Newform orbit 504.2.be.a

• Label: 504.2.be.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.be
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $184$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 2 q^{2} - 2 q^{4} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−1.41404	+	0.0218513i	−1.61755	−	0.619288i	1.99905	−	0.0617973i	0.694615	−	1.20311i	2.30083	+	0.840355i	−0.407789	−	2.61414i	−2.82539	+	0.131066i	2.23297	+	2.00346i	−0.955927	+	1.71643i
139.2	−1.41404	+	0.0218513i	1.61755	+	0.619288i	1.99905	−	0.0617973i	−0.694615	+	1.20311i	−2.30083	−	0.840355i	2.06001	+	1.66022i	−2.82539	+	0.131066i	2.23297	+	2.00346i	0.955927	−	1.71643i
139.3	−1.41366	−	0.0394185i	−1.54504	+	0.782839i	1.99689	+	0.111449i	1.11062	−	1.92366i	2.21503	−	1.04577i	−2.08899	+	1.62361i	−2.81854	−	0.236266i	1.77433	−	2.41904i	−1.64588	+	2.67563i
139.4	−1.41366	−	0.0394185i	1.54504	−	0.782839i	1.99689	+	0.111449i	−1.11062	+	1.92366i	−2.21503	+	1.04577i	−2.45058	+	0.997318i	−2.81854	−	0.236266i	1.77433	−	2.41904i	1.64588	−	2.67563i
139.5	−1.38885	−	0.266643i	−0.855137	+	1.50623i	1.85780	+	0.740655i	−1.56998	+	2.71929i	1.58928	−	1.86392i	1.10903	−	2.40210i	−2.38272	−	1.52403i	−1.53748	−	2.57607i	2.90555	−	3.35806i
139.6	−1.38885	−	0.266643i	0.855137	−	1.50623i	1.85780	+	0.740655i	1.56998	−	2.71929i	−1.58928	+	1.86392i	2.63479	+	0.240603i	−2.38272	−	1.52403i	−1.53748	−	2.57607i	−2.90555	+	3.35806i
139.7	−1.38351	+	0.293103i	−0.0384188	−	1.73162i	1.82818	−	0.811019i	−1.45134	+	2.51379i	0.560696	+	2.38445i	0.907665	−	2.48519i	−2.29159	+	1.65790i	−2.99705	+	0.133054i	1.27114	−	3.90323i
139.8	−1.38351	+	0.293103i	0.0384188	+	1.73162i	1.82818	−	0.811019i	1.45134	−	2.51379i	−0.560696	−	2.38445i	2.60607	+	0.456532i	−2.29159	+	1.65790i	−2.99705	+	0.133054i	−1.27114	+	3.90323i
139.9	−1.37416	−	0.334195i	−0.522528	−	1.65135i	1.77663	+	0.918475i	−0.649625	+	1.12518i	0.166162	+	2.44385i	−0.237312	+	2.63509i	−2.13442	−	1.85587i	−2.45393	+	1.72576i	1.26872	−	1.32908i
139.10	−1.37416	−	0.334195i	0.522528	+	1.65135i	1.77663	+	0.918475i	0.649625	−	1.12518i	−0.166162	−	2.44385i	−2.40071	−	1.11203i	−2.13442	−	1.85587i	−2.45393	+	1.72576i	−1.26872	+	1.32908i
139.11	−1.29729	+	0.563073i	−1.71842	−	0.216873i	1.36590	−	1.46093i	−1.92861	+	3.34045i	2.35140	−	0.686250i	1.79440	+	1.94426i	−0.949346	+	2.66435i	2.90593	+	0.745357i	0.621039	−	5.41946i
139.12	−1.29729	+	0.563073i	1.71842	+	0.216873i	1.36590	−	1.46093i	1.92861	−	3.34045i	−2.35140	+	0.686250i	−0.786580	−	2.52612i	−0.949346	+	2.66435i	2.90593	+	0.745357i	−0.621039	+	5.41946i
139.13	−1.28821	+	0.583540i	−0.476817	+	1.66513i	1.31896	−	1.50344i	−0.852648	+	1.47683i	−0.357427	−	2.42327i	−1.56588	+	2.13261i	−0.821782	+	2.70641i	−2.54529	−	1.58792i	0.236600	−	2.40002i
139.14	−1.28821	+	0.583540i	0.476817	−	1.66513i	1.31896	−	1.50344i	0.852648	−	1.47683i	0.357427	+	2.42327i	−2.62983	+	0.289791i	−0.821782	+	2.70641i	−2.54529	−	1.58792i	−0.236600	+	2.40002i
139.15	−1.20351	−	0.742667i	−1.22607	−	1.22342i	0.896891	+	1.78762i	0.417659	−	0.723407i	0.566998	+	2.38296i	2.60856	−	0.442060i	0.248186	−	2.81752i	0.00649163	+	2.99999i	−1.03991	+	0.560448i
139.16	−1.20351	−	0.742667i	1.22607	+	1.22342i	0.896891	+	1.78762i	−0.417659	+	0.723407i	−0.566998	−	2.38296i	1.68711	−	2.03805i	0.248186	−	2.81752i	0.00649163	+	2.99999i	1.03991	−	0.560448i
139.17	−1.19759	−	0.752180i	−1.58643	+	0.695160i	0.868451	+	1.80161i	0.0673890	−	0.116721i	2.42278	+	0.360761i	1.08501	+	2.41304i	0.315085	−	2.81082i	2.03351	−	2.20564i	−0.168500	+	0.0890957i
139.18	−1.19759	−	0.752180i	1.58643	−	0.695160i	0.868451	+	1.80161i	−0.0673890	+	0.116721i	−2.42278	−	0.360761i	−1.54725	−	2.14617i	0.315085	−	2.81082i	2.03351	−	2.20564i	0.168500	−	0.0890957i
139.19	−1.12708	+	0.854224i	−1.71594	+	0.235665i	0.540602	−	1.92555i	0.892228	−	1.54538i	1.73269	−	1.73141i	2.64346	−	0.110098i	1.03555	+	2.63204i	2.88892	−	0.808775i	0.314496	+	2.50393i
139.20	−1.12708	+	0.854224i	1.71594	−	0.235665i	0.540602	−	1.92555i	−0.892228	+	1.54538i	−1.73269	+	1.73141i	1.41708	−	2.23425i	1.03555	+	2.63204i	2.88892	−	0.808775i	−0.314496	−	2.50393i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.d	odd	2	1	inner
9.c	even	3	1	inner
56.e	even	2	1	inner
63.l	odd	6	1	inner
72.p	odd	6	1	inner
504.be	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.be.a	✓	184
7.b	odd	2	1	inner	504.2.be.a	✓	184
8.d	odd	2	1	inner	504.2.be.a	✓	184
9.c	even	3	1	inner	504.2.be.a	✓	184
56.e	even	2	1	inner	504.2.be.a	✓	184
63.l	odd	6	1	inner	504.2.be.a	✓	184
72.p	odd	6	1	inner	504.2.be.a	✓	184
504.be	even	6	1	inner	504.2.be.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.be.a	✓	184	1.a	even	1	1	trivial
504.2.be.a	✓	184	7.b	odd	2	1	inner
504.2.be.a	✓	184	8.d	odd	2	1	inner
504.2.be.a	✓	184	9.c	even	3	1	inner
504.2.be.a	✓	184	56.e	even	2	1	inner
504.2.be.a	✓	184	63.l	odd	6	1	inner
504.2.be.a	✓	184	72.p	odd	6	1	inner
504.2.be.a	✓	184	504.be	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).