Newform orbit 468.2.bp.d

• Label: 468.2.bp.d
• Level: $468$
• Weight: $2$
• Character orbit: 468.bp
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bp (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−1.41390	−	0.0297419i		0		1.99823	+	0.0841042i	1.08881				0		−0.981428	+	1.69988i	−2.82280	−	0.178346i		0		−1.53948	−	0.0323834i
179.2	−1.36720	−	0.361614i		0		1.73847	+	0.988798i	−0.652972				0		2.10984	−	3.65435i	−2.01927	−	1.98054i		0		0.892744	+	0.236124i
179.3	−1.36583	+	0.366771i		0		1.73096	−	1.00189i	−3.86412				0		0.450459	−	0.780219i	−1.99672	+	2.00327i		0		5.27771	−	1.41724i
179.4	−1.31781	+	0.513210i		0		1.47323	−	1.35262i	−1.48007				0		−0.806607	+	1.39708i	−1.24725	+	2.53857i		0		1.95045	−	0.759589i
179.5	−1.10336	+	0.884649i		0		0.434791	−	1.95217i	1.48007				0		0.806607	−	1.39708i	1.24725	+	2.53857i		0		−1.63305	+	1.30935i
179.6	−1.06568	−	0.929688i		0		0.271362	+	1.98151i	−1.80723				0		−1.99546	+	3.45623i	1.55300	−	2.36394i		0		1.92594	+	1.68016i
179.7	−1.00055	+	0.999454i		0		0.00218229	−	2.00000i	3.86412				0		−0.450459	+	0.780219i	1.99672	+	2.00327i		0		−3.86622	+	3.86201i
179.8	−0.681193	+	1.23934i		0		−1.07195	−	1.68847i	−1.08881				0		0.981428	−	1.69988i	2.82280	−	0.178346i		0		0.741693	−	1.34942i
179.9	−0.370433	+	1.36484i		0		−1.72556	−	1.01116i	0.652972				0		−2.10984	+	3.65435i	2.01927	−	1.98054i		0		−0.241882	+	0.891201i
179.10	−0.272292	−	1.38775i		0		−1.85171	+	0.755747i	−1.80723				0		1.99546	−	3.45623i	1.55300	+	2.36394i		0		0.492095	+	2.50799i
179.11	0.272292	+	1.38775i		0		−1.85171	+	0.755747i	1.80723				0		1.99546	−	3.45623i	−1.55300	−	2.36394i		0		0.492095	+	2.50799i
179.12	0.370433	−	1.36484i		0		−1.72556	−	1.01116i	−0.652972				0		−2.10984	+	3.65435i	−2.01927	+	1.98054i		0		−0.241882	+	0.891201i
179.13	0.681193	−	1.23934i		0		−1.07195	−	1.68847i	1.08881				0		0.981428	−	1.69988i	−2.82280	+	0.178346i		0		0.741693	−	1.34942i
179.14	1.00055	−	0.999454i		0		0.00218229	−	2.00000i	−3.86412				0		−0.450459	+	0.780219i	−1.99672	−	2.00327i		0		−3.86622	+	3.86201i
179.15	1.06568	+	0.929688i		0		0.271362	+	1.98151i	1.80723				0		−1.99546	+	3.45623i	−1.55300	+	2.36394i		0		1.92594	+	1.68016i
179.16	1.10336	−	0.884649i		0		0.434791	−	1.95217i	−1.48007				0		0.806607	−	1.39708i	−1.24725	−	2.53857i		0		−1.63305	+	1.30935i
179.17	1.31781	−	0.513210i		0		1.47323	−	1.35262i	1.48007				0		−0.806607	+	1.39708i	1.24725	−	2.53857i		0		1.95045	−	0.759589i
179.18	1.36583	−	0.366771i		0		1.73096	−	1.00189i	3.86412				0		0.450459	−	0.780219i	1.99672	−	2.00327i		0		5.27771	−	1.41724i
179.19	1.36720	+	0.361614i		0		1.73847	+	0.988798i	0.652972				0		2.10984	−	3.65435i	2.01927	+	1.98054i		0		0.892744	+	0.236124i
179.20	1.41390	+	0.0297419i		0		1.99823	+	0.0841042i	−1.08881				0		−0.981428	+	1.69988i	2.82280	+	0.178346i		0		−1.53948	−	0.0323834i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
13.e	even	6	1	inner
39.h	odd	6	1	inner
52.i	odd	6	1	inner
156.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.bp.d	✓	40
3.b	odd	2	1	inner	468.2.bp.d	✓	40
4.b	odd	2	1	inner	468.2.bp.d	✓	40
12.b	even	2	1	inner	468.2.bp.d	✓	40
13.e	even	6	1	inner	468.2.bp.d	✓	40
39.h	odd	6	1	inner	468.2.bp.d	✓	40
52.i	odd	6	1	inner	468.2.bp.d	✓	40
156.r	even	6	1	inner	468.2.bp.d	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.bp.d	✓	40	1.a	even	1	1	trivial
468.2.bp.d	✓	40	3.b	odd	2	1	inner
468.2.bp.d	✓	40	4.b	odd	2	1	inner
468.2.bp.d	✓	40	12.b	even	2	1	inner
468.2.bp.d	✓	40	13.e	even	6	1	inner
468.2.bp.d	✓	40	39.h	odd	6	1	inner
468.2.bp.d	✓	40	52.i	odd	6	1	inner
468.2.bp.d	✓	40	156.r	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{10} - 22T_{5}^{8} + 122T_{5}^{6} - 260T_{5}^{4} + 217T_{5}^{2} - 54 \)

T7^20 + 41*T7^18 + 1137*T7^16 + 17136*T7^14 + 185388*T7^12 + 1030476*T7^10 + 4077852*T7^8 + 9385632*T7^6 + 15232944*T7^4 + 10626032*T7^2 + 5326864