Properties

Label 756.2.k.b
Level 756
Weight 2
Character orbit 756.k
Analytic conductor 6.037
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -1 - 2 \zeta_{6} ) q^{7} + 5 q^{13} -8 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 7 - 7 \zeta_{6} ) q^{31} -11 \zeta_{6} q^{37} + 5 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + 13 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{67} + ( 10 - 10 \zeta_{6} ) q^{73} -17 \zeta_{6} q^{79} + ( -5 - 10 \zeta_{6} ) q^{91} + 5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} + 10q^{13} - 8q^{19} + 5q^{25} + 7q^{31} - 11q^{37} + 10q^{43} + 2q^{49} + 13q^{61} - 5q^{67} + 10q^{73} - 17q^{79} - 20q^{91} + 10q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −2.00000 1.73205i 0 0 0
541.1 0 0 0 0 0 −2.00000 + 1.73205i 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.b 2
3.b odd 2 1 CM 756.2.k.b 2
7.c even 3 1 inner 756.2.k.b 2
7.c even 3 1 5292.2.a.h 1
7.d odd 6 1 5292.2.a.e 1
9.c even 3 1 2268.2.i.f 2
9.c even 3 1 2268.2.l.d 2
9.d odd 6 1 2268.2.i.f 2
9.d odd 6 1 2268.2.l.d 2
21.g even 6 1 5292.2.a.e 1
21.h odd 6 1 inner 756.2.k.b 2
21.h odd 6 1 5292.2.a.h 1
63.g even 3 1 2268.2.i.f 2
63.h even 3 1 2268.2.l.d 2
63.j odd 6 1 2268.2.l.d 2
63.n odd 6 1 2268.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.b 2 1.a even 1 1 trivial
756.2.k.b 2 3.b odd 2 1 CM
756.2.k.b 2 7.c even 3 1 inner
756.2.k.b 2 21.h odd 6 1 inner
2268.2.i.f 2 9.c even 3 1
2268.2.i.f 2 9.d odd 6 1
2268.2.i.f 2 63.g even 3 1
2268.2.i.f 2 63.n odd 6 1
2268.2.l.d 2 9.c even 3 1
2268.2.l.d 2 9.d odd 6 1
2268.2.l.d 2 63.h even 3 1
2268.2.l.d 2 63.j odd 6 1
5292.2.a.e 1 7.d odd 6 1
5292.2.a.e 1 21.g even 6 1
5292.2.a.h 1 7.c even 3 1
5292.2.a.h 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5} \)
\( T_{13} - 5 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 5 T^{2} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \)
$67$ \( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )^{2} \)
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