Newform orbit 693.2.w.a

• Label: 693.2.w.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.w
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 6 q^{3} - 72 q^{4} + 6 q^{5} - 22 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
428.1	−1.38539	−	2.39956i	−0.807980	−	1.53205i	−2.83859	+	4.91658i	0.488851	+	0.282238i	−2.55687	+	4.06127i	0.866025	−	0.500000i	10.1886			−1.69434	+	2.47573i		−	1.56403i
428.2	−1.36423	−	2.36292i	−0.653554	+	1.60402i	−2.72227	+	4.71511i	3.54289	+	2.04549i	4.68177	−	0.643954i	−0.866025	+	0.500000i	9.39831			−2.14573	−	2.09662i		−	11.1621i
428.3	−1.36415	−	2.36278i	1.25139	+	1.19750i	−2.72183	+	4.71435i	−2.93836	−	1.69646i	1.12234	−	4.59035i	−0.866025	+	0.500000i	9.39538			0.131977	+	2.99710i			9.25696i
428.4	−1.27818	−	2.21387i	0.882640	−	1.49028i	−2.26747	+	3.92738i	0.665205	+	0.384056i	−4.42746	−	0.0492005i	−0.866025	+	0.500000i	6.48023			−1.44189	−	2.63077i		−	1.96357i
428.5	−1.25247	−	2.16934i	1.59978	−	0.663859i	−2.13736	+	3.70202i	2.70422	+	1.56128i	−3.44381	−	2.63900i	0.866025	−	0.500000i	5.69804			2.11858	−	2.12405i		−	7.82184i
428.6	−1.24493	−	2.15627i	−1.35406	+	1.08005i	−2.09968	+	3.63675i	−2.47886	−	1.43117i	4.01459	+	1.57515i	0.866025	−	0.500000i	5.47607			0.666980	−	2.92492i			7.12680i
428.7	−1.21640	−	2.10687i	1.24941	+	1.19958i	−1.95926	+	3.39353i	0.840774	+	0.485421i	1.00757	−	4.09150i	0.866025	−	0.500000i	4.66736			0.122035	+	2.99752i		−	2.36186i
428.8	−1.17973	−	2.04335i	0.247545	−	1.71427i	−1.78353	+	3.08917i	−2.45274	−	1.41609i	−3.79490	+	1.51655i	−0.866025	+	0.500000i	3.69743			−2.87744	−	0.848719i			6.68243i
428.9	−1.16533	−	2.01842i	0.233131	+	1.71629i	−1.71601	+	2.97222i	−0.240351	−	0.138767i	3.19252	−	2.47061i	0.866025	−	0.500000i	3.33757			−2.89130	+	0.800239i			0.646839i
428.10	−1.07122	−	1.85541i	−0.842708	−	1.51322i	−1.29504	+	2.24307i	−2.38557	−	1.37731i	−1.90493	+	3.18457i	0.866025	−	0.500000i	1.26420			−1.57969	+	2.55041i			5.90163i
428.11	−1.00965	−	1.74877i	−1.72413	−	0.165417i	−1.03879	+	1.79924i	0.546895	+	0.315750i	1.45150	+	3.18212i	−0.866025	+	0.500000i	0.156653			2.94527	+	0.570402i		−	1.27519i
428.12	−0.999533	−	1.73124i	−0.786674	−	1.54310i	−0.998131	+	1.72881i	3.81065	+	2.20008i	−1.88516	+	2.90430i	−0.866025	+	0.500000i	−0.00747069			−1.76229	+	2.42783i		−	8.79621i
428.13	−0.982554	−	1.70183i	1.72392	−	0.167597i	−0.930823	+	1.61223i	0.660301	+	0.381225i	−1.97907	−	2.76916i	−0.866025	+	0.500000i	−0.271881			2.94382	−	0.577850i		−	1.49830i
428.14	−0.977762	−	1.69353i	−1.31521	+	1.12704i	−0.912037	+	1.57969i	2.31754	+	1.33803i	3.19465	+	1.12537i	0.866025	−	0.500000i	−0.344027			0.459544	−	2.96459i		−	5.23310i
428.15	−0.961738	−	1.66578i	0.100851	+	1.72911i	−0.849880	+	1.47204i	0.954976	+	0.551356i	2.78333	−	1.83095i	−0.866025	+	0.500000i	−0.577505			−2.97966	+	0.348765i		−	2.12104i
428.16	−0.942384	−	1.63226i	0.615915	−	1.61884i	−0.776174	+	1.34437i	2.14838	+	1.24037i	−3.22279	+	0.520239i	0.866025	−	0.500000i	−0.843720			−2.24130	−	1.99414i		−	4.67561i
428.17	−0.924214	−	1.60079i	1.68947	+	0.381682i	−0.708344	+	1.22689i	−1.75597	−	1.01381i	−0.950444	−	3.05724i	0.866025	−	0.500000i	−1.07821			2.70864	+	1.28968i			3.74790i
428.18	−0.870788	−	1.50825i	−1.36375	−	1.06779i	−0.516543	+	0.894679i	−3.03651	−	1.75313i	−0.422952	+	2.98670i	−0.866025	+	0.500000i	−1.68395			0.719650	+	2.91241i			6.10641i
428.19	−0.764079	−	1.32342i	1.16874	+	1.27830i	−0.167635	+	0.290352i	1.60038	+	0.923982i	0.798717	−	2.52346i	−0.866025	+	0.500000i	−2.54397			−0.268083	+	2.98800i		−	2.82398i
428.20	−0.754308	−	1.30650i	−1.63895	−	0.560226i	−0.137962	+	0.238957i	−0.565828	−	0.326681i	0.504336	+	2.56387i	0.866025	−	0.500000i	−2.60097			2.37229	+	1.83636i			0.985672i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
11.b	odd	2	1	inner
99.g	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.w.a	✓	144
9.d	odd	6	1	inner	693.2.w.a	✓	144
11.b	odd	2	1	inner	693.2.w.a	✓	144
99.g	even	6	1	inner	693.2.w.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.w.a	✓	144	1.a	even	1	1	trivial
693.2.w.a	✓	144	9.d	odd	6	1	inner
693.2.w.a	✓	144	11.b	odd	2	1	inner
693.2.w.a	✓	144	99.g	even	6	1	inner

Hecke kernels

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