Embedded newform 21.7.h.a.2.14

• Label: 21.7.h.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	21.2
Dual form			21.7.h.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.1658 + 7.60129i) q^{2} +(-23.5594 + 13.1892i) q^{3} +(83.5593 + 144.729i) q^{4} +(-82.1651 - 47.4381i) q^{5} +(-410.434 - 5.43441i) q^{6} +(334.938 + 73.9300i) q^{7} +1567.67i q^{8} +(381.088 - 621.460i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.1658	+	7.60129i	1.64573	+	0.950161i	0.978743	+	0.205090i	\(0.0657488\pi\)
							0.666985	+	0.745071i	\(0.267585\pi\)
\(3\)	−23.5594	+	13.1892i	−0.872569	+	0.488490i
							\(5\)	−82.1651	−	47.4381i	−0.657321	−	0.379505i	0.133934	−	0.990990i	\(-0.457239\pi\)
−0.791255	+	0.611486i	\(0.790572\pi\)
\(7\)	334.938	+	73.9300i	0.976495	+	0.215539i
							\(11\)	735.924	−	424.886i	0.552910	−	0.319223i	−0.197385	−	0.980326i	\(-0.563245\pi\)
0.750295	+	0.661103i	\(0.229911\pi\)
\(13\)	1143.92			0.520674			0.260337	−	0.965518i	\(-0.416166\pi\)
							0.260337	+	0.965518i	\(0.416166\pi\)
\(17\)	−2380.94	+	1374.64i	−0.484621	+	0.279796i	−0.722340	−	0.691538i	\(-0.756934\pi\)
							0.237719	+	0.971334i	\(0.423600\pi\)
\(19\)	3551.65	−	6151.64i	0.517809	−	0.896871i	−0.481977	−	0.876184i	\(-0.660081\pi\)
							0.999786	−	0.0206875i	\(-0.00658549\pi\)
\(23\)	−4815.31	−	2780.12i	−0.395768	−	0.228497i	0.288888	−	0.957363i	\(-0.406714\pi\)
							−0.684656	+	0.728866i	\(0.740048\pi\)
\(29\)		−	22700.1i		−	0.930752i	−0.885113	−	0.465376i	\(-0.845919\pi\)
							0.885113	−	0.465376i	\(-0.154081\pi\)
\(31\)	−4260.11	−	7378.73i	−0.143000	−	0.247683i	0.785625	−	0.618703i	\(-0.212342\pi\)
							−0.928625	+	0.371020i	\(0.879008\pi\)
\(37\)	−35924.2	+	62222.6i	−0.709222	+	1.22841i	0.255924	+	0.966697i	\(0.417620\pi\)
							−0.965146	+	0.261712i	\(0.915713\pi\)
\(41\)			59009.1i			0.856184i	0.903735	+	0.428092i	\(0.140814\pi\)
							−0.903735	+	0.428092i	\(0.859186\pi\)
\(43\)	−37766.3			−0.475006			−0.237503	−	0.971387i	\(-0.576329\pi\)
							−0.237503	+	0.971387i	\(0.576329\pi\)
\(47\)	−97600.2	−	56349.5i	−0.940063	−	0.542746i	−0.0500830	−	0.998745i	\(-0.515949\pi\)
							−0.889980	+	0.455999i	\(0.849282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.7.h.a.2.14	yes	28
3.2	odd	2	inner	21.7.h.a.2.1	✓	28
7.2	even	3		147.7.b.e.50.1		14
7.4	even	3	inner	21.7.h.a.11.1	yes	28
7.5	odd	6		147.7.b.d.50.1		14
21.2	odd	6		147.7.b.e.50.14		14
21.5	even	6		147.7.b.d.50.14		14
21.11	odd	6	inner	21.7.h.a.11.14	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.h.a.2.1	✓	28	3.2	odd	2	inner
21.7.h.a.2.14	yes	28	1.1	even	1	trivial
21.7.h.a.11.1	yes	28	7.4	even	3	inner
21.7.h.a.11.14	yes	28	21.11	odd	6	inner
147.7.b.d.50.1		14	7.5	odd	6
147.7.b.d.50.14		14	21.5	even	6
147.7.b.e.50.1		14	7.2	even	3
147.7.b.e.50.14		14	21.2	odd	6