Properties

Label 1764.4.a.k
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(104.079369250\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{5} + 12 q^{11} + 82 q^{13} - 30 q^{17} - 68 q^{19} - 216 q^{23} - 89 q^{25} - 246 q^{29} + 112 q^{31} + 110 q^{37} - 246 q^{41} - 172 q^{43} + 192 q^{47} - 558 q^{53} + 72 q^{55} + 540 q^{59} - 110 q^{61} + 492 q^{65} + 140 q^{67} + 840 q^{71} + 550 q^{73} - 208 q^{79} + 516 q^{83} - 180 q^{85} - 1398 q^{89} - 408 q^{95} - 1586 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 6.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.a.k 1
3.b odd 2 1 196.4.a.b 1
7.b odd 2 1 252.4.a.c 1
7.c even 3 2 1764.4.k.e 2
7.d odd 6 2 1764.4.k.k 2
12.b even 2 1 784.4.a.n 1
21.c even 2 1 28.4.a.b 1
21.g even 6 2 196.4.e.c 2
21.h odd 6 2 196.4.e.d 2
28.d even 2 1 1008.4.a.f 1
84.h odd 2 1 112.4.a.c 1
105.g even 2 1 700.4.a.e 1
105.k odd 4 2 700.4.e.f 2
168.e odd 2 1 448.4.a.m 1
168.i even 2 1 448.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 21.c even 2 1
112.4.a.c 1 84.h odd 2 1
196.4.a.b 1 3.b odd 2 1
196.4.e.c 2 21.g even 6 2
196.4.e.d 2 21.h odd 6 2
252.4.a.c 1 7.b odd 2 1
448.4.a.d 1 168.i even 2 1
448.4.a.m 1 168.e odd 2 1
700.4.a.e 1 105.g even 2 1
700.4.e.f 2 105.k odd 4 2
784.4.a.n 1 12.b even 2 1
1008.4.a.f 1 28.d even 2 1
1764.4.a.k 1 1.a even 1 1 trivial
1764.4.k.e 2 7.c even 3 2
1764.4.k.k 2 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{11} - 12 \) Copy content Toggle raw display
\( T_{13} - 82 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T - 82 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T + 68 \) Copy content Toggle raw display
$23$ \( T + 216 \) Copy content Toggle raw display
$29$ \( T + 246 \) Copy content Toggle raw display
$31$ \( T - 112 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T + 246 \) Copy content Toggle raw display
$43$ \( T + 172 \) Copy content Toggle raw display
$47$ \( T - 192 \) Copy content Toggle raw display
$53$ \( T + 558 \) Copy content Toggle raw display
$59$ \( T - 540 \) Copy content Toggle raw display
$61$ \( T + 110 \) Copy content Toggle raw display
$67$ \( T - 140 \) Copy content Toggle raw display
$71$ \( T - 840 \) Copy content Toggle raw display
$73$ \( T - 550 \) Copy content Toggle raw display
$79$ \( T + 208 \) Copy content Toggle raw display
$83$ \( T - 516 \) Copy content Toggle raw display
$89$ \( T + 1398 \) Copy content Toggle raw display
$97$ \( T + 1586 \) Copy content Toggle raw display
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