Newform orbit 784.2.j.a

• Label: 784.2.j.a
• Level: $784$
• Weight: $2$
• Character orbit: 784.j
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{2} + 8 q^{4} + 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
195.1	−1.39496	+	0.232566i	−0.957039	−	0.957039i	1.89183	−	0.648841i	0.804316	+	0.804316i	1.55761	+	1.11246i		0		−2.48812	+	1.34508i		−	1.16815i	−1.30904	−	0.934931i
195.2	−1.39496	+	0.232566i	0.957039	+	0.957039i	1.89183	−	0.648841i	−0.804316	−	0.804316i	−1.55761	−	1.11246i		0		−2.48812	+	1.34508i		−	1.16815i	1.30904	+	0.934931i
195.3	−1.30774	−	0.538353i	−0.145010	−	0.145010i	1.42035	+	1.40805i	−1.33254	−	1.33254i	0.111568	+	0.267701i		0		−1.09942	−	2.60601i		−	2.95794i	1.02524	+	2.45999i
195.4	−1.30774	−	0.538353i	0.145010	+	0.145010i	1.42035	+	1.40805i	1.33254	+	1.33254i	−0.111568	−	0.267701i		0		−1.09942	−	2.60601i		−	2.95794i	−1.02524	−	2.45999i
195.5	−1.10953	+	0.876896i	−1.81900	−	1.81900i	0.462106	−	1.94588i	2.28693	+	2.28693i	3.61331	+	0.423158i		0		1.19362	+	2.56423i			3.61753i	−4.54282	−	0.532014i
195.6	−1.10953	+	0.876896i	1.81900	+	1.81900i	0.462106	−	1.94588i	−2.28693	−	2.28693i	−3.61331	−	0.423158i		0		1.19362	+	2.56423i			3.61753i	4.54282	+	0.532014i
195.7	−1.07857	−	0.914706i	−1.91404	−	1.91404i	0.326624	+	1.97315i	−0.329897	−	0.329897i	0.313640	+	3.81521i		0		1.45257	−	2.42694i			4.32709i	0.0540578	+	0.657575i
195.8	−1.07857	−	0.914706i	1.91404	+	1.91404i	0.326624	+	1.97315i	0.329897	+	0.329897i	−0.313640	−	3.81521i		0		1.45257	−	2.42694i			4.32709i	−0.0540578	−	0.657575i
195.9	−0.795020	+	1.16959i	−0.648349	−	0.648349i	−0.735886	−	1.85970i	−2.23699	−	2.23699i	1.27375	−	0.242853i		0		2.76013	+	0.617811i		−	2.15929i	4.39482	−	0.837912i
195.10	−0.795020	+	1.16959i	0.648349	+	0.648349i	−0.735886	−	1.85970i	2.23699	+	2.23699i	−1.27375	+	0.242853i		0		2.76013	+	0.617811i		−	2.15929i	−4.39482	+	0.837912i
195.11	−0.436725	−	1.34509i	−2.03150	−	2.03150i	−1.61854	+	1.17487i	−2.47931	−	2.47931i	−1.84535	+	3.61976i		0		2.28717	+	1.66399i			5.25398i	−2.25212	+	4.41768i
195.12	−0.436725	−	1.34509i	2.03150	+	2.03150i	−1.61854	+	1.17487i	2.47931	+	2.47931i	1.84535	−	3.61976i		0		2.28717	+	1.66399i			5.25398i	2.25212	−	4.41768i
195.13	0.0617395	−	1.41287i	−0.771824	−	0.771824i	−1.99238	−	0.174459i	1.02430	+	1.02430i	−1.13814	+	1.04283i		0		−0.369496	+	2.80419i		−	1.80857i	1.51044	−	1.38396i
195.14	0.0617395	−	1.41287i	0.771824	+	0.771824i	−1.99238	−	0.174459i	−1.02430	−	1.02430i	1.13814	−	1.04283i		0		−0.369496	+	2.80419i		−	1.80857i	−1.51044	+	1.38396i
195.15	0.266279	+	1.38892i	−0.180750	−	0.180750i	−1.85819	+	0.739679i	0.365541	+	0.365541i	0.202917	−	0.299177i		0		−1.52215	−	2.38392i		−	2.93466i	−0.410371	+	0.605043i
195.16	0.266279	+	1.38892i	0.180750	+	0.180750i	−1.85819	+	0.739679i	−0.365541	−	0.365541i	−0.202917	+	0.299177i		0		−1.52215	−	2.38392i		−	2.93466i	0.410371	−	0.605043i
195.17	0.714421	−	1.22049i	−1.43319	−	1.43319i	−0.979204	−	1.74389i	1.82200	+	1.82200i	−2.77309	+	0.725295i		0		−2.82797	−	0.0507629i			1.10805i	3.52541	−	0.922061i
195.18	0.714421	−	1.22049i	1.43319	+	1.43319i	−0.979204	−	1.74389i	−1.82200	−	1.82200i	2.77309	−	0.725295i		0		−2.82797	−	0.0507629i			1.10805i	−3.52541	+	0.922061i
195.19	0.945273	+	1.05188i	−1.68113	−	1.68113i	−0.212918	+	1.98863i	−0.611867	−	0.611867i	0.179226	−	3.35748i		0		−2.29308	+	1.65584i			2.65240i	0.0652315	−	1.22199i
195.20	0.945273	+	1.05188i	1.68113	+	1.68113i	−0.212918	+	1.98863i	0.611867	+	0.611867i	−0.179226	+	3.35748i		0		−2.29308	+	1.65584i			2.65240i	−0.0652315	+	1.22199i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
16.f	odd	4	1	inner
112.j	even	4	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^56 + 338*T3^52 + 46331*T3^48 + 3355716*T3^44 + 140044569*T3^40 + 3450490654*T3^36 + 49819891147*T3^32 + 412039923576*T3^28 + 1893041266203*T3^24 + 4621703550318*T3^20 + 5569119491625*T3^16 + 2988369796356*T3^12 + 560917329867*T3^8 + 3301228386*T3^4 + 4100625