Properties

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

Related objects

Downloads

Learn more about

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 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 14q^{5} + O(q^{10}) \) \( q + 14q^{5} - 4q^{11} - 54q^{13} - 14q^{17} - 92q^{19} + 152q^{23} + 71q^{25} + 106q^{29} + 144q^{31} + 158q^{37} - 390q^{41} - 508q^{43} - 528q^{47} - 606q^{53} - 56q^{55} - 364q^{59} - 678q^{61} - 756q^{65} + 844q^{67} + 8q^{71} + 422q^{73} + 384q^{79} - 548q^{83} - 196q^{85} + 1194q^{89} - 1288q^{95} + 1502q^{97} + O(q^{100}) \)

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 14.0000 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.l 1
3.b odd 2 1 588.4.a.a 1
7.b odd 2 1 252.4.a.a 1
7.c even 3 2 1764.4.k.c 2
7.d odd 6 2 1764.4.k.n 2
12.b even 2 1 2352.4.a.v 1
21.c even 2 1 84.4.a.b 1
21.g even 6 2 588.4.i.a 2
21.h odd 6 2 588.4.i.h 2
28.d even 2 1 1008.4.a.d 1
84.h odd 2 1 336.4.a.e 1
105.g even 2 1 2100.4.a.g 1
105.k odd 4 2 2100.4.k.g 2
168.e odd 2 1 1344.4.a.p 1
168.i even 2 1 1344.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.c even 2 1
252.4.a.a 1 7.b odd 2 1
336.4.a.e 1 84.h odd 2 1
588.4.a.a 1 3.b odd 2 1
588.4.i.a 2 21.g even 6 2
588.4.i.h 2 21.h odd 6 2
1008.4.a.d 1 28.d even 2 1
1344.4.a.b 1 168.i even 2 1
1344.4.a.p 1 168.e odd 2 1
1764.4.a.l 1 1.a even 1 1 trivial
1764.4.k.c 2 7.c even 3 2
1764.4.k.n 2 7.d odd 6 2
2100.4.a.g 1 105.g even 2 1
2100.4.k.g 2 105.k odd 4 2
2352.4.a.v 1 12.b even 2 1

Hecke kernels

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

\( T_{5} - 14 \)
\( T_{11} + 4 \)
\( T_{13} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -14 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 54 + T \)
$17$ \( 14 + T \)
$19$ \( 92 + T \)
$23$ \( -152 + T \)
$29$ \( -106 + T \)
$31$ \( -144 + T \)
$37$ \( -158 + T \)
$41$ \( 390 + T \)
$43$ \( 508 + T \)
$47$ \( 528 + T \)
$53$ \( 606 + T \)
$59$ \( 364 + T \)
$61$ \( 678 + T \)
$67$ \( -844 + T \)
$71$ \( -8 + T \)
$73$ \( -422 + T \)
$79$ \( -384 + T \)
$83$ \( 548 + T \)
$89$ \( -1194 + T \)
$97$ \( -1502 + T \)
show more
show less