Newform orbit 3276.2.bi.f

• Label: 3276.2.bi.f
• Level: $3276$
• Weight: $2$
• Character orbit: 3276.bi
• Analytic conductor: $26.159$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3276.bi (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ $32 q + 8 q^{7} - 72 q^{37} - 32 q^{67} + 8 q^{79} + 56 q^{85} + 32 q^{91}+O(q^{100})$

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
1945.1		0			0			0		−3.01582	+	3.01582i		0		−2.64107	+	0.157253i		0			0			0
1945.2		0			0			0		−3.01582	+	3.01582i		0		−0.157253	+	2.64107i		0			0			0
1945.3		0			0			0		−2.09589	+	2.09589i		0		2.62729	+	0.311986i		0			0			0
1945.4		0			0			0		−2.09589	+	2.09589i		0		−0.311986	−	2.62729i		0			0			0
1945.5		0			0			0		−1.55781	+	1.55781i		0		−2.36531	−	1.18546i		0			0			0
1945.6		0			0			0		−1.55781	+	1.55781i		0		1.18546	+	2.36531i		0			0			0
1945.7		0			0			0		−0.765058	+	0.765058i		0		2.21332	−	1.44956i		0			0			0
1945.8		0			0			0		−0.765058	+	0.765058i		0		1.44956	−	2.21332i		0			0			0
1945.9		0			0			0		0.765058	−	0.765058i		0		1.44956	−	2.21332i		0			0			0
1945.10		0			0			0		0.765058	−	0.765058i		0		2.21332	−	1.44956i		0			0			0
1945.11		0			0			0		1.55781	−	1.55781i		0		1.18546	+	2.36531i		0			0			0
1945.12		0			0			0		1.55781	−	1.55781i		0		−2.36531	−	1.18546i		0			0			0
1945.13		0			0			0		2.09589	−	2.09589i		0		−0.311986	−	2.62729i		0			0			0
1945.14		0			0			0		2.09589	−	2.09589i		0		2.62729	+	0.311986i		0			0			0
1945.15		0			0			0		3.01582	−	3.01582i		0		−0.157253	+	2.64107i		0			0			0
1945.16		0			0			0		3.01582	−	3.01582i		0		−2.64107	+	0.157253i		0			0			0
2449.1		0			0			0		−3.01582	−	3.01582i		0		−2.64107	−	0.157253i		0			0			0
2449.2		0			0			0		−3.01582	−	3.01582i		0		−0.157253	−	2.64107i		0			0			0
2449.3		0			0			0		−2.09589	−	2.09589i		0		2.62729	−	0.311986i		0			0			0
2449.4		0			0			0		−2.09589	−	2.09589i		0		−0.311986	+	2.62729i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
13.d	odd	4	1	inner
21.c	even	2	1	inner
39.f	even	4	1	inner
91.i	even	4	1	inner
273.o	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3276.2.bi.f	✓	32
3.b	odd	2	1	inner	3276.2.bi.f	✓	32
7.b	odd	2	1	inner	3276.2.bi.f	✓	32
13.d	odd	4	1	inner	3276.2.bi.f	✓	32
21.c	even	2	1	inner	3276.2.bi.f	✓	32
39.f	even	4	1	inner	3276.2.bi.f	✓	32
91.i	even	4	1	inner	3276.2.bi.f	✓	32
273.o	odd	4	1	inner	3276.2.bi.f	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3276.2.bi.f	✓	32	1.a	even	1	1	trivial
3276.2.bi.f	✓	32	3.b	odd	2	1	inner
3276.2.bi.f	✓	32	7.b	odd	2	1	inner
3276.2.bi.f	✓	32	13.d	odd	4	1	inner
3276.2.bi.f	✓	32	21.c	even	2	1	inner
3276.2.bi.f	✓	32	39.f	even	4	1	inner
3276.2.bi.f	✓	32	91.i	even	4	1	inner
3276.2.bi.f	✓	32	273.o	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3276, [\chi])$:

$T_{5}^{16} + 433T_{5}^{12} + 35744T_{5}^{8} + 649808T_{5}^{4} + 824464$

$T_{19}^{16} + 1977T_{19}^{12} + 364432T_{19}^{8} + 83200T_{19}^{4} + 1024$