Newform orbit 3276.2.gv.a

• Label: 3276.2.gv.a
• Level: $3276$
• Weight: $2$
• Character orbit: 3276.gv
• Analytic conductor: $26.159$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3276.gv (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (z - 3) * q^7 + (-3*z - 1) * q^13 + (5*z + 5) * q^19 - 5*z * q^25 + (z - 2) * q^31 + (3*z + 3) * q^37 - 13 * q^43 + (-5*z + 8) * q^49 + (-14*z + 14) * q^61 + (-9*z + 18) * q^67 + (z - 2) * q^73 + (-17*z + 17) * q^79 + (5*z + 6) * q^91 + (-16*z + 8) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 5 q^{7} - 5 q^{13} + 15 q^{19} - 5 q^{25} - 3 q^{31} + 9 q^{37} - 26 q^{43} + 11 q^{49} + 14 q^{61} + 27 q^{67} - 3 q^{73} + 17 q^{79} + 17 q^{91}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$.

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$-1$	$-1 + \zeta_{6}$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1117.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

0

0

−2.50000

−

0.866025i

0

0

0

2053.1

0

0

0

0

0

−2.50000

+

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
91.r	even	6	1	inner
273.w	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3276.2.gv.a	✓	2
3.b	odd	2	1	CM	3276.2.gv.a	✓	2
7.c	even	3	1		3276.2.gv.b	yes	2
13.b	even	2	1		3276.2.gv.b	yes	2
21.h	odd	6	1		3276.2.gv.b	yes	2
39.d	odd	2	1		3276.2.gv.b	yes	2
91.r	even	6	1	inner	3276.2.gv.a	✓	2
273.w	odd	6	1	inner	3276.2.gv.a	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3276.2.gv.a	✓	2	1.a	even	1	1	trivial
3276.2.gv.a	✓	2	3.b	odd	2	1	CM
3276.2.gv.a	✓	2	91.r	even	6	1	inner
3276.2.gv.a	✓	2	273.w	odd	6	1	inner
3276.2.gv.b	yes	2	7.c	even	3	1
3276.2.gv.b	yes	2	13.b	even	2	1
3276.2.gv.b	yes	2	21.h	odd	6	1
3276.2.gv.b	yes	2	39.d	odd	2	1

Hecke kernels

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

$T_{5}$

$T_{19}^{2} - 15T_{19} + 75$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2}$
$7$	$T^{2} + 5T + 7$
$11$	$T^{2}$