Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{5} + q^{7} + 3 q^{9} +O(q^{10})\) \( q + i q^{5} + q^{7} + 3 q^{9} + 2 i q^{11} -2 i q^{13} + 2 q^{17} + 4 i q^{19} - q^{25} + 4 i q^{29} + 4 q^{31} + i q^{35} -8 i q^{37} + 2 q^{41} + 6 i q^{43} + 3 i q^{45} + q^{49} -4 i q^{53} -2 q^{55} + 2 i q^{61} + 3 q^{63} + 2 q^{65} + 2 i q^{67} -8 q^{71} + 10 q^{73} + 2 i q^{77} -8 q^{79} + 9 q^{81} + 12 i q^{83} + 2 i q^{85} + 6 q^{89} -2 i q^{91} -4 q^{95} + 2 q^{97} + 6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{7} + 6q^{9} + 4q^{17} - 2q^{25} + 8q^{31} + 4q^{41} + 2q^{49} - 4q^{55} + 6q^{63} + 4q^{65} - 16q^{71} + 20q^{73} - 16q^{79} + 18q^{81} + 12q^{89} - 8q^{95} + 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
1.00000i
1.00000i
0 0 0 1.00000i 0 1.00000 0 3.00000 0
1121.2 0 0 0 1.00000i 0 1.00000 0 3.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.b.b yes 2
4.b odd 2 1 2240.2.b.a 2
8.b even 2 1 inner 2240.2.b.b yes 2
8.d odd 2 1 2240.2.b.a 2
16.e even 4 1 8960.2.a.i 1
16.e even 4 1 8960.2.a.k 1
16.f odd 4 1 8960.2.a.j 1
16.f odd 4 1 8960.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.a 2 4.b odd 2 1
2240.2.b.a 2 8.d odd 2 1
2240.2.b.b yes 2 1.a even 1 1 trivial
2240.2.b.b yes 2 8.b even 2 1 inner
8960.2.a.i 1 16.e even 4 1
8960.2.a.j 1 16.f odd 4 1
8960.2.a.k 1 16.e even 4 1
8960.2.a.l 1 16.f odd 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{23} \)
\( T_{31} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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