Properties

Label 784.4.f.c
Level $784$
Weight $4$
Character orbit 784.f
Analytic conductor $46.257$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -26 q^{9} +O(q^{10})\) \( q + q^{3} + ( -3 + 6 \zeta_{6} ) q^{5} -26 q^{9} + ( -15 + 30 \zeta_{6} ) q^{11} + ( 40 - 80 \zeta_{6} ) q^{13} + ( -3 + 6 \zeta_{6} ) q^{15} + ( -21 + 42 \zeta_{6} ) q^{17} -17 q^{19} + ( 81 - 162 \zeta_{6} ) q^{23} + 98 q^{25} -53 q^{27} + 90 q^{29} + 17 q^{31} + ( -15 + 30 \zeta_{6} ) q^{33} + 199 q^{37} + ( 40 - 80 \zeta_{6} ) q^{39} + ( -108 + 216 \zeta_{6} ) q^{41} + ( 146 - 292 \zeta_{6} ) q^{43} + ( 78 - 156 \zeta_{6} ) q^{45} + 567 q^{47} + ( -21 + 42 \zeta_{6} ) q^{51} -333 q^{53} -135 q^{55} -17 q^{57} + 801 q^{59} + ( -207 + 414 \zeta_{6} ) q^{61} + 360 q^{65} + ( 125 - 250 \zeta_{6} ) q^{67} + ( 81 - 162 \zeta_{6} ) q^{69} + ( -282 + 564 \zeta_{6} ) q^{71} + ( -233 + 466 \zeta_{6} ) q^{73} + 98 q^{75} + ( 689 - 1378 \zeta_{6} ) q^{79} + 649 q^{81} -468 q^{83} -189 q^{85} + 90 q^{87} + ( -111 + 222 \zeta_{6} ) q^{89} + 17 q^{93} + ( 51 - 102 \zeta_{6} ) q^{95} + ( 804 - 1608 \zeta_{6} ) q^{97} + ( 390 - 780 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 52q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 52q^{9} - 34q^{19} + 196q^{25} - 106q^{27} + 180q^{29} + 34q^{31} + 398q^{37} + 1134q^{47} - 666q^{53} - 270q^{55} - 34q^{57} + 1602q^{59} + 720q^{65} + 196q^{75} + 1298q^{81} - 936q^{83} - 378q^{85} + 180q^{87} + 34q^{93} + O(q^{100}) \)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 0 5.19615i 0 0 0 −26.0000 0
783.2 0 1.00000 0 5.19615i 0 0 0 −26.0000 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
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.c 2
4.b odd 2 1 784.4.f.b 2
7.b odd 2 1 784.4.f.b 2
7.c even 3 1 112.4.p.b 2
7.d odd 6 1 112.4.p.c yes 2
28.d even 2 1 inner 784.4.f.c 2
28.f even 6 1 112.4.p.b 2
28.g odd 6 1 112.4.p.c yes 2
56.j odd 6 1 448.4.p.b 2
56.k odd 6 1 448.4.p.b 2
56.m even 6 1 448.4.p.c 2
56.p even 6 1 448.4.p.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.b 2 7.c even 3 1
112.4.p.b 2 28.f even 6 1
112.4.p.c yes 2 7.d odd 6 1
112.4.p.c yes 2 28.g odd 6 1
448.4.p.b 2 56.j odd 6 1
448.4.p.b 2 56.k odd 6 1
448.4.p.c 2 56.m even 6 1
448.4.p.c 2 56.p even 6 1
784.4.f.b 2 4.b odd 2 1
784.4.f.b 2 7.b odd 2 1
784.4.f.c 2 1.a even 1 1 trivial
784.4.f.c 2 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 27 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 675 + T^{2} \)
$13$ \( 4800 + T^{2} \)
$17$ \( 1323 + T^{2} \)
$19$ \( ( 17 + T )^{2} \)
$23$ \( 19683 + T^{2} \)
$29$ \( ( -90 + T )^{2} \)
$31$ \( ( -17 + T )^{2} \)
$37$ \( ( -199 + T )^{2} \)
$41$ \( 34992 + T^{2} \)
$43$ \( 63948 + T^{2} \)
$47$ \( ( -567 + T )^{2} \)
$53$ \( ( 333 + T )^{2} \)
$59$ \( ( -801 + T )^{2} \)
$61$ \( 128547 + T^{2} \)
$67$ \( 46875 + T^{2} \)
$71$ \( 238572 + T^{2} \)
$73$ \( 162867 + T^{2} \)
$79$ \( 1424163 + T^{2} \)
$83$ \( ( 468 + T )^{2} \)
$89$ \( 36963 + T^{2} \)
$97$ \( 1939248 + T^{2} \)
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