Properties

Label 1764.4.a.p
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{5} +O(q^{10})\) \( q + ( -2 - \beta ) q^{5} + ( 26 + \beta ) q^{11} + ( -33 - 5 \beta ) q^{13} + ( 16 + 8 \beta ) q^{17} + ( -85 - \beta ) q^{19} + 96 q^{23} + ( 7 + 3 \beta ) q^{25} + ( 28 + 17 \beta ) q^{29} + ( 51 + 10 \beta ) q^{31} + ( -77 + 19 \beta ) q^{37} + ( 70 - 34 \beta ) q^{41} + ( -239 + 19 \beta ) q^{43} + ( 98 + 16 \beta ) q^{47} + ( -156 - 27 \beta ) q^{53} + ( -180 - 27 \beta ) q^{55} + ( -636 - 3 \beta ) q^{59} + ( -146 + 36 \beta ) q^{61} + ( 706 + 38 \beta ) q^{65} + ( -433 + 9 \beta ) q^{67} + ( 686 - 32 \beta ) q^{71} + ( 691 + 21 \beta ) q^{73} + ( -93 - 4 \beta ) q^{79} + ( -178 + 43 \beta ) q^{83} + ( -1056 - 24 \beta ) q^{85} + ( 400 - 22 \beta ) q^{89} + ( 298 + 86 \beta ) q^{95} + ( -410 + 21 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + O(q^{10}) \) \( 2q - 3q^{5} + 51q^{11} - 61q^{13} + 24q^{17} - 169q^{19} + 192q^{23} + 11q^{25} + 39q^{29} + 92q^{31} - 173q^{37} + 174q^{41} - 497q^{43} + 180q^{47} - 285q^{53} - 333q^{55} - 1269q^{59} - 328q^{61} + 1374q^{65} - 875q^{67} + 1404q^{71} + 1361q^{73} - 182q^{79} - 399q^{83} - 2088q^{85} + 822q^{89} + 510q^{95} - 841q^{97} + O(q^{100}) \)

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
4.27492
−3.27492
0 0 0 −12.8248 0 0 0 0 0
1.2 0 0 0 9.82475 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.p 2
3.b odd 2 1 588.4.a.h 2
7.b odd 2 1 1764.4.a.x 2
7.c even 3 2 1764.4.k.z 4
7.d odd 6 2 252.4.k.d 4
12.b even 2 1 2352.4.a.bp 2
21.c even 2 1 588.4.a.g 2
21.g even 6 2 84.4.i.b 4
21.h odd 6 2 588.4.i.i 4
84.h odd 2 1 2352.4.a.cb 2
84.j odd 6 2 336.4.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 21.g even 6 2
252.4.k.d 4 7.d odd 6 2
336.4.q.h 4 84.j odd 6 2
588.4.a.g 2 21.c even 2 1
588.4.a.h 2 3.b odd 2 1
588.4.i.i 4 21.h odd 6 2
1764.4.a.p 2 1.a even 1 1 trivial
1764.4.a.x 2 7.b odd 2 1
1764.4.k.z 4 7.c even 3 2
2352.4.a.bp 2 12.b even 2 1
2352.4.a.cb 2 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} - 126 \)
\( T_{11}^{2} - 51 T_{11} + 522 \)
\( T_{13}^{2} + 61 T_{13} - 2276 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -126 + 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 522 - 51 T + T^{2} \)
$13$ \( -2276 + 61 T + T^{2} \)
$17$ \( -8064 - 24 T + T^{2} \)
$19$ \( 7012 + 169 T + T^{2} \)
$23$ \( ( -96 + T )^{2} \)
$29$ \( -36684 - 39 T + T^{2} \)
$31$ \( -10709 - 92 T + T^{2} \)
$37$ \( -38816 + 173 T + T^{2} \)
$41$ \( -140688 - 174 T + T^{2} \)
$43$ \( 15454 + 497 T + T^{2} \)
$47$ \( -24732 - 180 T + T^{2} \)
$53$ \( -73188 + 285 T + T^{2} \)
$59$ \( 401436 + 1269 T + T^{2} \)
$61$ \( -139316 + 328 T + T^{2} \)
$67$ \( 181018 + 875 T + T^{2} \)
$71$ \( 361476 - 1404 T + T^{2} \)
$73$ \( 406522 - 1361 T + T^{2} \)
$79$ \( 6229 + 182 T + T^{2} \)
$83$ \( -197334 + 399 T + T^{2} \)
$89$ \( 106848 - 822 T + T^{2} \)
$97$ \( 120262 + 841 T + T^{2} \)
show more
show less