Newform orbit 896.2.z.a

• Label: 896.2.z.a
• Level: $896$
• Weight: $2$
• Character orbit: 896.z
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.z (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1		0		−0.825526	−	3.08091i		0		−1.86998	−	0.501060i		0		−2.17953	−	1.49988i		0		−6.21241	+	3.58674i		0
31.2		0		−0.700587	−	2.61463i		0		−0.450647	−	0.120751i		0		2.37326	+	1.16946i		0		−3.74737	+	2.16355i		0
31.3		0		−0.665801	−	2.48480i		0		3.12401	+	0.837076i		0		−1.56215	+	2.13534i		0		−3.13287	+	1.80877i		0
31.4		0		−0.388227	−	1.44888i		0		−2.81809	−	0.755106i		0		1.47726	+	2.19493i		0		0.649537	−	0.375010i		0
31.5		0		−0.282507	−	1.05433i		0		1.39922	+	0.374919i		0		0.298614	−	2.62885i		0		1.56627	−	0.904287i		0
31.6		0		−0.237312	−	0.885661i		0		−3.05579	−	0.818796i		0		0.640837	−	2.56697i		0		1.87000	−	1.07964i		0
31.7		0		−0.204601	−	0.763582i		0		3.81370	+	1.02188i		0		2.64575	−	0.00379639i		0		2.05688	−	1.18754i		0
31.8		0		−0.0661591	−	0.246909i		0		0.499339	+	0.133797i		0		−2.45011	+	0.998472i		0		2.54149	−	1.46733i		0
31.9		0		0.0530773	+	0.198087i		0		1.82029	+	0.487744i		0		−1.84933	−	1.89208i		0		2.56165	−	1.47897i		0
31.10		0		0.350301	+	1.30734i		0		−1.09872	−	0.294400i		0		1.56831	+	2.13082i		0		1.01165	−	0.584076i		0
31.11		0		0.449868	+	1.67893i		0		−0.731029	−	0.195879i		0		−2.52163	+	0.800849i		0		−0.0183525	+	0.0105958i		0
31.12		0		0.524583	+	1.95777i		0		−2.48890	−	0.666898i		0		2.38027	−	1.15512i		0		−0.959602	+	0.554026i		0
31.13		0		0.615336	+	2.29647i		0		0.835825	+	0.223959i		0		1.25761	−	2.32775i		0		−2.29704	+	1.32620i		0
31.14		0		0.743580	+	2.77508i		0		3.38680	+	0.907491i		0		−0.0791552	+	2.64457i		0		−4.55008	+	2.62699i		0
159.1		0		−3.08091	−	0.825526i		0		−0.501060	−	1.86998i		0		−2.17953	+	1.49988i		0		6.21241	+	3.58674i		0
159.2		0		−2.61463	−	0.700587i		0		−0.120751	−	0.450647i		0		2.37326	−	1.16946i		0		3.74737	+	2.16355i		0
159.3		0		−2.48480	−	0.665801i		0		0.837076	+	3.12401i		0		−1.56215	−	2.13534i		0		3.13287	+	1.80877i		0
159.4		0		−1.44888	−	0.388227i		0		−0.755106	−	2.81809i		0		1.47726	−	2.19493i		0		−0.649537	−	0.375010i		0
159.5		0		−1.05433	−	0.282507i		0		0.374919	+	1.39922i		0		0.298614	+	2.62885i		0		−1.56627	−	0.904287i		0
159.6		0		−0.885661	−	0.237312i		0		−0.818796	−	3.05579i		0		0.640837	+	2.56697i		0		−1.87000	−	1.07964i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
16.f	odd	4	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	896.2.z.a		56
4.b	odd	2	1		896.2.z.b		56
7.d	odd	6	1	inner	896.2.z.a		56
8.b	even	2	1		448.2.z.a		56
8.d	odd	2	1		112.2.v.a	✓	56
16.e	even	4	1		112.2.v.a	✓	56
16.e	even	4	1		896.2.z.b		56
16.f	odd	4	1		448.2.z.a		56
16.f	odd	4	1	inner	896.2.z.a		56
28.f	even	6	1		896.2.z.b		56
56.e	even	2	1		784.2.w.f		56
56.j	odd	6	1		448.2.z.a		56
56.k	odd	6	1		784.2.j.a		56
56.k	odd	6	1		784.2.w.f		56
56.m	even	6	1		112.2.v.a	✓	56
56.m	even	6	1		784.2.j.a		56
112.l	odd	4	1		784.2.w.f		56
112.v	even	12	1		448.2.z.a		56
112.v	even	12	1	inner	896.2.z.a		56
112.w	even	12	1		784.2.j.a		56
112.w	even	12	1		784.2.w.f		56
112.x	odd	12	1		112.2.v.a	✓	56
112.x	odd	12	1		784.2.j.a		56
112.x	odd	12	1		896.2.z.b		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.v.a	✓	56	8.d	odd	2	1
112.2.v.a	✓	56	16.e	even	4	1
112.2.v.a	✓	56	56.m	even	6	1
112.2.v.a	✓	56	112.x	odd	12	1
448.2.z.a		56	8.b	even	2	1
448.2.z.a		56	16.f	odd	4	1
448.2.z.a		56	56.j	odd	6	1
448.2.z.a		56	112.v	even	12	1
784.2.j.a		56	56.k	odd	6	1
784.2.j.a		56	56.m	even	6	1
784.2.j.a		56	112.w	even	12	1
784.2.j.a		56	112.x	odd	12	1
784.2.w.f		56	56.e	even	2	1
784.2.w.f		56	56.k	odd	6	1
784.2.w.f		56	112.l	odd	4	1
784.2.w.f		56	112.w	even	12	1
896.2.z.a		56	1.a	even	1	1	trivial
896.2.z.a		56	7.d	odd	6	1	inner
896.2.z.a		56	16.f	odd	4	1	inner
896.2.z.a		56	112.v	even	12	1	inner
896.2.z.b		56	4.b	odd	2	1
896.2.z.b		56	16.e	even	4	1
896.2.z.b		56	28.f	even	6	1
896.2.z.b		56	112.x	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^56 + 6*T3^55 + 18*T3^54 + 36*T3^53 - 115*T3^52 - 816*T3^51 - 2178*T3^50 - 4662*T3^49 + 9200*T3^48 + 73254*T3^47 + 188550*T3^46 + 404976*T3^45 - 310773*T3^44 - 3491760*T3^43 - 7964802*T3^42 - 18060798*T3^41 + 6073221*T3^40 + 115561212*T3^39 + 250737048*T3^38 + 566860548*T3^37 + 38030254*T3^36 - 2390667348*T3^35 - 4568020992*T3^34 - 10417638252*T3^33 - 3776679479*T3^32 + 32322150726*T3^31 + 72659486934*T3^30 + 127090569456*T3^29 + 75387600663*T3^28 - 223576534296*T3^27 - 628125804954*T3^26 - 1041565901022*T3^25 - 705876250815*T3^24 + 1130526690300*T3^23 + 3737477479680*T3^22 + 5816460589884*T3^21 + 4923177045966*T3^20 - 780493431996*T3^19 - 9244202955048*T3^18 - 15757821597132*T3^17 - 14068788515667*T3^16 - 1565035322778*T3^15 + 15867512543790*T3^14 + 28682989170696*T3^13 + 30567214565163*T3^12 + 22743343545480*T3^11 + 12193044798870*T3^10 + 4507392853026*T3^9 + 942111305208*T3^8 + 37212456726*T3^7 - 23957635842*T3^6 - 9608785704*T3^5 - 1621916379*T3^4 + 31886460*T3^3 + 26572050*T3^2 + 14762250*T3 + 4100625