Newform orbit 315.2.bz.c

• Label: 315.2.bz.c
• Level: $315$
• Weight: $2$
• Character orbit: 315.bz
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bz (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
73.1	−2.60306	−	0.697487i		0		4.55736	+	2.63119i	1.54516	−	1.61632i		0		−2.41104	−	1.08945i	−6.21670	−	6.21670i		0		−5.14949	+	3.12964i
73.2	−1.68784	−	0.452256i		0		0.912222	+	0.526671i	−1.82462	−	1.29258i		0		2.36415	−	1.18777i	1.16968	+	1.16968i		0		2.49508	+	3.00687i
73.3	−1.12243	−	0.300753i		0		−0.562663	−	0.324854i	0.706264	+	2.12160i		0		−1.60999	−	2.09951i	2.17719	+	2.17719i		0		−0.154651	−	2.59375i
73.4	−0.558216	−	0.149574i		0		−1.44282	−	0.833011i	2.11364	−	0.729749i		0		−0.0751776	+	2.64468i	1.49809	+	1.49809i		0		−1.28902	+	0.0912132i
73.5	0.558216	+	0.149574i		0		−1.44282	−	0.833011i	−2.11364	+	0.729749i		0		−0.0751776	+	2.64468i	−1.49809	−	1.49809i		0		−1.28902	+	0.0912132i
73.6	1.12243	+	0.300753i		0		−0.562663	−	0.324854i	−0.706264	−	2.12160i		0		−1.60999	−	2.09951i	−2.17719	−	2.17719i		0		−0.154651	−	2.59375i
73.7	1.68784	+	0.452256i		0		0.912222	+	0.526671i	1.82462	+	1.29258i		0		2.36415	−	1.18777i	−1.16968	−	1.16968i		0		2.49508	+	3.00687i
73.8	2.60306	+	0.697487i		0		4.55736	+	2.63119i	−1.54516	+	1.61632i		0		−2.41104	−	1.08945i	6.21670	+	6.21670i		0		−5.14949	+	3.12964i
82.1	−2.60306	+	0.697487i		0		4.55736	−	2.63119i	1.54516	+	1.61632i		0		−2.41104	+	1.08945i	−6.21670	+	6.21670i		0		−5.14949	−	3.12964i
82.2	−1.68784	+	0.452256i		0		0.912222	−	0.526671i	−1.82462	+	1.29258i		0		2.36415	+	1.18777i	1.16968	−	1.16968i		0		2.49508	−	3.00687i
82.3	−1.12243	+	0.300753i		0		−0.562663	+	0.324854i	0.706264	−	2.12160i		0		−1.60999	+	2.09951i	2.17719	−	2.17719i		0		−0.154651	+	2.59375i
82.4	−0.558216	+	0.149574i		0		−1.44282	+	0.833011i	2.11364	+	0.729749i		0		−0.0751776	−	2.64468i	1.49809	−	1.49809i		0		−1.28902	−	0.0912132i
82.5	0.558216	−	0.149574i		0		−1.44282	+	0.833011i	−2.11364	−	0.729749i		0		−0.0751776	−	2.64468i	−1.49809	+	1.49809i		0		−1.28902	−	0.0912132i
82.6	1.12243	−	0.300753i		0		−0.562663	+	0.324854i	−0.706264	+	2.12160i		0		−1.60999	+	2.09951i	−2.17719	+	2.17719i		0		−0.154651	+	2.59375i
82.7	1.68784	−	0.452256i		0		0.912222	−	0.526671i	1.82462	−	1.29258i		0		2.36415	+	1.18777i	−1.16968	+	1.16968i		0		2.49508	−	3.00687i
82.8	2.60306	−	0.697487i		0		4.55736	−	2.63119i	−1.54516	−	1.61632i		0		−2.41104	+	1.08945i	6.21670	−	6.21670i		0		−5.14949	−	3.12964i
208.1	−0.697487	−	2.60306i		0		−4.55736	+	2.63119i	−2.17235	−	0.529986i		0		1.08945	+	2.41104i	6.21670	+	6.21670i		0		0.135604	+	6.02441i
208.2	−0.452256	−	1.68784i		0		−0.912222	+	0.526671i	−0.207102	+	2.22646i		0		1.18777	−	2.36415i	−1.16968	−	1.16968i		0		3.85157	−	0.657372i
208.3	−0.300753	−	1.12243i		0		0.562663	−	0.324854i	1.48423	−	1.67244i		0		2.09951	+	1.60999i	−2.17719	−	2.17719i		0		−2.32358	−	1.16294i
208.4	−0.149574	−	0.558216i		0		1.44282	−	0.833011i	−1.68880	−	1.46559i		0		−2.64468	+	0.0751776i	−1.49809	−	1.49809i		0		−0.565516	+	1.16193i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
7.d	odd	6	1	inner
15.e	even	4	1	inner
21.g	even	6	1	inner
35.k	even	12	1	inner
105.w	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.bz.c	✓	32
3.b	odd	2	1	inner	315.2.bz.c	✓	32
5.c	odd	4	1	inner	315.2.bz.c	✓	32
7.d	odd	6	1	inner	315.2.bz.c	✓	32
15.e	even	4	1	inner	315.2.bz.c	✓	32
21.g	even	6	1	inner	315.2.bz.c	✓	32
35.k	even	12	1	inner	315.2.bz.c	✓	32
105.w	odd	12	1	inner	315.2.bz.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.bz.c	✓	32	1.a	even	1	1	trivial
315.2.bz.c	✓	32	3.b	odd	2	1	inner
315.2.bz.c	✓	32	5.c	odd	4	1	inner
315.2.bz.c	✓	32	7.d	odd	6	1	inner
315.2.bz.c	✓	32	15.e	even	4	1	inner
315.2.bz.c	✓	32	21.g	even	6	1	inner
315.2.bz.c	✓	32	35.k	even	12	1	inner
315.2.bz.c	✓	32	105.w	odd	12	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 64*T2^28 + 3484*T2^24 - 37240*T2^20 + 312748*T2^16 - 577168*T2^12 + 868096*T2^8 - 96400*T2^4 + 10000