Properties

Label 1260.1.bp.a
Level $1260$
Weight $1$
Character orbit 1260.bp
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.bp (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 6.2.13502538000.16

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{5} q^{2} + \zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{5} + q^{6} -\zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{12}^{5} q^{2} + \zeta_{12} q^{3} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{5} + q^{6} -\zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{3} q^{10} -\zeta_{12}^{5} q^{12} -\zeta_{12}^{2} q^{14} + \zeta_{12}^{5} q^{15} -\zeta_{12}^{2} q^{16} + \zeta_{12} q^{18} + \zeta_{12}^{2} q^{20} -\zeta_{12}^{4} q^{21} + \zeta_{12} q^{23} -\zeta_{12}^{4} q^{24} -\zeta_{12}^{2} q^{25} + \zeta_{12}^{3} q^{27} -\zeta_{12} q^{28} + ( -1 + \zeta_{12}^{4} ) q^{29} + \zeta_{12}^{4} q^{30} -\zeta_{12} q^{32} + \zeta_{12} q^{35} + q^{36} + \zeta_{12} q^{40} + \zeta_{12}^{4} q^{41} -\zeta_{12}^{3} q^{42} - q^{45} + q^{46} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{3} q^{48} - q^{49} -\zeta_{12} q^{50} + \zeta_{12}^{2} q^{54} - q^{56} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{58} + \zeta_{12}^{3} q^{60} + ( -1 + \zeta_{12}^{4} ) q^{61} -\zeta_{12}^{5} q^{63} - q^{64} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{67} + \zeta_{12}^{2} q^{69} + q^{70} -\zeta_{12}^{5} q^{72} -\zeta_{12}^{3} q^{75} + q^{80} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{3} q^{82} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{83} -\zeta_{12}^{2} q^{84} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{87} + q^{89} + \zeta_{12}^{5} q^{90} -\zeta_{12}^{5} q^{92} + ( -1 - \zeta_{12}^{2} ) q^{94} -\zeta_{12}^{2} q^{96} + \zeta_{12}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} - 2q^{14} - 2q^{16} + 2q^{20} + 2q^{21} + 2q^{24} - 2q^{25} - 6q^{29} - 2q^{30} + 4q^{36} - 2q^{41} - 4q^{45} + 4q^{46} - 4q^{49} + 2q^{54} - 4q^{56} - 6q^{61} - 4q^{64} + 2q^{69} + 4q^{70} + 4q^{80} - 2q^{81} - 2q^{84} + 4q^{89} - 6q^{94} - 2q^{96} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−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 −0.500000 + 0.866025i 1.00000 1.00000i 1.00000i 0.500000 + 0.866025i 1.00000i
419.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 1.00000i 1.00000i 0.500000 + 0.866025i 1.00000i
839.1 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.00000i 1.00000i 0.500000 0.866025i 1.00000i
839.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.00000i 1.00000i 0.500000 0.866025i 1.00000i
\(n\): 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.o even 6 1 inner
252.s odd 6 1 inner
315.z even 6 1 inner
1260.bp 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.bp.a 4
3.b odd 2 1 3780.1.bp.b 4
4.b odd 2 1 inner 1260.1.bp.a 4
5.b even 2 1 inner 1260.1.bp.a 4
7.b odd 2 1 1260.1.bp.b yes 4
9.c even 3 1 3780.1.bp.a 4
9.d odd 6 1 1260.1.bp.b yes 4
12.b even 2 1 3780.1.bp.b 4
15.d odd 2 1 3780.1.bp.b 4
20.d odd 2 1 CM 1260.1.bp.a 4
21.c even 2 1 3780.1.bp.a 4
28.d even 2 1 1260.1.bp.b yes 4
35.c odd 2 1 1260.1.bp.b yes 4
36.f odd 6 1 3780.1.bp.a 4
36.h even 6 1 1260.1.bp.b yes 4
45.h odd 6 1 1260.1.bp.b yes 4
45.j even 6 1 3780.1.bp.a 4
60.h even 2 1 3780.1.bp.b 4
63.l odd 6 1 3780.1.bp.b 4
63.o even 6 1 inner 1260.1.bp.a 4
84.h odd 2 1 3780.1.bp.a 4
105.g even 2 1 3780.1.bp.a 4
140.c even 2 1 1260.1.bp.b yes 4
180.n even 6 1 1260.1.bp.b yes 4
180.p odd 6 1 3780.1.bp.a 4
252.s odd 6 1 inner 1260.1.bp.a 4
252.bi even 6 1 3780.1.bp.b 4
315.z even 6 1 inner 1260.1.bp.a 4
315.bg odd 6 1 3780.1.bp.b 4
420.o odd 2 1 3780.1.bp.a 4
1260.bp odd 6 1 inner 1260.1.bp.a 4
1260.cq even 6 1 3780.1.bp.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bp.a 4 1.a even 1 1 trivial
1260.1.bp.a 4 4.b odd 2 1 inner
1260.1.bp.a 4 5.b even 2 1 inner
1260.1.bp.a 4 20.d odd 2 1 CM
1260.1.bp.a 4 63.o even 6 1 inner
1260.1.bp.a 4 252.s odd 6 1 inner
1260.1.bp.a 4 315.z even 6 1 inner
1260.1.bp.a 4 1260.bp odd 6 1 inner
1260.1.bp.b yes 4 7.b odd 2 1
1260.1.bp.b yes 4 9.d odd 6 1
1260.1.bp.b yes 4 28.d even 2 1
1260.1.bp.b yes 4 35.c odd 2 1
1260.1.bp.b yes 4 36.h even 6 1
1260.1.bp.b yes 4 45.h odd 6 1
1260.1.bp.b yes 4 140.c even 2 1
1260.1.bp.b yes 4 180.n even 6 1
3780.1.bp.a 4 9.c even 3 1
3780.1.bp.a 4 21.c even 2 1
3780.1.bp.a 4 36.f odd 6 1
3780.1.bp.a 4 45.j even 6 1
3780.1.bp.a 4 84.h odd 2 1
3780.1.bp.a 4 105.g even 2 1
3780.1.bp.a 4 180.p odd 6 1
3780.1.bp.a 4 420.o odd 2 1
3780.1.bp.b 4 3.b odd 2 1
3780.1.bp.b 4 12.b even 2 1
3780.1.bp.b 4 15.d odd 2 1
3780.1.bp.b 4 60.h even 2 1
3780.1.bp.b 4 63.l odd 6 1
3780.1.bp.b 4 252.bi even 6 1
3780.1.bp.b 4 315.bg odd 6 1
3780.1.bp.b 4 1260.cq even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{41}^{2} + T_{41} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

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