Properties

Label 2240.2.g.k
Level $2240$
Weight $2$
Character orbit 2240.g
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1120)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} + 2 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} + 2 q^{9} + \beta_{3} q^{11} -\beta_{2} q^{13} + \beta_{3} q^{15} + \beta_{2} q^{17} + 2 \beta_{3} q^{19} - q^{21} + 4 \beta_{1} q^{23} -5 q^{25} + 5 \beta_{1} q^{27} - q^{29} + \beta_{2} q^{33} + \beta_{3} q^{35} -4 \beta_{2} q^{37} + \beta_{3} q^{39} -6 \beta_{1} q^{43} -2 \beta_{2} q^{45} -3 \beta_{1} q^{47} - q^{49} -\beta_{3} q^{51} -2 \beta_{2} q^{53} -5 \beta_{1} q^{55} + 2 \beta_{2} q^{57} + 2 \beta_{3} q^{59} + 10 q^{61} + 2 \beta_{1} q^{63} -5 q^{65} + 2 \beta_{1} q^{67} -4 q^{69} + 4 \beta_{3} q^{71} + 6 \beta_{2} q^{73} -5 \beta_{1} q^{75} + \beta_{2} q^{77} + 7 \beta_{3} q^{79} + q^{81} + 4 \beta_{1} q^{83} + 5 q^{85} -\beta_{1} q^{87} + 14 q^{89} + \beta_{3} q^{91} -10 \beta_{1} q^{95} + \beta_{2} q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} - 4q^{21} - 20q^{25} - 4q^{29} - 4q^{49} + 40q^{61} - 20q^{65} - 16q^{69} + 4q^{81} + 20q^{85} + 56q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.61803i
0.618034i
0.618034i
1.61803i
0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.3 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 0
449.4 0 1.00000i 0 2.23607i 0 1.00000i 0 2.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.g.k 4
4.b odd 2 1 inner 2240.2.g.k 4
5.b even 2 1 inner 2240.2.g.k 4
8.b even 2 1 1120.2.g.a 4
8.d odd 2 1 1120.2.g.a 4
20.d odd 2 1 inner 2240.2.g.k 4
40.e odd 2 1 1120.2.g.a 4
40.f even 2 1 1120.2.g.a 4
40.i odd 4 1 5600.2.a.w 2
40.i odd 4 1 5600.2.a.bl 2
40.k even 4 1 5600.2.a.w 2
40.k even 4 1 5600.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.a 4 8.b even 2 1
1120.2.g.a 4 8.d odd 2 1
1120.2.g.a 4 40.e odd 2 1
1120.2.g.a 4 40.f even 2 1
2240.2.g.k 4 1.a even 1 1 trivial
2240.2.g.k 4 4.b odd 2 1 inner
2240.2.g.k 4 5.b even 2 1 inner
2240.2.g.k 4 20.d odd 2 1 inner
5600.2.a.w 2 40.i odd 4 1
5600.2.a.w 2 40.k even 4 1
5600.2.a.bl 2 40.i odd 4 1
5600.2.a.bl 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{11}^{2} - 5 \)
\( T_{19}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -5 + T^{2} )^{2} \)
$13$ \( ( 5 + T^{2} )^{2} \)
$17$ \( ( 5 + T^{2} )^{2} \)
$19$ \( ( -20 + T^{2} )^{2} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 80 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( ( -20 + T^{2} )^{2} \)
$61$ \( ( -10 + T )^{4} \)
$67$ \( ( 4 + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( 180 + T^{2} )^{2} \)
$79$ \( ( -245 + T^{2} )^{2} \)
$83$ \( ( 16 + T^{2} )^{2} \)
$89$ \( ( -14 + T )^{4} \)
$97$ \( ( 5 + T^{2} )^{2} \)
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