Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3288334336.1
Defining polynomial: \(x^{8} + 12 x^{6} + 30 x^{4} + 12 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{3} q^{11} + \beta_{6} q^{13} + 3 \beta_{1} q^{17} + ( \beta_{5} - 2 \beta_{6} ) q^{19} + ( -\beta_{3} - \beta_{4} ) q^{23} + ( -3 + \beta_{2} ) q^{25} + ( \beta_{3} - \beta_{4} ) q^{29} + ( 5 \beta_{5} + 4 \beta_{6} ) q^{31} + ( 16 + 3 \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{7} ) q^{41} + ( 22 - 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + \beta_{7} ) q^{47} + 2 \beta_{3} q^{53} + ( -\beta_{5} + 18 \beta_{6} ) q^{55} + ( -2 \beta_{1} - \beta_{7} ) q^{59} + ( 12 \beta_{5} - 13 \beta_{6} ) q^{61} + \beta_{3} q^{65} + ( -32 + 6 \beta_{2} ) q^{67} + ( \beta_{3} + 2 \beta_{4} ) q^{71} + ( 22 \beta_{5} + 21 \beta_{6} ) q^{73} + ( 54 + 8 \beta_{2} ) q^{79} + ( 2 \beta_{1} - 2 \beta_{7} ) q^{83} + ( 84 - 3 \beta_{2} ) q^{85} + ( 9 \beta_{1} - 2 \beta_{7} ) q^{89} + ( -2 \beta_{3} + \beta_{4} ) q^{95} + ( 2 \beta_{5} + 27 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} + 128q^{37} + 176q^{43} - 256q^{67} + 432q^{79} + 672q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 12 x^{6} + 30 x^{4} + 12 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 13 \nu^{5} + 43 \nu^{3} + 55 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} + 84 \nu^{4} + 203 \nu^{2} + 42 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{6} - 71 \nu^{4} - 174 \nu^{2} - 57 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -17 \nu^{6} - 193 \nu^{4} - 395 \nu^{2} - 39 \)\()/2\)
\(\beta_{5}\)\(=\)\( -2 \nu^{7} - 23 \nu^{5} - 49 \nu^{3} - 4 \nu \)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{7} - 107 \nu^{5} - 259 \nu^{3} - 81 \nu \)\()/4\)
\(\beta_{7}\)\(=\)\( 43 \nu^{7} + 510 \nu^{5} + 1212 \nu^{3} + 307 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 28 \beta_{6} - 7 \beta_{5} + 24 \beta_{1}\)\()/98\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - 11 \beta_{3} - 7 \beta_{2} - 147\)\()/49\)
\(\nu^{3}\)\(=\)\((\)\(-13 \beta_{7} - 280 \beta_{6} + 21 \beta_{5} - 116 \beta_{1}\)\()/98\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{3} + 12 \beta_{2} + 147\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(125 \beta_{7} + 2464 \beta_{6} + 21 \beta_{5} + 844 \beta_{1}\)\()/98\)
\(\nu^{6}\)\(=\)\((\)\(-29 \beta_{4} - 857 \beta_{3} - 791 \beta_{2} - 8379\)\()/49\)
\(\nu^{7}\)\(=\)\((\)\(-1121 \beta_{7} - 21532 \beta_{6} - 791 \beta_{5} - 6912 \beta_{1}\)\()/98\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.67825i
0.595859i
2.95189i
0.338766i
2.95189i
0.338766i
1.67825i
0.595859i
0 0 0 6.15626i 0 0 0 0 0
685.2 0 0 0 6.15626i 0 0 0 0 0
685.3 0 0 0 4.25447i 0 0 0 0 0
685.4 0 0 0 4.25447i 0 0 0 0 0
685.5 0 0 0 4.25447i 0 0 0 0 0
685.6 0 0 0 4.25447i 0 0 0 0 0
685.7 0 0 0 6.15626i 0 0 0 0 0
685.8 0 0 0 6.15626i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.8
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.d.g 8
3.b odd 2 1 inner 1764.3.d.g 8
7.b odd 2 1 inner 1764.3.d.g 8
7.c even 3 2 1764.3.z.n 16
7.d odd 6 2 1764.3.z.n 16
21.c even 2 1 inner 1764.3.d.g 8
21.g even 6 2 1764.3.z.n 16
21.h odd 6 2 1764.3.z.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.d.g 8 1.a even 1 1 trivial
1764.3.d.g 8 3.b odd 2 1 inner
1764.3.d.g 8 7.b odd 2 1 inner
1764.3.d.g 8 21.c even 2 1 inner
1764.3.z.n 16 7.c even 3 2
1764.3.z.n 16 7.d odd 6 2
1764.3.z.n 16 21.g even 6 2
1764.3.z.n 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 \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 56 T_{5}^{2} + 686 \)
\( T_{11}^{4} - 364 T_{11}^{2} + 1372 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 686 + 56 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 1372 - 364 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2 + 20 T^{2} + T^{4} )^{2} \)
$17$ \( ( 55566 + 504 T^{2} + T^{4} )^{2} \)
$19$ \( ( 4232 + 136 T^{2} + T^{4} )^{2} \)
$23$ \( ( 396508 - 1652 T^{2} + T^{4} )^{2} \)
$29$ \( ( 725788 - 2100 T^{2} + T^{4} )^{2} \)
$31$ \( ( 223112 + 1160 T^{2} + T^{4} )^{2} \)
$37$ \( ( -626 - 32 T + T^{2} )^{4} \)
$41$ \( ( 3655694 + 4928 T^{2} + T^{4} )^{2} \)
$43$ \( ( 92 - 44 T + T^{2} )^{4} \)
$47$ \( ( 6061496 + 4928 T^{2} + T^{4} )^{2} \)
$53$ \( ( 21952 - 1456 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2636984 + 5152 T^{2} + T^{4} )^{2} \)
$61$ \( ( 25589858 + 10388 T^{2} + T^{4} )^{2} \)
$67$ \( ( -2504 + 64 T + T^{2} )^{4} \)
$71$ \( ( 1318492 - 5964 T^{2} + T^{4} )^{2} \)
$73$ \( ( 122649122 + 24484 T^{2} + T^{4} )^{2} \)
$79$ \( ( -3356 - 108 T + T^{2} )^{4} \)
$83$ \( ( 86940896 + 19264 T^{2} + T^{4} )^{2} \)
$89$ \( ( 104876366 + 22792 T^{2} + T^{4} )^{2} \)
$97$ \( ( 235298 + 14308 T^{2} + T^{4} )^{2} \)
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