Newform orbit 784.2.w.e

• Label: 784.2.w.e
• Level: $784$
• Weight: $2$
• Character orbit: 784.w
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.w (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 112)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} + 8 q^{4} - 32 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.41028	+	0.105414i	−0.551303	+	2.05749i	1.97778	−	0.297327i	−3.47044	+	0.929901i	0.560603	−	2.95975i		0		−2.75787	+	0.627801i	−1.33126	−	0.768605i	4.79626	−	1.67725i
19.2	−1.41028	+	0.105414i	0.551303	−	2.05749i	1.97778	−	0.297327i	3.47044	−	0.929901i	−0.560603	+	2.95975i		0		−2.75787	+	0.627801i	−1.33126	−	0.768605i	−4.79626	+	1.67725i
19.3	−0.830359	−	1.14477i	−0.472168	+	1.76215i	−0.621007	+	1.90114i	0.818676	−	0.219363i	2.40933	−	0.922696i		0		2.69204	−	0.867721i	−0.284168	−	0.164064i	−0.930916	−	0.755047i
19.4	−0.830359	−	1.14477i	0.472168	−	1.76215i	−0.621007	+	1.90114i	−0.818676	+	0.219363i	−2.40933	+	0.922696i		0		2.69204	−	0.867721i	−0.284168	−	0.164064i	0.930916	+	0.755047i
19.5	0.757684	−	1.19412i	−0.817885	+	3.05239i	−0.851831	−	1.80953i	0.797811	−	0.213773i	3.02521	+	3.28940i		0		−2.80620	−	0.353863i	−6.05007	−	3.49301i	0.349219	−	1.11465i
19.6	0.757684	−	1.19412i	0.817885	−	3.05239i	−0.851831	−	1.80953i	−0.797811	+	0.213773i	−3.02521	−	3.28940i		0		−2.80620	−	0.353863i	−6.05007	−	3.49301i	−0.349219	+	1.11465i
19.7	1.11693	+	0.867450i	−0.379383	+	1.41588i	0.495063	+	1.93776i	2.30425	−	0.617422i	−1.65195	+	1.25234i		0		−1.12796	+	2.59378i	0.737300	+	0.425680i	3.10927	+	1.30921i
19.8	1.11693	+	0.867450i	0.379383	−	1.41588i	0.495063	+	1.93776i	−2.30425	+	0.617422i	1.65195	−	1.25234i		0		−1.12796	+	2.59378i	0.737300	+	0.425680i	−3.10927	−	1.30921i
227.1	−1.30970	+	0.533564i	−1.41588	+	0.379383i	1.43062	−	1.39762i	0.617422	−	2.30425i	1.65195	−	1.25234i		0		−1.12796	+	2.59378i	−0.737300	+	0.425680i	0.420830	+	3.34731i
227.2	−1.30970	+	0.533564i	1.41588	−	0.379383i	1.43062	−	1.39762i	−0.617422	+	2.30425i	−1.65195	+	1.25234i		0		−1.12796	+	2.59378i	−0.737300	+	0.425680i	−0.420830	−	3.34731i
227.3	0.613848	−	1.27404i	−2.05749	+	0.551303i	−1.24638	−	1.56414i	−0.929901	+	3.47044i	−0.560603	+	2.95975i		0		−2.75787	+	0.627801i	1.33126	−	0.768605i	3.85068	+	3.31506i
227.4	0.613848	−	1.27404i	2.05749	−	0.551303i	−1.24638	−	1.56414i	0.929901	−	3.47044i	0.560603	−	2.95975i		0		−2.75787	+	0.627801i	1.33126	−	0.768605i	−3.85068	−	3.31506i
227.5	0.655294	+	1.25323i	−3.05239	+	0.817885i	−1.14118	+	1.64247i	0.213773	−	0.797811i	−3.02521	−	3.28940i		0		−2.80620	−	0.353863i	6.05007	−	3.49301i	1.13993	−	0.254894i
227.6	0.655294	+	1.25323i	3.05239	−	0.817885i	−1.14118	+	1.64247i	−0.213773	+	0.797811i	3.02521	+	3.28940i		0		−2.80620	−	0.353863i	6.05007	−	3.49301i	−1.13993	+	0.254894i
227.7	1.40658	−	0.146726i	−1.76215	+	0.472168i	1.95694	−	0.412764i	0.219363	−	0.818676i	−2.40933	+	0.922696i		0		2.69204	−	0.867721i	0.284168	−	0.164064i	0.188432	−	1.18372i
227.8	1.40658	−	0.146726i	1.76215	−	0.472168i	1.95694	−	0.412764i	−0.219363	+	0.818676i	2.40933	−	0.922696i		0		2.69204	−	0.867721i	0.284168	−	0.164064i	−0.188432	+	1.18372i
411.1	−1.30970	−	0.533564i	−1.41588	−	0.379383i	1.43062	+	1.39762i	0.617422	+	2.30425i	1.65195	+	1.25234i		0		−1.12796	−	2.59378i	−0.737300	−	0.425680i	0.420830	−	3.34731i
411.2	−1.30970	−	0.533564i	1.41588	+	0.379383i	1.43062	+	1.39762i	−0.617422	−	2.30425i	−1.65195	−	1.25234i		0		−1.12796	−	2.59378i	−0.737300	−	0.425680i	−0.420830	+	3.34731i
411.3	0.613848	+	1.27404i	−2.05749	−	0.551303i	−1.24638	+	1.56414i	−0.929901	−	3.47044i	−0.560603	−	2.95975i		0		−2.75787	−	0.627801i	1.33126	+	0.768605i	3.85068	−	3.31506i
411.4	0.613848	+	1.27404i	2.05749	+	0.551303i	−1.24638	+	1.56414i	0.929901	+	3.47044i	0.560603	+	2.95975i		0		−2.75787	−	0.627801i	1.33126	+	0.768605i	−3.85068	+	3.31506i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
16.f	odd	4	1	inner
112.j	even	4	1	inner
112.u	odd	12	1	inner
112.v	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.w.e		32
7.b	odd	2	1	inner	784.2.w.e		32
7.c	even	3	1		112.2.j.d	✓	16
7.c	even	3	1	inner	784.2.w.e		32
7.d	odd	6	1		112.2.j.d	✓	16
7.d	odd	6	1	inner	784.2.w.e		32
16.f	odd	4	1	inner	784.2.w.e		32
28.f	even	6	1		448.2.j.d		16
28.g	odd	6	1		448.2.j.d		16
56.j	odd	6	1		896.2.j.h		16
56.k	odd	6	1		896.2.j.g		16
56.m	even	6	1		896.2.j.g		16
56.p	even	6	1		896.2.j.h		16
112.j	even	4	1	inner	784.2.w.e		32
112.u	odd	12	1		112.2.j.d	✓	16
112.u	odd	12	1	inner	784.2.w.e		32
112.u	odd	12	1		896.2.j.h		16
112.v	even	12	1		112.2.j.d	✓	16
112.v	even	12	1	inner	784.2.w.e		32
112.v	even	12	1		896.2.j.h		16
112.w	even	12	1		448.2.j.d		16
112.w	even	12	1		896.2.j.g		16
112.x	odd	12	1		448.2.j.d		16
112.x	odd	12	1		896.2.j.g		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.j.d	✓	16	7.c	even	3	1
112.2.j.d	✓	16	7.d	odd	6	1
112.2.j.d	✓	16	112.u	odd	12	1
112.2.j.d	✓	16	112.v	even	12	1
448.2.j.d		16	28.f	even	6	1
448.2.j.d		16	28.g	odd	6	1
448.2.j.d		16	112.w	even	12	1
448.2.j.d		16	112.x	odd	12	1
784.2.w.e		32	1.a	even	1	1	trivial
784.2.w.e		32	7.b	odd	2	1	inner
784.2.w.e		32	7.c	even	3	1	inner
784.2.w.e		32	7.d	odd	6	1	inner
784.2.w.e		32	16.f	odd	4	1	inner
784.2.w.e		32	112.j	even	4	1	inner
784.2.w.e		32	112.u	odd	12	1	inner
784.2.w.e		32	112.v	even	12	1	inner
896.2.j.g		16	56.k	odd	6	1
896.2.j.g		16	56.m	even	6	1
896.2.j.g		16	112.w	even	12	1
896.2.j.g		16	112.x	odd	12	1
896.2.j.h		16	56.j	odd	6	1
896.2.j.h		16	56.p	even	6	1
896.2.j.h		16	112.u	odd	12	1
896.2.j.h		16	112.v	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

T3^32 - 136*T3^28 + 14504*T3^24 - 466176*T3^20 + 10613040*T3^16 - 124611584*T3^12 + 1053039232*T3^8 - 4027719168*T3^4 + 11019960576

T5^32 - 200*T5^28 + 34408*T5^24 - 1107712*T5^20 + 30200368*T5^16 - 29365248*T5^12 + 21311104*T5^8 - 6925824*T5^4 + 1679616