Properties

Label 1764.4.k.bc
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.7027860559216896.42
Defining polynomial: \(x^{8} - 4 x^{7} - 88 x^{6} + 278 x^{5} + 3869 x^{4} - 8206 x^{3} - 71054 x^{2} + 75204 x + 945876\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{5} +O(q^{10})\) \( q -\beta_{5} q^{5} + ( \beta_{2} + \beta_{3} ) q^{11} + \beta_{6} q^{13} + ( 9 \beta_{4} + 9 \beta_{5} ) q^{17} + ( -\beta_{6} - \beta_{7} ) q^{19} -3 \beta_{2} q^{23} + ( 13 + 13 \beta_{1} ) q^{25} + 2 \beta_{3} q^{29} -\beta_{7} q^{31} + 78 \beta_{1} q^{37} -39 \beta_{4} q^{41} + 148 q^{43} + 44 \beta_{5} q^{47} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{53} -7 \beta_{6} q^{55} + ( 52 \beta_{4} + 52 \beta_{5} ) q^{59} + ( 7 \beta_{6} + 7 \beta_{7} ) q^{61} + 16 \beta_{2} q^{65} + ( 260 + 260 \beta_{1} ) q^{67} -13 \beta_{3} q^{71} + 8 \beta_{7} q^{73} + 664 \beta_{1} q^{79} -12 \beta_{4} q^{83} + 1008 q^{85} -83 \beta_{5} q^{89} + ( -16 \beta_{2} - 16 \beta_{3} ) q^{95} + 14 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 52q^{25} - 312q^{37} + 1184q^{43} + 1040q^{67} - 2656q^{79} + 8064q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 88 x^{6} + 278 x^{5} + 3869 x^{4} - 8206 x^{3} - 71054 x^{2} + 75204 x + 945876\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 122 \nu^{4} + 249 \nu^{3} + 4585 \nu^{2} - 4710 \nu - 88614 \)\()/46410\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 122 \nu^{4} + 249 \nu^{3} + 11215 \nu^{2} - 11340 \nu - 221214 \)\()/3315\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 57 \nu^{4} + 119 \nu^{3} + 1140 \nu^{2} - 1200 \nu + 8366 \)\()/455\)
\(\beta_{4}\)\(=\)\((\)\( -212 \nu^{7} + 742 \nu^{6} + 3310 \nu^{5} - 10130 \nu^{4} + 290614 \nu^{3} - 425420 \nu^{2} - 14728656 \nu + 7434876 \)\()/5610969\)
\(\beta_{5}\)\(=\)\((\)\( -172 \nu^{7} + 602 \nu^{6} + 16646 \nu^{5} - 43120 \nu^{4} - 834526 \nu^{3} + 1295210 \nu^{2} + 10396488 \nu - 5415564 \)\()/4007835\)
\(\beta_{6}\)\(=\)\((\)\( -688 \nu^{7} + 2408 \nu^{6} + 66584 \nu^{5} - 172480 \nu^{4} - 3338104 \nu^{3} + 5180840 \nu^{2} + 105711312 \nu - 53724936 \)\()/4007835\)
\(\beta_{7}\)\(=\)\((\)\( -13912 \nu^{7} + 48692 \nu^{6} + 1231676 \nu^{5} - 3200920 \nu^{4} - 37329676 \nu^{3} + 59219780 \nu^{2} + 90846408 \nu - 55401024 \)\()/28054845\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - 4 \beta_{5} + 8\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 4 \beta_{5} + 8 \beta_{2} - 112 \beta_{1} + 328\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(21 \beta_{7} + 28 \beta_{6} - 330 \beta_{5} - 28 \beta_{4} + 12 \beta_{2} - 168 \beta_{1} + 488\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(42 \beta_{7} + 55 \beta_{6} - 656 \beta_{5} - 56 \beta_{4} + 112 \beta_{3} + 448 \beta_{2} - 17696 \beta_{1} - 5944\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(1715 \beta_{7} + 566 \beta_{6} - 15474 \beta_{5} - 7504 \beta_{4} + 280 \beta_{3} + 1100 \beta_{2} - 43960 \beta_{1} - 15672\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(5040 \beta_{7} + 1561 \beta_{6} - 44784 \beta_{5} - 22372 \beta_{4} + 14504 \beta_{3} + 18288 \beta_{2} - 992880 \beta_{1} - 942072\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(71197 \beta_{7} - 21426 \beta_{6} - 533442 \beta_{5} - 654640 \beta_{4} + 49784 \beta_{3} + 60172 \beta_{2} - 3321416 \beta_{1} - 3241832\)\()/16\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−6.04303 2.29129i
4.39728 2.29129i
7.04303 + 2.29129i
−3.39728 + 2.29129i
−6.04303 + 2.29129i
4.39728 + 2.29129i
7.04303 2.29129i
−3.39728 2.29129i
0 0 0 −5.29150 9.16515i 0 0 0 0 0
361.2 0 0 0 −5.29150 9.16515i 0 0 0 0 0
361.3 0 0 0 5.29150 + 9.16515i 0 0 0 0 0
361.4 0 0 0 5.29150 + 9.16515i 0 0 0 0 0
1549.1 0 0 0 −5.29150 + 9.16515i 0 0 0 0 0
1549.2 0 0 0 −5.29150 + 9.16515i 0 0 0 0 0
1549.3 0 0 0 5.29150 9.16515i 0 0 0 0 0
1549.4 0 0 0 5.29150 9.16515i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.4
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 1764.4.k.bc 8
3.b odd 2 1 inner 1764.4.k.bc 8
7.b odd 2 1 inner 1764.4.k.bc 8
7.c even 3 1 1764.4.a.bb 4
7.c even 3 1 inner 1764.4.k.bc 8
7.d odd 6 1 1764.4.a.bb 4
7.d odd 6 1 inner 1764.4.k.bc 8
21.c even 2 1 inner 1764.4.k.bc 8
21.g even 6 1 1764.4.a.bb 4
21.g even 6 1 inner 1764.4.k.bc 8
21.h odd 6 1 1764.4.a.bb 4
21.h odd 6 1 inner 1764.4.k.bc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.4.a.bb 4 7.c even 3 1
1764.4.a.bb 4 7.d odd 6 1
1764.4.a.bb 4 21.g even 6 1
1764.4.a.bb 4 21.h odd 6 1
1764.4.k.bc 8 1.a even 1 1 trivial
1764.4.k.bc 8 3.b odd 2 1 inner
1764.4.k.bc 8 7.b odd 2 1 inner
1764.4.k.bc 8 7.c even 3 1 inner
1764.4.k.bc 8 7.d odd 6 1 inner
1764.4.k.bc 8 21.c even 2 1 inner
1764.4.k.bc 8 21.g even 6 1 inner
1764.4.k.bc 8 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 \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 112 T_{5}^{2} + 12544 \)
\( T_{11}^{4} + 3052 T_{11}^{2} + 9314704 \)
\( T_{13}^{2} - 6976 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 12544 + 112 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 9314704 + 3052 T^{2} + T^{4} )^{2} \)
$13$ \( ( -6976 + T^{2} )^{4} \)
$17$ \( ( 82301184 + 9072 T^{2} + T^{4} )^{2} \)
$19$ \( ( 48664576 + 6976 T^{2} + T^{4} )^{2} \)
$23$ \( ( 754491024 + 27468 T^{2} + T^{4} )^{2} \)
$29$ \( ( -12208 + T^{2} )^{4} \)
$31$ \( ( 48664576 + 6976 T^{2} + T^{4} )^{2} \)
$37$ \( ( 6084 + 78 T + T^{2} )^{4} \)
$41$ \( ( -170352 + T^{2} )^{4} \)
$43$ \( ( -148 + T )^{8} \)
$47$ \( ( 47016116224 + 216832 T^{2} + T^{4} )^{2} \)
$53$ \( ( 149035264 + 12208 T^{2} + T^{4} )^{2} \)
$59$ \( ( 91716911104 + 302848 T^{2} + T^{4} )^{2} \)
$61$ \( ( 116843646976 + 341824 T^{2} + T^{4} )^{2} \)
$67$ \( ( 67600 - 260 T + T^{2} )^{4} \)
$71$ \( ( -515788 + T^{2} )^{4} \)
$73$ \( ( 199330103296 + 446464 T^{2} + T^{4} )^{2} \)
$79$ \( ( 440896 + 664 T + T^{2} )^{4} \)
$83$ \( ( -16128 + T^{2} )^{4} \)
$89$ \( ( 595317178624 + 771568 T^{2} + T^{4} )^{2} \)
$97$ \( ( -1367296 + T^{2} )^{4} \)
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