Properties

Label 1764.2.b.a
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: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} - \beta_{2} q^{5} + (2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} - \beta_{2} q^{5} + (2 \beta_1 + 2) q^{8} + (\beta_{3} + \beta_{2}) q^{10} + \beta_1 q^{11} + 2 \beta_{2} q^{13} - 4 q^{16} + \beta_{2} q^{17} - 3 \beta_{3} q^{19} - 2 \beta_{3} q^{20} + ( - \beta_1 - 1) q^{22} - \beta_1 q^{23} + 2 q^{25} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{26} - 4 q^{29} - \beta_{3} q^{31} + ( - 4 \beta_1 + 4) q^{32} + ( - \beta_{3} - \beta_{2}) q^{34} + 3 q^{37} + (3 \beta_{3} - 3 \beta_{2}) q^{38} + (2 \beta_{3} - 2 \beta_{2}) q^{40} + 2 \beta_{2} q^{41} + 2 \beta_1 q^{43} + 2 q^{44} + (\beta_1 + 1) q^{46} - 5 \beta_{3} q^{47} + (2 \beta_1 - 2) q^{50} + 4 \beta_{3} q^{52} + q^{53} + \beta_{3} q^{55} + ( - 4 \beta_1 + 4) q^{58} + 3 \beta_{3} q^{59} + 3 \beta_{2} q^{61} + (\beta_{3} - \beta_{2}) q^{62} + 8 \beta_1 q^{64} + 6 q^{65} - 3 \beta_1 q^{67} + 2 \beta_{3} q^{68} + 14 \beta_1 q^{71} + 5 \beta_{2} q^{73} + (3 \beta_1 - 3) q^{74} + 6 \beta_{2} q^{76} + 9 \beta_1 q^{79} + 4 \beta_{2} q^{80} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{82} - 8 \beta_{3} q^{83} + 3 q^{85} + ( - 2 \beta_1 - 2) q^{86} + (2 \beta_1 - 2) q^{88} + 9 \beta_{2} q^{89} - 2 q^{92} + (5 \beta_{3} - 5 \beta_{2}) q^{94} + 9 \beta_1 q^{95} - 10 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} - 4 q^{22} + 8 q^{25} - 16 q^{29} + 16 q^{32} + 12 q^{37} + 8 q^{44} + 4 q^{46} - 8 q^{50} + 4 q^{53} + 16 q^{58} + 24 q^{65} - 12 q^{74} + 12 q^{85} - 8 q^{86} - 8 q^{88} - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

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
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.00000 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 2.00000i 0 −1.73205 + 1.73205i
1567.2 −1.00000 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 2.00000i 0 1.73205 1.73205i
1567.3 −1.00000 + 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 + 2.00000i 0 1.73205 + 1.73205i
1567.4 −1.00000 + 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 + 2.00000i 0 −1.73205 1.73205i
\(n\): 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.a 4
3.b odd 2 1 196.2.d.b 4
4.b odd 2 1 inner 1764.2.b.a 4
7.b odd 2 1 inner 1764.2.b.a 4
7.c even 3 1 252.2.bf.e 4
7.d odd 6 1 252.2.bf.e 4
12.b even 2 1 196.2.d.b 4
21.c even 2 1 196.2.d.b 4
21.g even 6 1 28.2.f.a 4
21.g even 6 1 196.2.f.a 4
21.h odd 6 1 28.2.f.a 4
21.h odd 6 1 196.2.f.a 4
24.f even 2 1 3136.2.f.e 4
24.h odd 2 1 3136.2.f.e 4
28.d even 2 1 inner 1764.2.b.a 4
28.f even 6 1 252.2.bf.e 4
28.g odd 6 1 252.2.bf.e 4
84.h odd 2 1 196.2.d.b 4
84.j odd 6 1 28.2.f.a 4
84.j odd 6 1 196.2.f.a 4
84.n even 6 1 28.2.f.a 4
84.n even 6 1 196.2.f.a 4
105.o odd 6 1 700.2.p.a 4
105.p even 6 1 700.2.p.a 4
105.w odd 12 1 700.2.t.a 4
105.w odd 12 1 700.2.t.b 4
105.x even 12 1 700.2.t.a 4
105.x even 12 1 700.2.t.b 4
168.e odd 2 1 3136.2.f.e 4
168.i even 2 1 3136.2.f.e 4
168.s odd 6 1 448.2.p.d 4
168.v even 6 1 448.2.p.d 4
168.ba even 6 1 448.2.p.d 4
168.be odd 6 1 448.2.p.d 4
420.ba even 6 1 700.2.p.a 4
420.be odd 6 1 700.2.p.a 4
420.bp odd 12 1 700.2.t.a 4
420.bp odd 12 1 700.2.t.b 4
420.br even 12 1 700.2.t.a 4
420.br even 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 21.g even 6 1
28.2.f.a 4 21.h odd 6 1
28.2.f.a 4 84.j odd 6 1
28.2.f.a 4 84.n even 6 1
196.2.d.b 4 3.b odd 2 1
196.2.d.b 4 12.b even 2 1
196.2.d.b 4 21.c even 2 1
196.2.d.b 4 84.h odd 2 1
196.2.f.a 4 21.g even 6 1
196.2.f.a 4 21.h odd 6 1
196.2.f.a 4 84.j odd 6 1
196.2.f.a 4 84.n even 6 1
252.2.bf.e 4 7.c even 3 1
252.2.bf.e 4 7.d odd 6 1
252.2.bf.e 4 28.f even 6 1
252.2.bf.e 4 28.g odd 6 1
448.2.p.d 4 168.s odd 6 1
448.2.p.d 4 168.v even 6 1
448.2.p.d 4 168.ba even 6 1
448.2.p.d 4 168.be odd 6 1
700.2.p.a 4 105.o odd 6 1
700.2.p.a 4 105.p even 6 1
700.2.p.a 4 420.ba even 6 1
700.2.p.a 4 420.be odd 6 1
700.2.t.a 4 105.w odd 12 1
700.2.t.a 4 105.x even 12 1
700.2.t.a 4 420.bp odd 12 1
700.2.t.a 4 420.br even 12 1
700.2.t.b 4 105.w odd 12 1
700.2.t.b 4 105.x even 12 1
700.2.t.b 4 420.bp odd 12 1
700.2.t.b 4 420.br even 12 1
1764.2.b.a 4 1.a even 1 1 trivial
1764.2.b.a 4 4.b odd 2 1 inner
1764.2.b.a 4 7.b odd 2 1 inner
1764.2.b.a 4 28.d even 2 1 inner
3136.2.f.e 4 24.f even 2 1
3136.2.f.e 4 24.h odd 2 1
3136.2.f.e 4 168.e odd 2 1
3136.2.f.e 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 \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 27 \) Copy content Toggle raw display
\( T_{29} + 4 \) Copy content Toggle raw display
\( T_{53} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
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