Embedded newform 21.7.h.a.2.6

• Label: 21.7.h.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	21.2
Dual form			21.7.h.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.88516 - 1.66575i) q^{2} +(-26.7007 + 4.00908i) q^{3} +(-26.4506 - 45.8137i) q^{4} +(166.359 + 96.0474i) q^{5} +(83.7139 + 32.9098i) q^{6} +(157.090 + 304.913i) q^{7} +389.456i q^{8} +(696.855 - 214.090i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.88516	−	1.66575i	−0.360645	−	0.208219i	0.308719	−	0.951153i	\(-0.400100\pi\)
							−0.669364	+	0.742935i	\(0.733433\pi\)
\(3\)	−26.7007	+	4.00908i	−0.988915	+	0.148484i
							\(5\)	166.359	+	96.0474i	1.33087	+	0.768379i	0.985433	−	0.170062i	\(-0.0543968\pi\)
0.345439	+	0.938441i	\(0.387730\pi\)
\(7\)	157.090	+	304.913i	0.457989	+	0.888958i
							\(11\)	−750.813	+	433.482i	−0.564097	+	0.325681i	−0.754788	−	0.655969i	\(-0.772260\pi\)
0.190692	+	0.981650i	\(0.438927\pi\)
\(13\)	3412.13			1.55309			0.776543	−	0.630064i	\(-0.216971\pi\)
							0.776543	+	0.630064i	\(0.216971\pi\)
\(17\)	−2065.25	+	1192.37i	−0.420364	+	0.242697i	−0.695233	−	0.718784i	\(-0.744699\pi\)
							0.274869	+	0.961482i	\(0.411366\pi\)
\(19\)	−1295.70	+	2244.22i	−0.188906	+	0.327194i	−0.944886	−	0.327401i	\(-0.893827\pi\)
							0.755980	+	0.654595i	\(0.227161\pi\)
\(23\)	6492.91	+	3748.68i	0.533649	+	0.308103i	0.742501	−	0.669845i	\(-0.233639\pi\)
							−0.208852	+	0.977947i	\(0.566973\pi\)
\(29\)			24538.1i			1.00611i	0.864253	+	0.503057i	\(0.167791\pi\)
							−0.864253	+	0.503057i	\(0.832209\pi\)
\(31\)	10218.4	+	17698.7i	0.343001	+	0.594096i	0.984989	−	0.172619i	\(-0.0552231\pi\)
							−0.641987	+	0.766715i	\(0.721890\pi\)
\(37\)	33989.3	−	58871.2i	0.671023	−	1.16225i	−0.306591	−	0.951841i	\(-0.599188\pi\)
							0.977614	−	0.210405i	\(-0.0674782\pi\)
\(41\)		−	87649.0i		−	1.27173i	−0.771800	−	0.635866i	\(-0.780643\pi\)
							0.771800	−	0.635866i	\(-0.219357\pi\)
\(43\)	11230.1			0.141247			0.0706236	−	0.997503i	\(-0.477501\pi\)
							0.0706236	+	0.997503i	\(0.477501\pi\)
\(47\)	−105944.	−	61166.8i	−1.02043	−	0.589145i	−0.106200	−	0.994345i	\(-0.533868\pi\)
							−0.914228	+	0.405200i	\(0.867202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.7.h.a.2.6	✓	28
3.2	odd	2	inner	21.7.h.a.2.9	yes	28
7.2	even	3		147.7.b.e.50.9		14
7.4	even	3	inner	21.7.h.a.11.9	yes	28
7.5	odd	6		147.7.b.d.50.9		14
21.2	odd	6		147.7.b.e.50.6		14
21.5	even	6		147.7.b.d.50.6		14
21.11	odd	6	inner	21.7.h.a.11.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.h.a.2.6	✓	28	1.1	even	1	trivial
21.7.h.a.2.9	yes	28	3.2	odd	2	inner
21.7.h.a.11.6	yes	28	21.11	odd	6	inner
21.7.h.a.11.9	yes	28	7.4	even	3	inner
147.7.b.d.50.6		14	21.5	even	6
147.7.b.d.50.9		14	7.5	odd	6
147.7.b.e.50.6		14	21.2	odd	6
147.7.b.e.50.9		14	7.2	even	3