Newform orbit 156.2.p.b

• Label: 156.2.p.b
• Level: $156$
• Weight: $2$
• Character orbit: 156.p
• Analytic conductor: $1.246$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	156.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.24566627153\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{4} + 2 q^{6} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.40167	−	0.187933i	−1.29017	−	1.15562i	1.92936	+	0.526840i			1.98084i	1.59121	+	1.86227i	2.42516	−	1.40017i	−2.60532	−	1.10105i	0.329066	+	2.98190i	0.372266	−	2.77649i
35.2	−1.39117	+	0.254263i	0.564863	−	1.63735i	1.87070	−	0.707445i		−	3.40345i	−0.369501	+	2.42146i	−2.05768	+	1.18800i	−2.42258	+	1.45983i	−2.36186	−	1.84976i	0.865372	+	4.73478i
35.3	−1.36641	−	0.364576i	−1.10270	+	1.33569i	1.73417	+	0.996323i			0.171572i	1.99370	−	1.42308i	−2.51891	+	1.45429i	−2.00636	−	1.99362i	−0.568109	−	2.94572i	0.0625510	−	0.234438i
35.4	−1.32222	+	0.501741i	1.71957	−	0.207570i	1.49651	−	1.32682i			3.37008i	−2.16949	+	1.13723i	−1.77448	+	1.02450i	−1.31299	+	2.50520i	2.91383	−	0.713863i	−1.69091	−	4.45597i
35.5	−1.06855	+	0.926389i	−1.63287	+	0.577710i	0.283607	−	1.97979i		−	1.45117i	1.20962	−	2.12998i	1.17936	−	0.680902i	1.53101	+	2.37824i	2.33250	−	1.88665i	1.34435	+	1.55065i
35.6	−0.998939	−	1.00106i	1.10270	−	1.33569i	−0.00424367	+	2.00000i			0.171572i	−2.43863	+	0.230399i	2.51891	−	1.45429i	2.00636	−	1.99362i	−0.568109	−	2.94572i	0.171754	−	0.171390i
35.7	−0.863590	−	1.11992i	1.29017	+	1.15562i	−0.508424	+	1.93430i			1.98084i	0.180026	−	2.44287i	−2.42516	+	1.40017i	2.60532	−	1.10105i	0.329066	+	2.98190i	2.21838	−	1.71064i
35.8	−0.475386	−	1.33192i	−0.564863	+	1.63735i	−1.54802	+	1.26635i		−	3.40345i	2.44935	−	0.0260236i	2.05768	−	1.18800i	2.42258	+	1.45983i	−2.36186	−	1.84976i	−4.53312	+	1.61795i
35.9	−0.268000	+	1.38859i	0.316122	+	1.70296i	−1.85635	−	0.744284i			1.45117i	−2.44943	−	0.0174310i	−1.17936	+	0.680902i	1.53101	−	2.37824i	−2.80013	+	1.07668i	−2.01508	−	0.388914i
35.10	−0.226588	−	1.39594i	−1.71957	+	0.207570i	−1.89732	+	0.632608i			3.37008i	0.679390	+	2.35339i	1.77448	−	1.02450i	1.31299	+	2.50520i	2.91383	−	0.713863i	4.70444	−	0.763619i
35.11	0.226588	+	1.39594i	−0.680023	−	1.59297i	−1.89732	+	0.632608i		−	3.37008i	2.06962	−	1.31022i	1.77448	−	1.02450i	−1.31299	−	2.50520i	−2.07514	+	2.16652i	4.70444	−	0.763619i
35.12	0.268000	−	1.38859i	1.63287	−	0.577710i	−1.85635	−	0.744284i		−	1.45117i	−0.364592	−	2.42220i	−1.17936	+	0.680902i	−1.53101	+	2.37824i	2.33250	−	1.88665i	−2.01508	−	0.388914i
35.13	0.475386	+	1.33192i	1.13556	−	1.30786i	−1.54802	+	1.26635i			3.40345i	2.28180	+	0.890732i	2.05768	−	1.18800i	−2.42258	−	1.45983i	−0.421012	−	2.97031i	−4.53312	+	1.61795i
35.14	0.863590	+	1.11992i	1.64588	+	0.539506i	−0.508424	+	1.93430i		−	1.98084i	0.817167	+	2.30916i	−2.42516	+	1.40017i	−2.60532	+	1.10105i	2.41787	+	1.77593i	2.21838	−	1.71064i
35.15	0.998939	+	1.00106i	−0.605388	+	1.62281i	−0.00424367	+	2.00000i		−	0.171572i	−2.22927	+	1.01506i	2.51891	−	1.45429i	−2.00636	+	1.99362i	−2.26701	−	1.96486i	0.171754	−	0.171390i
35.16	1.06855	−	0.926389i	−0.316122	−	1.70296i	0.283607	−	1.97979i			1.45117i	−1.91539	−	1.52685i	1.17936	−	0.680902i	−1.53101	−	2.37824i	−2.80013	+	1.07668i	1.34435	+	1.55065i
35.17	1.32222	−	0.501741i	0.680023	+	1.59297i	1.49651	−	1.32682i		−	3.37008i	1.69840	+	1.76506i	−1.77448	+	1.02450i	1.31299	−	2.50520i	−2.07514	+	2.16652i	−1.69091	−	4.45597i
35.18	1.36641	+	0.364576i	0.605388	−	1.62281i	1.73417	+	0.996323i		−	0.171572i	1.41885	−	1.99672i	−2.51891	+	1.45429i	2.00636	+	1.99362i	−2.26701	−	1.96486i	0.0625510	−	0.234438i
35.19	1.39117	−	0.254263i	−1.13556	+	1.30786i	1.87070	−	0.707445i			3.40345i	−1.24721	+	2.10819i	−2.05768	+	1.18800i	2.42258	−	1.45983i	−0.421012	−	2.97031i	0.865372	+	4.73478i
35.20	1.40167	+	0.187933i	−1.64588	−	0.539506i	1.92936	+	0.526840i		−	1.98084i	−2.20560	−	1.06553i	2.42516	−	1.40017i	2.60532	+	1.10105i	2.41787	+	1.77593i	0.372266	−	2.77649i

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.c	even	3	1	inner
39.i	odd	6	1	inner
52.j	odd	6	1	inner
156.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	156.2.p.b	✓	40
3.b	odd	2	1	inner	156.2.p.b	✓	40
4.b	odd	2	1	inner	156.2.p.b	✓	40
12.b	even	2	1	inner	156.2.p.b	✓	40
13.c	even	3	1	inner	156.2.p.b	✓	40
39.i	odd	6	1	inner	156.2.p.b	✓	40
52.j	odd	6	1	inner	156.2.p.b	✓	40
156.p	even	6	1	inner	156.2.p.b	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.p.b	✓	40	1.a	even	1	1	trivial
156.2.p.b	✓	40	3.b	odd	2	1	inner
156.2.p.b	✓	40	4.b	odd	2	1	inner
156.2.p.b	✓	40	12.b	even	2	1	inner
156.2.p.b	✓	40	13.c	even	3	1	inner
156.2.p.b	✓	40	39.i	odd	6	1	inner
156.2.p.b	✓	40	52.j	odd	6	1	inner
156.2.p.b	✓	40	156.p	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 29T_{5}^{8} + 279T_{5}^{6} + 991T_{5}^{4} + 1116T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\).