Newform orbit 693.2.bd.a

• Label: 693.2.bd.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.bd
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 80 q^{4} + 2 q^{7} + 8 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
419.1	−2.41842	+	1.39627i	−1.34383	+	1.09275i	2.89916	−	5.02150i	0.201463	−	0.348945i	1.72417	−	4.51909i	−2.63748	+	0.208984i			10.6070i	0.611782	−	2.93696i			1.12519i
419.2	−2.41842	+	1.39627i	1.34383	−	1.09275i	2.89916	−	5.02150i	−0.201463	+	0.348945i	−1.72417	+	4.51909i	1.49973	−	2.17964i			10.6070i	0.611782	−	2.93696i		−	1.12519i
419.3	−2.29950	+	1.32762i	−0.446721	−	1.67345i	2.52513	−	4.37366i	1.04761	−	1.81452i	3.24894	+	3.25503i	−0.947803	+	2.47016i			8.09916i	−2.60088	+	1.49513i			5.56331i
419.4	−2.29950	+	1.32762i	0.446721	+	1.67345i	2.52513	−	4.37366i	−1.04761	+	1.81452i	−3.24894	−	3.25503i	2.61312	+	0.414256i			8.09916i	−2.60088	+	1.49513i		−	5.56331i
419.5	−2.17855	+	1.25779i	−1.49370	−	0.876850i	2.16405	−	3.74825i	−1.54558	+	2.67703i	4.35698	+	0.0315080i	1.97956	+	1.75537i			5.85651i	1.46227	+	2.61950i		−	7.77605i
419.6	−2.17855	+	1.25779i	1.49370	+	0.876850i	2.16405	−	3.74825i	1.54558	−	2.67703i	−4.35698	−	0.0315080i	0.530412	+	2.59204i			5.85651i	1.46227	+	2.61950i			7.77605i
419.7	−2.09869	+	1.21168i	−0.120284	+	1.72787i	1.93634	−	3.35384i	1.90871	−	3.30598i	−1.84119	−	3.77201i	0.229094	−	2.63581i			4.53820i	−2.97106	−	0.415668i			9.25098i
419.8	−2.09869	+	1.21168i	0.120284	−	1.72787i	1.93634	−	3.35384i	−1.90871	+	3.30598i	1.84119	+	3.77201i	−2.39723	−	1.11951i			4.53820i	−2.97106	−	0.415668i		−	9.25098i
419.9	−2.01981	+	1.16614i	−1.51905	+	0.832152i	1.71977	−	2.97872i	−1.40542	+	2.43426i	2.09780	−	3.45222i	0.621796	−	2.57165i			3.35740i	1.61505	−	2.52817i		−	6.55567i
419.10	−2.01981	+	1.16614i	1.51905	−	0.832152i	1.71977	−	2.97872i	1.40542	−	2.43426i	−2.09780	+	3.45222i	−2.53801	−	0.747332i			3.35740i	1.61505	−	2.52817i			6.55567i
419.11	−1.95789	+	1.13039i	−1.57515	−	0.720352i	1.55555	−	2.69430i	1.57572	−	2.72923i	3.89824	−	0.370158i	−1.91779	−	1.82266i			2.51196i	1.96218	+	2.26932i			7.12471i
419.12	−1.95789	+	1.13039i	1.57515	+	0.720352i	1.55555	−	2.69430i	−1.57572	+	2.72923i	−3.89824	+	0.370158i	−0.619575	−	2.57218i			2.51196i	1.96218	+	2.26932i		−	7.12471i
419.13	−1.79587	+	1.03685i	−1.73093	−	0.0623852i	1.15010	−	1.99203i	−0.140492	+	0.243340i	3.17320	−	1.68267i	−1.37983	+	2.25745i			0.622507i	2.99222	+	0.215969i		−	0.582675i
419.14	−1.79587	+	1.03685i	1.73093	+	0.0623852i	1.15010	−	1.99203i	0.140492	−	0.243340i	−3.17320	+	1.68267i	2.64492	−	0.0662417i			0.622507i	2.99222	+	0.215969i			0.582675i
419.15	−1.64920	+	0.952166i	−1.66102	+	0.490944i	0.813239	−	1.40857i	0.677347	−	1.17320i	2.27189	−	2.39123i	2.53163	−	0.768675i		−	0.711311i	2.51795	−	1.63093i			2.57978i
419.16	−1.64920	+	0.952166i	1.66102	−	0.490944i	0.813239	−	1.40857i	−0.677347	+	1.17320i	−2.27189	+	2.39123i	−1.93151	+	1.80812i		−	0.711311i	2.51795	−	1.63093i		−	2.57978i
419.17	−1.57858	+	0.911394i	−0.892429	+	1.48444i	0.661279	−	1.14537i	1.18301	−	2.04903i	0.0558598	−	3.15667i	0.489054	+	2.60016i		−	1.23483i	−1.40714	−	2.64952i			4.31274i
419.18	−1.57858	+	0.911394i	0.892429	−	1.48444i	0.661279	−	1.14537i	−1.18301	+	2.04903i	−0.0558598	+	3.15667i	2.00728	+	1.72361i		−	1.23483i	−1.40714	−	2.64952i		−	4.31274i
419.19	−1.33414	+	0.770264i	−0.553028	+	1.64139i	0.186614	−	0.323224i	−1.81524	+	3.14410i	−0.526489	−	2.61582i	1.26973	+	2.32116i		−	2.50609i	−2.38832	−	1.81547i		−	5.59287i
419.20	−1.33414	+	0.770264i	0.553028	−	1.64139i	0.186614	−	0.323224i	1.81524	−	3.14410i	0.526489	+	2.61582i	1.37532	+	2.26020i		−	2.50609i	−2.38832	−	1.81547i			5.59287i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.d	odd	6	1	inner
63.o	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.bd.a	✓	160
7.b	odd	2	1	inner	693.2.bd.a	✓	160
9.d	odd	6	1	inner	693.2.bd.a	✓	160
63.o	even	6	1	inner	693.2.bd.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.bd.a	✓	160	1.a	even	1	1	trivial
693.2.bd.a	✓	160	7.b	odd	2	1	inner
693.2.bd.a	✓	160	9.d	odd	6	1	inner
693.2.bd.a	✓	160	63.o	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).