Properties

Label 1764.2.b.h
Level 1764
Weight 2
Character orbit 1764.b
Analytic conductor 14.086
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -2 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( -2 + \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -2 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( -2 + \beta_{2} ) q^{8} -\beta_{1} q^{10} + ( 1 + 2 \beta_{2} ) q^{11} -2 \beta_{3} q^{13} + ( 2 + 3 \beta_{2} ) q^{16} -4 \beta_{3} q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{20} + ( 4 + \beta_{2} ) q^{22} + ( 2 + 4 \beta_{2} ) q^{23} + 2 q^{25} + 2 \beta_{1} q^{26} + 5 q^{29} + ( -2 \beta_{1} - \beta_{3} ) q^{31} + ( 6 + \beta_{2} ) q^{32} + 4 \beta_{1} q^{34} + ( \beta_{1} - 2 \beta_{3} ) q^{40} -2 \beta_{3} q^{41} + ( -4 - 8 \beta_{2} ) q^{43} + ( 2 - 3 \beta_{2} ) q^{44} + ( 8 + 2 \beta_{2} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{47} -2 \beta_{2} q^{50} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{52} + 7 q^{53} + ( 2 \beta_{1} + \beta_{3} ) q^{55} -5 \beta_{2} q^{58} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{59} -6 \beta_{3} q^{61} + ( -\beta_{1} - 4 \beta_{3} ) q^{62} + ( 2 - 5 \beta_{2} ) q^{64} + 6 q^{65} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{68} + ( 2 + 4 \beta_{2} ) q^{71} + 4 \beta_{3} q^{73} + ( -3 - 6 \beta_{2} ) q^{79} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{80} + 2 \beta_{1} q^{82} + ( 2 \beta_{1} + \beta_{3} ) q^{83} + 12 q^{85} + ( -16 - 4 \beta_{2} ) q^{86} + ( -6 - 5 \beta_{2} ) q^{88} -6 \beta_{3} q^{89} + ( 4 - 6 \beta_{2} ) q^{92} + ( -2 \beta_{1} - 8 \beta_{3} ) q^{94} -5 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 6q^{4} - 10q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 6q^{4} - 10q^{8} + 2q^{16} + 14q^{22} + 8q^{25} + 20q^{29} + 22q^{32} + 14q^{44} + 28q^{46} + 4q^{50} + 28q^{53} + 10q^{58} + 18q^{64} + 24q^{65} + 48q^{85} - 56q^{86} - 14q^{88} + 28q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - x^{2} - 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} - \nu^{2} + \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.39564 + 0.228425i
−0.895644 + 1.09445i
−0.895644 1.09445i
1.39564 0.228425i
0.500000 1.32288i 0 −1.50000 1.32288i 1.73205i 0 0 −2.50000 + 1.32288i 0 −2.29129 0.866025i
1567.2 0.500000 1.32288i 0 −1.50000 1.32288i 1.73205i 0 0 −2.50000 + 1.32288i 0 2.29129 + 0.866025i
1567.3 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 1.73205i 0 0 −2.50000 1.32288i 0 2.29129 0.866025i
1567.4 0.500000 + 1.32288i 0 −1.50000 + 1.32288i 1.73205i 0 0 −2.50000 1.32288i 0 −2.29129 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.h 4
3.b odd 2 1 1764.2.b.b 4
4.b odd 2 1 inner 1764.2.b.h 4
7.b odd 2 1 inner 1764.2.b.h 4
7.c even 3 1 252.2.bf.a 4
7.d odd 6 1 252.2.bf.a 4
12.b even 2 1 1764.2.b.b 4
21.c even 2 1 1764.2.b.b 4
21.g even 6 1 252.2.bf.d yes 4
21.h odd 6 1 252.2.bf.d yes 4
28.d even 2 1 inner 1764.2.b.h 4
28.f even 6 1 252.2.bf.a 4
28.g odd 6 1 252.2.bf.a 4
84.h odd 2 1 1764.2.b.b 4
84.j odd 6 1 252.2.bf.d yes 4
84.n even 6 1 252.2.bf.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.a 4 7.c even 3 1
252.2.bf.a 4 7.d odd 6 1
252.2.bf.a 4 28.f even 6 1
252.2.bf.a 4 28.g odd 6 1
252.2.bf.d yes 4 21.g even 6 1
252.2.bf.d yes 4 21.h odd 6 1
252.2.bf.d yes 4 84.j odd 6 1
252.2.bf.d yes 4 84.n even 6 1
1764.2.b.b 4 3.b odd 2 1
1764.2.b.b 4 12.b even 2 1
1764.2.b.b 4 21.c even 2 1
1764.2.b.b 4 84.h odd 2 1
1764.2.b.h 4 1.a even 1 1 trivial
1764.2.b.h 4 4.b odd 2 1 inner
1764.2.b.h 4 7.b odd 2 1 inner
1764.2.b.h 4 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{11}^{2} + 7 \)
\( T_{19} \)
\( T_{29} - 5 \)
\( T_{53} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - 7 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 - 15 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 14 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{2}( 1 + 8 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 5 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 41 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 10 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 7 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 71 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2}( 1 + 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 98 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 95 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 145 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \)
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