Properties

Label 1764.2.e.a
Level 1764
Weight 2
Character orbit 1764.e
Analytic conductor 14.086
Analytic rank 1
Dimension 2
CM discriminant -4
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} -2 \beta q^{8} -6 q^{10} -6 q^{13} + 4 q^{16} -3 \beta q^{17} -6 \beta q^{20} -13 q^{25} -6 \beta q^{26} -7 \beta q^{29} + 4 \beta q^{32} + 6 q^{34} -2 q^{37} + 12 q^{40} + 9 \beta q^{41} -13 \beta q^{50} + 12 q^{52} -5 \beta q^{53} + 14 q^{58} -12 q^{61} -8 q^{64} -18 \beta q^{65} + 6 \beta q^{68} + 6 q^{73} -2 \beta q^{74} + 12 \beta q^{80} -18 q^{82} + 18 q^{85} -3 \beta q^{89} + 18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} - 12q^{10} - 12q^{13} + 8q^{16} - 26q^{25} + 12q^{34} - 4q^{37} + 24q^{40} + 24q^{52} + 28q^{58} - 24q^{61} - 16q^{64} + 12q^{73} - 36q^{82} + 36q^{85} + 36q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
1.41421i
1.41421i
1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
1079.2 1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 −6.00000
\(n\): 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 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b 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.e.a 2
3.b odd 2 1 inner 1764.2.e.a 2
4.b odd 2 1 CM 1764.2.e.a 2
7.b odd 2 1 1764.2.e.c yes 2
12.b even 2 1 inner 1764.2.e.a 2
21.c even 2 1 1764.2.e.c yes 2
28.d even 2 1 1764.2.e.c yes 2
84.h odd 2 1 1764.2.e.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.a 2 1.a even 1 1 trivial
1764.2.e.a 2 3.b odd 2 1 inner
1764.2.e.a 2 4.b odd 2 1 CM
1764.2.e.a 2 12.b even 2 1 inner
1764.2.e.c yes 2 7.b odd 2 1
1764.2.e.c yes 2 21.c even 2 1
1764.2.e.c yes 2 28.d even 2 1
1764.2.e.c yes 2 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 18 \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ 1
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 16 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( 1 + 40 T^{2} + 841 T^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 80 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 56 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 12 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 160 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )^{2} \)
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