes	28	3.2	odd	2	inner
21.7.h.a.11.4	yes	28	21.11	odd	6	inner
21.7.h.a.11.11	yes	28	7.4	even	3	inner
147.7.b.d.50.4		14	21.5	even	6
147.7.b.d.50.11		14	7.5	odd	6
147.7.b.e.50.4		14	21.2	odd	6
147.7.b.e.50.11		14	7.2	even	3