Newform orbit 315.2.bz.d

• Label: 315.2.bz.d
• Level: $315$
• Weight: $2$
• Character orbit: 315.bz
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

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:	no (minimal twist has level 105)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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	−2.17399	−	0.582519i		0		2.65485	+	1.53278i	1.96540	+	1.06640i		0		−0.660211	+	2.56205i	−1.69581	−	1.69581i		0		−3.65156	−	3.46322i
73.2	−1.72112	−	0.461174i		0		1.01754	+	0.587476i	1.95031	−	1.09375i		0		1.68076	−	2.04329i	1.03952	+	1.03952i		0		−3.86114	+	0.983040i
73.3	−1.49657	−	0.401003i		0		0.346853	+	0.200256i	−1.61490	+	1.54664i		0		−2.44171	+	1.01885i	1.75234	+	1.75234i		0		3.03701	−	1.66707i
73.4	−0.648264	−	0.173702i		0		−1.34198	−	0.774791i	−2.13259	+	0.672361i		0		2.57939	+	0.588837i	1.68450	+	1.68450i		0		1.49927	−	0.0654328i
73.5	0.394487	+	0.105703i		0		−1.58760	−	0.916603i	2.18897	+	0.456535i		0		−0.605712	−	2.57548i	−1.10697	−	1.10697i		0		0.815263	+	0.411477i
73.6	0.969545	+	0.259789i		0		−0.859523	−	0.496246i	−0.803857	−	2.08658i		0		2.42328	+	1.06195i	−2.12394	−	2.12394i		0		−0.237305	−	2.23187i
73.7	2.24814	+	0.602389i		0		2.95923	+	1.70851i	2.22726	−	0.198269i		0		−2.59417	−	0.519864i	2.33208	+	2.33208i		0		5.12664	+	0.895939i
73.8	2.42777	+	0.650518i		0		3.73883	+	2.15861i	−0.780598	−	2.09539i		0		1.61838	−	2.09305i	4.11829	+	4.11829i		0		−0.532020	−	5.59492i
82.1	−2.17399	+	0.582519i		0		2.65485	−	1.53278i	1.96540	−	1.06640i		0		−0.660211	−	2.56205i	−1.69581	+	1.69581i		0		−3.65156	+	3.46322i
82.2	−1.72112	+	0.461174i		0		1.01754	−	0.587476i	1.95031	+	1.09375i		0		1.68076	+	2.04329i	1.03952	−	1.03952i		0		−3.86114	−	0.983040i
82.3	−1.49657	+	0.401003i		0		0.346853	−	0.200256i	−1.61490	−	1.54664i		0		−2.44171	−	1.01885i	1.75234	−	1.75234i		0		3.03701	+	1.66707i
82.4	−0.648264	+	0.173702i		0		−1.34198	+	0.774791i	−2.13259	−	0.672361i		0		2.57939	−	0.588837i	1.68450	−	1.68450i		0		1.49927	+	0.0654328i
82.5	0.394487	−	0.105703i		0		−1.58760	+	0.916603i	2.18897	−	0.456535i		0		−0.605712	+	2.57548i	−1.10697	+	1.10697i		0		0.815263	−	0.411477i
82.6	0.969545	−	0.259789i		0		−0.859523	+	0.496246i	−0.803857	+	2.08658i		0		2.42328	−	1.06195i	−2.12394	+	2.12394i		0		−0.237305	+	2.23187i
82.7	2.24814	−	0.602389i		0		2.95923	−	1.70851i	2.22726	+	0.198269i		0		−2.59417	+	0.519864i	2.33208	−	2.33208i		0		5.12664	−	0.895939i
82.8	2.42777	−	0.650518i		0		3.73883	−	2.15861i	−0.780598	+	2.09539i		0		1.61838	+	2.09305i	4.11829	−	4.11829i		0		−0.532020	+	5.59492i
208.1	−0.650518	−	2.42777i		0		−3.73883	+	2.15861i	1.42436	−	1.72371i		0		2.09305	−	1.61838i	4.11829	+	4.11829i		0		−5.11135	−	2.33671i
208.2	−0.602389	−	2.24814i		0		−2.95923	+	1.70851i	1.28534	+	1.82973i		0		0.519864	+	2.59417i	2.33208	+	2.33208i		0		3.33923	−	3.99183i
208.3	−0.259789	−	0.969545i		0		0.859523	−	0.496246i	1.40510	−	1.73945i		0		−1.06195	−	2.42328i	−2.12394	−	2.12394i		0		−2.05151	−	0.910421i
208.4	−0.105703	−	0.394487i		0		1.58760	−	0.916603i	0.699113	+	2.12397i		0		2.57548	+	0.605712i	−1.10697	−	1.10697i		0		0.763981	−	0.500300i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.d	odd	6	1	inner
35.k	even	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.d		32
3.b	odd	2	1		105.2.u.a	✓	32
5.c	odd	4	1	inner	315.2.bz.d		32
7.d	odd	6	1	inner	315.2.bz.d		32
15.d	odd	2	1		525.2.bc.e		32
15.e	even	4	1		105.2.u.a	✓	32
15.e	even	4	1		525.2.bc.e		32
21.c	even	2	1		735.2.v.b		32
21.g	even	6	1		105.2.u.a	✓	32
21.g	even	6	1		735.2.m.c		32
21.h	odd	6	1		735.2.m.c		32
21.h	odd	6	1		735.2.v.b		32
35.k	even	12	1	inner	315.2.bz.d		32
105.k	odd	4	1		735.2.v.b		32
105.p	even	6	1		525.2.bc.e		32
105.w	odd	12	1		105.2.u.a	✓	32
105.w	odd	12	1		525.2.bc.e		32
105.w	odd	12	1		735.2.m.c		32
105.x	even	12	1		735.2.m.c		32
105.x	even	12	1		735.2.v.b		32

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 8*T2^29 - 56*T2^28 + 24*T2^27 + 32*T2^26 + 320*T2^25 + 2188*T2^24 - 80*T2^23 - 480*T2^22 - 7512*T2^21 - 42040*T2^20 - 656*T2^19 + 12480*T2^18 + 46560*T2^17 + 570252*T2^16 + 508096*T2^15 + 305536*T2^14 + 873680*T2^13 - 648848*T2^12 - 1321760*T2^11 - 251872*T2^10 - 735648*T2^9 + 1374784*T2^8 + 737376*T2^7 + 75008*T2^6 + 423008*T2^5 - 27856*T2^4 - 119680*T2^3 + 12800*T2^2 - 16000*T2 + 10000