Newform orbit 156.2.r.c

• Label: 156.2.r.c
• Level: $156$
• Weight: $2$
• Character orbit: 156.r
• 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.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{4} - 12 q^{6} - 14 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.41341	−	0.0475771i	−1.73176	−	0.0318611i	1.99547	+	0.134492i	−0.439505			2.44617	+	0.127425i	−0.124685	+	0.215961i	−2.81403	−	0.285032i	2.99797	+	0.110351i	0.621202	+	0.0209104i
23.2	−1.33472	+	0.467473i	0.255646	+	1.71308i	1.56294	−	1.24789i	2.71779			−1.14203	−	2.16697i	2.23503	−	3.87119i	−1.50272	+	2.39621i	−2.86929	+	0.875884i	−3.62748	+	1.27049i
23.3	−1.25324	−	0.655280i	−0.185336	+	1.72211i	1.14122	+	1.64245i	−2.19674			1.36073	−	2.03676i	−0.506035	+	0.876478i	−0.353954	−	2.80619i	−2.93130	−	0.638335i	2.75304	+	1.43948i
23.4	−1.07220	+	0.922162i	1.35575	+	1.07794i	0.299235	−	1.97749i	−2.71779			−2.44767	+	0.0944544i	−2.23503	+	3.87119i	1.50272	+	2.39621i	0.676108	+	2.92282i	2.91402	−	2.50624i
23.5	−1.06718	−	0.927975i	−0.666612	−	1.59863i	0.277727	+	1.98062i	2.96696			−0.772099	+	2.32462i	1.62713	−	2.81828i	1.54158	−	2.37140i	−2.11126	+	2.13134i	−3.16626	−	2.75326i
23.6	−0.953806	−	1.04415i	1.72739	+	0.126961i	−0.180510	+	1.99184i	−0.889920			−1.51503	−	1.92476i	1.15562	−	2.00159i	2.25195	−	1.71135i	2.96776	+	0.438624i	0.848811	+	0.929213i
23.7	−0.665504	+	1.24784i	0.838286	−	1.51568i	−1.11421	−	1.66088i	0.439505			1.33344	+	2.05474i	0.124685	−	0.215961i	2.81403	−	0.285032i	−1.59455	−	2.54114i	−0.292492	+	0.548432i
23.8	−0.427360	−	1.34810i	−1.72739	−	0.126961i	−1.63473	+	1.15224i	−0.889920			0.567062	+	2.38295i	−1.15562	+	2.00159i	2.25195	+	1.71135i	2.96776	+	0.438624i	0.380316	+	1.19970i
23.9	−0.270062	−	1.38819i	0.666612	+	1.59863i	−1.85413	+	0.749793i	2.96696			2.03918	−	1.35711i	−1.62713	+	2.81828i	1.54158	+	2.37140i	−2.11126	+	2.13134i	−0.801262	−	4.11870i
23.10	−0.0591302	+	1.41298i	1.58406	+	0.700548i	−1.99301	−	0.167099i	2.19674			−1.08352	+	2.19681i	0.506035	−	0.876478i	0.353954	−	2.80619i	2.01847	+	2.21941i	−0.129894	+	3.10394i
23.11	0.0591302	−	1.41298i	0.185336	−	1.72211i	−1.99301	−	0.167099i	−2.19674			−2.42234	−	0.363703i	0.506035	−	0.876478i	−0.353954	+	2.80619i	−2.93130	−	0.638335i	−0.129894	+	3.10394i
23.12	0.270062	+	1.38819i	−1.05115	−	1.37662i	−1.85413	+	0.749793i	−2.96696			1.62713	−	1.83097i	−1.62713	+	2.81828i	−1.54158	−	2.37140i	−0.790163	+	2.89407i	−0.801262	−	4.11870i
23.13	0.427360	+	1.34810i	−0.753744	+	1.55945i	−1.63473	+	1.15224i	0.889920			−2.42440	−	0.349674i	−1.15562	+	2.00159i	−2.25195	−	1.71135i	−1.86374	−	2.35084i	0.380316	+	1.19970i
23.14	0.665504	−	1.24784i	1.73176	+	0.0318611i	−1.11421	−	1.66088i	−0.439505			1.19225	−	2.13975i	0.124685	−	0.215961i	−2.81403	+	0.285032i	2.99797	+	0.110351i	−0.292492	+	0.548432i
23.15	0.953806	+	1.04415i	0.753744	−	1.55945i	−0.180510	+	1.99184i	0.889920			2.34722	−	0.700384i	1.15562	−	2.00159i	−2.25195	+	1.71135i	−1.86374	−	2.35084i	0.848811	+	0.929213i
23.16	1.06718	+	0.927975i	1.05115	+	1.37662i	0.277727	+	1.98062i	−2.96696			−0.155705	+	2.44454i	1.62713	−	2.81828i	−1.54158	+	2.37140i	−0.790163	+	2.89407i	−3.16626	−	2.75326i
23.17	1.07220	−	0.922162i	−0.255646	−	1.71308i	0.299235	−	1.97749i	2.71779			−1.85384	−	1.60102i	−2.23503	+	3.87119i	−1.50272	−	2.39621i	−2.86929	+	0.875884i	2.91402	−	2.50624i
23.18	1.25324	+	0.655280i	−1.58406	−	0.700548i	1.14122	+	1.64245i	2.19674			−1.52615	−	1.91595i	−0.506035	+	0.876478i	0.353954	+	2.80619i	2.01847	+	2.21941i	2.75304	+	1.43948i
23.19	1.33472	−	0.467473i	−1.35575	−	1.07794i	1.56294	−	1.24789i	−2.71779			−2.31345	−	0.804963i	2.23503	−	3.87119i	1.50272	−	2.39621i	0.676108	+	2.92282i	−3.62748	+	1.27049i
23.20	1.41341	+	0.0475771i	−0.838286	+	1.51568i	1.99547	+	0.134492i	0.439505			−1.25696	+	2.10239i	−0.124685	+	0.215961i	2.81403	+	0.285032i	−1.59455	−	2.54114i	0.621202	+	0.0209104i

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	156.2.r.c	✓	40
3.b	odd	2	1	inner	156.2.r.c	✓	40
4.b	odd	2	1	inner	156.2.r.c	✓	40
12.b	even	2	1	inner	156.2.r.c	✓	40
13.e	even	6	1	inner	156.2.r.c	✓	40
39.h	odd	6	1	inner	156.2.r.c	✓	40
52.i	odd	6	1	inner	156.2.r.c	✓	40
156.r	even	6	1	inner	156.2.r.c	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.r.c	✓	40	1.a	even	1	1	trivial
156.2.r.c	✓	40	3.b	odd	2	1	inner
156.2.r.c	✓	40	4.b	odd	2	1	inner
156.2.r.c	✓	40	12.b	even	2	1	inner
156.2.r.c	✓	40	13.e	even	6	1	inner
156.2.r.c	✓	40	39.h	odd	6	1	inner
156.2.r.c	✓	40	52.i	odd	6	1	inner
156.2.r.c	✓	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 \(S_{2}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{10} - 22T_{5}^{8} + 164T_{5}^{6} - 458T_{5}^{4} + 331T_{5}^{2} - 48 \)

T7^20 + 37*T7^18 + 955*T7^16 + 12238*T7^14 + 113164*T7^12 + 544984*T7^10 + 1850608*T7^8 + 1868464*T7^6 + 1456624*T7^4 + 90144*T7^2 + 5184