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: \(x^{4} - x^{2} + 1\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} + \zeta_{12}^{3} q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} -4 q^{16} + ( -1 + 2 \zeta_{12}^{2} ) q^{17} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} + ( -1 - \zeta_{12}^{3} ) q^{22} -\zeta_{12}^{3} q^{23} + 2 q^{25} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{26} -4 q^{29} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12}^{3} ) q^{32} + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{34} + 3 q^{37} + ( 3 + 6 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{38} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + ( -2 + 4 \zeta_{12}^{2} ) q^{41} + 2 \zeta_{12}^{3} q^{43} + 2 q^{44} + ( 1 + \zeta_{12}^{3} ) q^{46} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{47} + ( -2 + 2 \zeta_{12}^{3} ) q^{50} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} + q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( 4 - 4 \zeta_{12}^{3} ) q^{58} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( -3 + 6 \zeta_{12}^{2} ) q^{61} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + 6 q^{65} -3 \zeta_{12}^{3} q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + 14 \zeta_{12}^{3} q^{71} + ( -5 + 10 \zeta_{12}^{2} ) q^{73} + ( -3 + 3 \zeta_{12}^{3} ) q^{74} + ( -6 + 12 \zeta_{12}^{2} ) q^{76} + 9 \zeta_{12}^{3} q^{79} + ( -4 + 8 \zeta_{12}^{2} ) q^{80} + ( 2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + 3 q^{85} + ( -2 - 2 \zeta_{12}^{3} ) q^{86} + ( -2 + 2 \zeta_{12}^{3} ) q^{88} + ( -9 + 18 \zeta_{12}^{2} ) q^{89} -2 q^{92} + ( 5 + 10 \zeta_{12} - 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{94} + 9 \zeta_{12}^{3} q^{95} + ( 10 - 20 \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 8q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 8q^{8} - 16q^{16} - 4q^{22} + 8q^{25} - 16q^{29} + 16q^{32} + 12q^{37} + 8q^{44} + 4q^{46} - 8q^{50} + 4q^{53} + 16q^{58} + 24q^{65} - 12q^{74} + 12q^{85} - 8q^{86} - 8q^{88} - 8q^{92} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{11}^{2} + 1 \)
\( T_{19}^{2} - 27 \)
\( T_{29} + 4 \)
\( T_{53} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - 7 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 - 21 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 31 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 11 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 45 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 59 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 3 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 19 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - T + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 91 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 95 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 125 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 71 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 77 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 65 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 106 T^{2} + 9409 T^{4} )^{2} \)
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