Properties

Label 1260.1.u.b
Level $1260$
Weight $1$
Character orbit 1260.u
Analytic conductor $0.629$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -84
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.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.220500.3

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + q^{5} -\zeta_{8} q^{7} -\zeta_{8} q^{8} +O(q^{10})\) \( q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + q^{5} -\zeta_{8} q^{7} -\zeta_{8} q^{8} -\zeta_{8}^{3} q^{10} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} - q^{14} - q^{16} + ( -1 + \zeta_{8}^{2} ) q^{17} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{19} -\zeta_{8}^{2} q^{20} + ( -1 - \zeta_{8}^{2} ) q^{22} + q^{25} + \zeta_{8}^{3} q^{28} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{31} + \zeta_{8}^{3} q^{32} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{34} -\zeta_{8} q^{35} + ( 1 + \zeta_{8}^{2} ) q^{37} + ( 1 + \zeta_{8}^{2} ) q^{38} -\zeta_{8} q^{40} -2 \zeta_{8}^{2} q^{41} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{44} + \zeta_{8}^{2} q^{49} -\zeta_{8}^{3} q^{50} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{55} + \zeta_{8}^{2} q^{56} + ( 1 - \zeta_{8}^{2} ) q^{62} + \zeta_{8}^{2} q^{64} + ( 1 + \zeta_{8}^{2} ) q^{68} - q^{70} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{71} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{74} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{76} + ( -1 + \zeta_{8}^{2} ) q^{77} - q^{80} -2 \zeta_{8} q^{82} + ( -1 + \zeta_{8}^{2} ) q^{85} + ( -1 + \zeta_{8}^{2} ) q^{88} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{95} + \zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + O(q^{10}) \) \( 4q + 4q^{5} - 4q^{14} - 4q^{16} - 4q^{17} - 4q^{22} + 4q^{25} + 4q^{37} + 4q^{38} + 4q^{62} + 4q^{68} - 4q^{70} - 4q^{77} - 4q^{80} - 4q^{85} - 4q^{88} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i 0 1.00000i 1.00000 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 0.707107i
307.2 0.707107 0.707107i 0 1.00000i 1.00000 0 −0.707107 0.707107i −0.707107 0.707107i 0 0.707107 0.707107i
1063.1 −0.707107 0.707107i 0 1.00000i 1.00000 0 0.707107 0.707107i 0.707107 0.707107i 0 −0.707107 0.707107i
1063.2 0.707107 + 0.707107i 0 1.00000i 1.00000 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
4.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
140.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.u.b yes 4
3.b odd 2 1 1260.1.u.a 4
4.b odd 2 1 inner 1260.1.u.b yes 4
5.c odd 4 1 1260.1.u.a 4
7.b odd 2 1 1260.1.u.a 4
12.b even 2 1 1260.1.u.a 4
15.e even 4 1 inner 1260.1.u.b yes 4
20.e even 4 1 1260.1.u.a 4
21.c even 2 1 inner 1260.1.u.b yes 4
28.d even 2 1 1260.1.u.a 4
35.f even 4 1 inner 1260.1.u.b yes 4
60.l odd 4 1 inner 1260.1.u.b yes 4
84.h odd 2 1 CM 1260.1.u.b yes 4
105.k odd 4 1 1260.1.u.a 4
140.j odd 4 1 inner 1260.1.u.b yes 4
420.w even 4 1 1260.1.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.u.a 4 3.b odd 2 1
1260.1.u.a 4 5.c odd 4 1
1260.1.u.a 4 7.b odd 2 1
1260.1.u.a 4 12.b even 2 1
1260.1.u.a 4 20.e even 4 1
1260.1.u.a 4 28.d even 2 1
1260.1.u.a 4 105.k odd 4 1
1260.1.u.a 4 420.w even 4 1
1260.1.u.b yes 4 1.a even 1 1 trivial
1260.1.u.b yes 4 4.b odd 2 1 inner
1260.1.u.b yes 4 15.e even 4 1 inner
1260.1.u.b yes 4 21.c even 2 1 inner
1260.1.u.b yes 4 35.f even 4 1 inner
1260.1.u.b yes 4 60.l odd 4 1 inner
1260.1.u.b yes 4 84.h odd 2 1 CM
1260.1.u.b yes 4 140.j odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2 T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

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