Properties

Label 1260.2.d.a
Level $1260$
Weight $2$
Character orbit 1260.d
Analytic conductor $10.061$
Analytic rank $0$
Dimension $4$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} +O(q^{10})\) \( q - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} -\beta_{1} q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} -2 \beta_{3} q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} -\beta_{2} q^{23} + q^{25} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -\beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{3} ) q^{35} + ( 6 - 2 \beta_{3} ) q^{37} + ( 4 - 2 \beta_{3} ) q^{41} + ( 4 - 2 \beta_{3} ) q^{43} + ( 2 - 2 \beta_{3} ) q^{47} + ( 3 - 2 \beta_{2} ) q^{49} + ( -4 \beta_{1} - \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( -6 + 2 \beta_{3} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{65} + ( -\beta_{1} - 2 \beta_{2} ) q^{71} + ( -5 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -2 - \beta_{2} ) q^{77} + 4 \beta_{3} q^{79} + ( 10 - 2 \beta_{3} ) q^{83} + 2 \beta_{3} q^{85} + ( -2 - 4 \beta_{3} ) q^{89} + ( 2 - 5 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} + \beta_{2} ) q^{95} + ( 7 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{25} + 24q^{37} + 16q^{41} + 16q^{43} + 8q^{47} + 12q^{49} - 24q^{59} - 8q^{77} + 40q^{83} - 8q^{89} + 8q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 4 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
2.28825i
2.28825i
0.874032i
0.874032i
0 0 0 −1.00000 0 −2.23607 1.41421i 0 0 0
881.2 0 0 0 −1.00000 0 −2.23607 + 1.41421i 0 0 0
881.3 0 0 0 −1.00000 0 2.23607 1.41421i 0 0 0
881.4 0 0 0 −1.00000 0 2.23607 + 1.41421i 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
21.c 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.d.a 4
3.b odd 2 1 1260.2.d.b yes 4
4.b odd 2 1 5040.2.f.b 4
5.b even 2 1 6300.2.d.b 4
5.c odd 4 2 6300.2.f.c 8
7.b odd 2 1 1260.2.d.b yes 4
12.b even 2 1 5040.2.f.d 4
15.d odd 2 1 6300.2.d.a 4
15.e even 4 2 6300.2.f.a 8
21.c even 2 1 inner 1260.2.d.a 4
28.d even 2 1 5040.2.f.d 4
35.c odd 2 1 6300.2.d.a 4
35.f even 4 2 6300.2.f.a 8
84.h odd 2 1 5040.2.f.b 4
105.g even 2 1 6300.2.d.b 4
105.k odd 4 2 6300.2.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.d.a 4 1.a even 1 1 trivial
1260.2.d.a 4 21.c even 2 1 inner
1260.2.d.b yes 4 3.b odd 2 1
1260.2.d.b yes 4 7.b odd 2 1
5040.2.f.b 4 4.b odd 2 1
5040.2.f.b 4 84.h odd 2 1
5040.2.f.d 4 12.b even 2 1
5040.2.f.d 4 28.d even 2 1
6300.2.d.a 4 15.d odd 2 1
6300.2.d.a 4 35.c odd 2 1
6300.2.d.b 4 5.b even 2 1
6300.2.d.b 4 105.g even 2 1
6300.2.f.a 8 15.e even 4 2
6300.2.f.a 8 35.f even 4 2
6300.2.f.c 8 5.c odd 4 2
6300.2.f.c 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{41}^{2} - 8 T_{41} - 4 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 49 - 6 T^{2} + T^{4} \)
$11$ \( ( 2 + T^{2} )^{2} \)
$13$ \( 64 + 24 T^{2} + T^{4} \)
$17$ \( ( -20 + T^{2} )^{2} \)
$19$ \( 64 + 24 T^{2} + T^{4} \)
$23$ \( ( 10 + T^{2} )^{2} \)
$29$ \( 484 + 116 T^{2} + T^{4} \)
$31$ \( 64 + 24 T^{2} + T^{4} \)
$37$ \( ( 16 - 12 T + T^{2} )^{2} \)
$41$ \( ( -4 - 8 T + T^{2} )^{2} \)
$43$ \( ( -4 - 8 T + T^{2} )^{2} \)
$47$ \( ( -16 - 4 T + T^{2} )^{2} \)
$53$ \( 484 + 84 T^{2} + T^{4} \)
$59$ \( ( 16 + 12 T + T^{2} )^{2} \)
$61$ \( 1024 + 96 T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( 1444 + 84 T^{2} + T^{4} \)
$73$ \( 1600 + 280 T^{2} + T^{4} \)
$79$ \( ( -80 + T^{2} )^{2} \)
$83$ \( ( 80 - 20 T + T^{2} )^{2} \)
$89$ \( ( -76 + 4 T + T^{2} )^{2} \)
$97$ \( 64 + 376 T^{2} + T^{4} \)
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