Properties

Label 1764.3.bk.d
Level $1764$
Weight $3$
Character orbit 1764.bk
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.bk (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 2 \beta_{5} - \beta_{7} ) q^{11} + ( -2 - 2 \beta_{3} ) q^{13} + ( 5 \beta_{5} - 2 \beta_{7} ) q^{17} + ( 10 \beta_{2} - 3 \beta_{6} ) q^{19} + ( 2 \beta_{1} - 3 \beta_{4} ) q^{23} + ( -9 + 9 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{25} + ( -10 \beta_{1} - \beta_{4} - 10 \beta_{5} - \beta_{7} ) q^{29} + ( -10 + 10 \beta_{2} - \beta_{3} - \beta_{6} ) q^{31} + ( -8 \beta_{2} - 6 \beta_{6} ) q^{37} + ( -3 \beta_{1} - 3 \beta_{5} ) q^{41} + ( -4 + 6 \beta_{3} ) q^{43} + ( -6 \beta_{1} + 14 \beta_{4} ) q^{47} -11 \beta_{7} q^{53} + ( -26 - \beta_{3} ) q^{55} + ( -14 \beta_{5} - 2 \beta_{7} ) q^{59} + ( 68 \beta_{2} + \beta_{6} ) q^{61} + ( -6 \beta_{1} - 8 \beta_{4} ) q^{65} + ( -42 + 42 \beta_{2} - 14 \beta_{3} - 14 \beta_{6} ) q^{67} + ( 22 \beta_{1} - 15 \beta_{4} + 22 \beta_{5} - 15 \beta_{7} ) q^{71} + ( -36 + 36 \beta_{2} + \beta_{3} + \beta_{6} ) q^{73} + ( -58 \beta_{2} + 16 \beta_{6} ) q^{79} + ( 22 \beta_{1} - 20 \beta_{4} + 22 \beta_{5} - 20 \beta_{7} ) q^{83} + ( -68 - 4 \beta_{3} ) q^{85} + ( 3 \beta_{1} + 36 \beta_{4} ) q^{89} + ( -16 \beta_{5} - 12 \beta_{7} ) q^{95} + ( 24 + 17 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{13} + 40q^{19} - 36q^{25} - 40q^{31} - 32q^{37} - 32q^{43} - 208q^{55} + 272q^{61} - 168q^{67} - 144q^{73} - 232q^{79} - 544q^{85} + 192q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{6} - 296 \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 203 \nu \)\()/55\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{7} - 110 \nu^{5} + 880 \nu^{3} - 1152 \nu \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( -46 \nu^{6} + 440 \nu^{4} - 2530 \nu^{2} + 3312 \)\()/495\)
\(\beta_{7}\)\(=\)\((\)\( -31 \nu^{7} + 275 \nu^{5} - 1705 \nu^{3} + 2232 \nu \)\()/495\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{3} - 8 \beta_{2} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + 5 \beta_{5} + 2 \beta_{4} + 5 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(4 \beta_{6} - 23 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{7} + 31 \beta_{5}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-55 \beta_{3} - 296\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-110 \beta_{4} - 203 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−2.23256 + 1.28897i
−1.00781 + 0.581861i
1.00781 0.581861i
2.23256 1.28897i
−2.23256 1.28897i
−1.00781 0.581861i
1.00781 + 0.581861i
2.23256 + 1.28897i
0 0 0 −4.46512 + 2.57794i 0 0 0 0 0
557.2 0 0 0 −2.01563 + 1.16372i 0 0 0 0 0
557.3 0 0 0 2.01563 1.16372i 0 0 0 0 0
557.4 0 0 0 4.46512 2.57794i 0 0 0 0 0
1745.1 0 0 0 −4.46512 2.57794i 0 0 0 0 0
1745.2 0 0 0 −2.01563 1.16372i 0 0 0 0 0
1745.3 0 0 0 2.01563 + 1.16372i 0 0 0 0 0
1745.4 0 0 0 4.46512 + 2.57794i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1745.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.bk.d 8
3.b odd 2 1 inner 1764.3.bk.d 8
7.b odd 2 1 1764.3.bk.e 8
7.c even 3 1 252.3.c.a 4
7.c even 3 1 inner 1764.3.bk.d 8
7.d odd 6 1 1764.3.c.f 4
7.d odd 6 1 1764.3.bk.e 8
21.c even 2 1 1764.3.bk.e 8
21.g even 6 1 1764.3.c.f 4
21.g even 6 1 1764.3.bk.e 8
21.h odd 6 1 252.3.c.a 4
21.h odd 6 1 inner 1764.3.bk.d 8
28.g odd 6 1 1008.3.d.c 4
56.k odd 6 1 4032.3.d.e 4
56.p even 6 1 4032.3.d.h 4
63.g even 3 1 2268.3.bg.c 8
63.h even 3 1 2268.3.bg.c 8
63.j odd 6 1 2268.3.bg.c 8
63.n odd 6 1 2268.3.bg.c 8
84.n even 6 1 1008.3.d.c 4
168.s odd 6 1 4032.3.d.h 4
168.v even 6 1 4032.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.c.a 4 7.c even 3 1
252.3.c.a 4 21.h odd 6 1
1008.3.d.c 4 28.g odd 6 1
1008.3.d.c 4 84.n even 6 1
1764.3.c.f 4 7.d odd 6 1
1764.3.c.f 4 21.g even 6 1
1764.3.bk.d 8 1.a even 1 1 trivial
1764.3.bk.d 8 3.b odd 2 1 inner
1764.3.bk.d 8 7.c even 3 1 inner
1764.3.bk.d 8 21.h odd 6 1 inner
1764.3.bk.e 8 7.b odd 2 1
1764.3.bk.e 8 7.d odd 6 1
1764.3.bk.e 8 21.c even 2 1
1764.3.bk.e 8 21.g even 6 1
2268.3.bg.c 8 63.g even 3 1
2268.3.bg.c 8 63.h even 3 1
2268.3.bg.c 8 63.j odd 6 1
2268.3.bg.c 8 63.n odd 6 1
4032.3.d.e 4 56.k odd 6 1
4032.3.d.e 4 168.v even 6 1
4032.3.d.h 4 56.p even 6 1
4032.3.d.h 4 168.s odd 6 1

Hecke kernels

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

\( T_{5}^{8} - 32 T_{5}^{6} + 880 T_{5}^{4} - 4608 T_{5}^{2} + 20736 \)
\( T_{13}^{2} + 4 T_{13} - 108 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 20736 - 4608 T^{2} + 880 T^{4} - 32 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 8503056 - 338256 T^{2} + 10540 T^{4} - 116 T^{6} + T^{8} \)
$13$ \( ( -108 + 4 T + T^{2} )^{4} \)
$17$ \( 14666178816 - 85257216 T^{2} + 374512 T^{4} - 704 T^{6} + T^{8} \)
$19$ \( ( 23104 + 3040 T + 552 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$23$ \( 3111696 - 543312 T^{2} + 93100 T^{4} - 308 T^{6} + T^{8} \)
$29$ \( ( 1127844 + 3476 T^{2} + T^{4} )^{2} \)
$31$ \( ( 5184 + 1440 T + 328 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$37$ \( ( 891136 - 15104 T + 1200 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$41$ \( ( 11664 + 288 T^{2} + T^{4} )^{2} \)
$43$ \( ( -992 + 8 T + T^{2} )^{4} \)
$47$ \( 19007367745536 - 26995534848 T^{2} + 33981120 T^{4} - 6192 T^{6} + T^{8} \)
$53$ \( ( 4743684 - 2178 T^{2} + T^{4} )^{2} \)
$59$ \( 14281868906496 - 26786515968 T^{2} + 46460608 T^{4} - 7088 T^{6} + T^{8} \)
$61$ \( ( 21123216 - 625056 T + 13900 T^{2} - 136 T^{3} + T^{4} )^{2} \)
$67$ \( ( 13868176 - 312816 T + 10780 T^{2} + 84 T^{3} + T^{4} )^{2} \)
$71$ \( ( 32695524 + 15668 T^{2} + T^{4} )^{2} \)
$73$ \( ( 1607824 + 91296 T + 3916 T^{2} + 72 T^{3} + T^{4} )^{2} \)
$79$ \( ( 14470416 - 441264 T + 17260 T^{2} + 116 T^{3} + T^{4} )^{2} \)
$83$ \( ( 15116544 + 19328 T^{2} + T^{4} )^{2} \)
$89$ \( 361242176885473536 - 29772832905216 T^{2} + 1852781040 T^{4} - 49536 T^{6} + T^{8} \)
$97$ \( ( -7516 - 48 T + T^{2} )^{4} \)
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