Properties

Label 784.4.f.f
Level $784$
Weight $4$
Character orbit 784.f
Analytic conductor $46.257$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -5 \beta_{2} q^{5} -20 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -5 \beta_{2} q^{5} -20 q^{9} -15 \beta_{3} q^{11} -30 \beta_{2} q^{13} + 5 \beta_{3} q^{15} + 5 \beta_{2} q^{17} + 25 \beta_{1} q^{19} + 11 \beta_{3} q^{23} + 50 q^{25} -47 \beta_{1} q^{27} -168 q^{29} -85 \beta_{1} q^{31} + 105 \beta_{2} q^{33} -245 q^{37} + 30 \beta_{3} q^{39} -26 \beta_{2} q^{41} -66 \beta_{3} q^{43} + 100 \beta_{2} q^{45} -159 \beta_{1} q^{47} -5 \beta_{3} q^{51} -345 q^{53} + 225 \beta_{1} q^{55} + 175 q^{57} + 165 \beta_{1} q^{59} + 229 \beta_{2} q^{61} -450 q^{65} -97 \beta_{3} q^{67} -77 \beta_{2} q^{69} -10 \beta_{3} q^{71} + 555 \beta_{2} q^{73} + 50 \beta_{1} q^{75} -45 \beta_{3} q^{79} + 211 q^{81} -336 \beta_{1} q^{83} + 75 q^{85} -168 \beta_{1} q^{87} + 873 \beta_{2} q^{89} -595 q^{93} + 125 \beta_{3} q^{95} -250 \beta_{2} q^{97} + 300 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 80q^{9} + O(q^{10}) \) \( 4q - 80q^{9} + 200q^{25} - 672q^{29} - 980q^{37} - 1380q^{53} + 700q^{57} - 1800q^{65} + 844q^{81} + 300q^{85} - 2380q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

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

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
1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
−1.32288 + 2.29129i
0 −2.64575 0 8.66025i 0 0 0 −20.0000 0
783.2 0 −2.64575 0 8.66025i 0 0 0 −20.0000 0
783.3 0 2.64575 0 8.66025i 0 0 0 −20.0000 0
783.4 0 2.64575 0 8.66025i 0 0 0 −20.0000 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 inner
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.f 4
4.b odd 2 1 inner 784.4.f.f 4
7.b odd 2 1 inner 784.4.f.f 4
7.c even 3 1 112.4.p.e 4
7.d odd 6 1 112.4.p.e 4
28.d even 2 1 inner 784.4.f.f 4
28.f even 6 1 112.4.p.e 4
28.g odd 6 1 112.4.p.e 4
56.j odd 6 1 448.4.p.e 4
56.k odd 6 1 448.4.p.e 4
56.m even 6 1 448.4.p.e 4
56.p even 6 1 448.4.p.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.e 4 7.c even 3 1
112.4.p.e 4 7.d odd 6 1
112.4.p.e 4 28.f even 6 1
112.4.p.e 4 28.g odd 6 1
448.4.p.e 4 56.j odd 6 1
448.4.p.e 4 56.k odd 6 1
448.4.p.e 4 56.m even 6 1
448.4.p.e 4 56.p even 6 1
784.4.f.f 4 1.a even 1 1 trivial
784.4.f.f 4 4.b odd 2 1 inner
784.4.f.f 4 7.b odd 2 1 inner
784.4.f.f 4 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -7 + T^{2} )^{2} \)
$5$ \( ( 75 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 4725 + T^{2} )^{2} \)
$13$ \( ( 2700 + T^{2} )^{2} \)
$17$ \( ( 75 + T^{2} )^{2} \)
$19$ \( ( -4375 + T^{2} )^{2} \)
$23$ \( ( 2541 + T^{2} )^{2} \)
$29$ \( ( 168 + T )^{4} \)
$31$ \( ( -50575 + T^{2} )^{2} \)
$37$ \( ( 245 + T )^{4} \)
$41$ \( ( 2028 + T^{2} )^{2} \)
$43$ \( ( 91476 + T^{2} )^{2} \)
$47$ \( ( -176967 + T^{2} )^{2} \)
$53$ \( ( 345 + T )^{4} \)
$59$ \( ( -190575 + T^{2} )^{2} \)
$61$ \( ( 157323 + T^{2} )^{2} \)
$67$ \( ( 197589 + T^{2} )^{2} \)
$71$ \( ( 2100 + T^{2} )^{2} \)
$73$ \( ( 924075 + T^{2} )^{2} \)
$79$ \( ( 42525 + T^{2} )^{2} \)
$83$ \( ( -790272 + T^{2} )^{2} \)
$89$ \( ( 2286387 + T^{2} )^{2} \)
$97$ \( ( 187500 + T^{2} )^{2} \)
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