Newform orbit 312.2.h.c

• Label: 312.2.h.c
• Level: $312$
• Weight: $2$
• Character orbit: 312.h
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{3} + 20 q^{4} - 28 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
155.1	−1.38664	−	0.277882i	−0.662918	−	1.60017i	1.84556	+	0.770648i		−	2.38561i	0.474573	+	2.40308i	−2.63829			−2.34499	−	1.58146i	−2.12108	+	2.12156i	−0.662918	+	3.30799i
155.2	−1.38664	−	0.277882i	−0.662918	+	1.60017i	1.84556	+	0.770648i		−	2.38561i	1.36389	−	2.03465i	2.63829			−2.34499	−	1.58146i	−2.12108	−	2.12156i	−0.662918	+	3.30799i
155.3	−1.38664	+	0.277882i	−0.662918	−	1.60017i	1.84556	−	0.770648i			2.38561i	1.36389	+	2.03465i	2.63829			−2.34499	+	1.58146i	−2.12108	+	2.12156i	−0.662918	−	3.30799i
155.4	−1.38664	+	0.277882i	−0.662918	+	1.60017i	1.84556	−	0.770648i			2.38561i	0.474573	−	2.40308i	−2.63829			−2.34499	+	1.58146i	−2.12108	−	2.12156i	−0.662918	−	3.30799i
155.5	−1.30100	−	0.554443i	1.21097	−	1.23837i	1.38519	+	1.44266i			2.18412i	−2.26207	+	0.939697i	−0.621089			−1.00225	−	2.64490i	−0.0671052	−	2.99925i	1.21097	−	2.84153i
155.6	−1.30100	−	0.554443i	1.21097	+	1.23837i	1.38519	+	1.44266i			2.18412i	−0.888863	−	2.28253i	0.621089			−1.00225	−	2.64490i	−0.0671052	+	2.99925i	1.21097	−	2.84153i
155.7	−1.30100	+	0.554443i	1.21097	−	1.23837i	1.38519	−	1.44266i		−	2.18412i	−0.888863	+	2.28253i	0.621089			−1.00225	+	2.64490i	−0.0671052	−	2.99925i	1.21097	+	2.84153i
155.8	−1.30100	+	0.554443i	1.21097	+	1.23837i	1.38519	−	1.44266i		−	2.18412i	−2.26207	−	0.939697i	−0.621089			−1.00225	+	2.64490i	−0.0671052	+	2.99925i	1.21097	+	2.84153i
155.9	−1.05078	−	0.946498i	−1.47136	−	0.913837i	0.208283	+	1.98912i		−	1.55453i	0.681133	+	2.35288i	3.67000			1.66384	−	2.28727i	1.32980	+	2.68917i	−1.47136	+	1.63347i
155.10	−1.05078	−	0.946498i	−1.47136	+	0.913837i	0.208283	+	1.98912i		−	1.55453i	2.41102	+	0.432396i	−3.67000			1.66384	−	2.28727i	1.32980	−	2.68917i	−1.47136	+	1.63347i
155.11	−1.05078	+	0.946498i	−1.47136	−	0.913837i	0.208283	−	1.98912i			1.55453i	2.41102	−	0.432396i	−3.67000			1.66384	+	2.28727i	1.32980	+	2.68917i	−1.47136	−	1.63347i
155.12	−1.05078	+	0.946498i	−1.47136	+	0.913837i	0.208283	−	1.98912i			1.55453i	0.681133	−	2.35288i	3.67000			1.66384	+	2.28727i	1.32980	−	2.68917i	−1.47136	−	1.63347i
155.13	−0.728344	−	1.21224i	0.423309	−	1.67953i	−0.939031	+	1.76585i			0.349197i	−2.34430	+	0.710122i	−2.86091			2.82456	−	0.147818i	−2.64162	−	1.42192i	0.423309	−	0.254335i
155.14	−0.728344	−	1.21224i	0.423309	+	1.67953i	−0.939031	+	1.76585i			0.349197i	1.72767	−	1.73642i	2.86091			2.82456	−	0.147818i	−2.64162	+	1.42192i	0.423309	−	0.254335i
155.15	−0.728344	+	1.21224i	0.423309	−	1.67953i	−0.939031	−	1.76585i		−	0.349197i	1.72767	+	1.73642i	2.86091			2.82456	+	0.147818i	−2.64162	−	1.42192i	0.423309	+	0.254335i
155.16	−0.728344	+	1.21224i	0.423309	+	1.67953i	−0.939031	−	1.76585i		−	0.349197i	−2.34430	−	0.710122i	−2.86091			2.82456	+	0.147818i	−2.64162	+	1.42192i	0.423309	+	0.254335i
155.17	0.728344	−	1.21224i	0.423309	−	1.67953i	−0.939031	−	1.76585i			0.349197i	−1.72767	−	1.73642i	−2.86091			−2.82456	−	0.147818i	−2.64162	−	1.42192i	0.423309	+	0.254335i
155.18	0.728344	−	1.21224i	0.423309	+	1.67953i	−0.939031	−	1.76585i			0.349197i	2.34430	+	0.710122i	2.86091			−2.82456	−	0.147818i	−2.64162	+	1.42192i	0.423309	+	0.254335i
155.19	0.728344	+	1.21224i	0.423309	−	1.67953i	−0.939031	+	1.76585i		−	0.349197i	2.34430	−	0.710122i	2.86091			−2.82456	+	0.147818i	−2.64162	−	1.42192i	0.423309	−	0.254335i
155.20	0.728344	+	1.21224i	0.423309	+	1.67953i	−0.939031	+	1.76585i		−	0.349197i	−1.72767	+	1.73642i	−2.86091			−2.82456	+	0.147818i	−2.64162	+	1.42192i	0.423309	−	0.254335i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
13.b	even	2	1	inner
24.f	even	2	1	inner
39.d	odd	2	1	inner
104.h	odd	2	1	inner
312.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.2.h.c	✓	32
3.b	odd	2	1	inner	312.2.h.c	✓	32
4.b	odd	2	1		1248.2.h.c		32
8.b	even	2	1		1248.2.h.c		32
8.d	odd	2	1	inner	312.2.h.c	✓	32
12.b	even	2	1		1248.2.h.c		32
13.b	even	2	1	inner	312.2.h.c	✓	32
24.f	even	2	1	inner	312.2.h.c	✓	32
24.h	odd	2	1		1248.2.h.c		32
39.d	odd	2	1	inner	312.2.h.c	✓	32
52.b	odd	2	1		1248.2.h.c		32
104.e	even	2	1		1248.2.h.c		32
104.h	odd	2	1	inner	312.2.h.c	✓	32
156.h	even	2	1		1248.2.h.c		32
312.b	odd	2	1		1248.2.h.c		32
312.h	even	2	1	inner	312.2.h.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.h.c	✓	32	1.a	even	1	1	trivial
312.2.h.c	✓	32	3.b	odd	2	1	inner
312.2.h.c	✓	32	8.d	odd	2	1	inner
312.2.h.c	✓	32	13.b	even	2	1	inner
312.2.h.c	✓	32	24.f	even	2	1	inner
312.2.h.c	✓	32	39.d	odd	2	1	inner
312.2.h.c	✓	32	104.h	odd	2	1	inner
312.2.h.c	✓	32	312.h	even	2	1	inner
1248.2.h.c		32	4.b	odd	2	1
1248.2.h.c		32	8.b	even	2	1
1248.2.h.c		32	12.b	even	2	1
1248.2.h.c		32	24.h	odd	2	1
1248.2.h.c		32	52.b	odd	2	1
1248.2.h.c		32	104.e	even	2	1
1248.2.h.c		32	156.h	even	2	1
1248.2.h.c		32	312.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 13T_{5}^{6} + 54T_{5}^{4} + 72T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).