Properties

Label 1260.1.de.a
Level $1260$
Weight $1$
Character orbit 1260.de
Analytic conductor $0.629$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.de (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.661624362000.11

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} - q^{5} -\zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} - q^{5} -\zeta_{12}^{4} q^{6} + \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} -\zeta_{12}^{5} q^{10} + \zeta_{12}^{3} q^{12} - q^{14} -\zeta_{12}^{5} q^{15} -\zeta_{12}^{2} q^{16} + \zeta_{12}^{3} q^{18} + \zeta_{12}^{4} q^{20} - q^{21} -2 \zeta_{12}^{3} q^{23} -\zeta_{12}^{2} q^{24} + q^{25} + \zeta_{12}^{3} q^{27} -\zeta_{12}^{5} q^{28} + ( 1 + \zeta_{12}^{2} ) q^{29} + \zeta_{12}^{4} q^{30} + \zeta_{12} q^{32} -\zeta_{12} q^{35} -\zeta_{12}^{2} q^{36} -\zeta_{12}^{3} q^{40} + \zeta_{12}^{2} q^{41} -\zeta_{12}^{5} q^{42} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{43} + \zeta_{12}^{4} q^{45} + 2 \zeta_{12}^{2} q^{46} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{47} + \zeta_{12} q^{48} + \zeta_{12}^{2} q^{49} + \zeta_{12}^{5} q^{50} -\zeta_{12}^{2} q^{54} + \zeta_{12}^{4} q^{56} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{58} -\zeta_{12}^{3} q^{60} -\zeta_{12}^{5} q^{63} - q^{64} + 2 \zeta_{12}^{2} q^{69} + q^{70} + \zeta_{12} q^{72} + \zeta_{12}^{5} q^{75} + \zeta_{12}^{2} q^{80} -\zeta_{12}^{2} q^{81} -\zeta_{12} q^{82} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{83} + \zeta_{12}^{4} q^{84} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{86} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{87} + 2 \zeta_{12}^{4} q^{89} -\zeta_{12}^{3} q^{90} -2 \zeta_{12} q^{92} + ( -1 - \zeta_{12}^{2} ) q^{94} - q^{96} -\zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 4q^{5} + 2q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 4q^{5} + 2q^{6} + 2q^{9} - 4q^{14} - 2q^{16} - 2q^{20} - 4q^{21} - 2q^{24} + 4q^{25} + 6q^{29} - 2q^{30} - 2q^{36} + 2q^{41} - 2q^{45} + 4q^{46} + 2q^{49} - 2q^{54} - 2q^{56} - 4q^{64} + 4q^{69} + 4q^{70} + 2q^{80} - 2q^{81} - 2q^{84} - 4q^{89} - 6q^{94} - 4q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1\) \(-1\) \(-\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0.500000 + 0.866025i 0.866025 + 0.500000i
59.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i −0.866025 0.500000i
299.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.00000 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0.500000 0.866025i 0.866025 0.500000i
299.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.00000 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.500000 0.866025i −0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
63.s even 6 1 inner
252.bn odd 6 1 inner
315.u even 6 1 inner
1260.de odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.de.a yes 4
3.b odd 2 1 3780.1.de.b 4
4.b odd 2 1 inner 1260.1.de.a yes 4
5.b even 2 1 inner 1260.1.de.a yes 4
7.d odd 6 1 1260.1.br.a 4
9.c even 3 1 3780.1.br.b 4
9.d odd 6 1 1260.1.br.a 4
12.b even 2 1 3780.1.de.b 4
15.d odd 2 1 3780.1.de.b 4
20.d odd 2 1 CM 1260.1.de.a yes 4
21.g even 6 1 3780.1.br.b 4
28.f even 6 1 1260.1.br.a 4
35.i odd 6 1 1260.1.br.a 4
36.f odd 6 1 3780.1.br.b 4
36.h even 6 1 1260.1.br.a 4
45.h odd 6 1 1260.1.br.a 4
45.j even 6 1 3780.1.br.b 4
60.h even 2 1 3780.1.de.b 4
63.k odd 6 1 3780.1.de.b 4
63.s even 6 1 inner 1260.1.de.a yes 4
84.j odd 6 1 3780.1.br.b 4
105.p even 6 1 3780.1.br.b 4
140.s even 6 1 1260.1.br.a 4
180.n even 6 1 1260.1.br.a 4
180.p odd 6 1 3780.1.br.b 4
252.n even 6 1 3780.1.de.b 4
252.bn odd 6 1 inner 1260.1.de.a yes 4
315.u even 6 1 inner 1260.1.de.a yes 4
315.bn odd 6 1 3780.1.de.b 4
420.be odd 6 1 3780.1.br.b 4
1260.bc even 6 1 3780.1.de.b 4
1260.de odd 6 1 inner 1260.1.de.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.br.a 4 7.d odd 6 1
1260.1.br.a 4 9.d odd 6 1
1260.1.br.a 4 28.f even 6 1
1260.1.br.a 4 35.i odd 6 1
1260.1.br.a 4 36.h even 6 1
1260.1.br.a 4 45.h odd 6 1
1260.1.br.a 4 140.s even 6 1
1260.1.br.a 4 180.n even 6 1
1260.1.de.a yes 4 1.a even 1 1 trivial
1260.1.de.a yes 4 4.b odd 2 1 inner
1260.1.de.a yes 4 5.b even 2 1 inner
1260.1.de.a yes 4 20.d odd 2 1 CM
1260.1.de.a yes 4 63.s even 6 1 inner
1260.1.de.a yes 4 252.bn odd 6 1 inner
1260.1.de.a yes 4 315.u even 6 1 inner
1260.1.de.a yes 4 1260.de odd 6 1 inner
3780.1.br.b 4 9.c even 3 1
3780.1.br.b 4 21.g even 6 1
3780.1.br.b 4 36.f odd 6 1
3780.1.br.b 4 45.j even 6 1
3780.1.br.b 4 84.j odd 6 1
3780.1.br.b 4 105.p even 6 1
3780.1.br.b 4 180.p odd 6 1
3780.1.br.b 4 420.be odd 6 1
3780.1.de.b 4 3.b odd 2 1
3780.1.de.b 4 12.b even 2 1
3780.1.de.b 4 15.d odd 2 1
3780.1.de.b 4 60.h even 2 1
3780.1.de.b 4 63.k odd 6 1
3780.1.de.b 4 252.n even 6 1
3780.1.de.b 4 315.bn odd 6 1
3780.1.de.b 4 1260.bc even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 4 + T^{2} )^{2} \)
$29$ \( ( 3 - 3 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( 9 + 3 T^{2} + T^{4} \)
$47$ \( 9 + 3 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( 9 + 3 T^{2} + T^{4} \)
$89$ \( ( 4 + 2 T + T^{2} )^{2} \)
$97$ \( T^{4} \)
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