Properties

Label 1764.3.c.h
Level $1764$
Weight $3$
Character orbit 1764.c
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.224054542336.12
Defining polynomial: \(x^{8} + 199 x^{4} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{5} q^{11} + 3 \beta_{3} q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( 5 \beta_{3} + \beta_{7} ) q^{19} + ( -2 \beta_{4} - \beta_{5} ) q^{23} + ( -27 - \beta_{6} ) q^{25} + ( -4 \beta_{4} - \beta_{5} ) q^{29} + ( 3 \beta_{3} + \beta_{7} ) q^{31} + ( 16 + \beta_{6} ) q^{37} + ( 8 \beta_{1} + \beta_{2} ) q^{41} + ( -10 + \beta_{6} ) q^{43} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -9 \beta_{4} + 4 \beta_{5} ) q^{53} + ( 43 \beta_{3} + \beta_{7} ) q^{55} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -14 \beta_{3} - \beta_{7} ) q^{61} + ( -3 \beta_{4} - 3 \beta_{5} ) q^{65} + ( -48 + \beta_{6} ) q^{67} + ( -16 \beta_{4} - 3 \beta_{5} ) q^{71} + ( -44 \beta_{3} - \beta_{7} ) q^{73} + ( -70 - 2 \beta_{6} ) q^{79} + ( 18 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -70 - 2 \beta_{6} ) q^{85} + ( -8 \beta_{1} - 13 \beta_{2} ) q^{89} + ( -48 \beta_{4} - 14 \beta_{5} ) q^{95} + ( 50 \beta_{3} - 3 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 216q^{25} + 128q^{37} - 80q^{43} - 384q^{67} - 560q^{79} - 560q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 199 x^{4} + 14641\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{6} - 640 \nu^{2} \)\()/847\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{6} + 242 \nu^{4} + 960 \nu^{2} + 24079 \)\()/2541\)
\(\beta_{3}\)\(=\)\((\)\( 10 \nu^{7} + 121 \nu^{5} + 659 \nu^{3} + 10769 \nu \)\()/27951\)
\(\beta_{4}\)\(=\)\((\)\( -10 \nu^{7} + 121 \nu^{5} - 659 \nu^{3} + 10769 \nu \)\()/9317\)
\(\beta_{5}\)\(=\)\((\)\( 32 \nu^{7} + 121 \nu^{5} + 7699 \nu^{3} + 66671 \nu \)\()/27951\)
\(\beta_{6}\)\(=\)\((\)\( -6 \nu^{6} - 468 \nu^{2} \)\()/121\)
\(\beta_{7}\)\(=\)\((\)\( 32 \nu^{7} - 121 \nu^{5} + 7699 \nu^{3} - 66671 \nu \)\()/9317\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 21 \beta_{1}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 15 \beta_{5} + 16 \beta_{4} - 48 \beta_{3}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(42 \beta_{2} + 21 \beta_{1} - 398\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(89 \beta_{7} - 267 \beta_{5} + 551 \beta_{4} + 1653 \beta_{3}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-160 \beta_{6} + 819 \beta_{1}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-659 \beta_{7} - 1977 \beta_{5} - 7699 \beta_{4} + 23097 \beta_{3}\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
2.67196 1.96485i
−2.67196 + 1.96485i
−1.96485 2.67196i
1.96485 + 2.67196i
1.96485 2.67196i
−1.96485 + 2.67196i
−2.67196 1.96485i
2.67196 + 1.96485i
0 0 0 9.55744i 0 0 0 0 0
197.2 0 0 0 9.55744i 0 0 0 0 0
197.3 0 0 0 3.55744i 0 0 0 0 0
197.4 0 0 0 3.55744i 0 0 0 0 0
197.5 0 0 0 3.55744i 0 0 0 0 0
197.6 0 0 0 3.55744i 0 0 0 0 0
197.7 0 0 0 9.55744i 0 0 0 0 0
197.8 0 0 0 9.55744i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.c.h 8
3.b odd 2 1 inner 1764.3.c.h 8
7.b odd 2 1 inner 1764.3.c.h 8
7.c even 3 2 1764.3.bk.g 16
7.d odd 6 2 1764.3.bk.g 16
21.c even 2 1 inner 1764.3.c.h 8
21.g even 6 2 1764.3.bk.g 16
21.h odd 6 2 1764.3.bk.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.c.h 8 1.a even 1 1 trivial
1764.3.c.h 8 3.b odd 2 1 inner
1764.3.c.h 8 7.b odd 2 1 inner
1764.3.c.h 8 21.c even 2 1 inner
1764.3.bk.g 16 7.c even 3 2
1764.3.bk.g 16 7.d odd 6 2
1764.3.bk.g 16 21.g even 6 2
1764.3.bk.g 16 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 104 T_{5}^{2} + 1156 \)
\( T_{13}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1156 + 104 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 86 + T^{2} )^{4} \)
$13$ \( ( -18 + T^{2} )^{4} \)
$17$ \( ( 1444 + 248 T^{2} + T^{4} )^{2} \)
$19$ \( ( 524176 - 1648 T^{2} + T^{4} )^{2} \)
$23$ \( ( 196 + 316 T^{2} + T^{4} )^{2} \)
$29$ \( ( 40804 + 748 T^{2} + T^{4} )^{2} \)
$31$ \( ( 571536 - 1584 T^{2} + T^{4} )^{2} \)
$37$ \( ( -1292 - 32 T + T^{2} )^{4} \)
$41$ \( ( 3928324 + 4136 T^{2} + T^{4} )^{2} \)
$43$ \( ( -1448 + 20 T + T^{2} )^{4} \)
$47$ \( ( 1157776 + 4904 T^{2} + T^{4} )^{2} \)
$53$ \( ( 6724 + 5668 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1157776 + 4904 T^{2} + T^{4} )^{2} \)
$61$ \( ( 145924 - 2332 T^{2} + T^{4} )^{2} \)
$67$ \( ( 756 + 96 T + T^{2} )^{4} \)
$71$ \( ( 14699556 + 10764 T^{2} + T^{4} )^{2} \)
$73$ \( ( 9597604 - 9292 T^{2} + T^{4} )^{2} \)
$79$ \( ( -1292 + 140 T + T^{2} )^{4} \)
$83$ \( ( 104693824 + 21152 T^{2} + T^{4} )^{2} \)
$89$ \( ( 51638596 + 14696 T^{2} + T^{4} )^{2} \)
$97$ \( ( 3865156 - 23932 T^{2} + T^{4} )^{2} \)
show more
show less