Properties

Label 784.4.f.e
Level $784$
Weight $4$
Character orbit 784.f
Analytic conductor $46.257$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 7^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - 5 \beta_{2} ) q^{5} -27 q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - 5 \beta_{2} ) q^{5} -27 q^{9} + ( 4 \beta_{1} + 21 \beta_{2} ) q^{13} + ( 6 \beta_{1} - 29 \beta_{2} ) q^{17} + ( -125 - 23 \beta_{3} ) q^{25} -11 \beta_{3} q^{29} -13 \beta_{3} q^{37} + ( -51 \beta_{1} + 83 \beta_{2} ) q^{41} + ( 27 \beta_{1} + 135 \beta_{2} ) q^{45} + 572 q^{53} + ( -152 \beta_{1} - 41 \beta_{2} ) q^{61} + ( 1048 + 97 \beta_{3} ) q^{65} + ( 163 \beta_{1} + 193 \beta_{2} ) q^{73} + 729 q^{81} + ( -1332 - 157 \beta_{3} ) q^{85} + ( -226 \beta_{1} + 157 \beta_{2} ) q^{89} + ( -217 \beta_{1} + 257 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 108q^{9} + O(q^{10}) \) \( 4q - 108q^{9} - 500q^{25} + 2288q^{53} + 4192q^{65} + 2916q^{81} - 5328q^{85} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 7 \nu \)
\(\beta_{3}\)\(=\)\( 7 \nu^{2} + 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 14\)\()/7\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
0.765367i
1.84776i
1.84776i
0.765367i
0 0 0 21.8561i 0 0 0 −27.0000 0
783.2 0 0 0 4.72352i 0 0 0 −27.0000 0
783.3 0 0 0 4.72352i 0 0 0 −27.0000 0
783.4 0 0 0 21.8561i 0 0 0 −27.0000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.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 784.4.f.e 4
4.b odd 2 1 CM 784.4.f.e 4
7.b odd 2 1 inner 784.4.f.e 4
28.d even 2 1 inner 784.4.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.4.f.e 4 1.a even 1 1 trivial
784.4.f.e 4 4.b odd 2 1 CM
784.4.f.e 4 7.b odd 2 1 inner
784.4.f.e 4 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 10658 + 500 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 2913698 + 8788 T^{2} + T^{4} \)
$17$ \( 38596898 + 19652 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -11858 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( -16562 + T^{2} )^{2} \)
$41$ \( 18721512002 + 275684 T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -572 + T )^{4} \)
$59$ \( T^{4} \)
$61$ \( 11781432002 + 907924 T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 580595793698 + 1556068 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 1399267092962 + 2819876 T^{2} + T^{4} \)
$97$ \( 3254346436898 + 3650692 T^{2} + T^{4} \)
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