Properties

Label 784.4.f.a
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q -7 q^{3} + ( -9 + 18 \zeta_{6} ) q^{5} + 22 q^{9} +O(q^{10})\) \( q -7 q^{3} + ( -9 + 18 \zeta_{6} ) q^{5} + 22 q^{9} + ( 7 - 14 \zeta_{6} ) q^{11} + ( -40 + 80 \zeta_{6} ) q^{13} + ( 63 - 126 \zeta_{6} ) q^{15} + ( -19 + 38 \zeta_{6} ) q^{17} + 119 q^{19} + ( -77 + 154 \zeta_{6} ) q^{23} -118 q^{25} + 35 q^{27} + 210 q^{29} + 301 q^{31} + ( -49 + 98 \zeta_{6} ) q^{33} -77 q^{37} + ( 280 - 560 \zeta_{6} ) q^{39} + ( 68 - 136 \zeta_{6} ) q^{41} + ( 70 - 140 \zeta_{6} ) q^{43} + ( -198 + 396 \zeta_{6} ) q^{45} -357 q^{47} + ( 133 - 266 \zeta_{6} ) q^{51} + 327 q^{53} + 189 q^{55} -833 q^{57} + 609 q^{59} + ( -397 + 794 \zeta_{6} ) q^{61} -1080 q^{65} + ( 91 - 182 \zeta_{6} ) q^{67} + ( 539 - 1078 \zeta_{6} ) q^{69} + ( -126 + 252 \zeta_{6} ) q^{71} + ( 33 - 66 \zeta_{6} ) q^{73} + 826 q^{75} + ( -469 + 938 \zeta_{6} ) q^{79} -839 q^{81} -588 q^{83} -513 q^{85} -1470 q^{87} + ( -425 + 850 \zeta_{6} ) q^{89} -2107 q^{93} + ( -1071 + 2142 \zeta_{6} ) q^{95} + ( 180 - 360 \zeta_{6} ) q^{97} + ( 154 - 308 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{3} + 44q^{9} + O(q^{10}) \) \( 2q - 14q^{3} + 44q^{9} + 238q^{19} - 236q^{25} + 70q^{27} + 420q^{29} + 602q^{31} - 154q^{37} - 714q^{47} + 654q^{53} + 378q^{55} - 1666q^{57} + 1218q^{59} - 2160q^{65} + 1652q^{75} - 1678q^{81} - 1176q^{83} - 1026q^{85} - 2940q^{87} - 4214q^{93} + O(q^{100}) \)

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 −7.00000 0 15.5885i 0 0 0 22.0000 0
783.2 0 −7.00000 0 15.5885i 0 0 0 22.0000 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.a 2
4.b odd 2 1 784.4.f.d 2
7.b odd 2 1 784.4.f.d 2
7.c even 3 1 112.4.p.d yes 2
7.d odd 6 1 112.4.p.a 2
28.d even 2 1 inner 784.4.f.a 2
28.f even 6 1 112.4.p.d yes 2
28.g odd 6 1 112.4.p.a 2
56.j odd 6 1 448.4.p.d 2
56.k odd 6 1 448.4.p.d 2
56.m even 6 1 448.4.p.a 2
56.p even 6 1 448.4.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.a 2 7.d odd 6 1
112.4.p.a 2 28.g odd 6 1
112.4.p.d yes 2 7.c even 3 1
112.4.p.d yes 2 28.f even 6 1
448.4.p.a 2 56.m even 6 1
448.4.p.a 2 56.p even 6 1
448.4.p.d 2 56.j odd 6 1
448.4.p.d 2 56.k odd 6 1
784.4.f.a 2 1.a even 1 1 trivial
784.4.f.a 2 28.d even 2 1 inner
784.4.f.d 2 4.b odd 2 1
784.4.f.d 2 7.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 7 + T )^{2} \)
$5$ \( 243 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 147 + T^{2} \)
$13$ \( 4800 + T^{2} \)
$17$ \( 1083 + T^{2} \)
$19$ \( ( -119 + T )^{2} \)
$23$ \( 17787 + T^{2} \)
$29$ \( ( -210 + T )^{2} \)
$31$ \( ( -301 + T )^{2} \)
$37$ \( ( 77 + T )^{2} \)
$41$ \( 13872 + T^{2} \)
$43$ \( 14700 + T^{2} \)
$47$ \( ( 357 + T )^{2} \)
$53$ \( ( -327 + T )^{2} \)
$59$ \( ( -609 + T )^{2} \)
$61$ \( 472827 + T^{2} \)
$67$ \( 24843 + T^{2} \)
$71$ \( 47628 + T^{2} \)
$73$ \( 3267 + T^{2} \)
$79$ \( 659883 + T^{2} \)
$83$ \( ( 588 + T )^{2} \)
$89$ \( 541875 + T^{2} \)
$97$ \( 97200 + T^{2} \)
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