Properties

Label 756.2.b.a
Level $756$
Weight $2$
Character orbit 756.b
Analytic conductor $6.037$
Analytic rank $0$
Dimension $4$
CM discriminant -84
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{3} q^{7} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{3} q^{7} + 2 \beta_{2} q^{8} + ( -3 + \beta_{3} ) q^{10} + ( 3 \beta_{1} + \beta_{2} ) q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{14} + 4 q^{16} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( 6 + \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -1 + 3 \beta_{3} ) q^{22} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -3 + 3 \beta_{3} ) q^{25} + 2 \beta_{3} q^{28} + ( -3 - 2 \beta_{3} ) q^{31} -4 \beta_{2} q^{32} -2 \beta_{3} q^{34} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -4 + 3 \beta_{3} ) q^{37} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{38} + ( 6 - 2 \beta_{3} ) q^{40} + ( \beta_{1} + 8 \beta_{2} ) q^{41} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 5 + 3 \beta_{3} ) q^{46} + 7 q^{49} -6 \beta_{1} q^{50} + ( -12 + 5 \beta_{3} ) q^{55} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 \beta_{1} + 5 \beta_{2} ) q^{62} -8 q^{64} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -7 + 3 \beta_{3} ) q^{70} + ( 3 \beta_{1} + 7 \beta_{2} ) q^{71} + ( -6 \beta_{1} + \beta_{2} ) q^{74} + ( -12 - 2 \beta_{3} ) q^{76} + ( \beta_{1} - 10 \beta_{2} ) q^{77} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{80} + ( 15 + \beta_{3} ) q^{82} + ( 7 - 3 \beta_{3} ) q^{85} + ( 2 - 6 \beta_{3} ) q^{88} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -6 \beta_{1} - 8 \beta_{2} ) q^{92} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{95} -7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} - 12 q^{10} + 16 q^{16} + 24 q^{19} - 4 q^{22} - 12 q^{25} - 12 q^{31} - 16 q^{37} + 24 q^{40} + 20 q^{46} + 28 q^{49} - 48 q^{55} - 32 q^{64} - 28 q^{70} - 48 q^{76} + 60 q^{82} + 28 q^{85} + 8 q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 5 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
2.57794i
1.16372i
1.16372i
2.57794i
1.41421i 0 −2.00000 3.99215i 0 2.64575 2.82843i 0 −5.64575
55.2 1.41421i 0 −2.00000 0.250492i 0 −2.64575 2.82843i 0 −0.354249
55.3 1.41421i 0 −2.00000 0.250492i 0 −2.64575 2.82843i 0 −0.354249
55.4 1.41421i 0 −2.00000 3.99215i 0 2.64575 2.82843i 0 −5.64575
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.b.a 4
3.b odd 2 1 inner 756.2.b.a 4
4.b odd 2 1 756.2.b.b yes 4
7.b odd 2 1 756.2.b.b yes 4
12.b even 2 1 756.2.b.b yes 4
21.c even 2 1 756.2.b.b yes 4
28.d even 2 1 inner 756.2.b.a 4
84.h odd 2 1 CM 756.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.a 4 1.a even 1 1 trivial
756.2.b.a 4 3.b odd 2 1 inner
756.2.b.a 4 28.d even 2 1 inner
756.2.b.a 4 84.h odd 2 1 CM
756.2.b.b yes 4 4.b odd 2 1
756.2.b.b yes 4 7.b odd 2 1
756.2.b.b yes 4 12.b even 2 1
756.2.b.b yes 4 21.c even 2 1

Hecke kernels

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

\( T_{5}^{4} + 16 T_{5}^{2} + 1 \)
\( T_{19}^{2} - 12 T_{19} + 29 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1 + 16 T^{2} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 961 + 64 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 14 + T^{2} )^{2} \)
$19$ \( ( 29 - 12 T + T^{2} )^{2} \)
$23$ \( 361 + 88 T^{2} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -19 + 6 T + T^{2} )^{2} \)
$37$ \( ( -47 + 8 T + T^{2} )^{2} \)
$41$ \( 11881 + 232 T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( 841 + 184 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 6889 + 184 T^{2} + T^{4} \)
$97$ \( T^{4} \)
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