Properties

Label 1764.2.b.d
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 2 + 2 \beta_{2} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} -3 \beta_{2} q^{13} + ( -2 - 2 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + 5 q^{19} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{20} + ( -4 - 4 \beta_{2} ) q^{22} + ( \beta_{1} - \beta_{3} ) q^{23} -3 q^{25} -3 \beta_{1} q^{26} - q^{31} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{32} + ( 2 + 2 \beta_{2} ) q^{34} + 5 q^{37} + 5 \beta_{3} q^{38} + 8 q^{40} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{41} -3 \beta_{2} q^{43} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 2 + 2 \beta_{2} ) q^{46} + ( -\beta_{1} - 3 \beta_{3} ) q^{47} -3 \beta_{3} q^{50} + ( -9 - 3 \beta_{2} ) q^{52} + ( \beta_{1} + 3 \beta_{3} ) q^{53} + 16 q^{55} + ( \beta_{1} + 3 \beta_{3} ) q^{59} -4 \beta_{2} q^{61} -\beta_{3} q^{62} -8 q^{64} + ( 3 \beta_{1} + 9 \beta_{3} ) q^{65} -5 \beta_{2} q^{67} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( \beta_{1} - \beta_{3} ) q^{71} + \beta_{2} q^{73} + 5 \beta_{3} q^{74} + ( 5 - 5 \beta_{2} ) q^{76} -\beta_{2} q^{79} + 8 \beta_{3} q^{80} + ( -4 - 4 \beta_{2} ) q^{82} + ( 3 \beta_{1} + 9 \beta_{3} ) q^{83} -8 q^{85} -3 \beta_{1} q^{86} -16 q^{88} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -6 + 2 \beta_{2} ) q^{94} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} + 8q^{10} - 8q^{16} + 20q^{19} - 16q^{22} - 12q^{25} - 4q^{31} + 8q^{34} + 20q^{37} + 32q^{40} + 8q^{46} - 36q^{52} + 64q^{55} - 32q^{64} + 20q^{76} - 16q^{82} - 32q^{85} - 64q^{88} - 24q^{94} + O(q^{100}) \)

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

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

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
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 3.46410i
1567.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 + 3.46410i
1567.3 1.22474 0.707107i 0 1.00000 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 + 3.46410i
1567.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{2} + 8 \)
\( T_{11}^{2} + 32 \)
\( T_{19} - 5 \)
\( T_{29} \)
\( T_{53}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 5 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 + T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 5 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 59 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 82 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 94 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 59 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )^{2}( 1 + 17 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 155 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 50 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 50 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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