Properties

Label 1764.2.e.d
Level 1764
Weight 2
Character orbit 1764.e
Analytic conductor 14.086
Analytic rank 0
Dimension 4
CM discriminant -4
Inner twists 8

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{10} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{13} + 4 q^{16} + 8 \zeta_{8}^{2} q^{17} -8 \zeta_{8}^{2} q^{20} -11 q^{25} + 2 \zeta_{8}^{2} q^{26} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{29} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{34} -12 q^{37} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{40} -10 \zeta_{8}^{2} q^{41} + ( 11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{50} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{52} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} -14 q^{58} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{61} -8 q^{64} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} -16 \zeta_{8}^{2} q^{68} + ( -11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{73} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{74} + 16 \zeta_{8}^{2} q^{80} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{82} -32 q^{85} + 10 \zeta_{8}^{2} q^{89} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} - 44q^{25} - 48q^{37} - 56q^{58} - 32q^{64} - 128q^{85} + O(q^{100}) \)

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
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
1079.2 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
1079.3 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
1079.4 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
\(n\): 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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 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.e.d 4
3.b odd 2 1 inner 1764.2.e.d 4
4.b odd 2 1 CM 1764.2.e.d 4
7.b odd 2 1 inner 1764.2.e.d 4
12.b even 2 1 inner 1764.2.e.d 4
21.c even 2 1 inner 1764.2.e.d 4
28.d even 2 1 inner 1764.2.e.d 4
84.h odd 2 1 inner 1764.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.d 4 1.a even 1 1 trivial
1764.2.e.d 4 3.b odd 2 1 inner
1764.2.e.d 4 4.b odd 2 1 CM
1764.2.e.d 4 7.b odd 2 1 inner
1764.2.e.d 4 12.b even 2 1 inner
1764.2.e.d 4 21.c even 2 1 inner
1764.2.e.d 4 28.d even 2 1 inner
1764.2.e.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 \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{13}^{2} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2}( 1 + 2 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 + 11 T^{2} )^{4} \)
$13$ \( ( 1 + 24 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2}( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 12 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 + 120 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 96 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 144 T^{2} + 9409 T^{4} )^{2} \)
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