Properties

Label 1260.1.bt.b
Level $1260$
Weight $1$
Character orbit 1260.bt
Analytic conductor $0.629$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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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.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.11340.2
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.55566000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{5} + \zeta_{6}^{2} q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{5} + \zeta_{6}^{2} q^{7} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} -2 \zeta_{6} q^{13} + q^{15} + 2 q^{17} -\zeta_{6} q^{21} + \zeta_{6}^{2} q^{25} + q^{27} -\zeta_{6}^{2} q^{29} + \zeta_{6} q^{33} + q^{35} + 2 q^{39} + \zeta_{6}^{2} q^{45} -\zeta_{6}^{2} q^{47} -\zeta_{6} q^{49} + 2 \zeta_{6}^{2} q^{51} - q^{55} + q^{63} + 2 \zeta_{6}^{2} q^{65} - q^{71} - q^{73} -\zeta_{6} q^{75} + \zeta_{6} q^{77} -\zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} -\zeta_{6}^{2} q^{83} -2 \zeta_{6} q^{85} + \zeta_{6} q^{87} + 2 q^{91} -\zeta_{6}^{2} q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{5} - q^{7} - q^{9} + q^{11} - 2q^{13} + 2q^{15} + 4q^{17} - q^{21} - q^{25} + 2q^{27} + q^{29} + q^{33} + 2q^{35} + 4q^{39} - q^{45} + q^{47} - q^{49} - 2q^{51} - 2q^{55} + 2q^{63} - 2q^{65} - 2q^{71} - 2q^{73} - q^{75} + q^{77} + q^{79} - q^{81} + q^{83} - 2q^{85} + q^{87} + 4q^{91} + q^{97} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0
769.1 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
9.c even 3 1 inner
315.bg 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.bt.b 2
3.b odd 2 1 3780.1.bt.c 2
5.b even 2 1 1260.1.bt.d yes 2
7.b odd 2 1 1260.1.bt.d yes 2
9.c even 3 1 inner 1260.1.bt.b 2
9.d odd 6 1 3780.1.bt.c 2
15.d odd 2 1 3780.1.bt.a 2
21.c even 2 1 3780.1.bt.a 2
35.c odd 2 1 CM 1260.1.bt.b 2
45.h odd 6 1 3780.1.bt.a 2
45.j even 6 1 1260.1.bt.d yes 2
63.l odd 6 1 1260.1.bt.d yes 2
63.o even 6 1 3780.1.bt.a 2
105.g even 2 1 3780.1.bt.c 2
315.z even 6 1 3780.1.bt.c 2
315.bg odd 6 1 inner 1260.1.bt.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bt.b 2 1.a even 1 1 trivial
1260.1.bt.b 2 9.c even 3 1 inner
1260.1.bt.b 2 35.c odd 2 1 CM
1260.1.bt.b 2 315.bg odd 6 1 inner
1260.1.bt.d yes 2 5.b even 2 1
1260.1.bt.d yes 2 7.b odd 2 1
1260.1.bt.d yes 2 45.j even 6 1
1260.1.bt.d yes 2 63.l odd 6 1
3780.1.bt.a 2 15.d odd 2 1
3780.1.bt.a 2 21.c even 2 1
3780.1.bt.a 2 45.h odd 6 1
3780.1.bt.a 2 63.o even 6 1
3780.1.bt.c 2 3.b odd 2 1
3780.1.bt.c 2 9.d odd 6 1
3780.1.bt.c 2 105.g even 2 1
3780.1.bt.c 2 315.z even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)

Hecke characteristic polynomials

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