Properties

Label 784.3.s.a
Level 784
Weight 3
Character orbit 784.s
Analytic conductor 21.362
Analytic rank 0
Dimension 2
CM discriminant -7
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -9 \zeta_{6} q^{9} +O(q^{10})\) \( q -9 \zeta_{6} q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} + 18 \zeta_{6} q^{23} + ( -25 + 25 \zeta_{6} ) q^{25} -54 q^{29} + 38 \zeta_{6} q^{37} -58 q^{43} + ( 6 - 6 \zeta_{6} ) q^{53} + ( -118 + 118 \zeta_{6} ) q^{67} -114 q^{71} -94 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9q^{9} + O(q^{10}) \) \( 2q - 9q^{9} - 6q^{11} + 18q^{23} - 25q^{25} - 108q^{29} + 38q^{37} - 116q^{43} + 6q^{53} - 118q^{67} - 228q^{71} - 94q^{79} - 81q^{81} + 108q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 0 0 −4.50000 7.79423i 0
705.1 0 0 0 0 0 0 0 −4.50000 + 7.79423i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.3.s.a 2
4.b odd 2 1 49.3.d.a 2
7.b odd 2 1 CM 784.3.s.a 2
7.c even 3 1 112.3.c.a 1
7.c even 3 1 inner 784.3.s.a 2
7.d odd 6 1 112.3.c.a 1
7.d odd 6 1 inner 784.3.s.a 2
12.b even 2 1 441.3.m.a 2
21.g even 6 1 1008.3.f.a 1
21.h odd 6 1 1008.3.f.a 1
28.d even 2 1 49.3.d.a 2
28.f even 6 1 7.3.b.a 1
28.f even 6 1 49.3.d.a 2
28.g odd 6 1 7.3.b.a 1
28.g odd 6 1 49.3.d.a 2
56.j odd 6 1 448.3.c.b 1
56.k odd 6 1 448.3.c.a 1
56.m even 6 1 448.3.c.a 1
56.p even 6 1 448.3.c.b 1
84.h odd 2 1 441.3.m.a 2
84.j odd 6 1 63.3.d.a 1
84.j odd 6 1 441.3.m.a 2
84.n even 6 1 63.3.d.a 1
84.n even 6 1 441.3.m.a 2
140.p odd 6 1 175.3.d.a 1
140.s even 6 1 175.3.d.a 1
140.w even 12 2 175.3.c.a 2
140.x odd 12 2 175.3.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 28.f even 6 1
7.3.b.a 1 28.g odd 6 1
49.3.d.a 2 4.b odd 2 1
49.3.d.a 2 28.d even 2 1
49.3.d.a 2 28.f even 6 1
49.3.d.a 2 28.g odd 6 1
63.3.d.a 1 84.j odd 6 1
63.3.d.a 1 84.n even 6 1
112.3.c.a 1 7.c even 3 1
112.3.c.a 1 7.d odd 6 1
175.3.c.a 2 140.w even 12 2
175.3.c.a 2 140.x odd 12 2
175.3.d.a 1 140.p odd 6 1
175.3.d.a 1 140.s even 6 1
441.3.m.a 2 12.b even 2 1
441.3.m.a 2 84.h odd 2 1
441.3.m.a 2 84.j odd 6 1
441.3.m.a 2 84.n even 6 1
448.3.c.a 1 56.k odd 6 1
448.3.c.a 1 56.m even 6 1
448.3.c.b 1 56.j odd 6 1
448.3.c.b 1 56.p even 6 1
784.3.s.a 2 1.a even 1 1 trivial
784.3.s.a 2 7.b odd 2 1 CM
784.3.s.a 2 7.c even 3 1 inner
784.3.s.a 2 7.d odd 6 1 inner
1008.3.f.a 1 21.g even 6 1
1008.3.f.a 1 21.h odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 9 T^{2} )( 1 + 3 T + 9 T^{2} ) \)
$5$ \( ( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} ) \)
$7$ 1
$11$ \( 1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4} \)
$13$ \( ( 1 - 13 T )^{2}( 1 + 13 T )^{2} \)
$17$ \( ( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} ) \)
$19$ \( ( 1 - 19 T + 361 T^{2} )( 1 + 19 T + 361 T^{2} ) \)
$23$ \( 1 - 18 T - 205 T^{2} - 9522 T^{3} + 279841 T^{4} \)
$29$ \( ( 1 + 54 T + 841 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \)
$37$ \( 1 - 38 T + 75 T^{2} - 52022 T^{3} + 1874161 T^{4} \)
$41$ \( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \)
$43$ \( ( 1 + 58 T + 1849 T^{2} )^{2} \)
$47$ \( ( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} ) \)
$53$ \( 1 - 6 T - 2773 T^{2} - 16854 T^{3} + 7890481 T^{4} \)
$59$ \( ( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} ) \)
$61$ \( ( 1 - 61 T + 3721 T^{2} )( 1 + 61 T + 3721 T^{2} ) \)
$67$ \( 1 + 118 T + 9435 T^{2} + 529702 T^{3} + 20151121 T^{4} \)
$71$ \( ( 1 + 114 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 - 73 T + 5329 T^{2} )( 1 + 73 T + 5329 T^{2} ) \)
$79$ \( 1 + 94 T + 2595 T^{2} + 586654 T^{3} + 38950081 T^{4} \)
$83$ \( ( 1 - 83 T )^{2}( 1 + 83 T )^{2} \)
$89$ \( ( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} ) \)
$97$ \( ( 1 - 97 T )^{2}( 1 + 97 T )^{2} \)
show more
show less