Newform orbit 684.3.t.a

• Label: 684.3.t.a
• Level: $684$
• Weight: $3$
• Character orbit: 684.t
• Analytic conductor: $18.638$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.6376500822\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
265.1		0		−2.99885	−	0.0829166i		0		−2.98656	+	5.17287i		0		−4.37090	−	7.57062i		0		8.98625	+	0.497310i		0
265.2		0		−2.90955	−	0.731127i		0		−3.02121	+	5.23289i		0		6.41336	+	11.1083i		0		7.93091	+	4.25449i		0
265.3		0		−2.87088	−	0.870675i		0		1.28913	−	2.23284i		0		2.75015	+	4.76339i		0		7.48385	+	4.99920i		0
265.4		0		−2.83737	+	0.974328i		0		3.33339	−	5.77360i		0		4.83336	+	8.37163i		0		7.10137	−	5.52906i		0
265.5		0		−2.83582	+	0.978843i		0		−1.74243	+	3.01798i		0		−0.449830	−	0.779128i		0		7.08373	−	5.55164i		0
265.6		0		−2.79131	−	1.09935i		0		−2.57011	+	4.45156i		0		−2.98315	−	5.16697i		0		6.58286	+	6.13726i		0
265.7		0		−2.77003	−	1.15194i		0		3.04651	−	5.27672i		0		−4.19196	−	7.26070i		0		6.34609	+	6.38179i		0
265.8		0		−2.74539	+	1.20947i		0		3.78362	−	6.55342i		0		−5.25651	−	9.10454i		0		6.07437	−	6.64094i		0
265.9		0		−2.57066	+	1.54651i		0		1.45280	−	2.51632i		0		−0.663126	−	1.14857i		0		4.21662	−	7.95111i		0
265.10		0		−2.19396	−	2.04610i		0		2.17299	−	3.76373i		0		1.82107	+	3.15419i		0		0.626916	+	8.97814i		0
265.11		0		−1.74945	+	2.43709i		0		−1.72643	+	2.99026i		0		4.42424	+	7.66301i		0		−2.87882	−	8.52716i		0
265.12		0		−1.72468	+	2.45468i		0		−4.82077	+	8.34982i		0		−0.153959	−	0.266665i		0		−3.05094	−	8.46710i		0
265.13		0		−1.45850	−	2.62160i		0		0.370923	−	0.642458i		0		0.938087	+	1.62481i		0		−4.74554	+	7.64721i		0
265.14		0		−1.21404	−	2.74338i		0		−3.81342	+	6.60503i		0		1.86164	+	3.22446i		0		−6.05222	+	6.66113i		0
265.15		0		−1.08441	+	2.79715i		0		0.475788	−	0.824090i		0		−4.90114	−	8.48903i		0		−6.64810	−	6.06653i		0
265.16		0		−1.05443	−	2.80859i		0		−1.91223	+	3.31208i		0		−5.72810	−	9.92136i		0		−6.77637	+	5.92290i		0
265.17		0		−0.960366	−	2.84213i		0		4.33807	−	7.51375i		0		−2.48195	−	4.29886i		0		−7.15540	+	5.45897i		0
265.18		0		−0.913509	+	2.85753i		0		3.79806	−	6.57843i		0		3.58363	+	6.20703i		0		−7.33100	−	5.22077i		0
265.19		0		−0.775474	+	2.89804i		0		−0.775683	+	1.34352i		0		−0.749140	−	1.29755i		0		−7.79728	−	4.49471i		0
265.20		0		−0.0369083	+	2.99977i		0		−0.692448	+	1.19936i		0		5.80422	+	10.0532i		0		−8.99728	−	0.221433i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
19.b	odd	2	1	inner
171.o	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.3.t.a	✓	80
3.b	odd	2	1		2052.3.t.a		80
9.c	even	3	1	inner	684.3.t.a	✓	80
9.d	odd	6	1		2052.3.t.a		80
19.b	odd	2	1	inner	684.3.t.a	✓	80
57.d	even	2	1		2052.3.t.a		80
171.l	even	6	1		2052.3.t.a		80
171.o	odd	6	1	inner	684.3.t.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.3.t.a	✓	80	1.a	even	1	1	trivial
684.3.t.a	✓	80	9.c	even	3	1	inner
684.3.t.a	✓	80	19.b	odd	2	1	inner
684.3.t.a	✓	80	171.o	odd	6	1	inner
2052.3.t.a		80	3.b	odd	2	1
2052.3.t.a		80	9.d	odd	6	1
2052.3.t.a		80	57.d	even	2	1
2052.3.t.a		80	171.l	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).