Properties

Label 1764.1.co.a
Level 1764
Weight 1
Character orbit 1764.co
Analytic conductor 0.880
Analytic rank 0
Dimension 12
Projective image \(D_{42}\)
CM discriminant -3
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.co (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \(x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{42}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{42}^{16} q^{7} +O(q^{10})\) \( q -\zeta_{42}^{16} q^{7} + ( -\zeta_{42}^{8} + \zeta_{42}^{10} ) q^{13} + ( \zeta_{42}^{2} - \zeta_{42}^{12} ) q^{19} -\zeta_{42}^{5} q^{25} + ( \zeta_{42}^{3} + \zeta_{42}^{4} ) q^{31} + ( -\zeta_{42}^{6} - \zeta_{42}^{14} ) q^{37} + ( \zeta_{42}^{13} + \zeta_{42}^{17} ) q^{43} -\zeta_{42}^{11} q^{49} + ( \zeta_{42}^{7} - \zeta_{42}^{13} ) q^{61} + ( -\zeta_{42} + \zeta_{42}^{6} ) q^{67} + ( \zeta_{42} - \zeta_{42}^{9} ) q^{73} + ( -\zeta_{42}^{17} + \zeta_{42}^{18} ) q^{79} + ( -\zeta_{42}^{3} + \zeta_{42}^{5} ) q^{91} + ( \zeta_{42}^{9} + \zeta_{42}^{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - q^{7} + O(q^{10}) \) \( 12q - q^{7} + 3q^{19} + q^{25} + 3q^{31} + 8q^{37} - 2q^{43} + q^{49} + 7q^{61} - q^{67} - 3q^{73} - q^{79} - 3q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{42}^{19}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
0.365341 0.930874i
0.365341 + 0.930874i
0.826239 0.563320i
−0.988831 0.149042i
−0.733052 0.680173i
−0.733052 + 0.680173i
0.0747301 + 0.997204i
0.955573 + 0.294755i
0.826239 + 0.563320i
0.0747301 0.997204i
0.955573 0.294755i
−0.988831 + 0.149042i
0 0 0 0 0 −0.955573 + 0.294755i 0 0 0
145.1 0 0 0 0 0 −0.955573 0.294755i 0 0 0
397.1 0 0 0 0 0 0.988831 0.149042i 0 0 0
577.1 0 0 0 0 0 0.733052 0.680173i 0 0 0
649.1 0 0 0 0 0 −0.826239 + 0.563320i 0 0 0
829.1 0 0 0 0 0 −0.826239 0.563320i 0 0 0
1081.1 0 0 0 0 0 −0.365341 + 0.930874i 0 0 0
1153.1 0 0 0 0 0 −0.0747301 + 0.997204i 0 0 0
1333.1 0 0 0 0 0 0.988831 + 0.149042i 0 0 0
1405.1 0 0 0 0 0 −0.365341 0.930874i 0 0 0
1585.1 0 0 0 0 0 −0.0747301 0.997204i 0 0 0
1657.1 0 0 0 0 0 0.733052 + 0.680173i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1657.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
49.h odd 42 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.co.a 12
3.b odd 2 1 CM 1764.1.co.a 12
49.h odd 42 1 inner 1764.1.co.a 12
147.o even 42 1 inner 1764.1.co.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.co.a 12 1.a even 1 1 trivial
1764.1.co.a 12 3.b odd 2 1 CM
1764.1.co.a 12 49.h odd 42 1 inner
1764.1.co.a 12 147.o even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1764, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$7$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$11$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$13$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$17$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
$23$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$29$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
$37$ \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
$41$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$43$ \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
$47$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$53$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$59$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$61$ \( ( 1 - T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
$67$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
$71$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
$73$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$79$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
$83$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$89$ \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
show more
show less