Newform orbit 315.3.bi.e

• Label: 315.3.bi.e
• Level: $315$
• Weight: $3$
• Character orbit: 315.bi
• Analytic conductor: $8.583$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	315.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.58312832735\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 105)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 6 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−3.26825	−	1.88692i		0		5.12097	+	8.86977i	−0.907258	+	4.91700i		0		1.16497	+	6.90238i		−	23.5561i		0		12.2431	−	14.3580i
19.2	−2.95153	−	1.70407i		0		3.80769	+	6.59511i	4.81594	+	1.34415i		0		5.14807	−	4.74314i		−	12.3217i		0		−11.9239	−	12.1740i
19.3	−2.53945	−	1.46615i		0		2.29920	+	3.98234i	−4.74629	+	1.57250i		0		−6.93149	−	0.976973i		−	1.75471i		0		14.3585	+	2.96550i
19.4	−2.08943	−	1.20633i		0		0.910488	+	1.57701i	−4.15605	−	2.77979i		0		6.98737	−	0.420230i			5.25727i		0		5.33045	+	10.8218i
19.5	−1.71634	−	0.990928i		0		−0.0361245	−	0.0625695i	−0.266380	−	4.99290i		0		2.11222	+	6.67372i			8.07061i		0		−4.49040	+	8.83346i
19.6	−1.32532	−	0.765175i		0		−0.829016	−	1.43590i	−1.02421	+	4.89398i		0		−6.34933	−	2.94719i			8.65876i		0		5.10215	−	5.70239i
19.7	−0.952353	−	0.549841i		0		−1.39535	−	2.41681i	4.99803	+	0.140341i		0		6.08554	−	3.45923i			7.46761i		0		−4.68273	−	2.88178i
19.8	−0.428040	−	0.247129i		0		−1.87785	−	3.25254i	−2.23720	+	4.47157i		0		5.82770	−	3.87787i			3.83332i		0		2.06266	−	1.36113i
19.9	0.428040	+	0.247129i		0		−1.87785	−	3.25254i	2.75389	−	4.17326i		0		−5.82770	+	3.87787i		−	3.83332i		0		2.21011	−	1.10575i
19.10	0.952353	+	0.549841i		0		−1.39535	−	2.41681i	2.62055	+	4.25825i		0		−6.08554	+	3.45923i		−	7.46761i		0		0.154330	+	5.49625i
19.11	1.32532	+	0.765175i		0		−0.829016	−	1.43590i	3.72620	−	3.33398i		0		6.34933	+	2.94719i		−	8.65876i		0		7.48949	−	1.56739i
19.12	1.71634	+	0.990928i		0		−0.0361245	−	0.0625695i	−4.45717	+	2.26576i		0		−2.11222	−	6.67372i		−	8.07061i		0		−9.89520	−	0.527928i
19.13	2.08943	+	1.20633i		0		0.910488	+	1.57701i	−4.48539	−	2.20936i		0		−6.98737	+	0.420230i		−	5.25727i		0		−6.70671	−	10.0272i
19.14	2.53945	+	1.46615i		0		2.29920	+	3.98234i	−1.01132	−	4.89665i		0		6.93149	+	0.976973i			1.75471i		0		4.61104	−	13.9176i
19.15	2.95153	+	1.70407i		0		3.80769	+	6.59511i	3.57204	+	3.49865i		0		−5.14807	+	4.74314i			12.3217i		0		4.58104	+	16.4134i
19.16	3.26825	+	1.88692i		0		5.12097	+	8.86977i	3.80462	−	3.24421i		0		−1.16497	−	6.90238i			23.5561i		0		18.5560	−	3.42385i
199.1	−3.26825	+	1.88692i		0		5.12097	−	8.86977i	−0.907258	−	4.91700i		0		1.16497	−	6.90238i			23.5561i		0		12.2431	+	14.3580i
199.2	−2.95153	+	1.70407i		0		3.80769	−	6.59511i	4.81594	−	1.34415i		0		5.14807	+	4.74314i			12.3217i		0		−11.9239	+	12.1740i
199.3	−2.53945	+	1.46615i		0		2.29920	−	3.98234i	−4.74629	−	1.57250i		0		−6.93149	+	0.976973i			1.75471i		0		14.3585	−	2.96550i
199.4	−2.08943	+	1.20633i		0		0.910488	−	1.57701i	−4.15605	+	2.77979i		0		6.98737	+	0.420230i		−	5.25727i		0		5.33045	−	10.8218i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.d	odd	6	1	inner
35.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.3.bi.e		32
3.b	odd	2	1		105.3.r.a	✓	32
5.b	even	2	1	inner	315.3.bi.e		32
7.d	odd	6	1	inner	315.3.bi.e		32
15.d	odd	2	1		105.3.r.a	✓	32
15.e	even	4	1		525.3.o.p		16
15.e	even	4	1		525.3.o.q		16
21.g	even	6	1		105.3.r.a	✓	32
21.g	even	6	1		735.3.e.a		32
21.h	odd	6	1		735.3.e.a		32
35.i	odd	6	1	inner	315.3.bi.e		32
105.o	odd	6	1		735.3.e.a		32
105.p	even	6	1		105.3.r.a	✓	32
105.p	even	6	1		735.3.e.a		32
105.w	odd	12	1		525.3.o.p		16
105.w	odd	12	1		525.3.o.q		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.3.r.a	✓	32	3.b	odd	2	1
105.3.r.a	✓	32	15.d	odd	2	1
105.3.r.a	✓	32	21.g	even	6	1
105.3.r.a	✓	32	105.p	even	6	1
315.3.bi.e		32	1.a	even	1	1	trivial
315.3.bi.e		32	5.b	even	2	1	inner
315.3.bi.e		32	7.d	odd	6	1	inner
315.3.bi.e		32	35.i	odd	6	1	inner
525.3.o.p		16	15.e	even	4	1
525.3.o.p		16	105.w	odd	12	1
525.3.o.q		16	15.e	even	4	1
525.3.o.q		16	105.w	odd	12	1
735.3.e.a		32	21.g	even	6	1
735.3.e.a		32	21.h	odd	6	1
735.3.e.a		32	105.o	odd	6	1
735.3.e.a		32	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 48*T2^30 + 1386*T2^28 - 26088*T2^26 + 363039*T2^24 - 3759408*T2^22 + 30112866*T2^20 - 184309272*T2^18 + 876217365*T2^16 - 3148774992*T2^14 + 8619248448*T2^12 - 17019477432*T2^10 + 24475044780*T2^8 - 22431193920*T2^6 + 13745169936*T2^4 - 3060990000*T2^2 + 506250000