Properties

Label 756.1.d.b
Level $756$
Weight $1$
Character orbit 756.d
Analytic conductor $0.377$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 756.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.377293149551\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.4000752.4

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{7} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{13} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} + q^{25} + q^{37} -2 q^{43} -\zeta_{6} q^{49} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{61} - q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} - q^{79} + ( -1 - \zeta_{6} ) q^{91} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{7} + O(q^{10}) \) \( 2q + q^{7} + 2q^{25} + 2q^{37} - 4q^{43} - q^{49} - 2q^{67} - 2q^{79} - 3q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 0.500000 0.866025i 0 0 0
433.2 0 0 0 0 0 0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.1.d.b 2
3.b odd 2 1 CM 756.1.d.b 2
4.b odd 2 1 3024.1.f.a 2
7.b odd 2 1 inner 756.1.d.b 2
9.c even 3 1 2268.1.bc.b 2
9.c even 3 1 2268.1.bc.c 2
9.d odd 6 1 2268.1.bc.b 2
9.d odd 6 1 2268.1.bc.c 2
12.b even 2 1 3024.1.f.a 2
21.c even 2 1 inner 756.1.d.b 2
28.d even 2 1 3024.1.f.a 2
63.l odd 6 1 2268.1.bc.b 2
63.l odd 6 1 2268.1.bc.c 2
63.o even 6 1 2268.1.bc.b 2
63.o even 6 1 2268.1.bc.c 2
84.h odd 2 1 3024.1.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.d.b 2 1.a even 1 1 trivial
756.1.d.b 2 3.b odd 2 1 CM
756.1.d.b 2 7.b odd 2 1 inner
756.1.d.b 2 21.c even 2 1 inner
2268.1.bc.b 2 9.c even 3 1
2268.1.bc.b 2 9.d odd 6 1
2268.1.bc.b 2 63.l odd 6 1
2268.1.bc.b 2 63.o even 6 1
2268.1.bc.c 2 9.c even 3 1
2268.1.bc.c 2 9.d odd 6 1
2268.1.bc.c 2 63.l odd 6 1
2268.1.bc.c 2 63.o even 6 1
3024.1.f.a 2 4.b odd 2 1
3024.1.f.a 2 12.b even 2 1
3024.1.f.a 2 28.d even 2 1
3024.1.f.a 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{1}^{\mathrm{new}}(756, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 3 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 3 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 3 + T^{2} \)
$67$ \( ( 1 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 3 + T^{2} \)
$79$ \( ( 1 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 3 + T^{2} \)
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