Properties

Label 1764.2.b.f
Level 1764
Weight 2
Character orbit 1764.b
Analytic conductor 14.086
Analytic rank 0
Dimension 4
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + 2 q^{4} + \beta_{1} q^{5} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + 2 q^{4} + \beta_{1} q^{5} + 2 \beta_{2} q^{8} -\beta_{3} q^{10} + ( \beta_{1} + \beta_{3} ) q^{13} + 4 q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + 2 \beta_{1} q^{20} + ( -5 - \beta_{2} ) q^{25} + ( -2 \beta_{1} - \beta_{3} ) q^{26} -3 \beta_{2} q^{29} + 4 \beta_{2} q^{32} + ( 2 \beta_{1} - \beta_{3} ) q^{34} + 5 \beta_{2} q^{37} -2 \beta_{3} q^{40} + ( \beta_{1} + 2 \beta_{3} ) q^{41} + ( -2 - 5 \beta_{2} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{52} -4 q^{53} -6 q^{58} + ( 3 \beta_{1} - \beta_{3} ) q^{61} + 8 q^{64} + ( -8 + 9 \beta_{2} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{73} + 10 q^{74} + 4 \beta_{1} q^{80} + ( -4 \beta_{1} - \beta_{3} ) q^{82} + ( -12 - 11 \beta_{2} ) q^{85} + ( -\beta_{1} - 3 \beta_{3} ) q^{89} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} + 16q^{16} - 20q^{25} - 8q^{50} - 16q^{53} - 24q^{58} + 32q^{64} - 32q^{65} + 40q^{74} - 48q^{85} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{3} - 8 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/7\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} - 8 \beta_{1}\)\()/7\)

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
1.84776i
1.84776i
0.765367i
0.765367i
−1.41421 0 2.00000 2.93015i 0 0 −2.82843 0 4.14386i
1567.2 −1.41421 0 2.00000 2.93015i 0 0 −2.82843 0 4.14386i
1567.3 1.41421 0 2.00000 3.37849i 0 0 2.82843 0 4.77791i
1567.4 1.41421 0 2.00000 3.37849i 0 0 2.82843 0 4.77791i
\(n\): 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 \(\Q(\sqrt{-1}) \)
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.f 4
3.b odd 2 1 196.2.d.a 4
4.b odd 2 1 CM 1764.2.b.f 4
7.b odd 2 1 inner 1764.2.b.f 4
12.b even 2 1 196.2.d.a 4
21.c even 2 1 196.2.d.a 4
21.g even 6 2 196.2.f.c 8
21.h odd 6 2 196.2.f.c 8
24.f even 2 1 3136.2.f.c 4
24.h odd 2 1 3136.2.f.c 4
28.d even 2 1 inner 1764.2.b.f 4
84.h odd 2 1 196.2.d.a 4
84.j odd 6 2 196.2.f.c 8
84.n even 6 2 196.2.f.c 8
168.e odd 2 1 3136.2.f.c 4
168.i even 2 1 3136.2.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.a 4 3.b odd 2 1
196.2.d.a 4 12.b even 2 1
196.2.d.a 4 21.c even 2 1
196.2.d.a 4 84.h odd 2 1
196.2.f.c 8 21.g even 6 2
196.2.f.c 8 21.h odd 6 2
196.2.f.c 8 84.j odd 6 2
196.2.f.c 8 84.n even 6 2
1764.2.b.f 4 1.a even 1 1 trivial
1764.2.b.f 4 4.b odd 2 1 CM
1764.2.b.f 4 7.b odd 2 1 inner
1764.2.b.f 4 28.d even 2 1 inner
3136.2.f.c 4 24.f even 2 1
3136.2.f.c 4 24.h odd 2 1
3136.2.f.c 4 168.e odd 2 1
3136.2.f.c 4 168.i even 2 1

Hecke kernels

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

\( T_{5}^{4} + 20 T_{5}^{2} + 98 \)
\( T_{11} \)
\( T_{19} \)
\( T_{29}^{2} - 18 \)
\( T_{53} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ 1
$5$ \( 1 + 48 T^{4} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( 1 - 240 T^{4} + 28561 T^{8} \)
$17$ \( 1 - 480 T^{4} + 83521 T^{8} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 24 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( 1 + 1440 T^{4} + 2825761 T^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( 1 - 2640 T^{4} + 13845841 T^{8} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( 1 - 10560 T^{4} + 28398241 T^{8} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( 1 + 12480 T^{4} + 62742241 T^{8} \)
$97$ \( 1 - 18720 T^{4} + 88529281 T^{8} \)
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