Properties

Label 1260.2.k.a
Level $1260$
Weight $2$
Character orbit 1260.k
Analytic conductor $10.061$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - i ) q^{5} -i q^{7} +O(q^{10})\) \( q + ( -2 - i ) q^{5} -i q^{7} -4 q^{11} + 2 i q^{13} + 2 i q^{17} + 2 q^{19} + 6 i q^{23} + ( 3 + 4 i ) q^{25} + 6 q^{29} + 6 q^{31} + ( -1 + 2 i ) q^{35} + 4 i q^{37} -4 i q^{43} + 4 i q^{47} - q^{49} + 2 i q^{53} + ( 8 + 4 i ) q^{55} + 4 q^{59} -2 q^{61} + ( 2 - 4 i ) q^{65} + 12 i q^{67} + 8 q^{71} + 14 i q^{73} + 4 i q^{77} -16 q^{79} -16 i q^{83} + ( 2 - 4 i ) q^{85} + 16 q^{89} + 2 q^{91} + ( -4 - 2 i ) q^{95} + 14 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} - 8q^{11} + 4q^{19} + 6q^{25} + 12q^{29} + 12q^{31} - 2q^{35} - 2q^{49} + 16q^{55} + 8q^{59} - 4q^{61} + 4q^{65} + 16q^{71} - 32q^{79} + 4q^{85} + 32q^{89} + 4q^{91} - 8q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 0 0 −2.00000 1.00000i 0 1.00000i 0 0 0
1009.2 0 0 0 −2.00000 + 1.00000i 0 1.00000i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.k.a 2
3.b odd 2 1 420.2.k.b 2
4.b odd 2 1 5040.2.t.d 2
5.b even 2 1 inner 1260.2.k.a 2
5.c odd 4 1 6300.2.a.b 1
5.c odd 4 1 6300.2.a.r 1
12.b even 2 1 1680.2.t.g 2
15.d odd 2 1 420.2.k.b 2
15.e even 4 1 2100.2.a.i 1
15.e even 4 1 2100.2.a.n 1
20.d odd 2 1 5040.2.t.d 2
21.c even 2 1 2940.2.k.b 2
21.g even 6 2 2940.2.bb.f 4
21.h odd 6 2 2940.2.bb.a 4
60.h even 2 1 1680.2.t.g 2
60.l odd 4 1 8400.2.a.o 1
60.l odd 4 1 8400.2.a.bm 1
105.g even 2 1 2940.2.k.b 2
105.o odd 6 2 2940.2.bb.a 4
105.p even 6 2 2940.2.bb.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.b 2 3.b odd 2 1
420.2.k.b 2 15.d odd 2 1
1260.2.k.a 2 1.a even 1 1 trivial
1260.2.k.a 2 5.b even 2 1 inner
1680.2.t.g 2 12.b even 2 1
1680.2.t.g 2 60.h even 2 1
2100.2.a.i 1 15.e even 4 1
2100.2.a.n 1 15.e even 4 1
2940.2.k.b 2 21.c even 2 1
2940.2.k.b 2 105.g even 2 1
2940.2.bb.a 4 21.h odd 6 2
2940.2.bb.a 4 105.o odd 6 2
2940.2.bb.f 4 21.g even 6 2
2940.2.bb.f 4 105.p even 6 2
5040.2.t.d 2 4.b odd 2 1
5040.2.t.d 2 20.d odd 2 1
6300.2.a.b 1 5.c odd 4 1
6300.2.a.r 1 5.c odd 4 1
8400.2.a.o 1 60.l odd 4 1
8400.2.a.bm 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( -16 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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