Newform orbit 693.2.bj.a

• Label: 693.2.bj.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.bj
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q + 84 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
76.1	−2.37595	+	1.37175i	−0.394675	−	1.68649i	2.76342	−	4.78638i	−1.98905	−	1.14838i	3.25117	+	3.46560i	−2.33534	−	1.24345i			9.67591i	−2.68846	+	1.33123i	6.30117
76.2	−2.37595	+	1.37175i	0.394675	+	1.68649i	2.76342	−	4.78638i	1.98905	+	1.14838i	−3.25117	−	3.46560i	−0.0908102	+	2.64419i			9.67591i	−2.68846	+	1.33123i	−6.30117
76.3	−2.27083	+	1.31107i	−1.73110	−	0.0573827i	2.43778	−	4.22237i	3.20211	+	1.84874i	4.00627	−	2.13928i	1.24115	−	2.33657i			7.54012i	2.99341	+	0.198670i	−9.69528
76.4	−2.27083	+	1.31107i	1.73110	+	0.0573827i	2.43778	−	4.22237i	−3.20211	−	1.84874i	−4.00627	+	2.13928i	2.64410	+	0.0934146i			7.54012i	2.99341	+	0.198670i	9.69528
76.5	−2.24374	+	1.29542i	−1.14801	+	1.29695i	2.35625	−	4.08114i	−1.94673	−	1.12395i	0.895724	−	4.39718i	2.64239	+	0.133288i			7.02767i	−0.364167	−	2.97782i	5.82395
76.6	−2.24374	+	1.29542i	1.14801	−	1.29695i	2.35625	−	4.08114i	1.94673	+	1.12395i	−0.895724	+	4.39718i	1.20576	−	2.35502i			7.02767i	−0.364167	−	2.97782i	−5.82395
76.7	−2.17654	+	1.25663i	−0.993538	−	1.41876i	2.15822	−	3.73814i	0.156971	+	0.0906271i	3.94533	+	1.83948i	2.05858	+	1.66200i			5.82178i	−1.02576	+	2.81919i	−0.455537
76.8	−2.17654	+	1.25663i	0.993538	+	1.41876i	2.15822	−	3.73814i	−0.156971	−	0.0906271i	−3.94533	−	1.83948i	−0.410040	−	2.61378i			5.82178i	−1.02576	+	2.81919i	0.455537
76.9	−2.03940	+	1.17745i	−1.70730	−	0.291782i	1.77277	−	3.07052i	0.557844	+	0.322071i	3.82542	−	1.41519i	−2.57119	+	0.623686i			3.63956i	2.82973	+	0.996319i	−1.51689
76.10	−2.03940	+	1.17745i	1.70730	+	0.291782i	1.77277	−	3.07052i	−0.557844	−	0.322071i	−3.82542	+	1.41519i	−1.82572	+	1.91487i			3.63956i	2.82973	+	0.996319i	1.51689
76.11	−1.83080	+	1.05701i	−0.438972	+	1.67550i	1.23456	−	2.13832i	−1.79052	−	1.03376i	−0.967358	−	3.53151i	−1.77682	+	1.96033i			0.991721i	−2.61461	−	1.47099i	4.37079
76.12	−1.83080	+	1.05701i	0.438972	−	1.67550i	1.23456	−	2.13832i	1.79052	+	1.03376i	0.967358	+	3.53151i	−2.58611	+	0.558606i			0.991721i	−2.61461	−	1.47099i	−4.37079
76.13	−1.80043	+	1.03948i	−1.39930	+	1.02077i	1.16103	−	2.01097i	−1.20635	−	0.696484i	1.45827	−	3.29236i	−0.118565	−	2.64309i			0.669554i	0.916062	−	2.85672i	2.89592
76.14	−1.80043	+	1.03948i	1.39930	−	1.02077i	1.16103	−	2.01097i	1.20635	+	0.696484i	−1.45827	+	3.29236i	2.22970	+	1.42423i			0.669554i	0.916062	−	2.85672i	−2.89592
76.15	−1.74001	+	1.00460i	−1.34293	+	1.09386i	1.01843	−	1.76398i	3.26384	+	1.88438i	1.23782	−	3.25244i	−0.144013	+	2.64183i			0.0740665i	0.606919	−	2.93797i	−7.57217
76.16	−1.74001	+	1.00460i	1.34293	−	1.09386i	1.01843	−	1.76398i	−3.26384	−	1.88438i	−1.23782	+	3.25244i	−2.35990	−	1.19620i			0.0740665i	0.606919	−	2.93797i	7.57217
76.17	−1.72725	+	0.997228i	−1.48332	−	0.894300i	0.988929	−	1.71287i	−2.58642	−	1.49327i	3.45388	+	0.0654740i	0.616450	−	2.57293i		−	0.0441623i	1.40046	+	2.65306i	5.95652
76.18	−1.72725	+	0.997228i	1.48332	+	0.894300i	0.988929	−	1.71287i	2.58642	+	1.49327i	−3.45388	−	0.0654740i	2.53645	+	0.752605i		−	0.0441623i	1.40046	+	2.65306i	−5.95652
76.19	−1.70645	+	0.985221i	−0.537632	+	1.64650i	0.941319	−	1.63041i	1.99283	+	1.15056i	−0.704719	−	3.33935i	−1.96331	−	1.77353i		−	0.231255i	−2.42190	−	1.77042i	−4.53424
76.20	−1.70645	+	0.985221i	0.537632	−	1.64650i	0.941319	−	1.63041i	−1.99283	−	1.15056i	0.704719	+	3.33935i	0.554266	+	2.58704i		−	0.231255i	−2.42190	−	1.77042i	4.53424

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.c	even	3	1	inner
11.b	odd	2	1	inner
63.l	odd	6	1	inner
77.b	even	2	1	inner
99.h	odd	6	1	inner
693.bj	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.bj.a	✓	184
7.b	odd	2	1	inner	693.2.bj.a	✓	184
9.c	even	3	1	inner	693.2.bj.a	✓	184
11.b	odd	2	1	inner	693.2.bj.a	✓	184
63.l	odd	6	1	inner	693.2.bj.a	✓	184
77.b	even	2	1	inner	693.2.bj.a	✓	184
99.h	odd	6	1	inner	693.2.bj.a	✓	184
693.bj	even	6	1	inner	693.2.bj.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.bj.a	✓	184	1.a	even	1	1	trivial
693.2.bj.a	✓	184	7.b	odd	2	1	inner
693.2.bj.a	✓	184	9.c	even	3	1	inner
693.2.bj.a	✓	184	11.b	odd	2	1	inner
693.2.bj.a	✓	184	63.l	odd	6	1	inner
693.2.bj.a	✓	184	77.b	even	2	1	inner
693.2.bj.a	✓	184	99.h	odd	6	1	inner
693.2.bj.a	✓	184	693.bj	even	6	1	inner

Hecke kernels

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