Properties

Label 1764.3.bk.g
Level $1764$
Weight $3$
Character orbit 1764.bk
Analytic conductor $48.066$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.329365073333488765586374656.1
Defining polynomial: \(x^{16} - 199 x^{12} + 24960 x^{8} - 2913559 x^{4} + 214358881\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} -\beta_{13} q^{11} + 3 \beta_{9} q^{13} + ( \beta_{3} - \beta_{4} ) q^{17} + ( 5 \beta_{2} - 5 \beta_{9} + \beta_{14} - \beta_{15} ) q^{19} + ( -2 \beta_{8} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{23} + ( 27 \beta_{1} + \beta_{10} ) q^{25} + ( -\beta_{11} + 4 \beta_{12} ) q^{29} + ( -3 \beta_{2} + \beta_{15} ) q^{31} + ( -16 + 16 \beta_{1} - \beta_{5} ) q^{37} + ( -7 \beta_{4} + \beta_{6} + 7 \beta_{7} ) q^{41} + ( -10 + \beta_{5} + \beta_{10} ) q^{43} + ( -4 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} + 9 \beta_{7} ) q^{47} + ( 9 \beta_{8} - 4 \beta_{13} ) q^{53} + ( 43 \beta_{9} - \beta_{14} ) q^{55} + ( 4 \beta_{3} - 5 \beta_{4} ) q^{59} + ( -14 \beta_{2} + 14 \beta_{9} - \beta_{14} + \beta_{15} ) q^{61} + ( -3 \beta_{8} + 3 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{65} + ( 48 \beta_{1} - \beta_{10} ) q^{67} + ( -3 \beta_{11} + 16 \beta_{12} ) q^{71} + ( 44 \beta_{2} - \beta_{15} ) q^{73} + ( 70 - 70 \beta_{1} + 2 \beta_{5} ) q^{79} + ( -16 \beta_{4} + 2 \beta_{6} + 16 \beta_{7} ) q^{83} + ( -70 - 2 \beta_{5} - 2 \beta_{10} ) q^{85} + ( 13 \beta_{3} + 13 \beta_{4} + 13 \beta_{6} - 5 \beta_{7} ) q^{89} + ( 48 \beta_{8} + 14 \beta_{13} ) q^{95} + ( 50 \beta_{9} + 3 \beta_{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 216q^{25} - 128q^{37} - 160q^{43} + 384q^{67} + 560q^{79} - 1120q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 199 x^{12} + 24960 x^{8} - 2913559 x^{4} + 214358881\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -199 \nu^{12} + 24960 \nu^{8} - 4967040 \nu^{4} + 579798241 \)\()/ 365439360 \)
\(\beta_{2}\)\(=\)\((\)\( 15269 \nu^{15} + 2143031 \nu^{13} + 16448640 \nu^{11} - 268794240 \nu^{9} + 381114240 \nu^{7} + 9271891200 \nu^{5} - 44487132371 \nu^{3} - 2308430789489 \nu \)\()/ 10214395551360 \)
\(\beta_{3}\)\(=\)\((\)\( -363 \nu^{14} - 10319 \nu^{12} + 4967040 \nu^{8} - 257562240 \nu^{4} - 1841731683 \nu^{2} + 30065015321 \)\()/ 7674226560 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} + 5073641 \nu^{2} \)\()/10570560\)
\(\beta_{5}\)\(=\)\((\)\( 881 \nu^{14} - 1946880 \nu^{10} + 21989760 \nu^{6} - 2566845479 \nu^{2} \)\()/ 7369693760 \)
\(\beta_{6}\)\(=\)\((\)\( -95520 \nu^{14} - 1771561 \nu^{12} + 11980800 \nu^{10} - 1836020160 \nu^{6} + 102892262880 \nu^{2} + 761840320879 \)\()/ 464290706880 \)
\(\beta_{7}\)\(=\)\((\)\( -49039 \nu^{14} + 7987200 \nu^{10} - 1224013440 \nu^{6} + 142878019801 \nu^{2} \)\()/ 154763568960 \)
\(\beta_{8}\)\(=\)\((\)\( 15269 \nu^{15} - 2143031 \nu^{13} + 16448640 \nu^{11} + 268794240 \nu^{9} + 381114240 \nu^{7} - 9271891200 \nu^{5} - 44487132371 \nu^{3} + 2308430789489 \nu \)\()/ 3404798517120 \)
\(\beta_{9}\)\(=\)\((\)\( 121 \nu^{15} + 307 \nu^{13} - 222144 \nu^{9} + 7662720 \nu^{5} - 153512095 \nu^{3} - 894462613 \nu \)\()/ 8441649216 \)
\(\beta_{10}\)\(=\)\((\)\( \nu^{14} - 966679 \nu^{2} \)\()/503360\)
\(\beta_{11}\)\(=\)\((\)\( -121 \nu^{15} - 2969 \nu^{13} + 429780 \nu^{9} - 74106240 \nu^{5} - 374090981 \nu^{3} + 8650356671 \nu \)\()/ 2638015380 \)
\(\beta_{12}\)\(=\)\((\)\( -121 \nu^{15} + 307 \nu^{13} - 222144 \nu^{9} + 7662720 \nu^{5} + 153512095 \nu^{3} - 894462613 \nu \)\()/ 2813883072 \)
\(\beta_{13}\)\(=\)\((\)\(1063589 \nu^{15} - 13267529 \nu^{13} - 192167040 \nu^{11} + 1664108160 \nu^{9} + 26547181440 \nu^{7} - 286939361280 \nu^{5} - 3098829303251 \nu^{3} + 14291520955151 \nu\)\()/ 10214395551360 \)
\(\beta_{14}\)\(=\)\((\)\( -121 \nu^{15} + 2969 \nu^{13} - 429780 \nu^{9} + 74106240 \nu^{5} - 374090981 \nu^{3} - 8650356671 \nu \)\()/ 879338460 \)
\(\beta_{15}\)\(=\)\((\)\(1063589 \nu^{15} + 13267529 \nu^{13} - 192167040 \nu^{11} - 1664108160 \nu^{9} + 26547181440 \nu^{7} + 286939361280 \nu^{5} - 3098829303251 \nu^{3} - 14291520955151 \nu\)\()/ 3404798517120 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{14} - 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{9} + \beta_{8} - 3 \beta_{2}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} + 21 \beta_{4}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{14} + 16 \beta_{12} - 15 \beta_{11} - 48 \beta_{9}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-21 \beta_{7} + 42 \beta_{6} + 42 \beta_{4} + 42 \beta_{3} - 398 \beta_{1} + 398\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(89 \beta_{15} - 267 \beta_{13} + 551 \beta_{8} - 1653 \beta_{2}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-160 \beta_{10} - 819 \beta_{7} - 160 \beta_{5} + 819 \beta_{4}\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(659 \beta_{15} - 659 \beta_{14} + 1977 \beta_{13} + 7699 \beta_{12} - 1977 \beta_{11} - 23097 \beta_{9} + 7699 \beta_{8} + 23097 \beta_{2}\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(4179 \beta_{4} + 8358 \beta_{3} - 20638 \beta_{1}\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(1535 \beta_{14} - 47504 \beta_{12} - 4605 \beta_{11} - 142512 \beta_{9}\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(-18501 \beta_{7} - 49039 \beta_{5}\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(-15269 \beta_{15} - 45807 \beta_{13} + 1063589 \beta_{8} + 3190767 \beta_{2}\)\()/12\)
\(\nu^{12}\)\(=\)\(131040 \beta_{7} - 262080 \beta_{6} - 131040 \beta_{4} + 430039\)
\(\nu^{13}\)\(=\)\((\)\(692119 \beta_{15} - 692119 \beta_{14} - 2076357 \beta_{13} - 10839401 \beta_{12} + 2076357 \beta_{11} - 32518203 \beta_{9} - 10839401 \beta_{8} + 32518203 \beta_{2}\)\()/12\)
\(\nu^{14}\)\(=\)\((\)\(5073641 \beta_{10} + 20300259 \beta_{4}\)\()/12\)
\(\nu^{15}\)\(=\)\((\)\(-6343475 \beta_{14} - 49466576 \beta_{12} - 19030425 \beta_{11} + 148399728 \beta_{9}\)\()/6\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−0.365632 + 3.29641i
0.365632 3.29641i
−3.29641 0.365632i
3.29641 + 0.365632i
−1.33156 + 3.03759i
1.33156 3.03759i
−3.03759 1.33156i
3.03759 + 1.33156i
−0.365632 3.29641i
0.365632 + 3.29641i
−3.29641 + 0.365632i
3.29641 0.365632i
−1.33156 3.03759i
1.33156 + 3.03759i
−3.03759 + 1.33156i
3.03759 1.33156i
0 0 0 −8.27698 + 4.77872i 0 0 0 0 0
557.2 0 0 0 −8.27698 + 4.77872i 0 0 0 0 0
557.3 0 0 0 −3.08083 + 1.77872i 0 0 0 0 0
557.4 0 0 0 −3.08083 + 1.77872i 0 0 0 0 0
557.5 0 0 0 3.08083 1.77872i 0 0 0 0 0
557.6 0 0 0 3.08083 1.77872i 0 0 0 0 0
557.7 0 0 0 8.27698 4.77872i 0 0 0 0 0
557.8 0 0 0 8.27698 4.77872i 0 0 0 0 0
1745.1 0 0 0 −8.27698 4.77872i 0 0 0 0 0
1745.2 0 0 0 −8.27698 4.77872i 0 0 0 0 0
1745.3 0 0 0 −3.08083 1.77872i 0 0 0 0 0
1745.4 0 0 0 −3.08083 1.77872i 0 0 0 0 0
1745.5 0 0 0 3.08083 + 1.77872i 0 0 0 0 0
1745.6 0 0 0 3.08083 + 1.77872i 0 0 0 0 0
1745.7 0 0 0 8.27698 + 4.77872i 0 0 0 0 0
1745.8 0 0 0 8.27698 + 4.77872i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1745.8
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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.g 16
3.b odd 2 1 inner 1764.3.bk.g 16
7.b odd 2 1 inner 1764.3.bk.g 16
7.c even 3 1 1764.3.c.h 8
7.c even 3 1 inner 1764.3.bk.g 16
7.d odd 6 1 1764.3.c.h 8
7.d odd 6 1 inner 1764.3.bk.g 16
21.c even 2 1 inner 1764.3.bk.g 16
21.g even 6 1 1764.3.c.h 8
21.g even 6 1 inner 1764.3.bk.g 16
21.h odd 6 1 1764.3.c.h 8
21.h odd 6 1 inner 1764.3.bk.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.c.h 8 7.c even 3 1
1764.3.c.h 8 7.d odd 6 1
1764.3.c.h 8 21.g even 6 1
1764.3.c.h 8 21.h odd 6 1
1764.3.bk.g 16 1.a even 1 1 trivial
1764.3.bk.g 16 3.b odd 2 1 inner
1764.3.bk.g 16 7.b odd 2 1 inner
1764.3.bk.g 16 7.c even 3 1 inner
1764.3.bk.g 16 7.d odd 6 1 inner
1764.3.bk.g 16 21.c even 2 1 inner
1764.3.bk.g 16 21.g even 6 1 inner
1764.3.bk.g 16 21.h odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 104 T_{5}^{6} + 9660 T_{5}^{4} - 120224 T_{5}^{2} + 1336336 \)
\( T_{13}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 1336336 - 120224 T^{2} + 9660 T^{4} - 104 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 7396 - 86 T^{2} + T^{4} )^{4} \)
$13$ \( ( -18 + T^{2} )^{8} \)
$17$ \( ( 2085136 - 358112 T^{2} + 60060 T^{4} - 248 T^{6} + T^{8} )^{2} \)
$19$ \( ( 274760478976 + 863842048 T^{2} + 2191728 T^{4} + 1648 T^{6} + T^{8} )^{2} \)
$23$ \( ( 38416 - 61936 T^{2} + 99660 T^{4} - 316 T^{6} + T^{8} )^{2} \)
$29$ \( ( 40804 + 748 T^{2} + T^{4} )^{4} \)
$31$ \( ( 326653399296 + 905313024 T^{2} + 1937520 T^{4} + 1584 T^{6} + T^{8} )^{2} \)
$37$ \( ( 1669264 - 41344 T + 2316 T^{2} + 32 T^{3} + T^{4} )^{4} \)
$41$ \( ( 3928324 + 4136 T^{2} + T^{4} )^{4} \)
$43$ \( ( -1448 + 20 T + T^{2} )^{8} \)
$47$ \( ( 1340445266176 - 5677733504 T^{2} + 22891440 T^{4} - 4904 T^{6} + T^{8} )^{2} \)
$53$ \( ( 45212176 - 38111632 T^{2} + 32119500 T^{4} - 5668 T^{6} + T^{8} )^{2} \)
$59$ \( ( 1340445266176 - 5677733504 T^{2} + 22891440 T^{4} - 4904 T^{6} + T^{8} )^{2} \)
$61$ \( ( 21293813776 + 340294768 T^{2} + 5292300 T^{4} + 2332 T^{6} + T^{8} )^{2} \)
$67$ \( ( 571536 - 72576 T + 8460 T^{2} - 96 T^{3} + T^{4} )^{4} \)
$71$ \( ( 14699556 + 10764 T^{2} + T^{4} )^{4} \)
$73$ \( ( 92114002540816 + 89180936368 T^{2} + 76743660 T^{4} + 9292 T^{6} + T^{8} )^{2} \)
$79$ \( ( 1669264 + 180880 T + 20892 T^{2} - 140 T^{3} + T^{4} )^{4} \)
$83$ \( ( 104693824 + 21152 T^{2} + T^{4} )^{4} \)
$89$ \( ( 2666544596851216 - 758880806816 T^{2} + 164333820 T^{4} - 14696 T^{6} + T^{8} )^{2} \)
$97$ \( ( 3865156 - 23932 T^{2} + T^{4} )^{4} \)
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